I have this equation, and I'm not sure how to solve for the constant $\nu$, since everything else is known:
$$\begin{equation} a + \sqrt{a_i + 4 b_i \nu} + \sum^N_{j=1} (\sqrt{a_j + 4 b_j \nu}) p_{i,j}= 2\omega \end{equation}$$
I merely need to extract $\nu$ and $p_{i,j}$ are transition probabilities for a Markov chain. $a,b,p$ and $\omega$ are all known.