diff --git a/analysis.tex b/analysis.tex index e330cb8..da908ad 100644 --- a/analysis.tex +++ b/analysis.tex @@ -47,8 +47,31 @@ For a given $\wVec \in \pw$, substituting definitions we have \expect{\sum\limits_{\substack{\wVecPrime, \wVec \in \pw \st \\ \sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec} \\ \wVecPrime \neq \wVec}}\kMapParam{\wVecPrime}\sketchPolarParam{\wVecPrime}\sketchPolarParam{\wVec}}\label{eq:step-three}\\ +=&~\expect{\sum\limits_{\substack{\wVecPrime \in \pw \st \\ + \sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\ + \wVecPrime = \wVec}}\kMapParam{\wVecPrime}} \cdot \expect{\sum\limits_{\substack{\wVecPrime \in \pw \st \\ + \sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\ + \wVecPrime = \wVec}}\sketchPolarParam{\wVecPrime}^2} + \nonumber\\ +&\qquad\expect{\sum\limits_{\substack{\wVecPrime, \wVec \in \pw \st \\ + \sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec} \\ + \wVecPrime \neq \wVec}}\kMapParam{\wVecPrime}}\expect{\sum\limits_{\substack{\wVecPrime, \wVec \in \pw \st \\ + \sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec} \\ + \wVecPrime \neq \wVec}}\sketchPolarParam{\wVecPrime}\sketchPolarParam{\wVec}}\label{eq:step-three-a}\\ +=&~{\sum\limits_{\substack{\wVecPrime \in \pw \st \\ + \sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\ + \wVecPrime = \wVec}}\kMapParam{\wVecPrime}} \cdot \expect{\sum\limits_{\substack{\wVecPrime \in \pw \st \\ + \sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\ + \wVecPrime = \wVec}}\sketchPolarParam{\wVecPrime}^2} + \nonumber\\ +&\qquad\sum\limits_{\substack{\wVecPrime, \wVec \in \pw \st \\ + \sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec} \\ + \wVecPrime \neq \wVec}}\kMapParam{\wVecPrime}\expect{\sum\limits_{\substack{\wVecPrime, \wVec \in \pw \st \\ + \sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec} \\ + \wVecPrime \neq \wVec}}\sketchPolarParam{\wVecPrime}\sketchPolarParam{\wVec}}\label{eq:step-three-b}\\ %\] from which the last term evaluates to $0$ and we have %\[ +=&~\sum\limits_{\substack{\wVecPrime \in \pw \st \\ + \sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\ + \wVecPrime = \wVec}}\kMapParam{\wVecPrime} \label{eq:step-three-c}\\ =&~\expect{\sum\limits_{\substack{\wVecPrime\in \pw \st \\ \sketchHashParam{\wVecPrime} = \sketchHashParam{\wVec},\\ \wVecPrime = \wVec}}\kMapParam{\wVecPrime}\sketchPolarParam{\wVecPrime}^2}\label{eq:step-four}\\ diff --git a/main.synctex(busy) b/main.synctex(busy) new file mode 100644 index 0000000..d97dd35 Binary files /dev/null and b/main.synctex(busy) differ