December 2009 - mathfun@mailman.xmission.com

[math-fun] physics of particle collisions
by Henry Baker 29 Dec '09

29 Dec '09

If I take two identical "colliding" particles -- e.g., electrons -- which repel one another, then in the center-of-mass coordinate system the two particles can approach one another at any angle, including head-on. They may or may not actually "meet", because the repelling force eventually becomes large enough to cause them to bounce away from one another. Since the particles approaching head-on are identical, I can't really tell whether both particles bounced or simply went straight through. The angles after a "collision" are smoothly/continuously related to the angles prior to the collision. Are there any examples of real particles where this isn't so? It wouldn't necessarily violate conservation of energy or momentum, would it? For example, suppose two identical particles approached one another head-on, but instead of bouncing back or going straight through, they shot off at right angles to the original direction, but in equal & opposite directions & with the same energies as before. In 2D, this is well-defined, but in 3D the particles would somehow have to "choose" an arbitrary direction, and then agree on it. So perhaps such a physics might only work for 2D? Clearly, such a 3D physics would violate the conservation of information (it couldn't be reversed), but quantum physics would seem to do this all the time, so that can't be a show-stopper.

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[math-fun] Computational complexity of financial derivatives
by Henry Baker 28 Dec '09

28 Dec '09

If this new (Oct. 2009) paper is correct, there isn't enough computational power on the planet to properly price financial derivatives. This paper compares financial derivatives to used cars (!). Suppose that the buyer of a used car estimates that 20% of used cars are lemons & therefore is willing to pay only 80% of the value of a non-lemon. If the seller knows that he is selling a non-lemon, he is unwilling to sell it for 80% of its true value. The market seizes up due to the lack of information. The paper shows how the information to properly evaluate the derivative may be lying ("lie-ing" !) in plain sight, but due to computational complexity, cannot be utilized. The paper also contemplates a "factor derivative" that pays off on a specific date if the unemployment rate at that date (rounded to an integer) is the last digit of a factor of a large (public) integer. If the person selling such a derivative already knows the factorization of this integer, (s)he is in a much better position to evaluate the derivative than an arbitrary buyer. "Computational Complexity and Information Asymmetry in Financial Products": http://www.cs.princeton.edu/~rongge/ http://www.cs.princeton.edu/~rongge/derivativeFAQ.html http://www.cs.princeton.edu/~rongge/derivative.pdf

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[math-fun] "first derivative"
by Henry Baker 28 Dec '09

28 Dec '09

NYC Mayor Bloomberg used the phrase "first derivative" (correctly) on TV yesterday & I almost fell out of my chair. He didn't just use "derivative", but "first" derivative! I was very impressed that a politician would know calculus. According to Wikipedia, he has a BSEE from Johns Hopkins. http://en.wikipedia.org/wiki/Michael_Bloomberg He must be trying to appeal to the 1% (3% ?) (5% ?) of the U.S. population that took calculus & didn't hate it. Or the financial wizards on Wall Street.

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Re: [math-fun] discrete version of exp(d/dz)
by Dan Asimov 27 Dec '09

27 Dec '09

Mike Stay writes: << It's straightforward to show that exp(d/dz)f(z) = f(z+1). . . . . . . >> I'll just point out this holds for functions f defined by a convergent power series, but not necessarily for those that are not. E.g., the famous example f: R -> R via f(x) = exp(-1/x^2), x <> 0 and f(0) = 0 has exp(d/dx)f(x) = 1 at x = 0 but f(0+1) = f(1) = 1/e. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] Ray Solomonoff
by Dan Asimov 24 Dec '09

24 Dec '09

Rich wrote: << sad news from the e-grapevine ... Ray Solomonoff passed away peacefully on Tuesday 12/8/08, at the age of 83. >> I don't know who he is. If anyone would like to fill in a few biographical details, I'll be happy to be enlightened. Thanks, Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Ray Solomonoff
by rcs＠xmission.com 24 Dec '09

24 Dec '09

sad news from the e-grapevine ... Ray Solomonoff passed away peacefully on Tuesday 12/8/08, at the age of 83. Rich

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Re: [math-fun] Lie-sphere lurker
by Dan Asimov 23 Dec '09

23 Dec '09

Fred wrote: << . . . the rings involved are isomorphic. I'll denote this \H ~= Cl(0,2) . . . The quaternions have a natural order-3 automorphism i -> j -> k -> i, which Cl(0,2) lacks. This causes both computational inconvenience and conceptual confusion, when neophytes attmept to impose a symmetry which doesn't exist. . . . >> OK, so do I have it right: You're saying the quaternions and Cl(0,2) are isomorphic as rings, but unlike the quaternions, Cl(0,2) has no natural order-3 automorphism ? (But Cl(0,2) must have *some* order-3 isomorphism, even if not "natural", by virtue of being isomorphic to H.) I'm not sure I'd call i -> j -> k -> i a "natural" isomorphism of H, because its naturalness depends on a chosen basis for H. (More precisely, a choice of oriented 2-subspace of pure quaternions.) The automorphism group of H is SO(3), so without such a choice, any 120-degree rotation in SO(3) would be equally natural. I think. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] new oscillator from the Life list, period M607
by rcs＠xmission.com 23 Dec '09

23 Dec '09

Wed, 23 Dec 2009 13:42:31 -0000 [06:42:31 AM MST] From: calcyman <calcyman(a)yahoo.com> To: LifeCA(a)yahoogroups.com Subject: [LifeCA] 183-digit prime-period oscillator http://calcyman.awardspace.co.uk/life/basis17.lif The oscillator repeats every 2^607-1 generations. It is accomplished by a p8192 base loop (2^13) followed by 506 period doublers (2^506) and 44 period quadruplers (2^88) and a 1 generation glider advancer (-1). This corresponds to the Mersenne prime M607, which has 183 digits. This method can be extended to yield any Mersenne prime oscillator, including the recently discovered 12-million-digit megaprime. To avoid timing issues, the glider advancer is asynchronous. The input glider is allowed to arrive at any time (apart from a small 'window' where it collides with the clock glider). http://en.wikipedia.org/wiki/Mersenne_prime Merry Christmas and a Happy New Decade, Adam P. Goucher

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Re: [math-fun] Lie-sphere lurker
by Dan Asimov 23 Dec '09

23 Dec '09

Fred wrote: << Leading one to conclude that spending more time tuning the formatting would be unlikely to prove a worthwhile investment. Shame about that! >> That may be true for that particular website. But when notation is needlessly burdened with extra symbols, I'd guess 99% of mathematicians, including 100% of me, would be far less likely to bother to try to understand what you're trying to say. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Method for summing Zeta(even)
by Dan Asimov 22 Dec '09

22 Dec '09

I recently learned from mathematician Mikael Passare a nice way to see Zeta(2), as long as you accept two ways to express sin(x) / x: sin(x) / x = Sum{k=0..oo} (-1)^k x^(2k) / (2k+1)! and sin(x) / x = Prod{n=1..oo} (1 - x^2/(n pi)^2). The value of Zeta(2) then falls out immediately by equating coefficients of x^2. It soon became clear that equating coefficients of higher terms will, with a minor trick or two thrown in, reveal the sum for any Zeta(E) for E in 2Z+. (At least this also works for Zeta(4) and Zeta(6).) Along the way a few other interesting sums fall out. --Dan STAIRLIKE / TRISKELIA _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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