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\begin{eqnarray*}
 S_3(r) = \left\lbrace \begin{array}{ll}
                     1 & \quad\mathrm{if}\quad r < r_\mathrm{c} - w \\
                     3x^2 - 2x^3 & \quad\mathrm{if}\quad r < r_\mathrm{c} \quad\mathrm{with\quad} x=\frac{r_\mathrm{c} - r}{w} \\
                     0 & \quad\mathrm{if}\quad r >= r_\mathrm{c}
                 \end{array} \right.
\end{eqnarray*}

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