Living in university residence and on the college mailing list, you tend to be privy to quite a few interesting e-mails. E-mails written out of anger, however, tend to be divided into two categories: laundry rage (when someone steals your slot) and kitchen/food rage (when someone leaves mess or steals your food or kitchenware).

These e-mails also provide an interesting idea of cultural differences. Here’s two. The rambling, passive aggressive one is written by a British doctoral student regarding stolen kitchenware. The short, concise one is written by an American Marshall scholar from Los Angeles.

Enjoy.

From: The British Scholar
To: Everybody
Date: Sometime in June
Subject: Beware of Vanishing Objects

My 8 inch Wusthof-Trident Chef’s Knife, which I’ve owned since my restaurant days, seems to have disappeared from my carry-all in the North Wing Ground floor kitchen during the last week. Aside from the obvious sentimental value, its loss would be bearable had not my 10 inch Chef’s knife disappeared from the Martin Ground floor kitchen 3 days ago. Now we seem to be left with flimsy serrated knives which, if utilised to cut anything substantial, like say a butternut squash, will most likely snap and take out someone’s eye (and, of course, chopping anything is now out of the question).

I should also note that my toaster disappeared from our kitchen 2 weeks ago, and my industrial food processor sometime in Michaelmas–not to mention the loss of roughly 10 roasting pans in the last 2 years.

(1) If any of you have said items could you please return them

and,

(2) Is anyone else experiencing the same type of kitchen related separation anxiety?

Balliol MCR……effortlessly superior, or merely dirtbags?

It’s up to you.

- [The Brit]

From: The American Scholar
To: Everybody
Date: Sometime in June
Subject: Ice cream theft

To whoever helped themselves to my entire carton of ice cream in the Dellal ground floor kitchen — I hope you enjoyed it immensely. Because when I find you, I am going to remove your face.

- [The American]

A Review

exponential asymptotics

Exponential Asymptotics then, is the bridge between God’s picture-perfect, but wholly unrealizable representations, and the Devil’s ambiguous, but entirely necessary evils.

Last time, we left off talking about asymptotic approximations. Nature — we explained — is often too complicated to describe exactly, and so we turn to these approximate methods for help. Unfortunately, such approximations are divergent and mathematically badly-behaved, even though they provide excellent results.

Now, we will see that there are indeed consequences to using these devilish and divergent approximations.

Dancing with the Devil

Maybe the best thing we can do is look at a concrete example.

For this, we’re going to imagine the flow of water with speed U over a step which is relatively well submerged. To solve this problem, we propose the following asymptotic expansion for the surface of the water:

In this case, the perturbation parameter, ε corresponds to a measure of the inertial and gravitational forces. The idealized state, or first approximation corresponds to the case that ε = 0, or the case for which the force of gravity is infinite.

What that means is that our first order asymptotic approximation tells us the free surface should look like this:

That’s right. Completely flat.

Sure, this does make some sense: Imagine the step as being infinitely deep. Then in this ideal case, free surface wouldn’t feel the bump at all. Hence it would lie completely flat.

Unfortunately, this isn’t quite what happens.

If we use a computer and calculate the solution, it gives us something like this:

Somewhere on that free surface, something is happening.

The asymptotic approximation we calculated does a wonderful job initially, but then out of nowhere, a light flickers on and Pop!.

Waves appear where there were none.

Into the Mist

Maybe the first person who documented the first “WTF?!” moment upon encountering this strange phenomenon was none other than George Gabriel Stokes, one of the greats of British applied maths. He described the phenomenon as a new term, entering into view from behind a mist:

How do we clear the mist?

The argument goes like this: First, we have an approximation that works on the far left. Now, somewhere, along the surface, a switch is turned on, and an exponential (or a sinusoidal, if you will) is turned on.

Thus, there must be something that’s going on the complexified free surface.

Whoa there, Missy! What the heck does that mean?

By complexified free surface, we’re referring to extending the water surface (a real number) to the complex (or `imaginary’) numbers. In other words, if x measures the distance along the water surface (say, 1 meter, 2 meters, etc.), then we allow x to be an imaginary number (say, 1 + i or 2 - 2i)

Physically (and philosophically), it’s pretty nonsensical. We picture it like this:

So we’ve extended the surface by adding an extra dimension.

This, while baffling to outsiders, is a well known trick by mathematicians. Maybe there’s an easier way to explain it.

The Imaginary Route to School

Let’s suppose that Billy walks from his house (x = -10) to school (x = 10) every day.

Monday, however, he wakes up and hears on the radio that due to faulty engineering and a freak division-by-zero accident, the bridge (x = 0) on the way to school has collapsed. What can Billy (determined and diligent student that he is) do?

The answer is that Billy needs to find another way to school. He needs to complexify his usual route, enter into another dimension, and find a new path.

Billy’s adventure is the essence behind the mathematician’s method: Often, we will have some sort of singularity between two points. To deal with it, we seek a new path. It turns out that useful paths can be taken in the ‘imaginary’ plane. So while this is definitely unphysical and unintuitive, take a lesson from Billy.

If he can warp his mind to change his usual route, so can you.

Singularities and Stokes Lines

Let’s return to our complexified water surface.

Well, it turns out that on this extended surface, there are singularities: points where our asymptotic approximation (remember that?) goes terribly awry. This is the reason why the asymptotic expansions diverge — because of these singularities.

With a lot of work and advanced maths, it can be proven that there exists lines (called Stokes Lines) emanating from these singularities, across which a small exponential term is switched on.

Like into the mist, remember?

And so following the above picture, you can see that the asymptotic approximation (the flat surface) works well at the start. But then it hits the edge of the Stokes Line, and at that point a teensy exponential is turned on.

So actually, we should have,

Exponential Asymptotics is the name we give to the branch of mathematics which studies these small exponentials which seem to fly under the radar of ordinary methods. It’s these hidden exponentials which emerge from the divergent tails of asymptotic expansions.

From Heaven to Hell

It’s time to wrap up.

In each previous part, I ended by asking a crucial question. They were:

Part I: Sometimes, as mere mortals with a capacity for only understanding the finite, we’re forced to cut corners and make approximations. Most of the time, it won’t matter. When will it matter?

Part II: In order to describe nature’s oceans and lakes, mathematicians have had to use divergent (badly-behaved) series approximations. Although these give superb results, we are nonetheless representing perfect phenomenons using a far-from perfect representations. “Inventions of the devil”, said Niels Abel. But this is a practical necessity. How else are we to describe God’s infinity with our finite minds? The real question is, what’s being lost in translation?

In this part, we showed that while divergence is bad, there’s no need to disregard it. By using certain mathematical tools, we can unravel the divergence, and reveal what details were lost using the approximations.

Exponential Asymptotics then, is the bridge between God’s picture-perfect, but wholly unrealizable representations, and the Devil’s ambiguous, but entirely necessary evils.

This bridge is where I stand and this bridge is what I study.

A Review

abel’s hell

“Divergent series are the invention of the devil and it is shameful to base on them any demonstration whatsoever.”

niels abel

Last time, we left off talking about Pi.

We talked about how one can imagine Mother Nature using Pi to all its infinite glory, but for mere mortals like you and I, we have to truncate the number — to its tenth decimal place, hundredth decimal place, or even millionth decimal place.

But however we do it, we still miss infinitely many digits (infinity minus a finite number is still infinity, don’t-cha-know).

This is simply an example of our inability to describe the Universe to its fullest.

For the most part, that’s perfectly fine. But in what circumstance does our inability to capture the true nature of a phenomenon become a problem?

A Hopeless Challenge

Let’s talk oceans.

There are few things more majestic than a ship in water. Captain Ahab knew it. Jack and Rose knew it (well, at least until the end). So should you.

Just imagine it. The sparkling blue waters. The expanse of never-ending ocean. The beautiful wedge-shaped pattern that flows behind every ship, like a streaming cape.

And near the ship: turbulent white-water splashes, jets, and ripples, forming in a million different places with a million different patterns, often too quick and too elaborate for the eye to capture.

The mathematical equations which govern the flow of water in an ocean were known as early back as the 18th century. But for the next century and a bit, scientists were absolutely confounded by the equations.

“What in the world,” they asked together, “Can we do with that?”

The equations, as they stand, are much too complex for a direct analysis. Even today, there is a million dollar prize for anyone who can answer even the most basic questions about these equations.

So with no possibility of exact solutions in sight, the mathematicians and physicists turned to developing methods for approximating solutions.

Asymptotic Approximations

One method, in particular, traces its roots back to the time of Henri Poincare (1854-1912) who developed it in order to solve a rather terrifying problem in celestial mechanics.

The idea is to break up our solution into more manageable chunks. The first chunk describes the system — the ocean, in this case — in some idealized state. To this, we add the second chunk, which includes some kind of perturbation to this initial state. To this, we add a third and even smaller perturbation. And on to infinity.

For example, suppose we were to express the height of the water waves in this fashion,

Here, the greek epsilon, ε represents a small perturbation. It can represent, for example, the Froude number, or the surface tension, or some other small effect.

These sums or series are called asymptotic approximations because they get better and better in the asymptotic limit that ε tends to zero. That’s the idealized state.

The amazing thing with these approximations is that they tend to be very good. So good, in fact, that calculating only one or two chunks provides an excellent approximation to reality in most cases.

This was great. People were happy. We’d at least established some kind of systematic way of approaching these intractable formulae and, with a moderate amount of sweat and tears, they seemed to give excellent results.

The Physicists were happy.

So were the Applied Mathematicians.

The Pure Maths chaps, on the other hand, were incensed.

The Puries and their Partypooper Tendencies

The problem, the Puries were quick to point out, is that these asymptotic approximations are usually divergent.

What that means is that the approximations get better and better as we include more and more terms…but only up to a point. After that point, all hell breaks loose, and the whole thing blows up (to infinity).

Thus in effect, we have taken a system of water waves, perfectly well behaved and all very pretty, and approximated it using an infinite series of terms which, when added up, gives us completely nonsense.

The Pure Mathematicians, in all their need for rules and regulations, just weren’t comfortable toying with a concept that was so ill-defined.

But everybody else was all, “Fuggetaboutit. Take a look at all these purty graphs, yo.”

Should We Worry?

By now you’ve hopefully grappled with issue.

Nature in all its magnificence is woefully complicated.

In many cases, it’s so horrendously complicated that its mathematical description needs to be approximated. One such method of approximation involves using a divergent series.

Why do we use them? We use them because they work so nicely.

So the question becomes, in using these divergent approximations, are we ignoring anything useful? In particular, what is the nature of this divergence? Why does our approximation work so well initially, but behave so badly later on?

What’s the price we pay for dancing with the devil?

On Mother Nature

pi

“God, after all, can effortlessly spit out the infinitely long decimals of Pi. On a whim. While he’s sippin’ a brew and munchin’ some cheetos.

Because that’s how the Almighty rolls.”

Imagine a circle.

That’s right. Take that wonderful goody-two-shoes shape of two dimensional Euclidean geometry. It’s neat, it’s tidy, not as obtuse as a triangle (or god forbid, a square), compact, space efficient, and yes, undeniably sexy.

If Kylie Minogue was a Euclidean shape, she’d be a circle.

Now as we all know, the circle (and its three-dimensional brother, the sphere) is nature’s favourite shape. It occurs in everything, from the shape of soap bubbles in the bath to the wave expansion of sound disturbance at a concert. Everything from teensy atoms to humongous stars and planets are connected with the circle.

You know, in the Medieval Ages, they believed that the compass, with its magical ability to produce the most perfect of shapes — must have been one of God’s divine instruments.

Then again, would God really need a compass if he was God?

Never mind.

The point is, the circle is beautiful. It’s divine. It’s perfect.

On Mathematicians

Now we turn to the mathematicians.

It was known as early as the Egyptians that the ratio of a circle’s circumference to its diameter is constant. After that, more and more accurate approximations to this mysterious number proliferated through the years — 6 decimal places by Archimedes’ time and up to 100 decimal places by Newton’s time.

In 1706, William Jones — probably due to a lack of stationary and ink — began using the symbol Pi to represent the tedious number. Then in 1761, the German mathematician, Johannes Lambert proved Pi is irrational, and in 1882, Ferdinand von Lindemann (also German) proved Pi is transcendental.

Let’s talk about what these two facts imply.

Being irrational, Pi can never be written down as a terminating number, nor does it ever repeat. If you possessed an infinite amount of boredom, time, space, and pencil lead, and wrote out Pi all the way to infinity — you’d never see a pattern.

Being transcendental, Pi can never be constructed with a straight edge ruler and a compass — meaning, given theoretically perfect tools, you would never be able to measure out Pi or in particular, you would never be able to construct a square with the same area as a circle.

You put these two mathematical facts together and there is only one thing to conclude:

Pi is an awful, atrociously foul number.

On the Fundamental Question

And so we come to a fundamental, philosophical divide.

Nature has the ability to make perfect circles every time. God, after all, can effortlessly spit out the infinitely long decimals of Pi. On a whim. Just like that.

Because that’s how He rolls.

To the Almighty, Pi is indeed wonderful, beautiful, and simple.

But for us mere mortals with comparatively little brain space, we usually have to truncate Pi. So when we use calculators or a computers, programmed with only a finite number of decimal places, we’re not using Pi as nature intended it, but the imperfect, imprecise version we’ve constructed for our own evil schemes.

For the most part, this is okay.

Nobody (except for the clinically unhinged) loses sleep over the fact that we’re only using 16 decimal places. The resulting error between mathematical prediction and true phenomenon is so inconsequential, nobody cares.

But you see, all this buildup — it was just a metaphor.

An attempt to explain to you the difference between man, mathematics, and nature.

An attempt to explain that yes, nature is indeed wonderful and perfect, but sometimes, we aren’t always able to capture this magnificence completely. Especially when we impose our own mathematical rules to describe it.

So here’s a question: At what point does the difference between mathematical representation, pragmatic implementation, and true reality have severe consequences.

When it comes to maths, I don’t really consider myself a noob.

Of course, there’s a lot I don’t know. In fact, it’s a common belief that the last mathematician to know ‘everything’ (that is, to at least be familiar with everything) was David Hilbert — and that only really applies to the early 20th century.

Today, there is simply no way for you to be an expert in every field or even, in a substantial number of fields. There’s simply too much to know. You’d be spending the whole span of your life, playing catch-up with what’s been done and what’s being done.

But yeah. I’m not a noob, y’know?

For example, I think it’s safe to say I know my way around integration. Single, double, or triple integration — no problem. Laplace integrals, Fourier integrals, Elliptic integrals — piece of cake. Even more deviant and exotic creatures like Error functions, Fresnel integrals, Airy integrals, and so on and so forth I’ve encountered.

Today, however, I met a new beast. At first glance, I treated with disdain.

After longer, more protracted glance, I’m happy to report I’m still…uh…pretty repulsed by the whole thing.

The formula in question was derived by the Australian mathematician, John Henry Mitchell in 1898. The formula, which can be used to calculate the wave resistance for ships of different forms, was intended to revolutionize the Naval Engineering community.

It didn’t.

The popular theory is that the engineers took one look at the formula, went “Eeewww”, and promptly burnt the article.

People then go on to theorize that Mitchell’s short career (10 years?) in research mathematics was due to his disgust at the reception of this work.

But looking at the formula, can you really blame the community?

Yuck.

Things I Don’t Care to See #138: A quintuplet integration.

P.S. If you’re feeling particularly bad about the poor Aussie, don’t be. His work in the 1898 paper is now seen as one of the most important and revolutionary contributions to hydrodynamic ship resistance.

If only I can be so lucky.