Small fixes

1 files changed, 5 insertions, 5 deletions

diff --git a/mgr.tex b/mgr.tex
index 56ad569..3be81ed 100644
--- a/mgr.tex
+++ b/mgr.tex
@@ -802,7 +802,7 @@ This concludes the proof of the lemma, and Theorem
 
 \begin{theorem}\label{relabel-regular-queries-theorem}
     Problem \ref{relabel-regular-queries} can be solved in time
-    \qptime{$O(n)$}{$O(m \log m)$}
+    \qptime{$O(n)$}{$O(m \log m)$}.
 \end{theorem}
 
 The rest of this chapter is dedicated to proving Theorem
@@ -1099,9 +1099,9 @@ this formally, consider the alphabet $\Sigma' := Q \times \Sigma \times Q \times
 \{ \leftsymbol, \rightsymbol \}$. A word over this alphabet can be used to
 represent a subtree with a hole. A subtree with hole as described above would be
 represented by word $\alpha_0 \alpha_1 \ldots \alpha_p$ with each $\alpha_i$
-being a quadruple from $\Sigma'$, $\langle p_i, a_i, q_i, d_i \rangle$ (with
-$p_i, q_i \in Q$, $a_i \in \Sigma$, $d_i$ either $\leftsymbol$ or
-$\rightsymbol$) where:
+being a quadruple $\langle p_i, a_i, q_i, d_i \rangle \in \Sigma'$ (with $p_i,
+q_i \in Q$, $a_i \in \Sigma$, $d_i$ either $\leftsymbol$ or $\rightsymbol$)
+where:
 
 \begin{itemize}
     \item $a_i$ is the original label of $u_i$.
@@ -1134,7 +1134,7 @@ by the previously read letter, what state we arrived in at it and whether it was
 the current vertex's left or right child. The automaton works as follows:
 
 \begin{enumerate}
-    \item In the initial state $\mathsf{initial}$, read $\alph_0 = \langle p_0,
+    \item In the initial state $\mathsf{initial}$, read $\alpha_0 = \langle p_0,
         a_0, q_0, d_0 \rangle$, set the state to $\langle q, d_0 \rangle$.
     \item Now when reading letter $\alpha_i = \langle p_i, a_i, q_i, d_i
         \rangle$ in state $\langle r, d \rangle$, if $d$ if $\leftsymbol$, use