For this year’s annual Balliol Undergraduate Math Society (BUMS) dinner, Professor Robin Wilson, a former Balliol undergraduate himself, and now a professor at the Department of Mathematics from Open University, was supposed to address the audience — forty or so mathematicians, looking wholly uncomfortable in tuxedos and cocktail dresses.

Thing is, he never showed up.

The president of the society opened the evening’s talk.

“Professor Wilson had originally planned a talk centred around the 300 year old anniversary of Leonard Euler [pronounced as “Oiler”], but he was unable to attend.”

“So it’s fortunate that he was able to coax his good friend — Professor Euler himself — to come address us in his stead.”

And indeed, there he was.

Dining directly in front of me, swathed in elaborate gold robes and a funky hat with some furry creature on top — was Leonhard Euler himself.

“I realize that there’s some nasty rumours of my death floating around,” Euler said to us that night, “but they’re obviously false.”

“Now if my friend [Prof. R. Wilson] had been here, he’d undoubtedly spend all his time telling silly jokes — like the one about the geometry of kissing.”

“Oh,” he paused innocently, “do you all know the geometry of a kiss?”

We said nothing.

“Why, it’s an e-lip-twus of course!”

And in unison — perhaps fueled by the bad wine — we burst into an assortment of pained groans and wild cheering.

Euler was — and is still — known for his many great contributions to mathematics and physics. He is considered the most prolific author of all time (along with Paul Erdos), having published over 300 papers on a wide range of topics.

And today? What have you used recently that was discovered by Euler?

High school students will recognize his contributions to mathematical notation with the symbols of f(x) for functional notation, e (=2.7182..) for the base of the natural logarithm, i for the imaginary unit.

Scientists should also recognize his contributions to Calculus and Analysis which included, for example, his work on the analysis of infinite series’ and numerical methods of approximating functions.

But his most recreational problem was perhaps the one that involves the Seven Bridges of Königsberg:

The city of Königsberg, Prussia (now Kaliningrad, Russia) once included two large islands which were connected to each other and the mainland by seven bridges, thus separating the geography into four bodies of land.

The question many wondered was whether it was possible to travel in a route that passed each bridge once and only once (a route known today as an Eulerian Circuit).

Well Euler solved the stumper (in a fashion). And today, the Seven Bridges of Konigsberg problem is arguably considered the birth of the modern mathematical field known as Graph Theory, which is tied to a host of applications in, for example, computer science, network theory, and even sociology.

Later that night, I conversed openly with the esteemed mathematician.

“Professor, I was hoping that tonight, I’d bump into your friend Robin Wilson,” I said, feigning a sad look of ignorance.

“His book on Graph Theory saved me last year when I was taking a class on graphs.”

“Really?”, he said with a twinkle in his eye, “Well, I will definitely pass that on to my friend.”

And for the remainder of the dinner, the prolific and brilliant mathematician Leonhard Euler sat there — looking mighty pleased with himself.

My supervisor is strange.

Brilliant, but strange.

And it seems to be an inside joke with just about anybody in the math department.

“Who’s supervising you?”, they’d ask.

I’d reply.

Then they’d kind of half-gurgle, half-muffle a snicker. And say something totally ambiguous, like, “Oh, he’s quite the character, that one.”

What the hell does that mean?

So this afternoon…

I walked into his office.

“Hey,” I said casually, “I just realized I have a class tomorrow at eleven, so I may be a bit late for our meeting at twelve”.

“Phil, c’mere,” he said, gesturing.

I walked over to his desk.

On the desk — among the stacks of papers and mess of notebooks were, what appeared to be: two (2) steel magnum revolvers (replicas?), a bottle of vodka, and a large beer mug.

He picked up one of the revolvers and fingered it. Menacingly.

“Phil, this,” he gestured towards the revolver with his free hand, “is what we use when grad students get tensed up about silly issues like that.”

“Fair enough,” I croaked.

Then I scurried out, tail between my legs, trying furiously not to trip. Or sniffle.

So the last time we check on our venerable hero, he was starring in the fourth season of the hit Fox show, House, M.D., looking to catch the eye of a particular Dr. Allison Cameron…

Thing is, I left you all on a bit of a cliffhanger, right?

So let’s start with the basics.

The first thing we need is a love function.

Attraction has got to measured in some way, right?

A love function provides a measure of how attracted you are to someone else. You’ve all seen guys nudging each other subtly (or not so subtly) and whispering, “Dude, check out that nine”. Or something to that effect.

That’s a love function in nature, folks.

But we’ll go ahead and make the following definition:

P(t) = Phil’s attraction to Dr. Cameron as a function of the time.

A graphical depiction of such a love function would perhaps resemble the following,

As you can see, my attraction to Dr. Cameron goes through several phases as the seasons progressed: (1) an initial fast growth, when I was all smitten, (2) a fast decline into deep despair when she decided to grow out those stupid bangs in Season 3 (see right), and finally (3) a quick ascent back to blissful, puppy-like attraction.

Be sure to also distinguish the appropriate axis on the left. A positive value of the love function means you wanna get your mack on. A negative value means you’d rather go home and do some math.

And as you can imagine, one can come up with endless types of love functions, each of which can be analyzed mathematically. For example, that lustful adolescent love I’m sure we all felt when we met our first love can perhaps be modeled as an exponential growth:

And what of love after marriage? Well I assume after the honeymoon phase, the love function would be a sort of an exponential decay:

And according to my empirical analysis, most women exhibit love functions that are ‘randomly’ oscillatory — vague, unpredictable and seemingly chaotic:

Honestly, I’d throw in a discontinuity (or ten) and some other mathematical nastiness, but I try not to let my personal feelings run rampant.

But see, when we start to model our relationships, it’s not enough to know whether one person is in love with another. It’s equally important to know how fast or how slow this process is taking!

Compare, for example, the difference between a hot and heavy 2 week fling while vacationing in the Bahamas with say, a 10 year unrequited love from afar.

From a distance, both scenarios could very well resemble the above exponential growth — that is, a love that’s growing positively as time progresses. The real difference is found in the rate of change or speed that the couple falls in love.

A two week (or two day) affair would probably explosive, with a lightning fast rate of change. On the other hand, a 10 year unrequited romance would perhaps be a slow burning process — certainly not explosive.

Those of you who have taken Calculus or Physics will simply recognize this as the derivative of the love function. But if you haven’t (or more likely, if it’s been too long), all you need to know is that it’s vitally important we study both the level of attraction, and how fast or slow the attraction is growing (or dying).

P’(t) = How fast is Phil digging (or not digging) Dr. Cameron.

Notice the little prime (’) here which indicates that this function measures the rate or speed of attraction, not the amount. So in the above graph, P’(t) would be very large for an explosive attraction. But for a slow-burning and lengthy attraction, P’(t) would be quite small.

And of course, P’(t) could also be negative, which would indicate a growing dislike.

Phew! That’s enough mathematizing for one day, wouldn’t you say?

In the next part of this series, we’re going to finally get our hands dirty and start looking into the dynamics of relationships.

In fact, I’m going to mathematically prove why the Cameron-Chase relationship is doomed to fail.

Which totally leaves the door open for me.

What’s G103?

G103 is the code for a 4 year (MMath) undergraduate math program at British universities.

G103 is also the title of a short movie that was produced by the University of Warwick (pronounced, “Warrick”) Mathematics Institute.

It boasts a superb soundtrack, oddly high production value, and more than a few laugh-out-loud moments. There’s a hilarious scene about a professor trapped in his room and eating the chalk, a homage to The Matrix and Kill Bill movies, and even some ballroom dancing thrown in for variety.

What about the math? Is it rubbish?

“How do you use that Hausdorff thing?” an unseen character asks.

“I dunno,” the star of the movie mumbles, “it’s probably about balls — it’s almost always about balls.”

Which is, of course, true. Everything — absolutely everything — in math is about balls.

Any movie about mathematicians is going to ride the stereotypes. You know, the hermit-like behaviour, the disorganization, neuroticism, and oddball quirkiness. And so it’s really a matter of striking the right balance between wacky humour and grudging truthfulness. And that’s a balance the movie manages to strike, obviously thanks to the fact it was shot and directed by math students.

And yes, most of us do have this intense fascination with chalk and blackboards.

But I’m just wasting your time. Watch it.

Just skip the cheesy classroom shootout scene.

disclaimer

This post contains explicit math. In fact, there’s so much math, your head just might explode. If you don’t like math, you’re better off doing something else.

Like having a tea party with the girlfriends.

Wuss.

This post is going to be a little bit different.

For starters, there aren’t any pictures of cats.

Instead, I’m going to spend a little bit of time writing about what happens when the solution to a partial differential equation has a jump discontinuity.

Before we get started, let’s look at two cases.

Case A: If the above didn’t make any sense whatsoever…

You should read my last post. If you’re still bored, you can drool over my choco-licious lime biscotti. And if that doesn’t do it, you can laugh at how I looked as a kid.

Otherwise, come back tomorrow.

Case B: If the above gets your blood flowing and produces a strong tingle in your special place…

Read on.

Actually, this topic is pretty easy to understand. What’s odd is that most references I’ve managed to dig up on the problem explain it horrendously at best. But as long as you have a clear understanding of how to solve PDEs using the method of characteristics, you should be well on your way to bigger and better things after this lil’ talk.

Let’s begin with a generalized first-order quasi-linear equation,

We’re going to still be interested in finding solutions u(x,y), but now we’d like to relax the condition that solutions have to be differentiable everywhere.

Weak Discontinuity: Let u(x,y) be defined in some domain D. Let C be a curve parametrized by x = x(t), and y = y(t). Assume that u(x,y) is continuous and differentiable everywhere except on C, across which a first-order derivative of the solution may be discontinuous. This is a weak discontinuity.

Now we can look at the limtiing value of the functions as the curve is approached from either side, so from the chain rule,

(1)
(2)

But what does that mean? Be sure to distinguish the two types of derivatives above. The dot, as in , is the derivative along the curve (with respect to t). This is in contrast to the partial derivatives, which measure the rates of change of u(x,y) in the x and y directions.

But actually, because u is continuous,

Why? Imagine it this way: Cars are travelling along the road C. An observer to the right of the road measures the speed of the car as . But on the other side, his buddy measures the speed of the car as . Assuming that both observers are sufficiently close (continuity), they’ll obviously measure the same speed.

And so, subtracting (1) from (2) we have,

(3)

where the square bracket denotes the ‘jump’ in the expression across C.

Remember that u(x, y) should satisfy the quasi-linear PDE except on C, so by subtraction we have,

(4)

Notice the c disappears because of the continuity of u, and this also explains why a and b can be evaluated on C without ambiguity from any direction.

(3) and (4) provide us with a system of equations for the size of the jump discontinuities. And of course, since we’re interested in non-zero jump discontinuities, we’ll require the determinant,

But this is the precise equation for the characteristic curves of the PDE!

Wowsers!

Solutions with Weak Discontinuities: If the solution of a quasi-linear PDE has a weak discontinuity across some curve C, then C must be a characteristic projection.

Actually, we can squeeze a little bit more information out of this theory. It turns out the size of the jumps across C can be calculated from a first order ordinary differential equation. For example, in the case of a linear PDE,

where we follow the characteristic curves and ,

And so if the jump is non-zero at some point in space (x,y), it will continue to be non-zero along a characteristic. More importantly, if there is initially a weak discontinuity at t = 0, then this jump will propogate along the characteristic.

Propogation of Weak Discontinuities: Suppose that there is initially a jump in the derivatives of the initial boundary data (Cauchy problem). Then this jump will propogate along its characteristic curve. Furthermore, if there is no jump, then the solution will be well behaved (around the characteristic).

Let’s finish with a bang.

Let’s solve the following PDE

subject to the Cauchy conditions:

using the method of characteristics, it’s quite easy to show that the solution is given by,

Take a look at the graph.

Notice once you’ve pieced everything back together, the solution is continuous for t > 0. The important observation is of course that the solution is smooth everywhere, except along the two special characteristics , where the discontinuity from the Cauchy data propogates along the characteristic.

And that about completes our look into what happens when there’s a jump discontinuity in the solutions of a PDE.

But we’ve only gotten started!

The more interesting and deeper question is what happens when the solution itself is discontinuous. These solutions contain so-called shock waves and are termed weak solutions. Does this happen in real life? Sure. Take a look at this picture.

Notice that a description of the surface waves would be discontinuous in the region where the waves are about to break, or the region where the wave is about to ‘bend’ over itself.

The basic ideas behind studying discontinuous solutions remains the same as for the above problem. Except now we can’t really use expressions like,

Right? Because since u(x,y) is itself discontinuous at certain points, we can’t be so cavalier about taking derivatives! The key to approach these types of problems is to consider equivalent integral formulations, as we know that integration is more robust with respect to discontinuous solutions.

But that’s all for another day.

Phew. I’m going to go lie down now.