% template for seminar reports
% Clara L\"oh, 04/2023
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% a general comment:
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% Keep your LaTeX code as clean as possible;
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% Every LaTeX document needs a \documentclass declaration,
% which takes care of the general page layout, heading layout, etc.
\documentclass[a4paper,DIV=9,10pt]{scrartcl}
  
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  % importing further packages
  % include "ngerman" for German texts!
  %\usepackage{ngerman}          % for German hyphenation patterns
  \usepackage[utf8]{inputenc}    % enables utf8 input encoding 
  \usepackage{amsmath}           % contains useful math macros
  \usepackage{amsthm}            % for theorems, definitions, proofs, etc.
  \usepackage{amsfonts}          % some special math fonts
  \usepackage{tikz}              % for the creation of figures/illustrations
  \usepackage{scrlayer-scrpage}  % for the page headings
  \KOMAoptions{headinclude}
  \usepackage[pdfborder={0 0 0}]{hyperref} % PDF hyperlinks
  \urlstyle{sf}
  
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  % declaration of some math macros
  \newcommand{\N}{\ensuremath{\mathbb{N}}}   % natural numbers
  \newcommand{\Z}{\ensuremath{\mathbb{Z}}}   % integers
  \newcommand{\Q}{\ensuremath{\mathbb{Q}}}   % rationals
  \newcommand{\R}{\ensuremath{\mathbb{R}}}   % reals

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  % declaration of theorem/definition/proof environments via amsthm
  \newtheorem{thm}{Theorem}[section]
  \newtheorem{lemma}[thm]{Lemma}
  \newtheorem{corollary}[thm]{Corollary}
  \theoremstyle{definition}
  \newtheorem{definition}[thm]{Definition}
  \newtheorem{remark}[thm]{Remark}
  \newtheorem{exercise}[thm]{Exercise}
  \newtheorem{example}[thm]{Example}

  % for German texts:
  %\newtheorem{satz}{Satz}[section]
  %\newtheorem{lemma}[satz]{Lemma}
  %\newtheorem{korollar}[satz]{Korollar}
  %\theoremstyle{definition}
  %\newtheorem{definition}[satz]{Definition}
  %\newtheorem{bemerkung}[satz]{Bemerkung}
  %\newtheorem{aufgabe}[satz]{Aufgabe}
  %\newenvironment{beweis}%
  %  {\begin{proof}[Beweis]}
  %  {\end{proof}}
  %\newtheorem{beispiel}[satz]{Beispiel}

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  % further macros/settings
  \makeatletter                    % footnote without a symbol/label
    \def\blfootnote{\xdef\@thefnmark{}\@footnotetext}

  % title of the handout
  %   #1 name of the speaker
  %   #2 email address
  %   #3 date of the talk
  %   #4 title of the talk
  %   #5 seminar
  %   #6 semester
  \makeatletter
  \newcommand{\handouttitle}[6]
   {\begin{center}
      \Large\textbf{\sffamily #4}
    \end{center}

    \bigskip

    \noindent
    #1 (\textsf{#2})
    \hfill
    #3%
    \blfootnote{Seminar \emph{#5}, 
      #6, Universit\"at Regensburg}
  
    \noindent
    \rule{\linewidth}{.5pt}

    \bigskip

    \@afterindentfalse\@afterheading
   }
  \makeatother

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% start of the actual document
\begin{document}

  % title of the handout
  \handouttitle{N.~Imeta}
               {mail@spam.blorx}
               {30.~Februar~2010}
               {The House of Santa Claus}
               {The Intergalactic Santa Seminar}
               {WS~2088/89}


% The handout should be short (one or two pages)
% and thus should focus on the essential material.
               
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\section{Basics}

  \begin{definition}[House of Santa Claus]
    \label{def:housesanta}
    The \emph{House of Santa Claus} is the graph~$(V,E)$, defined as follows: 
    \begin{align*}
      V & := \{1, \dots, 5\}
      \\
      E & := \bigl\{ \{1,2\}
                   , \{1,5\}
                   , \{2,3\}
                   , \{2,4\}
                   , \{2,5\}
                   , \{3,4\}
                   , \{3,5\}
                   , \{4,5\}
             \bigr\}
    \end{align*}
  \end{definition}

  One can illustrate the House of Santa Claus as in
  Figure~\ref{fig:housesanta}; more information on Ti\textit{k}Z can
  be found in the documentation~\cite{tantau}. General information on
  \LaTeX\ can be found in the \LaTeX~Companion~\cite{companion}.

  \begin{figure}[h]
    \begin{center}
      \begin{tikzpicture}
         \begin{scope}[color=black!20]
           \draw[fill] (0,0) circle (2pt);
           \draw[fill] (1,0) circle (2pt);
           \draw[fill] (0,1) circle (2pt);
           \draw[fill] (0.5,1.5) circle (2pt);
           \draw[fill] (1,1) circle (2pt);
         \end{scope}
         \begin{scope}[color=blue]
           \draw (0.5,1.5) node[above] {$1$};
           \draw (1,1) node[right] {$2$};
           \draw (1,0) node[right] {$3$};
           \draw (0,0) node[left] {$4$};
           \draw (0,1) node[left] {$5$};
         \end{scope}
         \draw[->,rounded corners=0.1cm]
            (0,0) -- (1,0) -- (0,1) -- (1,1) -- (0,0)
         -- (0,1) -- (0.5,1.5) -- (1,1) -- (1,0.2);
      \end{tikzpicture}
    \end{center}
    \caption{House of Santa Claus}
    \label{fig:housesanta}
  \end{figure}

  When using material (including figures!) from external sources,
  copyright and plagiarism rules need to be respected. 
  In particular, also definitions, theorems, proof strategies,
  examples, etc.\ should be attributed appropriately. 
  Further suggestions on how to write mathematical texts are provided
  (among many other places \dots)
  in the notes \emph{Exercises in Academic Writing}~\cite{loeh}.
  
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\section{Properties of the House of Santa Claus}

  \begin{thm}[incompleteness theorem]
    The House of Santa Claus is not complete.
  \end{thm}
  \begin{proof}
    We use the notation from Definition~\ref{def:housesanta}.  The
    House of Santa Claus is not a complete graph because the
    edge~$\{1,3\}$ is not contained in the House of Santa Claus.
  \end{proof}


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\section{Examples}

  \begin{example}
    \hfil
    \begin{itemize}
      \item Here is an example
      \item \dots and another one
      \item \dots and another one
    \end{itemize}
  \end{example}

  \begin{exercise}
    Please do not forget to insert a few exercises -- so that the 
    participants can test their understanding of the topic.
  \end{exercise}

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% References
% * references can be entered "by hand" as below;
%   alternatively, also a tool such as biblatex can be used  
% * the bibliography may only contain references that are cited
%   within the text (with biblatex this is automatic)
\begin{thebibliography}{99}

   \bibitem{loeh}
     C.~L\"oh.
     Exercises in Academic Writing,
     2017.
     \\
     \url{https://loeh.app.uni-regensburg.de/seminars/eaw.pdf}
  
   \bibitem{companion}
     F.~Mittelbach, M.~Goossens, J.~Braams, D.~Carlisle, C.~Rowley.
     \emph{The \LaTeX\ Companion},
     second edition, Addison-Wesley, 2004.
     
   \bibitem{tantau} 
     T.~Tantau. \emph{The {\normalfont Ti\textit{k}Z} and
       {\normalfont PGF} Packages}, 
     \\
     \mbox{\url{http://www.ctan.org/tex-archive/graphics/pgf/base/doc/generic/pgf/pgfmanual.pdf}}
\end{thebibliography}

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