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| -rw-r--r-- | frsb_kac-rice.tex | 13 |
1 files changed, 7 insertions, 6 deletions
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index ab5d456..7e260f9 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -745,13 +745,14 @@ product formula \eqref{eq:cont.replica.prod} to write $CD$ and $R^2$, summing them, and finally using the $\ln\det$ formula \eqref{eq:replica.det.cont}. The saddle point equations take the form \begin{align} - 0&=\hat\mu c(x)+[(\hat\beta^2f'(c)+(2\hat\beta r-d)f''(c)+r^2f'''(c))\ast c](x)+(f'(c)\ast d)(x) \label{eq:extremum.c} \\ - 0&=-\mu^* c(x)+[(\hat\beta f'(c)+rf''(c))\ast c](x)+(f'(c)\ast r)(x) \label{eq:extremum.r} \\ - 0&=c(x)-\big(f'(c)\ast(c\ast d+r\ast r)\big)(x) \label{eq:extremum.d} + 0&=\hat\mu c(x)+\Big[\big(\hat\beta^2(f'\circ c)+(2\hat\beta r-d)(f''\circ c)+r^2(f'''\circ c)\big)\ast c\Big](x)+\big((f'\circ c)\ast d\big)(x) \label{eq:extremum.c} \\ + 0&=-\mu^* c(x)+\Big[\big(\hat\beta(f'\circ c)+r*(f''\circ c)\big)\ast c\Big](x)+\big((f'\circ c)\ast r\big)(x) \label{eq:extremum.r} \\ + 0&=c(x)-\big((f'\circ c)\ast(c\ast d+r\ast r)\big)(x) \label{eq:extremum.d} \end{align} -where $(a\ast b)(x)$ denotes the functional parameterization of the diagonal of -the product of hierarchical matrices $AB$, defined in -\eqref{eq:cont.replica.prod}. +where $(ab)(x)=a(x)b(x)$ denotes the hadamard product, $(a\ast b)(x)$ denotes +the functional parameterization of the diagonal of the product of hierarchical +matrices $AB$ defined in \eqref{eq:cont.replica.prod}, and $(a\circ +b)(x)=a(b(x))$ denotes composition. \subsection{Supersymmetric complexity} |