⟦92a64982b⟧

TextFile


Following this letter are three files (DIMACS.STY, DIMACS.TEX and DIMACS.
CHECKLIST) to be used when preparing a paper in AMS-LaTeX, for a volume in the
Proceedings of the DIMACS Workshop series.  To make it easier to load these
files separately on your system, they are separated by a row of asterisks
(***).


The .STY file is based on AMSART.STY. Instructions for using this file are
listed at the top.

The .TEX file is a sample of the macros defined in the .STY file.  Before
preparing your paper you should TeX this file and print it to test the macros
and to get instructions for their use.

The .CHECKLIST file should be printed and used to help you avoid the most
common problems.

Once your paper has been input and proofread, please send the .TEX file to our
Internet address: PUB-SUBMIT@MATH.AMS.COM.  When you send the file please be
sure to include a message that identifies both the publication series and the
editor of the volume.

Should you have any technical questions, please contact our Technical Support
Group at 800-321-4AMS or 401-455-4080.  They can also be reached through
electronic mail at Internet: tech-support@math.ams.com.

***************************************************************************

%% dimacs.sty based upon amsart10.sty (modeled after DIMACS specs)

\input amsart.sty

\textheight=49pc
\textwidth=29pc
\parskip=0pt

\def\large{\@setsize\large{14\p@}\xiipt\@xiipt}
\def\Large{\@setsize\Large{18\p@}\xivpt\@xivpt}

\def\@normalsize{\@setsize\normalsize{13\p@}\xpt\@xpt
  \abovedisplayskip 6\p@ \@plus 6\p@
  \belowdisplayskip \abovedisplayskip
  \abovedisplayshortskip \z@ \@plus 6\p@
  \belowdisplayshortskip 3\p@ \@plus 6\p@
  \let\@listi\@listI}

\def\footnotesize{\@setsize\footnotesize{10\p@}\viiipt\@viiipt%
  \abovedisplayskip 5\p@ \@plus 2.5\p@ minus2.5\p@%
  \belowdisplayskip \abovedisplayskip
  \abovedisplayshortskip \z@ \@plus 2\p@%
  \belowdisplayshortskip 4\p@ \@plus 2\p@ minus2\p@%
  \def\@listi{\topsep \z@ \parsep \z@ \itemsep \z@}}

\normalsize

%***************************************************************
%section 1
%%this section is left in here, although commented out for those
%%style files which do not have aheads centered (like LAM)
%%just comment out section 2 when this section is used
%%
%%\def\section{\@startsection{\@string\section}%
%%        1\z@{9\p@\@plus12\p@}{6\p@}%
%%        {\centering\defaultfont\sc}}
%%\def\subsection{\@startsection{\@string\subsection}%
%%        2\z@{6\p@\@plus9\p@}{-.5em}%
%%        {\defaultfont\bf}}
%%
%%\def\subsubsection{\@startsection{\@string\subsubsection}%
%%        3\z@{6\p@\@plus9\p@}{-.5em}%
%%        \defaultfont\it}
%%
%%\def\paragraph{\@startsection{\@string\paragraph}%
%%        4\z@\z@{-\spaceskip}\defaultfont}
%***********************************************************************

%***********************************************************************
%%section 2
%%
%M% We can't use \@startsection for AMS-style \section because it's not 
%M% designed for centered headings, so instead of something like
%M%
%M%\def\section{\@startsection {section}{1}{\z@}{12pt plus 3pt}{7pt}{\sc}}
%M%
%M% we use \secdef. If the user uses \section*, \@ssection will
%M% be called; otherwise \@section.

\def\section{\secdef\@section\@ssection}

%M% If the user does not use the square bracket option, #1 will be the same
%M% as #2 (done by \@dblarg, via \secdef).  If the square bracket option is
%M% used, then probably #1 will be a shortened form of #2, and it will only
%M% be used for the table of contents (since the article style does not
%M% put section headings into the running heads).

\def\@section[#1]#2{\par\addvspace{12\p@ \@plus 3\p@}%
%M% \sectionmark not needed because section text doesn't go into
%M% running heads in the AMS article style (only in the book style).
  \refstepcounter{section}%
%M% \begingroup\centering...\endgroup is the equivalent of
%M% \begin{center}...\end{center}, leaving out some unnecessary overhead.
  \begingroup\centering\bf
%M% The sectioning level of \section is 1.  So if the specified numbering
%M% depth (\c@secnumdepth) is 1 or more, then we typeset the section number,
%M% otherwise we omit it.
  \ifnum\c@secnumdepth>0\relax\thesection.\enspace\fi
  #2\@@par\endgroup\@xsect{7\p@}}

%M% ``starred \section''
\def\@ssection#1{\par\addvspace{12\p@\@plus3\p@}%
%M% \sectionmark not needed because section text doesn't go into
%M% running heads
%M%
%M% From the LaTeX book: ``The *-form suppresses the section
%M% number, does not increment the counter, does not affect the running
%M% head, and produces no table of contents entry.''  So this
%M% is nonstandard to have the \addcontentsline.
  \begingroup\centering\bf#1\@@par\endgroup\@xsect{7\p@}}

\def\subsection{\@startsection{subsection}% %M% counter name
{2}% %M% sectioning level
{\parindent}% %% indent to the left of the subsection title
{6\p@\@plus2\p@}% %M% vertical space above
{-.5em}% %M% following text is run in, after a horizontal space of this amount.
%M% The minussign is to get horizontal space instead of vertical space.
{\bf}}% %M% The font of the subsection title

\def\subsubsection{\@startsection {subsubsection}% %M% counter number
{3}% %M% sectioning level
{\parindent}% %% indent to the left of the subsubsection title
{6\p@\@plus2\p@}% %M%  vertical space above
{-.5em}% %M% horizontal space after (negative sign will be canceled)
{\it}}% %M% font of subsubsection title

%M% To save control sequence names, we don't provide \paragraph and
%M% \subparagraph.  We don't expect them to be used often enough to be worth
%M% it.

%M%\def\paragraph{\@startsection {paragraph}{4}{\z@}{\z@}{\z@}{\rm}}
%M%\def\subparagraph{\@startsection {subparagraph}{4}{\z@}{\z@}{\z@}{\rm}}
%***********************************************************************


\let\@ssect\relax % to save memory.  \@ssect not needed in amsart.sty.

\def\thesection{\@arabic\c@section}
\def\thesubsection{\thesection.\@arabic\c@subsection}
\def\thesubsubsection{\thesubsection.\@arabic\c@subsubsection}
\setcounter{secnumdepth}{3}
\def\@pnumwidth{1.6em}

\def\@listI{\leftmargin\leftmargini \parsep\z@
  \topsep\z@ \itemsep\z@}
\let\@listi\@listI
\@listi

\def\@cite#1#2{\rom{\mediumseries[{{\bf#1}\if@tempswa , #2\fi}]}}

\def\@afterheading{\global\@nobreaktrue
   \everypar{\if@nobreak \global\@nobreakfalse \clubpenalty\@M
               \if@afterindent \else {\setbox\z@\lastbox}\fi
             \else \clubpenalty\@clubpenalty \everypar{}\fi}}
\def\@hangfrom#1{\setboxz@h{#1}\hangindent\wdz@ \noindent\box\z@}


\theorembodyfont{}
\def\@begintheorem#1#2{\item[\hskip\labelsep \theorem@headerfont
       \theorem@indent#1\ \rom{#2}\unskip.]}
\def\@opargbegintheorem#1#2#3{\item[\hskip\labelsep {\theorem@headerfont
       \theorem@indent#1\ \rom{#2}}\unskip\ \rom(#3\rom).]}
\def\th@plain{\it}
\def\th@definition{\theorempreskipamount\z@
    \theorempostskipamount\theorempreskipamount
    \normalshape}
\def\th@remark{\theorempreskipamount6\p@\@plus6\p@
    \theorempostskipamount\z@\@plus6\p@
    \def\theorem@headerfont{\it}\normalshape}
\def\theorem@headerfont{\normalshape\sc}
\let\theorem@indent\indent
\theorempreskipamount=6pt plus9pt
\theorempostskipamount=\theorempreskipamount
\def\@endtheorem{\endtrivlist\@endpefalse }

\renewenvironment{pf}{\par\addvspace{6\p@\@plus6\p@}
  {\sc\proofname.}\enspace\ignorespaces}{\qed\par\addvspace{6\p@\@plus6\p@}}

\@namedef{pf*}#1{\par\begingroup\def\proofname{#1}\pf\endgroup\ignorespaces}
\expandafter\let\csname endpf*\endcsname=\endpf

\def\ps@headings{\let\@mkboth\@gobbletwo
  \let\partmark\@gobble\let\sectionmark\@gobble
  \let\subsectionmark\@gobble
  \let\@oddfoot\@empty\let\@evenfoot\@empty%
  \def\@evenhead{\defaultfont\scriptsize
      \rlap{\thepage}\hfil
      \expandafter\uppercasetext@\expandafter{\sh@rtauthor}\hfil}%
  \def\@oddhead{\defaultfont\scriptsize \hfil
      \expandafter\uppercasetext@\expandafter{\sh@rttitle}\hfil
      \llap{\thepage}}%
}

\def\ps@myheadings{\let\@mkboth\@gobbletwo
  \let\@oddfoot\@empty\let\@evenfoot\@empty
  \def\@oddhead{\null\defaultfont\scriptsize\rightmark \hfil\thepage}%
  \def\@evenhead{\defaultfont\scriptsize \thepage\hfil\leftmark\null}%
}

\def\ps@plain{\let\@mkboth\@gobbletwo
     \let\@oddhead\@empty \let\@evenhead\@empty \let\@evenfoot\@oddfoot
     \def\@oddfoot{\defaultfont\scriptsize \hfil\thepage\hfil}}


%       Some computations to make the \textheight come out
%       to an even multiple of baselineskip (after taking
%       \topskip etc. into account).  \normalbaselineskip
%       is set to 13pt in ams-l.sty.  MJD
\advance\textheight by -\headheight
\advance\textheight by -\headsep
\advance\textheight by -\normalbaselineskip
\advance\textheight by \topskip

%       We now have the right value for \textheight except that
%       we need to round it.
\advance\textheight by.5\normalbaselineskip
\divide\textheight by\normalbaselineskip
\multiply\textheight by\normalbaselineskip
%       One last note: if \baselinestretch is changed in an
%       actual document, this \textheight will most likely be
%       off and you'll get underfull vbox messages.

\advance\topmargin by -.25 true in

%***************************************************************************


%% ********** environment definitions ***********

\renewcommand{\labelenumi}{(\roman{enumi})}


\def\enumerate{\ifnum \@enumdepth >3 \@toodeep\else
      \advance\@enumdepth \@ne
      \edef\@enumctr{enum\romannumeral\the\@enumdepth}\list
      {\csname label\@enumctr\endcsname}{\usecounter
        {\@enumctr}\partopsep=6pt\def\makelabel##1{\hss\llap{\normalshape##1}}}\fi}

\def\itemize{\ifnum\@itemdepth>3 \@toodeep
  \else \advance\@itemdepth\@ne
  \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}%
  \list{\csname\@itemitem\endcsname}%
  {\partopsep=6pt\def\makelabel##1{\hss\llap{\normalshape##1}}}\fi}


\def\thebibliography#1{\section*{{\series{m}\sc References}\@mkboth
  {REFERENCES}{REFERENCES}}\normalshape\mediumseries\small\labelsep 0.5em
  \list{\arabic{enumi}.}{\settowidth\labelwidth{#1.}
  \leftmargin\labelwidth \advance\leftmargin\labelsep
  \usecounter{enumi}}%
    \def\bysame{\leavevmode\vbox to 1ex{\vskip 0pt plus 100pt
    \hbox to 3em{\leaders\hrule\hfil}\vskip 0pt plus 300pt}}
  \sloppy\clubpenalty4000\widowpenalty4000
  \sfcode`\.=1000\relax}
\let\endthebibliography=\endlist



%% ****** topmatter definitions **********
\def\@maketitle{%
  \defaultfont\normalsize
  \let\@makefnmark\relax  \let\@thefnmark\relax
  \ifx\@empty\thesubjclass@\else
   \@footnotetext{1980 {\it Mathematics Subject
     Classification} (1985 {\it Revision\/}).\enspace
        Primary \thesubjclass@.}\fi
  \ifx\@empty\thekeywords@\else
   \@footnotetext{{\it Key words and phrases.}\enspace \thekeywords@.}\fi
\ifx\@empty\thethanks@\else
   \@footnotetext{\thethanks@}\fi
%% Series logo needs to fall even with the top of the running heads.
  \null \kern-\topskip \kern-\headsep \kern-\headheight
  \nointerlineskip
  \serieslogo@
\kern 7.25pc%7.8pc%
  \vtop{\centering{\Large\bf
    \expandafter\expandafter{\thetitle@}\@@par}%
   \global\dimen@i\prevdepth}%
  \prevdepth\dimen@i
  \ifx\@empty\theauthors@
  \else
    \baselineskip32\p@
    \vtop{\@andify{ AND }\theauthors@
      \centering{%
  \expandafter\uppercasetext@\expandafter{\theauthors@}\@@par}%
         \global\dimen@i\prevdepth}\relax
    \prevdepth\dimen@i
  \fi
  \ifx\@empty\thededicatory@
  \else
    \baselineskip18\p@
  \vtop{\centering{\small\it\thededicatory@\@@par}%
      \global\dimen@i\prevdepth}\prevdepth\dimen@i
  \fi
  \ifx\@empty\thedate@\else
  \baselineskip24\p@
    \vtop{\centering\thedate@\@@par
      \global\dimen@i\prevdepth}\prevdepth\dimen@i
  \fi
  \normalbaselines
  \dimen@32\p@ \advance\dimen@-\baselineskip
  \vskip\dimen@\@plus14\p@
  } % end \@maketitle



%% This is not a vbox to height 0 (unlike amssty.sty) since the
%% outside design consultant hired by the AMS specifies the
%% sinkage of the title as a base-to-base distance from the
%% series logo.  21-MAR-1990 mjd

\def\serieslogo@{\vbox{%
  \parindent\z@ \size{6}{6.5\p@}\selectfont
	DIMACS Series in Discrete Mathematics\newline
        and Theoretical Computer Science\newline
	Volume {\bf\currentvolume}, \currentyear
  \vss}}


\def\copyrightyear{1990}
\def\copyrightyearmodC{90}
\def\ISSN{1052-1798}
\def\copyrightprice{\$1.00\;+\;\$.25 per page}
\def\currentvolume{00}
\def\currentyear{1990}

%% Doing the copyright info on the first page is a little tricky. We
%% want it to come at the bottom, after any footnotes and floating
%% inserts, but before the page number.  If we put it into \@oddfoot
%% (in \ps@plain) its height will not be subtracted from the height of
%% the text and then the page number will be lower than we want.
%% Instead we put the copyright info into \@textbottom, which is normally
%% used by \raggedbottom and \flushbottom.

%     If \raggedbottom is ever needed in CNM, then this will need to be
%     changed (from latex.tex):
% \def\raggedbottom{\def\@textbottom{\vskip \z@ plus.0001fil}%
%             \let\@texttop\relax}

\def\raggedbottom{\typeout{\string\raggedbottom\space is not
 usable in AMS LaTeX production; see the documentation for further details.}}

%% \baselineskip and \lineskip are set to 0 in LaTeX output routine,
%% so we don't need to worry about them in \@textbottom.  After the
%% first use of \@textbottom \copyrightbox@ is void, but \@textbottom
%% is still emptied out, to save a couple tokens of memory.

\def\@textbottom{\box\copyrightbox@ \global\let\@textbottom\@empty}


\newbox\copyrightbox@
\newdimen\pagetocopyright@      \pagetocopyright@=2pc

%% We set the copyright info in a box in order to measure the height,
%% because we want to subtract it from the vsize on the first page.
%% The actual setbox is done in the individual journal file.

\expandafter\def\expandafter\@preamblecmds\expandafter{%
   \@preamblecmds\advance\vsize-\ht\copyrightbox@}



%% We set the copyright info in a box in order to measure the height,
%% because we want to subtract it from the vsize on the first page.
%% This setbox operation has to be done in DIMWRK-L.STY instead of
%% AMS-L.STY because it freezes information like ISSN number.
\setbox\copyrightbox@=\vbox{\hsize\textwidth
  \parfillskip\z@ \leftskip\z@\@plus.9\textwidth
  \size{6}{6.5\p@}\selectfont
%% The spacing between the preceding text and the copyright info
%% is done with a strut of height \pagetocopyright@.
%% (\lineskip and \baselineskip are 0 in the LaTeX output routine.)
%% The \everypar{} turns off the LaTeX warning about setting
%% text before \begin{document}.
  \everypar{}\noindent\vrule\@width\z@\@height\pagetocopyright@
  \copyright\copyrightyear\ American Mathematical Society\break
  \ISSN/\copyrightyearmodC\ \copyrightprice\endgraf
%% This kern of 0pt forces the depth of the last line (if any) to
%% be added to the height of the box.
  \kern\z@}



\endinput
%% 
%% End of file `tex_inputs:amsart10.sty'.
*******************************************************************************

%%dimacs.tex                                       8/90
%
% This is a sample file for use with AMS-LaTeX. It provides an example of
% how to set up a file to be typeset with AMS-LaTeX.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\documentstyle{DIMACS}

\newtheorem{lemma}{Lemma}[section]
\newtheorem{definition}{Definition}[section]

\newcommand{\AmSLaTeX}{{\the\textfont2 A}\kern-.1667em\lower.5ex\hbox
 {\the\textfont2 M}\kern-.125em{\the\textfont2 S}-\LaTeX}

\begin{document}

\title[MAXIMAL IDEALS IN SUBALGEBRAS OF $C(X)$]{Sample Paper for DIMACS,\\
On Maximal Ideals in Subalgebras of $C(X)$}
\author[AUTHOR ONE AND AUTHOR TWO]{Author One and Author Two}
\address{Department of Mathematics, Northeastern University, Boston,
Massachusetts 02115} % Research address for author one
\email{XYZ\char`\@ Math.AMS.com}
\address{Mathematical Research Section, School of Mathematical Sciences,
Australian National University, Canberra ACT 2601, Australia} %address for 
%                                                             author two

\subjclass{Primary 54C40, 14E20; Secondary 46E25, 20C20}

%  \thanks will become a 1st page footnote.
%  Use \endgraf to indicate a new paragraph; a blank line or \par will
%  be recognized as an error.
%  Don't type a period at the end; it will be supplied.
\thanks{The first author was supported in part by NSF
Grant \#000000.\endgraf
The final version of this paper will be submitted for
publication elsewhere}

\maketitle

\begin{abstract}
This paper is a sample prepared to illustrate for authors
the use of the \AmSLaTeX{} Version~1.0 package.

The file used to prepare this sample is \verb+sample.tex+; an author
should use the coding in that file as a model.
\end{abstract}

\section{Introduction}

This sample paper illustrates the use of \AmSLaTeX{} Version~1.0. In this
sample paper, brief instructions to authors will be interspersed with
mathematical text extracted from (purposely unidentified) published papers. 
For instructions on preparing mathematical text, the author is referred to {\it
The Joy of \TeX}, by Michael Spivak \cite{spivak:jot} and {\it \LaTeX{}: A
Document Preparation System} by Leslie Lamport. \cite{lamport:latex}

\subsection{Top matter}
The input format and content of the top matter can be best understood
by examining the first part of the sample file \verb+sample.tex+, up
through the \verb+\begin{document}+ instruction.

The top matter includes both elements that must be input by the author and a
few that are provided automatically.  The author names and the title that are
to appear in the running heads should be input between square brackets as an
option to the \verb+\author+ and \verb+\title+ commands, respectively. The full
names and title should be used unless they require too much space; in that
event, abbreviated forms should be substituted. In the top matter, the title is
input in caps and lowercase and will be set that way.  The author names should
be input in caps and lowercase; they will automatically be set in all caps.

For each author an address should be input.    Following these addresses, an
address for electronic mail should be given, if one exists. Note that no
abbreviations are used in addresses, and complete addresses for each author
should be entered in the order that names appear on the title page.  Addresses
are considered part of the top matter but are set at the end of the paper,
following the references.

Subject classifications (\verb+\subjclass+) and acknowledgments
(\verb+\thanks+) are part of the top matter and will appear as footnotes at the
bottom of the first page.  Use the 1980 Mathematics Subject Classification
(1985 Revision) that appears in annual indexes of {\it Mathematical Reviews\/}
beginning in 1984.  (The two-digit code from the Contents is not sufficient.)

Use \verb+\thanks+ for the footnotes that appear on the first page.  It is
generally desirable not to attach footnote numbers or symbols to titles or
author names used as headings.  If a footnote applies to only one author then
include this information in the footnote.  Only one instance of
\verb+\thanks+ is permitted in a paper.  Multiple ``thanks'' footnotes may
be simulated by separating the appropriate parts from one another by
\verb+\endgraf+; this will begin a new paragraph in the footnote.

Papers published in proceedings of conferences are often abstracts or
preliminary versions.  In such a case, include the following in the
\verb+\thanks+ to appear at the bottom of the first page: ``The final
[detailed] version of this paper will be [has been] submitted for
publication elsewhere.''  Papers that are to be considered for review
by {\it Mathematical Reviews\/} should include the following statement:
``This paper is in final form and no version of it will be submitted
for publication elsewhere.''

\subsection{Fonts}
The fonts used in this paper are from the Computer Modern family; they
should be available to all authors preparing papers with these macros.
However, the final copy may be set by the AMS using other fonts.  

\subsection{A mathematical extract}
The mathematical content of this sample paper has been extracted from
published papers, with no effort made to retain any mathematical sense.
It is intended only to illustrate the recommended manner of input.

Mathematical symbols in text should always be input in math mode as
illustrated in the following paragraph.

A function is invertible in $C(X)$ if it is never zero, and in $C^*(X)$ if
it is bounded away from zero. In an arbitrary $A(X)$, of course, there
is no such description of invertibility which is independent of the 
structure of the algebra. Thus in \S 2 we associate to each noninvertible
$f\in A(X)$ a $z$-filter $\cal Z (f)$ that is a measure of where
$f$ is ``locally'' invertible in $A(X)$. This correspondence extends to
one between maximal ideals of $A(X)$ and $z$-ultrafilters on $X$.
In \S 3 we use the filters $\cal Z (f)$ to describe the intersection of 
the free maximal ideals in any algebra $A(X)$. Finally, our main result
allows us to introduce the notion of $A(X)$-compactness of which 
compactness and realcompactness are special cases. In \S 4 we show how
the Banach-Stone theorem extends to $A(X)$-compact spaces.

\section{Theorems, lemmas, and proofs}

Theorems and lemmas are varieties of \verb+theorem+ environments.  In this
document, a \verb+theorem+ environment called \verb+lemma+ has been created,
which is used below. Also, there is a  proof, which is in the predefined
\verb+pf+ environment.  The lemma and proof below illustrate the use of 
the \verb+enumerate+ environment. 

\begin{lemma}
{Lemma 1} Let $f, g\in  A(X)$ and let $E$, $F$ be cozero
sets in $X$.
\begin{enumerate}
\item If $f$ is $E$-regular and $F\subseteq E$, then $f$ is $F$-regular.

\item If $f$ is $E$-regular and $F$-regular, then $f$ is $E\cup F$-%
regular.

\item If $f(x)\ge c>0$ for all $x\in E$, then $f$ is $E$-regular.

\end{enumerate}
\end{lemma}

\begin{pf}
\begin{enumerate}

\item  Obvious.

\item Let $h, k\in A(X)$ satisfy $hf|_E=1$ and $kf|_F=1$. Let
$w=h+k-fhk$. Then $fw|_{E\cup F}=1$.

\item Let $h=\max\{c,f\}$. Then $h|_E=f|_E$ and $h\ge c$. So $0<h^{-1}
\le c^{-1}$. Hence $h^{-1} \in C^*(X)\subseteq A(X)$, and 
$h^{-1} f|_E=1$. 

\end{enumerate}
\end{pf}

Another \verb+theorem+-type environment was defined at the beginning of this
document, called \verb+definition+. Here is an example of it:

\begin{definition}
For $f\in A(X)$, we define
$$
\cal Z (f)=\{E\in Z[X]\: \text{$f$ is $E^c$-regular}\}.
%\tag 2.1
$$
\end{definition}

\section{Roman type}

Numbers, punctuation, (parentheses), [brackets], $\{$braces$\}$, and
symbols used as tags should always be set in roman type.  The following
sample theorem illustrates how to code for roman type within the
statement of a theorem.

\begin{lemma}
Let $\cal G$ be a free nilpotent-of-class-\rom{2} group of rank
$\ge 2$ with carrier $G$ and let
$$m : G\times G \to Z$$
satisfy \rom{(2.21)}, \rom{(2.22)}, and \rom{(2.24)}, and define
$\kappa$ by \rom{(2.23)}.  Then this kappa-group is kappa-nilpotent
of class \rom{2} and kappa-metabelian, that is to say, it satisfies
\rom{S2} and \rom{S3}, but it is kappa-abelian if, and only if,
$$m(x,y) = -1\quad\text{for all $x, y \notin G'$}.
%\tag 3.1
$$
\rom{(}Thus \rom{(3.1)} implies the trivial consequence
\rom{(2.1)}.\rom{)}  Assume now that \rom{(3.1)} does not hold,
so that the kappa-group is kappa-nonabelian.  Assume further that $m$
is not constant outside $G'$ \rom{(}inside $G'$ the values of $m$
clearly do not matter\rom{)}.  Then $\kappa$ is neither left nor right
linear, that is to say, neither \rom{S4} nor \rom{S5} holds:
\rom{I1} again holds, but none of \rom{I2--I5}.  As before,
\rom{I6} is equivalent to \rom{(2.25)}.  Now \rom{I7$'$}, however,
is equivalent to a condition similar to \rom{(2.25)}, namely
$$m(xz\sigma, yz\sigma) = m(x,y)\,.
%\tag 3.2
$$
\end{lemma}

Letters used as abbreviations rather than as variables or constants
are set in roman type.  Use the control sequences \cite[p.\ 99]{spivak:jot}
for common mathematical functions and operators like $\log$ and $\lim$,
and use \verb+\cite+ when citing a reference.  The reference tag
will be {\bf bold} automatically, but you will need to set any
additional information in roman type as illustrated by the coding of
the previous sentence.

\section{References}

When you are ready to begin the bibliography, type 
\verb+\begin{thebibliography}+
before the first reference.  This produces the heading for the
bibliography and changes the type to a smaller size.  Input each reference
as you would normally do in \LaTeX{} \cite{lamport:latex}
Use the abbreviations of names of journals as given in annual indexes of
{\it Mathematical Reviews}.

References are set with hanging indentation, The amount of indentation
is preset to accommodate the most common case, two-digit numbers.
It can be increased (or decreased) by specifying the widest number or
key used in the references. For example,
\verb+\widestnumber\no{999}\+ and
\verb+\widestnumber\key{GHMaR}+
will increase the indentation to accommodate a three-digit number or the key
\hbox{[GHMaR]} respectively.  Note that the parts of the formatting that
depend on the current journal style are taken into account automatically
(the period after a number or the [\dots]\ around a key, plus the usual
space).

After the last reference, type \verb+\end{thebibliography}+ to mark the end of
the bibliography.

\section{Figures}

Figures to be inserted later should be handled using \LaTeX's \verb+figure+
environment.  The amount of space left should equal the exact height of the
figure.  Extra space around the figure will be provided automatically.  The
positioning of figures may need to be changed to obtain the best possible page
layout.  

In most cases, figures will be rendered for consistency of style within a
book.  Please provide figure manuscript drawn in black ink with clean,
unbroken lines on nonabsorbent paper.

For the link in Figure 5a, the Massey product $\langle u_1, u_2, u_3,
u_4, u_5\rangle$ in $S^3-L$ is defined and consists of all
integer multiples of $\gamma_{1,5}$.  For the link in Figure 5b,
the Massey product $\langle u_1, u_2, u_3, u_4, u_5\rangle$ in
$S^3-L$ contains the single element $\gamma_{1,5}$.  Since the links
in Figures 5a and 5b are homotopic, the example indicates that Massey
products in $S^3-L$ with distinct $u_j$'s do not, in general,
determine homotopy invariants of the link.  For the link in Figure 5a
and the link in Figure 5b, the Massey product $\langle u_1, u_2, \dots,
u_5\rangle$ in $\{S^3-L_i\}_{i=1}^5$ contains the single element
$\gamma_{1,5}$.

%  art work measures 11.5pc for figure 5a, 7pc for figure 5b

\begin{figure}
\vskip 11.5pc
\caption{}
\end{figure}

\begin{figure}
\vskip 7pc
\caption{}
\end{figure}


\section{Other headings}

\subsection{A subsection} 
We conclude by noting that another characterization of $A$-compactness
follows from Mandelker \cite{coddington:orddiffeq}. We call a family $\cal S$ of closed sets in
$X\ A$-stable if every $f\in A(X)$ is bounded on some member of $\cal S$.
]Then one can show (as in \cite{coddington:orddiffeq}) that a space is $A$-compact if and only if 
every $A$-stable family of closed sets with the finite intersection property
has nonempty intersection.

%\subsubsection{A second-level subheading}
%
%This paragraph is included only to illustrate the appearance of a
%sub-subsection.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibliographystyle{amsplain}

\makeatletter \renewcommand{\@biblabel}[1]{\hfill#1.}\makeatother

\begin{thebibliography}{10}

\bibitem{bass:jacobian}
H. Bass, E. H. Connell, and D. Wright,
{\em The Jacobian conjecture}, Bull. Amer. Math. Soc.
 {\bf 7} (1982), 287--330.

\bibitem{bass:flows}
H. Bass and G. H. Meisters,
{\em Polynomial flows in the plane}, Adv. in Math.
 {\bf 55} (1985), 173--203.

\bibitem{coddington:orddiffeq}
E. A. Coddington and N. Levinson,
{\em Theory of ordinary differential equations},
McGraw-Hill, New York, 1955.

\bibitem{coomes:injectivity}
B. Coomes,
{\em Polynomial flows, symmetry groups, and conditions sufficient for 
        injectivity of maps},
 Ph.D. Thesis, Univ. Nebraska-Lincoln, 1988.

\bibitem{coomes:lorenz}
\bysame,
{\em The Lorenz system does not have a polynomial flow},
 {J. Differential Equations}, (to appear).

\bibitem{spivak:jot}
M. D. Spivak,
{\em The Joy of \TeX{}}, Amer. Math. Soc., Providence, R.~I., 1986.

\bibitem{lamport:latex} Leslie Lamport {\em \LaTeX{} -- A Document Preparation
System}, Addison-Wesley, Reading, Mass., 1986.

\end{thebibliography}

\end{document}

\endinput
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%***** END OF DIMACS.TEX
%


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% START OF DIMACS.CHECKLIST
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%
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