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means_._note [2015/06/20 17:05] nikolaj |
means_._note [2016/03/09 10:34] (current) nikolaj |
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| ===== Means . Note === | ===== Means . Note === | ||
| ==== Note ==== | ==== Note ==== | ||
| - | | @#55CCEE: context | @#55CCEE: $S\subseteq X$ | | + | | @#55CCEE: context | @#55CCEE: $S$ ... set | |
| | @#55CCEE: context | @#55CCEE: $G$ ... group | | | @#55CCEE: context | @#55CCEE: $G$ ... group | | ||
| | @#55CCEE: context | @#55CCEE: $w:S\to G$ | | | @#55CCEE: context | @#55CCEE: $w:S\to G$ | | ||
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| | @#FFBB00: definiendum | @#FFBB00: $\langle f\rangle:=I(f\cdot w)\cdot I(w)^{-1}$ | | | @#FFBB00: definiendum | @#FFBB00: $\langle f\rangle:=I(f\cdot w)\cdot I(w)^{-1}$ | | ||
| - | e.g. $\langle f\rangle_{[a,b]}:=\dfrac{\int_a^bf(x)\,{\mathrm dx}}{b-a}$ | + | Here $(f\cdot w)(s):=f(s)*w(s)$ where $*$ is the group operation. |
| - | where $[a,b]\subseteq{\mathbb R}$ and $w(x):=1$. | + | == Real functions == |
| - | == Note == | + | E.g. $\langle f\rangle_{[a,b]}:=\dfrac{\int_a^bf(x)\,{\mathrm dx}}{b-a}$ |
| - | I use this in the context of [[Minus twelve . Note]]. For $z\in(0,1)$, we find | + | where $[a,b]\subseteq{\mathbb R}$ and $w(x):=1$. |
| + | |||
| + | == Minus twelve == | ||
| + | For $z\in(0,1)$, we find | ||
| $\sum_{k=0}^\infty \langle q\mapsto q\,z^q\rangle_{[k,k+1]}=\dfrac{1}{\ln(z)^2}$, | $\sum_{k=0}^\infty \langle q\mapsto q\,z^q\rangle_{[k,k+1]}=\dfrac{1}{\ln(z)^2}$, | ||
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| $\sum_{k=0}^\infty \left(k\,z^k-\langle q\mapsto q\,z^q\rangle_{[k,k+1]}\right)=\dfrac{z}{(z-1)^2}-\dfrac{1}{\ln(z)^2}=-\dfrac{1}{12}+{\mathcal O}\left((z-1)^1\right)$ | $\sum_{k=0}^\infty \left(k\,z^k-\langle q\mapsto q\,z^q\rangle_{[k,k+1]}\right)=\dfrac{z}{(z-1)^2}-\dfrac{1}{\ln(z)^2}=-\dfrac{1}{12}+{\mathcal O}\left((z-1)^1\right)$ | ||
| + | |||
| + | See [[Minus twelve . Note]]. | ||
| ----- | ----- | ||
| === Requirements === | === Requirements === | ||
| - | [[Function integral]] | + | [[Function integral]], [[Classical probability density function]] |
