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some equalities

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@@ -157,6 +157,9 @@ So the correspondant one order tensor of $T_{\gamma}^{\{k+q\}}$ is $T_{\gamma}^{
&={\alpha}_{\left( \sum_{j=1}^{k} i_{j}\prod_{l=1}^{j-1}d_{i_{l}} \right)_{1}} \cdot b_{\left( \sum_{j=k+1}^{k+q} i_{j}\prod_{l=1}^{j-1}d_{i_{l}} \right)_{1} }\\ &={\alpha}_{\left( \sum_{j=1}^{k} i_{j}\prod_{l=1}^{j-1}d_{i_{l}} \right)_{1}} \cdot b_{\left( \sum_{j=k+1}^{k+q} i_{j}\prod_{l=1}^{j-1}d_{i_{l}} \right)_{1} }\\
&={\alpha}_{\left(i_{1},\ldots,i_{k}\right)_{k}} \cdot b_{\left( i_{k+1},\ldots,{k+q} \right)_{q} }\\ &={\alpha}_{\left(i_{1},\ldots,i_{k}\right)_{k}} \cdot b_{\left( i_{k+1},\ldots,{k+q} \right)_{q} }\\
\end{align*} \end{align*}
Then,
${\gamma}_{\left(\sum_{j=1}^{k+q} i_{j}\prod_{l=1}^{j-1}d_{i_{l}}\right)_{1} }=
{\alpha}_{\left( \sum_{j=1}^{k} i_{j}\prod_{l=1}^{j-1}d_{i_{l}} \right)_{1}} \cdot b_{\left( \sum_{j=k+1}^{k+q} i_{j}\prod_{l=1}^{j-1}d_{i_{l}} \right)_{1} }$
\subsection{tensor contraction} \subsection{tensor contraction}
@@ -249,6 +252,12 @@ then
&= {\gamma}_{\left(\sum_{j=1}^{k-k_0}i_{j}\prod_{u=1}^{j-1}d_{u}+\sum_{j=k_0+1}^k i_{j}\prod_{u=1}^{j-1}b_{u}\right)_{1}} \\ &= {\gamma}_{\left(\sum_{j=1}^{k-k_0}i_{j}\prod_{u=1}^{j-1}d_{u}+\sum_{j=k_0+1}^k i_{j}\prod_{u=1}^{j-1}b_{u}\right)_{1}} \\
&={\gamma}_{\left(i_{1},i_{2},\ldots,i_{k-k_0},i_{k_0+1},i_{k_0+2},\ldots,i_{q}\right)_{k+q-2\cdot k_0}} &={\gamma}_{\left(i_{1},i_{2},\ldots,i_{k-k_0},i_{k_0+1},i_{k_0+2},\ldots,i_{q}\right)_{k+q-2\cdot k_0}}
\end{align*} \end{align*}
Then
$$
{\gamma}_{\left(\sum_{j=1}^{k-k_0}i_{j}\prod_{u=1}^{j-1}d_{u}+\sum_{j=k_0+1}^k i_{j}\prod_{u=1}^{j-1}b_{u}\right)_{1}}
= \sum_{l=0}^{\prod_{t=1}^{k_0}b_{t}} {\alpha}_{\left(\sum_{j=1}^{k-k_0}i_{j}\prod_{u=1}^{j-1}d_{u}+l\cdot\prod_{s=1}^{k-k_0}d_{s}\right)_{1}} b_{\left(l+\sum_{j=k_0+1}^k i_{j}\prod_{u=1}^{j-1}b_{u}\right)_{1}}
$$
\section{Advantages using tensor order one} \section{Advantages using tensor order one}
The most advantage is computation, only one loop to browse all tensor elements! If we can compute in parallele, as we know the size and it is on a line, each threads or compute units available can have the same number of elements to compute! The most advantage is computation, only one loop to browse all tensor elements! If we can compute in parallele, as we know the size and it is on a line, each threads or compute units available can have the same number of elements to compute!