Difference between revisions of "Useful Examples"

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(Miscellaneous)
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   W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.
 
   W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.
 
\end{align}
 
\end{align}
 +
 
Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers.  
 
Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers.  
 
The reason for \eqref{eq:W3k} was  long a mystery, but it will be explained  
 
The reason for \eqref{eq:W3k} was  long a mystery, but it will be explained  
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plt.show()
 
plt.show()
 
</syntaxhighlight>
 
</syntaxhighlight>
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== Link to the LMD Generic GCM user manual ==
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[[https://web.lmd.jussieu.fr/~lmdz/planets/LMDZ.GENERIC/ManualGCM_GENERIC.pdf
 +
]]

Revision as of 08:40, 7 April 2022

To Edit Sidebar (as admin only!!!)

Go to MediaWiki:Sidebar

To write some LateX

See e.g. LMDZPedia And check out this page in "edit" mode! $$\newcommand{\Re}{\mathrm{Re}\,} \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}$$

We consider, for various values of $$s$$, the $$n$$-dimensional integral \begin{align} \label{def:Wns} W_n (s) &:= \int_{[0, 1]^n} \left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x} \end{align} which occurs in the theory of uniform random walk integrals in the plane, where at each step a unit-step is taken in a random direction. As such, the integral \eqref{def:Wns} expresses the $$s$$-th moment of the distance to the origin after $$n$$ steps.

By experimentation and some sketchy arguments we quickly conjectured and strongly believed that, for $$k$$ a nonnegative integer \begin{align} \label{eq:W3k} W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}. \end{align}

Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. The reason for \eqref{eq:W3k} was long a mystery, but it will be explained at the end of the paper.

Miscellaneous

  • HTML like syntax works! e.g. comments with
 <!-- This is a comment in the page -->
 which you see because between "pre" tags
  • Some special features:

Link to the internal page with all special features: Special:Version

  • Examples of Syntax Highlighting:

Some Fortran Code:

1 program truc
2 implicit none
3 integer :: i
4 do i=1,10
5   write(*,*) "i=",i
6 end do
7 end program

Some Python Code:

1 import numpy
2 import matplotlib as plt
3 for i in range(0,5,1):
4  print('hello planeto world")
5 plt.show()


Link to the LMD Generic GCM user manual

[[https://web.lmd.jussieu.fr/~lmdz/planets/LMDZ.GENERIC/ManualGCM_GENERIC.pdf ]]