Differences
This shows you the differences between two versions of the page.
multi-index_power [2013/09/17 00:24] nikolaj |
multi-index_power [2014/03/21 11:11] |
||
---|---|---|---|
Line 1: | Line 1: | ||
- | ===== Multi-index power ===== | ||
- | ==== Definition ==== | ||
- | | @#88DDEE: $ G $ ... group | | ||
- | | @#88DDEE: $ g \in \text{Sequence}(G) $ | | ||
- | | @#88DDEE: $ \alpha \in \text{Sequence}(\mathbb N) $ | | ||
- | | @#88DDEE: $ \mathrm{length}(g)=\mathrm{length}(\alpha) $ | | ||
- | | @#FFBB00: $ \langle g,\alpha\rangle \mapsto g^\alpha := \prod_{i=1}^{\mathrm{length}(\alpha)} g_i^{\alpha_i} $ | | ||
- | |||
- | We also write $|\gamma|=\sum_i^{\mathrm{length}(\gamma)} \gamma_i $. | ||
- | |||
- | ==== Discussion ==== | ||
- | In most cases, the base sequence is understood. E.g. if | ||
- | |||
- | $\gamma=\langle 3,1,0,0,2 \rangle$ | ||
- | |||
- | is taken to be a multiindex, then $|\gamma|=6$ and we write | ||
- | |||
- | $f^{(\gamma)}(x) \equiv \frac{\partial^{|\gamma|}}{\partial x\gamma} f \equiv \frac{\partial^3}{\partial x_1^3} \frac{\partial}{\partial x_2} \frac{\partial^2}{\partial x_5^2} f $ | ||
- | |||
- | === Reference === | ||
- | Wikipedia: [[http://en.wikipedia.org/wiki/Multi-index_notation|Multi-index notation]] | ||
- | ==== Parents ==== | ||
- | === Requirements === | ||
- | [[Group]], [[Integer]], [[Sequence length]] |