# Main Page

(Difference between revisions)
 Revision as of 22:03, 18 May 2011 (view source)Luis (Talk | contribs)← Older edit Revision as of 04:14, 27 May 2011 (view source)Luis (Talk | contribs) Newer edit → Line 10: Line 10: == what to do? == == what to do? == - In [[Mwiki:Current events|Current Events]] there is a '''to do''' list. Click on the links and edit the pages. So far most of the information is empty. + 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. ---- ----

## Revision as of 04:14, 27 May 2011

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## what to do?

In 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.

$\newcommand{\Re}{\mathrm{Re}\,} \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}$

We consider, for various values of $s$, the $n$-dimensional integral \begin{align} \tag{1} 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 (1) 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} \tag{2} W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}. \end{align} Appropriately defined, (2) also holds for negative odd integers. The reason for (2) was long a mystery, but it will be explained at the end of the paper.