Added comments in Sec 11

sop.tex

@@ -14,7 +14,7 @@ We now seek to bound the variance of a k-way join.
 \ex{\prod_{i = 1}^ks(w_i)\ind{h(w_i) = j}}\cdot \ex{\prod_{i = 1}^k\overline{s(w'_i)}\ind{h(w'_i) = j}} \right)\label{eq:sig-j-last}.
 \end{align}
 
-Before proceeding, we introduce some notation that will aid in communicating the bounds we are about to establish.  First note, that the only terms that survive the expectation above are mappings of $w_i = w'_i = w$ such that each $w_i$ has a match, i.e., no $w_i$ or $w'_i$ stands alone without a matching world in its complimentary set.  To help describe all possible matchings we use m-tuples and functions $f$ and $f'$.
+Before proceeding, we introduce some notation that will aid in communicating the bounds we are about to establish.  First note, that the only terms that survive the expectation above are mappings of $w_i = w'_i = w$ such that each $w_i$ has a match, i.e., no $w_i$ or $w'_i$ stands alone without a matching world in its complimentary set.\AR{Technically the way of you have stated it is not correct: a $w_i$ can be get matched to $w'_j$ for $j\ne i$-- I'm assuming this is what you mean but the statement does not say so.}  To help describe all possible matchings we use m-tuples and functions $f$ and $f'$.
 
 \subsection{M-tuples}
 \begin{Definition}
@@ -22,6 +22,8 @@ Given a $k$-way join, define $m \in [k]$.  An m-tuple then is a set of tuples, e
 \end{Definition}
 For example, when $k = 4$, $m = 2$, the m-tuple, denoted, $m_2$, would be$\left\{\left(1, 3\right), \left(2, 2\right)\right\}$.  Here, $m_{2_{1, 1}} = 1$, and while the tuple $\left(3, 1\right)$ sums up to $k = 4$, we do not include it since we have it's symmetrical term $\left(1, 3\right)$.
 
+\AR{Why is the definition of M-tuples needed? From what I understand you need this to define what kinds of $f$ and $f'$ are allowed but in that case why not state those properties directly in terms of $f$ and $f'$? Actually after reading the next section, I do not see why these properties are needed at all..}
+
 \subsection{f, f'}
 \begin{Definition}
 Functions f, f' are the set of surjective mappings from $k$ to $m$ elements: $f: [k] \rightarrow [m].$
@@ -34,10 +36,15 @@ f(i) = \begin{cases}
 		\widetilde{w_m}	&f(i) = m.
 	\end{cases}
 \end{equation*}
+
+\AR{The equation above does not make any sense to me. $f(i)$ is an integer in $[m]$: why is it getting mapped to a $\widetilde{w_j}$? Also why is the above needed? You just need $f$ to be be surjective and you are done, right? If you meant to say that $w_i\mapsto \widetilde{w_{f(i)}}$ and similarly for $f'$, then just state that and move on.}
+
 In particular, $f$ and $f'$ are machinery for mapping $k$ $\wElem$-world variables to $m$ distinct values.  We restrict $f$ and $f'$ in our results to belonging to the same $m_{t_i}$ tuple.  In the example above, $f$ mappings for $m_{2_1}$ may only cross product with $f'$ mappings for $m_{2_1}$ and not with those for $m_{2_2}$.  Likewise for $f, f'$ mappings of $m_{2_2}$.
 
 Using the above definitions, we can now present the variance bounds for $\sigsq_j$ based on \eqref{eq:sig-j-last}.
 
 \begin{equation*}
-\sigsq_j = \sum_{m \in [k]} \frac{1}{B^m} \sum_{\widetilde{w_1}\cdots\widetilde{w_m}} \sum{f, f'} \prod_{i = 1}^{k} v_i(\widetilde{w_{f(i)}}) v_i(\widetilde{w_{f'(i)}})
-\end{equation*}
+\sigsq_j = \sum_{m \in [k]} \frac{1}{B^m} \sum_{\widetilde{w_1}\cdots\widetilde{w_m}\in W} \sum_{f, f'} \prod_{i = 1}^{k} v_i(\widetilde{w_{f(i)}}) v_i(\widetilde{w_{f'(i)}})
+\end{equation*}
+
+\AR{You need to argue why the above follows from~\eqref{eq:sig-j-last}: either here or somewhere else. Basically how you go from a sum of $2k$ variables to this nested seem-- this needs to be fully argued.}