Exact calculation of S[i][j]

exact.tex

@@ -1,7 +1,7 @@
 % -*- root: main.tex -*-
 \section{Exact Results}
 \label{sec:exact}
-We turn to computing the exact values of $\sum\limits_{\wVec, \wVecPrime \in \pw} \sketchJParam{\sketchHashParam{\wVec}} \cdot \sketchPolarParam{\wVecPrime}.$  Starting with the term $\gIJ = \sum\limits_{\wVecPrime \in \pw}\sketchPolarParam{\wVecPrime}$, by the definition of $\sketchPolar$ and the property of associativity in addition, we can break the sum into
+We turn to computing the exact values of $\sum\limits_{\wVec \in \pw } \sketchJParam{\sketchHashParam{\wVec}} \cdot \sketchPolarParam{\wVecPrime}.$  Starting with the term $\gIJ = \sum\limits_{\wVecPrime \in \pw}\sketchPolarParam{\wVecPrime}$, by the definition of $\sketchPolar$ and the property of associativity in addition, we can break the sum into
 \begin{equation*}
 \gIJ = \sum_{\substack{\wVecPrime \in \pw \st\\
 				\sketchPolarParam{\wVecPrime} = 0}} 1 + \sum_{\substack{\wVecPrime \in \pw \st\\
@@ -28,4 +28,28 @@ T_1 \in \{0, 2^{\numTup - rank(\matrixH')}\}.
 \Else
 	$\gIJ = 0$
 \EndIf.
-\end{algorithmic}
+\end{algorithmic}
+For examining the first term of equation \eqref{eq:allWorlds-est}, we fix $\kMap{t}$ to be defined as
+\begin{equation*}
+\kMapParam{\wVec} = \begin{cases}
+					1,&\text{if } w_t = 1\\
+					0,		&\text{otherwise}.
+				  \end{cases}
+\end{equation*}
+Therefore, by definition we have
+\begin{equation*}
+\sum_{\wVec \in \pw}\sketchJParam{\sketchHashParam{\wVec}} = \sum_{\wVec \in \pw}\kMapParam{\wVec}\sketchPolarParam{\wVec},
+\end{equation*}
+and using the same argument as in $\gIJ$ yields
+\begin{equation*}
+\sum_{\wVec \in \pw \st \sketchPolarParam{\wVec} = 0}\kMapParam{\wVec} - \sum_{\wVec \in \pw \st \sketchPolarParam{\wVec} = 1}\kMapParam{\wVec}.
+\end{equation*}
+Setting $T_3 = \sum_{\wVec \in \pw \st \sketchPolarParam{\wVec} = 0}\kMapParam{\wVec}$, $T_4 = \sum_{\wVec \in \pw \st \sketchPolarParam{\wVec} = 1}\kMapParam{\wVec}$, 
+\begin{equation*}
+T_3 = | \{\wVec \st \matrixH \cdot \wVec = \buck^{(0)}, \kMapParam{\wVec} = 1\}\rightarrow T_3 \in [0, 2^{\numTup - rank(\matrixH')}]
+\end{equation*}
+\begin{equation*}
+T_4 = | \{\wVec \st \matrixH \cdot \wVec = \buck^{(1)}, \kMapParam{\wVec} = 1\}\rightarrow T_4 \in [0, 2^{\numTup - rank(\matrixH') - 1}]
+\end{equation*}
+
+\AH{Next: define the algorithm for initialization of $\sketchJParam{\sketchHashParam{\wVec}}$}

main.tex

@@ -2,6 +2,7 @@
 \documentclass[sigconf]{acmart}
 
 \usepackage{algpseudocode}
+
 \usepackage{comment}
 \usepackage{amsmath}
 \usepackage{amssymb}