User:Sohale/my favorite formulae

Here are the formula that make me nervous! :D Sohale (talk) 17:00, 25 March 2008 (UTC)

x^a^b
It's quite uncomfortable! :
 * $$\left(x^a\right)^b = x^{ab}$$

How could be that powers commutate?! $$(x^a)^b=(x^b)^a$$ but not with base?

nlogn
What art thou? :
 * $$n\log n$$.

e.g., in $$O\left(n\log n\right)$$. also similar to $$x\ln x$$ which appears in Shannon/Boltzmann entropy.

x|y
The holy bar! :
 * $$\Pr(x|y)$$

and the $$\Pr$$ itself. oof.

TIME^SPACE
veguely:
 * SPACE=log(TIME).

respectively: TIME=exp(SPACE)?!

Energy vs. probability duality

 * $$-\ln(\Pr(x)) \sim E_x$$

actually: $$\Pr(x)\times z_0 \sim \exp\left(-E_x \times \beta \right)$$ for some $$z_0$$ and $$\beta$$. or $$-\ln\left(z_0 \Pr(x)\right) \sim \beta E_x $$ for which $$z_0$$ is a normalization factor or partition function, and $$\beta$$ is Thermodynamic beta.

Misc.

 * Shannon's entropy: The heartburn formula! $$H(X)=\sum p_i\log p_i$$.
 * Log Gamma Function


 * $$\log$$ related formulae: $$\log(1-x)$$, $$\log(n!)$$, ...
 * $$f^{-1}(f(x))$$
 * Also $$;$$ notation in $$f(x,y;p,q)$$ and $$Div(P||Q)$$
 * $$\backslash$$, $$-$$
 * $$H_2=Pr(X_1=X_2)$$

See too

 * User:Sohale/mathasas