From 35f67091ec755281f38898aaff9b9d23610c43af Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 7 Feb 2022 12:22:03 +0100
Subject: Changed order of paragraphs.

---
 stokes.tex | 32 ++++++++++++++++----------------
 1 file changed, 16 insertions(+), 16 deletions(-)

diff --git a/stokes.tex b/stokes.tex
index ce01b52..3c8af84 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -442,22 +442,6 @@ eigenvalue problem. If we did not know the eigenvalues were real, we could
 still see it from the second implied equation, $(\beta\partial\partial\mathcal
 S)^*v^*=\lambda v$, which is the conjugate of the first if $\lambda^*=\lambda$.
 
-The effect of changing the phase of $\beta$ is revealed by
-\eqref{eq:generalized.eigenproblem}. Writing $\beta=|\beta|e^{i\phi}$ and
-dividing both sides by $|\beta|e^{i\phi/2}$, one finds
-\begin{equation}
-  \partial\partial\mathcal S(e^{i\phi/2}v)
-  =\frac{\lambda}{|\beta|}e^{-i\phi/2}v^*
-  =\frac{\lambda}{|\beta|}(e^{i\phi/2}v)^*
-\end{equation}
-Therefore, one only needs to consider solutions to the eigenproblem for the
-action alone, $\partial\partial\mathcal Sv_0=\lambda_0 v_0^*$, and then rotate the
-resulting vectors by a constant phase corresponding to half the phase of
-$\beta$ or $v(\phi)=v_0e^{-i\phi/2}$. One can see this in the examples of Figs. \ref{fig:1d.stokes} and
-\ref{fig:thimble.orientation}, where increasing the argument of $\beta$ from
-left to right produces a clockwise rotation in the thimbles in the
-complex-$\theta$ plane.
-
 Something somewhat hidden in the structure of the real hessian but more clear
 in its complex form is that each eigenvalue comes in a pair, since
 \begin{equation}
@@ -474,6 +458,22 @@ real eigenvectors which determine its index in the real problem are accompanied
 by $N$ purely imaginary eigenvectors, pointing into the complex plane and each
 with the negative eigenvalue of its partner.
 
+The effect of changing the phase of $\beta$ is revealed by
+\eqref{eq:generalized.eigenproblem}. Writing $\beta=|\beta|e^{i\phi}$ and
+dividing both sides by $|\beta|e^{i\phi/2}$, one finds
+\begin{equation}
+  \partial\partial\mathcal S(e^{i\phi/2}v)
+  =\frac{\lambda}{|\beta|}e^{-i\phi/2}v^*
+  =\frac{\lambda}{|\beta|}(e^{i\phi/2}v)^*
+\end{equation}
+Therefore, one only needs to consider solutions to the eigenproblem for the
+action alone, $\partial\partial\mathcal Sv_0=\lambda_0 v_0^*$, and then rotate the
+resulting vectors by a constant phase corresponding to half the phase of
+$\beta$ or $v(\phi)=v_0e^{-i\phi/2}$. One can see this in the examples of Figs. \ref{fig:1d.stokes} and
+\ref{fig:thimble.orientation}, where increasing the argument of $\beta$ from
+left to right produces a clockwise rotation in the thimbles in the
+complex-$\theta$ plane.
+
 These eigenvalues and vectors can be further related to properties of the
 complex symmetric matrix $\beta\partial\partial\mathcal S$. Suppose that
 $u\in\mathbb R^N$ satisfies the eigenvalue equation
-- 
cgit v1.2.3-70-g09d2