Interlude

[quasigeostrophy]

Noodling on Mathematics while I take a break from reading about the annihilator method of undetermined coefficients (there's a mouthful - don't click the link if you're scared ;-)), I have come up with two most likely unoriginal theories:

Sliding Window Memory

As one learns more advanced information regarding a particular topic, earlier, fundamental information on that topic is forgotten. Not completely, but recall is less likely or at least more error-prone. Knowing that, especially as I progress through mathematics education, I have a higher tendency to make sign and/or arithmetic errors, I think I have this to some degree. Watching the professor I have for this summer class, he has it as well. He's much more educated in mathematics, and makes more algebra mistakes than I do, providing evidence for my theory that some subject memory acts as a sliding window - the farther along one progresses, the more earlier stuff drops off the tail end. :-)

F*cking Magic

I've always jokingly referred to mathematics techniques I don't understand as being FM. As I learn more advanced techniques (and, from what I've seen, even what I'm learning now is hardly advanced by comparison to what's out there), I have enough evidence in my opinion not to stop regarding such things as FM but rather to confirm them as such. For example, not to bore or cause headaches with details, but the topic I'm currently reading for class seems to me to be just a way some folks in the 1800s or since developed to play with the numbers to get a solution to some ugly calculus equations they couldn't work out otherwise. Yes, they follow mathematical rules, but I think a lot of these techniques come from a lot of playing around, seeing what works, and developing tricks (only be sure always to call it, please, "research"[1]). Sure, it works. And I get it. But it's still FM.

Anyway, back to it...

----
[1] - From the song Lobachevsky by Tom Lehrer.

Comments

[hitchhiker.livejournal.com]

this post linked to this paper, which contained the following delightful fragment:

To see why seven is special, we first give an argument to establish Theorem 1 in the style of eighteenth-century analysis, where meaningless computations (e.g., manipulating divergent series as though they converged absolutely and uniformly) somehow gave correct results. This argument begins with the observation that a tree either is 0 or splits naturally into two subtrees (by removing the root). Thus the set T of trees satisfies T = 1 + T^2. (Of course equality here actually means an obvious isomorphism. Note that the same equation holds also for the variant notions of tree mentioned at the end of Section 1.) Solving this quadratic equation for T , we find T = (1/2) +/- (i sqrt(3)/2) . (The reader who objects that this is nonsense has not truly entered into the eighteenth-century spirit.) These complex numbers are primitive sixth roots of unity, so we have T^6 = 1 and T^7 = T. And this is why seven-tuples of trees can be coded as single trees.

Although this computation is nonsense, it has at least the psychological effect of suggesting that something like Theorem 1 has a better chance of being true for seven than for five or two. To improve the effect from psychological to mathematical, we attempt to remove the nonsense while keeping the essence of the computation.

[quasigeostrophy.livejournal.com]

Oh, that's fabulous. :-)

[hitchhiker.livejournal.com]

isn't it just? :) my favourite such "result" is the following:

(for notational convenience, let I f = integral f dx)

I e^x = e^x
e^x - I e^x = 0
(1 - I) e^x = 0
e^x = 1/(1-I) 0
e^x = (1 + I + I^2 + I^3 + ....) 0

digression:
I 0 = 1
I^2 0 = I I 0 = I 1 = x
I^3 0 = I I^2 0 = I x = x^2/2!
and so on

so we get
e^x = 1 + x + x^2/2! + x^3/3! + ....

which it indeed is :)

'proof' from Ian Stewart's very highly recommended "Game, Set and Math", where he notes that it can actually be made rigorous.

[quasigeostrophy.livejournal.com]

*snerk* I want to find the thing I saw pinned to an office door in the physics department a few months ago (it's no longer there and I don't know the occupant of the office) that was an elaborate derivation of "simplifying" something like 1+1=2 into about the most complicated thing one might put together, but I haven't been able to find it online as yet.

[djinnthespazz.livejournal.com]

Mark had a much beloved teeshirt, wherein a fellow was writing a proof out on a chalkboard, and one of the parts of the proof was 'and then a miracle occurs' - he wore it to shreds...

[quasigeostrophy.livejournal.com]

I love that comic. "I think you need more detail here." :-)