⧼exchistory⧽
5 exercise(s) shown, 0 hidden
BBot
Apr 22'25
[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.
Establish the Möbius inversion formula, namely
[[math]]
f(\sigma)=\sum_{\pi\leq\sigma}g(\pi)
\quad\implies\quad g(\sigma)=\sum_{\pi\leq\sigma}\mu(\pi,\sigma)f(\pi)
[[/math]]
for the functions on [math]P(p)[/math].
BBot
Apr 22'25
[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 the inverse of the adjacency matrix of [math]P(k)[/math], given by
[[math]]
A_k(\pi,\sigma)=\begin{cases}
1&{\rm if}\ \pi\leq\sigma\\
0&{\rm if}\ \pi\not\leq\sigma
\end{cases}
[[/math]]
is the Möbius matrix of [math]P[/math], given by [math]M_k(\pi,\sigma)=\mu(\pi,\sigma)[/math].
BBot
Apr 22'25
[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 given independent normal variables [math]x,y[/math], by setting
[[math]]
z=\frac{1}{\sqrt{2}}(x+iy)
[[/math]]
the even moments of the variable [math]|z|[/math] are given by the formula [math]\mathbb E(|z|^{2p})=p![/math].
BBot
Apr 22'25
[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.
Establish the following formulae,
[[math]]
F_{ix}F_{iy}=F_{i,x+y}\quad,\quad
\overline{F}_{ix}=F_{i,-x}\quad,\quad
\sum_xF_{ix}=N\delta_{i0}
[[/math]]
valid for any generalized Fourier matrix, [math]F=F_G[/math].
BBot
Apr 22'25
[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.
Compute the glow of the Walsh matrices
[[math]]
W_N=F_2^{\otimes n}
[[/math]]
with [math]N=2^n[/math], and check if this glow is polynomial or not in [math]N[/math].