update: finalize conditions and formulas transposition

onedimensiontensor.tex

@@ -600,6 +600,69 @@ m-\prod_{l=2}^kd_l&= -\prod_{l=2}^kd_l+ (d_3-1-i_3)\prod_{s=4}d_s+\sum_{j=4}^k(d
 \end{align*}
 
 
+\subsection{formulas}
+$$m=\sum_{j=1}^ki_j\prod_{l=j+1}^kd_l$$
+if $i_1<d_1-1$
+\begin{align*}
+m'&=m+\prod_{l=2}^kd_l \\
+&=(i_1+1)\prod_{l=2}^kd_l+\sum_{j=2}^ki_j\prod_{l=j+1}d_l
+\end{align*}
+if $i_1=d_1-1$ and $i_2<d_2-1$
+\begin{align*}
+m'&=m-(d_1-1)\prod_{l=2}^kd_l + \prod_{l=3}^kd_l \\
+ &=0\prod_{l=2}^k d_l+(i_2+1)\prod_{l=3}^k d_l+\sum_{j=3}^ki_j\prod_{l=j+1}d_l
+\end{align*}
+if $i_1=d_1-1$ and $i_2=d_2-1$ and $i_3<d_3-1$
+\begin{align*}
+m'&=m-(d_1-1)\prod_{l=2}^kd_l-(d_2-1)\prod_{l=3}^kd_l + \prod_{l=4}^kd_l \\
+ &=0\prod_{l=2}^k d_l+0\prod_{l=3}^k d_l+(i_3+1)\prod_{l=4}^k d_l+\sum_{j=4}^ki_j\prod_{l=j+1}d_l
+\end{align*}
+if $i_1=d_1-1$ and $i_2=d_2-1$ and $i_3=d_3-1$ and $i_4<d_4-1$
+\begin{align*}
+m'&=m-(d_1-1)\prod_{l=2}^kd_l-(d_2-1)\prod_{l=3}^kd_l -(d_3-1)\prod_{l=4}^kd_l + \prod_{l=5}^kd_l \\
+ &=0\prod_{l=2}^k d_l+0\prod_{l=3}^k d_l+0\prod_{l=4}^k d_l+(i_4+1)\prod_{l=5}^k d_l+\sum_{j=5}^ki_j\prod_{l=j+1}d_l
+\end{align*}
+
+and so on!!
+
+
+\subsection{conditions}
+
+if $i_1<d_1-1$
+\begin{align*}
+m'&=m+\prod_{l=2}^kd_l \\
+&=(i_1+1)\prod_{l=2}^kd_l+\sum_{j=2}^ki_j\prod_{l=j+1}d_l\\
+&\le\prod_{l=1}d_l-1
+\end{align*}
+if $i_1=d_1-1$ and $i_2<d_2-1$
+
+\begin{align*}
+m+\prod_{l=2}^kd_l  &= \prod_{l=1}^kd_l+\sum_{j=2}^ki_j\prod_{l=j+1}^kd_l\\
+&\ge \prod_{l=1}^kd_l \\
+m'&=m-(d_1-1)\prod_{l=2}^kd_l + \prod_{l=3}^kd_l \\
+ &=0\prod_{l=2}^k d_l+(i_2+1)\prod_{l=3}^k d_l+\sum_{j=3}^ki_j\prod_{l=j+1}d_l \\
+ & \le \prod_{l=2}^kd_l-1
+\end{align*}
+if $i_1=d_1-1$ and $i_2=d_2-1$ and $i_3<d_3-1$
+\begin{align*}
+m+\prod_{l=2}^kd_l  & \ge \prod_{l=1}^kd_l-1 \\
+m-(d_1-1)\prod_{l=2}^kd_l + \prod_{l=3}^kd_l  &= \prod_{l=2}^kd_l + \sum_{j=3}^ki_j\prod_{l=j+1}^kd_l \\
+& \ge \prod_{l=2}^kd_l \\
+m'&=m-(d_1-1)\prod_{l=2}^kd_l-(d_2-1)\prod_{l=3}^kd_l + \prod_{l=4}^kd_l \\
+ &=0\prod_{l=2}^k d_l+0\prod_{l=3}^k d_l+(i_3+1)\prod_{l=4}^k d_l+\sum_{j=4}^ki_j\prod_{l=j+1}d_l\\
+ & \le \prod_{l=3}d_l-1
+\end{align*}
+if $i_1=d_1-1$ and $i_2=d_2-1$ and $i_3=d_3-1$ and $i_4<d_4-1$
+\begin{align*}
+m+\prod_{l=2}^kd_l  & \ge \prod_{l=1}^kd_l-1 \\
+m-(d_1-1)\prod_{l=2}^kd_l + \prod_{l=3}^kd_l  &= \prod_{l=2}^kd_l + \sum_{j=3}^ki_j\prod_{l=j+1}^kd_l \\
+& \ge \prod_{l=2}^kd_l \\
+m-(d_1-1)\prod_{l=2}^kd_l -(d_2-1)\prod_{l=3}^kd_l + \prod_{l=4}^kd_l  &= \prod_{l=3}^kd_l + \sum_{j=4}^ki_j\prod_{l=j+1}^kd_l \\
+& \ge \prod_{l=3}^kd_l \\
+m'&=m-(d_1-1)\prod_{l=2}^kd_l-(d_2-1)\prod_{l=3}^kd_l -(d_3-1)\prod_{l=4}^kd_l + \prod_{l=5}^kd_l \\
+ &=0\prod_{l=2}^k d_l+0\prod_{l=3}^k d_l+0\prod_{l=4}^k d_l+(i_4+1)\prod_{l=5}^k d_l+\sum_{j=5}^ki_j\prod_{l=j+1}d_l\\
+ &\le \prod_{l=4}d_l-1
+\end{align*}