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\begin{document}

I
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current:

$V_K(t) = (t+t^3-t^4)(t^{\frac{1}{2}}+t^{-\frac{1}{2}})$
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or?

$V_K(t) = (t+t^3-t^4)(\sqrt{t}+\frac{\ds 1}{\ds \sqrt{t}})$
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or?

$V_K(t) = (t+t^3-t^4)(\sqrt{t}+\frac{1}{\sqrt{t}})$
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or?

$V_K(t) = (t+t^3-t^4)(\sqrt{t}+1/\sqrt{t})$
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or?

$V_K(t) = (t+t^3-t^4)(\sqrt{t}+\sqrt{t}^{\,-1})$
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$V_K(e^{\frac{2\pi i}{k+2}}) = -\frac{1}{Z}\int_{\mathcal A} 
\left (\mbox{Tr Pexp}\oint_KA\right )e^{\frac{ik}{4\pi}CS(A)}\mathcal{D}A $
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or?

$V_K(e^{2\pi i/(k+2)}) = -\frac{1}{Z}\int_{\mathcal A} 
\left (\mbox{Tr Pexp}\oint_KA\right )e^{(ik/4\pi)CS(A)}\mathcal{D}A $
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II
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$C_{ijk} \eta^{kl} C_{lmn} = C_{mjk} \eta^{kl} C_{lin}$
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III
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$R_{12} R_{23} R_{12} = R_{23} R_{12} R_{23}$
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IV
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{\it Equations for Lorenz attractor:} 
$$\frac{\ds dx}{\ds dt} = \sigma(y-x)$$
$$\frac{\ds dy}{\ds dt} = x(\rho-z) -y$$
$$\frac{\ds dz}{\ds dt} = xy - \beta z$$
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V
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$\p_t v_i + v_j \p_j v_i $
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$= -\p_i p + \nu \p_j \p_j v_i$
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VI 
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$\int_{C_1}\vec{A}\cdot d\vec{\ell} - \int_{C_2}\vec{A}\cdot d\vec{\ell} 
= \frac{1}{2\pi}\Phi$
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or?

$\int_{C_1}{\bf A}\cdot {\bf d \ell} - \int_{C_2}{\bf A}\cdot {\bf d \ell}
= \frac{1}{2\pi}\Phi$
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VII
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\noindent{\it Supergravity:}
$${\mathcal L}=R-\bar{\psi}_{\mu}\gamma^{\mu\rho\sigma}D_{\rho}\psi_{\sigma}$$
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VIII
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$r_S=2Gm/c^2$
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IX 
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$\chi = V - E + F$
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$2\pi\chi = \int_M K~dA$
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X. {\it Pythagoras:}
$$c^2 = a^2 + b^2$$
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XI. {\it Babylonian Tablet}
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XII. {\it Golden mean = partial fraction expansion:}
$$\lim_{n\rightarrow\infty}\frac{F_{n+1}}{F_n} = {1}+ \frac{\ds 1}{1+\dots} $$
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XIII. {\it Archimedes:}
$$v=\frac{2}{3}V$$
$$a=\frac{2}{3}A$$

\end{document}