deletions | additions       diff --git a/Trajectory spawning.tex b/Trajectory spawning.tex  index 9e93fa8..c200c25 100644  --- a/Trajectory spawning.tex  +++ b/Trajectory spawning.tex 
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\begin{equation}  \rho_i = \left| \vec{n}(x_{i}(0)) \cdot \vec{\nabla} U(x_i(0)) \right|  \left[  \int_0^{t_{spawn}} \int_0^{t_{death}}  J(x_i(t), t) dt + J(x_i(t_{spawn}), t_{spawn}) J(x_i(t_{death}), t_{death})  F( \{ \rho_{DS} \} ) \right]  \end{equation}  In the above equation $S_i^{start}$ 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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This can be re-formulated in the language of the message passing algorithm in which there are two types of nodes. The spawn surfaces are factor nodes, and the   trajectories are variable nodes (we'll call them trajectory nodes). The graph is bipartite, in that trajectory node are only connected to factor nodes and visa versa. Again, since we are only using local information, the graph will be a tree. The leaves will be the trajectories that encounter the basin boundary before spawning.  The messages from trajectory nodes to factor nodes will simply be the $\rho$ for that trajectory trajectory. But let's remove the projection that happens at the surface and make that part of the factor node  \begin{equation}  \mu_{t \to f} = \rho \int_0^{t_{death}} J(x_i(t), t) dt  + J(x_i(t_{death}), t_{death}) F( \{ \rho_{DS} \} )  \end{equation}  At a factor node (spawn surface) you combine all the incoming messages to estimate the relative downstream volume.  \begin{equation}  \mu_{f \to t} =   F \left( \left\{  \left| \vec{n}(x_{i}(0)) \cdot \vec{\nabla} U(x_i(0)) \right| \mu_{t \to f}  \right\} \right)  \end{equation}  This message is then passed upstream to the trajectory which initiated the spawning event. This allows us to re-write the $\mu_{t \to f}$ in terms of the single incoming message  \begin{equation}  \mu_{t \to f} = \int_0^{t_{death}} J(x_i(t), t) dt  + J(x_i(t_{death}), t_{death}) \mu_{f \to t} )  \end{equation}