From 96efa4aab61d26f673c725183d3da04a722da9ce Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 7 Dec 2020 18:52:17 +0100
Subject: Added a bunch of interesting equations with no context.

---
 bezout.tex | 35 +++++++++++++++++++++++++++++++++++
 1 file changed, 35 insertions(+)

diff --git a/bezout.tex b/bezout.tex
index 2b6512b..3b397e7 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -164,6 +164,41 @@ not be greater than the product over all singular values \cite{Weyl_1912_Das}.
 Therefore, the absence of zero eigenvalues implies the absence of zero singular
 values.
 
+% This is kind of a boring definition...
+\begin{equation} \label{eq:count.def.marginal}
+  \overline{\mathcal N}(\kappa,\epsilon)
+  =\int da\,\overline{\mathcal N}(\kappa,\epsilon,a)
+\end{equation}
+
+\begin{equation} \label{eq:count.zero.energy}
+  \overline{\mathcal N}(\kappa,0,a)
+  =\left[(p-1)a^{p-1}\sqrt{\frac{1-a^{-2}}{a^{2(p-1)}-|\kappa|^2}}\right]^N
+\end{equation}
+
+\begin{equation}
+  \overline{\mathcal N}(\kappa,\epsilon)
+  =\lim_{a\to\infty}\overline{\mathcal N}(\kappa,\epsilon,a)
+  =(p-1)^N
+\end{equation}
+
+For $|\kappa|<1$,
+\begin{equation}
+  \lim_{a\to1}\overline{\mathcal N}(\kappa,\epsilon,a)
+  =0
+\end{equation}
+
+\begin{equation}
+  \lim_{a\to1}\overline{\mathcal N}(1,0,a)
+  =(p-1)^{N/2}
+\end{equation}
+
+\begin{equation} \label{eq:threshold.energy}
+  |\epsilon_{\mathrm{th}}|^2
+  =\frac{p-1}{2p}\frac{(1-|\delta|^2)^2a^{p-2}}
+  {1+|\delta|^2-2|\delta|\cos(\arg\kappa+2\arg\epsilon)}
+\end{equation}
+for $\delta=\kappa a^{-(p-2)}$.
+
 \bibliographystyle{apsrev4-2}
 \bibliography{bezout}
 
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