⧼exchistory⧽
4 exercise(s) shown, 0 hidden
BBot
Apr 21'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
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Prove that the Bell numbers [math]B_k=|P(k)|[/math], which are the moments of the Poisson law [math]p_1[/math], have the following properties:
[[math]]
B_{k+1}=\sum_{r=0}^k\binom{k}{r}B_r\quad,\quad
B_k=\frac{1}{e}\sum_{r=0}^\infty\frac{r^k}{r!}
[[/math]]
[[math]]
\sum_{k=0}^\infty\frac{B_k}{k!}\,z^k=e^{e^z-1}\quad,\quad
B_k=\frac{k!}{2\pi ie}\int_{|z|=1}\frac{e^{e^z}}{z^{k+1}}\,dz
[[/math]]
Also, prove as well that we have [math]\log B_k/k\simeq\log k-\log\log k-1[/math].
BBot
Apr 21'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.
Prove that for the cyclic group [math]\mathbb Z_N\subset O_N[/math] we have
[[math]]
law(\chi)=\left(1-\frac{1}{N}\right)\delta_0+\frac{1}{N}\delta_N
[[/math]]
and look as well at truncated characters.
BBot
Apr 21'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.
Prove that for the dihedral group [math]D_N\subset S_N[/math] we have
[[math]]
law(\chi)=\begin{cases}
\left(\frac{3}{4}-\frac{1}{2N}\right)\delta_0+\frac{1}{4}\delta_2+\frac{1}{2N}\delta_N&(N\ even)\\
&\\
\left(\frac{1}{2}-\frac{1}{2N}\right)\delta_0+\frac{1}{2}\delta_1+\frac{1}{2N}\delta_N&(N\ odd)
\end{cases}
[[/math]]
and look as well at truncated characters.
BBot
Apr 21'25
[math]
\newcommand{\mathds}{\mathbb}[/math]
This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.
Prove that, if [math]g_{ij}[/math] are the standard coordinates of [math]S_N\subset O_N[/math],
[[math]]
{\rm law}(g_{11}+\ldots +g_{ss})=\frac{s!}{N!}\sum_{p=0}^s\frac{(N-p)!}{(s-p)!}
\cdot\frac{\left(\delta_1-\delta_0\right)^{*p}}{p!}
[[/math]]
and deduce from this that such variables become Poisson, with [math]N\to\infty[/math].