% This is a shell LaTeX input file.  (Version of 9 April 1986)

\documentclass[12pt]{article}    % Specifies the document style.
\pagestyle{empty}
\usepackage{amsfonts}
\hyphenation{equa-tion}

\def\Rl{\mathbb{R}}
\def\Cx{\mathbb{C}}
\def\Sp{\mathbb{S}}
\def\Ir{\mathbb{Z}}
\def\Nl{\mathbb{N}}

\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\beann}{\begin{eqnarray*}}
\newcommand{\eeann}{\end{eqnarray*}}

\begin{document}           


\centerline{{\sc Problem set 4}}
\centerline{Math 218A, Winter 2001}




\bigskip\noindent
{\bf 1.} Show that
\[
\frac{1}{1 - \left(x_1 + x_2 + \dots + x_n\right)}
= \sum_{\alpha \in \Ir_+^n} \frac{|\alpha|!
x^\alpha}{\alpha!}.
\]


\bigskip\noindent
{\bf 2.} Let $f$, $g$ be given smooth functions defined on $[0,\infty)$
such that
$f(0) = g(0)$. Find the solution $u(x,t)$ in
$\{(x,t) : |x| < t, t >0\}$ of the characteristic
initial value problem
\beann
&&u_{tt} - u_{xx} = 0,\\
&&u(t,t) = f(t),\\
&&u(-t,t) = g(t).
\eeann

\bigskip\noindent
{\bf 3.} Compute an equation for the characteristic variety of the
two-dimensional compressible Euler equations
\beann
&&\rho_t + (\rho u)_x + (\rho v)_y = 0,\\
&&u_t + uu_x + vu_y + \frac{1}{\rho} p_x = 0,\\
&&v_t + uv_x + vv_y + \frac{1}{\rho} p_y = 0,
\eeann
where $p = p(\rho)$ is a given function of $\rho$.


\bigskip\noindent
{\bf 4.} Prove that the series for the Tychonoff solution of the heat
equation that we discussed in class
converges. (See e.g. F.~John, \emph{Partial Differential Equations}
for help.)

\end{document}             % End of document.
