Throw away equations.

December 14, 2011 at 01:03
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    \[    \pi_p = 4\frac{\Gamma(\frac{1}{p})^2}{\Gamma(\frac{2}{p})p^2}  \]

    \[  \|x\|_p=\sqrt[p]{\left(|x_1|^p+|x_2|^p\right)} \]

    \[ \pi_p = 2\int_0^1 \left(1 + \left|-t^{p-1}(1-t^p)^{\frac{1}{p}-1}\right|^p\right)^{\frac{1}{p}} dt \]

    \[  \|x\|_p=\sqrt[p]{\left(|x_1|^p+|x_2|^p + |x_3|^p + \cdots + |x_n|^p\right)} \]

    \begin{align*} \pi_{-\infty} &= 2\\ \pi_{0^-} &= 1\\ \pi_{0^+} &= \infty\\ \pi_1 &= 4\\ \pi_{\mathrm{min}^+} &= \pi \\ \pi_{\infty} &= 4 \end{align*}

    \[ f(x) = U''(U')^{-\frac{2}{3}} + aU^\frac{1}{3} + b h'(x) = 0 \]

Pi_p vs p

    \[ \rho_c = \left( \begin{array}{ccc} \rho_{11} & \rho_{12} & \ldots \\ \rho_{21} & \rho_{22} & \ldots \\ \vdots & \vdots & \\ \end{array} \right) \]

    \begin{align*}  a_1 &= I_2 \otimes |0\rangle\langle 1|\\  a_2 &= |0\rangle\langle 1| \otimes I_2 \\ \end{align*}

    \[ H(\omega_i) = \left( \begin{array}{cccc} \omega & 0 & 0 & 0 \\ 0 & \omega_1 & 0 & 0 \\ 0 & 0 & \omega_2 & 0\\ 0 & 0 & 0  & (\omega_1 + \omega_2) - \omega \end{array}\right) \]

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