From b82d78af197676c90fa17e669bea7b7805b480c2 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 10 Dec 2020 15:32:16 +0100
Subject: Messy mathop/mathrm replaced by operatorname.

---
 bezout.tex | 44 ++++++++++++++++++++++----------------------
 1 file changed, 22 insertions(+), 22 deletions(-)

(limited to 'bezout.tex')

diff --git a/bezout.tex b/bezout.tex
index bfabdec..0c395c3 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -118,38 +118,38 @@ of random polynomials \cite{Bogomolny_1992_Distribution}, which are for $N=1$ an
 
 Since $H$ is holomorphic, a critical point of $\Re H_0$ is also a critical  point of $\Im H_0$. The number of
 critical points of $H$ is therefore the number of critical points of
-$\mathop{\mathrm{Re}}H$. From each critical point emerges a gradient line of $\Re H_0$, which is also one of constant phase $\Im H_0=const$.
+$\operatorname{Re}H$. From each critical point emerges a gradient line of $\Re H_0$, which is also one of constant phase $\Im H_0=const$.
 
-Writing $z=x+iy$, $\mathop{\mathrm{Re}}H$ can be
+Writing $z=x+iy$, $\operatorname{Re}H$ can be
 interpreted as a real function of $2N$ real variables. The number of critical
 points it has is given by the usual Kac--Rice formula:
 \begin{equation} \label{eq:real.kac-rice}
   \begin{aligned}
     \mathcal N_J(\kappa,\epsilon)
-      &= \int dx\,dy\,\delta(\partial_x\mathop{\mathrm{Re}}H)\delta(\partial_y\mathop{\mathrm{Re}}H) \\
+      &= \int dx\,dy\,\delta(\partial_x\operatorname{Re}H)\delta(\partial_y\operatorname{Re}H) \\
       &\qquad\times\left|\det\begin{bmatrix}
-          \partial_x\partial_x\mathop{\mathrm{Re}}H & \partial_x\partial_y\mathop{\mathrm{Re}}H \\
-          \partial_y\partial_x\mathop{\mathrm{Re}}H & \partial_y\partial_y\mathop{\mathrm{Re}}H
+          \partial_x\partial_x\operatorname{Re}H & \partial_x\partial_y\operatorname{Re}H \\
+          \partial_y\partial_x\operatorname{Re}H & \partial_y\partial_y\operatorname{Re}H
         \end{bmatrix}\right|.
   \end{aligned}
 \end{equation}
 The Cauchy--Riemann equations may be used to write \eqref{eq:real.kac-rice} in
 a manifestly complex way.  Using the Wirtinger derivative
 $\partial=\partial_x-i\partial_y$, one can write
-$\partial_x\mathop{\mathrm{Re}}H=\mathop{\mathrm{Re}}\partial H$ and
-$\partial_y\mathop{\mathrm{Re}}H=-\mathop{\mathrm{Im}}\partial H$. Carrying
+$\partial_x\operatorname{Re}H=\operatorname{Re}\partial H$ and
+$\partial_y\operatorname{Re}H=-\operatorname{Im}\partial H$. Carrying
 these transformations through, we have
 \begin{equation} \label{eq:complex.kac-rice}
   \begin{aligned}
     \mathcal N_J&(\kappa,\epsilon)
-      = \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H) \\
+      = \int dx\,dy\,\delta(\operatorname{Re}\partial H)\delta(\operatorname{Im}\partial H) \\
         &\qquad\qquad\qquad\times\left|\det\begin{bmatrix}
-            \mathop{\mathrm{Re}}\partial\partial H & -\mathop{\mathrm{Im}}\partial\partial H \\
-            -\mathop{\mathrm{Im}}\partial\partial H & -\mathop{\mathrm{Re}}\partial\partial H
+            \operatorname{Re}\partial\partial H & -\operatorname{Im}\partial\partial H \\
+            -\operatorname{Im}\partial\partial H & -\operatorname{Re}\partial\partial H
           \end{bmatrix}\right| \\
-      &= \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H)
+      &= \int dx\,dy\,\delta(\operatorname{Re}\partial H)\delta(\operatorname{Im}\partial H)
         \left|\det[(\partial\partial H)^\dagger\partial\partial H]\right| \\
-      &= \int dx\,dy\,\delta(\mathop{\mathrm{Re}}\partial H)\delta(\mathop{\mathrm{Im}}\partial H)
+      &= \int dx\,dy\,\delta(\operatorname{Re}\partial H)\delta(\operatorname{Im}\partial H)
         |\det\partial\partial H|^2.
   \end{aligned}
 \end{equation}
@@ -208,7 +208,7 @@ where
       d & b^* & \hat c & \hat a \\
       b & d^* & \hat a & \hat c^*
     \end{bmatrix}
-    +\mathop{\mathrm{Re}}\left\{\frac18\left[\hat aa^{p-1}+(p-1)|d|^2a^{p-2}+\kappa(\hat c^*+(p-1)b^2)\right]-\epsilon b\right\}
+    +\operatorname{Re}\left\{\frac18\left[\hat aa^{p-1}+(p-1)|d|^2a^{p-2}+\kappa(\hat c^*+(p-1)b^2)\right]-\epsilon b\right\}
     +\int d\lambda\,\rho(\lambda)\log|\lambda|^2
   \end{equation}
   where $\rho(\lambda)$, the distribution of eigenvalues $\lambda$ of
@@ -218,8 +218,8 @@ where
   \begin{equation} \label{eq:free.energy.a}
     \begin{aligned}
       f(a)&=1+\frac12\log\left(\frac4{p^2}\frac{a^2-1}{a^{2(p-1)}-\kappa^2}\right)+\int d\lambda\,\rho(\lambda)\log|\lambda|^2 \\
-          &\hspace{80pt}-\frac{a^p(1+p(a^2-1))-a^2\kappa}{a^{2p}+a^p(a^2-1)(p-1)-a^2\kappa^2}(\mathop{\mathrm{Re}}\epsilon)^2
-          -\frac{a^p(1+p(a^2-1))+a^2\kappa}{a^{2p}-a^p(a^2-1)(p-1)-a^2\kappa^2}(\mathop{\mathrm{Im}}\epsilon)^2,
+          &\hspace{80pt}-\frac{a^p(1+p(a^2-1))-a^2\kappa}{a^{2p}+a^p(a^2-1)(p-1)-a^2\kappa^2}(\operatorname{Re}\epsilon)^2
+          -\frac{a^p(1+p(a^2-1))+a^2\kappa}{a^{2p}-a^p(a^2-1)(p-1)-a^2\kappa^2}(\operatorname{Im}\epsilon)^2,
     \end{aligned}
   \end{equation}
 \end{widetext}
@@ -244,8 +244,8 @@ $\overline{|\partial_i\partial_j H_0|^2}=p(p-1)a^{p-2}/2N$ and
 $\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is
 constant inside the ellipse
 \begin{equation} \label{eq:ellipse}
-  \left(\frac{\mathop{\mathrm{Re}}(\lambda e^{i\theta})}{a^{p-2}+|\kappa|}\right)^2+
-  \left(\frac{\mathop{\mathrm{Im}}(\lambda e^{i\theta})}{a^{p-2}-|\kappa|}\right)^2
+  \left(\frac{\operatorname{Re}(\lambda e^{i\theta})}{a^{p-2}+|\kappa|}\right)^2+
+  \left(\frac{\operatorname{Im}(\lambda e^{i\theta})}{a^{p-2}-|\kappa|}\right)^2
   <\frac{p(p-1)}{2a^{p-2}}
 \end{equation}
 where $\theta=\frac12\arg\kappa$ \cite{Nguyen_2014_The}. The eigenvalue
@@ -291,7 +291,7 @@ the numerator of the Green function \cite{Livan_2018_Introduction} gives
     G(\sigma)=\lim_{n\to0}\int d\zeta\,d\zeta^*\,(\zeta_i^{(0)})^*\zeta_i^{(0)}
       \exp\left\{
       \frac12\sum_\alpha^n\left[(\zeta_i^{(\alpha)})^*\zeta_i^{(\alpha)}\sigma
-        -\Re\left(\zeta_i^{(\alpha)}\zeta_j^{(\alpha)}\partial_i\partial_jH\right)
+        -\operatorname{Re}\left(\zeta_i^{(\alpha)}\zeta_j^{(\alpha)}\partial_i\partial_jH\right)
       \right]
     \right\}
   \end{equation}
@@ -307,20 +307,20 @@ the numerator of the Green function \cite{Livan_2018_Introduction} gives
     \overline G(\sigma)=N\lim_{n\to0}\int d\alpha_0\,d\chi_0\,d\chi_0^*\,\alpha_0
     \exp\left\{nN\left[
       1+\frac{p(p-1)}{16}a^{p-2}\alpha_0^2-\frac{\alpha_0\sigma}2+\frac12\log(\alpha_0^2-|\chi_0|^2)
-      +\frac p4\mathop{\mathrm{Re}}\left(\frac{(p-1)}8\kappa^*\chi_0^2-\epsilon^*\chi_0\right)
+      +\frac p4\operatorname{Re}\left(\frac{(p-1)}8\kappa^*\chi_0^2-\epsilon^*\chi_0\right)
     \right]\right\}.
   \end{equation}
 \end{widetext}
 The argument of the exponential has several saddles. The solutions $\alpha_0$
 are the roots of a sixth-order polynomial, but the root with the
-smallest value of $\mathop{\mathrm{Re}}\alpha_0$ in all the cases we studied gives the correct
+smallest value of $\operatorname{Re}\alpha_0$ in all the cases we studied gives the correct
 solution. A detailed analysis of the saddle point integration is needed to
 understand why this is so. Given such $\alpha_0$, the density of singular
 values follows from the jump across the cut, or
 \begin{equation} \label{eq:spectral.density}
   \rho(\sigma)=\frac1{i\pi N}\left(
-    \lim_{\mathop{\mathrm{Im}}\sigma\to0^+}\overline G(\sigma)
-    -\lim_{\mathop{\mathrm{Im}}\sigma\to0^-}\overline G(\sigma)
+    \lim_{\operatorname{Im}\sigma\to0^+}\overline G(\sigma)
+    -\lim_{\operatorname{Im}\sigma\to0^-}\overline G(\sigma)
   \right)
 \end{equation}
 
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cgit v1.2.3-54-g00ecf

