465 lines
30 KiB
TeX
465 lines
30 KiB
TeX
\documentclass[a4paper,11pt]{report}
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%\usepackage[utf8]{inputenc}
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\usepackage[cal=boondox,scr=boondoxo]{mathalfa} %%% endrikan'ny mathcal sy mathscr
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\usepackage[a4paper,bindingoffset=0cm,%
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left=2cm,right=2cm,top=2cm,bottom=2cm,%
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footskip=0.8cm]{geometry}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage[french]{babel}
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\frenchbsetup{StandardLists=true} % àinclure si on utilise \usepackage[french]{babel}
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\usepackage{cite} % pour les citations multiples
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\usepackage{enumitem}
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%\bibliographystyle{fr-plain}
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\usepackage{amsmath, amsthm} % tr`es bon mode mathÂ?ematailletique
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\usepackage{amsfonts,amssymb}% permet la definition des ensembles
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\usepackage{mathtools}
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\usepackage{centernot}
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%\usepackage{tikz3d-fr}
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\usepackage{tikz,tkz-tab}
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\usetikzlibrary{3d}
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%\usepackage{pgfplots}
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%\usetikzlibrary{decorations.pathreplacing}
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\newtheorem{theorem}{Theorem}[section]
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\newtheorem{definition}{Definition}[section]
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\begin{document}
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\section{Goal: Write any tensor as vector !!}
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\begin{definition}
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A dimension of a tensor $T$ is a vector $d\in \left(\mathbb{N}^*\right)^k$ $(k<\infty)$. We have $$T=(a)_d=(a)_{d_1,\ldots,d_k} =({\alpha}_{i_1,\ldots,i_k}^{\{k\}})_{0\le i_1<d_1, \ldots , 0\le i_k< d_k} $$
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$$ {\alpha}_{i_1,\ldots,i_k}^{\{k\}}\in \mathbb{K} \ \ / \ \ 0\le i_j<d_j $$
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\end{definition}
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$\mathbb{K}$ is a field( for e.g $\mathbb{R}$ )\\
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$k>0$ is the order.\\
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The size of the Tensor is $d_1 \times \cdots \times d_k=\prod_{i=1}^kd_i$
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% Here $\mathbb{K} = \mathbb{R}$ to simplify but it works for all filds.
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We can write $$T\in \mathbb{K}^{d_1 \times d_2 \times \cdots \times d_k}$$
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\section{Bijection}
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For a given $d_1,\ldots,d_k\in \mathbb{N}^*$, there exists a bijective application between a set of tensor order 1 of size $n=d_1\cdot d_2 \cdots d_k$ $\mathbb{K}^{d_1\times d_2 \times \cdots \times d_k}$ and the set of tensor of order $k$ of the size $n= d_1\cdot d_2 \cdots d_k$ $\mathbb{K}^{d_1\times d_2 \times \cdots \times d_k}$.
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Little endian
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$$
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\begin{array}{ccc}
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\mathbb{K}^{d_1\times d_2 \times \cdots \times d_k} & \longrightarrow & \mathbb{K}^{d_1\cdot d_2\cdots d_k} \\
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\left({\alpha}_{(i_1,\ldots,i_k)_k}\right) & \longmapsto & \left({\alpha}_{\left(i_1+i_2\cdot d_1+\ldots+i_k \prod_{j=1}^{k-1}d_j\right)_1}\right)
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\end{array}
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$$
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Big endian
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$$
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\begin{array}{ccc}
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\mathbb{K}^{d_1\times d_2 \times \cdots \times d_k} & \longrightarrow & \mathbb{K}^{d_1\cdot d_2\cdots d_k} \\
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\left({\alpha}_{(i_1,\ldots,i_k)_k}\right) & \longmapsto & \left({\alpha}_{\left(i_1 \prod_{j=2}^kd_j+i_2\prod_{j=3}^k d_j+\ldots +i_{k-1} d_k +i_k \right)_1}\right)
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\end{array}
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$$
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For $a,b\in \mathbb{N}$, we note $ [\![ a,b ]\!] = [a.. b]=[a,b]\cap \mathbb{N}$
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and $ [\![ a,b [\![ = [\![ a,b ) = [a.. b) = [a.. b[=[a,b[\cap \mathbb{N}$
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If $a=0$ we note $[b]=[\![0,b]\!]$ and $[b)=[\![0,b[\![$ so here we defined a bijection of indexe, i.e,
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\begin{align*}
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f:[d_1)\times[d_2)\times\cdots\times [d_k) & \longrightarrow [d_1\cdot d_2\cdots d_k)\\
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(i_1,i_2,\ldots,i_k)_k & \longrightarrow f(i_1,i_2,\ldots,i_k)=(i)_1
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\end{align*}
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If little endian
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\begin{align*}
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i & =i_1 +i_2d_1+\cdots+i_k\prod_{j=1}^{k-1} d_j \\
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&=\sum_{j=1}^ki_j\prod_{l=1}^{j-1}d_l
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\end{align*}
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If big endian
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\begin{align*}
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i & =i_1 \prod_{j=2}^kd_j+i_2\prod_{j=3}^k d_j+\ldots +i_{k-1} d_k +i_k \\
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&=\sum_{j=1}^ki_j\prod_{l=i+1}^kd_l
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\end{align*}
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$ $
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\section{Inverse indexes}
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The inverse of Little endian :
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For any $i\in \mathbb{N} , i<n=\prod_{j=1}^kd_j \exists! (i_1, \ldots, i_k) / \forall 1 \le j \le k, 0\le i_j<d_j$ and $$ i=\sum_{j=1}^{k} i_j\left( \prod_{l=1}^{j-1} d_l \right) $$
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The inverse of big endian :
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For any $i\in \mathbb{N} , i<n=\prod_{j=1}^kd_j \exists! (i_1, \ldots, i_k) / \forall 1 \le j \le k, 0\le i_j<d_j$ and $$ i=\sum_{s=j}^k i_s\left( \prod_{l=s+1}^k d_l \right) $$
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Proof of the inverse of big endian, the Little endian is almost the same:
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By reccurence:
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by Euclide division $\exists ! (r_1,i_1)\in \mathbb{N}^2 / r_1 < \prod_{l=2}^k d_l $ and $i=i_1*\prod_{l=2}^k d_l + r_1$.
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And so on, there exists $(r_2, i_2)\in\mathbb{N}^2$ such that $r_2<\prod_{l=3}^k d_l$ and $r_1 = i_2*\prod_{l=3}^k d_l + r_2$.
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Suppose, for all $2<j< k$ there exists $(r_j,i_j)\in\mathbb{N}^2$ such that $r_j<\prod_{l=j+1}^kd_l$ and $r_{j-1}=i_j\prod_{l=j+1}^k d_l + r_j $
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By Euclide disvision on $r_j$ and $\prod_{l=j+2}^kd_l$ there exists $(r_{j+1}, i_{j+1})\in\mathbb{N}^2$ such that $r_j<\prod_{l=j+2}^k d_l$ and $r_j=i_{j+1}\prod_{l=j+2}^k d_l + r_{j+1}$
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One of the advantages using order 1 tensor than larger order only one loop to browse all elements of the tensor.
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\section{Operations on tensors}
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We prove below that all operations like tensor contraction and tensor multiplication work with these bijections.
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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
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$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} $
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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} $
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and $T_{\alpha}^{\{1\}}=\left({\alpha}_{\left(i_1\right)_1}\right)_{0\le i_1 < \prod_{j=1}^k d_{j}}$
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and $T_{\beta}^{\{1\}}=\left({\beta}_{\left(i_1\right)_1}\right)_{0\le i_1 < \prod_{j=1}^{q} b_{j}}$
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Here we use little endian so
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${\alpha}_{\left(\sum_{j=1}^k i_j\left(\prod_{l=1}^{j-1}d_{l}\right)\right)_1}={\alpha}_{\left(i_1,\ldots,i_k\right)_k}$ $\,\, /\,\, i_j\in[d_{j}) \,\,/\,\, 1\le j\le k$
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and
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${\beta}_{\left(\sum_{j=1}^{q} i_j\left(\prod_{l=1}^{j-1}b_{l}\right)\right)_1}={\beta}_{\left(i_1,\ldots,i_{q}\right)_q}$ $\,\, /\,\, i_j\in [b_{j}) \,\,/\,\, 1\le j\le q$
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\subsection{Tensor multiplication}
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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
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$${\gamma}_{\left(i_{1},\ldots, i_{k},i_{k+1},\ldots,i_{k+q}\right)_{k+q}}={\alpha}_{\left(i_{1},\ldots,i_{k}\right)_k} \cdot {\beta}_{\left(i_{k+1},\ldots,i_{k+q}\right)_q} $$
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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
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\begin{align*}
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{\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} }\\
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&={\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} }\\
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&={\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} }\\
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&={\alpha}_{\left(i_{1},\ldots,i_{k}\right)_{k}} \cdot b_{\left( i_{k+1},\ldots,{k+q} \right)_{q} }\\
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\end{align*}
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Then,
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${\gamma}_{\left(\sum_{j=1}^{k+q} i_{j}\prod_{l=1}^{j-1}d_{i_{l}}\right)_{1} }=
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{\alpha}_{\left( \sum_{j=1}^{k} i_{j}\prod_{l=1}^{j-1}d_{i_{l}} \right)_{1}} \cdot b_{\left( \sum_{j=k+1}^{k+q} i_{j}\prod_{l=1}^{j-1}d_{i_{l}} \right)_{1} }$
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\subsection{tensor contraction}
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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 $0<k_0 \le \min(k,q)$.
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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:
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\begin{align*}
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d_{i}&=b_{i-k + k_0 + 1} &\forall i\in [\![k- k_0 , k ]\!] \\
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p_{i}&=d_{i} &\forall i\in [\![ 1 , k- k_0]\!]\\
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p_{i}&=b_{i-k+2k_0} &\forall i\in [\![k-k_0+1, k+q-2k_0]\!] \\
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\end{align*}
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and
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\begin{align*}
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{\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}} \\
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&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}\\
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&=\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}} \\
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\end{align*}
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For example in little endian transformation:
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\begin{align*}
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{\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}}\\
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{\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}}\\
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\end{align*}
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We have to sum these terms $
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\forall l_{1}\in [\![0,b_{1}[\![, \forall l_{2}\in [\![0,b_{1}[\![\cdots \forall l_{k_0}\in[\![0,b_{k_0}[\![
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$
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So
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\begin{align*}
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0 \le l_{j} &\le b_{j}-1 \, \, , \forall 1 \le j \le k_0\\
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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\\
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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\\
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\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}\\
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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}\\
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% \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} \\
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% 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} \\
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% 0 \le \sum_{l_j=0}^{b_{j}-1} l_{j} \prod_{s=1}^{j-1}b_{s} &< \prod_{s=1}^{j}b_{s} \\
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\end{align*}
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if $k_0=1$
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$M_1+1 = (b_{1}-1) +1 = b_{1} $
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Suppose $M_{k_0-1}+1=\prod_{j=1}^{k_0-1}b_{j}$
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%$M_{k_0}+1 = \sum_{j=1}^{k_0} (b_{j}-1) \prod_{s=1}^{j-1}b_{s} +1$
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\begin{align*}
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M_{k_0}+1 &= \sum_{j=1}^{k_0} (b_{j}-1) \prod_{s=1}^{j-1}b_{s} +1 \\
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&= \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 \\
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&= \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} \\
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&=M_{k_0-1} + 1 + (b_{k_0}-1) \prod_{s=1}^{k_0-1}b_{s} \\
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&=\prod_{j=1}^{k_0-1}b_{j} + (b_{k_0}-1) \prod_{s=1}^{k_0-1}b_{s}\\
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&= \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}\\
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&= \prod_{s=1}^{k_0}b_{s} \\
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\end{align*}
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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 $\left[\!\!\left[0,\prod_{j=1}^{k_0}b_{j}\right[\!\!\right[$
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\begin{align*}
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&{\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}} \\
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&={\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}}\\
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& ( t=j-k+k_0 \Rightarrow j=t+k-k_0 )\\
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&(j=k-k_0+1\Rightarrow t=1; j=k\Rightarrow t=k_0)\\
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&= {\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} \\
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\end{align*}
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But $d_{s} = b_{s-k+k_0} \, \, \forall s\in[\![k-k_0+1,k]\!]$
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\begin{align*}
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&{\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}}\\
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&= {\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}} \\
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& (u=s-k+k_0 \Leftrightarrow s=u+k-k_0)\\
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& (s=k-k_0+1 \Rightarrow u=1)\\
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& (s=k-k_0+t-1 \Rightarrow u=t-1)\\
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&= {\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}} \\
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&= {\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} } \\
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\end{align*}
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as below
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$ \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 $1\le j\le k_0$
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As $l_j\in [\![0,b_{j}[\![ \forall 1\le j\le k_0$ and
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$$card([\![0,b_{j}[\![)=b_{j}$$
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also
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$$card\left(\prod_{j=1}^{k_0}[\![0,b_{j}[\![ \right)=\prod_{j=1}^{k_0}b_{j}$$
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then
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\begin{align*}
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&\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}}\\
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&= \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*}
|
||
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}
|
||
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!
|
||
|
||
|
||
\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( {\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( {\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( {\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( {\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({\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({\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({\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({\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({\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
|
||
\vspace{1cm}
|
||
little endian $k=4$
|
||
|
||
\begin{align*}
|
||
& \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({\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({\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({\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({\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({\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({\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({\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({\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({\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({\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({\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({\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({\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.
|
||
{\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
|
||
|
||
|
||
\iffalse
|
||
|
||
\begin{align*}
|
||
\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( {\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( {\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( {\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( {\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
|
||
|
||
|
||
\iffalse
|
||
\begin{EnvTikzEspace}
|
||
\draw[->,>=latex] (0,0,0)--(1,0,0) node[right] {$\vec{\imath}$} ;
|
||
\draw[->,>=latex] (0,0,0)--(0,1,0) node[right] {$\vec{\jmath}$} ;
|
||
\draw[->,>=latex] (0,0,0)--(0,0,1) node[above] {$\vec{k}$} ;
|
||
\end{EnvTikzEspace}
|
||
|
||
\begin{EmpilementCubes}
|
||
\BlocPetitsCubes{1,2,3,4,5 / 0,2,1,1,3 / 0,2,1,1,2 / 1,1,1,1,3}
|
||
\end{EmpilementCubes}
|
||
\fi
|
||
|
||
\iffalse
|
||
\begin{tikzpicture}
|
||
\draw (0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle; % face arrière
|
||
\draw (0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle; % face avant
|
||
% arêtes horizontales, de l’arrière vers l’avant
|
||
\draw (0,0,0) -- (0,0,1); % bas gauche
|
||
\draw (1,0,0) -- (1,0,1); % bas droit
|
||
\draw (1,1,0) -- (1,1,1); % haut droit
|
||
\draw (0,1,0) -- (0,1,1); % haut gauche
|
||
\end{tikzpicture}
|
||
|
||
\newcommand{\xangle}{13} %{15}
|
||
\newcommand{\yangle}{160}%{153}
|
||
\newcommand{\zangle}{90}
|
||
|
||
\newcommand{\xlength}{1}
|
||
\newcommand{\ylength}{1}
|
||
\newcommand{\zlength}{1}
|
||
|
||
\newcommand{\dimension}{1}% actually dimension-1
|
||
|
||
\pgfmathsetmacro{\xx}{\xlength*cos(\xangle)}
|
||
\pgfmathsetmacro{\xy}{\xlength*sin(\xangle)}
|
||
\pgfmathsetmacro{\yx}{\ylength*cos(\yangle)}
|
||
\pgfmathsetmacro{\yy}{\ylength*sin(\yangle)}
|
||
\pgfmathsetmacro{\zx}{\zlength*cos(\zangle)}
|
||
\pgfmathsetmacro{\zy}{\zlength*sin(\zangle)}
|
||
|
||
\begin{tikzpicture}
|
||
[ x={(\xx cm,\xy cm)},
|
||
y={(\yx cm,\yy cm)},
|
||
z={(\zx cm,\zy cm)},
|
||
]
|
||
\foreach \a in {0,...,\dimension}
|
||
{ \foreach \b in {0,...,\dimension}
|
||
{ \pgfmathsetmacro{\c}{100-\a*7-\b*7}
|
||
\draw[canvas is xy plane at z=\a, black!\c] (\b,0) -- (\b,\dimension) (0,\b) -- (\dimension,\b);
|
||
\draw[canvas is xz plane at y=\a, black!\c] (\b,0) -- (\b,\dimension) (0,\b) -- (\dimension,\b);
|
||
\draw[canvas is yz plane at x=\a, black!\c] (\b,0) -- (\b,\dimension) (0,\b) -- (\dimension,\b);
|
||
}
|
||
}
|
||
\pgfmathsetmacro{\dim}{\dimension-1}
|
||
\foreach \a in {0,...,\dimension}
|
||
{ \foreach \b in {0,...,\dimension}
|
||
{ \foreach \c in {0,...,\dimension}
|
||
{
|
||
\pgfmathsetmacro{\d}{\c+\b*\dimension+\a*\dimension*\dimension}
|
||
% \fill (\a,\b,\c) circle (0.05cm);
|
||
\node (1) at (\a,\b,\c) {$\d$};
|
||
}
|
||
}
|
||
}
|
||
\end{tikzpicture}
|
||
\fi
|
||
|
||
\end{document}
|