From e4a8e46b6a373ccf3b70644ea60374a5e3c9554c Mon Sep 17 00:00:00 2001
From: fanasina <rafanasin@gmail.com>
Date: Wed, 21 Jan 2026 01:14:32 +0100
Subject: [PATCH] change index notation

---
 onedimensiontensor.tex | 230 +++++++++++++++++++++--------------------
 1 file changed, 116 insertions(+), 114 deletions(-)

diff --git a/onedimensiontensor.tex b/onedimensiontensor.tex
index d1795e5..b18750f 100644
--- a/onedimensiontensor.tex
+++ b/onedimensiontensor.tex
@@ -39,8 +39,8 @@
 \section{Goal: Write any tensor as vector !!}
 
 \begin{definition}
-A dimension of a tensor $T$ is a vector $d\in \left(\mathbb{N}^*\right)^k$ $(k<\infty)$. We have $$T=(a)_d=(a)_{d_1,\ldots,d_k} =(a_{i_1,\ldots,i_k}^{\{k\}})_{0\le i_1<d_1, \ldots , 0\le i_k< d_k} $$
-$$ a_{i_1,\ldots,i_k}^{\{k\}}\in \mathbb{K} \ \ / \ \ 0\le i_j<d_j $$
+A dimension of a tensor $T$ is a vector $d\in \left(\mathbb{N}^*\right)^k$ $(k<\infty)$. We have $$T=(a)_d=(a)_{d_1,\ldots,d_k} =({\alpha}_{i_1,\ldots,i_k}^{\{k\}})_{0\le i_1<d_1, \ldots , 0\le i_k< d_k} $$
+$$ {\alpha}_{i_1,\ldots,i_k}^{\{k\}}\in \mathbb{K} \ \ / \ \ 0\le i_j<d_j $$
 \end{definition}
 
 $\mathbb{K}$ is a field( for e.g $\mathbb{R}$ )\\
@@ -61,7 +61,7 @@ Little endian
 $$
 \begin{array}{ccc}
 \mathbb{K}^{d_1\times  d_2 \times \cdots \times d_k} & \longrightarrow &  \mathbb{K}^{d_1\cdot d_2\cdots d_k}  \\
-\left(a_{(i_1,\ldots,i_k)_k}\right) & \longmapsto & \left(a_{\left(i_1+i_2\cdot d_1+\ldots+i_k \prod_{j=1}^{k-1}d_j\right)_1}\right)
+\left({\alpha}_{(i_1,\ldots,i_k)_k}\right) & \longmapsto & \left({\alpha}_{\left(i_1+i_2\cdot d_1+\ldots+i_k \prod_{j=1}^{k-1}d_j\right)_1}\right)
 \end{array}
 $$
 
@@ -70,7 +70,7 @@ Big endian
 $$
 \begin{array}{ccc}
 \mathbb{K}^{d_1\times  d_2 \times \cdots \times d_k} & \longrightarrow &  \mathbb{K}^{d_1\cdot d_2\cdots d_k}  \\
-\left(a_{(i_1,\ldots,i_k)_k}\right) & \longmapsto & \left(a_{\left(i_1 \prod_{j=2}^kd_j+i_2\prod_{j=3}^k d_j+\ldots +i_{k-1} d_k +i_k \right)_1}\right)
+\left({\alpha}_{(i_1,\ldots,i_k)_k}\right) & \longmapsto & \left({\alpha}_{\left(i_1 \prod_{j=2}^kd_j+i_2\prod_{j=3}^k d_j+\ldots +i_{k-1} d_k +i_k \right)_1}\right)
 \end{array}
 $$
 For $a,b\in \mathbb{N}$, we note $ [\![ a,b ]\!] = [a.. b]=[a,b]\cap \mathbb{N}$
@@ -128,125 +128,126 @@ We prove below that all operations like tensor contraction  and tensor multiplic
 
 Let $T_{\alpha}^{\{k\}}\in\mathbb{R}^{d_{1}\times\cdots\times d_{k}}$ and $T_{\beta}^{\{q\}}\in\mathbb{R}^{b_{1}\times\cdots\times b_{q}}$ and $T_{\alpha}^{\{1\}}\in\mathbb{R}^{d_{1}\cdots d_{k}}$ and $T_{\beta}^{\{1\}}\in\mathbb{R}^{b_{1}\cdots b_{q}}$ such that
 
-$T_{\alpha}^{\{k\}}=\left(\alpha_{\left(i_{1},\ldots,i_{k}\right)_k}\right)_{0\le i_{j}<d_{j} / 1\le j\le k} $
-and $T_{\beta}^{\{q\}}=\left(\beta_{\left(i_{1},\ldots,i_{q}\right)_q}\right)_{0\le i_{j}<b_{j} / 1\le j\le q} $
+$T_{\alpha}^{\{k\}}=\left({\alpha}_{\left(i_{1},\ldots,i_{k}\right)_k}\right)_{0\le i_{j}<d_{j} / 1\le j\le k} $
+and $T_{\beta}^{\{q\}}=\left({\beta}_{\left(i_{1},\ldots,i_{q}\right)_q}\right)_{0\le i_{j}<b_{j} / 1\le j\le q} $
 
-and $T_{\alpha}^{\{1\}}=\left(\alpha_{\left(i_1\right)_1}\right)_{0\le i_1 < \prod_{j=1}^k d_{j}}$
+and $T_{\alpha}^{\{1\}}=\left({\alpha}_{\left(i_1\right)_1}\right)_{0\le i_1 < \prod_{j=1}^k d_{j}}$
 
-and $T_{\beta}^{\{1\}}=\left(\beta_{\left(i_1\right)_1}\right)_{0\le i_1 < \prod_{j=1}^{q} b_{j}}$
+and $T_{\beta}^{\{1\}}=\left({\beta}_{\left(i_1\right)_1}\right)_{0\le i_1 < \prod_{j=1}^{q} b_{j}}$
 
 Here we use little endian so
 
-$\alpha_{\left(\sum_{j=1}^k i_j\left(\prod_{l=1}^{j-1}d_{l}\right)\right)_1}=\alpha_{\left(i_1,\ldots,i_k\right)_k}$ $\,\, /\,\, i_j\in[d_{j})  \,\,/\,\, 1\le j\le k$
+${\alpha}_{\left(\sum_{j=1}^k i_j\left(\prod_{l=1}^{j-1}d_{l}\right)\right)_1}={\alpha}_{\left(i_1,\ldots,i_k\right)_k}$ $\,\, /\,\, i_j\in[d_{j})  \,\,/\,\, 1\le j\le k$
 
 and
 
-$\beta_{\left(\sum_{j=1}^{q} i_j\left(\prod_{l=1}^{j-1}b_{l}\right)\right)_1}=\beta_{\left(i_1,\ldots,i_{q}\right)_q}$ $\,\, /\,\, i_j\in [b_{j}) \,\,/\,\, 1\le j\le q$
+${\beta}_{\left(\sum_{j=1}^{q} i_j\left(\prod_{l=1}^{j-1}b_{l}\right)\right)_1}={\beta}_{\left(i_1,\ldots,i_{q}\right)_q}$ $\,\, /\,\, i_j\in [b_{j}) \,\,/\,\, 1\le j\le q$
 
 \subsection{Tensor multiplication}
 
-tensor multiplication of $T_{\alpha}^{\{k\}} \times T_{\beta}^{\{q\}} = T_{\gamma}^{\{k+q\}}=\left(\gamma_{\left(i_{1},\ldots,i_{k+q}\right)_{k+q}}\right)\in \mathbb{R}^{d_{1}\times\cdots\times d_{k}\times b_{1}\times\cdots\times b_{q}}$ such that
-$$\gamma_{\left(i_{1},\ldots, i_{k},i_{k+1},\ldots,i_{k+q}\right)_{k+q}}=\alpha_{\left(i_{1},\ldots,i_{k}\right)_k} \cdot \beta_{\left(i_{k+1},\ldots,i_{k+q}\right)_q} $$
+tensor multiplication of $T_{\alpha}^{\{k\}} \times T_{\beta}^{\{q\}} = T_{\gamma}^{\{k+q\}}=\left({\gamma}_{\left(i_{1},\ldots,i_{k+q}\right)_{k+q}}\right)\in \mathbb{R}^{d_{1}\times\cdots\times d_{k}\times b_{1}\times\cdots\times b_{q}}$ such that
+$${\gamma}_{\left(i_{1},\ldots, i_{k},i_{k+1},\ldots,i_{k+q}\right)_{k+q}}={\alpha}_{\left(i_{1},\ldots,i_{k}\right)_k} \cdot {\beta}_{\left(i_{k+1},\ldots,i_{k+q}\right)_q} $$
 
 
-So the correspondant one order tensor of $T_c^{k+q}$ is $T_c^1=({\gamma}_{i}^{k+q})\in\mathbb{R}^{d_{(1,a)}\cdots d_{(k,a)}\cdot d_{(1,b)}\cdots d_{(q,b)}}$ such that
+So the correspondant one order tensor of $T_{\gamma}^{\{k+q\}}$ is $T_{\gamma}^{\{1\}}={\gamma}_{\left(i\right)_{1}}\in\mathbb{R}^{d_{1}\cdots d_{k}\cdot b_{1}\cdots b_{q}}$ such that
 
 \begin{align*}
-{\gamma}_{i_{(1,c)},\ldots, i_{(k,c)},i_{(k+1,c)}}^{k+q}&={\gamma}_{\sum_{j=1}^{k+q} i_{(j,c)}\prod_{l=1}^{j-1}d_{i_{(l,c)}} }^{1}\\
-&={\gamma}_{\sum_{j=1}^{k} i_{(j,c)}\prod_{l=1}^{j-1}d_{i_{(l,c)}} + \sum_{j=k+1}^{k+q} i_{(j,c)}\prod_{l=1}^{j-1}d_{i_{(l,c)}} }^{1}\\
-&=a_{\sum_{j=1}^{k} i_{(j,c)}\prod_{l=1}^{j-1}d_{i_{(l,c)}}}^1 \cdot b_{ \sum_{j=k+1}^{k+q} i_{(j,c)}\prod_{l=1}^{j-1}d_{i_{(l,c)}} }^{1}\\
-&=a_{i_{(1,c)},\ldots,i_{(k,c)}}^k \cdot b_{  i_{(k+1,c)},\ldots,{(k+q,c)} }^{q}\\
+{\gamma}_{\left(i_{1},\ldots, i_{k},i_{k+1}\right)_{k+q}}&={\gamma}_{\left(\sum_{j=1}^{k+q} i_{j}\prod_{l=1}^{j-1}d_{i_{l}}\right)_{1} }\\
+&={\gamma}_{\left( \sum_{j=1}^{k} i_{j}\prod_{l=1}^{j-1}d_{i_{l}} + \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} }\\
 \end{align*}
 
 
 \subsection{tensor contraction}
-Let $T_a^k\in\mathbb{R}^{d_{(1,a)}\times\cdots\times d_{(k,a)}}$ and $T_b^{q}\in\mathbb{R}^{d_{(1,b)}\times\cdots\times d_{(q,b)}}$ and $T_a^1\in\mathbb{R}^{d_{(1,a)}\cdots d_{(k,a)}}$ and $T_b^1\in\mathbb{R}^{d_{(1,b)}\cdots d_{(q,b)}}$ such that there exists $0<k_0 \le \min(k,q)$.
+Let $T_{\alpha}^{\{k\}}\in\mathbb{R}^{d_{1}\times\cdots\times d_{k}}$ and $T_{\beta}^{\{q\}}\in\mathbb{R}^{b_{1}\times\cdots\times b_{q}}$ and $T_{\alpha}^{\{1\}}\in\mathbb{R}^{d_{1}\cdots d_{k}}$ and $T_{\beta}^{\{1\}}\in\mathbb{R}^{b_{1}\cdots b_{q}}$ such that there exists $0<k_0 \le \min(k,q)$.
 
-tensor $k_0$ contraction of  $T_a^{k}$ and $T_b^{q}$ is $T_c^{k+q-2\cdot k_0}\in\mathbb{R}^{d_{(1,a)}\times\cdots\times d_{(k-k_0-1,c)} \times d_{(k_0+1+q,c)}\times\cdots\times d_{(k+q,c)} }$ such that:
+tensor $k_0$ contraction of  $T_{\alpha}^{\{k\}}$ and $T_{\beta}^{\{q\}}$ is $T_{\gamma}^{\{k+q-2\cdot k_0\}}\in\mathbb{R}^{p_{1}\times\cdots\times p_{k+q-2k_0} }$ such that:
 \begin{align*}
- d_{(i,c)}&=d_{(i,a)} &\forall 1\le i\le k- k_0\\
- d_{(i,c)}&=d_{(i-k,b)} &\forall k_0+k< i \le k+q\\
- d_{(i,a)}&=d_{(i-k+k_0,b)} &\forall i\in [k-k_0+1, k]\cap \mathbb{N}\\
+ d_{i}&=b_{i-k + k_0 + 1} &\forall i\in [\![k- k_0 , k ]\!] \\
+ p_{i}&=d_{i} &\forall  i\in [\![ 1 , k- k_0]\!]\\
+ p_{i}&=b_{i-k+2k_0} &\forall i\in [\![k-k_0+1, k+q-2k_0]\!] \\
 \end{align*}
 and
 \begin{align*}
-{\gamma}_{i_{(1,c)},\ldots,i_{k+q-2\cdot k_0,c}}^{k+q-2\cdot k_0}&={\gamma}_{i_{(1,a)},\ldots,i_{(k-k_0-1,a)},i_{(k_0+1,b)},\ldots,i_{(q,b)}}^{k+q-2\cdot k_0} \\
-&=\sum_{l_{1}=0}^{d_{(1,b)}-1}\sum_{l_{2}=0}^{d_{(1,b)}-1}\cdots \sum_{l_{k_0}=0}^{d_{k_0}-1} a_{i_{(1,a)},\ldots,i_{(k-k_0-1,a)},l_{1},l_{2},\ldots,l_{k_0}}^k b_{l_{1},l_{2},\ldots,l_{k_0},i_{(k_0+1,b)},i_{(k_0,b)},\ldots,i_{(q,b)}}^{q} \\
+{\gamma}_{\left(i_{n_1},\ldots,i_{n_{k+q-2\cdot k_0}}\right)_{k+q-2\cdot k_0}}&={\gamma}_{\left(i_{1},\ldots,i_{k-k_0-1},j_{k_0+1},\ldots,j_{q}\right)_{k+q-2\cdot k_0}} \\
+&i_{n_1}=i_1,\ldots,i_{n_{k-k_0-1}}=i_{k-k_0-1},i_{n_{k-k_0}}=j_{k_0+1},\ldots,i_{n_{k+q-2k_0}}=j_{q}\\
+&=\sum_{l_{1}=0}^{b_{1}-1}\sum_{l_{2}=0}^{b_{1}-1}\cdots \sum_{l_{k_0}=0}^{d_{k_0}-1} {\alpha}_{\left(i_{1},\ldots,i_{k-k_0-1},l_{1},l_{2},\ldots,l_{k_0}\right)_{k} } {\beta}_{\left(l_{1},l_{2},\ldots,l_{k_0},j_{k_0+1},j_{k_0+2},\ldots,j_{q}\right)_{q}} \\
 \end{align*}
 For example in little endian transformation:
 \begin{align*}
- a_{i_{(1,a)},\ldots,i_{(k-k_0,a)},l_{1},l_{2},\ldots,l_{k_0}}^k &= a_{\sum_{j=1}^{k-k_0}i_{(j,a)}\prod_{s=1}^{j-1}d_{(s,a)}+\sum_{j=k-k_0+1}^{k}l_{(j-k+k_0)}\prod_{s=1}^{j-1}d_{(s,a)}}^1\\
- b_{l_{1},l_{2},\ldots,l_{k_0},i_{(k_0+1,b)},i_{(k_0,b)},\ldots,i_{(q,b)}}^{q} &= b_{\sum_{j=1}^{k_0}l_j\prod_{s=1}^{j-1}d_{(s,b)}+\sum_{j=k_0+1}^{k}b_{(j,b)}\prod_{s=1}^{j-1}d_{(s,b)}}^1\\
+ {\alpha}_{\left(i_{1},\ldots,i_{k-k_0},l_{1},l_{2},\ldots,l_{k_0}\right)} &= {\alpha}_{\left(\sum_{j=1}^{k-k_0}i_{j}\prod_{s=1}^{j-1}d_{s}+\sum_{j=k-k_0+1}^{k}l_{(j-k+k_0)}\prod_{s=1}^{j-1}d_{s}\right)_{1}}\\
+ {\beta}_{\left(l_{1},l_{2},\ldots,l_{k_0},j_{k_0+1},j_{k_0+2},\ldots,j_{q}\right)_{q}} &= {\beta}_{\left(\sum_{t=1}^{k_0}l_t\prod_{s=1}^{t-1}b_{s}+\sum_{t=k_0+1}^{k}j_{t}\prod_{s=1}^{t-1}b_{s}\right)_{1}}\\
 \end{align*}
 We have to sum these terms $
-\forall l_{1}\in [0,d_{(1,b)}[\cap \mathbb{N}, \forall l_{2}\in [0,d_{(1,b)}[\cap\mathbb{N}\cdots \forall l_{k_0}\in[0,d_{(k_0,b)}[\cap\mathbb{N}
+\forall l_{1}\in [\![0,b_{1}[\![, \forall l_{2}\in [\![0,b_{1}[\![\cdots \forall l_{k_0}\in[\![0,b_{k_0}[\![
 $
 
 So
 \begin{align*}
- 0  \le l_{j} &\le d_{(j,b)}-1  \, \, ,  \forall 1 \le j \le k_0\\
- 0 \prod_{s=1}^{j-1}d_{(s,b)} \le l_{j} \prod_{s=1}^{j-1}d_{(s,b)} & \le (d_{(j,b)}-1) \prod_{s=1}^{j-1}d_{(s,b)}\, \, ,  \forall 1 \le j \le k_0\\
- 0 \le l_{j} \prod_{s=1}^{j-1}d_{(s,b)} & \le (d_{(j,b)}-1) \prod_{s=1}^{j-1}d_{(s,b)}\, \, ,  \forall 1 \le j \le k_0\\
-\sum_{j=1}^{k_0}  0 \le \sum_{j=1}^{k_0} l_{j} \prod_{s=1}^{j-1}d_{(s,b)} & \le \sum_{j=1}^{k_0} (d_{(j,b)}-1) \prod_{s=1}^{j-1}d_{(s,b)}\\
-  0 \le \sum_{j=1}^{k_0} l_{j} \prod_{s=1}^{j-1}d_{(s,b)} & \le \sum_{j=1}^{k_0} (d_{(j,b)}-1) \prod_{s=1}^{j-1}d_{(s,b)} = M_{k_0}\\
-%  \sum_{l_j=0}^{d_{(j,b)}-1} 0 \le \sum_{l_j=0}^{d_{(j,b)}-1} l_{j} \prod_{s=1}^{j-1}d_{(s,b)} &< \sum_{l_j=0}^{d_{(j,b)}-1}  \prod_{s=1}^{j}d_{(s,b)} \\
-%   0 \le \sum_{l_j=0}^{d_{(j,b)}-1} l_{j} \prod_{s=1}^{j-1}d_{(s,b)} &<  d_{(j,b)} \prod_{s=1}^{j-1}d_{(s,b)}=\prod_{s=1}^{j}d_{(s,b)} \\
-%   0 \le \sum_{l_j=0}^{d_{(j,b)}-1} l_{j} \prod_{s=1}^{j-1}d_{(s,b)} &<  \prod_{s=1}^{j}d_{(s,b)} \\
+ 0  \le l_{j} &\le b_{j}-1  \, \, ,  \forall 1 \le j \le k_0\\
+ 0 \prod_{s=1}^{j-1}b_{s} \le l_{j} \prod_{s=1}^{j-1}b_{s} & \le (b_{j}-1) \prod_{s=1}^{j-1}b_{s}\, \, ,  \forall 1 \le j \le k_0\\
+ 0 \le l_{j} \prod_{s=1}^{j-1}b_{s} & \le (b_{j}-1) \prod_{s=1}^{j-1}b_{s}\, \, ,  \forall 1 \le j \le k_0\\
+\sum_{j=1}^{k_0}  0 \le \sum_{j=1}^{k_0} l_{j} \prod_{s=1}^{j-1}b_{s} & \le \sum_{j=1}^{k_0} (b_{j}-1) \prod_{s=1}^{j-1}b_{s}\\
+  0 \le \sum_{j=1}^{k_0} l_{j} \prod_{s=1}^{j-1}b_{s} & \le \sum_{j=1}^{k_0} (b_{j}-1) \prod_{s=1}^{j-1}b_{s} = M_{k_0}\\
+%  \sum_{l_j=0}^{b_{j}-1} 0 \le \sum_{l_j=0}^{b_{j}-1} l_{j} \prod_{s=1}^{j-1}b_{s} &< \sum_{l_j=0}^{b_{j}-1}  \prod_{s=1}^{j}b_{s} \\
+%   0 \le \sum_{l_j=0}^{b_{j}-1} l_{j} \prod_{s=1}^{j-1}b_{s} &<  b_{j} \prod_{s=1}^{j-1}b_{s}=\prod_{s=1}^{j}b_{s} \\
+%   0 \le \sum_{l_j=0}^{b_{j}-1} l_{j} \prod_{s=1}^{j-1}b_{s} &<  \prod_{s=1}^{j}b_{s} \\
 \end{align*}
 
 if $k_0=1$
-$M_1+1 = (d_{(1,b)}-1) +1 = d_{(1,b)} $
+$M_1+1 = (b_{1}-1) +1 = b_{1} $
 
-Suppose $M_{k_0-1}+1=\prod_{j=1}^{k_0-1}d_{(j,b)}$
+Suppose $M_{k_0-1}+1=\prod_{j=1}^{k_0-1}b_{j}$
 
- %$M_{k_0}+1 = \sum_{j=1}^{k_0} (d_{(j,b)}-1) \prod_{s=1}^{j-1}d_{(s,b)} +1$
+ %$M_{k_0}+1 = \sum_{j=1}^{k_0} (b_{j}-1) \prod_{s=1}^{j-1}b_{s} +1$
 
 \begin{align*}
- M_{k_0}+1  &= \sum_{j=1}^{k_0} (d_{(j,b)}-1) \prod_{s=1}^{j-1}d_{(s,b)} +1 \\
- &= \left(\sum_{j=1}^{k_0-1} (d_{(j,b)}-1) \prod_{s=1}^{j-1}d_{(s,b)} \right) + (d_{(k_0,b)}-1) \prod_{s=1}^{k_0-1}d_{(s,b)} +1 \\
- &= \left(\sum_{j=1}^{k_0-1} (d_{(j,b)}-1) \prod_{s=1}^{j-1}d_{(s,b)} \right) +1 + (d_{(k_0,b)}-1) \prod_{s=1}^{k_0-1}d_{(s,b)}  \\
- &=M_{k_0-1} + 1 + (d_{(k_0,b)}-1) \prod_{s=1}^{k_0-1}d_{(s,b)} \\
- &=\prod_{j=1}^{k_0-1}d_{(j,b)} + (d_{(k_0,b)}-1) \prod_{s=1}^{k_0-1}d_{(s,b)}\\
- &=  \prod_{s=1}^{k_0-1}d_{(s,b)} \left( (d_{(k_0,b)}-1) + 1\right) = d_{(k_0,b)} \prod_{s=1}^{k_0-1}d_{(s,b)}\\
-&=  \prod_{s=1}^{k_0}d_{(s,b)} \\
+ M_{k_0}+1  &= \sum_{j=1}^{k_0} (b_{j}-1) \prod_{s=1}^{j-1}b_{s} +1 \\
+ &= \left(\sum_{j=1}^{k_0-1} (b_{j}-1) \prod_{s=1}^{j-1}b_{s} \right) + (b_{k_0}-1) \prod_{s=1}^{k_0-1}b_{s} +1 \\
+ &= \left(\sum_{j=1}^{k_0-1} (b_{j}-1) \prod_{s=1}^{j-1}b_{s} \right) +1 + (b_{k_0}-1) \prod_{s=1}^{k_0-1}b_{s}  \\
+ &=M_{k_0-1} + 1 + (b_{k_0}-1) \prod_{s=1}^{k_0-1}b_{s} \\
+ &=\prod_{j=1}^{k_0-1}b_{j} + (b_{k_0}-1) \prod_{s=1}^{k_0-1}b_{s}\\
+ &=  \prod_{s=1}^{k_0-1}b_{s} \left( (b_{k_0}-1) + 1\right) = b_{k_0} \prod_{s=1}^{k_0-1}b_{s}\\
+&=  \prod_{s=1}^{k_0}b_{s} \\
  \end{align*}
 
-So when $l_j$ browses $[0,d_{(j,b)}[\cap \mathbb{N}$ for all $1\le j\le k_0$ the indexe  $\sum_{j=1}^{k_0} l_{j} \prod_{s=1}^{j-1}d_{(s,b)}$ browses $$\left[0,\prod_{j=1}^{k_0}d_{(j,b)}\right[\cap\mathbb{N}$$
+So when $l_j$ browses $[\![0,b_{j}[\![ $ for all $1\le j\le k_0$ the indexe  $\sum_{j=1}^{k_0} l_{j} \prod_{s=1}^{j-1}b_{s}$ browses $\left[\!\!\left[0,\prod_{j=1}^{k_0}b_{j}\right[\!\!\right[$
 \begin{align*}
- &a_{\sum_{j=1}^{k-k_0}i_{(j,a)}\prod_{s=1}^{j-1}d_{(s,a)}+\sum_{j=k-k_0+1}^{k}l_{(j-k+k_0)}\prod_{s=1}^{j-1}d_{(s,a)}}^1 \\
- &=a_{\sum_{j=1}^{k-k_0}i_{(j,a)}\prod_{s=1}^{j-1}d_{(s,a)}+\sum_{j=k-k_0+1}^{k}l_{(j-k+k_0)}\prod_{s=1}^{k-k_0}d_{(s,a)}\prod_{s=k-k_0+1}^{j-1}d_{(s,a)}}^1\\
+ &{\alpha}_{\left(\sum_{j=1}^{k-k_0}i_{j}\prod_{s=1}^{j-1}d_{s}+\sum_{j=k-k_0+1}^{k}l_{(j-k+k_0)}\prod_{s=1}^{j-1}d_{s}\right)_{1}} \\
+ &={\alpha}_{\left(\sum_{j=1}^{k-k_0}i_{j}\prod_{s=1}^{j-1}d_{s}+\sum_{j=k-k_0+1}^{k}l_{(j-k+k_0)}\prod_{s=1}^{k-k_0}d_{s}\prod_{s=k-k_0+1}^{j-1}d_{s}\right)_{1}}\\
  & ( t=j-k+k_0 \Rightarrow j=t+k-k_0 )\\
  &(j=k-k_0+1\Rightarrow t=1; j=k\Rightarrow t=k_0)\\
- &= a_{\sum_{j=1}^{k-k_0}i_{(j,a)}\prod_{s=1}^{j-1}d_{(s,a)}+\sum_{t=1}^{k_0}l_{t}\prod_{s=k-k_0+1}^{k-k_0+t-1}d_{(s,a)} \left(\prod_{s=1}^{k-k_0}d_{(s,a)}\right)}^1 \\
+ &= {\alpha}_{\left(\sum_{j=1}^{k-k_0}i_{j}\prod_{s=1}^{j-1}d_{s}+\sum_{t=1}^{k_0}l_{t}\prod_{s=k-k_0+1}^{k-k_0+t-1}d_{s} \left(\prod_{s=1}^{k-k_0}d_{s}\right)\right)_1} \\
  \end{align*}
-But $d_{(s,a)} = d_{(s-k+k_0,b)} \, \, \forall s\in[k-k_0+1,k]\cap\mathbb{N}$
+But $d_{s} = b_{s-k+k_0} \, \, \forall s\in[\![k-k_0+1,k]\!]$
 
 \begin{align*}
-   &a_{\sum_{j=1}^{k-k_0}i_{(j,a)}\prod_{s=1}^{j-1}d_{(s,a)}+\sum_{t=1}^{k_0}l_{t}\prod_{s=k-k_0+1}^{k-k_0+t-1}d_{(s,a)} \left(\prod_{s=1}^{k-k_0}d_{(s,a)}\right)}^1\\
-    &= a_{\sum_{j=1}^{k-k_0}i_{(j,a)}\prod_{s=1}^{j-1}d_{(s,a)}+\sum_{t=1}^{k_0}l_{t}\prod_{s=k-k_0+1}^{k-k_0+t-1}d_{(s-k+k_0,b)} \left(\prod_{s=1}^{k-k_0}d_{(s,a)}\right)}^1 \\
+   &{\alpha}_{\left(\sum_{j=1}^{k-k_0}i_{j}\prod_{s=1}^{j-1}d_{s}+\sum_{t=1}^{k_0}l_{t}\prod_{s=k-k_0+1}^{k-k_0+t-1}d_{s} \left(\prod_{s=1}^{k-k_0}d_{s}\right)\right)_{1}}\\
+    &= {\alpha}_{\left(\sum_{j=1}^{k-k_0}i_{j}\prod_{s=1}^{j-1}d_{s}+\sum_{t=1}^{k_0}l_{t}\prod_{s=k-k_0+1}^{k-k_0+t-1}b_{s-k+k_0} \left(\prod_{s=1}^{k-k_0}d_{s}\right)\right)_{1}} \\
     & (u=s-k+k_0 \Leftrightarrow s=u+k-k_0)\\
     & (s=k-k_0+1 \Rightarrow u=1)\\
     & (s=k-k_0+t-1 \Rightarrow u=t-1)\\
-    &= a_{\sum_{j=1}^{k-k_0}i_{(j,a)}\prod_{s=1}^{j-1}d_{(s,a)}+\sum_{t=1}^{k_0}l_{t}\prod_{u=1}^{t-1}d_{(u,b)} \left(\prod_{s=1}^{k-k_0}d_{(s,a)}\right)}^1 \\
-    &= a_{\sum_{j=1}^{k-k_0}i_{(j,a)}\prod_{s=1}^{j-1}d_{(s,a)}+\left(\prod_{s=1}^{k-k_0}d_{(s,a)}\right)\sum_{t=1}^{k_0}l_{t}\prod_{u=1}^{t-1}d_{(u,b)} }^1 \\
+    &= {\alpha}_{\left(\sum_{j=1}^{k-k_0}i_{j}\prod_{s=1}^{j-1}d_{s}+\sum_{t=1}^{k_0}l_{t}\prod_{u=1}^{t-1}b_{u} \left(\prod_{s=1}^{k-k_0}d_{s}\right)\right)_{1}} \\
+    &= {\alpha}_{\left(\sum_{j=1}^{k-k_0}i_{j}\prod_{s=1}^{j-1}d_{s}+\left(\prod_{s=1}^{k-k_0}d_{s}\right)\sum_{t=1}^{k_0}l_{t}\prod_{u=1}^{t-1}b_{u}\right)_{1} } \\
 \end{align*}
 
 
 as below
-$ \sum_{t=1}^{k_0}l_{t}\prod_{u=1}^{t-1}d_{(u,b)}$ browse $ \left[0,\prod_{j=1}^{k_0}d_{(j,b)}\right[\cap\mathbb{N}$ when $l_j$  browses $[0,d_{(j,b)}[\cap\mathbb{N}$ for all $1\le j\le k_0$
+$ \sum_{t=1}^{k_0}l_{t}\prod_{u=1}^{t-1}b_{u}$ browse $ \left[\!\!\left[0,\prod_{j=1}^{k_0}b_{j}\right[\!\!\right[$ when $l_j$  browses $[\![0,b_{j}[\![$ for all $1\le j\le k_0$
 
-As $l_j\in[0,d_{(j,b)}[\cap\mathbb{N} \forall 1\le j\le k_0$ and
-$$card([0,d_{(j,b)}[\cap\mathbb{N})=d_{(j,b)}$$
+As $l_j\in [\![0,b_{j}[\![  \forall 1\le j\le k_0$ and
+$$card([\![0,b_{j}[\![)=b_{j}$$
 also
-$$card\left(\prod_{j=1}^{k_0}[0,d_{(j,b)}[\cap\mathbb{N}\right)=\prod_{j=1}^{k_0}d_{(j,b)}$$
+$$card\left(\prod_{j=1}^{k_0}[\![0,b_{j}[\![ \right)=\prod_{j=1}^{k_0}b_{j}$$
 
 then
 
 \begin{align*}
- &\sum_{l_{1}=0}^{d_{(1,b)}-1}\sum_{l_{2}=0}^{d_{(1,b)}-1}\cdots \sum_{l_{k_0}=0}^{d_{k_0}-1} a_{i_{(1,a)},\ldots,i_{(k-k_0,a)},l_{1},l_{2},\ldots,l_{k_0}}^k b_{l_{1},l_{2},\ldots,l_{k_0},i_{(k_0+1,b)},i_{(k_0,b)},\ldots,i_{(q,b)}}^{q}\\
-&= \sum_{l=0}^{\prod_{t=1}^{k_0}d_{(t,b)}} a_{\sum_{j=1}^{k-k_0}i_{(j,a)}\prod_{u=1}^{j-1}d_{(u,a)}+l\cdot\prod_{s=1}^{k-k_0}d_{(s,a)}}^1 b_{l+\sum_{j=k_0+1}^k i_{(j,b)}\prod_{u=1}^{j-1}d_{(u,b)}}^1 \\
-&= {\gamma}_{\sum_{j=1}^{k-k_0}i_{(j,a)}\prod_{u=1}^{j-1}d_{(u,a)}+\sum_{j=k_0+1}^k i_{(j,b)}\prod_{u=1}^{j-1}d_{(u,b)}}^1 \\
-&={\gamma}_{i_{(1,a)},i_{(2,a)},\ldots,i_{(k-k_0,a)},i_{(k_0+1,b)},i_{(k_0+2,b)},\ldots,i_{(q,b)}}^{k+q-2\cdot k_0}
+ &\sum_{l_{1}=0}^{b_{1}-1}\sum_{l_{2}=0}^{b_{1}-1}\cdots \sum_{l_{k_0}=0}^{d_{k_0}-1} {\alpha}_{\left(i_{1},\ldots,i_{k-k_0},l_{1},l_{2},\ldots,l_{k_0}\right)_{k}} b_{\left(l_{1},l_{2},\ldots,l_{k_0},i_{k_0+1},i_{k_0},\ldots,i_{q}\right)_{q}}\\
+&= \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}} \\
+&= {\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}}
 \end{align*}
 
 \section{Advantages using tensor order one}
@@ -254,45 +255,46 @@ The most advantage is computation, only one loop to browse all tensor elements!
 
 
 \section{Examples}
+In these examples, we note ${\alpha}_{(i_1,\ldots,i_k)}^{\left[n_{\left(i_1,\ldots,i_k\right)}\right]} $ such that $\left(i_1,\ldots,i_k\right)\in\mathbb{N}^k $ and $n_{(i_1,\ldots,i_k)}\in \mathbb{N} $, the index of tensor of order $k$ is $\left(i_1,\ldots,i_k \right)$ and the correspondant index in tensor of order $1$ is $\left[n_{\left(i_1,\ldots,i_k\right)}\right]$ 
 \hrule
 \vspace{1cm}
 Big endian $k=2$
 \begin{align*}
- \left( \left( a_{(0,0)}^{[0]},a_{(0,1)}^{[d_2]},\ldots, a_{(0,d_1-1)}^{[(d_1-1)d_2]} \right)\right. \\
-\left( a_{(1,0)}^{[1]},a_{(1,1)}^{[d_2+1]},\ldots, a_{(1,d_1-1)}^{[(d_1-1)d_2+1]} \right) \\
+ \left( \left( {\alpha}_{(0,0)}^{[0]},{\alpha}_{(0,1)}^{[d_2]},\ldots, {\alpha}_{(0,d_1-1)}^{[(d_1-1)d_2]} \right)\right. \\
+\left( {\alpha}_{(1,0)}^{[1]},{\alpha}_{(1,1)}^{[d_2+1]},\ldots, {\alpha}_{(1,d_1-1)}^{[(d_1-1)d_2+1]} \right) \\
    \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
-\left( a_{(d_2-1,0)}^{[d_2-1]},a_{(d_2-1,1)}^{[2d_2-1]},\ldots, a_{(d_2-1,d_1-1)}^{[(d_2-1)d_1 +d_1-1]} \right) \\
+\left( {\alpha}_{(d_2-1,0)}^{[d_2-1]},{\alpha}_{(d_2-1,1)}^{[2d_2-1]},\ldots, {\alpha}_{(d_2-1,d_1-1)}^{[(d_2-1)d_1 +d_1-1]} \right) \\
 \end{align*}
 
 \hrule
 \vspace{1cm}
 little endian $k=2$
 \begin{align*}
- \left( \left( a_{(0,0)}^{[0]},a_{(0,1)}^{[1]},\ldots, a_{(0,d_1-1)}^{[d_1-1]} \right)\right. \\
-\left( a_{(1,0)}^{[d_1]},a_{(1,1)}^{[d_1+1]},\ldots, a_{(1,d_1-1)}^{[2d_1-1]} \right) \\
+ \left( \left( {\alpha}_{(0,0)}^{[0]},{\alpha}_{(0,1)}^{[1]},\ldots, {\alpha}_{(0,d_1-1)}^{[d_1-1]} \right)\right. \\
+\left( {\alpha}_{(1,0)}^{[d_1]},{\alpha}_{(1,1)}^{[d_1+1]},\ldots, {\alpha}_{(1,d_1-1)}^{[2d_1-1]} \right) \\
    \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
-\left( a_{(d_2-1,0)}^{[(d_2-1)d_1]},a_{(d_2-1,1)}^{[(d_2-1)d_1+1]},\ldots, a_{(d_2-1,d_1-1)}^{[(d_2-1)d_1 +d_1-1]} \right) \\
+\left( {\alpha}_{(d_2-1,0)}^{[(d_2-1)d_1]},{\alpha}_{(d_2-1,1)}^{[(d_2-1)d_1+1]},\ldots, {\alpha}_{(d_2-1,d_1-1)}^{[(d_2-1)d_1 +d_1-1]} \right) \\
 \end{align*}
 
 \hrule
 \vspace{1cm}
 little endian $k=3$
 \begin{align*}
- \left( \left( \left(a_{(0,0,0)}^{[0]},a_{(0,0,1)}^{[1]},\ldots,a_{(0,0,d_1-1)}^{[d_1-1]} \right) \right. \right.\\
-  \left(a_{(0,1,0)}^{[d_1]},a_{(0,1,1)}^{[d_1+1]},\ldots,a_{(0,1,d_1-1)}^{[2d_1-1]} \right)  \\
+ \left( \left( \left({\alpha}_{(0,0,0)}^{[0]},{\alpha}_{(0,0,1)}^{[1]},\ldots,{\alpha}_{(0,0,d_1-1)}^{[d_1-1]} \right) \right. \right.\\
+  \left({\alpha}_{(0,1,0)}^{[d_1]},{\alpha}_{(0,1,1)}^{[d_1+1]},\ldots,{\alpha}_{(0,1,d_1-1)}^{[2d_1-1]} \right)  \\
   \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
-   \left. \left(a_{(0,d_2-1,0)}^{[(d_2-1)d_1]},a_{(0,d_2-1,1)}^{[(d_2-1)d_1+1]},\ldots,a_{(0,d_2-1,d_1-1)}^{[d_1 d_2-1]} \right) \right)  \\
- \left( \left(a_{(1,0,0)}^{[d_1 d_2]},a_{(1,0,1)}^{[d_1 d_2+1]},\ldots,a_{(1,0,d_1-1)}^{[d_1 d_2+d_1-1]} \right) \right. \\
-  \left(a_{(1,1,0)}^{[d_1 d_2+d_1]},a_{(1,1,1)}^{[d_1 d_2+d_1+1]},\ldots,a_{(1,1,d_1-1)}^{[d_1 d_2+2d_1-1]} \right)  \\
+   \left. \left({\alpha}_{(0,d_2-1,0)}^{[(d_2-1)d_1]},{\alpha}_{(0,d_2-1,1)}^{[(d_2-1)d_1+1]},\ldots,{\alpha}_{(0,d_2-1,d_1-1)}^{[d_1 d_2-1]} \right) \right)  \\
+ \left( \left({\alpha}_{(1,0,0)}^{[d_1 d_2]},{\alpha}_{(1,0,1)}^{[d_1 d_2+1]},\ldots,{\alpha}_{(1,0,d_1-1)}^{[d_1 d_2+d_1-1]} \right) \right. \\
+  \left({\alpha}_{(1,1,0)}^{[d_1 d_2+d_1]},{\alpha}_{(1,1,1)}^{[d_1 d_2+d_1+1]},\ldots,{\alpha}_{(1,1,d_1-1)}^{[d_1 d_2+2d_1-1]} \right)  \\
   \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
-   \left. \left(a_{(1,d_2-1,0)}^{[d_1 d_2+(d_2-1)d_1]},a_{(1,d_2-1,1)}^{[d_1 d_2+(d_2-1)d_1+1]},\ldots,a_{(1,d_2-1,d_1-1)}^{[d_1 d_2+(d_2-1)d_1+d_1-1]} \right) \right)  \\
+   \left. \left({\alpha}_{(1,d_2-1,0)}^{[d_1 d_2+(d_2-1)d_1]},{\alpha}_{(1,d_2-1,1)}^{[d_1 d_2+(d_2-1)d_1+1]},\ldots,{\alpha}_{(1,d_2-1,d_1-1)}^{[d_1 d_2+(d_2-1)d_1+d_1-1]} \right) \right)  \\
   \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
   \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
   \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
-   \left( \left(a_{(d_3-1,0,0)}^{[(d_3-1)d_1 d_2]},a_{(d_3-1,0,1)}^{[(d_3-1)d_1 d_2+1]},\ldots,a_{(d_3-1,0,d_1-1)}^{[(d_3-1)d_1 d_2+d_1-1]} \right) \right. \\
-  \left(a_{(d_3-1,1,0)}^{[(d_3-1)d_1 d_2+d_1]},a_{(d_3-1,1,1)}^{[(d_3-1)d_1 d_2+d_1+1]},\ldots,a_{(1,1,d_1-1)}^{[(d_3-1)d_1 d_2+2d_1-1]} \right)  \\
+   \left( \left({\alpha}_{(d_3-1,0,0)}^{[(d_3-1)d_1 d_2]},{\alpha}_{(d_3-1,0,1)}^{[(d_3-1)d_1 d_2+1]},\ldots,{\alpha}_{(d_3-1,0,d_1-1)}^{[(d_3-1)d_1 d_2+d_1-1]} \right) \right. \\
+  \left({\alpha}_{(d_3-1,1,0)}^{[(d_3-1)d_1 d_2+d_1]},{\alpha}_{(d_3-1,1,1)}^{[(d_3-1)d_1 d_2+d_1+1]},\ldots,{\alpha}_{(1,1,d_1-1)}^{[(d_3-1)d_1 d_2+2d_1-1]} \right)  \\
   \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
-   \left. \left. \left(a_{(d_3-1,d_2-1,0)}^{[(d_3-1)d_1 d_2+(d_2-1)d_1]},a_{(d_3-1,d_2-1,1)}^{[(d_3-1)d_1 d_2+(d_2-1)d_1+1]},\ldots,a_{(d_3-1,d_2-1,d_1-1)}^{[(d_3-1)d_1 d_2+(d_2-1)d_1+d_1-1]} \right)\right) \right)  \\
+   \left. \left. \left({\alpha}_{(d_3-1,d_2-1,0)}^{[(d_3-1)d_1 d_2+(d_2-1)d_1]},{\alpha}_{(d_3-1,d_2-1,1)}^{[(d_3-1)d_1 d_2+(d_2-1)d_1+1]},\ldots,{\alpha}_{(d_3-1,d_2-1,d_1-1)}^{[(d_3-1)d_1 d_2+(d_2-1)d_1+d_1-1]} \right)\right) \right)  \\
    \end{align*}
 
    \hrule
@@ -300,64 +302,64 @@ little endian $k=3$
 little endian $k=4$
 
 \begin{align*}
-& \left( \left( \left( \left(a_{(0,0,0,0)}^{[0]},a_{(0,0,0,1)}^{[1]},\ldots,a_{(0,0,0,d_1-1)}^{[d_1-1]} \right) \right. \right. \right. \\
-&  \left(a_{(0,0,1,0)}^{[d_1]},a_{(0,0,1,1)}^{[d_1+1]},\ldots,a_{(0,0,1,d_1-1)}^{[2d_1-1]} \right)  \\
+& \left( \left( \left( \left({\alpha}_{(0,0,0,0)}^{[0]},{\alpha}_{(0,0,0,1)}^{[1]},\ldots,{\alpha}_{(0,0,0,d_1-1)}^{[d_1-1]} \right) \right. \right. \right. \\
+&  \left({\alpha}_{(0,0,1,0)}^{[d_1]},{\alpha}_{(0,0,1,1)}^{[d_1+1]},\ldots,{\alpha}_{(0,0,1,d_1-1)}^{[2d_1-1]} \right)  \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
-&   \left. \left(a_{(0,0,d_2-1,0)}^{[(d_2-1)d_1]},a_{(0,0,d_2-1,1)}^{[(d_2-1)d_1+1]},\ldots,a_{(0,0,d_2-1,d_1-1)}^{[d_1 d_2-1]} \right) \right)  \\
-& \left( \left(a_{(0,1,0,0)}^{[d_1 d_2]},a_{(0,1,0,1)}^{[d_1 d_2+1]},\ldots,a_{(0,1,0,d_1-1)}^{[d_1 d_2+d_1-1]} \right) \right. \\
-&  \left(a_{(0,1,1,0)}^{[d_1 d_2+d_1]},a_{(0,1,1,1)}^{[d_1 d_2+d_1+1]},\ldots,a_{(0,1,1,d_1-1)}^{[d_1 d_2+2d_1-1]} \right)  \\
+&   \left. \left({\alpha}_{(0,0,d_2-1,0)}^{[(d_2-1)d_1]},{\alpha}_{(0,0,d_2-1,1)}^{[(d_2-1)d_1+1]},\ldots,{\alpha}_{(0,0,d_2-1,d_1-1)}^{[d_1 d_2-1]} \right) \right)  \\
+& \left( \left({\alpha}_{(0,1,0,0)}^{[d_1 d_2]},{\alpha}_{(0,1,0,1)}^{[d_1 d_2+1]},\ldots,{\alpha}_{(0,1,0,d_1-1)}^{[d_1 d_2+d_1-1]} \right) \right. \\
+&  \left({\alpha}_{(0,1,1,0)}^{[d_1 d_2+d_1]},{\alpha}_{(0,1,1,1)}^{[d_1 d_2+d_1+1]},\ldots,{\alpha}_{(0,1,1,d_1-1)}^{[d_1 d_2+2d_1-1]} \right)  \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
-&   \left. \left(a_{(0,1,d_2-1,0)}^{[d_1 d_2+(d_2-1)d_1]},a_{(0,1,d_2-1,1)}^{[d_1 d_2+(d_2-1)d_1+1]},\ldots,a_{(0,1,d_2-1,d_1-1)}^{[d_1 d_2+(d_2-1)d_1+d_1-1]} \right) \right)  \\
+&   \left. \left({\alpha}_{(0,1,d_2-1,0)}^{[d_1 d_2+(d_2-1)d_1]},{\alpha}_{(0,1,d_2-1,1)}^{[d_1 d_2+(d_2-1)d_1+1]},\ldots,{\alpha}_{(0,1,d_2-1,d_1-1)}^{[d_1 d_2+(d_2-1)d_1+d_1-1]} \right) \right)  \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
-&   \left( \left(a_{(0,d_3-1,0,0)}^{[(d_3-1)d_1 d_2]},a_{(0,d_3-1,0,1)}^{[(d_3-1)d_1 d_2+1]},\ldots,a_{(0,d_3-1,0,d_1-1)}^{[(0,d_3-1)d_1 d_2+d_1-1]} \right) \right. \\
-&  \left(a_{(0,d_3-1,1,0)}^{[(d_3-1)d_1 d_2+d_1]},a_{(0,d_3-1,1,1)}^{[(d_3-1)d_1 d_2+d_1+1]},\ldots,a_{(0,1,1,d_1-1)}^{[(d_3-1)d_1 d_2+2d_1-1]} \right)  \\
+&   \left( \left({\alpha}_{(0,d_3-1,0,0)}^{[(d_3-1)d_1 d_2]},{\alpha}_{(0,d_3-1,0,1)}^{[(d_3-1)d_1 d_2+1]},\ldots,{\alpha}_{(0,d_3-1,0,d_1-1)}^{[(0,d_3-1)d_1 d_2+d_1-1]} \right) \right. \\
+&  \left({\alpha}_{(0,d_3-1,1,0)}^{[(d_3-1)d_1 d_2+d_1]},{\alpha}_{(0,d_3-1,1,1)}^{[(d_3-1)d_1 d_2+d_1+1]},\ldots,{\alpha}_{(0,1,1,d_1-1)}^{[(d_3-1)d_1 d_2+2d_1-1]} \right)  \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
    %\\
-&    \left( \left( \left(a_{(1,0,0,0)}^{[d_1 d_2 d_3]},a_{(1,0,0,1)}^{[d_1 d_2 d_3+1]},\ldots,a_{(1,0,0,d_1-1)}^{[d_1 d_2 d_3+d_1-1]} \right) \right. \right.\\
-&  \left(a_{(1,0,1,0)}^{[d_1 d_2 d_3+d_1]},a_{(1,0,1,1)}^{[d_1 d_2 d_3+d_1+1]},\ldots,a_{(1,0,1,d_1-1)}^{[d_1 d_2 d_3+2d_1-1]} \right)  \\
+&    \left( \left( \left({\alpha}_{(1,0,0,0)}^{[d_1 d_2 d_3]},{\alpha}_{(1,0,0,1)}^{[d_1 d_2 d_3+1]},\ldots,{\alpha}_{(1,0,0,d_1-1)}^{[d_1 d_2 d_3+d_1-1]} \right) \right. \right.\\
+&  \left({\alpha}_{(1,0,1,0)}^{[d_1 d_2 d_3+d_1]},{\alpha}_{(1,0,1,1)}^{[d_1 d_2 d_3+d_1+1]},\ldots,{\alpha}_{(1,0,1,d_1-1)}^{[d_1 d_2 d_3+2d_1-1]} \right)  \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
-&   \left. \left(a_{(1,0,d_2-1,0)}^{[d_1 d_2 d_3+(d_2-1)d_1]},a_{(1,0,d_2-1,1)}^{[d_1 d_2 d_3+(d_2-1)d_1+1]},\ldots,a_{(1,0,d_2-1,d_1-1)}^{[d_1 d_2 d_3+d_1 d_2-1]} \right) \right)  \\
-& \left( \left(a_{(1,1,0,0)}^{[d_1 d_2 d_3+d_1 d_2]},a_{(1,1,0,1)}^{[d_1 d_2 d_3+d_1 d_2+1]},\ldots,a_{(1,1,0,d_1-1)}^{[d_1 d_2 d_3+d_1 d_2+d_1-1]} \right) \right. \\
-&  \left(a_{(1,1,1,0)}^{[d_1 d_2 d_3+d_1 d_2+d_1]},a_{(1,1,1,1)}^{[d_1 d_2 d_3+d_1 d_2+d_1+1]},\ldots,a_{(1,1,1,d_1-1)}^{[d_1 d_2 d_3+d_1 d_2+2d_1-1]} \right)  \\
+&   \left. \left({\alpha}_{(1,0,d_2-1,0)}^{[d_1 d_2 d_3+(d_2-1)d_1]},{\alpha}_{(1,0,d_2-1,1)}^{[d_1 d_2 d_3+(d_2-1)d_1+1]},\ldots,{\alpha}_{(1,0,d_2-1,d_1-1)}^{[d_1 d_2 d_3+d_1 d_2-1]} \right) \right)  \\
+& \left( \left({\alpha}_{(1,1,0,0)}^{[d_1 d_2 d_3+d_1 d_2]},{\alpha}_{(1,1,0,1)}^{[d_1 d_2 d_3+d_1 d_2+1]},\ldots,{\alpha}_{(1,1,0,d_1-1)}^{[d_1 d_2 d_3+d_1 d_2+d_1-1]} \right) \right. \\
+&  \left({\alpha}_{(1,1,1,0)}^{[d_1 d_2 d_3+d_1 d_2+d_1]},{\alpha}_{(1,1,1,1)}^{[d_1 d_2 d_3+d_1 d_2+d_1+1]},\ldots,{\alpha}_{(1,1,1,d_1-1)}^{[d_1 d_2 d_3+d_1 d_2+2d_1-1]} \right)  \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
-&   \left. \left(a_{(1,1,d_2-1,0)}^{[d_1 d_2 d_3+d_1 d_2+(d_2-1)d_1]},a_{(1,1,d_2-1,1)}^{[d_1 d_2 d_3+d_1 d_2+(d_2-1)d_1+1]},\ldots,a_{(1,1,d_2-1,d_1-1)}^{[d_1 d_2 d_3+d_1 d_2+(d_2-1)d_1+d_1-1]} \right) \right)  \\
+&   \left. \left({\alpha}_{(1,1,d_2-1,0)}^{[d_1 d_2 d_3+d_1 d_2+(d_2-1)d_1]},{\alpha}_{(1,1,d_2-1,1)}^{[d_1 d_2 d_3+d_1 d_2+(d_2-1)d_1+1]},\ldots,{\alpha}_{(1,1,d_2-1,d_1-1)}^{[d_1 d_2 d_3+d_1 d_2+(d_2-1)d_1+d_1-1]} \right) \right)  \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
       \end{align*}
 
    \begin{align*}
-&   \left( \left(a_{(1,d_3-1,0,0)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2]},a_{(1,d_3-1,0,1)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2+1]},\ldots,a_{(1,d_3-1,0,d_1-1)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2+d_1-1]} \right) \right. \\
-&  \left(a_{(1,d_3-1,1,0)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2+d_1]},a_{(1,d_3-1,1,1)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2+d_1+1]},\ldots,a_{(1,1,1,d_1-1)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2+2d_1-1]} \right)  \\
+&   \left( \left({\alpha}_{(1,d_3-1,0,0)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2]},{\alpha}_{(1,d_3-1,0,1)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2+1]},\ldots,{\alpha}_{(1,d_3-1,0,d_1-1)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2+d_1-1]} \right) \right. \\
+&  \left({\alpha}_{(1,d_3-1,1,0)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2+d_1]},{\alpha}_{(1,d_3-1,1,1)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2+d_1+1]},\ldots,{\alpha}_{(1,1,1,d_1-1)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2+2d_1-1]} \right)  \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
-&   \left. \left. \left(a_{(1,d_3-1,d_2-1,0)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2+(d_2-1)d_1]},a_{(1,d_3-1,d_2-1,1)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2+(d_2-1)d_1+1]},\ldots,a_{(1,d_3-1,d_2-1,d_1-1)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2+(d_2-1)d_1+d_1-1]} \right)\right) \right)  \\
+&   \left. \left. \left({\alpha}_{(1,d_3-1,d_2-1,0)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2+(d_2-1)d_1]},{\alpha}_{(1,d_3-1,d_2-1,1)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2+(d_2-1)d_1+1]},\ldots,{\alpha}_{(1,d_3-1,d_2-1,d_1-1)}^{[d_1 d_2 d_3+(d_3-1)d_1 d_2+(d_2-1)d_1+d_1-1]} \right)\right) \right)  \\
    %\\
 &   \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
       %\\
-&    \left( \left( \left(a_{(d_4-1,0,0,0)}^{[(d_4-1)d_1 d_2 d_3]},a_{(d_4-1,0,0,1)}^{[(d_4-1)d_1 d_2 d_3+1]},\ldots,a_{(d_4-1,0,0,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+d_1-1]} \right) \right. \right.\\
-&  \left(a_{(d_4-1,0,1,0)}^{[(d_4-1)d_1 d_2 d_3+d_1]},a_{(d_4-1,0,1,1)}^{[(d_4-1)d_1 d_2 d_3+d_1+1]},\ldots,a_{(d_4-1,0,1,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+2d_1-1]} \right)  \\
+&    \left( \left( \left({\alpha}_{(d_4-1,0,0,0)}^{[(d_4-1)d_1 d_2 d_3]},{\alpha}_{(d_4-1,0,0,1)}^{[(d_4-1)d_1 d_2 d_3+1]},\ldots,{\alpha}_{(d_4-1,0,0,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+d_1-1]} \right) \right. \right.\\
+&  \left({\alpha}_{(d_4-1,0,1,0)}^{[(d_4-1)d_1 d_2 d_3+d_1]},{\alpha}_{(d_4-1,0,1,1)}^{[(d_4-1)d_1 d_2 d_3+d_1+1]},\ldots,{\alpha}_{(d_4-1,0,1,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+2d_1-1]} \right)  \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
-&   \left. \left(a_{(d_4-1,0,d_2-1,0)}^{[(d_4-1)d_1 d_2 d_3+(d_2-1)d_1]},a_{(d_4-1,0,d_2-1,1)}^{[(d_4-1)d_1 d_2 d_3+(d_2-1)d_1+1]},\ldots,a_{(d_4-1,0,d_2-1,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2-1]} \right) \right)  \\
-& \left( \left(a_{(d_4-1,1,0,0)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2]},a_{(d_4-1,1,0,1)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2+1]},\ldots,a_{(d_4-1,1,0,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2+d_1-1]} \right) \right. \\
-&  \left(a_{(d_4-1,1,1,0)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2+d_1]},a_{(d_4-1,1,1,1)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2+d_1+1]},\ldots,a_{(d_4-1,1,1,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2+2d_1-1]} \right)  \\
+&   \left. \left({\alpha}_{(d_4-1,0,d_2-1,0)}^{[(d_4-1)d_1 d_2 d_3+(d_2-1)d_1]},{\alpha}_{(d_4-1,0,d_2-1,1)}^{[(d_4-1)d_1 d_2 d_3+(d_2-1)d_1+1]},\ldots,{\alpha}_{(d_4-1,0,d_2-1,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2-1]} \right) \right)  \\
+& \left( \left({\alpha}_{(d_4-1,1,0,0)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2]},{\alpha}_{(d_4-1,1,0,1)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2+1]},\ldots,{\alpha}_{(d_4-1,1,0,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2+d_1-1]} \right) \right. \\
+&  \left({\alpha}_{(d_4-1,1,1,0)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2+d_1]},{\alpha}_{(d_4-1,1,1,1)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2+d_1+1]},\ldots,{\alpha}_{(d_4-1,1,1,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2+2d_1-1]} \right)  \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
-&   \left. \left(a_{(d_4-1,1,d_2-1,0)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2+(d_2-1)d_1]},a_{(d_4-1,1,d_2-1,1)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2+(d_2-1)d_1+1]},\ldots,a_{(d_4-1,1,d_2-1,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2+(d_2-1)d_1+d_1-1]} \right) \right)  \\
+&   \left. \left({\alpha}_{(d_4-1,1,d_2-1,0)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2+(d_2-1)d_1]},{\alpha}_{(d_4-1,1,d_2-1,1)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2+(d_2-1)d_1+1]},\ldots,{\alpha}_{(d_4-1,1,d_2-1,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+d_1 d_2+(d_2-1)d_1+d_1-1]} \right) \right)  \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
-&   \left( \left(a_{(d_4-1,d_3-1,0,0)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2]},a_{(d_4-1,d_3-1,0,1)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2+1]},\ldots,a_{(d_4-1,d_3-1,0,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2+d_1-1]} \right) \right. \\
-&  \left(a_{(d_4-1,d_3-1,1,0)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2+d_1]},a_{(d_4-1,d_3-1,1,1)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2+d_1+1]},\ldots,a_{(d_4-1,1,1,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2+2d_1-1]} \right)  \\
+&   \left( \left({\alpha}_{(d_4-1,d_3-1,0,0)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2]},{\alpha}_{(d_4-1,d_3-1,0,1)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2+1]},\ldots,{\alpha}_{(d_4-1,d_3-1,0,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2+d_1-1]} \right) \right. \\
+&  \left({\alpha}_{(d_4-1,d_3-1,1,0)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2+d_1]},{\alpha}_{(d_4-1,d_3-1,1,1)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2+d_1+1]},\ldots,{\alpha}_{(d_4-1,1,1,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2+2d_1-1]} \right)  \\
 &  \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
-& \left.  \left. \left. \left(a_{(d_4-1,d_3-1,d_2-1,0)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2+(d_2-1)d_1]},
-%a_{(d_4-1,d_3-1,d_2-1,1)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2+(d_2-1)d_1+1]},
+& \left.  \left. \left. \left({\alpha}_{(d_4-1,d_3-1,d_2-1,0)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2+(d_2-1)d_1]},
+%{\alpha}_{(d_4-1,d_3-1,d_2-1,1)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2+(d_2-1)d_1+1]},
 \ldots,
 %\right.\right.\right.\\
 %&\ldots,\left.\left.\left.
-a_{(d_4-1,d_3-1,d_2-1,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2+(d_2-1)d_1+d_1-1]} \right)\right) \right) \right) \\
+{\alpha}_{(d_4-1,d_3-1,d_2-1,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2+(d_2-1)d_1+d_1-1]} \right)\right) \right) \right) \\
    \end{align*}
 \hrule
 
@@ -365,20 +367,20 @@ a_{(d_4-1,d_3-1,d_2-1,d_1-1)}^{[(d_4-1)d_1 d_2 d_3+(d_3-1)d_1 d_2+(d_2-1)d_1+d_1
 \iffalse
 
 \begin{align*}
- \left(  \left( \left( a_{(0,0,0,0)},a_{(0,0,0,1)},\ldots,a_{(0,0,0,d_1-1)} \right) \right. \right. \\
-  \left( a_{(0,0,1,0)},a_{(0,0,1,1)},\ldots,a_{(0,0,1,d_1-1)} \right) \\
+ \left(  \left( \left( {\alpha}_{(0,0,0,0)},{\alpha}_{(0,0,0,1)},\ldots,{\alpha}_{(0,0,0,d_1-1)} \right) \right. \right. \\
+  \left( {\alpha}_{(0,0,1,0)},{\alpha}_{(0,0,1,1)},\ldots,{\alpha}_{(0,0,1,d_1-1)} \right) \\
   \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
- \left. \left( a_{(0,0,d_2-1,0)},a_{(0,0,d_2-1,1)},\ldots,a_{(0,0,d_2-1,d_1-1)} \right) \right) \\
-\left( \left( a_{(0,1,0,0)},a_{(0,1,0,1)},\ldots,a_{(0,1,0,d_1-1)} \right) \right. \\
-  \left( a_{(0,1,1,0)},a_{(0,1,1,1)},\ldots,a_{(0,1,1,d_1-1)} \right) \\
+ \left. \left( {\alpha}_{(0,0,d_2-1,0)},{\alpha}_{(0,0,d_2-1,1)},\ldots,{\alpha}_{(0,0,d_2-1,d_1-1)} \right) \right) \\
+\left( \left( {\alpha}_{(0,1,0,0)},{\alpha}_{(0,1,0,1)},\ldots,{\alpha}_{(0,1,0,d_1-1)} \right) \right. \\
+  \left( {\alpha}_{(0,1,1,0)},{\alpha}_{(0,1,1,1)},\ldots,{\alpha}_{(0,1,1,d_1-1)} \right) \\
   \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
- \left. \left( a_{(0,1,d_2-1,0)},a_{(0,1,d_2-1,1)},\ldots,a_{(0,1,d_2-1,d_1-1)} \right) \right) \\
+ \left. \left( {\alpha}_{(0,1,d_2-1,0)},{\alpha}_{(0,1,d_2-1,1)},\ldots,{\alpha}_{(0,1,d_2-1,d_1-1)} \right) \right) \\
   \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
   \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
-\left( \left( a_{(0,d_3-1,0,0)},a_{(0,d_3-1,0,1)},\ldots,a_{(0,d_3-1,0,d_1-1)} \right) \right. \\
-  \left( a_{(0,d_3-1,1,0)},a_{(0,d_3-1,1,1)},\ldots,a_{(0,d_3-1,1,d_1-1)} \right) \\
+\left( \left( {\alpha}_{(0,d_3-1,0,0)},{\alpha}_{(0,d_3-1,0,1)},\ldots,{\alpha}_{(0,d_3-1,0,d_1-1)} \right) \right. \\
+  \left( {\alpha}_{(0,d_3-1,1,0)},{\alpha}_{(0,d_3-1,1,1)},\ldots,{\alpha}_{(0,d_3-1,1,d_1-1)} \right) \\
   \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots \hspace{1cm} \vdots   \\
- \left. \left( a_{(0,d_3-1,d_2-1,0)},a_{(0,d_3-1,d_2-1,1)},\ldots,a_{(0,d_3-1,d_2-1,d_1-1)} \right) \right) \\
+ \left. \left( {\alpha}_{(0,d_3-1,d_2-1,0)},{\alpha}_{(0,d_3-1,d_2-1,1)},\ldots,{\alpha}_{(0,d_3-1,d_2-1,d_1-1)} \right) \right) \\
 \end{align*}
 \fi