Most recent change of MultiIndex
Edit made on June 09, 2013 by ColinWright at 15:20:18
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WM
HEADERS_END
A multi-index is simply a tuple of natural numbers, for which the following operations are defined. For two n-dimensional multi-indices EQN:\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_n)\in\mathbb{N}_0^n EQN:\beta=(\beta_1,\beta_2,\ldots,\beta_n)\in\mathbb{N}_0^n , we define
* Partial ordering
** EQN:\beta\geq\alpha\Leftrightarrow\beta_i\geq\alpha_i\;\forall\;i\in\mathbb{N}^n\;,
* Component-wise summation
** EQN:\alpha\pm\beta=\left(\alpha_1\pm\beta_1,\alpha_2\pm\beta_2,\ldots,\alpha_n\pm\beta_n\right)\;,
* Absolute value
** EQN:\left|\alpha\right|=\sum_{i=1}^n\alpha_i\;,
* Factorial
** EQN:\alpha!=\prod_{i=1}^n\alpha_1!\;, EQN:\alpha!=\prod_{i=1}^n\alpha_i!\;,
* Power (for EQN:\b{x}\in\mathbb{R}^n\.)
** EQN:\b{x}^\alpha=\prod_{i=1}^nx_i\alpha_i\;.