% /usr/local/doc/tex-inputs/latex/notes/paper.tex % An example showing how to prepare an article using AMSLaTeX. % Stephen G. Simpson, Fall 1995. % Please send any comments or questions to simpson@math.psu.edu. % Lines beginning with a percent sign are comments. LaTeX ignores them. % Begin by declaring a document class and options. \documentclass[12pt,oneside]{amsart} % \documentclass{amsart} says to use the AMS article document class. % [12pt,oneside] says to use the 12pt and oneside options. % If you don't want these options, just say \documentstyle{amsart}. % After the document class declaration comes the preamble. % The preamble begins here. % First we activate any packages that we may need. % % The amssymb package provides \mathbb and other % math symbols. The amsmath package provides sophisticated math % constructions. The amsthm package provides \theoremstyle and % the \proof environment. % % The amsmath and amsthm packages are automatically activated by % \documentclass{amsart}, so there is no need to activate them here. \usepackage{amssymb} % Next we use \newtheorem to specify our theorem-like environments % (theorem, definition, etc.) and how to display and number them. % % Note: The \theoremstyle declarations affect the appearance of the % Theorems, Definitions, etc. \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} % The preamble is also a good place to define new commands and macros. % This part of the preamble is strictly optional according to your taste. \newcommand{\R}{{\mathbb R}} \newcommand{\nil}{\varnothing} % The following mysterious maneuver gets rid of AMS junk at the top % and bottom of the first page. \makeatletter \def\@setcopyright{} \def\serieslogo@{} \makeatother % This ends the preamble. We now proceed to the document itself. \begin{document} % First we specify the top matter (author, title, etc). % % Note: All of the top matter items are optional and can be omitted. % But you will probably want to specify at least the author and title % and perhaps an abstract. % author information % first author \author{David Hilbert} \address{University of G\"ottingen, G\"ottingen, Germany} \email{dh@math.gottingen.edu.de} % second author \author{Stephen G.~Simpson} % the address where the research was carried out \address{University of G\"ottingen, G\"ottingen, Germany} % current address, usually not needed because it is the same as the % regular address \curraddr{Department of Mathematics, Pennsylvania State University, University Park, State College PA 16802} \email{simpson@math.psu.edu} % title \title[The Infinite]{On \LaTeX{} and the Role of the Infinite\\ in Mathematical Reasoning} % Note that the short title for running heads goes in square % brackets. This is optional. The long title goes in curly % braces. In the long title, line breaks are indicated by \\. % abstract (optional) \begin{abstract} This is a sample document illustrating how to use \LaTeX{} in the Penn State Math Department. We also attempt to clarify the mysteries of the infinite. \end{abstract} % AMS subject classifications (used in AMS journals) \subjclass{Primary 00A30; Secondary 00A22, 03E20} % AMS keywords (used in AMS journals) \keywords{infinite, seminar} % acknowledge support, etc \thanks{This research was partially supported by NSF grant DOA-123456789.} \thanks{We would like to thank our colleagues for their helpful criticism.} % dedication \dedicatory{Dedicated to Professor Donald Knuth on the occasion of his $100$th birthday} % today's date, or fill in whatever date you prefer \date{\today} % This ends the top matter information. % We can now tell LaTeX to display the top matter. \maketitle % Having displayed the top matter, we now proceed to the body of the % article. % The body of the article is divided into sections. % Each section begins with a \section command. \section{Introduction} This document was prepared with \LaTeX{} using the AMS article style. By comparing this printed document with the raw document file from which it was generated, you can learn how to use \LaTeX{} to typeset mathematical documents. To start a new paragraph, skip a line. Many people regard the infinite as a great mystery. In this paper we shall try to end the mystery once and for all, so that mathematicians will never again have to bother about it. The set of real numbers is denoted $\R$. One of its subsets is $\nil$, \emph{i.e.} the empty set. We end this introductory section with a brief outline of the rest of the paper. Section~\ref{prelim} reviews a great number of important concepts and results. Then Section~\ref{main} presents our new contributions. At the end of the paper comes a bibliography, also known as a list of references. % Sections can be labeled for cross referencing. \section{Preliminaries: the many faces of $\infty$} \label{prelim} The purpose of this section is to review some well-known concepts and results. Most of these results are taken from the local newspaper \cite{CDT}. The O. J. Simpson trial is very important. Cousin OJ seems to be in a lot of trouble. See also the front page of \cite{CDT}. The infinite has many guises. One popular notation for the infinite is $\infty$. Cantor's aleph notation provides an important refinement: $\aleph_0$ denotes countable infinity, while other infinities are denoted $\aleph_\alpha$ where $\alpha$ is an ordinal number greater than $0$. \section{The main mathematical results} \label{main} We begin with the following lemma. \begin{lemma} For all ordinal numbers $\alpha$, we have $\aleph_\alpha<2^{\aleph_\alpha}$. \end{lemma} \begin{proof} The proof is obvious by diagonalization. \end{proof} An immediate consequence of the previous lemma is: % Theorems can be labeled for cross referencing. \begin{theorem} \label{continuum} We have $\aleph_0<2^{\aleph_0}$. \end{theorem} % A special theorem name can be specified in square brackets. \begin{corollary}[Cantor's Theorem] The real number system, $\mathbb{R}$, is uncountable. \end{corollary} \begin{proof}[Proof of Cantor's Theorem] Immediate from Theorem~\ref{continuum}. \end{proof} This completes our discussion of Cantor's Theorem. % Equations can be labeled for cross referencing. An interesting equation is \begin{equation} \label{bt} (A+B)^n = \sum_{i=0}^n \binom{n}{i} A^i B^{n-i} \,. \end{equation} Note that equation~\eqref{bt} is sometimes called the Binomial Theorem. \section{Some features of \LaTeX} \LaTeX{} has a lot of author-friendly features. Among them are: \begin{enumerate} \item a well-written reference manual; \item automatic numbering and cross referencing of \begin{enumerate} \item sections \item equations \item tables \item figures \item \emph{etc}. \end{enumerate} \item convenient bibliographic citations; \item special environments for tables and figures. \end{enumerate} For instance, Table~\ref{grades} is a simple table, produced using the \texttt{tabular} environment. % Tables and figures can be labeled for cross referencing, but % the \label must come after the \caption. \begin{table} \begin{center} \caption{Grades earned by political leaders} \label{grades} \begin{tabular}{|l|c|c|} \hline Last Name & Score & Letter Grade \\ \hline Clinton & 95 & A \\ \hline Dole & 95 & A \\ \hline Gingrich & 95 & A \\ \hline \end{tabular} \end{center} \end{table} \section{Conclusions} We hope that this article has illustrated how easy it is to use \LaTeX. It is also clear from Theorem~\ref{continuum} that the infinite is a subject that may or may not be worthy of the attention of all mathematicians. % This is the end of the last section. % Finally we create the bibliography or list of references. \begin{thebibliography}{99} \bibitem{DK} D. K\"onig, \emph{Theorie der Endlichen und Unendlichen Graphen}, Akademische Verlagsgesellschaft, Leipzig, 1936, reprinted by Chelsea Publishing Company, New York, 1950, 258 pages. \bibitem{CDT} The Centre Daily Times. \end{thebibliography} % Every LaTeX document must end with \end{document}. \end{document}