\documentclass[dvips]{beamer}%\documentclass[handout]{beamer}%\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 a Presentation}

\begin{frame}
  \frametitle{Slide Presentations}
  \framesubtitle{What you should not do...}
  \begin{itemize}
   \item In a slide presentation, you should not list too much information in a single slide. This causes the audience to spend too much time reading the slide and not paying any attention on what you are saying. You should never just read of the slide, otherwise why not just give a handout and leave? It is also \texttt{nice} {\large to use} ``fancy'' \textsl{letters} and \textrm{fonts} {\Large as well} {\footnotesize as clever}  \textcolor[rgb]{0.00,1.00,0.00}{coloring} \textcolor[rgb]{0.00,0.50,0.75}{schemes} \textcolor[rgb]{0.00,0.50,0.25}{but} \textcolor[rgb]{1.00,0.50,0.00}{they} \textcolor[rgb]{0.50,0.00,0.25}{can} \textcolor[rgb]{0.25,0.00,0.50}{be hard} \textcolor[rgb]{1.00,1.00,0.50}{to read}. Also, make sure you turn yourself towards the audience. Generally, the first part is to be understood by everyone, then you can begin to be more specific. It is common to only have the attention of a few ``experts''by the end of the presentation. This applies mostly to research talks, but also to Senior Project presentations.
  \end{itemize}

  \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{More Examples}
   \framesubtitle{ without displaystyle}

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

    \vfill %evenly space the lines above and below.

  \pause

   \item\verb|$\sum_{n=1}^{\infty}\frac{1}{n^2} = |

        \verb|\dfrac{\pi^2}{6}$|


  \end{itemize}

\end{frame}


\begin{frame}[fragile]
  \frametitle{More Examples}
   \framesubtitle{ with displaystyle}
  \begin{itemize}
    \item $\ds\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$

    \vfill %evenly space the lines above and below.

    \pause

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


  \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}
 \frametitle{Theorems}

\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{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}


  \begin{itemize}

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


     \vfill

     \pause

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

\end{verbatim}

  \end{itemize}

\end{frame}


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

  \begin{itemize}

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

     \vfill

     \pause

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

\end{verbatim}

  \end{itemize}

\end{frame}

\begin{frame}[fragile]
  \frametitle{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 Indeed,

 \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}

\pause

\vfill

\item The ``='' signs are aligned.

\vfill

\pause

\item Use \begin{verbatim} \begin{split} ... \end{split}
\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}
   \framesubtitle{Mountain Pass Landscape}
\begin{center}
  \includegraphics{Mountain_Pass.eps}
  \end{center}

\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}
\begin{figure}[h]
\begin{center}
\begin{psfrags}
\psfrag{p}{$p$} \psfrag{v}{$v$} \psfrag{q0}{$q_0$}
\psfrag{q1}{$q_1$} \psfrag{u0}{$u_0$} \psfrag{u1}{$u_1$}
\psfrag{U0}{$U_0$} \psfrag{U1}{$U_1$} \psfrag{H}{$H$}
\psfrag{f-1H}{$f^{-1}(H)$} \psfrag{ell}{$\ell$}
\psfrag{Wn1}{$\Gamma^{n-1}$} \psfrag{V}{$V$}
\psfrag{line}{$f^{-1}(\ell)$}
\includegraphics{INJ_sausage_in_intro.eps}
\end{psfrags}
\caption{Construction of $\Gamma^n$ by revolving affine hyperplanes} \label{fig:intro to sausage idea}
\end{center}
\end{figure}

\end{frame}






\begin{frame}[fragile]
 \frametitle{A Final Remark on LaTeX}
  \framesubtitle{Preamble}

  \begin{itemize}

  \item Preamble $\to$ ``Stuff'' on top of .tex file
\begin{verbatim}
%For an article using AMS template:
\documentclass[12pt]{amsart}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
...
\end{verbatim}
  \pause

  \vfill

  \item Don't worry about it!

  \pause

  \vfill

  \item With practice you can figure it out.

  \end{itemize}

\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}[fragile]
  \frametitle{How a Slide is done in Beamer}
   \framesubtitle{In this document...}

The option [fragile] is added to use verbatim

\begin{verbatim}

\begin{verbatim}
Anything you type $\Delta u = |u|^{p-2}$

Appears as is in the code

end{verbatim}

Note: if you correctly type \ end{verbatim} it assumes
      verbatim environment ended.

Hence ``fragile''

\end{verbatim}

\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}[fragile]
  \frametitle{Example of Figures}
  \emph{Taken from online Template}.
  \begin{itemize}
  \item<1-> Using \verb|\only<1>| to draw the first two Figures
  \item<2-> Using \verb|\only<2->| to draw the second two Figures
  \end{itemize}
  \only<1>{%
    \centerline{%
      \includegraphics[height=1.5in]{75_example_stabreg1_nonorm}
    }
  }
  \only<2->{%
    \centerline{%
      \includegraphics[height=1.5in]{75_example_stabreg1_weno}
    }
  }
\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}

  \uncover<1->{An idea from Calculus I:

  \jl}

  \uncover<2->{
  \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}



\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{Variational Calculus}

  \uncover<1->{
  Rolle's Theorem has the following landscape.
  }

  \uncover<2->{\begin{center}
  \includegraphics{rolle.eps}
  \end{center}
  }

\end{frame}

\begin{frame}[fragile]
  \frametitle{Variational Calculus - Code}
\begin{verbatim}
\begin{frame}
  \frametitle{Variational Calculus}
  \uncover<1->{
  Rolle's Theorem has the following landscape.
  }
  \uncover<2->{\begin{center}
  \includegraphics{rolle.eps}
  \end{center}
  }
\end{verbatim}
\verb|\end{frame}|

\end{frame}

\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{frame}


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

\begin{frame}[fragile]
  \frametitle{Mountain Pass Landscape}
\begin{center}
  \includegraphics{Mountain_Pass.eps}
  \end{center}

\end{frame}


\begin{frame}
  \frametitle{MPT - presentation}
  \framesubtitle{Not 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 $x_2$. Then $\varphi$ possesses a third critical point $x_3$ distinct from $x_1$ and $x_2$, characterized by
 \[
 \varphi(x_3)=\inf_{\Sigma\in\Gamma}\max_{x\in\Sigma} \varphi(x)
 \]
\noindent where $\ds\Gamma = \set{\Sigma\subset\mathbb{R}^n; \Sigma~
\textsf{is compact and connected and}~ x_1, x_2\in\Sigma}$.

\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}


\begin{frame}
 \frametitle{MPT - in parts}

 \begin{itemize}
 \item When displaying large info do in steps.
  \pause

  \jl

 \item Avoid audience reading ahead and not paying attention.

 \end{itemize}

\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}

\begin{frame}
  \frametitle{Geometry of MPT}
  \only<1>{
    \centerline{\includegraphics{MPFig/Mountain_Pass.eps}}
  }
  \only<2>{
    \centerline{\includegraphics{MPFig/Mountain_Pass1a.eps}}
  }
  \only<3>{
    \centerline{\includegraphics{MPFig/Mountain_Pass1b.eps}}
  }
  \only<4>{
    \centerline{\includegraphics{MPFig/Mountain_Pass1c.eps}}
  }
  \only<5>{
    \centerline{\includegraphics{MPFig/Mountain_Pass1d.eps}}
  }
  \only<6>{
    \centerline{\includegraphics{MPFig/Mountain_Pass1e.eps}}
  }
  \only<7>{
    \centerline{\includegraphics{MPFig/Mountain_Pass1f.eps}}
  }
  \only<8>{
    \centerline{\includegraphics{MPFig/Mountain_Pass1g.eps}}
  }
  \only<9>{
    \centerline{\includegraphics{MPFig/Mountain_Pass1h.eps}}
  }
  \only<10>{
    \centerline{\includegraphics{MPFig/Mountain_Pass2a.eps}}
  }
  \only<11>{
    \centerline{\includegraphics{MPFig/Mountain_Pass2b.eps}}
  }

\end{frame}


\section{A Proof}

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

 \begin{itemize}
 \item Good practice to have a proof

 \pause

 \vfill

 \item You only need the idea.

 \pause

 \vfill

 \item The actual proof will be in your final paper!

 \end{itemize}

\end{frame}


\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}[fragile]
 \frametitle{An Application of MPT}
\begin{verbatim}
\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:

\uncover<2->{\noindent \textbf{(i)} $\varphi'(x)$
             is invertible for all $x\in X$.
            }
\uncover<3->{\noindent \textbf{(ii)}
            $\norm{\varphi(x)}\to\infty$
            as $\norm{x}\to\infty$.
            }
\uncover<4->{\noindent Then $\varphi$ is a
             diffeomorphism of $X$ onto $Y$.
             }
 \end{thm}
\end{verbatim}
\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} 