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| -rw-r--r-- | marginal.tex | 14 |
1 files changed, 7 insertions, 7 deletions
diff --git a/marginal.tex b/marginal.tex index bb4a96d..b3896db 100644 --- a/marginal.tex +++ b/marginal.tex @@ -893,7 +893,7 @@ the contributions from the marginal pieces of the calculation, and is given by &\mathcal U_\mathrm{SSG}(\hat\lambda,Q,X,\hat X\mid\beta,\lambda^*,\mu,C) =\hat\lambda\lambda^* +\lim_{n\to0}\lim_{m_1\cdots m_n\to0}\frac1n\Bigg\{ - \frac12\log\det Q+ + \frac12\log\det Q- \sum_{a=1}^n\bigg( \sum_{\alpha=1}^{m_a}\beta\mu Q_{aa}^{\alpha\alpha} +\hat\lambda\mu Q_{aa}^{11} @@ -1124,7 +1124,7 @@ the complexity is likewise given by \begin{align} &\mathcal U_\mathrm{MSG}(\hat q,\hat\lambda,Q^{11},Q^{22},Q^{12}\mid\beta,\lambda^*,\omega_1,\omega_2) \notag \\ &\quad=\lim_{m\to0}\bigg\{\sum_{\alpha=1}^m\left[\hat q^\alpha(Q^{11,\alpha\alpha}+Q^{22,\alpha\alpha}-1)-\beta(\omega_1Q^{11,\alpha\alpha}+\omega_2Q^{22,\alpha\alpha}-2\epsilon Q^{12,\alpha\alpha})\right] - +\hat\lambda(\omega_1Q^{11,11}+\omega_2Q^{22,11}-2\epsilon Q^{12,11}) \notag \\ + -\hat\lambda(\omega_1Q^{11,11}+\omega_2Q^{22,11}-2\epsilon Q^{12,11}) \notag \\ &\qquad\qquad+\sum_{i=1,2}f_i''(1)\left[\beta^2\sum_{\alpha\gamma}^m(Q^{ii,\alpha\gamma})^2+2\beta\hat\lambda\sum_\alpha^m(Q^{ii,1\alpha})^2+\hat\lambda^2(Q^{ii,11})^2\right] +\frac12\log\det\begin{bmatrix} Q^{11}&Q^{12}\\ @@ -1162,12 +1162,12 @@ limit of $m\to0$ is taken, we find +2(q^{ii}_0)^2 -2(\tilde q^{ii}_0)^2 \right) - -2\beta\hat\lambda\left( + +2\beta\hat\lambda\left( (\tilde q^{ii}_d)^2-(\tilde q^{ii}_0))^2 \right) +\hat\lambda^2(\tilde q^{ii}_d)^2 \right] - +\hat\lambda\tilde q^{ii}_d\omega_i + -\hat\lambda\tilde q^{ii}_d\omega_i -\beta(\tilde q^{ii}_d-q^{ii}_d)\omega_i \right\} \notag \\ &+\frac12\log\bigg[ @@ -1191,9 +1191,9 @@ limit of $m\to0$ is taken, we find \bigg] \notag \\ &-\log\left[(q^{11}_d-q^{11}_0)(q^{22}_d-q^{22}_0)-(q^{12}_d-q^{12}_0)^2\right] - -2\epsilon\big[\hat\lambda\tilde q^{12}_d - -\beta(\tilde q^{12}_d-q^{12}_d)\big] - +\hat q(q^{11}_d+q^{22}_d-1)+\hat{\tilde q}(\tilde q^{11}_d+\tilde q^{22}_d-1) + +2\epsilon\big[\hat\lambda\tilde q^{12}_d + +\beta(\tilde q^{12}_d-q^{12}_d)\big] + -\hat q(q^{11}_d+q^{22}_d-1)+\hat{\tilde q}(\tilde q^{11}_d+\tilde q^{22}_d-1) \label{eq:multispherical.ansatz} \end{align} \end{widetext} |