Essai Latex - Revision history

Yd at 12:42, 3 April 2016

2016-04-03T12:42:22Z

2016-04-03T12:41:25Z

2016-04-03T12:27:57Z

2016-04-03T12:24:15Z

2016-04-03T12:23:38Z

2016-04-03T12:20:00Z

Replaced content with "<math>E=mc^2</math>"

2016-04-03T12:19:03Z

2016-04-03T12:16:05Z

2016-04-03T12:14:44Z

2016-04-03T12:14:11Z

← Older revision		Revision as of 12:42, 3 April 2016
Line 10:		Line 10:
	\left\| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right\|^s \mathrm{d}\boldsymbol{x}		\left\| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right\|^s \mathrm{d}\boldsymbol{x}
	\end{align}		\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 above integral expresses the <math>s</math>-th moment of the distance~~
−	~~to the origin after <math>n</math> steps.~~
−
−	~~By experimentation and some sketchy arguments we quickly conjectured and~~
−	~~strongly believed that, for <math>k</math> 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, this equation also holds for negative odd integers.~~

@@ Line 2: / Line 2: @@
-We consider, for various values of $s$, the $n$-dimensional integral
+We consider, for various values of <math>s</math>, the <math>n</math>-dimensional integral
 \begin{align}
    W_n (s)
    &:=
@@ Line 10: / Line 10: @@
      \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
+the above integral  expresses the <math>s</math>-th moment of the distance
-to the origin after $n$ steps.
+to the origin after <math>n</math> steps.
 By experimentation and some sketchy arguments we quickly conjectured and
-strongly believed that, for $k$ a nonnegative integer
+strongly believed that, for <math>k</math> 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
+Appropriately defined, this equation also holds for negative odd integers.
-at the end of the paper.

← Older revision		Revision as of 12:27, 3 April 2016
Line 24:		Line 24:
	The reason for \eqref{eq:W3k} was long a mystery, but it will be explained		The reason for \eqref{eq:W3k} was long a mystery, but it will be explained
	at the end of the paper.		at the end of the paper.
		+
		+
		+	<math>\begin{align}
		+	\nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\
		+	\nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\
		+	\nabla \cdot \vec{\mathbf{B}} & = 0
		+	\end{align}</math>

← Older revision		Revision as of 12:24, 3 April 2016
Line 1:		Line 1:
	<math>E=mc^2</math>		<math>E=mc^2</math>

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

	We consider, for various values of $s$, the $n$-dimensional integral		We consider, for various values of $s$, the $n$-dimensional integral

← Older revision		Revision as of 12:23, 3 April 2016
Line 1:		Line 1:
	<math>E=mc^2</math>		<math>E=mc^2</math>
		+
		+	$\newcommand{\Re}{\mathrm{Re}\,}
		+	\newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg\| {#5} \right)}$
		+
		+	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.

@@ Line 1: / Line 1: @@
-<!DOCTYPE html>
+<math>E=mc^2</math>
-<html>
-<head>
-<title>MathJax TeX Test Page</title>
-<script type="text/x-mathjax-config">
-  MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']]}});
-</script>
-<script type="text/javascript" async
-  src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_CHTML">
-</script>
-</head>
-<body>
-When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are
-$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
-</body>
-</html>

← Older revision		Revision as of 12:16, 3 April 2016
Line 1:		Line 1:
−	<~~math~~>\~~int_0^~~\~~infty~~ \~~frac~~{\~~sin~~(x)}{x} ~~dx </math>~~	+	<!-- 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.

← Older revision		Revision as of 12:14, 3 April 2016
Line 1:		Line 1:
−	<math>\int_0^\infty \frac{\sin(x)}{x} dx </math>	+	<math>\int_0^\infty \frac{\sin(x)}{x} dx </math>