%******************************************************************
%*********************    TeX Macros     **************************
%*********************      o.tex        **************************
%*********************  by Peter Olver   **************************
%*********************      v 6.3       **************************
%******************************************************************

\expandafter\ifx\csname omag\endcsname\relax\def\omag{\magstep1}\fi
\magnification=\omag

\def\pjo{\author
Peter J. Olver\\
School of Mathematics\\
University of Minnesota\\
Minneapolis, MN\quad 55455\\
\email olver\@umn.edu\\
\www math.umn.edu/$\sim$olver\\
\support NSF Grant DMS \hskip1pt\grantnumber.}

\def\pjoi{\author
Peter J. Olver\\
School of Mathematics\\
University of Minnesota\\
Minneapolis, MN\quad 55455\\
U.S.A.\\
\email olver\@umn.edu\\
\www math.umn.edu/$\sim$olver\\
\support NSF Grant DMS \ \grantnumber.}

\def\grantnumber{{\rm 11--08894}}  
       %08--07317 05�-05293 01--03944 98--03154 95--00931 

%%%%%%%%%%%%%%%%%% Macros for beginning a paper  %%%%%%%%%%%%%%%%%%

\def\Rough{\centerline{\bigit Rough\enskip Draft\enskip  
\bigrm --- \enskip \shortdate\enskip ---\enskip  
\bigit Do\enskip Not\enskip Distribute}\vglue 30pt}

\def\Title#1.{\def\\{\par}\ifproofmode\Rough\else\vglue 40pt\fi
   \atitle{#1}.\vskip20pt}

\def\Abstract#1\par{\par\bigskip\indent{\bf Abstract.}\enspace#1\par\bigskip}

\newdimen\shortlinedim \shortlinedim=3truein

\def\shortline#1{\hbox to \shortlinedim{#1}}

\newdimen\boxitdim \boxitdim=1.7in

\def\boxit#1{\def\\{\hfil\egroup\par\shortline\bgroup\relax\ignorespaces}%
\vbox to \boxitdim{\shortline\bgroup\ignorespaces#1\hfil\egroup\vfill}}

%\def\shorterline#1{\hbox to 2.5truein{#1}}
%
%\def\boxits#1{\def\\{\hfil\egroup\par\shorterline\bgroup\asis}%
%\vbox to 1.7in{\shorterline\bgroup#1\hfil\egroup\vfill}}

\newcount\autnumber  \autnumber=1

\def\email{\tt}
\def\www{\tt http://www.}
\def\author#1\\#2\support#3.{%
\expandafter\def\csname autx\romannumeral\the\autnumber\endcsname
   {\boxit{#1\fnotemark\\#2\\}
\vfootnote\fnotemark{\it Supported in part by #3.}
\global\advance\fnotenumber by 1}\global\advance\autnumber by 1}
\def\authorn#1\par{%
\expandafter\def\csname autx\romannumeral\the\autnumber\endcsname
   {\boxit{\ustrut9#1\\}}
\global\advance\autnumber by 1 }

\def\autformi{\vskip20pt\hfill \autxi \vskip 10pt minus 10pt}
\def\autformii{\vskip20pt\noindent\hfil \autxi \hfil \hfil 
   \autxii \hfil\vskip 10pt minus 10pt}
\def\autformiii{\vskip10pt\noindent\hfil \autxi \hfil \hfil 
   \autxii \hfil \vskip -25pt
\noindent\hfil \autxiii \hfil \vskip 10pt minus 10pt}
\def\autformiv{\vskip10pt\noindent\hfil \autxi \hfil \hfil 
   \autxii \hfil \vskip -25pt
\noindent\hfil \autxiii \hfil \hfil \autxiv \hfil \vskip 10pt minus 10pt}
\def\autformv{\vskip10pt\noindent\hfil \autxi \hfil \hfil 
   \autxii \hfil \vskip -25pt
\noindent\hfil \autxiii \hfil \hfil \autxiv \hfil \vskip 10pt minus 10pt
\noindent\hfil \autxv \hfil \vskip 10pt minus 10pt}

\def\printauthor{\global\advance\autnumber by -1
\csname autform\romannumber\the\autnumber\endcsname}

%%%%%%%%%%%%%%%%%%%%%%%%%%  Hyphenation  %%%%%%%%%%%%%%%%%%%%%%%%%%%%

\hyphenation{di-men-s-ion-al quad-ra-tic 
   par-a-m-e-t-ri-ze par-a-m-e-t-ri-zes par-a-m-e-tri-zed 
   par-a-m-e-t-ri-z-a-tion par-a-m-e-t-ri-z-a-tions
   re-par-a-m-e-t-ri-ze re-par-a-m-e-t-ri-zes re-par-a-m-e-tri-zed 
   re-par-a-m-e-t-ri-z-a-tion re-par-a-m-e-t-ri-z-a-tions
   La-g-ran-gian La-g-ran-gians}

%%%%%%%%%%%%%%%%%  Set size and paragraph skipping  %%%%%%%%%%%%%%%%%

\def\blackbox{\vrule height7pt width5pt depth0pt}
\def\noblackboxes{\overfullrule=0pt}

\parskip = 1pt plus 2pt
\abovedisplayskip = 9pt plus 3pt minus 3pt
\belowdisplayskip = 7pt plus 3pt minus 3pt

\newskip\refparskip \refparskip = 1pt plus 1pt

\def\doublespace{\refparskip = 6pt plus 1pt
\baselineskip = 12pt \lineskip = 7pt \lineskiplimit = 6pt}

\def\singlespace{\normalbaselines \refparskip = 1pt plus 1pt}

\long\def\ignore#1{} \def\asis#1{#1}

\def\hfb{\hfill\break} \let\hbf\hfb \def\hfbg#1{\hfill\break\hglue#1pt}
\def\hfp{\hfill\break\hglue\parindent}
\def\hfbs#1{\hfill\sstrut{#1}\break}
\def\hfbsg#1#2{\hfill\sstrut{#1}\break\hglue #2pt}



%%%%%%%%%%%%%%%% Output routine for complete pages %%%%%%%%%%%%%%%%%%

\output{\ooutput}
\def\ooutput{\shipout\vbox{\makeheadline\pagebody\makefootline}%
  \advancepageno\global\fnotenumber=0
  \ifnum\outputpenalty>-20000 \else\dosupereject\fi}

\def\page{\vfill\eject}\def\newpage{\vfill\eject}

%%%%%%%%%%%%%%%%%%% Messages on console %%%%%%%%%%%%%%%%%%%%%

\newif\ifwarn \warntrue
\def\nowarnings{\warnfalse} \def\warnings{\warntrue}

\def\msg#1{\immediate\write0{#1}}
\def\warn#1{\ifwarn\msg{** #1 **}\fi}
\def\warna#1{\ifwarn\msg{-> #1 <-}\fi}
\def\warnb#1{\ifwarn\msg{+> #1 <+}\fi}
\def\warnr#1{\ifwarn\msg{=> #1 <=}\fi}

%%%%%%%%%%%%%%%  To change category codes for numerals  %%%%%%%%%%%%%
%%%%%%%%%%%%%%%%    Used in referencing macros    %%%%%%%%%%%%%%%%%%%

\xdef\numcat{\catcode`0=11\catcode`1=11\catcode`2=11\catcode`3=11
\catcode`4=11\catcode`5=11\catcode`6=11\catcode`7=11\catcode`8=11
\catcode`9=11\catcode`*=11\relax}
\xdef\unnumcat{\catcode`0=12\catcode`1=12\catcode`2=12\catcode`3=12
\catcode`4=12\catcode`5=12\catcode`6=12\catcode`7=12\catcode`8=12
\catcode`9=12\catcode`*=12\relax}

{\numcat\gdef\0{0}\gdef\1{1}\gdef\2{2}\gdef\3{3}\gdef\4{4}\gdef\5{5}
\gdef\6{6}\gdef\7{7}\gdef\8{8}\gdef\9{9}
\gdef\10{10}\gdef\11{11}\gdef\12{12}\gdef\13{13}\gdef\14{14}
\gdef\15{15}\gdef\16{16}\gdef\17{17}\gdef\18{18}\gdef\19{19}
\gdef\20{20}\gdef\21{21}\gdef\22{22}\gdef\23{23}\gdef\24{24}
\gdef\25{25}\gdef\26{26}\gdef\27{27}\gdef\28{28}\gdef\29{29}
\gdef\30{30}\gdef\31{31}\gdef\32{32}\gdef\33{33}\gdef\34{34}
\gdef\35{35}\gdef\36{36}\gdef\37{37}\gdef\38{38}\gdef\39{39}
\gdef\40{40}\gdef\41{41}\gdef\42{42}\gdef\43{43}\gdef\44{44}
\gdef\45{45}\gdef\46{46}\gdef\47{47}\gdef\48{48}\gdef\49{49}
\gdef\91{A}\gdef\92{B}\gdef\93{C}\gdef\94{D}\gdef\95{E}
\gdef\96{F}\gdef\97{G}\gdef\98{H}\gdef\99{I}}

{\catcode`*=11\expandafter\gdef\csname *0\endcsname{0}
\expandafter\gdef\csname *1\endcsname{1}
\expandafter\gdef\csname *2\endcsname{2}
\expandafter\gdef\csname *3\endcsname{3}
\expandafter\gdef\csname *4\endcsname{4}
\expandafter\gdef\csname *5\endcsname{5}
\expandafter\gdef\csname *6\endcsname{6}
\expandafter\gdef\csname *7\endcsname{7}
\expandafter\gdef\csname *8\endcsname{8}
\expandafter\gdef\csname *9\endcsname{9}}

\def\hexnumber#1{\ifcase#1 0\or1\or2\or3\or4\or5\or6\or7\or8\or9\or
 A\or B\or C\or D\or E\or F\fi}
\def\romannumber#1{\romannumeral#1}
\def\alphanumber#1{\count255 = #1 \advance\count255 by 96 
\expandafter\char\count255}

\newif\ifroman\romanfalse
\def\rnumber{\ifroman\romannumeral\else\number\fi}
\def\rnumbert#1{\ifroman{\tenit\romannumeral#1}\else{\tenrm\number#1}\fi}
\def\rnumberq#1{\ifroman"\romannumeral#1"\else\number#1\fi}

%%%%%%%%%%%%%%%%%% Macros for footnotes and quotes  %%%%%%%%%%%%%%%%%

\newcount\fnotenumber  \fnotenumber=0
\def\fnotemark{\ifcase\the\fnotenumber$^\dagmath$\or$^\ddagmath$\or
                         $^\Smath$\or$^\Pmath$\or$^\|$\or$\Upstar$\fi}

\newdimen\fnoteskip \fnoteskip = 0pt

\def\fnoteadjust{\global\advance\fnotenumber by 1}
\def\fnotev#1{{\vfootnote\fnotemark
      {\hglue\fnoteskip\smallstyle #1\par\vskip-10pt}\fnoteadjust}}
\def\fnote#1{{\footnote\fnotemark
      {\hglue\fnoteskip\smallstyle #1\par\vskip-10pt}\fnoteadjust}}
\def\fnotea#1{{\fnoteadjust\footnote\fnotemark
      {\hglue\fnoteskip\smallstyle #1\par\vskip-10pt}\fnoteadjust}}

\def\fnotep(#1){\fnote{#1}}  \let\fnotex\ignore

\skip\footins=20pt plus 4pt minus 4pt

\def\footnoterule{\hbox{\raise 5pt \vbox{\kern-3pt
                       \hrule width 3 truein \kern 2.6pt}}}

\newdimen\qsize \qsize = 22 pc

\def\quote#1\qauthor#2\par{\medskip\noindent\hfil
  \vbox{\hsize=\qsize \smallstyle #1\par\hfill --- #2\par\vfil}\hfill
   \par\medskip}


%%%%%%%%%%%%%% Restore the idiotic AMS-TeX omissions  %%%%%%%%%%%%%%%

\def\amstexname{AmS-TeX}
\ifx\fmtname\amstexname
   %\let\proclaim\plainproclaim
   \let\footnote\plainfootnote\def\cal{\fam2}%
   \def\>{\mskip\medmuskip}\def\=#1{{\accent22 #1}}%
   \def\mit{\fam1} \def\oldstyle{\fam1\teni}
\fi
\def\@{\char64 }% character @ won't work in AMS-TeX - use \@ instead

%%%%%%%%%%%%%%%%%%%%% Macros for input-output %%%%%%%%%%%%%%%%%%%%%%%
%
%      \keyout  write \jobname.key file for referencing
%      \keyin   read  \jobname.key file
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newif\ifeqnwrite  \eqnwritefalse
\newif\ifeqnread  \eqnreadfalse
\newif\iflocal \localfalse

\def\keyout{\eqnwritetrue\immediate\openout1=\jobname.key }
\def\keyin{\openin2=\jobname.key \relax
\ifeof2 \warn{I can't find the file \jobname.key}\closein2 \else
\closein2 \numcat \input \jobname.key \unnumcat \fi}

\def\keys#1{\openin2=#1.key \relax
\ifeof2 \warn{I can't find the file #1.key}\closein2 \else
\closein2 \numcat \input #1.key \unnumcat \fi}

\newtoks\outtoks
\def\wdef#1{\outtoks=\expandafter{\csname#1\endcsname}%
\immediate\write1{\global\def\the\outtoks{\csname#1\endcsname}}}

% \ends  macro for ending document, which closes appropriate files  %

\outer\def\ends{\ifeqnwrite\closeout1\fi
\ifrefnum\closeout3\fi
\ifindex\closeout4\closeout5\closeout6\fi
\ifcontents\closeout7\fi
\vfill\supereject\end}

%%%%%%%%%%%%%%%%%% Macros for proof mode  %%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%             \proofmodetrue  for writing referencing in margins
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newif\ifproofmode \proofmodefalse

\def\strutdepth{\dp\strutbox}
\def\margintagleft#1{\vadjust{\kern-\strutdepth\vtop to \strutdepth
   {\baselineskip\strutdepth\vss\llap{\sevenrm#1\quad}\null}}}
\def\prooflabel#1{\ifproofmode\margintagleft{#1}\fi}

\def\folio{\ifproofmode\jobname\quad\shortdate\quad\fi{\tenrm \number\pageno}}

\def\date{\ifcase\month\or January\or February\or March\or April\or May\or
June\or July\or August\or September\or October\or November\or December\fi
\space\number\day, \number\year}
\def\gobble#1#2{}
\def\shortdate{\number\month/\number\day/\expandafter\gobble\number\year}

\def\dated{\footnote\ {\vskip-3pt\hfill\bigrm \date}}
\def\dateo#1{\footnote\ {\vskip-3pt\hfill\bigrm #1}}

%%%%%%%%% Macros for paragraph formats and section headings  %%%%%%%%

\def\firstpar{\relax}
\def\pari{\par}\def\parin{\pari\nobreak}

\def\hbq{\hfill\break\quad} \def\hbqq{\hfill\break\qquad}

\def\xskip{\vskip-10pt}\def\yskip{\vskip-6pt}
\def\fskip{\smallskip}

\def\center{\parindent=0pt \parfillskip = 0pt 
\leftskip=0pt plus 1fil \rightskip = \leftskip \spaceskip=.3333em 
\xspaceskip = .5em \pretolerance=9999 \tolerance=9999 
\hyphenpenalty=9999 \exhyphenpenalty=9999\relax}

\def\atitle#1.{{\baselineskip=20pt \center \Bigrm #1\par\nobreak}}
\def\btitle#1.{{\center \bf #1\par\nobreak}}
\def\ctitle#1.{{\center \bigrm #1\par\nobreak}}
\def\ititle#1.{{\center\it #1\par\nobreak}}
\def\rtitle#1.{{\center #1\par\nobreak}}

\def\Headfont{\Bigbf} \def\headfont{\bf} \def\Subheadfont{\it}

\def\Head#1{\bigbreak\heading{#1}\Headfont}
\def\head#1{\bigbreak\heading{#1}\headfont}
\def\Subhead#1{\medskip\heading{#1}\Subheadfont}
\def\Headtagleft#1#2{\bigbreak{\parindent = 0pt\noindent
      \prooflabel{#2}\heading{#1}\Headfont}}
\def\headtagleft#1#2{\bigbreak{\parindent = 0pt\noindent\prooflabel{#2}%
    \rightskip = 0pt plus 1fil \spaceskip=.3333em \xspaceskip=.5em 
    \hyphenpenalty=9999 \exhyphenpenalty=9999 \headfont
    \hangindent=8pc #1\par\nobreak}\medskip}
\def\heading#1#2{{\noindent \rightskip = 0pt plus 1fil 
\spaceskip=.3333em \xspaceskip=.5em \hyphenpenalty=9999 
\exhyphenpenalty=9999 #2\hangindent=8pc #1\par\nobreak}\medskip}

%%%%%%%%%%%%%%%  Macros for section numbering %%%%%%%%%%%%%%%%%%%%%%%
%
%   The following flags give different numbering systems:  
%     for example, consider the equation which is 4th in its section 
%     and 9th in its chapter, and the statement which is 5th in 
%     its section and 10th in its chapter.
%   Suppose  Chapter number = 3,  Section number = 2
%
%  \bookmode   \longmode      equation #     statement #
%
%    false       false           (9)         Theorem 10
%    false       true           (2.4)        Theorem 2.5  --- default 
%    true        false          (3.9)        Theorem 3.10
%    true        true          (3.2.4)       Theorem 3.2.5
%
%  \runonmode --- equations and statements are numbered consecutively 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcount\chapternumber  \chapternumber=1
\newcount\sectionnumber  \sectionnumber=1
\newcount\equationnumber  \equationnumber=1
\newcount\statementnumber \statementnumber=1
\newcount\figurenumber \figurenumber=1
\newcount\itemnumber \itemnumber=1

\newif\ifbeginningdocument \beginningdocumenttrue
\newif\iflongmode  \longmodetrue
\newif\ifbookmode \bookmodefalse
\newif\ifappendixmode \appendixmodefalse

\def\runonmode{\def\statementnumber{\equationnumber}}

\def\chapterkey{cxyz}  %  default chapter name
\def\sectionkey{sxyz}  %  default section name

\def\chapnum{\the\chapternumber}
\def\secnum{\the\sectionnumber}
\def\eqnum{\the\equationnumber}
\def\stnum{\the\statementnumber}
\def\fgnum{\the\figurenumber}
\def\secxnum{\chapnum.\secnum.}

\def\acsname#1{\expandafter\aftergroup\csname#1\endcsname}
\def\xcsname#1{\expandafter\zcsname\the#1.\aftergroup.}
\def\zcsname#1{\ifx#1.\let\next=\relax\else\expandafter
\aftergroup\csname#1\endcsname\let\next=\zcsname\fi\next}
\def\ncsname#1{\ifx#1.\let\next=\relax\else\expandafter
\aftergroup\csname*#1\endcsname\let\next=\zcsname\fi\next}

\def\cxnum{\xcsname\chapternumber}
\def\scxnum{\xcsname\chapternumber\xsectionnumber}
\def\eqxnum{\scxnum\xcsname\equationnumber}
\def\stxnum{\scxnum\xcsname\statementnumber}
\def\fgxnum{\cxnum\xcsname\figurenumber}
\def\xsectionnumber{\expandafter\acsname{\the\sectionnumber}\aftergroup.}
\def\pgnum{\the\pageno}
\def\setcsnum#1.#2.{\begingroup\aftergroup\global\aftergroup
  \chapternumber\ncsname#1.\aftergroup\global\aftergroup\sectionnumber
  \ncsname#2.\endgroup}

\def\secmode#1.#2.{\ifbookmode#1.\fi#2}
\def\sekmode#1.#2.{\ifbookmode#1.\fi\iflongmode#2.\fi}
\def\eqmode#1.#2.#3.{\ifbookmode#1.\fi\iflongmode#2.\fi#3}
\def\eqmod#1.#2.#3.{#3}
\def\stmode#1 #2.#3.#4.{#1~\ifbookmode#2.\fi\iflongmode#3.\fi#4}
\def\stnmode#1 #2.#3.#4.{\ifbookmode#2.\fi\iflongmode#3.\fi#4}
\def\fgmode#1.#2.{\ifbookmode#1.\fi#2}

\def\chaptertop{20}\def\chaptermid{15}\def\chapterbot{30}\def\introbot{35}

\def\Chapter#1.#2.#3.#4.{\Chapt#1.#2.#3.#4.#4.}

\def\Chapt#1.#2.#3.#4.#5.{{\numcat\global\beginningdocumentfalse
\global\pageno=#3
\global\chapternumber=#2 \global\sectionnumber=0
\global\equationnumber=1 \global\statementnumber=1
\expandafter\xdef\csname c*#1\endcsname{#2}
\gdef\chapterkey{#1} \gdef\sectionkey{#1}%
\ifeqnwrite\wdef{c*#1}\fi
\msg{Chapter \csname c*#1\endcsname.  Key = \chapterkey}%
\ifcontents
\write7{\csname Chapp\endcsname #2. #4.#3.}\fi
\page
\vglue\chaptertop pt \center \baselineskip=20pt \bigbf Chapter\kern.4em
\the\chapternumber\par\vglue\chaptermid pt \Bigbf\ignorespaces#5\par
\vglue\chapterbot pt}\HeaderNumber=0 \LeftHead #4.\RightHead #4.\firstpar}

\def\chapter#1{Chapter~\chapternum{#1}}
\def\chapters#1#2{Chapters~\chapternum{#1} and~\chapternum{#2}}
\def\chapternum#1{{\numcat\expandafter\ifx\csname c*#1\endcsname\relax
  \warn{Chapter #1 not defined}{\bf #1}\relax\else \csname c*#1\endcsname\fi}}

\def\Introduction{{\numcat\global\beginningdocumentfalse
\global\pageno=1
\global\chapternumber=0 \global\sectionnumber=0
\global\equationnumber=1 \global\statementnumber=1
\gdef\chapterkey{intro} \gdef\sectionkey{intro}%
\page
\vglue\chaptertop pt \center \Bigbf Introduction\par
\vglue\introbot pt}\HeaderNumber=0 
\LeftHead Introduction.\RightHead Introduction.\romantrue\firstpar}

\def\Section#1#2.{{\numcat\gdef\sectionkey{#1}\ifbeginningdocument
   \else\global\advance\sectionnumber by 1\fi \global\beginningdocumentfalse
\begingroup\aftergroup\xdef\aftergroup\nuxxx\aftergroup{
\scxnum\aftergroup}\endgroup
\expandafter\ifx\csname s*#1\endcsname\relax\else
\edef\eqxxx{\csname s*#1\endcsname}%
\ifx\eqxxx\nuxxx\else\warna{Section #1 already exists}\fi\fi
\iflongmode\global\equationnumber=1\global\statementnumber=1\fi
\expandafter\xdef\csname s*#1\endcsname{\nuxxx} \ifeqnwrite\wdef{s*#1}\fi  
\msg{Section \sectionkey}\headtagleft{\expandafter\secmode\nuxxx. #2.}{#1}}%
\def\Subsecthead{#2}\Rightheadset\firstpar}

\def\section#1{Section~\sectionnum{#1}}  \let\sect=\section
\def\sections#1#2{Sections~\sectionnum{#1} and~\sectionnum{#2}}
\def\sectionnum#1{{\numcat\expandafter\ifx\csname s*#1\endcsname\relax
  \warn{Section #1 not defined}{\bf #1}\else
  \edef\nuxxx{\csname s*#1\endcsname}\expandafter\secmode\nuxxx\fi}}

\def\Subsection#1.{\def\Subsecthead{#1}\Subhead{#1}\ifcontents
\write7{\csname Subsec\endcsname #1.\the\pageno.}\fi\Rightheadset\firstpar}

\def\Subsect#1.{\def\Subsecthead{#1}\Subhead{#1}\ifcontents
\write7{\csname Subsec\endcsname #1.\the\pageno.}\fi\firstpar}

\def\Rightheadset{\expandafter\RightHead \Subsecthead.}

\def\Appendix#1#2.{{\numcat\gdef\sectionkey{#1}\ifappendixmode
\global\advance\sectionnumber by 1\else
\global\sectionnumber = 91\global\appendixmodetrue\fi
\begingroup\aftergroup\xdef\aftergroup\nuxxx\aftergroup{
\scxnum\aftergroup}\endgroup
\expandafter\ifx\csname s*#1\endcsname\relax\else
\edef\eqxxx{\csname s*#1\endcsname}%
\ifx\eqxxx\nuxxx\else\warna{Section #1 already exists}\fi\fi
\iflongmode\global\equationnumber=1\global\statementnumber=1\fi
\expandafter\xdef\csname s*#1\endcsname{\nuxxx} \ifeqnwrite\wdef{s*#1}\fi  
\msg{Appendix \sectionkey}\headtagleft{Appendix \expandafter
\secmode\nuxxx. #2.}{#1}}\firstpar}

\def\appendix#1{Appendix~\appendixnum{#1}}  \let\appendx=\appendix
\def\appendixs#1#2{Appendices~\appendixnum{#1} and~\appendixnum{#2}}
\let\appendices=\appendixs
\def\appendixnum#1{{\numcat \expandafter\ifx\csname s*#1\endcsname\relax
  \warn{Section #1 not defined}{\bf #1}\else
  \edef\nuxxx{\csname s*#1\endcsname}\expandafter\secmode\nuxxx\fi}}

\def\addkey#1{\addkeyz#1.\addkeyz}
\def\addkeyz#1.#2\addkeyz{\ifx.#2.\sectionkey.#1.\else#1.#2\fi}

\def\noindentsecpar{\def\firstpar{\par\nobreak\noindent\eatspacepar}}

%%%%%%%%%%%%%% Macros for equation numbering %%%%%%%%%%%%%%%%%%%%%%%%
%
%   General format:
%        $$ x = y  \Eq{label}$$   (plain TeX version)
%    or
%        \Eq{label} $$ x = y  $$  (LaTeX version)
%
%   Note:  label can contain upper & lower case letters and numerals
%
%   Cite in text as  \eq{label}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\def\Eq{\eqtag}
\def\Eq{\ifmmode\let\eqzzz\eqtag\else\let\eqzzz\StEq\fi\eqzzz}
\def\StEq#1$$#2$${\StEqq#1  \EnEqq{#2}}
\def\StEqq#1 #2\EnEqq#3{$$#3\Eq{#1}$$}

\def\eqnostar{\eqno{(*)}} 
\def\Eqstar$$#1$${$$#1\eqnostar$$}
\def\eqstar{$(*)$}  
\def\eqnostars{\eqno{(**)}} 
\def\Eqstars$$#1$${$$#1\eqnostars$$}
\def\eqstars{$(**)$}  

\def\eqtag#1{\newequation\eqno{#1}()}
\def\nqtag#1{\hbox{\rm\newequation\relax{#1}()}\ignorespaces}
\def\Eqx#1{\newequation\hfill{#1}()}
\def\Eqa#1#2{\newequation\eqno{#1}({#2)}}
\def\Eqpm#1{\Eqa{#1}\pm}
\def\Eql#1#2$${\Eq{#1}\,\kern -\displaywidth
     \rlap{\quad $\displaystyle#2$}\kern \displaywidth \kern -1pt$$}
\def\El#1$${\eqno\,\kern -\displaywidth
     \rlap{\quad $\displaystyle#1$}\kern \displaywidth \kern -1pt$$}

\def\zlap#1{\llap{#1\kern-3pt}}

\def\newequation#1#2#3#4{\prooflabel{#2}{\numcat
\begingroup\aftergroup\xdef\aftergroup\nuxxx\aftergroup{
\eqxnum\aftergroup}\endgroup
\expandafter\ifx\csname e*#2\endcsname\relax\else
\edef\eqxxx{\csname e*#2\endcsname}%
\ifx\eqxxx\nuxxx\else\warna{Equation #2 already exists}\fi\fi
\expandafter\xdef\csname e*#2\endcsname{\nuxxx} \ifeqnwrite\wdef{e*#2}\fi}%
#1{#3\expandafter\eqmode\nuxxx#4}\global\advance\equationnumber by 1}

\def\eq#1{\eqcite{#1}()}
\def\eqn#1{\eqcite{#1}\null\null}
\def\eqr#1#2{\eqcite{#1}(\null--\eqcite{#2}\null)}
\def\eqp#1{\eqcite{#1}({$+$)}}\def\eqm#1{\eqcite{#1}({$-$)}}
\def\eqpm#1{\eqcite{#1}({$\pm $)}}

\def\eqcite#1#2#3{{\numcat\expandafter\ifx\csname e*#1\endcsname\relax
\warn{Equation #1 not defined.}\edef\eqxxx{#2#1\blackbox#3}\else
\edef\nuxxx{\csname e*#1\endcsname}%
\edef\eqxxx{#2\expandafter\eqmode\nuxxx#3}\fi{\rm\eqxxx}}}

\def\eqxx{{\rm (\blackbox.\blackbox)}}

\def\eqe{equation~\eq} \def\eqse{equations~\eq} \def\eqE{Equation~\eq} 
\def\eqs#1#2{\eq{#1},~\eq{#2}}
\def\eqas#1#2{\eq{#1} and \eq{#2}}
\def\eqes#1#2{equations~\eq{#1},~\eq{#2}}
\def\eqeas#1#2{equations~\eq{#1} and \eq{#2}}
\def\eqss#1#2#3{\eq{#1},~\eq{#2},~\eq{#3}}
\def\eqass#1#2#3{\eq{#1},~\eq{#2} and \eq{#2}}
\def\eqess#1#2#3{equations~\eq{#1},~\eq{#2},~\eq{#3}}
\def\eqeass#1#2#3{equations~\eq{#1},~\eq{#2} and \eq{#3}}
\def\eqf{formula~\eq} \def\eqF{Formula~\eq} \def\eqfe{formulae~\eq}
\def\eqfs#1#2{formulae~\eq{#1},~\eq{#2}}

%\def\eqdef{:=}

\def\eqcit#1#2#3{{\numcat\expandafter\ifx\csname e*#1\endcsname\relax
\warn{Equation #1 not defined.}\edef\eqxxx{#2#1\blackbox#3}\else
\edef\nuxxx{\csname e*#1\endcsname}%
\edef\eqxxx{#2\expandafter\eqmod\nuxxx#3}\fi{\rm\eqxxx}}}

\def\eqrr#1#2{\eqcite{#1}(\null--\eqcit{#2}\null)} \let\eqr\eqrr
\def\eqc#1#2{\eqcite{#1}(,\hskip2pt\eqcit{#2}\null)}
\def\eqcc#1#2#3{\eqcite{#1}(,\hskip2pt\eqcit{#2}\null
        ,\hskip2pt\eqcit{#3}\null)}
\def\eqccc#1#2#3#4{\eqcite{#1}(,\hskip2pt\eqcit{#2}\null
        ,\hskip2pt\eqcit{#3}\null,\hskip2pt\eqcit{#4}\null)}
\def\eqcccc#1#2#3#4#5{\eqcite{#1}(,\hskip2pt\eqcit{#2}\null
        ,\hskip2pt\eqcit{#3}\null,\hskip2pt\eqcit{#4}\null
        ,\hskip2pt\eqcit{#5}\null)}

%%%%%%%%%%%%%%%% Macros for equation alignment %%%%%%%%%%%%%%%%%%%%%%
%
%  In all cases you use  \\   instead of  \cr  
%       and no final  \\ is required 
%
%   \aeq  -   same as plain tex  \eqalign  with  &  tabs
%   \eaeq  -   aligns on first  =  sign, and uses prescribed spacing
%      \eeq \feq or \heq  -  \eaeq with  \qquad \quad or no spacing
%   \iaeq  -  aligns at start, indenting all but first a prescribed 
%                                                         amount
%      \ieq \jeq or \keq  -  \iaeq with  \qquad \quad or no indenting
%   \qaeq  -   separates equations on one line by prescribed spacing 
%      \qeq \xeq \yeq or \zeq  --  \qaeq with  \qquad \quad \ or none 
%   \saeq -   aligns several equations at start, and uses prescribed 
%                                                    spacing
%      \seq \teq or \ueq  -  \saeq with  \qquad \quad or no spacing
%   \caeq -   centers several equations, and uses prescribed spacing 
%      \ceq \deq or \req -   \saeq with  \qquad \quad or no spacing 
%   \eeeq  -   aligns on second  =  sign, and then uses \qquad spacing
%
%   \ceqn  - centers several equations, with separate equation numbers 
%   \eeqn  - aligns equations on first  =  sign, with equation numbers
%   \seqn  - aligns several equations at start, with equation numbers
%
%   \meq \mzeq \myeq \mxeq \mqeq \mbeq - 
%               used for inline spacing of formulas
%
%   \table  -   makes a table of entries with  &  tabs
%   \ntable  -   makes a table of numbered entries with  &  tabs
%   \ltable  -   makes a table of entries with  &  tabs, 
%                                     first one flush left roman
%   \rtable  -   makes a table of entries with  &  tabs, 
%                                     first one flush right roman
%   \datatable -  data table with lines
%   \Rtable -  specialized table with title:  
%                         roman, centered math, left math, roman
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

\newdimen\eqjot \eqjot = 1\jot
\def\openupeq{\openup \the\eqjot}
\newdimen\tablejot \tablejot = 1\jot
\def\openuptable{\openup \the\tablejot}

\def\addtab#1={#1\;&=}
\def\addtabe#1=#2={#1=#2\;&=}

\def\aeq#1{{\def\\{\cr}\eqalign{#1\cr}}}

\def\qaeq#1#2{{\def\\{&}\vcenter{\openupeq\halign{$\displaystyle
   ##\hfil$&&\hskip#1pt$\displaystyle##\hfil$\cr #2\cr}}}}
\def\qqeq{\qaeq{40}} \def\qeqq{\qaeq{40}}
\def\qxeq{\qaeq{30}} \def\qeqx{\qaeq{30}}
\def\qeq{\qaeq{20}} \def\weq{\qaeq{15}}
\def\xeq{\qaeq{10}} \def\yeq{\qaeq{7}} \def\zeq{\qaeq{3}}

\def\caeq#1#2{{\def\\{\cr}\vcenter{\openupeq \halign
    {$\hfil\displaystyle ##\hfil$&&$\hfil\hskip#1pt\displaystyle ##\hfil
         $\cr#2\cr}}}}
\def\ceq{\caeq{20}} \def\deq{\caeq{10}} \def\req{\caeq{3}}

\def\ezeq#1#2#3{{\def\\{\cr#1}\vcenter{\openupeq \halign{$\displaystyle 
   \hfil##$&$\displaystyle##\hfil$&&\hskip#2pt$\displaystyle##\hfil
        $\cr#1#3\cr}}}}
\def\eaeq{\ezeq\addtab} \def\eeeq{\ezeq\addtabe{20}}
\def\eqeq{\eaeq{40}} \def\eeqq{\eaeq{40}} 
\def\exeq{\eaeq{30}} \def\eeqx{\eaeq{30}}
\def\eeq{\eaeq{20}} \def\feq{\eaeq{10}} \def\heq{\eaeq{3}}

\def\iaeq#1#2#3{{\def\\{\cr\hskip#1pt}\vcenter{\openupeq
   \halign{$\displaystyle ##\hfil$&&\hskip#2pt$\displaystyle##\hfil
   $\cr #3\cr}}}}
\def\ibeq#1{\iaeq{#1}{#1}}
\def\iqeq{\iaeq{40}{40}}  \def\iQeq{\iaeq{80}{80}}
\def\ieq{\iaeq{20}{20}} \def\jeq{\iaeq{10}{10}} \def\keq{\iaeq{10}{3}}

\def\saeq#1#2{{\def\\{\cr}\vcenter{\openupeq\halign{$\displaystyle
   ##\hfil$&&\hskip#1pt$\displaystyle##\hfil$\cr #2\cr}}}}
\def\sqeq{\saeq{40}} \def\seqq{\saeq{40}}
\def\sxeq{\saeq{30}} \def\seqx{\saeq{30}}
\def\seq{\saeq{20}} \def\teq{\saeq{10}} \def\ueq{\saeq{3}}

\def\meq#1{\relax${\def\\{$ $\dsty}$\dsty#1$}$\relax}
\def\mzeq#1{\relax${\def\\{$ \hskip1pt $\dsty}$\dsty#1$}$\relax}
\def\myeq#1{\relax${\def\\{$ \ $\dsty}$\dsty#1$}$\relax}
\def\myyeq#1{\relax${\def\\{$ \ \ $\dsty}$\dsty#1$}$\relax}
\def\mxeq#1{\relax${\def\\{$ \quad $\dsty}$\dsty#1$}$\relax}
\def\mqeq#1{\relax${\def\\{$ \qquad $\dsty}$\dsty#1$}$\relax}
\def\mbeq#1{\relax${\def\\{$ \hfill\break $\dsty}$\dsty#1$}$\relax}

\def\macases#1#2{\left\{\enspace\saeq{#1}{#2}\right.}
\def\mcases{\macases{20}}
\def\xcases{\macases{15}}
\def\tcases{\macases{10}}
\def\ucases{\macases{3}}

\def\mjacases#1#2#3{{\eqjot = #1\jot\left
                        \{\enspace\saeq{#2}{#3}\right.}}
\def\mjcases#1{\mjacases{#1}{20}}
\def\xjcases#1{\mjacases{#1}{15}}
\def\tjcases#1{\mjacases{#1}{10}}
\def\ujcases#1{\mjacases{#1}{3}}

\def\occases#1{\mcases{#1&\otherwise,}}
\def\opcases#1{\mcases{#1&\otherwise.}}
\def\oopcases#1{\opcases{#1\\0,}} 
\def\ooccases#1{\occases{#1\\0,}}
\def\ocucases#1{\ucases{#1&\otherwise,}}
\def\opucases#1{\ucases{#1&\otherwise.}}
\def\oopucases#1{\opucases{#1\\0,}} 
\def\oocucases#1{\ocucases{#1\\0,}}
\def\octcases#1{\tcases{#1&\otherwise,}}
\def\optcases#1{\tcases{#1&\otherwise.}}
\def\ooptcases#1{\optcases{#1\\0,}} 
\def\ooctcases#1{\octcases{#1\\0,}}

%\def\occases#1{\left\{\enspace\seq{#1&\otherwise,}\right.}
%\def\opcases#1{\left\{\enspace\seq{#1&\otherwise.}\right.}
%\def\oopcases#1{\opcases{#1\\0,}} \def\ooccases#1{\occases{#1\\0,}}
%\def\ocucases#1{\left\{\enspace\ueq{#1&\otherwise,}\right.}
%\def\opucases#1{\left\{\enspace\ueq{#1&\otherwise.}\right.}
%\def\oopucases#1{\opucases{#1\\0,}} \def\oocucases#1{\ocucases{#1\\0,}}
%\def\octcases#1{\left\{\enspace\teq{#1&\otherwise,}\right.}
%\def\optcases#1{\left\{\enspace\teq{#1&\otherwise.}\right.}
%\def\ooptcases#1{\optcases{#1\\0,}} \def\ooctcases#1{\octcases{#1\\0,}}

\def\crh#1{{}\cr\hskip#1pt{}}
\def\crhz{\crh0} \def\crhx{\crh{10}} \def\crhq{\crh{20}} \def\crhQ{\crh{40}}
\def\crr#1#2{#2{}\cr\hskip#1pt{}#2}
\def\crz{\crr0} \def\crx{\crr{10}} \def\crq{\crr{20}} \def\crQ{\crr{40}}
\def\cth#1#2{#2{}\cr&\hskip#1pt{}#2}
\def\ctz{\cth0} \def\ctx{\cth{10}} \def\ctq{\cth{20}} \def\ctQ{\cth{40}}
\def\creq{\ctq}\def\crpm{\crq}
\def\cthn#1#2{{}\cr&\hskip#1pt{}#2}
\def\ctnz{\cthn0} \def\ctnx{\cthn{10}} \def\ctnq{\cthn{20}} \def\ctnQ{\cthn{40}}
\def\creqn{\ctnq}
\def\crs{\cr\noalign{\medskip}}

\def\addtabby#1&#2={#1&#2\;&=}
\def\waeq#1#2{\def\\{\cr\addtabby}\vcenter{\openupeq \halign{$\displaystyle 
   \hfil##\hfil$&\hskip#1pt$\hfil\displaystyle##$&$\displaystyle
   ##\hfil$\cr\addtabby#2\cr}}}
\def\wweq{\waeq{50}}

\def\eqsas#1#2#3{\eq{#1}, \eq{#2}, and \eq{#3}}
\def\eqss#1#2#3{\eq{#1},~\eq{#2},~\eq{#3}}
\def\eqess#1#2#3{equations~\eq{#1},~\eq{#2},~\eq{#3}}

\def\ceqn#1{\def\\{\cr}\def\Eq{&\nqtag}\openupeq\tabskip=\centering 
  \halign to \displaywidth{$\hfil\displaystyle
   ##\hfil$\tabskip=\centering&\hfill $##$\tabskip=0pt\cr#1\cr}\def\Eq{\eqtag}}
\def\seqn#1{\def\\{\cr}\def\Eq{&\nqtag}\openupeq\tabskip=\centering 
  \halign to \displaywidth{$\displaystyle
   ##\hfil$\tabskip=\centering&\hfill $##$\tabskip=0pt\cr#1\cr}\def\Eq{\eqtag}}
\def\eeqn#1{\def\\{\cr\addtab}\def\Eq{&\nqtag}\openupeq\tabskip=\centering 
  \halign to \displaywidth{$\hfil\displaystyle
   ##$\tabskip=0pt&$\displaystyle##\hfil$\tabskip=\centering
   &##\tabskip=0pt\cr\addtab#1\cr}\def\Eq{\eqtag}}

\def\table#1{\def\\{\cr}\vcenter{\openuptable 
   \halign{$\displaystyle{}##\hfil$&&\qquad$\displaystyle{}##\hfil$\cr#1\cr}}}
\def\ltable#1{\def\\{\cr}\vcenter{\openuptable 
   \halign{##\hfil&&\qquad$\displaystyle{}##\hfil$\cr#1\cr}}}
\def\rtable#1{\def\\{\cr}\vcenter{\openuptable 
   \halign{\hfil##&&\qquad$\displaystyle{}##\hfil$\cr#1\cr}}}
\def\ntable#1{\itemnumber=1 
   \def\\{\cr \the\itemnumber.\global\advance
   \itemnumber by 1 &}\vcenter{\openuptable 
   \halign{##&&\quad $\displaystyle{}##\hfil$\cr 
   \the\itemnumber.\global\advance\itemnumber by 1 &#1\cr}}}
\def\atable#1{\itemnumber=1 
   \def\\{\cr \alphanumber{\the\itemnumber}\/\ro )\global\advance
   \itemnumber by 1 &}\vcenter{\openuptable 
   \halign{##&\enspace##\hfill&&\quad $\displaystyle{}##\hfil$\cr 
   \alphanumber{\the\itemnumber}\/\ro )\global\advance
   \itemnumber by 1 &#1\cr}}}
\def\Rtable#1#2#3#4{\def\\{\cr}\hfill$$\vcenter{\openuptable
   \halign{\hfil\hskip #1pt##&\qquad$\hfil\displaystyle{}##\hfil$
     &\qquad$\displaystyle{}##\hfil$&\qquad##\hfil\hskip#2pt\cr
          \noalign{\hrule}\noalign{\medskip}
          \multispan4\hfil\it #3\hfil\cr\noalign{\medskip}
   \noalign{\hrule}\noalign{\vskip2pt}\noalign{\hrule}
          \noalign{\medskip}#4\cr
          \noalign{\medskip}\noalign{\hrule}}}$$}
\def\datable#1#2{\def\\{\cr\noalign{\vskip2pt}}\vskip1pt
   \noindent\hfil \vbox{\halign{\quad \hfil$##$\hfil\quad 
   \vrr{12}&&\hskip15pt\hfil$##$\quad \hfil\cr
   \noalign{\vskip7pt\hrule\medskip}#1\\
   \noalign{\vskip6pt\hrule\vskip2pt\hrule\medskip}#2\\
   \noalign{\vskip10pt\hrule\vskip7pt}}}}

%%%%%%%%%%%%%%%%% Macros for itemized lists %%%%%%%%%%%%%%%%%%%%%%%%%
%
%   In all cases you use  \\   to separate items; 
%                                 no final \\ is required 
%
%   \items  -   makes a numbered list of items
%   \ritems  -  same as \items, but uses roman numbers
%   \aitems  -  same as \items, but uses letters
%   \bitems  -  same as \items, but uses bullets
%   \zitems  -  same as \items, but no prefix
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

\newdimen\itemstart \itemstart = 0pt%\parindent
\newdimen\itemindent \itemindent = 35pt
\def\hangitem{\hskip0pt\hangindent\itemindent}
\def\itemo{\par\hangitem\textinden}
\def\textinden#1{\hskip\itemstart\llap{#1\enspace}\ignorespaces}

\newcount\ritemnumber \ritemnumber=1
\newcount\aitemnumber \aitemnumber=1
\newcount\nitemnumber \nitemnumber=1

\def\itemadv{\advance\itemnumber by 1}

\def\zitemo{\par\hangitem\textindenz}
\def\zitemno{\hangitem\textindennz}
\def\textindenz#1{\hskip\itemstart\llap{#1\enspace}\ignorespaces}
\def\textindennz#1{#1\enspace}

\def\bitems#1{{\def\\{\par \itemo{$\bullet$}}\\ #1\par}}
\def\zitems#1{{\def\\{\par \itemo{}\enspace}\\ #1\par}}
\def\bitemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\bitems{#3}}}
\def\zitemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\zitems{#3}}}
\def\bnitems#1{\bitems{#1}\noindent\ignorespaces}
\def\znitems#1{\zitems{#1}\noindent\ignorespaces}
\def\bnitemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\bitems{#3}}
      \noindent\ignorespaces}
\def\znitemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\zitems{#3}}
      \noindent\ignorespaces}

\def\aitema{\ro({\sl\alphanumber{\the\aitemnumber}\/}\ro )\enspace
      \advance\aitemnumber by 1\ignorespaces}
\def\ritema{\ro({\it\romannumber{\the\ritemnumber}\/}\ro )\enspace
      \advance\ritemnumber by 1\ignorespaces}
\def\nitema{\ro({\the\nitemnumber}\/\ro )\enspace
      \advance\nitemnumber by 1\ignorespaces}

\def\aitemaq{\quad\aitema}\def\aitemax{\ \aitema}
\def\ritemaq{\quad\ritema}\def\ritemax{\ \ritema}
\def\nitemaq{\quad\nitema}\def\nitemax{\ \nitema}

\def\aitemo{\zitemo{\ro (\sl \alphanumber{\the\aitemnumber}\/\ro )}
      \advance\aitemnumber by 1\ignorespaces}
\def\ritemo{\zitemo{\ro (\it\romannumber{\the\ritemnumber}\/\ro )}
      \advance\ritemnumber by 1\ignorespaces}
\def\nitemo{\zitemo{\ro(\the\nitemnumber\ro)}
      \advance\nitemnumber by 1\ignorespaces}

\def\aitemno{\zitemno{\ro ({\sl \alphanumber{\the\aitemnumber}\/}\ro )}
      \advance\aitemnumber by 1\ignorespaces}
\def\ritemno{\zitemno{\ro ({\it\romannumber{\the\ritemnumber}\/}\ro )}
      \advance\ritemnumber by 1\ignorespaces}
\def\nitemno{\zitemno{\ro(\the\nitemnumber\ro)}
      \advance\nitemnumber by 1\ignorespaces}

\def\itemsz#1{\def\\{\par \nitemo}\\#1\par}
\def\qitemsz#1{\def\\{\par \qquad \nitemo}\\#1\par}
\def\ritemsz#1{\def\\{\par \ritemo}\\#1\par}
\def\rqitemsz#1{\def\\{\par \qquad \ritemo}\\#1\par}
\def\aitemsz#1{\def\\{\par \aitemo}\\#1\par}
\def\aqitemsz#1{\def\\{\par \qquad \aitemo}\\#1\par}


\def\itemnsz#1{\def\\{\par \nitemo}\nitemno#1\par}
\def\ritemnsz#1{\def\\{\par \ritemo}\ritemno#1\par}
\def\aitemnsz#1{\def\\{\par \aitemo}\aitemno#1\par}

\def\itemtz#1{\def\\{ \nitema}\\#1}
\def\ritemtz#1{\def\\{ \ritema}\\#1}
\def\aitemtz#1{\def\\{ \aitema}\\#1}
\def\zitemtz#1{\def\\{}#1}

\def\items#1{\nitemnumber=1\itemsz{#1}}
\def\qitems#1{\nitemnumber=1\qitemsz{#1}}
\def\ritems#1{\ritemnumber=1\ritemsz{#1}}
\def\rqitems#1{\ritemnumber=1\rqitemsz{#1}}
\def\aitems#1{\aitemnumber=1\aitemsz{#1}}
\def\aqitems#1{\aitemnumber=1\aqitemsz{#1}}

\def\nitems#1{\items{#1}\noindent\ignorespaces}
\def\rnitems#1{\ritems{#1}\noindent\ignorespaces}
\def\anitems#1{\aitems{#1}\noindent\ignorespaces}
\def\bnitems#1{\bitems{#1}\noindent\ignorespaces}
\def\znitems#1{\zitems{#1}\noindent\ignorespaces}

%\def\itemns#1{\itemnumber=1\itemnsz{#1}}
%\def\ritemns#1{\ritemnumber=1\ritemnsz{#1}}
%\def\aitemns#1{\aitemnumber=1\aitemnsz{#1}}

\def\itemr#1{\nitemnumber=1\def\\{\par \nitemo}\nitema#1\par}
\def\ritemr#1{\ritemnumber=1\def\\{\par \ritemo}\ritema#1\par}
\def\rqitemr#1{\ritemnumber=1\def\\{\par \qquad \ritemo}\ritema#1\par}
\def\aitemr#1{\aitemnumber=1\def\\{\par \aitemo}\aitema#1\par}
\def\aqitemr#1{\aitemnumber=1\def\\{\par \qquad \aitemo}\aitema#1\par}

\def\itemp#1{\hfill\break\nitemnumber=1\def\\{ \nitema}\nitema#1}
\def\ritemp#1{\hfill\break\ritemnumber=1\def\\{ \ritema}\ritema#1}
\def\aitemp#1{\hfill\break\aitemnumber=1\def\\{ \aitema}\aitema#1}

\def\itemt#1{{\nitemnumber=1\def\\{ \nitema}\nitema#1}}
\def\ritemt#1{{\ritemnumber=1\def\\{ \ritema}\ritema#1}}
\def\aitemt#1{{\aitemnumber=1\def\\{ \aitema}\aitema#1}}
\def\zitemt#1{{\def\\{}#1}}

\def\itemb#1{{\nitemnumber=1\def\\{\hfill\break\nitema}\\#1}}
\def\ritemb#1{{\ritemnumber=1\def\\{\hfill\break\ritema}\\#1}}
\def\aitemb#1{{\aitemnumber=1\def\\{\hfill\break\aitema}\\#1}}
\def\zitemb#1{{\def\\{\hfill\break}\\#1}}

\def\itembt#1{{\nitemnumber=1\def\\{ \nitema}\hfill\break\nitema#1}}
\def\ritembt#1{{\ritemnumber=1\def\\{ \ritema}\hfill\break\ritema#1}}
\def\aitembt#1{{\aitemnumber=1\def\\{ \aitema}\hfill\break\aitema#1}}
\def\zitembt#1{{\def\\{}\hfill\break#1}}

\def\itemx#1{{\nitemnumber=1\def\\{ \nitemax}\nitema#1}}
\def\ritemx#1{{\ritemnumber=1\def\\{ \ritemax}\ritema#1}}
\def\aitemx#1{{\aitemnumber=1\def\\{ \aitemax}\aitema#1}}
\def\zitemx#1{{\def\\{\quad }\\#1}}

\def\itembx#1{{\nitemnumber=1\def\\{ \nitemax}\hfill\break\nitema#1}}
\def\ritembx#1{{\ritemnumber=1\def\\{ \ritemax}\hfill\break\ritema#1}}
\def\aitembx#1{{\aitemnumber=1\def\\{ \aitemax}\hfill\break\aitema#1}}
\def\zitembx#1{{\def\\{\quad }\hfill\break#1}}

\def\itembxg#1#2{{\nitemnumber=1\def\\{ \nitemax}\hfbg{#1}\nitema#2}}
\def\ritembxg#1#2{{\ritemnumber=1\def\\{ \ritemax}\hfbg{#1}\ritema#2}}
\def\aitembxg#1#2{{\aitemnumber=1\def\\{ \aitemax}\hfbg{#1}\aitema#2}}
\def\zitembxg#1#2{{\def\\{\quad }\hfbg{#1}#2}}

\def\itemq#1{{\nitemnumber=1\def\\{ \nitemaq}\nitema#1}}
\def\ritemq#1{{\ritemnumber=1\def\\{ \ritemaq}\ritema#1}}
\def\aitemq#1{{\aitemnumber=1\def\\{ \aitemaq}\aitema#1}}
\def\zitemq#1{{\def\\{\qquad }\\#1}}

\def\itembq#1{{\nitemnumber=1\def\\{ \nitemaq}\hfill\break\nitema#1}}
\def\ritembq#1{{\ritemnumber=1\def\\{ \ritemaq}\hfill\break\ritema#1}}
\def\aitembq#1{{\aitemnumber=1\def\\{ \aitemaq}\hfill\break\aitema#1}}
\def\zitembq#1{{\def\\{\qquad }\hfill\break#1}}

\def\itembqg#1#2{{\nitemnumber=1\def\\{ \nitemaq}\hfbg{#1}\nitema#2}}
\def\ritembqg#1#2{{\ritemnumber=1\def\\{ \ritemaq}\hfbg{#1}\ritema#2}}
\def\aitembqg#1#2{{\aitemnumber=1\def\\{ \aitemaq}\hfbg{#1}\aitema#2}}
\def\zitembqg#1#2{{\def\\{\qquad }\hfbg{#1}#2}}

\def\itemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\items{#3}}}
\def\aitemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\aitems{#3}}}
\def\ritemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\ritems{#3}}}
\def\nitemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\items{#3}}\noindent\ignorespaces}
\def\anitemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\aitems{#3}}\noindent\ignorespaces}
\def\rnitemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\ritems{#3}}\noindent\ignorespaces}

\def\itemns#1{\itemnumber=1\def\itemlab{{\ro(\the\itemnumber\ro)}}% 
\def\\{\par\textinden\itemlab\itemadv\hangitem\hskip0pt}\hangitem
\itemlab\enspace\itemadv#1\par}
\def\ritemns#1{\itemnumber=1\def\itemlab{{\ro (\it
      \romannumber{\the\itemnumber}\/\ro )}}%
\def\\{\par\textinden\itemlab\itemadv\hangitem\hskip0pt}\hangitem
      \itemlab\enspace\itemadv#1\par}
\def\aitemns#1{\itemnumber=1\def\itemlab{{\ro (\sl
      \alphanumber{\the\itemnumber}\/\ro )}}%
\def\\{\par\textinden\itemlab\itemadv\hangitem\hskip0pt}\hangitem
      \itemlab\enspace\itemadv#1\par}
\def\zitemns#1{\def\\{\par\textinden\hangitem\hskip0pt}\hangitem
      \enspace#1\par}


\def\itemn#1#2#3{{\itemstart=#1pt\itemindent = #2pt\itemns{#3}}}
\def\aitemn#1#2#3{{\itemstart=#1pt\itemindent = #2pt\aitemns{#3}}}
\def\ritemn#1#2#3{{\itemstart=#1pt\itemindent = #2pt\ritemns{#3}}}
\def\zitemn#1#2#3{{\itemstart=#1pt\itemindent = #2pt\zitemns{#3}}}

\def\nitemn#1#2#3{{\itemstart=#1pt\itemindent = #2pt
      \itemns{#3}}\noindent\ignorespaces}
\def\anitemn#1#2#3{{\itemstart=#1pt\itemindent = #2pt
      \aitemns{#3}}\noindent\ignorespaces}
\def\rnitemn#1#2#3{{\itemstart=#1pt\itemindent = #2pt
      \ritemns{#3}}\noindent\ignorespaces}

\def\itembi#1#2{{\nitemnumber=1\def\\{\hfbg{#1}\nitema}\\#2}}
\def\ritembi#1#2{{\ritemnumber=1\def\\{\hfbg{#1}\ritema}\\#2}}
\def\aitembi#1#2{{\aitemnumber=1\def\\{\hfbg{#1}\aitema}\\#2}}
\def\zitembi#1#2{{\def\\{\hfbg{#1}}\\#2}}


\def\itemmz#1#2#3{{\itemstart=#1pt\itemindent = #2pt\itemsz{#3}}}
\def\aitemmz#1#2#3{{\itemstart=#1pt\itemindent = #2pt\aitemsz{#3}}}
\def\ritemmz#1#2#3{{\itemstart=#1pt\itemindent = #2pt\ritemsz{#3}}}
\def\nitemmz#1#2#3{{\itemstart=#1pt\itemindent = #2pt\itemsz{#3}}
      \noindent\ignorespaces}
\def\anitemmz#1#2#3{{\itemstart=#1pt\itemindent = #2pt\aitemsz{#3}}
      \noindent\ignorespaces}
\def\rnitemmz#1#2#3{{\itemstart=#1pt\itemindent = #2pt\ritemsz{#3}}
      \noindent\ignorespaces}

%\def\items#1{\itemnumber=1 
%   \def\\{\par \itemo{\ro(\the\itemnumber\ro)}\enspace\advance\itemnumber by 1}\\#1\par}
%\def\ritems#1{\itemnumber=1 \def\\{\par \itemo{\ro (\it  
%   \romannumber{\the\itemnumber}\/\ro )}\enspace\advance\itemnumber by 1}\\#1\par}
%\def\aitems#1{\itemnumber=1  \def\\{\par \itemo{\ro (\sl 
%  \alphanumber{\the\itemnumber}\/\ro )}\enspace\advance\itemnumber by 1}\\#1\par}
%\def\bitems#1{{\def\\{\par \itemo{$\bullet$}}\\ #1\par}}
%\def\zitems#1{{\def\\{\par \itemo{}\enspace}\\ #1\par}}

%\def\aitemz#1{{\def\\{\par \itemo{\ro (\sl 
%  \alphanumber{\the\itemnumber}\/\ro )}\enspace\advance\itemnumber by 1}\\#1\par}}

%%\def\rqitems#1{{\itemnumber=1 \def\\{\par \qquad \itemo{\ro (\it  
%%   \romannumber{\the\itemnumber}\/\ro )}\enspace\advance\itemnumber by 1}\\#1\par}}
%%\def\aqitems#1{\itemnumber=1  \def\\{\par \qquad \itemo{\ro (\sl 
%%  \alphanumber{\the\itemnumber}\/\ro )}\enspace\advance\itemnumber by 1}\\#1\par}
%%\def\zitems#1{\itemnumber=1\def\\{\par \itemo{\ro (\sl 
%%   \alphanumber{\the\itemnumber}\/\ro )}\enspace\advance\itemnumber by 1}{\ro(\sl
%%   \alphanumber{\the\itemnumber}\/\ro )}\enspace\advance\itemnumber by 1#1\par
%%   \noindent\ignorespaces}

%%\def\nitems#1{\itemnumber=1 
%%   \def\\{\par \itemo{\ro(\the\itemnumber\ro)}\enspace\advance\itemnumber by 1}\\#1\par
%%   \noindent\ignorespaces}
%%\def\rnitems#1{\itemnumber=1 \def\\{\par \itemo{\ro (\it  
%%   \romannumber{\the\itemnumber}\/\ro )}\enspace\advance\itemnumber by 1}\\#1\par
%%   \noindent\ignorespaces}
%%\def\anitems#1{\itemnumber=1  \def\\{\par \itemo{\ro (\sl 
%%   \alphanumber{\the\itemnumber}\/\ro )}\enspace\advance\itemnumber by 1}\\#1\par
%%   \noindent\ignorespaces}

%\def\itemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\items{#3}}}
%\def\aitemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\aitems{#3}}}
%\def\ritemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\ritems{#3}}}
%\def\bitemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\bitems{#3}}}
%\def\zitemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\zitems{#3}}}

%\def\nitemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\items{#3}}\noindent\ignorespaces}
%\def\anitemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\aitems{#3}}\noindent\ignorespaces}
%\def\rnitemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\ritems{#3}}\noindent\ignorespaces}
%\def\bnitemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\bitems{#3}}\noindent\ignorespaces}
%\def\znitemm#1#2#3{{\itemstart=#1pt\itemindent = #2pt\zitems{#3}}\noindent\ignorespaces}

%\def\mitems#1{\itemnumber=1  \def\\{$\par \itemo{\ro (\sl 
%   \alphanumber{\the\itemnumber}\/\ro )}\quad \advance\itemnumber by 1$}\par 
%   \itemo{\ro (\sl  \alphanumber{\the\itemnumber}\/\ro )}\quad
%   \advance\itemnumber by 1$#1$\par\noindent\ignorespaces}
%\def\mmitems#1{\itemnumber=1  \def\\{$$\par \itemo{\ro (\sl 
%   \alphanumber{\the\itemnumber}\/\ro )}\quad \advance\itemnumber by 1$$}\par 
%   \itemo{\ro (\sl  \alphanumber{\the\itemnumber}\/\ro )}\quad 
%   \advance\itemnumber by 1$$#1$$\par\noindent\ignorespaces}

\def\taitems#1#2{\bigskip\centerline{\it #1\hglue 1in}\medskip\aitems{#2}\bigskip}
\def\tritems#1#2{\bigskip\centerline{\it #1\hglue 1in}\medskip\ritems{#2}\bigskip}
\def\tanitems#1#2{\bigskip\centerline{\it #1\hglue 1in}\medskip\aitems{#2}\bigskip
      \noindent\ignorespaces}
\def\trnitems#1#2{\bigskip\centerline{\it #1\hglue 1in}\medskip\ritems{#2}\bigskip
      \noindent\ignorespaces}

%%%%%%%%%%%%%%%% Macros for statement numbering %%%%%%%%%%%%%%%%%%%%%
%
%  General format:
%   \Th{label} Text of theorem.\par
%       (Must end with paragraph; use \pari for embedded paragraphs)
%    Note:  label can contain upper & lower case letters and numerals
%  
%   Cite in text by  \th{label}.
%
%     Statement                          Citation
%
%  \Th  =  \Theorem                    \th  =  \theorem
%  \Pr  =  \Proposition                \pr  =  \proposition
%  \Lm  =  \Lemma                      \lm  =  \lemma
%  \Co  =  \Corollary                  \co  =  \corollary
%  \Df  =  \Definition                 \df  =  \definition
%  \Ex  =  \Example                    \ex  =  \example
%  \Ez  =  \Exercise                   \ez  =  \exercise
%  \Cj  =  \Conjecture                 \cj  =  \conjecture
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\declare#1#2{\medbreak\indent{\bf#1.\kern.2em}#2\par\medbreak}
\def\exdeclare#1#2#3\par{\declare{\Statement{#1}{#2}}{#3}}
\def\stdeclare#1#2#3\par{\exdeclare{#1}{#2}{{\sl #3}}\par}

\def\Statement#1#2{\prooflabel{#2}{\numcat\begingroup\aftergroup
\xdef\aftergroup\nuxxx\aftergroup{\stxnum\aftergroup}\endgroup 
\edef\stxxx{#1 \nuxxx}%
\expandafter\ifx\csname t*#2\endcsname\relax\else
\edef\eqxxx{\csname t*#2\endcsname}%
\ifx\stxxx\eqxxx\else\warna{Statement #2 already exists}\fi\fi
\expandafter\xdef\csname t*#2\endcsname{\stxxx}%
\ifeqnwrite\wdef{t*#2}\fi}%
#1\kern.4em\expandafter\eqmode\nuxxx\global\advance\statementnumber by 1}

\def\stcite#1#2{{\numcat\expandafter\ifx\csname t*#2\endcsname\relax
\warn{#1 #2 not defined.}\edef\stxxx{#1~#2\blackbox}\else
\edef\nuxxx{\csname t*#2\endcsname}%
\edef\stxxx{\expandafter\stmode\nuxxx}\fi{\stxxx}}}
\def\stncite#1#2{{\numcat\expandafter\ifx\csname t*#2\endcsname\relax
\warn{#1 #2 not defined.}\edef\stxxx{#2\blackbox}\else
\edef\nuxxx{\csname t*#2\endcsname}%
\edef\stxxx{\expandafter\stnmode\nuxxx}\fi{\stxxx}}}

\def\st{\stcite{Statement}}
\def\stn{\stncite{Statement}}
\def\stx{\blackbox.\blackbox}

\def\Ex#1{\exdeclare{Example}{#1}} \def\ex{\stcite{Example}}
\def\exs#1#2{Examples \stn{#1} and~\stn{#2}} \let\examples\exs
\def\Example{\Ex} \def\example{\ex}
\def\Exc#1#2\par{\par{\declare{\ex{#1}}{({\it continued\/})\enspace #2}}}
\def\Ez#1{\exdeclare{Exercise}{#1}} \def\ez{\stcite{Exercise}}
\def\ezs#1#2{Exercises \stn{#1} and~\stn{#2}}
\def\Th#1{\stdeclare{Theorem}{#1}} \def\th{\stcite{Theorem}}
\def\Theorem{\Th} \def\theorem{\th}
\def\ths#1#2{Theorems \stn{#1} and~\stn{#2}} \let\theorems\ths  
\def\Lm#1{\stdeclare{Lemma}{#1}} \def\lm{\stcite{Lemma}}
\def\Lemma{\Lm} \def\lemma{\lm}
\def\lms#1#2{Lemmas \stn{#1} and~\stn{#2}} \let\lemmas\lms
\def\Pr#1{\stdeclare{Proposition}{#1}} \def\pr{\stcite{Proposition}}
\def\Proposition{\Pr} \def\proposition{\pr}
\def\prs#1#2{Propositions \stn{#1} and~\stn{#2}}
\def\Co#1{\stdeclare{Corollary}{#1}} \def\co{\stcite{Corollary}}
\def\Corollary{\Co} \def\corollary{\co}
\def\cors#1#2{Corollarys \stn{#1} and~\stn{#2}}
\def\Df#1{\exdeclare{Definition}{#1}} \def\df{\stcite{Definition}}
\def\Definition{\Df} \def\definition{\df}
\def\dfs#1#2{Definitions \stn{#1} and~\stn{#2}}
\def\Cj#1{\stdeclare{Conjecture}{#1}} \def\cj{\stcite{Conjecture}}
\def\Conjecture{\Cj} \def\conjecture{\cj}
\def\cjs#1#2{Conjectures \stn{#1} and~\stn{#2}}

\def\Labl#1{{\it#1\/}:\enspace}
\def\Label#1{\medbreak\indent\Labl{#1}}
\def\Labelpar #1.#2\par{\Label{#1}#2\par\medbreak}

\def\Hint{\Labl{Hint}} 
\def\Note{\Labl{Note}}
\def\Important{\Labl{Important}} 
\def\Challenge{\Labl{Challenge}}

\def\Proof{\Label{Proof}} \let\Pf\Proof
\def\Sketch{\Label{Sketch of Proof}}
%\def\Proof{\par\indent{\it Proof.}\par\nobreak}
\def\Proofth#1{\Label{Proof of \th{#1}}}

\def\Remark #1\par{\Labelpar Remark.#1\par} \let \Rmk\Remark
\def\Warning #1\par{\Labelpar Warning.#1\par}
\def\Acknowledgment #1\par{\Labelpar Acknowledgments.#1\par} 
\let\Ack\Acknowledgment

\def\Subtopic#1{\smallskip\is{#1}: \hskip 5pt}
\def\Subtopicn#1{\smallskip\noindent\is{#1}: \hskip 10pt}

\def\qed{\hfill\parfillskip=0pt {\it Q.E.D.}\par\parfillskip=0pt plus 1fil\medbreak}

%%%%%%%%%%%%%%%%%%%% Reference macros %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%   They are listed in alphabetical order in file 
%     jobname.ref (in paper mode)  or  book.ref (in book mode) as
%    \key XXX \type YYY
%    where XXX is the abbreviated name,
%           which can contain upper & lower case letters and numerals
%    \type is either \book  \paper  \preprint \upreprint \inbook  
%        \thesis \uthesis \appx \personal \submit \prepare or \other
%
%  With refmode true, references are printed out
%  with refmode false, the references are numbered, and \refxXXX 
%   becomes a macro for [N],  where N  is the reference number.
%  Use \rf{XXX} to cite it in the text, or \rf{XXX; Y}, which gives
%     [N; Y], so Y  can refer to page numbers, section numbers, etc.
%  Several refences can be put in the same bracket 
%      by using \rf{XXX,YYY,ZZZ}.
%
%    \refsin at the start of the paper reads the references in
%    \References at the end of the paper prints them
%
%    \refnum  at the beginning of the file paper.tex will output  
%         a file called paper.rfn that contains a list of the numbers
%         of all references that are cited
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcount\refnumber \refnumber=1
\newif\ifrefmode \refmodefalse \newif\ifrefs
\newif\ifrefcheck \refchecktrue
 
\def\refsin{{\refmodefalse \input\jobname.ref }\refstrue}
\def\norefsin{\refsfalse}

\def\refstyle{\normalbaselines
\parskip=\refparskip \frenchspacing
\rightskip=0pt plus 4em \spaceskip = .3333em \xspaceskip = .5em
\pretolerance=10000 \tolerance=10000
\hyphenpenalty=10000 \exhyphenpenalty=10000
\interlinepenalty=10000 \raggedbottom}

\newdimen\refskip \refskip = 15pt

\def\References{\msg{References}{\refmodetrue {\center{\bigbf References}\par
    \nobreak} \vskip \refskip{\refstyle \input \jobname.ref }}}

\def\endname{,}
\def\book#1;#2;#3\par{\ifrefmode\namez#1;\endname{\it #2},#3.\par\fi}
\def\paper#1;#2;#3; #4(#5)#6\par{\ifrefmode\namez#1;\endname#2,{\it #3}\if a#4, to appear.\else
     { \bf #4}(#5),#6.\fi\par\fi}
\def\inbook#1;#2;#3;#4\par{\ifrefmode\namez#1;\endname#2, {\sl in\/}:{\it #3},#4.\par\fi}
\def\appx#1;#2;#3;#4\par{\ifrefmode\namez#1;\endname#2, {\sl appendix in\/}:{\it #3},#4.\par \fi}
\def\thesis#1;#2;#3\par{\ifrefmode\namez#1;\endname{\it #2}, Ph.D.~Thesis,#3.\par\fi}
\def\uthesis#1;#2;#3\par{\ifrefmode\namez#1;\endname{\it #2}, Ph.D.~Thesis, University of Minnesota,#3.\par\fi}
\def\preprint#1;#2;#3\par{\ifrefmode\namez#1;\endname#2, preprint,#3.\par\fi}
\def\upreprint#1;#2;#3\par{\ifrefmode\namez#1;\endname#2, preprint, University of Minnesota,#3.\par\fi}
\def\prepare#1;#2;#3\par{\ifrefmode\namez#1;\endname#2, in preparation.\par\fi}
\def\personal#1;#2\par{\ifrefmode\namez#1;\endname\ personal communication,#2.\par\fi}
\def\submit#1;#2;#3\par{\ifrefmode\namez#1;\endname#2, submitted.\par\fi}
\def\other#1\par{\ifrefmode#1.\par\fi}

\def\key#1 {\ifrefmode\ifrefcheck
{\numcat\expandafter\ifx\csname w*#1\endcsname\relax\warnr{Reference #1 not cited in text.}\else\fi}\fi
\hangindent2\parindent \textindent
{\ifrefs\ifproofmode{\fiverm #1} \fi\fi \rf{#1}}\else
{\numcat\expandafter\xdef\csname r*#1\endcsname{\the\refnumber}%
\global\advance\refnumber by 1\ifeqnwrite\wdef{r*#1}\fi}\fi}

\newif\ifnames \namesfalse
\newif\ifflname \flnamefalse
\def\namez#1;{\namesfalse\namezz#1,;} 
\def\namezz#1,#2,#3;{\ifx,#3,\ifnames \ and\fi\fi 
    \ifflname #2 \hskip-1pt\ignorespaces#1\else #1,#2\fi,\ifx,#3,\else\namestrue\namezz #3;\fi}
\def\namezz#1,#2,#3;{\ifx,#3,\ifnames\ifflname \else ,\fi\ and\fi\else\ifnames,\fi\fi
    \ifflname #2 \ignorespaces#1\else #1,#2\fi\ifx,#3,\else\namestrue\namezz #3;\fi}

\def\namew#1;{\expandafter\nameww#1,,;} 
\def\nameww#1,#2,#3#4;{\write5{"#1,#2",\space\rnumberq\pageno}%
\ifx,#3\else\nameww#3#4;\fi}

\newskip\refspaceskip \refspaceskip=.25em
\newskip\refbrakskip \refbrakskip=.5pt
\newskip\refcommaskip \refcommaskip=2pt
\newskip\refminskip \refminskip=1pt

\def\rf#1{{\rm[\hglue\refbrakskip\rfn{#1}\hglue\refbrakskip]}}
\def\rfn#1{\rfvvv#1,\rfvvv}
\def\rfvvv#1,#2\rfvvv{\rfzzz#1;\rfzzz\ifx,#2,\else ,\hglue\refcommaskip\rfvvv#2\rfvvv\fi}
%\def\rfvvv#1,#2\rfvvv{\rfwww#1-\rfwww\ifx,#2,\else ,\hglue\refcommaskip\rfvvv#2\rfvvv\fi}
%\def\rfwww#1-#2\rfwww{\rfzzz#1;\rfzzz\ifx-#2-\else 
%                      \hglue\refminskip--\hglue\refminskip\rfwww#2\rfwww\fi}
\def\rfzzz#1;#2\rfzzz{\def\rfyyy{\eatspace#1}%
{\sfcode`\.=1000\sfcode`\;=1000\sfcode`\:=1000\sfcode`\,=1000%
\spaceskip=\refspaceskip\numcat\edef\rfxxx{\rfyyy}\ifrefs\expandafter\ifx
\csname r*\rfyyy\endcsname\relax\warnr{Reference  [\rfyyy]  not provided.}\else
\ifindex\expandafter\ifx\csname a*\eatspace#1\endcsname\relax
\warnr{Authors for reference  [\rfyyy]  not provided.}\else
\expandafter\namew\csname a*\eatspace#1\endcsname;\fi\fi
\xdef\rfxxx{\csname r*\rfyyy\endcsname}\expandafter\xdef\csname
w*\rfyyy\endcsname{x}\ifrefmode\else
\ifrefnum\immediate\write3{[\rfxxx]}\fi\fi\fi\fi{\reffont \rfxxx}\ifx;#2;\else;\nosemiz#2\fi}}

\def\rfr#1{[\hglue\refbrakskip\rfrn{#1}\hglue\refbrakskip]}
\def\rfrn#1{\rfrrr#1-}
\def\rfrrr#1-#2-{\rfn{#1}\hglue\refminskip--\hglue\refminskip\rfn{#2}}
\let\reffont=\bf

\def\nosemiz#1;{#1}

\newif\ifrefnum \refnumfalse
\def\refnum{\immediate\openout3=\jobname.rfn\refnumtrue}

\newcount\rfnote \rfnote=1
\newcount\rgnote \rgnote=1

\def\orderref{
\def\key##1 ##2\par{{\numcat\expandafter\gdef\csname x*##1\endcsname{##2\par}}}
\def\rfzzz##1;##2\rfzzz{\def\rfyyy{\eatspace##1}%
{\sfcode`\.=1000\sfcode`\;=1000\sfcode`\:=1000\sfcode`\,=1000%
\spaceskip=\refspaceskip\numcat\edef\rfxxx{\rfyyy}\ifrefs\expandafter\ifx
\csname x*\rfyyy\endcsname\relax\warnr{Reference  [\rfyyy]  not provided.}\fi
\expandafter\ifx\csname r*\rfyyy\endcsname\relax
\xdef\rfxxx{\the\rfnote}\expandafter\xdef\csname r*\rfyyy\endcsname{\the\rfnote}%
\expandafter\xdef\csname y*\the\rfnote\endcsname{x*\rfyyy}%
\global\advance\rfnote by 1\else\edef\rfxxx{\csname r*\rfyyy\endcsname}\fi
{\reffont \rfxxx}\ifx;##2;\else,\nosemiz##2\fi\fi}}
\def\refsin{{\refmodetrue \input\jobname.ref }\refstrue}
\def\References{{\refmodetrue {\center{\bigbf References}\par\nobreak} \vskip 15pt
{\refstyle \everypar={\hangindent 3.5em}\parindent=0pt
\def\skx{\quad\phantom{\bf 0}}%
\loop\ifnum\rgnote < \the\rfnote
\ifnum\rgnote = 10 \def\skx{\quad} \fi
[\hglue\refbrakskip{\reffont\the\rgnote}\hglue\refbrakskip]\skx {\numcat\csname\csname y*\the\rgnote\endcsname\endcsname}%.\endgraf
\global\advance\rgnote by 1\repeat}}}
}

%%%%%%%%%%%%%%    Macros for Figures and Pictures    %%%%%%%%%%%%%%%%
%
%   Uses \epsf.tex  macros.
%   Figure "fig" must be in epsf file labelled  fig.ps
%   To insert it use:   
%      \Figure{fig}{Label}  ---  includes  "Figure n"  in text
%      \Fig{fig}{Label}       ---  no extra text
%   To refer to it use:   \figure{fig}  or  \fig{fig}
%
%   Set \figsfalse to disable reading in and drawing figures
%   For Textures version, use \texturesepsf
%   For dvips version, use \dvipsepsf  
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%% Textures version of \epsf.tex.
%% Added \epsfscale to change the size of pictures.
%% The effect of \epsfscale is implemented using \epsfsize,
%% so don't change it.
%%
%% The form \epsfbox[0 0 30 50]{filename.ps} works differently with Textures
%% and dvips because Textures automatically translates the origin to the
%% lower left corner of the picture.
%%
%% Ari Lehtonen, 19 Feb 1992.
%%
%   EPSF.TEX macro file:
%   Written by Tomas Rokicki of Radical Eye Software, 29 Mar 1989.
%   Revised by Don Knuth, 3 Jan 1990.
%   Revised by Tomas Rokicki to accept bounding boxes with no
%      space after the colon, 18 Jul 1990.
%
%   TeX macros to include an Encapsulated PostScript graphic.
%   Works by finding the bounding box comment,
%   calculating the correct scale values, and inserting a vbox
%   of the appropriate size at the current position in the TeX document.
%
%   To use with the center environment of LaTeX, preface the \epsffile
%   call with a \leavevmode.  (LaTeX should probably supply this itself
%   for the center environment.)
%
%   To use, simply say
%   \input epsf           % somewhere early on in your TeX file
%   \epsfbox{filename.ps} % where you want to insert a vbox for a figure
%
%   Alternatively, you can type
%
%   \epsfbox[0 0 30 50]{filename.ps} % to supply your own BB
%
%   which will not read in the file, and will instead use the bounding
%   box you specify.
%
%   The effect will be to typeset the figure as a TeX box, at the
%   point of your \epsfbox command. By default, the graphic will have its
%   `natural' width (namely the width of its bounding box, as described
%   in filename.ps). The TeX box will have depth zero.
%
%   You can enlarge or reduce the figure by saying
%     \epsfxsize=<dimen> \epsfbox{filename.ps}
%   (or
%     \epsfysize=<dimen> \epsfbox{filename.ps})
%   instead. Then the width of the TeX box will be \epsfxsize and its
%   height will be scaled proportionately (or the height will be
%   \epsfysize and its width will be scaled proportiontally).  The
%   width (and height) is restored to zero after each use.
%
%   A more general facility for sizing is available by defining the
%   \epsfsize macro.    Normally you can redefine this macro
%   to do almost anything.  The first parameter is the natural x size of
%   the PostScript graphic, the second parameter is the natural y size
%   of the PostScript graphic.  It must return the xsize to use, or 0 if
%   natural scaling is to be used.  Common uses include:
%
%      \epsfxsize  % just leave the old value alone
%      0pt         % use the natural sizes
%      #1          % use the natural sizes
%      \hsize      % scale to full width
%      0.5#1       % scale to 50% of natural size
%      \ifnum#1>\hsize\hsize\else#1\fi  % smaller of natural, hsize
%% Changed:
%% \epsfxsize should not leave a value; it should compute the 
%% appropriate quantities and assign the values.
%
%  If you want TeX to report the size of the figure (as a message on 
%    your terminal when it processes each figure),
%    say `\epsfverbosetrue'.
%
\newread\epsffilein    % file to \read
\newif\ifepsffileok    % continue looking for the bounding box?
\newif\ifepsfbbfound   % success?
\newif\ifepsfverbose   % report what you're making?
\newdimen\epsfxsize    % horizontal size after scaling
\newdimen\epsfysize    % vertical size after scaling
\newdimen\epsftsize    % horizontal size before scaling
\newdimen\epsfrsize    % vertical size before scaling
\newdimen\epsftmp      % register for arithmetic manipulation
\newdimen\pspoints     % conversion factor
%
\pspoints=1bp          % Adobe points are `big'
\epsfxsize=0pt         % Default value, means `use natural size'
\epsfysize=0pt         % ditto
%
%
\def\texturesepsf{\def\specialstuff##1{illustration ##1 scaled \number\epsfscale}}
\def\dvipsepsf{\def\specialstuff##1{PSfile=##1 llx=\epsfllx\space lly=\epsflly\space
          urx=\epsfurx\space ury=\epsfury\space rwi=\number\epsftmp}}
%
\def\epsfbox#1{\global\def\epsfllx{72}\global\def\epsflly{72}%
   \global\def\epsfurx{540}\global\def\epsfury{720}%
   \def\lbracket{[}\def\testit{#1}\ifx\testit\lbracket
   \let\next=\epsfgetlitbb\else\let\next=\epsfnormal\fi\next{#1}}%
%
\def\epsfgetlitbb#1#2 #3 #4 #5]#6{\epsfgrab #2 #3 #4 #5 .\\%
   \epsfsetgraph{#6}}%
%
\def\epsfnormal#1{\epsfgetbb{#1}\epsfsetgraph{#1}}%
%
\def\epsfgetbb#1{%
%
%   The first thing we need to do is to open the
%   PostScript file, if possible.
%
%\openin\epsffilein=#1
%\ifeof\epsffilein\errmessage{I couldn't open #1, will ignore it}\else
%
%   Okay, we got it. Now we'll scan lines until we find one that 
%   doesn't start with %. We're looking for the bounding box comment.
%
   {\epsffileoktrue \chardef\other=12
    \def\do##1{\catcode`##1=\other}\dospecials \catcode`\ =10
% Read first line without doing anything 
%      (because Mathematica does not start first line with %)
       \read\epsffilein to \epsffileline 
    \loop
       \read\epsffilein to \epsffileline 
       \ifeof\epsffilein\epsffileokfalse\else
%
%   We check to see if the first character is a % sign;
%   if not, we stop reading (unless the line was entirely blank);
%   if so, we look further and stop only if the line begins with
%   `%%BoundingBox:'.
%
        \expandafter\epsfaux\epsffileline:. \\
        \fi
   \ifepsffileok\repeat
   \ifepsfbbfound\else
    \ifepsfverbose\message{No bounding box comment in #1; using defaults}\fi\fi}}%
%
%   Now we have to calculate the scale and offset values to use.
%   First we compute the natural sizes.
%
\def\epsfsetgraph#1{%
   \epsfrsize=\epsfury\pspoints
   \advance\epsfrsize by-\epsflly\pspoints
   \epsftsize=\epsfurx\pspoints
   \advance\epsftsize by-\epsfllx\pspoints
%
%   If `epsfxsize' is 0, we default to the natural size of the 
%   picture. Otherwise we scale the graph to be \epsfxsize wide.
%
%%   \epsfxsize\epsfsize\epsftsize\epsfrsize
   \epsfsize\epsftsize\epsfrsize
   \ifnum\epsfxsize=0 \ifnum\epsfysize=0
      \epsfxsize=\epsftsize \epsfysize=\epsfrsize
%
%   We have a sticky problem here:  TeX doesn't do floating point 
%   arithmetic! Our goal is to compute y = rx/t. The following loop 
%   does this reasonably fast, with an error of at most 
%   about 16 sp (about 1/4000 pt).
% 
     \else\epsftmp=\epsftsize \divide\epsftmp\epsfrsize
       \epsfxsize=\epsfysize \multiply\epsfxsize\epsftmp
       \multiply\epsftmp\epsfrsize \advance\epsftsize-\epsftmp
       \epsftmp=\epsfysize
       \loop \advance\epsftsize\epsftsize \divide\epsftmp 2
       \ifnum\epsftmp>0
          \ifnum\epsftsize<\epsfrsize\else
             \advance\epsftsize-\epsfrsize \advance\epsfxsize\epsftmp \fi
       \repeat
     \fi
   \else\epsftmp=\epsfrsize \divide\epsftmp\epsftsize
     \epsfysize=\epsfxsize \multiply\epsfysize\epsftmp   
     \multiply\epsftmp\epsftsize \advance\epsfrsize-\epsftmp
     \epsftmp=\epsfxsize
     \loop \advance\epsfrsize\epsfrsize \divide\epsftmp 2
     \ifnum\epsftmp>0
        \ifnum\epsfrsize<\epsftsize\else
           \advance\epsfrsize-\epsftsize \advance\epsfysize\epsftmp \fi
     \repeat     
   \fi
%
%  Finally, we make the vbox and stick in a \special that  
%  dvips can parse.The \special command depends on whether
%  \texturesepsf or \dvipsepsf has been set
%
   \ifepsfverbose\message{#1: width=\the\epsfxsize, height=\the\epsfysize}\fi
   \epsftmp=10\epsfxsize \divide\epsftmp\pspoints
   \vbox to\epsfysize{\vfil\hbox to\epsfxsize{%
%%      \special{PSfile=#1 llx=\epsfllx\space lly=\epsflly\space
%%          urx=\epsfurx\space ury=\epsfury\space rwi=\number\epsftmp}%
%%      \hfil}}%
%%      \special{illustration #1 scaled \number\epsfscale}%% Textures
    \special{\specialstuff{#1}}\hfil}}}%
%\epsfxsize=0pt\epsfysize=0pt\epsfscale=1000 }%
%
%   We still need to define the tricky \epsfaux macro. This requires
%   a couple of magic constants for comparison purposes.
%
%%% Old version in double comments.
%%{\catcode`\%=12 \global\let\epsfpercent=%\global\def\epsfbblit{%BoundingBox}}%
%%%
%%%   So we're ready to check for `%BoundingBox:' and to grab the
%%%   values if they are found.
%%%
%%\long\def\epsfaux#1#2:#3\\{\ifx#1\epsfpercent
%%   \def\testit{#2}\ifx\testit\epsfbblit
%%      \epsfgrab #3 . . . \\%
%%      \epsffileokfalse
%%      \global\epsfbbfoundtrue
%%   \fi\else\ifx#1\par\else\epsffileokfalse\fi\fi}%
%%
% New version also checks for %%BoundingBox: (atend) in file
%
{\catcode`\%=12 \global\let\epsfpercent=%\global\def\epsfbblit{%BoundingBox}\global\def\epsfatend{ (atend) :. }}%
%
%   So we're ready to check for `%BoundingBox:' and to grab the
%   values if they are found.
%
\long\def\epsfaux#1#2:#3\\{\ifx#1\epsfpercent
   \def\testit{#2} \def\testiz{#3}
   \ifx\testit\epsfbblit
   \ifx\testiz\epsfatend\else
      \epsfgrab #3 . . . \\%
      \fi\epsffileokfalse
      \global\epsfbbfoundtrue
   \fi\else\ifx#1\par\else\epsffileokfalse\fi\fi}%
%
%   Here we grab the values and stuff them in the appropriate definitions.
%
\def\epsfgrab #1 #2 #3 #4 #5\\{%
   \global\def\epsfllx{#1}\ifx\epsfllx\empty
      \epsfgrab #2 #3 #4 #5 .\\\else
   \global\def\epsflly{#2}%
   \global\def\epsfurx{#3}\global\def\epsfury{#4}\fi}%
%
%   Finally, another definition for compatibility with older macros.
%
\let\epsffile=\epsfbox
%%
%% Introduce scaling factor a la TeX.
%%
\newcount\epsfscale    % computed scaling factor
\newdimen\epsftmpp     % register for arithmetic manipulation
\newdimen\epsftmppp    % register for arithmetic manipulation
\newdimen\epsfM        % scale=1000 means natural size (cf. TeX's \mag)
\newdimen\sppoints     % compute in units of \sppoints
%
%\epsfscale=1000        % default value
\epsfscale = 833  %  Value to offset \magstep1 = 1200, i.e. (1000/1200)*1000 = 833
\sppoints=1000sp       % TeX's unit is a scaled point
\epsfM=1000\sppoints
%
% \computescale{a<dim>}{b<dim>} calculates scaling factor
%   s = c a/b with c=1000 in the following way:  integer division: 
%      a = p b + q  (p=[a/b], [x]=integer part, q=a mod b)
%   c a/b = c p + c q/b
%   c q/b = c/2 2q/b
%   if 2q >= b then c q/b = c/2 (2q-b)/b + c/2 =: c' q'/b + c/2
%                  c a/b = c p + c/2 + c' q'/b
%   repeat until c'/2 < 1
\def\computescale#1#2{%
  \epsftmpp=#1 \epsftmppp=#2
  \epsftmp=\epsftmpp \divide\epsftmp\epsftmppp  % p=[a/b]
  \epsfscale=\epsfM \multiply\epsfscale\epsftmp % s=c p
  \multiply\epsftmp\epsftmppp                   % p b
  \advance\epsftmpp-\epsftmp                    % q=a-p b
  \epsftmp=\epsfM                               % c=1000
  \loop \advance\epsftmpp\epsftmpp              % q'=2q
    \divide\epsftmp 2                           % c'=c/2
    \ifnum\epsftmp>0
      \ifnum\epsftmpp<\epsftmppp\else           % q' >= b  =>
        \advance\epsftmpp-\epsftmppp            %   q'=2q-b
        \advance\epsfscale\epsftmp \fi          %   s=s+[c/2]
  \repeat
  \divide\epsfscale\sppoints}
%
\def\epsfsize#1#2{%
  \ifnum\epsfscale=1000
    \ifnum\epsfxsize=0
      \ifnum\epsfysize=0
      \else \computescale{\epsfysize}{#2}% returns \epsfscale
      \fi
    \else \computescale{\epsfxsize}{#1}% returns \epsfscale
    \fi
  \else
    \epsfxsize=#1
    \divide\epsfxsize by 1000 \multiply\epsfxsize by \epsfscale
  \fi}

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

\newif\iffigs \figstrue
\newif\iffigread

\newdimen\figtopskip
\newdimen\figbotskip
\newdimen\figlabelskip
\newdimen\prefigskip
\newdimen\interfigskip
\newdimen\postfigskip
\newdimen\figboxl
\newdimen\figboxlii \figboxlii = 3in
\newdimen\figboxliii \figboxliii = 1.7in
\newdimen\figlabl
\newdimen\figlablii \figlablii = 2.2in
\newdimen\figlabliii \figlabliii = 1.4in
\newdimen\interfiglablii \interfiglablii = -5pt
\newdimen\interfiglabliii \interfiglabliii = 5pt
\newdimen\prefiglabl \prefiglabl = -10pt
\newdimen\prefigskipi \prefigskipi = -30pt
\newdimen\figunderskip \figunderskip=0pt

\figtopskip=-10pt \figbotskip=5pt \figlabelskip=20pt
\prefigskip= 0pt \interfigskip=-10pt \postfigskip = -1in

\newcount\smallfigscale\smallfigscale=400
\newcount\medfigscale\medfigscale=600
\newcount\figscale\figscale=833

\def\figure{\fg}
\def\fg#1{Figure~\fgcite{#1}}
\def\fgg#1{Figure~\fgciten{#1}}
\def\fgs#1#2{Figures~\fgcite{#1},~\fgcite{#2}}
\def\fgas#1#2{Figures \fgcite{#1} and~\fgcite{#2}}

\def\fgx{Figure~\blackbox}

\def\fgciten#1{{\numcat\expandafter\ifx\csname f*#1\endcsname\relax\edef\eqxxx{#1\blackbox}\else
\edef\nuxxx{\csname f*#1\endcsname}\edef\eqxxx{\expandafter\fgmode\nuxxx}\fi\eqxxx}}

\def\fgcite#1{{\numcat\expandafter\ifx\csname f*#1\endcsname\relax
\warn{Figure #1 not defined.}\edef\eqxxx{#1\blackbox}\else
\edef\nuxxx{\csname f*#1\endcsname}\edef\eqxxx{\expandafter\fgmode\nuxxx}\fi\eqxxx}}

\def\Figread#1{\openin\epsffilein=#1.ps \ifeof\epsffilein
\warnb{Postscript file #1.ps not found}\figreadfalse\else\figreadtrue\fi}

\def\Figname#1{{\numcat
\begingroup\aftergroup\xdef\aftergroup\nuxxx\aftergroup{\fgxnum\aftergroup}\endgroup 
\expandafter\ifx\csname f*#1\endcsname\relax\else
\edef\eqxxx{\csname f*#1\endcsname}%
\ifx\nuxxx\eqxxx\else\warna{Figure #1 already exists}\fi\fi
\expandafter\xdef\csname f*#1\endcsname{\nuxxx}\ifeqnwrite\wdef{f*#1}\fi
\global\advance\figurenumber by 1}}

\def\Figinsert#1#2{{\epsfscale=#1\insert\topins{\penalty100 \splittopskip=0pt \splitmaxdepth=\maxdimen
\floatingpenalty=0\vglue\figtopskip\hglue\prefigskip#2\vglue\figlabelskip}}}

\def\Figlabel#1#2{\center{{\bf \fgg{#1}.} \quad #2.}}
\def\Figprooflabel#1#2{\Figlabel{#1}{#2}\prooflabel{#1}}
\def\Figinsertlabel#1#2#3#4{\Figname{#1}%
\iffigs\Figinsert{#3}{#4\parin\vglue\figbotskip\Figprooflabel{#1}{#2}\pari}\fi}
\def\Fignamelabel#1#2{\Figname{#1} \Figlabel{#1}{#2}}

\def\Figdraw#1#2{\Figname{#1}\Figinsertlabel{\hfil\epsfbox{#1.ps}\hfill}{\Figprooflabel{#1}{#2}}}

\def\Fg#1#2{\Fig{#1}{#2}\fgg{#1}} \let\Figure=\Fg
\def\Fig#1#2{\Figread{#1}\iffigread\Figi{#1}{#2}\closein\epsffilein\fi}
\def\Figi#1#2{\Figinsertlabel{#1}{#2}\figscale{\hfil\epsfbox{#1.ps}\hfill}}

\def\Fgii#1#2#3#4{\Figii{#1}{#2}{#3}{#4}\fgg{#1}}
\def\Figii#1#2#3#4{\Figinsertlabel{#1}{#2}\medfigscale{\Figboxi{#3}\Figboxii{#4}}}

\def\Fgiii#1#2#3#4#5{\Figiii{#1}{#2}{#3}{#4}{#5}\fgg{#1}}
\def\Figiii#1#2#3#4#5{\Figinsertlabel{#1}{#2}\smallfigscale{\hbox to \hsize{\Figboxi{#3}\Figboxii{#4}\Figboxii{#5}}}}
%\def\Figiii#1#2#3#4#5{\Figinsertlabel{#1}{#2}\smallfigscale{\Figboxi{#3}\Figboxii{#4}\Figboxii{#5}}}

\def\Fgiv#1#2#3#4#5#6{\Figiv{#1}{#2}{#3}{#4}{#5}{#6}\fgg{#1}}
\def\Figiv#1#2#3#4#5#6{\Figinsertlabel{#1}{#2}\smallfigscale{\parin\Figboxi{#3}\Figboxii{#4}\parin
\vglue\figbotskip\Figboxi{#5}\Figboxii{#6}\parin}}

\def\Fgvi#1#2#3#4#5#6#7#8{\Figvi{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}\fgg{#1}}
\def\Figvi#1#2#3#4#5#6#7#8{\Figinsertlabel{#1}{#2}\smallfigscale{\parin\Figboxi{#3}\Figboxii{#4}\Figboxii{#5}\parin
\vglue\figbotskip\Figboxi{#6}\Figboxii{#7}\Figboxii{#8}\parin}}

\def\Fgiil#1#2#3#4#5#6{\Figiil{#1}{#2}{#3}{#4}{#5}{#6}\fgg{#1}}
\def\Figiil#1#2#3#4#5#6{{\figboxl=\figboxlii\figlabl=\figlablii\interfigskip=\interfiglablii
\prefigskipi=\prefiglabl\Figinsertlabel{#1}{#2}\medfigscale{\Figboxli{#3}{#5}\Figboxlii{#4}{#6}}}}

\def\Fgiiil#1#2#3#4#5#6#7#8{\Figiiil{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}\fgg{#1}}
\def\Figiiil#1#2#3#4#5#6#7#8{{\figboxl=\figboxliii\figlabl=\figlabliii\interfigskip=\interfiglabliii
\prefigskipi=\prefiglabl
\Figinsertlabel{#1}{#2}\smallfigscale{\Figboxli{#3}{#6}\Figboxlii{#4}{#7}\Figboxlii{#5}{#8}}}}

\def\pofbox#1 #2$#3${\setbox0=\hbox{$#3$}\ht0=0pt\dp0=0pt\wd0=0pt\hskip-#1pt\raise#2pt\box0\hskip#1pt}

%\def\Figbox#1{\Figread{#1}\iffigread\hfil\epsfbox{#1.ps}\hfil\closein\epsffilein\fi} 
%\def\Figboxs#1#2{{\epsfscale=#1\Figbox{#2}}}

\def\Figboxglue#1#2{\hglue#1\Figbox{#2}}
\def\Figboxi#1{\Figboxglue\prefigskipi{#1}} 
\def\Figboxii#1{\Figboxglue\interfigskip{#1}} 

\def\Figboxlglue#1#2#3{\hglue#1\Figboxl{#2}{#3}}
\def\Figboxli#1#2{\Figboxlglue\prefigskipi{#1}{#2}} 
\def\Figboxlii#1#2{\Figboxlglue\interfigskip{#1}{#2}} 

\def\Figbox#1{\Figread{#1}\iffigread\hfil\epsfbox{#1.ps}\hfil\closein\epsffilein\else\blackbox#1\blackbox \fi}
\def\Figboxd#1#2{\Figread{#2}\iffigread\hfil\lower #1pt\hbox{\epsfbox{#2.ps}}\hfil\closein\epsffilein
   \else\blackbox#2\blackbox \fi}
\def\Figboxs#1#2{{\epsfscale=#1\Figbox{#2}}}
\def\Figboxsd#1#2#3{{\epsfscale=#1\Figboxd{#2}{#3}}}
\def\Figboxh#1#2#3{\hskip#1pt\Figbox{#3}\hskip#2pt}
\def\Figo{\Figboxh00}
\def\Figboxhd#1#2#3#4{\hskip#1pt\Figboxd{#3}{#4}\hskip#2pt}
\def\Figboxsh#1#2#3#4{{\epsfscale=#1\Figboxh{#2}{#3}{#4}}}
\def\Figboxshd#1#2#3#4#5{{\epsfscale=#1\Figboxhd{#2}{#3}{#4}{#5}}}

\def\Figboxl#1#2{\Figread{#1}%
\iffigread\hbox to \figboxl{\vbox{\hbox to \figboxl{\hfil\epsfbox{#1.ps}\hfil}\vglue\figunderskip
\hbox to \figlabl{\hfil #2\hfil}}\hfil}\closein\epsffilein\fi} 
\def\Figboxr#1#2{\hbox{\vbox{\vskip #1pt\Figbox{#2}}}}
 
\let\Fx\Figbox \let\Fxs\Figboxs \let\Fxd\Figboxd \let\Fxsd\Figboxsd 
\let\Fxh\Figboxh \let\Fxhd\Figboxhd \let\Fxsh\Figboxsh \let\Fxshd\Figboxshd

\def\Fgx#1#2#3{\Figx{#1}{#2}{#3}\fgg{#1}}
\def\Figx#1#2#3{\Figread{#1}\iffigread\Figxi{#1}{#2}{#3}\closein\epsffilein\else\blackbox#1\blackbox \fi}
\def\Figxi#1#2#3{\Figinsertlabel{#1}{#2}\figscale{\hfil\hbox{\Figbox{#1}#3}\hfill}}

\def\Fgs#1#2#3{{\figscale=#1\Fig{#2}{#3}\fgg{#2}}}
\def\Figs#1#2#3{{\figscale=#1\Figread{#2}\iffigread\Figi{#2}{#3}\closein\epsffilein\fi}}
\def\Fgsx#1#2#3#4{{\figscale=#1\Figx{#2}{#3}{#4}\fgg{#2}}}
\def\Figsx#1#2#3#4{{\figscale=#1\Figread{#2}\iffigread\Figxi{#2}{#3}{#4}\closein\epsffilein\else\blackbox#2\blackbox \fi}}

\def\Fgxii#1#2#3#4#5#6{\Figxii{#1}{#2}{#3}{#4}{#5}{#6}\fgg{#1}}
\def\Figxii#1#2#3#4#5#6{\Figinsertlabel{#1}{#2}\medfigscale{\Figboxxi{#3}{#4}\Figboxxii{#5}{#6}}}
\def\Figboxxglue#1#2{\hglue#1\Figboxx{#2}}
\def\Figboxxi#1#2{\Figboxxglue\prefigskipi{#1}{#2}} 
\def\Figboxxii#1#2{\Figboxxglue\interfigskip{#1}{#2}} 
\def\Figboxx#1#2{\Figread{#1}\iffigread\hfil\hbox{\epsfbox{#1.ps}#2}\hfil\closein\epsffilein\else\blackbox#1\blackbox \fi} 
\def\Figboxxd#1#2#3{\Figread{#2}\iffigread\hfil\lower #1pt\hbox{\epsfbox{#2.ps}#3}\hfil\closein\epsffilein\else\blackbox#1\blackbox \fi} 
\def\Figboxxs#1#2#3{{\epsfscale=#1\Figboxx{#2}{#3}}}
\def\Figboxxsd#1#2#3#4{{\epsfscale=#1\Figboxxd{#2}{#3}{#4}}}

\def\Fgxiil#1#2#3#4#5#6#7#8{\Figxiil{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}\fgg{#1}}
\def\Figxiil#1#2#3#4#5#6#7#8{{{\figboxl=\figboxlii\figlabl=\figlablii\interfigskip=\interfiglablii
\prefigskipi=\prefiglabl\Figinsertlabel{#1}{#2}\medfigscale{\Figboxxli{#3}{#4}{#5}\Figboxxlii{#6}{#7}{#8}}}}}
\def\Figboxxli#1#2#3{\Figboxxlglue\prefigskipi{#1}{#2}{#3}} 
\def\Figboxxlii#1#2#3{\Figboxxlglue\interfigskip{#1}{#2}{#3}} 
\def\Figboxxlglue#1#2#3#4{\hglue#1\Figboxxl{#2}{#3}{#4}}
\def\Figboxxl#1#2#3{\Figread{#1}%
\iffigread\hbox to \figboxl{\vbox{\hbox to \figboxl{\hfil\epsfbox{#1.ps}\hfil#2}\vglue\figunderskip
\hbox to \figlabl{\hfil #3\hfil}}\hfil}\closein\epsffilein\fi}

\def\Fgxiii#1#2#3#4#5#6#7#8{\Figxiii{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}\fgg{#1}}
\def\Figxiii#1#2#3#4#5#6#7#8{\Figinsertlabel{#1}{#2}\smallfigscale{\Figboxxi{#3}{#4}\Figboxxii{#5}{#6}\Figboxxii{#7}{#8}}}
 




%%%%%%%%%%%%%%%%%%%%%   Indexing Macros    %%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  To write out index use:    \indexout
%
%  Subjects are written in file  \jobname.ind
%  Authors are written in file   \jobname.aut
%  Symbols are written in file   \jobname.sym
%  Reference authors are automatically included via file \jobname.rfa
%  which is generated by running program  autref.tex  on  \jobname.ref
%
%\ic{word} places word in subject index; current font is used in text
%\is{word} places word in subject index; emphasis font is used in text
%\ir{word} places word in subject index; roman font is used in text
%\ii{word} places word in subject index; italic font is used in text
%\ix{word} places word in subject index; no text is printed
%\iz{word} prints word using emphasis font; NO index entry added
%
%\ia{name} places name in author index; current font is used in text
%\iu{name} places name in author index; no text is printed
%
%\il{symb} places symb in symbol index; symbol is printed
%\iy{symb} places symb in symbol index; no text is printed
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newif\ifindex \indexfalse
\newif\ifindexproof \indexprooffalse
\newif\ifcontents \contentsfalse

\def\contentsout{\contentstrue\immediate\openout7=\jobname.con}

\def\indexout{\indextrue
\immediate\openout4=\jobname.ind%
\openout5=\jobname.aut%
\openout6=\jobname.sym%
\numcat\input ref.aut\unnumcat}

\def\indexmargin#1{\ifindexproof\margintagleft{#1}\fi}

\def\ix#1{\ifindex\write4{"#1",\space\rnumberq\pageno}\fi\indexmargin{#1}}
\def\iu#1{\ifindex\write5{"#1",\space\rnumberq\pageno}\fi\indexmargin{#1}}
\def\iy#1{\ifindex\write6{$#1$,\space\rnumberq\pageno}\fi\indexmargin{$#1$}}
\def\iz#1{\expandafter\ifx\the\fam0{\it #1\/}\else {\rm #1}\fi}

\def\is#1{\iz{#1}\ix{#1}}
\def\ic#1{#1\ix{#1}}
\def\ia#1{#1\iu{#1}}
\def\il#1{#1\iy{#1}}
\def\ii#1{{\it #1\/}\ix{#1}}
\def\ir#1{{\rm #1}\ix{#1}}

\def\izp#1{\iz{#1}\ro)} \def\izq#1{\ro(\iz{#1}\ro)}
\def\zlp#1{\zl{#1}\ro)} \def\zlq#1{\ro(\zl{#1}\ro)}

%********************************************************************
%*************    TeX Fonts and Special Symbols   *******************
%********************************************************************

%%%%%%%%%%%%%% Adjust subscripts %%%%%%%%%%%%%%%%%%%

\fontdimen16\tensy=2.7pt \fontdimen17\tensy=2.7pt 

\def\supbox#1{\raise 2.7pt \hbox{\sevenrm #1}}
\def\subbox#1{\lower 2.7pt \hbox{\sevenrm #1}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%   Roman, italic, slanted, and bold face font macros
%
%    \iz{#1}  uses emphasis font, which is italic if current font
%                 is roman, and roman otherwise.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\ro#1{{\rm #1}} \def\rbox#1{\hbox{\rm #1}}
\def\roh{\rbox}
\def\roz#1{\  \rbox{#1}\ }
\def\rox#1{\quad \rbox{#1}\quad }
\def\roy#1{\ \ \rbox{#1}\ \ }
\def\roq#1{\qquad \rbox{#1}\qquad }
\def\roqx#1{\qquad \quad \rbox{#1}\quad \qquad }
\def\roqq#1{\qquad \qquad \rbox{#1}\qquad \qquad }

\def\sbox#1{\hbox{\sl #1}}
\def\sox#1{\quad \sbox{#1}\quad }
\def\soq#1{\qquad \sbox{#1}\qquad }
\def\soqx#1{\quad \qquad \sbox{#1}\qquad \quad }
\def\soqq#1{\qquad \qquad \sbox{#1}\qquad \qquad }

\def\rof$$#1$${\roq{#1}}
\def\rofz$$#1$${\roz{#1}}
\def\rofx$$#1$${\rox{#1}}
\def\rofq$$#1$${\roq{#1}}
\def\rofqx$$#1$${\roqx{#1}}
\def\rofqq$$#1$${\roqq{#1}}

\def\ibox#1{\hbox{\it #1}}
\def\bo#1{{\bf #1}}  \def\bbox#1{\hbox{\bf #1}}
\def\zl#1{{\sl #1\/}} 

\font\bigrm=cmr12
\font\Bigrm=cmr10 scaled\magstep2
\font\bigit=cmmi12
\font\Bigit=cmmi10 scaled\magstep2
\font\bigbf=cmbx12
\font\Bigbf=cmbx10 scaled\magstep2

%%%%%%%%%%%%%%%%%%%%%%%%%%%% Small caps %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\font\tensmc=cmcsc10  \def\smc{\tensmc}
\font\ninesmc=cmcsc9

\def\Mathematica{{\smc Mathematica}} \def\Maple{{\smc Maple}} 
\def\Java{{\smc Java}} \def\Matlab{{\smc Matlab}} 
\def\Cpp{C$++$} \def\Fortran{{\smc Fortran}}

%%%%%%%%%%%%%%%%%%%%%%%%% Bold face symbols %%%%%%%%%%%%%%%%%%%%%%%%%

\newfam\ibfam
\font\tenib=cmmib10 \textfont\ibfam=\tenib  
\font\sevenib=cmmib7 \scriptfont\ibfam=\sevenib
\font\fiveib=cmmib5 \scriptscriptfont\ibfam=\fiveib

\def\ibf{\fam\ibfam\relax}
\def\ibo#1{{\ibf#1}}
\edef\ibhx{\hexnumber\ibfam}

\mathchardef\alphaB="0\ibhx0B
\mathchardef\betaB="0\ibhx0C
\mathchardef\gammaB="0\ibhx0D
\mathchardef\deltaB="0\ibhx0E
\mathchardef\epsilonB="0\ibhx0F
\mathchardef\zetaB="0\ibhx10
\mathchardef\etaB="0\ibhx11
\mathchardef\thetaB="0\ibhx12
\mathchardef\iotaB="0\ibhx13
\mathchardef\kappaB="0\ibhx14
\mathchardef\lambdaB="0\ibhx15
\mathchardef\muB="0\ibhx16
\mathchardef\nuB="0\ibhx17
\mathchardef\xiB="0\ibhx18
\mathchardef\piB="0\ibhx19
\mathchardef\rhoB="0\ibhx1A
\mathchardef\sigmaB="0\ibhx1B
\mathchardef\tauB="0\ibhx1C
\mathchardef\upsilonB="0\ibhx1D
\mathchardef\phiB="0\ibhx1E
\mathchardef\chiB="0\ibhx1F
\mathchardef\psiB="0\ibhx20
\mathchardef\omegaB="0\ibhx21
\mathchardef\varepsilonB="0\ibhx22
\mathchardef\varthetaB="0\ibhx23
\mathchardef\varpiB="0\ibhx24
\mathchardef\varrhoB="0\ibhx25
\mathchardef\varsigmaB="0\ibhx26
\mathchardef\varphiB="0\ibhx27
\mathchardef\partialB="0\ibhx40

\mathchardef\GammaBI="0\ibhx00
\mathchardef\DeltaBI="0\ibhx01
\mathchardef\ThetaBI="0\ibhx02
\mathchardef\LambdaBI="0\ibhx03
\mathchardef\XiBI="0\ibhx04
\mathchardef\PiBI="0\ibhx05
\mathchardef\SigmaBI="0\ibhx06
\mathchardef\UpsilonBI="0\ibhx07
\mathchardef\PhiBI="0\ibhx08
\mathchardef\PsiBI="0\ibhx09
\mathchardef\OmegaBI="0\ibhx0A

\def\GammaB{\bo \Gamma }
\def\DeltaB{\bo \Delta }
\def\ThetaB{\bo \Theta }
\def\LambdaB{\bo \Lambda }
\def\XiB{\bo \Xi }
\def\PiB{\bo \Pi }
\def\SigmaB{\bo \Sigma }
\def\UpsilonB{\bo \Upsilon }
\def\PhiB{\bo \Phi }
\def\PsiB{\bo \Psi }
\def\OmegaB{\bo \Omega }

%%%%%%%%%%%%%%%%%%%%%%%%% Euler Fraktur font.%%%%%%%%%%%%%%%%%%%%%%%%

\newfam\euffam
\font\teneuf=eufm10 \textfont\euffam=\teneuf 
\font\seveneuf=eufm7 \scriptfont\euffam=\seveneuf 
%\font\fiveeuf=eufm5 \scriptscriptfont\euffam=\fiveeuf

\def\frak#1{{\fam\euffam\relax#1}}

\def\a{\frak a} \def\g{\frak g} \def\h{\frak h} \def\k{\frak k}

%%%%%%%%%%%%%%%%%%%%%%%%% Capital script letters %%%%%%%%%%%%%%%%%%%%

\def\Cal#1{{\cal#1}}

\def\CA{\Cal A} \def\CB{\Cal B} \def\CC{\Cal C} \def\CD{\Cal D}
\def\CE{\Cal E} \def\CF{\Cal F} 
\def\CG{\Cal G} \def\CH{\Cal H} \def\CI{\Cal I}
\def\CJ{\Cal J} \def\CK{\Cal K} \def\CL{\Cal L}
\def\CM{\Cal M} \def\CN{\Cal N} \def\CO{\Cal O}
\def\CP{\Cal P} \def\CQ{\Cal Q} \def\CR{\Cal R}
\def\CS{\Cal S} \def\CT{\Cal T} \def\CU{\Cal U}
\def\CV{\Cal V} \def\CW{\Cal W} \def\CX{\Cal X} 
\def\CY{\Cal Y} \def\CZ{\Cal Z}

%%%%%%%%%%%%%%%%%%%%%%%   Accented letters   %%%%%%%%%%%%%%%%%%%%%%%%

\def\ctxt#1{{\accent'30 #1}}  % for use if \c is redefined
\chardef\itxt="10             % for use if \i is redefined
\chardef\otxt="1C             % for use if \o is redefined
\chardef\Otxt="1F             % for use if \O is redefined
\def\utxt#1{{\accent"15 #1}}  % for use if \u is redefined
\def\vtxt#1{{\accent"14 #1}}  % for use if \v is redefined

\def\Abar{\,\overline{\!A}{}}\def\Bbar{\,\overline{\!B}{}}
\def\Cbar{\,\overline{\!C}{}}\def\Dbar{\,\overline{\!D}{}}
\def\Ebar{\,\overline{\!E}{}}\def\Fbar{\,\overline{\!F}{}}
\def\Gbar{\kern.1em \overline{\kern-.1em G}{}}
\def\Hbar{\,\overline{\!H}{}}\def\Ibar{\,\overline{\!I}{}}
\def\Jbar{\,\overline{\!J}{}}\def\Kbar{\,\overline{\!K}{}}
\def\Lbar{\,\overline{\!L}{}}\def\Mbar{\,\overline{\!M}{}}
\def\Nbar{\,\overline{\!N}{}}\def\Obar{\,\overline{\!O}{}}
\def\Pbar{\,\overline{\!P}{}}\def\Qbar{\,\overline{\!Q}{}}
\def\Rbar{\,\overline{\!R}{}}\def\Sbar{\,\overline{\!S}{}}
\def\Tbar{\,\overline {\!T}{}}\def\Ubar{\,\overline{\!U}{}}
\def\Vbar{\,\overline{\!V}{}}\def\Wbar{\,\overline{\!W}{}}
\def\Xbar{\,\overline{\!X}{}}\def\Ybar{\,\overline{\!Y}{}}
\def\Zbar{\,\overline{\!Z}{}}

\def\Ahat{\widehat A{}}\def\Bhat{\widehat B{}}\def\Chat{\widehat C{}}
\def\Dhat{\widehat D{}}\def\Ehat{\widehat E{}}\def\Fhat{\widehat F{}}
\def\Ghat{\widehat G{}}\def\Hhat{\widehat H{}}\def\Ihat{\widehat I{}}
\def\Jhat{\widehat J{}}\def\Khat{\widehat K{}}\def\Lhat{\widehat L{}}
\def\Mhat{\widehat M{}}\def\Nhat{\widehat N{}}\def\Ohat{\widehat O{}}
\def\Phat{\widehat P{}}\def\Qhat{\widehat Q{}}\def\Rhat{\widehat R{}}
\def\Shat{\widehat S{}}\def\That{\widehat T{}}\def\Uhat{\widehat U{}}
\def\Vhat{\widehat V{}}\def\What{\widehat W{}}
\def\Xhat{\widehat X{}}\def\Yhat{\widehat Y{}}\def\Zhat{\widehat Z{}}

\def\Atilde{\widetilde A{}}\def\Btilde{\widetilde B{}}\def\Ctilde{\widetilde C{}}
\def\Dtilde{\widetilde D{}}\def\Etilde{\widetilde E{}}\def\Ftilde{\widetilde F{}}
\def\Gtilde{\widetilde G{}}\def\Htilde{\widetilde H{}}\def\Itilde{\widetilde I{}}
\def\Jtilde{\widetilde J{}}\def\Ktilde{\widetilde K{}}\def\Ltilde{\widetilde L{}}
\def\Mtilde{\widetilde M{}}\def\Ntilde{\widetilde N{}}\def\Otilde{\widetilde O{}}
\def\Ptilde{\widetilde P{}}\def\Qtilde{\widetilde Q{}}\def\Rtilde{\widetilde R{}}
\def\Stilde{\widetilde S{}}\def\Ttilde{\widetilde T{}}\def\Utilde{\widetilde U{}}
\def\Vtilde{\widetilde V{}}\def\Wtilde{\widetilde W{}}
\def\Xtilde{\widetilde X{}}\def\Ytilde{\widetilde Y{}}\def\Ztilde{\widetilde Z{}}

\def\hats{
\def\ahat{\hat a{}}\def\bhat{\hat b{}}\def\chat{\hat c{}}
\def\dhat{\hat d{}}\def\ehat{\hat e{}}\def\fhat{\hat f{}}
\def\ghat{\hat g{}}\def\hhat{\hat h{}}
\def\ihat{\hat \imath{}}\def\jhat{\hat \jmath{}}
\def\khat{\hat k{}}\def\lhat{\hat l{}}
\def\mhat{\hat m{}}\def\nhat{\hat n{}}\def\ohat{\hat o{}}
\def\phat{\hat p{}}\def\qhat{\hat q{}}\def\rhat{\hat r{}}
\def\shat{\hat s{}}\def\that{\hat t{}}\def\uhat{\hat u{}}
\def\vhat{\hat v{}}\def\what{\hat w{}}
\def\xhat{\hat x{}}\def\yhat{\hat y{}}\def\zhat{\hat z{}}}

\def\widehats{
\def\ahat{\widehat a{}}\def\bhat{\widehat b{}}\def\chat{\widehat c{}}
\def\dhat{\widehat d{}}\def\ehat{\widehat e{}}\def\fhat{\widehat f{}}
\def\ghat{\widehat g{}}\def\hhat{\widehat h{}}
\def\ihat{\widehat \imath{}}\def\jhat{\widehat \jmath{}}
\def\khat{\widehat k{}}\def\lhat{\widehat l{}}
\def\mhat{\widehat m{}}\def\nhat{\widehat n{}}\def\ohat{\widehat o{}}
\def\phat{\widehat p{}}\def\qhat{\widehat q{}}\def\rhat{\widehat r{}}
\def\shat{\widehat s{}}\def\that{\widehat t{}}\def\uhat{\widehat u{}}
\def\vhat{\widehat v{}}\def\what{\widehat w{}}
\def\xhat{\widehat x{}}\def\yhat{\widehat y{}}\def\zhat{\widehat z{}}}

\def\tildes{
\def\atilde{\tilde a{}}\def\btilde{\tilde b{}}\def\ctilde{\tilde c{}}
\def\dtilde{\tilde d{}}\def\etilde{\tilde e{}}\def\ftilde{\tilde f{}}
\def\gtilde{\tilde g{}}\def\htilde{\tilde h{}}
\def\itilde{\tilde \imath{}}\def\jtilde{\tilde \jmath{}}
\def\ktilde{\tilde k{}}\def\ltilde{\tilde l{}}
\def\mtilde{\tilde m{}}\def\ntilde{\tilde n{}}\def\otilde{\tilde o{}}
\def\ptilde{\tilde p{}}\def\qtilde{\tilde q{}}\def\rtilde{\tilde r{}}
\def\stilde{\tilde s{}}\def\ttilde{\tilde t{}}\def\utilde{\tilde u{}}
\def\vtilde{\tilde v{}}\def\wtilde{\tilde w{}}
\def\xtilde{\tilde x{}}\def\ytilde{\tilde y{}}\def\ztilde{\tilde z{}}}

\def\widetildes{
\def\atilde{\widetilde a{}}\def\btilde{\widetilde b{}}\def\ctilde{\widetilde c{}}
\def\dtilde{\widetilde d{}}\def\etilde{\widetilde e{}}\def\ftilde{\widetilde f{}}
\def\gtilde{\widetilde g{}}\def\htilde{\widetilde h{}}
\def\itilde{\widetilde \imath{}}\def\jtilde{\widetilde \jmath{}}
\def\ktilde{\widetilde k{}}\def\ltilde{\widetilde l{}}
\def\mtilde{\widetilde m{}}\def\ntilde{\widetilde n{}}\def\otilde{\widetilde o{}}
\def\ptilde{\widetilde p{}}\def\qtilde{\widetilde q{}}\def\rtilde{\widetilde r{}}
\def\stilde{\widetilde s{}}\def\ttilde{\widetilde t{}}\def\utilde{\widetilde u{}}
\def\vtilde{\widetilde v{}}\def\wtilde{\widetilde w{}}
\def\xtilde{\widetilde x{}}\def\ytilde{\widetilde y{}}\def\ztilde{\widetilde z{}}}

\def\bars{
\def\abar{\bar a{}}\def\bbar{\bar b{}}\def\cbar{\bar c{}}
\def\dbar{\bar d{}}\def\ebar{\bar e{}}\def\fbar{\bar f{}}
\def\gbar{\bar g{}}\def\hbar{\bar h{}}
\def\ibar{\bar \imath{}}\def\jbar{\bar \jmath{}}
\def\kbar{\bar k{}}\def\lbar{\bar l{}}
\def\mbar{\bar m{}}\def\nbar{\bar n{}}\def\obar{\bar o{}}
\def\pbar{\bar p{}}\def\qbar{\bar q{}}\def\rbar{\bar r{}}
\def\sbar{\bar s{}}\def\tbar{\bar t{}}\def\ubar{\bar u{}}
\def\vbar{\bar v{}}\def\wbar{\mkern2mu \overline{\mkern-2mu w\mkern-1mu }\mkern1mu{}}
\def\xbar{\bar x{}}\def\ybar{\bar y{}}\def\zbar{\bar z{}}}

\def\barr#1{\overline{#1}{}}

\def\barrs{
\def\abar{\barr a{}}\def\bbar{\barr b{}}\def\cbar{\barr c{}}
\def\dbar{\barr d{}}\def\ebar{\barr e{}}\def\fbar{\barr f{}}
\def\gbar{\barr g{}}\def\hbar{\barr h{}}
\def\ibar{\barr \imath{}}\def\jbar{\barr \jmath{}}
\def\kbar{\barr k{}}\def\lbar{\barr l{}}
\def\mbar{\barr m{}}\def\nbar{\barr n{}}\def\obar{\barr o{}}
\def\pbar{\barr p{}}\def\qbar{\barr q{}}\def\rbar{\barr r{}}
\def\sbar{\barr s{}}\def\tbar{\barr t{}}\def\ubar{\barr u{}}
\def\vbar{\barr v{}}\def\wbar{\barr w{}}
\def\xbar{\barr x{}}\def\ybar{\barr y{}}\def\zbar{\barr z{}}}

\def\narrowaccents{\tildes\hats\bars}
\def\wideaccents{\widetildes\widehats\barrs}
\narrowaccents

%%%%%%%%%%%%% Open face fonts for R^n, etc. %%%%%%%%%%%%

\newfam\msbfam 
\font\tenmsb=msbm10 \textfont\msbfam=\tenmsb  
\font\sevenmsb=msbm7 \scriptfont\msbfam=\sevenmsb
%\font\fivemsb=msbm5 \scriptscriptfont\msbfam=\fivemsb

\edef\msbhx{\hexnumber\msbfam}

\def\Bbb#1{{\fam\msbfam\relax#1}}

\def\R{\Bbb R}  \def\C{\Bbb C}
\def\N{\Bbb N}  \def\P{\Bbb P}
\def\Q{\Bbb Q}  \def\Z{\Bbb Z}

\def\Rx{\R\mkern1mu^}  \def\Cx{\C\mkern1mu^}
\def\Nx{\N\mkern1mu^}  \def\Px{\P\mkern1mu^}
\def\Qx{\Q\mkern1mu^}  \def\Zx{\Z\mkern1mu^}
\def\Sx{\Bbb S\mkern1mu^}    

\def\Rn{\Rx n} \def\Rxn{\Rn} \def\Rd{\Rx d} 
\def\Rm{\Rx m} \def\Rp{\Rx p} \def\Rk{\Rx k}
\def\Cn{\Cx n} \def\Cm{\Cx m} \def\Ck{\Cx k}
\def\Sn{\Sx n} \def\Sm{\Sx m}

\def\RP{\Bbb{RP}} \def\RPx{\RP\mkern1mu^}  
\def\CP{\Bbb{CP}} \def\CPx{\CP\mkern1mu^}  
\def\RPn{\RPx n} \def\RPm{\RPx m} 
\def\CPm{\CPx m} \def\CPn{\CPx n}

%%%%%%%%%%%%% Extended symbol font %%%%%%%%%%%%%%%%

\newfam\exsfam
\font\eightex=cmex8 \textfont\exsfam=\eightex 
\font\sixex=cmex7 scaled 857 \scriptfont\exsfam=\sixex 
%\font\fourex=cmex10 scaled 400 \scriptscriptfont\exsfam=\fourex

\edef\exshx{\hexnumber\exsfam}


%%%%%%%%%%%%% msam font %%%%%%%%%%%%%%%%

\newfam\msafam 
\font\tenmsa=msam10 \textfont\msafam=\tenmsa  
\font\sevenmsa=msam7 \scriptfont\msafam=\sevenmsa
\edef\msahx{\hexnumber\msafam}

\mathchardef\square="1\msahx03
\mathchardef\bigdot="1\msahx05

%%%%%%%%%%%%%%%%%%  Additional math symbols  %%%%%%%%%%%%%%%%%

\def\pii{\msk2\pi}

\mathchardef\hbarq="0\msbhx7E   % quantum mechanical  h-bar
\mathchardef\varkappa="0\msbhx7B

\mathchardef\semidirect="2\msbhx6E
\mathchardef\directsemi="2\msbhx6F
\mathchardef\emptyset="0\msbhx3F
\mathchardef\subsetneq="2\msbhx28
\mathchardef\supsetneq="2\msbhx29

\mathchardef\dagmath="0279
\mathchardef\ddagmath="027A
\mathchardef\Smath="0278
\mathchardef\Pmath="027B 
\def\Pback{$\Pmath$}
\mathchardef\Fracx="031E
\mathchardef\Slashx="031F
\def\Frac{\raise 14pt \hbox{$\Fracx$}}
\def\Slash{\raise 14pt \hbox{$\Slashx$}}

\mathchardef\odotsy="220C
\mathchardef\oplussy="2208
\mathchardef\bigvee="1\exshx57
\mathchardef\bigwedge="1\exshx56
\mathchardef\bigcap="1\exshx54
\mathchardef\bigcup="1\exshx53
\mathchardef\bigotimes="1\exshx4E
\mathchardef\bigoplus="1\exshx4C
\mathchardef\bigodot="1\exshx4A
\mathchardef\Bigotimes="1\exshx4F
\mathchardef\Bigoplus="1\exshx4D
\mathchardef\Bigodot="1\exshx4B
\mathchardef\Bigwedge="1\exshx5E

\def\odot{\raise 1pt \hbox{$\,\scriptscriptstyle\odotsy\,$}}
\def\oplus{\raise 1pt \hbox{$\,\scriptscriptstyle\oplussy\,$}}
\def\tensor{\raise 1pt \hbox{$\,\scriptscriptstyle\otimes\,$}}
\def\Union{\bigcup} \def\Intersection{\bigcap}
\def\Wedge{\raise 1pt \hbox{$\bigwedge$}}
\def\Odot{\raise 1pt \hbox{$\bigodot$}}
\def\Oplus{\raise 1pt \hbox{$\bigoplus$}}
\def\Tensor{\raise 1pt \hbox{$\bigotimes$}}
\def\SymmetricProduct{\Bigodot}
\def\DirectSum{\Bigoplus}
\def\TensorProduct{\Bigotimes}
\def\WedgeProduct{\Bigwedge}

\def\cupy{\>\cup\>} \def\capz{\>\cap\>}
\def\cupz{\,\cup\,} \def\capz{\,\cap\,}
\def\cdotz{\!\cdot\!} \def\cdotx{\,\cdot\,}
\def\comp{\raise 1pt \hbox{$\,\scriptstyle\circ\,$}}
\def\degree{^\circ}
\def\upstar{^*}
\def\Upstar{^{\displaystyle *}}
\def\nequiv{\not\equiv}
\def\circstar{\bigcirc\hskip-7.4pt*}

\def\ldotsx{\ \ldots\ } \def\cdotsx{\ \cdots\ }
\def\ldotsz{\ldots\>} \def\cdotsz{\cdots\>}

\def\longrightarrowq{\quad\longrightarrow\quad}  
\def\longrightarrowx{\ \longrightarrow\ } 
\def\longrightarrowz{\,\longrightarrow\,} 
\def\longmapstoq{\quad\longmapsto\quad} 
\def\longmapstox{\ \longmapsto\ } 
\def\longmapstoz{\,\longmapsto\,} 

\def\longrightharpoonup{\relbar\joinrel\rightharpoonup}
\def\weakto{\rightharpoonup}
\def\longweakto{\longrightharpoonup}

\def\simarrow{\setbox0 = \hbox{$\longrightarrow$} 
\setbox2 = \hbox to \wd0{\hfil$\widetilde {\hbox to .8 \wd0{\hfill}}\,$\hfil}
\wd2 = \wd0 \wd0=0pt \; \box0 \lower 1pt \box2\>}

\def\notarrow{\hskip 7pt\not{\hskip-7pt \longrightarrow}}

\def\approxq{\quad  \approx \quad } \def\approxx{\  \approx \ } 
\def\approxz{\,\approx\,}

\def\eqx{\ =\ } \def\leqx{\ \leq\ }\def\geqx{\ \geq\ }
\def\eqy{\>=\>} \def\leqy{\>\leq\>}\def\geqy{\>\geq\>}
\def\eqz{\,=\,} \def\leqz{\,\leq\,}\def\geqz{\,\geq\,}
\def\equivx{\ \equiv \ } \def\simeqx{\ \simeq\ } \def\simx{\ \sim\ }
\def\equivy{\>\equiv \>} \def\simeqy{\>\simeq\>} \def\simy{\>\sim\>}
\def\equivz{\,\equiv \,} \def\simeqz{\,\simeq\,} \def\simz{\,\sim\,}

\def\angles{\hbox{$\,<\kern-7.5pt\raise .7pt\hbox{$\scriptstyle)$}\,$}}
\def\angly{\hskip1pt<\hskip-5pt)\hskip3pt}

\def\interior{\mathbin{\hbox{\hbox{{\vbox
    {\hrule height.4pt width6pt depth0pt}}}\vrule height6pt width.4pt depth0pt}\,}}

\def\Id{\font\elevenrm=cmr10 at 11pt\hbox{\tenrm 1\kern-3.8pt \elevenrm1}} 
\def\id{\ro I}

\def\rddots{\mathinner{\mkern1mu\raise1pt\vbox{\kern7pt\hbox{.}}\mkern2mu
    \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}}

%\def\trademark{\raise 4pt \hbox{\ooalign{\hfil\raise.04ex\hbox{$\scriptscriptstyle \rm T$}\hfil\crcr\scriptstyle\mathhexbox20D}}}
\def\trademark{$^{\bigcirc\hskip-6.5pt \scriptscriptstyle \rm T\hskip1pt}$}

%%%%%%%%%%%%%%%%%%%%%% Nine Point Style  %%%%%%%%%%%%%%%%%%%%%%%%%%%

\newif\ifsmallstyle \smallstylefalse

\def\smallstyle{\smallstyletrue
\font\ninerm=cmr9\font\ninei=cmmi9%
\font\ninett=cmtt9\font\ninesl=cmsl9\font\nineit=cmti9%
\font\ninesy=cmsy9\font\ninebf=cmbx9\font\ninesmc=cmcsc9%
\font\nineeuf=eufm9\font\ninemsb=msbm9\font\sevenex=cmex7% 
\textfont0=\ninerm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm
\textfont1=\ninei \scriptfont1=\seveni \scriptscriptfont1=\fivei
\textfont2=\ninesy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy
\textfont3=\sevenex \scriptfont3=\sevenex \scriptscriptfont3=\sevenex
\def\rm{\fam0\ninerm}%
\def\it{\fam\itfam\nineit}\textfont\itfam=\nineit
\def\sl{\fam\slfam\ninesl}\textfont\slfam=\ninesl
\def\bf{\fam\bffam\ninebf}\textfont\bffam=\ninebf 
\scriptfont\bffam=\sevenbf\scriptscriptfont\bffam=\fivebf
\def\tt{\fam\ttfam\ninett}\textfont\ttfam=\ninett
\textfont\msbfam=\ninemsb\textfont\euffam=\teneuf
\def\smc{\ninesmc}
\setbox\strutbox=\hbox{\vrule height 8pt depth 3pt width 0pt}%
\lineskip=0pt\normalbaselineskip=10pt\parskip=2pt\normalbaselines
\ninerm}

%%%%%%%%%%%%%%%%%%%%%%  Twelve Point Style  %%%%%%%%%%%%%%%%%%%%%%%%

\newif\ifbigstyle \bigstylefalse

\def\bigstyle{\bigstyletrue
\font\bigi=cmmi12\font\bigsy=cmsy10 scaled 1200%
\font\bigsl=cmsl12\font\bigit=cmti12\font\bigtt=cmtt12%
\font\bigex=cmex10\font\ninerm=cmr9\font\ninei=cmmi9%
\font\ninesy=cmsy9\font\ninebf=cmbx9%
\textfont0=\bigrm \scriptfont0=\ninerm \scriptscriptfont0=\sevenrm
\def\rm{\fam0\bigrm}\textfont1=\bigi \scriptfont1=\ninei \scriptscriptfont1=\seveni
\textfont2=\bigsy \scriptfont2=\ninesy \scriptscriptfont2=\sevensy
\textfont3=\bigex \scriptfont3=\eightex \scriptscriptfont3=\sixex
\def\it{\fam\itfam\bigit}\textfont\itfam=\bigit
\def\sl{\fam\slfam\bigsl}\textfont\slfam=\bigsl
\def\bf{\fam\bffam\bigbf}\textfont\bffam=\bigbf
\def\tt{\fam\ttfam\bigtt}\textfont\ttfam=\bigtt
\lineskip=0pt\normalbaselineskip=12pt\parskip=6pt\normalbaselines\bigrm}

%********************************************************************
%******************** Special TeX Macros ****************************
%********************************************************************

\def\dsty{\displaystyle} 

\def\ts#1 {{\tt #1} }
\def\tz#1{{\obeylines \tt #1} }

\def\msk#1{\mskip #1 mu} \def\:{\mskip2mu}

\def\phsub{_{\vphantom x}}

%%%%%%%%%%%%%%%%%%%%  struts and boxes %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\vrr#1{\setbox0 = \hbox{\vrule height 15pt depth #1pt} \ht0 = 9pt \dp0 = 0pt\box0}
\def\vrrr#1#2{\setbox0 = \hbox{\vrule height #1pt depth #2pt} \ht0 = 0pt \dp0 = 0pt \box0}

\def\ostrut#1#2{\hbox{\vrule height #1pt depth #2pt width 0pt}}
\def\frstrut{\ostrut{11}3} \def\Bstrut{\ostrut{11}{4}}
\def\Cstrut{\ostrut{10}{5}} \def\Frstrut{\ostrut{15}{10}}

\def\sstrut{\ostrut0} \def\ustrut#1{\ostrut{#1}0}

\def\bigstrut{\vphantom{\big|}}
\def\Bigstrut{\vphantom{\Big|}}
\def\Biggstrut{\vphantom{\Bigg|}}

\def\ophantom#1#2{\setbox0=\hbox{$#1#2$}\setbox2 = \null
                 \ht2 = \ht0 \dp2 = \dp0 \box2}
\def\odphantom#1{\ophantom\displaystyle{#1}}
\def\otphantom#1{\ophantom\textstyle{#1}}

\def\fbox$#1${\hbox{$#1$}}
\def\mbox#1{\hbox{$\dsty #1$}}

\def\dpbox#1#2{{\setbox0 = \hbox{#2}\dp0=#1pt\box0}}
\def\htbox#1#2{{\setbox0 = \hbox{#2}\ht0=#1pt\box0}}
\def\phbox#1{{\setbox0 = \hbox{#1}\wd0=0pt\box0}}
\def\ovbox#1#2#3{{\setbox0 = \hbox{#3}\ht0=#1pt\dp0=#2pt\box0}}
\def\pvbox{\ovbox00}
\def\ombox#1#2#3{{\setbox0 = \hbox{$\dsty #3$}\ht0=#1pt\dp0=#2pt\box0}}
\def\pmbox{\ombox00}
\def\hmbox#1#2{{\setbox0 = \hbox{$\dsty #2$}\ht0=#1pt\box0}}
\def\dmbox#1#2{{\setbox0 = \hbox{$\dsty #2$}\dp0=#1pt\box0}}
\def\ommbox#1#2#3$${{\setbox0 = \hbox{$\dsty #3$}\ht0=#1pt\dp0=#2pt\box0}$$}
\def\hmmbox#1#2$${{\setbox0 = \hbox{$\dsty #2$}\ht0=#1pt\box0}$$}
\def\dmmbox#1#2$${{\setbox0 = \hbox{$\dsty #2$}\dp0=#1pt\box0}$$}
\def\pmmbox{\ommbox00}
\def\ofbox#1#2$#3${{\setbox0 = \hbox{$#3$}\ht0=#1pt\dp0=#2pt\box0}}
\def\pfbox{\ofbox00}
\def\hfbox#1$#2${{\setbox0 = \hbox{$#2$}\ht0=#1pt\box0}}
\def\dfbox#1$#2${{\setbox0 = \hbox{$#2$}\dp0=#1pt\box0}}

\def\rofz$$#1$${\roz{#1}}
\def\rofq$$#1$${\roq{#1}}
\def\rofqx$$#1$${\roqx{#1}}
\def\rofqq$$#1$${\roqq{#1}}

\def\soqx#1{\quad \qquad \sbox{#1}\qquad \quad }
\def\soqq#1{\qquad \qquad \sbox{#1}\qquad \qquad }


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

\def\ie{i.e., }
\def\eg{e.g., }
\def\cf{cf.~} \def \crf{\cf \rf}
\def\etal{{\it et.~al.}}

\def\upth{\supbox{th} }  \def\upst{\supbox{st} }  \def\upnd{\supbox{nd} }
\def\xth #1{$#1$\upth}  \def\xst #1{$#1$\upst} \def\xnd #1{$#1$\upnd}
\def\xths #1{$(#1)$\upth} \def\xsts #1{$(#1)$\upst} \def\xnds #1{$(#1)$\upnd}

\def\ith{\xth i}  \def\jth{\xth j}  \def\kth{\xth k} \def\lth{\xth l}
\def\mth{\xth m}  \def\nth{\xth n}  \def\rth{\xth r} \def\sth{\xth s} \def\tth{\xth t}
\def\noth{\xsts{n+1}}\def\nmth{\xsts{n-1}}\def\ntth{\xnds{n+2}}
\def\zeroth{zero\upth}

%%%%%%%%%%%%%%%%%%%%%%%%  Fraction macros %%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\f#1{{\textstyle {1\over #1}}}
\def\fr#1#2{{\textstyle {#1\over #2}}}
\def\frac#1#2{{#1\over #2}}
\def\fra#1{{1\over #1}}
\def\frax#1/#2 {\frac{#1}{#2}} \def\frx#1/#2 {\fr{#1}{#2}}
\def\fraz#1/#2{\frac{#1}{#2}} \def\frz#1/#2{\fr{#1}{#2}}
\def\fraco#1#2{{#1/#2}} \def\frao#1{{1/#1}}
\def\fracp#1#2{{#1/(#2)}} \def\frap#1{{1/(#1)}}
\def\fram#1{#1^{-1}} \def\framp#1{(#1)^{-1}}
\def\supfr#1#2{\raise 6pt \hbox{$\scriptstyle #1\above.1pt
     \raise .4pt\hbox{$\scriptstyle #2$}$}}
\def\suppbox#1{\raise 7pt \hbox{$\scriptstyle #1$}}

\def\sfrac#1#2#3#4{{\dsty #3\sstrut{#1}\over \dsty #4\ustrut{#2}}} \def\sfra#1#2#3{\sfrac{#1}{#2}1{#3}}

\def\dfrac{\sfrac28} \def\dfra#1{\dfrac1{#1}}
\def\efrac{\sfrac26} \def\efra#1{\efrac1{#1}}
\def\tfrac{\sfrac24} \def\tfra#1{\tfrac1{#1}}

\def\zfrac#1#2#3#4{{\textstyle {\dsty #3 \sstrut{#1}\over \dsty #4\ustrut{#2}}}} \def\zfra#1#2#3{\zfrac{#1}{#2}1{#3}}
\def\xfrac{\zfrac37} \def\xfra#1{\xfrac1{#1}}
\def\yfrac{\zfrac36} \def\yfra#1{\yfrac1{#1}}

\def\fsqrt#1{\fr1{\sqrt {#1}}} \def\frsqrt#1#2{\fr{\sqrt {#1}}{\sqrt {#2}}} \let\sqfr\frsqrt
\def\fnsqrt#1#2{\fr{{#1}}{\sqrt {#2}}}  \def\fnacsqrt#1#2{\frac{{#1}}{\sqrt{#2}}} 
\def\fdsqrt#1#2{\fr{\sqrt {#1}}{{#2}}}  \def\fdacsqrt#1#2{\frac{\sqrt{#1}}{{#2}}} 
\def\frasqrt#1{\frac1{\sqrt {#1}}} \def\fracsqrt#1#2{\sqrt {\frac{#1}{#2}}} \let\sqfrac\fracsqrt
\def\sqrtfr#1#2{\sqrt {\fr{#1}{#2}}} 

%%%%%%%%%%%%%%%%%%%%%%  Lists and vectors %%%%%%%%%%%%%%%%%%%%%%%%%%

\def\rg#1#2{#1=1,\ldots,#2} \def\rgo#1#2{#1=0,\ldots,#2} 
\def\rgi#1{#1=1,2,\ldots\>} \def\rgoi#1{#1=0,1,2,\ldots\>}
\def\ran#1#2{1\leq  #1 \leq #2} \def\rano#1#2{0\leq  #1 \leq #2}

\def\subs #1#2{#1_1,\ldots,#1_{#2}}
\def\psubs #1#2{(#1_1,\ldots,#1_{#2})}
\def\bsubs #1#2{\{#1_1,\ldots,#1_{#2}\}}
\def\csubs #1#2{[#1_1,\ldots,#1_{#2}]}
\def\ksubs #1#2{[#1_1,\ldots,#1_{#2}]}
\def\sups #1#2{#1^1,\ldots,#1^{#2}}
\def\psups #1#2{(#1^1,\ldots,#1^{#2})}
\def\bsups #1#2{\{#1^1,\ldots,#1^{#2}\}}
\def\csups #1#2{[#1^1,\ldots,#1^{#2}]}
\def\ksups #1#2{[#1^1,\ldots,#1^{#2}]}
\def\supps #1#2{#1^{(1)},\ldots,#1^{(#2)}}
\def\psupps #1#2{(#1^{(1)},\ldots,#1^{(#2)})}
\def\bsupps #1#2{\{#1^{(1)},\ldots,#1^{(#2)}\}}
\def\csupps #1#2{[#1^{(1)},\ldots,#1^{(#2)}]}
\def\ksupps #1#2{[#1^{(1)},\ldots,#1^{(#2)}]}

\def\subos #1#2{#1_0,\ldots,#1_{#2}}
\def\psubos #1#2{(#1_0,\ldots,#1_{#2})}
\def\bsubos #1#2{\{#1_0,\ldots,#1_{#2}\}}
\def\csubos #1#2{[#1_0,\ldots,#1_{#2}]}
\def\ksubos #1#2{[#1_0,\ldots,#1_{#2}]}
\def\supos #1#2{#1^0,\ldots,#1^{#2}}
\def\psupos #1#2{(#1^0,\ldots,#1^{#2})}
\def\bsupos #1#2{\{#1^0,\ldots,#1^{#2}\}}
\def\csupos #1#2{[#1^0,\ldots,#1^{#2}]}
\def\ksupos #1#2{[#1^0,\ldots,#1^{#2}]}
\def\suppos #1#2{#1^{(0)},\ldots,#1^{(#2)}}
\def\psuppos #1#2{(#1^{(0)},\ldots,#1^{(#2)})}
\def\bsuppos #1#2{\{#1^{(0)},\ldots,#1^{(#2)}\}}
\def\csuppos #1#2{[#1^{(0)},\ldots,#1^{(#2)}]}
\def\ksuppos #1#2{[#1^{(0)},\ldots,#1^{(#2)}]}

\def\subf #1#2#3{#1_1(#3),\ldots,#1_{#2}(#3)}
\def\psubf #1#2#3{(#1_1(#3),\ldots,#1_{#2}(#3))}
\def\bsubf #1#2#3{\{#1_1(#3),\ldots,#1_{#2}(#3)\}}
\def\csubf #1#2#3{[#1_1(#3),\ldots,#1_{#2}(#3)]}
\def\ksubf #1#2#3{[#1_1(#3),\ldots,#1_{#2}(#3)]}
\def\supf #1#2#3{#1^1(#3),\ldots,#1^{#2}(#3)}
\def\psupf #1#2#3{(#1^1(#3),\ldots,#1^{#2}(#3))}
\def\bsupf #1#2#3{\{#1^1(#3),\ldots,#1^{#2}(#3)\}}
\def\csupf #1#2#3{[#1^1(#3),\ldots,#1^{#2}(#3)]}
\def\ksupf #1#2#3{[#1^1(#3),\ldots,#1^{#2}(#3)]}
\def\suppf #1#2#3{#1^{(1)}(#3),\ldots,#1^{(#2)}(#3)}
\def\psuppf #1#2#3{(#1^{(1)}(#3),\ldots,#1^{(#2)}(#3))}
\def\bsuppf #1#2#3{\{#1^{(1)}(#3),\ldots,#1^{(#2)}(#3)\}}
\def\csuppf #1#2#3{[#1^{(1)}(#3),\ldots,#1^{(#2)}(#3)]}
\def\ksuppf #1#2#3{[#1^{(1)}(#3),\ldots,#1^{(#2)}(#3)]}

\def\subof #1#2#3{#1_0(#3),\ldots,#1_{#2}(#3)}
\def\psubof #1#2#3{(#1_0(#3),\ldots,#1_{#2}(#3))}
\def\bsubof #1#2#3{\{#1_0(#3),\ldots,#1_{#2}(#3)\}}
\def\csubof #1#2#3{[#1_0(#3),\ldots,#1_{#2}(#3)]}
\def\ksubof #1#2#3{[#1_0(#3),\ldots,#1_{#2}(#3)]}
\def\supof #1#2#3{#1^0(#3),\ldots,#1^{#2}(#3)}
\def\psupof #1#2#3{(#1^0(#3),\ldots,#1^{#2}(#3))}
\def\bsupof #1#2#3{\{#1^0(#3),\ldots,#1^{#2}(#3)\}}
\def\csupof #1#2#3{[#1^0(#3),\ldots,#1^{#2}(#3)]}
\def\ksupof #1#2#3{[#1^0(#3),\ldots,#1^{#2}(#3)]}
\def\suppof #1#2#3{#1^{(0)}(#3),\ldots,#1^{(#2)}(#3)}
\def\psuppof #1#2#3{(#1^{(0)}(#3),\ldots,#1^{(#2)}(#3))}
\def\bsuppof #1#2#3{\{#1^{(0)}(#3),\ldots,#1^{(#2)}(#3)\}}
\def\csuppof #1#2#3{[#1^{(0)}(#3),\ldots,#1^{(#2)}(#3)]}
\def\ksuppof #1#2#3{[#1^{(0)}(#3),\ldots,#1^{(#2)}(#3)]}

\def\subx #1#2{\subf {#1}{#2}x}
\def\psubx #1#2{\psubf {#1}{#2}x}
\def\bsubx #1#2{\bsubf {#1}{#2}x}
\def\csubx #1#2{\csubf {#1}{#2}x}
\def\ksubx #1#2{\ksubf {#1}{#2}x}
\def\supx #1#2{\supf {#1}{#2}x}
\def\psupx #1#2{\psupf {#1}{#2}x}
\def\bsupx #1#2{\bsupf {#1}{#2}x}
\def\csupx #1#2{\csupf {#1}{#2}x}
\def\ksupx #1#2{\ksupf {#1}{#2}x}
\def\suppx #1#2{\suppf {#1}{#2}x}
\def\psuppx #1#2{\psuppf {#1}{#2}x}
\def\bsuppx #1#2{\bsuppf {#1}{#2}x}
\def\csuppx #1#2{\csuppf {#1}{#2}x}
\def\ksuppx #1#2{\ksuppf {#1}{#2}x}

\def\subox #1#2{\subof {#1}{#2}x}
\def\psubox #1#2{\psubof {#1}{#2}x}
\def\bsubox #1#2{\bsubof {#1}{#2}x}
\def\csubox #1#2{\csubof {#1}{#2}x}
\def\ksubox #1#2{\ksubof {#1}{#2}x}
\def\supox #1#2{\supof {#1}{#2}x}
\def\psupox #1#2{\psupof {#1}{#2}x}
\def\bsupox #1#2{\bsupof {#1}{#2}x}
\def\csupox #1#2{\csupof {#1}{#2}x}
\def\ksupox #1#2{\ksupof {#1}{#2}x}
\def\suppox #1#2{\suppof {#1}{#2}x}
\def\psuppox #1#2{\psuppof {#1}{#2}x}
\def\bsuppox #1#2{\bsuppof {#1}{#2}x}
\def\csuppox #1#2{\csuppof {#1}{#2}x}
\def\ksuppox #1#2{\ksuppof {#1}{#2}x}

\def\subt #1#2{\subf {#1}{#2}t}
\def\psubt #1#2{\psubf {#1}{#2}t}
\def\bsubt #1#2{\bsubf {#1}{#2}t}
\def\csubt #1#2{\csubf {#1}{#2}t}
\def\ksubt #1#2{\ksubf {#1}{#2}t}
\def\supt #1#2{\supf {#1}{#2}t}
\def\psupt #1#2{\psupf {#1}{#2}t}
\def\bsupt #1#2{\bsupf {#1}{#2}t}
\def\csupt #1#2{\csupf {#1}{#2}t}
\def\ksupt #1#2{\ksupf {#1}{#2}t}
\def\suppt #1#2{\suppf {#1}{#2}t}
\def\psuppt #1#2{\psuppf {#1}{#2}t}
\def\bsuppt #1#2{\bsuppf {#1}{#2}t}
\def\csuppt #1#2{\csuppf {#1}{#2}t}
\def\ksuppt #1#2{\ksuppf {#1}{#2}t}

\def\subot #1#2{\subof {#1}{#2}t}
\def\psubot #1#2{\psubof {#1}{#2}t}
\def\bsubot #1#2{\bsubof {#1}{#2}t}
\def\csubot #1#2{\csubof {#1}{#2}t}
\def\ksubot #1#2{\ksubof {#1}{#2}t}
\def\supot #1#2{\supof {#1}{#2}t}
\def\psupot #1#2{\psupof {#1}{#2}t}
\def\bsupot #1#2{\bsupof {#1}{#2}t}
\def\csupot #1#2{\csupof {#1}{#2}t}
\def\ksupot #1#2{\ksupof {#1}{#2}t}
\def\suppot #1#2{\suppof {#1}{#2}t}
\def\psuppot #1#2{\psuppof {#1}{#2}t}
\def\bsuppot #1#2{\bsuppof {#1}{#2}t}
\def\csuppot #1#2{\csuppof {#1}{#2}t}
\def\ksuppot #1#2{\ksuppof {#1}{#2}t}

%\def\set#1#2{\mathchoice{\left \{#1\odphantom{#2}\;\right |
%\left .\;#2\odphantom{#1}\right \}}{\{#1\otphantom{#2}\,|\,#2\otphantom{#1} \}}
%{\{#1\otphantom{#2}\,|\,#2\otphantom{#1} \}}
%{\{#1\otphantom{#2}\,|\,#2\otphantom{#1} \}}}
\def\set#1#2{\mathchoice{\left \{\>#1\odphantom{#2}\;\right |
\left .\;#2\odphantom{#1}\>\right \}}{\{\,#1\otphantom{#2}\,|\,#2\otphantom{#1}\,\}}
{\{\,#1\otphantom{#2}\,|\,#2\otphantom{#1}\,\}}
{\{\,#1\otphantom{#2}\,|\,#2\otphantom{#1}\,\}}}
\def\setb#1{\{#1\}} \def\sete{\setb e}
\def\szero{\setb0} \def\mzero{\setminus\szero}

\def\vectors#1#2{{#1\choose#2}}
\def\vectorb#1#2{\pmatrix{#1\cr#2\cr}}
\def\vectorbf#1#2{\pmatrix{\frstrut#1\cr\frstrut#2\cr}}
\def\vectorc#1#2#3{\pmatrix{#1\cr#2\cr#3\cr}}
\def\vectorcf#1#2#3{\pmatrix{\frstrut#1\cr\frstrut#2\cr\frstrut#3\cr}}
\def\vectorB#1#2{\pmatrix{\displaystyle{#1}&\displaystyle{#2}\cr}}
\def\vectorC#1#2#3{\pmatrix{\displaystyle{#1}&\displaystyle{#2}\displaystyle{#3}\cr}}

\def\mx#1{#1\times #1} \def\mxp#1{(#1)\times(#1)} 
\def\Mx#1{\CM_{\mx{#1}}} \def\Mn{\Mx n}
\def\Mat#1#2{\CM_{#1\times #2}} \def\Mmn{\Mat mn}  \def\Mnn{\Mat nn}

\def\matrixs#1#2#3#4{{#1\,#2\choose#3\,#4}}
\def\matrixb#1#2#3#4{\pmatrix{#1&#2\cr#3&#4\cr}}
\def\matrixc#1#2#3#4#5#6#7#8#9{\pmatrix{#1&#2&#3\cr#4&#5&#6\cr#7&#8&#9\cr}}
\def\matrixbf#1#2#3#4{\pmatrix{\frstrut#1&#2\cr\frstrut#3&#4\cr}}
\def\matrixcf#1#2#3#4#5#6#7#8#9{\pmatrix{\frstrut#1&#2&#3\cr\frstrut#4&#5&#6\cr\frstrut#7&#8&#9\cr}}
\def\matrixB#1#2#3#4{\pmatrix{\displaystyle{#1}&\displaystyle{#2}\cr
                              \displaystyle{#3}&\displaystyle{#4}\cr}}
\def\matrixC#1#2#3#4#5#6#7#8#9{\pmatrix
        {\displaystyle{#1}&\displaystyle{#2}&\displaystyle{#3}\cr
         \displaystyle{#4}&\displaystyle{#5}&\displaystyle{#6}\cr
         \displaystyle{#7}&\displaystyle{#8}&\displaystyle{#9}\cr}}
\def\rmatrix#1{\pmatrix{\cos#1&-\sin#1\cr \sin#1&\hphantom-\cos#1\cr}}
\def\rmatrixs#1{{\cos#1\>-\sin#1\choose\sin#1\>\cos#1}}

\def\determinant#1{\det\;\left |\matrix{#1}\right |}
\def\pdetb#1#2#3#4{\left |\matrix{#1&#2\cr#3&#4\cr}\right |}
\def\pdetc#1#2#3#4#5#6#7#8#9{\left |\matrix
                   {#1&#2&#3\cr#4&#5&#6\cr#7&#8&#9\cr}\right |}
\def\detb#1#2#3#4{\det \left |\matrix{#1&#2\cr#3&#4\cr}\right |}
\def\detc#1#2#3#4#5#6#7#8#9{\det \left |\matrix
                   {#1&#2&#3\cr#4&#5&#6\cr#7&#8&#9\cr}\right |}
\def\detB#1#2#3#4{\det \left |\matrix
                   {\displaystyle{#1}&\displaystyle{#2}\cr
                    \displaystyle{#3}&\displaystyle{#4}\cr}\right |}
\def\detC#1#2#3#4#5#6#7#8#9{\det \left |\matrix
          {\displaystyle{#1}&\displaystyle{#2}\displaystyle{#3}\cr
           \displaystyle{#4}&\displaystyle{#5}\displaystyle{#6}\cr
  \displaystyle{#7}&\displaystyle{#8}\displaystyle{#9}\cr}\right |}


\def\omatrix#1{{\def\\{\cr}\pmatrix{#1\cr}}}
\def\omatrixs#1{{\def\\{\cr}\left (\ \matrix{#1\cr}\ \right )}}
\def\omatrixf#1{{\def\\{\cr\frstrut}\pmatrix{\frstrut#1\cr}}}
\def\omatrixsf#1{{\def\\{\cr\frstrut}\left (\ \matrix{#1\cr}\ \right )}}
\def\nmatrix#1{{\def\\{\cr}\matrix{#1\cr}}}
\def\nmatrixf#1{{\def\\{\cr\frstrut}\matrix{\frstrut#1\cr}}}
\def\nmatrixb#1#2#3#4{\matrix{#1&#2\cr#3&#4\cr}}
\def\nmatrixbf#1#2#3#4{\matrix{\frstrut#1&#2\cr\frstrut#3&#4\cr}}
\def\nmatrixc#1#2#3#4#5#6#7#8#9{\matrix{#1&#2&#3\cr#4&#5&#6\cr#7&#8&#9\cr}}
\def\nmatrixcf#1#2#3#4#5#6#7#8#9{\matrix{\frstrut#1&#2&#3\cr\frstrut#4&#5&#6\cr\frstrut#7&#8&#9\cr}}

\def\odet#1{\det \omatrix{#1}}
\def\odets#1{\det \omatrixs{#1}}
\def\odetf#1{\det \omatrixf{#1}}
\def\odetsf#1{\det \omatrixsf{#1}}

\def\nvectorb#1#2{\matrix{#1\cr#2\cr}}
\def\nvectorbf#1#2{\matrix{\frstrut#1\cr\frstrut#2\cr}}
\def\nvectorc#1#2#3{\matrix{#1\cr#2\cr#3\cr}}
\def\nvectorcf#1#2#3{\matrix{\frstrut#1\cr\frstrut#2\cr\frstrut#3\cr}}

\def\amatrix#1#2{\left (\nmatrix{#1}\ \>\right |\,\left .\zvector{#2}\right )}
\def\amatrixf#1#2{\left (\nmatrixf{#1}\ \>\right |\,\left .\zvectorf{#2}\right )}
\def\amatrixb#1 #2 {\left (\nmatrixb#1\ \>\right |\,\left .\nvectorb#2\right )}
\def\amatrixbf#1 #2 {\left (\nmatrixbf#1\ \>\right |\,\left .\nvectorbf#2\right )}
\def\amatrixc#1 #2 {\left (\nmatrixc#1\ \>\right |\,\left .\nvectorc#2\right )}
\def\amatrixcf#1 #2 {\left (\nmatrixcf#1\ \>\right |\,\left .\nvectorcf#2\right )}
\def\bmatrixb#1 #2 {\left (\nmatrixb#1\ \>\right |\,\left .\nmatrixb#2\right )}
\def\bmatrixbf#1 #2 {\left (\nmatrixbf#1\ \>\right |\,\left .\nmatrixbf#2\right )}
\def\bmatrixc#1 #2 {\left (\nmatrixc#1\ \>\right |\,\left .\nmatrixc#2\right )}
\def\bmatrixcf#1 #2 {\left (\nmatrixcf#1\ \>\right |\,\left .\nmatrixcf#2\right )}

\def\amat#1#2{\bigl(\,#1\mid #2\,\bigr)}
\def\diag#1{{\rm diag}\,(#1)}
\def\Diag#1{{\rm diag}\,\left (\,#1\,\right )}

\def\commacr#1{\commacs#1,\commacz}
\def\commacs#1,#2\commacz{{#1}\ifx,#2,\else \\\commacs#2\commacz\fi}
\def\nocommas#1{\nocommaz#1,\nocommae}
\def\nocommaz#1,#2\nocommae{{#1}\ifx,#2,\else\nocomskip\nocommaz#2\nocommae\fi}
\def\nocomskip{\ }
\def\skipcommas#1{\skipcommaz#1,\skipcommae}
\def\skipcommaz#1,#2\skipcommae{{#1}\ifx,#2,\else\nocomskip,\nocomskip\skipcommaz#2\skipcommae\fi}

\def\rvector#1{\left (\,#1\,\right )}
\def\tvector#1{\rvector{#1}^T}
%\def\cvector#1{\xdef\cvectz{\commacr{#1}}\pmatrix\cvectz}
\def\cvector#1{{\def\\{\cr}\xdef\cvectz{\commacr{#1}}\pmatrix\cvectz}}
\def\cvectorf#1{{\def\\{\cr\frstrut}\xdef\cvectz{\commacr{\frstrut#1}}\pmatrix\cvectz}}
\def\zvector#1{{\def\\{\cr}\xdef\cvectz{\commacr{#1}}\matrix\cvectz}}
\def\zvectorf#1{{\def\\{\cr\frstrut}\xdef\cvectz{\commacr{\frstrut#1}}\matrix\cvectz}}
\def\nvector#1{{\def\nocomskip{\ }\left (\,\nocommas{#1}\,\right )}}
\def\Nvector#1{{\def\nocomskip{\quad }\left (\,\nocommas{#1}\,\right )}}
\def\uvector#1{\nvector{#1}^T}
\def\Uvector#1{\Nvector{#1}^T}
\def\Rvector#1{{\def\nocomskip{,\quad }\left (\,\nocommas{#1}\,\right )}}
\def\Tvector#1{\Rvector{#1}^T}
\def\ptvector#1{\pa{#1}^T}
\def\btvector#1{\bpa{#1}^T}
\def\cvectorF#1{{\def\\{\cr\Frstrut}\xdef\cvectz{\commacr{\Frstrut#1}}\pmatrix\cvectz}}

\def\rvectors#1#2{\rvector{#1_1,#1_2,\ldots,#1_{#2}}}
\def\cvectors#1#2{\cvector{#1_1,#1_2,\vdots,#1_{#2}}}
\def\tvectors#1#2{\rvectors{#1}{#2}^T}
\def\zvectors#1#2{\zvector{#1_1,#1_2,\ldots,#1_{#2}}}
\def\nvectors#1#2{\nvector{#1_1,#1_2,\ldots,#1_{#2}}}
\def\uvectors#1#2{\nvectors{#1}{#2}^T}
\def\rvectorso#1#2{\rvector{#1_0,#1_1,\ldots,#1_{#2}}}
\def\cvectorso#1#2{\cvector{#1_0,#1_1,\vdots,#1_{#2}}}
\def\tvectorso#1#2{\rvectorso{#1}{#2}^T}
\def\zvectorso#1#2{\zvector{#1_0,#1_1,\ldots,#1_{#2}}}
\def\nvectorso#1#2{\nvector{#1_0,#1_1,\ldots,#1_{#2}}}
\def\uvectorso#1#2{\nvectorso{#1}{#2}^T}
\def\Nvectors#1#2{\Nvector{#1_1,#1_2,\ldots,#1_{#2}}}
\def\Uvectors#1#2{\Uvector{#1_1,#1_2,\ldots,#1_{#2}}}

\def\nsvector#1{{\def\nocomskip{\ }(\,\nocommas{#1}\,)}}
\def\Nsvector#1{{\def\nocomskip{\quad }(\,\nocommas{#1}\,)}}
\def\usvector#1{\nsvector{#1}^T}
\def\Usvector#1{\Nsvector{#1}^T}


\def\matrixgroup#1#2{\left \{\pmatrix{#1}\;\right | \left.\quad 
          \vphantom{\pmatrix{#1}}#2\;\right \}}
\def\matrixgrouprank#1#2{\left \{\; \rank \pmatrix{#1}\;\right | \left.\quad 
          \vphantom{\pmatrix{#1}}#2\;\right \}}

\def\array#1{\null\,\vcenter{\normalbaselines\mathsurround=0pt
    \ialign{$##$\hfil&&\quad$##$\hfil\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}
      #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,}
\def\Array#1{\null\,\vcenter{\normalbaselines\mathsurround=0pt
    \ialign{$##$\hfil&&\qquad$##$\hfil\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}
      #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,}
\def\parray#1{\left(\array{#1}\right)}
\def\arrayb#1#2#3#4{\array{#1&#2\cr#3&#4\cr}}
\def\arrayc#1#2#3#4#5#6#7#8#9{\array{#1&#2&#3\cr#4&#5&#6\cr#7&#8&#9\cr}}


\def\comm#1#2{[\,{#1},{#2}\,]}

\def\norm#1{|\,{#1}\,|} \let\abs\norm
\def\nnorm#1{ \|\,{#1}\,\|} 
\def\nnnorm#1{ |\mskip-1.5mu\|\,{#1}\,\|\mskip-1.5mu|} 
\def\Norm#1{\left |\,{#1}\,\right |} \let\Abs\Norm
\def\Nnorm#1{\left \|\,{#1}\,\right \|} 

\def\ipsep{;}

\def\ip#1#2{\langle \,{#1}\,\ipsep{#2}\,\rangle}
\def\iq#1#2{\langle {#1}\,,{#2}\rangle}
\def\Ip#1#2{\left \langle \,{#1}\,\ipsep\,{#2}\,\right \rangle}
\def\ipq#1#2{\langle\! \langle\,{#1}\,\ipsep{#2}\,\rangle\!\rangle}
\def\Ipq#1#2{\left \langle\msk{-7.5}\left \langle\,{#1}\,\ipsep{#2}\,\right \rangle\msk{-7.5}\right\rangle}
\def\ipr#1#2{\langle\!\langle\!\langle\,{#1}\,\ipsep{#2}\,\rangle\!\rangle\!\rangle}
\def\ipd#1#2{#1\cdot #2} \def\ippd#1#2{(#1)\cdot #2}
\def\ipdp#1#2{#1\cdot (#2)} \def\ipdpp#1#2{(#1)\cdot (#2)}

\def\com#1#2{\brackets{\:#1,#2\:}} \def\Com#1#2{\Brackets{#1,\ #2}}

\def\parens#1{({#1})}                       \let\pa\parens
\def\parenk#1{(\,{#1}\,)}                   \let\pak\parenk
\def\parenz#1{\bigl({#1}\bigr)}             \let\paz\parenz
\def\bparens#1{\bigl(\,{#1}\,\bigr)}        
                           \let\bpa\bparens \let\pab\bparens
\def\Parens#1{\left (\,{#1}\,\right )}      \let\Pa\Parens
\def\Parenz#1{\left ({#1}\right )}          \let\Paz\Parenz
\def\brackets#1{[\mskip2mu{#1}\mskip 2mu]}  
                             \let\br\brackets \let\bk\brackets
\def\bracketz#1{\bigl [{#1}\bigr ]}         
                             \let\brz\bracketz \let\bkz\bracketz
\def\bbrackets#1{\bigl [\,{#1}\,\bigr ]}    
           \let\brb\bbrackets \let\bbr\bbrackets \let\bbk\bbrackets
\def\Brackets#1{\left [\,{#1}\,\right ]}  
                             \let\Br\Brackets \let\Bk\Brackets
\def\Bracketz#1{\left [{#1}\right ]}  
                             \let\Brz\Bracketz \let\Bkz\Bracketz 
\def\braces#1{\{\mskip2mu{#1}\mskip 2mu\}}  \let\bc\braces
\def\bracez#1{\bigl \{{#1}\bigr \}}         \let\bcz\bracez
\def\bbraces#1{\bigl \{\,{#1}\,\bigr \}}  
                             \let\bcb\bbraces \let\bbc\bbraces
\def\Braces#1{\left \{\,{#1}\,\right \}}    \let\Bc\Braces
\def\Bracez#1{\left \{{#1}\right \}}        \let\Bcz\Bracez

\def\Pastrut{\ostrut{13}3}
\def\Parenw#1{\left ({#1}\right )\hskip-2pt\Pastrut}  \let\Paw\Parenw
\def\Bracketw#1{\left [{#1}\right ]\hskip-2pt\Pastrut} 
   \let\Brw\Bracketw \let\Bkw\Bracketw 
\def\Bracew#1{\left \{{#1}\right \}\hskip-2pt\Pastrut} \let\Bcw\Bracew

\def\pai#1{\pa{#1}^{-1}}
\def\npa#1{#1\,}

%%%%%%%%%%%%%%%%%%%%%   Derivative macros   %%%%%%%%%%%%%%%%%%%%%%%

\def\od#1#2{\mathchoice{d#1 \over d#2}{d#1/d#2}{d#1/d#2}{d#1/d#2}}
\def\odw#1#2{\mathchoice{d^2#1 \over d#2^2}%
{d^2#1 /d#2^2}{d^2#1 /d#2^2}{d^2#1 /d#2^2}}
\def\odt{\odw}
\def\ods#1#2#3{\mathchoice{d^{#3}#1 \over d#2^{#3}}%
{d^{#3}#1/d#2^{#3}}{d^{#3}#1/d#2^{#3}}{d^{#3}#1/d#2^{#3}}}

\def\odo#1{\mathchoice{d \over d#1}{d/d#1}{d/d#1}{d/d#1}}
\def\odow#1{\mathchoice{d^2 \over d#1^2}%
{d^2 /d#1^2}{d^2 /d#1^2}{d^2 /d#1^2}}
\def\odos#1#2{\mathchoice{d^{#2} \over d#1^{#2}}%
{d^{#2}/d#1^{#2}}{d^{#2}/d#1^{#2}}{d^{#2}/d#1^{#2}}}

\def\odx#1{\od{#1}x} \def\odsx#1#2{\ods{#1}{#2}x} \def\odwx#1{\odw{#1}x}
\def\odtx{\odwx}
\def\odox{\odo x} \def\odosx#1{\odos{#1}x} \def\odowx{\odow x}
\def\odz#1{\od{#1}z} \def\odoz{\odo z}

\def\sd#1#2{#1_{#2}} \def\sdt#1#2{#1_{#2#2}} \def\sdw#1#2#3{#1_{#2#3}}
\def\sdo#1{\partial _{#1}} \def\sdot#1{\partial _{#1}^2}

\def\pd#1#2{\mathchoice{\partial #1 \over \partial #2}%
{\partial #1/\partial #2}{\partial #1/\partial #2}{\partial #1/\partial #2}}
\def\pdt#1#2{\mathchoice{\partial^2 #1 \over \partial #2^2}%
{\partial^2 #1/\partial #2^2}{\partial^2 #1/\partial #2^2}{\partial^2 #1/\partial #2^2}}
\def\pdtp#1#2{\mathchoice{\partial^2 #1 \over (\partial #2)^2}%
{\partial^2 #1/(\partial #2)^2}{\partial^2 #1/(\partial #2)^2}{\partial^2 #1/(\partial #2)^2}}
\def\pdw#1#2#3{\mathchoice{\partial^2 #1 \over \partial #2\,\partial #3}%
{\partial^2 #1/\partial #2\,\partial #3}{\partial^2 #1/\partial #2\,\partial #3}%
{\partial^2 #1/\partial #2\,\partial #3}}
\def\pds#1#2#3{\mathchoice{\partial^{#3} #1 \over \partial #2^{#3}}%
{\partial^{#3} #1/\partial #2^{#3}}{\partial^{#3} #1/\partial #2^{#3}}
{\partial^{#3} #1/\partial #2^{#3}}}
\def\pdsp#1#2#3{\mathchoice{\partial^{#3} #1 \over (\partial #2)^{#3}}%
{\partial^{#3} #1/(\partial #2)^{#3}}{\partial^{#3} #1/(\partial #2)^{#3}}
{\partial^{#3} #1/(\partial #2)^{#3}}}

\def\pdz#1#2{\partial _{#2}#1} \def\pdoz#1{\partial _{#1}}
\def\pdtz#1#2{\partial _{#2#2}#1} \def\pdtoz#1{\partial _{#1#1}} 
\def\pdtw#1#2#3{\partial _{#2#3}#1} \def\pdtow#1#2{\partial _{#1#2}} 

\def\pdo#1{\mathchoice{\partial \over \partial #1}%
{\partial /\partial #1}{\partial /\partial #1}{\partial /\partial #1}}
\def\pdot#1{\mathchoice{\partial^2 \over \partial #1^2}%
{\partial^2 /\partial #1^2}{\partial^2 /\partial #1^2}{\partial^2 /\partial #1^2}}
\def\pdotp#1{\mathchoice{\partial^2 \over (\partial #1)^2}%
{\partial^2 /(\partial #1)^2}{\partial^2 /(\partial #1)^2}{\partial^2 /(\partial #1)^2}}
\def\pdow#1#2{\mathchoice{\partial^2 \over \partial #1 \partial #2}%
{\partial^2 /\partial #1 \partial #2}{\partial^2 /\partial #1 \partial #2}%
{\partial^2 /\partial #1 \partial #2}}
\def\pdos#1#2{\mathchoice{\partial^{#2} \over \partial #1^{#2}}%
{\partial^{#2}/\partial #1^{#2}}{\partial^{#2}/\partial #1^{#2}}%
{\partial^{#2}/\partial #1^{#2}}}
\def\pdosp#1#2{\mathchoice{\partial^{#2} \over (\partial #1)^{#2}}%
{\partial^{#2}/(\partial #1)^{#2}}{\partial^{#2}/(\partial #1)^{#2}}%
{\partial^{#2}/(\partial #1)^{#2}}}

\def\pdx#1{\pd{#1}x} \def\pdtx#1{\pdt{#1}x}
\def\pdox{\pdo x} \def\pdotx{\pdot x}

\def\pdxx#1#2{\mathchoice{\partial #1 \over \partial {x^{#2}}}%
{\partial #1/\partial {x^{#2}}}{\partial #1/\partial {x^{#2}}}{\partial #1/\partial {x^{#2}}}}
\def\pdoxx#1{\mathchoice{\partial \over \partial {x^{#1}}}%
{\partial /\partial {x^{#1}}}{\partial /\partial {x^{#1}}}{\partial /\partial {x^{#1}}}}
\def\pdtxx#1#2{\mathchoice{\partial^2 #1 \over (\partial x^{#2})^2}%
{\partial^2 #1/(\partial x^{#2})^2}{\partial^2 #1/(\partial x^{#2})^2}%
{\partial^2 #1/(\partial x^{#2})^2}}
\def\pdwxx#1#2#3{\mathchoice{\partial^2 #1 \over \partial {x^{#2}} \partial {x^{#3}}}%
{\partial^2 #1/\partial {x^{#2}} \partial {x^{#3}}}%
{\partial^2 #1/\partial {x^{#2}} \partial {x^{#3}}}%
{\partial^2 #1/\partial {x^{#2}} \partial {x^{#3}}}}
\def\pdotxx#1{\mathchoice{\partial^2 \over (\partial x^{#1})^2}{\partial^2 /(\partial x^{#1})^2}%
{\partial^2 /(\partial x^{#1})^2}{\partial^2 /(\partial x^{#1})^2}}
\def\pdowxx#1#2{\mathchoice{\partial^2 \over \partial {x^{#1}} \partial {x^{#2}}}%
{\partial^2 /\partial {x^{#1}} \partial {x^{#2}}}{\partial^2 /\partial {x^{#1}} \partial {x^{#2}}}%
{\partial^2 /\partial {x^{#1}} \partial {x^{#2}}}}

\def\jd#1#2{\mathchoice{\partial({#1})\over \partial({#2})}%
{\partial({#1})/\partial({#2})}{\partial({#1})/\partial({#2})}{\partial({#1})/\partial({#2})}}

\def\cvd#1#2{\mathchoice{\partial #1 \over \partial \theta ^{#2}}%
{\partial #1/\partial \theta ^{#2}}{\partial #1/\partial \theta ^{#2}}%
{\partial #1/\partial \theta ^{#2}}}
\def\cvdo#1{\mathchoice{\partial \over \partial \theta ^{#1}}%
{\partial /\partial \theta ^{#1}}{\partial /\partial \theta ^{#1}}%
{\partial /\partial \theta ^{#1}}}

\def\vd#1#2{\mathchoice{\delta #1 \over \delta #2}%
{\delta #1/\delta #2}{\delta #1/\delta #2}{\delta #1/\delta #2}}

\def\d{\, d}
\def\dx{\d x}\def\dy{\d y}\def\dz{\d z}\def\dt{\d t}\def\du{\d u}\def\ds{\d s}\def\dr{\d r}

\def\mstrut{\vphantom{1_x}}
\def\sqrtm#1{\sqrt{\mstrut #1}}
\def\sqsub#1_#2{\sqrt {\smash{#1_{#2}}\mstrut}}
\def\cusub#1_#2{\root 3 \of {\smash{#1_{#2}}\mstrut}}
\def\squx{\sqsub u_{x}}
\def\squxx{\sqsub u_{xx}}
\def\cuuxx{\cusub u_{xx}}

\def\fpsub#1_#2#3#4{{\displaystyle #1_{#2}^{#3/#4}}}
\def\uxf#1#2{\fpsub u_{x}{#1}{#2}}
\def\uxxf#1#2{\fpsub u_{xx}{#1}{#2}}

\def\regbox#1{\setbox1 = \hbox{$#1$}\ht1=9pt\dp1 = 2pt\box1}
\def\regsqrt#1{\hbox{$\textstyle\sqrt{\regbox{#1}}$}}

%%%%%%%%%%%%% Math Operators (set in roman type) %%%%%%%%%%%

\def\operator#1{\expandafter\def\csname#1\endcsname{\mathop{\rm #1}\nolimits}}

\operator{sign} \operator{mod}
\operator{Re} \operator{Im}
\operator{order} \operator{Order} \operator{ord}
\operator{Degree}
\operator{wt} \operator{weight}
\operator{tr} \operator{rank}
\operator{im} \operator{dom} \operator{supp}  
\operator{Span} \operator{dist}
\operator{Ad} \operator{ad}
\operator{Hom} \operator{Ker}
\operator{Div} \operator{Grad} \operator{Curl}
\operator{div} \operator{grad} \operator{curl}
\operator{vol} \operator{Vol}
\operator{sup} \operator{inf}
\operator{ker} \operator{coker} \operator{sgn}
\operator{sech} \operator{csch}
\operator{arcsinh} \operator{arccosh}
\operator{erf} \operator{erfc} \operator{erfi}
\operator{Ai} \operator{Bi}  
\operator{codim}

\def\supp{\ro{supp}\,}
\def\spn{\ro{span}\;}
\def\rng{\ro{ran}\;} \let\ran\rng
\def\corng{\ro{coran}\;} \let\coran\corng
\def\area{\ro{area}\;}

\def\Re{\ro{Re}\;} \def\Im{\ro{Im}\;}
\def\Repa#1{\ro{Re}\,\pa{#1}} \def\Impa#1{\ro{Im}\,\pa{#1}}

\def\Res#1{\mathop{\rm Res}_{#1}\;}
\def\Resa{\Res a} \def\Resaf{\Res af} \def\Resafz{\Res af(z)}
\def\Resz#1{\Res{z\:=\:#1}} \def\Resza{\Resz a} \def\Reszaf{\Resza f} \def\Reszafz{\Resza f(z)}

%%%%%%%%%%%%%% Lie groups and algebras %%%%%%%%%%%%%%%

\def\GL#1{\ro{GL}(#1)} \def\gl#1{\frak{gl}(#1)}
\def\GLR#1{\GL{#1,\R}} \def\glR#1{\gl{#1,\R}}
\def\GLC#1{\GL{#1,\C}} \def\glC#1{\gl{#1,\C}}
\def\SL#1{\ro{SL}(#1)} \def\sL#1{\frak{sl}(#1)}
\def\SLR#1{\SL{#1,\R}} \def\slR#1{\sL{#1,\R}} \def\sLR#1{\sL{#1,\R}}
\def\SLC#1{\SL{#1,\C}} \def\slC#1{\sL{#1,\C}} \def\sLC#1{\sL{#1,\C}}
\def\PSL#1{\ro{PSL}(#1)} \def\PSLR#1{\PSL{#1,\R}} \def\PSLC#1{\SL{#1,\C}}
\def\SO#1{\ro{SO}(#1)} \def\so#1{\frak{so}(#1)}
\def\SU#1{\ro{SU}(#1)} \def\su#1{\frak{su}(#1)}
\def\GO#1{\ro{O}(#1)}  \def\go#1{\frak{o}(#1)} \def\OG#1{\ro{O}(#1)} \def\og#1{\frak{o}(#1)}
\def\AG#1{\ro{A}(#1)}  \def\ag#1{\frak{a}(#1)}
\def\AGR#1{\AG{#1,\R}} \def\agR#1{\ag{#1,\R}}
\def\SA#1{\ro{SA}(#1)} \def\sa#1{\frak{sa}(#1)}
\def\SAR#1{\SA{#1,\R}} \def\saR#1{\sa{#1,\R}}
\def\EU#1{\ro{E}(#1)}  \def\eu#1{\frak{e}(#1)}
\def\SE#1{\ro{SE}(#1)} \def\se#1{\frak{se}(#1)}

\def\GLn{\GL n} \def\gln{\gl n}
\def\GLnR{\GL{n,\R}} \def\glnR{\gl{n,\R}}
\def\GLRn{\GL{n,\R}} \def\glRn{\gl{n,\R}}
\def\GLnC{\GL{n,\C}} \def\glnC{\gl{n,\C}}
\def\GLCn{\GL{n,\C}} \def\glCn{\gl{n,\C}}
\def\SLn{\SL n} \def\sln{\sL n} \def\sLn{\sL n}  
\def\SLnR{\SL{n,\R}} \def\slnR{\sL{n,\R}}  \def\sLnR{\sL{n,\R}}
\def\SLRn{\SL{n,\R}} \def\slRn{\sL{n,\R}}  \def\sLRn{\sL{n,\R}}
\def\SLnC{\SL{n,\C}} \def\slnC{\sL{n,\C}}  \def\sLnC{\sL{n,\C}}
\def\SLCn{\SL{n,\C}} \def\slCn{\sL{n,\C}}  \def\sLCn{\sL{n,\C}}
\def\PSLn{\ro{PSL}(n)} \def\PSLnR{\PSL{n,\R}} \def\PSLnC{\PSL{n,\C}} 
\def\PSLRn{\PSL{n,\R}} \def\PSLCn{\PSL{n,\C}} 
\def\SOn{\ro{SO}(n)} \def\son{\frak{so}(n)}
\def\SUn{\ro{SU}(n)} \def\sun{\frak{su}(n)}
\def\On{\ro{O}(n)}   \def\on{\frak{o}(n)}  \def\GOn{\ro{O}(n)}  \def\gon{\frak{o}(n)}
\def\An{\ro{A}(n)}   \def\an{\frak{a}(n)}  \def\AGn{\ro{A}(n)}  \def\agn{\frak{a}(n)}
\def\AnR{\AG{n,\R}}  \def\anR{\ag{n,\R}}   \def\ARn{\AG{n,\R}}  \def\aRn{\ag{n,\R}}    
\def\AGnR{\AG{n,\R}} \def\agnR{\ag{n,\R}}  \def\AGRn{\AG{n,\R}} \def\agRn{\ag{n,\R}}
\def\SAn{\SA n} \def\san{\sa n} \def\SAnR{\SA{n,\R}} \def\sanR{\sa{n,\R}}
\def\SARn{\SA{n,\R}} \def\saRn{\sa{n,\R}}
\def\En{\EU n}  \def\en{\eu n}  \def\EUn{\EU n}  \def\eun{\eu n}
\def\SEn{\SE n} \def\sen{\se n}
\def\PGL#1#2{\ro{GL}\ps{#1}(#2)}
\def\PGLn#1{\PGL n{#1}}\def\PGLnp{\PGL np}

%%%%%%%%%%%%%%%%%% Integrals, sums, products %%%%%%%%%%%%%%%%%%%%%%%

\mathchardef\intsign="1352 \def\int{\intsign\nolimits}

\def\Intn#1#2 {\int\>{#2}\d{#1}}
\def\intnx{\Intn x} \def\intny{\Intn y} \def\intnz{\Intn z} 
\def\Into#1#2{\int _{#1}^{#2}} %\def\Into#1#2{\int _{#1}^{#2}\>} 
\def\Intu#1{\int _{#1}\>} \def\Intz#1 {\int\>{#1}} 
%\def\Intm#1#2 {\int\>{#2}\d{#1}} \def\intmx{\Intm x} 
%\def\intmy{\Intm y} \def\intmz{\Intm z} 

\def\Int#1#2#3#4 {\Into{#2}{#3}{#4}\d{#1}} 
\def\Intb#1#2#3#4 {\Into{#2}{#3}\bbr{#4}\d{#1}} 
\def\IntB#1#2#3#4 {\Into{#2}{#3}\Br{#4}\d{#1}} 
\def\Intso#1{\Into{-\:#1}{#1}} 
\def\Ints#1#2#3 {\Intso{#2}{#3}\d{#1}} \def\intsx{\Ints x} 
\def\Intsb#1#2#3 {\Intso{#2}\bbr{#3}\d{#1}}

\def\intt{\Int t} \def\intbt{\Intb t} 
\def\intx{\Int x} \def\intbx{\Intb x} 
\def\inty{\Int y} \def\intby{\Intb y} 
\def\intz{\Int z} \def\intbz{\Intb z} 
\def\intab{\Into ab} \def\intabt{\intt ab}
\def\intabx{\intx ab} \def\intbabx{\intbx ab} \def\intaby{\inty ab} \def\intabz{\intz ab} \def\intabxi{\Int\xi ab}

\def\intoi{\Into{0}{\infty }} \def\intoii{\Into{-\infty }{\infty }}
\def\intop{\Into{0}{\pi }} \def\intopp{\Into{-\pi }{\pi }} \def\intotp{\Into{0}{2\pii }} 
\def\inti#1{\Int{#1}{0}{\infty }} \def\intii#1{\Int{#1}{-\infty }{\infty }}
\def\intp#1{\Int{#1}{0}{\pi }} \def\intpp#1{\Int{#1}{-\pi }{\pi }} 
\def\inttp#1{\Int{#1}{0}{2\pii }}
\def\intbpp#1{\Intb{#1}{-\pi }{\pi }} \def\intBpp#1{\IntB{#1}{-\pi }{\pi }}

\def\intix{\inti x} \def\intik{\inti k} \def\intit{\inti t} 
\def\intiix{\intii x} \def\intiiy{\intii y} \def\intiik{\intii k}\def\intiiz{\intii z}
\def\intiit{\intii t} 
\def\intpx{\intp x} \def\intppx{\intpp x} \def\inttpx{\inttp x}
\def\intpy{\intp y} \def\intppy{\intpp y} \def\inttpy{\inttp y}
\def\intpq{\intp \theta } \def\intppq{\intpp \theta } \def\inttpq{\inttp \theta }
 
\def\dintn{\int\hskip-4pt\int \>} 
\def\dintu#1{\int\hskip-4pt\int _{#1} \>}
\def\dintO{\dintu\Omega} \def\dintD{\dintu D} \def\dintS{\dintu S} 
\def\dint#1#2#3#4 {\dintu{#3}{#4}\d{#1}\d{#2}}
\def\dintm#1#2#3 {\dintn{#3}\d{#1}\d{#2}}
\def\intmxy#1 {\dintm xy{#1} }
\def\intbxy#1#2 {\dint xy{#1}{\bbr{#2}} }
\def\intxy#1#2 {\dint xy{#1}{#2} }
\def\intOxy{\intxy\Omega} \def\intDxy{\intxy D} 
\def\intObxy#1 {\intOxy\bbr{#1} }
\def\intpxy#1#2 {\dint xy{#1}{\pa{#2}} } 
\def\intOpxy#1 {\intOxy\pa{#1} }

\def\tintn{\int\hskip-4pt\int\hskip-4pt\int \>} 
\def\tintu#1{\int\hskip-4pt\int\hskip-4pt\int _{#1} \>}
\def\tintO{\tintu\Omega} \def\tintD{\tintu D}
\def\tint#1#2#3#4#5 {\tintu{#4}{#5}\d{#1}\d{#2}\d{#3}}
\def\tintm#1#2#3#4 {\tintn{#4}\d{#1}\d{#2}\d{#3}}
\def\intxyz#1#2 {\tint xyz{#1}{#2} } \def\intmxyz#1 {\tintm xyz{#1} }
\def\intOxyz{\intxyz\Omega} \def\intObxyz#1 {\intOxyz\bbr{#1} }

\def\intC{\Intu C} \def\intO{\Intu\Omega} 
\def\intOx#1 {\intO {#1}\d\x}
\def\ints#1#2 {\Intu{#1}{#2}\ds} \def\intCs{\ints C}
\def\intCz#1 {\intC{#1}\dz}

\def\Oint#1{\oint_{#1}\>} 
\def\Ointz#1#2 {\Oint{#1}{#2}\dz} \def\Oints#1#2 {\Oint{#1}{#2}\ds} 
\def\OintC{\Oint C} \def\OintCs#1 {\OintC{#1}\ds} \def\OintCz#1 {\OintC{#1}\dz}
\let\ointC\OintC \let\ointCs\OintCs \let\ointCz\OintCz
\def\bint#1{\oint_{\partial #1}} 
\def\bintO{\bint\Omega } \def\bintD{\bint D} 
\def\bints#1#2 {\bint{#1}{#2}\ds}  \def\bintz#1#2 {\bint{#1}{#2}\dz} 
\def\bintOs#1 {\bintO{#1}\ds}  \def\bintOz#1 {\bintO{#1}\dz}

\def\Sint#1#2 {\dintu{#1}{#2}\d S} 
\def\SintS{\Sint S} \def\SintO{\Sint{\partial \Omega }}
%\def\intS{\Sint S} 

\def\at#1{|_{#1}} \def\bigat#1{\big|_{#1}} \def\Bigat#1{\Big|_{#1}}
\def\Atu#1#2#3 {\left .{#3}\; \right |_{#1={#2}}}
\def\At#1#2#3#4 {\left .{#4}\; \right |_{#1={#2}}^{#3}}
\def\Atb#1#2#3#4 {\left [{#4}\right ]_{#1={#2}}^{#3}}
\def\Atz#1#2 {\Atu z{#1}{#2}}
\def\atx{\At x} \def\att{\At t} \def\atbx{\Atb x} \def\atbt{\Atb t} 
\def\atabx{\atx ab} \def\atbabx{\atx ab}
\def\atpx{\atx0{\pi }} \def\atppx{\atx{-\pi }{\pi }}
\def\atbpx{\atbx0{\pi }} \def\atbppx{\atbx{-\pi }{\pi }}
\def\atix{\atx{0}{\infty }} \def\atiix{\atx{-\infty }{\infty }} 
\def\atbix{\atbx{0}{\infty }} \def\atbiix{\atbx{-\infty }{\infty }}
\def\atit{\att{0}{\infty }} \def\atbit{\atbt{0}{\infty }} 

%\def\Ato#1#2{\left . #2\; \right |_{\displaystyle #1=0}}
%\def\ato#1#2{\left . #2\; \right |_{#1=0}}
%\def\Atto{\Ato t} \def\atto{\ato t}
%\def\Dto#1#2{\left . \odo{#1} #2\; \right |_{\displaystyle #1=0}}
%\def\dto#1#2{\left . \odo{#1} #2\; \right |_{#1=0}}
%\def\Dtto{\Dto t} \def\dtto{\dto t}

\def\Atb#1#2#3#4 {\left [\,{#4}\,\right ]_{#1\:=\:{#2}}^{#3}}
\def\Atu#1#2#3 {\left .{#3}\; \right |_{#1\:=\:{#2}}}
\def\At#1#2#3#4 {\left .{#4}\; \right |_{#1\:=\:{#2}}^{#3}}
\def\Atb#1#2#3#4 {\left [{#4}\right ]_{#1\:=\:{#2}}^{#3}}
\def\Atz#1#2 {\Atu z{#1}{#2}}
\def\atx{\At x} \def\att{\At t} \def\atbx{\Atb x} \def\atbt{\Atb t} 
\def\atabx{\atx ab} \def\atbabx{\atx ab}
\def\atpx{\atx0{\pi }} \def\atppx{\atx{-\pi }{\pi }}
\def\atbpx{\atbx0{\pi }} \def\atbppx{\atbx{-\pi }{\pi }}
\def\atix{\atx{0}{\infty }} \def\atiix{\atx{-\infty }{\infty }} 
\def\atbix{\atbx{0}{\infty }} \def\atbiix{\atbx{-\infty }{\infty }}
\def\atit{\att{0}{\infty }} \def\atbit{\atbt{0}{\infty }} 

\def\Ato#1#2{\left . #2\; \right |_{\displaystyle #1\:=\:0}}
\def\ato#1#2{\left . #2\; \right |_{#1\:=\:0}}
\def\Atto{\Ato t} \def\atto{\ato t}
\def\Dto#1#2{\left . \odo{#1} #2\; \right |_{\displaystyle #1\:=\:0}}
\def\dto#1#2{\left . \odo{#1} #2\; \right |_{#1\:=\:0}}
\def\Dtto{\Dto t} \def\dtto{\dto t}

\def\Sumu#1{\sum _{#1}\>}
\def\Summ#1#2#3{\sum _{#1\:=\:#2}^{#3}\>}
\def\Sum#1#2{\Summ{#1}1{#2}} \def\Sumo#1#2{\Summ{#1}0{#2}} 
\def\Sumin{\Sum in} \def\Sumjn{\Sum jn} \def\Sumkn{\Sum kn}
\def\Sumim{\Sum im} \def\Sumjm{\Sum jm} \def\Sumkm{\Sum km}
\def\Sumijn{\Sum{i,j}n} \def\Sumijm{\Sum{i,j}m}
\def\Sumnn#1{\Summ{#1}{-n}{n}}\def\Sumknn{\Sumnn k} 
\def\Sumoi#1{\Summ{#1}{0}{\infty }} \def\Sumkoi{\Sumoi k} \def\Sumnoi{\Sumoi n} 
\def\Sumi#1{\Summ{#1}{1}{\infty }} \def\Sumki{\Sumi k} \def\Sumni{\Sumi n}
\def\Sumii#1{\Summ{#1}{-\infty }{\infty }} 
\def\Sumiio#1{\sum_{\scriptstyle #1=-\:\infty \atop\scriptstyle #1\ne 0}^\infty} 
\def\Sumiii{\Sumii i}  \def\Sumjii{\Sumii j} \def\Sumkii{\Sumii k} \def\Sumnii{\Sumii n}
\def\Sumiiio{\Sumiio i} \def\Sumjiio{\Sumiio j} 
\def\Sumkiio{\Sumiio k} \def\Sumniio{\Sumiio n}
\def\Sumioi{\Sumoi i}\def\Sumjoi{\Sumoi j} \def\Sumji{\Sumi j} 
\def\Summoi{\Sumoi m} \def\Summi{\Sumi m}
\def\Sumoin{\Sumo in} \def\Sumojn{\Sumo jn} \def\Sumoijn{\Sumo{i,j}n} 
\def\Sumion{\Sumo in} \def\Sumjon{\Sumo jn} \def\Sumijon{\Sumo{i,j}n} 
\def\Sumojno{\Sumo j{n-1}} 

\def\Suminj{\sum_{i\ne j}\>} \def\Sumjni{\sum_{j\ne i}\>}
\def\Suminjn{\sum_{\ssty i=1\atop\ssty i\ne j}^n\>} 
\def\Sumjnin{\sum_{\ssty j=1\atop\ssty j\ne i}^n\>}

\def\Produ#1{\prod _{#1}\>}
\def\Prodd#1#2#3{\prod _{#1=#2}^{#3}\>}
\def\Prod#1#2{\Prodd{#1}1{#2}} \def\Prodin{\Prod in}
\def\Prodo#1#2{\Prodd{#1}0{#2}} \def\Prodion{\Prodo in} \def\Prodoin{\Prodo in}

\def\Limu#1{\lim_{#1}\>} \def\Lim#1#2{\Limu{#1\,\to \,#2}}
\def\Limi#1{\Lim{#1}\infty }  \def\Limim#1{\Lim{#1}{-\:\infty}} \def\Limo#1{\Lim{#1}0}
\def\Limni{\Lim n\infty } \def\Limki{\Lim k\infty } \def\Limxi{\Lim x\infty }
\def\Limmi{\Lim m\infty }  \def\Limti{\Lim t\infty }

%%%%%%%%%%%%%%%%%%% More math macros %%%%%%%%%%%%%%%%%%%%%%%%%%

\def\ivl#1#2{[\:#1,#2\:]} \def\iab{\ivl ab} \def\iol{\ivl0\ell}
\def\ivls#1{\ivl{-\:#1}{#1}} 
\def\iop{\ivl0\pi} \def\ipp{\ivls\pi } \def\iopp{\ivl0{2\pii}}
\def\ivlo#1#2{(\:#1,#2\:)} \def\ioab{\ivlo ab}
\def\ivlos#1{\ivlo{-\:#1}{#1}} 
\def\ioi{[\:0,\infty \:)}\def\iii{\ivlos \infty }
\def\ivlr#1#2{[\:#1,#2\:)} \def\Ivlr#1#2{\left [\:{#1}\,,{#2}\:\right )}
\def\ivll#1#2{(\:#1,#2\:]} \def\Ivll#1#2{\left (\:{#1}\,,{#2}\:\right ]}
\def\Ivl#1#2{\left [\:{#1}\,,{#2}\:\right ]}
\def\Ivlo#1#2{\left (\:{#1}\,,{#2}\:\right )}


\def\Cc#1{{\rm C}^{#1}} \def\Ccn{\Cc n} \def\Cci{\Cc \infty }  \def\Ca{\CA}
\def\Cab{\Cc0\iab} \def\Coab{\Cc0\ioab} \def\Col{\Cc0\iol}  
\def\Ceab{\Cc1\iab} \def\Ctab{\Cc2\iab} \def\Cnab{\Cc n\iab}  \def\Cnoab{\Cc n\ioab} 
\def\Ciab{\Cci\iab} \def\Cioab{\Cci\ioab} \def\Caab{\CA\iab} \def\Caoab{\CA\ioab} 
\def\Cabv{\Cc0(\iab,\Rx2)} \def\Ceabv{\Cc1(\iab,\Rx2)} 
\def\Ctabv{\Cc2(\iab,\Rx2)} \def\Ciabt{\Cci(\iab,\Rx2)}
\def\Ceoab{\Cc1\ioab} \def\Ctoab{\Cc2\ioab} 

\def\dex{\Delta x} \def\ldex{\lim_{\dex\to 0}}
 
\def\ni{n\to \infty } \def\ki{k\to \infty } \def\ti{t\to \infty } 
\def\nil{n\longrightarrow \infty } \def\kil{k\longrightarrow \infty }
\def\til{t\longrightarrow \infty }

\def\andq{\roq{and}} \def\andx{\rox{and}} \def\andqx{\roqx{and}} \def\andqq{\roqq{and}} \def\andz{\roz{and}}
\def\forq{\roq{for}}  \def\forx{\rox{for}}  \def\forqx{\roqx{for}}  \def\forqq{\roqq{for}}   
\def\forall{\roq{for all}} \def\forallx{\rox{for all}} \def\forallqx{\roqx{for all}}
\def\forevery{\roq{for every}}
\def\iffq{\roq{ if and only if }} \def\iffx{\rox{ if and only if }}
\def\iffqq{\roqq{ if and only if }} \let\iff\iffq
\def\orq{\roq{or}} \def\orqx{\roqx{or}} \def\orx{\rox{or}} 
\def\orz{\roz{or}}  \def\orqq{\roqq{or}}
\def\sothat{\roq{so that}}  \def\sothatqx{\roqx{so that}}
\def\thenq{\roq{then}} \def\thenx{\rox{then}} \def\thenqx{\roqx{then}} \def\thenqq{\roq{then}} \let\then\thenq
\def\whereq{\roq{where}} \def\wherex{\rox{where}} \def\whereqx{\roqx{where}} \def\whereqq{\roqq{where}} \let\where\whereq
\def\whileq{\roq{while}} \def\whilex{\rox{while}} \def\whileqx{\roqx{while}} \def\whileqq{\roqq{while}} \let\while\whileq
\def\whenq{\roq{when}} \def\whenx{\rox{when}} \def\whenqx{\roqx{when}} \def\whenqq{\roqq{when}} \let\when\whenq
\def\withq{\roq{with}} \def\withx{\rox{with}} \def\withqx{\roqx{with}} \def\withqq{\roqq{with}} \let\with\withq
\def\whenever{\roq{whenever}} \def\whereas{\roq{whereas}} \def\whereby{\roq{whereby}} \def\whence{\roq{whence}} 
\def\since{\roq{since}} \def\otherwise{\roh{otherwise}}

\def\const{\ro{const.}} \def\constant{\ro{constant}}

\def\ps#1{^{(#1)}} \def\pb#1{^{[#1]}}

\def\un{u\ps n} \def\uno{u\ps{n+1}} \def\ui{u\ps\infty }
\def\xun{(x,\un)} \def\xuno{(x,\uno)} \def\xui{(x,\ui)}

\def\J#1{\ro J{}^{#1}} \def\Jn{\J n}  \def\Jk{\J k} \def\Ji{\J\infty }
\def\j#1{{\rm j}_{#1}} \def\jn{\j n} \def\jk{\j k} \def\ji{\j \infty }

\def\pro#1{\ro{pr}^{(#1)}\,} \def\prn{\pro n} \def\prv{\ro{pr}\;\bo v}
\def\bov{\bo v}\def\bow{\bo w}

\def\Diff#1{\CD\hbox{\it i\char'13}\,(#1)}

\def\tv#1#2#3{(#1,\,#2)^{(#3)}}

\def\tc#1#2#3{T^{\,#1}_{#2#3}} \def\tcbar#1#2#3{\Tbar{}^{\,#1}_{#2#3}}

%%%%%%%%%%%%%%%%%%%% Dotted variables %%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\def\raisebox#1{\setbox0 = \hbox{$#1$}\dimen0=.75\ht0\dimen1=.2\dp0
%\ifinner\advance\dimen1by.2pt\fi\advance\dimen1by-.5pt\ht0=\dimen0\raise\dimen1\box0}
\def\raisebox#1{\setbox0 = \hbox{$#1$}\dimen0=.75\ht0\dimen1=-1.5\dp0\advance\dimen1 by -.6pt\ht0=\dimen0\raise\dimen1\box0}
\def\overscript#1#2{\raisebox{\mathop{#1}\limits^{\scriptstyle#2}}{}}

\def\bigddot{{\bigdot\!\bigdot}}
\def\bigdddot{{\bigdot\!\bigdot\!\bigdot}}
\def\dott#1{\overscript{#1}\bigdot}
\def\ddott#1{\overscript{#1}\bigddot}
\def\dddott#1{\overscript{#1}\bigdddot}
\let\dot=\dott \let\ddot=\ddott \let\dddot=\dddott

%\def\idot{\dot \i} \def\jdot{\dot \j} 
%\def\kdot{\dot k} \def\ldot{\dot l}
%\def\udot{\dot u} \def\vdot{\dot v} 
%\def\xdot{\dot x} \def\ydot{\dot y} \def\zdot{\dot z}
%\def\xddot{\ddot x} \def\uddot{\ddot u} \def\yddot{\ddot y} \def\zddot{\ddot z}

\def\overbox#1#2#3{\setbox0 = \hbox{$\mathop{#3}\limits^{\scriptstyle#2}$}%
\dimen0=.75\ht0\dimen1=#1pt\ht0=\dimen0\raise\dimen1\box0{}}
\def\overboxh#1#2#3#4{\setbox0 = \hbox{$\mathop{#4}\limits^{\mskip#2mu\scriptstyle#3}$}%
\dimen0=.75\ht0\dimen1=#1pt\ht0=\dimen0\raise\dimen1\box0{}}

\def\dotbox#1#2{\overbox{#1}\bigdot{#2}}
\def\dotboxh#1#2#3{\overboxh{#1}{#2}\bigdot{#3}}
\def\ddotbox#1#2{\overbox{#1}\bigddot{#2}}
\def\dddotbox#1#2{\overbox{#1}\bigdddot{#2}}

\def\qdot{\dotbox{-1.4}q} \def\udot{\dotbox{-.4}u} 
\def\vdot{\dotbox{-.4}v} \def\wdot{\dotbox{-.4}w} 
%\def\xdot{\dotbox{-.4}x} \def\ydot{\dotbox{-1.4}y} 
%\def\zdot{\dotbox{-.4}z}  
\def\xdot{\dotboxh{-.4}2x} \def\ydot{\dotboxh{-1.4}{1.3}y} 
\def\zdot{\dotboxh{-.4}{.6}z}
\def\xddot{\ddotbox{-.4}x} \def\uddot{\ddotbox{-.4}u} 
\def\yddot{\ddotbox{-1.4}y} \def\zddot{\ddotbox{-.4}z} 
\def\vddot{\ddotbox{-.4}v} \def\wddot{\ddotbox{-.4}w} 

\def\bxdot{\dotbox{-.3}{\bo x}} \def\bxddot{\ddotbox{-.3}{\bo x}}
\def\budot{\dotbox{-.3}{\bo u}} \def\buddot{\ddotbox{-.3}{\bo u}} 
%\let\ud\budot \let\udd\buddot
\def\bvdot{\dotbox{-.3}{\bo v}}\def\bvddot{\ddotbox{-.3}{\bo v}} 
%\let\vd\bvdot \let\vdd\bvddot
\def\bzdot{\dotbox{-.3}{\bo z}} \def\bzddot{\ddotbox{-.3}{\bo z}} 
%\let\zd\bzdot \let\zdd\bzddot
\def\bydot{\dotbox{-1.4}{\bo y}} \def\byddot{\ddotbox{-1.4}{\bo y}}
\def\bpdot{\dotbox{-1.4}{\bo p}} \def\bqdot{\dotbox{-1.4}{\bo q}} 

\def\Udot{\dotbox{1}U} \def\Vdot{\dotbox{1}V}

\def\zetadot{\dotbox0\zeta } 
\def\thetadot{\dotbox1\theta } \def\thetaddot{\ddotbox1\theta }
\def\xidot{\dotbox0\xi } \def\etadot{\dotbox{-.4}\eta }

%\def\qdot{\dotbox{-1.45}q} \def\udot{\dotbox{-.45}u} \def\vdot{\dotbox{-.45}v} 
%\def\xdot{\dotbox{-.45}x} \def\ydot{\dotbox{-1.45}y} \def\zdot{\dotbox{-.45}z}  
%\def\xddot{\ddotbox{-.45}x} \def\uddot{\ddotbox{-.45}u} \def\yddot{\ddotbox{-1.45}y} 
%\def\zddot{\ddotbox{-.45}z} \def\vddot{\ddotbox{-.45}v} \def\wddot{\ddotbox{-.45}w} 
%\def\bxdot{\dotbox{-.3}{\bo x}}\def\bxddot{\ddotbox{-.3}{\bo x}}
%\def\budot{\dotbox{-.3}{\bo u}}\def\buddot{\ddotbox{-.3}{\bo u}}
%\def\zetadot{\dotbox0\zeta }

%\def\qdot{\dotbox{-1.35}q} \def\udot{\dotbox{-.25}u} \def\vdot{\dotbox{-.25}v} 
%\def\xdot{\dotbox{-.25}x} \def\ydot{\dotbox{-1.35}y} \def\zdot{\dotbox{-.25}z}  
%\def\xddot{\ddotbox{-.25}x} \def\uddot{\ddotbox{-.25}u} \def\yddot{\ddotbox{-1.35}y} 
%\def\zddot{\ddotbox{-.25}z} \def\vddot{\ddotbox{-.25}v} \def\wddot{\ddotbox{-.25}w} 
%\def\bxdot{\dotbox{-.25}{\bo x}}\def\bxddot{\ddotbox{-.25}{\bo x}}
%\def\budot{\dotbox{-.25}{\bo u}}\def\buddot{\ddotbox{-.25}{\bo u}}
%\def\zetadot{\dotbox0\zeta }


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   
%                   Cambridge University Press Book Macros
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   
%              Running headline macros courtesy of Alan Harvey
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newdimen\headerskip \newbox\headerbox
\newdimen\headskip \headskip=7.25pt
\newdimen\headerdim
\newcount\HeaderNumber \HeaderNumber=0

\def\pagenobox{\hbox{\rnumbert\pageno}}

\newtoks\LeftHeader
\def\LeftHead#1.{\LeftHeader{#1}}% left page head
\newtoks\RightHeader
\def\RightHead#1.{\RightHeader{#1}}% right page head

\def\MakeHeaderbox{\setbox0 = \pagenobox
                   \setbox2 = \hbox to \wd0{\hfill}%
\ifodd\pageno
   \setbox\headerbox=\hbox to \hsize{\box2\hfill\tenit\the\RightHeader\hfill\box0}%
\else
   \setbox\headerbox=\hbox to \hsize{\box0\hfill\tenit\the\LeftHeader\hfill\box2}%
\fi
\ifnum\HeaderNumber=0 \setbox\headerbox=\hbox{\hfil} \fi
 \headerskip=\headskip \advance\headerskip by -\ht\headerbox
 \vbox to \headerdim{\vskip\headerskip\noindent\unhbox\headerbox\vskip-\headskip\vfill}%
 \nointerlineskip}

\def\MakeFooterbox{\ifproofmode\baselineskip = 24pt 
\line{\hss \tenrm \jobname\quad\shortdate\hss}\fi}


\def\PageBody{\vbox to\vsize{\boxmaxdepth\maxdepth \pagecontents}}
\def\OutputPage{
 \shipout\vbox{\MakeHeaderbox\PageBody\MakeFooterbox}
 \advancepageno\global\fnotenumber=0
 \ifnum\outputpenalty>-1000000 \else \dosupereject\fi
 \global\HeaderNumber=1}

\def\eatspace#1{\ifx#1 \eatspace\else#1\fi}

\long\def\eatspacepar#1{\ifx#1\par\let\next=\eatspacepar\else
\ifx#1 \let\next=\eatspacepar\else\let\next=#1\fi\fi\next}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%      Macro  \bookit  sets page sizes 
%                      and new output routine for complete pages
%
%             \bookref  inputs all book references 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\long\def\bookit{
\bookmodetrue
\longmodefalse

\def\Subheadfont{\bf}
\def\heading##1##2{\medskip{\center ##2 ##1\par}\nobreak\bigskip}
\def\firstpar{\par\nobreak\noindent\eatspacepar}

\def\refstyle{\parskip=1pt \frenchspacing \baselineskip = 12pt  
\rightskip=0pt plus 4em \spaceskip = .3333em \xspaceskip = .5em
\pretolerance=10000 \tolerance=10000
\hyphenpenalty=10000 \exhyphenpenalty=10000
\interlinepenalty=10000 \raggedbottom}

\parskip = 0pt plus 1pt
\hsize = 26 pc
\baselineskip = 13pt  
\vsize = 530pt
\topskip = 26pt
\hoffset = .5in
\voffset = .2in
\headerdim = 10pt
\headskip = 12pt

\abovedisplayskip = 6pt plus 1pt minus 1pt
\belowdisplayskip = 6pt plus 1pt minus 1pt

\widehats \widetildes
\output{\OutputPage}

}

\def\bookref{{\numcat \input book.key \global\eqnreadtrue 
\input ref.key \global\refstrue}}


%%%%%%%%%%%%  Set Postscript Figure Processing Version %%%%%%%%%%%%%%

%\texturesepsf     % Default for Textures
\dvipsepsf       % Default for unix using dvips
%\epsfverbosetrue

\endinput

