\typeout{ Document Style `tac7' <00-1-2>. Modified
from `article' by Michael Barr and Robert Rosebrugh. }
 
\hyphenation{pro-mon-oid-al}

\def\abstractname{Abstract}

\font\tenit=cmti10

\lineskip 1pt
\normallineskip 1pt
\def\baselinestretch{1}
 
\def\@normalsize{\@setsize\normalsize{14.5pt}\xiipt\@xiipt
\abovedisplayskip 12\p@ plus3\p@ minus7\p@
\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip  \z@ plus3\p@
\belowdisplayshortskip  6.5\p@ plus3.5\p@ minus3\p@
\let\@listi\@listI}
 
\def\small{\@setsize\small{13.6pt}\xipt\@xipt
\abovedisplayskip 11\p@ plus3\p@ minus6\p@
\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip  \z@ plus3\p@
\belowdisplayshortskip  6.5\p@ plus3.5\p@ minus3\p@
\def\@listi{\leftmargin\leftmargini
 \parsep 4.5\p@ plus2\p@ minus\p@ \itemsep \parsep
            \topsep 9\p@ plus3\p@ minus5\p@}}
 
\def\footnotesize{\@setsize\footnotesize{12pt}\xpt\@xpt
\abovedisplayskip 10\p@ plus2\p@ minus5\p@
\belowdisplayskip \abovedisplayskip
\abovedisplayshortskip  \z@ plus3\p@
\belowdisplayshortskip  6\p@ plus3\p@ minus3\p@
\def\@listi{\leftmargin\leftmargini
\topsep 6\p@ plus2\p@ minus2\p@\parsep 3\p@ plus2\p@ minus\p@
\itemsep \parsep}}
 
\def\scriptsize{\@setsize\scriptsize{9.5pt}\viiipt\@viiipt}
\def\tiny{\@setsize\tiny{7pt}\vipt\@vipt}
\def\large{\@setsize\large{18pt}\xivpt\@xivpt}
\def\Large{\@setsize\Large{22pt}\xviipt\@xviipt}
\def\LARGE{\@setsize\LARGE{25pt}\xxpt\@xxpt}
\def\huge{\@setsize\huge{30pt}\xxvpt\@xxvpt}
\let\Huge=\huge
 
\normalsize
 
\marginparsep 10pt
 
\topmargin 0pt
 
\headheight 12pt
\headsep 15pt
\topskip = 0pt
 
\footskip 20pt
\footheight=12pt
 
\textheight = 8.5in
\textwidth 6.3in
 
 
\footnotesep 8.4pt
 
\skip\footins 10.8pt plus 4pt minus 2pt
 
\floatsep 14pt plus 2pt minus 4pt
 
\textfloatsep 20pt plus 2pt minus 4pt
 
\intextsep 14pt plus 4pt minus 4pt
 
\@maxsep 20pt
 
\dblfloatsep 14pt plus 2pt minus 4pt
 
\dbltextfloatsep 20pt plus 2pt minus 4pt
 
\@dblmaxsep 20pt
 
\@fptop 0pt plus 1fil
 
\@fpsep 10pt plus 2fil
\@fpbot 0pt plus 1fil
 
\@dblfptop 0pt plus 1fil
 
\@dblfpsep 10pt plus 2fil% Space between floats on float page.
\@dblfpbot 0pt plus 1fil
 
\marginparpush 7pt
 
\parskip 0pt plus 1pt
 
\parindent 1.5em
 
\partopsep 3pt plus 2pt minus 2pt
 
\@lowpenalty   51
\@medpenalty  151
\@highpenalty 301
 
\@beginparpenalty -\@lowpenalty
 
\@endparpenalty   -\@lowpenalty
 
\@itempenalty     -\@lowpenalty
 
\def\section{\@startsection {section}{1}{\z@}{-4ex plus -1ex minus
-.2ex}{1.75ex plus .2ex}{\large}}
\def\subsection{\@startsection{subsection}{2}{\z@}{-3.2ex plus -1ex minus
 -.2ex}{-1em plus 1em minus .4em}{\normalsize}}
\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{-3.25ex plus
 -1ex minus-.2ex}{1.5ex plus.2ex}{\reset@font\normalsize}}
\def\paragraph{\@startsection
     {paragraph}{4}{\z@}{3.25ex plus1ex minus.2ex}{-1em}{\reset@font
     \normalsize\bf}}
\def\subparagraph{\@startsection
     {subparagraph}{4}{\parindent}{3.25ex plus1ex minus
     .2ex}{-1em}{\reset@font\normalsize\bf}}
 
\setcounter{secnumdepth}{3}
 
\def\appendix{\par
  \setcounter{section}{0}
  \setcounter{subsection}{0}
  \def\thesection{\Alph{section}}}
 
\leftmargini 2.5em
\leftmarginii 2.2em
\leftmarginiii 1.87em
\leftmarginiv 1.7em
\leftmarginv 1em
\leftmarginvi 1em
 
\leftmargin\leftmargini
\labelsep .5em
\labelwidth\leftmargini\advance\labelwidth-\labelsep
 
\def\@listI{\leftmargin\leftmargini \parsep 5\p@ plus2.5\p@ minus\p@
\topsep 10\p@ plus4\p@ minus6\p@
\itemsep 5\p@ plus2.5\p@ minus\p@}
 
\let\@listi\@listI
\@listi
 
\def\@listii{\leftmargin\leftmarginii
   \labelwidth\leftmarginii\advance\labelwidth-\labelsep
   \topsep 5\p@ plus2.5\p@ minus\p@
   \parsep 2.5\p@ plus\p@ minus\p@
   \itemsep \parsep}
 
\def\@listiii{\leftmargin\leftmarginiii
    \labelwidth\leftmarginiii\advance\labelwidth-\labelsep
    \topsep 2.5\p@ plus\p@ minus\p@
    \parsep \z@ \partopsep \p@ plus\z@ minus\p@
    \itemsep \topsep}
 
\def\@listiv{\leftmargin\leftmarginiv
     \labelwidth\leftmarginiv\advance\labelwidth-\labelsep}
 
\def\@listv{\leftmargin\leftmarginv
     \labelwidth\leftmarginv\advance\labelwidth-\labelsep}
 
\def\@listvi{\leftmargin\leftmarginvi
     \labelwidth\leftmarginvi\advance\labelwidth-\labelsep}
 
 
 
\def\labelenumi{\theenumi.}
\def\theenumi{\arabic{enumi}}
\def\labelenumii{(\theenumii)}
\def\theenumii{\alph{enumii}}
\def\p@enumii{\theenumi}
\def\labelenumiii{\theenumiii.}
\def\theenumiii{\roman{enumiii}}
\def\p@enumiii{\theenumi(\theenumii)}
\def\labelenumiv{\theenumiv.}
\def\theenumiv{\Alph{enumiv}}
\def\p@enumiv{\p@enumiii\theenumiii}
 
\def\labelitemi{$\bullet$}
\def\labelitemii{\bf --}
\def\labelitemiii{$\ast$}
\def\labelitemiv{$\cdot$}
 
\def\verse{\let\\=\@centercr
 \list{}{\itemsep\z@ \itemindent -1.5em\listparindent \itemindent
 \rightmargin\leftmargin\advance\leftmargin 1.5em}\item[]}
\let\endverse\endlist
\def\quotation{\list{}{\listparindent 1.5em
 \itemindent\listparindent
 \rightmargin\leftmargin \parsep \z@ plus 1\p@}\item[]}
\let\endquotation=\endlist
\def\quote{\list{}{\rightmargin\leftmargin}\item[]}
\let\endquote=\endlist
 
\def\descriptionlabel#1{\hspace\labelsep \bf #1}
\def\description{\list{}{\labelwidth\z@ \itemindent-\leftmargin
 \let\makelabel\descriptionlabel}}
\let\enddescription\endlist
 
 
\def\theequation{\arabic{equation}}
 
 
\def\titlepage{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
 \else \newpage \fi \thispagestyle{empty}\c@page\z@}
\def\endtitlepage{\if@restonecol\twocolumn \else \newpage \fi}
 
\arraycolsep 5\p@ \tabcolsep 6\p@ \arrayrulewidth .4\p@ \doublerulesep 2\p@
\tabbingsep \labelsep
 
\skip\@mpfootins = \skip\footins
\fboxsep = 3\p@ \fboxrule = .4\p@
 
\newcounter{part}
\newcounter {section}
\newcounter {subsection}[section]
\newcounter {subsubsection}[subsection]
\newcounter {paragraph}[subsubsection]
\newcounter {subparagraph}[paragraph]
 
\def\thepart{\Roman{part}} \def\thesection {\arabic{section}}
\def\thesubsection {\thesection.\arabic{subsection}}
\def\thesubsubsection {\thesubsection .\arabic{subsubsection}}
\def\theparagraph {\thesubsubsection.\arabic{paragraph}}
\def\thesubparagraph {\theparagraph.\arabic{subparagraph}}
 
\def\@pnumwidth{1.55em}
\def\@tocrmarg {2.55em}
\def\@dotsep{4.5}
\setcounter{tocdepth}{3}
 
\def\tableofcontents{\section*{Contents\@mkboth{CONTENTS}{CONTENTS}}
 \@starttoc{toc}}
\def\l@part#1#2{\addpenalty{\@secpenalty}
 \addvspace{2.25em plus 1\p@} \begingroup
 \@tempdima 3em \parindent \z@ \rightskip \@pnumwidth \parfillskip
-\@pnumwidth
 {\large \bf \leavevmode #1\hfil \hbox to\@pnumwidth{\hss #2}}\par
 \nobreak \endgroup}
\def\l@section#1#2{\addpenalty{\@secpenalty} \addvspace{1.0em plus 1\p@}
\@tempdima 1.5em \begingroup
 \parindent \z@ \rightskip \@pnumwidth
 \parfillskip -\@pnumwidth
 \bf \leavevmode \advance\leftskip\@tempdima \hskip -\leftskip #1\nobreak\hfil
\nobreak\hbox to\@pnumwidth{\hss #2}\par
 \endgroup}
 
\def\l@subsection{\@dottedtocline{2}{1.5em}{2.3em}}
\def\l@subsubsection{\@dottedtocline{3}{3.8em}{3.2em}}
\def\l@paragraph{\@dottedtocline{4}{7.0em}{4.1em}}
\def\l@subparagraph{\@dottedtocline{5}{10em}{5em}}
\def\listoffigures{\section*{List of Figures\@mkboth
 {LIST OF FIGURES}{LIST OF FIGURES}}\@starttoc{lof}}
\def\l@figure{\@dottedtocline{1}{1.5em}{2.3em}}
\def\listoftables{\section*{List of Tables\@mkboth
 {LIST OF TABLES}{LIST OF TABLES}}\@starttoc{lot}}
\let\l@table\l@figure
 
 
 
\long\def\@makefntext#1{\parindent 1em\noindent
 \hbox to 1.8em{\hss$^{\@thefnmark}$}#1}
 
 
\setcounter{topnumber}{2}
\def\topfraction{.7}
\setcounter{bottomnumber}{1}
\def\bottomfraction{.3}
\setcounter{totalnumber}{3}
\def\textfraction{.2}
\def\floatpagefraction{.5}
\setcounter{dbltopnumber}{2}
\def\dbltopfraction{.7}
\def\dblfloatpagefraction{.5}
 
\long\def\@makecaption#1#2{
 \vskip 10\p@
 \setbox\@tempboxa\hbox{#1: #2}
 \ifdim \wd\@tempboxa >\hsize #1: #2\par \else \hbox
to\hsize{\hfil\box\@tempboxa\hfil}
 \fi}
 
 
\newcounter{figure}
\def\thefigure{\@arabic\c@figure}
\def\fps@figure{tbp}
\def\ftype@figure{1}
\def\ext@figure{lof}
\def\fnum@figure{Figure \thefigure}
\def\figure{\@float{figure}}
\let\endfigure\end@float
\@namedef{figure*}{\@dblfloat{figure}}
\@namedef{endfigure*}{\end@dblfloat}
\newcounter{table}
\def\thetable{\@arabic\c@table}
\def\fps@table{tbp}
\def\ftype@table{2}
\def\ext@table{lot}
\def\fnum@table{Table \thetable}
\def\table{\@float{table}}
\let\endtable\end@float
\@namedef{table*}{\@dblfloat{table}}
\@namedef{endtable*}{\end@dblfloat}
 
\pagenumbering{arabic}
 
 
\let\then\iftrue
\def\ifnull#1\then{\def\P@rtoch@ck{#1}\ifx\P@rtoch@ck\@empty}
 
% The following is to add periods:
 
\def\@sect#1#2#3#4#5#6[#7]#8{\ifnum #2>\c@secnumdepth
     \def\@svsec{}\else
     \refstepcounter{#1}\ifnull{#8}\then
     \edef\@svsec{\csname the#1\endcsname} \else
     \edef\@svsec{\csname the#1\endcsname.\hskip6\p@} \fi\fi
     \@tempskipa #5\relax
      \ifdim \@tempskipa>\z@
        \begingroup #6\relax
          \@hangfrom{\hskip #3\relax\@svsec}{\interlinepenalty \@M #8\par}
        \endgroup
       \csname #1mark\endcsname{#7}\addcontentsline
         {toc}{#1}{\ifnum #2>\c@secnumdepth \else
                      \protect\numberline{\csname the#1\endcsname}\fi
                    #7}\else
        \def\@svsechd{#6\hskip #3\@svsec #8.\csname #1mark\endcsname
                      {#7}\addcontentsline
                           {toc}{#1}{\ifnum #2>\c@secnumdepth \else
                             \protect\numberline{\csname the#1\endcsname}\fi
                       #7}}\fi
     \@xsect{#5}}
 
\def\@ssect#1#2#3#4#5{\@tempskipa #3\relax
   \ifdim \@tempskipa>\z@
     \begingroup #4\@hangfrom{\hskip #1}{\interlinepenalty \@M #5\par}\endgroup
   \else \def\@svsechd{#4\hskip #1\relax #5.}\fi
    \@xsect{#3}}
 
 
\def\subsection{\@startsection{subsection}{2}{\z@}{\medskipamount}
{-7\p@}{\normalsize}}
\def\subsubsection{\@startsection{subsection}{3}{\z@}{\medskipamount}
{-7\p@}{\normalsize}}
 


% new on 99-5-30: 
\def\@credit[#1]{{\rm [#1]}}
\def\state#1{\subsection{\sc#1}\begingroup\em\@ifnextchar[{\@credit}{}}
\def\staterm#1{\subsection{\sc#1}\begingroup\@ifnextchar[{\@credit}{}}

\def\newtheorem#1#2{\expandafter\def\csname #1\endcsname{%
\state{#2}\@ifnextchar[{\@credit}{}}%
 \expandafter\def\csname
     end#1\endcsname{\endgroup \par\addvspace{\medskipamount}}}

\def\newtheoremrm#1#2{\expandafter\def\csname #1\endcsname{%
\staterm{#2}\@ifnextchar[{\@credit}{}}%
 \expandafter\def\csname
     end#1\endcsname{\endgroup \par\addvspace{\medskipamount}}}

\newtheorem{theorem}{Theorem}
\newtheorem{proposition}{Proposition}
\newtheorem{corollary}{Corollary}
\newtheorem{lemma}{Lemma}
\newtheorem{definition}{Definition}        
\newtheorem{notation}{Notation}
\newtheorem{scholium}{Scholium}
\newtheorem{assumption}{Assumption}


\newtheoremrm{remark}{Remark}
\newtheoremrm{example}{Example}
\newtheoremrm{remarks}{Remarks}
\newtheoremrm{examples}{Examples}
        



\def\endtheorem{\endgroup \par\addvspace{\medskipamount}}
\let\enddefinition=\endtheorem
\let\endnotation=\endtheorem
\let\endscholium=\endtheorem
\let\endproposition=\endtheorem
\let\endlemma=\endtheorem
\let\endcorollary=\endtheorem
\def\endremark{\endgroup \par\medskip} % from \addvspace{\medskipamount}}
\let\endremarks=\endremark
\let\endexample=\endremark
\let\endexamples=\endremark
\def\substate#1{\subsubsection*{\it#1}}
 
% The following replaces the latex \Box, whose .tfm appears to be
% corrupted in some fonts, including the ones distributed by PC-tex.
 
%\def\Box{\vbox{\hrule\hbox{\vrule\kern.5ex
%   \vbox{\kern1ex}\kern.5ex\vrule}\hrule}}

% modified 97-2-28 to allow parameter, but \ifnextchar undef
%\def\Pr@@f{\subsubsection*{\sc Proof}}
%\def\pr@@f[#1]{\subsection*{{\sc Proof} #1}}
%\def\proof{\ifnextchar[{\pr@@f}{\Pr@@f}}

\def\proof{\subsubsection*{\sc Proof}}

% added 96-10-26
\def\pushright#1{{%        set up
   \parfillskip=0pt            % so \par doesnt push \square to left
   \widowpenalty=10000         % so we dont break the page before \square
   \displaywidowpenalty=10000  % ditto
   \finalhyphendemerits=0      % TeXbook exercise 14.32
  %
  %                 horizontal
   \leavevmode                 % \nobreak means lines not pages
   \unskip                     % remove previous space or glue
   \nobreak                    % don't break lines
   \hfil                       % ragged right if we spill over
   \penalty50                  % discouragement to do so
   \hskip.2em                  % ensure some space
   \null                       % anchor following \hfill
   \hfill                      % push \square to right
   {#1}                        % the end-of-proof mark (or whatever)
  %
  %                   vertical
   \par}}                      % build paragraph


\newbox\@endproofbox
\setbox\@endproofbox\hbox{\vrule width5pt height 5pt depth 0pt}
\def\endproofbox{\copy\@endproofbox}
\def\endproof{\pushright{\endproofbox}%
\penalty-700 \par\addvspace{\medskipamount}}      
 
% The following are to produce the page one headers and footnotes
% Note that there must be two passes to get the page numbers correct
% AMS classification, thanks, keywords and revision dates are optional.
% The others are required.
 
\newif\ifthanks
\newif\ifkeywords
\newif\ifamsclass
\newif\ifdedication
\newif\ifeaddress
 
\def\@title{\relax}
\def\@author{\relax}
\def\@yer{\relax}
\def\@copryear{\relax}
\def\@address{\relax}
\def\@eaddress{\relax}
\def\@thanks{\relax}
\def\@keywords{\relax}
\def\@amsclass{\relax}
\def\@runningoddhead{\relax}
\def\@runningevenhead{\relax}
\def\@dedication{\relax}
 
\def\title#1{\def\@title{{\let\cr=\\ \def\\{\cr[1\p@]}
\large\uppercase{#1}}}}
\def\dedication#1{\def\@dedication{{\let\cr=\\ \def\\{\cr[1\p@]}
\large\it#1}}\dedicationtrue}
\long\def\address#1{\global\def\@address{{\par\addvspace{12\p@}
\baselineskip11.5pt\parindent0pt
% \let\\=\par
\let\newline=\par
\noindent\tenit#1\\ %\vspace{5pt}
\ifeaddress Email: \tt \@eaddress\fi }}}
\def\eaddress#1{\def\@eaddress{#1}\eaddresstrue}
\def\author#1{\def\@author{\uppercase{#1}}}
\def\tacyear#1{\def\@yer{#1}}
\def\copyrightyear#1{\def\@copryear{#1}}
\def\keywords#1{\def\@keywords{Key words and phrases: #1.}\keywordstrue}
\def\amsclass#1{\def\@amsclass{2000 Mathematics Subject
Classification: #1.}\amsclasstrue}
\def\startpage#1{\setcounter{page}{#1}}
\def\thanks#1{\def\@thanks{#1}\thankstrue}
 
\def\ps@tac{\let\@mkboth\markboth
\def\@oddfoot{}
\let\@evenfoot\@oddfoot
\def\@evenhead{\hfil\thepage}
\let\@oddhead\@evenhead}
\pagestyle{tac}
 
\def\ps@taci{\let\@mkboth\markboth
\def\@oddfoot{\hfil\rm\thepage\hfil}
\let\@evenfoot\@oddfoot
\def\@oddhead{\hfil}
\let\@evenhead\@oddhead}
 
 
\def\maketitle{\par
 \begingroup
 \def\thefootnote{\fnsymbol{footnote}}
 \def\@makefnmark{\hbox
 to\z@{$^{\@thefnmark}$\hss}}
 \global\@topnum\z@ \@maketitle \thispagestyle{taci}
 \endgroup
 \setcounter{footnote}{0}
 \let\@maketitle\relax
{\def\thempfn{}
\ifthanks\footnotetext{\@thanks}\fi
\ifamsclass\footnotetext{\@amsclass}\fi
\ifkeywords\footnotetext{\@keywords}\fi
}}
 
\def\@maketitle{\newpage\vspace*{12\p@}
{\topsep\z@ \partopsep\z@
\begin{center}
\ifdedication
   \@title\\[12\p@]
   \@dedication\\[18\p@]
\else
   \@title\\[14\p@]
\fi
\@author\\
\end{center}}
\vskip24\p@}

\def\abstract{\par
\footnotesize\begin{list}{}{\rightmargin=13mm \leftmargin=13mm
\parsep5\p@ \listparindent=\z@ 
\topsep=\z@\partopsep=\z@}
\item\leavevmode
{A{\scriptsize BSTRACT}.}\hskip 1em\baselineskip 10pt}
\def\endabstract{\end{list}\par}
 

 
%Note that this is a redefinition.  Its purpose is to compile the
%label lastpage so that we can get the page numbers right.
%It also adds the authors' address
 
\def\enddocument{
\nopagebreak[4]
{\@address}
\label{lastpage} 
\nopagebreak[4]
\@checkend{document}\clearpage\begingroup
\if@filesw \immediate\closeout\@mainaux
\def\global\@namedef##1##2{}\def\newlabel{\@testdef r}%
\def\bibcite{\@testdef b}\@tempswafalse \makeatletter\input \jobname.aux
\if@tempswa \@warning{Label(s) may have changed.  Rerun to get
cross-references right}\fi\fi\endgroup\deadcycles\z@\@@end}
 
 
\def\thebibliography#1{\section*{References\@mkboth
 {REFERENCES}{REFERENCES}}\list
 {[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth
 \advance\leftmargin\labelsep
 \usecounter{enumi}}
 \def\newblock{\hskip .11em plus .33em minus .07em}
 \sloppy\clubpenalty4000\widowpenalty4000
 \sfcode`\.=1000\relax}
\let\endthebibliography=\endlist



% Below are the codes for the two reference styles.  They can be called by
% \begin{references} ... \end{references} and \begin{references*} ...
% \end{references*} .  In addition, the first can be called by \references
% ... \endreferences .
 
\newdimen\refindent \refindent 8mm
\def\references{%
\fontdimen2\twlrm=4.3pt% space instead of 3.91663
\fontdimen3\twlrm=4.2pt%stretch instead of 1.95831
\fontdimen4\twlrm=1.7pt%shrink instead of 1.30554
\section*{References}
\sloppy\footnotesize
\begin{list}{}
{\leftmargin\refindent \itemindent-\refindent \itemsep2pt \topsep0pt}}
\def\endreferences{\end{list}}
 
\newcounter {reflister}
\expandafter\def\csname references*\endcsname {%
\fontdimen2\twlrm=4.3pt% space instead of 3.91663
\fontdimen3\twlrm=4.2pt%stretch instead of 1.95831
\fontdimen4\twlrm=1.7pt%shrink instead of 1.30554
\section*{References}
\sloppy\footnotesize
\begin{list}{{\rm[\arabic{reflister}]}}{\usecounter{reflister}}
{\leftmargin\refindent \itemindent-\refindent \itemsep2pt \topsep0pt}}
\expandafter\let\csname endreferences*\endcsname=\endreferences
 
 
% Modification of the `plain' definition, taken over by latex.  Note the
% failure to parametrize the definition, one of the many such that testify
% to Knuth's conception of plain as a provisional package.
 
\def\footnoterule{%\kern-3\p@
  \hrule width 1truein \kern 2.6\p@} % the \hrule is .4\p@ high
 
% Follows are a small number of procedures I have personally found very
% useful.  The first three define new math operations like Hom.  In the
% first, the word is spaced like a mathop and is used if there are to be
% arguments.  It is better not used if, like Hom, it is followed by an
% opening paranthesis.  The next two are without the spacing and are used
% for boldface, roman face, resp.  The usage is: \mathrm{Hom} that defines
% a new macro \Hom that puts the word Hom in ordinary type.
 
\def\mathopdef#1{\expandafter\def\csname#1\endcsname{{\mathop{\rm#1}}}}
\def\mathbfdef#1{\expandafter\def\csname#1\endcsname{{\hbox{\rm\bf#1}}}}
\def\mathrmdef#1{\expandafter\def\csname#1\endcsname{{\hbox{\rm#1}}}}
 
\def\mld#1$${\null\,\vcenter\bgroup\def\\{\cr&}\openup9pt\m@th
\ialign\bgroup\strut\hfil$\displaystyle{##}$%
&$\displaystyle{{}##}$\hfil\crcr #1
\crcr\egroup\egroup\,$$}
\let\emld\relax % for compatibility with old code
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Here is a typical usage of \mld (multi-line display)          %%
%%                                                               %%
%% The pattern is                                                %%
%% $$\mld formula 1 & formula 2 \\                               %%
%%                    formula 3 \\                               %%
%%                    ..........\\                               %%
%%                    formula n $$                               %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
\endinput