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In a project of mine I've encountered the following set of equations:

$$ \sum_{i=1}^N \left(\frac{1}{M}\sum_{\alpha=1}^Mg_{ij}^\alpha - u_{ij}^* \right) = 0 \qquad \forall: 1\leq j \leq N$$

$$\sum_{i<j}\left( (u_{ij}^*)^2 - \frac{2}{M^2}\sum_{\alpha < \beta}^Mg_{ij}^\alpha g_{ij}^\beta \right) = 0 $$

and ##u_{ij}^*## is the ##u_{ij}## that solves the equation

$$ u_{ij}= \frac{1}{2} + \frac{1}{2}\tanh{\left( - \frac{\theta_i + \theta_j}{2} + 2\cdot J\cdot M \cdot u_{ij}\right)} $$

where the ##\theta_i (i=1...N)## and ##J## are the variables to be solved for (which are ##N+1## different variables) and ##M,N,g_{ij}^\alpha## are known values (observables).

Do you think it is possible to solve this set of equations for ##\theta_i, J##? I would naively say yes, since we have ##N+1## equations (the first two) and ##N+1## variables, but I'm not sure how I would solve this numerically.

I have been stuck on this for a long time so some thoughts from you smart guys would be greatly appreciated!!