Thursday, March 05, 2009

King Leopard II

King Leopard (properly pronounced Lee-oh-pard) ruled the Belgian Congo. His reign was a beneficient one; for, after all, he had come to educate the native peoples, teaching them of Leopardism, the One True Religion. (Its primary tenets were the primacy of the Leopard in all creation, the duty for lesser species to abase themselves before Leopards, and the delight and joy that is Roquefort cheese.*) King Leopard educated the people, King Leopard improved the people, King Leopard protected the people from those who would exploit them; and all he demanded in return was a life of backbreaking labour, absolute subjugation before his royal majesty, and a willingness to, at any given moment, be devoured within his fanged** maw. Who could possibly oppose such a kind ruler; especially when there were so many crueler tyrants lurking, just waiting to pounce?

Unbelievably yet, there were some who challenged the rule of King Leopard. But they were unable to surmount the challenges that he set - for King Leopard was as cunning as he was wise. For any who would depose him from his rule over the Belgian Congo, he set this challenge: to solve a very special riddle, a riddle of unparalled complexity. Should they solve it, he would resign peacefully; many had tried. But none succeeded. (This provided a splendid source of food for the ever-hungry King Leopard, delighting him further.)

The riddle was as follows:

For the First Leopard, at any given time, the sum of his acceleration summed with five times his position is equal to four times his velocity summed with one metre for each second since he had first begun to move. When he began, he was five metres from the centre of the earth, and moving away from it at a speed of exactly one metre per second; he always moves along a line exactly fixed between the centre of the earth and its tallest mountain. Where, then, is the First Leopard at any given time, in terms of time and time only?

This riddle seemed impenetrable; many were cowed by it, and none surpassed it. Until one day, a traveller came to the court of King Leopard. His hat was large, and his tailcoat dark; a feather-plume jutted from his shoulder. "I will solve your riddle, Your Majesty," the stranger said, "And I will solve it with mathematics."

"Just try it!" cried King Leopard, laughing a leopard-like laugh. "And I shall enjoy devouring you, from your fresh-pressed boots to the crown of your hat."

"With my apologies, Your Majesty," the stranger said deferentially, "You shan't have the pleasure today."

"So! If I am to solve this problem in the space of a week or less, we had best establish some short-hand for our variables. Acceleration will be called 'a'; velocity 'v', and position 'p'. (Time, of course, will be 't'.) Thus, the equation is a + 5p = 6v + t. Are we agreed thus far?"

"Entirely so," King Leopard said, yawning.

"But we don't really have four variables," the stranger said: "We have just two. For velocity is just the derivative of position with respect to time, and acceleration is the derivative of velocity with respect to time. We may then rephrase the equation, with p' being the first derivative of position with respect to time, and p' being the second such, as p''+ 5p = 6p' + t. And of course we may readily subtract 4p' from both sides, resulting in the equation p'' - 6p' +5p = t."

"Derivatives?" King Leopard said, startled. "I have never heard of such a thing."

"Curious artifacts of a device called the calculus," one of his advisors whispered, "a cumbersome and baroque device. Fear not: I know its limits well, and it cannot crack this nut, not with derivative nor integral."

"Very well," King Leopard said, with a gracious wave of his paw. "Continue."

"I will digress a moment," the traveller said. "Let us imagine that the equation for the First Leopard's position, with respect only to time, is of the form of a constant multiplying the exponential of another constant times time; that is, succinctly, p = c*e^(rt). Continuing the hypothetical, we may take the first derivative of this position function, naturally finding it equal to p' = cr*e^(rt), and the second derivative, that being p'' = cr^2+e^(rt). Is this all clear?"

During this time, the King's advisors had been frentically whispering in his ears, explaining the basics of the calculus to him. Now he lifted his head, gazing majestically at the challenger. A glob of saliva dropped from the corner of his mouth. "Positively crystalline," the King replied. "Continue."

"So we now have - hypothetically - some values for p, p', and p''," the stranger continued. "Substituting them into the original equation, and momentarily neglecting the term on the right side, we have cr^2+e^(rt) - 6*cr*e^(rt) + 5* c*e^(rt) = 0. We know that e^(rt) is never 0 - it is an exponential! - so we may divide both sides by it, leaving ourselves with cr^2 - 6cr + 5c = 0. We may further assume that our constant c is never zero, and divide by that as well, with the equation now being r^2 - 6r + 5 = 0. Can you see what we will do next, Your Majesty?" the stranger asked.

"Of course I can!" King Leopard replied. "This is a simple quadratic equation - factorable, even! - so we can say that (r - 1)(r - 5) = 0, and therefore r equals either 5 or 1. But what can be done with this queer, hypothetical constant? It even has two values!"

"It's quite simple, Your Majesty," the stranger replied. "We put both values into the equation we derived earlier - so, for two still unknown constants c and d, p = ce^t + de^(5t), plus another term or two to deal with the term we neglected earlier, the t on the right side."

"Oh?" asked King Leopard, his voice sneering. "And just how will you do that?"

"I'm happy to demonstrate," the stranger said. "Let's split our function for 'p' into two parts: call the first 'r', and let it be the exponentials that we've already dealt with, and let the second be 'q', which we don't know quite yet. Now - it seems logical that if 'o' deals with the case in which a - 6v +5p = 0, then summing some other function to it would be all that is necessary to account for a function summed to that 0, yes?"

"Ah - for the moment, I... think," King Leopard said, his mind working furiously. "Go on."

"So let - again, purely hypothetically - let our 'q' be composed of two terms, a constant A multiplying t and another constant B, standing alone. In this case, q' would be equal simply to A, and q'' would be nothing at all. Does that make sense?"

"Constants upon constants!" the King moaned. "Hypotheticals upon hypotheticals!"

"I will proceed, then," the traveller said mercilessly. "Once more substituting q into our original expression (in place of p), we find that -6A + 5At + 5B = t. Logically, the terms containing no t must sum to 0; so 5B - 6A = 0, and B = 6A/5. Equally so, the t term on the left must be equal to that on the right; 5At = t, so, for this to hold for all t, A must equal one-fifth, and B must be six twenty-fifths. Simple algebra, yes?"

"...yes..." King Leopard agreed, nervously. He had a bad feeling. "...so?"

"So, Your Majesty," the stranger said, "We are nearly done. Our equation for p, the sum of o and q, is ce^t + de^(5t) + t/5 + 6/25. You may note that we still have two constants left to eliminate, c and d. Luckily, we have saved ourselves a pair of equations to deal with them: 'when he began, he was five metres from the centre of the earth, and moving away from it at a speed of exactly one metre per second.' When t is 0, p = 5, and v, or p', = 1. We may readily substitute this into our equations, finding that p(0) = c + d + 6/25 = 5, and p'(0) = c + 5d + 1/5 = 1. Therefore c = 4/5 - 5d, which we may substitute into the first equation to find d - 5d + 4/5 + 6/25 = 5, or 99/500 = d, and that back again to see that c = -19/100. And so we are done: p = -19e^t/100 + 99e^(5t)/500 + t/5 + 6/25. You may check this if you wish; be assured that it is perfectly correct."

King Leopard turned to his advisors; but they were in shock, furiously scribbling into their notebooks and muttering among themselves. "I can't see any flaw!" one cried. "His mathematics seem impeccable!" said another. King Leopard turned to the stranger; who simply stood there, leaning against a tree, smirking faintly.

King Leopard knew better than anyone that desperate times demand desperate measures; and what circumstance was more desperate than the thought that he might have to give up his rulership of the Belgian Congo? He tensed in decision; and then he leapt at the stranger, fanged*** maw held wide. The traveller needed to do no more than to draw five lines in the air, and thrust the Harmonic Series squarely between King Leopard's jaws; and so, in this understated way, he put an end to our tale, of the tyranny of King Leopard in the Belgian Congo, and of the brave mathematician who brought it to an unbounded conclusion.




*It prevents indigestion! It cures toothache! It serves as a viable alternative to grout!

**Within the Belgian Congo, this was properly pronounced "fang-ed". Two syllables.

***Please remember the earlier note.

1 comment:

Calvacadeofcats said...

how renimiscent of the malphigian tubules of a summers day, and a the glomerulus of harsh wintres