Eikonal Blog

2010.01.22

Chuck Norris superman meme

Filed under: martial arts, memetics, mind & brain — Tags: , , — sandokan65 @ 16:50

Various cultural heroes have assigned to them list of “facts” which are extreme hyperboles of the virtues exposed by the specific hero. The extremity of these “facts” makes them infectious and memorizeable, i.e. good material to be memes.

The most familiar example in the web culture is Chuck Norris, martial artist and actor (in quite bad martial arts movies and series). Use Google to find various collections of “Chuck Norris facts”, for example:


Related here: Martial arts sites – https://eikonal.wordpress.com/2010/09/21/martial-arts-sites/ | Martial Arts articles – https://eikonal.wordpress.com/2010/09/08/martial-arts-articles/ | Martial Arts magazines and other sources – https://eikonal.wordpress.com/2010/02/05/martial-arts-magazines-and-other-sources/.

BCS formula on computer

Filed under: mathematics — Tags: , , — sandokan65 @ 16:06

Def: e^C :\equiv e^A e^B

C = A + \int_0^1 dt \psi(e^{a}e^{t b}) B,
where a:\equiv ad(A), b:\equiv ad(B) and
\psi(z):\equiv \frac{z \ln(z)}{z-1} = 1 + \frac{w}{1\cdot 2} - \frac{w^2}{2\cdot 3} + \frac{w^3}{3\cdot 4} - \cdots for w:\equiv z-1.

Richtmyer and Greenspan obtained the 512 initial terms of that expansion, but here I present only several of them:

C = A + B + \frac12 a B + \frac1{12} a^2 B - \frac1{12} b a B + \frac1{12,096} a^4 b^2 a B - \frac1{6,048} a^3 b a^3 B - \frac1{3,780} a^3 b a b a B + ...

Not all terms are independent, due to Jacobi identities.

Sources:

  • T1267: R.D. Richtmyer and S. Greenspan: “Expansion of the Campbell-Baker-Hausdorff formula by Computer”; Communications on Pure and Applied Mathematics, Vol XVIII 107-108 (1965).

Guenin on commutators

Filed under: Uncategorized — Tags: , — sandokan65 @ 15:46

A B^n = \sum_{j=0}^n \binom{n}{j} (-)^j B^{n-j} \ ad(B)^j A.

B^n A = \sum_{j=0}^n \binom{n}{j} (-)^j  \ (ad(B)^j A) B^{n-j}.

\partial_s f(F(s)) = \sum_{j=0}^\infty \frac{(-)^j}{(j+1)!} f^{(j+1)}(F(s)) \ ad(F(s)^j \partial_s F(s) = \sum_{j=0}^\infty \frac{1}{(j+1)!}  \ (ad(F(s))^j \partial_s F(s) ) f^{(j+1)}(F(s)).

[H, f(A)] = \sum_{j=0}^\infty \frac{(-)^j}{j!} f^{(j)}(A) \ ad(A)^j H = -  \sum_{j=0}^\infty \frac1{j!}\ (ad(A)^j H) f^{(j)}(A).
—-
Sources:

  • T1266: M. Guenin: “On the Derivation and Commutation of Operator Functionals”; Helv. Phys. Act. 41, 75-76, 1968.

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