Diferenças entre edições de "Utilizador:NunoOliveira"

Edição atual desde as 22h21min de 9 de abril de 2019

Uma página inicial de teste.

@@ Linha 1: / Linha 1: @@
+Uma página inicial de teste.
-$\newcommand{\Re}{\mathrm{Re}\,}
-\newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}$
-Teste onde <math>x</math> é definido por:
+<br />
-<math>\begin{align}
-\dot{x} & = \sigma(y-x) \label{eq:1}\\
-\dot{y} & = \rho x - y - xz \\
-\dot{z} & = -\beta z + xy
-\end{align}</math>
-Equation \eqref{eq:1} above. 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.