Minor changes from supervision

main.tex

@@ -50,8 +50,8 @@ This dichotomy does not only hold for real walls, realised by actual objects in
 $\bddderived(X)$, but also for pseudowalls. Here pseudowalls are defined as
 `potential' walls, induced by hypothetical Chern characters of destabilizers
 which satisfy certain numerical conditions which would be satisfied by any real
-destabilizer, regardless of whether they are realised by actual elements of
-$\bddderived(X)$.
+destabilizer, regardless of whether they are realised by actual semistabilizers
+in $\bddderived(X)$.
 
 Since real walls are a subset of pseudowalls, the irrational $\beta_{-}$ case
 follows immediately from the corresponding case for real walls.
@@ -991,7 +991,7 @@ proof of theorem \ref{thm:rmax_with_uniform_eps}:
 
 \begin{lemmadfn}[
 	Finding better alternatives to $\epsilon_F$:
-	$\epsilon_q^1$ and $\epsilon_q^2$
+	$\epsilon_{q,1}$ and $\epsilon_{q,2}$
 ]
 \label{lemdfn:epsilon_q}
 Suppose $d \in \frac{1}{m}\ZZ$ satisfies the condition in
@@ -1007,16 +1007,16 @@ Then we have:
 
 \begin{equation*}
 	d - \frac{(\aa r + 2\bb)\aa}{2n^2}
-	\geq \epsilon_q^2 \geq \epsilon_q^1 > 0
+	\geq \epsilon_{q,2} \geq \epsilon_{q,1} > 0
 \end{equation*}
 
-Where $\epsilon_q^1$ and $\epsilon_q^2$ are defined as follows:
+Where $\epsilon_{q,1}$ and $\epsilon_{q,2}$ are defined as follows:
 
 \begin{equation*}
-	\epsilon_q^1 :=
+	\epsilon_{q,1} :=
 	\frac{k_q^1}{2mn^2}
 	\qquad
-	\epsilon_q^2 :=
+	\epsilon_{q,2} :=
 	\frac{k_q^2}{2mn^2}
 \end{equation*}
 \begin{align*}
@@ -1033,7 +1033,8 @@ Where $\epsilon_q^1$ and $\epsilon_q^2$ are defined as follows:
 	
 \end{lemmadfn}
 
-It's worth noting that $\epsilon_q^2$ is potentially larger than $\epsilon_q^2$
+It is worth noting that $\epsilon_{q,2}$ is potentially larger than
+$\epsilon_{q,2}$
 but calculating it involves a $\gcd$, a modulo reduction, and a modulo $n$
 inverse, for each $q$ considered.
 
@@ -1077,11 +1078,12 @@ eqn \ref{eqn:finding_better_eps_problem}.
 Since such a $k$ must also satisfy eqn \ref{eqn:better_eps_problem_k_mod_n},
 we can pick the smallest $k_q^1 \in \ZZ_{>0}$ which satisfies this new condition
 (a computation only depending on $q$ and $\beta$, but not $r$).
-We are then guaranteed that the gap $\frac{k}{2mn^2}$ is at least $\epsilon_q^1$.
+We are then guaranteed that the gap $\frac{k}{2mn^2}$ is at least
+$\epsilon_{q,1}$.
 Furthermore, $k$ also satisfies
 eqn \ref{eqn:better_eps_problem_k_mod_gcd2n2_a2mn}
 so we can also pick the smallest $k_q^2 \in \ZZ_{>0}$ satisfying this condition,
-which also guarantees that the gap $\frac{k}{2mn^2}$ is at least $\epsilon_q^2$.
+which also guarantees that the gap $\frac{k}{2mn^2}$ is at least $\epsilon_{q,2}$.
 
 \end{proof}
 
@@ -1094,7 +1096,7 @@ which also guarantees that the gap $\frac{k}{2mn^2}$ is at least $\epsilon_q^2$.
 	are bounded above by the following expression (with $i=1$ or 2).
 
 	\begin{equation*}
-		\frac{1}{2 \epsilon_q^i}
+		\frac{1}{2 \epsilon_{q,i}}
 			\min
 			\left(
 				q^2,
@@ -1104,10 +1106,10 @@ which also guarantees that the gap $\frac{k}{2mn^2}$ is at least $\epsilon_q^2$.
 				-2Cq
 				+q^2
 				+\frac{R}{\lcm(m,2n^2)}
-				+R \epsilon_q^i
+				+R \epsilon_{q,i}
 			\right)
 	\end{equation*}
-	Where $\epsilon_q^i$ is defined as in definition/lemma \ref{lemdfn:epsilon_q}.
+	Where $\epsilon_{q,i}$ is defined as in definition/lemma \ref{lemdfn:epsilon_q}.
 \end{theorem}