diff --git a/g.tex b/g.tex
index 854cbf9..941fdd9 100644
--- a/g.tex
+++ b/g.tex
@@ -543,29 +543,19 @@ This probability is given by
 	= \sum_{\ket{i_1} \in \cD_{I_0}} \cdots \sum_{\ket{i_{n-1}} \in \cD_{I_0}} 
 	p_{I_0 \to i_1} \ldots p_{i_{n-2} \to i_{n-1}} p_{i_{n-1} \to I}.
 \ee
-Since the sums are restricted to states belonging to the domain it is convenient to introduce a projector for over each domain
+Since the sums are restricted to states belonging to the domain, it is convenient to introduce a projector over each domain
 \be
 \label{eq:pi}
-	P_I = \sum_{\ket{i} \in \cD_I} \dyad{i}{i}.
+	P_I = \sum_{\ket{i} \in \cD_I} \dyad{i}{i},
 \ee
-and, also,
-the projection of the operator $T^+$ over the domain, \ie,
-\be
-	T^+_I= P_I T^+ P_I.
-\ee
-Th operator $T^+_I$  governs the dynamics of the walkers trapped in the domain $\cD_{I}$,
-see Eq.(\ref{eq:pij}) where $T^+$ is now restricted to the domain.
+as well as the projection of $T^+$ over $\cD_I$, \ie, $T^+_I = P_I T^+ P_I$, which governs the dynamics of the walkers trapped in this domain.
+%see Eq.~\eqref{eq:pij} where $T^+$ is now restricted to the domain.
 Using Eqs.~\eqref{eq:pij} and \eqref{eq:eq1C}, the probability can be rewritten as
 \be
 \label{eq:eq3C}
 	\cP_{I_0 \to I}(n) = \frac{1}{\PsiG_{I_0}} \mel{I_0}{\qty(T^+_{I_0})^{n-1} F^+_{I_0}}{I} \PsiG_{I},
 \ee
-where the operator 
-\be
-\label{eq:Fi}
-	F^+_I =  P_I T^+ (1-P_I),
-\ee
-corresponding to the last move connecting the inside and outside regions of the domain, has the following matrix elements:
+where the operator $F^+_I =  P_I T^+ (1-P_I)$, corresponding to the last move connecting the inside and outside regions of the domain, has the following matrix elements:
  \be
 	(F^+_I)_{ij} = 
 	\begin{cases}
@@ -582,9 +572,7 @@ the probability of being trapped $n$ times in $\cD_{I}$, just by summing over al
 \label{eq:PiN}
 	P_{I}(n) = \frac{1}{\PsiG_{I}} \mel{ I }{ \qty(T^+_{I})^{n-1} F^+_{I} }{ \PsiG }.
 \ee
-The normalization to one of this probability can be verified 
-by using 
-the fact that 
+The normalization of this probability can be verified using the fact that 
 \be
 \label{eq:relation}
 	\qty(T^+_{I})^{n-1} F^+_I = \qty(T^+_{I})^{n-1} T^+ - \qty(T^+_I)^n,