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q^{\uparrow}_n (t) +  \tau_jn \cdot  \frac{ d q^{\uparrow}_n}{ dt }= \sum_m f_{nm} \cdot q^{\downarrow}_m(t)  - \beta_n  \cdot  \frac{d p_n}{dt}

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q^{\uparrow}_j=n (t) =\tau_j^n^{-1} \exp(-t/\tau_jn) \cdot \int_0^t \exp(s/\tau_j) \left[ \sum_im f_{ijnm} q^{\uparrow}_jm(s) - \beta_jn \frac{dp_jn}{ds} \right] ds 


The objective function is:

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