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% -*- root: main.tex -*-
\onecolumn
\section { Bounding $ \sigsq $ }
\label { sec:var_ est}
We wish to prove that
\[
\sigsq \leq \sum _ j \sigsq _ j.
\]
Therefore, substituting in the definition of variance for complex numbers,
\begin { align}
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\sigsq & = \ex { \sum _ j \est _ j \cdot \conj { \sum _ { j'} \est _ j'} } - \ex { \sum _ j \est _ j} \cdot \ex { \conj { \sum _ { j'} \est _ { j'} } } \nonumber \\
& = \ex { \sum _ j \est _ j \cdot \sum _ { j'} \conj { \est _ j'} } - \ex { \sum _ j \est _ j} \cdot \ex { \sum _ { j'} \conj { \est _ { j'} } } \nonumber \\
& = \sum _ { j, j'} \left (\ex { \est _ j \cdot \overline { \est _ j'} } - \ex { \est _ j} \ex { \overline { \est _ { j'} } } = \cvar { j, j'} \right )\nonumber \\
& = \sum _ j\ex { \est _ j \cdot \overline { \est _ j'} } - \ex { \est _ j} \ex { \overline { \est _ j} } + \sum _ { j \neq j'} \cvar { j, j'} \nonumber \\
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& = \sum _ j \sigsq _ j + \sum _ { j \neq j'} \cvar { j, j'} \label { eq:sigsq-jneqj} \\
& \Rightarrow \sum _ { j \neq j'} \cvar { j, j'} \leq 0. \nonumber
\end { align}
\subsection { Bounding $ \sum _ { j \neq j' } \cvar { j, j' } $ }
\begin { align*}
\sum _ { j \neq j'} \cvar { j, j'} & = \sum _ { j \neq j'} \ex { \est _ j \cdot \conj { \est _ { j'} } } - \ex { \est _ j} \cdot \ex { \conj { \est _ { j'} } } \\
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& =\ex { \prod _ { i = 1} ^ { \prodsize } \sum _ { \wElem \in W} v_ i(\wElem )s(\wElem )\ind { h(\wElem ) = j} \cdot \prod _ { i = 1} ^ { \prodsize } \sum _ { \wElem ' \in W} v_ i(\wElem ')\conj { s(\wElem ')} \ind { h(\wElem ') = j'} } - \ex { \prod _ { i = 1} ^ { \prodsize } \sum _ { \wElem \in W} v_ i(\wElem )s(\wElem )\ind { h(\wElem ) = j} } \cdot \ex { \prod _ { i = 1} ^ { \prodsize } \sum _ { \wElem ' \in W} v_ i(\wElem ')\conj { s(\wElem ')} \ind { h(\wElem ') = j'} } \\
& =\ex { \sum _ { \substack { \wElem _ 1,\cdots ,\wElem _ \prodsize ,\\ \wElem '_ 1,\cdots ,\wElem '_ \prodsize \\ \in W} } \prod _ { i = 1} ^ { \prodsize } v_ i(\wElem _ i)s(\wElem _ i)v_ i(\wElem '_ i)s(\wElem '_ i) \ind { h(\wElem _ i) = j} \ind { h(\wElem '_ i) = j'} } - \ex { \sum _ { \substack { \wElem _ 1,\cdots , \wElem _ \prodsize \\ \in W} } \prod _ { i = 1} ^ { \prodsize } v_ i(\wElem _ i)s(\wElem _ i) \ind { h(\wElem _ i) = j} } \cdot \ex { \sum _ { \substack { \wElem '_ 1,\cdots , \wElem '_ \prodsize \\ \in W} } \prod _ { i = 1} ^ { \prodsize } v_ i(\wElem '_ i)s(\wElem '_ i) \ind { h(\wElem '_ i) = j'} } \\
& =\sum _ { \substack { \wElem _ 1,\cdots ,\wElem _ \prodsize ,\\ \wElem '_ 1,\cdots ,\wElem '_ \prodsize \\ \in W} } \ex { \prod _ { i = 1} ^ { \prodsize } v_ i(\wElem _ i)s(\wElem _ i)v_ i(\wElem '_ i)s(\wElem '_ i)\ind { h(\wElem _ i) = j} \ind { h(\wElem '_ i) = j'} } - \ex { \prod _ { i = 1} ^ { \prodsize } v_ i(\wElem _ i)s(\wElem _ i) \ind { h(\wElem _ i) = j} } \cdot \ex { \prod _ { i = 1} ^ { \prodsize } v_ i(\wElem '_ i)s(\wElem '_ i)\ind { h(\wElem '_ i) = j'} } \\
& = \sum _ { \substack { \wElem _ 1,\cdots ,\wElem _ \prodsize ,\\ \wElem '_ 1,\cdots ,\wElem '_ \prodsize \\ \in W} } \prod _ { i = 1} ^ { \prodsize } v_ i(\wElem _ i)v_ i(\wElem '_ i)\ex { \prod _ { i = 1} ^ { \prodsize } s(\wElem _ i)s(\wElem '_ i)\ind { h(\wElem _ i) = j} \ind { h(\wElem '_ i) = j'} } - \prod _ { i = 1} ^ { \prodsize } v_ i(\wElem _ i)\ex { \prod _ { i = 1} ^ { \prodsize } s(\wElem _ i)\ind { h(\wElem _ i) = j} } \cdot \prod _ { i = 1} ^ { \prodsize } v_ i(\wElem '_ i)\ex { \prod _ { i = 1} ^ { \prodsize } s(\wElem '_ i)\ind { h(\wElem _ i') = j'} } \\
& = \sum _ { \substack { \wElem _ 1,\cdots ,\wElem _ \prodsize ,\\ \wElem '_ 1,\cdots ,\wElem '_ \prodsize \\ \in W} } \prod _ { i = 1} ^ { \prodsize } v_ i(\wElem _ i)v_ i(\wElem '_ i)\left (\ex { \prod _ { i = 1} ^ { \prodsize } s(\wElem _ i)s(\wElem '_ i)\ind { h(\wElem _ i) = j} \ind { h(\wElem '_ i) = j'} } - \ex { \prod _ { i = 1} ^ { \prodsize } s(\wElem _ i)\ind { h(\wElem _ i) = j} } \cdot \ex { \prod _ { i = 1} ^ { \prodsize } s(\wElem '_ i)\ind { h(\wElem _ i') = j'} } \right ).
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\end { align*}
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For $ T _ 1 = \ex { \prod _ { i = 1 } ^ { \prodsize } s ( \wElem _ i ) s ( \wElem ' _ i ) \ind { h ( \wElem _ i ) = j } \ind { h ( \wElem ' _ i ) = j' } } $ , because hash function $ h $ cannot bucket the same world to two different buckets, the only surviving terms occur when there is no overlap between the $ \wElem _ i $ and $ \wElem ' _ i $ variables. Given the condition of no overlap, the only terms that survive are when $ \forall i \in [ \prodsize ] , \wElem _ i = \wElem , \wElem ' _ i = \wElem ', \wElem \neq \wElem ' $ . Notice, however, that in such a case, the product of the remaining expectations will cancel this out. Looking at the remaining two expectations, each can only survive when $ \forall i \in [ \prodsize ] , \wElem _ i = \wElem , \wElem ' _ i = \wElem ' $ . Such constraints leave us with only one surviving case, when all variables are the same world. Thus,
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\begin { align}
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& \sum _ { j \neq j'} \cvar { j, j'} = - \frac { 1} { B^ 2} \sum _ { \wElem \in W} \prod _ { i = 1} ^ { \prodsize } v_ i^ 2(\wElem )\label { eq:cvar-bound} .
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\end { align}
\subsection { Bounding $ \sigsq _ j $ }
We now seek to bound the remaining term in ~\eqref { eq:sigsq-jneqj} . We take a look at the variance of a single bucket estimate.
\begin { align*}
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& \sigsq _ j = \ex { \est _ j \cdot \overline { \est _ j} } - \ex { \est _ j} \cdot \ex { \overline { \est _ j} } \\
& = \ex { \prod _ { i = 1} ^ { \prodsize } \sum _ { \wElem \in W_ j} v_ i(\wElem )s(\wElem ) \cdot \prod _ { i = 1} ^ \prodsize \sum _ { \wElem ' \in W_ j} v_ i(\wElem ')\overline { s(\wElem ')} } -
\ex { \prod _ { i = 1} ^ { \prodsize } \sum _ { \wElem \in W_ j} v_ i(\wElem )s(\wElem )} \cdot \ex { \prod _ { i = 1} ^ \prodsize \sum _ { \wElem ' \in W_ j} v_ i(\wElem ')\overline { s(\wElem ')} } \\
& = \ex { \sum _ { \substack { \wElem _ 1...\wElem _ \prodsize \\ \wElem '_ 1...\wElem '_ \prodsize \\ \in W} } \prod _ { i = 1} ^ \prodsize v_ i(\wElem _ i)v(\wElem '_ i)s(\wElem _ i)\overline { s(\wElem '_ i)} \ind { h(\wElem _ i) = j} \ind { h(\wElem '_ i) = j} } -
\ex { \sum _ { \wElem _ 1...\wElem _ \prodsize \in W} \prod _ { i = 1} ^ \prodsize v_ i(\wElem _ i)s(\wElem _ i)\ind { h(\wElem _ i) = j} } \cdot
\ex { \sum _ { \wElem '_ 1...\wElem '_ \prodsize \in W} \prod _ { i = 1} ^ \prodsize v_ i(\wElem '_ i)\overline { s(\wElem '_ i)} \ind { h(\wElem '_ i) = j} } \\
=& \sum _ { \substack { \wElem _ 1...\wElem _ \prodsize \\ \wElem '_ 1...\wElem '_ \prodsize \\ \in W} } \ex { \prod _ { i = 1} ^ \prodsize v_ i(\wElem _ i)v_ i(\wElem '_ i)s(\wElem _ i)\overline { s(\wElem '_ i)} \ind { h(\wElem _ i) = j} \ind { h(\wElem '_ i) = j} } -
\ex { \prod _ { i = 1} ^ kv_ i(\wElem _ i)s(\wElem _ i)\ind { h(\wElem _ i) = j} } \cdot \ex { \prod _ { i = 1} ^ \prodsize v_ i(\wElem '_ i)\overline { s(\wElem '_ i)} \ind { h(\wElem '_ i) = j} } \\
& = \sum _ { \substack { \wElem _ 1...\wElem _ \prodsize \\ \wElem '_ 1...\wElem '_ \prodsize \\ \in W} } \prod _ { i = 1} ^ \prodsize v_ i(\wElem _ i)v_ i(\wElem '_ i)\cdot \left ( \ex { \prod _ { i = 1} ^ \prodsize s(\wElem _ i)\overline { s(\wElem '_ i)} \ind { h(\wElem _ i) = j} \ind { h(\wElem '_ i) = j} } -
\ex { \prod _ { i = 1} ^ ks(\wElem _ i)\ind { h(\wElem _ i) = j} } \cdot \ex { \prod _ { i = 1} ^ \prodsize \overline { s(\wElem '_ i)} \ind { h(\wElem '_ i) = j} } \right ).
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\end { align*}
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\subsection { Non-generic $ \prodsize $ }
\subsubsection { $ \prodsize = 2 $ }
Taking $ \prodsize = 2 $ and looking at $ T _ 1 = \ex { \prod \limits _ { i = 1 } ^ \prodsize s ( \wElem _ i ) \overline { s ( \wElem ' _ i ) } \ind { h ( \wElem _ i ) = j } \ind { h ( \wElem ' _ i ) = j } } $ , it can be seen that only specific combinations of $ \wElem $ can survive. First, when $ \forall i \in [ \prodsize ] , \wElem _ i = \wElem , \wElem ' _ i = \wElem ' $ , then we end up with $ s ( \wElem ) ^ \prodsize = 1 $ and $ s ( \wElem ' ) ^ \prodsize = 1 $ . This translates into:
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\begin { align*}
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\frac { 1} { B} \sum _ { \wElem \in W} \prod _ { i = 1} ^ { 2} v_ i^ 2(\wElem ) +\frac { 1} { B^ 2} \sum _ { \wElem \neq \wElem ' \in W} \prod _ { i = 1} ^ { 2} v_ i(\wElem )v_ i(\wElem '_ i).
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\end { align*}
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Taking into account that for $ \omega \in \mathbb { C } , \omega \cdot \conj { \omega } = 1 $ , terms in $ T _ 1 $ also survive the expectation when all $ \wElem _ i $ have a matching counterpart in $ \wElem ' _ i $ , yielding
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\begin { align*}
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\frac { 1} { B^ 2} \sum _ { \substack { \wElem _ 1 \neq \wElem _ 2\\ \in W} } \prod _ { i = 1} ^ { 2} v_ i^ 2(\wElem _ i) + \frac { 1} { B^ 2} \sum _ { \wElem \neq \wElem ' \in W} \prod _ { i = 1} ^ { 2} v_ i(\wElem )v_ i(\wElem ').
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\end { align*}
Putting all cases together we have that
\begin { align*}
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T_ 1 = \frac { 1} { B} \sum _ { \wElem \in W} \prod _ { i = 1} ^ { 2} v_ i^ 2(\wElem ) + \frac { 1} { B^ 2} \left (2\sum _ { \wElem \neq \wElem ' \in W} \prod _ { i = 1} ^ { 2} v_ i(\wElem )v_ i(\wElem ') + \sum _ { \substack { \wElem _ 1 \neq \wElem _ 2\\ \in W} } \prod _ { i = 1} ^ { 2} v_ i^ 2(\wElem _ i)\right ).
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\end { align*}
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For $ T _ 2 = \ex { \prod _ { i = 1 } ^ ks ( \wElem _ i ) \ind { h ( \wElem _ i ) = j } } $ and $ T _ 3 = \ex { \prod _ { i = 1 } ^ \prodsize \overline { s ( \wElem ' _ i ) } \ind { h ( \wElem ' _ i ) = j } } $ , we get
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\begin { align*}
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& T_ 2 = \frac { 1} { B} \sum _ { \wElem \in W} \prod _ { i = 1} ^ { 2} v_ i(\wElem ),\\
& T_ 3 = \frac { 1} { B} \sum _ { \wElem ' \in W} \prod _ { i = 1} ^ { 2} v_ i(\wElem '),\\
& T_ 2 \cdot T_ 3 = \frac { 1} { B^ 2} \sum _ { \wElem , \wElem ' \in W} \prod _ { i = 1} ^ { 2} v_ i(\wElem )v_ i(\wElem ') = \frac { 1} { B^ 2} \left (\sum _ { \wElem \in W} \prod _ { i = 1} ^ { 2} v_ i^ 2(\wElem ) + \sum _ { \wElem \neq \wElem ' \in W} \prod _ { i = 1} ^ { 2} v_ i(\wElem )v_ i(\wElem ')\right ).
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\end { align*}
Combining all $ T _ i $ ,
\begin { align*}
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\sigsq _ j = T_ 1 - T_ 2 \cdot T_ 3 = \frac { B - 1} { B^ 2} \sum _ { \wElem \in W} \prod _ { i = 1} ^ { 2} v_ i^ 2(\wElem ) + \frac { 1} { B^ 2} \left (\sum _ { \substack { \wElem _ 1\neq \wElem _ 2 \\ \in W} } \prod _ { i = 1} ^ { 2} v_ i(\wElem _ 1)v_ i(\wElem _ 2) + v_ i^ 2(\wElem _ i)\right ).%+ \sum_{\wElem}\prod_{i = 1}^{2}v_i(\wElem)^2\right)
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\end { align*}
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Recall ~\eqref { eq:cvar-bound} , that $ \sum \limits _ { j \neq j' } \cvar { j, j' } = - \frac { 1 } { B ^ 2 } \sum \limits _ { \wElem \in W } \prod _ { i = 1 } ^ { 2 } v _ i ^ 2 ( \wElem ) $ . Thus, for $ \prodsize = 2 $ we can compute ~\eqref { eq:sigsq-jneqj}
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\begin { align*}
& \sigsq = \sum _ { j \in B} \sigsq _ j + \sum _ { j \neq j'} \cvar { j, j'} \\
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& =B \cdot \left (\frac { B - 1} { B^ 2} \sum _ { \wElem \in W} \prod _ { i = 1} ^ { 2} v_ i^ 2(\wElem ) + \frac { 1} { B^ 2} \left (\sum _ { \substack { \wElem _ 1\neq \wElem _ 2 \\ \in W} } \prod _ { i = 1} ^ { 2} v_ i(\wElem _ 1)v_ i(\wElem _ 2) + v_ i^ 2(\wElem _ i)\right )
\right ) - \frac { B\left (B - 1\right )} { B^ 2} \sum _ { \wElem \in W} \prod _ { i = 1} ^ { 2} v_ i^ 2(\wElem )\\
& = \frac { 1} { B} \left (\sum _ { \substack { \wElem _ 1\neq \wElem _ 2 \\ \in W} } \prod _ { i = 1} ^ { 2} v_ i(\wElem _ 1)v_ i(\wElem _ 2) + v_ i^ 2(\wElem _ i)\right )\\
& = \frac { 1} { B} \left (\left (\sum _ { \wElem \in W} v_ 1(\wElem )\right )^ 2\left (\sum _ { \wElem \in W} v_ 2(\wElem )\right )^ 2 + \left (\sum _ { \wElem \in W} v_ 1(\wElem )v_ 2(\wElem )\right )^ 2\right )\\
& = \frac { 1} { B} \left (\norm { v_ 1} _ 2^ 2\norm { v_ 2} _ 2^ 2 + \left (\sum _ { \wElem \in W} v_ 1(\wElem )v_ 2(\wElem )\right )^ 2\right )\\
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& \leq \frac { 1} { B} \left (\norm { v_ 1} _ 2^ 2\norm { v_ 2} _ 2^ 2 + \norm { v_ 1} _ 2^ 2\norm { v_ 2} _ 2^ 2\right )\\
& \leq \frac { 2} { B} \left (\norm { v_ 1} _ 2^ 2\norm { v_ 2} _ 2^ 2\right ).
\end { align*}
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\subsubsection { $ \prodsize = 3 $ }
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\begin { align*}
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& = \sum _ { \substack { \wElem _ 1...\wElem _ 3\\ \wElem '_ 1...\wElem '_ 3\\ \in W} } \prod _ { i = 1} ^ 3 v_ i(\wElem _ i)v_ i(\wElem '_ i)\cdot \left ( \ex { \prod _ { i = 1} ^ 3 s(\wElem _ i)\overline { s(\wElem '_ i)} \ind { h(\wElem _ i) = j} \ind { h(\wElem '_ i) = j} } -
\ex { \prod _ { i = 1} ^ 3s(\wElem _ i)\ind { h(\wElem _ i) = j} } \cdot \ex { \prod _ { i = 1} ^ 3\overline { s(\wElem '_ i)} \ind { h(\wElem '_ i) = j} } \right )
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\end { align*}
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In the above expression, we seek to know which combinations of $ \wElem _ i $ and $ \wElem ' _ i $ variables will survive the expectation calculations. We can divide the possibilities up into several different cases.
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First, for roots of unity, we have that $ \omega ^ \prodsize = 1 $ if $ \omega $ is a kth root of unity. This gives our first case.
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\underline { Case 1:}
\begin { align*}
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& \wElem _ 1 = \wElem _ 2 = \wElem _ 3 = \wElem \\
& \wElem '_ 1 = \wElem '_ 2 = \wElem '_ 3 = \wElem '\\
& 1.1)~ \wElem = \wElem '\qquad 1.2)~ \wElem \neq \wElem '
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\end { align*}
The remaining cases take into account the property for roots of unity that $ \omega \cdot \conj { \omega } = 1 $ . Note that we omit the case of all variables being equal because that has already been covered above.
\underline { Case 2:}
\begin { align*}
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& \wElem _ 1 = \wElem '_ 1 = \wElem \\
& \wElem _ 2 = \wElem '_ 2 = \wElem '\\
& \wElem _ 3 = \wElem '_ 3 = \wElem ''\\
& 2.1)~ \wElem = \wElem ' \neq \wElem ''\qquad 2.2)~ \wElem \neq \wElem '= \wElem ''\qquad 2.3)\wElem = \wElem '' \neq \wElem '\qquad 2.4) \wElem \neq \wElem ' \neq \wElem ''
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\end { align*}
\underline { Case 3:}
\begin { align*}
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& \wElem _ 1 = \wElem '_ 2 = \wElem \\
& \wElem _ 2 = \wElem '_ 3 = \wElem '\\
& \wElem _ 3 = \wElem '_ 1 = \wElem ''\\
& 3.1)~ \wElem = \wElem ' \neq \wElem ''\qquad 3.2)~ \wElem \neq \wElem '= \wElem ''\qquad 3.3)\wElem = \wElem '' \neq \wElem '\qquad 3.4) \wElem \neq \wElem ' \neq \wElem ''
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\end { align*}
\underline { Case 4:}
\begin { align*}
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& \wElem _ 1 = \wElem '_ 3 = \wElem \\
& \wElem _ 2 = \wElem '_ 1 = \wElem '\\
& \wElem _ 3 = \wElem '_ 2 = \wElem ''\\
& 4.1)~ \wElem = \wElem ' \neq \wElem ''\qquad 4.2)~ \wElem \neq \wElem '= \wElem ''\qquad 4.3)\wElem = \wElem '' \neq \wElem '\qquad 4.4) \wElem \neq \wElem ' \neq \wElem ''
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\end { align*}
\underline { Case 5:}
\begin { align*}
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& \wElem _ 1 = \wElem '_ 2 = \wElem \\
& \wElem _ 2 = \wElem '_ 1 = \wElem '\\
& \wElem _ 3 = \wElem '_ 3 = \wElem ''\\
& 5.1)~ \wElem = \wElem ' \neq \wElem ''\qquad 5.2)~ \wElem \neq \wElem '= \wElem ''\qquad 5.3)\wElem = \wElem '' \neq \wElem '\qquad 5.4) \wElem \neq \wElem ' \neq \wElem ''
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\end { align*}
\underline { Case 6:}
\begin { align*}
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& \wElem _ 1 = \wElem '_ 1 = \wElem \\
& \wElem _ 2 = \wElem '_ 3 = \wElem '\\
& \wElem _ 3 = \wElem '_ 2 = \wElem ''\\
& 6.1)~ \wElem = \wElem ' \neq \wElem ''\qquad 6.2)~ \wElem \neq \wElem '= \wElem ''\qquad 6.3)\wElem = \wElem '' \neq \wElem '\qquad 6.4) \wElem \neq \wElem ' \neq \wElem ''
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\end { align*}
\underline { Case 7:}
\begin { align*}
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& \wElem _ 1 = \wElem '_ 3 = \wElem \\
& \wElem _ 2 = \wElem '_ 2 = \wElem '\\
& \wElem _ 3 = \wElem '_ 1 = \wElem ''\\
& 7.1)~ \wElem = \wElem ' \neq \wElem ''\qquad 7.2)~ \wElem \neq \wElem '= \wElem ''\qquad 7.3)\wElem = \wElem '' \neq \wElem '\qquad 7.4) \wElem \neq \wElem ' \neq \wElem ''
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\end { align*}
The surviving terms are:
\begin { align*}
& \text { Case 1:} \\
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& \frac { B - 1} { B^ 2} \left (\sum _ \wElem v_ 1^ 2(\wElem ) v_ 2^ 2(\wElem ) v_ 3^ 2(\wElem )\right ) + \\
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& Case 2:\\
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& \frac { 1} { B^ 2} \left (\sum _ { \wElem \neq \wElem '} v_ 1^ 2(\wElem )\left (v_ 2^ 2(\wElem )v_ 3^ 2(\wElem ') + v_ 2^ 2(\wElem ')v_ 3^ 2(\wElem ') + v_ 2^ 2(\wElem ')v_ 3^ 2(\wElem )\right )\right ) + \frac { 1} { B^ 3} \sum _ { \wElem \neq \wElem ' \neq \wElem ''} v_ 1^ 2(\wElem )v_ 2^ 2(\wElem ')v_ 3^ 2(\wElem '') +\\
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& \text { Case 3 and 4:} \\
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& \frac { 2} { B^ 2} \left (\sum _ { \wElem \neq \wElem '} v_ 1(\wElem )v_ 1(\wElem ')v_ 2^ 2(\wElem )v_ 3(\wElem )v_ 3(\wElem ') + v_ 1(\wElem )v_ 1(\wElem ')v_ 2(\wElem )v_ 2(\wElem ')v_ 3^ 2(\wElem ') + v_ 1^ 2(\wElem )v_ 2(\wElem )v_ 2(\wElem ')v_ 3(\wElem )v_ 3(\wElem ')\right ) + \\
& \qquad \qquad \frac { 1} { B^ 3} \left (\sum _ { \wElem \neq \wElem ' \neq \wElem ''} v_ 1(\wElem )v_ 1(\wElem '')v_ 2(\wElem ')v_ 2(\wElem )v_ 3(\wElem '')v_ 3(\wElem ') + v_ 1(\wElem )v_ 1(\wElem ')v_ 2(\wElem ')v_ 2(\wElem '')v_ 2(\wElem )v_ 3(\wElem '')\right )\\
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& \text { Case 5, 6, 7:} \\
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& \frac { 1} { B^ 3} \left (\sum _ { \wElem \neq \wElem ' \neq \wElem ''} v_ 1(\wElem )v_ 1(\wElem ')v_ 2(\wElem )v_ 2(\wElem ')v_ 3^ 2(\wElem '') + v_ 1^ 2(\wElem )v_ 2(\wElem ')v_ 2(\wElem '')v_ 3(\wElem ')v_ 3(\wElem '') + v_ 1(\wElem )v_ 1(\wElem '')v_ 2^ 2(\wElem ')v_ 3(\wElem )v_ 3(\wElem '')\right )
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\end { align*}