\def\iint{\mathrel{\mathop{{\int\!\!\ldots\!\!\int}}\limits_{V\ldots V}}}
\def\ssum{\mathrel{\mathop{{\sum\limits^N_{i=1}\ \sum\limits^N_{j=1}}}
     \limits_{i\not=j}}}
\def\Det{\,{\hbox{Det}}\,}
\def\llambda{\lambda\!\!\!\!^-}
\def\gboxit#1{\hbox{\vrule\vbox{\hrule\kern3pt\vtop
{\hbox{\kern3pt#1\kern3pt}
\kern3pt\hrule}}\vrule}}
\def\cslash#1{\rlap{$#1$}/}
\def\ll{\left\langle}
\def\rr{\right\rangle}
\def\Diff{{\cal D}\!{\it iff}}
\def\ttilde#1{\raise2ex\hbox{${\scriptscriptstyle(}\!
\sim\scriptscriptstyle{)}$}\mkern-16.5mu #1}
\def\dddots#1{\raise1ex\hbox{$^{\ldots}$}\mkern-16.5mu #1}
\def\siton#1#2{\raise1.5ex\hbox{$\scriptscriptstyle{#2}$}\mkern-16.5mu #1}
\def\upleftarrow#1{\raise1.5ex\hbox{$\leftarrow$}\mkern-16.5mu #1}
\def\uprightarrow#1{\raise1.5ex\hbox{$\rightarrow$}\mkern-16.5mu #1}
\def\upleftrightarrow#1{\raise1.5ex\hbox{$\leftrightarrow$}\mkern-16.5mu #1}
\def\bx#1#2{\vcenter{\hrule \hbox{\vrule height #2in \kern #1in\vrule}\hrule}}
 \def\Im{\,{\rm Im}\,}
\def\Re{\,{\rm Re}\,}
\def\tr{\,{\rm tr}\,}
\def\TR{\,{\rm TR}\,}
\def\Tr{\,{\rm Tr}\,}
\def\mod{\,{\rm mod}\,}
\def\diag{\,{\rm diag}\,}
\def\bookref{\hangindent=60pt\hangafter=1}
\def\brvec{\mathrel{\mathop{\vert_{{}_{\!\!\rightarrow}}}}}
\def\neath#1#2{\mathrel{\mathop{#1}\limits_{#2}}}
\def\underl{\mathrel{\mathop{{\cal L}t}\limits_{(x)\to\infty}}}
\def\updownbigcup#1#2{\mathrel{\mathop{\bigcup}\limits^{#1}_{#2}}}
\def\underbigoplus#1{\mathrel{\mathop{\bigoplus}\limits_{#1}}}
\def\undermax#1{\mathrel{\mathop{\max}\limits_{#1}}}
\def\undersigma#1{\mathrel{\mathop{\Sigma}\limits_{#1}}}
\def\underbigcup#1{\mathrel{\mathop{\bigcup}\limits_{#1}}}
\def\underbigoplus#1{\mathrel{\mathop{\bigoplus}\limits_{#1}}}
\def\undercap#1{\mathrel{\mathop{\cap}\limits_{#1}}}
\def\undercup#1{\mathrel{\mathop{\cup}\limits_{#1}}}
\def\updowncup#1#2{\mathrel{\mathop{\cup}\limits^{#1}_{#2}}}
%\def\uplrarrow#1{#1^{{}^{\!\!\!\!\!\leftrightarrow}}}
\def\lleftarrow#1{#1^{{}^{\!\!\!\!\!\!\leftarrow}}}
\def\underMin#1{\mathrel{\mathop{\hbox{Min}}\limits_{#1}}}
\def\undermin#1{\mathrel{\mathop{\hbox{min}}\limits_{#1}}}
\def\underPi#1{\mathrel{\mathop{\Pi}\limits_{#1}}}
\def\undertr#1{\mathrel{\mathop{\hbox{tr}}\limits_{#1}}}
\def\Sum#1{\mathrel{\mathop{\sum\!\!\!\!\!\!\bigg|}\limits_{#1}}}
\def\underpi#1{\mathrel{\mathop{\pi}\limits_{#1}}}
\def\overapprox#1{\mathrel{\mathop{\approx}\limits^{#1}}}
\def\underapprox#1{\mathrel{\mathop{\approx}\limits_{#1}}}
\def\squiggle#1{\lower1.5ex\hbox{$\sim$}\mkern-14mu #1}%I changed 16 to 14?
%\def\squiggle#1{\mathrel{\mathop{#1}\limits_{{}_\sim}}}
\def\under#1{\mathrel{\mathop{#1}\limits_{{}^\sim}}}
\def\onlongarrow#1{\mathrel{\mathop{\longrightarrow}\limits^{#1}}}
%\def\onbigarrow#1{\mathrel{\mathop{\left.\Bigg\rightarrow}\limits^{#1}}}
\def\bigarrowdown{\mathrel{\mathop{\Bigg\downarrow}}}
\def\onbigarrow#1{\mathrel{\mathop{\left.\Biggr\rightarrow}\limits^{#1}}}
\def\underapprox#1{\mathrel{\mathop{\approx}\limits_{#1}}}
\def\tequal{\mathrel{\mathop{=}\limits^{{\cal T}}}}
\def\onequal#1{\mathrel{\mathop{=}\limits^{#1}}}
\def\underequal#1{\mathrel{\mathop{=}\limits_{#1}}}
\def\underpropto#1{\mathrel{\mathop{\propto}\limits_{#1}}}
\def\overapprox#1{\mathrel{\mathop{\approx}\limits^{#1}}}
\def\overcong#1{\mathrel{\mathop{\cong}\limits^{#1}}}
\def\oversim#1{\mathrel{\mathop{\sim}\limits^{#1}}}
\def\undersim#1{\mathrel{\mathop{\sim}\limits_{#1}}}
\def\oversim#1{\mathrel{\mathop{\sim}\limits^{#1}}}
\def\undersimeq#1{\mathrel{\mathop{\simeq}\limits_{#1}}}
\def\onleftarrow#1{\mathrel{\mathop{\longleftarrow}\limits^{#1}}}
 
\def\onarrow#1{\mathrel{\mathop{\longrightarrow}\limits^{#1}}}
%\def\siton#1#2{\mathrel{\mathop{#1}\limits^{\scriptscriptstyle #2}}}
%\def\under#1#2{\mathrel{\mathop{#1}\limits_{#2}}}
\def\thrupork#1{\mathrel{\mathop{#1\!\!\!/}}}
\def\pork#1{\mathrel{\mathop{#1\!\!\!/}}}
\def\thrux#1{\mathrel{\mathop{#1\!\!\!/}}}
\def\thrub#1{\mathrel{\mathop{#1\!\!\!/}}}
\def\thru#1{\mathrel{\mathop{#1\!\!\!/}}}
\def\thrupartial{\mathrel{\mathop{\partial\!\!\!/}}}
\def\thrud#1{\mathrel{\mathop{#1\!\!\!\!/}}}
\def\thrui#1{\mathrel{\mathop{#1\!\!\!/}}}
\def\thruf#1{\mathrel{\mathop{#1\!\!\!/}}}
\def\thrun#1{\mathrel{\mathop{#1\!\!\!/}}}
\def\timestar{\mathrel{\mathop{\leftarrow \!\!\!\!\!\!\times}}}
\def\same#1#2{\mathrel{\mathop{#1\!\!\!\!\!\!\!\!#2}}}
\def\narrower{\advance\leftskip by\parindent \advance\rightskip by\parindent}
\def\underleftrightarrow#1{\mathrel{\mathop{\longleftrightarrow}\limits_{#1}}}
\def\underarrow#1{\mathrel{\mathop{\longrightarrow}\limits_{#1}}}
\def\onunderarrow#1#2{\mathrel{\mathop{\longleftrightarrow}\limits^{#1}_{#2}}}
\def\underleftarrow#1{\mathrel{\mathop{\longleftarrow}\limits_{#1}}}
\def\under#1{\mathrel{\mathop{\lim}\limits_{#1}}}
\def\figcapt#1#2{\narrower\smallskip\noindent Fig. #1
                \quad #2\smallskip}
\def\tabcapt#1#2{\narrower\smallskip\noindent Table #1
                \quad #2\smallskip}
\def\cfigcapt#1#2{\narrower\smallskip\centerline{ Fig. #1
                \quad #2}\smallskip}
\def\fbox#1#2{$\vcenter{\hrule width#1in\hbox{\vrule height#2in
   \hskip#1in\vrule height#2in}\hrule width#1in}$}
 
\def\mbox#1#2{\vcenter{\hrule width#1in\hbox{\vrule height#2in
   \hskip#1in\vrule height#2in}\hrule width#1in}}
\def\eqsquare #1:#2:{\vcenter{\hrule width#1\hbox{\vrule height#2
   \hskip#1\vrule height#2}\hrule width#1}}
\def\inbox#1#2#3{\vcenter to #2in{\vfil\hbox to #1in{$$\hfil#3\hfil$$}\vfil}}
       %margin bullet
\def\strutdepth{\dp\strutbox}
\def\marbul{\strut\vadjust{\kern-\strutdepth\specialbul}}
\def\specialbul{\vtop to \strutdepth{
    \baselineskip\strutdepth\vss\llap{$\bullet$\qquad}\null}}
\def\Bcomma{\lower6pt\hbox{$,$}}    % Big commutator
\def\bcomma{\lower3pt\hbox{$,$}}    % commutator
\def\normalor#1{\;{:}#1{:}\;}
\def\tnormalor#1{\hbox{$\;{:}#1{:}\;$}}
\def\sl{\scrsf}
\def\underarrow#1{\mathrel{\mathop{\longrightarrow}\limits_{#1}}}
\def\updots{\mathinner{\mskip 1mu\raise 1pt\hbox{.}
    \mskip 2mu\raise 4pt\hbox{.}\mskip 2mu
    \raise 7pt\vbox{\kern 7pt\hbox{.}}\mskip 1mu}}
\def\svec#1{\skew{-2}\vec#1}
 
\def\littlesquare#1#2{\vcenter{\hrule width#1in\hbox{\vrule height#2in
   \hskip#1in\vrule height#2in}\hrule width#1in}}
\def\undersup#1{\mathrel{\mathop{\hbox{sup}}\limits_{#1}}}
\def\square{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt
   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt}
\def\ssquare{\kern1pt\vbox{\hrule height .6pt\hbox{\vrule width .6pt\hskip 3pt
   \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt}
\def\lege{\hbox{$ {     \lower.40ex\hbox{$>$}
                   \atop \raise.20ex\hbox{$<$}
                   }     $}  }
 
\def\rege{\hbox{$ {     \lower.40ex\hbox{$<$}
                   \atop \raise.20ex\hbox{$>$}
                   }     $}  }
 
 
\def\lapp{\hbox{$ {     \lower.40ex\hbox{$<$}
                   \atop \raise.20ex\hbox{$\sim$}
                   }     $}  }
\def\rapp{\hbox{$ {     \lower.40ex\hbox{$>$}
                   \atop \raise.20ex\hbox{$\sim$}
                   }     $}  }
 
\def\tridots{\hbox{$ {     \lower.40ex\hbox{$.$}
                   \atop \raise.20ex\hbox{$.\,.$}
                   }     $}  }
\def\Times{\times\hskip-2.3pt{\raise.25ex\hbox{{$\scriptscriptstyle|$}}}}
 
 
\def\rightonleft{\hbox{$ {     \lower.40ex\hbox{$\longrightarrow$}
                   \atop \raise.20ex\hbox{$\longleftarrow$}
                   }     $}  }
 
 
\def\pmb#1{\setbox0=\hbox{#1}%
\kern-.025em\copy0\kern-\wd0
\kern.05em\copy0\kern-\wd0
\kern-.025em\raise.0433em\box0 }

%\input roger
%\input roger.tex

\font\fivebf=cmbx5
\font\sixbf=cmbx6
\font\sevenbf=cmbx7
\font\eightbf=cmbx8
\font\ninebf=cmbx9
\font\tenbf=cmbx10
\font\bfmhalf=cmbx10 scaled\magstephalf
\font\bfmone=cmbx10 scaled\magstep1
\font\bfmtwo=cmbx10 scaled\magstep2
\font\bfmthree=cmbx10 scaled\magstep3
\font\bfmfour=cmbx10 scaled\magstep4
%
\font\sevenit=cmti7
\font\eightit=cmti8
\font\nineit=cmti9
\font\tenit=cmti10
\font\itmhalf=cmti10 scaled\magstephalf
\font\itmone=cmti10 scaled\magstep1
\font\itmtwo=cmti10 scaled\magstep2
\font\itmthree=cmti10 scaled\magstep3
\font\itmfour=cmti10 scaled\magstep4
%
\font\fiverm=cmr5
\font\sixrm=cmr6
\font\sevenrm=cmr7
\font\eightrm=cmr8
\font\ninerm=cmr9
\font\tenrm=cmr10
\font\rmmhalf=cmr10 scaled\magstephalf
\font\rmmone=cmr10 scaled\magstep1
\font\rmmtwo=cmr10 scaled\magstep2
\font\rmmthree=cmr10 scaled\magstep3
\font\rmmfour=cmr10 scaled\magstep4
 
\def\fontone{\def\rm{\fcm0\rmmone}%
  \textfont0=\rmmone \scriptfont0=\tenrm \scriptscriptfont0=\sevenrm
  \textfont1=\itmone \scriptfont1=\tenit \scriptscriptfont1=\sevenit
  \def\it{\fcm\itfcm\itmone}%
  \textfont\itfcm=\itmone
  \def\bf{\fcm\bffcm\bfmone}%
  \textfont\bffcm=\bfmone \scriptfont\bffcm=\tenbf
   \scriptscriptfont\bffcm=\sevenbf
  \tt \ttglue=.5em plus.25em minus.15em
  \normalbaselineskip=25pt
  \let\sc=\tenrm
  \let\big=\tenbig
  \setbox\strutbox=\hbox{\vrule height10.2pt depth4.2pt width\z@}%
  \normalbaselines\rm}
 
 
%\font\tentex=cmtex10
 
%%GAC%%\font\inchhigh=cminch
%\font\titlefont=cmssmc40
%\font\titlefont=cmssmc10
 
\font\ninerm=cmr9
\font\eightrm=cmr8
\font\sixrm=cmr6
 
\font\ninei=cmmi9
\font\eighti=cmmi8
\font\sixi=cmmi6
\skewchar\ninei='177 \skewchar\eighti='177 \skewchar\sixi='177
 
\font\ninesy=cmsy9
\font\eightsy=cmsy8
\font\sixsy=cmsy6
\skewchar\ninesy='60 \skewchar\eightsy='60 \skewchar\sixsy='60
 
\font\eightss=cmssq8
 
\font\eightssi=cmssqi8
 
\font\ninebf=cmbx9
\font\eightbf=cmbx8
\font\sixbf=cmbx6
 
\font\ninett=cmtt9
\font\eighttt=cmtt8
 
\hyphenchar\tentt=-1 % inhibit hyphenation in typewriter type
\hyphenchar\ninett=-1
\hyphenchar\eighttt=-1
 
\font\ninesl=cmsl9
\font\eightsl=cmsl8
 
\font\nineit=cmti9
\font\eightit=cmti8
 
\font\tenu=cmu10 % unslanted text italic
 
\newskip\ttglue
\def\tenpoint{\def\rm{\fcm0\tenrm}%
  \textfont0=\tenrm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm
  \textfont1=\teni \scriptfont1=\seveni \scriptscriptfont1=\fivei
  \textfont2=\tensy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy
  \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
  \def\it{\fcm\itfcm\tenit}%
  \textfont\itfcm=\tenit
  \def\sl{\fcm\slfcm\tensl}%
  \textfont\slfcm=\tensl
  \def\bf{\fcm\bffcm\tenbf}%
  \textfont\bffcm=\tenbf \scriptfont\bffcm=\sevenbf
   \scriptscriptfont\bffcm=\fivebf
  \def\tt{\fcm\ttfcm\tentt}%
  \textfont\ttfcm=\tentt
  \tt \ttglue=.5em plus.25em minus.15em
  \normalbaselineskip=16pt
  \let\sc=\eightrm
  \let\big=\tenbig
  \setbox\strutbox=\hbox{\vrule height8.5pt depth3.5pt width\z@}%
  \normalbaselines\rm}
 
\def\ninepoint{\def\rm{\fcm0\ninerm}%
  \textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
  \textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei
  \textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
  \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
  \def\it{\fcm\itfcm\nineit}%
  \textfont\itfcm=\nineit
  \def\sl{\fcm\slfcm\ninesl}%
  \textfont\slfcm=\ninesl
  \def\bf{\fcm\bffcm\ninebf}%
  \textfont\bffcm=\ninebf \scriptfont\bffcm=\sixbf
   \scriptscriptfont\bffcm=\fivebf
  \def\tt{\fcm\ttfcm\ninett}%
  \textfont\ttfcm=\ninett
  \tt \ttglue=.5em plus.25em minus.15em
  \normalbaselineskip=11pt
  \let\sc=\sevenrm
  \let\big=\ninebig
  \setbox\strutbox=\hbox{\vrule height8pt depth3pt width\z@}%
  \normalbaselines\rm}
 
\def\eightpoint{\def\rm{\fcm0\eightrm}%
  \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm
  \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei
  \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy
  \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex
  \def\it{\fcm\itfcm\eightit}%
  \textfont\itfcm=\eightit
  \def\sl{\fcm\slfcm\eightsl}%
  \textfont\slfcm=\eightsl
  \def\bf{\fcm\bffcm\eightbf}%
  \textfont\bffcm=\eightbf \scriptfont\bffcm=\sixbf
   \scriptscriptfont\bffcm=\fivebf
  \def\tt{\fcm\ttfcm\eighttt}%
  \textfont\ttfcm=\eighttt
  \tt \ttglue=.5em plus.25em minus.15em
  \normalbaselineskip=9pt
  \let\sc=\sixrm
  \let\big=\eightbig
  \setbox\strutbox=\hbox{\vrule height7pt depth2pt width\z@}%
  \normalbaselines\rm}

%\input font.tex

\font\tiny=cmbx8
\font\sf=cmss10                    %San-Serif 10
\font\ssf=cmssq8                   %San-Serif 8
\font\scrsf=cmssi10                %Italic San-Serif 10
\font\tf=cmb10
\font\ttf=cmb10 scaled 1200
\font\ninebf=cmbx9
\font\tsf=cmssbx10                 %Bold San-Serif 10
\font\ttsf=cmssbx10 scaled 1200    %Bold San-Serif 12
\font\rg=cmfib8

%\input font.3

 \def\bfmath{\loadbfmath \gdef\loadbfmath{}\fcm0
     \textfont0=\tenmbf \scriptfont0=\eightmbf
     \scriptscriptfont0=\sevenmbf
     \textfont1=\tenbi \scriptfont1=\eightbi
     \scriptscriptfont1=\sevenbi \textfont2=\tenbsy
     \scriptfont2=\eightbsy \scriptscriptfont2=\sevenbsy}
\def\bigbfmath{\loadbigbfmath \gdef\loadbigbfmath{}\fcm0
     \textfont0=\twelvembbf \scriptfont0=\eightmbbf
     \scriptscriptfont0=\sevenmbbf
     \textfont1=\twelvebbi \scriptfont1=\eightbbi
     \scriptscriptfont1=\sevenbbi \textfont2=\twelvebbsy
     \scriptfont2=\eightbbsy \scriptscriptfont2=\sevenbbsy
     \textfont3=\twelveex \scriptfont3=\twelveex
     \scriptscriptfont3=\twelveex}
\def\bigbigbfmath{\loadbigbigbfmath \gdef\loadbigbigbfmath{}%
     \fcm0 \textfont0=\fourteenmbbbf \scriptfont0=\tenmbbbf
     \scriptscriptfont0=\sevenmbbbf
     \textfont1=\fourteenbbbi \scriptfont1=\tenbbbi
     \scriptscriptfont1=\sevenbbbi \textfont2=\fourteenbbbsy
     \scriptfont2=\tenbbbsy \scriptscriptfont2=\sevenbbbsy
     \textfont3=\fourteenex \scriptfont3=\fourteenex
     \scriptscriptfont3=\fourteenex}
\def\loadbfmath{%
     \global\font\tenmbf=cmbx10
     \global\font\eightmbf=cmbx10 scaled 800
     \global\font\sevenmbf=cmbx10 scaled 667
     \global\font\tenbi=cmbi10
     \global\font\eightbi=cmbi10 scaled 800
     \global\font\sevenbi=cmbi10 scaled 667
     \global\font\tenbsy=cmbsy10
     \global\font\eightbsy=cmbsy10 scaled 800
     \global\font\sevenbsy=cmbsy10 scaled 667}
\def\loadbigbfmath{%
     \global\font\twelvembbf=cmbx10 scaled 1200
     \global\font\eightmbbf=cmbx10 scaled 800
     \global\font\sevenmbbf=cmbx10 scaled 667
     \global\font\twelvebbi=cmbi10 scaled 1200
     \global\font\eightbbi=cmbi10 scaled 800
     \global\font\sevenbbi=cmbi10 scaled 667
     \global\font\twelvebbsy=cmbsy10 scaled 1200
     \global\font\eightbbsy=cmbsy10 scaled 800
     \global\font\sevenbbsy=cmbsy10 scaled 667
     \global\font\twelveex=cmex10 scaled 1200}
%The following font is not actually boldface, but it is bigbig.
\def\loadbigbigbfmath{%
     \global\font\fourteenmbbbf=cmr10 scaled 1440
     \global\font\tenmbbbf=cmr7 scaled 1440
     \global\font\sevenmbbbf=cmr5 scaled 1440
     \global\font\fourteenbbbi=cmmi10 scaled 1440
     \global\font\tenbbbi=cmmi7 scaled 1440
     \global\font\sevenbbbi=cmmi5 scaled 1440
     \global\font\fourteenbbbsy=cmsy10 scaled 1440
     \global\font\tenbbbsy=cmsy7 scaled 1440
     \global\font\sevenbbbsy=cmsy5 scaled 1440
     \global\font\fourteenex=cmex10 scaled 1440}
\font\rbigbf=cmbx10 scaled 1200
\font\rbigbigbf=cmbx10 scaled 1440
\font\rtitlefont=cmbx10 scaled 2073
\font\rlargetitlefont=cmbx10 scaled 2488

\font\ssnormal=cmss10
\font\ssbf=cmssbx10
\font\ssit=cmssi10
\font\sssl=cmssqi8 scaled 1200
\font\ssbigbf=cmssbx10 scaled 1200

%\input nmacro.tex
\magnification=1200
\newbox\leftpage
\newdimen\fullhsize
\newdimen\hstitle
\newdimen\hsbody
\tolerance=600\hfuzz=2pt
%\hoffset=1truein \voffset=1truein
\hsbody=\hsize \hstitle=\hsize
%---------------------------------------------------------------------
\font\titlefont=cmr10 scaled\magstep3
\font\secfont=cmbx10 scaled\magstep1
\font\bodyfont=cmr10
%use \nolabels to get rid of eqn and ref labels in draft mode
\def\nolabels{\def\eqnlabel##1{}\def\eqlabel##1{}\def\reflabel##1{}}
\def\writelabels{\def\eqnlabel##1{%
{\escapechar=` \hfill\rlap{\hskip.09in\string##1}}}%
\def\eqlabel##1{{\escapechar=` \rlap{\hskip.09in\string##1}}}%
\def\reflabel##1{\noexpand\llap{\string\string\string##1\hskip.31in}}}
\nolabels
\global\newcount\meqno \global\meqno=1
\global\meqno=1
%---------------------------------------------------------------------
%       \eqn\label{a+b=c}       gives a displayed equation with number
%                               chosen consecutively within sections.
%     \eqnn and \eqna define labels in advance
 
\def\eqnn#1{\xdef #1{(\the\meqno)}%
\global\advance\meqno by1\eqnlabel#1}
\def\eqna#1{\xdef #1##1{\hbox{$(\the\meqno##1)$}}%
\global\advance\meqno by1\eqnlabel{#1$\{\}$}}
\def\eqn#1#2{\xdef #1{(\the\meqno)}\global\advance\meqno by1%
$$#2\eqno#1\eqlabel#1$$}
%---------------------------------------------------------------------
%     \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#1#2{\the\refno\nref#1{#2}}
\def\nref#1#2{\xdef#1{\the\refno}%
\ifnum\refno=1\immediate\openout\rfile=refs.tmp\fi%
\immediate\write\rfile{\noexpand\item{#1\ }\reflabel{#1}#2}%
\global\advance\refno by1}
\def\addref#1{\immediate\write\rfile{\noexpand\item{}#1}}
\def\listrefs{\vfill\eject\immediate\closeout\rfile
\centerline{{\bf References}}\bigskip{
\catcode`\@=11\escapechar=` %
\input refs.tmp\vfill\eject}}
\def\pacs#1{\medskip\noindent{\bf PACS numbers:} #1
\vfill\eject\footline={\hss\tenrm\folio\hss}}
\def\figures{\centerline{{\bf Figure Captions}}\medskip\parindent=40pt}
\def\fig#1#2{\medskip\item{Fig.~#1:  }#2}
%---------------------------------------------------------------------
\def\frac#1#2{{#1\over#2}}
\def\pmb#1{\setbox0=\hbox{#1}%
\kern-.025em\copy0\kern-\wd0
\kern.05em\copy0\kern-\wd0
\kern-.025em\raise.0433em\box0 }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\e{\space}
\pageno=0
\centerline{{\ttsf Comment on ``Partition Function of Anyon Gas"}}
\vskip 60pt
\bigskip
\centerline{G. Amelino-Camelia}
\vskip 12pt
\centerline{Department of Physics}
\centerline{Boston University}
\centerline{590 Commonwealth Avenue}
\centerline{Boston, Massachusetts 02215 U.S.A.}
\vskip 12pt
\centerline{and}
\vskip 12pt
\centerline{Long Hua}
\vskip 12pt
\baselineskip 12pt plus 0.2pt minus 0.2pt
\centerline{Center for Theoretical Physics}
\centerline{Laboratory for Nuclear Science}
\centerline{and Department of Physics}
\centerline{Massachusetts Institute of Technology}
\centerline{Cambridge, Massachusetts 02139 U.S.A.}
\centerline{and}
\centerline{Department of Physics} 
\centerline{New York University} 
\centerline{New York, NY 10003 U.S.A.} 
\vskip 20pt
\centerline{{\it Physical Review Letters} 69 (1992) 2875}
\vfill
\noindent{BUHEP-92-13}
\noindent{CTP\#2089\hfill }
\eject
\vfill 
\baselineskip 24pt plus 0.2pt minus 0.2pt
In a recent Letter$^{\ref\cr{K. H. Cho  and C. Rim,
{\it Phys. Rev. Lett. } {\bf  68}, 1621 (1992).}}$ Cho and Rim 
calculated the partition function of 
N anyons in an external harmonic well 
and/or in a constant external magnetic field.
The calculation
is based on the conjecture that
all the energy eigenvalues depend linearly on the statistical 
parameter $\delta$ and that they belong to one of the ``classes   
of eigenvalues" $E^I$\e and $E^{II}$\e given in eq.(5) of ref.[1]. 
This conjecture has been 
recently proven to be incorrect by several authors using different arguments.
\medskip
In perturbative
studies of 3 anyons in an external harmonic well, 
it was found that there are
energy eigenvalues with nonlinear dependence on $\delta$. 
In ref.[$ \ref\chou{C. Chou,
{\it Phys. Rev.} {\bf D 44}, 2533 (1991), {\bf E} (in press).}$] 
using  ``fermionic end" perturbation
theory (assuming  $\delta \simeq 1$) it is shown analytically that 
there is an energy eigenvalue given to order $(1-\delta)^2$\e 
by (we set $\hbar=c=1$)
\eqn\ppp{ E^{ferm.}=\omega\{5+ {9 \over 2} \ln({4\over 3})(1-\delta)^2
+O((1-\delta)^3) \}~~.}
\indent
In ref.[$ \ref\gac{ C. Chou, L. Hua and G. Amelino-Camelia,
{\it Phys. Lett.} {\bf B} (in press). }$] 
using ``improved bosonic end" perturbation 
theory (assuming  $\delta \simeq 0$) two more
energy eigenvalues have been calculated analytically to order $\delta^2$ 
$$E^{bos.1}=\omega\{5 + {9 \over 2} \ln({4\over 3})\delta^2
+O(\delta^3) \} $$
\eqn\vc{E^{bos.2}=\omega \{6 - {3 \over 2}\delta + 
{9\over 8}(3\ln({4\over 3})-1)\delta^2
+O(\delta^3) \}~~. }
\medskip
Further indications that not all eigenvalues belong to 
the classes $E^I$\e and $E^{II}$\e
have been given by
numerical calculations of the low-lying energy levels for systems of
three$^{\ref\spoa{M. Sporre,
J. J. M. Verbaarschot, and I. Zahed, {\it Phys. Rev. Lett.}
{\bf 67}, 1813 (1991); M. V. N. Murthy, J. Law, M. Brack, and R. K.
Bhaduri, {\it Phys. Rev. Lett.}
{\bf 67}, 1817 (1991).}}$ and four$^{\ref\spob{M. Sporre,
J. J. M. Verbaarschot, and I. Zahed, SUNY-NTG-91/40.}}$ anyons
in external harmonic well and constant magnetic field
(these results are also mentioned in ref.[1]), and
by the study$^{\ref\hot{T. Awaji  and M. Hotta,
Tohuku University preprint TU-387, (1991).}}$
of the implications of the ``modified conformal symmetry"
which exists when an external magnetic field is applied to a system of 
anyons. We also note that in ref.[$ \ref\dun{G. V. Dunne, A. Lerda, S. Sciuto,
and C. A. Trugenberger, {\it Nucl. Phys.} {\bf B} 370, 601 (1992).}$],
where these classes are also 
discussed, it is suggested that $E^I$\e and $E^{II}$\e do not include
all the energy eigenvalues.
\medskip
In their Reply$^{\ref\rep{K. H. Cho and C. Rim, Reply submitted to
{\it Phys. Rev. Lett.}, (1992).}}$
Cho and Rim conjecture that the energies obtained in ref.[2-5] correspond
to eigenfunctions which do not verify the hard core boundary condition.
However, there seems to be no physical 
reason
to impose this boundary condition
to anyonic wave functions$^{\ref\har{M. Bourdeau and R.D. Sorkin,
BRX-TH-307, (1991).}}$
and, anyway, both in ``fermionic end" and in ``improved bosonic end" 
perturbation
theory, the 0-th order eigenfunctions do verify the hard core boundary 
condition, so that at any given perturbative order the anyonic wave functions
also verify it. (The necessity 
of improving the 0-th order bosonic end eigenfunctions by imposing 
an hard core is discussed in ref.[2,3].)
\medskip
Based on the results of ref.[2-\dun] it is easy to understand that the 
partition function obtained in ref.[1] is neither exact nor
an approximation to the true partition function.
In fact, among the energy eigenvalues which are ignored in the calculation
there are some low-lying eigenvalues, 
which cannot be neglected even at low temperatures.
\medskip
Our final comment concerns the equation of state obtained in ref.[1] .
At present we cannot calculate the exact equation of state because the exact 
expressions for the eigenvalues which do not belong to $E^I$\e and $E^{II}$\e
are not available. However, the results
in ref.[2-\dun] indicate that, varying $\delta$\e between 0 and 1,
the complete set of anyonic energy eigenvalues interpolates
continuously between the bosonic and the fermionic complete sets of eigenvalues;
this would imply that the equation of state for a gas of anyons 
with $\delta \simeq 0$\e 
($\delta \simeq 1$) should be very similar to the one of a gas of bosons
(fermions).
On the contrary, in ref.[1] it is claimed that, for large N,
a gas of anyons 
exhibits quantum behavior up to temperatures much higher than in the
bosonic or fermionic case.
\medskip
\bigskip
We thank C. Chou, R. Jackiw, and S.Y.-Pi for useful discussions. 
This work is
supported in part by funds provided by the U.S. Department of Energy (D.O.E.)
under contract \#DE-AC02-89ER40509. G. A.-C. is supported
partly by funds provided by the ``Fondazioni Angelo Della
Riccia", Firenze, Italy.
L. H. would like to thank New York University for
the Dean's Dissertation Fellowship and the Center for Theoretical
Physics at MIT
for the hospitality during this work.  
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