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\end{equation}  M is a lower-triangular matrix where the diagonal elements are the eigenvalues  $$ \mu_i = 1 - \frac{\alpha^{PA} \alpha^{AP}}{\lambda_i^q} > 0 )$$ $$   since in our model $ \frac{\alpha^{PA} \alpha^{AP}}{\lambda_i^q} = \frac{1-0.00032}{(2i-1)^{1.64}}$     
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