From 8bc5cec374682cf79d78f12e37ad2519285b58bb Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 13 Jul 2022 13:25:53 +0200
Subject: More clear notation for the functional extremal conditions.

---
 frsb_kac-rice.tex | 13 +++++++------
 1 file 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}
 
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