deletions | additions       diff --git a/Trajectory spawning.tex b/Trajectory spawning.tex  index d52c974..6ec5674 100644  --- a/Trajectory spawning.tex  +++ b/Trajectory spawning.tex 
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The recursive equation for $\rho_i$ becomes  \begin{equation}  \rho_i = \left| \vec{n}(x_{i}) \cdot \vec{b}(x_i) \right|  \left[  \int_0^{t_{end}} J(x_i, t) dt  + \chi(x_i, t_{end}) \left< \rho(x) \right>_{x \in \sigma_{i}^{end}}  \right]  \end{equation}  In the above equation $\sigma_i^{start}$ does not appear because $\rho$ only computes the factor by which the surface increases, not the actual number. That will reappear when we get back to the reference volume (the central sphere) 
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\begin{equation}  \mu_{t \to f} =   \left| \vec{n}(x) \cdot \vec{b}(x) \right|  \left[  \int_0^{t_{end}} J(x, t) dt  + \chi(x, t_{end}) \left< \rho(x) \right>_{x \in \sigma_{i}^{end}}  \right]  \end{equation}  At a factor node (spawn surface) you combine all the incoming messages to estimate the relative downstream volume.  \begin{equation} 
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\begin{equation}  \mu_{t \to f} =   \left| \vec{n}(x) \cdot \vec{b}(x) \right|  \left[  \int_0^{t_{death}} J(x, t) dt  + \chi(x, t_{death}) \mu_{f \to t} )  \right]  \end{equation}  note: