\documentclass[12pt]{amsart} % Standard LaTeX; 12 point for old eyes \usepackage{amssymb} % For better support of math \usepackage{graphicx} % Enable for eps figures, if they occur % Commands to force sequential numbering: \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{examples}[theorem]{Examples} \newtheorem{remarks}[theorem]{Remarks} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \title{The 18.821 Mathematics Project Lab Report} \author{Your names here.} \date{\today} \begin{document} \def\CC{\mathbb C} % blackboard math , for ``complex,'' etc \def\qed{$\Box$} % box indicating end of proof. \begin{abstract} In this note we give some advice about and examples of mathematical writing and \LaTeX\ typography for use in preparing written reports for 18.821. % An abstract is a short summary of the main points in the project. It % does not mention background material or detail. \end{abstract} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction}\label{sec_intro} Your introduction section should provide an outline of your project. Include the motivation, methods used, and your conclusions. Keep technical information out of your introduction. You must use use \LaTeX\ to format your document. \LaTeX\ is the typesetting system universally used to communicate matheamtics. It \LaTeX\ provides the best and (after you get used to it) the easiest way to typeset all manner of mathematical expressions. Learning \LaTeX\ is an important part of your mathematical education because it is the standard means of producing math papers and books. %Wherever %possible, use the \AmS-\LaTeX\ package commands. They are %specifically designed for mathematics as opposed to other science %disciplines. The \AmS-\LaTeX\ packages are especially useful for %symbols and typesetting equations: the standard \LaTeX\ commands for %equations just aren't as good. One free and particularly good reference for standard \LaTeX\ is given at \cite{oetiker}. A short reference for \AmS-\LaTeX\ is given at \cite{downes}. The course website has links to these sources. The standard reference book for \LaTeX, which includes a section on \AmS-\LaTeX, is \cite{goossens}. To conclude your introduction, give a road map describing the sections in the rest of your document. This road map provides a quick reference for the reader interested in a particular topic or aspect of your project. In Section \ref{sec_content}, we provide a list of topics that you should include in your report. In Section \ref{sec_eg}, we give a few examples of common commands that you may find useful along the way. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Content of the Report}\label{sec_content} For 18.821, each project is to be written up as a group report, but each member must write an approximately equal portion of the report. Your individual grades will be determined as follows: \begin{enumerate} \item sixty percent of the total grade will be based on the overall quality of the project and the final report; \item forty percent of the total grade will be based on the portion of the final report that has been written by you. \end{enumerate} The preliminary report will be marked up for revision. It will not be graded. Since your grade is affected by the overall report as well as by your individual work, the team should read and comment on the entire report. Try to make the sections flow as naturally as possible and always choose clarity above all else. Assume that the readers of the report will be fellow 18.821 students. Add the background information that they will need to understand the report. The following topics must be included in your report, but if you feel you need to add more information for clarity, please do so. If you have consulted any literature, you must review it all here and give precise citations. You must also acknowledge any substantive conversations about your project that you have had with people other than the course staff. Describe all avenues of exploration that you considered. These avenues include the different ways that one might interpret the project, the different methods that could be applied, and the different parts of the problem that one might concentrate on. Then describe the avenues that you did explore. If you used any computational analysis, include a conceptual explanation of the computer program. Do not include code in the body of the report: add all programs as appendices. Describe the conclusions you have drawn, whether computationally or theoretically. Include explanations, heuristic reasoning and proofs when you have them. Include enough computational data as an appendix. You should also discuss all possible errors in the data, explain why they are there, and explain the possible consequences of the errors. Finally, by way of conclusion, describe any further work you would have liked to have done, and pose questions for potential future work. There are no specific limits on the length of your report. The style grade will be based in part on the appropriateness of your choice of the amount of detail to include. Read Professor Munkres's {\em Comments on Style}, available at the course website, on this point. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \section{\LaTeX\ Examples}\label{sec_eg} Here are a few examples of \LaTeX\ goodies. You should view the {\tt .tex} file as you read this section. %Remember to use \AmS-\LaTeX\ commands where possible. \LaTeX\ will indent after a blank line unless told not to do so. Some symbols, such as the dollar sign and the percent sign, have special meanings in a TeX document. To tell the computer that want such a symbol printed in the text instead of interpreted as an instruction to the computer precede the symbol by a backslash: \$, \%. The percent sign, \% when used without the backslash, tells the computer to ignore the rest of that line. %This sentence will not appear. There are many ways to write formulas in \LaTeX. Inline formulas are deliminated by dollar signs \$. %without the backslash For example, $z^2 = \sin \left(\theta_0 \right)$, or $2 \times 2 \neq 5.$ Here are three ways to display equations without a reference number: $$\int_1^x \frac{dt}{t} = \ln x.$$ \begin{equation*} \prod_{p\,\, \mathrm{prime}}(1-p^{-s})^{-1}=\sum_{n=1}^\infty\, n^{-s} \end{equation*} \[ \int_1^x \frac{dt}{t} = \ln x. \] If you want to reference an equation, don't add a star $\ast$ at the end of {\tt equation}: \begin{equation}\label{myfirsteqn} \lim_{n \to \infty}\, \sum_{k = 1}^n \frac{1}{k^2} = \frac{\pi}{6}. \end{equation} % To group several equations, aligning on the $=$ sign, use the \verb+align+ command. The computer aligns at the \&: % \begin{align*} x_1 + 2x_2 + 3x_3 &= 7 \\ y &= mx + c \\ &= 4x - 9. \end{align*} % For more examples of equation alignments, see \cite{downes}. Figure 1 is an example of a table given within a floating table box. Figure 2 is an example of {\tt .eps} (encapsulated postscript) figure. To convert images to {\tt .eps}, use your favorite graphics program such as {\tt xfig} or {\tt gimp}. %\begin{figure} %{\epsfysize=2.5in \epsfbox{sinusoid.eps}} %\caption{Parameters of a sinusoidal function} %\label{fig-sinusoid} %\end{figure} \begin{figure} \label{myfirsttable} \begin{center} \begin{tabular}{|r|l c|} 7C0 & hexadecimal & * \\ 3700 & octal & **** \\ 111110000000 & binary & ******* \\ \hline 1978 & decimal & ** \\ \end{tabular} \end{center} \vspace{.2in} \caption{My first table} \end{figure} %line \begin{tabular} line tells the computer: three columns % | inserts a vertical line % r justifies the first column on the right % l justifies the second column on the left % c centers the third column % & between columns % \\ at the end of a column % no * was used, so the table gets numbered \begin{figure}\label{myfirstfig} \begin{center} \includegraphics[width=4in]{spiral.eps} %\includegraphics[width=5cm]{smiley.png} \end{center} \vspace{-.2in} % corrects bad spacing \caption{My first figure.} \end{figure} An example of a ``conjecture environment'' is given in below, in Conjecture \ref{myfirstconj}. Theorems, lemmas, propositions, definitions, and such all use the same command with the appropriate name changed. The block of commands at the top of this {\tt .tex} file causes them to be numbered sequentially within sections, which is the preferred choice. \begin{conjecture}[Vaught's Conjecture]\label{myfirstconj} Let $T$ be a countable complete theory. If $T$ has fewer than $2^{\aleph_0}$ many countable models (up to isomorphism), then it has countably many countable models. \end{conjecture} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Filetype translations}\label{sec_translations} Produce a {\tt .tex} file using any text editor that produces ASCII files. Tex files conventionally bear the extension {\tt .tex}. The command \centerline {\tt latex filename.tex} \noindent produces either error reports or a file {\tt filename.dvi}. Many systems have screen {\tt .dvi} viewers. Under Linux for example you can use {\tt xdvi filename}. Most printers print postscript files. Under Linux, files can be converted from {\tt .dvi} to {\tt .ps} using \centerline {{\tt dvips -f filename.dvi > filename.ps}.} \noindent and from {\tt .ps} to {\tt .pdf} using \centerline {{\tt ps2pdf filename.ps}.} \noindent If the file does not contain {\tt .eps} elements, {\tt .pdf} output can be obtained directly using \centerline {{\tt pdflatex filename.tex}.} Matlab can be made to produce {\tt .eps} files by typing \centerline {\tt print -deps filename} \noindent at the prompt. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{9} % ``9'' specifies that the bibliography % has just one-digit reference numbers. \bibitem{downes} Downes, M., ``Short Math Guide to \LaTeX,'' \\ \verb+http://www.ams.org/tex/short-math-guide.html+ \bibitem{goossens} Goossens, Mittelbach, and Samarin, ``The \LaTeX\ Companion,'' Addison--Wesley, USA, 1994. \bibitem{oetiker} Oetiker, T., et. al., ``The Not So Short Introduction to \LaTeXe,'' \\ \verb+http://www.ctan.org/tex-archive/info/lshort/english/lshort.pdf+ \end{thebibliography} \end{document}