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$$ 
k_2 := k_2 (\delta_{max}) = \left\{\begin{matrix}
\hat k_2 & \textup{if } \delta_{max} \geq \delta^*_{max} \\ 
k_1 + (\hat k_2 - k_1 ) \frac{\delta_{max}}{\delta^*_{max}} & \textup{if } \delta_{max} < \delta^*_{max}
\end{matrix}\right.
$$

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