\begin{frame}{Contents}

\begin{frame}{Contents}
\tableofcontents
\end{frame}

\section{Motivation}

\subsection{The Basic Problem That We Studied}

\begin{frame}{Make Titles Informative. Use Uppercase Letters. Long Titles
are Split Automatically.}{Subtitles are optional.}
\begin{itemize}
\item
Use \texttt{itemize} a lot.
\item
Use very short sentences or short phrases.
\end{itemize}
\end{frame}

\begin{frame}{Make Titles Informative.}

You can create overlays\dots
\begin{itemize}
\item using the \texttt{pause} command:
\begin{itemize}
\item
First item.
\pause
\item
Second item.
\end{itemize}
\item
using overlay specifications:
\begin{itemize}
\item<3->
First item.
\item<4->
Second item.
\end{itemize}
\item
using the general \texttt{uncover} command:
\begin{itemize}
\uncover<5->{\item
First item.}
\uncover<6->{\item
Second item.}
\end{itemize}
\end{itemize}
\end{frame}

\subsection{Previous Work}

\begin{frame}[fragile]{An old algorithm}
% NB. listings is quite powerful, but not well suited to be used with beamer
% consider using semiverbatim or the like, see below
\begin{semiverbatim}
int main (void)
{
std::vector<bool> is_prime (100, true);
for (int i = 2; i < 100; i++)
if (is_prime[i])
{
std::cout << i << " ";
for (int j = i; j < 100;
is_prime [j] = false, j+=i);
}
return 0;
}
\end{semiverbatim}
\end{frame}

\begin{frame}[fragile]{An Algorithm For Finding Primes Numbers.}
\begin{semiverbatim}
\uncover<1->{\alert<0>{int main (void)}}
\uncover<1->{\alert<0>{\{}}
\uncover<1->{\alert<1>{ \alert<4>{std::}vector<bool> is_prime (100, true);}}
\uncover<1->{\alert<1>{ for (int i = 2; i < 100; i++)}}
\uncover<2->{\alert<2>{ if (is_prime[i])}}
\uncover<2->{\alert<0>{ \{}}
\uncover<3->{\alert<3>{ \alert<4>{std::}cout << i << " ";}}
\uncover<3->{\alert<3>{ for (int j = i; j < 100;}}
\uncover<3->{\alert<3>{ is_prime [j] = false, j+=i);}}
\uncover<2->{\alert<0>{ \}}}
\uncover<1->{\alert<0>{ return 0;}}
\uncover<1->{\alert<0>{\}}}
\end{semiverbatim}
\visible<4->{Note the use of \alert{\texttt{std::}}.}
\end{frame}

\section{Our Results/Contribution}

\subsection{Main Results}

\begin{frame}{Make Titles Informative.}
\begin{example}
\begin{itemize}
\item 2 is prime (two divisors: 1 and 2).
\item 3 is prime (two divisors: 1 and 3).
\item 4 is not prime (\alert{three} divisors: 1, 2, and 4).
\end{itemize}
\end{example}
\end{frame}

\begin{frame}{Make Titles Informative.}
\begin{theorem}
There is no largest prime number and, in addition, $$\int_\Omega \nabla u \cdot \nabla v = - \int_\Omega u \Delta v + \int_{\partial\Omega} u v n$$
\end{theorem}
\begin{proof}
\begin{enumerate}
\item<1-> Suppose $p$ were the largest prime number.
\item<2-> Let $q$ be the product of the first $p$ numbers.
\item<3-> Then $q + 1$ is not divisible by any of them.
\item<1-> Thus $q + 1$ is also prime and greater than $p$.\qedhere
\end{enumerate}
\end{proof}
\uncover<4->{The proof used \textit{reductio ad absurdum}.}
\end{frame}

\begin{frame}{Make Titles Informative.}
\end{frame}

\subsection{Basic Ideas for Proofs/Implementation}

\begin{frame}{Make Titles Informative.}
\end{frame}

\begin{frame}{Make Titles Informative.}
\end{frame}

\begin{frame}{Make Titles Informative.}
\end{frame}

\section*{Summary}

\begin{frame}{Summary}

% Keep the summary *very short*.
\begin{itemize}
\item
The \alert{first main message} of your talk in one or two lines.
\item
The \alert{second main message} of your talk in one or two lines.
\item
Perhaps a \alert{third message}, but not more than that.
\end{itemize}

% The following outlook is optional.
\vskip0pt plus.5fill
\begin{itemize}
\item
Outlook
\begin{itemize}
\item
Something you haven't solved.
\item
Something else you haven't solved.
\end{itemize}
\end{itemize}
\end{frame}

% All of the following is optional and typically not needed.
\appendix
\section<presentation>*{\appendixname}
\subsection<presentation>*{For Further Reading}

\begin{frame}[allowframebreaks]
\frametitle<presentation>{For Further Reading}

\begin{thebibliography}{10}

\beamertemplatebookbibitems
% Start with overview books.

\bibitem{Author1990}
A.~Author.
\newblock {\em Handbook of Everything}.
\newblock Some Press, 1990.

\beamertemplatearticlebibitems
% Followed by interesting articles. Keep the list short.

\bibitem{Someone2000}
S.~Someone.
\newblock On this and that.
\newblock {\em Journal of This and That}, 2(1):50--100,
2000.
\end{thebibliography}
\end{frame}