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 [1]

• $\log$ 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