text:
\section*{Abstract (Not appropriate in this style!)}%
   \else \small
   \begin{center}{\bf Abstract\vspace{-.5em}\vspace{\z@}}\end{center}%
   \quotation
   \fi
  }%
 }{%
 }%
\@ifundefined{endabstract}{\def\endabstract
  {\if@twocolumn\else\endquotation\fi}}{}%
\@ifundefined{maketitle}{\def\maketitle#1{}}{}%
\@ifundefined{affiliation}{\def\affiliation#1{}}{}%
\@ifundefined{proof}{\def\proof{\noindent{\bfseries Proof. }}}{}%
\@ifundefined{endproof}{\def\endproof{\mbox{\ \rule{.1in}{.1in}}}}{}%
\@ifundefined{newfield}{\def\newfield#1#2{}}{}%
\@ifundefined{chapter}{\def\chapter#1{\par(Chapter head:)#1\par }%
 \newcount\c@chapter}{}%
\@ifundefined{part}{\def\part#1{\par(Part head:)#1\par }}{}%
\@ifundefined{section}{\def\section#1{\par(Section head:)#1\par }}{}%
\@ifundefined{subsection}{\def\subsection#1%
 {\par(Subsection head:)#1\par }}{}%
\@ifundefined{subsubsection}{\def\subsubsection#1%
 {\par(Subsubsection head:)#1\par }}{}%
\@ifundefined{paragraph}{\def\paragraph#1%
 {\par(Subsubsubsection head:)#1\par }}{}%
\@ifundefined{subparagraph}{\def\subparagraph#1%
 {\par(Subsubsubsubsection head:)#1\par }}{}%
\@ifundefined{therefore}{\def\therefore{}}{}%
\@ifundefined{backepsilon}{\def\backepsilon{}}{}%
\@ifundefined{yen}{\def\yen{\hbox{\rm\rlap=Y}}}{}%
\@ifundefined{registered}{%
   \def\registered{\relax\ifmmode{}\r@gistered
                    \else$\m@th\r@gistered$\fi}%
 \def\r@gistered{^{\ooalign
  {\hfil\raise.07ex\hbox{$\scriptstyle\rm\RIfM@\expandafter\text@\else\expandafter\mbox\fi{R}$}\hfil\crcr
  \mathhexbox20D}}}}{}%
\@ifundefined{Eth}{\def\Eth{}}{}%
\@ifundefined{eth}{\def\eth{}}{}%
\@ifundefined{Thorn}{\def\Thorn{}}{}%
\@ifundefined{thorn}{\def\thorn{}}{}%
\def\TEXTsymbol#1{\mbox{$#1$}}%
\@ifundefined{degree}{\def\degree{{}^{\circ}}}{}%
\newdimen\theight
\def\Column{%
 \vadjust{\setbox\z@=\hbox{\scriptsize\quad\quad tcol}%
  \theight=\ht\z@\advance\theight by \dp\z@\advance\theight by \lineskip
  \kern -\theight \vbox to \theight{%
   \rightline{\rlap{\box\z@}}%
   \vss
   }%
  }%
 }%
\def\qed{%
 \ifhmode\unskip\nobreak\fi\ifmmode\ifinner\else\hskip5\p@\fi\fi
 \hbox{\hskip5\p@\vrule width4\p@ height6\p@ depth1.5\p@\hskip\p@}%
 }%
\def\cents{\hbox{\rm\rlap/c}}%
\def\miss{\hbox{\vrule height2\p@ width 2\p@ depth\z@}}%
\def\vvert{\Vert
\def\tcol#1{{\baselineskip=6\p@ \vcenter{#1}} \Column}  %
\def\dB{\hbox{{}}
\def\mB#1{\hbox{$#1$}
\def\nB#1{\hbox{#1}
\def\note{$^{\dag}}%

\defLaTeX2e{LaTeX2e}
\def\chkcompat{%
   \if@compatibility
   \else
     \usepackage{latexsym}
   \fi
}

\ifx\fmtnameLaTeX2e
  \DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
  \DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
  \DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
  \DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
  \DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
  \DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl}
  \DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc}
  \chkcompat
\fi

\def\alpha{\Greekmath 010B }%
\def\beta{\Greekmath 010C }%
\def\gamma{\Greekmath 010D }%
\def\delta{\Greekmath 010E }%
\def\epsilon{\Greekmath 010F }%
\def\zeta{\Greekmath 0110 }%
\def\eta{\Greekmath 0111 }%
\def\theta{\Greekmath 0112 }%
\def\iota{\Greekmath 0113 }%
\def\kappa{\Greekmath 0114 }%
\def\lambda{\Greekmath 0115 }%
\def\mu{\Greekmath 0116 }%
\def\nu{\Greekmath 0117 }%
\def\xi{\Greekmath 0118 }%
\def\pi{\Greekmath 0119 }%
\def\rho{\Greekmath 011A }%
\def\sigma{\Greekmath 011B }%
\def\tau{\Greekmath 011C }%
\def\upsilon{\Greekmath 011D }%
\def\phi{\Greekmath 011E }%
\def\chi{\Greekmath 011F }%
\def\psi{\Greekmath 0120 }%
\def\omega{\Greekmath 0121 }%
\def\varepsilon{\Greekmath 0122 }%
\def\vartheta{\Greekmath 0123 }%
\def\varpi{\Greekmath 0124 }%
\def\varrho{\Greekmath 0125 }%
\def\varsigma{\Greekmath 0126 }%
\def\varphi{\Greekmath 0127 }%

\def\Greekmath 0272 {\Greekmath 0272 }

\def\Greekmath#1#2#3#4{%
    \if@compatibility
        \ifnum\mathgroup=\symbold
           \mathchoice{\mbox{\boldmath$\displaystyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\textstyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\scriptstyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\scriptscriptstyle\mathchar"#1#2#3#4$}}%
        \else
           \mathchar"#1#2#3#
        \fi
    \else
        \ifnum\mathgroup=5
           \mathchoice{\mbox{\boldmath$\displaystyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\textstyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\scriptstyle\mathchar"#1#2#3#4$}}%
                      {\mbox{\boldmath$\scriptscriptstyle\mathchar"#1#2#3#4$}}%
        \else
           \mathchar"#1#2#3#
        \fi
	  \fi}

\newif\ifGreekBold  \GreekBoldfalse
\let\SAVEPBF=\pbf
\def\pbf{\GreekBoldtrue\SAVEPBF}%

\@ifundefined{theorem}{\newtheorem{theorem}{Theorem}}{}
\@ifundefined{lemma}{\newtheorem{lemma}[theorem]{Lemma}}{}
\@ifundefined{corollary}{\newtheorem{corollary}[theorem]{Corollary}}{}
\@ifundefined{conjecture}{\newtheorem{conjecture}[theorem]{Conjecture}}{}
\@ifundefined{proposition}{\newtheorem{proposition}[theorem]{Proposition}}{}
\@ifundefined{axiom}{\newtheorem{axiom}{Axiom}}{}
\@ifundefined{remark}{\newtheorem{remark}{Remark}}{}
\@ifundefined{example}{\newtheorem{example}{Example}}{}
\@ifundefined{exercise}{\newtheorem{exercise}{Exercise}}{}
\@ifundefined{definition}{\newtheorem{definition}{Definition}}{}

\@ifundefined{mathletters}{%

  \newcounter{equationnumber}
  \def\mathletters{%
     \addtocounter{equation}{1}
     \edef\@currentlabel{\arabic{equation}}%
     \setcounter{equationnumber}{\c@equation}
     \setcounter{equation}{0}%
     \edef\arabic{equation}{\@currentlabel\noexpand\alph{equation}}%
  }
  \def\endmathletters{%
     \setcounter{equation}{\value{equationnumber}}%
  }
}{}

\@ifundefined{BibTeX}{%
    \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
                 T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}}{}%
\@ifundefined{AmS}%
    {\def\AmS{{\protect\usefont{OMS}{cmsy}{m}{n}%
                A\kern-.1667em\lower.5ex\hbox{M}\kern-.125emS}}}{}%
\@ifundefined{AmSTeX}{\def\AmSTeX{\protect\AmS-\protect\TeX\@}}{}%

\ifx\ds@amstex\relax
   \message{amstex already loaded}\makeatother\endinpu
\else
   \@ifpackageloaded{amstex}%
      {\message{amstex already loaded}\makeatother\endinput}
      {}
\fi
\let\DOTSI\relax
\def\RIfM@{\relax\ifmmode}%
\def\FN@{\futurelet\next}%
\newcount\intno@
\def\iint{\DOTSI\intno@\tw@\FN@\ints@}%
\def\iiint{\DOTSI\intno@\thr@@\FN@\ints@}%
\def\iiiint{\DOTSI\intno@4 \FN@\ints@}%
\def\idotsint{\DOTSI\intno@\z@\FN@\ints@}%
\def\ints@{\findlimits@\ints@@}%
\newif\iflimtoken@
\newif\iflimits@
\def\findlimits@{\limtoken@true\ifx\next\limits\limits@true
 \else\ifx\next\nolimits\limits@false\else
 \limtoken@false\ifx\ilimits@\nolimits\limits@false\else
 \ifinner\limits@false\else\limits@true\fi\fi\fi\fi}%
\def\multint@{\int\ifnum\intno@=\z@\intdots@
 \else\intkern@\fi
 \ifnum\intno@>\tw@\int\intkern@\fi
 \ifnum\intno@>\thr@@\int\intkern@\fi
 \int
\def\multintlimits@{\intop\ifnum\intno@=\z@\intdots@\else\intkern@\fi
 \ifnum\intno@>\tw@\intop\intkern@\fi
 \ifnum\intno@>\thr@@\intop\intkern@\fi\intop}%
\def\intic@{%
    \mathchoice{\hskip.5em}{\hskip.4em}{\hskip.4em}{\hskip.4em}}%
\def\negintic@{\mathchoice
 {\hskip-.5em}{\hskip-.4em}{\hskip-.4em}{\hskip-.4em}}%
\def\ints@@{\iflimtoken@
 \def\ints@@@{\iflimits@\negintic@
   \mathop{\intic@\multintlimits@}\limits
  \else\multint@\nolimits\fi
  \eat@
 \else
 \def\ints@@@{\iflimits@\negintic@
  \mathop{\intic@\multintlimits@}\limits\else
  \multint@\nolimits\fi}\fi\ints@@@}%
\def\intkern@{\mathchoice{\!\!\!}{\!\!}{\!\!}{\!\!}}%
\def\plaincdots@{\mathinner{\cdotp\cdotp\cdotp}}%
\def\intdots@{\mathchoice{\plaincdots@}%
 {{\cdotp}\mkern1.5mu{\cdotp}\mkern1.5mu{\cdotp}}%
 {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}%
 {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}}%
\def\RIfM@{\relax\protect\ifmmode}
\def\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\RIfM@\expandafter\RIfM@\expandafter\text@\else\expandafter\mbox\fi@\else\expandafter\mbox\fi}
\let\nfss@text\RIfM@\expandafter\text@\else\expandafter\mbox\fi
\def\RIfM@\expandafter\text@\else\expandafter\mbox\fi@#1{\mathchoice
   {\textdef@\displaystyle\f@size{#1}}%
   {\textdef@\textstyle\tf@size{\firstchoice@false #1}}%
   {\textdef@\textstyle\sf@size{\firstchoice@false #1}}%
   {\textdef@\textstyle \ssf@size{\firstchoice@false #1}}%
   \glb@settings}

\def\textdef@#1#2#3{\hbox{{%
                    \everymath{#1}%
                    \let\f@size#2\selectfont
                    #3}}}
\newif\iffirstchoice@
\firstchoice@true
\def\Let@{\relax\iffalse{\fi\let\\=\cr\iffalse}\fi}%
\def\vspace@{\def\vspace##1{\crcr\noalign{\vskip##1\relax}}}%
\def\multilimits@{\bgroup\vspace@\Let@
 \baselineskip\fontdimen10 \scriptfont\tw@
 \advance\baselineskip\fontdimen12 \scriptfont\tw@
 \lineskip\thr@@\fontdimen8 \scriptfont\thr@@
 \lineskiplimit\lineskip
 \vbox\bgroup\ialign\bgroup\hfil$\m@th\scriptstyle{##}$\hfil\crcr}%
\def\Sb{_\multilimits@}%
\def\endSb{\crcr\egroup\egroup\egroup}%
\def\Sp{^\multilimits@}%
\let\endSp\endSb
\newdimen\ex@
\ex@.2326ex
\def\rightarrowfill@#1{$#1\m@th\mathord-\mkern-6mu\cleaders
 \hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill
 \mkern-6mu\mathord\rightarrow$}%
\def\leftarrowfill@#1{$#1\m@th\mathord\leftarrow\mkern-6mu\cleaders
 \hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill\mkern-6mu\mathord-$}%
\def\leftrightarrowfill@#1{$#1\m@th\mathord\leftarrow
\mkern-6mu\cleaders
 \hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill
 \mkern-6mu\mathord\rightarrow$}%
\def\overrightarrow{\mathpalette\overrightarrow@}%
\def\overrightarrow@#1#2{\vbox{\ialign{##\crcr\rightarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}%
\let\overarrow\overrightarrow
\def\overleftarrow{\mathpalette\overleftarrow@}%
\def\overleftarrow@#1#2{\vbox{\ialign{##\crcr\leftarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}%
\def\overleftrightarrow{\mathpalette\overleftrightarrow@}%
\def\overleftrightarrow@#1#2{\vbox{\ialign{##\crcr
   \leftrightarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}%
\def\underrightarrow{\mathpalette\underrightarrow@}%
\def\underrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil
  $\crcr\noalign{\nointerlineskip}\rightarrowfill@#1\crcr}}}%
\let\underarrow\underrightarrow
\def\underleftarrow{\mathpalette\underleftarrow@}%
\def\underleftarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil
  $\crcr\noalign{\nointerlineskip}\leftarrowfill@#1\crcr}}}%
\def\underleftrightarrow{\mathpalette\underleftrightarrow@}%
\def\underleftrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th
  \hfil#1#2\hfil$\crcr
 \noalign{\nointerlineskip}\leftrightarrowfill@#1\crcr}}}%

\def\qopnamewl@#1{\mathop{\operator@font#1}\nlimits@}
\let\nlimits@\displaylimits
\def\setboxz@h{\setbox\z@\hbox}

\def\varlim@#1#2{\mathop{\vtop{\ialign{##\crcr
 \hfil$#1\m@th\operator@font lim$\hfil\crcr
 \noalign{\nointerlineskip}#2#1\crcr
 \noalign{\nointerlineskip\kern-\ex@}\crcr}}}}

 \def\rightarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
  $#1\copy\z@\mkern-6mu\cleaders
  \hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
  \mkern-6mu\mathord\rightarrow$}
\def\leftarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
  $#1\mathord\leftarrow\mkern-6mu\cleaders
  \hbox{$#1\mkern-2mu\copy\z@\mkern-2mu$}\hfill
  \mkern-6mu\box\z@$}

\def\qopnamewl@{proj\,lim}{\qopnamewl@{proj\,lim}}
\def\qopnamewl@{inj\,lim}{\qopnamewl@{inj\,lim}}
\def\mathpalette\varlim@\rightarrowfill@{\mathpalette\varlim@\rightarrowfill@}
\def\mathpalette\varlim@\leftarrowfill@{\mathpalette\varlim@\leftarrowfill@}
\def\mathpalette\varliminf@{}{\mathpalette\mathpalette\varliminf@{}@{}}
\def\mathpalette\varliminf@{}@#1{\mathop{\underline{\vrule\@depth.2\ex@\@width\z@
   \hbox{$#1\m@th\operator@font lim$}}}}
\def\mathpalette\varlimsup@{}{\mathpalette\mathpalette\varlimsup@{}@{}}
\def\mathpalette\varlimsup@{}@#1{\mathop{\overline
  {\hbox{$#1\m@th\operator@font lim$}}}}

\def\tfrac#1#2{{\textstyle {#1 \over #2}}}%
\def\dfrac#1#2{{\displaystyle {#1 \over #2}}}%
\def\binom#1#2{{#1 \choose #2}}%
\def\tbinom#1#2{{\textstyle {#1 \choose #2}}}%
\def\dbinom#1#2{{\displaystyle {#1 \choose #2}}}%
\def\QATOP#1#2{{#1 \atop #2}}%
\def\QTATOP#1#2{{\textstyle {#1 \atop #2}}}%
\def\QDATOP#1#2{{\displaystyle {#1 \atop #2}}}%
\def\QABOVE#1#2#3{{#2 \above#1 #3}}%
\def\QTABOVE#1#2#3{{\textstyle {#2 \above#1 #3}}}%
\def\QDABOVE#1#2#3{{\displaystyle {#2 \above#1 #3}}}%
\def\QOVERD#1#2#3#4{{#3 \overwithdelims#1#2 #4}}%
\def\QTOVERD#1#2#3#4{{\textstyle {#3 \overwithdelims#1#2 #4}}}%
\def\QDOVERD#1#2#3#4{{\displaystyle {#3 \overwithdelims#1#2 #4}}}%
\def\QATOPD#1#2#3#4{{#3 \atopwithdelims#1#2 #4}}%
\def\QTATOPD#1#2#3#4{{\textstyle {#3 \atopwithdelims#1#2 #4}}}%
\def\QDATOPD#1#2#3#4{{\displaystyle {#3 \atopwithdelims#1#2 #4}}}%
\def\QABOVED#1#2#3#4#5{{#4 \abovewithdelims#1#2#3 #5}}%
\def\QTABOVED#1#2#3#4#5{{\textstyle
   {#4 \abovewithdelims#1#2#3 #5}}}%
\def\QDABOVED#1#2#3#4#5{{\displaystyle
   {#4 \abovewithdelims#1#2#3 #5}}}%

\def\tint{\mathop{\textstyle \int}}%
\def\tiint{\mathop{\textstyle \iint }}%
\def\tiiint{\mathop{\textstyle \iiint }}%
\def\tiiiint{\mathop{\textstyle \iiiint }}%
\def\tidotsint{\mathop{\textstyle \idotsint }}%
\def\toint{\mathop{\textstyle \oint}}%
\def\tsum{\mathop{\textstyle \sum }}%
\def\tprod{\mathop{\textstyle \prod }}%
\def\tbigcap{\mathop{\textstyle \bigcap }}%
\def\tbigwedge{\mathop{\textstyle \bigwedge }}%
\def\tbigoplus{\mathop{\textstyle \bigoplus }}%
\def\tbigodot{\mathop{\textstyle \bigodot }}%
\def\tbigsqcup{\mathop{\textstyle \bigsqcup }}%
\def\tcoprod{\mathop{\textstyle \coprod }}%
\def\tbigcup{\mathop{\textstyle \bigcup }}%
\def\tbigvee{\mathop{\textstyle \bigvee }}%
\def\tbigotimes{\mathop{\textstyle \bigotimes }}%
\def\tbiguplus{\mathop{\textstyle \biguplus }}%

\def\dint{\mathop{\displaystyle \int}}%
\def\diint{\mathop{\displaystyle \iint }}%
\def\diiint{\mathop{\displaystyle \iiint }}%
\def\diiiint{\mathop{\displaystyle \iiiint }}%
\def\didotsint{\mathop{\displaystyle \idotsint }}%
\def\doint{\mathop{\displaystyle \oint}}%
\def\dsum{\mathop{\displaystyle \sum }}%
\def\dprod{\mathop{\displaystyle \prod }}%
\def\dbigcap{\mathop{\displaystyle \bigcap }}%
\def\dbigwedge{\mathop{\displaystyle \bigwedge }}%
\def\dbigoplus{\mathop{\displaystyle \bigoplus }}%
\def\dbigodot{\mathop{\displaystyle \bigodot }}%
\def\dbigsqcup{\mathop{\displaystyle \bigsqcup }}%
\def\dcoprod{\mathop{\displaystyle \coprod }}%
\def\dbigcup{\mathop{\displaystyle \bigcup }}%
\def\dbigvee{\mathop{\displaystyle \bigvee }}%
\def\dbigotimes{\mathop{\displaystyle \bigotimes }}%
\def\dbiguplus{\mathop{\displaystyle \biguplus }}%
\def\stackunder#1#2{\mathrel{\mathop{#2}\limits_{#1}}}%
\begingroup \catcode `|=0 \catcode `[= 1
\catcode`]=2 \catcode `\{=12 \catcode `\}=12
\catcode`\\=12
|gdef|@alignverbatim#1\end{align}[#1|end[align]]
|gdef|@salignverbatim#1\end{align*}[#1|end[align*]]

|gdef|@alignatverbatim#1\end{alignat}[#1|end[alignat]]
|gdef|@salignatverbatim#1\end{alignat*}[#1|end[alignat*]]

|gdef|@xalignatverbatim#1\end{xalignat}[#1|end[xalignat]]
|gdef|@sxalignatverbatim#1\end{xalignat*}[#1|end[xalignat*]]

|gdef|@gatherverbatim#1\end{gather}[#1|end[gather]]
|gdef|@sgatherverbatim#1\end{gather*}[#1|end[gather*]]

|gdef|@gatherverbatim#1\end{gather}[#1|end[gather]]
|gdef|@sgatherverbatim#1\end{gather*}[#1|end[gather*]]

|gdef|@multilineverbatim#1\end{multiline}[#1|end[multiline]]
|gdef|@smultilineverbatim#1\end{multiline*}[#1|end[multiline*]]

|gdef|@arraxverbatim#1\end{arrax}[#1|end[arrax]]
|gdef|@sarraxverbatim#1\end{arrax*}[#1|end[arrax*]]

|gdef|@tabulaxverbatim#1\end{tabulax}[#1|end[tabulax]]
|gdef|@stabulaxverbatim#1\end{tabulax*}[#1|end[tabulax*]]

|endgroup

\def\align{\@verbatim \frenchspacing\@vobeyspaces \@alignverbatim
You are using the "align" environment in a style in which it is not defined.}
\let\endalign=\endtrivlist

\@namedef{align*}{\@verbatim\@salignverbatim
You are using the "align*" environment in a style in which it is not defined.}
\expandafter\let\csname endalign*\endcsname =\endtrivlist

\def\alignat{\@verbatim \frenchspacing\@vobeyspaces \@alignatverbatim
You are using the "alignat" environment in a style in which it is not defined.}
\let\endalignat=\endtrivlist

\@namedef{alignat*}{\@verbatim\@salignatverbatim
You are using the "alignat*" environment in a style in which it is not defined.}
\expandafter\let\csname endalignat*\endcsname =\endtrivlist

\def\xalignat{\@verbatim \frenchspacing\@vobeyspaces \@xalignatverbatim
You are using the "xalignat" environment in a style in which it is not defined.}
\let\endxalignat=\endtrivlist

\@namedef{xalignat*}{\@verbatim\@sxalignatverbatim
You are using the "xalignat*" environment in a style in which it is not defined.}
\expandafter\let\csname endxalignat*\endcsname =\endtrivlist

\def\gather{\@verbatim \frenchspacing\@vobeyspaces \@gatherverbatim
You are using the "gather" environment in a style in which it is not defined.}
\let\endgather=\endtrivlist

\@namedef{gather*}{\@verbatim\@sgatherverbatim
You are using the "gather*" environment in a style in which it is not defined.}
\expandafter\let\csname endgather*\endcsname =\endtrivlist

\def\multiline{\@verbatim \frenchspacing\@vobeyspaces \@multilineverbatim
You are using the "multiline" environment in a style in which it is not defined.}
\let\endmultiline=\endtrivlist

\@namedef{multiline*}{\@verbatim\@smultilineverbatim
You are using the "multiline*" environment in a style in which it is not defined.}
\expandafter\let\csname endmultiline*\endcsname =\endtrivlist

\def\arrax{\@verbatim \frenchspacing\@vobeyspaces \@arraxverbatim
You are using a type of "array" construct that is only allowed in AmS-LaTeX.}
\let\endarrax=\endtrivlist

\def\tabulax{\@verbatim \frenchspacing\@vobeyspaces \@tabulaxverbatim
You are using a type of "tabular" construct that is only allowed in AmS-LaTeX.}
\let\endtabulax=\endtrivlist

\@namedef{arrax*}{\@verbatim\@sarraxverbatim
You are using a type of "array*" construct that is only allowed in AmS-LaTeX.}
\expandafter\let\csname endarrax*\endcsname =\endtrivlist

\@namedef{tabulax*}{\@verbatim\@stabulaxverbatim
You are using a type of "tabular*" construct that is only allowed in AmS-LaTeX.}
\expandafter\let\csname endtabulax*\endcsname =\endtrivlist

\def\@@eqncr{\let\@tempa\relax
    \ifcase\@eqcnt \def\@tempa{& & &}\or \def\@tempa{& &}%
      \else \def\@tempa{&}\fi
     \@tempa
     \if@eqnsw
        \iftag@
           \@taggnum
        \else
           \@eqnnum\stepcounter{equation}%
        \fi
     \fi
     \global\@ifnextchar*{\@tagstar}{\@tag}@false
     \global\@eqnswtrue
     \global\@eqcnt\z@\cr}

 \def\endequation{%
     \ifmmode\ifinner
      \iftag@
        \addtocounter{equation}{-1}
        $\hfil
           \displaywidth\linewidth\@taggnum\egroup \endtrivlist
        \global\@ifnextchar*{\@tagstar}{\@tag}@false
        \global\@ignoretrue
      \else
        $\hfil
           \displaywidth\linewidth\@eqnnum\egroup \endtrivlist
        \global\@ifnextchar*{\@tagstar}{\@tag}@false
        \global\@ignoretrue
      \fi
     \else
      \iftag@
        \addtocounter{equation}{-1}
        \eqno \hbox{\@taggnum}
        \global\@ifnextchar*{\@tagstar}{\@tag}@false%
        $$\global\@ignoretrue
      \else
        \eqno \hbox{\@eqnnum
        $$\global\@ignoretrue
      \fi
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\begin{document}

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{{\small E}\norm{N}{\large S}{\Large L}{\large A}\norm{P}{\small P}}
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\begin{center}
{\Large {\bf REPRESENTATIVES OF THE THOM CLASS\\[0.24cm]OF A VECTOR
BUNDLE$^{\dagger}$}}%
\\[1cm]

\vspace{10mm}

{\large Michel Bauer$^{1}$ and Frank Thuillier$^{2}$}

{\em Laboratoire de Physique Th\'eorique }{\small E}{{\normalsize {N}}}%
{\large S}{\Large L}{\large A}{{\normalsize {P}}}{\small P}\footnote{%
URA 14-36 du CNRS, associ\'ee \`a l'Ecole Normale Sup\'erieure de Lyon,
\par
\noindent
$^{\dagger}$ Work partially supported by European Community Contract
ERBCHRXCT920069
\par
\noindent
$^1$ Groupe d'Annecy: LAPP, Chemin de Bellevue BP 110, F-74941
Annecy-le-Vieux Cedex, France.
\par
\noindent
$^2$ Groupe de Lyon: ENS Lyon, 46 all\'ee d'Italie, F-69364 Lyon Cedex
07,France.}\\
\end{center}

\vspace{20mm}

\centerline{ {\bf Abstract}}

\vspace{5mm}

\indent After a review of several methods designed to produce equivariant
cohomology classes, we apply one introduced by Berline, Getzler
and Vergne to get a family of representatives of the universal Thom
class of a vector bundle. Surprisingly, this family does not contain
the representative given by Matha\"{\i} and Quillen. However it
contains the particularly simple and symmetric representative of Harvey
and Lawson.

\vfill
\rightline{\LAP-A-574/96} \rightline{ 1996}

\newpage
\pagestyle{plain} \renewcommand{\thefootnote}{\arabic{footnote}} \newpage

\pagestyle{plain} \pagenumbering{arabic}

\section{Introduction.}

In a recent paper \cite{STW94} it has been shown how equivariant
cohomology is related to the so-called (cohomological) topological
models \cite{Bl92,BS88,BS91,OSB89,W88,WBS88}. In the same work, a way
to compute some representatives of equivariant cohomology classes
(i.e. observables of the corresponding topological model) was exhibited.

Here, we shall use this method in order to generate a  family of
representatives of the Thom class of a vector bundle depending on two
arbitrary functions. As we shall see, these
representatives are quite different from the Matha\"{\i}--Quillen
representative. They offer a good deal of
flexibility at the price of being slightly complicated. Special
choices allow to find a very special representative with remarkable
symmetry properties. However, its slow decrease at infinity makes it
necessary to consider a cohomology theory with coefficients with
sufficiently fast decrease (instead of compact). Some of these
representatives (in particular the most symmetric one) already appeared in
a quite different framework in the work of Harvey and Lawson on singular
connections \cite{hl93}, a fact we learned after this work was completed.

This work is divided into three parts. In the first section we recall
basic facts about equivariant cohomology as well as the way
to compute representatives of equivariant cohomology classes. This
section parallels the explanations given in \cite{STW94}. The second
section is devoted to the Matha\"{\i}--Quillen representative of the
Thom class. Finally, the last section exhibits a large family of
representatives of the Thom class.
\\

\section{Equivariant cohomology.} \label{sec:equicohom}

Let us consider the following setting~: ${\cal M}$ is a smooth manifold
and ${\cal G}$ a connected Lie group acting smoothly on ${\cal M}$.
We would like to define a cohomology of the quotient space ${\cal M}/{\cal G}$
which coincides with the De Rham cohomology when this quotient is a smooth
manifold but which also exists when it is not, i.e. when ${\cal G}$ acts with fixed
points. Equivariant cohomology solves this problem.

Let ${\cal M}$ be a smooth manifold and $\Omega^* ({\cal M})$ the exterior
 algebra of differential forms on ${\cal M}$ endowed with the differential
$d_{{\cal M}}$. A Lie group ${\cal G}$ is assumed to be acting on ${\cal
M}$ as well as its Lie algebra, denoted $Lie{\cal G}$. For any $\lambda
\in Lie {\cal G}$ there is a vector field $\lambda _{{\cal M}}$
representing the infinitesimal action of $\lambda $ on ${\cal M}$.
This vector field $\lambda _{{\cal M}}$ is usually called the
fundamental vector field associated with $\lambda $. We shall denote
by $i_{{\cal M}}(\lambda )=i_{{\cal M}}(\lambda _{{\cal M}})$ and
$l_{{\cal M}}(\lambda )=l_{{\cal M}}(\lambda _{{\cal M}})=\left[
d_{{\cal M}},i_{{\cal M}}(\lambda )\right] _{+}$ the contraction
(or inner derivative) and Lie derivative acting on $\Omega^* ({\cal M})$.
Let us recall that $i_{{\cal M}}(\lambda )$ takes n-forms into (n-1)-forms
while $l_{{\cal M}}(\lambda )$ acts on forms without changing the degree.
Elements of $\Omega^* ({\cal M})$ which are annihilated by both $i_{{\cal
M}}(\lambda )$ and $l_{{\cal M}}(\lambda )$, for any $\lambda \in Lie{\cal
G}$, are the so-called {\bf basic} elements of $\Omega^* ({\cal M})$ for the
action of ${\cal G}$. As $d_{{\cal M}}$ maps basic elements into basic
elements, this leads to the definition of the {\bf basic
cohomology} of ${\cal M}$ for the action of ${\cal G}$ \cite{C50}.

We now consider the Weil algebra ${\cal W(\QTR{mathcal}{G})}$ of $Lie{\cal
G}$ \footnote{This is a harmless abuse of notation, but it is to be
remembered that equivariant cohomology deals only with the local
structure of ${\cal G}$.}. It is a graded differential algebra
generated by two $Lie{\cal G}$-valued indeterminates, the
``connection'' $\omega $, of degree 1, and its
``curvature'' $\Omega $, of degree 2, such that~:
\begin{equation}
\Omega =d_{{\cal W}}\omega +\dfrac 12\left[ \omega ,\omega \right]
\label{2.0}
\end{equation}
where $d_{{\cal W}}$ is the differential of ${\cal
W(\QTR{mathcal}{G})}$. Of course, one has the Bianchi identity~:
\begin{equation}
d_{{\cal W}}\Omega +\left[ \omega ,\Omega \right] =0 \label{2.1}
\end{equation}
There is an action $i_{{\cal W}}(\lambda )$, $l_{{\cal
W}}(\lambda )$ for $\lambda \in Lie{\cal G}$~:
\begin{eqnarray}
i_{{\cal W}}(\lambda )\omega &=&\lambda \qquad l_{{\cal W}}(\lambda )\omega
=-[\lambda ,\omega ] \label{2.3} \\
i_{{\cal W}}(\lambda )\Omega &=&0 \qquad l_{{\cal W}}(\lambda )\Omega =-[\lambda
,\Omega ] \label{2.4}
\end{eqnarray}
For instance, $\omega $ may be a connection on a principal
${\cal G}$-bundle $\Pi $ and $\Omega $ its curvature. In that case
$i_{{\cal W}}(\lambda )$ and $l_{{\cal W}}(\lambda )$ are generated by
the action of ${\cal G}$ on $\Pi $, and in this case
 ${\cal W(\QTR{mathcal}{G})}$ will be referred to as ${\cal W}_\Pi $.

We now consider the graded differential algebra $(\Omega^* ({\cal
M})\otimes {\cal W(\QTR{mathcal}{G})},d_{{\cal
M}}+d_{{\cal W}})$, on which the operations $i_{{\cal M}}+i_{{\cal
W}}$ and $l_{{\cal M}}+l_{{\cal W}}(\lambda )$ for any $\lambda
\in Lie{\cal G}$ are well-defined. There common
kernel is a graded differential subalgebra of $\Omega^* ({\cal M})\otimes {\cal
W(\QTR{mathcal}{G})}$. By definition, the so-called {\bf
equivariant cochains} are the elements of this subalgebra annihilated
by the differential $d_{{\cal M}}+d_{{\cal W}}$, leading to the
{\bf equivariant cohomology} of ${\cal M}$ for the action of ${\cal G}$~:
this is the so-called  {\bf Weil model} for equivariant cohomology.

Equivariant cohomology can be alternatively described in the so-called
{\bf intermediate model}, which was introduced in \cite{K93} and which will be
repeatedly used in the sequel. It is obtained from the Weil model via of
the following algebra isomorphism \footnote{See \cite{DV93} for a more general theorem.}~:
\begin{equation}
x\longmapsto \exp \left\{ -i_{{\cal M}}(\lambda )\right\} x \label{2.5}
\end{equation}
for any $x\in $ $\Omega^* ({\cal M})\otimes {\cal W(\QTR{mathcal}{G})}$.
This isomorphism changes the original differential and operations on
$\Omega^* ({\cal M})\otimes {\cal W(\QTR{mathcal}{G})}$ by conjugation~:
\begin{eqnarray}
d_{{\cal M}}+d_{{\cal W}} &\longrightarrow &D_{int}^{}=d_{{\cal
M}}+d_{{\cal W}}+l_{{\cal M}}(\omega )-i_{{\cal M}}(\Omega )) \label{2.6} \\
(i_{{\cal M}}+i_{{\cal W}})(\lambda ) &\longrightarrow &i_{{\cal W}}(\lambda
)=e^{-i_{{\cal M}}(\lambda )}(i_{{\cal M}}+i_{{\cal W}})(\lambda )e^{i_{
{\cal M}}(\lambda )} \label{2.7} \\
(l_{{\cal M}}+l_{{\cal W}})(\lambda ) &\longrightarrow &(l_{{\cal M}}+l_{
{\cal W}})(\lambda )=e^{-i_{{\cal M}}(\lambda )}(l_{{\cal M}}+l_{{\cal W}
})(\lambda )e^{i_{{\cal M}}(\lambda )} \label{2.8}
\end{eqnarray}

Finally, the so-called {\bf Cartan model} is obtained from the
intermediate model by putting $\omega =0$ so that $D_{int}^2\mid
_{\omega =0}$ vanishes when restricted to invariant cochains. This is
the most popular model, although many calculations are better
automatized in the intermediate model.

Another item which will be repeatedly used is ''Cartan's theorem 3'' \cite
{C50}~: let us assume that $(\Omega^* ({\cal M}),d_{{\cal M}},i_{{\cal
M}},l_{{\cal M}})$ admits a ${\cal G}$-connection $\theta$ \footnote{
that is to say a $Lie{\cal G}$-valued 1-form on ${\cal M}$ such that $i_{
{\cal M}}(\lambda )\theta =\lambda $ and $l_{{\cal M}}(\lambda )\theta
=-\left[ \lambda ,\theta \right] $ for any $\lambda \in Lie{\cal G}$.},
with curvature $\Theta $. Then any equivariant cohomology class of $\Omega^*
({\cal M})\otimes {\cal W(\QTR{mathcal}{G})}$ with representative $P(\omega
,\Omega )$ gives rise canonically to a basic cohomology class of $\Omega
({\cal M})$ with representative $P(\theta ,\Theta )$. There is a
simple proof using the homotopy that expresses the triviality of the
cohomology of the Weil algebra \cite{MSZ85}. It follows from the
construction that the cohomology class of $P(\theta ,\Theta )$ does not
depend on $\theta $.

One convenient way to produce equivariant cohomology classes is as follows
\cite{BGV91}~: we consider an $H$-bundle ${\cal P}({\cal M},H)$ over {\cal M}
on which there exists an action of ${\cal G}$ which lifts the action of ${\cal G}$
on ${\cal M}$. In general, the Lie group $H$ has nothing to do with the Lie
group ${\cal G}$. As before, ${\cal P}({\cal M},H)$ is endowed with a
differential $d_{{\cal P}}$, a contraction $i_{{\cal P}}$ and a Lie
derivative $l_{{\cal P}}$.

Next, let $\Gamma $ be a ${\cal G}$-invariant $H$-connection on ${\cal
P}({\cal M},H)$~:
\begin{equation}
l_{{\cal P}}(\lambda )\Gamma =0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\quad ,\quad for any }\lambda \in
Lie%
{\cal G} \label{2.9}
\end{equation}

The pull-back $\hat{\Gamma }$ of $\Gamma $ on $\Omega^* ({\cal M})\otimes
{\cal W(\QTR{mathcal}{G})}$ is a 1-form on ${\cal P}({\cal M},H)$ and
a 0-form in ${\cal W(\QTR{mathcal}{G})}$. It follows that~:
\begin{equation}
i_{{\cal W}}(\lambda )\hat{\Gamma }=0 \label{2.10}
\end{equation}
for any $\lambda \in Lie{\cal G}$.

In $\Omega^* ({\cal M})\otimes {\cal W(\QTR{mathcal}{G})}$, the equivariant
curvature of $\hat{\Gamma }$ is defined by~:
\begin{equation}
R_{int}^{eq}(\hat{\Gamma },\omega ,\Omega )=D_{int}^{}\hat{\Gamma }+\dfrac
12\left[ \hat{\Gamma },\hat{\Gamma }\right] \label{2.11}
\end{equation}
where $D_{int}^{}=d_{{\cal W}}+d_{{\cal P}}+l_{{\cal P}}(\omega
)-i_{{\cal P}}(\Omega )$. Then, if $I_H$ is a symmetric invariant
polynomial on $LieH$, we consider the $H$-characteristic class
$I_{H,int}^{eq}(\hat{\Gamma },\omega ,\Omega )=I_H\left(
R_{H,int}^{eq}(\hat{\Gamma },\omega ,\Omega )\right) $. It is defined
on ${\cal M}$ and fulfills~:
\begin{eqnarray}
\left( d_{{\cal W}}+d_{{\cal M}}+l_{{\cal M}}(\omega )-i_{{\cal M}}(\Omega
)\right) I_{H,int}^{eq}(\hat{\Gamma },\omega ,\Omega ) &=&0 \label{2.12}
\\
i_{{\cal W}}(\lambda )\ I_{H,int}^{eq}(\hat{\Gamma },\omega ,\Omega ) &=&0
\label{2.13} \\
(l_{{\cal W}}+l_{{\cal M}})(\lambda )\ I_{H,int}^{eq}(\hat{\Gamma },\omega
,\Omega ) &=&0 \label{2.14}
\end{eqnarray}
for any $\lambda \in Lie{\cal G}$.

In the Weil model, the equivariant curvature is defined by~:
\begin{equation}
R_W^{eq}(\hat{\Gamma },\omega ,\Omega )=(d_{{\cal W}}+d_{{\cal
P}})\hat{\Gamma }+\dfrac 12\left[ \hat{\Gamma }+i_{{\cal P}}(\Omega
)\hat{\Gamma },\hat{\Gamma } +i_{{\cal P}}(\Omega )\hat{\Gamma }\right]
\label{2.15}
\end{equation}

We may similarly consider~:
\begin{equation}
I_{H,W}^{eq}(\hat{\Gamma },\omega ,\Omega )=I_H\left(
R_{H,W}^{eq}(\hat{\Gamma },\omega ,\Omega )\right) =e^{-i_{{\cal M}}(\lambda
)}I_{H,int}^{eq}(\hat{\Gamma },\omega ,\Omega ) \label{2.16}
\end{equation}
which fulfills~:
\begin{eqnarray}
(d_{{\cal W}}+d_{{\cal M}})\ I_{H,W}^{eq}(\hat{\Gamma },\omega ,\Omega )
&=&0 \label{2.17} \\
(i_{{\cal W}}+i_{{\cal M}})(\lambda )\ I_{H,W}^{eq}(\hat{\Gamma },\omega
,\Omega ) &=&0 \label{2.18} \\
(l_{{\cal W}}+l_{{\cal M}})(\lambda )\ I_{H,W}^{eq}(\hat{\Gamma },\omega
,\Omega ) &=&0 \label{2.19}
\end{eqnarray}
for any $\lambda \in Lie{\cal G}$.

Finally, if ${\cal M}$ admits a ${\cal G}$-connection $\theta $ with
curvature $\Theta $, we can apply ''Cartan's theorem 3'', and substitute
$\theta $ and $\Theta $ instead of $\omega $ and $\Omega $ in
$I_{H,W}^{eq}(\hat{\Gamma },\omega ,\Omega )$, so that~:
\begin{eqnarray}
d_{{\cal M}}\ I_{H,W}^{eq}(\hat{\Gamma },\theta ,\Theta ) &=&0
\label{2.20} \\
i_{{\cal M}}(\lambda )\ I_{H,W}^{eq}(\hat{\Gamma },\theta ,\Theta ) &=&0
\label{2.21} \\
l_{{\cal M}}(\lambda )\ I_{H,W}^{eq}(\hat{\Gamma },\theta ,\Theta ) &=&0
\label{2.22}
\end{eqnarray}
for any $\lambda \in Lie{\cal G}$.

By standard arguments, these cohomology classes do not depend either on $%
\hat{\Gamma }$ or on $\theta $.

\section{The Thom Class of a Vector Bundles~: the Matha\"{\i}--Quillen
strategy \protect\cite{MQ86}.}

Let $V$ be a real oriented Euclidean vector space of dimension $n=2d$ with
scalar product $(\ ,\ )_V$. On $V$, we choose a canonical basis
$\left\{ \vec{e}_k\right\} $ orthonormal with respect to $(\ ,\ )_V$~:
\begin{equation}
(\vec{e}_i,\vec{e}_j)_V=\delta _{ij} \label{3.1}
\end{equation}
Any vector on $V$ can be  decomposed as~:
\begin{equation}
\vec{v}=v^k\vec{e}_k \label{3.2}
\end{equation}
Such a decomposition gives a coordinates system $(v^k)$ on $V$,
turning $V$ into a manifold. Due to the linear space structure of $V$, only
$GL(n,{\Bbb R})$ transformations define allowed coordinate changes. The
group of isometries of $V$, with respect to $(\ ,\ )_V$, is
$SO(n)\subset GL(n,{\Bbb R})$, with Lie algebra $so(n)$ and Weil
algebra ${\cal W}(SO(n))$. Finally, we endow $V$ and ${\cal W}(SO(n))$
with the standard differential operations $d_V$, $i_V$, $l_V$, $d_{{\cal W}}$,
$i_{{\cal W}}$ and $l_{{\cal W}}$.

Now, let $E({\cal M},V)$ be a vector bundle over a smooth manifold ${\cal
M}$ with typical fiber $V$, equipped with differential operations~: $d_E$,
$i_E$ and $l_E$,. We denote $\Omega _{rdv}^n(E)$ the space of n-forms on $E$
whose restriction to each fiber of $E$ is rapidly decreasing. The
corresponding cohomology space is written $H_{rdv}^n(E)$. The Thom
Class of $E$ is the element T$(E)$ of $H_{rdv}^n(E)$ such that~:
\begin{equation}
\int_VT(E)=1 \label{3.3}
\end{equation}
which means that integration of T$(E)$ along the fiber produces
the constant function 1 on ${\cal M}$.

Actually, following Matha\"{\i} and Quillen \cite{MQ86}, we would like to
exhibit a representative of T$(E)$ in the form of an integral representation.
Then, we consider $V^{*}$, the
dual space of $V$, equipped with the scalar product $(\ ,\ )_{V^{*}}$,
dual to $(\ ,\ )_V$ on $V$. Moreover, we introduce coordinates $(\varpi _k)$
for the Grassmann algebra $\Lambda V^{*}$ of $V^{*}$ together with the
differential operations $\delta $, $I$ and $L$, dual to those on
$V$.

We take as structure equations~:
\begin{equation}
\begin{array}{rcl} \label{eq:struceq}
s^{top}v^k &=&\Psi ^k+L^{top}(\omega )v^k  \\
s^{top}\Psi ^k &=&-L^{top}(\Omega )v^k+L^{top}(\omega )\Psi ^k
 \\
s^{top}\varpi _k &=&b_k+L^{top}(\omega )\varpi _k  \\
s^{top}b_k &=&-L^{top}(\Omega )\varpi _k+L^{top}(\omega )b_k
\\
s^{top}\omega &=&\Omega -\dfrac 12\left[ \omega ,\omega \right]
 \\
s^{top}\Omega &=&-\left[ \omega ,\Omega \right]
\end{array}
\end{equation}
with~:
\begin{eqnarray}
s^{top} &=&d_{{\cal W}}+(d_V+\delta )+(l_V+L)(\omega )-(i_V+I)(\Omega )
\label{3.10} \\
\Psi ^k &=&d_Vv^k\equiv \Psi _{int}^k \label{3.11}
\end{eqnarray}
in the intermediate model, and~:
\begin{eqnarray}
s^{top} &=&d_{{\cal W}}+d_V+\delta \label{3.12} \\
\Psi ^k &=&(d_V-L^{top}(\omega ))v^k\equiv \Psi _W^k \label{3.13}
\end{eqnarray}
in the Weil model, while~:
\begin{equation}
L^{top}=l_V+L \label{3.14}
\end{equation}
in any model.

The null section $s_0$ of $E({\cal M},V)$ that sends any point of ${\cal
M}$ into the null vector, diffeomorphically maps ${\cal M}$ into $s_0({\cal M}
)\subset E$. Then, the Thom Class T$(E)$ of $E$ is nothing but the
Poincar\'{e} dual of $s_0({\cal M})$ in $E$ \cite{BT82}, and the Dirac
form on $E$~:
\begin{equation}
\delta (\vec{v})dv^1\wedge ...\wedge dv^n \label{3.15}
\end{equation}
represents the Poincar\'{e} dual of $s_0({\cal M})$ in $E$. This
form can be written as a Fourier transform~:
\begin{equation}
\frac 1{(2\pi )^n}\int dbd\varpi \exp i\left\{ b.\vec{v}+\varpi .\Psi
\right\} =\frac 1{(2\pi )^n}\int dbd\varpi \exp i\left\{ b_kv^k+\varpi
_k\Psi ^k\right\} \label{3.16}
\end{equation}
From the structure equations (\ref{eq:struceq}), we deduce~:
\begin{equation}
b.\vec{v}+\varpi .\Psi =s^{top}(\varpi .\vec{v}) \label{3.17}
\end{equation}
However, we can consider a smoother representative, with a
gaussian behavior for instance. That means that we must insert a term of
the form~:
\begin{equation}
i(b,b)_{V^{*}} \label{3.18}
\end{equation}
into (\ref{3.17}). Now, we can try to write the new argument as a
$s^{top}$-exact term~:
\begin{equation}
s^{top}(\varpi .\vec{v}+i(\varpi ,b)_{V_{*}})=b.\vec{v}+\varpi .\Psi
+i(b,b)_{V^{*}}-i(L^{top}(\Omega )\varpi ,\varpi )_{V^{*}} \label{3.19}
\end{equation}
so that we are led to define~:
\begin{equation}
U=\frac 1{(2\pi )^n}\int dbd\varpi \exp i\left\{ s^{top}(\varpi .\vec{v}%
+i(\varpi ,b)_{V_{*}})\right\} \label{3.20}
\end{equation}
Note that U is an element of ${\cal W}(SO(n))\otimes \Omega^* (V)$.

In order to prove that U maps into a representative of T$(E)$, let us proceed in
the intermediate model where we write $U_{int}$ instead of $U$. Then,
since in (\ref{3.20}) $\omega \in {\cal W}(SO(n))$ does not appear, we
immediately conclude that $U_{int}$ does not explicitly depend on
$\omega $, that is to say~:
\begin{equation}
\forall \lambda \in so(n)\qquad i_{{\cal W}}(\lambda )U_{int}=0
\label{3.21}
\end{equation}
which express the basicity condition within the intermediate
model. Now, there remains to show that $U_{int}$ is closed with respect to
$D_{int}=d_{{\cal W}}+d_V+l_V(\omega )-i_V(\Omega )$. Indeed~:
\begin{eqnarray}
D_{int}U_{int} &=&\frac 1{(2\pi )^n}D_{int}\int dbd\varpi \exp \left\{
i.s^{top}(\varpi .\vec{v}+i(\varpi ,b)_{V_{*}})\right\} \label{3.22} \\
&=&\frac 1{(2\pi )^n}\int dbd\varpi \left( s^{top}-D_{V^{*}}\right) \exp
\left\{ i.s^{top}(\varpi .\vec{v}+i(\varpi ,b)_{V_{*}})\right\}
\label{3.23}
\end{eqnarray}
where $D_{V^{*}}=\delta +L(\omega )-I(\Omega )$. Hence~:
\begin{equation}
D_{int}U_{int}=-\frac 1{(2\pi )^n}\int dbd\varpi \left[ D_{V^{*}}\exp
\left\{ i.s^{top}(\varpi .\vec{v}+i(\varpi ,b)_{V_{*}})\right\} \right]
\label{3.24}
\end{equation}
Now, from the structure equations (\ref{eq:struceq}), we get~:
\begin{eqnarray}
D_{V^{*}} &=&\left( b_k+L^{top}(\omega )\varpi _k\right) \dfrac \partial
{\partial \varpi _k}+\left( -L^{top}(\Omega )\varpi _k+L^{top}(\omega
)b_k\right) \dfrac \partial {\partial b_k} \label{3.25} \\
&=&\left( L^{top}(\omega )\varpi _k\dfrac \partial {\partial \varpi
_k}+L^{top}(\omega )b_k\dfrac \partial {\partial b_k}\right) +\left(
b_k\dfrac \partial {\partial \varpi _k}-L^{top}(\Omega )\varpi _k\dfrac
\partial {\partial b_k}\right) \label{3.26}
\end{eqnarray}
The first term in $D_{V^{*}}$ corresponds to an
$so(n)$-transformation. Due to the $so(n)$-invariance of the
measure $dbd\varpi $, it does not contribute to (\ref{3.24}). The last
term in (\ref{3.26}) vanishes upon integration by parts. Then~:
\begin{equation}
D_{int}U_{int}=0 \label{3.27}
\end{equation}
Finally, combining equations (\ref{3.21}) and (\ref{3.27}), we deduce that~:

\begin{equation}
\forall \lambda \in so(n)\qquad (l_{{\cal W}}+l_V)(\lambda )U_{int}=0
\label{3.28}
\end{equation}
and conclude that $U_{int}$ is a representative in ${\cal W}
(SO(n))\otimes \Omega^* (V)$ of the Thom Class of $E({\cal M},V)$. The
corresponding representative in the Weil model is obtained by setting~:
\begin{eqnarray}
s^{top} &=&d_{{\cal W}}+d_V+\delta \label{3.29} \\
\Psi ^k &=&(d_V-L^{top}(\omega ))v^k\equiv \Psi _W^k \label{3.30}
\end{eqnarray}
within equation (\ref{3.20}).

\indent

Actually, it can be easily shown that Fourier transform (denoted ${\cal F}$) commutes with
equivariant differential operations. More precisely :

\begin{eqnarray}
{\cal F}\left[ \left( d_{{\cal W}}+\delta +L(\omega )-I(\Omega )\right) \Phi
\right] &=&\left( d_{{\cal W}}+d_V+l_V(\omega )-i_V(\Omega )\right)
{\cal F}\left[
\Phi \right] \label{3.31} \\
{\cal F}\left[ i_{{\cal W}}(\lambda )\Phi \right] &=&i_{{\cal W}}(\lambda
){\cal F}\left[ \Phi \right] \\
{\cal F}\left[ (l_{{\cal W}}+L)(\lambda )\Phi \right] &=&(l_{{\cal W}%
}+l_V)(\lambda ){\cal F}\left[ \Phi \right]
\end{eqnarray}
in the intermediate model. The same holds in the Weil model with
suitable differentials. Let us point out that this mainly relies on the
identity $b.\vec{v}+\varpi .\Psi =s^{top}(\varpi .\vec{v})$.

Then, since $\phi =(b,b)_{V^{*}}+(L^{top}(\Omega )\varpi ,\varpi
)_{V^{*}}$ is equivariant, it is straightforward to find that its
Fourier transform is also equivariant. This simple remark allows to
construct representatives of equivariant cohomology classes using
Fourier transform of functions of $\phi$.

\indent

Finally, we can consider a principal $SO(n)$-bundle $P$ over ${\cal M}$.
It is well known that $P\times _{SO(n)}V$ is a vector bundle isomorphic to
$E$, and $P\times V$ is called the principal $SO(n)$-bundle associated with
$E({\cal M},V)$. Hence, as an n-form on $E$, any representative of the Thom
Class T$(E)$ of $E$ comes from a closed $SO(n)$-basic $n$-form on the
associated bundle $P\times V$ of $E$. In order to produce such a representative of
T$(E)$, we use Cartan's Theorem 3, that is to say we replace $(\omega
,\Omega ) $ (in the representative $U$) by $(\theta ,\Theta )$, a
connection and its curvature on $P({\cal M},SO(n))$.

\section{Construction of Representatives of Thom Class of Vector Bundles~:
the Berline--Getzler--Vergne strategy \protect\cite{BGV91}.}

In this section, we shall use the strategy explained in section 2) in
order to produce representatives of T$(E)$.

To begin with, we are going to turn $V$ into a Riemannian manifold, i.e. a
manifold $V$ with a metric. The tangent bundle of $V$, denoted by $TV$, is
obviously isomorphic to $V\times V$. The only $SO(n)$-invariants formed
with $\vec{v}$ and $d\vec{v}$ are the three scalar products,
so that the general $SO(n)$-invariant metric on $V$ is~:

\begin{equation}
ds^2(\vec{v})=e^{\varphi }\left( (dv^i)^2+\sigma (v^idv^i)^2\right)
\label{4.1}
\end{equation}
where $\varphi $ and $\sigma $ are smooth functions of
$t=(\vec{v},\vec{v})_V$ only. The above expression is positive
definite if and only if $1+\sigma (t)t > 0$ for $t \geq 0$. One can
assume if convenient that the metric is asymptotically flat (i.e. that
the curvature vanishes at infinity).

We can consider the principal $GL(n,{\Bbb R})$-bundle associated with
$TV$, i.e. the frame bundle $R(V)$ of $V$. It is made of the points
$(\vec{v},b_{\vec{v}})$ where $b_{\vec{v}}$ is a frame (i.e. a basis)
at $\vec{v}$. Coordinates for $b_{\vec{v}}$ are defined as
follows. We denoted by $(\partial _k)$ the natural basis of $T_{
\vec{v}}V$ defined by the canonical coordinates $(v^k)$ of $V$~: $\partial
_k=\frac \partial {\partial v^k}$. Then, the coordinates of $b_{\vec{v}}$
are the components $b_k^j$ of the decomposition of $b_{\vec{v}}$ with
respect to the natural basis $(\partial _k)$~:
\begin{equation}
b_{\vec{v}\ k}=b_k^j\partial _j \label{4.2}
\end{equation}
with $b_{\vec{v}\ k}$ the $k$-th frame vector of the frame
$b_{\vec{v}}$. The isometry group of $(V,(\ ,\ )_V)$, namely $SO(n)$,
acts both on elements of $V$ and on frames, that is to say on $R(V)$.
This goes as follows. For any $\Phi \in SO(n)$~:
\begin{equation}
\Phi ^k(\vec{v})=\Phi _m^kv^m \label{4.3}
\end{equation}

At the infinitesimal level, if we write $\Phi _m^k=\delta _m^k+\varphi
_m^k$, we get~:
\begin{equation}
\Phi ^k(\vec{v})=\left( \delta _m^k+\varphi _m^k\right) v^m=v_{}^k+\varphi
_m^kv^m=v_{}^k+\xi ^k \label{4.4}
\end{equation}
where $\xi ^k=$ $\varphi _m^kv^m$ defines a vector field on $V$,
the so-called fundamental vector field associated with the action of $
\varphi \in so(n)$~:
\begin{equation}
\xi =\xi ^k\dfrac \partial {\partial v^k} \label{4.5}
\end{equation}
The natural action of $\Phi \in SO(n)$ on $T_{\vec{v}}V$,
is given by the so-called differential of $\Phi $ at $\vec{v}$,
$d_{\vec{v}}\Phi :T_{\vec{v}}V\rightarrow T_{\Phi (\vec{v})}V$~:
\begin{equation}
\forall X_{\vec{v}}\in T_{\vec{v}}V\quad ,\quad \forall f\in C^\infty
(V)\quad \quad d_{\vec{v}}\Phi X_{\vec{v}}\ (f)=X_{\vec{v}}(f\circ \Phi )
\label{4.6}
\end{equation}
Applying this definition to the frame vectors $b_{\vec{v}\ k}$,
one gets~:
\begin{equation}
\tilde b_j^i=b_j^m\left( \partial _m\Phi ^i(\vec{v})\right) =b_j^m\Phi
_m^i
\label{4.7}
\end{equation}
where $\tilde b_j^i$ are the coordinates of the transformed frame
at $\Phi (\vec{v})$. At the infinitesimal level, for $\varphi \in so(n)$~:

\begin{equation}
\tilde b_j^i=b_j^m\left( \delta _m^i+\varphi _m^i\right)
=b_j^i+b_j^m\varphi _m^i=b_j^i+b_j^m\varphi _m^i=b_j^i+\Xi _j^i
\label{4.8}
\end{equation}
Combining equations (\ref{4.4}) and (\ref{4.8}), we deduce that the fundamental
vector field associated with the action of $\varphi \in so(n)$ on $R(V)$
reads~:
\begin{equation}
\lambda _R=\xi ^k\dfrac \partial {\partial v^k}+\Xi _q^p\dfrac \partial
{\partial b_q^p}=\varphi _m^kv^m\dfrac \partial {\partial
v^k}+b_q^m\varphi
_m^p\dfrac \partial {\partial b_q^p} \label{4.9}
\end{equation}

Now, let $P({\cal M},SO(n))$ be some principal $SO(n)$-bundle over a
smooth manifold ${\cal M}$. It is well known that there is a vector
bundle over ${\cal M}$ associated with $P$ for the action of $SO(n)$
on $V$. The group $SO(n)$ acts on the right on $P$ and on the left on
$V$. We first define a right-action of $SO(n) $ on $P\times V$ by setting~:
\begin{equation}
(p,\vec{v})^\Phi =(p.\Phi ,\Phi ^{-1}(\vec{v})) \label{4.10}
\end{equation}
so that, the fundamental vector field representing the action of
$\varphi \in so(n)$ on $P\times V$ reads~:
\begin{equation}
\lambda _{P\times V}=\lambda _P-\xi ^k\dfrac \partial {\partial
v^k}=\lambda
_P-\varphi _m^kv^m\dfrac \partial {\partial v^k} \label{4.11}
\end{equation}
where $\lambda _P$ is the fundamental vector field representing
the action of $\varphi $ on $P$.

Finally, the action of any $\varphi \in so(n)$ on the $GL(n,{\Bbb
R})$-principal bundle $P\times R(V)$, is given by following
fundamental vector field~:
\begin{equation}
\lambda =\lambda _P-\lambda _R \label{4.12}
\end{equation}
with $\lambda _R$ defined in equation (\ref{4.9}).

In the following, $V$, $R(V)$ and $P$ are equipped with the following
differential operations~: $d_V$, $d_R$, $d_P$, $i_V$, $i_R$, $i_P$, $l_V$,
$l_R$ and $l_P$, respectively exterior differentials, inner products and
Lie derivatives.

Now, since we are looking for representatives of equivariant cohomology
classes, we can mimic the construction made in \cite{STW94} in the case of
two-dimensional Gravity. We first look for a $GL(n,{\Bbb R})$-connection
on ${\cal P}(P\times V,GL(n,{\Bbb R}))=P\times R(V)$ invariant under the
action of $SO(n)$. If we notice that, by construction, the metric {\bf
g} on $V$ is $SO(n)$-invariant, we can consider the Levi-Cevita
connection $^{LC}\Gamma $ associated with {\bf g}. Due to the
$SO(n)$-invariance of {\bf g}, $^{LC}\Gamma $ is an $SO(n)$-invariant
connection. More precisely, the lift of $^{LC}\Gamma $ into a
connection one form $\Gamma $ on $R(V)$ according to~:
\begin{equation}
\Gamma =b^{-1}(^{LC}\Gamma )b+b^{-1}d_Rb \label{4.13}
\end{equation}
is invariant under the action of $SO(n)$. The fundamental vector
field for the action of $so(n)$ was given before, so that~:
\begin{equation}
\left( i_{{\cal P}}(\lambda )\Gamma \right) _\tau ^\sigma =(b^{-1})_\nu
^\sigma \left( -\ ^{LC}D_\mu \xi ^\nu \right) b_\tau ^\mu \label{4.14}
\end{equation}
where $i_{{\cal P}}(\lambda )=(i_P+i_R)(\lambda )$, and~:
\begin{equation}
l_{{\cal P}}(\lambda )\Gamma =0 \label{4.15}
\end{equation}
with $l_{{\cal P}}(\lambda )=(l_P+l_R)(\lambda )$.

The next step is to consider the Weil algebra ${\cal W}(SO(n))$ of
$so(n)$. The relevant formul\ae\ were given in section
\ref{sec:equicohom}. We recall that  the equivariant curvature of $\Gamma
$ in the intermediate model is~:

\begin{equation}
R_{int}^{eq}(\Gamma ,\omega ,\Omega )=\left( d_{{\cal W}}+d_{{\cal
P}}+l_{{\cal P}}(\omega )-i_{{\cal P}}(\Omega )\right) \Gamma +\dfrac
12\left[\Gamma ,\Gamma \right] \label{4.20}
\end{equation}
while the corresponding curvature in the Weil model is obtained as~:
\begin{equation}
R_W^{eq}(\Gamma ,\omega ,\Omega )=e^{i_{{\cal P}}(\omega
)}R_{int}^{eq}(\Gamma ,\omega ,\Omega ) \label{4.21}
\end{equation}
which gives~:
\begin{equation}
R_W^{eq}(\Gamma ,\omega ,\Omega )=R(\Gamma )+i_{{\cal P}}(\omega )R(\Gamma
)+\dfrac 12i_{{\cal P}}(\omega )i_{{\cal P}}(\omega )R(\Gamma )-i_{{\cal
P}}(\Omega )\Gamma \label{4.22}
\end{equation}
The Weil equivariant Euler class is defined by~:

\begin{equation}
E_W^{eq}=\dfrac{\varepsilon ^{\mu _1\rho _1...\mu _d\rho _d}}{\sqrt{{\bf
g}}}g_{\rho _1\upsilon _1}...g_{\rho _d\upsilon _d}\left( R_W^{eq}\right)
_{\mu_1}^{\upsilon _1}\wedge ...\wedge \left( R_W^{eq}\right) _{\mu
d}^{\upsilon_d} \label{4.23}
\end{equation}
which after normalization gives rise to a representative of T$(E)$ in
$P({\cal M},SO(n))\times V$.

It is now time to use the explicit form of the metric to get a formula
for the Thom class. Surprisingly, we shall see there is no choice of
metric that allows to recover the Matha\"{\i}--Quillen representative of
T$(E)$. From now on, the computations, if painful, are
straightforward. We use the intermediate model so that $d_V v^i\equiv
\Psi^i$. As (\ref{4.13}) looks formally like a change of
coordinates in the fiber, we know that its effect on curvature will
be a simple conjugation which disappears completely on the Thom class.
So we can forget it in the computation. From (\ref{4.1}) we find that
the metric is~:

\begin{equation}
g_{ij}=e^{\varphi }(\delta_{ij}+\sigma v_iv_j)
\end{equation}

Our notations need some comment~: we start with global coordinates
$v^i$ on $V$, so the exponent $i$ is not a tensor component but just a
label. The metric is expressed with respect to this particular
coordinate system. However it is convenient to deal consistently with
formal lower and upper indices in the Einstein summation
convention. So we define $v_i \equiv v^i$ and
$\delta_{ij}\equiv\delta^{ij}\equiv\delta^i_j\equiv\delta_i^j=1$ if
$i=j$ and $0$ else. For instance we use the notation $v_i$ and
$\delta_{ij}$ in $g_{ij}$ and we write $t=v_iv^i$. This becomes
slightly less formal if we restrict the diffeomorphism group of $V$ to
linear orthogonal transformations.

A simple computation shows that the inverse metric is

\begin{equation}
g^{ij}=e^{-\varphi }(\delta^{ij}+\tilde{\sigma} v^iv^j)
\end{equation}
where $\tilde{\sigma}$ is defined by $(1+t\tilde{\sigma} )(1+t\sigma )=1$.

First, we need a formula for the connection and curvature. The fact
that $\varphi$ and $\sigma$ depend only on $t$ leads to many simplifications
in the computation. We use dots for derivatives with respect to $t$.

With the expression of $\bf g$, we get for the
connection \footnote{Remember that $\Gamma ^k_{ij}\equiv
\frac{1}{2}g^{kl}(\partial_i g_{lj} + \partial_j
g_{il} - \partial_l g_{ij})$.}~:

\begin{equation}
\Gamma ^k_{ij}=(1+t\tilde{\sigma})v^k[(\sigma-\dot{\varphi})\delta_{ij} +
(\dot{\sigma}-\sigma\dot{\varphi}) v_iv_j]+\dot{\varphi}(v_i\delta_j^k+v_j\delta_i^k).
\end{equation}
so that the connection matrix is
\begin{equation}
\Gamma _i^j\equiv \Psi^k\Gamma _{ik}^j=Av^jv_iv_k\Psi^k
+Bv^j\Psi_i+C(v_i\Psi^j+\delta^j_iv_k\Psi^k)
\end{equation}
where we have set
\begin{equation}
\left\{ \begin{array}{l} A=(1+t\tilde{\sigma})(\dot{\sigma}-C\sigma) \\ B=(1+t\tilde{\sigma})(\sigma -C) \\
C=\dot{\varphi}\end{array} \right. .
\end{equation}

The curvature matrix is given by

\begin{equation}
R_i^j \equiv d\Gamma _i^j+\Gamma ^j_k \wedge \Gamma ^k_i.
\end{equation}

A tedious computation leads to

\begin{equation}
R^{ij}\equiv g^{ik}R^j_k=e^{-\varphi}(1+t\tilde{\sigma}) \left[ M_{11} \Psi^i\Psi^j
+ M_{12} \left(v^i(v_k\Psi^k)\Psi^j-v^j(v_k\Psi^k)\Psi^i\right)\right]
\end{equation}
where
\begin{equation}
\left\{ \begin{array}{l} M_{11}=C^2t+2C-\sigma \\
M_{12}=2\dot{C}-C^2-\dot{\sigma }+(1+t\tilde{\sigma})(\sigma -C)\dot{t\sigma} \end{array} \right. .
\end{equation}

To get the full equivariant curvature, we need the part involving
$\Omega $. In accordance with our convention on indices, we define
$\Omega ^{ij}\equiv \Omega _i^j$. By definition $\Omega ^{ij}$ is antisymmetric.
According to formul\ae\ (\ref{4.14},\ref{4.20}), the part of the
equivariant curvature containing $\Omega $
is the covariant derivative of $\Omega _k^jv^k$, the $so(n)$ vector
field associated to $\Omega $. Consequently

\begin{equation}
\left(-i_{{\cal P}}(\Omega ) \Gamma \right)_k^j=\Omega _k^j+\Omega _m^lv^m\Gamma ^j_{lk}.
\end{equation}

The antisymmetry of $\Omega $ leads to further simplifications.
The outcome is~:

\begin{equation}
g^{ik}\left(\Omega _k^j+\Omega _m^lv^m\Gamma ^j_{lk}\right)=e^{-\varphi} (1+t\tilde{\sigma})\left[
M_{21} \Omega ^{ij} + M_{22} \left(v^iv_k\Omega ^{kj}-v^jv_k\Omega ^{ki}
\right)\right]
\end{equation}
where
\begin{equation}
\left\{ \begin{array}{l} M_{21}=1+t\sigma \\
M_{22}=C-\sigma \end{array} \right. .
\end{equation}

We note the striking similarity between the two contributions.
If we define a two-by-two matrix $N^{ij}$ by

\begin{equation}
N^{ij}=\left(\begin{array}{cc} \Psi^i\Psi^j & \Omega ^{ij} \\
v^i(v_k\Psi^k)\Psi^j-v^j(v_k\Psi^k)\Psi^i & v^iv_k\Omega ^{kj}-v^jv_k\Omega
^{ki}\end{array} \right)
\end{equation}
the equivariant curvature can be written as a trace ~:
\begin{equation}
(R_{int}^{eq})^{ij}=e^{-\varphi}(1+t\tilde{\sigma}){\rm Tr}\;   MN^{ij}
\end{equation}

The equivariant Euler class is
\begin{equation}
E_{int}^{eq}=2^{n/2}\sqrt{\bf g}\, {\rm Pfaff}\;  (R_{int}^{eq})^{ij}.
\end{equation}
with the usual definition of the Pfaffian. Note that $\sqrt{\bf
g}=e^{n\varphi/2}(1+t\sigma)^{1/2}$.

This is the explicit formula for the universal Thom class that we
were after. It involves two arbitrary functions of $t$, $\varphi$ and
$\sigma$ (with the mild restriction $1+t\sigma > 0$) which may be localized
at will thus so leaving a fair amount of flexibility.

The first comment to make is that apparently the above representative,
which is of course $so(n)$ invariant when $so(n)$ acts on $V$, $\Omega $ and
$\Psi$ at the same time, is not invariant when $so(n)$ acts only on
$V$. To state it more simply, the $V$ dependence of the Thom class is
not only through $t$. This is to be contrasted with the
Matha\"{\i}--Quillen representative.

Let us deal with a special case first. When $n =2$, it is easy
to see that

\begin{equation}
\varepsilon _{ij} N^{ij} =2\left(\begin{array}{cc} \Psi^1\Psi^2 & \Omega ^{12} \\
t\Psi^1\Psi^2 & t\Omega ^{12}\end{array} \right)
\end{equation}
so we have some hope to recover the Matha\"{\i}--Quillen formula
as a special case. After some manipulations one finds

\begin{equation} \label{eq:Ed=2}
E_{int}^{eq}=4\dot{F}\Psi^1\Psi^2+2F\Omega ^{12}
\end{equation}
where

\begin{equation} \label{eq:F}
F \equiv \frac{1+tC}{(1+t\sigma)^{1/2}}.
\end{equation}
So the Thom class depends only on one arbitrary function of $t$,
namely $F$, which can easily be adjusted to recover the
Matha\"{\i}--Quillen representative. The correct choice is
$F=-{\textstyle{1 \over {(4\pi )}}}\exp (-{t \mathord{\left/
{\vphantom {t {4)}}} \right. \kern-\nulldelimiterspace} {4)}}$.

When $n > 2$ the situation is more complicated. We shall use a trick
to see how much the symmetry of the $so(n)$ action on $V$ is broken.

The first observation is that under a similarity, the Pfaffian has a
simple behavior~: if $A$ is a square antisymmetric matrix and $S$
an arbitrary square matrix of the same size with transpose $S^t$,
$S^tAS$ is again antisymmetric and ${\rm Pfaff}\;  S^tAS ={\rm Det}\;  S \; {\rm Pfaff}\;  A$. The square
of this equation just follows from the
multiplicative property of the determinant, and the sign is fixed
by the case when $S$ is the identity matrix. So if we can find a
matrix $S^{ij}$ (independent of $\Psi$ and $\Omega$) such
that $S^tR_{int}^{eq}S$ simplifies, we shall end with a simpler formula
for the Thom class.

Define a symmetric matrix $S(D)$ of parameter $D$ by

\begin{equation}
S(D)^i_j=\delta^i_j+Dv^iv_j.
\end{equation}
This matrix is easily diagonalized : the vectors orthogonal to $v^i$
are left invariant and $v^i$ is multiplied by $1+tD$. So

\begin{equation} {\rm Det}\;  S(D)=1+tD \end{equation}
and

\begin{equation} S(D)S(E)=S(D+E+tDE). \end{equation}

Moreover, if $A^{ij}$ is any antisymmetric matrix,

\begin{equation}
(S(D)AS(D))^{ij}= A^{ij}+D(v^iv_kA^{kj}-v^jv_kA^{ki}).
\end{equation}

We apply this identity to the four antisymmetric objects building
the two-by-two matrix $N^{ij}$ to get

\begin{equation} S(D)NS(D)=\bar{S}(D)N \label{ij22} \end{equation}
where $\bar{S}(D)$ is the  two-by-two matrix
\begin{equation}  \left(\begin{array}{cc} 1 & D \\ 0 & 1+tD \end{array} \right) .\end{equation}

In equation (\ref{ij22}), the left-hand side
involves a product of $n \times n$ matrices, and the $2 \times 2$
indices are spectators whereas on the right-hand side the opposite
occurs.

So we can write

\begin{equation}
S(D)R_{int}^{eq}S(D)=e^{-\varphi}(1+t\tilde{\sigma}){\rm Tr}\;  M(D)N
\end{equation}
with $M(D)\equiv M\bar{S}(D)$, and
the Thom class is

\begin{equation}
E_{int}^{eq}=(1+t\sigma)^{(1-n)/2}(1+tD)^{-1}{\rm Pfaff}\;  ({\rm Tr}\;  M(D)N).
\end{equation}

We can choose $D$ to simplify the expression of $E_{int}^{eq}$.

First we take $D=D_1$ where $M(D_1)=\left(\begin{array}{cc} * & 0 \\ {*} & * \end{array}
\right)$.
This makes it easy to compute the term in $E_{int}^{eq}$ that does
not involve $\Omega$. The outcome is

\begin{equation}
E_{int}^{eq}=n!(1+t\sigma)^{(1-n)/2}(1+t\frac{M_{12}}{M_{11}})
M_{11}^{n/2} \Psi^1\cdots\Psi^n + \mbox{\rm terms involving } \Omega .
\end{equation}
One can check that this is compatible with (\ref{eq:Ed=2}) for $n=2$.

Second, we take $D=D_2$ where $M(D_2)=\left(\begin{array}{cc} * & * \\ {*} & 0 \end{array}
\right)$. This makes it easy to compute the term in $E_{int}^{eq}$
that does not involve $\Psi $. The outcome is

\begin{equation}
E_{int}^{eq}=2^{n/2}(1+t\sigma)^{(1-n)/2}(1+t\frac{M_{22}}{M_{21}})
M_{21}^{n/2} {\rm Pfaff}\;  \Omega  + \mbox{\rm terms involving } \Psi
\end{equation}
a result which is again compatible with (\ref{eq:Ed=2}) for $n=2$.

Those two terms in $E_{int}^{eq}$ automatically depend only on $t$.
On the other hand, the other terms are not scalars for the $so(n)$
action on $V$. To see this we keep $D=D_2$, set
$A^{ij}=M_{11}\Psi^i\Psi^j+M_{21} \Omega ^{ij}$ and
$B^{ij}=v^i(v_k\Psi^k)\Psi^j-v^j(v_k\Psi^k)\Psi^i$. Using the fact that
$x_k\Psi ^{k}$ squares to $0$ we get

\begin{equation} \label{eq:eulerclass}
\begin{array}{rcl}
E_{int}^{eq}& =& \\ & & \hspace{-.7cm}(1+t\sigma)^{(1-n)/2}\left(
2^{n/2}(1+t\frac{M_{22}}{M_{21}}){\rm Pfaff}\;  A - n/2 \frac{{\rm Det}\;
M}{M_{21}}\varepsilon_{i_1j_1\cdots i_nj_n}B^{i_1j_1}A^{i_2j_2}\cdots
A^{i_nj_n}\right).
\end{array}
\end{equation}

As $\Omega$ and $\Psi$ are independent families of indeterminates, the
matrix elements of $A^{ij}$ are independent of each other (except for
antisymmetry) and of the matrix elements of $B^{ij}$. So in the expansion
of

\begin{equation}
\varepsilon_{i_1j_1\cdots i_nj_n}B^{i_1j_1}A^{i_2j_2}\cdots A^{i_nj_n}
\end{equation}
no compensation can occur between $A$-factors and
$B$-factors or between different $B$-factors. Moreover $B$-factors
contain the full non $so(n)$ invariant part of the $V$ dependence
of the Thom class. So we have the following three possibilities. Either ${\rm Det}\;
M$ is $0$, or $B^{ij}$ is invariant for the action of $so(n)$ on $V$,
or the representative of the Thom class is not invariant for the
action of $so(n)$ on $V$. The first term of the alternative depends on
our choice of $\varphi$ and $\sigma$. The second is easily checked to occur
if and only if $n=2$, a case we have already treated.

So finally, we have shown that if $n > 2$ the representative of the
Thom class is invariant for the $so(n)$ action on $V$ if and only if
${\rm Det}\;  M =0$.

We shall now see that despite the fact that apparently our
representative of the Thom class depends on two arbitrary functions,
the single condition ${\rm Det}\;  M =0$ fixes it completely. This can be seen as
a manifestation of the topological character of the Thom class. We
shall also see that the representative we end up with is not the
Matha\"{\i}--Quillen representative.

From now on, we set ${\rm Det}\;  M =0$. Explicit computation shows that this
equation has a first integral. Namely ${\rm Det}\;  M =0$ is equivalent to

\begin{equation} \label{eq:detm=0}
(1+tC)^3\frac{d}{dt}
\left(\frac{\sigma}{(1+tC)^2}+\frac{1}{t(1+tC)^2}-\frac{1}{t} \right)=0 .
\end{equation}
The term in parenthesis can be written as

\begin{equation} \label{eq:secfact}
-\frac{C^2t+2C-\sigma}{(1+tC)^2} \qquad \mbox{\rm  or } \qquad \frac{(1+t\sigma)-(1+tC)^2}{t(1+tC)^2}
\end{equation}
Now, we distinguish two cases.

Suppose first that for some value of
$t$ the function $1+tC$ vanishes together with its first derivative.
Then

\begin{equation}
M=\frac{1+t\sigma}{t}\left(\begin{array}{cc} -1 & 1/t \\ t & -1 \end{array}\right).
\end{equation}

As a byproduct, $M_{21}+tM_{22}$ vanishes, and the equivariant Euler
class vanishes. So clearly, the function $1+tC$ cannot vanish
everywhere if we are to find a non-trivial class. Anyway, the
vanishing of $1+tC$ would mean that $e^{\varphi}=t_0/t$ for some constant
$t_0$ leading to a metric singular at the origin. It is likely that in
this case, a careful computation with distributions would give a
curvature concentrated at the origin, but we are not interested in
this anyway.

On the open intervals where $1+tC \neq 0$ the second factor of
(\ref{eq:detm=0}) has to vanish. We get

\begin{equation}
\frac{\sigma}{(1+tC)^2}+\frac{1}{t(1+tC)^2}-\frac{1}{t}=\frac{1}{t_0}
\end{equation}
for some constant $t_0$. Using (\ref{eq:secfact}), one obtains

\begin{equation}
M=\frac{1+tC}{t_0}\left(\begin{array}{cc} -(1+tC) & (1+(t_0+t)C)(t_0+t)^{-1} \\
(1+tC)(t_0+t)  & -(1+(t_0+t)C) \end{array}\right)
\end{equation}
leading to a remarkable simplification of (\ref{eq:eulerclass})~:

\begin{equation} \label{eq:final}
E_{int}^{eq}=2^{n/2}\left(\frac{t_0}{t_0+t}\right)^{1/2} {\rm Pfaff}\;  \left(\Omega
^{ij}-\frac{1}{t_0+t}\Psi ^i \Psi ^j\right).
\end{equation}

Now, as $M_{21}=1+t\sigma$, which has to remain strictly positive, $1+tC$
cannot vanish at the boundary of an open interval where it is nonzero.
This means that $1+Ct$ vanishes nowhere, and that formula
(\ref{eq:final}) is valid everywhere. This is our final formula for the
equivariant Euler class if we decide to trade flexibility (arbitrary
choice of $\varphi$ and $\sigma$) for simplicity ($so(n)$ invariance on $V$,
leading to a simple Pfaffian). The Matha\"{\i}--Quillen representative
never shows up for $n > 2$.

Some comments are in order. Usually the Thom class is defined by using
function with compact support  (differential topology) or rapid
decrease at infinity (quantum field theory) on $V$. The
Matha\"{\i}--Quillen representative belongs to this second
category. With the general formula, the freedom on $\varphi$ and $\sigma$
allows us to impose any behavior at infinity \footnote{In fact to localize
the Thom class on arbitrary spherical shells if this proves useful.}. On
the other hand our rigid proposal for the Thom class does not decrease
fast at infinity. Despite the fact that this may be
inconvenient in certain applications, we would like to point that it
makes sense nevertheless. To define the Thom class, the crucial point
is that the cohomology of
$V$ with coefficients having compact support or rapid decrease at infinity is
concentrated in the dimension of $V$ and one dimensional there. It
seems clear that a cohomology of $V$ can be build such as to retain
this property and accept our rigid representative as a well-defined
cohomology class. For instance $k$-forms on $V$
such that for any non-negative integer $l$ the partial derivatives
of order $l$ of the coefficients
exist and are $O(t^{-\frac{k+l+1}{2}})$ at infinity, endowed with
the usual exterior derivative, should work.

In particular, we can normalize things in such a way that the integral
on $V$ of the term independent of $\Omega $ is 1, as is usual for the
Thom class. A simple calculation gives for the normalized Thom class

\begin{equation}
T_V=\frac{1}{(2\pi)^d}\left(\frac{t_0}{t_0+t}\right)^{1/2}
{\rm Pfaff}\;  \left(\frac{1}{t_0+t}\Psi ^i \Psi ^j - \Omega ^{ij}\right).
\end{equation}
Playing with the value of $t_0$ allows to localize around the zero section.
This formula already appears in \cite{hl93} as a specialization of another
formula for the Thom class.

\section{Conclusion.}

In these notes, we have obtained formul\ae\ for the universal Thom
class of a vector bundle. A special choice leads to a rigid
representative involving Cauchy-type kernels.
It would be very interesting to know whether the Matha\"{\i}--Quillen
representative, with its Gaussian-type kernel, is also a rigid member
of some natural family of representatives.

\noindent

\section*{\large \bf Acknowledgments}

\noindent We are very grateful to Raymond Stora and Georges Girardi
for many fruitful discussions. We thank Reese Harvey for making us aware of
ref. \cite{hl93}, which contains material closely related to this work.

redpajama_set_name: RedPajamaArXiv


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text:
Long-term traffic air and noise pollution in relation to mortality and hospital readmission among myocardial infarction survivors.
Tonne, Cathryn; Halonen, Jaana I; Beevers, Sean D; Dajnak, David; Gulliver, John; Kelly, Frank J; Wilkinson, Paul; Anderson, H Ross; (2015) Long-term traffic air and noise pollution in relation to mortality and hospital readmission among myocardial infarction survivors. International journal of hygiene and environmental health, 219 (1). pp. 72-78. ISSN 1438-4639 DOI: https://doi.org/10.1016/j.ijheh.2015.09.003
Use this Digital Object Identifier when citing or linking to this resource.
https://doi.org/10.1016/j.ijheh.2015.09.003
BACKGROUND: There is relatively little evidence of health effects of long-term exposure to traffic-related pollution in susceptible populations. We investigated whether long-term exposure to traffic air and noise pollution was associated with all-cause mortality or hospital readmission for myocardial infarction (MI) among survivors of hospital admission for MI. METHODS: Patients from the Myocardial Ischaemia National Audit Project database resident in Greater London (n = 1 8,138) were followed for death or readmission for MI. High spatially-resolved annual average air pollution (11 metrics of primary traffic, regional or urban background) derived from a dispersion model (resolution 20 m × 20 m) and road traffic noise for the years 2003-2010 were used to assign exposure at residence. Hazard ratios (HR, 95% confidence interval (CI)) were estimated using Cox proportional hazards models. RESULTS: Most air pollutants were positively associated with all-cause mortality alone and in combination with hospital readmission. The largest associations with mortality per interquartile range (IQR) increase of pollutant were observed for non-exhaust particulate matter (PM(10)) (HR = 1.05 (95% CI 1.00, 1.10), IQR = 1.1 μg/m(3)); oxidant gases (HR = 1.05 (95% CI 1.00, 1.09), IQR = 3.2 μg/m(3)); and the coarse fraction of PM (HR = 1.05 (95% CI 1.00, 1.10), IQR = 0.9 μg/m(3)). Adjustment for traffic noise only slightly attenuated these associations. The association for a 5 dB increase in road-traffic noise with mortality was HR = 1.02 (95% CI 0.99, 1.06) independent of air pollution. CONCLUSIONS: These data support a relationship of primary traffic and regional/urban background air pollution with poor prognosis among MI survivors. Although imprecise, traffic noise appeared to have a modest association with prognosis independent of air pollution.
Faculty of Public Health and Policy > Public Health, Environments and Society
Published Version Download
Licence: Creative Commons: Attribution 3.0
Filename: TRAFFIC_MINAP_07092015.docx

redpajama_set_name: RedPajamaCommonCrawl


111 W. 57th St., New York.
The story of how this album came about makes a fascinating read. The producers contacted Laurent Guenoun, custodian of Marcel Romano's archives. Romano had been the manager of Barney Wilen, the French saxophonist. They were in search of unreleased material. With Wilen they drew a blank, but their endeavours unearthed other treasures. Guenoun had a cache of tapes marked 'Thelonious Monk' which turned out to be the soundtrack and recording sessions for Roger Vadim's 1960 film Les Liaisons Dangereuses, an adaptation of Pierre Choderlos de Laclos' 18th century novel. Monk had recorded about 30 minutes of music for the film, never issued as a stand-alone album. Remarkably it had been set down in New York on a single day - July 27, 1959. This first official release, courtesy of Sam Records, has the imprimatur of the Thelonious Monk Estate. This year marks the centenary of Monk's birth, and I can think of no better way to mark this significant milestone.
Monk wrote nothing new for the film, but utilized music that already existed in his repertoire; he seemed disinterested in financial gain. Although an avid filmgoer, he had no previous experience of film scores. At the time he was approached, he was busy, in constant demand, overworked and beset by personal tragedies. Mental health issues, he suffered from bipolar disorder, had a bearing on the situation. After some prevarications, he saw the film and eventually signed a contract. He didn't make it to Paris, where the recording sessions were originally intended to take place, so the producers and engineers came to him in New York. These recorded gems capture Monk at the height of his powers.
The Quintet consists of Monk on piano, Charlie Rouse and Barney Wilen on tenor sax, Sam Jones on double bass and Art Taylor on drums. "Rhythm-A-Ning" , "Well You Needn't" and " Six in One" are spirited and animated and one senses a real joy in the music-making. "Crepuscule with Nellie" (his wife) and " By and By" offer an element of contrast by softening the pace. The latter is a gospel hymn and seems a fitting end to the soundtrack. " Pannonica", named after Baroness Pannonica de Koenigswarter, a friend of Monk, exists in three versions, two for solo piano and one for quintet. Disc 2 offers alternate takes. "Pannonica" and " Light Blue" are 45 masters, but the most substantial offering is a 14 minute rehearsal sequence of the making of "Light Blue" containing plenty of dialogue. It makes for an absorbing listen.
I'm astonished how well these recordings sound. The source material for the album, the master tapes in this case, have obviously been well-preserved. François Lê Xuân's superb transfers and remasterings breathe new life into these nearly sixty year old recordings, and they emerge with the freshness and vitality as the day they were set down.
Sam Records have gone to town on this release. The whole package is aesthetically appealing. The two CDs are housed in a sturdy, attractive tri-fold cardboard sleeve. Also included is a 56 page booklet featuring several interesting articles by various contributors, providing context and background to the making of these recordings. There are both black and white and colour photos of the New York sessions and a couple of still shots from the film itself. An outer card case supports both.
It all constitutes a most desirable acquisition.


The Angelika Pallas Hotel is situated opposite of the new international port of Igoumenitsa, the main entrance to western Greece by boat, and just a few hundred meters from the Egnatia Motorway (from Igoumenitsa to Alexandroupolis). The hotel is the ideal place for all those who are travelling due professional reasons, but also for those who wish to spend their holidays at the gorgeous Thesprotia, at a comfortable and discreet place.Experience Greek hospitality combined with luxury interior and stunning panoramic views over the city, sea and the mountains. From the grand and elegant facade and entrance to the intimate and relaxed atmosphere of the designer-furnished suites, Angelika Pallas is the first choice for discerning travellers seeking the very best in comfort and convenience.Enjoy your rich continental breakfast in bed. This delicious breakfast will be served in your room and will get your day off to a great start. At Angelika Pallas we offer great services to our customers by adding great value to our already affordable rates.
Nice room with great sea view.
"The half-board meal option is worth the money."
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"No fitness centre or swimming pool just could not delete once clicked on in error. Our room was overlooking the sea which was lovely but in July it was to hot to sit on the balcony, so perhaps ask for a shady room."
"Second floor was perfect, lift or stairs."


Today's furniture the way YOU imagined it! Today's furniture the way YOU imagined it!
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Today's furniture the way YOU imagined it!


Citation Ferrer, J., J.M. Wingert and L.R. Malabarba, 2014. Description of a new species and phylogenetic analysis of the subtribe Cynopoecilina, including continous characters without discretization (Cyprinodontiformes: Rivulidae). Zool. J. Linn. Soc. 172(4):846-866.


[S6] G.E. Cokayne; with Vicary Gibbs, H.A. Doubleday, Geoffrey H. White, Duncan Warrand and Lord Howard de Walden, editors, The Complete Peerage of England, Scotland, Ireland, Great Britain and the United Kingdom, Extant, Extinct or Dormant, new ed., 13 volumes in 14 (1910-1959; reprint in 6 volumes, Gloucester, U.K.: Alan Sutton Publishing, 2000), volume XI, page 97. Hereinafter cited as The Complete Peerage.
[S6] Cokayne, and others, The Complete Peerage, volume XI, page 95.
[S6] Cokayne, and others, The Complete Peerage, volume XI, page 96.
[S6] G.E. Cokayne; with Vicary Gibbs, H.A. Doubleday, Geoffrey H. White, Duncan Warrand and Lord Howard de Walden, editors, The Complete Peerage of England, Scotland, Ireland, Great Britain and the United Kingdom, Extant, Extinct or Dormant, new ed., 13 volumes in 14 (1910-1959; reprint in 6 volumes, Gloucester, U.K.: Alan Sutton Publishing, 2000), volume XI, page 95. Hereinafter cited as The Complete Peerage.
[S6] Cokayne, and others, The Complete Peerage, volume II, page 302.
[S6] Cokayne, and others, The Complete Peerage, volume VII, page 211.
[S62] Douglas Richardson, Magna Carta Ancestry (Baltimore, Maryland, U.S.A.: Genealogical Publishing Co, 2005), page 447. Hereinafter cited as Magna Carta Ancestry.
[S6] G.E. Cokayne; with Vicary Gibbs, H.A. Doubleday, Geoffrey H. White, Duncan Warrand and Lord Howard de Walden, editors, The Complete Peerage of England, Scotland, Ireland, Great Britain and the United Kingdom, Extant, Extinct or Dormant, new ed., 13 volumes in 14 (1910-1959; reprint in 6 volumes, Gloucester, U.K.: Alan Sutton Publishing, 2000), volume XI, page 96. Hereinafter cited as The Complete Peerage.


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text:
Home of the 2016 College Football Playoff National Championship: Phoenix
By Jaclyn Harris
After making its debut in Texas during January, resulting in Ohio State being named national champions, the College Football Playoff is making its way to Phoenix, Arizona for the 2016 College Football Playoff National Championship.
Having recently played host to Super Bowl XLIX, the University of Phoenix Stadium and the city itself are prepared to host crowds of the caliber expected for the 2016 national championship event. Read on to learn about the University of Phoenix Stadium, home of the 2016 College Football Playoff National Championship!
The University of Phoenix Stadium: "Unlike Any Other Stadium in America"
As stated on their website, the University of Phoenix Stadium is "unlike any other stadium in America because it's truly a multi-purpose facility." Built to be able to host football, basketball, soccer, consumer shows, motorsports, rodeos, concerts, and corporate events, this stadium can encompass events of all sizes and calibers.
Located in Glendale, Arizona, right outside Phoenix, the stadium holds 63,400 seats (expandable up to 72,200 seats), boasts a "Go Green" mission and took three years to build. The University of Phoenix Stadium is 160,000 square feet of contiguous exhibit space and 20,000 square feet of meeting room space. To put into perspective its size, if it was stretched in a straight line, it would be 18 miles long. In fact, the amount of concrete used on the stadium is equavilent to 900 miles of sidewalk.
Its massive size isn't the only thing that makes the University of Phoenix Stadium unique. On each level, there is a mural designed to represent Arizona's environmental attractions and the 400 Level is designed to represent the night sky. Additionally, there are four columns in the main concourse which show hieroglyphics of records and significant moments of the Arizona Cardinals.
To get the fans engaged during events, there are two large "thermometers" at the south end zone to gauge fan noise.
For more fun facts on the University of Phoenix Stadium, visit its website>>
Plan Your Trip to AZ for the 2016 College Football Playoff National Championship!
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Before handing over your shipments to Australia Post, you'll need to create an Order Summary Report. The report must be printed, signed and submitted alongside your shipments when they are lodged.
You can create an Order Summary Report by selecting the shipments you wish to manifest, then clicking Generate Order Summary Report under More on the top right corner of the Manage Shipments page.
You can also find previously generated reports by clicking this button.
An Order Summary Report can only be generated once a day, so make sure that all the shipments you want to handover that day are selected.
Note: To make it easier to look up your unmanifested Australia Post shipments, you can use the filters. On the Manage Shipments page, click on Filters and select Unmanifested Shipments under Manifest.


The medium Waxing Crescent measures 17.5" x 10.5" x 6" and is rated at a capacity of 150 cigars.
Pricing is outlined on the "Waxing Crescent" page. The example shown below is Cherry and is shown with the crumb catcher upgrade.
Capacity display for a 17.5" x 10.5" x 6" - 150 count - Medium with 5-1/2" x 42RG Corona Cigars.


I teach HypnoBirthing in CT. I love sitting with new couples with that very powerful and hopeful look in there eyes. They come to Hypnobirthing because they hear wonderful reviews but mostly I believe they come because something inside that mother feels strong and capable, that she wants the best for her baby and her body. She feels healthy and happy and knows that she is not sick- and the hospital makes us feel sick, but pregnancy is not an illness it is gift that we should enjoy. I teach parents to smile and relax, enjoy the changes in her body. Learn to accept that a woman is strong and capable, that her body is made to give birth.


Meet Michelle aka my Maid of Honor!!
After a lot of planning and pining over how to ask my college roommate, Michelle, to be my Maid of Honor, I finally did it!
It wasn't as glamorous as I imagined, since my original thoughts were to spend the day at the spa, or do something (expensive and) grandiose like that. However, it was marvelous and I'm so excited that I have a Maid of Honor now!
We got together at California Pizza Kitchen (not very glamorous, I know), in our sweats (even less glamorous), and I finally got to pop the question! There was no hesitation on her part, as she's just as excited about the wedding as I am. I turned over about a half-a-dozen (small and mostly funny) gifts, we discussed the wedding plans (or lack thereof), and left anxious to get this show on the road!
While we didn't get into what exactly she's going to do as MOH, I am 110 percent confident that her wheels are turning! Plus, if you knew Michelle, you'd know you don't have to ask or tell her to do anything--she's the type to just get it done!
She's a Leo and very outgoing and friendly. She's career oriented and successful. She's a planner and has great leadership skills. She loves red bull (sometimes with vodka) and she also has a pit bull, Kia, who is just as sweet as her!
Michelle and I went to the same High School and lived together in college. Not only has she helped me through countless crises in my life, I've learned a lot from her too. Plus, she's always been non-judgmental, easy to get along with, and supportive.
She was also the first to know that Aaron and I were dating. She's been there (literally, in the same house) during arguments Aaron and I had, as well as, during the many, many good times we've shared.
Overall, Michelle has always been a very reliable friend. We always have a blast when we hang out and the truth is, she has never let me down and I know she never will.
Annnnnnnnnnnnnnd... She's going to be my Maid of Honor! Woo hoo!
Michelle's favorite drink Sticky notes on everything!! Tissues--for when you cry (tears of joy).
A notebook--for when you get wedding ideas.
Bingo cards--for when you feel like just goofing off.
A whoopy cushion--for when I need a good laugh.
Socks--for when I get cold feet (in the meantime, you stay warm).
A receipt book--for when this MOH thing gets too expensive (bill me back).
A tee shirt that says Maid of Honor--for when you feel like showing off!
I wanna be a blogger!


2017 LEGO Justice League Super Heroes 76087 Cyborg minifigure closer look.
LEGO Justice League Sets Revealed! 3 Sets! NEW Aquaman! Cyborg! Steppenwolf! Pictures & My Thoughts!
Genuine Lego DC comics cyborg justice League movie Super Heroes Minifigures.


text:
Q: Why do we need singletons to be closed in the definition of Normal/Regular spaces? In the book of Topology by Munkres, at page 193, it is given that

Definition. Suppose that one-point sets are closed in X. Then X is said to be regular if for each pair consisting of a point x and a
  closed set B disjoint from x, there exist disjoint open sets
  containing x and B, respectively. The space X is said to be normal if
  for each pair A, B of disjoint closed sets of X, there exist disjoint
  open sets containing A and B, respectively.
It is clear that a regular space is Hausdorff, and that a normal space
  is regular. (We need to include the condition that one-point sets be
  closed as part of the definition of regularity and normality in order
  for this to be the case. A two-point space in the indiscrete topology
  satisfies the other part of the definitions of regularity and
  normality, even though it is not Hausdorff.)

But, how does assuming one-point sets are closed allow regular (normal) space to be Hausdorff(regular) ?

A: Further considerations:
(i). Consider Sierpinski space $S=\{x,y\}$ with $x\ne y,$ where $S,\emptyset,$ and $\{x\}$ are open but $\{y\}$ is not open. If $A, B$ are disjoint closed subsets of $S$ then at least one of $A, B$ is empty so $A,B$ are covered by disjoint open sets. But $S$ does not satisfy the condition for regularity:  $x$ does not belong to the closed subset $\{y\}$ but the only open set covering $\{y\}$ is the whole space $S$.
(ii). Let $X$ be a normal space. Let $p\in X$ and let $Y$ be a closed subset of $X$ with $p\not \in Y.$ Since $X$ is a $T_1$ space (one-point subsets are closed), the sets $\{p\}, Y$ are closed and disjoint. So, since $X$ is normal, there are disjoint open sets with $\{p\}\subset U$ and $Y\subset V. $ That is, $p\in U$ and $Y\subset V.$ So $X$ is regular.

A: If $x \neq y$ then $\{y\}$ is a closed set adn $x \notin \{y\}$ so there exist disjoint open sets $U,V$ such that $x \in U$ and $\{y\} \subset V$. This shows that the space is Hausdorff.

redpajama_set_name: RedPajamaStackExchange


text:
passer
======

Converts GPU torch7 models to CPU equivalent models.
The current version is very limited in what it is able to convert, although the code is easy to extend.

Installation
============
To install via luarocks, run:

    luarocks install --server=http://luarocks.org/manifests/williford passer

Or clone the repository:

    git clone git@github.com:williford/passer.torch.git

and then copy the passer.lua file to your working folder.

In order to use:

    local passer = require 'passer'

Usage
=====

The conversion needs to be done on a system that has a GPU (that is able to load the required model).

```lua
require 'nn';
require 'cudnn';

-- do not put "local" if in interactive session!
local passer = require 'passer'

model = torch.load('/path/to/model_xx.t7')
cpu_model = passer.tocpu(model)
torch.save('cpu_model_xx.t7')
```

redpajama_set_name: RedPajamaGithub