diff options
author | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
---|---|---|
committer | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
commit | e0c6872cf40896c7be36b11dcc744620f10adf1d (patch) | |
tree | 60335e10d2f4354b0674ec22d7b53f0f8abee672 /macros/generic/occam |
Initial commit
Diffstat (limited to 'macros/generic/occam')
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 @@ +(This file must be converted with BinHex 4.0) +:%84PCP0dFQP`,9&9484$E@4c!&&03804483a!3!!!!!!!!!TJ)bi!!!!!!%!!!! +S#3!!*`N!!!&h!!$`)[rr!!!N2!!!!!$`-!!R-!!S'J!!!!!!!2rr)!!44'9Q8h4 +bDA!Y899&4%0YC(-#!!!!88e"3e&&4$%"!!!!88e"3e&&4$%"!!!F!,N!!!!!!!! +!!!!!!!!!!!!!!!#XPK,M!!!!!!!!+B#ffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff +fffffffffffff!!!!3%9NDA3J6@&MFQr*$8YPH5"NC@CcG(*TF'KXF!e,CANJ$3e +$E'PMDb!a-M3X)#dh$80XD@0V)$%b0#`J,6F0$3d!!!$B6h"PEL!LBA9NDA3ZE(0 +d)L!J$80[F(NJ)MJL)#)k2PYH35eDB5ekA5)J)!e'D@jN)#)k2&`UA&aFA&Y",9T +K,ATG+b)J)Le"Ch48)Je$Eh"j)#)j)L!06h"PEL!LGfpbDb)03fKKEQGP)#GF3cP +F3cJZ+VdR)#"dEb!J)LDp)#!P*5"E9N969%P(46mrA5)J)N&KCh3L$80SB@jRC5! 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+!*r%!!#Ea!!!"BJ!-0!`-)!!!!"`!`J!!88e"3`!-!!SYEJ!!)!!313!--p`"r!! +0)!!*Y!!--q"!#`!C)!!,Q3!--q4'D!!Q)!!2j3!--qJfN3!b)!!")!!--qadJJ! +r)!!)h!!--r"Y83"+)!!!!!!--r4`e3"6)!!Fb`!-0!"HN3"I)!!!4!!--pKD#J# +')!!#&J!-T2!3rJ"X)!!*J`!-T3"Jr`"i)!!'SJ!-T1JZU3#6)!!6G3!-T13-C'9 +QFh4bDA"SC@&N#f4PCR0dFQP`F'9b$'4PCR0dFQP`G'PNH3XU+LTNC@CcG(*TF!a +NC@CcG(*TF("bCA!+Fh4TCfeKG'PkC3JU+LSUD'9XF!YNC@CcG(*TF'KXF!aNC@C +cG(*TF(4KD@`,Fh0dD@GYBA4THQ80C'4PCR0dFQP`G'&TE!`U+QpMBbedEbedCAJ +-C'9QFh4bDA!ZD'a`$3i: 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 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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 Binary files differnew file mode 100644 index 0000000000..4d0d8a8fec --- /dev/null +++ b/macros/generic/occam/Occam95.pdf diff --git a/macros/generic/occam/PlainEx1/Example.readme b/macros/generic/occam/PlainEx1/Example.readme new file mode 100644 index 0000000000..5591c08d97 --- /dev/null +++ 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 + \vbox\bgroup\let\cr\DeepCR\let\\\DeepCR + \parindent=0 pt\parskip= 0 pt + \Titlefont\baselineskip=\fontdimen6\font + \advance\baselineskip by \TitleLineskip + \null\vskip22pt plus 15pt + \bgroup \hfil}% + + \def \endTitle {\unskip\hfil\egroup\par \egroup\medskip}% + + \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 \Subheading#1{\medskip\bigskip \goodbreak + \par \noindent {\Subheadingfont #1.}% + \nobreak \vskip 3pt \nobreak}% + + \def \Remark#1{\bigskip \goodbreak + \par\noindent{\Remarkfont #1.}}% + + \def \endRemark{\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}% + + \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}}% + + \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\preLinefigure[#1*#2]_{% + \Wd=#1\Ht=#2\catcode`\w=11 \catcode`\h=11 \LLinefigure} + \catcode`\w=11 \catcode`\h=11 % + + \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 \Footnote #1#2{\footnote{\raise.4ex\hbox{\Admin{ + #1}}}{\Smallfonts #2}} %% \Smallfonts problematic here + %% \scriptstyle? % + + \def \ItemStyle{\bf}% + + \def \Item #1{% + \smallskip\nointerlineskip{\parskip=4pt\noindent + {\def\Bf{}\hbox{$\ItemStyle #1$\relax}}\enskip}}% + + \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\,{\ifmmode\mskip \thinmuskip\else\hskip1pt\fi }% + + \def \Eqno #1$${\eqno \Admin{#1}$$\relax}% + + \def \bigMidvert{\kern4pt \big \vert \kern4pt}% + + \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 \Lim {\lim\limits}% + + \def \llongrightarrow {\kern-2pt\mathop + {\kern3pt\longrightarrow\kern3pt}\limits}% + +\catcode`\@=12 + + \endinput + 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 --- /dev/null +++ b/macros/generic/occam/PlainEx2/W.sty @@ -0,0 +1,196 @@ + +%% 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) + +%%% apple lw parameters by default + \def \unredoffs{}% + + \newbox\leftpage + \newdimen\fullhsize + \newdimen\hstitle + \newdimen\hsbody + \tolerance=1000\hfuzz=2pt + +\catcode`\@=11 + + \def \usedBIGans{}%% sic! % + + \magnification=1200 + %% Adjust: + \unredoffs %% set predefined offsets + %% but maybe adjust: + %\voffset=0truein + %\hoffset=0truein + \edef\tfontsize{scaled\magstep3}% + \def \abstractfont{\tenpoint}% + \baselineskip=16pt plus 2pt minus 1pt + \hsbody=\hsize \hstitle=\hsize + %% take default values for unreduced format + +\newcount\yearltd\yearltd=\year\advance\yearltd by -1900% + + \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 \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. + |