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authorNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
committerNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
commite0c6872cf40896c7be36b11dcc744620f10adf1d (patch)
tree60335e10d2f4354b0674ec22d7b53f0f8abee672 /macros/generic/occam
Initial commit
Diffstat (limited to 'macros/generic/occam')
-rw-r--r--macros/generic/occam/DefStrip-QUEDCmds.hqx224
-rw-r--r--macros/generic/occam/DefStrip-QUEDCmds.hqx_old223
-rw-r--r--macros/generic/occam/Euro92.sty296
-rw-r--r--macros/generic/occam/Occam94-old/DefStrip-QEDMacros.hqx125
-rw-r--r--macros/generic/occam/Occam94-old/auditor.tex599
-rw-r--r--macros/generic/occam/Occam94-old/defstrip.hlp56
-rw-r--r--macros/generic/occam/Occam94-old/occam.pub173
-rw-r--r--macros/generic/occam/Occam94-old/occam.spc209
-rw-r--r--macros/generic/occam/Occam94-old/sample.dir/Kohler-aud.sty493
-rw-r--r--macros/generic/occam/Occam94-old/sample.dir/Kohler-ori.sty247
-rw-r--r--macros/generic/occam/Occam94-old/sample.dir/Kohler.sty277
-rw-r--r--macros/generic/occam/Occam94-old/sample.dir/Kohler.tex1169
-rw-r--r--macros/generic/occam/Occam95.pdfbin0 -> 145590 bytes
-rw-r--r--macros/generic/occam/PlainEx1/Example.readme6
-rw-r--r--macros/generic/occam/PlainEx1/RESOURCESimple.occ648
-rw-r--r--macros/generic/occam/PlainEx1/newton.sty212
-rw-r--r--macros/generic/occam/PlainEx1/newton.tex1281
-rw-r--r--macros/generic/occam/PlainEx2/W.sty196
-rw-r--r--macros/generic/occam/PlainEx2/W.tex1362
-rw-r--r--macros/generic/occam/PlainEx2/harvmac.occ571
-rw-r--r--macros/generic/occam/PlainEx2/harvmac.tex327
-rw-r--r--macros/generic/occam/PlainEx2/harvtst.doc170
-rw-r--r--macros/generic/occam/PlainEx2/witten94.tex1402
-rw-r--r--macros/generic/occam/auditor.tex598
-rw-r--r--macros/generic/occam/defstrip.hlp56
-rw-r--r--macros/generic/occam/occam.pub272
-rw-r--r--macros/generic/occam/occam.spc415
27 files changed, 11607 insertions, 0 deletions
diff --git a/macros/generic/occam/DefStrip-QUEDCmds.hqx b/macros/generic/occam/DefStrip-QUEDCmds.hqx
new file mode 100644
index 0000000000..2331b08789
--- /dev/null
+++ b/macros/generic/occam/DefStrip-QUEDCmds.hqx
@@ -0,0 +1,224 @@
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diff --git a/macros/generic/occam/DefStrip-QUEDCmds.hqx_old b/macros/generic/occam/DefStrip-QUEDCmds.hqx_old
new file mode 100644
index 0000000000..598dbb0174
--- /dev/null
+++ b/macros/generic/occam/DefStrip-QUEDCmds.hqx_old
@@ -0,0 +1,223 @@
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diff --git a/macros/generic/occam/Euro92.sty b/macros/generic/occam/Euro92.sty
new file mode 100644
index 0000000000..555cf8fee7
--- /dev/null
+++ b/macros/generic/occam/Euro92.sty
@@ -0,0 +1,296 @@
+
+
+% Call this file euro92.sty and use it in LaTeX file as
+% \documentstyle[euro92]{article}
+% Also specify
+% \titlehead{Short title for paper}
+% \paperhead{Author Name(s)}
+% The fields of \maketitle are further redefined: \date is eliminated and
+% \affiliation is added.
+%
+% Richard Furuta December 9, 1987
+% (adapte pour les "Cahiers GUTenberg" - Ph Louarn 88-89)
+% (adaptation for EuroTeX '92 Prague conference - Petr Sojka
+% <sojka@aci.cvut.cs>)
+%
+
+% hyphenation commands (For MLTeX)
+\ifx\fhyph\undefined\relax\else % si \fhyph est connu, on suppose \ehyph connu
+ \ifx\enhyph\undefined\let\enhyph=\ehyph\fi
+ \ifx\frhyph\undefined\let\frhyph=\fhyph\fi
+\fi
+
+\def\NoC#1{\gdef\N@C{#1}}
+
+\def\DateC#1{\gdef\D@teC{#1}}
+
+% size requirements
+\def\@normalsize{\@setsize\normalsize{14pt}\xipt\@xipt
+\abovedisplayskip 10pt plus2pt minus5pt\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip \z@ plus3pt\belowdisplayshortskip 6pt plus3pt minus3pt}
+\def\small{\@setsize\small{11pt}\ixpt\@ixpt
+\abovedisplayskip 8.5pt plus 3pt minus 4pt\belowdisplayskip \abovedisplayskip
+\abovedisplayshortskip \z@ plus2pt\belowdisplayshortskip 4pt plus2pt minus 2pt
+\def\@listi{\topsep 4pt plus 2pt minus 2pt\parsep 2pt plus 1pt minus 1pt
+\itemsep \parsep}}
+
+% baseline for section is set at 12 because I need 13 and can't get it
+% easily another way (see redefinition of section, etc.)
+\def\sectsize{\@setsize\sectsize{21pt}\xivpt\@xivpt} % for section heads
+\def\subsectsize{\@setsize\subsectsize{14pt}\xipt\@xipt} % for section heads
+\def\abstsize{\@setsize\abstsize{11pt}\viiipt\@viiipt} % for abstract
+\def\captsize{\@setsize\captsize{12pt}\xpt\@xpt} % for captions
+\def\titlesize{\@setsize\titlesize{25pt}\xxpt\@xxpt} % title large
+\def\authsize{\@setsize\authsize{14pt}\xipt\@xipt} % author size
+\def\affilsize{\@setsize\affilsize{12pt}\xpt\@xpt} % affiliation
+
+% - redefine the citation style to put spaces after the commas
+% redefine the citation style to put spaces after the commas
+\def\@citex[#1]#2{\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi
+ \def\@citea{}\@cite{\@for\@citeb:=#2\do
+ {\@citea\def\@citea{, }\@ifundefined
+ {b@\@citeb}{{\bf ?}\@warning
+ {Citation `\@citeb' on page \thepage \space undefined}}%
+\hbox{\csname b@\@citeb\endcsname}}}{#1}}
+
+% new field for the title
+\def\affiliation#1{\gdef\@affiliation{#1}}
+
+%title's spacing
+\def\titlesp{\list{}{\leftmargin0pc}\item[]}
+\let\endtitlesp=\endlist
+
+% title needs adjusting, no date in title
+\def\@maketitle{\newpage
+ \null
+ \vspace*{-28pt}
+ \begin{titlesp}
+ {\titlesize \@title \par}
+ \vskip 8pt % Vertical space after title. 8pt
+ \hrule % one point rule
+ \vskip 14pt % 14 points below rule (adjusted for baseline)
+ {\authsize \@author \par}
+% there should be 18 pts between author and affiliation baselines
+ \vskip 6pt % Vertical space after author. 18pt - 6pt
+ {\affilsize\it \@affiliation \par}
+\end{titlesp}
+ \par
+ \vskip 16pt} % Vertical space after title. 26pt - 10pt
+
+\def\maketitle{\par
+\begingroup
+\def\thefootnote{\fnsymbol{footnote}}
+\def\@makefnmark{\hbox
+to 0pt{$^{\@thefnmark}$\hss}}
+\newpage \global\@topnum\z@ \@maketitle \thispagestyle{titre}\@thanks
+\endgroup
+\setcounter{footnote}{0}
+\let\maketitle\relax
+\let\@maketitle\relax
+\gdef\@thanks{}\gdef\@author{}\gdef\@title{}\let\thanks\relax}
+
+\def\abstract{\small
+\begin{center}
+{\bf Abstract\vspace{-.5em}\vspace{0pt}}
+\end{center}
+\quotation
+\rm\ifx\enhyph\undefined\relax\else\enhyph\fi}
+\def\endabstract{\endquotation\normalsize\rm}
+
+\def\resume#1{\small
+\begin{center}
+{\bf #1\vspace{-.5em}\vspace{0pt}}
+\end{center}
+\quotation
+\em\ifx\frhyph\undefined\relax\else\frhyph\fi}
+\def\endresume{\endquotation\normalsize\ifx\enhyph\undefined\relax\else\enhyph\f
+i}
+
+\def\keywords{\small\quotation
+\noindent\rm {\bf Key words:\ }}
+\def\endkeywords{\endquotation\normalsize}
+
+% modify the sectioning commands as is appropriate for style. Adjusting
+% for baselines again to get 35 pt above and 15 pt below
+\def\section{\@startsection {section}{1}{\z@}{-12pt plus -2pt minus
+-2pt}{10pt plus 1pt minus 1pt}{\sectsize\raggedright}}
+\def\subsection{\@startsection{subsection}{2}{\z@}{-10pt plus -2pt minus
+-1pt}{8pt plus 1pt minus 1pt}{\subsectsize\bf\raggedright}}
+\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{-8pt plus
+-2pt minus -1pt}{6pt plus 1pt minus 1 pt}{\subsectsize\it\raggedright}}
+\def\paragraph{\@startsection
+{paragraph}{4}{\z@}{6pt plus 2pt minus 2pt}{-1em}{\normalsize\bf}}
+\def\subparagraph{\@startsection
+{subparagraph}{4}{\parindent}{6pt plus 2pt minus
+2pt}{-1em}{\normalsize\bf}}
+
+\gdef\thesection{\arabic{section}}
+\gdef\thesubsection{\thesection.\arabic{subsection}}
+\gdef\thesubsubsection{\thesubsection.\arabic{subsubsection}}
+\gdef\theparagraph{\thesubsubsection.\arabic{paragraph}}
+\gdef\thesubparagraph{\theparagraph.\arabic{subparagraph}}
+
+% define a new environment for use in setting the abstract
+\def\abst{\list{}{\leftmargin0pc}\abstsize\item[]}
+\def\endabst{\endlist\vskip-7pt}
+
+\def\listelabel{$\bullet$}
+\settowidth{\labelwidth}{\listelabel}
+\def\liste{\list{}{\settowidth{\labelwidth}{\listelabel}
+\leftmargin\labelwidth\advance\leftmargin\labelsep\rightmargin0pt\labelsep0pt
+\itemsep0pt \let\makelabel\listelabel}\small}
+\let\endliste=\endlist
+
+\def\enumeration{\list{{\small \arabic{enumi}.\ }}%
+{\settowidth{\labelwidth}{{\small 1.\ }}
+\leftmargin\labelwidth\advance\leftmargin\labelsep\rightmargin0pt\labelsep0pt
+\itemsep0pt \usecounter{enumi}}\small}
+\let\endenumeration=\endlist
+
+% bibliography stuff
+
+\def\thebibliography#1{\section*{References}%
+\list{[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth
+\advance\leftmargin\labelsep\usecounter{enumi}}\small}
+\let\endthebibliography=\endlist \newif\if@restonecol
+
+% modifying the format of the footnote display
+% A simpler macro is used, in which the footnote text is
+% set like an ordinary text paragraph, with no indentation except
+% on the first line of a paragraph, and the first line of the
+% footnote. Thus, all the macro must do is set \parindent
+% to the appropriate value for succeeding paragraphs and put the
+% proper indentation before mark.
+
+\long\def\@makefntext#1{\parindent 1em\noindent
+ \hbox to 1.8em{\hss$^{\@thefnmark}$}#1}
+
+
+% figure spacings
+% ONE-COLUMN MODE OR SINGLE-COLUMN FLOATS IN TWO-COLUMN MODE:
+\textfloatsep 13pt plus 2pt minus 4pt % Space between main text and floats
+ % at top or bottom of page.
+\@maxsep 13pt % The maximum of \floatsep,
+ % \textfloatsep and \intextsep (minus
+ % the stretch and shrink).
+
+\long\def\@caption#1[#2]#3{\addcontentsline{\csname
+ ext@#1\endcsname}{#1}{\protect\numberline{\csname
+ the#1\endcsname}{\ignorespaces #2}}\par
+ \begingroup
+ \@parboxrestore
+ \captsize
+ \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par
+ \endgroup}
+
+\long\def\@makecaption#1#2{
+ \vskip 1pt % almost no skip at all
+ \setbox\@tempboxa\hbox{#1: #2}
+ \ifdim \wd\@tempboxa >\hsize % IF longer than one line:
+ \unhbox\@tempboxa\par % THEN set as ordinary paragraph.
+ \else % ELSE center.
+ \hbox to\hsize{\hfil\box\@tempboxa\hfil}
+ \fi}
+
+% figure placement modifications
+\setcounter{topnumber}{1}
+\def\bottomfraction{0.01}
+
+% running headers
+\def\titlehead#1{\gdef\@titlehead{#1}}
+\def\authorhead#1{\gdef\@authorhead{#1}}
+
+% pagestyle
+\def\ps@gut{\let\@mkboth\@gobbletwo%
+ \def\@evenhead{\normalsize\it{%
+\ifx\@authorhead\u@ndefined\@author\else\@authorhead\fi}\hfil}%
+ \def\@oddfoot{\rm\hfil\thepage}\def\@oddhead{\normalsize\it\hfil{%
+\ifx\@titlehead\u@ndefined\@title\else\@titlehead\fi}}%
+ \def\@evenfoot{\rm\thepage\hfil}}
+
+\def\ps@titre{\let\@mkboth\@gobbletwo%
+ \def\@evenhead{\hfill{\footnotesize Euro\TeX\ \N@C{} --- \D@teC}}%
+ \def\@oddhead{{\footnotesize Euro\TeX\ \N@C{} --- \D@teC}\hfill}%
+ \def\@oddfoot{\rm\hfil\thepage}%
+ \def\@evenfoot{\rm\thepage\hfil}}
+
+\ds@twoside
+
+\def\@outputpage{\begingroup\catcode`\ =10 \if@specialpage
+ \global\@specialpagefalse\@nameuse{ps@\@specialstyle}\fi
+ \if@twoside
+ \ifodd\count\z@ \let\@thehead\@oddhead \let\@thefoot\@oddfoot
+ \let\@themargin\oddsidemargin
+ \else \let\@thehead\@evenhead
+ \let\@thefoot\@evenfoot \let\@themargin\evensidemargin
+ \fi\fi
+ \shipout
+ \vbox{\normalsize \baselineskip\z@ \lineskip\z@
+ \vskip \topmargin \moveright\@themargin
+ \vbox{\setbox\@tempboxa
+ \vbox to\headheight{\vfil \hbox to\textwidth{\@thehead}
+ \vskip 10pt \hbox to\textwidth{\hrulefill}}
+ \dp\@tempboxa\z@
+ \box\@tempboxa
+ \vskip \headsep
+ \box\@outputbox
+ \baselineskip\footskip
+ \hbox to\textwidth{\@thefoot}}}\global\@colht\textheight
+ \endgroup\stepcounter{page}\let\firstmark\botmark}
+
+% heading needs a bar
+
+\oddsidemargin3mm \evensidemargin36mm
+
+\marginparwidth 2cm \marginparsep 10pt
+\topmargin 3cm \headheight 1cm \headsep 14pt \footheight .3cm \footskip
+1cm
+
+\textheight 546pt \textwidth12.7cm \columnsep 10pt \columnseprule 0pt
+
+\footnotesep 8.4pt
+\skip\footins 10.8pt plus 4pt minus 2pt
+\floatsep 14pt plus 2pt minus 4pt \textfloatsep 20pt plus 2pt minus 4pt
+\intextsep 14pt plus 4pt minus 4pt \@maxsep 20pt \dblfloatsep 14pt plus 2pt
+minus 4pt \dbltextfloatsep 20pt plus 2pt minus 4pt \@dblmaxsep 20pt
+\@fptop 0pt plus 1fil \@fpsep 10pt plus 2fil \@fpbot 0pt plus 1fil
+\@dblfptop 0pt plus 1fil \@dblfpsep 10pt plus 2fil \@dblfpbot 0pt plus 1fil
+\marginparpush 7pt
+
+\parskip 1.1ex plus 1pt \parindent 1.5em \topsep 10pt plus 4pt minus 6pt
+\partopsep 3pt plus 2pt minus 2pt \itemsep 5pt plus 2.5pt minus 1pt
+\@lowpenalty 51 \@medpenalty 151 \@highpenalty 301
+\@beginparpenalty -\@lowpenalty \@endparpenalty -\@lowpenalty \@itempenalty
+-\@lowpenalty
+
+\hfuzz2pt
+\tolerance3500
+\pretolerance3500
+\hbadness10000
+\hyphenpenalty50
+\exhyphenpenalty50
+
+%\sloppy
+\frenchspacing
+
+\pagestyle{gut}
+\thispagestyle{titre}
+
+\def\at{{\char '100}}
+\def\boi{{\tt\char '134}}
+\def\circonflexe{{\char '136}}
+\def\tild{{\char '176}}
+\def\degre{{\char '027}}
+%% remove the 3 following lines if you don't the guill font.
+%\font\trom=guill
+%\def\oguill{{\trom A}\nobreak\mbox{\hglue.25em}\nobreak}
+%\def\fguill{\nobreak\mbox{\hglue.25em}\nobreak{\trom B}\,}
+
+\NoC{92}
+\DateC{September 14--18, Prague, Czechoslovakia}
+\let\twocolumn\relax % Don't use twocolumn, please
+\ifx\enhyph\undefined\relax\else\enhyph\fi
+
+
+
+
diff --git a/macros/generic/occam/Occam94-old/DefStrip-QEDMacros.hqx b/macros/generic/occam/Occam94-old/DefStrip-QEDMacros.hqx
new file mode 100644
index 0000000000..2473008235
--- /dev/null
+++ b/macros/generic/occam/Occam94-old/DefStrip-QEDMacros.hqx
@@ -0,0 +1,125 @@
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diff --git a/macros/generic/occam/Occam94-old/auditor.tex b/macros/generic/occam/Occam94-old/auditor.tex
new file mode 100644
index 0000000000..8d92c2f91b
--- /dev/null
+++ b/macros/generic/occam/Occam94-old/auditor.tex
@@ -0,0 +1,599 @@
+
+ %% auditor.tex of 6-94 (alpha)
+ %% "audits macro use"
+ %% By laurent siebenmann, lcs@topo.math.u-psud.fr (comments please!)
+ %% Used mostly for Plain and amstex; LaTeX use not debugged.
+ %% Documentation after endinput.
+ %% Alpha posting on ftp ftp.math.u-psud.fr in pub/TeX/Occam.dir
+ %% auditor.tex is part of the Occam system for macro management.
+ %% But it can also be used alone.
+
+ \ifx\undefined\auditortex\def\auditortex{}
+ \else
+ \immediate\write16{}%
+ \errmessage{%
+ The auditor.tex macro \noexpand\Def already loaded\string!}%
+ \EX@\endinput
+ \fi
+
+ \chardef\auditAt=\catcode`\@
+ \catcode`\@=11
+
+ %%temporarily suppress Plain's logging of allocations
+ \let\auditorwlog@ld\wlog
+ \def\wlog#1{\relax}
+
+ \def\WrSc@{\immediate\write16}
+ \def\WrOut@{\immediate\write\unusedout@}
+
+ \WrSc@{}%
+ \WrSc@{%
+ *** The auditor.tex macro-auditing system}%
+ \WrSc@{%
+ *** is being installed for a test run of your typescript.}%
+ \WrSc@{%
+ *** Its report will appear after typesetting.}%
+ \WrSc@{%
+ *** See documentation in auditor.tex.}%
+ \WrSc@{%
+ *** Hit return to get audit.}%
+ \WrSc@{%
+ *** Hit \space x \space and return to typeset without audit.}%
+ \def\temp{\par}%
+ \read16 to \YourChoice
+ %\def\YourChoice{\par}%%%%%%%%%% comment out?
+ \ifx\YourChoice\temp
+ \def\Modify@Audit{}%
+ \else
+ \def\Modify@Audit{\let\Def\def \let\gDef\gdef \let\Let\let
+ \def\gLet{\global\let}\let\The@Audit@\relax}%
+ \WrSc@{%
+ *** Auditor apparatus switched off\string.}%
+ \fi
+ \WrSc@{}%
+
+
+ \newwrite \unusedout@
+ \newtoks\temptoks@
+ \newtoks\nibbletoks@
+ \newtoks\resttoks@
+ \newtoks\deftoks@
+ \newtoks\mactoks@
+ \newtoks\unusedtoks@
+ \newtoks\AuditDepth
+
+ \let\EX@\expandafter
+
+ \def\Nibbl@#1#2\endNibbl@{\nibbletoks@{#1}\resttoks@{#2}}
+
+ \def\gobble#1{}
+
+ \def\WrDef@#1{\EX@
+ \ifx\csname\string_%
+ \string#1\string_\endcsname\@Used
+ \else
+ \EX@\global\EX@\let\csname\string_%
+ \string#1\string_\endcsname\@Defed
+ \edef\@tmp@{\global
+ \noexpand\deftoks@{\the\deftoks@\noexpand#1}}%
+ \@tmp@
+ \fi
+ }
+
+ \def\WrDDef@#1{\EX@
+ \ifx\csname\string_%
+ \string#1\string_\endcsname\@Used
+ \else
+ \EX@\global\EX@\let\csname\string_%
+ \string#1\string_\endcsname\@DDefed
+ \edef\@tmp@{\global
+ \noexpand\deftoks@{\the\deftoks@\noexpand#1}}%
+ \@tmp@
+ \fi
+ }
+
+ \def\@UsedMathSym{@UsedMathSym}
+ \newtoks\Def@toks@
+ \newtoks\Def@@toks@
+ \newtoks\Font@toks@
+ \newtoks\Mathchar@toks@
+ \newtoks\MATHchar@toks@
+ \newtoks\MATHchars@toks@
+
+\def\Deff@{%
+ \edef\@tmp@ {\@d@f\the\Def@toks@ {%\noexpand
+ \global\let \EX@\noexpand\csname\string_%
+ \EX@\string\the\nibbletoks@\string_\endcsname
+ \noexpand\@Used
+ \the\Def@@toks@ }}%
+ \@tmp@
+ \EX@\WrDef@\the\nibbletoks@
+ }
+
+\def\DDeff@{%
+ \edef\@tmp@ {\@d@f\the\Def@toks@ {%\noexpand
+ \global\let \EX@\noexpand\csname\string_%
+ \EX@\string\the\nibbletoks@\string_\endcsname
+ \noexpand\@Used
+ \the\Def@@toks@ }}%
+ \@tmp@
+ \EX@\WrDDef@\the\nibbletoks@
+ }
+
+ \def\Def#1#{\Def@toks@{#1}%
+ \Nibbl@#1\endNibbl@\let\@d@f\def
+ \afterassignment\Deff@\Def@@toks@
+ }
+
+ \def\DDef#1#{\Def@toks@{#1}%
+ \Nibbl@#1\endNibbl@\let\@d@f\def
+ \afterassignment\DDeff@\Def@@toks@
+ }
+
+ \def\gDef#1#{\Def@toks@{#1}%
+ \Nibbl@#1\endNibbl@\let\@d@f\gdef
+ \afterassignment\Deff@\Def@@toks@
+ }
+
+ %% \Let \gLet require two *macros* following, without =
+ %% Only then enhances \let
+ \def\Let#1#2{%
+ \EX@\Def\EX@#1\EX@{#2}%
+ }
+ \def\gLet#1#2{%
+ \EX@\gDef\EX@#1\EX@{#2}%
+ }
+
+ \def\WrMATHchardef@{%
+ \EX@\edef\the\MATHchar@toks@{%
+ \noexpand\ifx
+ \csname\string_\EX@\string\the\MATHchar@toks@\string_\endcsname
+ \noexpand\@UsedMathSym
+ \noexpand\else
+ \global\MATHchars@toks@\noexpand\EX@{%
+ \noexpand\the\MATHchars@toks@\the\MATHchar@toks@}%
+ \global\let\csname\string_\EX@\string\the\MATHchar@toks@\string_\endcsname
+ \noexpand\@UsedMathSym
+ \noexpand\fi
+ \csname \EX@\string\the\MATHchar@toks@\string_\endcsname}
+ %\EX@\show\the\MATHchar@toks@
+ }
+
+ \def\MATHchardef#1{\MATHchar@toks@{#1}%
+ %\showthe\MATHchar@toks@
+ \afterassignment\WrMATHchardef@
+ \EX@\mathchardef\csname \string#1\string_\endcsname}
+
+ \def\AfterMathchardef@{%
+ \edef\@@temp@{%
+ \noexpand\Def\the\Mathchar@toks@{%
+ \csname \EX@\string\the\Mathchar@toks@\string_\endcsname}}
+ %\show\@@temp@
+ \@@temp@}
+
+ \def\Mathchardef#1{\Mathchar@toks@{#1}%
+ \afterassignment\AfterMathchardef@
+ \EX@\mathchardef\csname \string#1\string_\endcsname}
+
+ %%% \Newsymbol, \NEWsymbol for \newsymbol of amssym.def
+ \def\Newsymbol#1#2#3#4#5{\let\next@\relax
+ \ifnum#2=\@ne\let\next@\msafam@\else
+ \ifnum#2=\tw@\let\next@\msbfam@\fi\fi
+ \Mathchardef#1="#3\next@#4#5}
+
+ \def\NEWsymbol#1#2#3#4#5{\let\next@\relax
+ \ifnum#2=\@ne\let\next@\msafam@\else
+ \ifnum#2=\tw@\let\next@\msbfam@\fi\fi
+ \MATHchardef#1="#3\next@#4#5}
+
+ \def\@F@nt@{\edef\@@temp@{%
+ \noexpand\Def\the\Font@toks@{%
+ \csname \EX@\string\the\Font@toks@\string_\endcsname}}%
+ \@@temp@}
+
+ \def\Font#1{\Font@toks@{#1}\afterassignment\@F@nt@
+ \EX@\font\csname \string#1\string_\endcsname}
+
+ \def\@FF@nt@{\edef\@@temp@{%
+ \noexpand\DDef\the\Font@toks@{%
+ \csname \EX@\string\the\Font@toks@\string_\endcsname}}%
+ \@@temp@}
+
+ \def\FFont#1{\Font@toks@{#1}\afterassignment\@FF@nt@
+ \EX@\font\csname \string#1\string_\endcsname}
+
+ \def\Loop@#1\Repeat@{%
+ \def\Iterate@{#1\EX@\Iterate@\fi}%
+ \Iterate@}
+
+ \bgroup\catcode`\%=12
+ \global\def\Pct@{ %% }\egroup
+
+ \def\WriteToToks@{\edef\@tmp@{\global\noexpand
+ \unusedtoks@{\the\unusedtoks@\the\nibbletoks@}}%
+ \@tmp@}
+
+ %\def\@Defed{@Defed}
+ %\def\@Used{@Used}
+ \def\@Used{\WrOut@{ \the\nibbletoks@}}%
+ \def\@Defed{\WrOut@{ *\the\nibbletoks@}\WriteToToks@}%
+ \def\@DDefed{\WrOut@{ **\the\nibbletoks@}\WriteToToks@}%
+ \let\@Filler\relax
+ \def\@Tail{@Tail}
+
+ \def\List@M@cs{%
+ \Loop@
+ %\message{ x }
+ \EX@\Nibbl@\the\deftoks@\@Tail\endNibbl@
+ \deftoks@\resttoks@ %\showthe\resttoks@
+ \edef\@Temp@{\EX@\noexpand\csname\string_%
+ \EX@\string\the\nibbletoks@\string_\endcsname}%
+ \EX@\let\EX@\@Temp\@Temp@ %
+ %% \@Temp is x-equal and let-equal to
+ %% \@Defed or \@DDefed or \@Used or \@Filler or @
+ %\showthe\nibbletoks@
+ %\show\@Temp@
+ %\show\@Temp
+ \EX@
+ \ifx \the\nibbletoks@\@Tail
+ %% exit if next token \@Tail
+ \else
+ \@Temp %% write appropriate stuff to file and log
+ \Repeat@
+ }
+
+ \newtoks\hrct@
+
+
+ {\catcode`\#=12\gdef\StringSharp{\string#}}
+
+ \def\The@Audit@{%\show\patience
+ \def\AuditSheet@{audit.lst}
+ %%%
+ \count255=\time\divide\count255 by 60\relax
+ \edef\temp@{\the\count255}
+ \multiply\count255 by -60\relax
+ \advance\count255 by \time
+ \immediate\openout\unusedout@ \AuditSheet@
+ \WrOut@{\Pct@ auditor.tex output, date
+ \the\day-\the\month-\the\year,
+ time \temp@\string:\the\count255.}%
+ \WrOut@{}
+ \edef\@temp@{\the\MATHchars@toks@}\def\empty{}%
+ \ifx\@temp@\empty\else
+ \WrOut@{\Pct@ Math characters defined via \noexpand\MATHchardef }
+ \WrOut@{\Pct@ or \noexpand\NEWsymbol and really used were\string:}
+ \WrOut@{\the\MATHchars@toks@}
+ \WrOut@{\Pct@ Beware lack of wordwrap\string!}
+ \WrOut@{}
+ \fi
+ \WrOut@{\Pct@ Macros (if any) defined by \string\Def, \string\Let, etc.}%
+ \WrOut@{\Pct@ are listed in order defined\string:}
+ \WrOut@{\Pct@ *Unused* macros among these are marked by *.}%
+ \WrOut@{\Pct@ And \StringSharp\space
+ indicates nesting (hence delayed action).}%
+ \WrSc@{}%
+ \WrOut@{}%
+ \List@M@cs
+ %\showthe\unusedtoks@
+ %\showthe\deftoks@
+ \def\empty{}%
+ \edef\@tmp@{\the\unusedtoks@}%
+ \ifx\@tmp@\empty
+ \EX@\def\EX@\@tmp@\EX@{\the\deftoks@}%
+ \ifx\@tmp@\empty
+ \WrSc@{*** No macros have been defined via \string\Def,
+ \string\Let, etc.}%
+ \else
+ \WrSc@{*** All macros defined via \string\Def,
+ \string\Let, etc. have been used.}%
+ \fi
+ \else
+ \WrSc@{*** The following macros defined via \string\Def,
+ \string\Let, etc. have not been used\string;}%
+ \WrSc@{%
+ *** --- you can probably delete their definitions\string:}%
+ \WrSc@{*** }%
+ \WrSc@{***** \the\unusedtoks@}%
+ \fi
+ \edef\@tmp@{\the\MATHchars@toks@}%
+ \ifx\@tmp@\empty\else
+ \WrSc@{ }%
+ \WrSc@{*** The following math chars defined by \noexpand\MATHchardef}%
+ \WrSc@{*** or \noexpand\NEWsymbol are really used\string:}%
+ \WrSc@{***** \the\MATHchars@toks@}%
+ \fi
+ \WrSc@{}%
+ \WrSc@{*** See the file \string"\AuditSheet@\string" for details.}%
+ \WrSc@{*** See the DefStrip utility for cleanup.}%
+ \WrOut@{}
+ \WrOut@{\the\AuditDepth\noexpand\ITERATIONCOUNTER}
+ \WrOut@{\Pct@ PLEASE iterate ***defstrip macro of QUEDM }
+ \WrOut@{\Pct@ until asterisks disappear from iteration counter line.}
+ \WrOut@{\Pct@ Name successive output files output1, output2, ...}
+ }
+
+ \ifx\undefined\@@end
+ \let\audprim@end@\end
+ \def\end{\The@Audit@\audprim@end@}
+ \else %%% LaTeX
+ \let\audprim@end@\@@end
+ \def\@@end{\The@Audit@\audprim@end@}
+ \fi
+
+ \Modify@Audit
+
+ \let\wlog\auditorwlog@ld
+ \catcode`\@=\auditAt
+
+ %\let\DDef\Def
+
+ \endinput %% comment out for tests
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end code
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% begin tests
+
+ \input amssym.def
+ \input amssym.aud
+
+$\digamma\digamma\varkappa$
+
+\Font\cmr cmr10 scaled 1500
+%\cmr
+
+\end
+
+ %\catcode`\@=11
+
+
+ %\documentstyle{article} %% LaTeX only
+ %\begin{document} %% LaTeX only
+ %\let\Def\def
+ %\let\Let\let
+
+ \Def\Rm#1{\mathop{\fam0#1}}
+
+ \Def \End {\Rm {End}}
+ \Def \Hom {\Rm {Hom}}
+ \Def \ind {\Rm {ind}}
+ \Def \Re {\Rm {Re}}
+ \Def \Tr {\Rm {Tr}}
+ \Def \rk {\Rm {rk}}
+ \Def \rg {\Rm {rg}}
+ \Def \Td {\Rm{Td}}
+ \Def \ch {\Rm{ch}}
+ \Def \T{\Rm{T}}
+ \Def \R{\Rm{R}}
+ \Def \e{\Rm{e}}
+ \Def \odd {\Rm{odd}}
+ \Def \even {\Rm{even}}
+ \Def \Ker {\Rm{Ker}}
+ \Def \id {\Rm{id}}
+ \Def \Pf {\Rm{Pf}}
+
+ \gDef\filler{filler}
+ \gLet\Filler\filler
+
+ %\Filler
+
+ \Font\myfont=cmr10 at 11.5pt
+ %\myfont
+
+ $\End
+ %\Hom
+ \ind
+ \Re
+ \Tr
+ \rk
+ \rg
+ \Td
+ \ch
+ \T
+ \R
+ \e
+ %\odd
+ %\even
+ \Ker
+ \id
+ \Pf $
+
+ \end{document}
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end tests
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% begin doc
+
+
+DOCUMENTATION FOR auditor.tex.
+
+
+ Warnings : These macros are not bulletproof. I believe no such
+macros could be bulletproof. But in practice that should be of
+no importance; hopefully, any significant annoyances will be
+routinely reported and remedied. Use of this alpha version
+with an imbricated format like LaTeX is probably for hardened
+texperts only. Plain and AmSTeX seem to respond well.
+
+ INSTRUCTIONS.
+
+ Suppose you have a macro file x.sty used for typesetting the
+typescript x.tex. The aim is to find out, quickly and easily,
+which of the macro definitions in x.sty are necessary to typeset
+x.tex.
+
+ 1) Modify a *copy* of x.sty to x-aud.sty by altering
+definitions as follows:
+
+ (a) replace \def by \Def for definitions of macros you suspect
+ might be unused in x.tex. Similarly use \gDef in place
+ of \gdef or \global\def.
+
+ (b) similarly replace \mathchardef by \Mathchardef, and so forth
+ as indicated in the list of currently possible substitutions
+ indicated in occam.spc.
+
+Provisional Warnings:
+ --- Do not modify \def's or \let's within other definitions.
+This would be pointless and perhaps dangerous. Be ready to
+revert to \def or \let if trouble ensues.
+ --- Avoid "\outer" and "\long" macros; also \alloc@
+
+ (c) Place the lines
+
+ %%% auditor.tex audits use of macros in a typescript.
+ \input auditor.tex %% Keep this file available to TeX!
+
+at the head of x-aud.sty. (Watch the order!)
+
+ 2) Modify x.tex temporarily replacing
+
+ \input x.sty by \input x-aud.sty
+
+in the header. Typeset as usual. The audit should now proceed
+with an explanatory dialog.
+
+ 3) If there is trouble in step 2), repeat it choosing (by
+dialog) to compose *without* an audit. There should be no
+change from the original behavior of x.tex. Correct any
+misbehavior --- probably arising from a malformation in
+x-aud.sty. Sometimes, here and there in x-aud.sty, one has to
+change \Def back to \def etc.
+
+ 4) Delete the unused definitions in another copy of x.sty, say
+x-min.sty. The list in the "audit.lst" output file is designed to
+make this easy, in fact so easy that a utility called "defstrip" can
+do the job automatically. Then replace x-aud.sty by x-min.sty in
+the header of x.tex. Typeset x.tex and check a proof copy.
+
+ 5) You can now send x.tex and x-min.sty to a colleague or
+publisher, without burdening him/her with useless macros present in
+x.sty.
+
+
+
+
+IMPROVEMENTS CURRENT AND FUTURE
+ A) In all, the list of prepared macros is currently (July
+1994):
+
+ \Def, \Let, \gDef, \gLet, \Font, \Mathchardef, \Newsymbol
+
+Unless the contrary is indicated, each is to be used in analogy
+with \Def replacing in x.sty a corresponding uncapitalized TeX
+primitive.
+
+ [Exception: \newsymbol (from amssym.def) is *not*
+primitive; it is used copiously for declaration of symbols from
+the AmS math fonts msam and msbm, as in amssym.tex.]
+
+ When the file amssym.tex is \input, huge numbers of math
+characters are defined via \newsymbol but only few are used. In
+this case it is appropriate to use variants \MATHchardef,
+\NEWsymbol. which signal math characters only if they are used
+--- and *not* if they are unused. For this, try temporarily
+replacing
+
+ \input amssym.tex
+
+in x-aud.sty (after \input auditor.tex please) by
+
+ \input amssym.aud
+
+where amssym.aud is obtained from amssym.tex by replacing
+\newsymbol by \NEWsymbol; then you get a list of the symbols really
+needed in x.tex.
+
+ In using \Let and \gLet, avoid macros with parameters and
+note that the condition that the second argument be a macro is
+often not satisfied. Do not use = in the syntax.
+
+ \Let, \gLet, and \Font tend to be troublesome; while
+\Mathchardef, \MATHchardef, \Newsymbol, \NEWsymbolseem tend to
+be troublefree.
+
+ The usefullness of the above list can be stretched by the
+user. For example, before \Font was introduced
+
+\font\myfont=cmr10 scaled 11.5pt
+
+would have been replaced by
+
+ \font\myfont@=cmr10 at 11.5pt \Def\myfont{\myfont@}
+
+to report use of \myfont.
+
+ In many formats, user definitions are made via macros not
+available in Plain TeX; for example, \define and \redefine in
+AmSTeX. In practice, they can usually be replaced by \Def for
+the audit. But the programmer may want to invent new macros,
+say \Define and \Redefine for AmSTeX.
+ (B) In case x.sty is of permanent interest, it is a good idea
+to begin to use x-aud.sty instead of x.sty after commenting
+out the line \input auditor.tex and replacing it by something
+like
+
+ %% Audit.tex apparatus
+ \let\Def\def \let\gDef\gdef \let\Let\let
+ \let\Font\font \def\gLet{\global\let}
+ \let\Mathchardef\mathchardef %% extend as necessary
+ % \input auditor.tex %% comment out to suppress audit function
+
+This will make steps (1) to (4) superfluous for your next
+typescript y.tex.
+
+ (C) DefStrip automates the generation of x-min.sty.
+(auditor.tex plus DefStrip make up the "Occam" utility.)
+Its starting materials are "x-aud.sty" plus the "list audit.lst"
+of unused macros provided by auditor.tex. This depends on a
+special arrangement of x-aud.sty described in occam.spc.
+
+ Ultimately, an auxiliary ".tex" program "defstrip.tex"
+will rewrite x-aud.sty omitting the inused definitions. This
+auxiliary program resembles the "docstrip" utility of LaTeX
+fame.
+
+ At the present time defstrip.tex is unavailable. But there
+exists a QUEDM script called Auditor-QUEDM-Macros; QUEDM is a
+editor with convenient "macro" (=composite command) capabilities
+that is available on Macintosh computers at prices as low as $60
+The instructions for Auditor-QUEDM-Macros are found in
+defstrip.hlp.
+
+
+ *** How "auditor.tex" functions or fails to function.
+
+ "auditor.tex" prefixes a reporting device to the expansion
+of macros defined by \Def; this device reports the use of the
+macro by defining a tell-tale auxilliary macro that is then
+polled after typesetting. But it may cause strange behavior or
+even stop TeX.
+
+ As has been mentioned auditor.tex is not bullet-proof. Any
+change whatever in the expansion of a macro can in principle
+alter its behavior. For example TeX can use \ifx and many other
+means to examine the expansion of a macro; it can detect any
+tampering with definitions.
+
+
+ *** The "watchman" mechanism.
+
+ This mechanism is capable of deleting any collection of
+lines of a macro file in response to the non-use of a single
+macro called the "watchman". This "watchman" may be a macro that
+is specially defined for the purpose. The mechanism is
+definitely only for macro files carefully formatted for Occam.
+This mechanism is more powerful but more cumbersome than that
+for \Def etc. It is still to be implemented.
+
+
+CAVEAT LaTeX : LaTeX environments tend to define their user
+macros locally; indeed their definitions are not set up while
+the macro file x.sty is being read but when the environment is
+entered. Thus the modus operandi indicated above is suspect.
+However, normally, the meanings assigned upon entering the
+environment are stored in macros whose name involves @ ; these
+are possibly the macros to spy on with \Def etc. Alternatively,
+the "watchman" mechanism may prove effective. The cleanup based
+on audit.lst is then still a delicate matter requiring texpert
+attention. Further, LaTeX's fragility must be attended to...
+
+
diff --git a/macros/generic/occam/Occam94-old/defstrip.hlp b/macros/generic/occam/Occam94-old/defstrip.hlp
new file mode 100644
index 0000000000..2691b39397
--- /dev/null
+++ b/macros/generic/occam/Occam94-old/defstrip.hlp
@@ -0,0 +1,56 @@
+ Part of the Occam utility.
+ Laurent Siebenmann<lcs@topo.math.u-psud.fr>
+ Master posting 1994, ftp ftp.math.u-psud.fr
+
+
+ June 1994 alpha version
+
+ *************************************************
+ ******
+ ****** DefStrip-QUEDM-Macros Help
+ ******
+ *************************************************
+
+ PURPOSE OF THESE MACROS.
+
+ The main purpose is to use a file "audit.lst" derived from
+audit.tex, and a TeX macro file, x-aud.sty say, in the
+"Occam" format (defined in occam.spc) to derive a new
+simplified ".tex" macro file in which selected unwanted material is
+suppressed. This material is designated by "audit.lst" in
+conjunction with internal formating involving %^, %_, \Def, \Let,
+\gDef, \gLet, \Font, and some others. The user macro
+***defstrip governs this function.
+
+ The secondary purpose is to remove Occam formatting.
+The macro **aud-to-tex governs this function.
+
+
+ DIRECTIONS FOR USE.
+
+ Put together into one folder a copy of DefStrip-QUEDM-Macros
+macro file, the file audit.lst, and the macro file to be simplified.
+
+ Open the DefStrip-QUEDM-Macros macro file. If it is not
+visible, launch QUEDM using it.
+
+ Open the TeX macro file under QUEDM and *activate* its window.
+Then launch the macro ***defstrip from the QUEDM macros menu.
+
+ An audit.lst window will now open and much activity will be
+seen. Finally, a cleaned-up version of the ".tex" macro file will
+appear in a window called "output".
+
+ Only the ***help, ***defstrip, and **aud-to-tex macros
+from the QUEDM macros menu should be launched
+by the user. The others are subroutines.
+
+ See occam.spc for further details, and occam.pub for an
+overview.
+
+
+POSSIBLE PROBLEMS....(please report bugs to to author)
+
+
+
+
diff --git a/macros/generic/occam/Occam94-old/occam.pub b/macros/generic/occam/Occam94-old/occam.pub
new file mode 100644
index 0000000000..37333dbb32
--- /dev/null
+++ b/macros/generic/occam/Occam94-old/occam.pub
@@ -0,0 +1,173 @@
+Part of the Occam utility.
+Laurent Siebenmann <lcs@topo.math.u-psud.fr>
+Master posting 1994, ftp ftp.math.u-psud.fr
+
+
+ *************************************************
+ ******
+ ****** OCCAM'S RAZOR AND MACRO MANAGEMENT
+ ******
+ ****** Laurent Siebenmann
+ ******
+ ****** lcs@topo.math.u-psud.fr
+ ******
+ *************************************************
+
+THE APHORISM:
+
+ entia non sunt multiplicanda praeter necessitatem
+
+ entities should not be multiplied beyond necessity
+
+ William of Occam 1285-1349(?)
+
+\footnote{Occam's Razor is not just a trendy novel of this
+decade; it is the aphorism quoted! Experts believe that Occam
+did not formulated it in exactly these famous words, but rather
+as
+
+ What can be done with fewer assumptions
+ is done in vain with more.
+
+or
+
+ Plurality is not to be assumed without necessity.
+
+The term Principle of Parcimony is also used for the Razor.}
+
+
+THE SOFTWARE:
+ --- occam.pub
+ --- auditor.tex
+ --- DefStrip-QUEDM-Macros (Macintosh only)
+ --- defstrip.hlp help file for above
+ --- occam.spc specifications for source macro files to be minimized
+ --- defstrip.tex (is mere wishful thinking in summer 1994)
+
+MASTER POSTING in 1994:
+ --- ftp ftp.math.u-psud.fr directory pub/TeX/defstrip.dir/
+
+ Have you ever felt guilty about burdening a friend with
+macros that are not really necessary for composing your
+typescript? I certainly have; and would ideally like to
+follow Knuth's example of using macro files which define
+exactly what is necessary for a document and nothing more.
+
+ However, pruning a macro file that has served for
+another purpose is a pain. Most of us respond to this pain
+by adopting a rather messy maximalist approach in which all
+the macros that have a geneology related to the necessary
+macros are transmitted.
+
+ But there is another approach! One can seek efficient
+mechanisms to ease the task of weeding out unnecessary macros.
+
+ One such mechanism is auditor.tex, which sets up a list of
+names of those macros of macro file that turn out to be
+*unnecessary* in a given typescript.
+
+ A complementary tool is the utility DefStrip which
+combines a specially arranged version of the macro file to be
+"cleaned up" with the list of unused macros provided by
+auditor.tex to delete the unneeded macros listed together
+with some related material.
+
+ Ultimately, DefStrip will hopefully be a ".tex" program
+"defstrip.tex" resembling the "docstrip.cmd" utility of LaTeX
+fame. At the present time there exists instead a QUEDM script
+called DefStrip-QUEDM-Macros; QUEDM is a editor with convenient
+"macro" (=composite command) capabilities that is available on
+Macintosh computers.
+
+ DefStrip and Auditor together make up a system called
+"Occam".
+
+ Let us consider two plausible examples of use of the Occam
+system. Only the first has proved genuinely useful thus far.
+
+ (A) Many TeX users build up a cumulative personal macro file
+through composing many articles with TeX. A time inevitably
+comes when it is embarrassing, cumbersome, or confusing to
+submit the whole macro file along with the article. The Occam
+system makes the pruning of the macro file painless. It is
+advisable to tidy up the macro file and maintain it in "Occam"
+format as explained in "occam.spc"; then and only then will
+auditor.tex and DefStrip collaborate to *automatically*
+produce a minimal version of the macro file suitable for the
+article at hand.
+
+ (B) Suppose that one proposes to post in electronic ".tex"
+form an article prepared using the TUGboat macro package.
+(The alternative ".dvi" form is less flexible; for instance the
+".tex" version can be reformatted to be read in comfort on any
+computer screen whereas a ".dvi" version often does not have an
+appropriate width.) Such a macro package is not immune to
+alteration with time and unfortunately the principles of
+upward compatibility are pious hopes, not laws. Consequently,
+one is well-advised to post, along with the article the macros
+necessary to compile it --- especially if modifications to
+the macros have been used. Unfortunately, the TUGboat macros
+are more voluminous than most articles. This is an unfortunate
+obstacle to electronic posting of ".tex" typescripts.
+
+ An attractive solution would be to have a version of the
+TUGboat macros that have been set out in a form suitable for
+use with Occam. Then the necessary macros for a
+given article can quickly be extracted from the total package
+to make the total posting both compact and archival.
+
+ The archival nature of such TeX postings still depends on
+Knuth's Plain format being archival. Plain probably will be at
+least upwards compatible in the best sense. However, the
+article (or a book, say) could perhaps be made archival on the
+scale of many decades by subjecting the Plain macros to the same
+process as the TUGboat macros; this incidentally seems necessary
+to realize best economy. The article would then have its own
+format built with initex. This may seem needlessly radical to
+an English speaking user. But I consider bootstrapping from
+initex the best approach for fully archival ".tex" postings
+where other languages are concerned.
+
+
+Afterthoughts
+
+ 1) Occam as presently realized does not make much sense in the
+LaTeX world. The LaTeX group is building official macro modules
+that cover all needs and are univerally available. I suspect this
+will require continuing exponential growth of the LaTeX project,
+of the computers that run LaTeX, and of the of the time invested
+by LaTeX users. I may be wrong.
+
+ 2) Bootstrapping on initex as mentioned above seems to have a
+parallel in classical programming, namely the use of assembly
+language --- whereas the LaTeX approach is parallel to the
+use of a big and constantly evolving interpreter.
+
+3) Occam's razor was one of the guiding principles of
+scientific thought in the era before the coming of age of
+computers. I suspect the philosophy of Aristotle or Descartes
+is far more likely to appeal to computer scientists. One might
+go so far as to say that Occam's razor has by now been
+discarded --- indeed object oriented programming consciously
+cultivates the art of multiplication of entities, and this
+sort of thing is to be found in TeX in commands such as
+\newheading of LaTeX. What can the minimalism of Occam's
+razor offer TeX users at this late date? Probably just a few
+things.
+
+ (a) Friendliness to human beings. Unnecessary entities
+that cost a microprocessor only a fraction of a second can cost
+the human mind a significant amount of time.
+ (b) Extra storage space and computing power. Both are in a
+period of exponential growth. But so is the TeX software we
+use. Where performance in a fixed task is concerned these
+growths may even cancel one another. When this happens the
+old-fashioned minimalism of Occam's razor can
+help derive pleasure from progress.
+
+
+
+ Laurent Siebenmann
+ <lcs@topo.math.u-psud.fr>
+
+
diff --git a/macros/generic/occam/Occam94-old/occam.spc b/macros/generic/occam/Occam94-old/occam.spc
new file mode 100644
index 0000000000..dbd61b999d
--- /dev/null
+++ b/macros/generic/occam/Occam94-old/occam.spc
@@ -0,0 +1,209 @@
+Part of the Occam utility.
+Laurent Siebenmann <lcs@topo.math.u-psud.fr>
+Master posting 1994, ftp ftp.math.u-psud.fr
+Alpha version 6-94 subject to change.
+
+
+ ====== Occam Syntax and Specifications ======
+
+ Occam is a system for extracting from a large macro file
+exactly those macros required by a given typescript. Its
+active parts are auditor.tex that determines which macros
+are necessary, and DefStrip a utility to deletes unnecessary
+macros.
+
+ The first created DefStrip utility is a QUEDM script; QUEDM
+is a editor with macro capabilities that is available on
+Macintosh computers. (Hopefully a version of this utility
+which is a ".tex" program like auditor.tex will follow in due
+time; it would be very analogous to the LaTeX "docstrip.cmd"
+utility.)
+
+(I) About the QUEDM Version of Summer 1994 (preliminary)
+
+ The Occam syntax is for TeX macro files. Its purpose is to let
+the "DefStrip" utility delete selected lines of the file with the
+help of a list audit.lst of "unused" control sequences (mostly
+macros).
+
+ These lines come in blocks of roughly two sorts:
+
+ (a) Material that is to be unconditionally deleted.
+
+ (b) Blocks surrounding lines beginning (modulo spaces) with
+one of:
+
+ \Def (variant of \def)
+ \gDef (variant of \gdef or \global\def)
+ \Let (variant of \let)
+ \gLet (variant of \global\let)
+ \Font (variant of \font)
+ \Mathchardef (variant of \mathchardef)
+ \Newsymbol (variant of \newsymbol)
+
+This list may be extended. A particular such block is to be deleted
+precisely if the macro name is designated in an external list called
+"audit.lst". output by the TeX utility "auditor.tex".
+
+
+MAIN SPECIFICATIONS of the Occam syntax.
+
+ ASCII (7-bit) text files only. No tab characters please.
+
+ The names of macro files conforming to this syntax should
+involve the suffix "aud" in some form if at all possible. For
+example, "x.sty" might become "x-aud.sty" or "x.aud", say
+"x-aud.sty" for future reference.
+
+ See internal documentation of "audit.tex" to generate a
+list of macros in "x-aud.sty" that are unnecessary in a
+given typesetting job "x.tex".
+
+ In x-aud.sty, the lines
+
+ %^ This file is formatted by <programmer>, <date>, <email>
+ % for use of the Occam utility posted on the CTAN archives
+ % (master posting 1994 on ftp ftp.math.u-psud.fr)
+ %% DO NOT ALTER "OCCAM" SIGNS <percent>^ or <percent>_ , ^_
+ %% UNLESS YOU UNDERSTAND THEM!
+ \let\Def\def \let\gDef\gdef \let\Let\let
+ \def\gLet{\global\let} \let\Font\font
+ \let\Mathchardef\mathchardef\let\Newsymbol\newsymbol
+ \let\MATHchardef\mathchardef\let\NEWsymbol\newsymbol
+ % \input auditor.tex %% keep auditor.tex available
+ %% comment out above line to suppress audit function. %_
+
+should appear in the header.
+
+ Two composite symbols %^ and %_ are employed to
+designate possible deletions. On its line %^ is always
+preceeded by spaces only (zero or more); similarly %_ is
+always followed by spaces only.
+
+ (A) Unconditionally deleted material:
+
+ %%^_ <delete me>
+
+Everything from %%^_ to the end of file is then deleted.
+To delete just a segment use
+
+ %^ <delete me> %_
+
+The deleted material can span many lines, but must include no
+blank line. We have just seen a block of such material above!
+Note that it may well contain \Def etc. but not %^, %_.
+
+ The unconditional deletions will occur in the order
+described, and before conditional deletions are considered.
+
+
+ (B) Conditionally deleted material:
+
+ \Def \somemacro<maybe delete me>%_
+
+may cause deletion of the block of lines beginning with \Def
+etc. and ending with %_. This material is really deleted
+precisely if the macro \somemacro is marked for deletion in
+the the file "audit.lst".
+
+ The material <maybe delete me> must contain no blank line nor %^,
+%_, \Def etc; but it is otherwise arbitrary; in particular, macro
+arguments, comments, and auxiliary definitions are OK.
+
+ Along with this material some additional preceding material
+is deleted, namely contiguous preceeding lines (if any) that (a)
+are nonempty and (b) contain no %_ (but \Def etc; are allowed).
+Typically, such preceeding material might be comments or commands
+"owned" by the macro being deleted. For example the whole block
+
+ %_
+ \ifx\undefined\eightpoint
+ \Def\eightpoint{}
+ \fi %_
+
+will be deleted precisely in case \eightpoint is marked as unused
+in audit.lst. (The first %_ could be replaced by a blank line.)
+
+ Note that %_ is not really a closing delimiter since it can
+exist in arbitrary numbers without belonging to a matching pair.
+For another example, consider:
+
+ \Def\amacro ...%_
+ \newtoks\btoks %_
+ \Def\cmacro ...%_
+
+Here, the the first two %_ prevent \newtoks\btoks being deleted
+--- in all circumstances.
+
+ The example
+
+ \Def\amacro ...
+ \Def\bmacro ...%_
+
+is incorrect because the block beginning with \Def\amacro ...
+contains \Def\bmacro.
+
+ There is a second type of conditional deletion. Suppose
+\amacro is not used and is so designated in audit.lst. It
+often occurs that several *disjoint* blocks of lines should be
+deleted along with \amacro. These blocks should be
+designated as follows:
+
+ %/^\amacro
+ <stuff>
+ %/_
+
+\amacro is called the sentinel (watchman).
+The sentinel's line %/^... must contain nothing more than
+%/^\amacro and blank space. The initial and terminal
+lines will vanish along with <stuff>.
+
+IN SUMMARY: the blocks %^...%_ are unconditionally deleted,
+while a block signalled by \Def, \gDef, etc. with the help of
+%_ and/or blank lines is deleted or not according as the macro
+following \Def etc. is marked for deletion in "audit.lst".
+Similarly for blocks with sentinel macro. None of these blocks
+for conditional or unconditional deletion is allowed to contain
+an empty line nor any extraneous %^,%_,%/^,%/_,%%^_,\Def,
+\gDef, etc. The blocks introduced by \Def, \gDef, etc. include
+material extending backward as far as (but not including) a
+preceding line that is blank or terminated by one of %_,%/_. No
+such extension for blocks introduced by %^, %/^ is allowed ---
+nor would it be helpful.
+
+ Beyond these primary deletions, the utility DefStrip
+performs a few auxiliary tasks:
+
+ --- All remaining \Def, \gDef, etc. are converted to \def,
+\global\def, etc.. Also, if a remaining %_ is alone on its line
+(spaces ignored), the whole line disappears. And each remaining %_
+*not* alone on its line becomes % (this is the only deletion that
+can affect a line that survives.)
+
+ --- any empty line sequence (usually created by the deletion of
+blocks of lines) is reduced to a single empty line.
+
+ --- Residual appearances in x-aud.sty of macros marked for
+deletion in audit.lst will be marked by %%[VESTIGE] (on a new
+following line). They should be considered a failure of the current
+Occam format".
+
+ Users may find the vestiges mentioned above hard to deal with.
+(Can they simply be deleted?) Thus programmers should attempt to
+set up "Occam" formatting so as to assure that vestiges
+never occur; for their part, users should report vestiges to the
+programmers along with the involved audit.lst file from
+auditor.tex.
+
+ It is the programmer's or the user's responsibility to
+assure that the deletions made by the DefStrip utility result
+in a useful TeX macro file. The DefStrip utility is of little
+help here since it does not understand the macros. Thus it is
+expected that programmers take on the task of preparing macro
+files in Occam format. In most cases, anyone who programs TeX
+macros at an intermediate level will find it an easy task to
+put a macro file in Occam format. Beware that a good deal of
+testing and a bit of cleverness is usually necessary to assure
+that the Occam formatting does the job desired and in the most
+efficient way.
+
diff --git a/macros/generic/occam/Occam94-old/sample.dir/Kohler-aud.sty b/macros/generic/occam/Occam94-old/sample.dir/Kohler-aud.sty
new file mode 100644
index 0000000000..439e0f1d4a
--- /dev/null
+++ b/macros/generic/occam/Occam94-old/sample.dir/Kohler-aud.sty
@@ -0,0 +1,493 @@
+
+ %^% RESOURCEamspptSimple ;
+ % Assumes AmSTeX(version) loaded, but NOT AmSppt.sty
+ % These macros are intentionally simple-minded and have
+ % simple syntax; this permits easy alteration (TeX macros are usually
+ % very devious). Don't hesitate to make alterations,
+ % but in that case rename this file!
+ %% This file will become an %_
+ %% auxilliary macros file named
+ %% Kohler.sty
+ %% needed by the .tex file: Kohler.sty
+ %% Date: 29 June, 1994
+ %% Contact: Martine, email: sectop@matups.matups.fr
+
+ %^%% DefStrip apparatus:
+ %% Master posting 1994 on ftp matups.matups.fr
+ %% DO NOT ALTER DEFSTRIP SIGNS <percent>^ or <percent>_
+ %% UNLESS YOU UNDERSTAND THEM!
+ \let\Def\def \let\gDef\gdef \let\Let\let
+ \def\gLet{\global\let} \let\Font\font
+ \input auditor.tex %% comment out to suppress audit function. %_
+
+ %%% Avoid double input
+ %%
+ \bgroup
+ \catcode`\!=12
+ \ifx\auxmacros\relax
+ \immediate\write16{}%
+ \message{ !!! auxmacros already defined !!!}
+ \gdef\auxmacros{\endinput}%
+ \else \global\let\auxmacros\relax
+ \fi
+ \egroup
+ \auxmacros
+
+ \catcode`\@=11
+
+ %%% The following macros will make you call upon auxiliary files
+ %% when needed. %_
+ \bgroup
+ \catcode`\!=12 %_
+ \gDef\Acc{%\show\Acc
+ \errmessage{ !!! You need to input RESOURCEamsAcc.tex !!!}\end
+ }%_
+ \gDef\amsCD{%
+ \errmessage{ !!! You need to input RESOURCEamsCD.tex !!!}\end
+ }%_
+ \gDef\gLinefigure{%
+ \errmessage{ !!! You need to input RESOURCEfig.tex !!!}\end
+ }%_
+ \egroup
+
+ %^%%% Input following as needed
+ %%% or better append them to a copy of this file.
+ %%% before using the DefStrip utility to clean up
+ % \input RESOURCEamsAcc.tex
+ % \input RESOURCEamsAcc.tex
+ % \input RESOURCEamsAcc.tex%_
+
+
+ %%% FONTDEFams stuff defines your fonts;
+ %% depends on your TeX printing system;
+
+ \Font \Bigbf=cmr17 %_
+ %%TimesB scaled 1500
+ \Font \bigbf=cmbx12 %_
+ \Font \biggbf=cmbx12 scaled \magstep 2 %_
+ \Font \bigbxi=cmbxti10 %at 12pt %_
+
+ \Font \textbf=cmb10 %_
+ \Font \bi=TimesBI%cmbti10 %cmbi10 %_
+ \Font \smc=cmcsc10 % %_
+
+ \Font \smallrm=cmr9 %_
+ \Font \smallbf=cmbx9 %=cmb9 %_
+ \Font \smallit=cmti9 %_
+ %\Font \smallbi=cmbti9 %rare %_
+ %\Font \smallsmc=cmcsc9 %rare %_
+
+ \Font \foliofont=cmr10 %_
+
+ %\Font \ss=cmss10 %_
+ %\Font \ssb=cmssbx10 %_
+ %\Font \ssi=cmssi10 %_
+ %\Font \ssbi=cmssbi10 %rare %_
+
+ %\def \bi{\textbf}
+
+ %\Def \Smc#1{{\smc #1}}%_
+ \Def \Bbd#1{{\Bbb #1}}%_
+ \Def \Calig#1{{\Cal #1}}%_
+ \Def \Frak#1{{\frak #1}}%_
+ %\Def \Bold#1{{\bold #1}}%_
+
+ %%% Improve following
+ %\Def \Smallfonts {\def\rm{\smallrm} %_
+ %\Def\it{\smallit}\def\bf{\smallbf}%_
+ % \def\smc{\smallsmc}}%_
+ %\Def \Smallfonts {\eightpoint}%_
+ \Def \Smallfonts {\relax}%_
+
+ \Def \Titlefont {\bigbf}%_
+ \Def \Authorfont {\smc}%_
+ \Def \Headingfont {\bf}%_
+ \Def \Subheadingfont {\bf}%_
+ \Def \Theoremfont {\smc}%_
+ \Def \TheoremTextfont{\it}%_
+ \Def \Prooffont {\smc}%_
+ \Def \Remarkfont {\smc}%_
+ \Def \Diagramfont {\smc}%_
+ \Def \Captionfont{\it}%_
+
+
+ % Font macros
+
+ \long\def \It #1{{\it{#1}\unskip\/}}%_
+
+ \Def \Bi #1{{bi{#1}\unskip\/}}%_
+
+ \Def \Bold #1{{\bold #1}}%_
+ \Def \BF #1{{\bf{#1}\unskip}}%_
+
+ \Def \Bf #1{%
+ {\ifmmode \let\this\Bold
+ \else \let\this\BF \fi
+ \this {#1}}}%
+ %% in math, transcoder provides surrounding brackets,
+ %% in case of preceding accent, so one superfluous layer here? %_
+
+ % \Def\Bbb#1{{\relaxnext@\ifmmode\let\next\Bbb@\else
+ % \def\next{
+ % \Err@{Use \string\Bbb\space only in mathmode}}\fi
+ % \next{#1}}}%_
+
+ \Def \Smc#1{{\smc{#1}\unskip}}%_
+
+ %\def \leftAdmin{\begingroup\mathsurround=0 pt$}%_
+ %\def \rightAdmin{$\endgroup}%_
+
+ \Def \Admin #1{\begingroup\mathsurround=0 pt
+ \leavevmode%p222-3: \hskip -\lastskip for \unskip
+ \ifmmode\hbox{$\roman{{#1}}$}\else$\roman{{#1}}$\fi
+ \endgroup}%
+ %%{{}} prevents misinterpretation of naked numbers %_
+
+ \Def \Rm #1{{\text{\rm\kern 1pt #1\kern .5pt}}}%
+ %%outer {} for SwtCD %_
+
+ \Def \Displaystyle {\displaystyle}%_
+
+ %%% Logical formatting
+ %%
+
+ %mild revision of an AmSTeX definition
+ \Def \endheading{\cr \egroup \egroup \egroup \nobreak\unvbox
+ \headingbox@ \nobreak\medskip\nobreak}%_
+
+ \Def \Medskip{\vskip 6pt plus 10pt minus 2pt}%_
+
+ \Def \Medbreak{\par \ifdim \lastskip <\bigskipamount \removelastskip
+ \penalty -100\Medskip \fi}%_
+
+ \newif\ifAfterHead
+
+ \Def \hRule{}%%=\hrule for diagnosis %_
+
+ \everypar={\global\AfterHeadfalse}%_
+
+ %% \Title block begins
+\newskip\TitleLineskip
+ \TitleLineskip=4pt
+ \def \DeepCr{\unskip\hfil\egroup\par\hfil\bgroup}%_
+ \def \\#1{%
+ \def\test{#1}%
+ \ifx \test\space\def\this{\DeepCr}\else\def\this{\DeepCr #1}\fi
+ \this}
+ \Def \Title{\goodbreak
+ \ifdim\pagetotal>.70\pagegoal
+ \def\this{\vfill\eject}
+ \else\def\this{}\fi \this
+ \vbox\bgroup\Titlefont\def\cr{\DeepCr}
+ \parindent=0 pt\parskip= 0 pt
+ \Titlefont\baselineskip=\fontdimen6\font
+ \lineskip=\TitleLineskip%
+ \lineskiplimit=\baselineskip%
+ \advance\lineskiplimit by -1ex %
+ \hbox to 0pt{}\vskip15pt plus 15pt\bigskip
+ \bgroup \hfil}
+ \def \endTitle {\unskip\hfil\egroup\par \egroup\medskip}
+ %% end of \Title block %_
+
+ %% \Author block begins
+ \newskip\AuthorLineskip
+ \AuthorLineskip=5pt plus 5pt
+ \Def \Author{\nobreak\vskip 20Pt plus 10pt minus 5pt\nobreak
+ \vbox\bgroup\Authorfont\def\cr{\DeepCr}
+ \parindent=0 pt\parskip= 0 pt
+ \Titlefont\baselineskip=\fontdimen6\font
+ \advance\baselineskip by \AuthorLineskip%
+ \bgroup \hfil}%
+ \def \endAuthor {\unskip\hfil\egroup\par \egroup\medskip}%_
+
+ %\Def \Heading{\redefine\headingfont{\Headingfont}\heading}%_
+ %ams ppt sty, not yet used
+ %\Def \endHeading{\endheading \redefine\headingfont{\smc}}%_
+
+ \Def \BoldItemTags{\def\ItemStyle{\Bold}}%_
+ \Def \PlainItemTags{\def\ItemStyle{\Rm}}%_
+ \Def \ItemStyle{\Bold}%_
+ \Def \Item #1{\ifAfterHead\def\this{\nobreak\hRule\nobreak}%\vskip1pt
+ \else\def\this{\smallskip}\fi
+ \ifhmode\else\nointerlineskip\fi
+ \this{\parskip=4pt\noindent
+ \hbox{$\let\bf\relax
+ \ItemStyle{{#1}}$}\enskip}\AfterHeadfalse}%_
+
+ \Def \Itemitem #1{\itemitem{\hbox{$\def\let\relax\let\bf\relax%
+ \ItemStyle{\show\bf{#1}}$}}}%_
+
+ %\def \Item #1{\item{\def\Bf{}\hbox{$\Bold{{#1}}$}}}%_
+ %%simpler def with snugness to heading above
+ % \Def \Itemitem #1{\itemitem{\def\Bf{}\hbox{$\Bold{{#1}}$}}}%
+ %%The \Bf kill necessitated by some nightmare in AmSTeX?? %_
+
+ \Def \Subheading #1{\SubheadingBoldPar{#1}\AfterHeadtrue}%_
+
+ \Def \SubheadingA #1{\SubheadingBoldNopar{#1}\AfterHeadfalse}%_
+
+ \Def \SubheadingB #1{\SubheadingIndBoldNopar{#1}\AfterHeadfalse}%_
+
+ \Def \SubheadingC #1{\SubheadingRmNopar{#1}\AfterHeadfalse}%_
+
+ \edef\everymathholder{\noexpand\AfterHeadfalse\the\everymath}%
+ \expandafter\everymath\expandafter{\everymathholder}%_
+
+ \Def \SubheadingBoldPar #1{%
+ \ifAfterHead\def\this{\nobreak\hRule\nobreak}%
+ \else\def\this{\goodbreak\vskip 10pt plus 2pt\goodbreak
+ \vbox to 0pt{\vss}\smallskip\nobreak}\fi\this
+ \nobreak\noindent{\unskip\bf#1\hfill}\nobreak\vskip0pt
+ \nobreak
+ }%_
+
+ \Def \SubheadingBoldNopar #1{%
+ \ifAfterHead\def\this{\nobreak \hRule\nobreak}%
+ \else\def\this{\Medbreak}\fi
+ \this\vbox to 0pt{\vss}%
+ \vskip 4pt\nobreak\nointerlineskip
+ {\parskip=0pt\noindent\unskip\bf\ignorespaces#1\unskip.\enskip}}%
+ %%\nointerlineskip is nec! %_
+
+ \Def \SubheadingIndBoldNopar #1{\ifAfterHead
+ \def\this{\nobreak \hRule\nobreak}%
+ \else\def\this{\Medbreak}\fi
+ \this\vbox to 0pt{\vss}%
+ \vskip 4pt\nobreak\nointerlineskip
+ {\parskip=0pt\noindent\unskip\indent\bf#1\unskip.\enskip}}%
+ %%\nointerlineskip is nec! %_
+
+ \Def \SubheadingRmNopar #1{%
+ \ifAfterHead\def\this{\nobreak\hRule\nobreak}%
+ \else\let\this\goodbreak\medskip\fi
+ \this\nobreak\noindent{\rm #1\unskip.}\enskip}%
+ %%changeable! %_
+
+ %\Def \Theoremfont{\bf}%_
+ \Def \Theorem #1{\goodbreak\bigskip\par\noindent\Theoremfont #1.
+ \hskip 2pt plus 1pt minus 1pt
+ \begingroup\it
+ \everymath={\ifdim\mathsurround=0pt\def\this{}%
+ \else\def\this{\kern1.5pt}\fi\this}%
+ }%_
+
+ \Def \endTheorem {\endgroup \rm \goodbreak \smallskip}%_
+
+ \Def \Proof#1{\goodbreak \medskip
+ \par\noindent \Prooffont #1\hskip .7pt:\hskip 3pt\rm}%_
+
+ %% A Box for the Quod est demonstrandum:
+\Def\qedbox{\hbox{\vbox{
+ \hrule width0.2cm height0.2pt
+ \hbox to 0.2cm{\vrule height 0.2cm width 0.2pt
+ \hfil\vrule height0.2cm width 0.2pt}
+ \hrule width0.2cm height 0.2pt}\kern1pt}}%_
+
+ %% Typing in \qed makes the qedbox right justified:
+\Def\qed{\ifmmode\qedbox
+ \else\unskip\ \hglue0mm\hfill\qedbox\medskip
+ \goodbreak\fi}%_
+
+ \Def \endProof{\qed\goodbreak\vskip10pt}
+
+ \Def \Remark#1{\medskip \goodbreak\par\noindent{\Remarkfont #1.}}%_
+
+ \Def \endRemark{\medskip \goodbreak}%_
+
+ \Def \Example#1{\Remark {#1}}%_
+ \Def \endExample{\medskip \goodbreak}%_
+
+ \Def \Definition#1{\Remark {#1}}%_
+ \Def \endDefinition{\medskip \goodbreak}%_
+
+ \Def \Cite#1{{\rm\cite{#1}}}%_
+
+ \Def \cite#1{\cite@#1,\endcite@}%_
+ \Def \cite@@#1,{#1}%_
+ \Def \cite@#1,#2\endcite@{\def\temp{#2}%
+ \ifx\temp\empty\relax
+ \def \temp{{\bf[#1]}}%
+ \else\relax
+ \def\temp{{[\bf #1,\rm\ \cite@@#2]}}%
+ \fi\temp}%_
+
+ \def \Benchmark { }
+ \Def \References#1{\begingroup \leftskip=25 pt \parskip=4 pt plus 2 pt
+ \goodbreak \hbox to 1 pt{} \vskip 15 pt plus 10 pt minus 5 pt
+ \centerline{\Headingfont #1}%
+ \frenchspacing \Smallfonts \def \Benchmark{\Refmark}%
+ \def \Refmark##1##2{\par\noindent \llap {##1{##2}\kern 12 pt}\kern 0pt}%
+ \nobreak\vskip 8pt \nobreak}%
+ %%##1##2 secretly is \Cite ##2 %_
+
+ \Def \endReferences {~\unskip\par\endgroup \medskip\goodbreak}%_
+
+
+ \Def \Footnote #1#2{\plainfootnote{#1}{#2}}%_
+
+ %\hbox nec for AMS!!! not Plain
+
+ %Symbols macros
+
+ \Def \preXbox{\hbox{$
+ \vcenter{\hbox{
+ \vrule\vbox to 6.7 pt{
+ \hrule \vfil \hbox to 12 pt{
+ \hfil}%
+ \vfil\hrule}%
+ \vrule}}\hskip 4pt%
+ $}}%_
+
+ \def \Xbox{\raise -.25pt\hbox{\preXbox}}%
+ \def \Nonsense {{~\unskip \kern-3.5 pt %
+ \hbox{\mathsurround=0 pt\Xbox \kern -16.5 pt $>\kern-3pt<$}}}%_
+
+ \Def \Blackbox
+ {\leavevmode\hskip .3pt \vbox{\hrule height 6.9pt
+ \hbox{\hskip 4.5pt}}\hskip .5pt}%
+
+ \Def \<{{$\mathsurround= 0 pt\raise 1.4
+ pt\hbox{$\scriptscriptstyle
+ \langle \kern -1 pt \langle\hskip 2 pt $}$}}%_
+
+ \Def \>{{\mathsurround= 0 pt$\hskip 2 pt \raise 1.4
+ pt\hbox{$\scriptscriptstyle
+ \rangle \kern -1 pt \rangle$}$}}%_
+
+ \Def \Matrix #1{\matrix#1\endmatrix}%_
+
+ \Def \amsMatrix #1\endamsMatrix{\matrix#1\endmatrix}%_
+
+ \Def \amsCases #1\endamsCases{\cases#1\endcases}%_
+
+ %\TagsOnRight
+
+ \def \tagform@#1{\hbox{\rm\Pretag\ignorespaces#1\unskip\Posttag}}%
+ \def \Pretag{(}%
+ \def \Posttag{)}%
+ \def \Eqno #1$${\def\Pretag{}\def\Posttag{}%
+ \tag\Admin{#1}$$}%
+ %^%% too subtle for auditor.tex %_
+
+ \Def \Multline#1\endMultline#2$${\def\Pretag{}\def
+ \Posttag{}\def\Eqno{}
+ \multline#1\endmultline\tag\Admin{#2\hbox to 1sp{}}$$}%_
+
+ \Def \bigMidvert{\kern4pt \big \vert \kern4pt}%_
+
+ \Def \Midvert{\kern3pt \vert \kern3pt}%_
+
+ \Def \Sharp {\mathord{\#}}%_
+
+ \Def \bigConnectedsum {\mathop{\#}\limits}%_
+
+ \Def \Coprod {\mathop{\raise 1.2pt \hbox{$\coprod$}}\limits}%_
+
+ \Def \Lim {\lim\limits}%_
+
+ \Def \Lbrack {{$[\![$}}%_
+
+ \Def \Rbrack {{$]\!]$}}%_
+
+
+ %%% Binomial coef trick macros mentioned in Sweet-teX manual (rare)
+ \Def \Rparen {\right )}%_
+ \Def \Lparen {\futurelet\next \Lptaupe}%
+ \def \Lptaupe{\ifx \next ^ \let\this\LLparen
+ \else \let\this\LLLparen \fi\this}%
+ \def \LLparen {\left ( \Atop}%
+ \def \Atop ^#1_#2{{#1\atop#2}}%
+ \def \LLLparen {\left (}%_
+
+
+ %^%%%%%%%%%%%%%%%% Accents were here%_
+
+ \Def \Cdot{\mathbin{\raise .4 ex \hbox to 3pt {\hss\bf .\hss}}}%_
+
+ \Def \llonguparrow{\bigg\uparrow}%_
+
+ \Def \llongdownarrow{\bigg\downarrow}%_
+
+ %\def \Limgadget #1{\mathrel {\kern-2p\mathop
+ % {\kern3pt #1\kern3pt}\limits}}
+ %%no; redo correctly to replace following? %_
+
+ \Def \llongrightarrow{\kern1pt\mathop
+ {\kern0pt\longrightarrow\kern1pt}\limits}%_
+
+ \Def\llongleftarrow
+ {\kern-2pt\mathop{\kern3pt\longleftarrow\kern3pt}\limits}%_
+
+ \Def \llongtwoheadrarrow {\kern-2pt
+ \mathop{\kern3pt\longrightarrow \kern-14pt
+ \longrightarrow\kern3pt}\limits}%_
+
+ \Def \llongleftrightarrow {\kern-2pt
+ \mathop{\kern3pt\longleftrightarrow\kern3pt}\limits}%_
+
+ \Def \llongmapsto {\kern-2pt
+ \mathop{\kern3pt\longmapsto\kern3pt}\limits}%_
+
+ \Def \rarrow{\rightarrow}%_
+ \Def \larrow{\leftarrow}%_
+
+ \def\Matrix#1{\matrix #1 \endmatrix}
+ %tabs inactivated too! %_
+
+ %^%%%%%%%%%%%%%%%%%%%%%%%%%% CD macros were here%_
+
+ \Def\Trademark{\hbox{\Admin{{}^{\scriptscriptstyle TM}}}}%_
+
+ \Def \AmS{{\textfontii A}\kern -.1667em\lower .5ex\hbox {%
+ \textfontii M}\kern -.125em{\textfontii S}}%_
+
+ %\def\LaTeX{{\rm L\kern-.36em\raise.3ex\hbox{\sc a}\kern-.15em
+ % T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
+
+ \Def\LaTeX{{\rm L\kern-.36em\raise.3ex\hbox
+ {\smc a}\kern-.15em\TeX}}%_
+
+ \Def \Enskip{{\hskip 4pt plus3pt minus2pt}}%_
+
+ \def\Undef{\Nonsense}
+
+ %The following correspond to rare Sweet-teX symbols,
+ %and will produce a X'ed box on your TeX printout
+ %until something better is devised
+ \Def\Bigasterisk{\Undef}%_
+ \Def\Control{\Undef}%_
+ \Def\llongrightarrowtail{\Undef}%_
+ \Def\llongswarrow{\Undef}%_
+ \Def\llongsearrow{\Undef}%_
+ \Def\llongnwarrow{\Undef}%_
+ \Def\llongnearrow{\Undef}%_
+ \Def\llongtwoheadrightarrow{\mathbin
+ {{\longrightarrow} \kern -1850\mu {\rightarrow}}}%_
+
+ \Def \Longeq {\mathop{=\kern-5pt=}\limits}%_
+
+ \Def\Break{\break}%_
+
+ \output{\plainoutput} %%% change if amsppt.sty used
+ \headline{\hss}
+
+ \ifx\undefined\eightpoint
+ \Def\eightpoint{}
+ \fi %_
+
+\catcode`\@=13
+
+
+\Def\dim{\text{\,dim\,}}
+\Def\ch{\text{\,ch\,}}
+\Def\Td{\text{Td}}
+\Def\max{\text{\,max\,}}
+\Def\Pf{\text{\,Pf\,}}
+\Def\End{\text{\,End\,}}
+\Def\loc{\text{\,loc\,}}
+\Def\vol {\text{\,vol\,}}
+\Def\Tr {\text{\,Tr\,}}
+
+
+\endinput
+
diff --git a/macros/generic/occam/Occam94-old/sample.dir/Kohler-ori.sty b/macros/generic/occam/Occam94-old/sample.dir/Kohler-ori.sty
new file mode 100644
index 0000000000..a3009632ad
--- /dev/null
+++ b/macros/generic/occam/Occam94-old/sample.dir/Kohler-ori.sty
@@ -0,0 +1,247 @@
+ % RESOURCEamspptSimple ;
+ % Assumes AmSTeX(version2) loaded, but NOT AmSppt.sty
+ % These macros are intentionally simple minded and have
+ % simple syntax; this permits easy alteration (TeX macros are usually
+ % very devious). Don't hesitate to make alterations,
+ % but in that case rename this file!
+
+ %%% avoid double input
+
+ %avoid double input
+ \bgroup
+ \catcode`\!=12
+ \ifx\RESOURCEams\relax
+ \immediate\write16{}%
+ \message{ !!! RESOURCEams macros already defined !!!}
+ \gdef\RESOURCEams{\endinput}%
+ \else \global\let\RESOURCEams\relax
+ \fi
+ \egroup
+ \RESOURCEams
+
+ \catcode`\@=11
+
+ %%% The following macros will make you call upon auxiliary files
+ %% when needed.
+ \bgroup
+ \catcode`\!=12
+ \gdef\Acc{%\show\Acc%
+ \errmessage{ !!! You need to input RESOURCEamsAcc.tex !!!}\end
+ }
+ \gdef\amsCD{%
+ \errmessage{ !!! You need to input RESOURCEamsCD.tex !!!}\end
+ }
+ \gdef\gLinefigure{%
+ \errmessage{ !!! You need to input RESOURCEfig.tex !!!}\end
+ }
+ \egroup
+
+ %%% Font macros
+ %%
+
+ \def\Bf#1{{\bf#1}}
+
+ \def \Bi #1{{\bi{#1}\unskip\/}}
+
+ \def \Admin #1{\begingroup\mathsurround=0 pt
+ \leavevmode%p222-3: \hskip -\lastskip for \unskip
+ \ifmmode\hbox{$\roman{{#1}}$}\else$\roman{{#1}}$\fi
+ \endgroup }%{{}} prevents misinterpretation of naked numbers
+
+ %%% Logical formatting
+ %%
+
+ \font \titlefont=cmbx10 scaled 1200
+ \long\def\Title#1\endTitle{\heading\titlefont#1\endheading}
+
+\long\def\Author#1\endAuthor{\heading\smc #1\endheading}
+
+\def\Subheading{\medskip\goodbreak\subheading}
+\let\SubheadingA\Subheading
+\let\SubheadingB\Subheading
+\let\SubheadingC\Subheading
+
+ \let \Theorem\proclaim
+ \let \endTheorem\endproclaim
+
+ %\let\Proof\demo %% make substitution by hand
+ %\let\endProof\enddemo
+
+ \def\Proof{\medskip\goodbreak\noindent
+ \hbox{\it Proof.\kern 1em}\ignorespaces}
+
+ %% A Box for the Quod est demonstrandum:
+ \def\qedbox{\hbox{\vbox{
+ \hrule width0.2cm height0.2pt
+ \hbox to 0.2cm{\vrule height 0.2cm width 0.2pt
+ \hfil\vrule height0.2cm width 0.2pt}
+ \hrule width0.2cm height 0.2pt}\kern1pt}}
+
+ %% Typing in \qed makes the qedbox right justified:
+ \def\qed{\ifmmode\qedbox
+ \else\unskip\ \hglue0mm\hfill\qedbox\medskip
+ \goodbreak\fi}
+
+ \def \Remark#1{\medskip \goodbreak\par\noindent{\smc #1.}}
+ \def \endRemark{\medskip \goodbreak}
+
+ \def \Example#1{\Remark {#1}}
+ \def \endExample{\medskip \goodbreak}
+
+ \def \Definition#1{\Remark {#1}}
+ \def \endDefinition{\medskip \goodbreak}
+
+ \def\ItemStyle{\bold}%
+ \def \Item #1{\smallskip
+ \ifhmode\else\nointerlineskip\fi
+ {\parskip=4pt\noindent
+ \hbox{$\def\let\relax\let\bf\relax\ItemStyle{{#1}}$}\enskip}}
+
+ \def \Itemitem #1{\itemitem{\hbox{$\def\let\relax\let\bf\relax%
+ \ItemStyle{#1}$}}}
+
+ \def \Benchmark { }
+ \def \References#1{\begingroup \leftskip=25 pt \parskip=4 pt plus 2 pt
+ \goodbreak \hbox to 1 pt{} \vskip 15 pt plus 10 pt minus 5 pt
+ \centerline{\bf #1}
+ \frenchspacing \def \Benchmark{\Refmark }
+ \def \Refmark##1\cite ##2{%
+ \par\noindent \llap{\null\cite{##2}\quad}\kern 0pt}
+ \nobreak\vskip 8pt \nobreak}
+
+ \def \endReferences {~\unskip\par\endgroup \medskip\goodbreak }
+
+ \def \Footnote #1#2{\plainfootnote{#1}{#2}}
+
+ %Symbols macros
+
+ \def \preXbox{\hbox{$
+ \vcenter{\hbox{
+ \vrule\vbox to 6.7 pt{
+ \hrule \vfil \hbox to 12 pt{
+ \hfil}%
+ \vfil\hrule}%
+ \vrule}}\hskip 4pt%
+ $}}
+
+ \def \Xbox{\raise -.25pt\hbox{\preXbox}}
+
+ \def \Nonsense {{~\unskip \kern-3.5 pt %
+ \hbox{\mathsurround=0 pt\Xbox \kern -16.5 pt $>\kern-3pt<$}}}
+
+ \def \Blackbox
+ {\leavevmode\hskip .3pt \vbox{\hrule height 6.9pt
+ \hbox{\hskip 4.5pt}}\hskip .5pt}%
+
+ \def \<{{$\mathsurround= 0 pt\raise 1.4
+ pt\hbox{$\scriptscriptstyle
+ \langle \kern -1 pt \langle\hskip 2 pt $}$}}
+
+ \def \>{{\mathsurround= 0 pt$\hskip 2 pt \raise 1.4
+ pt\hbox{$\scriptscriptstyle
+ \rangle \kern -1 pt \rangle$}$}}
+
+ \def \Matrix #1{\matrix#1\endmatrix}
+
+ \def \amsMatrix #1\endamsMatrix{\matrix#1\endmatrix}
+
+ \def\amsCases #1\endamsCases{\cases#1\endcases}
+
+ \def\tagform@#1{\hbox{\rm\Pretag\ignorespaces#1\unskip\Posttag}}
+ \def\Pretag{(}
+ \def\Posttag{)}
+
+ \def\Eqno #1$${\def\Pretag{}\def\Posttag{}
+ \tag"\Admin{#1}"$$}
+
+ \def \Multline#1\endMultline#2$${\def\Pretag{}\def\Posttag{}
+ \def\Eqno{}\multline#1\endmultline\tag"\Admin{#2\hbox to 1sp{}}"$$}
+
+ \def \bigMidvert{\kern4pt \big \vert \kern4pt}
+
+ \def \Midvert{\kern3pt \vert \kern3pt}
+
+ \def \Sharp {\mathord{\#}}
+
+ \def \bigConnectedsum {\mathop{\#}\limits}
+
+ \def \Coprod {\mathop{\raise 1.2pt \hbox{$\coprod$}}\limits}
+
+ \def \Lim {\lim\limits}
+
+ \def \Lbrack {{$[\![$}}
+
+ \def \Rbrack {{$]\!]$}}
+
+ %%% Binomial coef trick macros mentioned in Sweet-teX manual (rare)
+ %%
+ \def \Rparen {\right ) }
+ \def \Lparen {\futurelet\next \Lptaupe}
+ \def\Lptaupe{\ifx \next ^ \let\this\LLparen
+ \else \let\this\LLLparen \fi\this}
+ \def\LLparen {\left ( \Atop }
+ \def \Atop ^#1_#2{{#1\atop#2}}
+ \def\LLLparen {\left (}
+
+ %%%%%%%%%%%%%%%% Accents were here
+
+ \def \circleover #1{\overset\circ\to {#1}}
+
+ \def \Cdot{\mathbin{\raise .4 ex \hbox to 3pt {\hss\bf .\hss}}}
+
+ \def \llonguparrow{\bigg\uparrow }
+
+ \def \llongdownarrow{\bigg\downarrow}
+
+ \def \llongrightarrow{\kern 1pt\mathop
+ {\kern0pt\longrightarrow\kern1pt}\limits}
+
+ \def \llongleftarrow{\kern-2pt\mathop
+ {\kern3pt\longleftarrow\kern3pt}\limits}
+
+ \def \llongtwoheadrarrow {\kern-2pt
+ \mathop{\kern3pt\longrightarrow \kern-14pt
+ \longrightarrow\kern3pt}\limits}
+
+ \def \llongleftrightarrow {\kern-2pt
+ \mathop{\kern3pt\longleftrightarrow\kern3pt}\limits}
+
+ \def \llongmapsto {\kern-2pt
+ \mathop{\kern3pt\longmapsto\kern3pt}\limits}
+
+
+ %%%%%%%%%%%%%%%%% CD macros were here
+
+ \def\Trademark{\hbox{\Admin{{}^{\scriptscriptstyle TM}}}}
+
+ \def \AmS{{\textfontii A}\kern -.1667em\lower .5ex\hbox {\textfontii
+ M}\kern -.125em{\textfontii S}}
+
+ \def\LaTeX{{\rm L\kern-.36em\raise.3ex\hbox{\smc a}\kern-.15em
+ \TeX}}
+
+ \def\Undef{\Nonsense }
+
+ %The following correspond to rare Sweet-teX symbols,
+ %and will produce a X'ed box on your TeX printout
+ %until something better is devised
+ \def\Bigasterisk{\Undef}
+ \def\Control{\Undef}
+ \def\llongrightarrowtail{\Undef}
+ \def\llongswarrow{\Undef }
+ \def\llongsearrow{\Undef }
+ \def\llongnwarrow{\Undef }
+ \def\llongnearrow{\Undef }
+ \def\llongtwoheadrightarrow{\mathbin
+ {{\longrightarrow} \kern -1850\mu {\rightarrow}}}
+
+ \def \Longeq {\mathop{=\kern-5pt=}\limits }
+
+%%%% 4 lines in case cmex pointsizes missing
+ \font\eightex=cmex10 at 8pt \skewchar\eightex='60
+ \font\sevenex=cmex10 at 7pt \skewchar\sevenex='60
+ \font\sixex=cmex10 at 6pt \skewchar\sixex='60
+ \tenpoint
+
+ \catcode`\@=13 %active
+
diff --git a/macros/generic/occam/Occam94-old/sample.dir/Kohler.sty b/macros/generic/occam/Occam94-old/sample.dir/Kohler.sty
new file mode 100644
index 0000000000..bfac833af1
--- /dev/null
+++ b/macros/generic/occam/Occam94-old/sample.dir/Kohler.sty
@@ -0,0 +1,277 @@
+
+ %% auxilliary macros file named
+ %% Kohler.sty
+ %% needed by the .tex file: Kohler.sty
+ %% Date: 29 June, 1994
+ %% Contact: Martine, email: sectop@matups.matups.fr
+
+ %%% Avoid double input
+ %%
+ \bgroup
+ \catcode`\!=12
+ \ifx\auxmacros\relax
+ \immediate\write16{}%
+ \message{ !!! auxmacros already defined !!!}
+ \gdef\auxmacros{\endinput}%
+ \else \global\let\auxmacros\relax
+ \fi
+ \egroup
+ \auxmacros
+
+ \catcode`\@=11
+
+ %%% The following macros will make you call upon auxiliary files
+ %% when needed. %
+ \bgroup
+ \catcode`\!=12 %
+ \egroup
+
+ %%% FONTDEFams stuff defines your fonts;
+ %% depends on your TeX printing system;
+
+ %%TimesB scaled 1500
+ \font \bigbf=cmbx12 %
+
+ \font \smc=cmcsc10 % %
+
+ %\Font \smallbi=cmbti9 %rare %
+ %\Font \smallsmc=cmcsc9 %rare %
+
+ %\Font \ss=cmss10 %
+ %\Font \ssb=cmssbx10 %
+ %\Font \ssi=cmssi10 %
+ %\Font \ssbi=cmssbi10 %rare %
+
+ %\def \bi{\textbf}
+
+ %\Def \Smc#1{{\smc #1}}%
+ %\Def \Bold#1{{\bold #1}}%
+
+ %%% Improve following
+ %\Def \Smallfonts {\def\rm{\smallrm} %
+ %\Def\it{\smallit}\def\bf{\smallbf}%
+ % \def\smc{\smallsmc}}%
+ %\Def \Smallfonts {\eightpoint}%
+ \def \Smallfonts {\relax}%
+
+ \def \Titlefont {\bigbf}%
+ \def \Authorfont {\smc}%
+ \def \Headingfont {\bf}%
+ \def \Theoremfont {\smc}%
+ \def \Prooffont {\smc}%
+ \def \Remarkfont {\smc}%
+
+ % Font macros
+
+ \long\def \It #1{{\it{#1}\unskip\/}}%
+
+ \def \Bold #1{{\bold #1}}%
+
+ % \Def\Bbb#1{{\relaxnext@\ifmmode\let\next\Bbb@\else
+ % \def\next{
+ % \Err@{Use \string\Bbb\space only in mathmode}}\fi
+ % \next{#1}}}%
+
+ %\def \leftAdmin{\begingroup\mathsurround=0 pt$}%
+ %\def \rightAdmin{$\endgroup}%
+
+ \def \Admin #1{\begingroup\mathsurround=0 pt
+ \leavevmode%p222-3: \hskip -\lastskip for \unskip
+ \ifmmode\hbox{$\roman{{#1}}$}\else$\roman{{#1}}$\fi
+ \endgroup}%
+ %%{{}} prevents misinterpretation of naked numbers %
+
+ %%% Logical formatting
+ %%
+
+ \newif\ifAfterHead
+
+ \def \hRule{}%%=\hrule for diagnosis %
+
+ \everypar={\global\AfterHeadfalse}%
+
+ %% \Title block begins
+\newskip\TitleLineskip
+ \TitleLineskip=4pt
+ \def \DeepCr{\unskip\hfil\egroup\par\hfil\bgroup}%
+ \def \\#1{%
+ \def\test{#1}%
+ \ifx \test\space\def\this{\DeepCr}\else\def\this{\DeepCr #1}\fi
+ \this}
+ \def \Title{\goodbreak
+ \ifdim\pagetotal>.70\pagegoal
+ \def\this{\vfill\eject}
+ \else\def\this{}\fi \this
+ \vbox\bgroup\Titlefont\def\cr{\DeepCr}
+ \parindent=0 pt\parskip= 0 pt
+ \Titlefont\baselineskip=\fontdimen6\font
+ \lineskip=\TitleLineskip%
+ \lineskiplimit=\baselineskip%
+ \advance\lineskiplimit by -1ex %
+ \hbox to 0pt{}\vskip15pt plus 15pt\bigskip
+ \bgroup \hfil}
+ \def \endTitle {\unskip\hfil\egroup\par \egroup\medskip}
+ %% end of \Title block %
+
+ %% \Author block begins
+ \newskip\AuthorLineskip
+ \AuthorLineskip=5pt plus 5pt
+ \def \Author{\nobreak\vskip 20Pt plus 10pt minus 5pt\nobreak
+ \vbox\bgroup\Authorfont\def\cr{\DeepCr}
+ \parindent=0 pt\parskip= 0 pt
+ \Titlefont\baselineskip=\fontdimen6\font
+ \advance\baselineskip by \AuthorLineskip%
+ \bgroup \hfil}%
+ \def \endAuthor {\unskip\hfil\egroup\par \egroup\medskip}%
+
+ %\Def \Heading{\redefine\headingfont{\Headingfont}\heading}%
+ %ams ppt sty, not yet used
+ %\Def \endHeading{\endheading \redefine\headingfont{\smc}}%
+
+ \def \ItemStyle{\Bold}%
+ \def \Item #1{\ifAfterHead\def\this{\nobreak\hRule\nobreak}%\vskip1pt
+ \else\def\this{\smallskip}\fi
+ \ifhmode\else\nointerlineskip\fi
+ \this{\parskip=4pt\noindent
+ \hbox{$\let\bf\relax
+ \ItemStyle{{#1}}$}\enskip}\AfterHeadfalse}%
+
+ %\def \Item #1{\item{\def\Bf{}\hbox{$\Bold{{#1}}$}}}%
+ %%simpler def with snugness to heading above
+ % \Def \Itemitem #1{\itemitem{\def\Bf{}\hbox{$\Bold{{#1}}$}}}%
+ %%The \Bf kill necessitated by some nightmare in AmSTeX?? %
+
+ \def \Subheading #1{\SubheadingBoldPar{#1}\AfterHeadtrue}%
+
+ \edef\everymathholder{\noexpand\AfterHeadfalse\the\everymath}%
+ \expandafter\everymath\expandafter{\everymathholder}%
+
+ \def \SubheadingBoldPar #1{%
+ \ifAfterHead\def\this{\nobreak\hRule\nobreak}%
+ \else\def\this{\goodbreak\vskip 10pt plus 2pt\goodbreak
+ \vbox to 0pt{\vss}\smallskip\nobreak}\fi\this
+ \nobreak\noindent{\unskip\bf#1\hfill}\nobreak\vskip0pt
+ \nobreak
+ }%
+
+ %\Def \Theoremfont{\bf}%
+ \def \Theorem #1{\goodbreak\bigskip\par\noindent\Theoremfont #1.
+ \hskip 2pt plus 1pt minus 1pt
+ \begingroup\it
+ \everymath={\ifdim\mathsurround=0pt\def\this{}%
+ \else\def\this{\kern1.5pt}\fi\this}%
+ }%
+
+ \def \endTheorem {\endgroup \rm \goodbreak \smallskip}%
+
+ \def \Proof#1{\goodbreak \medskip
+ \par\noindent \Prooffont #1\hskip .7pt:\hskip 3pt\rm}%
+
+ %% A Box for the Quod est demonstrandum:
+\def\qedbox{\hbox{\vbox{
+ \hrule width0.2cm height0.2pt
+ \hbox to 0.2cm{\vrule height 0.2cm width 0.2pt
+ \hfil\vrule height0.2cm width 0.2pt}
+ \hrule width0.2cm height 0.2pt}\kern1pt}}%_
+
+ %% Typing in \qed makes the qedbox right justified:
+\def\qed{\ifmmode\qedbox
+ \else\unskip\ \hglue0mm\hfill\qedbox\medskip
+ \goodbreak\fi}%_
+
+ \def \endProof{\qed\goodbreak\vskip10pt}
+
+
+ \def \Remark#1{\medskip \goodbreak\par\noindent{\Remarkfont #1.}}%
+
+ \def \Definition#1{\Remark {#1}}%
+ \def \endDefinition{\medskip \goodbreak}%
+
+ \def \cite#1{\cite@#1,\endcite@}%
+ \def \cite@@#1,{#1}%
+ \def \cite@#1,#2\endcite@{\def\temp{#2}%
+ \ifx\temp\empty\relax
+ \def \temp{{\bf[#1]}}%
+ \else\relax
+ \def\temp{{[\bf #1,\rm\ \cite@@#2]}}%
+ \fi\temp}%
+
+ \def \Benchmark { }
+ \def \References#1{\begingroup \leftskip=25 pt \parskip=4 pt plus 2 pt
+ \goodbreak \hbox to 1 pt{} \vskip 15 pt plus 10 pt minus 5 pt
+ \centerline{\Headingfont #1}%
+ \frenchspacing \Smallfonts \def \Benchmark{\Refmark}%
+ \def \Refmark##1##2{\par\noindent \llap {##1{##2}\kern 12 pt}\kern 0pt}%
+ \nobreak\vskip 8pt \nobreak}%
+ %%##1##2 secretly is \Cite ##2 %
+
+ \def \endReferences {~\unskip\par\endgroup \medskip\goodbreak}%
+
+ %\hbox nec for AMS!!! not Plain
+
+ %Symbols macros
+
+ \def \Xbox{\raise -.25pt\hbox{\preXbox}}%
+ \def \Nonsense {{~\unskip \kern-3.5 pt %
+ \hbox{\mathsurround=0 pt\Xbox \kern -16.5 pt $>\kern-3pt<$}}}%
+
+ \def \Blackbox
+ {\leavevmode\hskip .3pt \vbox{\hrule height 6.9pt
+ \hbox{\hskip 4.5pt}}\hskip .5pt}%
+
+ \def \<{{$\mathsurround= 0 pt\raise 1.4
+ pt\hbox{$\scriptscriptstyle
+ \langle \kern -1 pt \langle\hskip 2 pt $}$}}%
+
+ \def \>{{\mathsurround= 0 pt$\hskip 2 pt \raise 1.4
+ pt\hbox{$\scriptscriptstyle
+ \rangle \kern -1 pt \rangle$}$}}%
+
+ %\TagsOnRight
+
+ \def \tagform@#1{\hbox{\rm\Pretag\ignorespaces#1\unskip\Posttag}}%
+ \def \Pretag{(}%
+ \def \Posttag{)}%
+ \def \Eqno #1$${\def\Pretag{}\def\Posttag{}%
+ \tag\Admin{#1}$$}%
+
+ \def \Multline#1\endMultline#2$${\def\Pretag{}\def
+ \Posttag{}\def\Eqno{}
+ \multline#1\endmultline\tag\Admin{#2\hbox to 1sp{}}$$}%
+
+ %\def \Limgadget #1{\mathrel {\kern-2p\mathop
+ % {\kern3pt #1\kern3pt}\limits}}
+ %%no; redo correctly to replace following? %
+
+ \def \llongrightarrow{\kern1pt\mathop
+ {\kern0pt\longrightarrow\kern1pt}\limits}%
+
+ \def\Matrix#1{\matrix #1 \endmatrix}
+ %tabs inactivated too! %
+
+ %\def\LaTeX{{\rm L\kern-.36em\raise.3ex\hbox{\sc a}\kern-.15em
+ % T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
+
+ \def\Undef{\Nonsense}
+
+ \output{\plainoutput} %%% change if amsppt.sty used
+ \headline{\hss}
+
+ \ifx\undefined\eightpoint
+ \def \eightpoint{}
+ \fi %
+
+\catcode`\@=13
+
+ \def \dim{\text{\,dim\,}}
+ \def \ch{\text{\,ch\,}}
+ \def \Td{\text{Td}}
+ \def \max{\text{\,max\,}}
+ \def \Pf{\text{\,Pf\,}}
+ \def \End{\text{\,End\,}}
+ \def \loc{\text{\,loc\,}}
+ \def \vol {\text{\,vol\,}}
+ \def \Tr {\text{\,Tr\,}}
+
+\endinput
+
diff --git a/macros/generic/occam/Occam94-old/sample.dir/Kohler.tex b/macros/generic/occam/Occam94-old/sample.dir/Kohler.tex
new file mode 100644
index 0000000000..9ba9d3e171
--- /dev/null
+++ b/macros/generic/occam/Occam94-old/sample.dir/Kohler.tex
@@ -0,0 +1,1169 @@
+
+%\input amstex
+%\input amsppt.sty
+
+\input Kohler.sty
+
+ \hsize= 14truecm
+
+\Title
+Analytic torsion forms on torus fibrations
+\endTitle
+\bigskip
+\bigskip
+
+\Author
+Kai K{\"O}HLER
+\endAuthor
+\vskip 20mm
+
+{
+{\noindent\smc abstract} : \eightpoint We construct
+analytic torsion forms on holomorphic Torus
+fibrations, which are not necessarily K{\"a}hler
+fibrations. This is done by doubly transgressing the
+top Chern class. Also we establish a corresponding
+double transgression formula and an anomaly formula.
+}
+
+
+\Subheading {0. Introduction} The purpose of this
+paper is to construct analytic torsion forms for torus
+fibrations, which are not necessarily K{\"a}hler
+fibrations. These forms are needed to construct direct
+images in the hermitian $K$-theory, which was
+developped by Gillet and Soul\'e \cite{GS1} in the
+context of Arakelov geometry.
+
+Let $\pi :M\rightarrow B$ be a holomorphic submersion
+with compact basis $B$, compact fibres $Z$ and a
+K{\"a}hler metric $g^{TZ}$ on the fibres. Let $\xi $
+be a holomorphic vector bundle on $M$, equipped with a
+hermitian metric $h^{\xi }$. Then one could try to
+define analytic torsion forms $T$ associated to $\pi
+$, i.e. real forms on $B$, sums of forms of type
+$(p,p)$, defined modulo~$\partial $- and $\overline
+\partial $-coboundaries. They have to satisfy a
+particular double transgression formula and when the
+metrics $g^{TZ}$ and $h^{\xi }$ change, they have to
+change in a special way to make the forms ``natural''
+in Arakelov geometry. They must not depend on metrics
+on $B$, and their component in degree zero should be
+the logarithm of the ordinary Ray-Singer torsion
+\cite{RS}.
+
+Such forms were first constructed by Bismut, Gillet
+and Soul\'e \cite{BGS2, Th.2.20} for locally
+K{\"a}hler fibrations and $H^{*}(Z_{b},\xi
+\vert_{Z_{b}})=0\enskip \forall b\in B$. Gillet and
+Soul\'e \cite{GS2} and after them Faltings \cite{F}
+suggested definitions for more general cases. Then
+Bismut and the author gave in \cite{BK} an explicit
+construction of torsion forms $T$ for K{\"a}hler
+fibrations with $\dim H^{*}(Z_{b},\xi
+\vert_{Z_{b}})=\text{const. on }B$. $T$ satisfies the
+double transgression formula
+$$
+{\overline \partial \partial \over 2\pi i}T=
+\ch\big(H^{*}(Z,\xi \vert_{Z}),h^{H^{*}(Z,\xi
+\vert_{Z})}\big)
+ -\displaystyle \int
+_{Z}\Td(TZ,g^{TZ})\ch(\xi ,h^{\xi })
+\Eqno (0.0)$$
+and for two pairs of metrics $(g_{0}^{TZ},h_{0}^{\xi
+})$ and $(g_{1}^{TZ},h_{1}^{\xi })$,
+ $T$ satisfies the anomaly formula
+$$\Multline
+T(g_{1}^{TZ},h_{1}^{\xi })-T(g_{0}^{TZ},h_{0}^{\xi })
+ =\widetilde {\ch}(H^{*}(Z,\xi
+\vert_{Z}),h_{0}^{H^{*}(Z,\xi
+\vert_{Z})},h_{1}^{H^{*}(Z,\xi \vert_{Z})}) \\
+-\displaystyle \int _{Z}\left(\widetilde
+{\Td}(TZ,g_{0}^{TZ},g_{1}^{TZ})
+ \ch(\xi ,h^{\xi
+}_{0})+\Td(TZ,g_{1}^{TZ})\widetilde {\ch}(\xi
+,h_{0}^{\xi },h_{1}^{\xi })\right)
+\endMultline
+\Eqno (0.1)$$
+modulo $\partial $- and $\overline \partial
+$-coboundaries. Here $\int _{Z}$ denotes the integral
+along the fibres, $\Td$ and $\ch$ are the Chern-Weil
+forms associated to the corresponding holomorphic
+hermitian connections and $\widetilde {\Td}$ and
+$\widetilde {\ch}$ denote Bott-Chern forms as
+constructed in \cite{BGS1, {\S }1f}.
+
+In this paper, we shall construct analytic torsion
+forms $T$ in the following situation: consider a
+holomorphic hermitian vector bundle $\pi
+:(E^{1,0},g^{E})\rightarrow B$ on a compact complex
+manifold. Let $\Lambda $ be a lattice, spanning the
+underlying real bundle $E$ of $E^{1,0}$, so that local
+sections of $\Lambda $ are holomorphic sections of
+$E^{1,0}$. Then the fibration $E/\Lambda \rightarrow
+B$ is a holomorphic torus fibration which is not
+necessarily flat as a complex fibration.
+
+In this situation, $H^{*}(Z,{\Cal O}_{Z})=\Lambda
+^{*}E^{*0,1}$. Classically, the formula
+$$
+\ch(\Lambda ^{*}E^{*0,1})={c_{\max}\over \Td}(E^{0,1})
+\Eqno (0.2)
+$$
+holds on the cohomological level (see e.g. \cite{H,
+Th.10.11}). If one assumes supplementary that the
+volume of the fibres $Z$ is equal to 1, (0.2) holds
+also on the level of forms for the associated
+Chern-Weil forms. Thus, (0.1) suggests that $T$ should
+satisfy
+$$
+{\overline \partial \partial \over 2\pi i}T(E/\Lambda
+,g^{E})={c_{\max}\over \Td}(E^{0,1},g^{E})\enskip
+\enskip .\Eqno (0.3)
+$$
+
+For two hermitian structures $g_{0}^{E}$ and
+$g_{1}^{E}$ on $E$, one should find the following
+anomaly formula
+$$
+T(E/\Lambda ,g_{1}^{E})-T(E/\Lambda ,g_{0}^{E})
+=\widetilde
+{\Td^{-1}}(g_{0}^{E},g_{1}^{E})c_{\max}(g_{0}^{E})+%'
+\Td^{-1}(g_{1}^{E})\widetilde
+{\ch}(g_{0}^{E},g_{1}^{E})\enskip . \Eqno (0.4)
+$$
+
+In this paper, such a $T$ will be constructed by
+explicitly doubly transgressing the top Chern class of
+$E^{0,1}$, which was proven to be 0 in cohomology by
+Sullivan \cite{S}.
+
+Our method is closely following an article of Bismut
+and Cheeger \cite{BC}, in which they investigate eta
+invariants on real {\Blackbox}{\Blackbox} $(2n,{\Bbb
+Z})$ vector bundles. In this article, they are
+considering a quotient of a Riemannian vector bundle
+by a lattice bundle. Then they found a Fourier
+decomposition of the infinite dimensional bundle of
+sections on the fibres $Z$, which allowed them to
+transgress the Euler class explicitly via an
+Eisenstein series $\gamma $, i.e.
+$$
+d\gamma =\Pf\left({\Omega ^{E}\over 2\pi
+}\right)\enskip \enskip ,$$
+where $\Pf$ denotes the Pfaffian and $\Omega ^{E}$ the
+curvature.
+
+The case considered here is a bit more sophisticated
+because not only the metric but also the complex
+structure has not to have any direct relation with the
+flat structure. It turns out that the right choice for
+the holomorphic structure on $E^{0,1}$ is not, as in
+\cite{BK}, the by the metric induced structure, but an
+exotic holomorphic structure canonically induced by
+the flat structure on $E$ and the holomorphic
+structure on $E^{1,0}$.
+
+We want to emphasize that here, as in \cite{BC}, the
+use of certain formulas in the Mathai-Quillen calculus
+\cite{MQ} is crucial. The formulas which we are using
+were established by Bismut, Gillet and Soul\'e in
+\cite{BGS5}.
+
+\Subheading {I. Definitions} Let $\pi
+:E^{1,0}\rightarrow B$ be a $n$-dimensional
+holomorphic vector bundle on a compact complex
+manifold $B$, with underlying real bundle $E$. Assume
+a lattice bundle $\Lambda \subset E$, spanning the
+realisation of $E^{1,0}$, so that a local section of
+$\Lambda $ induces a holomorphic section of $E^{1,0}$.
+Let $M$ be the total space of the fibration $E/\Lambda
+$, where the fibre $Z_{x}$ over a point $x\in B$ is
+given by the torus $E_{x}/\Lambda _{x}$. We call $J$
+the different complex structures acting on $E$, $TM$
+or $TB$ with $J\circ J=-1$.
+
+Let $E^{*}$ be the dual bundle to $E$, equipped with
+the complex structure
+$$
+(J\mu )(\lambda ):=\mu (J\lambda )\enskip \enskip
+\enskip \forall \mu \in E^{*}\enskip ,\enskip \enskip
+\lambda \in E\enskip \enskip . \Eqno (1.0)
+$$
+
+In the same way, one defines $T^{*}B$ and $T^{*}M$. We
+get
+$$E^{1,0}=\lbrace \lambda \in E\otimes {\Bbb C}\vert
+J\lambda =i\lambda \rbrace \enskip \enskip , \Eqno
+(1.1)
+$$
+$$
+E^{0,1}=\lbrace \lambda \in E\otimes {\Bbb C}\vert
+J\lambda =-i\lambda \rbrace \enskip \enskip , \Eqno
+(1.2)
+$$
+and similar equations for $E^{*\,1,0}$, $E^{*\,0,1}$,
+$T^{1,0}M$, $T^{0,1}M$, etc.
+
+
+For $\lambda \in E$, we define
+$$
+ \lambda ^{1,0}:={\textstyle {1\over 2}}(\lambda
+-iJ\lambda )\enskip \enskip \enskip \text{and}\enskip
+\enskip \enskip \lambda ^{0,1}:={\textstyle {1\over
+2}}(\lambda +iJ\lambda )\enskip \enskip , \Eqno (1.3)
+$$
+and in the same manner maps $E^{*}\rightarrow
+E^{*\,1,0}$, $TB\rightarrow T^{1,0}B$, etc. Let
+$\Lambda ^{*}\in E^{*}$ be the dual lattice bundle
+$$
+ \Lambda ^{*}:=\lbrace \mu \in E^{*}\vert \mu (\lambda
+)\in 2\pi {\Bbb Z}\enskip \forall \lambda \in \Lambda
+\rbrace \enskip \enskip . \Eqno (1.4)
+$$
+We set $\Lambda ^{1,0}:=\lbrace \lambda ^{1,0}\vert
+\lambda \in \Lambda \rbrace $, similar for $\Lambda
+^{0,1}$, $\Lambda ^{*\,1,0}$ and $\Lambda ^{*\,0,1}$.
+Also we fix a Hermitian metric $g^{E}=\left \langle
+\enskip ,\enskip \right \rangle $ on $E$, i.e. a
+Riemannian metric with the property
+$$
+ \left \langle J\lambda ,J\eta \right \rangle =\left
+\langle \lambda ,\eta \right \rangle \enskip \enskip
+\forall \lambda ,\eta \in E\enskip \enskip . \Eqno
+(1.5)
+$$
+This induces a Hermitian metric canonically on
+$E^{*}$. We assume the volumes of the fibres $Z$ of
+$M$ to be equal to $1$.
+
+\Subheading {II. Some connections} Now one finds
+several canonical connections on $E$. First, the
+lattices $\Lambda $ and $\Lambda ^{*}$ induce
+(compatible) flat connections $\nabla $ on $E$ and
+$E^{*}$ by $\nabla \lambda :=0$ for all local sections
+ $\lambda $ of $\Lambda $ (resp. $\nabla \mu :=0$ for
+$\mu \in \Gamma ^{\text{loc}}(\Lambda )$). We shall
+always use the same symbol for a connection on
+$E^{1,0}$, its conjugate on $E^{0,1}$, its realisation
+on $E$ and by duality induced connections on
+$E^{*\,1,0}$, $E^{*\,0,1}$ and $E^{*}$.
+
+Generally, the connection $\nabla $ is not compatible
+with the complex structure $J$ (i.e. $\nabla
+J\mathbin{\not =}0$), so it does not extend to
+$E^{1,0}$. $\nabla $ induces a splitting
+$$
+ TM=\pi ^{*}E\oplus T^{H}M \Eqno (2.0)
+$$
+of the tangent space of $M$.
+
+\Theorem {Proposition} $T^{H}M$ is a complex
+subbundle of $TM$.\endTheorem
+
+\Proof {Proof} At a point $(x,\Sigma \alpha
+_{i}\lambda _{i})\in M$, $x\in B$, $\alpha _{i}\in
+{\Bbb R}$, $\lambda _{i}\in \Lambda _{x}$, $T^{H}M$ is
+equal to the image of the homomorphism
+$$
+ \Sigma \alpha _{i}\,T_{x}\lambda
+_{i}\,:\,TB\llongrightarrow TM\enskip \enskip .
+$$
+The latter commutes with $J$ by the holomorphy
+condition on $\Lambda $. Thus, $T^{H}M$ is invariant
+by $J$.\qed
+
+The horizontal lift of $Y\in TB$ to $T^{H}M$ will be
+denoted by $Y^{H}$. Let $\overline \partial ^{E}$ be
+the Dolbeault operator on $E^{1,0}$. Now one can use
+$\nabla $ to construct a canonical holomorphic
+connection $\nabla ^{h}$ on $E^{1,0}$, not depending
+on the metric; furthermore, we will see that $\nabla
+^{h}$ induces a canonical holomorphic structure
+$\overline \partial ^{\overline E}$ on $E^{*\,0,1}$
+with the property
+$$
+\overline \partial ^{\overline E}\mu ^{0,1}=0\enskip
+\enskip \enskip \forall \mu \in \Lambda ^{*}\enskip
+\enskip . \Eqno (2.1)
+$$
+
+Let us denote by $\nabla '\lambda $, $\nabla ''\lambda
+$ the restrictions of $\nabla .\lambda :TB\otimes
+{\Bbb C}\llongrightarrow E\otimes {\Bbb C}$ to
+$T^{1,0}B$ and $T^{0,1}B$ (we will use the same
+convention for all connections and for $\End(E\otimes
+{\Bbb C})$-valued one forms on $B$).
+
+\Theorem {Lemma 1} $\nabla '$ maps $\Gamma
+(E^{1,0})$ into $\Gamma (T^{1,0}B\otimes E^{1,0})$.
+The connection on $E^{1,0}$
+$$
+ \nabla ^{h}:=\nabla '+\overline \partial ^{E} \Eqno
+(2.2)
+$$
+is a holomorphic connection. Its curvature $(\nabla
+^{h})^{2}$ is a $(1,1)$-form.
+
+The dual connection on $E^{*}$ satisfies
+$$
+ \nabla ^{h''}\mu ^{0,1}=0\enskip \enskip \forall
+\lambda \in \Lambda ^{*}\enskip \enskip ; \Eqno (2.3)
+$$
+hence it induces a canonical holomorphic structure
+$\overline \partial ^{\overline E}$ on $E^{*\,0,1}$,
+depending only on the flat structure on $E$ and the
+holomorphic structure on $E^{1,0}$.\endTheorem
+
+\Proof {Proof} The lift of $\nabla $ to $M$ is given
+by
+$$
+ (\pi ^{*}\nabla )_{Y^{H}}Z=[Y^{H},Z]\enskip \enskip
+\forall Z\in \Gamma (TZ)\cong \Gamma (TE),Y\in \Gamma
+(TB)\enskip \enskip , \Eqno (2.4)
+$$
+in particular
+$$
+(\pi ^{*}\nabla )_{Y^{H\,1,0}}(\pi ^{*}\lambda
+^{1,0})=[Y^{H^{1,0}},\pi ^{*}\lambda ^{1,0}]\enskip
+\enskip \forall \lambda \in \Gamma (E)\enskip \enskip
+. \Eqno (2.5)
+$$
+The r.h.s. of (2.5) takes values in $T^{1,0}Z$, hence
+$\nabla '$ maps in fact $E^{1,0}$ to $E^{1,0}$ (this
+is equivalent to the equation
+$$
+ \nabla _{JY}J=J\nabla _{Y}J\enskip \enskip \forall
+Y\in TB\enskip \enskip )\enskip . \Eqno (2.6)
+$$
+This proves the first part of the Lemma. Now one
+computes for $\mu \in \Gamma ^{\text{loc}}(\Lambda
+^{*})$, $\lambda \in \Gamma ^{\text{loc}}(\Lambda )$
+$$\Multline
+ 0=\overline \partial (\mu (\lambda ))=(\nabla
+^{h''}\mu ^{0,1})(\lambda ^{0,1})+(\nabla ^{h''}\mu
+^{1,0})(\lambda ^{1,0})\\
+ +\mu ^{0,1}(\nabla ^{h''}\lambda ^{0,1})+\mu
+^{1,0}(\nabla ^{h''}\lambda ^{1,0})\enskip
+.\endMultline
+\Eqno (2.7)$$
+By condition, $\nabla ^{h''}\lambda ^{1,0}=0$; also
+$0=\nabla ''\mu =\nabla ''\mu ^{1,0}+\nabla ''\mu
+^{0,1}$, so
+$$\Multline
+ 0=-\overline \partial (\mu ^{0,1}(\lambda ^{1,0}))
+ =(-\nabla ''\mu ^{0,1})(\lambda ^{1,0})+\mu
+^{0,1}(-\nabla ''\lambda ^{1,0})\\
+ =(\nabla ^{h''}\mu ^{1,0})(\lambda ^{1,0})+\mu
+^{0,1}(\nabla ^{h''}\lambda ^{0,1})\enskip .
+\endMultline
+ \Eqno (2.8)$$
+
+This proves the second part of the Lemma.\qed
+
+In fact, one could simply verify that $\nabla ^{h}$ is
+just the ``complexification'' of $\nabla $
+$$
+ \nabla ^{h}=\nabla -{\textstyle {1\over 2}}J\nabla J
+\Eqno (2.9)
+$$
+both on $E$ and $E^{*}$.
+
+The metric $\left \langle \cdot ,\cdot \right \rangle
+$ induces an isomorphism of real vector bundles
+\hbox{${\frak i}:E\rightarrow E^{*}$,} so that ${\frak
+i}\circ J=-J\circ {\frak i}$.
+
+\Definition {Definition} Let $\nabla ^{\overline E}$
+be the hermitian holomorphic connection on
+$E^{*\,0,1}$ associated to the canonical holomorphic
+structure in Lemma~1. We denote by ${}^{t}\theta
+^{*}:TB\otimes {\Bbb C}\rightarrow \End(E^{*}\otimes
+{\Bbb C})$ the one-form given by
+$$
+ {}^{t}\theta ^{*}:=\nabla -\nabla ^{\overline E}
+\Eqno (2.10)
+$$
+and by $\vartheta $ the one-form on $B$ with
+coefficients in $\End(E^{*})$
+$$
+ \vartheta _{Y}:={\frak i}^{-1}\nabla {\frak
+i}\enskip \enskip \forall Y\in TB\enskip \enskip .
+\Eqno (2.11)
+$$\endDefinition
+ $\nabla ^{\overline E}$ should not be confused with
+the hermitian holomorphic connection on $E^{1,0}$
+associated to its original holomorphic structure,
+which we shall not use in this article.
+
+The transposed of ${}^{t}\theta ^{*}$ with respect to
+the natural pairing $E\otimes E^{*}\rightarrow {\Bbb
+R}$ will be denoted by $\theta ^{*}$, thus
+$$
+({}^{t}\theta ^{*}\mu )(\lambda )=\mu (\theta
+^{*}\lambda )\enskip \enskip \forall \mu \in
+E^{*}\enskip ,\enskip \enskip \lambda \in E\enskip
+\enskip . \Eqno (2.12)
+$$
+The duals of ${}^{t}\theta ^{*}$ and $\theta ^{*}$
+will be denoted by ${}^{t}\theta $ and $\theta $. This
+notation is chosen to be compatible with the notation
+in \cite{BC}. By definition, ${}^{t}\theta ^{*}$
+satisfies
+$$\aligned
+{}^{t}\theta ^{*}{}'' & :E\otimes {\Bbb
+C}\llongrightarrow E^{1,0}\enskip \enskip , \\
+{}^{t}\theta ^{*}{}' &: E\otimes {\Bbb
+C}\llongrightarrow E^{0,1}\enskip \enskip .
+\endaligned \Eqno (2.13)
+$$
+Notice that the connection $\nabla +{\Cal V}$ on
+$E^{*}$ is just the pullback of $\nabla $ by the
+isomorphism ${\frak i}^{-1}$.
+
+\Theorem {Lemma 2} The hermitian connection $\nabla
+^{\overline E}$ on $E^{*\,0,1}$ is given by
+$$
+ \nabla ^{\overline E}=(\nabla +\vartheta )'+\overline
+\partial ^{\overline E}=\nabla ^{h}+\vartheta '\enskip
+\enskip . \Eqno (2.14)
+$$
+Its curvature on $E^{*\,0,1}$ is given by
+$$
+\Omega ^{\overline E}= \overline \partial ^{\overline
+E}\vartheta '\enskip \enskip , \Eqno (2.15)
+$$
+and it is characterized by the equation
+$$
+\left \langle (\Omega ^{\overline E}+\theta \theta
+^{*})\mu ,\nu \right \rangle =i\partial \overline
+\partial \left \langle \mu ,J\nu \right \rangle
+\enskip \enskip \forall \mu ,\nu \in \Gamma
+^{\loc}(\Lambda ^{*})\enskip \enskip . \Eqno (2.16)
+$$\endTheorem
+
+\Proof {Proof} The first part is classical, but we
+shall give a short proof to illustrate our notations.
+For all $\mu \in \Gamma ^{\loc}(\Lambda ^{*})$, $\nu
+\in \Gamma (E^{*})$
+$$
+\overline \partial \left \langle \mu ^{0,1},\nu
+^{1,0}\right \rangle
+ =\overline \partial (({\frak i}^{-1}\mu )(\nu
+^{1,0}))=({\frak i}^{-1}\mu )((\nabla +\vartheta
+)''\nu ^{1,0})\enskip \enskip ; \Eqno (2.17)
+$$
+but also
+$$
+\overline \partial \left \langle \mu ^{0,1},\nu
+^{1,0}\right \rangle
+ =\left \langle \mu ^{0,1},\nabla ^{\overline
+E}{}''\nu ^{1,0}\right \rangle =({\frak i}^{-1}\mu
+)(\nabla ^{\overline E}{}''\nu ^{1,0})\enskip \enskip
+, \Eqno (2.18)
+$$
+hence $(\nabla +\vartheta )'=\nabla ^{\overline E}{}'$
+on $E^{*0,1}$. To see the second part, one calculates
+for $\mu ,\nu \in \Gamma ^{\loc}(\Lambda ^{*})$
+$$\aligned
+\partial \overline \partial \left \langle \mu
+^{0,1},\nu ^{1,0}\right \rangle &=\left \langle
+\nabla ^{\overline E}{}'\mu ^{0,1},\nabla ^{\overline
+E}{}''\nu ^{1,0}\right \rangle +\left \langle \mu
+^{0,1},\nabla ^{\overline E}{}'\nabla ^{\overline
+E}{}''\nu ^{1,0}\right \rangle \\
+ &= \left \langle \nabla ^{\overline E}{}'\mu
+,\nabla ^{\overline E}{}''\nu \right \rangle +\left
+\langle \mu ^{0,1},\Omega ^{\overline E}\nu
+^{1,0}\right \rangle \\
+ &= -\left \langle {}^{t}\theta ''{}^{t}\theta
+^{*}{}'\mu ,\nu \right \rangle - \left \langle \Omega
+^{\overline E}\mu ^{0,1},\nu ^{1,0}\right \rangle
+\enskip \enskip ; \endaligned
+\Eqno (2.19)$$
+but also
+$$
+\partial \overline \partial \left \langle \mu
+^{1,0},\nu ^{0,1}\right \rangle =\left \langle
+{}^{t}\theta '{}^{t}\theta ^{*}{}''\mu ,\nu \right
+\rangle +\left \langle \Omega ^{\overline E}\mu
+^{0,1},\nu ^{1,0}\right \rangle \enskip \enskip .
+\Eqno (2.20)$$
+Taking the difference and using (2.13), one finds
+$$\aligned
+i\partial \overline \partial \left \langle \mu ,J\nu
+\right \rangle &=\partial \overline \partial \left
+\langle \mu ^{1,0},\nu ^{0,1}\right \rangle -\partial
+\overline \partial \left \langle \mu ^{0,1},\nu
+^{1,0}\right \rangle \\
+ & = \left \langle \Omega ^{\overline E}\mu
+,\nu \right \rangle +\left \langle ({}^{t}\theta
+'{}^{t}\theta ^{*}{}''+{}^{t}\theta ''{}^{t}\theta
+^{*}{}'),\mu ,\nu \right \rangle \\
+ & = \left \langle (\Omega ^{\overline
+E}+{}^{t}\theta {}^{t}\theta ^{*})\mu ,\nu \right
+\rangle \enskip \enskip .
+\endaligned \Eqno (2.21)$$
+
+Notice that $i\partial \overline \partial \left
+\langle \mu ,J\nu \right \rangle = \overline
+{i\partial \overline \partial \left \langle \mu ,J\nu
+\right \rangle }$ is in fact a real form.\qed
+
+\Subheading {III. Computation of the Levi-Civita
+superconnection} The analytic torsion forms of a
+fibration are defined using a certain superconnection,
+acting on the infinite dimensional bundle of forms on
+the fibres. In this section, this superconnection will
+be investigated for the torus fibration $\smallmatrix
+M\\\pi \,\downarrow \\B\endsmallmatrix$.
+
+Let $F:=\Gamma (Z,\Lambda T^{*\,0,1}Z)$ be the
+infinite dimensional bundle on $B$ with the
+antiholomorphic forms on $Z$ as fibres. By using the
+holomorphic hermitian connection $\nabla ^{\overline
+E}$ on $E^{*\,0,1}$, one can define a connection
+$\widetilde \nabla $ on $F$ setting
+$$
+ \widetilde \nabla _{Y}h:=(\pi _{*}\nabla ^{\overline
+E})_{Y^{H}}h\enskip \enskip \forall Y\in \Gamma
+(TB)\enskip ,\enskip \enskip h\in \Gamma (B,F)\enskip
+\enskip . \Eqno (3.0)
+$$
+
+The metric $\left \langle \enskip ,\enskip \right
+\rangle $ on $E$ induces a metric on $Z$. Then $F$ has
+a natural $TZ\otimes {\Bbb C}$ Clifford module
+structure, given by the actions of
+$$
+c(Z^{1,0}):=\sqrt 2 {\frak i}(Z^{1,0})\Lambda \enskip
+\enskip \text{and}\enskip \enskip c(Z^{0,1}):=-\sqrt
+2\iota _{Z^{0,1}}\enskip \enskip \enskip \forall z\in
+TZ\enskip \enskip . \Eqno (3.1)
+$$
+ $\iota _{Z^{0,1}}$ denotes here interior
+multiplication. Clearly $$c(Z)c(Z')+c(Z')c(Z)=-2\left
+\langle Z,Z'\right \rangle \enskip \forall Z,Z'\in
+TZ\otimes {\Bbb C}\enskip \enskip .\Eqno (3.3)$$
+
+ Let $\overline \partial ^{Z}$, $\overline \partial
+^{Z*}$ be the Dolbeault operator and its dual on $Z$,
+and let
+$$
+ D:=\overline \partial ^{Z}+\overline \partial ^{Z*}
+\Eqno (3.3)
+$$
+denote the Dirac operator action on $F$. In fact, for
+an orthonormal basis $(e_{i})$ of $TZ\otimes {\Bbb C}$
+and the hermitian connection $\nabla ^{Z}$ on $Z$
+$$
+ D={1\over \sqrt 2} \sum c(e_{i})\nabla
+^{Z}_{e_{i}}\enskip \enskip . \Eqno (3.4)
+$$
+
+A form $\mu =\mu ^{1,0}+\mu ^{0,1}\in \Lambda ^{*}$
+can be identified with a ${\Bbb R}/2\pi {\Bbb
+Z}$-valued function on $Z$. In particular, the ${\Bbb
+C}$-valued function $e^{i\mu }$ is welldefined on $Z$.
+Then one finds the analogue of Theorem~2.7 in
+\cite{BC}.
+
+\Theorem {Lemma 3} For $x\in B$, $F_{x}$ has the
+orthogonal decomposition in Hilbert spaces
+$$
+ F_{x} = \bigoplus\limits _{\mu \in \Lambda ^{*}_{x}}
+\Lambda E_{x}^{*\,0,1}\otimes \lbrace e^{i\mu }\rbrace
+\enskip \enskip . \Eqno (3.5)
+$$
+
+For $\mu \in \Lambda ^{*}_{x}$, $\alpha \in \Lambda
+\,E_{x}^{*\,0,1}$, $D$ acts on $\Lambda
+\,E_{x}^{*\,0,1}\otimes \lbrace e^{i\mu }\rbrace $ as
+$$
+ D(\alpha \otimes e^{i\mu })={ic({\frak i}^{-1}\mu
+)\over \sqrt 2}\alpha \otimes e^{i\mu } \Eqno (3.6)
+$$
+and
+$$
+ D^{2}(\alpha \otimes e^{i\mu })={\textstyle {1\over
+2}} \left \vert \mu \right \vert ^{2}\alpha \otimes
+e^{i\mu }\enskip \enskip . \Eqno (3.7)
+$$\endTheorem
+
+\Proof {Proof} The first part of the lemma is standard
+Fourier analysis, using that $\text{vol}(\Lambda )=1$.
+The second part is obtained by calculating
+$$\aligned
+\overline \partial ^{Z}(\alpha \otimes e^{i\mu
+^{1,0}})&=0\enskip ,\enskip \enskip \enskip \overline
+\partial ^{Z}(\alpha \otimes e^{i'\mu ^{0,1}})=i\,\mu
+^{0,1}\wedge \alpha \otimes e^{i\mu ^{0,1}}\enskip ,\\
+\overline \partial ^{*\,Z}(\alpha \otimes e^{i\mu
+^{0,1}})&=0\enskip ,\enskip \enskip \enskip
+\overline \partial ^{Z\,*}(\alpha \otimes e^{i\mu
+^{1,0}})=-i\,\iota _{{\frak i}^{-1}\mu ^{1,0}}\alpha
+\otimes e^{i\mu ^{1,0}}\enskip , .
+\endaligned
+\Eqno (3.8)$$
+\qed
+
+Now one can determine the action of $\widetilde
+\nabla $ with respect to this splitting. Define a
+connection on the infinite dimensional bundle
+$C^{\infty }(Z,{\Bbb C})$ by setting
+$$
+ \nabla ^{\infty }_{Y}f:=Y^{H}.f\enskip \enskip
+\forall Y\in TB\enskip ,\enskip \enskip f\in C^{\infty
+}(Z,{\Bbb C})\enskip \enskip .
+\Eqno (3.9)$$
+
+\Theorem {Lemma 3.10} The connection $\widetilde
+\nabla $ acts on $F=\Lambda E^{*\,0,1}\otimes
+C^{\infty }(Z,{\Bbb C})$ as
+$$
+ \widetilde \nabla =\nabla ^{\overline E}\otimes
+1+1\otimes \nabla ^{\infty }\enskip \enskip ; \Eqno
+(3.10)
+$$
+hence it acts on local sections of $\Lambda
+E^{*\,0,1}\otimes \lbrace e^{i\mu }\rbrace $ for $\mu
+\in \Gamma ^{\loc}(\Lambda ^{*})$ as $\nabla
+^{E}\otimes 1$. In particular,
+$$
+\widetilde \nabla ^{2}=\Omega ^{\overline E}\otimes
+1\enskip \enskip . \Eqno (3.11)
+$$\endTheorem
+
+\Proof {Proof} This is obvious because $\mu $ is a
+flat local section.
+
+\Definition {Definition} The superconnection $A_{t}$
+on $\smallmatrix F\\\downarrow \\B\endsmallmatrix$,
+depending on $t\in {\Bbb R}$, $t\geq 0$, given by
+$$
+ A_{t}:=\widetilde \nabla +\sqrt tD \Eqno (3.12)
+$$
+is called the Levi-Civita
+superconnection.\endDefinition
+
+In fact, this definition is the analogue to the
+Definition~2.1 in \cite{BGS2}; the torsion term
+appearing there vanishes in the case mentioned here.
+By Lemma~3 and Lemma~4, it is clear that $A^{2}_{t}$
+acts on $\Lambda E^{*\,0,1}\otimes \lbrace e^{i\mu
+}\rbrace $, $\mu \in \Gamma ^{\loc}(\Lambda ^{*})$, as
+$$
+A^{2}_{t}=(\nabla ^{\overline E}+i\sqrt {{t\over
+2}}c({\frak i}^{-1}\mu ))^{2}\otimes 1\enskip \enskip
+. \Eqno (3.13)
+$$
+\Subheading {IV. A transgression of the top Chern
+class}
+
+In this section, a form $\vartheta $ on $B$ will be
+constructed using the superconnection $A_{t}$, which
+transgresses the top Chern class $c_{n}({-\Omega
+^{\overline E}\over 2\pi i})$ of $E^{0,1}$. $\vartheta
+$, divided by the Todd class, will define the torsion
+form in section V. We will use the Mathai-Quillen
+calculus \cite{MQ}, in its version described and used
+by \cite{BGS5}. Mathai and Quillen observed that for
+$A\in \End(E)$ skew and invertible and $\Pf(A)$ its
+Pfaffian, the forms $\Pf(A) (A^{-1})^{k}$ are
+polynomial functions in $A$, so they can be extended
+to arbitrary skew elements of $\End(E)$. An
+endomorphism $A\in \End(E^{0,1})$, i.e. $A\in \End(E)$
+with $J \circ A = A \circ J$, may be turned into a
+skew endomorphism of $E \otimes {\Bbb C}$ by replacing
+$$
+ A \mapsto {\textstyle {1\over 2}} (A-A^{*}) +
+{\textstyle {1\over 2}} iJ(A+A^{*})\,\,.\Eqno (4.0)
+$$
+That means, $A$ is replaced by the operator which acts
+on $E^{1,0}$ as $-A^{*}$ and on $E^{0,1}$ as $A$. This
+is the convention of \cite{BGS5, p. 288} adapted to
+the fact that we are handling with $E^{0,1}$ and not
+with $E^{1,0}$. The same conventions will be applied
+to $\End(TM)$.
+
+With $I_{\overline E} \in \End(E^{0,1})$ the identity
+map, we consider at $Y\in E$ and $b\in {\Bbb R}$
+$$
+ \alpha _{t} := \text{det}_{T^{0,1} E}\left({-\pi
+^{*}\Omega ^{\overline E}\over 2\pi i} - b
+I_{\overline E}\right) e^{-t({ \left \vert Y\right
+\vert \over 2} + (\pi ^{*} \Omega ^{\overline E}-2\pi
+b J)^{-1})} \Eqno (4.1)
+$$
+by antisymmetrization as a form on the total space of
+$E$.
+\Definition {Definition} Let $\widetilde \beta _{t}
+\in \Lambda T^{*}B$ be the form
+$$
+ \widetilde \beta _{t}:= \sum _{\mu \in \Lambda ^{*}}
+({\frak i}^{-1}\mu )^{*} {\partial \over \partial
+b}\Big\vert_{b=0} \alpha _{t} \Eqno (4.2)
+$$
+and $\beta _{t} \in \Lambda T^{*}B$ be the form
+$$
+ \beta _{t} := \sum _{\mu \in \Lambda ^{*}} ({\frak
+i}^{-1}\mu )^{*} \alpha _{t}\vert_{b=0}\,\,.\Eqno
+(4.3)
+$$
+\endDefinition
+The geometric meaning of $\beta _{t}$ will become
+clear in the proof of Lemma 8. We recall that $\theta
+^{*} = \nabla ^{\overline E} - \nabla $ on $E$, hence
+for $\mu \in \Gamma ^{\loc}(\Lambda ^{*})$
+$$
+ \nabla ^{\overline E}({\frak i}^{-1}\mu ) = -\theta
+{\frak i}^{-1}\mu \Eqno (4.4)
+$$
+and one obtains
+$$
+ ({\frak i}^{-1}\mu )^{*} (\pi ^{*} \Omega ^{\overline
+E} - 2\pi bJ)^{-1} = {\textstyle {1\over 2}} \left
+\langle {\frak i}^{-1}\mu , \theta ^{*}(\Omega
+^{\overline E} - 2\pi bJ)^{-1} \theta {\frak
+i}^{-1}\mu \right \rangle \,\,.\Eqno (4.5)
+$$
+Hence one obtains
+\Theorem {Lemma 5} $\widetilde \beta _{t}$ is given
+by
+$$
+ \widetilde \beta _{t} = {\partial \over \partial
+b}\Big\vert_{b=0} \text{det}_{E^{0,1}} \left({-\Omega
+^{\overline E}\over 2\pi i} - bI_{\overline E}\right)
+\sum _{\mu \in \Lambda ^{*}} e^{-{t\over 2} \left
+\langle {\frak i}^{-1}\mu ,(1+\theta ^{*} (\Omega
+^{\overline E}-2\pi bJ)^{-1}\theta ){\frak i}^{-1}\mu
+\right \rangle } \Eqno (4.6)
+$$
+and
+$$
+ \widetilde \beta _{t} = {\partial \over \partial
+b}\Big\vert_{b=0} {\text{det}_{E^{0,1}}({-\Omega
+^{\overline E}\over 2\pi i} - bI_{\overline E})\over
+\text{det}^{1/2}_{E}(1+\theta ^{*}(\Omega ^{\overline
+E} -2\pi bJ)^{-1}\theta )} \sum _{\lambda \in \Lambda
+} e^{-{1\over 2t}\left \langle \lambda ,(1+\theta
+^{*}(\Omega ^{\overline E} -2\pi bJ)^{-1}\theta
+)\lambda \right \rangle } \,\,.\Eqno (4.7)
+$$
+It has the asymptotics
+$$
+ \widetilde \beta _{t} = - c_{n-1} \left({-\Omega
+^{\overline E}\over 2\pi i}\right) + {\Cal
+O}_{t\nearrow \infty }(e^{-t}) \Eqno (4.8)
+$$
+for $t\nearrow \infty $ and
+$$
+ \widetilde \beta _{t} = -(2\pi t)^{-n} c_{n-1}
+\left({-\Omega ^{\overline E}-\theta \theta ^{*}\over
+2\pi i}\right) + {\Cal O}_{t\searrow 0}(e^{-{1\over
+t}}) \Eqno (4.9)
+$$
+for $t\searrow 0$.
+\endTheorem
+\Proof {Proof} The second equation follows by the
+Poisson summation formula (recall $\vol(\Lambda ) =
+1$). The first asymptotic (4.8) is clear. The second
+asymptotic (4.9) may be proved by using formula (1.40)
+in \cite{BC}, which is obtained by a nontrivial result
+on Brezinians in \cite{Ma, pp. 166-167}. One finds
+$$
+ \aligned
+{\text{det}_{E^{0,1}}({-\Omega ^{\overline E}\over
+2\pi i} - bI_{\overline E})\over
+\text{det}^{1/2}_{E}(1+\theta ^{*}(\Omega ^{\overline
+E} -2\pi bJ)^{-1}\theta )}
+&=
+{(-1)^{n} \Pf({\Omega ^{\overline E}\over 2\pi
+}-bJ)\over \text{det}^{1/2}_{E}(1+\theta ^{*}(\Omega
+^{\overline E} -2\pi bJ)^{-1}\theta )}\\
+&=
+(-1)^{n} \Pf \left({-\Omega ^{\overline E}-\theta
+\theta ^{*}\over 2\pi }-bJ\right) \\
+&=
+\text{det}_{E^{0,1}}\left({-\Omega ^{\overline E}-
+\theta \theta ^{*}\over 2\pi i} - bI_{\overline
+E}\right)\,\,.\endaligned\Eqno (4.10)
+$$
+\qed
+In the same manner one obtains
+\Theorem {Lemma 6} $\beta _{t}$ is given by
+$$
+ \beta _{t} = \text{det}_{E^{0,1}} \left({-\Omega
+^{\overline E}\over 2\pi i}\right) \sum _{\mu \in
+\Lambda ^{*}} e^{-{t\over 2} \left \langle {\frak
+i}^{-1} \mu ,(1+\theta ^{*} \Omega ^{\overline
+E-1}\theta ){\frak i}^{-1}\mu \right \rangle } \Eqno
+(4.10)
+$$
+and
+$$
+\beta _{t} =(2\pi t)^{-n}
+{\text{det}_{E^{0,1}}({-\Omega ^{\overline E}\over
+2\pi i})\over \text{det}_{E}^{1/2}(1+\theta ^{*}\Omega
+^{\overline E-1}\theta )} \sum _{\lambda \in \Lambda }
+e^{-{1\over 2t}\left \langle \lambda ,(1+\theta
+^{*}\Omega ^{\overline E-1}\theta )\lambda \right
+\rangle }\,\,.\Eqno (4.11)
+$$
+It has the asymptotics
+$$
+ \beta _{t} = c_{n} \left({-\Omega ^{\overline E}\over
+2\pi i}\right) + {\Cal O}_{t\nearrow \infty
+}(e^{-t})\Eqno (4.12)
+$$
+for $t\nearrow \infty $ and for $t\searrow 0$
+$$
+ \beta _{t} = (2\pi t)^{-n} c_{n} \left({-\Omega
+^{\overline E}-\theta \theta ^{*}\over 2\pi i}\right)
++ {\Cal O}_{t\searrow 0}(e^{-{1\over t}})\,\,.\Eqno
+(4.13)
+$$
+\endTheorem
+We define the Epstein $\zeta $-function for $s >n$
+$$
+ \zeta (s) := - {1\over \Gamma (s)} \displaystyle \int
+^{\infty }_{0} t^{s-1} \left(\widetilde \beta _{t} +
+c_{n-1}\big({-\Omega ^{\overline E}\over 2\pi i}\big)
+\right) dt\,\,.\Eqno (4.14)
+$$
+Classically, $\zeta $ has a holomorphic continuation
+to $0[E]$. Hence we may define
+\Definition {Definition} Let $\vartheta $ be the
+form on $B$
+$$
+ \vartheta := \zeta '(0)\,\,.\Eqno (4.15)
+$$
+\endDefinition
+Then $\vartheta $ transgresses the top Chern class :
+\Theorem {Theorem 7} $\vartheta $ permits the
+double-transgression formula
+$$
+ {\overline \partial \partial \over 2\pi i} \vartheta
+= c_{n} \left({-\Omega ^{\overline E}\over 2\pi
+i}\right)\,\,.\Eqno (4.16)
+$$
+\endTheorem
+\Proof {Proof} By \cite{BGS5, Th. 3.10}, one knows
+that
+$$
+ - t {\partial \over \partial t} \alpha
+_{t}\big\vert_{b=0} = {\overline \partial \partial
+\over 2\pi i} {\partial \over \partial
+b}\Big\vert_{b=0} \alpha _{t}\,\,.\Eqno (4.17)
+$$
+The minus sign occuring here contrary to \cite{BGS5}
+is caused by the different sign of $J = -i
+I_{\overline E}$ in our formulas.
+
+We define $\beta ^{0}$ by $\beta _{t} = t^{-n} \beta
+^{0} + {\Cal O}_{t\searrow 0}(e^{-1/t})$ as in Lemma
+6. Then one obtains for $s > n$
+$$
+ \Multline
+ {\overline \partial \partial \over 2\pi i} \zeta (s)
+= {1\over \Gamma (s)} \displaystyle \int ^{\infty
+}_{0} t^{s} {\partial \beta _{t}\over \partial t} dt\\
+ = {1\over \Gamma (s)} \displaystyle \int ^{1}_{0}
+t^{s} {\partial \over \partial t} (\beta _{t}-t^{-n}
+\beta ^{0})dt - {n\over \Gamma (s)} \displaystyle \int
+^{1}_{0} t^{s-1-n} \beta ^{0} dt + {1\over \Gamma (s)}
+\displaystyle \int ^{\infty }_{1} t^{s} {\partial
+\over \partial t} \beta _{t} dt\\
+ = {1\over \Gamma (s)} \displaystyle \int ^{1}_{0}
+t^{s} {\partial \over \partial t} (\beta _{t}-t^{-n}
+\beta ^{0})dt + {1\over \Gamma (s)} {n\over n-s} \beta
+^{0} + {1\over \Gamma (s)} \displaystyle \int
+^{\infty }_{1} t^{s} {\partial \over \partial t} \beta
+_{t} dt\endMultline \Eqno (4.18)
+$$
+and hence for the holomorphic continuation of $\zeta $
+to 0
+$$
+ {\overline \partial \partial \over 2\pi i} \zeta
+'(0) = \lim_{t\nearrow \infty } \beta _{t} = c_{n}
+\left({-\Omega ^{\overline E}\over 2\pi
+i}\right)\,\,.\Eqno (4.19)
+$$
+\qed
+
+
+
+\Subheading { V. The analytic torsion form}
+
+Let $N_{H}$ be the number operator on $B$ acting on
+$\Lambda ^{p} T^{*}B\otimes F$ by multiplication with
+$p$ $\Tr_{s}\bullet $ will denote the supertrace
+$\Tr(-1)^{N_{H}}\bullet $. Let $\varphi $ be the map
+acting on $\Lambda ^{2p}T^{*}B$ by multiplication with
+$(2\pi i)^{-p}$.
+\Theorem {Lemma 8} Up to a cboundary,
+$$
+ \varphi \Tr_{s} N_{H} e^{-A^{2}_{t}} =
+\Td^{-1}\left({-\Omega ^{\overline E}\over 2\pi
+i}\right) \widetilde \beta _{t}\,\,,\Eqno (5.0)
+$$
+where $\Td^{-1}$ denotes the inverse of the Todd
+genus.
+\endTheorem
+\Proof {Proof} Define a form $\widehat \alpha _{t}$
+on the total space of $E$ with value
+$$
+ \widehat \alpha _{t} := \varphi \Tr_{s} N_{H}
+\exp\left(-(\nabla ^{\overline E} + i \sqrt {{t\over
+2}} c(\lambda ))^{2}\right)\Eqno (5.1)
+$$
+at $\lambda \in E$. Then one observes
+$$
+ \varphi \Tr_{s} N_{H} e^{-A^{2}_{t}} = \sum _{\mu
+\in \Lambda ^{*}} ({\frak i}^{-1}\mu )^{*} \widehat
+\alpha _{t}\,\,.\Eqno (5.2)
+$$
+But one knows that
+$$
+ \widehat \alpha _{t} = {\partial \over \partial
+b}\Big\vert_{b=0} \Td^{-1} \left({-\pi ^{*} \Omega
+^{\overline E}\over 2\pi i} - b I_{E}\right) \alpha
+_{t}\Eqno (5.3)
+$$
+by \cite{BGS5, Proof of Th. 3.3}. The result follows.
+\qed
+
+Now we define the analytic torsion form $T(M, \left
+\langle i\right \rangle )$ in \cite{BK} via the $\zeta
+$-function to $\varphi \Tr_{s} N_{H} e^{-A^{2}_{t}}$,
+modulo $\partial -$ and $\overline \partial
+-$coboundaries.
+\Definition {Definition} The analytic torsion form
+$T(M, g^{E})$ is defined by
+$$
+ T(M, g^{E}) := \Td^{-1}\left({-\Omega ^{\overline
+E}\over 2\pi i}\right) \vartheta \,\,.\Eqno (5.4)
+$$
+\endDefinition
+In particular, we deduce from Theorem 7
+$$
+ {\overline \partial \partial \over 2\pi i} T(M,
+g^{E}) = \left({c_{n}\over \Td}\right) \left({-\Omega
+^{\overline E}\over 2\pi i}\right) \,\,\,\,.\Eqno
+(5.5)
+$$
+Now we shall investigate the dependence of $T$ on the
+metric $g^{E}$. For a charactersitic class $\phi $,
+we shall denote by $\phi (g^{E})$ its evaluation for
+the hermitian holomorphic connection $ \nabla ^{E}$ on
+$E^{0,1}$ with respect to $\overline \partial $. For
+two Hermitian metrics $g^{E}_{0}, g^{E}_{1}$ on $E$,
+let $\widetilde \phi (g^{E}_{0}, g^{E}_{1})$ denote
+the axiomatically defined Bott-Chern classes of
+\cite{BGS1, Sect. 1f)}. $\widetilde \phi $ is living
+in the space of sums of $(p,p)$-forms modulo $\partial
+-$ and $\overline \partial -$coboundaries. It has the
+following property
+$$
+ {\overline \partial \partial \over 2\pi i}\widetilde
+\phi (g^{E}_{0}, g^{E}_{1}) = \phi (g^{E}_{1}) -
+\phi (g^{E}_{0})\,\,.\Eqno (5.6)
+$$
+\Theorem {Theorem 9} Let $g^{E}_{0}, g^{E}_{1}$ be
+two Hermitian metrics on $E$. Then the associated
+analytic torsion forms change by
+$$
+ T(M, g^{E}_{1})-T(M,g^{E}_{0}) =
+\widetilde{\Td^{-1}}(g^{E}_{0}, g^{E}_{1})
+c_{n}(g^{E}_{0}) + \Td^{-1}(g^{E}_{1})
+\widetilde{c_{n}}(g^{E}_{0}, g^{E}_{1})\Eqno (5.7)
+$$
+modulo $\partial -$ and $\overline \partial
+-$coboundaries.
+\endTheorem
+\Proof {Proof} This follow by the uniqueness of the
+Bott-Chern classes. Using (5.5) and the
+characterization of Bott-Chern classes in \cite{BGS1,
+Th. 1.29}, it is clear that
+$$
+ T(M,g^{E}_{0}) - T(M, g^{E}_{1}) =
+\left({\widetilde{c_{n}}\over \Td}\right) (g^{E}_{0},
+g^{E}_{1})\,\,.\Eqno (5.8)
+$$
+The result follows.\qed
+
+\Subheading {VI. The K{\"a}hler condition}
+
+The analytic torsion forms were only constructed in
+\cite{BK} for the case were the fibration is
+K{\"a}hler. That means, there had to exist a
+K{\"a}hler metric on the total space $M$, so that the
+decomposition (2.0) is an orthogonal decomposition.
+Hence it is interesting to see when this happens for
+the case investigated here.
+\Theorem {Lemma 10} The fibration $\smallmatrix
+M\\\downarrow \\B\endsmallmatrix$ is K{\"a}hler iff
+the base $B$ is K{\"a}hler and there exists a falt
+symplectic structure $\omega ^{E}_{0}$ on $E$, which
+is a positive $(1,1)$-form with respect to $J$, i.e.
+$$
+ \alignat 3
+ \text{I)} &\quad\nabla \omega ^{E}_{0} = 0\,, \tag
+6.0\\
+ \text{II)} &\quad \omega ^{E}_{0}(JX,JY) = \omega
+^{E}_{0}(X,Y) &\qquad \forall & \,X,Y \in E \,, \tag
+6.1\\
+ \text{III)} &\quad \omega ^{E}_{0}(X,JX) > 0 &\qquad
+\forall & \,X\in E\,\,. \tag 6.2
+\endalignat
+$$
+\endTheorem
+It follows easily that $\overline \partial ^{\overline
+E}$ is the by the metric and $\overline \partial
+^{\overline E}$ induced holomorphic structure if $M$
+is K{\"a}hler. Thus, $T$ coincides with the torsion
+form in \cite{BK} in this case. Furthermore, $\Omega
+^{\overline E} + \theta \theta ^{*} = 0$, so the
+asymptotic terms in (4.9), (4.13) vanish.
+\Proof {Proof} Let $g$ any Hermitian metric on $TM$,
+so that $g(T^{H}M, TZ) = 0$. Let $\omega := g(\bullet
+,J\bullet )$ be the corresponding K{\"a}hler form. By
+$\omega ^{H}$ and $\omega ^{Z}$ we denote the
+horizontal and the vertical part of $\omega $. Using
+the decomposition (2.0), the condition $d\omega =0$
+splits into four parts :
+\Item {{\bf I)}} For $Y_{1}, Y_{2}, Y_{3} \in TB$
+:
+$$
+ 0 = d\omega (Y^{H}_{1}, Y^{H}_{2}, Y^{H}_{3}) =
+d\omega ^{H} (Y^{H}_{1}, Y^{H}_{2}, Y^{H}_{3})\,,\Eqno
+(6.3)
+$$
+\Item {{\bf II)}} for $Y_{2}, Y_{2} \in TB$, $Z\in
+TZ$ :
+$$
+ 0 = d\omega (Y^{H}_{1}, Y^{H}_{2}, Z) = 2 . \omega
+^{H}(Y^{H}_{1}, Y^{H}_{2})\,,\Eqno (6.4)
+$$
+\Item {{\bf III)}} for $Y\in TB, Z_{1}, Z_{2} \in
+TZ$ :
+$$
+ 0 = d\omega (Y^{H}, Z_{1}, Z_{2}) = (L_{Y^{H}} \omega
+^{Z})(Z_{1}, Z_{2})\,,\Eqno (6.5)
+$$
+\Item {{\bf IV)}} for $Z_{1}, Z_{2}; Z_{3} \in TZ$
+:
+$$
+ 0 = d\omega (Z_{1}, Z_{2},Z_{3}) = d\omega
+^{Z}(Z_{1}, Z_{2}, Z_{3})\,.\Eqno (6.5)
+$$
+Conditions I) and II) just mean that
+$g\vert_{T^{H}M\times T^{H}M}$ is he horizonal lift of
+a K{\"a}hler metric on $B$. If there is a form
+$\omega ^{Z}$ satisfying condition III), then its
+restriction to the zero section of $E$ induces a
+K{\"a}hler form $\omega ^{E}$ on $E$, so that the left
+$\pi ^{*} \omega ^{E}$ satisfies conditions III) and
+IV). Only the following necessary condition remains
+\Item {{\bf III\,\,\alpha )}} There exists a
+Hermitian metric $g^{E}$ on $E$, so that for the
+corresponding K{\"a}hler, form $\omega ^{E}$ and all
+$\lambda _{1}, \lambda _{2}\in \Gamma ^{\loc}(\Lambda
+)$
+$$
+ \omega ^{E}(\lambda _{1},\lambda _{2}) =
+\text{const}\,.\Eqno (6.7)
+$$
+On the other hand, $M$ is clearly K{\"a}hler if this
+condition is satisfied. This proves the Lemma.\qed
+
+\noindent One may also investigate the local
+K{\"a}hler condition as posed in \cite{BGS1},
+\cite{BGS2}. Because $B$ is always locally K{\"a}hler,
+the same proof as above shows
+\Theorem {Lemma 11} The fibration $\smallmatrix
+M\\\downarrow \\B\endsmallmatrix$ is locally
+K{\"a}hler at $x_{0} \in B$ iff there exists locally
+on $B$ at $x_{0}$ a flat symplectic structure $\omega
+^{E}_{0}$ on $E$, so that
+$$
+ \alignat 2
+\text{I)} \qquad &\omega ^{E}_{0}(JX,JY) = \omega
+^{E}_{0}(X,Y) & \qquad \forall &X,Y \in E\,, \tag
+6.8\\
+\text{II)} \qquad & \omega ^{E}_{0}(X,JX) >0 \,\,
+\text{ at } x_{0} & \qquad \forall &X \in
+E_{x_{0}}\,.\tag 6.9
+\endalignat
+$$
+\endTheorem
+
+
+\References {
+References
+}
+\Benchmark
+\cite{BC} J.-M. Bismut and J. Cheeger, {\it
+Transgressed Euler Classes of $SL(2n, {\Bbb Z})$
+vector bundles, adiabatic limits if eta invariants and
+special values of $L$-functions\/}, Ann. Scient. Ec.
+Norm. Sup. 4e s\'erie, t. 25 (1992), 335--391.
+\Benchmark
+\cite{BGS1} J.-M. Bismut, H. Gillet and C. Soul\'e,
+{\it Analytic torsion and holomorphic determinant
+bundles I\/}, Comm. Math. Phys. {\bf 115} (1988),
+49--78.
+\Benchmark
+\cite{BGS2} ---, {\it Analytic torsion and holomorphic
+determinant bundles II\/}, Comm. Math. Phys. {\bf 115}
+(1988), 79--126.
+\Benchmark
+\cite{BGS5} ---, Complex immersions and Arakelov
+geometry, The Grothendieck Festschrift vol. 1,
+Birkh{\"a}user 1990.
+\Benchmark
+\cite{BK} J.-M. Bismut and Kai K{\"o}hler, {\it Higher
+analytic torsion forms for direct images and anomaly
+formulas\/}, J. Alg. Geom. {\bf 1} (1992, 647--684.
+\Benchmark
+\cite{E} P. Epstein, {\it Zur Theoric allg{\Blackbox}
+Zetafunktionen\/}, Math. Ann. {\bf 56} (1903),
+615--644.
+\Benchmark
+\cite{F} G. Faltings, Lectures on the arithmetic
+Riemann-Roch theorem, Princeton 1992.
+\Benchmark
+\cite{GS1} H. Gillet, S. Soul\'e, {\it Characteristic
+classes for algebraic vector bundles with Hermitian
+metrics I, II\/}, Ann. Math. {\bf 131} (1990),
+163--203, 205--238.
+\Benchmark
+\cite{GS2} H. Gillet and C. Soul\'e,{\it Analytic
+torsion and the arithmetic Todd genus\/}, with an
+appendix by D. Zagier, Topology {\bf 30} (1991),
+21--54.
+\Benchmark
+\cite{H} F. Hirzebruch, Tpological Methods in
+Algebraic Geometry, 3. ed. 1978.
+\Benchmark
+\cite{MQ} V. Mathai and D. Quillen, {\it
+Superconnections, Thom classes and equivariant
+differential forms\/}, Topology {\bf 25} (1986),
+85--110.
+\Benchmark
+\cite{RS} D.B. Ray and I.M. Singer, {\it Analytic
+torsion for complex manifolds\/}, Ann. of Math. {\bf
+98} (1973), 154--177.
+\Benchmark
+\cite{S} D. Sullivan, {\it La classe d'Euler r\'eelle
+d'un fibr\'e vectoriel \`a groupe structural
+$SL_{n}(x)$ est nulle\/}, C.R. Acad. Sci. Paris, {\bf
+281}, S\'erie A, (1975), 17--18.
+\Benchmark
+\cite{V} I. Vaisman, Symplectic Geometry and Secondary
+Characteristic Classes, Birkh{\"a}user
+1987.\endReferences
+
+
+
+\end
diff --git a/macros/generic/occam/Occam95.pdf b/macros/generic/occam/Occam95.pdf
new file mode 100644
index 0000000000..4d0d8a8fec
--- /dev/null
+++ b/macros/generic/occam/Occam95.pdf
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diff --git a/macros/generic/occam/PlainEx1/Example.readme b/macros/generic/occam/PlainEx1/Example.readme
new file mode 100644
index 0000000000..5591c08d97
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+++ b/macros/generic/occam/PlainEx1/Example.readme
@@ -0,0 +1,6 @@
+
+ Here is a routine example where a macro file reduces
+from 19Ko to 6Ko by use of Occam.
+
+ The macro file RESOURCESimple.occ is maintained in
+Occam format to make this weeding process convenient. \ No newline at end of file
diff --git a/macros/generic/occam/PlainEx1/RESOURCESimple.occ b/macros/generic/occam/PlainEx1/RESOURCESimple.occ
new file mode 100644
index 0000000000..aa4f61ea95
--- /dev/null
+++ b/macros/generic/occam/PlainEx1/RESOURCESimple.occ
@@ -0,0 +1,648 @@
+ %^%% RESOURCESimple simplified macro file for Sweet-teX 1985--
+ %% Composes typing as transcoded in default mode.
+ %% Use with Plain TeX.
+ %% Includes includes FONTDEF stuff.
+ %% Occam formatted for minimization.
+ %% Last update: 7-94
+ %% Author L. Siebenmann <ftp ftp.math.u-psud.fr>
+ %% Documentation after \endinput.
+ %% This file will become an %_
+ %% auxiliary macros file
+ %% derived from RESOURCESimple.occ
+ %% named <macro-file-name>
+ %% needed by the ".tex" typescript: <typescript-name>
+ %% Date: ????
+ %% Contact: ????, email: ????
+
+ %^% This version is in Occam format
+ %% and includes FONTDEF.tex%_
+
+ %^% This file is formatted by LS, 7-94
+ %% for use of the DefStrip utility posted on the CTAN archives
+ %% (master posting 1994 on ftp ftp.math.u-psud.fr)
+ %% DO NOT ALTER DEFSTRIP SIGNS %^, %_ , %%^_
+ %% UNLESS YOU UNDERSTAND THEM!
+ \let\Def\def \let\gDef\gdef \let\Let\let
+ \def\gLet{\global\let} \let\Font\font
+ \let\Mathchardef\mathchardef\let\Newsymbol\newsymbol
+ \let\MATHchardef\mathchardef\let\NEWsymbol\newsymbol
+ \input auditor.tex %% keep auditor.tex available
+ %% comment out above line to suppress audit function. %_
+
+ %%% Avoid double input of these macros
+ %%
+ \bgroup
+ \catcode`\!=12
+ \ifx\auxmacros\relax
+ \immediate\write16{}%
+ \message{ !!! auxmacros already defined !!!}
+ \gdef\auxmacros{\endinput}%
+ \else \global\let\auxmacros\relax
+ \fi
+ \egroup
+ \auxmacros
+
+ %^%%% avoid double input of Sweet-teX macros
+ \bgroup
+ \catcode`\!=12
+ \ifx\RESOURCE\relax
+ \immediate\write16{}%
+ \message{ !!! Sweet-teX RESOURCE macros already defined !!!}
+ \gdef\RESOURCE{\endinput}%
+ \else \global\let\RESOURCE\relax
+ \fi
+ \egroup
+ \RESOURCE %_
+
+\catcode`\@=11
+
+ \newcount\Ht
+ %¬% pg121; Height register, used in Linefigure & accents %_
+ \newcount\Wd
+ %^% Width %_
+
+ %^%% FONTDEF stuff defines your fonts;
+ %% depends on your TeX printing system;
+ %
+ % FONFDEF/lcd file for Sweet-teX. These are lowest common denominator
+ % settings; please adapt by enriching the font palate with whatever
+ % you happen to have and like!
+ %
+ \ifx\FONTDEF\undefined
+ \let\FONTDEF\relax
+ \else
+ \errmessage{ FONTDEF stuff already loaded\string!}
+ \expandafter\endinput
+ \fi %_
+
+ \def\cm{cm}
+ %\def\cm{dm} %% alternatives...
+ %\def\cm{dc}
+
+ \Font\bigbf=\cm bx12 %_
+
+ \Font \smallrm=\cm r7 %_
+ \Font \smallbf=\cm bx7 %_
+ \Font\tenbi=\cm bxti10 %TimesBI %cmbi10 %_
+ \Font \tensmc=\cm csc10 %_
+
+ \Font\foliofont=\cm r10 %_
+ %^\font\tenrm=\cm r10
+ %% restore \cmr10 for messages %_
+
+
+ \Def \bi{\tenbi}
+ %\def \bi{\bf}%_
+
+ \Def\smc{\tensmc }
+ %\def \smc{\bf}%_
+
+ \Def \Bbd#1{{\bf #1}}%_
+ \Def \Calig#1{{\cal #1}}%_
+ \Def \Cal#1{{\cal #1}}%_
+ \Def \Frak#1{{\bf #1}}%_
+
+ \Def \Smallfonts {}%_
+
+ \Def \Titlefont {\bigbf}%_
+ \Def \Authorfont {\bf}%_
+ \Def \Headingfont {\bf}%_
+ \Def \Subheadingfont {\bf}%_
+ \Def \Theoremfont {\bf}%_
+ \Def \TheoremTextfont {\bf}%_
+ \Def \Prooffont {\it}%_
+ \Def \Remarkfont {\bf}%_
+ \Def \Diagramfont {\bf}%_
+ \Def \Captionfont{\it}%_
+
+
+ %^%% Font macros %_
+
+ \Def\rm{\everymath={}\fam0\tenrm}%_
+ %^% allows \rm inside \Theorem
+ %% becomes difficult for amsTeX %_
+
+ %^\def\bf{\everymath={}\fam \bffam \tenbf}%_
+
+ %^% Various alternatives for italic
+ %\def \It #1 \endIt{{\it #1\/}}
+ %\def \It #1{{\it{\unskip#1}\unskip\/}}
+ %% now no double spaces; fault: space forced before (or use \unskip in typing)
+ %% also \Bi \Bf, \Smc
+ %\def \It #1{{\it{#1}\unskip\/}}
+ %%_
+
+ \long
+ \def \It #1{{\it
+ \ifdim\mathsurround=0pt
+ \else\advance\mathsurround by 1.2pt \fi\ignorespaces
+ #1\unskip\/}}
+ %^% Provided mathsurround is being used at all, gives better
+ %% spacing of math in (bold-)italic; also in \Theorem %_
+
+ %^%\def \Bi #1 \endIt{{\bi #1\/}}%_
+ %^%\def \Bi #1{{\bi{#1}\unskip\/}}%_
+ \Def \Bi #1{{\bi
+ \ifdim\mathsurround=0pt
+ \else\advance\mathsurround by 1.2pt \fi\ignorespaces
+ #1\unskip\/}}%_
+
+ \Def \Bf #1{{\bf#1\unskip}}%_
+
+ \Def \Smc#1{{\smc#1\unskip}}%_
+
+ \Def \Admin #1{\begingroup\mathsurround=0 pt
+ \leavevmode
+ %% p222-3:~ then \hskip -\lastskip for \unskip
+ \ifmmode\hbox{$\rm #1$\relax}\else$\rm #1$\relax\fi
+ \endgroup}
+ %% \relax's anticipate mathsurround mechanisms %_
+
+ \Def \Rm #1{\hbox{\kern 1pt \rm #1\kern 1pt}}%_
+
+ \Def\Displaystyle {\displaystyle}%_
+
+
+ %^%% Logical formatting %_
+
+ \Def\Medskip{\medskip}%_
+
+ %^%% \Title block begins%_
+ \newskip\TitleLineskip
+ \TitleLineskip=4pt
+ \def \DeepCr{\unskip\hfil\egroup\par\hfil\bgroup}%
+ \def \\#1{%
+ \def\test{#1}%
+ \ifx \test\space
+ \def\this{\DeepCr}\else\def\this{\DeepCr #1}%
+ \fi
+ \this}%
+ \Def \Title{\goodbreak
+ \ifdim\pagetotal>.70\pagegoal
+ \def\this{\vfill\eject}
+ \else\def\this{}\fi \this
+ \vbox\bgroup\Titlefont\def\cr{\DeepCr}%
+ \parindent=0 pt\parskip= 0 pt
+ \baselineskip=\fontdimen6\font
+ \lineskip=\TitleLineskip%
+ \lineskiplimit=\baselineskip%
+ \advance\lineskiplimit by -1ex %
+ \hbox to 0pt{}\vskip15pt plus 15pt\bigskip
+ \bgroup \hfil}
+ \def \endTitle {\unskip\hfil\egroup\par \egroup\medskip}
+ %^%% \Title block ends%_
+
+ \newskip\AuthorLineskip
+ \AuthorLineskip=5pt plus 5pt
+ \Def\Author{\nobreak\vskip 20Pt plus 10pt minus 5pt\nobreak
+ \vbox\bgroup\let\cr\DeepCR\let\\\DeepCR
+ \parindent=0 pt\parskip= 0 pt
+ \Authorfont\baselineskip=\fontdimen6\font
+ \advance\baselineskip by \AuthorLineskip
+ \bgroup \hfil}%_
+
+ \Def \endAuthor {\unskip\hfil\egroup\par \egroup\medskip}%_
+
+ \Def \Heading#1{\hbox{}\hfil \goodbreak \bigskip \medskip
+ \centerline{\Headingfont #1}}%_
+
+ \Def \Subheading#1{\medskip\bigskip \goodbreak
+ \par \noindent {\Subheadingfont #1.}%
+ \nobreak \vskip 3pt \nobreak}%_
+
+ \Def \SubheadingA#1{\medskip \goodbreak \par
+ {\Subheadingfont #1.}\quad}%_
+
+ \Def \SubheadingB#1{\medskip \goodbreak \par
+ {\Subheadingfont\quad #1.}\quad}%_
+
+ \Def \SubheadingC#1{\medskip \goodbreak \par
+ {\bi\quad #1.}\quad}%_
+
+ \Def \Remark#1{\bigskip \goodbreak
+ \par\noindent{\Remarkfont #1.}}%_
+
+ \Def \endRemark{\medskip \goodbreak}%_
+
+ \Def \Example#1{\Remark {#1}}%_
+ \Def \endExample{\medskip \goodbreak}%_
+
+ \Def \Definition#1{\Remark {#1}}%_
+ \Def \endDefinition{\medskip \goodbreak}%_
+
+ \Def \Theorem #1{\goodbreak\bigskip\par\noindent\Theoremfont #1.
+ \hskip 2pt plus 1pt minus 1pt
+ \begingroup\it
+ \everymath={\ifdim\mathsurround=0pt\def\this{}
+ \else\def\this{\kern1.5pt}\fi\this}
+ }%_
+
+ \Def \endTheorem {\endgroup \rm \goodbreak \smallskip}%_
+
+ \Def \Proof#1{\goodbreak \medskip
+ \par\noindent \Prooffont #1\hskip .7pt:\hskip 3pt\rm}%_
+
+ \Def \endProof{\qed\goodbreak\vskip10pt}%_
+
+ %^% A Box for the Quod est demonstrandum: %_
+ \Def\qedbox{\hbox{\vbox{
+ \hrule width0.2cm height0.2pt
+ \hbox to 0.2cm{\vrule height 0.2cm width 0.2pt
+ \hfil\vrule height0.2cm width 0.2pt}
+ \hrule width0.2cm height 0.2pt}\kern1pt}}%_
+
+ %^% Typing in \qed makes the qedbox right justified: %_
+ \Def\qed{\ifmmode\qedbox
+ \else\unskip\ \hglue0mm\hfill\qedbox\medskip
+ \goodbreak\fi}%_
+
+ \Def\Endphase {}%_
+
+ \def \Benchmark { }
+ \Def \References#1{\begingroup \leftskip=25 pt
+ \parskip=4 pt plus 2 pt
+ \goodbreak \hbox to 1 pt{}%
+ \vskip 15 pt plus 10 pt minus 5 pt
+ \centerline{\Headingfont #1}%
+ \frenchspacing \Smallfonts \def \Benchmark{\Refmark}%
+ \def \Refmark##1##2{\par\noindent \llap {##1{##2}\kern 12 pt}\kern 0pt}%
+ \nobreak\vskip 8pt \nobreak}
+ %##1##2 secretly is \Cite ##2 %_
+
+ \Def \endReferences {~\unskip\par\endgroup \medskip\goodbreak}%_
+
+ \Def \Phantom{}%_
+ \Def\NoLineFigureBoxes {\gdef \Phantom{\phantom}}%_
+ \Def\LineFigureBoxes {\gdef \Phantom{}}%_
+
+ \catcode`\w=\active \catcode`\h=\active
+ \Def \Linefigure{\begingroup \catcode`\w=\active
+ \catcode`\h=\active \def w{30 }\def h{12}\preLinefigure}%
+ \def\preLinefigure[#1*#2]_{%
+ \Wd=#1\Ht=#2\catcode`\w=11 \catcode`\h=11 \LLinefigure}
+ \catcode`\w=11 \catcode`\h=11 %_
+
+ \Def \LLinefigure#1{%
+ \setbox1=\hbox{#1}%
+ \Phantom
+ %% make \phantom for final printing; empty meaning before!!!!
+ {\hskip 0 pt\hbox{\mathsurround=0 pt$%
+ \vcenter{\hbox{%
+ \vrule \vbox to \Ht pt{%
+ \hrule \vfil \hbox to \Wd pt{%
+ \hfil\unhbox1\hss}%
+ \vfil\hrule}%
+ \vrule}}\hskip 0 pt
+ $\relax}}%
+ \endgroup}% stray spaces would make this mushy %_
+
+ \Def\OldTexturesLinefigure[#1*#2scaled#3]_#4{%
+ \dimen1=#1 pc\dimen2=#2 pc
+ \divide\dimen1 by 1000 \multiply\dimen1 by #3
+ \divide\dimen2 by 1000 \multiply\dimen2 by #3
+ \noindent\hbox{\mathsurround=0 pt$%
+ \vcenter{\hbox{%
+ \vbox to \dimen2{%
+ \vfil \hbox to \dimen1{%
+ \special{picture #4 scaled #3}\hfil}%
+ }%
+ }}%
+ $\relax}}%_
+
+ \Def\TexturesLinefigure{\OldTexturesLinefigure}%_
+
+ \Let\gLinefigure\TexturesLinefigure %_
+
+
+ \Def \metaDiagram#1#2@{%
+ \def\SetHt##1{\def\test{##1}\def\Test{h} \ifx \test\Test \Ht=40
+ \else \Ht=##1 \fi}%
+ \SetHt{#1}\goodbreak\midinsert\vskip -8pt
+ \vbox to \Ht pt{\vfil \noindent\hfil\Diagramfont#2 \hfil}%
+ \vskip-8pt\endinsert}%_
+
+ \Def \Diagram#1{\metaDiagram#1@}%_
+
+ \Def \vDiagram#1{%
+ \Diagram{#1\hfill\hfill\vfil\vskip -\baselineskip}}%
+ %% note restored parens %_
+
+ \Def \AutoMetaDiagram#1#2@{\goodbreak\midinsert
+ \offinterlineskip\vbox to 0pt{}
+ %\vskip-1.5\abovedisplayskip %% adjust -1.5?
+ \line{\hfil #1\hfil}%
+ \vskip\bigskipamount
+ \line{\hfil\Diagramfont #2\hfil}%
+ %\vskip-1.5\belowdisplayskip
+ \endinsert}%_
+
+ \Def \AutoDiagram#1{\AutoMetaDiagram#1@}%_
+
+
+ \Def \Footnote #1#2{\footnote{\raise.4ex\hbox{\Admin{
+ #1}}}{\Smallfonts #2}} %% \Smallfonts problematic here
+ %% \scriptstyle? %_
+
+ \Def\BoldItemTags{\def\ItemStyle{\bf}}%_
+ \Def\PlainItemTags{\def\ItemStyle{\rm}}%_
+ \Def\ItemStyle{\bf}%_
+
+ %^\Def \Item #1{\item {$\bf {{#1}} $\relax}}%_
+ \Def \Item #1{%
+ \smallskip\nointerlineskip{\parskip=4pt\noindent
+ {\def\Bf{}\hbox{$\ItemStyle #1$\relax}}\enskip}}%_
+
+ %^\Def \Itemitem #1{\itemitem {$\bf {{#1}} $\relax}}%_
+ \Def \Itemitem #1{\itemitem{\def\Bf{}\hbox
+ {$\ItemStyle #1$\relax}}}%_
+
+ \Def \Cite#1{{\rm\cite{#1}}}%_
+
+ \Def \cite#1{\cite@#1,\endcite@}%_
+ \Def \cite@@#1,{#1}%_
+ \Def \cite@#1,#2\endcite@{\def\temp{#2}%
+ \ifx\temp\empty\relax
+ \def \temp{{\bf[#1]}}%
+ \else\relax
+ \def\temp{{[\bf #1,\rm\ \cite@@#2]}}%
+ \fi\temp}%_
+
+
+ %^%%% Symbols macros %
+
+ \def\,{\ifmmode\mskip \thinmuskip\else\hskip1pt\fi }%_
+
+ \Def \@{\char '100}%_
+
+ \Def \preXbox{\hbox{$
+ \vcenter{\hbox{%
+ \vrule\vbox to 6.7 pt{%
+ \hrule \vfil \hbox to 12 pt{%
+ \hfil}%
+ \vfil\hrule}%
+ \vrule}}\hskip 4pt
+ $\relax}}%_
+
+ \Def \Xbox{\raise -.25pt\hbox{\preXbox}}%_
+
+ \Def \Nonsense {{~\unskip \kern-3.5 pt \mathsurround=0 pt
+ \hbox{\Xbox \kern -16.5 pt $>\kern-3pt<$\relax}}}%_
+
+ \Def \Blackbox
+ {\leavevmode\hskip .3pt \vbox{%
+ \hrule height 6.9pt\hbox{\hskip 4.5pt}}\hskip .5pt}%_
+
+ \Def \Eqno #1$${\eqno \Admin{#1}$$\relax}%_
+ \Def \Rparen {\right )}%_
+
+ \Def \bigMidvert{\kern4pt \big \vert \kern4pt}%_
+
+ \Def \Midvert{\kern3pt \vert \kern3pt}%_
+
+ \Def\lvert{\left\vert}% defaults %_
+ \Def\rvert{\right\vert}%_
+ \Def\lVert{\left\Vert}%_
+ \Def\rVert{\right\Vert}%_
+
+ \def \proseSharp {{\mathsurround=0pt\kern1pt
+ \hbox{$\vcenter{\hbox{$\scriptstyle
+ \# $\relax}\vskip.7pt}$\relax}\kern1pt}}
+ \def \mathSharp {\mathord{\#}}
+ \Def\Sharp{\ifmmode\expandafter\mathSharp\else
+ \expandafter\proseSharp\fi}%_
+
+ \Def \bigConnectedsum {\mathop{\#}\limits}%_
+
+ %^\Def \Cup{\bigcup}%_
+ %^\Def \Cap{\bigcap}%_
+
+ \Def\littlecup{\mathchoice%{}
+ {\scriptstyle\cup}
+ {\scriptstyle\cup}
+ {\scriptscriptstyle\cup}
+ {\scriptscriptstyle\cup}
+ }%_
+
+ \Def\littlecap{\mathchoice%{}
+ {\scriptstyle\cap}
+ {\scriptstyle\cap}
+ {\scriptscriptstyle\cap}
+ {\scriptscriptstyle\cap}
+ }%_
+
+ \Def \Otimes{\mathbin{\kern-2pt\raise 1.2pt
+ \hbox{\mathsurround\z@$\scriptstyle \otimes$\relax}\kern-2pt}}%_
+
+ \Def \Oplus{\mathbin{\kern-2pt\raise 1.2pt
+ \hbox{\mathsurround=0pt$\scriptstyle \oplus$\relax}\kern-2pt}}%_
+
+ \Def \Amalg{\mathbin{\raise .5pt
+ \hbox{$\scriptstyle \amalg$\relax}}}%_
+
+ \Def\Circ {\mathchoice%
+ {\hbox{\raise .8pt
+ \hbox{\mathsurround=2pt$\scriptstyle \circ$\relax}}}
+ {\hbox{\raise .8pt
+ \hbox{\mathsurround=2pt$\scriptstyle \circ$\relax}}}
+ {\hbox{\raise .5pt
+ \hbox{\mathsurround=.7pt$\scriptscriptstyle \circ$\relax}}}
+ {\hbox{\raise .5pt
+ \hbox{\mathsurround=.7pt$\scriptscriptstyle \circ$\relax}}}
+ }
+ %\def \Circ {\hbox{\kern-1pt\raise 1pt
+ % \hbox{$\scriptstyle \circ$\relax}\kern-1pt}}
+ %\def \Circ {\circ}
+ %_
+
+ \Def \Coprod {\mathop{\raise 1.2pt
+ \hbox{\mathsurround=0pt$\coprod$\relax}}}%_
+
+ \Def \Lim {\lim\limits}%_
+
+ \Def \Lbrack {{\mathsurround=0pt$[\![$\relax}}%_
+
+ \Def \Rbrack {{\mathsurround=0pt$]\!]$\relax}}%_
+
+ \Def \Smash#1{\vbox to 0 pt{%
+ \vss\hbox{\mathsurround=0pt${#1}$\relax}\vss}}%_
+
+ %% The following tentative macros for accents
+ %% will misbehave in subscript position
+ %% and also with unslanted characters.
+ %% In such cases resort to other pre-existing macros.
+ \Def \Acc{\relax\expandafter}%_
+
+ \Def\swthat{\raise -1.1 ex\hbox{%
+ \mathsurround=0pt$\widehat{}$\relax}}%_
+ \Def\swttilde{\raise -1.2 ex\hbox{%
+ \mathsurround=0pt$\widetilde{}$\relax}}%_
+ \Def \overdot{{\raise .2 ex \hbox to 0pt {\hss\bf\smash{.}\hss}}}%_
+ \Def \overcircle{{\raise .1 ex \hbox to 0pt
+ {\mathsurround=0pt$\scriptscriptstyle\hss\circ\hss$\relax}}}%_
+
+ \Def \Mathaccent#1#2{{\mathsurround=0 pt %% E.g. #1=\widehat
+ \setbox4=\hbox{$\vphantom{#2}$\relax}
+ \Ht=\ht4 %% pg120, understood unit sp
+ \setbox5=\hbox{${#1}$\relax}
+ \setbox6=\hbox{${#2}$\relax}
+ \setbox7=\hbox to .5\wd6{}
+ \copy7\kern .1\Ht sp\raise\Ht sp\hbox{\copy5}\kern-.1\Ht sp
+ %% some missing sp 1-93 (unnecessary?)
+ \copy7\llap{\box6}
+ }}%_
+ %^% italic test or other needed; oops, index height slightly raised
+ %% extra braces 9-90
+ %_
+
+
+ \Def \Overdot #1{\Mathaccent {\overdot} {#1}}%_
+
+ \Def \Overcircle #1{\Mathaccent {\overcircle} {#1}}%_
+
+ \Def \SwtHat #1{\Mathaccent {\swthat} {#1}}%_
+
+ \Def \SwtTilde #1{\Mathaccent {\swttilde} {#1}}%_
+
+ \Def\SwtCheck #1{%
+ \ifmmode \check{#1}%
+ \else \v {#1}%
+ \fi}%_
+
+ \Def \ChOline#1{\setbox1=\hbox{\mathsurround=0pt${#1}$\relax}%
+ \ifdim \wd1 > 7pt
+ \kern .15\ht1 \kern .9 pt
+ \overline {\kern -.15\ht1 \kern -.9 pt#1\kern-.9 pt}%
+ \kern .9 pt
+ \else
+ \ifdim \wd1 > 4pt
+ \kern .3\ht1
+ \overline {\kern -.3\ht1 {#1}}%
+ \else
+ \kern .3\ht1 \kern-.9 pt
+ \overline {\kern -.3\ht1 \kern .9 pt{#1}\kern .9 pt }%
+ \kern-.9 pt
+ \fi
+ \fi}%_
+
+
+ \Def \ChUline#1{%
+ {\kern .5pt \underline {\kern -.5pt#1\kern-2.2pt}\kern1.3pt}
+ }%% single underline for char %_
+
+ \Def \Uuline#1{\closerunderline{.9pt}{\closerunderline {-.3pt}{#1}}}
+ %% close dble underline
+ \Def \ChUuline#1{
+ {\kern .5pt \Uuline {\kern -.5pt#1\kern-2.2pt}\kern1.3pt}
+ }%% double underline for char %_
+
+ \Def \Cdot{\mathbin{\raise .4 ex \hbox to 0pt {\hss\bf .\hss}}}%_
+
+ \Def \llonguparrow{\bigg\uparrow}%_
+ \Def \llongdownarrow{\bigg\downarrow}%_
+
+ %^\Def \Limgadget #1 {\mathrel
+ %{\kern-2p\mathop{\kern3pt #1\kern3pt}\limits}}
+ %% no; redo correctly to replace following?
+ %_
+
+ \Def \llongrightarrow {\kern-2pt\mathop
+ {\kern3pt\longrightarrow\kern3pt}\limits}%_
+
+ \Def \llongleftarrow{\kern-2pt\mathop
+ {\kern3pt\longleftarrow\kern3pt}\limits}%_
+
+ \Def \llongtwoheadrarrow {\kern-2pt
+ \mathop{\kern3pt\longrightarrow
+ \kern-14pt \longrightarrow\kern3pt}\limits}%_
+
+ \Def \llongleftrightarrow {\kern-2pt
+ \mathop{\kern3pt\longleftrightarrow\kern3pt}\limits}%_
+
+ \Def \llongmapsto {\kern-2pt
+ \mathop{\kern3pt\longmapsto\kern3pt}\limits}%_
+
+ \Def\rarrow{\rightarrow}%_
+ \Def\larrow{\leftarrow}%_
+
+ \Def\ProseBullet{{\mathsurround=0pt$\bullet$\relax}}%_
+
+ \Def\Bullet{\ifmmode\let\this\bullet
+ \else\let\this\ProseBullet\fi\this}%_
+
+ \Def\Trademark{\hbox{\Admin{{}^{\scriptscriptstyle TM}}}}%_
+
+ \Def\LaTeX{{\rm L\kern-.34em
+ \raise.47ex\hbox{\mathsurround=0pt
+ $\scriptstyle\rm A$\relax}\kern-.15em \TeX}}
+ %% here 7pt A not \sc a %_
+
+ \Def \AmS{{\tensy A}\kern -.1667em\lower .5ex\hbox {\tensy
+ M}\kern -.125em{\tensy S}}%_
+
+ \Def \AmSTeX{\AmS-\TeX}%_
+
+ \Def \Enskip{{\hskip 4pt plus3pt minus2pt}}%_
+
+ \Def\Matrix#1{\matrix{#1}}%_
+
+ \Def\Undef{\Nonsense}%_
+ %^% The following correspond to rare Sweet-teX symbols,
+ %% and will produce a X'ed box on your TeX printout
+ %% until something better is devised %_
+ \Def\Bigasterisk{\Undef}%_
+ \Def\Control{\Undef}%_
+ \Def\llongrightarrowtail{\Undef}%_
+ \Def\llongswarrow{\Undef}%_
+ \Def\llongsearrow{\Undef}%_
+ \Def\llongnwarrow{\Undef}%_
+ \Def\llongnearrow{\Undef}%_
+ \Def\llongtwoheadrightarrow{\mathbin
+ {{\longrightarrow} \kern -1850\mu {\rightarrow}}}%_
+
+ \Def\complement{\Undef}%_
+
+ \Def\Break{\break}%_
+
+
+\catcode`\@=12
+
+ \endinput
+
+ %%^_
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ %%% RESOURCE.tex documentation
+
+ The typogaphy afforded by default is in effect an informal
+"preprint style" that has the minimal virtue of being easy to
+change, whereas TeX macros are usually very devious. It is best to
+make changes and enhancements through a personal style file to
+be loaded after RESOURCE.tex.
+
+ Sweet-teX typing is readily adapted to the sophisticated
+typography of monograph styles, for example to osudeG.sty whose
+master posting is on ftp shape.mps.ohio-state.edu [128.146.7.200].
+Such adaptation should not necessitate any alteration of
+RESOURCE.tex, provided one takes care to load the monograph
+style after RESOURCE.tex. An model adaptation is provided with
+the posting of osudeG.sty on ftp ftp.math.u-psud.fr. The order of
+loading is then plain.tex, RESOURCE.tex, osudeG.sty, mymacros.sty.
+
+ If alteration of RESOURCE.tex proves truly necessary please
+rename it! One sound motive for such alteration is to produce a
+simple submission to a journal that proposes to rework your typescript
+using its own TeX methods. See a posting about Journal submission on
+ftp ftp.math.u-psud.fr and an appendix to the Sweet-teX manual
+(versions >= 0.98).
+
+ If one attempts to read RESOURCE.tex a second time, it is bypassed.
+This means that if you incorporate RESOURCE.tex in a format, then
+whenever RESOURCE.tex is updated, then the format must be recompiled
+to take account of the update.
+
+ Sweet-teX macros tend to begin with a capital. Unfortunately a
+few of Knuth's (involving doubling of arrows etc.) do also.
+
+
+
+
diff --git a/macros/generic/occam/PlainEx1/newton.sty b/macros/generic/occam/PlainEx1/newton.sty
new file mode 100644
index 0000000000..ee5e8135de
--- /dev/null
+++ b/macros/generic/occam/PlainEx1/newton.sty
@@ -0,0 +1,212 @@
+ %% auxiliary macros file
+ %% derived from RESOURCESimple.occ
+ %% named newton.sty
+ %% needed by the ".tex" typescript: newton.tex
+ %% Date: 1986
+ %% Contact: ????, email: ????
+
+ %%% Avoid double input of these macros
+ %%
+ \bgroup
+ \catcode`\!=12
+ \ifx\auxmacros\relax
+ \immediate\write16{}%
+ \message{ !!! auxmacros already defined !!!}
+ \gdef\auxmacros{\endinput}%
+ \else \global\let\auxmacros\relax
+ \fi
+ \egroup
+ \auxmacros
+
+\catcode`\@=11
+
+ \newcount\Ht %% pg121; Height register, used in Linefigure & accents
+ \newcount\Wd %% Width
+
+ \def\cm{cm}
+ %\def\cm{dm} %% alternatives...
+ %\def\cm{dc}
+
+ \font \bigbf=\cm bx12 %
+
+ \font \tenbi=\cm bxti10 %TimesBI %cmbi10 %
+ \font \tensmc=\cm csc10 %
+
+ \def \bi{\tenbi}
+ %\def \bi{\bf}%
+
+ \def \smc{\tensmc }
+ %\def \smc{\bf}%
+
+ \def \Bbd#1{{\bf #1}}%
+ \def \Cal#1{{\cal #1}}%
+
+ \def \Smallfonts {}%
+
+ \def \Titlefont {\bigbf}%
+ \def \Authorfont {\bf}%
+ \def \Headingfont {\bf}%
+ \def \Subheadingfont {\bf}%
+ \def \Theoremfont {\bf}%
+ \def \Prooffont {\it}%
+ \def \Remarkfont {\bf}%
+ \def \Diagramfont {\bf}%
+
+ \def \rm{\everymath={}\fam0\tenrm}%
+
+ \long
+ \def \It #1{{\it
+ \ifdim\mathsurround=0pt
+ \else\advance\mathsurround by 1.2pt \fi\ignorespaces
+ #1\unskip\/}}
+
+ \def \Bi #1{{\bi
+ \ifdim\mathsurround=0pt
+ \else\advance\mathsurround by 1.2pt \fi\ignorespaces
+ #1\unskip\/}}%
+
+ \def \Admin #1{\begingroup\mathsurround=0 pt
+ \leavevmode
+ %% p222-3:~ then \hskip -\lastskip for \unskip
+ \ifmmode\hbox{$\rm #1$\relax}\else$\rm #1$\relax\fi
+ \endgroup}
+ %% \relax's anticipate mathsurround mechanisms %
+
+ \def \Rm #1{\hbox{\kern 1pt \rm #1\kern 1pt}}%
+
+ \newskip\TitleLineskip
+ \TitleLineskip=15pt plus 5pt
+ \def \DeepCR{\unskip\hfil
+ \egroup\par\hfil\bgroup\ignorespaces}%
+
+ \def \Title{\goodbreak
+ \ifdim\pagetotal>.70\pagegoal
+ \def\this{\vfill\eject}
+ \else\def\this{}\fi \this
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diff --git a/macros/generic/occam/PlainEx1/newton.tex b/macros/generic/occam/PlainEx1/newton.tex
new file mode 100644
index 0000000000..5e5e084523
--- /dev/null
+++ b/macros/generic/occam/PlainEx1/newton.tex
@@ -0,0 +1,1281 @@
+
+%%% newton.tex
+
+%Sem. Bourbaki / A. Marin / 1986
+%The following is for Plain TeX
+%Save this Sweet-Write file as "Text only" file ...'
+%Then transcode to Bourbaki/ Marin0-3.tex using...'
+
+\input newton.sty %% after export
+%\input RESOURCESimple.occ %% during export
+
+\magnification=1200 %adjust! %for print run only
+\hsize=15truecm %for print
+%\vsize=480 pt %for print
+\nopagenumbers
+
+
+\headline{\ifnum \pageno>1 \hss\raise -5pt
+\hbox{\tenrm
+670-\folio}\hss\fi}
+\widowpenalty=5000
+\emergencystretch=20pt
+
+\csname Francais\endcsname %%Ok of not there
+
+\par \noindent S\'eminaire BOURBAKI\hfill
+Novembre 1986 \break
+\Admin {19^{e}} ann\'ee, 1986-87, \Admin {n^{o} 670}
+\vskip 35pt plus 5pt minus 5pt
+
+\Title
+--- G\'eom\'etrie des polyn\^omes ---\\
+Co\^ut global moyen de la m\'ethode de Newton
+\endTitle \medskip
+\Author
+(d'apr\`es M. Shub et S. Smale)
+\endAuthor \vskip 20pt plus 5pt minus 5pt
+\Author
+par Alexis MARIN
+\endAuthor \vskip 25pt plus 5pt minus 5pt
+%
+
+ La suite $1,\enskip 3/2,\enskip 17/12,\enskip
+\dots{}\,\,$ d\'efinie par $x_{0}=1$ et la relation de
+r\'ecurrence $x_{n+1}={\textstyle {1\over
+2}}(x_{n}+2/x_{n})$ converge tr\`es rapidement vers
+$\sqrt 2 $. Le troisi\`eme terme $x_{3}=577/408\sim
+1,414215\,\dots{}$ poss\`ede d\'ej\`a les six premiers
+chiffres du d\'eveloppement d\'ecimal de $\sqrt 2\sim
+1,414213\,\dots{}$ alors que $x_{2}=17/12\sim
+1,416\,\dots{} $ n'en avait que trois. Cette suite,
+connue des Babyloniens, est produite par la {\it
+m\'ethode de Newton\/}. Pour approcher une solution
+d'une \'equation $f(x)=0$, o\`u $f$ est de classe
+$C^{2}$, Newton prend une {\it valeur initiale\/}
+$x_{0}$ et, it\'erant l'application $x\mapsto
+N_{f}(x)=x-f(x)/f'(x)$, il engendre la suite $x_{n}=
+N_{f}^{n}(x_{0})$. (Notez que $N_{f}(x)={\textstyle
+{1\over 2}}(x+2/x)$ si $f(x)=x^{2}-2$.) Les points
+fixes de $N=N_{f}$ sont les z\'eros de $f$, et puisque
+$N'=ff''/(f')^{2}$, un {\it z\'ero simple\/} $w$ de
+$f$ (i.e. $f(w)=0\mathbin{\not =}f'(w))$ est un {\it
+point fixe superattractif\/} de $N$ (i.e. $N(w)=w$ et
+$N'(w)=0)$, en particulier il y a une constante $K$ et
+un voisinage $V$ de $w$ dont tout point $x$ v\'erifie
+$ \left \vert N(x)-w\right \vert \leq K \left \vert
+x-w\right \vert ^{2}$. Cette {\it convergence
+quadratique\/} explique le ``doublement de
+pr\'ecision'' \`a chaque it\'eration observ\'e
+ci-dessus.
+
+ Cependant, si $x_{0}$ n'est pas suffisamment proche
+d'une racine de $f$, la suite de Newton peut avoir un
+comportement chaotique ou s'accumuler sur un cycle, ce
+dernier ph\'enom\`ene se produisant dans un ouvert de
+l'espace $P_{d}$ des polyn\^omes de degr\'e $d$, d\`es
+le degr\'e 3, et ce pour un ouvert de valeurs
+initiales. Pour trouver une bonne valeur initiale, on
+peut explorer syst\'ematiquement le domaine de $f $,
+mais le co\^ut d'un tel balayage est prohibitif; on
+pr\'ef\`ere {\it tirer au sort\/} une valeur initiale,
+et it\'erer quatre ou cinq fois l'application de
+Newton; avec un peu de chance, la valeur de $f$ sur
+le dernier it\'er\'e est pratiquement nulle, sinon on
+relance les d\'es \dots{} . L'exp\'erience confirme
+que, sauf pour les polyn\^omes pathologiques et {\it
+rares\/}, les joueurs de d\'es sont gagnants.
+
+ Pour justifier cette pratique, il faut munir les
+espaces de poly\-n\^omes et de valeurs initiales d'une
+mesure de probabilit\'e. Smale consid\`ere l'espace
+$$
+ P_{d}(1)=\left \lbrace
+p(v)=v^{d}+a_{d-1}v^{d-1}+\dots{}+a_{0}\enskip
+\bigMidvert \enskip a_{i}\in {\Bbd C}\,,\enskip \left
+\vert a_{i}\right \vert \leq 1\right \rbrace
+$$
+ muni de la mesure de Lebesgue $\mu $ sur les
+coefficients, normalis\'ee par $\mu (P_{d}(1))=1$. On
+ne perd rien \`a se limiter \`a $P_{d}(1)$ car, pour
+tout polyn\^ome $q$ de degr\'e $d>1$, il existe $a$ et
+$b$ positifs avec $p(v)=aq(bv)$ dans $P_{d}(1)$;
+d'autre part (voir 5.4), les racines de $p\in
+P_{d}(1)$ sont de module inf\'erieur \`a 2.
+Connaissant $p(v_{0})$ et $p'(v_{0})$, Smale a un
+crit\`ere pour savoir si, partant de $v_{0}$, la
+m\'ethode de Newton converge quadratiquement
+(\Cite{Sm5}, cf. {\S }6). Il vient de montrer
+(\Cite{Sm6}) que, si pour un polyn\^ome $p$ de
+$P_{d}(1)$ la moyenne du nombre de tirages dans le
+disque $\lbrace \left \vert v\right \vert \leq
+3\rbrace $ pour obtenir une valeur initiale $v_{0}$
+v\'erifiant son crit\`ere est sup\'erieur \`a $n$
+alors $p$ se trouve dans un ensemble exceptionnel
+${\Cal E}_{n,d}$ de mesure born\'ee par $(\Rm
+{constante})(d^{5}/n)$. Dans \Cite{ShSm1}, il y
+avait un r\'esultat dans le m\^eme sens, mais sans
+crit\`ere permettant de savoir si on a gagn\'e (cf.
+2.3).
+
+ D\`es 1976, lors de ses travaux d'\'economie
+math\'ematique (\Cite{Sm1}), puis avec Hirsch
+(\Cite{HiSm}), Smale avait introduit des {\it
+m\'ethodes de Newton glo\-bales\/} (cf. {\S }{\S }3 et
+5) n'ayant qu'une convergence lin\'eaire mais dont le
+co\^ut global peut \^etre estim\'e plus finement. En
+1981, il d\'egageait une jolie d\'emonstration du
+th\'eor\`eme de d'Alembert-Gauss (\Cite{Sm2} et {\S
+}3) et prouvait le r\'esultat suivant: {\it Pour
+tout $0<\mu <1$, il y a un ensemble exceptionnel
+$U_{d}(\mu )$ de mesure $\mu $ dans $P_{d}(1)$ tel
+que, pour tout $p$ hors de $U_{d}(\mu )$, une
+m\'ethode de Newton glo\-bale partant de $v_{0}=0$
+fournit en moins de $(100(d+2))^{9}/\mu ^{7}$
+it\'erations\/} une valeur \`a partir de laquelle le
+``doublement de pr\'ecision'' \`a chaque it\'eration
+de Newton se produit. En 1983 avec Shub (\Cite{ShSm1}
+et \Cite{ShSm2}), il introduit des m\'ethodes d'ordre
+sup\'erieur (dont chaque it\'eration n\'ecessite le
+calcul de $\log{d}$ d\'eriv\'ees de $p$ si $p\in
+P_{d}(1)$). Partant d'un point $v_{0}$ pris au hasard
+sur un cercle de grand rayon, Shub et Smale obtiennent
+un bon z\'ero approch\'e en $N_{d}(\mu )$ it\'erations
+o\`u $N_{d}(\mu )$ cro\^\i t lin\'eairement en $d$ :
+$$
+ N_{d}(\mu ) = L_{1}d\,( \left \vert \log\mu
+\right \vert /\mu )^{1+1/\log{d}}+ L_{2}\enskip
+\enskip \Rm {avec}\enskip \enskip
+20>L_{1}>L_{2}\enskip \enskip ,
+$$
+et ils estiment le nombre d'op\'erations
+arithm\'etiques n\'ecessaires. En 1984, dans un
+survol (\Cite{Sm4}) o\`u, {\it sans pr\'etendre
+produire des algorithmes rivalisant avec ceux
+couramment utilis\'es\/}, il cherche \`a expliquer
+pourquoi {\it les algorithmes efficaces de l'analyse
+sont rapides en moyenne\/} bien qu'ils pi\'etinent
+sur des cas d\'eg\'en\'er\'es, et il d\'ecrit une
+m\'ethode glo\-bale du premier ordre qui permet de
+retrouver avec de meilleures bornes la plupart des
+r\'esultats de \Cite{ShSm1} et \Cite{ShSm2} en
+n'ayant besoin \`a chaque it\'eration de n'\'evaluer
+que $p$ et $p'$ (cf. {\S }4).
+
+ Le grand avantage de la m\'ethode de Newton sur les
+autres proc\'ed\'es de r\'esolution approch\'ee d'une
+\'equation d'une variable r\'eelle (dichotomie,
+s\'ecante, \dots{}) est qu'elle s'applique en toute
+dimension et m\^eme dans un Banach, ce qui permet de
+traiter des \'equations fonctionnelles (\Cite{KA},
+chap. 18). Renegar a \'etendu aux syst\`emes de $n$
+\'equations polynomiales en $n$ inconnues complexes
+les r\'esultats de Shub-Smale \Cite{R1}, et \Cite{Sm6}
+contient des r\'esultats valables dans un Banach.
+
+ Dans le cas d'un polyn\^ome d'une variable complexe,
+l'application de Newton a une extension rationnelle
+$(x,p)\mapsto N_{p}(x)$ \`a la sph\`ere de Gauss $S$~:
+ La m\'ethode de Newton est un algorithme {\it
+purement it\'eratif rationnel\/} et l'on peut se
+demander s'il existe un autre algorithme purement
+it\'eratif rationnel $G : S\times P_{d}
+\llongrightarrow S$ qui soit de plus
+\Bi{g\'en\'eriquement convergent }, i.e.
+$$
+ \lbrace (x,p)\in S\times P_{d}\enskip \bigMidvert
+\enskip G_{p}^{n}(x)\enskip \,\Rm {ne converge pas
+vers une racine de}\,\enskip p\rbrace
+$$
+est de mesure nulle. Pour $d>3$, la th\`ese de
+McMullen (\Cite{McM}) r\'epond: ``non''. Ceci
+n'arr\^eta pas Shub et Smale: avec la conjugaison
+complexe en plus des op\'erations arithm\'etiques, ils
+ont construit un algorithme purement it\'eratif
+g\'en\'eriquement convergent pour les syst\`emes de
+$n$ polyn\^omes en $n$variables complexes, et, pour
+$n=1$, un tel algorithme qui, pr\`es des racines, est
+la m\'ethode de Newton (\Cite{ShSm3} et {\S }7).
+
+ D'autre part, Dantzig (cf. \Cite{Sm3}) avait
+conjectur\'e que: $\rho (m,n)$ {\it le nombre moyen
+d'it\'erations n\'ecessaires pour r\'esoudre par la
+m\'e\-thode du simplexe un probl\`eme lin\'eaire \`a
+$m$ contraintes et $n$ variables est, pour $m$
+fix\'e, lin\'eaire en\/} $n$. En 1982, Smale
+r\'esolvait par les m\^emes m\'ethodes ce fameux
+probl\`eme; il donne une fonction $K(\varepsilon ,m)$
+telle que, pour tout $\varepsilon >0$,\enskip $\rho
+(m,n)\leq Kn^{\varepsilon }$ (\Cite{Sm3}). Depuis,
+des algorithmes polynomiaux en $m$ et $n$ ont \'et\'e
+trouv\'es (\Cite{R2}, \Cite{Sm6}).
+
+ Tous ces r\'esultats illustrent l'int\'er\^et de
+l'introduction d'id\'ees \'el\'ementaires de topologie
+et de r\'esultats g\'en\'eraux comme le th\'eor\`eme
+de la vari\'et\'e stable dans les probl\`emes concrets
+de calcul (cf. \Cite{Sh} et {\S }7).
+
+ Dans cet expos\'e, je me limiterai aux polyn\^omes
+d'une variable complexe.
+
+ Mes remerciements vont \`a J.J.~Risler qui m'a
+convaincu d'\'etudier les travaux de Smale pour son
+s\'eminaire, ainsi qu'\`a L. Guillou pour ses
+commentaires perspicaces sur ce texte.
+
+
+\Subheading {1. M\'ethode de Newton, chaos, et non
+convergence g\'en\'erique}
+%
+ Si un polyn\^ome r\'eel $f(t) = \prod _{i=1}^{d}
+(t-x_{i})$ a toutes ses racines r\'eelles et
+distinctes, il en est de m\^eme de sa d\'eriv\'ee
+$f'(t)=d\prod _{j=1}^{d-1}(t-c_{j})$, et z\'eros et
+points critiques alternent
+$x_{1}<c_{1}<x_{2}<\dots{}<x_{d}$ dans ${\Bbd R}$.
+Soit $f$ un tel polyn\^ome de degr\'e $d\geq 3$ et
+notons $N=N_{f}$. Pour l'orbite $\lbrace
+t_{n}=N^{n}(t)\rbrace $ d'un point $t$, il y a trois
+possibilit\'es:
+\Item {1)} $t_{n}$ converge vers un point fixe de
+$N$ (un z\'ero de $f$);
+\Item {2)} $t_{n}$ n'est pas d\'efini pour
+$n>n_{0}$ car $t_{n_{0}}$ est un point critique;
+\Item {3)} $t_{n}$ est d\'efini pour tout $n$, mais
+ne converge pas.
+\medskip\nobreak
+ Soit $S_{1}$, $S_{2}$, $S_{3}$ la partition
+correspondante de ${\Bbd R}$. Comme les points fixes
+$x_{i}$ de $N$ sont attractifs ($ \left \vert
+N'(x_{i})\right \vert =0<1$), $S_{1}$ est ouvert.
+Soient $b_{1},\dots{},b_{d}$ les bandes
+$]c_{i-1},c_{i}[$ \enskip (ici $c_{0}=-\infty $ et
+$c_{d}=+\infty $). Un coup d'oeil \`a la figure~1 nous
+assure que $N$ envoie des points proches des $c_{i}$
+dans $b_{1}$ ou $b_{d}$ et que $S_{1}$ contient ces
+deux bandes, donc $S_{2}$ est disjoint de $\overline
+S_{3}$, et $S_{3}$ est un ferm\'e invariant par $N$
+inclus dans $b_{2}\cup \dots{}\cup b_{d-1}$.
+
+
+
+\Diagram {{140}\hfill \raise 0 pt \hbox{Figure 1}
+\hfill \hfill Figure 2\hfill }
+
+ Soit $\Omega $ l'espace des suites infinies en les
+symboles $b_{2},\dots{},b_{d-1}$ muni de la topologie
+produit; le d\'ecalage $D$ agit sur $\Omega $.
+L'application $h:S_{3}\rightarrow \Omega $ d\'efinie
+par $N^{n}(x)\in h(b)_{n}$ est continue et envoie le
+syst\`eme dynamique $(S_{3},N)$ dans $(\Omega ,D)$.
+
+\Theorem {Th\'eor\`eme 1.1 \Rm {(Barna, Saari et
+Urenho \Cite{Ba}, \Cite{SU})}} Si le polyn\^ome
+$f$ de degr\'e $d\geq 3$ a tous ses z\'eros r\'eels
+et distincts, alors $S_{3}$ est un Cantor de mesure
+nulle, l'application $h$ est surjective et toute
+orbite p\'eriodique non constante de $D$ se rel\`eve
+en une orbite de $h$ de m\^eme p\'eriode, les orbites
+constantes se relevant en des orbites de p\'eriode
+2.\endTheorem
+
+\Proof {D\'emonstration} Pour $1<i<d$, soit $\beta
+_{i}=\,\,]e_{i},f_{i}[$ la composante connexe de
+$x_{i}$ dans $S_{1}$, alors $S_{3}\subset B = \bigcup
+\left \lbrace b_{i}\setminus \beta _{i} \bigMidvert
+1<i<d\right \rbrace $ et $N$ \'echange $e_{i}$ et
+$f_{i}$ (cf. Figure~2).
+
+ Comme $\Lim_{t\rightarrow c_{i}}N'(t) = -\infty $ et
+ $N'(x_{i})=0$, l'\'equation
+ $$N'(x)=-a^{2} \Eqno (*)$$
+a au moins une solution dans chacun des $2d-2$
+intervalles de $B$ et il n'y a qu'une solution par
+intervalle car \Admin {(*)} est de degr\'e $2d-2$,
+donc $N'$ est strictement monotone sur chaque
+intervalle sur lequel elle est n\'egative, en
+particulier croissante sur $]c_{i-1},e_{i}]$ et
+d\'ecroissante sur $[f_{i},c_{i}[$. Par un calcul
+astucieux (\Cite{Ba}, p.202-206), Barna obtient
+$N'(e_{i})<-1$ et $N'(f_{i})<-1$; donc $N$ est
+dilatante sur $B$, et a fortiori sur $S_{3}\subset B$.
+Comme $S_{3}$ est invariant, sa mesure doit \^etre
+nulle. Le reste du th\'eor\`eme est facile.
+\endProof
+
+ Dans \Cite{SU} et \Cite{HM}, on trouvera d'autres
+informations sur la dynamique de $N$. Hurley et
+Martin ont aussi observ\'e que si $T_{f}$ est une
+application $C^{2}$ d\'efinie dans le compl\'ementaire
+d'un ensemble fini $J$, tendant vers l'infini pr\`es
+de $J$ et dont les points fixes sont attractifs, alors
+deux points fixes cons\'ecutifs sont s\'epar\'es par
+une asymptote du graphe de $T_{f}$, et $T_{f}$
+pr\'esente le m\^eme chaos que $N_{f}$.
+
+ Le th\'eor\`eme de Barna reste vrai s'il y a des
+racines multiples. En ce cas, comme $S_{2}$ est
+d\'enombrable, la mesure de ${\Bbd R}\setminus S_{1}$
+est nulle, ce qui semble donner raison aux joueurs.
+Cependant, pour tout $d>2$, il y a des polyn\^omes $p$
+de degr\'e $d$ avec $p''(0)=0$ et
+$p(0)=-p'(0)\mathbin{\not =}0\mathbin{\not
+=}p(1)=p'(1)$ et donc pour lesquels $\lbrace
+0,1\rbrace $ est un cycle superattractif d'ordre 2 de
+$N$ (par exemple \hskip 1pt \Footnote{(\dag
+)}{D'apr\`es le th\'eor\`eme de Barna, un tel
+polyn\^ome a des racines complexes.} le polyn\^ome
+$p(z)={\textstyle {1\over 2}}z^{3}-z+1$). Tout
+polyn\^ome $q$ proche de $p$ poss\`ede un cycle
+attractif d'ordre $2$. Il s'en suit que {\it la
+m\'ethode de Newton n'est pas un algorithme
+g\'en\'eriquement convergent\/}.
+
+ La dynamique holomorphe donne d'autres informations
+sur les orbites de $N$ (voir \Cite{Sm4}, \Cite{DH} et
+\Cite{F}).
+
+
+
+
+
+\Subheading {2. Les m\'ethodes d'Euler-Newton, un
+crit\`ere au but de convergence}
+%
+ Soient $p : {\Bbd C}\rightarrow {\Bbd C}$ un
+polyn\^ome et $v$ un point r\'egulier de $p$ (i.e.
+$p'(v)\mathbin{\not =}0$); il y a, d\'efini au
+voisinage de $z=p(v)$, un unique inverse \`a droite de
+$p$ envoyant $z$ sur $v$, notons le $p_{v}^{-1}$, de
+rayon de convergence $r_{v}$ en $v$. Notons
+$D_{v}=D(z,r_{v})$. Si $r_{v}> \left \vert z\right
+\vert $, Euler calcule une racine $w$ de $p$ gr\^ace
+au d\'eveloppement de Taylor de $p_{v}^{-1}$ en $z$ :
+$$
+ w = p_{v}^{-1}(0) = p_{v}^{-1}(z-z)
+ = \displaystyle \sum _{n=0}^{\infty } {1\over n!}\,
+\left \lbrace {d^{n}\over dz^{n}}p_{v}^{-1 }\right
+\rbrace _{z}\,(-z)^{n } \Eqno (2.1)
+$$
+ Le calcul des termes de (2.1) co\^utant de plus en
+plus cher, Euler pr\'ef\`ere tronquer cette s\'erie
+\`a l'ordre $k$ et it\'erer l'application $v\mapsto
+E_{k}(v)$ ainsi obtenue. Remarquons que
+$E_{1}(v)=v-p(v)/p'(v)$ est l'application de Newton.
+
+\Theorem {Proposition 2.1 \Rm {(\Cite{ShSm1},
+\Cite{Sm4})}} Pour chaque $k$, il y a une
+constante $c_{k}<1$ telle que si $w$ est une racine
+simple de $p$ et $v_{0}\in p_{w}^{-1}(c_{k}D_{w})$,
+alors pour tout $n$ l'it\'er\'e
+$v_{n}=E_{k}^{n}(v_{0})$ est dans
+$p_{w}^{-1}(D_{w})$, la suite $v_{n}$ converge vers
+$w$, et
+$$
+ \left \vert p(v_{n})\right \vert \leq
+b^{(k+1)^{n}-1} \left \vert p(v_{0})\right \vert
+\enskip \enskip ,\enskip \enskip \Rm {o\`u }\enskip
+\enskip \enskip b= \left \vert p(v_{0})\right \vert
+/c_{k}r_{w}<1\enskip .
+$$
+La suite $c_{k}$ est croissante, tend vers $(2-\sqrt
+2)/4$ quand $k\rightarrow \infty $, et
+$$
+ c_{1} = 1/9 < c_{2} < 1/8 < c_{3} < 1/7 < c_{4} <
+(2-\sqrt 2)/4 < 1/6\enskip \enskip .
+$$\endTheorem
+ Un \Bi{bon z\'ero approch\'e } (pour la m\'ethode
+d'Euler d'ordre $k$) est un $v_{0}$ auquel on peut
+appliquer la proposition 2.1. Nous ne d\'emontrerons
+et n'utiliserons 2.1 qu'avec $k=1$. Pour $k>1$, voir
+\Cite{ShSm1, \Rm {p.115-121}}.
+
+\Remark {Remarque 2.2} La connaissance de $r_{w}$
+est inaccessible \`a un observateur en $v=v_{0}$; {\it
+on peut cependant minorer $r_{w}$ par $\rho _{p}$ le
+module de la plus petite valeur critique\/}, qui peut
+s'estimer \`a l'aide du discri\-mi\-nant. Le
+crit\`ere obtenu a peu d'int\'er\^et du point de vue
+num\'erique, car le calcul du discriminant co\^ute
+trop cher; il sera cependant utile pour donner des
+estim\'ees probabilistes.\endRemark
+
+\Theorem {Proposition 2.3 \Rm {(\Cite{ShSm1})}} Soit
+$D$ le disque unit\'e; la probabilit\'e dans $D\times
+P_{d}(1)$ pour que $v_{0}$ dans $D$ soit un bon
+z\'ero approch\'e d'un polyn\^ome $p$ de $P_{d}(1)$
+est minor\'ee par $c/d^{5}$ avec $c>4\cdot
+10^{-4}$.\endTheorem
+
+\Proof {Preuve de 2.3} Un point $z$ est une valeur
+critique de $p$ si et seulement si le polyn\^ome $p-z$
+est de discriminant z\'ero. Comme le discriminant
+${\Cal D} : P_{d}\rightarrow {\Bbd C}$ est de degr\'e
+$d-1$ en le terme constant, on a, en posant ${\Cal
+E}_{d}(\rho )=\lbrace p\in P_{d}(1)\bigMidvert \rho
+_{p}<\rho \rbrace $, le lemme suivant:
+
+\Theorem {Lemme 2.4} La mesure de ${\Cal E}_{d}(\rho
+)$ est major\'ee par $(d-1)\rho ^{2}$. \endProof
+\endTheorem
+
+ Comme le produit des racines de $p$ est $a_{0}$, un
+polyn\^ome $p$ de $P_{d}(1)$ (qui v\'erifie $ \left
+\vert a_{0}\right \vert \leq 1$) a une racine $w$ dans
+$D$. Si de plus $p$ est hors de ${\Cal E}_{d}(\rho )$
+pour un $\rho >0$, $w$ est un z\'ero simple et $ \left
+\vert p'(w)\right \vert \leq 1+2+\dots{}+d
+={\textstyle {1\over 2}} d(d+1)$; donc, gr\^ace au
+th\'eor\`eme du quart (cf. \Cite{O},~p.\,\,3), le
+disque $D'$ centr\'e en $w$ et de rayon $r=\rho
+/18d(d+1)$ est inclus dans $ p_{w}^{-1}(1/9D_{w})$, et
+donc constitu\'e de bons z\'eros approch\'es. L'aire
+de $D\cap D'$ est sup\'erieure \`a $\sigma =r^{2}\sqrt
+{1-r^{2}/4}$\enskip \enskip (cf. Figure~3). Soit
+$0<\mu <1$. En prenant $\rho =\sqrt {\mu /(d-1)}$
+dans 2.4, la probabilit\'e cherch\'ee est minor\'ee
+par
+${1\over \raise .5pt\hbox{$\pi $}}\int _{0}^{1}\sigma
+(\mu )d\mu \geq c/d^{5}$. \endProof
+
+
+
+\vDiagram{ {75}\hskip\parindent Figure 3}
+
+
+
+\Remark {Notations 2.5} Le polyn\^ome $p$ \'etant
+fix\'e et $v$ r\'egulier pour $p$, on pose: $\beta
+(v)= \left \vert p(v)/p'(v)\right \vert $, $\gamma
+(v)= \left \vert p^{(k)}(v)/k!\,p'(v)\right \vert
+^{1/k-1}$ et $\alpha (v)=\beta (v)\gamma (v)$. Si
+l'on a besoin de pr\'eciser $p$, on \'ecrit $\beta
+(v,p)$, $\gamma (v,p)$, $\alpha (v,p)$.\endRemark
+
+\Proof {D\'emonstration de 2.1 lorsque $k=1$}
+Calculons $p(v_{1})$ par la s\'erie de Taylor de $p$
+en $v=v_{0}$; comme $k=1$, $v_{1}-v=\beta (v)$ et les
+deux premiers termes s'annulent, les autres \'etant
+major\'es par $ \left \vert p(v)\right \vert (\alpha
+(v))^{k-1}$ d'o\`u le
+
+\Theorem {Lemme 2.6} Si $v$ est r\'egulier pour
+$p$ et $\alpha (v)<\alpha $, alors $ \left \vert
+p(v_{1})\right \vert <{\alpha \over 1-\alpha } \left
+\vert p(v)\right \vert $. \endProof \endTheorem
+
+\Theorem {Lemme 2.7} Si $v$ est r\'egulier pour
+$p$, alors $\alpha (v)<4 \left \vert p(v)\right \vert
+/r_{v}$.\endTheorem
+
+\Proof {D\'emonstration de 2.7} Comme $p$ est
+l'inverse de la fonction univalente
+$p_{v}^{-1}:D_{v}\rightarrow {\Bbd C}$, les
+coefficients $a_{k}$ de son d\'eveloppement de Taylor
+en $v$ sont born\'es par un th\'eor\`eme de L{\"o}wner
+(\Cite{H},~p.\,\,137):
+ $$
+ \left \vert a_{k}\right \vert \leq B_{k} \left
+\vert a_{i}\right \vert ^{k}r_{v}^{k-1} \enskip
+\enskip \enskip \enskip \Rm {o\`u}
+ \enskip \enskip \enskip \enskip B_{k} = 2^{k
+}{1\cdot 3\,\dots{}(2k-1) \over 1\cdot
+2\,\dots{}k\cdot (k+1)} \leq 4^{k-1}
+$$
+d'o\`u le lemme. \endProof
+
+ Sous les hypoth\`eses de 2.1, comme $r_{v}>r_{w}-
+\left \vert p(v)\right \vert $, on d\'eduit de ces
+deux lemmes $ \left \vert p(v)\right \vert \leq A
+\left \vert p(v)\right \vert ^{2}$ o\`u $A \left \vert
+p(v)\right \vert <b$. D'autre part, comme sur
+$D(0,r_{w}/9)$ les inverses $p_{v}^{-1}$ et
+$p_{w}^{-1}$ co\"\i ncident, et $$
+ \left \vert v_{1}-v\right \vert = \left \vert
+p(v)/p'(v)\right \vert \leq r_{w}\,/\,9 \left \vert
+p'(v)\right \vert \leq r_{v}\,/\,8 \left \vert
+p'(v)\right \vert <r_{v}\,/\,4 \left \vert p'(v)\right
+\vert \enskip \enskip ,
+$$
+le th\'eor\`eme du quart assure que $v_{1}$ est dans
+$p_{w}^{-1}(D_{w})$. Donc, par r\'ecurrence, $ \left
+\vert p(v_{n+1})\right \vert ^{2}<A \left \vert
+p(v_{n})\right \vert ^{2}$ avec $A \left \vert
+p(v_{0})\right \vert <b$, d'o\`u la proposition.
+\endProof
+
+
+\Subheading {3. Le th\'eor\`eme de d'Alembert-Gauss,
+une m\'ethode de Newton globale}
+%
+ L'ensemble $\Sigma $ des valeurs critiques d'un
+polyn\^ome $p$ de degr\'e $d>0$ est fini ($ \left
+\vert \Sigma \right \vert \leq d-1$); comme $p$ est
+une application ouverte, il y a un $v$ tel que le
+rayon $R_{z}=\lbrace tz\enskip \vert \enskip 0<t\leq
+1\rbrace $ ($z=p(v)$) \'evite $\Sigma $ (cf.
+Figure~4); le point $v$ est donc r\'egulier,
+$p_{v}^{-1}$ se prolonge analytiquement \`a un
+voisinage de $R_{z}$ et $w=\Lim_{t\rightarrow 0}
+p_{v}^{-1}(tz)$
+est une racine de $p$. Par des majorations faciles,
+on peut obtenir un ensemble de $d$ points de grand
+module contenant un tel $v$: cette d\'emonstration est
+constructive. Elle conduisit Smale (\Cite{Sm2}) \`a
+\'etendre $p_{v}^{-1}$ \`a des secteurs angulaires.
+
+
+
+\vDiagram{ {50}Figure 4 \hfill Figure 5 }
+
+
+
+
+ L'\Bi{\'etroitesse } $e(v)$ (relativement \`a $p$)
+d'un point $v$ non racine de $p$ est le plus grand
+$\theta \geq 0$ tel que $p_{v}^{-1}$ se prolonge au
+secteur:
+ $$ W(z,\theta ) = \lbrace z'\in {\Bbd
+C}\enskip \bigMidvert \enskip \left \vert \Rm
+{Arg}(z'/z)\right \vert <\theta \rbrace $$
+(rappelons que $z=p(v)$ et convenons que
+$W(z,0)=\lbrace z\rbrace $ (cf. \Cite{G})).
+
+\Theorem {Th\'eor\`eme 3.1} Il y a des constantes
+$M$ et $L$ d\'ependant de $\theta $ ($L>0<M<1$)
+telles que, pour $1-M\leq K<1$ et $v_{0}$ un point
+d'\'etroitesse $\theta >0$ relativement \`a un
+polyn\^ome non constant $p$, et si l'on note
+$t_{i}=K^{i}p(v_{0})$ la suite $v_{i}$ d\'efinie par:
+$$
+ v_{i} = N_{p-t_{i}}(v_{i-1}) =
+v_{i-1}-{p(v_{i-1})-t_{i} \over p'(v_{i-1})}
+\Eqno ({\Sharp})
+$$
+converge vers une racine de $p$. De plus:
+$$
+ \left \vert p(v_{n})\right \vert \leq K^{n}(1+L)
+\left \vert p(v_{0})\right \vert \Eqno
+({\Sharp}{\Sharp})
+$$
+$M=\sin\theta /(24+\sin\theta )$ et $L={2\over 3}M$
+conviennent.\endTheorem
+\penalty -1000
+\Theorem {Corollaire 3.2} Soit $p$ un polyn\^ome de
+ $P_{d}(1)$ et $\varepsilon >0$. Si $v_{0}$ est un
+point d'\'etroitesse $\theta >0$ et de module $R>1$,
+et $v_{n}$ est la suite d\'efinie par ({\Sharp})
+avec $K=1-M$, alors, si $n\geq N(d,\theta
+,R,\varepsilon )$, on a $ \left \vert p(v_{n})\right
+\vert \leq \varepsilon $, o\`u $N(d,\theta
+,R,\varepsilon )=K_{1}d+K_{2}+K_{3}\log(1/\varepsilon
+)$ avec $K_{1}=K_{3}\log{R}$,\,\,
+$K_{2}=K_{3}\log{R(1+L)\over R-1}$ et $K_{3} =
+{1\over \left \vert \log(1-M)\right \vert }<{1\over
+M}$.
+\endProof \endTheorem
+
+\Proof {D\'emonstration de 3.1} Posons
+$z_{n}=p(v_{n})$, $r_{n}=r_{v_{n}}$ et
+$D_{n}=D(t_{n},t_{n}\sin\theta )$. Comme $t_{n}=K^{n}
+\left \vert p(v_{0})\right \vert $, l'in\'egalit\'e
+({\Sharp}{\Sharp}) suit de:
+\medskip\nobreak
+\Item {{\bf (a_{n})} } $ \left \vert
+z_{n}-t_{n}\right \vert \leq L \left \vert t_{n}\right
+\vert $.
+\medskip\nobreak
+ \Admin {(a_{0})} est vraie car $t_{0}=z_{0}$. De
+\Admin {(a_{n})}, il vient:
+\medskip\nobreak
+\Item {{\bf (b_{n})}} $ \left \vert
+z_{n}-t_{n+1}\right \vert \leq \left \vert
+z_{n}-t_{n}\right \vert + \left \vert
+t_{n}-t_{n+1}\right \vert \leq (L+M) \left \vert
+t_{n}\right \vert $.
+\medskip\nobreak
+ Comme $D_{n}$ est inclus dans $W(t_{0},\theta )$ (cf.
+Figure~5), on tire de \Admin {(a_{n})} et \Admin
+{(b_{n})} :
+$$
+ r_{n}\geq \left \vert t_{n}\right \vert \sin\theta -
+\left \vert z_{n}-t_{n}\right \vert \geq (\sin\theta
+-L) \left \vert t_{n}\right \vert \geq J \left \vert
+z_{n}-t_{n+1}\right \vert \enskip \enskip ,
+$$
+o\`u $J = (\sin\theta -L)/(L+M)$. Supposons
+$$
+ J>4 \enskip \enskip . \Eqno (*)
+$$
+Les lemmes 2.6 et 2.7 appliqu\'es au polyn\^ome
+$p-t_{n+1}$ donnent:
+$$
+ \left \vert z_{n+1}-t_{n+1}\right \vert \leq
+{ 4 \left \vert z_{n}-t_{n+1}\right \vert ^{2} \over
+r_{n}-4 \left \vert z_{n}-t_{n+1}\right \vert } \leq
+I \left \vert t_{n+1}\right \vert \enskip \enskip ,
+$$
+o\`u $I=4(L+M)/M(J-4)$. Donc, si de plus
+$$
+ I<\sin\theta \enskip \enskip , \Eqno (*')
+$$
+le point $z_{n+1}$ est dans $W(z_{0},\theta )$, et si
+enfin
+$$
+ I<L\enskip \enskip , \Eqno (*'')
+$$
+la condition \Admin {(a_{n+1})} est v\'erifi\'ee. Que
+les valeurs de $M$ et $L$ donn\'ees dans 3.1
+v\'erifient les hypoth\`eses (*) n'est plus qu'un
+exercice de calcul.~\endProof
+%\input RESOURCE% for test run
+%\input Bourbaki/Marin0-3% for final run
+
+\Subheading {4. Les algorithmes probabilistes}
+%
+ Soit $S_{R}$ le cercle $\lbrace v\,\,\bigMidvert \,\,
+\left \vert v\right \vert =R\rbrace $ muni de la
+mesure de Lebesgue normalis\'ee par $\mu (S_{R})=1$.
+Soit $p$ un polyn\^ome de $P_{d}(1)$ et posons
+$E_{R}(\theta )=\lbrace v\in S_{R}\bigMidvert e(v)\leq
+\theta \rbrace $; on \'etablira au {\S }5 la:
+
+\Theorem {Proposition 4.1} L'ensemble $E_{R}(\theta )$
+est inclus dans la r\'eunion de $2(d-1)$ arcs de
+$S_{R}$ chacun de mesure: ${1\over \pi d}\left
+\lbrack \theta +2\arcsin{1\over R-1}\right \rbrack
+$.\endTheorem
+
+On notera $\mu _{\theta ,R}$ la somme de leur mesure.
+
+\Theorem {Algorithme $ {\Cal A}(\theta ,R)$ 4.2 \Rm
+{(\Cite{Sm4})}} {\rm
+Soit $p$ un polyn\^ome de $P_{d}(1)$ et $\varepsilon
+>0$. Les constantes $M$,\enskip \enskip $K=1-M$, et
+$N$ (d\'ependant de $d$, $\theta $, $R$, et
+$\varepsilon $) \'etant celles de 3.2, proc\'eder
+comme suit:}
+\Item {1^{\circ }} Choisir au hasard un $v_{0}$ sur
+ $S_{R}$.
+\Item {2^{\circ }} Calculer $v_{N}$ en it\'erant
+$N$ fois (3.1).
+\Item {3^{\circ }} Si $ \left \vert
+p(v_{N})\right \vert \leq \varepsilon $ fin; sinon
+retourner au \Admin {1^{\circ }}.
+
+\par \noindent {\rm La moyenne du nombre de cycles
+dans cet algorithme est major\'ee par} $t = t_{\theta
+,R} = 1/(1-\mu _{\theta ,R})$. \qed \endTheorem
+
+
+\Subheading {Quelques valeurs num\'eriques}
+\vskip-12pt
+$$\matrix{
+ & & & M& t& tK_{1}& tK_{2}& tK_{3} \Endphase \cr
+ R = 3,& \theta = \pi /12& : & 1/94& 6& 615&
+231& 560 \Endphase \cr
+R = 6,& \theta = \pi /6& : & 1/49& 2,44& 212& 24&
+119 \Endphase \cr
+}$$
+
+ Lorsque, pour $R$ donn\'e, on cherche \`a minorer
+$tK_{1}$, l'optimum semble \^etre autour de $R=8$ et
+$\theta =7\pi /36$ et est assez stable (Pour $6\leq
+R\leq 16$,\enskip $tK_{1}$ varie entre 206 et 223.)
+Assez paradoxalement, il est plus \'economique de
+partir d'une grande valeur initiale (cf. {\S }5).
+
+ Si vous ne faites pas confiance au hasard, soit
+$k=2(d-1)\,\ell \,\,t_{\theta ,R}$ un entier avec
+$\ell >1$. Placez alors $k$ points \'egalement
+r\'epartis sur $S_{R}$. Comme $E_{k}(\theta )$ est
+dans une r\'eunion de $2(d-1)$ arcs, il ne peut
+contenir plus de $T_{k}=2(d-1)+k\mu _{\theta ,R}$ de
+ces points. Tirant au sort les valeurs initiales dans
+cette urne de $k$ points, on est s\^ur que
+l'algorithme 4.2 n'a pas plus de $T_{k}$ cycles; la
+moyenne du nombre de tirages est major\'ee par
+$t_{k}=t_{\theta ,k}\,\ell /(\ell -1)$. Le choix de
+$\ell $ est affaire de go\^ut: un grand $\ell $
+am\'eliore la moyenne, mais recule la barri\`ere de
+s\'ecurit\'e $T_{k}$.
+
+
+\Theorem {Th\'eor\`eme 4.3} Soit $0<\mu <1$. Il y
+a une fonction $N(\mu ,d)$ telle que, si
+$R=1+4d^{\,2}/\pi \mu $ et $(v_{0},p)$ est hors d'un
+ensemble exceptionnel de mesure $\mu $ dans
+$S_{R}\times P_{d}(1)$, il suffit, partant de $v_{0}$
+et fixant $\theta ={\textstyle {1\over 2}}\pi \mu $
+de $N(\mu ,d)$ it\'erations de la formule \Admin
+{3.1({\Sharp})} pour obtenir un bon z\'ero approch\'e
+de $p$. On a :
+$$
+ N(\mu ,d) = K_{1}(\mu )(d+{\textstyle {\textstyle
+{1\over 2}}}) {\log(1/\mu )+ K_{2}( d) \over \mu } +
+K_{3}(d)
+$$
+o\`u $K_{1}$ est croissante,
+$$
+ 15,2\leq K_{1}(\mu )\leq 25\enskip \enskip ,
+\enskip \enskip \enskip \enskip \enskip
+K_{1}(1/4)<15,5<K_{1}(1/2)<18\enskip \enskip ,
+$$$$
+ (d+{\textstyle {1\over
+2}})K_{2}=3\log{d}+\log(36/\pi )\enskip ,\enskip
+\enskip \Rm { {\it et\/}}
+ \enskip \enskip K_{3}(d)=2/3+K_{1}\pi
+/4d^{2}\enskip \enskip .
+$$\endTheorem
+
+\Proof {D\'emonstration de 4.3} La mesure des
+$(v_{0},p)$ tels que, soit $\rho _{p}<\rho $, soit
+$e_{p}(v_{0})<\theta $, est major\'ee par
+$$
+ \mu = (d-1)\rho ^{2}+{2(d-1)\over \pi d} \left
+\lbrack \theta +2\arcsin{1\over R-1}\right \rbrack
+\enskip \enskip .
+$$
+{\smc }Il suffit de poser $\rho =\sqrt \mu /d$ et
+d'appliquer 3.2 avec $\varepsilon =\rho /9$, et, pour
+$\theta $ et $R$, les valeurs donn\'ees dans 4.3.
+\endProof
+
+
+
+\Subheading {5. La g\'eom\'etrie des polyn\^omes \Rm
+{(cf. \Cite{ShSm1})}}
+%
+ Une id\'ee naturelle pour trouver un z\'ero d'un
+polyn\^ome $p$ est de minimiser la fonction r\'eelle
+$g(v)={\textstyle {1\over 2}} \left \vert p(v)\right
+\vert ^{2}$ en descendant le long des lignes de
+gradient. Or, $X(v)=\Rm {grad}g(v)=\overline
+{p'(v)}p(v)$ est colin\'eaire au champ de Newton
+$n(v)=p(v)/p'(v)$ ($X(v)= \left \vert p'(v)\right
+\vert ^{2}n(v)$) et donc $X$ a m\^emes courbes
+int\'egrales que l'\'equation diff\'erentielle de
+Newton $v'=-n(v)$. La m\'ethode d'Euler, avec le pas
+1 pour r\'esoudre cette \'equation diff\'erentielle,
+produit la m\^eme it\'eration que la m\'ethode de
+Newton. Remarquons aussi que, comme
+$p'(v)n(v)=p(v)=z$, le polyn\^ome $p$ envoie son champ
+de Newton sur le champ radial du but. Donc les lignes
+de gradient que nous cherchons \`a descendre sont
+aussi les feuilles du feuilletage ${\Cal F}_{p}$,
+pr\'eimage par $p$ des rayons de ${\Bbd C}$ \`a savoir
+les courbes que Smale suivait dans sa d\'emonstration
+du th\'eor\`eme fondamental de l'alg\`ebre. Dans
+\Cite{Sm2} (et aussi \Cite{ShSm1}), Smale avait suivi
+ces lignes de gradient de plus pr\`es en relaxant le
+pas de la m\'ethode d'Euler, i.e. en it\'erant
+$v\mapsto v-hp(v)/p'(v)$ o\`u $0<h<1$.
+
+ De mani\`ere imag\'ee, si le graphe de $g$ est une
+montagne que l'on cherche \`a descendre, au lieu \`a
+chaque pas de viser le refuge attendu, Smale
+\Cite{Sm2} vise une \'etape interm\'ediaire, disons
+\`a mi-distance. La m\'ethode du {\S }3 (\Cite{Sm4})
+est plus s\^ure: avant de partir, on d\'etermine
+toutes ses \'etapes, ce qui permet d'\'eviter de
+mauvais embranchements aux cols. Pour expliquer le
+paradoxe du {\S }4, on peut dire que, de plus haut, la
+vue permet de distinguer la vall\'ee principale des
+vall\'ees secondaires et donc on peut faire un
+meilleur plan de route.
+
+ Le feuilletage ${\Cal F}_{p}$, orient\'e par $X$, a
+des singularit\'es de deux types (cf. Figure~6): des
+sources aux racines de $p$ et des selles de singe \`a
+$m-1$ pattes aux points critiques de multiplicit\'e
+$m$ et de valeur critique non nulle
+($0=p'(c)=\dots{}=p^{(m-1)}(c)$ et
+$p(c)p^{(m)}(c)\mathbin{\not =}0$). Soit $\Gamma
+(p)$ la r\'eunion des vari\'et\'es instables des
+points critiques de ${\Cal F}_{p}$; c'est un arbre
+plong\'e dans ${\Bbd C}$, \Bi{l' arbre de Shub-Smale
+du polyn\^ome } $p$. Posons $\Gamma (p)^{+}$
+l'union de $\Gamma (p)$ et des vari\'et\'es stables
+des singularit\'es de type selle (\'eventuellement
+multiples) de ${\Cal F}_{p}$. Les composantes de
+$\Gamma (p)^{+}/\Gamma (p)$ {\it sont en nombre
+inf\'erieur ou \'egal \`a\/} $2(d-1)$ (ce maximum est
+atteint quand les racines de $p$ sont simples, les
+points critiques sont de multiplicit\'e 2 et n'ont pas
+de liaisons entre eux dans ${\Cal F}_{p}$).
+
+
+
+\vDiagram{ {66}Figure 6}
+
+
+
+
+\Remark {Remarque 5.1} Les arbres plong\'es dans
+${\Bbd C}$ qui peuvent se r\'ealiser comme arbres de
+Shub-Smale d'un polyn\^ome peuvent \^etre facilement
+d\'etermin\'es et, \`a l'aide des arbres de
+Shub-Smale, on peut {\it param\'etrer les polyn\^omes
+par leurs valeurs critiques\/} (cf. \Cite{M} et
+\Cite{STW}).\endRemark
+
+\Theorem {Lemme 5.2 \Rm {(Gauss)}} Soit $p$ un
+polyn\^ome dont toutes les racines sont \`a
+l'int\'erieur d'un cercle $S$, alors le champ de
+Newton est transverse \`a $S$.\endTheorem
+
+\Theorem {Corollaire 5.3} Soit $S$ un cercle
+contenant dans son int\'erieur tous les z\'eros d'un
+polyn\^ome $p$, alors:
+\Item {(i)} l'arbre $\Gamma (p)$ est dans
+l'int\'erieur de $S$;
+\Item {(ii)} $\Gamma (p)^{+}\cap S$ contient au plus
+ $2(d-1)$ points. \endProof \endTheorem
+
+\Theorem {Lemme 5.4} Soit $p$ un polyn\^ome de
+$P_{d}(1)$, $R\geq 2$, $S_{R}$ le cercle $ \left
+\vert v\right \vert =R$ et $v_{1}$, $v_{2}$ deux
+points de $S_{R}$. Alors :
+\medskip
+\Item {(i)} $p(v_{i})$ est non nul;
+\medskip
+\Item {(ii)} $\displaystyle d \left \vert \Rm
+{Arg}{v_{1}\over v_{2}}\right \vert - 2\Rm
+{arcsin}{1\over R-1}\leq \left \vert \Rm
+{Arg}{p(v_{1})\over p(v_{2})}\right \vert \leq $
+ $$\leq d \left \vert \Rm {Arg}{v_{1}\over
+v_{2}}\right \vert + 2\Rm {arcsin}{1\over R-1}\enskip
+\enskip .$$
+\endTheorem
+\Proof {D\'emonstration de 4.1} Soit $R\geq 2$ et
+$p$ dans $P_{d}(1)$; remarquons qu'un point $v_{0}$ de
+$S_{R}$ est dans $E_{R}(\theta )$ si et seulement s'il
+y a un $v'$ dans $S_{R}\cap \Gamma (p)^{+}$ avec $~
+\left \vert \Rm {Arg}(p(v)/p(v')\right \vert <\theta
+$. D'apr\`es 5.4(i) et 5.2, ${\Cal F}_{p}$ est
+transverse \`a $S_{R}$ et l'un des arcs limit\'es par
+$v$ et $v'$ sur $S_{R}$ a tous ses points $u$
+v\'erifiant $ \left \vert \Rm {Arg}(p(u)/p(v')\right
+\vert <\theta $. La proposition~4.1 suit alors de
+l'in\'egalit\'e de gauche dans 5.4~(ii). \endProof
+
+
+
+
+
+
+
+\Diagram {{75}\hfill Figure 7 \hfill \hfill Figure 8
+\hfill }
+\Proof {D\'emonstration de 5.2 (voir Figure 7)}
+Soient $w_{i}$ les racines de $p$ et soit $v$ sur
+$S$. On a:
+$$
+ \left \vert n(v)\right \vert ^{^{-2}}v\overline n(v)
+= v\,{p'(v)\over p(v)} = \sum _{i=1}^{d }{v\over
+v-w_{i}}\enskip \enskip .
+$$
+Or, comme $w_{i}$ est int\'erieur \`a $S$, on a $\Rm
+{Re}(v(\overline {v-w_{i}}))>0$; donc $\Rm {Re}(
+v/(v-w_{i}))>0$ et $\Rm {Re}( v\,\,\overline n(v))>0$.
+ \endProof
+
+\Proof {D\'emonstration de 5.4 (voir Figure 8)} Soit
+$p(v)=v^{d}+a_{1}v^{d-1}+\dots{}+a_{d}$ pour $ \left
+\vert v_{j}\right \vert =R$. On a:
+$$
+ \left \vert 1-{p(v_{j})\over v_{j}^{d}}\right
+\vert \leq \sum _{i=1}^{d }{ \left \vert
+a_{i}\right \vert \over \left \vert v_{j}\right \vert
+^{d}} < {1\over \left \vert v_{j}\right \vert
+}\,\,{1\over 1-1/ \left \vert v_{j}\right \vert }
+ = {1\over R-1} \leq 1
+$$
+car $R\geq 2$, d'o\`u (i); et pour (ii):
+$$
+ \left \vert \Rm {Arg}{p(v_{1})\over p(v_{2})} -
+\Rm {Arg}\left({v_{1}\over v_{2}}\right)^{d} \right
+\vert
+ = \Rm {Arg} \left \vert {p(v_{1})\over
+v_{1}^{d}}\,{v_{2}^{d}\over p(v_{2})}\right \vert
+\leq
+$$$$
+\hfil \leq \left \vert \Rm {Arg}{p(v_{1})\over
+v_{1}^{d}}\right \vert + \left \vert \Rm
+{Arg}{p(v_{2})\over v_{2}^{d}}\right \vert
+ \leq 2\Rm {arcsin}{1\over R-1}\enskip \enskip
+\enskip .
+$$ \endProof
+
+
+\Subheading {6. Un crit\`ere de convergence
+calculable en $\Headingfont v_{0}$}
+%
+ Soit $v$ un point r\'egulier d'un polyn\^ome $p$. En
+2.5, on a d\'efini $\alpha (v,p)=\beta (v,p)\gamma
+(v,p)$ o\`u $\beta (v,p)$ est la norme du vecteur de
+Newton $p(v)/p'(v)$, alors que le calcul de $\gamma
+(v)$ n\'ecessite la connaissance de toutes les
+d\'eriv\'ees de $p$ en $v$. La proposition suivante
+permet de majorer $\gamma (v)$ (et donc $\alpha (v)$)
+en n'ayant \`a \'evaluer que $p$ et $p'$ en $v$:
+
+\Theorem {Proposition 6.1 \Rm {(\Cite{Sm5},
+\Cite{Sm6})}} Soit $\varphi (r)=1+\dots{}+r^{d}$
+et $p(v)=a_{0}+a_{1}v+\dots{}+a_{d}v^{d}$ un
+polyn\^ome de degr\'e $d$, alors :
+$$
+\gamma (v,p) \leq {\left \Vert p\right \Vert (\varphi
+'( \left \vert v\right \vert ))^{2}\over \left \vert
+p'(v)\right \vert \varphi ( \left \vert v\right \vert
+)}\enskip \enskip ,\enskip \enskip \enskip \Rm {o\`u
+}\enskip \enskip \enskip \left \Vert p\right \Vert =
+\Rm {Max} \left \vert a_{i}\right \vert \enskip
+\enskip .
+$$\endTheorem
+
+\Theorem {Th\'eor\`eme 6.2 \Rm {(\Cite{Sm5})}} Soit
+$p$ un polyn\^ome et $v_{0}$ un point r\'egulier tel
+que $\alpha (v_{0},p)<3-2\sqrt 2$, alors la suite de
+Newton $v_{n+1}=v_{n}-p(v_{n})/p'(v_{n})$ converge
+vers une racine de $p$. De plus, il y a une
+bijection croissante $K:\left \lbrack \right
+.0,3-2\sqrt 2\left \lbrack \right . \llongrightarrow
+ [0,1[$ telle que: \enskip Si $\alpha
+(v_{0},p)<\alpha $, alors $\left \Vert
+v_{n+1}-v_{n}\right \Vert \leq K(\alpha
+)^{2^{n}-1}\left \Vert v_{1}-v_{0}\right \Vert $ et
+$\alpha _{0}=K^{-1}({\textstyle {1\over
+2}})>0,14$.\endTheorem
+
+ Pour $p$ dans $P_{d}(1)$, le nombre $\alpha (0,p)$
+est major\'e par $ \left \vert a_{0}\right \vert /
+\left \vert a_{1}\right \vert ^{2}$ qui est
+ind\'ependant de $d$. Ainsi, {\it l'ensemble des
+polyn\^omes de $P_{d}(1)$, dont la suite de Newton
+partant de z\'ero converge, a une mesure minor\'ee
+ind\'ependamment de\/} $d$.
+
+
+\Proof {D\'emonstration de 6.1} En majorant les
+coefficients de $p$ par $\left \Vert p\right \Vert $,
+on obtient $ \left \vert p^{(k)}(v)/k!\right \vert
+\leq \left \Vert p\right \Vert \varphi ^{(k)}( \left
+\vert v\right \vert )/k!$, d'o\`u en faisant $k=1$ on
+obtient $\left \Vert p\right \Vert \varphi '( \left
+\vert v\right \vert )/ \left \vert p'(v)\right \vert
+\geq 1$. Un jeu sur les coefficients du bin\^ome
+(\Cite{Sm6}, p.12-14) permet d'\'etablir que $\alpha
+(r,\varphi )\leq 1$ pour tout $r\geq 0$, d'o\`u
+$$
+ \left \vert p^{(k)}(v)/k!\right \vert \leq \left
+\Vert p\right \Vert \varphi ^{(k)}( \left \vert
+v\right \vert )/k! \leq \left \Vert p\right \Vert
+(\varphi '( \left \vert v\right \vert ))^{k}/(\varphi
+( \left \vert v\right \vert ))^{k-1}\enskip \enskip
+,\enskip \Rm {et}
+$$$$
+ \gamma (v,p) \leq \sup _{k>1}\left \lbrack
+\left \Vert p\right \Vert (\varphi '( \left \vert
+v\right \vert ))^{k}\over \left \vert p'(v)\right
+\vert (\varphi ( \left \vert v\right \vert ))^{k-1}
+\right \rbrack ^{1/k-1}\leq \,{\varphi '\over \varphi
+} \,\,\sup _{k>1}\left \lbrack \left \Vert p\right
+\Vert \varphi '\over \left \vert p'\right \vert
+\right \rbrack ^{1/k-1}\enskip \enskip .
+$$
+\par \noindent Le dernier supr\'emum est atteint pour
+$k=2$, d'o\`u la proposition. \endProof
+
+\Proof {D\'emonstration de 6.2} Pour $0\leq t<\gamma
+^{-1}$, soit $\psi $ la fonction $\psi (t)={1\over
+1-\gamma t}-2\gamma t+\alpha -1$; comme en 2.6, on
+\'etablit que, pour $\left \Vert v-v_{0}\right \Vert
+\leq t\gamma ^{-1} $ on a:
+ $$ \left \vert p(v_{0})\over p'(v_{0})\right \vert
+\leq {\psi (0)\over -\psi '(0)}\enskip \enskip
+\enskip ,\enskip \enskip \enskip \Rm {et}\enskip
+\enskip \enskip
+ \left \vert p''(v)\over p'(v_{0})\right \vert
+\leq {\psi ''(t)\over -\psi '(0)}\enskip \enskip
+\enskip .$$
+
+
+ Kantorovitch (\Cite{KA}, th\'eor\`eme 3, p.234) nous
+dit que, si $\psi (t)=0$ a une racine dans $\left
+[0,\gamma ^{-1}\right [\,$, alors la suite $v_{n}$
+converge vers une racine $w$ de $p$, que la suite de
+Newton $t_{n}$ de $\psi $ partant de $t_{0}=0$
+converge vers la plus petite racine $t_{*}$ de $\psi
+(t)=0$ et $ \left \vert v_{n+1}-v_{n}\right \vert \leq
+t_{n+1}-t_{n}$.
+
+ Il ne reste plus qu'\`a estimer $t_{n+1}-t_{n}$ pour
+obtenir les majorations du th\'eor\`eme 6.2.
+\endProof
+
+\Remark {Remarques}
+%
+\par \noindent Comme Kantorovitch \'enonce son
+th\'eor\`eme dans un Banach, le th\'eor\`eme~6.2 est
+valable pour une fonction analytique d\'efinie sur un
+Banach (il en est bien s\^ur de m\^eme pour 6.1).
+
+\par \noindent Smale d\'emontre 6.2 plus directement
+en majorant \`a chaque it\'eration de Newton les
+quantit\'es $\beta $ et $\gamma $. N'utilisant pas le
+th\'eor\`eme de Kantorovitch, cette approche se
+g\'en\'eralise aux m\'ethodes d'ordre sup\'erieur
+(\Cite{C}).\endRemark
+
+
+
+\Subheading {7. Un algorithme g\'en\'eriquement
+convergent \Rm {(\Cite{ShSm3})}}
+%
+ Soit $G_{d}$ l'ouvert de $P_{d}$ form\'e des
+polyn\^omes $p(v)=a_{0}+a_{1}v+\dots{} +a_{d}v^{d}$
+ayant leurs racines et leurs points critiques
+distincts: $P_{d}\setminus G_{d}$ \'etant
+alg\'ebrique, $G_{d}$ est un ouvert dense de mesure
+pleine. Soient $p$ dans $G_{d}$ et $v$ dans la
+sph\`ere de Gauss $S={\Bbd C}\cup \infty $.
+D\'efinissons alors:
+$$
+k_{p}(v)={\varphi ( \left \vert v\right \vert ) \left
+\vert p'(v)\right \vert ^{2}\over 2(\varphi '( \left
+\vert v\right \vert ))^{2} \left \vert p(v)\right
+\vert \left \Vert p\right \Vert }\enskip \enskip ,
+h_{p}(v)=\Rm {Min}(1,k_{p}(v))\enskip \enskip , $$
+et $T_{p}(v)=v-h_{p}(v)p(v)/p'(v)$ \enskip (o\`u
+$\varphi $ et $\left \Vert p\right \Vert $ sont
+d\'efinis au {\S }6).
+
+ Les points fixes de $T_{p}$ sont les racines et les
+points critiques de $p$. L'application $\widetilde T
+: S\times G_{d}\rightarrow S$ ainsi d\'efinie est
+continue et $C^{\infty }$ pr\`es des points critiques
+de $p$. Quand $p$ est fix\'e par le contexte, nous
+\'ecrirons $T$ au lieu de $T_{p}$.
+
+\Remark {Remarque} Les formules ci-dessus
+d\'efinissent une application continue $S\times
+P_{d}\rightarrow S$, mais si $p$ a des racines
+multiples, cette application n'est pas lisse pr\`es
+des racines multiples. Remarquons d'autre part que
+l'application de Newton $\widetilde N : S\times
+P_{d}\rightarrow S$ n'est pas continue (en $P_{d}$)
+aux racines multiples, car quand le polyn\^ome varie,
+un point critique se rapprochant d'une racine voit son
+image passer brusquement de l'infini \`a cette
+racine.\endRemark
+
+{\bf }\Theorem {Th\'eor\`eme 7.1} Pour tout $p$ de
+$G_{d}$, il y a un ferm\'e $V_{p}$ de mesure nulle
+dans $S$ tel que, pour tout $v$ hors de $V_{p}$ la
+suite des it\'er\'es $T^{k}(v)$, $k\geq 0$, converge
+vers un z\'ero de $p$.
+
+ De plus, $T$ est la m\'ethode de Newton dans un
+voisinage de chaque racine de $p$.\endTheorem
+
+\Theorem {Proposition 7.2} Soit $p$ dans $G_{d}$ et
+ $c$ un point critique de $p$, alors
+ $$V^{s}(c)=\lbrace v\bigMidvert T^{k}(v)\rightarrow
+c\enskip \enskip \Rm {quand}\enskip \enskip
+k\rightarrow \infty \rbrace $$
+est un ferm\'e de mesure nulle dans $S$.\endTheorem
+
+\Proof {D\'emonstration de 7.2} Comme le jacobien de
+$T$ est presque partout non nul, il suffit de montrer
+qu'un voisinage de $c$ dans $V^{s}(c)$ est de mesure
+nulle. Comme $T$ est lisse au voisinage de $c$, il
+suffit, selon le th\'eor\`eme de la vari\'et\'e stable
+(\Cite{HPS}), de prouver que $T'(c)$ est hyperbolique
+(en effet ce th\'eor\`eme impliquera alors que
+$V^{s}(c)$ est, pr\`es de $c$, un arc lisse). On a
+$$T'(c)(V) = V-\lambda \overline V\enskip \enskip
+\enskip ,\enskip \enskip \enskip \Rm {o\`u}\enskip
+\enskip \enskip
+ \lambda = {p(c)\,\overline p''(c)\,\varphi ( \left
+\vert c\right \vert )\over 2\left(\varphi '( \left
+\vert c\right \vert )\right)^{2}\left \Vert p\right
+\Vert \left \vert p(c)\right \vert }\enskip \enskip
+\enskip .$$
+Les valeurs propres de $T'(c)$ sont donc $1\pm \left
+\vert \lambda \right \vert $; or, comme on a
+\break$\varphi (r)\varphi ''(r)\leq 2(\varphi
+'(r))^{2}$,
+ $$ \left \vert \lambda \right \vert \leq { \left
+\vert p''(c)\right \vert \over \left \Vert p\right
+\Vert }\,\,{1\over \varphi ''( \left \vert c\right
+\vert )} \leq 1\enskip \enskip .$$
+\endProof
+
+\Proof {D\'emonstration de 7.1} Posons $V_{p}=\bigcup
+\left \lbrace V^{s}(c)\bigMidvert p'(c)=0\right
+\rbrace $. Soit $v$ hors de $V_{p}$ et
+$v_{n}=T^{n}(v)$, alors $p'(v)\mathbin{\not =}0$ et
+comme, d'apr\`es 6.1, $k(v)<{\textstyle {1\over
+2}}\alpha (v)$, on obtient par une majoration analogue
+\`a celle de 2.6~:
+ $$ \left \vert p(v_{n+1})\right \vert \leq
+h(v_{n}) \left \vert p(v_{n})\right \vert < \left
+\vert p(v_{n})\right \vert \enskip \enskip \enskip ,
+$$
+o\`u l'application continue $h : {\Bbd C}\rightarrow
+[0,1]$ n'est nulle qu'aux points critiques de $p$.
+
+ Comme $v$ n'est dans aucun des $V^{s}(c)$, cela
+implique que la suite d\'ecroissante $ \left \vert
+p(v_{n})\right \vert $ tend vers z\'ero et $v_{n}$
+tend vers une racine de $p$ (deux racines de $p$ ne
+peuvent \^etre adh\'erentes \`a $v_{n}$ puisque, comme
+$T$ co\"\i ncide avec l'application de Newton pr\`es
+des racines de $p$, ces derni\`eres sont des points
+fixes attractifs de $T$).\endProof
+
+
+\References {
+Bibliographie
+}
+%
+\Benchmark
+\Cite{Ba} {\smc B. Barna}, {\it Uber Divergenzpunkte
+des Newtonshe Verfahrens zur Bestimmung von
+W{\"u}rzeln algebraischer Gleichungen III\/}{\smc ,}
+Publ. Math. Debrecen {\bf 8} (1961), 193-207.
+\Benchmark
+\Cite{C} {\smc J. Curry}, {\it On zero finding methods
+of higher order from data at one point\/}, MSRI
+Berkeley (1986).
+\Benchmark
+\Cite{DH} {\smc A. Douady and J.H. Hubbard}, {\it On
+the dynamics of polynomial like mappings\/}, Ann.
+Sci. E.N.S., 4e s\'erie, {\bf t.18} (1985), 287-343.
+\Benchmark
+\Cite{F} {\smc J.-P. Francoise}, {\it Estimations
+uniformes pour les domaines de convergence de la
+m\'ethode de Newton\/}, S\'eminaire de G\'eom\'etrie
+Alg\'ebrique R\'eelle (J.J. Risler), Publ. Univ. Paris
+VII, {\bf 24} (1986).
+\Benchmark
+\Cite{G} {\smc O. Gabber}, {\it Diverses interventions
+au S\'eminaire Bourbaki\/}, tradition orale.
+\Benchmark
+\Cite{H} {\smc W. Hayman}, ``Multivalent functions'',
+Cambridge Univ. Press, Cambridge, 1958.
+\Benchmark
+\Cite{HPS} {\smc M. Hirsch, C. Pugh and M. Shub}, {\it
+Invariant manifolds\/}, Lectures Notes in Math. {\bf
+583} (1977), Springer, New-York.
+\Benchmark
+\Cite{HM} {\smc M. Hurley and C. Martin}, {\it
+Newton's algorithm and chaotic dynamical systems\/},
+SIAM J. Math. Anal. {\bf 15} (1984), 238-252.
+\Benchmark
+\Cite{Ka} {\smc L. Kantorovich et G. Akilov}, Analyse
+fonctionnelle, {\bf t.2}, Editions MIR, Moscou, 1981.
+\Benchmark
+\Cite{McM} {\smc C. McMullen}, ``Families of
+Rationals Maps and Iterative Root-Finding
+Algorithms'', Ph. D., Harvard Univ., Mai 1985, \`a
+para\^\i tre.
+\Benchmark
+\Cite{M} {\smc A. Marin}, {\it Les arbres de
+Shub-Smale\/}, S\'eminaire de G\'eom\'etrie
+alg\'ebrique r\'eelle (J.J. Risler), Publ. Univ. Paris
+VII {\bf 24} (1986).
+\Benchmark
+\Cite{O} {\smc J. Oesterl\'e}, {\it D\'emonstration de
+la conjecture de Bieberbach\/}, expos\'e \Admin
+{n^{o}649} du S\'eminaire Bourbaki (juin 1985).
+\Benchmark
+\Cite{R1} {\smc J. Renegar}, {\it On the efficiency of
+Newton's method in approximating all zeros of a system
+of complex polynomials\/}, \`a para\^\i tre dans
+Mathematics of Operations Research.
+\Benchmark
+\Cite{R2} {\smc J. Renegar}, {\it A polynomial-time
+algorithm based on Newton's method for linear
+programming\/}, MSRI Berkeley (1986).
+\Benchmark
+\Cite{SU} {\smc D. Saari and J. Urenko}, {\it Newton's
+method, circle maps and chaotic motions\/}, Amer.
+Math. Monthly {\bf 91} (1984), 3-17.
+\Benchmark
+\Cite{Sh} {\smc M. Shub}, {\it The geometry and
+topology of dynamical systems, and algorithms for
+numerical problems\/}, notes pr\'epar\'ees pour des
+conf\'erences donn\'ees \`a D.D.A, Universit\'e de
+Peking, Bejing, Chine, Ao\^ut-Septembre 1983.
+\Benchmark
+\Cite{{\bf ShSm1\&2}} {\smc M. Shub and S. Smale},
+{\it Computational complexity on the geometry of
+polynomials and a theory of cost\/}, PartI, Ann. Sci.
+E.N.S. ({\bf 4}) {\bf t.18} (1985), 107-161; Part II,
+SIAM J. Computing {\bf 15} (1986), 145-161.
+\Benchmark
+\Cite{ShSm3} {\smc M. Shub and S. Smale}, {\it On the
+existence of generally convergent algorithms\/},
+Jour. of Complexity {\bf 2} (1986), 2-11.
+\Benchmark
+\Cite{STW} {\smc M. Shub, D. Tischler and R.
+Williams}, {\it The Newtonian graph of a complex
+polynomial\/}, soumis \`a SIAM J. of Math. Analysis.
+\Benchmark
+\Cite{Sm1} {\smc S. Smale}, {\it A Convergent process
+of price adjustment and global Newton methods\/}, J.
+Math. Econom. {\bf 3} (1976), 107-120.
+\Benchmark
+\Cite{Sm2} {\smc S. Smale}, {\it The fundamental
+theorem of algebra and complexity theory\/}, Bull.
+Amer. Math. Soc. {\bf 4} (1981), 107-120.
+\Benchmark
+\Cite{Sm3} {\smc S. Smale}, {\it The Problem of the
+average speed of the simplex method\/}, in
+``Mathematical Programming: the state of the Art, Bonn
+1982'' (editors Bachem et al.), Springer, 1983.
+\Benchmark
+\Cite{Sm4} {\smc S. Smale}, {\it On the efficiency of
+algorithms of analysis\/}, Bull. Amer. Math. Soc.
+{\bf 13} (1985), 87-121.
+\Benchmark
+\Cite{Sm5} {\smc S. Smale}, {\it Newton's method
+estimates from data at one point\/}, \`a para\^\i tre
+dans: Proceedings of a Conference at Laramie in Honor
+of Gail S. Young, Springer New York (1986).
+\Benchmark
+\Cite{Sm6} {\smc S. Smale}, {\it Algorithms for
+solving equations\/}, MSRI Berkeley, 1986 (texte
+\'ecrit pour le congr\`es mondial de Berkeley).
+\endReferences
+\bigskip
+\goodbreak
+
+
+ Vous trouverez de nombreuses autres r\'ef\'erences
+dans les bibliographies de \Cite{Sm2}, \Cite{Sm4},
+\Cite{Sm6}; en plus, \Cite{Sm6} contient une
+discussion de l'\'etat actuel des probl\`emes pos\'es
+dans \Cite{Sm2} et \Cite{Sm4}.
+
+
+
+\medskip
+\vfill
+\leftskip = .55 \hsize
+\parindent=0pt \parskip=0pt
+\def \Par{\nobreak\par}
+ Alexis MARIN
+ \nobreak\vskip 1.5pt \nobreak\Par
+ UA 1169 du CNRS\Par
+ Universit\'e Paris-Sud\Par
+ Math\'ematiques, b\^at 425\Par
+ 91405 ORSAY\Par
+
+\vfill
+\vfill
+\eject
+\end
diff --git a/macros/generic/occam/PlainEx2/W.sty b/macros/generic/occam/PlainEx2/W.sty
new file mode 100644
index 0000000000..8f366dec57
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+++ b/macros/generic/occam/PlainEx2/W.sty
@@ -0,0 +1,196 @@
+
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+
+% use \nolabels to get rid of eqn, ref, and fig labels in draft mode
+\def\nolabels{\def\wrlabeL##1{}\def\eqlabeL##1{}\def\reflabeL##1{}}
+\nolabels
+
+% tagged sec numbers
+\global\newcount\secno \global\secno=0
+\global\newcount\meqno \global\meqno=1%
+
+ \def \newsec#1{\global\advance\secno by1\message{(\the\secno. #1)}%
+ \global\subsecno=0
+ \eqnres@t\noindent{\bf\the\secno. #1}%
+ \writetoca{{\secsym} {#1}}\par\nobreak\medskip\nobreak}%
+
+ \def \eqnres@t{\xdef\secsym{\the\secno.}%
+ \global\meqno=1\bigbreak\bigskip}%
+
+ \xdef\secsym{}%
+
+ \global\newcount\subsecno \global\subsecno=0%
+
+ \def \eqn#1#2{\xdef #1{(\secsym
+ \the\meqno)}\writedef{#1\leftbracket#1}%
+ \global\advance\meqno by1$$#2\eqno#1\eqlabeL#1$$}%
+
+%% Footnotes
+ \newskip\footskip\footskip14pt plus 1pt minus 1pt
+ %% sets footnote baselineskip
+ \def \footnotefont{\ninepoint}
+ \def\f@t#1{\footnotefont #1\@foot}
+ \def\f@@t{\baselineskip\footskip\bgroup
+ \footnotefont\aftergroup\@foot\let\next}
+ \setbox\strutbox=\hbox{\vrule height9.5pt depth4.5pt width0pt}%
+%
+ \global\newcount\ftno \global\ftno=0
+ \def \foot{\global\advance\ftno by1\footnote{$^{\the\ftno}$}}%
+
+ \def \footatend{}%
+
+%% References
+%
+% \ref\label{text}
+% generates a number, assigns it to \label, generates an entry.
+% To list the refs on a separate page, \listrefs
+%
+\global\newcount\refno \global\refno=1
+\newwrite\rfile
+ \def \nref#1{\xdef#1{[\the\refno]}\writedef{#1\leftbracket#1}%
+ \ifnum\refno=1\immediate\openout\rfile=refs.tmp\fi
+ \global\advance\refno by1\chardef\wfile=\rfile\immediate
+ \write\rfile{\noexpand\item{#1\ }%
+ \reflabeL{#1\hskip.31in}\pctsign}\findarg}
+ %
+ % horrible hack to sidestep tex \write limitation
+ % FRAGILE!!!
+ \def\findarg#1#{\begingroup\obeylines\newlinechar=`\^^M\pass@rg}
+ {\obeylines\gdef\pass@rg#1{\writ@line\relax #1^^M\hbox{}^^M}%
+ \gdef\writ@line#1^^M{\expandafter\toks0\expandafter{\striprel@x #1}%
+ \edef\next{\the\toks0}\ifx\next\em@rk\let\next=\endgroup\else\ifx\next\empty%
+ \else\immediate\write\wfile{\the\toks0}\fi\let\next=\writ@line\fi\next\relax}}
+ \def\striprel@x#1{} \def\em@rk{\hbox{}}%
+ %
+
+ \def \ref{[\the\refno]\nref}%
+
+ \def \listrefs{\footatend\vfill
+ \supereject\immediate\closeout\rfile\writestoppt
+ \baselineskip=14pt\centerline
+ {{\bf References}}\bigskip{\frenchspacing
+ \parindent=20pt\escapechar=` \input refs.tmp
+ \vfill\eject}\nonfrenchspacing}%
+
+ \def \xref{\expandafter\xr@f}\def\xr@f[#1]{#1}%
+
+ \def \refs#1{\count255=1[\r@fs #1{\hbox{}}]}
+ \def\r@fs#1{\ifx\und@fined#1\message{reflabel \string#1 is undefined.}%
+ \nref#1{need to supply reference \string#1.}\fi%
+ \vphantom{\hphantom{#1}}\edef\next{#1}\ifx\next\em@rk\def\next{}%
+ \else\ifx\next#1\ifodd\count255\relax\xref#1\count255=0\fi%
+ \else#1\count255=1\fi\let\next=\r@fs\fi\next}%
+
+%% Files
+
+\newwrite\lfile
+
+{\escapechar-1\xdef\pctsign{\string\%}\xdef\leftbracket{\string\{}
+ \xdef\rightbracket{\string\}}\xdef\numbersign{\string\#}}%
+
+ \def \writestoppt{}%
+ \def \writedef#1{}%
+
+\newwrite\tfile
+
+ \def \writetoca#1{}%
+
+\catcode`\@=12 % at signs are no longer letters
+
+ %%% Fonts 10pt, Title, 9pt
+
+ %% Stop frivolous math family waste
+ \let \TextFontInMathBad\undefined
+ \def\TextWarning{\ifmmode\TextFontInMathBad\fi}
+
+ % Unpleasantness in calling in abstract and title fonts
+
+ %%% Title fonts
+
+ \font\titlerm=cmr10 \tfontsize
+ \def \titlefont{\textfont0=\titlerm
+ \def\rm{\fam0\titlerm}%
+ \rm
+ }%
+
+ %%% Tenpoint Fonts
+ %
+ \def \tenpoint{%
+ \def\rm{\fam0\tenrm}%
+ \rm
+ \textfont0=\tenrm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm
+ \textfont1=\teni \scriptfont1=\seveni \scriptscriptfont1=\fivei
+ \textfont2=\tensy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy
+ \textfont\bffam=\tenbf
+ \def\bf{\fam\bffam\tenbf}%
+ \def\it{\TextWarning\tenit}%
+ \def\sl{\TextWarning\tensl}%
+ }%
+
+ %%% Ninepoint Fonts
+ \font\sixrm=cmr6%
+ \font\ninei=cmmi9 \skewchar\ninei='177
+ \font\sixi=cmmi6 \skewchar\ninei='177
+ \font\ninesy=cmsy9 \skewchar\ninesy='60
+ \font\sixsy=cmsy6 \skewchar\sixsy='60
+
+ \font\ninerm=cmr9 %
+
+ \def \ninepoint{%
+ \textfont0=\ninerm
+ \def\rm{\fam0\ninerm}%
+ \rm
+ \def\it{\TextWarning\nineit}%
+ \def\sl{\TextWarning\ninesl}%
+ \textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
+ \textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei
+ \textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
+ }%
+
+ %% Hyphenation exceptions
+
+\hyphenation{anom-aly anom-alies coun-ter-term coun-ter-terms}
+
+ \def \Tr{{\rm Tr}}%
+
+\endinput \ No newline at end of file
diff --git a/macros/generic/occam/PlainEx2/W.tex b/macros/generic/occam/PlainEx2/W.tex
new file mode 100644
index 0000000000..05d0af917e
--- /dev/null
+++ b/macros/generic/occam/PlainEx2/W.tex
@@ -0,0 +1,1362 @@
+%Paper: ftp://xxx.lanl.gov//hep-th/9411102
+%From: WITTEN@sns.ias.edu
+%Date: 14 Nov 1994 22:09:04 -0400 (EDT)
+
+ %\input harvmac.tex %% leading % added for Occam
+ %\input harvmac.occ %% line added for Occam processing
+ \input W.sty %% line added for Occam
+
+ %% header simplified by Occam
+ %% original header about 1600 bytes
+ \def \underarrow#1{\vbox{\ialign{##\crcr$\hfil\displaystyle
+ {#1}\hfil$\crcr\noalign{\kern1pt
+ \nointerlineskip}$\longrightarrow$\crcr}}}
+ % use of underarrow A~~~\underarrow{a}~~~B
+
+ \overfullrule=0pt
+
+ \def \tilde{\widetilde}%
+ \def \bar{\overline}%
+
+ %% no further changes for Occam
+
+\Title{hep-th/9411102, IASSNS-HEP-94-96}
+{\vbox{\centerline{MONOPOLES AND FOUR-MANIFOLDS}}}
+\smallskip
+\centerline{Edward Witten}
+\smallskip
+\centerline{\it School of Natural Sciences, Institute for Advanced Study}
+\centerline{\it Olden Lane, Princeton, NJ 08540, USA}\bigskip
+\baselineskip 18pt
+
+\medskip
+
+\noindent
+%write abstract here
+Recent developments in the understanding of $N=2$ supersymmetric Yang-Mills
+theory in four dimensions suggest a new point of view about Donaldson
+theory of four manifolds: instead of defining four-manifold invariants
+by counting $SU(2)$ instantons, one can define equivalent four-manifold
+invariants by counting solutions of a non-linear equation with an
+abelian gauge group. This is a ``dual'' equation in which the gauge
+group is the dual of the maximal torus of $SU(2)$.
+The new viewpoint suggests many new results about
+the Donaldson invariants.
+\Date{November, 1994}
+%text of paper
+
+\newsec{Introduction}
+\nref\witten{E. Witten, ``Topological Quantum Field Theory,'' Commun. Math.
+Phys. {\bf 117} (1988) 353.}
+For some years now it has been known that Donaldson theory is equivalent
+to a quantum field theory, in fact, a twisted version of $N=2$ supersymmetric
+Yang-Mills theory \witten.
+The question therefore arises of whether this viewpoint
+is actually useful for computing Donaldson invariants \ref\doninv{S. Donaldson,
+``Polynomial Invariants For Smooth Four-Manifolds,'' Topology, {\bf 29}
+(1990) 257.} or understanding
+their properties.
+
+\nref\floer{A. Floer, ``An Instanton Invariant For 3-Manifolds,''
+Commun. Math. Phys. {\bf 118} 215.}
+One standard physical technique is to cut and sum over
+physical states. In the context of Donaldson theory, such methods
+have been extensively developed by mathematicians,
+starting with the work of Floer \floer.
+So far, despite substantial efforts,
+the physical reformulation has not given any essentially new insight
+about these methods.
+
+Another approach to using physics to illuminate Donaldson theory
+starts with the fact that the $N=2$ gauge theory is
+asymptotically free; therefore, it is weakly coupled in the ultraviolet
+and strongly coupled in the infrared. Since the Donaldson invariants
+-- that is, the correlation functions of the twisted theory -- are metric
+independent, they can be computed in the ultraviolet or the infrared,
+as one wishes. Indeed, the weak coupling in the
+ultraviolet is used to show that the quantum field theory correlation
+functions do coincide with the Donaldson invariants.
+
+\nref\newwitten{E. Witten, ``Supersymmetric Yang-Mills Theory On A
+Four-Manifold,'' J. Math. Phys. {\bf 35} (1994) 5101.}
+If one could
+understand the infrared behavior of the $N=2$ theory, one might get
+a quite different description and, perhaps, a quite different way to
+compute the Donaldson invariants. Until recently, this line of thought was
+rather hypothetical for general four-manifolds
+since the infrared behavior of $N=2$ super Yang-Mills
+theory in the strong coupling region was unknown.
+Previous work along these lines was therefore limited to K\"ahler manifolds,
+where one can reduce the discussion to the $N=1$ theory, whose infrared
+behavior was known. This led to an almost complete determination \newwitten\
+of the Donaldson invariants of K\"ahler manifolds with $H^{2,0}\not= 0$.
+
+\nref\sw{N. Seiberg and E. Witten, ``Electric-Magnetic Duality,
+Monopole Condensation, And Confinement In $N=2$ Supersymmetric
+Yang-Mills Theory,'' Nucl. Phys. {\bf B426} (1994) 19,
+``Monopoles, Duality, And Chiral Symmetry
+Breaking In $N=2$ Supersymmetric QCD,'' hep-th/9408099,
+to appear in Nucl. Phys. B.}
+\nref\seiberg{N. Seiberg, ``The Power Of Holomorphy -- Exact Results
+In $4d$ SUSY Field Theories,'' hep-th/9408013.}
+The purpose of the present paper is to exploit recent work by Seiberg
+and the author
+\sw\ in which the infrared behavior of the $N=2$ theory was
+determined using methods somewhat akin to methods that have shed light
+on various $N=1$ theories (for a survey see \seiberg).
+The answer turned out to be quite surprising: the
+infrared limit of the $N=2$ theory in the ``strongly coupled'' region
+of field space is equivalent to a weakly coupled theory of abelian
+gauge fields coupled to ``monopoles.'' The monopole theory is
+dual to the original theory in the sense that, for instance, the
+gauge group is the dual of the maximal torus of the original gauge group.
+
+
+\nref\km{P. Kronheimer and T. Mrowka, ``Recurrence Relations And
+Asymptotics For Four-Manifold Invariants,'' Bull. Am. Math. Soc. {\bf 30}
+(1994) 215, ``Embedded Surfaces And The Structure Of Donaldson's
+Polynomial Invariants,'' preprint (1994).}
+\nref\arg{P. Argyres and A. Faraggi, ``The Vacuum Structure And Spectrum
+Of $N=2$ Supersymmetric $SU(N)$ Gauge Theory,'' hep-th/9411057.}
+\nref\yank{A. Klemm, W. Lerche, S. Yankielowicz, and S. Theisen,
+``Simple Singularities and $N=2$ Supersymmetric Yang-Mills Theory,''
+hep-th/9411048.}
+Since the dual theory is weakly coupled in the infrared,
+everything is computable in that region, and
+in particular for gauge group $SU(2)$,
+one does get an alternative formulation of the usual Donaldson
+invariants. Instead of computing the Donaldson invariants by counting
+$SU(2)$ instanton solutions, one can obtain the same invariants
+by counting the solutions of the dual equations, which involve
+$U(1)$ gauge fields and monopoles.\foot{In this paper, we only consider
+Donaldson theory with
+gauge group $SU(2)$ or $SO(3)$, but an analogous dual description
+by abelian gauge fields and monopoles will hold for
+any compact Lie group, the gauge group of the dual theory being always
+the dual of the maximal torus of the original gauge group. For example,
+most of the results needed to write the precise monopole equations for
+$SU(N)$ have been obtained recently \refs{\arg,\yank}.}
+
+This formulation makes manifest various properties of the Donaldson
+invariants. For instance, one can get new proofs of some of the
+classic results of Donaldson theory; one gets
+a new description of the basic classes of Kronheimer and Mrowka \km, and some
+new results about them; one gets a new understanding
+of the ``simple type'' condition for four-manifolds;
+one finds new types of vanishing theorems
+that severely limit the behavior of Donaldson theory on manifolds
+that admit a metric of positive scalar curvature;
+and one gets a complete determination of the Donaldson invariants
+of K\"ahler manifolds with $H^{2,0}\not= 0$, eliminating the
+assumptions made in \newwitten\ about the canonical divisor.
+
+It should be possible to justify
+directly the claims sketched in this paper about the consequences
+of the monopole equations even if the relation to Donaldson theory
+is difficult to prove. The reformulation may make the problems
+look quite different
+as the gauge group is abelian and the most relevant moduli spaces
+are zero dimensional.
+{}From a physical point of view the dual description via monopoles and abelian
+gauge fields should be simpler than the microscopic $SU(2)$ description
+since in the renormalization group sense
+it arises by ``integrating out the irrelevant degrees of freedom.''
+
+
+\nref\vw{C. Vafa and E. Witten, ``A Strong Coupling Test Of $S$-Duality,''
+hep-th/9408074, to appear in Nucl. Phys. B.}
+\nref\om{C. Montonen and D. Olive, Phys. Lett. {\bf B72} (1977) 117;
+P. Goddard, J. Nuyts, and D. Olive, Nucl. Phys. {\bf B125} (1977) 1.}
+\nref\sen{A. Sen, ``Strong-Weak Coupling Duality In Four Dimensional
+String Theory,'' hep-th/9402002.}
+The monopole equations are close cousins of equations studied in section two
+of \vw; the reason for the analogy is that in each case one is studying
+$N=2$ theories of hypermultiplets coupled to vector multiplets.
+The investigation in \vw\ dealt with microscopic Montonen-Olive duality
+\refs{\om,\sen}, while the duality in Donaldson theory \sw\ is a sort of
+phenomenological analog of this.
+
+
+The monopole equations, definition of four-manifold invariants,
+and relation to Donaldson theory are stated
+in section two of this paper. Vanishing theorems are used in section three
+to deduce some basic properties. Invariants
+of K\"ahler
+manifolds are computed in section four.
+A very brief sketch of the origin in physics
+is in section five. A fuller account of the contents of section five
+will appear elsewhere \ref\nsw{N. Seiberg and E. Witten, to appear.}.
+
+
+
+
+\newsec{The Monopole Equations}
+
+Let $X$ be an oriented, closed four-manifold on which we pick a Riemannian
+structure with metric tensor $g$.
+$\Lambda^pT^*X$, or simply $\Lambda^p$,
+will denote the bundle of real-valued $p$-forms,
+and $\Lambda^{2,\pm}$ will be the sub-bundle of $\Lambda^2$ consisting
+of self-dual or anti-self-dual forms.
+
+The monopole equations relevant to $SU(2)$ or $SO(3)$
+Donaldson theory can be described
+as follows. If $w_2(X)=0$,
+then $X$ is a spin manifold and one can pick
+positive and negative spin bundles
+$S^+$ and $S^-$, of rank two. (If there is more than one spin structure,
+the choice of a spin structure will not matter as we will ultimately
+sum over twistings by line bundles.) In that case, introduce a complex line
+bundle $L$; the data in the monopole equation will be a connection $A$
+on $L$ and a section $M$ of $S^+\otimes L$. The curvature two-form
+of $A$ will
+be called $F$ or $F(A)$; its self-dual and anti-self-dual
+projections will be called $F^+$ and $F^-$.
+
+If $X$ is not spin, the $S^{\pm}$ do not exist,
+but their projectivizations ${\bf P}S^{\pm}$ do exist (as bundles with fiber
+isomorphic to ${\bf CP}^1$). A ${\rm Spin}_c$ structure (which exists
+on any oriented four-manifold \ref\hirz{F. Hirzebruch and H. Hopf,
+``Felder von Flachenelementen in 4-dimensionalen Mannigfaltigkeiten,''
+Math. Annalen {\bf 136} (1958) 156.})
+can be described as a choice of
+a rank two complex vector bundle, which we write as $S^+\otimes L$,
+whose projectivization is isomorphic to ${\bf P}S^+$. In this situation, $L$
+does not exist as a line bundle, but $L^2$ does\foot{One might
+be tempted to call this bundle $L$ and write the ${\rm Spin}_c$
+bundle as $S^+\otimes L^{1/2}$; that amounts to assigning magnetic
+charge $1/2$ to the monopole and seems unnatural physically.};
+the motivation for
+writing the ${\rm Spin}_c$ bundle as $S^+\otimes L$ is that the tensor
+powers of this bundle obey isomorphisms suggested by the notation.
+For instance, $(S^+\otimes L)^{\otimes 2}\cong L^2\otimes(\Lambda^0\oplus
+\Lambda^{2,+})$.
+The data of the monopole equation
+are now a section $M$ of $S^+\otimes L$ and a connection on $S^+\otimes L$
+that projects to the Riemannian connection on ${\bf P}S^+$. The symbol
+$F(A)$ will now denote $1/2$ the trace of the curvature form of $S^+\otimes L$.
+
+Since $L^2$ is an ordinary line bundle, one has an integral
+cohomology class
+$x=-c_1(L^2)\in H^2(X,{\bf Z})$. (The minus sign makes some
+later formulas come out in a standard form.) Note that $x$ reduces modulo two
+to $w_2(X)$; in particular, if $w_2(X)=0$, then $L$ exists as a line
+bundle and $x=-2c_1(L)$.
+
+To write the monopole equations, recall that $S^+$ is symplectic or
+pseudo-real, so that
+if $M$ is a section of $S^+\otimes L$, then the complex conjugate $\bar M$
+is a section of $S^+\otimes L^{-1}$. The product $M\otimes \bar M$
+would naturally lie in $(S^+\otimes L)\otimes (S^+\otimes L^{-1})\cong
+\Lambda^0\oplus\Lambda^{2,+}$.
+$F^{+}$ also takes values in $\Lambda^{2,+}$ making it possible to
+write the following equations.
+Introduce Clifford matrices $\Gamma_i$
+(with anticommutators $\{\Gamma_i,\Gamma_j\}=2g_{ij}$), and
+set $\Gamma_{ij}={1\over 2}[\Gamma_i,\Gamma_j]$. Then
+the equations are\foot{
+To physicists the connection form $A$ on a unitary line bundle is
+real; the covariant derivative is $d_A=d+iA$ and the curvature is
+$F=dA$ or in components $F_{ij}=\partial_iA_j-\partial_jA_i$.}
+\eqn\noneq{\eqalign{F^+_{ij}&=-{i\over 2}\bar M\Gamma_{ij}M \cr
+ \sum_i\Gamma^iD_iM & = 0.\cr}}
+In the second equation, $\sum_i\Gamma^iD_i$ is the Dirac operator
+$D$ that maps sections of $S^+\otimes L$ to sections of $S^-\otimes L$.
+We will sometimes abbreviate the first as $F^+=(M\bar M)^+$.
+Alternatively,
+if positive spinor indices are written $A,B,C$, and
+negative spinor indices as $A',B',C'$,
+\foot{Spinor indices are raised and lowered using the invariant
+tensor in $\Lambda^2 S^+$. In components, if $M^A=(M^1,M^2)$,
+then $M_A= (-M_2,M_1)$. One uses the same operation in interpreting
+$\bar M$ as a section of $S^+\otimes L$, so $\bar M^A=(\bar M^2,-\bar M^1)$.
+Also $F_{AB}={1\over 4}F_{ij}\Gamma^{ij}_{AB}$.}
+the equations can be written
+\eqn\indeq{\eqalign{F_{AB}& = {i\over 2}\left(M_A\bar M_B+M_B\bar M_A\right)\cr
+ D_{AA'}M^A & = 0.\cr}}
+
+As a first step in understanding these equations, let us work out
+the virtual dimension of the moduli space ${\cal M}$
+of solutions of the
+equations up to gauge transformation.
+The linearization of the monopole equations fits into
+an elliptic complex
+\eqn\pindeq{0\to \Lambda^0\underarrow{s}\Lambda^1
+\oplus (S^+\otimes L)\underarrow{t}\Lambda^{2,+}
+\oplus (S^-\otimes L) \to 0.}
+Here $t$ is the linearization of the monopole equations, and $s$
+is the map from zero forms to deformations in $A,M$ given by the infinitesimal
+action of the gauge group. Since we wish to work with real operators
+and determine the real dimension
+of ${\cal M}$, we temporarily think of $S^\pm\otimes L$ as
+real vector bundles (of rank four).
+Then an elliptic operator
+\eqn\pxxx{T:\Lambda^1\oplus(S^+\otimes L)\to \Lambda^0\oplus \Lambda^{2,+}
+\oplus (S^-\otimes L)}
+is defined by $T=s^*\oplus t$.
+The virtual dimension of the moduli space is given by the index of $T$.
+By dropping terms in $T$ of order zero,
+$T$ can be deformed to the direct sum of the operator $d+d^*$
+\foot{What is meant here is of course a projection of the $d+d^*$ operator
+to self-dual forms.}
+from $\Lambda^1$ to $\Lambda^0\oplus \Lambda^{2,+}$ and the Dirac
+operator from $S^+\otimes L$ to $S^-\otimes L$.
+The index of $T$ is
+the index of $d+d^*$ plus twice what is usually called the index of the Dirac
+operator; the factor of two comes from looking at $S^{\pm}\otimes L$
+as real bundles of twice the dimension.
+Let $\chi$ and $\sigma$ be the Euler
+characteristic and signature of $X$. Then the index of $d+d^*$ is
+$-(\chi+\sigma)/2$, while twice the Dirac index is $-\sigma/4+c_1(L)^2$.
+The virtual dimension of the moduli space is the sum of these or
+\eqn\hurf{W= -{2\chi+3\sigma\over 4} +c_1(L)^2.}
+
+When this number is negative, there are generically no solutions of
+the monopole equations. When $W=0$, that is, when $x=-c_1(L^2)=-2c_1(L)$ obeys
+\eqn\burf{x^2=2\chi+3\sigma,}
+then the virtual dimension is zero and the moduli space generically
+consists of a finite set of points $P_{i,x}$, $i=1\dots t_x$.
+With each such point, one can associate
+a sign $\epsilon_{i,x}=\pm 1$ -- the sign of the determinant of $T$ as we
+discuss momentarily.
+Once this is done, define for each $x$ obeying \burf\ an integer $n_x$ by
+\eqn\gurofo{n_x=\sum_i\epsilon_{i,x}.}
+We will see later that
+$n_x=0$ -- indeed, the moduli space is empty -- for all but finitely many $x$.
+Under certain conditions that we will discuss in a moment, the $n_x$
+are topological invariants.
+
+Note that $W=0$ if and only if the index of the Dirac operator
+is
+\eqn\inxxon{\Delta={\chi+\sigma\over 4}.}
+In particular, $\Delta$ must be an integer to have non-trivial $n_x$.
+Similarly, $\Delta$ must be integral for the Donaldson invariants
+to be non-trivial (otherwise $SU(2)$ instanton moduli space is odd
+dimensional).
+
+
+For the sign of the determinant of $T$ to make sense one must trivialize
+the determinant line of $T$. This can be done by deforming $T$ as above
+to the direct sum of $d+d^*$ and the Dirac operator. If the Dirac operator,
+which naturally has a non-trivial {\it complex} determinant line, is regarded
+as a real operator, then its determinant line is naturally trivial -- as a
+complex line has a natural orientation. The $d+d^*$ operator is
+independent of $A$ and $M$ (as the gauge group is abelian), and its
+deterinant line is trivialized once and for all by picking an orientation
+of $H^1(X,{\bf R})\oplus H^{2,+}(X,{\bf R})$. Note that this is the
+same data needed by Donaldson
+\ref\donor{S. Donaldson, ``The Orientation Of Yang-Mills Moduli
+Spaces And Four-Manifold Topology,'' J. Diff. Geom. {\bf 26} (1987) 397.}
+to orient instanton moduli spaces for $SU(2)$;
+this is an aspect of the relation between the two theories.
+
+If one replaces $L$ by $L^{-1}$, $A$ by $-A$, and $M$ by $\bar M$, the
+monopole equations are invariant, but the trivialization of the
+determinant line is multiplied by $(-1)^\delta$ with $\delta$ the Dirac
+index. Hence the invariants for $L$ and $L^{-1}$ are related by
+\eqn\pixxx{n_{-x}=(-1)^\Delta n_x.}
+
+For $W<0$, the moduli space is generically empty. For $W>0$ one can
+try, as in Donaldson theory, to define topological invariants that involve
+integration over the moduli space. Donaldson theory does not detect those
+invariants at least in known situations.
+We will see in section three that even when $W>0$, the
+moduli space is empty for almost all $x$.
+
+
+\bigskip
+\noindent{\it Topological Invariance}
+
+In general, the number of solutions
+of a system of equations
+weighted by the sign of the determinant of the operator analogous to $T$
+ is always a topological invariant if a suitable compactness
+holds.
+If as in the case at hand one has a gauge invariant system of equations, and
+one wishes to count gauge orbits of solutions up to gauge transformations,
+then one requires (i) compactness; and (ii) free action
+of the gauge group on the space of solutions.
+
+
+Compactness fails if a field or its derivatives can go to
+infinity.
+The Weitzenbock formula used in section three to discuss vanishing
+theorems indicates that these phenomena
+do not occur for the monopole equations.
+To explain the contrast with Donaldson theory, note that
+for $SU(2)$ instantons
+compactness fails precisely
+\ref\uhl{K. Uhlenbeck, ``Removable Singularities In Yang-Mills Fields,''
+Commun. Math. Phys. {\bf 83} (1982) 11.}
+because an instanton can shrink to zero size. This is
+possible because (i) the equations are conformally invariant, (ii) they
+have non-trivial solutions on a flat ${\bf R}^4$, and (iii) embedding
+such a solution, scaled to very small size,
+on any four-manifold gives a highly localized approximate
+solution of the instanton equations (which can sometimes
+\ref\taubes{C. H. Taubes, ``Self-Dual Yang-Mills Connections Over
+Non-Self-Dual 4-Manifolds,'' J. Diff. Geom. {\bf 19} (1982) 517.}
+be perturbed to
+an exact solution). The monopole equations by contrast
+are scale invariant but
+(as follows immediately from the Weitzenbock formula) they have
+no non-constant $L^2$ solutions on flat ${\bf R}^4$ (or after dimensional
+reduction on flat ${\bf R}^n$ with $1\leq n \leq 3$).
+So there is no analog for the monopole equations of the phenomenon
+where an instanton shrinks to zero size.
+
+On the other hand, an obstruction does arise, just as in Donaldson
+theory (in what follows we imitate some arguments in
+\ref\dono{S. Donaldson, ``Irrationality And The $h$-Cobordism
+Conjecture,'' J. Diff. Geom. {\bf 26} (1987) 141.})
+from the question of whether the gauge group acts freely on the
+space of solutions of the monopole equations. The only way for the gauge
+group to fail to act freely is that there might be a solution with $M=0$,
+in which case a constant gauge transformation acts trivially.
+A solution with $M=0$ necessarily has $F^+=0$, that is, it is an abelian
+instanton.
+
+Since $F/2\pi$ represents the first Chern class of the line bundle $L$,
+it is integral; in particular if $F^+=0$ then $F/2\pi$ lies in the intersection
+of the integral lattice in $H^2(X,{\bf R})$ with the anti-self-dual subspace
+$H^{2,-}(X,{\bf R})$.
+As long as $b_2^+\geq 1$, so that the self-dual part of $H^2(X,{\bf R})$ is
+non-empty, the intersection of the anti-self-dual part and the integral
+lattice generically consists only of the zero vector.
+ In this case,
+for a generic metric on $X$, there are no abelian instantons (except for
+$x=0$, which we momentarily exclude) and $n_x $ is well-defined.
+
+
+To show that the $n_x$ are topological invariants, one must further show
+that any two generic metrics on $X$ can be joined by a path along which
+there is never an abelian instanton. As in Donaldson theory, this can
+fail if $b_2^+=1$. In that case, the self-dual part
+of $H^2(X,{\bf R})$ is one dimensional, and in a generic
+one parameter family of metrics on $X$, one may meet a metric for
+which there is an abelian instanton. When this occurs, the $n_x$ can jump.
+Let us analyze how this happens, assuming for simplicity that $b_1=0$.
+Given $b_1=0$ and
+$b_2{}^+=1$, one has $W=0$ precisely if the index of the Dirac
+equation is 1. Therefore, there is generically a single solution $M_0$
+of the Dirac equation $DM=0$.
+
+The equation $F^+(A)=0$ cannot be obeyed for a generic metric on $X$,
+but we want to look at the behavior near a special metric for which it does
+have a solution.
+ Consider a one parameter family of metrics parametrized
+by a real parameter $\epsilon$, such that at $\epsilon=0$ the
+self-dual subspace in $H^2(X,{\bf R})$ crosses a ``wall''
+and a solution $A_0$ of
+$F^+(A)=0$ appears. Hence for $\epsilon=0$, we can solve the monopole
+equations with $A=A_0, \,M=0$. Let us see what happens to this solution
+when $\epsilon $ is very small but non-zero. We set $M=mM_0$, with $m$
+a complex number, to obey $DM=0$, and we write $A=A_0+\epsilon \delta A$.
+The equation $F^+(A)-(M\bar M)^+=0$ becomes
+\eqn\nurk{F^+(A_0)+(d\delta A)^+-|m|^2 (M_0\bar M_0)^+=0.}
+As the cokernel of $A\to F^+(A)$
+is one dimensional, $\delta A$ can be chosen
+to project the left hand side of equation \nurk\ into a one dimensional
+subspace. (As $b_1=0$, this can be done in a unique way up to a gauge
+transformation.)
+The remaining equation looks near $\epsilon=0$ like
+\eqn\modlik{c \epsilon - \,m\bar m=0}
+with $c$ a constant.
+The $\epsilon$ term on the left comes from the fact that $F^+(A_0)$ is
+proportional to $\epsilon$.
+
+Now we can see what happens for $\epsilon\not= 0$ to the solution that
+at $\epsilon=0$ has $A=A_0$, $M=M_0$.
+Depending on the sign of $c$,
+there is a solution for $m$, uniquely
+determined up to gauge transformation, for $\epsilon>0$ and no solution
+for $\epsilon<0$, or vice-versa. Therefore $n_x$ jumps by $\pm 1$, depending
+on the sign of $c$,
+in passing through $\epsilon=0$.
+To compare this precisely to the similar behavior of Donaldson
+theory, one would also need to understand the
+role of the $u$ plane, discussed in section five.
+
+The trivial abelian instanton with $x=0$ is an exception to the
+above discussion,
+since it cannot be removed by perturbing the metric. To define $n_0$,
+perturb the equation $F_{AB}={i\over 2}(M_A\bar M_B+M_B\bar M_A)$
+to
+\eqn\hinnoc{F_{AB}={i\over 2}(M_A\bar M_B+M_B\bar M_A)-p_{AB},}
+with $p$ a self-dual
+harmonic two-form; with this perturbation, the gauge group acts
+freely on the solution space.
+Then define $n_0$ as the number of gauge orbits of solutions of the
+perturbed equations
+weighted by sign in the usual way. This is invariant under continuous
+deformations of $p$ for $p\not=0$;
+as long as $b_2^+>1$, so that
+the space of possible $p$'s is connected, the integer $n_0$ defined
+this way is a topological invariant.
+
+The perturbation just
+pointed out will be used later in the case that $p$ is the real part
+of a holomorphic two-form to compute the invariants of K\"ahler manifolds
+with $b_2^+>1$. It probably has other applications; for instance, the
+case that $p$ is a symplectic form is of interest.
+
+\bigskip
+\noindent{\it Relation To Donaldson Theory}
+
+With an appropriate restriction on $b_2^+$, the $n_x$ have
+(by an argument sketched in section five) a relation to the Donaldson
+invariants that will now be stated.
+
+Let us recall that in $SU(2)$ Donaldson theory, one wishes to compute
+the integrals or expectation values of certain cohomology classes
+or quantum field operators: for every Riemann surface
+(or more generally every
+two-dimensional homology cycle) $\Sigma$ in $X$, one has an operator
+$I(\Sigma)$ of dimension (or $R$ charge or
+ghost number) two; there is one additional
+operator ${\cal O}$, of dimension four.
+For every value of the instanton number, one computes the expectation value
+of a suitable product of these operators by integration over instanton
+moduli space using a recipe due to Donaldson, or by evaluating a suitable
+quantum field theory correlation function as in \witten.
+It is natural to organize this data in the form of a generating function
+\eqn\jurn{\left\langle
+\exp\left(\sum_a\alpha_aI(\Sigma_a)+\lambda {\cal O}\right)
+\right\rangle,}
+summed over instanton numbers; here
+the $\Sigma_a$ range over a basis of $H_2(X,{\bf R})$ and $\lambda,
+\,\alpha_a$ are complex numbers.
+
+Let $v= \sum_a\alpha_a[\Sigma_a]$,
+with $[\Sigma_a]$ the cohomology class that is Poincar\'e dual to $\Sigma_a$.
+So for instance $v^2=\sum_{a,b}\alpha_a\alpha_b\,\,\Sigma_a\cdot\Sigma_b$
+(here $\Sigma_a\cdot \Sigma_b$ is the intersection number of $\Sigma_a$ and
+$\Sigma_b$), and for any $x\in H^2(X,{\bf Z})$, $v\cdot x=\sum_a\alpha_a
+(\Sigma_a,x)$. Let as before $\Delta=(\chi+\sigma)/4$.
+
+A four-manifold is said to be of simple type if the generating function
+in \jurn\ is annihilated by $\partial^2/\partial\lambda^2-4$; all known
+simply-connected four-manifolds with $b_2^+>1$ have this property.
+The relation of the simple type condition to physics is discussed in
+section five.
+I claim that for manifolds of simple type
+\eqn\jimmo{\eqalign{\left
+\langle\exp\left(\sum_a\alpha_aI(\Sigma_a)+\lambda {\cal O}\right)
+\right\rangle = 2^{1+{1\over 4}(7\chi+11\sigma)}&\left(\exp\left(
+{v^2\over 2}+2\lambda\right)
+\sum_x
+n_x e^{v\cdot x}\right.\cr&\left.
+ +i^{\Delta} \exp\left(-{v^2\over 2}-2\lambda\right)\sum_xn_x
+e^{-iv\cdot x}\right).\cr}}
+That the expression is real follows from \pixxx.
+
+As sketched in section five, this formula is a sort of corollary of the
+analysis of $N=2$ supersymmetric Yang-Mills theory in \sw. Here I will
+just make a few remarks:
+
+(1) The structure in \jimmo\ agrees with the general form
+proved by Kronheimer and Mrowka \km.
+The classes $x\in H^2(X,{\bf Z})$ such that $n_x\not= 0$ are the basic
+classes in their terminology. From the properties by which $x$ and $n_x$
+were defined, we have that $x$ is congruent to $w_2(X)$ modulo 2 and
+that $x^2=2\chi+3\sigma$. The first assertion is a result of Kronheimer
+and Mrowka and the second was conjectured by them.
+
+
+
+(2) The prefactor $2^{1+{1\over 4}(7\chi+11\sigma)}$ has the following
+origin, as in \newwitten. One factor of two comes because, even though the
+center of $SU(2)$ acts trivially on the $SU(2)$ instanton moduli space,
+the Donaldson invariants are usually defined without dividing by two.
+The remaining factor of $2^{{1\over 4}(7\chi+11\sigma)}$ is a $c$-number
+renormalization factor that arises in comparing the microscopic $SU(2)$
+theory to the dual description with monopoles.
+(In \nsw\ a more general function of the form
+$e^{a(u)\chi+b(u)\sigma}$ that arises on the complex $u$ plane will be
+calculated.) Some coefficients in the formula such as the $7/4$ and $11/4$
+were fixed in \newwitten\ to agree with
+calculations of special cases of Donaldson invariants.
+
+
+(3) Most fundamentally, in the above formula, the first term, that is
+\eqn\kdn{\exp\left({v^2\over 2}+2\lambda\right)
+\sum_xn_x e^{v\cdot x},}
+is the contribution from one vacuum at $u=\Lambda^2$, and the second
+term,
+\eqn\hkn{i^\Delta \exp\left(-{v^2\over 2}-2\lambda\right)\sum_xn_x
+e^{-iv\cdot x},}
+is the contribution of a second vacuum at $u=-\Lambda^2$.
+These terms are analogous to the two terms in equation (2.66) of \newwitten.
+The factor of $i^\Delta$ arises, as there, because
+of a global anomaly in the discrete symmetry that exchanges the two
+vacua. This factor
+can be written in the form $e^{a\chi+b\sigma}$ and so means
+that the two vacua have different values
+of the renormalization mentioned in the last paragraph. The
+replacement of $e^{v\cdot x}$ in the first vacuum by $e^{-iv\cdot x}$
+in the second is likewise determined by the symmetries, as in \newwitten,
+and can be seen microscopically.
+For a general simple compact gauge group, the analogous sum will have
+$h$ terms ($h$ the dual Coxeter number) associated with $h$ vacua.
+
+(4) This formula generalizes as follows for the case that the
+gauge group is $SO(3)$ rather than $SU(2)$. Consider an
+$SO(3)$ bundle $E$ with, say, $ w_2(E)=z$.
+Define a generating functional of correlation functions
+summed over bundles with
+ all values of the first Pontryagin class
+but fixed $w_2$. Pick an integral lift of $w_2(X)$, and, using
+the fact that the $x$'s are congruent to $w_2(X)$ mod two, let $x'$ be
+such that $2x'=x+ w_2(X)$. Then $w_2(E)\not= 0$ modifies
+the derivation of \jimmo\ only by certain minus signs that
+appear in the duality transformation that relates the microscopic
+and macroscopic descriptions; the result is
+\eqn\himmo{
+\eqalign{
+\left\langle\exp\left(\sum_a\alpha_aI(\Sigma_a)+\lambda {\cal O}\right)
+\right\rangle_z
+=& 2^{1+{1\over 4}(7\chi+11\sigma)}\left(\exp({v^2\over 2}+2\lambda)
+\sum_x(-1)^{x'\cdot z}
+n_x e^{v\cdot x} \right.\cr &\left.
++i^{\Delta-z^2} \exp(-{v^2\over 2}-2\lambda)\sum_x
+(-1)^{x'\cdot z
+}n_xe^{-iv\cdot x}\right).\cr}}
+The replacement of $i^\Delta$ by $i^{\Delta-z^2}$ arises, as in equation
+(2.79) of \newwitten\ (where $w_2(E)$ is written as $x$),
+because the global anomaly has an extra term that depends on $z$.
+(Note that as $z$ is defined modulo two, $z^2$ is well-defined modulo four.)
+The factor of $(-1)^{x'\cdot z}$ was obtained in \km\ for manifolds
+of simple type and in \newwitten\ for K\"ahler manifolds.
+If the integral lift of $w_2(X)$ used in defining $x'$
+is shifted by $w_2(X)\to w_2(X)+2y$, then \himmo\
+is multiplied by $(-1)^{y\cdot z}$. The reason for this factor
+is that \himmo\ is reproducing the conventional Donaldson invariants,
+whose sign depends on the orientation of the instanton moduli spaces.
+A natural orientation \donor\ depends on an integral lift of $w_2(X)$
+and transforms as \himmo\ does if this lift is changed.
+
+(5) For K\"ahler manifolds with $b_2^+>1$, the quantities entering in \jimmo\
+will be completely computed in section four.
+We will find that, letting $\eta$ be a holomorphic two-form, the sum in \jimmo\
+can be interpreted as a sum over factorizations $\eta=\alpha\beta$
+with $\alpha$ and $\beta$ holomorphic sections of $K^{1/2}\otimes L^{\pm 1}$.
+Each such factorization contributes $\pm 1$ to $n_x$ with
+$x=-2c_1(L)$ provided $x^2=c_1(K)^2$; the contribution is $+1$ or $-1$
+according to a formula computed at the end of section four.
+
+\bigskip
+\noindent{\it Imitating Arguments From Donaldson Theory}
+
+Apart from relating Donaldson theory to the monopole equations,
+one can simply try to adapt familiar arguments about Donaldson theory
+to the monopole equations. We have already seen some examples.
+
+As another example, consider Donaldson's theorem \doninv\
+asserting that the Donaldson invariants vanish for a connected sum $X\# Y$
+of four-manifolds $X$ and $Y$ which each have $b_2{}^+>0$. The theorem
+is proved by considering a metric on $X\# Y$
+in which $X$ and $Y$ are joined by
+a long neck of the form ${\bf S}^3\times I$, with $I$ an interval in ${\bf R}$.
+Take the metric on the neck to be the product of the standard metric
+on ${\bf S}^3$ and a metric that assigns length $t$ to $I$, and consider
+the monopole equations on this space. For $t\to \infty$, any solution
+of the monopole equations will vanish in the neck because of the positive
+scalar curvature of ${\bf S}^3$ (this follows from the Weitzenbock
+formula of the next section).
+This lets one define a $U(1)$ action on the moduli space ${\cal M}$
+(analogous to the $SO(3)$ action used by Donaldson)
+by gauge transforming the solutions on $Y$ by a constant gauge transformation,
+leaving fixed the data on $X$. A fixed point of this $U(1)$ action
+would be a solution for which $M$ vanishes on $X$ or on $Y$. But
+as $X$ and $Y$ both have $b_2{}^+>0$, there is no such solution if
+generic metrics are used on the two sides. A zero dimensional moduli
+space with a free $U(1)$ action is empty, so the basic invariants would
+be zero for such connected sums. (A free $U(1)$ action also leads
+to vanishing of the higher invariants.)
+Since we will see in section four
+that the invariants are non-zero for K\"ahler manifolds
+(analogous to another basic result of Donaldson), one gets a proof
+directly from the monopole equations and independent of the equivalence to
+Donaldson theory that algebraic surfaces do not have connected
+sum decompositions with $b_2^+>0$ on both sides.
+
+If one considers instead a
+situation with $b_2^+$ positive for $X$ but zero for $Y$, there will
+be fixed points consisting of solutions with $M=0$ on $Y$, and one will get
+a formula expressing invariants of $X\# Y$ in terms
+of invariants of $X$ and elementary data concerning $Y$.
+
+\newsec{Vanishing Theorems}
+
+Some of the main properties of the monopole equations
+can be
+understood by means of vanishing theorems. The general strategy in
+deriving such vanishing theorems is quite standard, but as in section two
+of \vw, some unusual cancellations (required by the Lorentz invariance
+of the underlying untwisted theory) lead to unusually strong results.
+
+If we set $s=F^+-M\bar M$, $k=DM$,
+then a small calculation gives
+\eqn\highor{\eqalign{\int_Xd^4x\sqrt g\left({1\over 2}|s|^2+|k|^2\right)
+=\int_Xd^4x\sqrt g&\left({1\over 2}|F^+|^2+g^{ij}D_iM^AD_j\bar M_A \right.
+\cr & \left.+{1\over 2}|M|^4
++{1\over 4}R|M|^2\right) .\cr}} Here $g$ is the metric of $X$, $R$ the scalar
+curvature, and $d^4x\sqrt g$ the Riemannian measure.
+A salient feature here is that a term $F_{AB}M^A\bar M{}^B$, which appears
+in either $|s|^2 $ or $|k|^2$, cancels in the sum.
+This sharpens the implications of the formula, as we will see.
+One can also consider the effect here of the perturbation in \hinnoc;
+the sole effect of this is to replace
+${1\over 2}|M|^4$ in \highor\ by
+\eqn\bihor{\int_Xd^4x\sqrt g\left(
+F^+\wedge p+\sum_{A,B}\left|{1\over 2}(M_A\bar M_B+M_B\bar M_A)-p_{AB}\right|^2
+\right).}
+The second term is non-negative, and the first is simply the intersection
+pairing
+\eqn\juhor{2\pi c_1(L)\cdot [p].}
+
+An obvious inference from \highor\ is that if $X$ admits a metric
+whose scalar curvature is positive,
+then for such a metric any solution
+of the monopole equations must have $M=0$ and $F^+=0$. But
+if $b_2{}^+>0$, then after a generic small perturbation of the metric
+(which will preserve the fact that the scalar curvature is positive),
+there are no abelian solutions of $F^+=0$ except flat connections.
+Therefore,
+for such manifolds and metrics, a solution of the monopole equations
+is a flat connection with $M=0$. These too can be eliminated
+using the perturbation in \hinnoc.\foot{
+Flat connections can only arise if $c_1(L)$ is torsion; in that case,
+$c_1(L)\cdot [p]=0$. The vanishing argument
+therefore goes through, the modification in \highor\ being that which
+is indicated in \bihor.}
+Hence a four-manifold
+for which $b_2^+>0$ and $n_x\not= 0$ for some $x$
+does not admit a metric of
+positive scalar curvature.
+
+We can extend this to determine the possible four-manifolds $X$ with $b_2^+>0$,
+some $n_x\not= 0$, and a metric of {\it non-negative}
+scalar curvature.\foot{If $b_2^+=1$, the $n_x$ are not all topological
+invariants, and we interpret the hypothesis to mean that with at least
+one sign of the perturbation in \hinnoc, the $n_x$ are not all zero.}
+If $X$ obeys those conditions, then for any metric of $R\geq 0$,
+any basic class $x$ is in $H^{2,-}$ modulo torsion
+(so that $L$ admits a connection
+with $F^+=0$, enabling \highor\ to vanish);
+in particular if $x$ is not torsion then $x^2<0$.
+Now consider the effect of the perturbation \hinnoc. As $x\in H^{2,-}$,
+\juhor\ vanishes; hence if $R\geq 0$, $R$ must
+be zero, $M$
+must be covariantly constant and $(M\bar M)^+=p$ (from \bihor).
+For $ M$ covariantly constant,
+$(M\bar M)^+=p$ implies
+that $p$ is covariantly constant also; but for all $p\in H^{2,+}$
+to be covariantly constant implies that $X$ is K\"ahler with $b_2^+=1$
+or is hyper-K\"ahler. Hyper-K\"ahler metrics certainly have $R=0$,
+and there are examples of metrics with $R=0$
+on K\"ahler manifolds with $b_2^+=1$ \ref\lebrun{C. LeBrun, ``Scalar-Flat
+K\"ahler Metrics On Blown-Up Ruled Surfaces,'' J. Reine Angew
+Math. {\bf 420} (1991) 161.}.
+
+As an example,
+for a K\"ahler manifold with $b_2^+\geq 3$, the canonical divisor
+$K$ always arises as a basic class, as we will see in section four, so
+except in the hyper-K\"ahler case,
+such manifolds do not admit a metric of non-negative
+scalar curvature.
+
+Even if the scalar curvature is not positive, we can get an explicit
+bound from \highor\ showing that there are only finitely many basic classes.
+Since
+\eqn\gegor{\int_Xd^4x\sqrt g\left({1\over 2}|M|^4+{1\over 4}R|M|^2\right)
+\geq -{1\over 32}\int_Xd^4x\sqrt g R^2,}
+it follows from \highor, even if we throw away the term $|D_iM|^2$,
+that
+\eqn\egor{\int_Xd^4x\sqrt g |F^+|^2\leq {1\over 16}\int_Xd^4x\sqrt g R^2.}
+On the other hand, basic classes correspond to line bundles
+$L$ with $c_1(L)^2=(2\chi+3\sigma)/4$, or
+\eqn\negor{{1\over (2\pi)^2}\int d^4x\sqrt g\left(|F^+|^2-|F^-|^2\right)
+ ={2\chi+3\sigma\over 4}.}
+Therefore, for a basic class both $I^+=\int d^4x\sqrt g |F^+|^2$
+and $I^-=\int d^4x\sqrt g |F^-|^2$ are bounded. For a given metric,
+there are only finitely
+many isomorphism classes of line bundles
+admitting connections with given bounds on both $I^+$ and $I^-$, so
+the set of basic classes is finite. This is a result
+proved by Kronheimer and Mrowka with their definition of the basic classes.
+
+The basic classes correspond, as indicated in section three,
+to line bundles on which
+the moduli space of solutions of the monopole equations is of zero virtual
+dimension.
+We can analyze in a similar way components of the moduli space of positive
+dimension. Line bundles $L$ such that $c_1(L)^2<(2\chi+3\sigma)/4$ are not
+of much interest in that connection, since for such line bundles the
+moduli space has negative virtual dimension and is generically empty.
+But if $c_1(L)^2>(2\chi+3\sigma)/4$, then \negor\ is simply replaced by
+the stronger bound
+\eqn\unegor{{1\over (2\pi)^2}\int d^4x\sqrt g\left(|F^+|^2-|F^-|^2\right)
+ >{2\chi+3\sigma\over 4}.}
+The set of isomorphism classes of line bundles admitting a connection
+obeying this inequality as well as \egor\ is once again finite.
+So we conclude that for any given metric on $X$, the set of isomorphism
+classes of line bundles for which
+the moduli space is non-empty and of non-negative virtual dimension
+is finite; for a generic metric on $X$, there are only finitely many
+non-empty components of the moduli space.
+
+For further consequences of \highor, we turn to a basic case in the study of
+four-manifolds: the case that $X$ is K\"ahler.
+
+\newsec{Computation On K\"ahler Manifolds}
+
+If $X$ is K\"ahler and spin, then $S^+\otimes L$ has a decomposition
+$S^+\otimes L\cong (K^{1/2}\otimes L)\oplus (K^{-1/2}\otimes L)$,
+where $K$ is the canonical bundle and $K^{1/2}$ is a square root.
+If $X$ is K\"ahler but not spin, then $S^+\otimes L$, defined as before,
+has a similar decomposition where now $K^{1/2}$ and $L$ are not defined
+separately and $K^{1/2}\otimes L$ is characterized
+as a square root of the line bundle $K\otimes L^2$.
+
+We denote the components of $M$ in $K^{1/2}\otimes L$ and
+in $K^{-1/2}\otimes L$ as $\alpha$ and $-i\bar \beta$, respectively.
+The equation $F^+(A)=M\bar M$ can now be decomposed
+\eqn\juffy{\eqalign{F^{2,0} & = \alpha\beta \cr
+ F_\omega^{1,1} & =-{\omega\over 2}
+ \left(|\alpha|^2-|\beta|^2\right)\cr
+ F^{0,2} & =\bar\alpha\bar\beta.\cr}}
+Here $\omega$ is the K\"ahler form and $F_\omega^{1,1}$ is the $(1,1)$
+part of $F^+$.
+\highor\ can be rewritten
+\eqn\nohighor{\eqalign{\int_Xd^4x\sqrt g\left({1\over 2}|s|^2+|k|^2\right)
+=\int_Xd^4x\sqrt g & \left({1\over 2}|F^+|^2
++g^{ij}D_i\bar\alpha D_j\alpha+g^{ij}D_i\bar\beta
+D_j\beta\right.\cr & \left. +{1\over 2}(|\alpha|^2+|\beta|^2)^2
++{1\over 4}R(|\alpha|^2+|\beta|^2)\right) .\cr}}
+
+The right hand side of \nohighor\ is not manifestly non-negative (unless
+$R\geq 0$), but the fact that it is equal to the left hand side shows that
+it is non-negative and zero precisely for solutions of the monopole
+equations. Consider the operation
+\eqn\pohighor{\eqalign{A & \to A\cr
+ \alpha & \to \alpha \cr
+ \beta & \to -\beta.\cr}}
+This is not a symmetry of the monopole equations. But it is a symmetry
+of the right hand side of \nohighor. Therefore, given a zero of the right
+hand side of \nohighor\ -- that is, a solution of the monopole equations --
+the operation \pohighor\ gives another zero of the right hand side of
+\nohighor\ -- that is, another solution of the monopole equations.
+So, though not a symmetry of the monopole equations, the transformation
+\pohighor\ maps solutions of those equations to other solutions.
+
+Given that any solution of \juffy\ is mapped to another solution by
+\pohighor, it follows that such a solution has
+\eqn\tohighor{0=F^{2,0}=F^{0,2}=\alpha\beta=\bar\alpha\bar\beta.}
+Vanishing of $F^{0,2}$ means that the connection $A$ defines a holomorphic
+structure on $L$.
+The basic classes (which are first Chern classes of $L$'s that are such that
+\juffy\ has a solution) are therefore of type $(1,1)$ for any K\"ahler
+structure
+on $X$, a severe constraint.
+
+
+Vanishing of $\alpha\beta$ means that $\alpha=0$
+or $\beta=0$. If $\alpha=0$, then the Dirac equation for $M$ reduces
+to
+\eqn\jipp{\bar\partial_A \beta=0,}
+where $\bar\partial_A$ is the $\bar\partial $ operator on $L$. Similarly,
+if $\beta=0$, then the Dirac equation gives
+\eqn\ipp{\bar\partial_A\alpha= 0.}
+
+Knowing that either $\alpha$ or $\beta$ is zero, we can deduce which it is.
+Integrating the $(1,1)$ part of \juffy\ gives
+\eqn\jippo{{1\over 2\pi}\int_X\omega\wedge F=-{1\over 4\pi}\int_X\omega\wedge
+\omega\left(|\alpha|^2-|\beta|^2\right).}
+The left hand side of \jippo\ is a topological invariant which can be
+interpreted as
+\eqn\ippo{J= [\omega]\cdot c_1(L).}
+The condition that there are no non-trivial abelian instantons is
+that $J$ is non-zero; we only wish to consider metrics for which this
+is so. If $J<0$, we must have $\alpha\not= 0$, $\beta=0$, and if
+$J>0$, we must have $\alpha=0$, $\beta\not= 0$.
+
+The equation that we have not considered so far is the $(1,1)$ part of \juffy.
+This equation can be interpreted
+as follows. Suppose for example that we are in the situation with $\beta=0$.
+The space of connections $A$
+and sections $\alpha$ of $K^{1/2}\otimes L$
+can be interpreted as a symplectic manifold,
+the symplectic structure being defined by
+\eqn\defby{\eqalign{\langle\delta_1A,\delta_2A\rangle & =\int_X\omega
+ \wedge \delta_1A\wedge\delta_2 A\cr
+ \langle \delta_1\alpha,\delta_2\alpha
+\rangle & =-i\int_X\omega\wedge\omega
+\left(\delta_1\overline \alpha\delta_2\alpha-\delta_2\bar \alpha
+\delta_1\alpha\right).\cr}}
+On this symplectic manifold acts the group of $U(1)$ gauge transformations.
+The moment map $\mu$ for this action is the quantity that appears in the
+$(1,1)$ equation that we have not yet exploited, that is
+\eqn\hefby{ \mu\omega= F_\omega^{1,1}+\omega|\alpha|^2.}
+By analogy with many similar problems, setting
+to zero the moment map and dividing by the group of $U(1)$ gauge
+transformations
+should be
+equivalent to dividing by the complexification of the group of gauge
+transformations.\foot{In such comparisons of symplectic and complex quotients,
+one usually needs a stability condition on the complex side.
+In the present case, this is the condition discussed in
+connection with \ippo.} In the present case, the complexification of the
+group of gauge transformations acts by $\alpha\to t\alpha$,
+$\bar\partial_A\to t\bar\partial_At^{-1}$, where $t$ is a map from
+$X$ to ${\bf C}^*$.
+
+Conjugation by $t$ has the effect of identifying any two $A$'s that
+define the same complex structure on $L$. This can be done almost
+uniquely: the ambiguity is that conjugation by a constant $t$ does
+not change $A$. Of course, a gauge transformation by
+a constant $t$ rescales
+$\alpha$ by a constant. The result therefore, for $J<0$, is that the moduli
+space of solutions of the monopole equations is the moduli space of
+pairs consisting of a complex structure on $L$ and a non-zero
+holomorphic section, defined
+up to scaling, of $K^{1/2}\otimes L$. For $J>0$, it is instead
+$\beta$ that is non-zero, and $K^{1/2}\otimes L$ is replaced by
+$K^{1/2}\otimes L^{-1}$.
+
+This result can be stated particularly nicely if $X$ has $b_1=0$.
+Then the complex structure on $L$, assuming that it exists, is unique.
+The moduli space of solutions of the monopole equations is
+therefore simply a complex projective space, ${\bf P}H^0(X,K^{1/2}\otimes L)$
+or ${\bf P}H^0(X,K^{1/2}\otimes L^{-1})$, depending on the sign of $J$.
+
+\bigskip
+\noindent{\it Identifying The Basic Classes}
+
+We would now like to identify the basic classes.
+The above description of the moduli space gives considerable information:
+basic classes are of the form $x=-2c_1(L)$, where $L$ is such that
+$J<0$ and $H^0(X,K^{1/2}\otimes L)$ is non-empty, or $J>0$
+and $H^0(X,K^{1/2}\otimes L^{-1})$ is non-empty. This, however,
+is not a sharp result.
+
+That is closely related to the fact that the moduli spaces ${\bf P}H^0(X,
+K^{1/2}\otimes L^{\pm 1})$ found above very frequently have a dimension
+bigger than the ``generic'' dimension of the moduli space as predicted
+by the index theorem. In fact, K\"ahler metrics
+are far from being generic. In case the expected dimension
+is zero, one would have always $n_x>0$ if the moduli spaces behaved
+``generically'' (given the complex orientation, an isolated point on the
+moduli space would always contribute $+1$ to $n_x$; this is a special
+case of a discussion below). Since the $n_x$
+are frequently negative (as in the examples of Kronheimer
+and Mrowka or equation (2.66) of \newwitten), moduli spaces of
+non-generic dimension must appear.
+
+When the moduli space has greater than the generically expected dimension,
+one can proceed by integrating over
+the bosonic and fermionic collective
+coordinates in the path integral. This gives a result that can be
+described topologically: letting $T$ be the operator that arises in linearizing
+the monopole equations, the cokernel of $T$ is a vector bundle $V$
+(the ``bundle of antighost zero modes'') over the moduli space ${\cal M}$;
+its Euler class integrated over ${\cal M}$ is the desired $n_x$.
+
+Alternatively, one can perturb the equations to more generic ones.
+We use the same perturbation as before.
+For a K\"ahler manifold $X$, the condition $b_2^+>1$ is equivalent
+to $H^{2,0}(X)\not= 0$, so we can pick a non-zero holomorphic two-form
+$\eta$.\foot{In \newwitten, where essentially the same perturbation
+was made, the two-form was called $\omega$, but
+here we reserve that name for the K\"ahler form.}
+We perturb the monopole equations \juffy\
+to
+\eqn\ojuffy{\eqalign{F^{2,0} & = \alpha\beta -\eta\cr
+ F_\omega^{1,1} & = -\omega\left(|\alpha|^2-|\beta|^2\right)\cr
+ F^{0,2} & =\bar\alpha\bar\beta-\bar\eta,\cr}}
+leaving unchanged the Dirac equation for $M$.
+
+It suffices to consider the case that the first Chern class of $L$
+is of type $(1,1)$, since the unperturbed moduli space vanishes otherwise.
+That being so, we have
+\eqn\ijuffy{0=\int_XF^{2,0}\wedge\bar\eta=\int_XF^{0,2}\wedge \eta.}
+Using this, one finds that \nohighor\ generalizes to
+\eqn\onohighor{\eqalign{\int_Xd^4x\sqrt g\left({1\over 2}|s|^2+|k|^2\right)
+=\int_Xd^4x &\left({1\over 2}
+|F^+|^2+g^{ij}D_i\bar\alpha D_j\alpha+g^{ij}D_i\bar\beta
+D_j\beta\right.\cr & \left.
+ +{1\over 2}(|\alpha|^2-|\beta|^2)^2+{2}|\alpha\beta-\eta|^2
++{R\over 4}(|\alpha|^2+|\beta|^2)\right) .\cr}}
+We can now make an argument of a sort we have already seen: the transformation
+\eqn\hoggy{\eqalign{A & \to A\cr
+ \alpha & \to \alpha \cr
+ \beta & \to -\beta \cr
+ \eta & \to -\eta, \cr}}
+though not a symmetry of \ojuffy, is a symmetry of the right hand side of
+\onohighor. As solutions of \ojuffy\ are the same as zeroes of the right
+hand side of \onohighor, we deduce that the solutions of \ojuffy\ with
+a two-form $\eta$ are transformed by \hoggy\ to the solutions with $-\eta$.
+The terms in \ojuffy\ even or odd under the transformation must therefore
+separately vanish, so
+any solution of \ojuffy\ has
+\eqn\goggy{0= F^{0,2}=F^{2,0}=\alpha\beta-\eta.}
+The condition $F^{0,2}=0$ means that the connection still defines
+a holomorphic structure on $L$.
+
+The condition
+\eqn\jipoggy{ \alpha\beta =\eta}
+gives our final criterion for determining the basic classes: they are
+of the form
+$x=-2c_1(L)$ where, for any choice of $\eta\in H^0(X,K)$, one has
+a factorization $\eta=\alpha\beta$ with
+holomorphic sections $\alpha$ and $\beta$ of $K^{1/2}\otimes L^{\pm 1}$,
+and $x^2=c_1(K)^2$.
+
+To make this completely explicit, suppose
+the divisor of $\eta$ is a union of irreducible
+components $C_i$ of multiplicity $r_i$.
+Thus the canonical divisor is
+\eqn\rufu{c_1(K)=\sum_ir_i[C_i],}
+where $[C_i]$ denotes the cohomology class that is
+Poincar\'e dual to the curve $C_i$.
+The existence of the factorization $\eta=\alpha\beta$
+means that the divisor of $K^{1/2}\otimes L$ is
+\eqn\jufu{c_1(K^{1/2}\otimes L)=\sum_is_i[C_i],}
+where $s_i$ are integers with $0\leq s_i\leq r_i$.
+The first Chern class of $L$ is therefore
+\eqn\tufu{c_1(L)=\sum_i(s_i-{1\over 2}r_i)[C_i].}
+And the basic classes are of the form $x=-2c_1(L)$ or
+\eqn\pufu{x=-\sum_i(2s_i-r_i)[C_i].}
+
+An $x$ of this form is is of type $(1,1)$ and congruent to $c_1(K)$
+modulo two, but may not obey $x^2=c_1(K)^2$.
+It is actually possible to prove using the Hodge index theorem
+that for $x$ as above, $x^2\leq c_1(K)^2$.\foot{Such an argument
+was pointed out by D. Morrison.} This is clear from the monopole
+equations: perturbed to $\eta\not=0$, these equations have
+ at most isolated solutions
+(from the isolated factorization $\eta=\alpha\beta$) and not a moduli
+space of solutions of positive dimension. So for K\"ahler manifolds,
+the non-empty perturbed moduli spaces are at most of dimension zero; invariants
+associated with monopole moduli spaces of higher dimension vanish.
+
+Our final conclusion about the basic classes, then, is that they
+are classes of the form \pufu\ such that $x^2=c_1(K)^2$.
+Each factorization $\eta=\alpha\beta$ contributes
+$\pm 1$ to $n_x$ with the corresponding $x$.
+Since several factorizations might give the same $x$, cancellations
+may be possible, making it possible to write the invariant in
+the Kronheimer-Mrowka form, with a shorter list of basic classes.
+Such cancellations can be effectively found since the signs of the various
+contributions are computed below. In any event the classes $x=\pm K$
+arise only from $s_i=0$ or $s_i=r_i$, respectively, and so always
+arise as basic classes with $n_x=\pm 1$.
+\foot{G. Tian and S.-T. Yau, P. Kronheimer and T. Mrowka, D. Morrison,
+and R. Friedman and J. Morgan pointed out that it actually follows
+from these conditions (or related arguments)
+that if $X$ is a minimal surface of general
+type, then the only basic classes are $\pm K$ (so that $K$ is a differentiable
+invariant up to sign). Indeed, according to Lemma 4 in \ref\kodaira{
+K. Kodaira, ``Pluricanonical Systems On Algebraic Surfaces Of General
+Type,'' J. Math. Soc. Japan {\bf 20} (1968) 170.}, on such a surface, if
+$K={\cal O}(C_1)\otimes {\cal O}(C_2)$
+with non-zero effective divisors $C_1,C_2$, then $C_1\cdot C_2>0$.
+This means that a factorization $\eta=\alpha\beta$ with $\alpha,\beta$
+sections of $K^{1/2}\otimes L^{\pm 1}$ and $x^2=c_1(K)^2$
+implies that $K^{1/2}\otimes L^{\pm 1}$ is trivial with one choice of sign,
+and hence that $x=\pm c_1(K)$.}
+
+\bigskip
+\noindent{\it Comparison To Previous Results}
+
+Let us compare these statements to previous results. The main case considered
+in \newwitten\ was that in which the $C_i$ were disjoint
+with multiplicities $r_i=1$. The allowed values of the $s_i$ are
+then $0$ and 1, so the basic classes are
+\eqn\ppufu{x_{\vec \rho}=\sum_i\rho_i[C_i],}
+with each $\rho_i=\pm 1$, as claimed in \newwitten.
+Notice that all of these classes have $x_{\vec\rho}^2=c_1(K)^2$.
+
+The most important case in which the $r_i$ are not all one is the case
+of an elliptic surface with multiple fibers. A fiber of multiplicity $n$
+appears in the canonical divisor with weight $r=n-1$. For elliptic
+surfaces, one has $C_i\cdot C_j=0$ for all $i,j$,
+so the classes in \pufu\ actually do all have
+$x^2=c_1(K)^2=0$. The formulas of
+Kronheimer and Mrowka for the Donaldson invariants of these surfaces show
+that the basic classes, in their sense,
+are indeed the classes given in \pufu.
+
+
+
+
+\bigskip
+\noindent{\it Determination Of The Sign}
+
+To complete this story, we must compute, for each factorization,
+the sign of $\det T$.
+Let us first explain in an abstract setting the strategy that will be
+used.
+Suppose that $E$ and $F$ are real vector spaces of equal even dimension
+with given complex structures, and $T:E\to F$ is an invertible linear map that
+commutes with the complex structure. Then $\det T$ is naturally
+defined as an element of $\det F\otimes \det E^{-1}$.
+If $\det E$ and $\det F$ are trivialized using the complex orientations of $E$
+and $F$, then $\det T>0$ roughly because the complex structure gives a pairing
+of eigenvalues. If $T$ {\it reverses} the complex structures then
+the sign of $\det T$ is $(-1)^w$ with $w=\dim_{\bf C}E$. For instance,
+by reversing the complex structure of $E$ one could reduce to the case
+in which $T$ preserves the complex structures, but reversing the complex
+structure of $E$ multiplies its orientation by $(-1)^w$.
+
+One can combine the two cases as follows. Suppose that $T$ preserves
+the complex structures but is not invertible. Let $T':E\to F$
+be a map that reverses the complex structures and maps ${\rm ker}\,T$
+invertibly to $F/T(E)$. Then for small real $\epsilon$ (of any sign)
+the sign of $\det(T\oplus \epsilon T')$ is $(-1)^w$ where now
+$w={\rm dim}_{\bf C}{\rm ker}\,T$. The same formula holds if
+$U$ and $V$ are vector bundles,
+$E=\Gamma(U)$, $F=\Gamma(V)$, $T:E\to F$
+is an elliptic operator with zero index, $T'$ is a sufficiently mild
+perturbation, and $\det (T+\epsilon T')$ is
+understood as the Ray-Singer-Quillen
+determinant.
+
+Our problem is of this form with $T$ understood as the linearization
+of the monopole equations at $\eta=0$ and $T'$ as the correction
+proportional to $\eta$ (which enters the linearization because of the shift
+it induces in $\alpha$ or $\beta$).
+As in \pxxx, one has $U=\Lambda^1\oplus (S^+\otimes L)$,
+with $S^+\otimes L$ now regarded as a real vector bundle of rank four.
+If $J<0$ (so $\beta=0$ for $\eta=0$), then
+give $U$ a complex structure that acts naturally on
+$S^+\otimes L$ and multiplies $\Lambda^{0,1}$ and $\Lambda^{1,0}$ by
+$i$ and $-i$, respectively. Likewise
+give $V=\Lambda^0\oplus\Lambda^{2,+}\oplus (S^-\otimes L)$
+a complex structure that acts naturally on $S^-\otimes L$; multiplies
+$\Lambda^{0,2}$ and $\Lambda^{2,0}$ by $i$ and $-i$; and exchanges the
+$(1,1)$ part of $\Lambda^{2,+}$ with $\Lambda^0$.
+Then $T$ preserves the
+complex structures on these bundles and $T'$ reverses them.
+
+The sign of the contribution to $n_x$ from a factorization $\eta=\alpha\beta$
+is therefore $(-1)^w$ with $w={\rm dim}_{\bf C}{\rm \ker}\,T$.
+The kernel of $T$ can be described as follows. There is an exact
+sequence
+\eqn\immo{0\to {\cal O}\underarrow{\alpha}K^{1/2}\otimes L\to R\to 0,}
+with some sheaf $R$. The kernel of $T$ has the same dimension as
+$H^0(X,R)$, as explained below. So the sign of the contribution to $n_x$
+is
+\eqn\uddu{(-1)^{{\rm dim}\,H^0(X,R)}.}
+If instead $J>0$, so the unperturbed solution has $\alpha=0$, $\beta\not=0$,
+then first of all we reverse the complex structures on $S^\pm\otimes L$;
+this multiplies the determinant by $(-1)^\Delta$ where $\Delta=-\sigma/8
++c_1(L)^2/2=(\chi+\sigma)/4$ is the Dirac index. The rest of
+the discussion goes through with
+\immo\ replaced by
+\eqn\dimmo{0\to {\cal O}\underarrow{\beta}K^{1/2}\otimes L^{-1}
+\to \tilde R\to 0,}
+so the sign is
+\eqn\duddu{(-1)^{\Delta+{\rm dim}\,H^0(X,\tilde R)}.}
+(It can be verified using the classification of surfaces that \uddu\
+and \duddu\ are equal.)
+With these signs, \jimmo\ becomes completely explicit: the sum in \jimmo\ is
+a sum over factorizations $\eta=\alpha\beta$; each such factorization
+determines a class $x$ and contributes to $n_x$ an amount $\pm 1$
+as just determined.
+
+Before justifying the claim about $\ker T$, let us
+check that the sign just determined agrees with what has been computed
+by other methods. Suppose as in \newwitten\ that the divisor of $\eta$
+is a union of disjoint smooth curves $C_i$. Then $R$ is a sum of
+sheaves $R_i$ supported on $C_i$; $R_i$ is trivial if $s_i$
+(defined in \jufu) is 0 and is isomorphic to a spin bundle of $C_i$
+(determined by $\eta $ and independent of the factorization $\eta=\alpha\beta$)
+if $s_i=1$.
+Let $t_i=1$ if this spin bundle is even, that is, if ${\rm dim}\,H^0(C_i,R_i)$
+is even, and $-1$ if it is odd. Then \uddu\ becomes
+\eqn\nurfo{(-1)^{{\rm dim}\,H^0(X,R)}=\prod_{i|s_i=1} t_i.}
+This is the result claimed in equation (2.66)
+of \newwitten. One can similarly check that \jimmo\ when evaluated
+with the signs given above agrees with the formulas of Kronheimer
+and Mrowka for Donaldson invariants of elliptic surfaces with multiple
+fibers.
+
+It remains to justify the claimed structure of $\ker\, T$. Suppose, for
+instance, that
+we are linearizing around a solution with $\beta=0$, $\alpha\not= 0$.
+Let $\delta A$, $\delta \alpha$, and $\delta \beta$ denote
+first order variations of $A,\alpha,$ and $\beta$. The argument
+that proves the vanishing theorem shows that for $\delta A,\delta\alpha,
+\delta\beta$ to be annihilated by $T$, one must
+have $\alpha\delta\beta=0$ and hence $\delta\beta=0$. The remaining
+equations can be written
+\eqn\remeq{\eqalign{\bar\partial \,\,\delta A^{0,1} & = 0 \cr
+ i\delta A^{0,1}\alpha +\bar\partial_A\delta\alpha & = 0.
+\cr}}
+One must divide the space of solutions of \remeq\ by solutions that arise
+from complex gauge transformations of $A,\alpha$.
+If $\delta A^{0,1}=0$, then the second equation says that $\delta\alpha
+\in H^0(X, K^{1/2}\otimes L)$; however, upon dividing by complex
+gauge transformations (which include rescalings of $\alpha$ by a constant)
+we should regard $\delta\alpha$ as an element of $H^0(X,K^{1/2}\otimes L)/
+{\bf C}\alpha$. The first equation says that $\delta A^{0,1}$ defines
+an element of $H^1(X,{\cal O})$, and the second equation says that
+multiplication by $\alpha$ maps this element to zero in $H^1(X,K^{1/2}\otimes
+L)$. So if ${\rm ker}\,\alpha$ is the kernel of
+$H^1(X,{\cal O})\underarrow{\alpha}H^1(X,K^{1/2}\otimes L)$, then there
+is an exact sequence
+\eqn\imoc{0\to H^0(X,K^{1/2}\otimes L)/{\bf C}\alpha
+\to {\rm ker}\,T\to {\rm ker}\,\alpha\to
+0.}
+This can be compared to the exact sequence
+\eqn\nimoc{0\to H^0(X,K^{1/2}\otimes L)/{\bf C}\alpha
+\to H^0(X,R)\to {\rm ker}\,\alpha\to
+0}
+that comes from \immo. Comparison of these sequences shows
+that ${\rm ker}\,T$ and $H^0(X,R)$ have the same dimension,
+as asserted above; one should be able to identify these spaces canonically.
+
+\newsec{A Short Sketch Of The Physics}
+
+To sketch the relation of these ideas to quantum field theory,
+let us first recall the analysis of $N=2$ supersymmetric Yang-Mills
+theory in \sw. To begin with we work on flat ${\bf R}^4$.
+It has long been known that this theory has a family of quantum vacuum
+states parametrized by a complex variable $u$, which corresponds
+to the four dimensional class in Donaldson theory. For $u\to\infty$,
+the gauge group is spontaneously broken down to the maximal torus,
+the effective coupling is small, and everything can be computed
+using asymptotic freedom. For small $u$, the effective coupling is strong.
+Classically, at $u=0$, the full $SU(2)$ gauge symmetry is restored.
+But the classical approximation is not valid near $u=0$.
+
+Quantum mechanically, as explained in \sw, the $u$ plane turns out to
+parametrize a family of elliptic curves,
+\foot{If $SU(2)$ is replaced by
+a Lie group of rank $r$, elliptic curves are replaced
+by abelian varieties of rank $r$; the analog of
+the simple type condition is that the commutative
+algebra of operators obtained by evaluating the Chern classes of the universal
+bundle at a point in a four-manifold has a spectrum consisting of $h$
+points ($h$, which is $N$ for $SU(N)$, is the dual Coxeter number of the
+Lie group) where these varieties degenerate maximally.}
+in fact, the modular curve
+of the group $\Gamma(2)$. The family can be described by the equation
+\eqn\urmo{y^2=(x^2-\Lambda^4)(x-u),}
+where $\Lambda$ is the analog of a parameter that often goes by the same
+name in the theory of strong interactions. (The fact that $\Lambda\not= 0$
+means that the quantum theory does not have the conformal invariance
+of the classical theory.)
+The curve \urmo\ is smooth for generic $u$, but degenerates to
+a rational curve for $u=\Lambda^2,-\Lambda^2$, or $\infty$. Near each
+degeneration, the theory becomes weakly coupled, and everything is calculable,
+if the right variables are used. At $u=\infty$, the weak coupling is
+(by asymptotic freedom) in terms of the original field
+variables. Near $u=\pm \Lambda^2$
+a magnetic monopole (or a dyon, that is a particle carrying both
+electric and magnetic charge) becomes massless; the light degrees
+of freedom are the monopole or dyon and a dual photon or $U(1)$ gauge
+boson. In terms of the dyon and dual photon, the theory is weakly
+coupled and controllable near $u=\pm \Lambda^2$.
+
+Notice that quantum mechanically on flat ${\bf R}^4$,
+the full $SU(2)$ gauge symmetry is never restored. The only really
+exceptional behavior that
+occurs anywhere is that magnetically charged particles
+become massless.
+
+Now, for any $N=2$ supersymmetric field theory, a standard twisting
+procedure \witten\ gives a topological field theory. In many cases,
+these topological field theories are related to the counting of
+solutions of appropriate equations. For instance, the procedure,
+applied to the underlying $SU(2)$ gauge theory, gives Donaldson theory
+(that is, the problem of counting $SU(2)$ instantons); applied to
+the quantum theory near $u=\pm \Lambda^2$, it gives the problem of
+counting the solutions of the monopole
+equations; applied at a generic point on the $u$ plane, it gives, roughly,
+the problem of counting {\it abelian} instantons.
+
+Now let us apply this experience, to work
+on a general oriented four-manifold $X$.
+The structure of the argument is analogous to the heat kernel proof
+of the index theorem, in which one considers the trace of the heat kernel
+$\Tr (-1)^Fe^{-tH}$. This is independent of $t$ but can be evaluated
+in different ways for $t\to 0$ or for $t\to \infty$;
+for small $t$, one sees local geometry and gets
+a cohomological formula, while for large $t$, one gets a description
+in terms of the physical ground states (harmonic spinors).
+
+In the four-manifold problem, letting $g$ be any Riemannian
+metric on $X$, we consider the one parameter family of metrics $g_t=tg$,
+with $t>0$. Correlation functions of the twisted topological field
+theory are metric independent and so independent of $t$.
+For $t\to 0$, using asymptotic freedom,
+the classical description becomes valid,
+and one recovers Donaldson's definition of four-manifold invariants
+from the $N=2 $ theory. In particular, for four-manifolds on which
+there are no abelian instantons, the main contribution comes from
+$u=0$ where for small $t$ one computes in the familiar fashion
+with the full $SU(2)$ gauge theory.
+
+What happens for large $t$? Once the scale of the four-manifold
+is much greater than $1/\Lambda$, the good description is in terms of
+the degrees of freedom of the vacuum states on ${\bf R}^4$. At first
+sight, it might appear that the answer will come by integration over
+the $u$ plane. That is apparently so for some classes of problems.
+
+However, for four-manifolds with $b_2^+>1$, one can show
+that the contribution of any region of the $u$ plane bounded away
+from $u=\pm \Lambda^2$ vanishes as a power of $t$ for $t\to \infty$.
+This is roughly because in the abelian theory that prevails away
+from $u=\pm \Lambda^2$, there are too many fermion zero modes
+and no sufficiently efficient way to lift
+them. (It is not clear if the gap in the
+argument for non-K\"ahler manifolds with $b_2^+=3$ is significant,
+or could be removed with a more precise treatment.)
+
+Under the above condition on $b_2^+$, a contribution that survives
+for $t\to\infty$ can therefore come only from a neighborhood of
+$u=\pm \Lambda^2$ that shrinks to zero as $t$ grows. The contribution
+from this region does survive for $t\to \infty$; it can be computed
+using the monopole equations since those are the relevant equations
+in the topologically twisted theory near $u=\pm \Lambda^2$.
+In computing a correlation function of operators of the twisted theory
+near $u=\pm \Lambda^2$, one can expand all operators of the microscopic
+theory in terms of operators of successively higher dimension in the
+macroscopic, monopole theory.
+
+For $u$, the most relevant term (that is, the term of lowest dimension)
+is the $c$-number $u=\Lambda^2$ or $u=-\Lambda^2$.
+The simple type condition -- which asserts that $u$ is semi-simple
+with a spectrum consisting of two points -- arises when one may
+replace $u$ by this $c$-number. For the operator
+related to the two-dimensional classes of Donaldson theory, the most
+relevant term is again a $c$-number,
+measuring the first Chern class of the dual line bundle $L$ of the monopole
+problem. Keeping only these terms, since the operators are replaced
+by $c$-numbers, correlation
+functions can be computed by simply
+counting solutions weighted by the sign of the fermion determinant; only zero
+dimensional moduli spaces contribute. Upon fixing the normalizations
+by comparing to known special cases, one arrives at \jimmo.
+
+This in fact appears to be justified since as usual in such
+problems operators of higher
+dimension give contributions that vanish as negative powers of $t$.
+This would give a quantum field theory proof that
+all oriented four-manifolds with $b_2^+>3$ are of simple type.
+If, however, higher terms in the expansion of the operators survive
+on some four-manifolds with $b_2^+>3$, the consequences would be as follows.
+Then the higher monopole invariants of $W\not= 0$ can be detected in
+Donaldson theory, and \jimmo\ will be replaced by a more general
+formula involving the expansion near $u=\pm \Lambda^2$ of some of the
+functions computed in \sw. The number $s$ of higher terms that one would have
+to keep in the expansion would be one half
+the maximum value of $W$ that contributes. $u$ will still have a spectrum
+consisting of two points, but instead of $u^2-\Lambda^4=0$, one would
+get $(u^2-\Lambda^4)^{s+1}=0$. Such a situation has in fact been
+analyzed by Kronheimer and Mrowka.
+
+\listrefs
+\end
diff --git a/macros/generic/occam/PlainEx2/harvmac.occ b/macros/generic/occam/PlainEx2/harvmac.occ
new file mode 100644
index 0000000000..f108b9cc9c
--- /dev/null
+++ b/macros/generic/occam/PlainEx2/harvmac.occ
@@ -0,0 +1,571 @@
+ %^
+ %%%% harvmac.occ
+ %% is a shrinkable version of harvmac.tex (from ftp://xxx.lanl.gov).
+ %% This file was formatted by LS, March 1995, lcs@topo.math.u-psud.fr
+ %% for use with the Occam utility
+ %% (master posting 1994 on ftp ftp.math.u-psud.fr)
+ %
+ %%% DO NOT ALTER "OCCAM" SIGNS <percent>^, <percent>_ , etc.
+ %% UNLESS YOU UNDERSTAND THEM!
+ \let\DDef\def \let\Def\def \let\gDef\gdef
+ \let\Let\let \def\gLet{\global\let}%
+ \let\Font\font \let\FFont\font
+ \let\Mathchardef\mathchardef\let\Newsymbol\newsymbol
+ \let\MATHchardef\mathchardef\let\NEWsymbol\newsymbol
+ \input auditor.tex %% keep auditor.tex available
+ %% comment out above line to suppress audit function.
+ %
+ %%% Remarks on the Occam formatting of harvmac.tex
+ %% ---" big typ"e choice assumed and prechosen.
+ %% --- Slight printer differences ignored.
+ %% --- Reference macros \ref, \nref very fragile
+ %% --- Volume reduction 15 ko to 6ko is typical
+ %% --- You can easils do still better by simplifying the mss.
+ %%%_
+
+%^ A comment for the shrunken macro file: %_
+%% The following are macros extracted from harvmac.tex
+%% to make an ad hoc header file for <FILENAME>
+%% See ftp://xxx.lanl.gov for original of harvmac.tex
+%%%%%%%%%%%%%%%%%% tex macros for preprints, cm version %%%%%%%%%%%%%%
+% (P. Ginsparg, last updated 9/91)
+%^ if confused, type `b' in response to query
+%---------------------------------------------------------------------%
+%_
+%^% site dependent options:
+%% \unredoffs and \redoffs define horizontal and vertical offsets
+%% respectively for unreduced and reduced modes. \speclscape defines
+%% the \special{} call that sets printer to landscape (sideways) mode.
+%% from standard set below, leave uncommented as appropriate or redefine
+%_
+%^%% next 400dpi
+%\def\unredoffs{} \def\redoffs{\voffset=-.31truein\hoffset=-.48truein}
+%\def\speclscape{\special{landscape}}%_
+
+%%% apple lw parameters by default
+ \Def\unredoffs{}%_
+ \Def\redoffs{\voffset=-.31truein \hoffset=-.59truein}%%_
+ \Def\speclscape{\special{ps: landscape}}%
+ %% above special for dvips ?? %_
+
+%^%% qms lasergrafix:
+%\def\unredoffs{} \def\redoffs{\voffset=-.4truein\hoffset=.125truein}
+%\def\speclscape{\special{qms: landscape}}
+%_
+%^%% saclay A4 paper:
+%\def\unredoffs{\hoffset-.14truein\voffset-.2truein}
+%\def\redoffs{\voffset=-.45truein\hoffset=-.21truein}
+%\def\speclscape{\special{landscape}}
+%---------------------------------------------------------------------%
+%_
+
+ \newbox\leftpage
+ \newdimen\fullhsize
+ \newdimen\hstitle
+ \newdimen\hsbody
+ \tolerance=1000\hfuzz=2pt
+
+ %^
+ \def\fontflag{cm} %% never used
+ %_
+
+\catcode`\@=11
+
+%#^
+\Def\usedLITTLEans{}%_
+\Def\usedBIGans{}%% sic! %_
+%#_
+
+%^ % This allows us to modify PLAIN macros.
+\def\bigans{b }
+\def\answ{b }
+\message{ big or little pointsize (b/l)? }\read-1 to\answ
+\ifx
+ \answ\bigans\message{(This will come out unreduced.}%
+ \usedBIGans
+ %_
+ %/^\usedBIGans
+ \magnification=1200
+ %% Adjust:
+ \unredoffs %% set predefined offsets
+ %% but maybe adjust:
+ %\voffset=0truein
+ %\hoffset=0truein
+ \edef\tfontsize{scaled\magstep3}%#_
+ \FFont\authorfont=cmcsc10 %#_
+ \DDef\abstractfont{\tenpoint}%#_
+ \DDef\lspace{{}}%#_
+ \DDef\lbspace{{}}%% $$\lbspace...$$ %#_
+ \baselineskip=16pt plus 2pt minus 1pt
+ \hsbody=\hsize \hstitle=\hsize
+ %% take default values for unreduced format
+ %/_
+ %^
+ \fi
+ \ifx \answ\bigans
+ \else
+ \usedLITTLEans
+ %_
+ %/^\usedLITTLEans
+ \message{(This will be reduced.} \let\l@r=L
+ \magnification=1000\baselineskip=16pt plus 2pt minus 1pt
+ \redoffs %% set predefined offsets
+ %% but maybe adjust:
+ %\voffset=-.31truein
+ %\hoffset=-.59truein
+ %% above sample for dvips and apple laserwriter
+ \vsize=7truein
+ \hstitle=8truein\hsbody=4.75truein
+ \fullhsize=10truein\hsize=\hsbody
+ \edef\tfontsize{scaled\magstep4}%#_
+ \FFont\authorfont=cmcsc10 scaled \magstep1 %#_
+ \DDef\lspace{\qquad}%#_
+ \DDef\lbspace{\hskip-.2in}%% $$\lbspace...$$ %#_
+ %
+\output={\ifnum\pageno=0 %%% This is the HUTP version
+ \shipout\vbox{\speclscape{\hsize\fullhsize\makeheadline}
+ \hbox to \fullhsize{\hfill\pagebody\hfill}}\advancepageno
+ \else
+ \almostshipout{\leftline{\vbox{\pagebody\makefootline}}}\advancepageno
+ \fi}
+\def\almostshipout#1{\if L\l@r \count1=1 \message{[\the\count0.\the\count1]}
+ \global\setbox\leftpage=#1 \global\let\l@r=R
+ \else \count1=2
+ \shipout\vbox{\speclscape{\hsize\fullhsize\makeheadline}
+ \hbox to\fullhsize{\box\leftpage\hfil#1}} \global\let\l@r=L\fi}
+%/_
+%^
+\fi
+%---------------------------------------------------------------------
+%_
+
+\newcount\yearltd\yearltd=\year\advance\yearltd by -1900%_
+
+\Def\HUTP#1#2{\Title{HUTP-\number\yearltd/A#1}{#2}}%_
+
+\Def\Title#1#2{\nopagenumbers
+ \abstractfont\hsize=\hstitle\rightline{#1}%
+ \vskip 1in\centerline{\titlefont #2}
+ \abstractfont\vskip .5in\pageno=0}%_
+
+\Def\Date#1{\vfill\leftline{#1}\tenpoint
+ \supereject\global\hsize=\hsbody%
+ \footline={\hss\tenrm\folio\hss}}% restores pagenumbers%_
+
+% use following instead of \Date on the preliminary draft,
+% puts date/time on each page in big mode, writes labels in margins
+\Def\draft{\draftmode\Date{\draftdate}}%_
+\Def\draftmode{\message{ DRAFTMODE }%
+ \def\draftdate{{\rm preliminary draft:
+ \number\month/\number\day/\number\yearltd\ \ \hourmin}}%
+ \headline={\hfil\draftdate}\writelabels
+ \baselineskip=20pt plus 2pt minus 2pt
+ {\count255=\time\divide\count255 by 60 \xdef\hourmin{\number\count255}
+ \multiply\count255 by-60\advance\count255 by\time
+ \xdef\hourmin{\hourmin:\ifnum\count255<10 0\fi\the\count255}}}%_
+
+\Def\writelabels{\def\wrlabeL##1{\leavevmode\vadjust{\rlap{\smash%
+{\line{{\escapechar=` \hfill\rlap{\sevenrm\hskip.03in\string##1}}}}}}}%
+\def\eqlabeL##1{{\escapechar-1\rlap{\sevenrm\hskip.05in\string##1}}}%
+\def\reflabeL##1{\noexpand\llap{%
+ \noexpand\sevenrm\string\string\string##1}}}%_
+
+% use \nolabels to get rid of eqn, ref, and fig labels in draft mode
+\def\nolabels{\def\wrlabeL##1{}\def\eqlabeL##1{}\def\reflabeL##1{}}
+\nolabels
+
+% tagged sec numbers
+\global\newcount\secno \global\secno=0
+\global\newcount\meqno \global\meqno=1%_
+
+\DDef\newsec#1{\global\advance\secno by1\message{(\the\secno. #1)}%
+ %^\ifx\answ\bigans \vfill\eject
+ %\else \bigbreak\bigskip \fi %if desired %_
+ \global\subsecno=0
+ \eqnres@t\noindent{\bf\the\secno. #1}%
+ \writetoca{{\secsym} {#1}}\par\nobreak\medskip\nobreak}%#_
+
+ \Def\eqnres@t{\xdef\secsym{\the\secno.}%
+ \global\meqno=1\bigbreak\bigskip}%_
+
+ \Def\sequentialequations{\def\eqnres@t{\bigbreak}}%_
+
+ \xdef\secsym{}%_
+
+ \global\newcount\subsecno \global\subsecno=0%_
+
+\Def\subsec#1{\global\advance\subsecno by1
+ \message{(\secsym\the\subsecno. #1)}
+ \ifnum\lastpenalty>9000\else\bigbreak\fi
+ \noindent{\it\secsym\the\subsecno. #1}\writetoca{\string\quad
+ {\secsym\the\subsecno.} {#1}}\par\nobreak\medskip\nobreak}%_
+
+\Def\appendix#1#2{\global\meqno=1
+ \global\subsecno=0\xdef\secsym{\hbox{#1.}}
+ \bigbreak\bigskip\noindent{\bf Appendix #1. #2}\message{(#1. #2)}
+ \writetoca{Appendix {#1.} {#2}}\par\nobreak\medskip\nobreak}%_
+
+% \eqn\label{a+b=c} gives displayed equation, numbered
+% consecutively within sections.
+% \eqnn and \eqna define labels in advance (of eqalign?)
+%
+\Def\eqnn#1{\xdef #1{(\secsym
+ \the\meqno)}\writedef{#1\leftbracket#1}%
+ \global\advance\meqno by1\wrlabeL#1}%_
+\Def\eqna#1{\xdef #1##1{\hbox{$(\secsym\the\meqno##1)$}}
+ \writedef{#1\numbersign1\leftbracket#1{\numbersign1}}%
+ \global\advance\meqno by1\wrlabeL{#1$\{\}$}}%_
+\Def\eqn#1#2{\xdef #1{(\secsym
+ \the\meqno)}\writedef{#1\leftbracket#1}%
+ \global\advance\meqno by1$$#2\eqno#1\eqlabeL#1$$}%_
+
+%% Footnotes
+ \newskip\footskip\footskip14pt plus 1pt minus 1pt
+ %% sets footnote baselineskip
+ \Def\footnotefont{\ninepoint}
+ \def\f@t#1{\footnotefont #1\@foot}
+ \def\f@@t{\baselineskip\footskip\bgroup
+ \footnotefont\aftergroup\@foot\let\next}
+ \setbox\strutbox=\hbox{\vrule height9.5pt depth4.5pt width0pt}%_
+%
+ \global\newcount\ftno \global\ftno=0
+ \Def\foot{\global\advance\ftno by1\footnote{$^{\the\ftno}$}}%_
+
+%
+\newwrite\ftfile
+ %% say \footend to put footnotes at end
+ %% will cause problems if \ref used inside
+ %% \foot, instead use \nref before
+\Def\footend{\def\foot{\global\advance\ftno by1\chardef\wfile=\ftfile
+ $^{\the\ftno}$\ifnum\ftno=1\immediate\openout\ftfile=foots.tmp\fi%
+ \immediate\write\ftfile{\noexpand\smallskip%
+ \noexpand\item{f\the\ftno:\ }\pctsign}\findarg}%
+ \def\footatend{\vfill\eject\immediate\closeout\ftfile{\parindent=20pt
+ \centerline{\bf Footnotes}\nobreak\bigskip\input foots.tmp }}}%_
+ \Def\footatend{}%_
+
+%% References
+%
+% \ref\label{text}
+% generates a number, assigns it to \label, generates an entry.
+% To list the refs on a separate page, \listrefs
+%
+\global\newcount\refno \global\refno=1
+\newwrite\rfile
+\Def\nref#1{\xdef#1{[\the\refno]}\writedef{#1\leftbracket#1}%
+ \ifnum\refno=1\immediate\openout\rfile=refs.tmp\fi
+ \global\advance\refno by1\chardef\wfile=\rfile\immediate
+ \write\rfile{\noexpand\item{#1\ }%
+ \reflabeL{#1\hskip.31in}\pctsign}\findarg}
+ %
+ % horrible hack to sidestep tex \write limitation
+ % FRAGILE!!!
+ \def\findarg#1#{\begingroup\obeylines\newlinechar=`\^^M\pass@rg}
+ {\obeylines\gdef\pass@rg#1{\writ@line\relax #1^^M\hbox{}^^M}%
+ \gdef\writ@line#1^^M{\expandafter\toks0\expandafter{\striprel@x #1}%
+ \edef\next{\the\toks0}\ifx\next\em@rk\let\next=\endgroup\else\ifx\next\empty%
+ \else\immediate\write\wfile{\the\toks0}\fi\let\next=\writ@line\fi\next\relax}}
+ \def\striprel@x#1{} \def\em@rk{\hbox{}}%_
+ %
+
+\Def\ref{[\the\refno]\nref}%_
+
+\Def\lref{\begingroup\obeylines\lr@f}
+ \def\lr@f#1#2{\gdef#1{\ref#1{#2}}\endgroup\unskip}%_
+
+\Def\semi{;\hfil\break}%_
+
+\Def\addref#1{\immediate\write\rfile{\noexpand\item{}#1}}
+ %now unnecessary%_
+
+\Def\listrefs{\footatend\vfill
+ \supereject\immediate\closeout\rfile\writestoppt
+ \baselineskip=14pt\centerline
+ {{\bf References}}\bigskip{\frenchspacing
+ \parindent=20pt\escapechar=` \input refs.tmp
+ \vfill\eject}\nonfrenchspacing}%_
+
+\Def\startrefs#1{\immediate\openout\rfile=refs.tmp\refno=#1}%_
+
+\Def\xref{\expandafter\xr@f}\def\xr@f[#1]{#1}%_
+
+\Def\refs#1{\count255=1[\r@fs #1{\hbox{}}]}
+ \def\r@fs#1{\ifx\und@fined#1\message{reflabel \string#1 is undefined.}%
+ \nref#1{need to supply reference \string#1.}\fi%
+ \vphantom{\hphantom{#1}}\edef\next{#1}\ifx\next\em@rk\def\next{}%
+ \else\ifx\next#1\ifodd\count255\relax\xref#1\count255=0\fi%
+ \else#1\count255=1\fi\let\next=\r@fs\fi\next}%_
+
+ %% Figures
+ %
+\Def\figures{\centerline{{\bf Figure Captions}}\medskip\parindent=40pt%
+ \def\fig##1##2{\medskip\item{Fig.~##1. }##2}}%_
+
+% this is ugly, but moore insists
+ \newwrite\ffile\global
+ \newcount\figno \global\figno=1
+\Def\fig{fig.~\the\figno\nfig}%_
+
+\Def\nfig#1{\xdef#1{fig.~\the\figno}%
+ \writedef{#1\leftbracket fig.\noexpand~\the\figno}%
+ \ifnum\figno=1
+ \immediate\openout\ffile=figs.tmp
+ \fi
+ \chardef\wfile=\ffile%
+ \immediate\write\ffile{\noexpand\medskip\noexpand
+ \item{Fig.\ \the\figno. }
+ \reflabeL{#1\hskip.55in}\pctsign}%
+ \global\advance\figno by1\findarg}%_
+
+\Def\listfigs{\vfill\eject\immediate\closeout\ffile{\parindent40pt
+ \baselineskip14pt\centerline{{\bf Figure Captions}}\nobreak\medskip
+ \escapechar=` \input figs.tmp\vfill\eject}}%_
+
+\Def\xfig{\expandafter\xf@g}\def\xf@g fig.\penalty\@M\ {}%_
+\Def\figs#1{figs.~\f@gs #1{\hbox{}}}%
+\def\f@gs#1{\edef\next{#1}%
+ \ifx\next\em@rk
+ \def\next{}
+ \else
+ \ifx\next#1\xfig #1
+ \else
+ #1
+ \fi
+ \let\next=\f@gs
+ \fi\next}%_
+
+%% Files
+
+\newwrite\lfile
+
+{\escapechar-1\xdef\pctsign{\string\%}\xdef\leftbracket{\string\{}
+ \xdef\rightbracket{\string\}}\xdef\numbersign{\string\#}}%_
+
+\Def\writedefs{\immediate\openout
+ \lfile=labeldefs.tmp \def\writedef##1{%
+ \immediate\write\lfile{\string\def\string##1\rightbracket}}}%_
+
+\Def\writestop{\def\writestoppt{\immediate\write\lfile
+ {\string\pageno \the\pageno
+ \string\startrefs\leftbracket\the\refno\rightbracket
+ \string\def\string\secsym\leftbracket\secsym\rightbracket
+ \string\secno\the\secno\string\meqno\the\meqno
+ }\immediate\closeout\lfile}}%_
+
+\Def\writestoppt{}%_
+\Def\writedef#1{}%_
+
+\Def\seclab#1{\xdef #1{\the\secno}%
+ \writedef{#1\leftbracket#1}\wrlabeL{#1=#1}}%_
+\Def\subseclab#1{\xdef #1{\secsym\the\subsecno}%
+ \writedef{#1\leftbracket#1}\wrlabeL{#1=#1}}%_
+
+\newwrite\tfile
+
+\Def\writetoca#1{}%_
+
+\Def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill}
+ % use this to write file with table of contents%_
+
+\Def\writetoc{\immediate\openout\tfile=toc.tmp
+ \def\writetoca##1{{\edef\next{\write\tfile{\noindent ##1
+ \string\leaderfill {\noexpand\number\pageno} \par}}\next}}}
+% and this lists table of contents on second pass%_
+
+\Def\listtoc{\centerline{\bf Contents}\nobreak
+ \medskip{\baselineskip=12pt
+ \parskip=0pt\catcode`\@=11
+ \input toc.tex \catcode`\@=12 \bigbreak\bigskip}}%_
+
+\catcode`\@=12 % at signs are no longer letters
+
+ %%% Fonts 10pt, Title, 9pt
+ %^% The font loading has been rearranged to suit Occam%_
+
+ %% Stop frivolous math family waste
+ \let\TextFontInMathBad\undefined
+ \def\TextWarning{\ifmmode\TextFontInMathBad\fi}
+
+ % Unpleasantness in calling in abstract and title fonts
+
+ %%% Title fonts
+ %#/^\TitlepointMathTest
+ \font\titlerms=cmr7 \tfontsize
+ \font\titlermss=cmr5 \tfontsize
+ \font\titlei=cmmi10 \tfontsize\relax \skewchar\titlei='177
+ \font\titleis=cmmi7 \tfontsize\relax \skewchar\titleis='177
+ \font\titleiss=cmmi5 \tfontsize\relax \skewchar\titleiss='177
+ \font\titlesy=cmsy10 \tfontsize\relax \skewchar\titlesy='60
+ \font\titlesys=cmsy7 \tfontsize\relax \skewchar\titlesys='60
+ \font\titlesyss=cmsy5 \tfontsize\relax \skewchar\titlesyss='60
+ %#/_
+ %^
+ \DDef \TitlepointMathTest{\relax}%
+ %% diagnostic that never survives%#_
+ %_
+
+ \font\titlerm=cmr10 \tfontsize
+ \Def\titlefont{\textfont0=\titlerm
+ \def\rm{\fam0\titlerm}%
+ \rm
+ %#/^\TitlepointMathTest
+ \textfont0=\titlerm \scriptfont0=\titlerms
+ \scriptscriptfont0=\titlermss
+ \textfont1=\titlei \scriptfont1=\titleis
+ \scriptscriptfont1=\titleiss
+ \textfont2=\titlesy \scriptfont2=\titlesys
+ \scriptscriptfont2=\titlesyss
+ %#/_
+ %#^
+ \everymath{\TitlepointMathTest}%
+ %#_
+ }%_
+
+ %^
+ \ifx\answ\bigans
+ \else
+ %_
+ %/^\usedLITTLEans
+ %%% Abstract fonts
+ %
+ \font\bigit=cmti10 scaled \magstep1
+ \font\abssl=cmsl10 scaled \magstep1
+ \font\absrm=cmr10 scaled\magstep1 \relax\textfont0=\absrm
+ \font\absrms=cmr7 scaled\magstep1 \relax\scriptfont0=\absrms
+ \font\absrmss=cmr5 scaled\magstep1 \relax\scriptscriptfont0=\absrmss
+ \font\absi=cmmi10 scaled\magstep1
+ \relax\skewchar\absi='177\textfont1=\absi
+ \font\absis=cmmi7 scaled\magstep1
+ \relax\skewchar\absis='177\scriptfont1=\absis
+ \font\absiss=cmmi5 scaled\magstep1
+ \relax\skewchar\absiss='177\scriptscriptfont1=\absiss
+ \font\abssy=cmsy10 scaled\magstep1
+ \relax\skewchar\abssy='60\textfont2=\abssy
+ \font\abssys=cmsy7 scaled\magstep1
+ \relax\skewchar\abssys='60\scriptfont2=\abssys
+ \font\abssyss=cmsy5 scaled\magstep1
+ \relax\skewchar\abssyss='60\scriptscriptfont2=\abssyss
+ \font\absbf=cmbx10 scaled\magstep1
+ %
+ \def\abstractfont{\def\rm{\fam0\absrm}% switch to abstract font
+ %
+ \textfont\itfam=\bigit
+ \def\it{\fam\itfam\bigit}%
+ \textfont\slfam=\abssl \def\sl{\fam\slfam\abssl}%
+ \textfont\bffam=\absbf \def\bf{\fam\bffam\absbf}\rm}%
+ %/_
+ %^
+\fi
+ %_
+
+ %%% Tenpoint Fonts
+ %
+\Def\tenpoint{%
+ \def\rm{\fam0\tenrm}%
+ \rm
+ \textfont0=\tenrm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm
+ \textfont1=\teni \scriptfont1=\seveni \scriptscriptfont1=\fivei
+ \textfont2=\tensy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy
+ \textfont\bffam=\tenbf
+ \def\bf{\fam\bffam\tenbf}%
+ \def\it{\TextWarning\tenit}%
+ \def\sl{\TextWarning\tensl}%
+ }%_
+
+ %%% Ninepoint Fonts
+ %#/^\NinepointMathTest
+ \font\sixrm=cmr6%
+ \font\ninei=cmmi9 \skewchar\ninei='177
+ \font\sixi=cmmi6 \skewchar\ninei='177
+ \font\ninesy=cmsy9 \skewchar\ninesy='60
+ \font\sixsy=cmsy6 \skewchar\sixsy='60
+ %#/_
+ %^
+ \DDef\NinepointMathTest{\relax}%
+ %% diagnostic only%#_
+ %_
+
+ %^% Autonomous 9pt prose fonts
+ %% \ninerm also math%_
+ \font\ninerm=cmr9 %_
+ \Font\nineit=cmti9 %_
+ \Font\ninesl=cmsl9 %_
+
+ \Def\ninepoint{%
+ \textfont0=\ninerm
+ \def\rm{\fam0\ninerm}%
+ \rm
+ \def\it{\TextWarning\nineit}%
+ \def\sl{\TextWarning\ninesl}%
+ %#/^\NinepointMathTest
+ \textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
+ \textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei
+ \textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
+ %#/_
+ %#^
+ \everymath{\NinepointMathTest}%
+ %#_
+ }%_
+
+
+ %% Hyphenation exceptions
+
+\hyphenation{anom-aly anom-alies coun-ter-term coun-ter-terms}
+
+\Def\noblackbox{\overfullrule=0pt}%_
+
+\Def\inv{^{\raise.15ex\hbox{${\scriptscriptstyle -}$}\kern-.05em 1}}%_
+\Def\dup{^{\vphantom{1}}}%_
+\Def\Dsl{\,\raise.15ex\hbox{/}\mkern-13.5mu D}
+ %this one can be subscripted%_
+\Def\dsl{\raise.15ex\hbox{/}\kern-.57em\partial}%_
+\Def\del{\partial}%_
+\Def\Psl{\dsl}%_
+\Def\tr{{\rm tr}}%_
+\Def\Tr{{\rm Tr}}%_
+
+\font\bigit=cmti10 scaled \magstep1
+\Def\biglie{\hbox{\bigit\$}} %pound sterling%_
+
+\Def\boxeqn#1{\vcenter{\vbox{\hrule\hbox{\vrule\kern3pt\vbox{\kern3pt
+ \hbox{${\displaystyle #1}$}\kern3pt}\kern3pt\vrule}\hrule}}}%_
+\Def\mbox#1#2{\vcenter{\hrule \hbox{\vrule height#2in
+ \kern#1in \vrule} \hrule}} %e.g. \mbox{.1}{.1}%_
+
+%^ matters of taste%_
+%^\Def\tilde{\widetilde}%_
+%^\Def\bar{\overline}%_
+%^\Def\hat{\widehat}%_
+
+% some sample definitions
+\Def\CAG{{\cal A/\cal G}}%_
+\Def\CO{{\cal O}} % curly letters%_
+\Def\CA{{\cal A}}%_
+\Def\CC{{\cal C}}%_
+\Def\CF{{\cal F}}%_
+\Def\CG{{\cal G}}%_
+\Def\CL{{\cal L}}%_
+\Def\CH{{\cal H}}%_
+\Def\CI{{\cal I}}%_
+\Def\CU{{\cal U}}%_
+\Def\CB{{\cal B}}%_
+\Def\CR{{\cal R}}%_
+\Def\CD{{\cal D}}%_
+\Def\CT{{\cal T}}%_
+\Def\e#1{{\rm e}^{^{\textstyle#1}}}%_
+\Def\grad#1{\,\nabla\!_{{#1}}\,}%_
+\Def\gradgrad#1#2{\,\nabla\!_{{#1}}\nabla\!_{{#2}}\,}%_
+\Def\ph{\varphi}%_
+\Def\psibar{\overline\psi}%_
+\Def\om#1#2{\omega^{#1}{}_{#2}}%_
+\Def\vev#1{\langle #1 \rangle}%_
+\Def\lform{\hbox{$\sqcup$}\llap{\hbox{$\sqcap$}}}%_
+\Def\darr#1{\raise1.5ex\hbox{$\leftrightarrow$}\mkern-16.5mu #1}%_
+\Def\lie{\hbox{\it\$}} %pound sterling%_
+\Def\ha{{1\over2}}%_
+\Def\half{{\textstyle{1\over2}}} %puts a small half in a displayed eqn%_
+\Def\roughly#1{\raise.3ex\hbox{$#1$\kern-.75em\lower1ex\hbox{$\sim$}}}%_
+
+%^
+\AuditDepth{**}%_
+\endinput \ No newline at end of file
diff --git a/macros/generic/occam/PlainEx2/harvmac.tex b/macros/generic/occam/PlainEx2/harvmac.tex
new file mode 100644
index 0000000000..e223d269c0
--- /dev/null
+++ b/macros/generic/occam/PlainEx2/harvmac.tex
@@ -0,0 +1,327 @@
+%%%%%%%%%%%%%%%%%% tex macros for preprints, cm version %%%%%%%%%%%%%%
+% (P. Ginsparg, last updated 9/91)
+% if confused, type `b' in response to query
+%
+%---------------------------------------------------------------------%
+%% site dependent options:
+%% \unredoffs and \redoffs define horizontal and vertical offsets
+%% respectively for unreduced and reduced modes. \speclscape defines
+%% the \special{} call that sets printer to landscape (sideways) mode.
+%% from standard set below, leave uncommented as appropriate or redefine
+%
+%%% next 400dpi
+%\def\unredoffs{} \def\redoffs{\voffset=-.31truein\hoffset=-.48truein}
+%\def\speclscape{\special{landscape}}
+%
+%%% apple lw
+\def\unredoffs{} \def\redoffs{\voffset=-.31truein\hoffset=-.59truein}
+\def\speclscape{\special{ps: landscape}}
+%
+%%% qms lasergrafix:
+%\def\unredoffs{} \def\redoffs{\voffset=-.4truein\hoffset=.125truein}
+%\def\speclscape{\special{qms: landscape}}
+%
+%%% saclay A4 paper:
+%\def\unredoffs{\hoffset-.14truein\voffset-.2truein}
+%\def\redoffs{\voffset=-.45truein\hoffset=-.21truein}
+%\def\speclscape{\special{landscape}}
+%
+%---------------------------------------------------------------------%
+%
+\newbox\leftpage
+\newdimen\fullhsize
+\newdimen\hstitle
+\newdimen\hsbody
+\tolerance=1000\hfuzz=2pt\def\fontflag{cm}
+%
+\catcode`\@=11 % This allows us to modify PLAIN macros.
+\def\bigans{b }
+\message{ big or little (b/l)? }\read-1 to\answ
+%
+\ifx\answ\bigans\message{(This will come out unreduced.}
+\magnification=1200\unredoffs\baselineskip=16pt plus 2pt minus 1pt
+\hsbody=\hsize \hstitle=\hsize %take default values for unreduced format
+%
+\else\message{(This will be reduced.} \let\l@r=L
+\magnification=1000\baselineskip=16pt plus 2pt minus 1pt \vsize=7truein
+\redoffs \hstitle=8truein\hsbody=4.75truein\fullhsize=10truein\hsize=\hsbody
+%
+\output={\ifnum\pageno=0 %%% This is the HUTP version
+ \shipout\vbox{\speclscape{\hsize\fullhsize\makeheadline}
+ \hbox to \fullhsize{\hfill\pagebody\hfill}}\advancepageno
+ \else
+ \almostshipout{\leftline{\vbox{\pagebody\makefootline}}}\advancepageno
+ \fi}
+\def\almostshipout#1{\if L\l@r \count1=1 \message{[\the\count0.\the\count1]}
+ \global\setbox\leftpage=#1 \global\let\l@r=R
+ \else \count1=2
+ \shipout\vbox{\speclscape{\hsize\fullhsize\makeheadline}
+ \hbox to\fullhsize{\box\leftpage\hfil#1}} \global\let\l@r=L\fi}
+\fi
+%---------------------------------------------------------------------
+%
+\newcount\yearltd\yearltd=\year\advance\yearltd by -1900
+\def\HUTP#1#2{\Title{HUTP-\number\yearltd/A#1}{#2}}
+\def\Title#1#2{\nopagenumbers\abstractfont\hsize=\hstitle\rightline{#1}%
+\vskip 1in\centerline{\titlefont #2}\abstractfont\vskip .5in\pageno=0}
+%
+\def\Date#1{\vfill\leftline{#1}\tenpoint\supereject\global\hsize=\hsbody%
+\footline={\hss\tenrm\folio\hss}}% restores pagenumbers
+%
+% use following instead of \Date on the preliminary draft,
+% puts date/time on each page in big mode, writes labels in margins
+\def\draft{\draftmode\Date{\draftdate}}
+\def\draftmode{\message{ DRAFTMODE }\def\draftdate{{\rm preliminary draft:
+\number\month/\number\day/\number\yearltd\ \ \hourmin}}%
+\headline={\hfil\draftdate}\writelabels\baselineskip=20pt plus 2pt minus 2pt
+ {\count255=\time\divide\count255 by 60 \xdef\hourmin{\number\count255}
+ \multiply\count255 by-60\advance\count255 by\time
+ \xdef\hourmin{\hourmin:\ifnum\count255<10 0\fi\the\count255}}}
+% use \nolabels to get rid of eqn, ref, and fig labels in draft mode
+\def\nolabels{\def\wrlabeL##1{}\def\eqlabeL##1{}\def\reflabeL##1{}}
+\def\writelabels{\def\wrlabeL##1{\leavevmode\vadjust{\rlap{\smash%
+{\line{{\escapechar=` \hfill\rlap{\sevenrm\hskip.03in\string##1}}}}}}}%
+\def\eqlabeL##1{{\escapechar-1\rlap{\sevenrm\hskip.05in\string##1}}}%
+\def\reflabeL##1{\noexpand\llap{\noexpand\sevenrm\string\string\string##1}}}
+\nolabels
+%
+% tagged sec numbers
+\global\newcount\secno \global\secno=0
+\global\newcount\meqno \global\meqno=1
+%
+\def\newsec#1{\global\advance\secno by1\message{(\the\secno. #1)}
+%\ifx\answ\bigans \vfill\eject \else \bigbreak\bigskip \fi %if desired
+\global\subsecno=0\eqnres@t\noindent{\bf\the\secno. #1}
+\writetoca{{\secsym} {#1}}\par\nobreak\medskip\nobreak}
+\def\eqnres@t{\xdef\secsym{\the\secno.}\global\meqno=1\bigbreak\bigskip}
+\def\sequentialequations{\def\eqnres@t{\bigbreak}}\xdef\secsym{}
+%
+\global\newcount\subsecno \global\subsecno=0
+\def\subsec#1{\global\advance\subsecno by1\message{(\secsym\the\subsecno. #1)}
+\ifnum\lastpenalty>9000\else\bigbreak\fi
+\noindent{\it\secsym\the\subsecno. #1}\writetoca{\string\quad
+{\secsym\the\subsecno.} {#1}}\par\nobreak\medskip\nobreak}
+%
+\def\appendix#1#2{\global\meqno=1\global\subsecno=0\xdef\secsym{\hbox{#1.}}
+\bigbreak\bigskip\noindent{\bf Appendix #1. #2}\message{(#1. #2)}
+\writetoca{Appendix {#1.} {#2}}\par\nobreak\medskip\nobreak}
+%
+% \eqn\label{a+b=c} gives displayed equation, numbered
+% consecutively within sections.
+% \eqnn and \eqna define labels in advance (of eqalign?)
+%
+\def\eqnn#1{\xdef #1{(\secsym\the\meqno)}\writedef{#1\leftbracket#1}%
+\global\advance\meqno by1\wrlabeL#1}
+\def\eqna#1{\xdef #1##1{\hbox{$(\secsym\the\meqno##1)$}}
+\writedef{#1\numbersign1\leftbracket#1{\numbersign1}}%
+\global\advance\meqno by1\wrlabeL{#1$\{\}$}}
+\def\eqn#1#2{\xdef #1{(\secsym\the\meqno)}\writedef{#1\leftbracket#1}%
+\global\advance\meqno by1$$#2\eqno#1\eqlabeL#1$$}
+%
+% footnotes
+\newskip\footskip\footskip14pt plus 1pt minus 1pt %sets footnote baselineskip
+\def\footnotefont{\ninepoint}\def\f@t#1{\footnotefont #1\@foot}
+\def\f@@t{\baselineskip\footskip\bgroup\footnotefont\aftergroup\@foot\let\next}
+\setbox\strutbox=\hbox{\vrule height9.5pt depth4.5pt width0pt}
+%
+\global\newcount\ftno \global\ftno=0
+\def\foot{\global\advance\ftno by1\footnote{$^{\the\ftno}$}}
+%
+%say \footend to put footnotes at end
+%will cause problems if \ref used inside \foot, instead use \nref before
+\newwrite\ftfile
+\def\footend{\def\foot{\global\advance\ftno by1\chardef\wfile=\ftfile
+$^{\the\ftno}$\ifnum\ftno=1\immediate\openout\ftfile=foots.tmp\fi%
+\immediate\write\ftfile{\noexpand\smallskip%
+\noexpand\item{f\the\ftno:\ }\pctsign}\findarg}%
+\def\footatend{\vfill\eject\immediate\closeout\ftfile{\parindent=20pt
+\centerline{\bf Footnotes}\nobreak\bigskip\input foots.tmp }}}
+\def\footatend{}
+%
+% \ref\label{text}
+% generates a number, assigns it to \label, generates an entry.
+% To list the refs on a separate page, \listrefs
+%
+\global\newcount\refno \global\refno=1
+\newwrite\rfile
+%
+\def\ref{[\the\refno]\nref}
+\def\nref#1{\xdef#1{[\the\refno]}\writedef{#1\leftbracket#1}%
+\ifnum\refno=1\immediate\openout\rfile=refs.tmp\fi
+\global\advance\refno by1\chardef\wfile=\rfile\immediate
+\write\rfile{\noexpand\item{#1\ }\reflabeL{#1\hskip.31in}\pctsign}\findarg}
+% horrible hack to sidestep tex \write limitation
+\def\findarg#1#{\begingroup\obeylines\newlinechar=`\^^M\pass@rg}
+{\obeylines\gdef\pass@rg#1{\writ@line\relax #1^^M\hbox{}^^M}%
+\gdef\writ@line#1^^M{\expandafter\toks0\expandafter{\striprel@x #1}%
+\edef\next{\the\toks0}\ifx\next\em@rk\let\next=\endgroup\else\ifx\next\empty%
+\else\immediate\write\wfile{\the\toks0}\fi\let\next=\writ@line\fi\next\relax}}
+\def\striprel@x#1{} \def\em@rk{\hbox{}}
+%
+\def\lref{\begingroup\obeylines\lr@f}
+\def\lr@f#1#2{\gdef#1{\ref#1{#2}}\endgroup\unskip}
+%
+\def\semi{;\hfil\break}
+\def\addref#1{\immediate\write\rfile{\noexpand\item{}#1}} %now unnecessary
+%
+\def\listrefs{\footatend\vfill\supereject\immediate\closeout\rfile\writestoppt
+\baselineskip=14pt\centerline{{\bf References}}\bigskip{\frenchspacing%
+\parindent=20pt\escapechar=` \input refs.tmp\vfill\eject}\nonfrenchspacing}
+%
+\def\startrefs#1{\immediate\openout\rfile=refs.tmp\refno=#1}
+%
+\def\xref{\expandafter\xr@f}\def\xr@f[#1]{#1}
+\def\refs#1{\count255=1[\r@fs #1{\hbox{}}]}
+\def\r@fs#1{\ifx\und@fined#1\message{reflabel \string#1 is undefined.}%
+\nref#1{need to supply reference \string#1.}\fi%
+\vphantom{\hphantom{#1}}\edef\next{#1}\ifx\next\em@rk\def\next{}%
+\else\ifx\next#1\ifodd\count255\relax\xref#1\count255=0\fi%
+\else#1\count255=1\fi\let\next=\r@fs\fi\next}
+%
+\def\figures{\centerline{{\bf Figure Captions}}\medskip\parindent=40pt%
+\def\fig##1##2{\medskip\item{Fig.~##1. }##2}}
+%
+% this is ugly, but moore insists
+\newwrite\ffile\global\newcount\figno \global\figno=1
+%
+\def\fig{fig.~\the\figno\nfig}
+\def\nfig#1{\xdef#1{fig.~\the\figno}%
+\writedef{#1\leftbracket fig.\noexpand~\the\figno}%
+\ifnum\figno=1\immediate\openout\ffile=figs.tmp\fi\chardef\wfile=\ffile%
+\immediate\write\ffile{\noexpand\medskip\noexpand\item{Fig.\ \the\figno. }
+\reflabeL{#1\hskip.55in}\pctsign}\global\advance\figno by1\findarg}
+%
+\def\listfigs{\vfill\eject\immediate\closeout\ffile{\parindent40pt
+\baselineskip14pt\centerline{{\bf Figure Captions}}\nobreak\medskip
+\escapechar=` \input figs.tmp\vfill\eject}}
+%
+\def\xfig{\expandafter\xf@g}\def\xf@g fig.\penalty\@M\ {}
+\def\figs#1{figs.~\f@gs #1{\hbox{}}}
+\def\f@gs#1{\edef\next{#1}\ifx\next\em@rk\def\next{}\else
+\ifx\next#1\xfig #1\else#1\fi\let\next=\f@gs\fi\next}
+%
+\newwrite\lfile
+{\escapechar-1\xdef\pctsign{\string\%}\xdef\leftbracket{\string\{}
+\xdef\rightbracket{\string\}}\xdef\numbersign{\string\#}}
+\def\writedefs{\immediate\openout\lfile=labeldefs.tmp \def\writedef##1{%
+\immediate\write\lfile{\string\def\string##1\rightbracket}}}
+%
+\def\writestop{\def\writestoppt{\immediate\write\lfile{\string\pageno%
+\the\pageno\string\startrefs\leftbracket\the\refno\rightbracket%
+\string\def\string\secsym\leftbracket\secsym\rightbracket%
+\string\secno\the\secno\string\meqno\the\meqno}\immediate\closeout\lfile}}
+%
+\def\writestoppt{}\def\writedef#1{}
+%
+\def\seclab#1{\xdef #1{\the\secno}\writedef{#1\leftbracket#1}\wrlabeL{#1=#1}}
+\def\subseclab#1{\xdef #1{\secsym\the\subsecno}%
+\writedef{#1\leftbracket#1}\wrlabeL{#1=#1}}
+%
+\newwrite\tfile \def\writetoca#1{}
+\def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill}
+% use this to write file with table of contents
+\def\writetoc{\immediate\openout\tfile=toc.tmp
+ \def\writetoca##1{{\edef\next{\write\tfile{\noindent ##1
+ \string\leaderfill {\noexpand\number\pageno} \par}}\next}}}
+% and this lists table of contents on second pass
+\def\listtoc{\centerline{\bf Contents}\nobreak\medskip{\baselineskip=12pt
+ \parskip=0pt\catcode`\@=11 \input toc.tex \catcode`\@=12 \bigbreak\bigskip}}
+%
+\catcode`\@=12 % at signs are no longer letters
+%
+% Unpleasantness in calling in abstract and title fonts
+\edef\tfontsize{\ifx\answ\bigans scaled\magstep3\else scaled\magstep4\fi}
+\font\titlerm=cmr10 \tfontsize \font\titlerms=cmr7 \tfontsize
+\font\titlermss=cmr5 \tfontsize \font\titlei=cmmi10 \tfontsize
+\font\titleis=cmmi7 \tfontsize \font\titleiss=cmmi5 \tfontsize
+\font\titlesy=cmsy10 \tfontsize \font\titlesys=cmsy7 \tfontsize
+\font\titlesyss=cmsy5 \tfontsize \font\titleit=cmti10 \tfontsize
+\skewchar\titlei='177 \skewchar\titleis='177 \skewchar\titleiss='177
+\skewchar\titlesy='60 \skewchar\titlesys='60 \skewchar\titlesyss='60
+%
+\def\titlefont{\def\rm{\fam0\titlerm}% switch to title font
+\textfont0=\titlerm \scriptfont0=\titlerms \scriptscriptfont0=\titlermss
+\textfont1=\titlei \scriptfont1=\titleis \scriptscriptfont1=\titleiss
+\textfont2=\titlesy \scriptfont2=\titlesys \scriptscriptfont2=\titlesyss
+\textfont\itfam=\titleit \def\it{\fam\itfam\titleit}\rm}
+%
+\font\authorfont=cmcsc10 \ifx\answ\bigans\else scaled\magstep1\fi
+%
+\ifx\answ\bigans\def\abstractfont{\tenpoint}\else
+\font\abssl=cmsl10 scaled \magstep1
+\font\absrm=cmr10 scaled\magstep1 \font\absrms=cmr7 scaled\magstep1
+\font\absrmss=cmr5 scaled\magstep1 \font\absi=cmmi10 scaled\magstep1
+\font\absis=cmmi7 scaled\magstep1 \font\absiss=cmmi5 scaled\magstep1
+\font\abssy=cmsy10 scaled\magstep1 \font\abssys=cmsy7 scaled\magstep1
+\font\abssyss=cmsy5 scaled\magstep1 \font\absbf=cmbx10 scaled\magstep1
+\skewchar\absi='177 \skewchar\absis='177 \skewchar\absiss='177
+\skewchar\abssy='60 \skewchar\abssys='60 \skewchar\abssyss='60
+%
+\def\abstractfont{\def\rm{\fam0\absrm}% switch to abstract font
+\textfont0=\absrm \scriptfont0=\absrms \scriptscriptfont0=\absrmss
+\textfont1=\absi \scriptfont1=\absis \scriptscriptfont1=\absiss
+\textfont2=\abssy \scriptfont2=\abssys \scriptscriptfont2=\abssyss
+\textfont\itfam=\bigit \def\it{\fam\itfam\bigit}\def\footnotefont{\tenpoint}%
+\textfont\slfam=\abssl \def\sl{\fam\slfam\abssl}%
+\textfont\bffam=\absbf \def\bf{\fam\bffam\absbf}\rm}\fi
+%
+\def\tenpoint{\def\rm{\fam0\tenrm}% switch back to 10-point type
+\textfont0=\tenrm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm
+\textfont1=\teni \scriptfont1=\seveni \scriptscriptfont1=\fivei
+\textfont2=\tensy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy
+\textfont\itfam=\tenit \def\it{\fam\itfam\tenit}\def\footnotefont{\ninepoint}%
+\textfont\bffam=\tenbf \def\bf{\fam\bffam\tenbf}\def\sl{\fam\slfam\tensl}\rm}
+%
+\font\ninerm=cmr9 \font\sixrm=cmr6 \font\ninei=cmmi9 \font\sixi=cmmi6
+\font\ninesy=cmsy9 \font\sixsy=cmsy6 \font\ninebf=cmbx9
+\font\nineit=cmti9 \font\ninesl=cmsl9 \skewchar\ninei='177
+\skewchar\sixi='177 \skewchar\ninesy='60 \skewchar\sixsy='60
+%
+\def\ninepoint{\def\rm{\fam0\ninerm}% switch to footnote font
+\textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
+\textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei
+\textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
+\textfont\itfam=\ninei \def\it{\fam\itfam\nineit}\def\sl{\fam\slfam\ninesl}%
+\textfont\bffam=\ninebf \def\bf{\fam\bffam\ninebf}\rm}
+%
+%---------------------------------------------------------------------
+%
+\def\noblackbox{\overfullrule=0pt}
+\hyphenation{anom-aly anom-alies coun-ter-term coun-ter-terms}
+%
+\def\inv{^{\raise.15ex\hbox{${\scriptscriptstyle -}$}\kern-.05em 1}}
+\def\dup{^{\vphantom{1}}}
+\def\Dsl{\,\raise.15ex\hbox{/}\mkern-13.5mu D} %this one can be subscripted
+\def\dsl{\raise.15ex\hbox{/}\kern-.57em\partial}
+\def\del{\partial}
+\def\Psl{\dsl}
+\def\tr{{\rm tr}} \def\Tr{{\rm Tr}}
+\font\bigit=cmti10 scaled \magstep1
+\def\biglie{\hbox{\bigit\$}} %pound sterling
+\def\lspace{\ifx\answ\bigans{}\else\qquad\fi}
+\def\lbspace{\ifx\answ\bigans{}\else\hskip-.2in\fi} % $$\lbspace...$$
+\def\boxeqn#1{\vcenter{\vbox{\hrule\hbox{\vrule\kern3pt\vbox{\kern3pt
+ \hbox{${\displaystyle #1}$}\kern3pt}\kern3pt\vrule}\hrule}}}
+\def\mbox#1#2{\vcenter{\hrule \hbox{\vrule height#2in
+ \kern#1in \vrule} \hrule}} %e.g. \mbox{.1}{.1}
+% matters of taste
+%\def\tilde{\widetilde} \def\bar{\overline} \def\hat{\widehat}
+%
+% some sample definitions
+\def\CAG{{\cal A/\cal G}} \def\CO{{\cal O}} % curly letters
+\def\CA{{\cal A}} \def\CC{{\cal C}} \def\CF{{\cal F}} \def\CG{{\cal G}}
+\def\CL{{\cal L}} \def\CH{{\cal H}} \def\CI{{\cal I}} \def\CU{{\cal U}}
+\def\CB{{\cal B}} \def\CR{{\cal R}} \def\CD{{\cal D}} \def\CT{{\cal T}}
+\def\e#1{{\rm e}^{^{\textstyle#1}}}
+\def\grad#1{\,\nabla\!_{{#1}}\,}
+\def\gradgrad#1#2{\,\nabla\!_{{#1}}\nabla\!_{{#2}}\,}
+\def\ph{\varphi}
+\def\psibar{\overline\psi}
+\def\om#1#2{\omega^{#1}{}_{#2}}
+\def\vev#1{\langle #1 \rangle}
+\def\lform{\hbox{$\sqcup$}\llap{\hbox{$\sqcap$}}}
+\def\darr#1{\raise1.5ex\hbox{$\leftrightarrow$}\mkern-16.5mu #1}
+\def\lie{\hbox{\it\$}} %pound sterling
+\def\ha{{1\over2}}
+\def\half{{\textstyle{1\over2}}} %puts a small half in a displayed eqn
+\def\roughly#1{\raise.3ex\hbox{$#1$\kern-.75em\lower1ex\hbox{$\sim$}}}
diff --git a/macros/generic/occam/PlainEx2/harvtst.doc b/macros/generic/occam/PlainEx2/harvtst.doc
new file mode 100644
index 0000000000..db5134f2f3
--- /dev/null
+++ b/macros/generic/occam/PlainEx2/harvtst.doc
@@ -0,0 +1,170 @@
+
+
+ %%%%% harvtst.doc
+
+
+HARVTST.DIR CONTENTS AND DOCUMENTATION:
+
+harvmac.occ 20K
+harvmac.tex 20K
+harvtst.doc 2K
+W.sty 6K
+W.tex 68K
+witten94.tex 68K
+
+Other tools needed (from Occam)
+
+auditor.tex 18K
+defstrip
+occam.spc
+
+This is a sample application of the Occam utility.
+
+A) INPUT files were
+ --- harvmac.tex Paul Ginsparg's preprint macro package based on
+Plain TeX and designed for physics preprints.
+ --- witten.tex a preprint of Edward Witten, posted in November 1994
+on Ginsparg's preprint server tp://xxx.lanl.gov.
+
+B) OUTPUT files were:
+ --- W.sty essentially a subset of those macros in harvmac.tex
+that are needed in witten.tex
+ --- W.tex essentially witten.tex with unused macros eliminated
+from the header; there is no change at all in the body of the article.
+
+C) TOOL files (from the Occam distribution) were:
+--- auditor.tex TeX utility
+--- "defstrip" TeX utility currently a Mac QUEDM macro package.
+--- harvmac.occ a version of harvmac.tex in Occam format as
+described in occam.spc and below. This file would normally be
+supplied by the macro file's author, since it should be carefully
+prepared, and can serve all users of harvmac.tex
+
+ The aim is to to get (C) from (A) + (B). What good is that?
+--- If a publisher wants to use macros from harvmac.tex
+but is not familiar with that package, then use of W.sty
+will save him time.
+--- W.sty is so simple and small (10% of the whole or less than
+average sales tax in the 1990's) that it can be archived with
+W.tex; then future changes to harvmac.tex will not cause
+disruption.
+
+ Here is the transformation recipe.
+
+ --- Make a copy WW.sty of harvmac.occ and append the macro
+header of witten94.tex to it. Save witten94.tex as W.tex after
+replacing its header by
+
+ ------------
+
+ %Paper: ftp://xxx.lanl.gov//hep-th/9411102
+ %From: WITTEN@sns.ias.edu
+ %Date: 14 Nov 1994 22:09:04 -0400 (EDT)
+
+ \input WW.sty %% unsimplified style file
+ %\input W.sty %% simplified style file
+ %% input one of two
+
+ \Title{hep-th/9411102, IASSNS-HEP-94-96}
+ etc.
+
+ ------------------------
+
+ --- append to WW.sty format, the header macros of
+witten94.tex at the end, formatting as as occam.spc
+indicates, namely as follows:
+
+ ------------------------
+
+\Def\underarrow#1{\vbox{\ialign{##\crcr$\hfil\displaystyle
+ {#1}\hfil$\crcr\noalign{\kern1pt
+ \nointerlineskip}$\longrightarrow$\crcr}}}%_
+ % use of underarrow
+ %A~~~\underarrow{a}~~~B
+ \newcount\figno \figno=0
+
+ \Def\fig#1#2#3{%
+ \par\begingroup\parindent=0pt\leftskip=1cm
+ \rightskip=1cm\parindent=0pt
+ \baselineskip=11pt
+ \global\advance\figno by 1
+ \midinsert
+ \epsfxsize=#3
+ \centerline{\epsfbox{#2}}%
+ \vskip 12pt
+ {\bf Fig. \the\figno:} #1\par
+ \endinsert\endgroup\par
+ }%_
+
+ \Def\figlabel#1{\xdef#1{\the\figno}}%_
+ \Def\encadremath#1{\vbox{\hrule\hbox{\vrule\kern8pt\vbox{\kern8pt
+ \hbox{$\displaystyle #1$}\kern8pt}
+ \kern8pt\vrule}\hrule}}%_
+
+ \overfullrule=0pt
+
+ %macros
+ %
+ \Def\tilde{\widetilde}%_
+ \Def\bar{\overline}%_
+ \Def\np#1#2#3{Nucl. Phys. {\bf B#1} (#2) #3}%_
+ \Def\pl#1#2#3{Phys. Lett. {\bf #1B} (#2) #3}%_
+ \Def\prl#1#2#3{Phys. Rev. Lett.{\bf #1} (#2) #3}%_
+ \Def\physrev#1#2#3{Phys. Rev. {\bf D#1} (#2) #3}%_
+ \Def\ap#1#2#3{Ann. Phys. {\bf #1} (#2) #3}%_
+ \Def\prep#1#2#3{Phys. Rep. {\bf #1} (#2) #3}%_
+ \Def\rmp#1#2#3{Rev. Mod. Phys. {\bf #1} (#2) #3}%_
+ \Def\cmp#1#2#3{Comm. Math. Phys. {\bf #1} (#2) #3}%_
+ \Def\mpl#1#2#3{Mod. Phys. Lett. {\bf #1} (#2) #3}%_
+ \Def\Lam#1{\Lambda_{#1}}%_
+ \Def\pf{{\rm Pf ~}}%_
+ \Font\zfont = cmss10 %scaled \magstep1 %_
+ \Font\litfont = cmr6 %_
+ \Font\fvfont=cmr5 %_
+ \Def\bigone{\hbox{1\kern -.23em {\rm l}}}%_
+ \Def\ZZ{\hbox{\zfont Z\kern-.4emZ}}%_
+ \Def\half{{\litfont {1 \over 2}}}%_
+ \Def\mx#1{m_{\hbox{\fvfont #1}}}%_
+ \Def\gx#1{g_{\hbox{\fvfont #1}}}%_
+ \Def\gG{{\cal G}}%_
+ \Def\lamlam#1{\langle S_{#1}\rangle}%_
+ \Def\CM{{\cal M}}%_
+ \Def\Re{{\rm Re ~}}%_
+ \Def\Im{{\rm Im ~}}%_
+ \Def\lfm#1{\medskip\noindent\item{#1}}%_
+
+ ------------------------
+
+Headers like this usually contain a high proportion unused macros.
+This example is no exception.
+
+ --- compose under Plain TeX the file W.tex. Note that the file
+auditor.tex is called to examine which macros of WW.sty
+are really used.
+
+ --- "defstrip" the file WW.sty in the presence of the resulting list
+of macros audit.lst. (The "defstrip" macros currently operate under
+QEDM; ultimately "defstrip" will be a TeX utility and TeX will replace
+QEDM.) The result is a simplified macro file; call it output1. Put
+output1 back through "defstrip" once to get output2. The second pass
+gains an extra 10% of compactness and perfect tidyness; it uses
+the file audit.lst as modified by the first pass; beware that
+audit.lst is changing slightly as some * become # but one can
+always get back to the original by changing all # to *. Save the final
+output as W.sty. In general, the number of passes needed is the number
+of stars on the final macro \ITERATIONCOUNTER in audit.lst.
+
+
+ --- change the header of W.tex to input W.sty rather than WW.sty:
+
+ %\input WW.sty
+ \input W.sty
+
+Then compose the file W.tex under Plain TeX, checking that the
+printed result is unchanged.
+
+ --- the (tiny!) shrunken form of the header material from
+witten94.tex can now be restored to the header of W.tex; it is indeed
+best to separate the Ginsparg macros from IAS macros.
+
+
diff --git a/macros/generic/occam/PlainEx2/witten94.tex b/macros/generic/occam/PlainEx2/witten94.tex
new file mode 100644
index 0000000000..dbfc175bcd
--- /dev/null
+++ b/macros/generic/occam/PlainEx2/witten94.tex
@@ -0,0 +1,1402 @@
+%Paper: ftp://xxx.lanl.gov//hep-th/9411102
+%From: WITTEN@sns.ias.edu
+%Date: 14 Nov 1994 22:09:04 -0400 (EDT)
+
+
+\input harvmac
+\def\underarrow#1{\vbox{\ialign{##\crcr$\hfil\displaystyle
+{#1}\hfil$\crcr\noalign{\kern1pt
+\nointerlineskip}$\longrightarrow$\crcr}}}
+% use of underarrow
+%A~~~\underarrow{a}~~~B
+\newcount\figno
+\figno=0
+\def\fig#1#2#3{
+\par\begingroup\parindent=0pt\leftskip=1cm\rightskip=1cm\parindent=0pt
+\baselineskip=11pt
+\global\advance\figno by 1
+\midinsert
+\epsfxsize=#3
+\centerline{\epsfbox{#2}}
+\vskip 12pt
+{\bf Fig. \the\figno:} #1\par
+\endinsert\endgroup\par
+}
+\def\figlabel#1{\xdef#1{\the\figno}}
+\def\encadremath#1{\vbox{\hrule\hbox{\vrule\kern8pt\vbox{\kern8pt
+\hbox{$\displaystyle #1$}\kern8pt}
+\kern8pt\vrule}\hrule}}
+
+\overfullrule=0pt
+
+%macros
+%
+\def\tilde{\widetilde}
+\def\bar{\overline}
+\def\np#1#2#3{Nucl. Phys. {\bf B#1} (#2) #3}
+\def\pl#1#2#3{Phys. Lett. {\bf #1B} (#2) #3}
+\def\prl#1#2#3{Phys. Rev. Lett.{\bf #1} (#2) #3}
+\def\physrev#1#2#3{Phys. Rev. {\bf D#1} (#2) #3}
+\def\ap#1#2#3{Ann. Phys. {\bf #1} (#2) #3}
+\def\prep#1#2#3{Phys. Rep. {\bf #1} (#2) #3}
+\def\rmp#1#2#3{Rev. Mod. Phys. {\bf #1} (#2) #3}
+\def\cmp#1#2#3{Comm. Math. Phys. {\bf #1} (#2) #3}
+\def\mpl#1#2#3{Mod. Phys. Lett. {\bf #1} (#2) #3}
+%
+\def\Lam#1{\Lambda_{#1}}
+\def\pf{{\rm Pf ~}}
+\font\zfont = cmss10 %scaled \magstep1
+\font\litfont = cmr6
+\font\fvfont=cmr5
+\def\bigone{\hbox{1\kern -.23em {\rm l}}}
+\def\ZZ{\hbox{\zfont Z\kern-.4emZ}}
+\def\half{{\litfont {1 \over 2}}}
+\def\mx#1{m_{\hbox{\fvfont #1}}}
+\def\gx#1{g_{\hbox{\fvfont #1}}}
+\def\gG{{\cal G}}
+\def\lamlam#1{\langle S_{#1}\rangle}
+\def\CM{{\cal M}}
+\def\Re{{\rm Re ~}}
+\def\Im{{\rm Im ~}}
+\def\lfm#1{\medskip\noindent\item{#1}}
+
+\Title{hep-th/9411102, IASSNS-HEP-94-96}
+{\vbox{\centerline{MONOPOLES AND FOUR-MANIFOLDS}}}
+\smallskip
+\centerline{Edward Witten}
+\smallskip
+\centerline{\it School of Natural Sciences, Institute for Advanced Study}
+\centerline{\it Olden Lane, Princeton, NJ 08540, USA}\bigskip
+\baselineskip 18pt
+
+\medskip
+
+\noindent
+%write abstract here
+Recent developments in the understanding of $N=2$ supersymmetric Yang-Mills
+theory in four dimensions suggest a new point of view about Donaldson
+theory of four manifolds: instead of defining four-manifold invariants
+by counting $SU(2)$ instantons, one can define equivalent four-manifold
+invariants by counting solutions of a non-linear equation with an
+abelian gauge group. This is a ``dual'' equation in which the gauge
+group is the dual of the maximal torus of $SU(2)$.
+The new viewpoint suggests many new results about
+the Donaldson invariants.
+\Date{November, 1994}
+%text of paper
+
+\newsec{Introduction}
+\nref\witten{E. Witten, ``Topological Quantum Field Theory,'' Commun. Math.
+Phys. {\bf 117} (1988) 353.}
+For some years now it has been known that Donaldson theory is equivalent
+to a quantum field theory, in fact, a twisted version of $N=2$ supersymmetric
+Yang-Mills theory \witten.
+The question therefore arises of whether this viewpoint
+is actually useful for computing Donaldson invariants \ref\doninv{S. Donaldson,
+``Polynomial Invariants For Smooth Four-Manifolds,'' Topology, {\bf 29}
+(1990) 257.} or understanding
+their properties.
+
+\nref\floer{A. Floer, ``An Instanton Invariant For 3-Manifolds,''
+Commun. Math. Phys. {\bf 118} 215.}
+One standard physical technique is to cut and sum over
+physical states. In the context of Donaldson theory, such methods
+have been extensively developed by mathematicians,
+starting with the work of Floer \floer.
+So far, despite substantial efforts,
+the physical reformulation has not given any essentially new insight
+about these methods.
+
+Another approach to using physics to illuminate Donaldson theory
+starts with the fact that the $N=2$ gauge theory is
+asymptotically free; therefore, it is weakly coupled in the ultraviolet
+and strongly coupled in the infrared. Since the Donaldson invariants
+-- that is, the correlation functions of the twisted theory -- are metric
+independent, they can be computed in the ultraviolet or the infrared,
+as one wishes. Indeed, the weak coupling in the
+ultraviolet is used to show that the quantum field theory correlation
+functions do coincide with the Donaldson invariants.
+
+\nref\newwitten{E. Witten, ``Supersymmetric Yang-Mills Theory On A
+Four-Manifold,'' J. Math. Phys. {\bf 35} (1994) 5101.}
+If one could
+understand the infrared behavior of the $N=2$ theory, one might get
+a quite different description and, perhaps, a quite different way to
+compute the Donaldson invariants. Until recently, this line of thought was
+rather hypothetical for general four-manifolds
+since the infrared behavior of $N=2$ super Yang-Mills
+theory in the strong coupling region was unknown.
+Previous work along these lines was therefore limited to K\"ahler manifolds,
+where one can reduce the discussion to the $N=1$ theory, whose infrared
+behavior was known. This led to an almost complete determination \newwitten\
+of the Donaldson invariants of K\"ahler manifolds with $H^{2,0}\not= 0$.
+
+\nref\sw{N. Seiberg and E. Witten, ``Electric-Magnetic Duality,
+Monopole Condensation, And Confinement In $N=2$ Supersymmetric
+Yang-Mills Theory,'' Nucl. Phys. {\bf B426} (1994) 19,
+``Monopoles, Duality, And Chiral Symmetry
+Breaking In $N=2$ Supersymmetric QCD,'' hep-th/9408099,
+to appear in Nucl. Phys. B.}
+\nref\seiberg{N. Seiberg, ``The Power Of Holomorphy -- Exact Results
+In $4d$ SUSY Field Theories,'' hep-th/9408013.}
+The purpose of the present paper is to exploit recent work by Seiberg
+and the author
+\sw\ in which the infrared behavior of the $N=2$ theory was
+determined using methods somewhat akin to methods that have shed light
+on various $N=1$ theories (for a survey see \seiberg).
+The answer turned out to be quite surprising: the
+infrared limit of the $N=2$ theory in the ``strongly coupled'' region
+of field space is equivalent to a weakly coupled theory of abelian
+gauge fields coupled to ``monopoles.'' The monopole theory is
+dual to the original theory in the sense that, for instance, the
+gauge group is the dual of the maximal torus of the original gauge group.
+
+
+\nref\km{P. Kronheimer and T. Mrowka, ``Recurrence Relations And
+Asymptotics For Four-Manifold Invariants,'' Bull. Am. Math. Soc. {\bf 30}
+(1994) 215, ``Embedded Surfaces And The Structure Of Donaldson's
+Polynomial Invariants,'' preprint (1994).}
+\nref\arg{P. Argyres and A. Faraggi, ``The Vacuum Structure And Spectrum
+Of $N=2$ Supersymmetric $SU(N)$ Gauge Theory,'' hep-th/9411057.}
+\nref\yank{A. Klemm, W. Lerche, S. Yankielowicz, and S. Theisen,
+``Simple Singularities and $N=2$ Supersymmetric Yang-Mills Theory,''
+hep-th/9411048.}
+Since the dual theory is weakly coupled in the infrared,
+everything is computable in that region, and
+in particular for gauge group $SU(2)$,
+one does get an alternative formulation of the usual Donaldson
+invariants. Instead of computing the Donaldson invariants by counting
+$SU(2)$ instanton solutions, one can obtain the same invariants
+by counting the solutions of the dual equations, which involve
+$U(1)$ gauge fields and monopoles.\foot{In this paper, we only consider
+Donaldson theory with
+gauge group $SU(2)$ or $SO(3)$, but an analogous dual description
+by abelian gauge fields and monopoles will hold for
+any compact Lie group, the gauge group of the dual theory being always
+the dual of the maximal torus of the original gauge group. For example,
+most of the results needed to write the precise monopole equations for
+$SU(N)$ have been obtained recently \refs{\arg,\yank}.}
+
+This formulation makes manifest various properties of the Donaldson
+invariants. For instance, one can get new proofs of some of the
+classic results of Donaldson theory; one gets
+a new description of the basic classes of Kronheimer and Mrowka \km, and some
+new results about them; one gets a new understanding
+of the ``simple type'' condition for four-manifolds;
+one finds new types of vanishing theorems
+that severely limit the behavior of Donaldson theory on manifolds
+that admit a metric of positive scalar curvature;
+and one gets a complete determination of the Donaldson invariants
+of K\"ahler manifolds with $H^{2,0}\not= 0$, eliminating the
+assumptions made in \newwitten\ about the canonical divisor.
+
+It should be possible to justify
+directly the claims sketched in this paper about the consequences
+of the monopole equations even if the relation to Donaldson theory
+is difficult to prove. The reformulation may make the problems
+look quite different
+as the gauge group is abelian and the most relevant moduli spaces
+are zero dimensional.
+{}From a physical point of view the dual description via monopoles and abelian
+gauge fields should be simpler than the microscopic $SU(2)$ description
+since in the renormalization group sense
+it arises by ``integrating out the irrelevant degrees of freedom.''
+
+
+\nref\vw{C. Vafa and E. Witten, ``A Strong Coupling Test Of $S$-Duality,''
+hep-th/9408074, to appear in Nucl. Phys. B.}
+\nref\om{C. Montonen and D. Olive, Phys. Lett. {\bf B72} (1977) 117;
+P. Goddard, J. Nuyts, and D. Olive, Nucl. Phys. {\bf B125} (1977) 1.}
+\nref\sen{A. Sen, ``Strong-Weak Coupling Duality In Four Dimensional
+String Theory,'' hep-th/9402002.}
+The monopole equations are close cousins of equations studied in section two
+of \vw; the reason for the analogy is that in each case one is studying
+$N=2$ theories of hypermultiplets coupled to vector multiplets.
+The investigation in \vw\ dealt with microscopic Montonen-Olive duality
+\refs{\om,\sen}, while the duality in Donaldson theory \sw\ is a sort of
+phenomenological analog of this.
+
+
+The monopole equations, definition of four-manifold invariants,
+and relation to Donaldson theory are stated
+in section two of this paper. Vanishing theorems are used in section three
+to deduce some basic properties. Invariants
+of K\"ahler
+manifolds are computed in section four.
+A very brief sketch of the origin in physics
+is in section five. A fuller account of the contents of section five
+will appear elsewhere \ref\nsw{N. Seiberg and E. Witten, to appear.}.
+
+
+
+
+\newsec{The Monopole Equations}
+
+Let $X$ be an oriented, closed four-manifold on which we pick a Riemannian
+structure with metric tensor $g$.
+$\Lambda^pT^*X$, or simply $\Lambda^p$,
+will denote the bundle of real-valued $p$-forms,
+and $\Lambda^{2,\pm}$ will be the sub-bundle of $\Lambda^2$ consisting
+of self-dual or anti-self-dual forms.
+
+The monopole equations relevant to $SU(2)$ or $SO(3)$
+Donaldson theory can be described
+as follows. If $w_2(X)=0$,
+then $X$ is a spin manifold and one can pick
+positive and negative spin bundles
+$S^+$ and $S^-$, of rank two. (If there is more than one spin structure,
+the choice of a spin structure will not matter as we will ultimately
+sum over twistings by line bundles.) In that case, introduce a complex line
+bundle $L$; the data in the monopole equation will be a connection $A$
+on $L$ and a section $M$ of $S^+\otimes L$. The curvature two-form
+of $A$ will
+be called $F$ or $F(A)$; its self-dual and anti-self-dual
+projections will be called $F^+$ and $F^-$.
+
+If $X$ is not spin, the $S^{\pm}$ do not exist,
+but their projectivizations ${\bf P}S^{\pm}$ do exist (as bundles with fiber
+isomorphic to ${\bf CP}^1$). A ${\rm Spin}_c$ structure (which exists
+on any oriented four-manifold \ref\hirz{F. Hirzebruch and H. Hopf,
+``Felder von Flachenelementen in 4-dimensionalen Mannigfaltigkeiten,''
+Math. Annalen {\bf 136} (1958) 156.})
+can be described as a choice of
+a rank two complex vector bundle, which we write as $S^+\otimes L$,
+whose projectivization is isomorphic to ${\bf P}S^+$. In this situation, $L$
+does not exist as a line bundle, but $L^2$ does\foot{One might
+be tempted to call this bundle $L$ and write the ${\rm Spin}_c$
+bundle as $S^+\otimes L^{1/2}$; that amounts to assigning magnetic
+charge $1/2$ to the monopole and seems unnatural physically.};
+the motivation for
+writing the ${\rm Spin}_c$ bundle as $S^+\otimes L$ is that the tensor
+powers of this bundle obey isomorphisms suggested by the notation.
+For instance, $(S^+\otimes L)^{\otimes 2}\cong L^2\otimes(\Lambda^0\oplus
+\Lambda^{2,+})$.
+The data of the monopole equation
+are now a section $M$ of $S^+\otimes L$ and a connection on $S^+\otimes L$
+that projects to the Riemannian connection on ${\bf P}S^+$. The symbol
+$F(A)$ will now denote $1/2$ the trace of the curvature form of $S^+\otimes L$.
+
+Since $L^2$ is an ordinary line bundle, one has an integral
+cohomology class
+$x=-c_1(L^2)\in H^2(X,{\bf Z})$. (The minus sign makes some
+later formulas come out in a standard form.) Note that $x$ reduces modulo two
+to $w_2(X)$; in particular, if $w_2(X)=0$, then $L$ exists as a line
+bundle and $x=-2c_1(L)$.
+
+To write the monopole equations, recall that $S^+$ is symplectic or
+pseudo-real, so that
+if $M$ is a section of $S^+\otimes L$, then the complex conjugate $\bar M$
+is a section of $S^+\otimes L^{-1}$. The product $M\otimes \bar M$
+would naturally lie in $(S^+\otimes L)\otimes (S^+\otimes L^{-1})\cong
+\Lambda^0\oplus\Lambda^{2,+}$.
+$F^{+}$ also takes values in $\Lambda^{2,+}$ making it possible to
+write the following equations.
+Introduce Clifford matrices $\Gamma_i$
+(with anticommutators $\{\Gamma_i,\Gamma_j\}=2g_{ij}$), and
+set $\Gamma_{ij}={1\over 2}[\Gamma_i,\Gamma_j]$. Then
+the equations are\foot{
+To physicists the connection form $A$ on a unitary line bundle is
+real; the covariant derivative is $d_A=d+iA$ and the curvature is
+$F=dA$ or in components $F_{ij}=\partial_iA_j-\partial_jA_i$.}
+\eqn\noneq{\eqalign{F^+_{ij}&=-{i\over 2}\bar M\Gamma_{ij}M \cr
+ \sum_i\Gamma^iD_iM & = 0.\cr}}
+In the second equation, $\sum_i\Gamma^iD_i$ is the Dirac operator
+$D$ that maps sections of $S^+\otimes L$ to sections of $S^-\otimes L$.
+We will sometimes abbreviate the first as $F^+=(M\bar M)^+$.
+Alternatively,
+if positive spinor indices are written $A,B,C$, and
+negative spinor indices as $A',B',C'$,
+\foot{Spinor indices are raised and lowered using the invariant
+tensor in $\Lambda^2 S^+$. In components, if $M^A=(M^1,M^2)$,
+then $M_A= (-M_2,M_1)$. One uses the same operation in interpreting
+$\bar M$ as a section of $S^+\otimes L$, so $\bar M^A=(\bar M^2,-\bar M^1)$.
+Also $F_{AB}={1\over 4}F_{ij}\Gamma^{ij}_{AB}$.}
+the equations can be written
+\eqn\indeq{\eqalign{F_{AB}& = {i\over 2}\left(M_A\bar M_B+M_B\bar M_A\right)\cr
+ D_{AA'}M^A & = 0.\cr}}
+
+As a first step in understanding these equations, let us work out
+the virtual dimension of the moduli space ${\cal M}$
+of solutions of the
+equations up to gauge transformation.
+The linearization of the monopole equations fits into
+an elliptic complex
+\eqn\pindeq{0\to \Lambda^0\underarrow{s}\Lambda^1
+\oplus (S^+\otimes L)\underarrow{t}\Lambda^{2,+}
+\oplus (S^-\otimes L) \to 0.}
+Here $t$ is the linearization of the monopole equations, and $s$
+is the map from zero forms to deformations in $A,M$ given by the infinitesimal
+action of the gauge group. Since we wish to work with real operators
+and determine the real dimension
+of ${\cal M}$, we temporarily think of $S^\pm\otimes L$ as
+real vector bundles (of rank four).
+Then an elliptic operator
+\eqn\pxxx{T:\Lambda^1\oplus(S^+\otimes L)\to \Lambda^0\oplus \Lambda^{2,+}
+\oplus (S^-\otimes L)}
+is defined by $T=s^*\oplus t$.
+The virtual dimension of the moduli space is given by the index of $T$.
+By dropping terms in $T$ of order zero,
+$T$ can be deformed to the direct sum of the operator $d+d^*$
+\foot{What is meant here is of course a projection of the $d+d^*$ operator
+to self-dual forms.}
+from $\Lambda^1$ to $\Lambda^0\oplus \Lambda^{2,+}$ and the Dirac
+operator from $S^+\otimes L$ to $S^-\otimes L$.
+The index of $T$ is
+the index of $d+d^*$ plus twice what is usually called the index of the Dirac
+operator; the factor of two comes from looking at $S^{\pm}\otimes L$
+as real bundles of twice the dimension.
+Let $\chi$ and $\sigma$ be the Euler
+characteristic and signature of $X$. Then the index of $d+d^*$ is
+$-(\chi+\sigma)/2$, while twice the Dirac index is $-\sigma/4+c_1(L)^2$.
+The virtual dimension of the moduli space is the sum of these or
+\eqn\hurf{W= -{2\chi+3\sigma\over 4} +c_1(L)^2.}
+
+When this number is negative, there are generically no solutions of
+the monopole equations. When $W=0$, that is, when $x=-c_1(L^2)=-2c_1(L)$ obeys
+\eqn\burf{x^2=2\chi+3\sigma,}
+then the virtual dimension is zero and the moduli space generically
+consists of a finite set of points $P_{i,x}$, $i=1\dots t_x$.
+With each such point, one can associate
+a sign $\epsilon_{i,x}=\pm 1$ -- the sign of the determinant of $T$ as we
+discuss momentarily.
+Once this is done, define for each $x$ obeying \burf\ an integer $n_x$ by
+\eqn\gurofo{n_x=\sum_i\epsilon_{i,x}.}
+We will see later that
+$n_x=0$ -- indeed, the moduli space is empty -- for all but finitely many $x$.
+Under certain conditions that we will discuss in a moment, the $n_x$
+are topological invariants.
+
+Note that $W=0$ if and only if the index of the Dirac operator
+is
+\eqn\inxxon{\Delta={\chi+\sigma\over 4}.}
+In particular, $\Delta$ must be an integer to have non-trivial $n_x$.
+Similarly, $\Delta$ must be integral for the Donaldson invariants
+to be non-trivial (otherwise $SU(2)$ instanton moduli space is odd
+dimensional).
+
+
+For the sign of the determinant of $T$ to make sense one must trivialize
+the determinant line of $T$. This can be done by deforming $T$ as above
+to the direct sum of $d+d^*$ and the Dirac operator. If the Dirac operator,
+which naturally has a non-trivial {\it complex} determinant line, is regarded
+as a real operator, then its determinant line is naturally trivial -- as a
+complex line has a natural orientation. The $d+d^*$ operator is
+independent of $A$ and $M$ (as the gauge group is abelian), and its
+deterinant line is trivialized once and for all by picking an orientation
+of $H^1(X,{\bf R})\oplus H^{2,+}(X,{\bf R})$. Note that this is the
+same data needed by Donaldson
+\ref\donor{S. Donaldson, ``The Orientation Of Yang-Mills Moduli
+Spaces And Four-Manifold Topology,'' J. Diff. Geom. {\bf 26} (1987) 397.}
+to orient instanton moduli spaces for $SU(2)$;
+this is an aspect of the relation between the two theories.
+
+If one replaces $L$ by $L^{-1}$, $A$ by $-A$, and $M$ by $\bar M$, the
+monopole equations are invariant, but the trivialization of the
+determinant line is multiplied by $(-1)^\delta$ with $\delta$ the Dirac
+index. Hence the invariants for $L$ and $L^{-1}$ are related by
+\eqn\pixxx{n_{-x}=(-1)^\Delta n_x.}
+
+For $W<0$, the moduli space is generically empty. For $W>0$ one can
+try, as in Donaldson theory, to define topological invariants that involve
+integration over the moduli space. Donaldson theory does not detect those
+invariants at least in known situations.
+We will see in section three that even when $W>0$, the
+moduli space is empty for almost all $x$.
+
+
+\bigskip
+\noindent{\it Topological Invariance}
+
+In general, the number of solutions
+of a system of equations
+weighted by the sign of the determinant of the operator analogous to $T$
+ is always a topological invariant if a suitable compactness
+holds.
+If as in the case at hand one has a gauge invariant system of equations, and
+one wishes to count gauge orbits of solutions up to gauge transformations,
+then one requires (i) compactness; and (ii) free action
+of the gauge group on the space of solutions.
+
+
+Compactness fails if a field or its derivatives can go to
+infinity.
+The Weitzenbock formula used in section three to discuss vanishing
+theorems indicates that these phenomena
+do not occur for the monopole equations.
+To explain the contrast with Donaldson theory, note that
+for $SU(2)$ instantons
+compactness fails precisely
+\ref\uhl{K. Uhlenbeck, ``Removable Singularities In Yang-Mills Fields,''
+Commun. Math. Phys. {\bf 83} (1982) 11.}
+because an instanton can shrink to zero size. This is
+possible because (i) the equations are conformally invariant, (ii) they
+have non-trivial solutions on a flat ${\bf R}^4$, and (iii) embedding
+such a solution, scaled to very small size,
+on any four-manifold gives a highly localized approximate
+solution of the instanton equations (which can sometimes
+\ref\taubes{C. H. Taubes, ``Self-Dual Yang-Mills Connections Over
+Non-Self-Dual 4-Manifolds,'' J. Diff. Geom. {\bf 19} (1982) 517.}
+be perturbed to
+an exact solution). The monopole equations by contrast
+are scale invariant but
+(as follows immediately from the Weitzenbock formula) they have
+no non-constant $L^2$ solutions on flat ${\bf R}^4$ (or after dimensional
+reduction on flat ${\bf R}^n$ with $1\leq n \leq 3$).
+So there is no analog for the monopole equations of the phenomenon
+where an instanton shrinks to zero size.
+
+On the other hand, an obstruction does arise, just as in Donaldson
+theory (in what follows we imitate some arguments in
+\ref\dono{S. Donaldson, ``Irrationality And The $h$-Cobordism
+Conjecture,'' J. Diff. Geom. {\bf 26} (1987) 141.})
+from the question of whether the gauge group acts freely on the
+space of solutions of the monopole equations. The only way for the gauge
+group to fail to act freely is that there might be a solution with $M=0$,
+in which case a constant gauge transformation acts trivially.
+A solution with $M=0$ necessarily has $F^+=0$, that is, it is an abelian
+instanton.
+
+Since $F/2\pi$ represents the first Chern class of the line bundle $L$,
+it is integral; in particular if $F^+=0$ then $F/2\pi$ lies in the intersection
+of the integral lattice in $H^2(X,{\bf R})$ with the anti-self-dual subspace
+$H^{2,-}(X,{\bf R})$.
+As long as $b_2^+\geq 1$, so that the self-dual part of $H^2(X,{\bf R})$ is
+non-empty, the intersection of the anti-self-dual part and the integral
+lattice generically consists only of the zero vector.
+ In this case,
+for a generic metric on $X$, there are no abelian instantons (except for
+$x=0$, which we momentarily exclude) and $n_x $ is well-defined.
+
+
+To show that the $n_x$ are topological invariants, one must further show
+that any two generic metrics on $X$ can be joined by a path along which
+there is never an abelian instanton. As in Donaldson theory, this can
+fail if $b_2^+=1$. In that case, the self-dual part
+of $H^2(X,{\bf R})$ is one dimensional, and in a generic
+one parameter family of metrics on $X$, one may meet a metric for
+which there is an abelian instanton. When this occurs, the $n_x$ can jump.
+Let us analyze how this happens, assuming for simplicity that $b_1=0$.
+Given $b_1=0$ and
+$b_2{}^+=1$, one has $W=0$ precisely if the index of the Dirac
+equation is 1. Therefore, there is generically a single solution $M_0$
+of the Dirac equation $DM=0$.
+
+The equation $F^+(A)=0$ cannot be obeyed for a generic metric on $X$,
+but we want to look at the behavior near a special metric for which it does
+have a solution.
+ Consider a one parameter family of metrics parametrized
+by a real parameter $\epsilon$, such that at $\epsilon=0$ the
+self-dual subspace in $H^2(X,{\bf R})$ crosses a ``wall''
+and a solution $A_0$ of
+$F^+(A)=0$ appears. Hence for $\epsilon=0$, we can solve the monopole
+equations with $A=A_0, \,M=0$. Let us see what happens to this solution
+when $\epsilon $ is very small but non-zero. We set $M=mM_0$, with $m$
+a complex number, to obey $DM=0$, and we write $A=A_0+\epsilon \delta A$.
+The equation $F^+(A)-(M\bar M)^+=0$ becomes
+\eqn\nurk{F^+(A_0)+(d\delta A)^+-|m|^2 (M_0\bar M_0)^+=0.}
+As the cokernel of $A\to F^+(A)$
+is one dimensional, $\delta A$ can be chosen
+to project the left hand side of equation \nurk\ into a one dimensional
+subspace. (As $b_1=0$, this can be done in a unique way up to a gauge
+transformation.)
+The remaining equation looks near $\epsilon=0$ like
+\eqn\modlik{c \epsilon - \,m\bar m=0}
+with $c$ a constant.
+The $\epsilon$ term on the left comes from the fact that $F^+(A_0)$ is
+proportional to $\epsilon$.
+
+Now we can see what happens for $\epsilon\not= 0$ to the solution that
+at $\epsilon=0$ has $A=A_0$, $M=M_0$.
+Depending on the sign of $c$,
+there is a solution for $m$, uniquely
+determined up to gauge transformation, for $\epsilon>0$ and no solution
+for $\epsilon<0$, or vice-versa. Therefore $n_x$ jumps by $\pm 1$, depending
+on the sign of $c$,
+in passing through $\epsilon=0$.
+To compare this precisely to the similar behavior of Donaldson
+theory, one would also need to understand the
+role of the $u$ plane, discussed in section five.
+
+The trivial abelian instanton with $x=0$ is an exception to the
+above discussion,
+since it cannot be removed by perturbing the metric. To define $n_0$,
+perturb the equation $F_{AB}={i\over 2}(M_A\bar M_B+M_B\bar M_A)$
+to
+\eqn\hinnoc{F_{AB}={i\over 2}(M_A\bar M_B+M_B\bar M_A)-p_{AB},}
+with $p$ a self-dual
+harmonic two-form; with this perturbation, the gauge group acts
+freely on the solution space.
+Then define $n_0$ as the number of gauge orbits of solutions of the
+perturbed equations
+weighted by sign in the usual way. This is invariant under continuous
+deformations of $p$ for $p\not=0$;
+as long as $b_2^+>1$, so that
+the space of possible $p$'s is connected, the integer $n_0$ defined
+this way is a topological invariant.
+
+The perturbation just
+pointed out will be used later in the case that $p$ is the real part
+of a holomorphic two-form to compute the invariants of K\"ahler manifolds
+with $b_2^+>1$. It probably has other applications; for instance, the
+case that $p$ is a symplectic form is of interest.
+
+\bigskip
+\noindent{\it Relation To Donaldson Theory}
+
+With an appropriate restriction on $b_2^+$, the $n_x$ have
+(by an argument sketched in section five) a relation to the Donaldson
+invariants that will now be stated.
+
+Let us recall that in $SU(2)$ Donaldson theory, one wishes to compute
+the integrals or expectation values of certain cohomology classes
+or quantum field operators: for every Riemann surface
+(or more generally every
+two-dimensional homology cycle) $\Sigma$ in $X$, one has an operator
+$I(\Sigma)$ of dimension (or $R$ charge or
+ghost number) two; there is one additional
+operator ${\cal O}$, of dimension four.
+For every value of the instanton number, one computes the expectation value
+of a suitable product of these operators by integration over instanton
+moduli space using a recipe due to Donaldson, or by evaluating a suitable
+quantum field theory correlation function as in \witten.
+It is natural to organize this data in the form of a generating function
+\eqn\jurn{\left\langle
+\exp\left(\sum_a\alpha_aI(\Sigma_a)+\lambda {\cal O}\right)
+\right\rangle,}
+summed over instanton numbers; here
+the $\Sigma_a$ range over a basis of $H_2(X,{\bf R})$ and $\lambda,
+\,\alpha_a$ are complex numbers.
+
+Let $v= \sum_a\alpha_a[\Sigma_a]$,
+with $[\Sigma_a]$ the cohomology class that is Poincar\'e dual to $\Sigma_a$.
+So for instance $v^2=\sum_{a,b}\alpha_a\alpha_b\,\,\Sigma_a\cdot\Sigma_b$
+(here $\Sigma_a\cdot \Sigma_b$ is the intersection number of $\Sigma_a$ and
+$\Sigma_b$), and for any $x\in H^2(X,{\bf Z})$, $v\cdot x=\sum_a\alpha_a
+(\Sigma_a,x)$. Let as before $\Delta=(\chi+\sigma)/4$.
+
+A four-manifold is said to be of simple type if the generating function
+in \jurn\ is annihilated by $\partial^2/\partial\lambda^2-4$; all known
+simply-connected four-manifolds with $b_2^+>1$ have this property.
+The relation of the simple type condition to physics is discussed in
+section five.
+I claim that for manifolds of simple type
+\eqn\jimmo{\eqalign{\left
+\langle\exp\left(\sum_a\alpha_aI(\Sigma_a)+\lambda {\cal O}\right)
+\right\rangle = 2^{1+{1\over 4}(7\chi+11\sigma)}&\left(\exp\left(
+{v^2\over 2}+2\lambda\right)
+\sum_x
+n_x e^{v\cdot x}\right.\cr&\left.
+ +i^{\Delta} \exp\left(-{v^2\over 2}-2\lambda\right)\sum_xn_x
+e^{-iv\cdot x}\right).\cr}}
+That the expression is real follows from \pixxx.
+
+As sketched in section five, this formula is a sort of corollary of the
+analysis of $N=2$ supersymmetric Yang-Mills theory in \sw. Here I will
+just make a few remarks:
+
+(1) The structure in \jimmo\ agrees with the general form
+proved by Kronheimer and Mrowka \km.
+The classes $x\in H^2(X,{\bf Z})$ such that $n_x\not= 0$ are the basic
+classes in their terminology. From the properties by which $x$ and $n_x$
+were defined, we have that $x$ is congruent to $w_2(X)$ modulo 2 and
+that $x^2=2\chi+3\sigma$. The first assertion is a result of Kronheimer
+and Mrowka and the second was conjectured by them.
+
+
+
+(2) The prefactor $2^{1+{1\over 4}(7\chi+11\sigma)}$ has the following
+origin, as in \newwitten. One factor of two comes because, even though the
+center of $SU(2)$ acts trivially on the $SU(2)$ instanton moduli space,
+the Donaldson invariants are usually defined without dividing by two.
+The remaining factor of $2^{{1\over 4}(7\chi+11\sigma)}$ is a $c$-number
+renormalization factor that arises in comparing the microscopic $SU(2)$
+theory to the dual description with monopoles.
+(In \nsw\ a more general function of the form
+$e^{a(u)\chi+b(u)\sigma}$ that arises on the complex $u$ plane will be
+calculated.) Some coefficients in the formula such as the $7/4$ and $11/4$
+were fixed in \newwitten\ to agree with
+calculations of special cases of Donaldson invariants.
+
+
+(3) Most fundamentally, in the above formula, the first term, that is
+\eqn\kdn{\exp\left({v^2\over 2}+2\lambda\right)
+\sum_xn_x e^{v\cdot x},}
+is the contribution from one vacuum at $u=\Lambda^2$, and the second
+term,
+\eqn\hkn{i^\Delta \exp\left(-{v^2\over 2}-2\lambda\right)\sum_xn_x
+e^{-iv\cdot x},}
+is the contribution of a second vacuum at $u=-\Lambda^2$.
+These terms are analogous to the two terms in equation (2.66) of \newwitten.
+The factor of $i^\Delta$ arises, as there, because
+of a global anomaly in the discrete symmetry that exchanges the two
+vacua. This factor
+can be written in the form $e^{a\chi+b\sigma}$ and so means
+that the two vacua have different values
+of the renormalization mentioned in the last paragraph. The
+replacement of $e^{v\cdot x}$ in the first vacuum by $e^{-iv\cdot x}$
+in the second is likewise determined by the symmetries, as in \newwitten,
+and can be seen microscopically.
+For a general simple compact gauge group, the analogous sum will have
+$h$ terms ($h$ the dual Coxeter number) associated with $h$ vacua.
+
+(4) This formula generalizes as follows for the case that the
+gauge group is $SO(3)$ rather than $SU(2)$. Consider an
+$SO(3)$ bundle $E$ with, say, $ w_2(E)=z$.
+Define a generating functional of correlation functions
+summed over bundles with
+ all values of the first Pontryagin class
+but fixed $w_2$. Pick an integral lift of $w_2(X)$, and, using
+the fact that the $x$'s are congruent to $w_2(X)$ mod two, let $x'$ be
+such that $2x'=x+ w_2(X)$. Then $w_2(E)\not= 0$ modifies
+the derivation of \jimmo\ only by certain minus signs that
+appear in the duality transformation that relates the microscopic
+and macroscopic descriptions; the result is
+\eqn\himmo{
+\eqalign{
+\left\langle\exp\left(\sum_a\alpha_aI(\Sigma_a)+\lambda {\cal O}\right)
+\right\rangle_z
+=& 2^{1+{1\over 4}(7\chi+11\sigma)}\left(\exp({v^2\over 2}+2\lambda)
+\sum_x(-1)^{x'\cdot z}
+n_x e^{v\cdot x} \right.\cr &\left.
++i^{\Delta-z^2} \exp(-{v^2\over 2}-2\lambda)\sum_x
+(-1)^{x'\cdot z
+}n_xe^{-iv\cdot x}\right).\cr}}
+The replacement of $i^\Delta$ by $i^{\Delta-z^2}$ arises, as in equation
+(2.79) of \newwitten\ (where $w_2(E)$ is written as $x$),
+because the global anomaly has an extra term that depends on $z$.
+(Note that as $z$ is defined modulo two, $z^2$ is well-defined modulo four.)
+The factor of $(-1)^{x'\cdot z}$ was obtained in \km\ for manifolds
+of simple type and in \newwitten\ for K\"ahler manifolds.
+If the integral lift of $w_2(X)$ used in defining $x'$
+is shifted by $w_2(X)\to w_2(X)+2y$, then \himmo\
+is multiplied by $(-1)^{y\cdot z}$. The reason for this factor
+is that \himmo\ is reproducing the conventional Donaldson invariants,
+whose sign depends on the orientation of the instanton moduli spaces.
+A natural orientation \donor\ depends on an integral lift of $w_2(X)$
+and transforms as \himmo\ does if this lift is changed.
+
+(5) For K\"ahler manifolds with $b_2^+>1$, the quantities entering in \jimmo\
+will be completely computed in section four.
+We will find that, letting $\eta$ be a holomorphic two-form, the sum in \jimmo\
+can be interpreted as a sum over factorizations $\eta=\alpha\beta$
+with $\alpha$ and $\beta$ holomorphic sections of $K^{1/2}\otimes L^{\pm 1}$.
+Each such factorization contributes $\pm 1$ to $n_x$ with
+$x=-2c_1(L)$ provided $x^2=c_1(K)^2$; the contribution is $+1$ or $-1$
+according to a formula computed at the end of section four.
+
+\bigskip
+\noindent{\it Imitating Arguments From Donaldson Theory}
+
+Apart from relating Donaldson theory to the monopole equations,
+one can simply try to adapt familiar arguments about Donaldson theory
+to the monopole equations. We have already seen some examples.
+
+As another example, consider Donaldson's theorem \doninv\
+asserting that the Donaldson invariants vanish for a connected sum $X\# Y$
+of four-manifolds $X$ and $Y$ which each have $b_2{}^+>0$. The theorem
+is proved by considering a metric on $X\# Y$
+in which $X$ and $Y$ are joined by
+a long neck of the form ${\bf S}^3\times I$, with $I$ an interval in ${\bf R}$.
+Take the metric on the neck to be the product of the standard metric
+on ${\bf S}^3$ and a metric that assigns length $t$ to $I$, and consider
+the monopole equations on this space. For $t\to \infty$, any solution
+of the monopole equations will vanish in the neck because of the positive
+scalar curvature of ${\bf S}^3$ (this follows from the Weitzenbock
+formula of the next section).
+This lets one define a $U(1)$ action on the moduli space ${\cal M}$
+(analogous to the $SO(3)$ action used by Donaldson)
+by gauge transforming the solutions on $Y$ by a constant gauge transformation,
+leaving fixed the data on $X$. A fixed point of this $U(1)$ action
+would be a solution for which $M$ vanishes on $X$ or on $Y$. But
+as $X$ and $Y$ both have $b_2{}^+>0$, there is no such solution if
+generic metrics are used on the two sides. A zero dimensional moduli
+space with a free $U(1)$ action is empty, so the basic invariants would
+be zero for such connected sums. (A free $U(1)$ action also leads
+to vanishing of the higher invariants.)
+Since we will see in section four
+that the invariants are non-zero for K\"ahler manifolds
+(analogous to another basic result of Donaldson), one gets a proof
+directly from the monopole equations and independent of the equivalence to
+Donaldson theory that algebraic surfaces do not have connected
+sum decompositions with $b_2^+>0$ on both sides.
+
+If one considers instead a
+situation with $b_2^+$ positive for $X$ but zero for $Y$, there will
+be fixed points consisting of solutions with $M=0$ on $Y$, and one will get
+a formula expressing invariants of $X\# Y$ in terms
+of invariants of $X$ and elementary data concerning $Y$.
+
+\newsec{Vanishing Theorems}
+
+Some of the main properties of the monopole equations
+can be
+understood by means of vanishing theorems. The general strategy in
+deriving such vanishing theorems is quite standard, but as in section two
+of \vw, some unusual cancellations (required by the Lorentz invariance
+of the underlying untwisted theory) lead to unusually strong results.
+
+If we set $s=F^+-M\bar M$, $k=DM$,
+then a small calculation gives
+\eqn\highor{\eqalign{\int_Xd^4x\sqrt g\left({1\over 2}|s|^2+|k|^2\right)
+=\int_Xd^4x\sqrt g&\left({1\over 2}|F^+|^2+g^{ij}D_iM^AD_j\bar M_A \right.
+\cr & \left.+{1\over 2}|M|^4
++{1\over 4}R|M|^2\right) .\cr}} Here $g$ is the metric of $X$, $R$ the scalar
+curvature, and $d^4x\sqrt g$ the Riemannian measure.
+A salient feature here is that a term $F_{AB}M^A\bar M{}^B$, which appears
+in either $|s|^2 $ or $|k|^2$, cancels in the sum.
+This sharpens the implications of the formula, as we will see.
+One can also consider the effect here of the perturbation in \hinnoc;
+the sole effect of this is to replace
+${1\over 2}|M|^4$ in \highor\ by
+\eqn\bihor{\int_Xd^4x\sqrt g\left(
+F^+\wedge p+\sum_{A,B}\left|{1\over 2}(M_A\bar M_B+M_B\bar M_A)-p_{AB}\right|^2
+\right).}
+The second term is non-negative, and the first is simply the intersection
+pairing
+\eqn\juhor{2\pi c_1(L)\cdot [p].}
+
+An obvious inference from \highor\ is that if $X$ admits a metric
+whose scalar curvature is positive,
+then for such a metric any solution
+of the monopole equations must have $M=0$ and $F^+=0$. But
+if $b_2{}^+>0$, then after a generic small perturbation of the metric
+(which will preserve the fact that the scalar curvature is positive),
+there are no abelian solutions of $F^+=0$ except flat connections.
+Therefore,
+for such manifolds and metrics, a solution of the monopole equations
+is a flat connection with $M=0$. These too can be eliminated
+using the perturbation in \hinnoc.\foot{
+Flat connections can only arise if $c_1(L)$ is torsion; in that case,
+$c_1(L)\cdot [p]=0$. The vanishing argument
+therefore goes through, the modification in \highor\ being that which
+is indicated in \bihor.}
+Hence a four-manifold
+for which $b_2^+>0$ and $n_x\not= 0$ for some $x$
+does not admit a metric of
+positive scalar curvature.
+
+We can extend this to determine the possible four-manifolds $X$ with $b_2^+>0$,
+some $n_x\not= 0$, and a metric of {\it non-negative}
+scalar curvature.\foot{If $b_2^+=1$, the $n_x$ are not all topological
+invariants, and we interpret the hypothesis to mean that with at least
+one sign of the perturbation in \hinnoc, the $n_x$ are not all zero.}
+If $X$ obeys those conditions, then for any metric of $R\geq 0$,
+any basic class $x$ is in $H^{2,-}$ modulo torsion
+(so that $L$ admits a connection
+with $F^+=0$, enabling \highor\ to vanish);
+in particular if $x$ is not torsion then $x^2<0$.
+Now consider the effect of the perturbation \hinnoc. As $x\in H^{2,-}$,
+\juhor\ vanishes; hence if $R\geq 0$, $R$ must
+be zero, $M$
+must be covariantly constant and $(M\bar M)^+=p$ (from \bihor).
+For $ M$ covariantly constant,
+$(M\bar M)^+=p$ implies
+that $p$ is covariantly constant also; but for all $p\in H^{2,+}$
+to be covariantly constant implies that $X$ is K\"ahler with $b_2^+=1$
+or is hyper-K\"ahler. Hyper-K\"ahler metrics certainly have $R=0$,
+and there are examples of metrics with $R=0$
+on K\"ahler manifolds with $b_2^+=1$ \ref\lebrun{C. LeBrun, ``Scalar-Flat
+K\"ahler Metrics On Blown-Up Ruled Surfaces,'' J. Reine Angew
+Math. {\bf 420} (1991) 161.}.
+
+As an example,
+for a K\"ahler manifold with $b_2^+\geq 3$, the canonical divisor
+$K$ always arises as a basic class, as we will see in section four, so
+except in the hyper-K\"ahler case,
+such manifolds do not admit a metric of non-negative
+scalar curvature.
+
+Even if the scalar curvature is not positive, we can get an explicit
+bound from \highor\ showing that there are only finitely many basic classes.
+Since
+\eqn\gegor{\int_Xd^4x\sqrt g\left({1\over 2}|M|^4+{1\over 4}R|M|^2\right)
+\geq -{1\over 32}\int_Xd^4x\sqrt g R^2,}
+it follows from \highor, even if we throw away the term $|D_iM|^2$,
+that
+\eqn\egor{\int_Xd^4x\sqrt g |F^+|^2\leq {1\over 16}\int_Xd^4x\sqrt g R^2.}
+On the other hand, basic classes correspond to line bundles
+$L$ with $c_1(L)^2=(2\chi+3\sigma)/4$, or
+\eqn\negor{{1\over (2\pi)^2}\int d^4x\sqrt g\left(|F^+|^2-|F^-|^2\right)
+ ={2\chi+3\sigma\over 4}.}
+Therefore, for a basic class both $I^+=\int d^4x\sqrt g |F^+|^2$
+and $I^-=\int d^4x\sqrt g |F^-|^2$ are bounded. For a given metric,
+there are only finitely
+many isomorphism classes of line bundles
+admitting connections with given bounds on both $I^+$ and $I^-$, so
+the set of basic classes is finite. This is a result
+proved by Kronheimer and Mrowka with their definition of the basic classes.
+
+The basic classes correspond, as indicated in section three,
+to line bundles on which
+the moduli space of solutions of the monopole equations is of zero virtual
+dimension.
+We can analyze in a similar way components of the moduli space of positive
+dimension. Line bundles $L$ such that $c_1(L)^2<(2\chi+3\sigma)/4$ are not
+of much interest in that connection, since for such line bundles the
+moduli space has negative virtual dimension and is generically empty.
+But if $c_1(L)^2>(2\chi+3\sigma)/4$, then \negor\ is simply replaced by
+the stronger bound
+\eqn\unegor{{1\over (2\pi)^2}\int d^4x\sqrt g\left(|F^+|^2-|F^-|^2\right)
+ >{2\chi+3\sigma\over 4}.}
+The set of isomorphism classes of line bundles admitting a connection
+obeying this inequality as well as \egor\ is once again finite.
+So we conclude that for any given metric on $X$, the set of isomorphism
+classes of line bundles for which
+the moduli space is non-empty and of non-negative virtual dimension
+is finite; for a generic metric on $X$, there are only finitely many
+non-empty components of the moduli space.
+
+For further consequences of \highor, we turn to a basic case in the study of
+four-manifolds: the case that $X$ is K\"ahler.
+
+\newsec{Computation On K\"ahler Manifolds}
+
+If $X$ is K\"ahler and spin, then $S^+\otimes L$ has a decomposition
+$S^+\otimes L\cong (K^{1/2}\otimes L)\oplus (K^{-1/2}\otimes L)$,
+where $K$ is the canonical bundle and $K^{1/2}$ is a square root.
+If $X$ is K\"ahler but not spin, then $S^+\otimes L$, defined as before,
+has a similar decomposition where now $K^{1/2}$ and $L$ are not defined
+separately and $K^{1/2}\otimes L$ is characterized
+as a square root of the line bundle $K\otimes L^2$.
+
+We denote the components of $M$ in $K^{1/2}\otimes L$ and
+in $K^{-1/2}\otimes L$ as $\alpha$ and $-i\bar \beta$, respectively.
+The equation $F^+(A)=M\bar M$ can now be decomposed
+\eqn\juffy{\eqalign{F^{2,0} & = \alpha\beta \cr
+ F_\omega^{1,1} & =-{\omega\over 2}
+ \left(|\alpha|^2-|\beta|^2\right)\cr
+ F^{0,2} & =\bar\alpha\bar\beta.\cr}}
+Here $\omega$ is the K\"ahler form and $F_\omega^{1,1}$ is the $(1,1)$
+part of $F^+$.
+\highor\ can be rewritten
+\eqn\nohighor{\eqalign{\int_Xd^4x\sqrt g\left({1\over 2}|s|^2+|k|^2\right)
+=\int_Xd^4x\sqrt g & \left({1\over 2}|F^+|^2
++g^{ij}D_i\bar\alpha D_j\alpha+g^{ij}D_i\bar\beta
+D_j\beta\right.\cr & \left. +{1\over 2}(|\alpha|^2+|\beta|^2)^2
++{1\over 4}R(|\alpha|^2+|\beta|^2)\right) .\cr}}
+
+The right hand side of \nohighor\ is not manifestly non-negative (unless
+$R\geq 0$), but the fact that it is equal to the left hand side shows that
+it is non-negative and zero precisely for solutions of the monopole
+equations. Consider the operation
+\eqn\pohighor{\eqalign{A & \to A\cr
+ \alpha & \to \alpha \cr
+ \beta & \to -\beta.\cr}}
+This is not a symmetry of the monopole equations. But it is a symmetry
+of the right hand side of \nohighor. Therefore, given a zero of the right
+hand side of \nohighor\ -- that is, a solution of the monopole equations --
+the operation \pohighor\ gives another zero of the right hand side of
+\nohighor\ -- that is, another solution of the monopole equations.
+So, though not a symmetry of the monopole equations, the transformation
+\pohighor\ maps solutions of those equations to other solutions.
+
+Given that any solution of \juffy\ is mapped to another solution by
+\pohighor, it follows that such a solution has
+\eqn\tohighor{0=F^{2,0}=F^{0,2}=\alpha\beta=\bar\alpha\bar\beta.}
+Vanishing of $F^{0,2}$ means that the connection $A$ defines a holomorphic
+structure on $L$.
+The basic classes (which are first Chern classes of $L$'s that are such that
+\juffy\ has a solution) are therefore of type $(1,1)$ for any K\"ahler
+structure
+on $X$, a severe constraint.
+
+
+Vanishing of $\alpha\beta$ means that $\alpha=0$
+or $\beta=0$. If $\alpha=0$, then the Dirac equation for $M$ reduces
+to
+\eqn\jipp{\bar\partial_A \beta=0,}
+where $\bar\partial_A$ is the $\bar\partial $ operator on $L$. Similarly,
+if $\beta=0$, then the Dirac equation gives
+\eqn\ipp{\bar\partial_A\alpha= 0.}
+
+Knowing that either $\alpha$ or $\beta$ is zero, we can deduce which it is.
+Integrating the $(1,1)$ part of \juffy\ gives
+\eqn\jippo{{1\over 2\pi}\int_X\omega\wedge F=-{1\over 4\pi}\int_X\omega\wedge
+\omega\left(|\alpha|^2-|\beta|^2\right).}
+The left hand side of \jippo\ is a topological invariant which can be
+interpreted as
+\eqn\ippo{J= [\omega]\cdot c_1(L).}
+The condition that there are no non-trivial abelian instantons is
+that $J$ is non-zero; we only wish to consider metrics for which this
+is so. If $J<0$, we must have $\alpha\not= 0$, $\beta=0$, and if
+$J>0$, we must have $\alpha=0$, $\beta\not= 0$.
+
+The equation that we have not considered so far is the $(1,1)$ part of \juffy.
+This equation can be interpreted
+as follows. Suppose for example that we are in the situation with $\beta=0$.
+The space of connections $A$
+and sections $\alpha$ of $K^{1/2}\otimes L$
+can be interpreted as a symplectic manifold,
+the symplectic structure being defined by
+\eqn\defby{\eqalign{\langle\delta_1A,\delta_2A\rangle & =\int_X\omega
+ \wedge \delta_1A\wedge\delta_2 A\cr
+ \langle \delta_1\alpha,\delta_2\alpha
+\rangle & =-i\int_X\omega\wedge\omega
+\left(\delta_1\overline \alpha\delta_2\alpha-\delta_2\bar \alpha
+\delta_1\alpha\right).\cr}}
+On this symplectic manifold acts the group of $U(1)$ gauge transformations.
+The moment map $\mu$ for this action is the quantity that appears in the
+$(1,1)$ equation that we have not yet exploited, that is
+\eqn\hefby{ \mu\omega= F_\omega^{1,1}+\omega|\alpha|^2.}
+By analogy with many similar problems, setting
+to zero the moment map and dividing by the group of $U(1)$ gauge
+transformations
+should be
+equivalent to dividing by the complexification of the group of gauge
+transformations.\foot{In such comparisons of symplectic and complex quotients,
+one usually needs a stability condition on the complex side.
+In the present case, this is the condition discussed in
+connection with \ippo.} In the present case, the complexification of the
+group of gauge transformations acts by $\alpha\to t\alpha$,
+$\bar\partial_A\to t\bar\partial_At^{-1}$, where $t$ is a map from
+$X$ to ${\bf C}^*$.
+
+Conjugation by $t$ has the effect of identifying any two $A$'s that
+define the same complex structure on $L$. This can be done almost
+uniquely: the ambiguity is that conjugation by a constant $t$ does
+not change $A$. Of course, a gauge transformation by
+a constant $t$ rescales
+$\alpha$ by a constant. The result therefore, for $J<0$, is that the moduli
+space of solutions of the monopole equations is the moduli space of
+pairs consisting of a complex structure on $L$ and a non-zero
+holomorphic section, defined
+up to scaling, of $K^{1/2}\otimes L$. For $J>0$, it is instead
+$\beta$ that is non-zero, and $K^{1/2}\otimes L$ is replaced by
+$K^{1/2}\otimes L^{-1}$.
+
+This result can be stated particularly nicely if $X$ has $b_1=0$.
+Then the complex structure on $L$, assuming that it exists, is unique.
+The moduli space of solutions of the monopole equations is
+therefore simply a complex projective space, ${\bf P}H^0(X,K^{1/2}\otimes L)$
+or ${\bf P}H^0(X,K^{1/2}\otimes L^{-1})$, depending on the sign of $J$.
+
+\bigskip
+\noindent{\it Identifying The Basic Classes}
+
+We would now like to identify the basic classes.
+The above description of the moduli space gives considerable information:
+basic classes are of the form $x=-2c_1(L)$, where $L$ is such that
+$J<0$ and $H^0(X,K^{1/2}\otimes L)$ is non-empty, or $J>0$
+and $H^0(X,K^{1/2}\otimes L^{-1})$ is non-empty. This, however,
+is not a sharp result.
+
+That is closely related to the fact that the moduli spaces ${\bf P}H^0(X,
+K^{1/2}\otimes L^{\pm 1})$ found above very frequently have a dimension
+bigger than the ``generic'' dimension of the moduli space as predicted
+by the index theorem. In fact, K\"ahler metrics
+are far from being generic. In case the expected dimension
+is zero, one would have always $n_x>0$ if the moduli spaces behaved
+``generically'' (given the complex orientation, an isolated point on the
+moduli space would always contribute $+1$ to $n_x$; this is a special
+case of a discussion below). Since the $n_x$
+are frequently negative (as in the examples of Kronheimer
+and Mrowka or equation (2.66) of \newwitten), moduli spaces of
+non-generic dimension must appear.
+
+When the moduli space has greater than the generically expected dimension,
+one can proceed by integrating over
+the bosonic and fermionic collective
+coordinates in the path integral. This gives a result that can be
+described topologically: letting $T$ be the operator that arises in linearizing
+the monopole equations, the cokernel of $T$ is a vector bundle $V$
+(the ``bundle of antighost zero modes'') over the moduli space ${\cal M}$;
+its Euler class integrated over ${\cal M}$ is the desired $n_x$.
+
+Alternatively, one can perturb the equations to more generic ones.
+We use the same perturbation as before.
+For a K\"ahler manifold $X$, the condition $b_2^+>1$ is equivalent
+to $H^{2,0}(X)\not= 0$, so we can pick a non-zero holomorphic two-form
+$\eta$.\foot{In \newwitten, where essentially the same perturbation
+was made, the two-form was called $\omega$, but
+here we reserve that name for the K\"ahler form.}
+We perturb the monopole equations \juffy\
+to
+\eqn\ojuffy{\eqalign{F^{2,0} & = \alpha\beta -\eta\cr
+ F_\omega^{1,1} & = -\omega\left(|\alpha|^2-|\beta|^2\right)\cr
+ F^{0,2} & =\bar\alpha\bar\beta-\bar\eta,\cr}}
+leaving unchanged the Dirac equation for $M$.
+
+It suffices to consider the case that the first Chern class of $L$
+is of type $(1,1)$, since the unperturbed moduli space vanishes otherwise.
+That being so, we have
+\eqn\ijuffy{0=\int_XF^{2,0}\wedge\bar\eta=\int_XF^{0,2}\wedge \eta.}
+Using this, one finds that \nohighor\ generalizes to
+\eqn\onohighor{\eqalign{\int_Xd^4x\sqrt g\left({1\over 2}|s|^2+|k|^2\right)
+=\int_Xd^4x &\left({1\over 2}
+|F^+|^2+g^{ij}D_i\bar\alpha D_j\alpha+g^{ij}D_i\bar\beta
+D_j\beta\right.\cr & \left.
+ +{1\over 2}(|\alpha|^2-|\beta|^2)^2+{2}|\alpha\beta-\eta|^2
++{R\over 4}(|\alpha|^2+|\beta|^2)\right) .\cr}}
+We can now make an argument of a sort we have already seen: the transformation
+\eqn\hoggy{\eqalign{A & \to A\cr
+ \alpha & \to \alpha \cr
+ \beta & \to -\beta \cr
+ \eta & \to -\eta, \cr}}
+though not a symmetry of \ojuffy, is a symmetry of the right hand side of
+\onohighor. As solutions of \ojuffy\ are the same as zeroes of the right
+hand side of \onohighor, we deduce that the solutions of \ojuffy\ with
+a two-form $\eta$ are transformed by \hoggy\ to the solutions with $-\eta$.
+The terms in \ojuffy\ even or odd under the transformation must therefore
+separately vanish, so
+any solution of \ojuffy\ has
+\eqn\goggy{0= F^{0,2}=F^{2,0}=\alpha\beta-\eta.}
+The condition $F^{0,2}=0$ means that the connection still defines
+a holomorphic structure on $L$.
+
+The condition
+\eqn\jipoggy{ \alpha\beta =\eta}
+gives our final criterion for determining the basic classes: they are
+of the form
+$x=-2c_1(L)$ where, for any choice of $\eta\in H^0(X,K)$, one has
+a factorization $\eta=\alpha\beta$ with
+holomorphic sections $\alpha$ and $\beta$ of $K^{1/2}\otimes L^{\pm 1}$,
+and $x^2=c_1(K)^2$.
+
+To make this completely explicit, suppose
+the divisor of $\eta$ is a union of irreducible
+components $C_i$ of multiplicity $r_i$.
+Thus the canonical divisor is
+\eqn\rufu{c_1(K)=\sum_ir_i[C_i],}
+where $[C_i]$ denotes the cohomology class that is
+Poincar\'e dual to the curve $C_i$.
+The existence of the factorization $\eta=\alpha\beta$
+means that the divisor of $K^{1/2}\otimes L$ is
+\eqn\jufu{c_1(K^{1/2}\otimes L)=\sum_is_i[C_i],}
+where $s_i$ are integers with $0\leq s_i\leq r_i$.
+The first Chern class of $L$ is therefore
+\eqn\tufu{c_1(L)=\sum_i(s_i-{1\over 2}r_i)[C_i].}
+And the basic classes are of the form $x=-2c_1(L)$ or
+\eqn\pufu{x=-\sum_i(2s_i-r_i)[C_i].}
+
+An $x$ of this form is is of type $(1,1)$ and congruent to $c_1(K)$
+modulo two, but may not obey $x^2=c_1(K)^2$.
+It is actually possible to prove using the Hodge index theorem
+that for $x$ as above, $x^2\leq c_1(K)^2$.\foot{Such an argument
+was pointed out by D. Morrison.} This is clear from the monopole
+equations: perturbed to $\eta\not=0$, these equations have
+ at most isolated solutions
+(from the isolated factorization $\eta=\alpha\beta$) and not a moduli
+space of solutions of positive dimension. So for K\"ahler manifolds,
+the non-empty perturbed moduli spaces are at most of dimension zero; invariants
+associated with monopole moduli spaces of higher dimension vanish.
+
+Our final conclusion about the basic classes, then, is that they
+are classes of the form \pufu\ such that $x^2=c_1(K)^2$.
+Each factorization $\eta=\alpha\beta$ contributes
+$\pm 1$ to $n_x$ with the corresponding $x$.
+Since several factorizations might give the same $x$, cancellations
+may be possible, making it possible to write the invariant in
+the Kronheimer-Mrowka form, with a shorter list of basic classes.
+Such cancellations can be effectively found since the signs of the various
+contributions are computed below. In any event the classes $x=\pm K$
+arise only from $s_i=0$ or $s_i=r_i$, respectively, and so always
+arise as basic classes with $n_x=\pm 1$.
+\foot{G. Tian and S.-T. Yau, P. Kronheimer and T. Mrowka, D. Morrison,
+and R. Friedman and J. Morgan pointed out that it actually follows
+from these conditions (or related arguments)
+that if $X$ is a minimal surface of general
+type, then the only basic classes are $\pm K$ (so that $K$ is a differentiable
+invariant up to sign). Indeed, according to Lemma 4 in \ref\kodaira{
+K. Kodaira, ``Pluricanonical Systems On Algebraic Surfaces Of General
+Type,'' J. Math. Soc. Japan {\bf 20} (1968) 170.}, on such a surface, if
+$K={\cal O}(C_1)\otimes {\cal O}(C_2)$
+with non-zero effective divisors $C_1,C_2$, then $C_1\cdot C_2>0$.
+This means that a factorization $\eta=\alpha\beta$ with $\alpha,\beta$
+sections of $K^{1/2}\otimes L^{\pm 1}$ and $x^2=c_1(K)^2$
+implies that $K^{1/2}\otimes L^{\pm 1}$ is trivial with one choice of sign,
+and hence that $x=\pm c_1(K)$.}
+
+\bigskip
+\noindent{\it Comparison To Previous Results}
+
+Let us compare these statements to previous results. The main case considered
+in \newwitten\ was that in which the $C_i$ were disjoint
+with multiplicities $r_i=1$. The allowed values of the $s_i$ are
+then $0$ and 1, so the basic classes are
+\eqn\ppufu{x_{\vec \rho}=\sum_i\rho_i[C_i],}
+with each $\rho_i=\pm 1$, as claimed in \newwitten.
+Notice that all of these classes have $x_{\vec\rho}^2=c_1(K)^2$.
+
+The most important case in which the $r_i$ are not all one is the case
+of an elliptic surface with multiple fibers. A fiber of multiplicity $n$
+appears in the canonical divisor with weight $r=n-1$. For elliptic
+surfaces, one has $C_i\cdot C_j=0$ for all $i,j$,
+so the classes in \pufu\ actually do all have
+$x^2=c_1(K)^2=0$. The formulas of
+Kronheimer and Mrowka for the Donaldson invariants of these surfaces show
+that the basic classes, in their sense,
+are indeed the classes given in \pufu.
+
+
+
+
+\bigskip
+\noindent{\it Determination Of The Sign}
+
+To complete this story, we must compute, for each factorization,
+the sign of $\det T$.
+Let us first explain in an abstract setting the strategy that will be
+used.
+Suppose that $E$ and $F$ are real vector spaces of equal even dimension
+with given complex structures, and $T:E\to F$ is an invertible linear map that
+commutes with the complex structure. Then $\det T$ is naturally
+defined as an element of $\det F\otimes \det E^{-1}$.
+If $\det E$ and $\det F$ are trivialized using the complex orientations of $E$
+and $F$, then $\det T>0$ roughly because the complex structure gives a pairing
+of eigenvalues. If $T$ {\it reverses} the complex structures then
+the sign of $\det T$ is $(-1)^w$ with $w=\dim_{\bf C}E$. For instance,
+by reversing the complex structure of $E$ one could reduce to the case
+in which $T$ preserves the complex structures, but reversing the complex
+structure of $E$ multiplies its orientation by $(-1)^w$.
+
+One can combine the two cases as follows. Suppose that $T$ preserves
+the complex structures but is not invertible. Let $T':E\to F$
+be a map that reverses the complex structures and maps ${\rm ker}\,T$
+invertibly to $F/T(E)$. Then for small real $\epsilon$ (of any sign)
+the sign of $\det(T\oplus \epsilon T')$ is $(-1)^w$ where now
+$w={\rm dim}_{\bf C}{\rm ker}\,T$. The same formula holds if
+$U$ and $V$ are vector bundles,
+$E=\Gamma(U)$, $F=\Gamma(V)$, $T:E\to F$
+is an elliptic operator with zero index, $T'$ is a sufficiently mild
+perturbation, and $\det (T+\epsilon T')$ is
+understood as the Ray-Singer-Quillen
+determinant.
+
+Our problem is of this form with $T$ understood as the linearization
+of the monopole equations at $\eta=0$ and $T'$ as the correction
+proportional to $\eta$ (which enters the linearization because of the shift
+it induces in $\alpha$ or $\beta$).
+As in \pxxx, one has $U=\Lambda^1\oplus (S^+\otimes L)$,
+with $S^+\otimes L$ now regarded as a real vector bundle of rank four.
+If $J<0$ (so $\beta=0$ for $\eta=0$), then
+give $U$ a complex structure that acts naturally on
+$S^+\otimes L$ and multiplies $\Lambda^{0,1}$ and $\Lambda^{1,0}$ by
+$i$ and $-i$, respectively. Likewise
+give $V=\Lambda^0\oplus\Lambda^{2,+}\oplus (S^-\otimes L)$
+a complex structure that acts naturally on $S^-\otimes L$; multiplies
+$\Lambda^{0,2}$ and $\Lambda^{2,0}$ by $i$ and $-i$; and exchanges the
+$(1,1)$ part of $\Lambda^{2,+}$ with $\Lambda^0$.
+Then $T$ preserves the
+complex structures on these bundles and $T'$ reverses them.
+
+The sign of the contribution to $n_x$ from a factorization $\eta=\alpha\beta$
+is therefore $(-1)^w$ with $w={\rm dim}_{\bf C}{\rm \ker}\,T$.
+The kernel of $T$ can be described as follows. There is an exact
+sequence
+\eqn\immo{0\to {\cal O}\underarrow{\alpha}K^{1/2}\otimes L\to R\to 0,}
+with some sheaf $R$. The kernel of $T$ has the same dimension as
+$H^0(X,R)$, as explained below. So the sign of the contribution to $n_x$
+is
+\eqn\uddu{(-1)^{{\rm dim}\,H^0(X,R)}.}
+If instead $J>0$, so the unperturbed solution has $\alpha=0$, $\beta\not=0$,
+then first of all we reverse the complex structures on $S^\pm\otimes L$;
+this multiplies the determinant by $(-1)^\Delta$ where $\Delta=-\sigma/8
++c_1(L)^2/2=(\chi+\sigma)/4$ is the Dirac index. The rest of
+the discussion goes through with
+\immo\ replaced by
+\eqn\dimmo{0\to {\cal O}\underarrow{\beta}K^{1/2}\otimes L^{-1}
+\to \tilde R\to 0,}
+so the sign is
+\eqn\duddu{(-1)^{\Delta+{\rm dim}\,H^0(X,\tilde R)}.}
+(It can be verified using the classification of surfaces that \uddu\
+and \duddu\ are equal.)
+With these signs, \jimmo\ becomes completely explicit: the sum in \jimmo\ is
+a sum over factorizations $\eta=\alpha\beta$; each such factorization
+determines a class $x$ and contributes to $n_x$ an amount $\pm 1$
+as just determined.
+
+Before justifying the claim about $\ker T$, let us
+check that the sign just determined agrees with what has been computed
+by other methods. Suppose as in \newwitten\ that the divisor of $\eta$
+is a union of disjoint smooth curves $C_i$. Then $R$ is a sum of
+sheaves $R_i$ supported on $C_i$; $R_i$ is trivial if $s_i$
+(defined in \jufu) is 0 and is isomorphic to a spin bundle of $C_i$
+(determined by $\eta $ and independent of the factorization $\eta=\alpha\beta$)
+if $s_i=1$.
+Let $t_i=1$ if this spin bundle is even, that is, if ${\rm dim}\,H^0(C_i,R_i)$
+is even, and $-1$ if it is odd. Then \uddu\ becomes
+\eqn\nurfo{(-1)^{{\rm dim}\,H^0(X,R)}=\prod_{i|s_i=1} t_i.}
+This is the result claimed in equation (2.66)
+of \newwitten. One can similarly check that \jimmo\ when evaluated
+with the signs given above agrees with the formulas of Kronheimer
+and Mrowka for Donaldson invariants of elliptic surfaces with multiple
+fibers.
+
+It remains to justify the claimed structure of $\ker\, T$. Suppose, for
+instance, that
+we are linearizing around a solution with $\beta=0$, $\alpha\not= 0$.
+Let $\delta A$, $\delta \alpha$, and $\delta \beta$ denote
+first order variations of $A,\alpha,$ and $\beta$. The argument
+that proves the vanishing theorem shows that for $\delta A,\delta\alpha,
+\delta\beta$ to be annihilated by $T$, one must
+have $\alpha\delta\beta=0$ and hence $\delta\beta=0$. The remaining
+equations can be written
+\eqn\remeq{\eqalign{\bar\partial \,\,\delta A^{0,1} & = 0 \cr
+ i\delta A^{0,1}\alpha +\bar\partial_A\delta\alpha & = 0.
+\cr}}
+One must divide the space of solutions of \remeq\ by solutions that arise
+from complex gauge transformations of $A,\alpha$.
+If $\delta A^{0,1}=0$, then the second equation says that $\delta\alpha
+\in H^0(X, K^{1/2}\otimes L)$; however, upon dividing by complex
+gauge transformations (which include rescalings of $\alpha$ by a constant)
+we should regard $\delta\alpha$ as an element of $H^0(X,K^{1/2}\otimes L)/
+{\bf C}\alpha$. The first equation says that $\delta A^{0,1}$ defines
+an element of $H^1(X,{\cal O})$, and the second equation says that
+multiplication by $\alpha$ maps this element to zero in $H^1(X,K^{1/2}\otimes
+L)$. So if ${\rm ker}\,\alpha$ is the kernel of
+$H^1(X,{\cal O})\underarrow{\alpha}H^1(X,K^{1/2}\otimes L)$, then there
+is an exact sequence
+\eqn\imoc{0\to H^0(X,K^{1/2}\otimes L)/{\bf C}\alpha
+\to {\rm ker}\,T\to {\rm ker}\,\alpha\to
+0.}
+This can be compared to the exact sequence
+\eqn\nimoc{0\to H^0(X,K^{1/2}\otimes L)/{\bf C}\alpha
+\to H^0(X,R)\to {\rm ker}\,\alpha\to
+0}
+that comes from \immo. Comparison of these sequences shows
+that ${\rm ker}\,T$ and $H^0(X,R)$ have the same dimension,
+as asserted above; one should be able to identify these spaces canonically.
+
+\newsec{A Short Sketch Of The Physics}
+
+To sketch the relation of these ideas to quantum field theory,
+let us first recall the analysis of $N=2$ supersymmetric Yang-Mills
+theory in \sw. To begin with we work on flat ${\bf R}^4$.
+It has long been known that this theory has a family of quantum vacuum
+states parametrized by a complex variable $u$, which corresponds
+to the four dimensional class in Donaldson theory. For $u\to\infty$,
+the gauge group is spontaneously broken down to the maximal torus,
+the effective coupling is small, and everything can be computed
+using asymptotic freedom. For small $u$, the effective coupling is strong.
+Classically, at $u=0$, the full $SU(2)$ gauge symmetry is restored.
+But the classical approximation is not valid near $u=0$.
+
+Quantum mechanically, as explained in \sw, the $u$ plane turns out to
+parametrize a family of elliptic curves,
+\foot{If $SU(2)$ is replaced by
+a Lie group of rank $r$, elliptic curves are replaced
+by abelian varieties of rank $r$; the analog of
+the simple type condition is that the commutative
+algebra of operators obtained by evaluating the Chern classes of the universal
+bundle at a point in a four-manifold has a spectrum consisting of $h$
+points ($h$, which is $N$ for $SU(N)$, is the dual Coxeter number of the
+Lie group) where these varieties degenerate maximally.}
+in fact, the modular curve
+of the group $\Gamma(2)$. The family can be described by the equation
+\eqn\urmo{y^2=(x^2-\Lambda^4)(x-u),}
+where $\Lambda$ is the analog of a parameter that often goes by the same
+name in the theory of strong interactions. (The fact that $\Lambda\not= 0$
+means that the quantum theory does not have the conformal invariance
+of the classical theory.)
+The curve \urmo\ is smooth for generic $u$, but degenerates to
+a rational curve for $u=\Lambda^2,-\Lambda^2$, or $\infty$. Near each
+degeneration, the theory becomes weakly coupled, and everything is calculable,
+if the right variables are used. At $u=\infty$, the weak coupling is
+(by asymptotic freedom) in terms of the original field
+variables. Near $u=\pm \Lambda^2$
+a magnetic monopole (or a dyon, that is a particle carrying both
+electric and magnetic charge) becomes massless; the light degrees
+of freedom are the monopole or dyon and a dual photon or $U(1)$ gauge
+boson. In terms of the dyon and dual photon, the theory is weakly
+coupled and controllable near $u=\pm \Lambda^2$.
+
+Notice that quantum mechanically on flat ${\bf R}^4$,
+the full $SU(2)$ gauge symmetry is never restored. The only really
+exceptional behavior that
+occurs anywhere is that magnetically charged particles
+become massless.
+
+Now, for any $N=2$ supersymmetric field theory, a standard twisting
+procedure \witten\ gives a topological field theory. In many cases,
+these topological field theories are related to the counting of
+solutions of appropriate equations. For instance, the procedure,
+applied to the underlying $SU(2)$ gauge theory, gives Donaldson theory
+(that is, the problem of counting $SU(2)$ instantons); applied to
+the quantum theory near $u=\pm \Lambda^2$, it gives the problem of
+counting the solutions of the monopole
+equations; applied at a generic point on the $u$ plane, it gives, roughly,
+the problem of counting {\it abelian} instantons.
+
+Now let us apply this experience, to work
+on a general oriented four-manifold $X$.
+The structure of the argument is analogous to the heat kernel proof
+of the index theorem, in which one considers the trace of the heat kernel
+$\Tr (-1)^Fe^{-tH}$. This is independent of $t$ but can be evaluated
+in different ways for $t\to 0$ or for $t\to \infty$;
+for small $t$, one sees local geometry and gets
+a cohomological formula, while for large $t$, one gets a description
+in terms of the physical ground states (harmonic spinors).
+
+In the four-manifold problem, letting $g$ be any Riemannian
+metric on $X$, we consider the one parameter family of metrics $g_t=tg$,
+with $t>0$. Correlation functions of the twisted topological field
+theory are metric independent and so independent of $t$.
+For $t\to 0$, using asymptotic freedom,
+the classical description becomes valid,
+and one recovers Donaldson's definition of four-manifold invariants
+from the $N=2 $ theory. In particular, for four-manifolds on which
+there are no abelian instantons, the main contribution comes from
+$u=0$ where for small $t$ one computes in the familiar fashion
+with the full $SU(2)$ gauge theory.
+
+What happens for large $t$? Once the scale of the four-manifold
+is much greater than $1/\Lambda$, the good description is in terms of
+the degrees of freedom of the vacuum states on ${\bf R}^4$. At first
+sight, it might appear that the answer will come by integration over
+the $u$ plane. That is apparently so for some classes of problems.
+
+However, for four-manifolds with $b_2^+>1$, one can show
+that the contribution of any region of the $u$ plane bounded away
+from $u=\pm \Lambda^2$ vanishes as a power of $t$ for $t\to \infty$.
+This is roughly because in the abelian theory that prevails away
+from $u=\pm \Lambda^2$, there are too many fermion zero modes
+and no sufficiently efficient way to lift
+them. (It is not clear if the gap in the
+argument for non-K\"ahler manifolds with $b_2^+=3$ is significant,
+or could be removed with a more precise treatment.)
+
+Under the above condition on $b_2^+$, a contribution that survives
+for $t\to\infty$ can therefore come only from a neighborhood of
+$u=\pm \Lambda^2$ that shrinks to zero as $t$ grows. The contribution
+from this region does survive for $t\to \infty$; it can be computed
+using the monopole equations since those are the relevant equations
+in the topologically twisted theory near $u=\pm \Lambda^2$.
+In computing a correlation function of operators of the twisted theory
+near $u=\pm \Lambda^2$, one can expand all operators of the microscopic
+theory in terms of operators of successively higher dimension in the
+macroscopic, monopole theory.
+
+For $u$, the most relevant term (that is, the term of lowest dimension)
+is the $c$-number $u=\Lambda^2$ or $u=-\Lambda^2$.
+The simple type condition -- which asserts that $u$ is semi-simple
+with a spectrum consisting of two points -- arises when one may
+replace $u$ by this $c$-number. For the operator
+related to the two-dimensional classes of Donaldson theory, the most
+relevant term is again a $c$-number,
+measuring the first Chern class of the dual line bundle $L$ of the monopole
+problem. Keeping only these terms, since the operators are replaced
+by $c$-numbers, correlation
+functions can be computed by simply
+counting solutions weighted by the sign of the fermion determinant; only zero
+dimensional moduli spaces contribute. Upon fixing the normalizations
+by comparing to known special cases, one arrives at \jimmo.
+
+This in fact appears to be justified since as usual in such
+problems operators of higher
+dimension give contributions that vanish as negative powers of $t$.
+This would give a quantum field theory proof that
+all oriented four-manifolds with $b_2^+>3$ are of simple type.
+If, however, higher terms in the expansion of the operators survive
+on some four-manifolds with $b_2^+>3$, the consequences would be as follows.
+Then the higher monopole invariants of $W\not= 0$ can be detected in
+Donaldson theory, and \jimmo\ will be replaced by a more general
+formula involving the expansion near $u=\pm \Lambda^2$ of some of the
+functions computed in \sw. The number $s$ of higher terms that one would have
+to keep in the expansion would be one half
+the maximum value of $W$ that contributes. $u$ will still have a spectrum
+consisting of two points, but instead of $u^2-\Lambda^4=0$, one would
+get $(u^2-\Lambda^4)^{s+1}=0$. Such a situation has in fact been
+analyzed by Kronheimer and Mrowka.
+
+\listrefs
+\end
diff --git a/macros/generic/occam/auditor.tex b/macros/generic/occam/auditor.tex
new file mode 100644
index 0000000000..20b40cf983
--- /dev/null
+++ b/macros/generic/occam/auditor.tex
@@ -0,0 +1,598 @@
+
+ %% auditor.tex of 6-94 (alpha)
+ %% "audits macro use"
+ %% By laurent siebenmann, lcs@topo.math.u-psud.fr (comments please!)
+ %% Used mostly for Plain and amstex; LaTeX use not debugged.
+ %% Documentation after endinput.
+ %% Alpha posting on ftp ftp.math.u-psud.fr in pub/TeX/Occam.dir
+ %% auditor.tex is part of the Occam system for macro management.
+ %% But it can also be used alone.
+
+ \ifx\undefined\auditortex\def\auditortex{}
+ \else
+ \immediate\write16{}%
+ \errmessage{%
+ The auditor.tex macro \noexpand\Def already loaded\string!}%
+ \EX@\endinput
+ \fi
+
+ \chardef\auditAt=\catcode`\@
+ \catcode`\@=11
+
+ %%temporarily suppress Plain's logging of allocations
+ \let\auditorwlog@ld\wlog
+ \def\wlog#1{\relax}
+
+ \def\WrSc@{\immediate\write16}
+ \def\WrOut@{\immediate\write\unusedout@}
+
+ \WrSc@{}%
+ \WrSc@{%
+ *** The auditor.tex macro-auditing system}%
+ \WrSc@{%
+ *** is being installed for a test run of your typescript.}%
+ \WrSc@{%
+ *** Its report will appear after typesetting.}%
+ \WrSc@{%
+ *** See documentation in auditor.tex.}%
+ \WrSc@{%
+ *** Hit return to get audit.}%
+ \WrSc@{%
+ *** Hit \space x \space and return to typeset without audit.}%
+ \def\temp{\par}%
+ \read16 to \YourChoice
+ %\def\YourChoice{\par}%%%%%%%%%% comment out?
+ \ifx\YourChoice\temp
+ \def\Modify@Audit{}%
+ \else
+ \def\Modify@Audit{\let\Def\def \let\gDef\gdef \let\Let\let
+ \def\gLet{\global\let}\let\The@Audit@\relax}%
+ \WrSc@{%
+ *** Auditor apparatus switched off\string.}%
+ \fi
+ \WrSc@{}%
+
+
+ \newwrite \unusedout@
+ \newtoks\temptoks@
+ \newtoks\nibbletoks@
+ \newtoks\resttoks@
+ \newtoks\deftoks@
+ \newtoks\mactoks@
+ \newtoks\unusedtoks@
+ \newtoks\AuditDepth
+
+ \let\EX@\expandafter
+
+ \def\Nibbl@#1#2\endNibbl@{\nibbletoks@{#1}\resttoks@{#2}}
+
+ \def\gobble#1{}
+
+ \def\WrDef@#1{\EX@
+ \ifx\csname\string_%
+ \string#1\string_\endcsname\@Used
+ \else
+ \EX@\global\EX@\let\csname\string_%
+ \string#1\string_\endcsname\@Defed
+ \edef\@tmp@{\global
+ \noexpand\deftoks@{\the\deftoks@\noexpand#1}}%
+ \@tmp@
+ \fi
+ }
+
+ \def\WrDDef@#1{\EX@
+ \ifx\csname\string_%
+ \string#1\string_\endcsname\@Used
+ \else
+ \EX@\global\EX@\let\csname\string_%
+ \string#1\string_\endcsname\@DDefed
+ \edef\@tmp@{\global
+ \noexpand\deftoks@{\the\deftoks@\noexpand#1}}%
+ \@tmp@
+ \fi
+ }
+
+ \def\@UsedMathSym{@UsedMathSym}
+ \newtoks\Def@toks@
+ \newtoks\Def@@toks@
+ \newtoks\Font@toks@
+ \newtoks\Mathchar@toks@
+ \newtoks\MATHchar@toks@
+ \newtoks\MATHchars@toks@
+
+\def\Deff@{%
+ \edef\@tmp@ {\@d@f\the\Def@toks@ {%\noexpand
+ \global\let \EX@\noexpand\csname\string_%
+ \EX@\string\the\nibbletoks@\string_\endcsname
+ \noexpand\@Used
+ \the\Def@@toks@ }}%
+ \@tmp@
+ \EX@\WrDef@\the\nibbletoks@
+ }
+
+\def\DDeff@{%
+ \edef\@tmp@ {\@d@f\the\Def@toks@ {%\noexpand
+ \global\let \EX@\noexpand\csname\string_%
+ \EX@\string\the\nibbletoks@\string_\endcsname
+ \noexpand\@Used
+ \the\Def@@toks@ }}%
+ \@tmp@
+ \EX@\WrDDef@\the\nibbletoks@
+ }
+
+ \def\Def#1#{\Def@toks@{#1}%
+ \Nibbl@#1\endNibbl@\let\@d@f\def
+ \afterassignment\Deff@\Def@@toks@
+ }
+
+ \def\DDef#1#{\Def@toks@{#1}%
+ \Nibbl@#1\endNibbl@\let\@d@f\def
+ \afterassignment\DDeff@\Def@@toks@
+ }
+
+ \def\gDef#1#{\Def@toks@{#1}%
+ \Nibbl@#1\endNibbl@\let\@d@f\gdef
+ \afterassignment\Deff@\Def@@toks@
+ }
+
+ %% \Let \gLet require two *macros* following, without =
+ %% Only then enhances \let
+ \def\Let#1#2{%
+ \EX@\Def\EX@#1\EX@{#2}%
+ }
+ \def\gLet#1#2{%
+ \EX@\gDef\EX@#1\EX@{#2}%
+ }
+
+ \def\WrMATHchardef@{%
+ \EX@\edef\the\MATHchar@toks@{%
+ \noexpand\ifx
+ \csname\string_\EX@\string\the\MATHchar@toks@\string_\endcsname
+ \noexpand\@UsedMathSym
+ \noexpand\else
+ \global\MATHchars@toks@\noexpand\EX@{%
+ \noexpand\the\MATHchars@toks@\the\MATHchar@toks@}%
+ \global\let\csname\string_\EX@\string\the\MATHchar@toks@\string_\endcsname
+ \noexpand\@UsedMathSym
+ \noexpand\fi
+ \csname \EX@\string\the\MATHchar@toks@\string_\endcsname}
+ %\EX@\show\the\MATHchar@toks@
+ }
+
+ \def\MATHchardef#1{\MATHchar@toks@{#1}%
+ %\showthe\MATHchar@toks@
+ \afterassignment\WrMATHchardef@
+ \EX@\mathchardef\csname \string#1\string_\endcsname}
+
+ \def\AfterMathchardef@{%
+ \edef\@@temp@{%
+ \noexpand\Def\the\Mathchar@toks@{%
+ \csname \EX@\string\the\Mathchar@toks@\string_\endcsname}}
+ %\show\@@temp@
+ \@@temp@}
+
+ \def\Mathchardef#1{\Mathchar@toks@{#1}%
+ \afterassignment\AfterMathchardef@
+ \EX@\mathchardef\csname \string#1\string_\endcsname}
+
+ %%% \Newsymbol, \NEWsymbol for \newsymbol of amssym.def
+ \def\Newsymbol#1#2#3#4#5{\let\next@\relax
+ \ifnum#2=\@ne\let\next@\msafam@\else
+ \ifnum#2=\tw@\let\next@\msbfam@\fi\fi
+ \Mathchardef#1="#3\next@#4#5}
+
+ \def\NEWsymbol#1#2#3#4#5{\let\next@\relax
+ \ifnum#2=\@ne\let\next@\msafam@\else
+ \ifnum#2=\tw@\let\next@\msbfam@\fi\fi
+ \MATHchardef#1="#3\next@#4#5}
+
+ \def\@F@nt@{\edef\@@temp@{%
+ \noexpand\Def\the\Font@toks@{%
+ \csname \EX@\string\the\Font@toks@\string_\endcsname}}%
+ \@@temp@}
+
+ \def\Font#1{\Font@toks@{#1}\afterassignment\@F@nt@
+ \EX@\font\csname \string#1\string_\endcsname}
+
+ \def\@FF@nt@{\edef\@@temp@{%
+ \noexpand\DDef\the\Font@toks@{%
+ \csname \EX@\string\the\Font@toks@\string_\endcsname}}%
+ \@@temp@}
+
+ \def\FFont#1{\Font@toks@{#1}\afterassignment\@FF@nt@
+ \EX@\font\csname \string#1\string_\endcsname}
+
+ \def\Loop@#1\Repeat@{%
+ \def\Iterate@{#1\EX@\Iterate@\fi}%
+ \Iterate@}
+
+ \bgroup\catcode`\%=12
+ \global\def\Pct@{ %% }\egroup
+
+ \def\WriteToToks@{\edef\@tmp@{\global\noexpand
+ \unusedtoks@{\the\unusedtoks@\the\nibbletoks@}}%
+ \@tmp@}
+
+ %\def\@Defed{@Defed}
+ %\def\@Used{@Used}
+ \def\@Used{\WrOut@{ \the\nibbletoks@}}%
+ \def\@Defed{\WrOut@{ *\the\nibbletoks@}\WriteToToks@}%
+ \def\@DDefed{\WrOut@{ **\the\nibbletoks@}\WriteToToks@}%
+ \let\@Filler\relax
+ \def\@Tail{@Tail}
+
+ \def\List@M@cs{%
+ \Loop@
+ %\message{ x }
+ \EX@\Nibbl@\the\deftoks@\@Tail\endNibbl@
+ \deftoks@\resttoks@ %\showthe\resttoks@
+ \edef\@Temp@{\EX@\noexpand\csname\string_%
+ \EX@\string\the\nibbletoks@\string_\endcsname}%
+ \EX@\let\EX@\@Temp\@Temp@ %
+ %% \@Temp is x-equal and let-equal to
+ %% \@Defed or \@DDefed or \@Used or \@Filler or @
+ %\showthe\nibbletoks@
+ %\show\@Temp@
+ %\show\@Temp
+ \EX@
+ \ifx \the\nibbletoks@\@Tail
+ %% exit if next token \@Tail
+ \else
+ \@Temp %% write appropriate stuff to file and log
+ \Repeat@
+ }
+
+ \newtoks\hrct@
+
+
+ {\catcode`\#=12\gdef\StringSharp{\string#}}
+
+ \def\The@Audit@{%\show\patience
+ \def\AuditSheet@{audit.lst}
+ %%%
+ \count255=\time\divide\count255 by 60\relax
+ \edef\temp@{\the\count255}
+ \multiply\count255 by -60\relax
+ \advance\count255 by \time
+ \immediate\openout\unusedout@ \AuditSheet@
+ \WrOut@{\Pct@ auditor.tex output, date
+ \the\day-\the\month-\the\year,
+ time \temp@\string:\the\count255.}%
+ \WrOut@{}
+ \edef\@temp@{\the\MATHchars@toks@}\def\empty{}%
+ \ifx\@temp@\empty\else
+ \WrOut@{\Pct@ Math characters defined via \noexpand\MATHchardef }
+ \WrOut@{\Pct@ or \noexpand\NEWsymbol and really used were\string:}
+ \WrOut@{\the\MATHchars@toks@}
+ \WrOut@{\Pct@ Beware lack of wordwrap\string!}
+ \WrOut@{}
+ \fi
+ \WrOut@{\Pct@ Macros (if any) defined by \string\Def, \string\Let, etc.}%
+ \WrOut@{\Pct@ are listed in order defined\string:}
+ \WrOut@{\Pct@ *Unused* macros among these are marked by * or \StringSharp.}%
+ \WrOut@{\Pct@ And **, ***, etc. indicate nesting (hence delayed action).}%
+ \WrSc@{}%
+ \WrOut@{}%
+ \List@M@cs
+ %\showthe\unusedtoks@
+ %\showthe\deftoks@
+ \def\empty{}%
+ \edef\@tmp@{\the\unusedtoks@}%
+ \ifx\@tmp@\empty
+ \EX@\def\EX@\@tmp@\EX@{\the\deftoks@}%
+ \ifx\@tmp@\empty
+ \WrSc@{*** No macros have been defined via \string\Def,
+ \string\Let, etc.}%
+ \else
+ \WrSc@{*** All macros defined via \string\Def,
+ \string\Let, etc. have been used.}%
+ \fi
+ \else
+ \WrSc@{*** The following macros defined via \string\Def,
+ \string\Let, etc. have not been used\string;}%
+ \WrSc@{%
+ *** --- you can probably delete their definitions\string:}%
+ \WrSc@{*** }%
+ \WrSc@{***** \the\unusedtoks@}%
+ \fi
+ \edef\@tmp@{\the\MATHchars@toks@}%
+ \ifx\@tmp@\empty\else
+ \WrSc@{ }%
+ \WrSc@{*** The following math chars defined by \noexpand\MATHchardef}%
+ \WrSc@{*** or \noexpand\NEWsymbol are really used\string:}%
+ \WrSc@{***** \the\MATHchars@toks@}%
+ \fi
+ \WrSc@{}%
+ \WrSc@{*** See the file \string"\AuditSheet@\string" for details.}%
+ \WrSc@{*** See the DefStrip utility for cleanup.}%
+ \WrOut@{}
+ \WrOut@{\the\AuditDepth\noexpand\ITERATIONCOUNTER}
+ \WrOut@{\Pct@ PLEASE iterate ***defstrip macro of QUEDM }
+ \WrOut@{\Pct@ until asterisks disappear from iteration counter line.}
+ \WrOut@{\Pct@ Name successive output files output1, output2, ...}
+ }
+
+ \ifx\undefined\@@end
+ \let\audprim@end@\end
+ \def\end{\The@Audit@\audprim@end@}
+ \else %%% LaTeX
+ \let\audprim@end@\@@end
+ \def\@@end{\The@Audit@\audprim@end@}
+ \fi
+
+ \Modify@Audit
+
+ \let\wlog\auditorwlog@ld
+ \catcode`\@=\auditAt
+
+ %\let\DDef\Def
+
+ \endinput %% comment out for tests
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end code
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% begin tests
+
+ \input amssym.def
+ \input amssym.occ
+
+$\digamma\digamma\varkappa$
+
+\Font\cmr cmr10 scaled 1500
+%\cmr
+
+\end
+
+ %\catcode`\@=11
+
+
+ %\documentstyle{article} %% LaTeX only
+ %\begin{document} %% LaTeX only
+ %\let\Def\def
+ %\let\Let\let
+
+ \Def\Rm#1{\mathop{\fam0#1}}
+
+ \Def \End {\Rm {End}}
+ \Def \Hom {\Rm {Hom}}
+ \Def \ind {\Rm {ind}}
+ \Def \Re {\Rm {Re}}
+ \Def \Tr {\Rm {Tr}}
+ \Def \rk {\Rm {rk}}
+ \Def \rg {\Rm {rg}}
+ \Def \Td {\Rm{Td}}
+ \Def \ch {\Rm{ch}}
+ \Def \T{\Rm{T}}
+ \Def \R{\Rm{R}}
+ \Def \e{\Rm{e}}
+ \Def \odd {\Rm{odd}}
+ \Def \even {\Rm{even}}
+ \Def \Ker {\Rm{Ker}}
+ \Def \id {\Rm{id}}
+ \Def \Pf {\Rm{Pf}}
+
+ \gDef\filler{filler}
+ \gLet\Filler\filler
+
+ %\Filler
+
+ \Font\myfont=cmr10 at 11.5pt
+ %\myfont
+
+ $\End
+ %\Hom
+ \ind
+ \Re
+ \Tr
+ \rk
+ \rg
+ \Td
+ \ch
+ \T
+ \R
+ \e
+ %\odd
+ %\even
+ \Ker
+ \id
+ \Pf $
+
+ \end{document}
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end tests
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% begin doc
+
+
+DOCUMENTATION FOR auditor.tex.
+
+
+ Warnings : These macros are not bulletproof. I believe no such
+macros could be bulletproof. But in practice that should be of
+no importance; hopefully, any significant annoyances will be
+routinely reported and remedied. Use of this alpha version
+with an imbricated format like LaTeX is probably for hardened
+texperts only. Plain and AmSTeX seem to respond well.
+
+ INSTRUCTIONS.
+
+ Suppose you have a macro file x.sty used for typesetting the
+typescript x.tex. The aim is to find out, quickly and easily,
+which of the macro definitions in x.sty are necessary to typeset
+x.tex.
+
+ 1) Modify a *copy* of x.sty to x-occ.sty by altering
+definitions as follows:
+
+ (a) replace \def by \Def for definitions of macros you suspect
+ might be unused in x.tex. Similarly use \gDef in place
+ of \gdef or \global\def.
+
+ (b) similarly replace \mathchardef by \Mathchardef, and so forth
+ as indicated in the list of currently possible substitutions
+ indicated in occam.spc.
+
+Provisional Warnings:
+ --- Do not modify \def's or \let's within other definitions.
+This would be pointless and perhaps dangerous. Be ready to
+revert to \def or \let if trouble ensues.
+ --- Avoid "\outer" and "\long" macros; also \alloc@
+
+ (c) Place the lines
+
+ %%% auditor.tex audits use of macros in a typescript.
+ \input auditor.tex %% Keep this file available to TeX!
+
+at the head of x-occ.sty. (Watch the order!)
+
+ 2) Modify x.tex temporarily replacing
+
+ \input x.sty by \input x-occ.sty
+
+in the header. Typeset as usual. The audit should now proceed
+with an explanatory dialog.
+
+ 3) If there is trouble in step 2), repeat it choosing (by
+dialog) to compose *without* an audit. There should be no
+change from the original behavior of x.tex. Correct any
+misbehavior --- probably arising from a malformation in
+x-occ.sty. Sometimes, here and there in x-occ.sty, one has to
+change \Def back to \def etc.
+
+ 4) Delete the unused definitions in another copy of x.sty, say
+x-min.sty. The list in the "audit.lst" output file is designed to
+make this easy, in fact so easy that a utility called "defstrip" can
+do the job automatically. Then replace x-occ.sty by x-min.sty in
+the header of x.tex. Typeset x.tex and check a proof copy.
+
+ 5) You can now send x.tex and x-min.sty to a colleague or
+publisher, without burdening him/her with useless macros present in
+x.sty.
+
+
+
+
+IMPROVEMENTS CURRENT AND FUTURE
+ A) In all, the list of prepared macros is currently (July
+1994):
+
+ \Def, \Let, \gDef, \gLet, \Font, \Mathchardef, \Newsymbol
+
+Unless the contrary is indicated, each is to be used in analogy
+with \Def replacing in x.sty a corresponding uncapitalized TeX
+primitive.
+
+ [Exception: \newsymbol (from amssym.def) is *not*
+primitive; it is used copiously for declaration of symbols from
+the AmS math fonts msam and msbm, as in amssym.tex.]
+
+ When the file amssym.tex is \input, huge numbers of math
+characters are defined via \newsymbol but only few are used. In
+this case it is appropriate to use variants \MATHchardef,
+\NEWsymbol. which signal math characters only if they are used
+--- and *not* if they are unused. For this, try temporarily
+replacing
+
+ \input amssym.tex
+
+in x-occ.sty (after \input auditor.tex please) by
+
+ \input amssym.occ
+
+where amssym.occ is obtained from amssym.tex by replacing
+\newsymbol by \NEWsymbol; then you get a list of the symbols really
+needed in x.tex.
+
+ In using \Let and \gLet, avoid macros with parameters and
+note that the condition that the second argument be a macro is
+often not satisfied. Do not use = in the syntax.
+
+ \Let, \gLet, and \Font tend to be troublesome; while
+\Mathchardef, \MATHchardef, \Newsymbol, \NEWsymbolseem tend to
+be troublefree.
+
+ The usefullness of the above list can be stretched by the
+user. For example, before \Font was introduced
+
+\font\myfont=cmr10 scaled 11.5pt
+
+would have been replaced by
+
+ \font\myfont@=cmr10 at 11.5pt \Def\myfont{\myfont@}
+
+to report use of \myfont.
+
+ In many formats, user definitions are made via macros not
+available in Plain TeX; for example, \define and \redefine in
+AmSTeX. In practice, they can usually be replaced by \Def for
+the audit. But the programmer may want to invent new macros,
+say \Define and \Redefine for AmSTeX.
+ (B) In case x.sty is of permanent interest, it is a good idea
+to begin to use x-occ.sty instead of x.sty after commenting
+out the line \input auditor.tex and replacing it by something
+like
+
+ %% Audit.tex apparatus
+ \let\Def\def \let\gDef\gdef \let\Let\let
+ \let\Font\font \def\gLet{\global\let}
+ \let\Mathchardef\mathchardef %% extend as necessary
+ % \input auditor.tex %% comment out to suppress audit function
+
+This will make steps (1) to (4) superfluous for your next
+typescript y.tex.
+
+ (C) DefStrip automates the generation of x-min.sty.
+(auditor.tex plus DefStrip make up the "Occam" utility.)
+Its starting materials are "x-occ.sty" plus the "list audit.lst"
+of unused macros provided by auditor.tex. This depends on a
+special arrangement of x-occ.sty described in occam.spc.
+
+ Ultimately, an auxiliary ".tex" program "defstrip.tex"
+will rewrite x-occ.sty omitting the inused definitions. This
+auxiliary program resembles the "docstrip" utility of LaTeX
+fame.
+
+ At the present time defstrip.tex is unavailable. But there
+exists a QUEDM script called Auditor-QUEDM-Macros; QUEDM is a
+editor with convenient "macro" (=composite command) capabilities
+that is available on Macintosh computers at prices as low as $60
+The instructions for Auditor-QUEDM-Macros are found in
+defstrip.hlp.
+
+
+ *** How "auditor.tex" functions or fails to function.
+
+ "auditor.tex" prefixes a reporting device to the expansion
+of macros defined by \Def; this device reports the use of the
+macro by defining a tell-tale auxilliary macro that is then
+polled after typesetting. But it may cause strange behavior or
+even stop TeX.
+
+ As has been mentioned auditor.tex is not bullet-proof. Any
+change whatever in the expansion of a macro can in principle
+alter its behavior. For example TeX can use \ifx and many other
+means to examine the expansion of a macro; it can detect any
+tampering with definitions.
+
+
+ *** The "watchman" mechanism.
+
+ This mechanism is capable of deleting any collection of
+lines of a macro file in response to the non-use of a single
+macro called the "watchman". This "watchman" may be a macro that
+is specially defined for the purpose. The mechanism is
+definitely only for macro files carefully formatted for Occam.
+This mechanism is more powerful but more cumbersome than that
+for \Def etc. It is still to be implemented.
+
+
+CAVEAT LaTeX : LaTeX environments tend to define their user
+macros locally; indeed their definitions are not set up while
+the macro file x.sty is being read but when the environment is
+entered. Thus the modus operandi indicated above is suspect.
+However, normally, the meanings assigned upon entering the
+environment are stored in macros whose name involves @ ; these
+are possibly the macros to spy on with \Def etc. Alternatively,
+the "watchman" mechanism may prove effective. The cleanup based
+on audit.lst is then still a delicate matter requiring texpert
+attention. Further, LaTeX's fragility must be attended to...
+
+
diff --git a/macros/generic/occam/defstrip.hlp b/macros/generic/occam/defstrip.hlp
new file mode 100644
index 0000000000..2691b39397
--- /dev/null
+++ b/macros/generic/occam/defstrip.hlp
@@ -0,0 +1,56 @@
+ Part of the Occam utility.
+ Laurent Siebenmann<lcs@topo.math.u-psud.fr>
+ Master posting 1994, ftp ftp.math.u-psud.fr
+
+
+ June 1994 alpha version
+
+ *************************************************
+ ******
+ ****** DefStrip-QUEDM-Macros Help
+ ******
+ *************************************************
+
+ PURPOSE OF THESE MACROS.
+
+ The main purpose is to use a file "audit.lst" derived from
+audit.tex, and a TeX macro file, x-aud.sty say, in the
+"Occam" format (defined in occam.spc) to derive a new
+simplified ".tex" macro file in which selected unwanted material is
+suppressed. This material is designated by "audit.lst" in
+conjunction with internal formating involving %^, %_, \Def, \Let,
+\gDef, \gLet, \Font, and some others. The user macro
+***defstrip governs this function.
+
+ The secondary purpose is to remove Occam formatting.
+The macro **aud-to-tex governs this function.
+
+
+ DIRECTIONS FOR USE.
+
+ Put together into one folder a copy of DefStrip-QUEDM-Macros
+macro file, the file audit.lst, and the macro file to be simplified.
+
+ Open the DefStrip-QUEDM-Macros macro file. If it is not
+visible, launch QUEDM using it.
+
+ Open the TeX macro file under QUEDM and *activate* its window.
+Then launch the macro ***defstrip from the QUEDM macros menu.
+
+ An audit.lst window will now open and much activity will be
+seen. Finally, a cleaned-up version of the ".tex" macro file will
+appear in a window called "output".
+
+ Only the ***help, ***defstrip, and **aud-to-tex macros
+from the QUEDM macros menu should be launched
+by the user. The others are subroutines.
+
+ See occam.spc for further details, and occam.pub for an
+overview.
+
+
+POSSIBLE PROBLEMS....(please report bugs to to author)
+
+
+
+
diff --git a/macros/generic/occam/occam.pub b/macros/generic/occam/occam.pub
new file mode 100644
index 0000000000..1fef5aee33
--- /dev/null
+++ b/macros/generic/occam/occam.pub
@@ -0,0 +1,272 @@
+ Part of the Occam utility. Laurent Siebenmann
+<lcs@topo.math.u-psud.fr>
+ Master posting 1994, ftp ftp.math.u-psud.fr
+
+Title: Occam.pub (1994--)
+This Text file has beem assimilated in a longer article
+corresponding to a talk at EuroTeX 95, Papendal, Netherlands.
+The extended article is posted ".dvi" form as
+Occam95.dvi.
+
+ ****************************************************
+ ****** OCCAM'S RAZOR AND MACRO MANAGEMENT
+ ****** (a text abstract)
+ ****** Laurent Siebenmann
+ ****** lcs@topo.math.u-psud.fr
+ ****************************************************
+
+THE APHORISM:
+
+
+ entia non sunt multiplicanda praeter necessitudinem
+
+ entities should not be multiplied beyond necessity
+
+ William of Occam 1285-1349(?)
+
+Occam's Razor is the aphorism quoted!
+Experts believe that Occam did not formulate it in
+exactly these famous words, but rather as
+
+ What can be done with fewer assumptions is done in
+ vain with more.
+
+or
+
+ Plurality is not to be assumed without necessity.
+
+The term Principle of Parcimony is also used for the
+Razor. At this European congress, let me remind you that
+Occam worked successively in Cambridge, Avignon, and
+M"unich.}
+
+ THE SOFTWARE:
+ --- occam.pub
+ --- auditor.tex
+ --- DefStrip-QUEDM-Cmds (Macintosh only)
+ --- defstrip.hlp help file for above
+ --- occam.spc specifications for source macro files to be minimized
+ --- defstrip.tex (is mere wishful thinking in summer 1995)
+
+MASTER POSTING in 1994:
+ --- ftp ftp.math.u-psud.fr
+directory pub/TeX/defstrip.dir/ }
+
+ Have you ever felt guilty about burdening a friend
+with macros that are not really necessary for composing
+your typescript? I certainly have; and would ideally
+like to follow Knuth's example of using macro files
+which define exactly what is necessary for a document
+and nothing more.
+
+ However, pruning a macro file that has served for
+another purpose is a pain. Most of us respond to this
+pain by adopting a rather messy maximalist approach in
+which all the macros that have a geneology related to
+the necessary macros are transmitted.
+
+ But there is another approach! One can seek
+efficient mechanisms to ease the task of weeding out
+unnecessary macros.
+
+ One such mechanism is auditor.tex, which sets up
+a list of names of those macros of macro file that turn
+out to be *unnecessary* in a given typescript.
+
+ A complementary tool is the utility DefStrip which
+combines a specially arranged version of the macro file
+to be "cleaned up" with the list of unused macros
+provided by auditor.tex to delete the unneeded macros
+listed together with some related material.
+
+ Ultimately, DefStrip will hopefully be a ".tex"
+program "defstrip.tex" resembling the "docstrip.cmd"
+utility of LaTeX fame. At the present time, there exists
+instead a QUEDM script called DefStrip-QUEDM-Macros;
+QUEDM is a editor with convenient "macro" (=composite
+command) capabilities that is available on Macintosh
+computers.
+
+ DefStrip and Auditor together make up a system
+called "Occam".
+
+ Let us consider two plausible examples of use of
+the Occam system. Only the first has proved genuinely
+useful thus far.
+
+ (A) Weeding out one's personal macro files
+
+Many TeX users build up a cumulative personal macro file
+through composing many articles with TeX. A time
+inevitably comes when it is embarrassing, cumbersome, or
+confusing to submit the whole macro file along with the
+article. The Occam system makes the pruning of the macro
+file painless. It is advisable to tidy up the total
+macro file and maintain it in "Occam" format\footnote{
+The extension distinguishing this format is usually
+".occ" for "audit".} as explained in "occam.spc"; then
+and only then will "auditor.tex" and DefStrip collaborate
+to *automatically* produce a minimal version of the
+macro file suitable for the article at hand.
+
+ (B) Autonomous and archival electronic ".tex"
+postings.
+
+ Suppose that one proposes to post in electronic
+".tex" form an article prepared using a more or less
+standard package such as the "harvmac.tex" macro package
+of Paul Ginsparg for Plain TeX. (The alternative ".dvi"
+form undeniably convenient, but also less flexible; for
+instance the ".tex" version can be reformatted to be read
+in comfort on any computer screen whereas a ".dvi"
+version often does not have an small enough text width.)
+Such a macro package is not immune to alteration with
+time, and unfortunately the principles of upward
+compatibility are just pious hopes, not laws.
+Consequently, one is well-advised to post, along with
+the article the macros necessary to compile it ---
+especially if modifications to the macros have been
+used. Unfortunately, the "harvmac.tex" macros are as
+voluminous as a 10 page article. This is an unfortunate
+obstacle to electronic posting of ".tex" typescripts.
+
+ The solution proposed is to have a version
+"harvmac.occ" of the "harvmac.tex" macros set out in a
+form designed for use with Occam. Then the necessary
+macros for a given article can quickly be extracted from
+the total package to make the total posting both compact
+and archival.
+
+ This is illustrated in the Occam distribution for a
+famous article of Edward Witten posted electronically in
+November 1994. The resulting archival posting (Plain
+based) requires only 6 Ko of macros rather than the
+original 20 Ko. The 68 Ko body of the article is
+untouched.
+
+ Recently "harvmac.tex" has been enhanced by inputting
+the hyper-reference macros "hyperbasics.tex" macros of
+Tanmoy Bhattacharya and the new name is "lanlmac.tex".
+The example will shortly be enhanced to cover this
+development.
+
+ The archival nature of TeX postings minimized using
+Occam still depends on Knuth's Plain format being
+archival. Plain probably will be at least upwards
+compatible in the best sense. However, the article (or a
+book, say) could perhaps be made archival on the scale
+of many decades by subjecting the Plain macros to Occam
+discipline; this incidentally seems necessary to realize
+best economy. The article would then have its own
+format built with initex. This may seem needlessly
+radical to an English speaking user. But I consider
+bootstrapping from initex the best approach for fully
+archival ".tex" postings where other languages are
+concerned.
+
+ There are many macro packages that might benefit from
+being put in Occam format. The picture macro package of
+LaTeX is an example; interestingly these macros run on
+Plain. The "amssym.tex" math symbol definition package
+for AmS fonts is another; it defines several hundred
+control sequences of which precious few are used in any
+given article. These two will also be included in ".occ"
+form in the Occam distribution.
+
+ Would it be reasonable to convert AmSTeX into a
+Plain macro package in ".occ" format, much as it has been
+converted by the AmS into a LaTeX package? This
+involves `repealing' AmSTeX's status as a full-fledged
+format. In particular the AmS-Plain package would have
+no influence outside of math mode.
+
+ In the long term, the formatting of a macro package
+for use with Occam will be the responsibility of the
+author of the package. Clearly this will catch on only
+if Occam performs well and becomes very stable.
+
+
+AFTERTHOUGHTS
+
+ Occam as presently realized does not make much sense
+in the LaTeX world. The LaTeX group is building official
+macro modules that cover all needs and are universally
+available. I suspect this will require continuing
+exponential growth of the LaTeX project, of the
+computers that run LaTeX, and of the of the time
+invested by LaTeX users.
+
+ I may be wrong. Occam is nevertheless to some extent
+my remedy for this real or imaginary catastrophic malady
+of LaTeX. Don Knuth seems to have had similar
+premonitions. I like to repeat what he said on the
+occasion of the 10th anniversary celebration of TeX
+82.
+
+ Suppose you were allowed to rewrite all
+the world's literature; should you try to put it all
+into the same format? I~doubt it. I~tend to think such
+unification is a dream that's not going to work.
+
+ [TUGboat, vol~13 (1992), page 424]
+
+ The happiest outcome would be for both approaches
+to work well. If this comes about, I expect the "look
+and feel" of Plain and LaTeX to steadily diverge.
+
+ Bootstrapping on initex as mentioned above seems to
+have a parallel in classical programming, namely the use
+of assembly language --- whereas the LaTeX approach is
+parallel to the use of a big and constantly evolving
+interpreter.
+
+ Occam's razor was one of the guiding principles of
+scientific thought for several hundred years before the
+coming of age of computers. I suspect the philosophy of
+Aristotle or Descartes is far more likely to appeal to
+computer scientists. One might go so far as to say that
+Occam's razor has by now been discarded --- indeed,
+object oriented programming consciously cultivates the
+art of multiplication of entities, and this sort of
+thing is to be found in of LaTeX commands such as
+\newheading. What can the minimalism of Occam's razor
+offer TeX users at this late date? Probably just a few
+things.
+
+ (a) Friendliness to human beings. Unnecessary
+entities that cost a microprocessor only a fraction of a
+second can cost the human mind a significant amount of
+time.
+
+ (b) Extra storage space and computing power. Both
+are in a period of exponential growth. But so is the
+TeX related software we use. Where performance in a
+fixed task is concerned, these growths may even cancel
+one another. When this happens, the old-fashioned
+minimalism of Occam's razor can help derive pleasure and
+profit from progress.
+
+ In summary, Occam is a TeX application offering
+possible new direction in programming for Plain TeX,
+and an answer to the most debilitating problem of Plain
+TeX, namely the confusion and incoherence that come
+from continual macro additions. Naturally, for this
+weakness, I have a quotation from Leslie Lamport (1995).
+
+
+ Because Plain TeX is fixed, it seems
+likely that the Plain TeX community will fragment into
+numerous small islands in a sea of incompatibility.
+
+
+ Lamport's answer in his LaTeX is to
+systematize indefinite growth with constantly improving
+infrastructure, whereas Occam's answer is to prune the
+macro collection to restore simplicity to every
+document.
+
+
+ Laurent Siebenmann
+ <lcs@topo.math.u-psud.fr>
+
+
diff --git a/macros/generic/occam/occam.spc b/macros/generic/occam/occam.spc
new file mode 100644
index 0000000000..fc9b01db1d
--- /dev/null
+++ b/macros/generic/occam/occam.spc
@@ -0,0 +1,415 @@
+Part of the Occam utility.
+Laurent Siebenmann <lcs@topo.math.u-psud.fr>
+Master posting 1994, ftp ftp.math.u-psud.fr
+Alpha version 6-94 subject to change.
+
+
+ ====== Occam Syntax and Specifications ======
+
+ Occam is a system for extracting from a large macro file
+exactly those macros required by a given typescript. Its
+active parts are auditor.tex that determines which macros
+are necessary, and DefStrip a utility to deletes unnecessary
+macros.
+
+ The first created DefStrip utility is a QUEDM script; QUEDM
+is a editor with macro capabilities that is available on
+Macintosh computers. (Hopefully a version of this utility
+which is a ".tex" program like auditor.tex will follow in due
+time; it would be very analogous to the LaTeX "docstrip.cmd"
+utility.)
+
+(I) About the QUEDM Version of Summer 1994 (preliminary)
+
+ The Occam syntax is for TeX macro files. Its purpose is to let
+the "DefStrip" utility delete selected lines of the file with the
+help of a list audit.lst of "unused" control sequences (mostly
+macros).
+
+ These lines come in blocks of roughly two sorts:
+
+ (a) Material that is to be unconditionally deleted.
+
+ (b) Blocks surrounding lines beginning (modulo spaces) with
+one of:
+
+ \Def (variant of \def)
+ \gDef (variant of \gdef or \global\def)
+ \Let (variant of \let)
+ \gLet (variant of \global\let)
+ \Font (variant of \font)
+ \Mathchardef (variant of \mathchardef)
+ \Newsymbol (variant of \newsymbol)
+
+This list may be extended. A particular such block is to be deleted
+precisely if the macro name is designated in an external list called
+"audit.lst". output by the TeX utility "auditor.tex".
+
+
+MAIN SPECIFICATIONS of the Occam syntax.
+
+ ASCII (7-bit) text files only. No tab characters please.
+
+ The names of macro files conforming to this syntax should
+involve the suffix "aud" in some form if at all possible. For
+example, "x.sty" might become "x-aud.sty" or "x.occ", say
+"x-aud.sty" for future reference.
+
+ See internal documentation of "audit.tex" to generate a
+list of macros in "x-aud.sty" that are unnecessary in a
+given typesetting job "x.tex".
+
+ In x-aud.sty, the lines
+
+ %^ This file is formatted by <programmer>, <date>, <email>
+ % for use of the Occam utility posted on the CTAN archives
+ % (master posting 1994 on ftp ftp.math.u-psud.fr)
+ %% DO NOT ALTER "OCCAM" SIGNS <percent>^ or <percent>_ , ^_
+ %% UNLESS YOU UNDERSTAND THEM!
+ \let\Def\def \let\gDef\gdef \let\Let\let
+ \def\gLet{\global\let} \let\Font\font
+ \let\Mathchardef\mathchardef\let\Newsymbol\newsymbol
+ \let\MATHchardef\mathchardef\let\NEWsymbol\newsymbol
+ % \input auditor.tex %% keep auditor.tex available
+ %% comment out above line to suppress audit function. %_
+
+should appear in the header.
+
+ Two composite symbols %^ and %_ are employed to
+designate possible deletions. On its line %^ is always
+preceeded by spaces only (zero or more); similarly %_ is
+always followed by spaces only.
+
+ (A) Unconditionally deleted material:
+
+ %%^_ <delete me>
+
+Everything from %%^_ to the end of file is then deleted.
+To delete just a segment use
+
+ %^ <delete me> %_
+
+The deleted material can span many lines, but must include no
+blank line. We have just seen a block of such material above!
+Note that it may well contain \Def etc. but not %^, %_.
+
+ The unconditional deletions will occur in the order
+described, and before conditional deletions are considered.
+
+
+ (B) Conditionally deleted material:
+
+ \Def \somemacro<maybe delete me>%_
+
+may cause deletion of the block of lines beginning with \Def
+etc. and ending with %_. This material is really deleted
+precisely if the macro \somemacro is marked for deletion in
+the the file "audit.lst".
+
+ The material <maybe delete me> must contain no blank line nor %^,
+%_, \Def etc; but it is otherwise arbitrary; in particular, macro
+arguments, comments, and auxiliary definitions are OK.
+
+ Along with this material some additional preceding material
+is deleted, namely contiguous preceeding lines (if any) that (a)
+are nonempty and (b) contain no %_ (but \Def etc; are allowed).
+Typically, such preceeding material might be comments or commands
+"owned" by the macro being deleted. For example the whole block
+
+ %_
+ \ifx\undefined\eightpoint
+ \Def\eightpoint{}
+ \fi %_
+
+will be deleted precisely in case \eightpoint is marked as unused
+in audit.lst. (The first %_ could be replaced by a blank line.)
+
+ Note that %_ is not really a closing delimiter since it can
+exist in arbitrary numbers without belonging to a matching pair.
+For another example, consider:
+
+ \Def\amacro ...%_
+ \newtoks\btoks %_
+ \Def\cmacro ...%_
+
+Here, the the first two %_ prevent \newtoks\btoks being deleted
+--- in all circumstances.
+
+ The example
+
+ \Def\amacro ...
+ \Def\bmacro ...%_
+
+is incorrect because the block beginning with \Def\amacro ...
+contains \Def\bmacro.
+
+ There is a second type of conditional deletion. Suppose
+\amacro is not used and is so designated in audit.lst. It
+often occurs that several *disjoint* blocks of lines should be
+deleted along with \amacro. These blocks should be
+designated as follows:
+
+ %/^\amacro
+ <stuff>
+ %/_
+
+\amacro is called the sentinel (watchman).
+The sentinel's line %/^... must contain nothing more than
+%/^\amacro and blank space. The initial and terminal
+lines will vanish along with <stuff>.
+
+IN SUMMARY: the blocks %^...%_ are unconditionally deleted,
+while a block signalled by \Def, \gDef, etc. with the help of
+%_ and/or blank lines is deleted or not according as the macro
+following \Def etc. is marked for deletion in "audit.lst".
+Similarly for blocks with sentinel macro. None of these blocks
+for conditional or unconditional deletion is allowed to contain
+an empty line nor any extraneous %^,%_,%/^,%/_,%%^_,\Def,
+\gDef, etc. The blocks introduced by \Def, \gDef, etc. include
+material extending backward as far as (but not including) a
+preceding line that is blank or terminated by one of %_,%/_. No
+such extension for blocks introduced by %^, %/^ is allowed ---
+nor would it be helpful.
+
+ Beyond these primary deletions, the utility DefStrip
+performs a few auxiliary tasks:
+
+ --- All remaining \Def, \gDef, etc. are converted to \def,
+\global\def, etc.. Also, if a remaining %_ is alone on its line
+(spaces ignored), the whole line disappears. And each remaining %_
+*not* alone on its line becomes % (this is the only deletion that
+can affect a line that survives.)
+
+ --- any empty line sequence (usually created by the deletion of
+blocks of lines) is reduced to a single empty line.
+
+ --- Residual appearances in x-aud.sty of macros marked for
+deletion in audit.lst will be marked by %%[VESTIGE] (on a new
+following line). They should be considered a failure of the current
+Occam format".
+
+ Users may find the vestiges mentioned above hard to deal with.
+(Can they simply be deleted?) Thus programmers should attempt to
+set up "Occam" formatting so as to assure that vestiges
+never occur; for their part, users should report vestiges to the
+programmers along with the involved audit.lst file from
+auditor.tex.
+
+ It is the programmer's or the user's responsibility to
+assure that the deletions made by the DefStrip utility result
+in a useful TeX macro file. The DefStrip utility is of little
+help here since it does not understand the macros. Thus it is
+expected that programmers take on the task of preparing macro
+files in Occam format. In most cases, anyone who programs TeX
+macros at an intermediate level will find it an easy task to
+put a macro file in Occam format. Beware that a good deal of
+testing and a bit of cleverness is usually necessary to assure
+that the Occam formatting does the job desired and in the most
+efficient way.
+
+
+ ---------------------------------
+
+
+ The following is documentation for additions made in 1995.
+Example 2 (with harvmac.occ) illustrates these features.
+
+EXPLANATIONS AND EXTRAPOLATIONS.
+
+ (a) Unnecessary macros nested within macros can also be be
+eliminated. Currently this is is achieved quite trivially by making
+several passes through the "defstrip" utility and the example below
+is conveniently explained in terms of several passes. However, the
+TeX version of "defstrip" will almost certainly reduce this to a
+single pass; the audit.tex utility already acts in a single pass.)
+
+ Here is a generic "example". The original
+macro file contains:
+
+\def\MACRO{<stuff1>%
+ <stuff2>
+ \def\macro{<stuff3>}%
+ <stuff4>
+ <stuff5>}
+
+An Occam formatted version is:
+
+\Def\MACRO{<stuff1>%#_
+ <stuff2>
+ \DDef\macro{<stuff3>}%
+ <stuff4>%#_
+ <stuff5>}%_
+
+ In relation to audit.tex, the macro \DDef behaves much like \Def
+except that the associated tag distinguishing unused macros in
+audit.lst is ** in place of *.
+
+ Note that if \MACRO is unused then the whole block
+vanishes.
+
+ Suppose not. Then, on first pass of the macro file through
+"defstrip", \DDef is converted to \Def. There is an accounting
+procedure set up in audit.lst in terms of * and #. First off all **
+become *# (and *** would become *## if there were any, etc.).
+At the close of the first pass the marks %#_ in WW.sty are
+converted %_; at the same time the file audit.lst undergoes changes
+** ==> *# ==> #* and * ==> #\, i.e. asterisks move right or die on
+backslash. On the second pass through "defstrip", one is treating:
+
+\def\MACRO{<stuff1>%_
+ <stuff2>
+ \Def\macro{<stuff3>}%
+ <stuff4>%_
+ <stuff5>}%
+
+and in response to an entry #*\macro in audit.lst
+"defstrip" will conditionally delete the block
+
+ <stuff2>
+ \Def\macro{<stuff3>}%
+ <stuff4>%_
+
+i.e this block is deleted precisely if \macro is not "used".
+(Only a programmer can guess whether this elimination is
+safe!)
+
+ (b) Often one wants to delete other material along with the
+block surrounding \macro; for that, the following "sentinel"
+approach mentioned elsewhere is useful. The provisional
+syntax for a block to be eliminated with \MACRO is
+
+%/^\MACRO
+<stuff 6>
+%/_
+
+and for \macro it would be
+
+%#/^\macro
+<stuff 6>
+%#/_.
+
+It is OK to use _ in place of /_ in the above syntax;
+But not ^ in place of /^ since that would give an
+unconditional deletion.
+
+ (c) There is also a notion of nested *un*conditional
+deletion useful for deleting nested diagnostic macros as
+in (d) below. The syntax is:
+
+%#^
+<stuff6>
+%#_
+
+ (d) For text fonts, \Font works reasonably well. But it
+fails badly for math font systems. The latter are
+particularly difficult to minimize because TeX seems not to
+readily indicate which fonts it is using for math. At a
+given pointsize the following \everymath device is used in
+harvmac.occ; it manages to tell whether math mode has been
+called.
+
+ %%% Title fonts
+ %#/^\TitlepointMathTest
+ \font\titlerms=cmr7 \tfontsize
+ \font\titlermss=cmr5 \tfontsize
+ \font\titlei=cmmi10 \tfontsize\relax \skewchar\titlei='177
+ \font\titleis=cmmi7 \tfontsize\relax \skewchar\titleis='177
+ \font\titleiss=cmmi5 \tfontsize\relax \skewchartitleiss='177
+ \font\titlesy=cmsy10 \tfontsize\relax \skewchar\titlesy='60
+ \fonttitlesys=cmsy7 \tfontsize\relax \skewchar\titlesys='60
+ \font\titlesyss=cmsy5 \tfontsize\relax \skewchar\titlesyss='60
+ %#/_
+ %^
+ \DDef \TitlepointMathTest{\relax}
+ %% a diagnostic that never survives%_
+ %% note the peculiar nesting: %#^ would be illegal.
+ %% The nesting below is the other way about.
+ %_
+
+ \font\titlerm=cmr10 \tfontsize
+ \Def\titlefont{\textfont0=\titlerm
+ \def\rm{\fam0\titlerm}%
+ \rm
+ %#^\TitlepointMathTest
+ \textfont0=\titlerm \scriptfont0=\titlerms
+ \scriptscriptfont0=\titlermss
+ \textfont1=\titlei \scriptfont1=\titleis
+ \scriptscriptfont1=\titleiss
+ \textfont2=\titlesy \scriptfont2=\titlesys
+ \scriptscriptfont2=\titlesyss
+ %#_
+ %#^
+ \everymath{\TitlepointMathTest}%
+ %#_
+ }%_
+
+For witten94.tex this device allowed suppression of the math
+system at title pointsize, an appreciable saving.
+\TitlepointMathTest is a diagnostic macro that in no event
+will survive. Nevertheless it controls the inclusion
+or omission of two separated code segments.
+
+ Essentially all the features of the Occam syntax so far
+mentioned are put to work in these few lines so it is worth
+pausing to think about it all.
+
+ (e) This test piece has caused second thoughts about the
+policy of systematically notifying the user of "vestigeal"
+macros remaining in the pruned macro file, i.e. unused macros
+that remain in W.tex but whose definitions have been deleted
+by "defstrip"; it can be helpful to stumble on an undefined
+macro with a suggestive name. The notification will be
+probably be in some sense optional.
+
+ (f) It is perhaps unwise to wring the last unused font out of
+an article's font system --- as that may interfere with
+revision. However, ridiculously wasteful font systems abound,
+and the problem of analysing math font use is in principle
+challenging. Below I venture to point out two approaches that
+were *not* employed in harvmac.occ. Are there others worth
+trying?
+
+ (i) To test whether a given math font is really used,
+replace it by a math font with no characters \nullmath and look
+for missing character complaints by TeX. This approach is
+powerful but clumsy and slow because a huge .log file must
+examined.
+
+ Here is a construction via initex:
+
+ \font\nullmath=\nullfont
+ \fontdimen23\nullmath=1pt \skewchar\nullmath='60
+ \input plain\dump
+
+Second construction: *create* font "empty" beforehand
+as a .dvi file.
+
+ \font\nullmath=empty at 1 pt
+ \fontdimen23\nullmath=1pt \skewchar\nullmath='60
+
+Making do otherwise:
+
+ \ifx\undefined\nullmath \font\nullmath=logo10 at 1 pt \fi
+ \fontdimen23\nullmath=1pt \skewchar\nullmath='60
+
+This last way is easy but not perfect as logo has a few
+characters. (Math fonts are formally required to have up to
+23 dimensions).
+
+ (ii) To discover whether any math font \myfont is needed,
+examine the ".dvi" file. A list of the used fonts from the .dvi
+file (.tfm names and sizes) can then be assimilated first off by
+audit.tex and used to decide whether \myfont should bear an
+asterisk in audit.lst. Then the power of \ffont to eliminate math
+(and other) fonts would be comparable to the power of \Def to
+eliminate unused macros. Perfection seems never attainable: in
+principle, one can use a font without it showing up in output to
+the .dvi!.
+
+
+NATURAL THINGS THAT ARE NOT (YET) IN PLACE
+ --- nesting beyond depth two is not yet supported. The only
+problem is to program for all levels at once.
+ --- at depth two, \FFont and \DDef exist, but \LLet and others do
+not yet exist.
+