\documentclass[dvips]{beamer}
% [dvips] allows use of latex->dvips->ps2pdf instead of pdflatex

\usepackage{beamerthemesplit}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
%\usepackage{graphicx,psfrag} %only include if using pictures with psfrag



\mode<presentation>
{
  %\usetheme{Berkeley}   % left bar
  %\usetheme{Antibes} % tree header
  %\usetheme{Boadilla}  % very plain
  %\usetheme{Warsaw}
  %\usetheme{Rochester} %
  %%\usetheme{Goettingen}  % right bar
  %\usetheme{Ilmenau}

  %\usepackage{beamerouterthemeshadow}
  %\usepackage{beamerouterthemesmoothtree}
  %\usepackage{beamercolorthemeorchid}
  %\usepackage{beamerinnerthemerounded}

  %\setbeamercovered{transparent}
  % comment out to completely hide covered material
}

% **** All themes: remove comment and find out which one you like!

%\usetheme{Warsaw}
\usetheme{Frankfurt} % I like it.
%\usetheme{Szeged}
%\usetheme{Singapore}
%\usetheme{Rochester}  % no menus at top
%\usetheme{Pittsburgh}
%\usetheme{Berlin}   % too much header/footer stuff
%\usetheme{Antibes}
%\usetheme{Bergen} % very large side margins
%\usetheme{Boadilla} % nice and simple
%\usetheme{boxes}
%\usetheme{Copenhagen}
%\usetheme{Darmstadt}
%\usetheme{default}
%\usetheme{Dresden}
%\usetheme{Goettingen}
%\usetheme{Hannover}
%\usetheme{Ilmenau}
%\usetheme{JuanLesPins}
%\usetheme{Luebeck}
%\usetheme{Madrid}
%\usetheme{Malmoe}
%\usetheme{Marburg}
%\usetheme{Montpellier}
%\usetheme{PaloAlto}

%You can also pick a color theme in addition to theme above.

%\usecolortheme{albatross}
%\usecolortheme{beetle}
%\usecolortheme{crane}
%\usecolortheme{dolphin}
%\usecolortheme{dove}
%\usecolortheme{fly}
%\usecolortheme{lily}
%\usecolortheme{seagull}
%\usecolortheme{seahorse}
%\usecolortheme{rose}


%\usefonttheme[onlylarge]{structuresmallcapsserif}


%\usepackage{times}
%\usepackage[T1]{fontenc}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.


\newtheorem{thm}{Theorem}




% ************************ space ************************************
\newcommand{\hhs}[1]{\hspace{#1mm}}
\newcommand{\hs}{\hspace{5mm}}
\newcommand{\vp}{\vspace{1mm}}
\newcommand{\vs}{\vspace{5mm}}
\newcommand{\jl}{$\frac{}{}$} %User defined for empty symbol to jump line
% ********************** newcommand *********************************
\newcommand{\mbf}[1]{\mbox{\boldmath $#1$}}
\newcommand{\hb}[1]{\hspace{-#1 mm}}
\newcommand{\ds}{\displaystyle}
\newcommand{\QED}{\hfill $\Box$}
% *********************** frequently used math symbols from AMS *******
\newcommand{\norm}[1]{\left\Vert#1\right\Vert}
\newcommand{\abs}[1]{\left\vert#1\right\vert}
\newcommand{\set}[1]{\left\{#1\right\}}


\definecolor{Maroon}{rgb}{.5,.1,.25}% Used for Trinity University


\title[Presentations in \LaTeX]{A Guide to Presentations in \LaTeX-beamer}
%\title[Short Title]{Long Title}
\subtitle{with a detour to Geometric Analysis}


%\author[Name to appear in every slide]{Author name to appear in Title}
\author[Balreira]{Eduardo Balreira}


\institute[Trinity University]{%
   \includegraphics[width=1.0cm]{ltr-logo}\\%remove this line if no logo.
   \textcolor{Maroon}{{\Large Trinity University}}\\
   Mathematics Department
}

\date[Major Seminar] % (optional, should be abbreviation of conference name)
{Major Seminar, Fall 2008}
% - Either use conference name or its abbreviation.



\begin{document}

\begin{frame}
  \titlepage
\end{frame}

%\section{Outline}
%\frame{\tableofcontents}



\begin{frame}
  \frametitle{Outline}
  \tableofcontents[pausesections]
  % You might wish to add the option [pausesections] to display one at a time
\end{frame}
%
%

\section{Intro to \LaTeX}


\begin{frame}[fragile] %fragile option needed when using verb or verbatim
  \frametitle{Some Symbols}
  LaTeX is a mathematics typesetting program.

  \jl

  \begin{itemize}
    \item Standard Language to Write Mathematics
    \pause

    \jl %just to add an extra space between items...

    \item $(M^2, g) \leftrightarrow$ \verb|$(M^2,g)$|
    \pause

    \jl

    \item $\Delta u -K(x) - e^{2u} = 0 \leftrightarrow$ \verb|$\Delta u -K(x) - e^{2u} = 0$|
    \pause

    \jl

    \item $\ds\inf_{n\in\mathbb{N}}\set{\dfrac{1}{n}}=0$

    \jl

    \verb|$\ds\inf_{n\in\mathbb{N}}\set{\dfrac{1}{n}}=0$|

  \end{itemize}

\end{frame}



\begin{frame}[fragile]
  \frametitle{Compare displaystyle}
  \begin{itemize}
    \item $\sum_{n=1}^{\infty}\frac{1}{n^2} = \dfrac{\pi^2}{6}$ versus $\ds\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$

    \pause

    \vfill

   \item\begin{verbatim}
$\sum_{n=1}^{\infty}\frac{1}{n^2}=\dfrac{\pi^2}{6}$

   and

$\ds\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$

   \end{verbatim}

  \end{itemize}

\end{frame}




\begin{frame}[fragile]
  \frametitle{Common functions}

  \begin{itemize}

    \item $\cos x$ $\to$ \verb|$\cos x$|

    \pause

    \vfill

    \item $\arctan x$ $\to$ \verb|$\arctan x$|

    \pause

    \vfill

    \item $f(x) = \sqrt{x^2+1}$ $\to$ \verb|$f(x) = \sqrt{x^2+1}$|

    \pause

    \vfill

    \item $f(x) = \sqrt[n]{x^2+1}$ $\to$ \verb|$f(x) = \sqrt[n]{x^2+1}$|

  \end{itemize}

\end{frame}


\begin{frame}[fragile]
 \frametitle{Theorems - code}

 \begin{thm}[Poincar\'{e} Inequality]
If $|\Omega| < \infty$, then
\[
 \lambda_1(\Omega) =  \inf_{u\neq 0} \dfrac{|\nabla u|^2_2}{\|u\|^2} > 0
\]

is achieved.
\end{thm}

\begin{verbatim}
\begin{thm}[Poincar\'{e} Inequality]
If $|\Omega| < \infty$, then
\[
 \lambda_1(\Omega) =
 \inf_{u\neq 0} \dfrac{|\nabla u|^2_2}{\|u\|^2} > 0
\]
is achieved.
\end{thm}

\end{verbatim}

\end{frame}








\begin{frame}[fragile]
  \frametitle{Example - Arrays}
   \framesubtitle{Change centering}

  \begin{itemize}


    \item $\left\{
            \begin{array}{rcll}
              -\Delta u +\lambda u &= & |u|^{p-2}, &\textrm{ in } \Omega \\
              u &\geq & 0, & u\in H_0^1(\Omega)
            \end{array}
           \right.$

     \vfill

     \pause

     \item \begin{verbatim}
$\left\{
  \begin{array}{rcll}
  -\Delta u +\lambda u &= & |u|^{p-2}, &\textrm{ in }
  \Omega \\
  u &\geq & 0, & u\in H_0^1(\Omega)
  \end{array}
\right.$

\end{verbatim}

  \end{itemize}

\end{frame}


\begin{frame}[fragile]
  \frametitle{More Examples}

  \begin{itemize}

    \item $\ds \varphi (u) = \int_{\Omega} \left[ \dfrac{\|\nabla u\|^2}{2} + \lambda\dfrac{u^2}{2} - \dfrac{(u^+)^p}{p}\right] d\mu $

     \vfill

     \pause

     \item \begin{verbatim}
$\ds \varphi (u) = \int_{\Omega} \left[
 \dfrac{\|\nabla u\|^2}{2} +
 \lambda\dfrac{u^2}{2} -
 \dfrac{(u^+)^p}{p} \right] d\mu $

\end{verbatim}

  \end{itemize}

\end{frame}


\begin{frame}[fragile]
  \frametitle{Even More Examples}

De Morgan's Law

  \begin{itemize}

    \item $\ds \left(\bigcup_{i=1}^{n} A_i\right)^c =
     \bigcap_{i=1}^n A_i^c$

     \vfill

     \pause

     \item \begin{verbatim}
$\ds \left(\bigcup_{i=1}^{n} A_i\right)^c =
     \bigcap_{i=1}^n A_i^c$
\end{verbatim}

     \vfill

     \pause

     \item $A\times B = \set{(a,b)|a\in A, b\in B}$

     \vfill

     \pause

     \item \begin{verbatim}
$A\times B = \set{(a,b)|a\in A, b\in B}$
\end{verbatim}


  \end{itemize}

\end{frame}

\begin{frame}[fragile]
  \frametitle{Equations}

  \begin{itemize}

\item Consider the equation of Energy below.

 \begin{equation}\label{eq:energy}
     E(u) = \int |\nabla u|^2 dx
 \end{equation}

This is how we refer to \eqref{eq:energy}.

\pause

\vfill

\item\begin{verbatim}
 \begin{equation}\label{eq:energy}
     E(u) = \int |\nabla u|^2 dx
 \end{equation}

This is how we refer to \eqref{eq:energy}.
\end{verbatim}

\end{itemize}

\end{frame}


\begin{frame}[fragile]
  \frametitle{Equations}

  \begin{itemize}

\item Consider the equation without a number below.

 \begin{equation}
     E(u) = \int |\nabla u|^2 dx \nonumber
 \end{equation}


\pause

\vfill

\item\begin{verbatim}
 \begin{equation}\label{eq:energy}
     E(u) = \int |\nabla u|^2 dx \nonumber
 \end{equation}
\end{verbatim}

\end{itemize}

\end{frame}


\begin{frame}[fragile]
  \frametitle{Equations}
   \framesubtitle{Tag an equation}
  \begin{itemize}

\item Consider the equation with a tag

 \begin{equation}\label{eq:energytag}
     E(u) = \int |\nabla u|^2 dx \tag{E}
 \end{equation}

 If $u$ is harmonic, \eqref{eq:energytag} is preserved.

\pause

\vfill

\item\begin{verbatim}
 \begin{equation}\label{eq:energytag}
     E(u) = \int |\nabla u|^2 dx \tag{E}
 \end{equation}

If $u$ is harmonic, \eqref{eq:energytag} is preserved.
\end{verbatim}

\end{itemize}

\end{frame}

\begin{frame}[fragile]
  \frametitle{Equations}
   \framesubtitle{a small proof}
  \begin{itemize}

\item \begin{verbatim}
 \begin{equation}
  \begin{split}
   \dfrac{d}{dt} E(u) & = 2 \int \langle\nabla u,
                                 \nabla u\rangle\\
   & = -2 \int\langle\Delta u,u\rangle = 0
  \end{split}
 \end{equation}

\end{verbatim}


\end{itemize}

\end{frame}




\begin{frame}[fragile]
  \frametitle{Equations}
   \framesubtitle{in an array}
  \begin{itemize}

\item Consider the expression below

 \begin{equation}
  \begin{split}
   (a+b)^2 &  = (a+b)(a+b) \\
           &  = a^2 +2ab +b^2
   \end{split}
 \end{equation}


\pause

\vfill

\item\begin{verbatim}
 \begin{equation}
  \begin{split}
   (a+b)^2 &  = (a+b)(a+b) \\
           &  = a^2 +2ab +b^2
  \end{split}
 \end{equation}
\end{verbatim}

\end{itemize}

\end{frame}




\begin{frame}[fragile]
  \frametitle{Environments}

  In LaTeX, environments must match:

  \begin{itemize}

  \item\begin{verbatim}
  \begin{...}
  .
  .
  .
  \end{...}
  \end{verbatim}

  \pause

  \item \$ ...\$ $\to$ for math symbols

  \pause

  \item \verb|\[ ... \]| $\to$ for centering expressions

  \pause

  \item \verb|\left( ... \right)| $\to$ match size of parentheses

  \end{itemize}

\end{frame}


\begin{frame}[fragile]
  \frametitle{Environments}
   \framesubtitle{delimiters}
  \begin{itemize}

  \item $(\ds\int|\nabla u|^p d\mu)^p$ versus $\left(\ds\int|\nabla u|^p d\mu\right)^p$

  \pause

  \vfill

  \item \begin{verbatim}
$(\ds\int|\nabla u|^p d\mu)^p$
  \end{verbatim}


  \item \begin{verbatim}
$\left(\ds\int|\nabla u|^p d\mu\right)^p$
  \end{verbatim}



\end{itemize}

\end{frame}




\begin{frame}[fragile]
  \frametitle{Tables}

  Consider the truth table:

  \jl

  \begin{tabular}{c c c | c}
$P$ & $Q$ & $\neg P$ & $\neg P\to (P \vee Q)$ \\ \hline
T & T & F & T \\
T & F & F & T \\
F & T & T & T \\
F & F & T & F
\end{tabular}


\end{frame}


\begin{frame}[fragile]
  \frametitle{Tables - code}
\begin{verbatim}

  \begin{tabular}{c c c | c}
$P$ & $Q$ & $\neg P$ & $\neg P\to (P \vee Q)$ \\ \hline
T & T & F & T \\
T & F & F & T \\
F & T & T & T \\
F & F & T & F

\end{tabular}

\end{verbatim}


\end{frame}



\begin{frame}[fragile]
  \frametitle{Inserting Pictures - code}
\begin{verbatim}
\begin{center}
  \includegraphics{Mountain_Pass.eps}
\end{center}

\end{verbatim}

\end{frame}

\begin{frame}[fragile]
  \frametitle{{Inserting Pictures - code} - psfrags - Code}
\begin{verbatim}
\begin{frame}
  \frametitle{Variational Calculus - psfrags}
  \uncover<1->{Rolle's Theorem has the following landscape.}
  \uncover<2->{\begin{figure}[h]
\begin{center}
\begin{psfrags}
\psfrag{x1}{$x_1$}\psfrag{x2}{$x_2$}
\psfrag{x3}{$x_3$}\psfrag{x3'}{$x_3'$}
\psfrag{y=f(x)}{$y=f(x)$}
\includegraphics{rolle.eps}
\end{psfrags}
\end{center}
\end{figure}
  }
\end{verbatim}
\verb|\end{frame}|

\end{frame}



% ******** End of \section{Intro to \LaTeX} *************************


\section{Intro to Beamer}

\begin{frame}
  \frametitle{How a Slide is done in Beamer}
  \framesubtitle{my subtitle}  % optional

This is a slide


  \begin{itemize}

    \item First Item

     \jl

    \item Second Item


  \end{itemize}

\end{frame}


\begin{frame}[fragile]
  \frametitle{How a Slide is done in Beamer}
   \framesubtitle{my subtitle}  % optional

The code should look like:

\begin{verbatim}
\begin{frame}

  \frametitle{How a Slide is done in Beamer}
  \framesubtitle{my subtitle}  % optional
This is a slide
  \begin{itemize}
    \item First Item
    \item Second Item
  \end{itemize}
\end{verbatim}
\verb|\end{frame}|

\end{frame}

\begin{frame}
  \frametitle{How a Slide with pause is done in Beamer}

This is a slide


  \begin{itemize}

    \item First Item

     \jl

    \pause

    \item Second Item


  \end{itemize}

\end{frame}


\begin{frame}[fragile]
  \frametitle{How a Slide with pause is done in Beamer}

The code should look like:

\begin{verbatim}
\begin{frame}

  \frametitle{How a Slide with pause is done in Beamer}
This is a slide
  \begin{itemize}
    \item First Item
    \pause
    \item Second Item
  \end{itemize}
\end{verbatim}
\verb|\end{frame}|

\end{frame}

\begin{frame}[fragile]
  \frametitle{Overlay example}

  \begin{itemize}
    \only<1->{\item First item}

    \jl

    \uncover<2->{\item Second item}

    \jl

    \uncover<3->{\item Third item}

    \jl

    \only<1->{\item Fourth item}
  \end{itemize}
\end{frame}



\begin{frame}[fragile]
  \frametitle{Overlay example}

The code should look like:

\begin{verbatim}
\begin{frame}[fragile]
  \frametitle{Overlay example}

  \begin{itemize}
    \only<1->{\item First item}
    \uncover<2->{\item Second item}
    \uncover<3->{\item Third item}
    \only<1->{\item Fourth item}
  \end{itemize}
\end{verbatim}
\verb|\end{frame}|

\end{frame}




\begin{frame}[plain]
   Need a plain slide?

   \jl

   \jl

   Add [plain] option to the slide.
\end{frame}


\section{Geometric Analysis}

\begin{frame}
  \frametitle{Variational Calculus}

  A simple Idea to solve equations:

  \begin{itemize}

  \item<1-> Solve $f(x) = 0$

  \jl

  \item<2-> Suppose we know that $F' = f$.

  \jl

  \item<3-> Critical points of $F$ are solutions of $f(x)=0$.
  \end{itemize}

\end{frame}


\begin{frame}[fragile]
  \frametitle{Variational Calculus}

  An idea from Calculus I:

  \jl

  \begin{thm}[Rolle]
    Let $f\in C^1([x_1,x_2];\mathbb{R})$. If $f(x_1)=f(x_2)$, then there exists $x_3\in(x_1,x_2)$ such that $f'(x_3) = 0$.
  \end{thm}
  \jl


\begin{verbatim}
\begin{thm}[Rolle]
Let $f\in C^1([x_1,x_2];\mathbb{R})$. If $f(x_1)=f(x_2)$,
then there exists $x_3\in(x_1,x_2)$
such that $f'(x_3) = 0$.
\end{thm}
\end{verbatim}


\end{frame}


\begin{frame}
  \frametitle{MPT - presentation}
  \framesubtitle{A friendly introduction}
\begin{thm}[Finite Dimensional MPT, Courant]
Suppose that $\varphi\in C^1(\mathbb{R}^n,\mathbb{R})$ is coercive
and possesses two distinct strict relative minima $x_1$ and $x_2$. \uncover<2->{Then $\varphi$ possesses a third critical point $x_3$ distinct from $x_1$ and $x_2$}\uncover<3->{, characterized by
 \[
 \varphi(x_3)=\inf_{\Sigma\in\Gamma}\max_{x\in\Sigma} \varphi(x)
 \]}
\uncover<4->{\noindent where $\ds\Gamma = \set{\Sigma\subset\mathbb{R}^n; \Sigma~
\textsf{is compact and connected and}~ x_1, x_2\in\Sigma}$.}

\uncover<5->{\noindent Moreover, $x_3$ is not a relative minimizer, that it, in every neighborhood of $x_3$ there exists a point $x$ such that $\varphi(x)<\varphi(x_3)$.}
\end{thm}

\end{frame}

\section{A Proof}


\begin{frame}
 \frametitle{An Application of MPT}

 \begin{thm}[Hadamard]Let $X$ and $Y$ be finite dimensional Euclidean spaces, and let $\varphi:X\to Y$ be a $C^1$ function such that:

 \jl

\uncover<2->{\noindent \textbf{(i)} $\varphi'(x)$ is invertible for all $x\in X$.}

\jl

\uncover<3->{\noindent \textbf{(ii)} $\norm{\varphi(x)}\to\infty$ as $\norm{x}\to\infty$.}

\jl

\uncover<4->{\noindent Then $\varphi$ is a diffeomorphism of $X$ onto $Y$.}

 \end{thm}

\end{frame}


\begin{frame}
 \frametitle{An Application of MPT}
  \framesubtitle{Hadamard's Theorem - Idea of Proof}

\begin{itemize}

   \item Check that $\varphi$ is onto.

   \pause

   \vfill

   \item Prove injectivity by contradiction.

   \pause

   \vfill

   \item Suppose $\varphi(x_1)=\varphi(x_2)=y$, then define
    \[
    f(x) = \frac{1}{2}\norm{\varphi(x)-y}^2
    \]

   \pause

   \vfill

   \item Check the MPT geometry for $f$.

   \pause

   \vfill

   \item $\exists x_3$, $f(x_3)>0$ (i.e., $\norm{\varphi(x_3)-y}>0$.)

   \pause

   \vfill

   \item $f'(x_3) = \nabla^{T}\varphi(x_3)\cdot(\varphi(x_3)-y)=0$

   \end{itemize}

\end{frame}




\end{document} 