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'''MediaWiki has been successfully installed.'''
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<div style="background:#FFEEDD;border-style:ridge;font-size:150%;">
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To see the contents of this wiki, start on [[starting page|this page]].
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Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.
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For the impatient, there is a [[list of equations]].
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</div>
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== Getting started ==
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Please check the [[frequently asked questions]] including how to register and edit the content.
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* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]
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* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]
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* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]
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== what to do? ==
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== Purpose ==
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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.
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The purpose of this wiki is to create a clear reference for the methods, difficulties, [[open problems]], and [[applications]] of elliptic and parabolic nonlocal equations.
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----
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Especially since this is a relatively new subject, it is important to clearly point out where the fundamental mathematical challenges are, and what are the applications to the equations studied.
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<!-- some LaTeX macros we want to use: -->
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We would like to write a reference that is easy to read and navigate. It should be convenient to easily find the precise assumptions for which several theorems are proved (for example Harnack inequality or $C^{1,\alpha}$ estimates). It would also be desirable to have some indication of the methods used in the proofs, which may be a non rigorous idea, or an analogy with a classical theorem.
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$
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  \newcommand{\Re}{\mathrm{Re}\,}
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The emphasis of this wiki will be on nonlocal nonlinear (nonbananas?) equations of elliptic and parabolic type. There are several hyperbolic nonlocal equations that are described in the [http://wiki.math.toronto.edu/DispersiveWiki/index.php/Main_Page Dispersive wiki].
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  \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}
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$
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The ultimate goal of this wiki would be to prove for all equations that they are either well posed in the classical sense, or a weak solution and their possible singularities are well understood. The emphasis should then be on regularity results. We may also include some topics only indirectly related to regularity estimates (for example homogenization).
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We consider, for various values of $s$, the $n$-dimensional integral
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We should make an effort to negate the [[myths about nonlocal equations]]
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\begin{align}
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  \label{def:Wns}
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  W_n (s)
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  &:=
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  \int_{[0, 1]^n}
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    \left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x}
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\end{align}
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which occurs in the theory of uniform random walk integrals in the plane,  
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where at each step a unit-step is taken in a random direction. As such,
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the integral \eqref{def:Wns} expresses the $s$-th moment of the distance
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to the origin after $n$ steps.
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By experimentation and some sketchy arguments we quickly conjectured and
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strongly believed that, for $k$ a nonnegative integer
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\begin{align}
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  \label{eq:W3k}
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  W_3(k) &= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.
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\end{align}
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Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers.  
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The reason for \eqref{eq:W3k} was  long a mystery, but it will be explained
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at the end of the paper.
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Latest revision as of 16:22, 17 November 2012

To see the contents of this wiki, start on this page.

For the impatient, there is a list of equations.

Please check the frequently asked questions including how to register and edit the content.

Purpose

The purpose of this wiki is to create a clear reference for the methods, difficulties, open problems, and applications of elliptic and parabolic nonlocal equations.

Especially since this is a relatively new subject, it is important to clearly point out where the fundamental mathematical challenges are, and what are the applications to the equations studied.

We would like to write a reference that is easy to read and navigate. It should be convenient to easily find the precise assumptions for which several theorems are proved (for example Harnack inequality or $C^{1,\alpha}$ estimates). It would also be desirable to have some indication of the methods used in the proofs, which may be a non rigorous idea, or an analogy with a classical theorem.

The emphasis of this wiki will be on nonlocal nonlinear (nonbananas?) equations of elliptic and parabolic type. There are several hyperbolic nonlocal equations that are described in the Dispersive wiki.

The ultimate goal of this wiki would be to prove for all equations that they are either well posed in the classical sense, or a weak solution and their possible singularities are well understood. The emphasis should then be on regularity results. We may also include some topics only indirectly related to regularity estimates (for example homogenization).

We should make an effort to negate the myths about nonlocal equations

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