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\begin{document}
\begin{eqnarray*}
\xi^0 & = & \frac{\xi_O}{m} = \frac{\xi_C}{n} \\
\frac{\beta_O}{\sqrt{m}} & = & \frac{\beta_C}{\sqrt{n}} = \xi^0 \frac{\sqrt{m}+\sqrt{n}}{2}\\
\end{eqnarray*}
\end{document}