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multi-index_power [2013/09/17 00:23] nikolaj |
multi-index_power [2014/03/21 11:11] (current) |
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| ===== Multi-index power ===== | ===== Multi-index power ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $ G $ ... group | | + | | @#55CCEE: context | @#55CCEE: $ G $ ... group | |
| - | | @#88DDEE: $ g \in \text{Sequence}(G) $ | | + | | @#55CCEE: context | @#55CCEE: $ g \in \text{Sequence}(G) $ | |
| - | | @#88DDEE: $ \alpha \in \text{Sequence}(\mathbb N) $ | | + | | @#55CCEE: context | @#55CCEE: $ \alpha \in \text{Sequence}(\mathbb N) $ | |
| - | | @#88DDEE: $ \mathrm{length}(g)=\mathrm{length}(\alpha) $ | | + | | @#55CCEE: context | @#55CCEE: $ \mathrm{length}(g)=\mathrm{length}(\alpha) $ | |
| - | | @#FFBB00: $ \langle g,\alpha\rangle \mapsto g^\alpha := \prod_{i=1}^{\mathrm{length}(\alpha)} g_i^{\alpha_i} $ | | + | | @#FFBB00: definiendum | @#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_i $. | + | We also write $|\gamma|=\sum_i^{\mathrm{length}(\gamma)} \gamma_i $. |
| ==== Discussion ==== | ==== Discussion ==== | ||
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| is taken to be a multiindex, then $|\gamma|=6$ and we write | 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 $ | + | $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 === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Multi-index_notation|Multi-index notation]] | Wikipedia: [[http://en.wikipedia.org/wiki/Multi-index_notation|Multi-index notation]] | ||
| ==== Parents ==== | ==== Parents ==== | ||
| - | === Requirements === | + | === Context === |
| [[Group]], [[Integer]], [[Sequence length]] | [[Group]], [[Integer]], [[Sequence length]] | ||
