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| | | Welcome to personal web site. |
| __TOC__
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| <strong>MediaWiki has been installed.</strong>
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| Consult the [//meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.
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| Let's add a link to my [[Blog|blog]]
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| $\newcommand{\Re}{\mathrm{Re}\,}
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| \newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}
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| \newcommand{\ket}[1]{\vert #1\rangle}$
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| Take a look at this beautifully typesetted equation:
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| \begin{equation}
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| p(n) = \mathcal{C}_n\times
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| {}_1\!F_2\left(n+1;\frac{2n+1}{2}+\frac{\Gamma_\sigma}{\Gamma},
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| n+\frac{\Gamma_\sigma}{\Gamma};
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| -\frac{P_\sigma} {2\Gamma}
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| \right)\, .
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| \end{equation}
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| We consider, for various values of $s$, the $n$-dimensional integral
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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 \, \pFq 32{\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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| == First Section ==
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| === First Subsection ===
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| ==== First subsubsection ====
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| <m>x^2</m>
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| <i class="fa fa-file-pdf-o fa-2x" aria-hidden="true"></i> CV file
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| <i class="fa fa-camera-retro"></i> fa-camera-retro
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| <a class="list-group-item" href="#"><i class="fa fa-home fa-fw" aria-hidden="true"></i> Home</a>
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| <a class="list-group-item" href="#"><i class="fa fa-book fa-fw" aria-hidden="true"></i> Library</a>
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| <a class="list-group-item" href="#"><i class="fa fa-pencil fa-fw" aria-hidden="true"></i> Applications</a>
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| <a class="list-group-item" href="#"><i class="fa fa-cog fa-fw" aria-hidden="true"></i> Settings</a>
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| <div class="list-group">
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| <a class="list-group-item" href="#"><i class="fa fa-home fa-fw" aria-hidden="true"></i> Home</a>
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| <a class="list-group-item" href="#"><i class="fa fa-book fa-fw" aria-hidden="true"></i> Library</a>
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| <a class="list-group-item" href="#"><i class="fa fa-pencil fa-fw" aria-hidden="true"></i> Applications</a>
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| <a class="list-group-item" href="#"><i class="fa fa-cog fa-fw" aria-hidden="true"></i> Settings</a>
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| </div>
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| <ul class="fa-ul">
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| <li><i class="fa-li fa fa-check-square"></i>List icons</li>
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| <li><i class="fa-li fa fa-check-square"></i>can be used</li>
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| <li><i class="fa-li fa fa-spinner fa-spin"></i>as bullets</li>
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| <li><i class="fa-li fa fa-square"></i>in lists</li>
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| </ul>
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| <i class="fa fa-quote-left fa-3x fa-pull-left fa-border" aria-hidden="true"></i>
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| ...tomorrow we will run faster, stretch out our arms farther...
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| And then one fine morning— So we beat on, boats against the
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| current, borne back ceaselessly into the past.
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