imported>Luis |
|
| (35 intermediate revisions by 3 users not shown) |
| Line 1: |
Line 1: |
| '''MediaWiki has been successfully installed.'''
| | <strong>MediaWiki has been installed.</strong> |
|
| |
|
| Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software. | | Consult the [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Contents User's Guide] for information on using the wiki software. |
|
| |
|
| == Getting started == | | == Getting started == |
| * [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list] | | * [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list] |
| * [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ] | | * [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ] |
| * [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list] | | * [https://lists.wikimedia.org/postorius/lists/mediawiki-announce.lists.wikimedia.org/ MediaWiki release mailing list] |
| | | * [https://www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language] |
| == what to do? ==
| | * [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Combating_spam Learn how to combat spam on your wiki] |
| | |
| In [[Mwiki:Current events|Current Events]] there is a '''to do''' list. Click on the links and edit the pages.
| |
| | |
| if you don't know how to start, you can use the pages that are already written as a sample.
| |
| | |
| ----
| |
| | |
| The following is a sample of LeTeX writing.
| |
| | |
| <!-- some LaTeX macros we want to use: -->
| |
| $
| |
| \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.
| |