User:Sohale/my favorite formulae

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

Contents

[edit] x^a^b

It's quite uncomfortable! :

\left(x^a\right)^b = x^{ab}

How could be that powers commutate?! (xa)b = (xb)a

[edit] nlogn

What art thou? :

nlogn.

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

[edit] x|y

The holy bar! :

\Pr(x|y)

and the \Pr itself. oof.

[edit] TIME^SPACE

veguely:

SPACE=log(TIME).

respectively: TIME=exp(SPACE)?!

[edit] 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 z0 and β. or -\ln\left(z_0 \Pr(x)\right) \sim \beta E_x for which z0 is a normalization factor or partition function, and β is Thermodynamic beta.

[edit] 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 , −
  • H2 = Pr(X1 = X2)