\NeedsTeXFormat{LaTeX2e}[1995/12/01]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%This file is a class file for RIMS Kokyuroku Bessatsu.
%This is a modification of rims27.cls (modified by T. A.
%and S. Y. on April 18, 2007).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\ProvidesClass{article}
%              [1998/05/05 v1.3y
% Standard LaTeX document class]
%\ProvidesClass{rims25-2000}
%\ProvidesClass{rims25-2}
\ProvidesClass{rims-bessatsu}
%%%              [2000/05/11 v0.01 LaTeX document class]
%%%              [2000/06/20 v0.02 LaTeX document class]
%%%              [2000/07/14 v0.03 LaTeX document class]
%%%              [2000/09/11 v3.01 LaTeX document class]
%%%              [2000/09/25 v4.01 LaTeX document class]
%%%              [2000/10/10 v2.1 LaTeX document class]
%%%              [2000/11/10 v2.2 LaTeX document class]
%%%              [2000/12/15 v2.3 LaTeX document class]
%%%              [2001/01/15 v2.4 LaTeX document class]
              [2001/10/19 v2.5 LaTeX document class]
\newif\if@restonecol

\newif\if@cropon
\@croponfalse

\@twosidetrue  \@mparswitchtrue

%\setlength\paperheight {20.99cm}
%\setlength\paperwidth  {29.70cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mag= \magstep1



\setcounter{secnumdepth}{3}
\newcounter {part}
\newcounter {section}
\newcounter {subsection}[section]
\newcounter {subsubsection}[subsection]
\newcounter {paragraph}[subsubsection]
\newcounter {subparagraph}[paragraph]

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




\DeclareOption{draft}{\setlength\overfullrule{5pt}}
\DeclareOption{final}{\setlength\overfullrule{0pt}}

\DeclareOption{onecolumn}{\@twocolumnfalse}
\DeclareOption{twocolumn}{\@twocolumntrue}
\DeclareOption{leqno}{\PassOptionsToPackage{leqno}{amsmath}}
\DeclareOption{fleqn}{\input{fleqn.clo}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\DeclareOption{secnum}{%
	\renewcommand\theequation {\thesection.\@arabic\c@equation}
	\@addtoreset{equation}{section}}
\DeclareOption{ssecnum}{%
	\renewcommand\theequation {%
		\@arabic\c@section.%
		\ifnum \the\c@subsection = 0\else\@arabic\c@subsection.\fi%
		\@arabic\c@equation}
	\@addtoreset{equation}{subsection}}
\DeclareOption{nsecnum}{%
	\renewcommand\theequation {\@arabic\c@equation}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%Only for Nakanishi Printing\DeclareOption{cropon}{\@cropontrue}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\ExecuteOptions{onecolumn,final,leqno}
\ProcessOptions

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%Only for Nakanishi Printing
%\if@cropon \usepackage[12pt,cam,axes,noinfo,a4center]{crop}
	%info option contains tribial error (size mismatch)
%	\else \usepackage[12pt,a4center,noinfo]{crop} \fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%leqno.clo
\renewcommand\@eqnnum{\hb@xt@.01\p@{}%
                      \rlap{\normalfont\normalcolor
                        \hskip -\displaywidth(\theequation)}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%\ProvidesFile{size10.clo}
%%%              [1998/05/05 v1.3y
%%%      Standard LaTeX file (size option)]
\renewcommand\normalsize{%
   \@setfontsize\normalsize\@xpt{13.5}
   \abovedisplayskip 10\p@ \@plus2\p@ \@minus5\p@
   \abovedisplayshortskip \z@ \@plus3\p@
   \belowdisplayshortskip 6\p@ \@plus3\p@ \@minus3\p@
   \belowdisplayskip \abovedisplayskip
   \let\@listi\@listI}
\normalsize
\newcommand\small{%
   \@setfontsize\small\@ixpt{11}%
   \abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@
   \abovedisplayshortskip \z@ \@plus2\p@
   \belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@
   \def\@listi{\leftmargin\leftmargini
               \topsep 4\p@ \@plus2\p@ \@minus2\p@
               \parsep 2\p@ \@plus\p@ \@minus\p@
               \itemsep \parsep}%
   \belowdisplayskip \abovedisplayskip
}
\newcommand\footnotesize{%
   \@setfontsize\footnotesize\@viiipt{9.5}%
   \abovedisplayskip 6\p@ \@plus2\p@ \@minus4\p@
   \abovedisplayshortskip \z@ \@plus\p@
   \belowdisplayshortskip 3\p@ \@plus\p@ \@minus2\p@
   \def\@listi{\leftmargin\leftmargini
               \topsep 3\p@ \@plus\p@ \@minus\p@
               \parsep 2\p@ \@plus\p@ \@minus\p@
               \itemsep \parsep}%
   \belowdisplayskip \abovedisplayskip
}
\newcommand\scriptsize{\@setfontsize\scriptsize\@viipt\@viiipt}
\newcommand\tiny{\@setfontsize\tiny\@vpt\@vipt}
\newcommand\large{\@setfontsize\large\@xiipt{14}}
\newcommand\Large{\@setfontsize\Large\@xivpt{18}}
\newcommand\LARGE{\@setfontsize\LARGE\@xviipt{22}}
\newcommand\huge{\@setfontsize\huge\@xxpt{25}}
\newcommand\Huge{\@setfontsize\Huge\@xxvpt{30}}

\setlength\smallskipamount{3\p@ \@plus 1\p@ \@minus 1\p@}
\setlength\medskipamount{6\p@ \@plus 2\p@ \@minus 2\p@}
\setlength\bigskipamount{12\p@ \@plus 4\p@ \@minus 4\p@}


%\setlength\topmargin{-60pt}%88bp-1inch
\setlength{\topmargin}{0truecm}
%\setlength{\oddsidemargin}{0.02 true cm}
%\setlength{\evensidemargin}{0.02 true cm}
%\setlength\oddsidemargin  {170pt}%92bp - 1inch
%\setlength\evensidemargin {170pt}%84bp - 1inch

\setlength\headheight{8\p@}
\setlength\headsep   {15\p@}

\setlength\topskip   {12\p@}
%\setlength\footskip{30\p@}
\setlength\footskip{11mm}
\setlength\maxdepth{.5\topskip}

\setlength\textwidth{16true cm}
\setlength\textheight{23true cm}
\setlength\parindent{2 em}

\setlength\marginparwidth {50bp}%marginpar will not used
\setlength\marginparpush{5\p@}


\setlength\footnotesep{6.65\p@}
\setlength{\skip\footins}{9\p@ \@plus 4\p@ \@minus 2\p@}
\setlength\floatsep    {12\p@ \@plus 2\p@ \@minus 2\p@}
\setlength\textfloatsep{20\p@ \@plus 2\p@ \@minus 4\p@}
\setlength\intextsep   {12\p@ \@plus 2\p@ \@minus 2\p@}
\setlength\dblfloatsep    {12\p@ \@plus 2\p@ \@minus 2\p@}
\setlength\dbltextfloatsep{20\p@ \@plus 2\p@ \@minus 4\p@}
\setlength\@fptop{0\p@ \@plus 1fil}
\setlength\@fpsep{8\p@ \@plus 2fil}
\setlength\@fpbot{0\p@ \@plus 1fil}
\setlength\@dblfptop{0\p@ \@plus 1fil}
\setlength\@dblfpsep{8\p@ \@plus 2fil}
\setlength\@dblfpbot{0\p@ \@plus 1fil}
\setlength\partopsep{2\p@ \@plus 1\p@ \@minus 1\p@}
\def\@listi{\leftmargin\leftmargini
            \parsep 4\p@ \@plus2\p@ \@minus\p@
            \topsep 8\p@ \@plus2\p@ \@minus4\p@
            \itemsep4\p@ \@plus2\p@ \@minus\p@}
\let\@listI\@listi
\@listi
\def\@listii {\leftmargin\leftmarginii
              \labelwidth\leftmarginii
              \advance\labelwidth-\labelsep
              \topsep    4\p@ \@plus2\p@ \@minus\p@
              \parsep    2\p@ \@plus\p@  \@minus\p@
              \itemsep   \parsep}
\def\@listiii{\leftmargin\leftmarginiii
              \labelwidth\leftmarginiii
              \advance\labelwidth-\labelsep
              \topsep    2\p@ \@plus\p@\@minus\p@
              \parsep    \z@
              \partopsep \p@ \@plus\z@ \@minus\p@
              \itemsep   \topsep}
\def\@listiv {\leftmargin\leftmarginiv
              \labelwidth\leftmarginiv
              \advance\labelwidth-\labelsep}
\def\@listv  {\leftmargin\leftmarginv
              \labelwidth\leftmarginv
              \advance\labelwidth-\labelsep}
\def\@listvi {\leftmargin\leftmarginvi
              \labelwidth\leftmarginvi
              \advance\labelwidth-\labelsep}
%%%\endinput
%%
%% End of file `size10.clo'.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\setlength\lineskip{1\p@}
\setlength\normallineskip{1\p@}
\renewcommand\baselinestretch{1.04}
\setlength\parskip{0\p@}

\@lowpenalty   51
\@medpenalty  151
\@highpenalty 301

\setcounter{topnumber}{2}
\renewcommand\topfraction{.7}
\setcounter{bottomnumber}{1}
\renewcommand\bottomfraction{.3}
\setcounter{totalnumber}{3}
\renewcommand\textfraction{.2}
\renewcommand\floatpagefraction{.5}
\setcounter{dbltopnumber}{2}
\renewcommand\dbltopfraction{.7}
\renewcommand\dblfloatpagefraction{.5}



\def\title#1{\gdef\@title{#1}
	\@ifundefined{@TitleHead}{\TitleHead{#1}}{}}
\def\TitleHead#1{\def\@TitleHead{#1}}
\def\dedicatory#1{\gdef\@dedicatory{#1}}

\def\author#1{\gdef\@author{\large #1}
	\@ifundefined{@AuthorHead}{\AuthorHead{#1}}{}}
\def\AuthorHead#1{\def\@AuthorHead{#1}}



\def\VolumeNo#1{%
  \def\@VolumeNo{#1}
}
\def\YearNo#1{%
  \def\@YearNo{#1}
}
\def\PagesNo#1{%
  \def\@PagesNo{#1}
}
\VolumeNo{\null}
\YearNo{\null}
\PagesNo{\null}


\def\communication#1{\def\@comm{#1}}
\communication{\null}

\def\classification#1{\def\@class{#1}}
\classification{\null}

\def\keywords#1{\gdef\@keywords{\textit{Key Words}:  #1}}

\def\support#1{\gdef\@support{#1}}

%\def\email#1{\gdef\@email{\texttt{e-mail:  #1}}}
%\email{\null}

%%%\def\keywords#1{%
%%%  \def\@keywords{#1}%
%%%}


\newif\ifCopyNRes
\global\CopyNResfalse%

\def\volyr{202x}%

\gdef\FooterText{\ifnum\currentpageno=\c@page%
\ifCopyNRes%
  \vbox{\footnotesize%
    \hbox to\textwidth{\hfill Published by Research Institute for Mathematical Sciences, Kyoto University.}%
    \hbox to\textwidth{\hfill}%
       }%
\else%
  \vbox{\footnotesize%
    \hbox to\textwidth{\hfill $\copyright$ \volyr\ Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.}%
       }%
\fi%
\else%
\fi}%

  \def\ps@headings{%
      \def\@oddfoot{\FooterText}%
      \def\@evenfoot{\FooterText}%
      \def\@evenhead{\footnotesize\thepage\hfil\scshape\@AuthorHead\hfil}%
      \def\@oddhead{\footnotesize\hfil{\scshape\@TitleHead}\hfil\thepage}}
\def\ps@myheadings{%
      \def\@oddfoot{\FooterText}%
      \def\@evenfoot{\FooterText}%
      \def\@evenhead{\footnotesize\thepage\hfil\scshape\@AuthorHead\hfil}%
%%%from fancyheadings.sty%%%
      \def\@oddhead{\scriptsize %
                   \hbox to\textwidth{\vbox{\hbox
                   {\rlap{\parbox[b]{\textwidth}%
                   {\raggedright{\it{\small RIMS K\^oky\^uroku Bessatsu }}
                   \strut}}}%
                  { {\bf{\small B\@VolumeNo}}
\small (\@YearNo ), \@PagesNo \hfil}}}}%
    \let\@mkboth\@gobbletwo
    \let\sectionmark\@gobble
    \let\subsectionmark\@gobble
    }



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\@@fnsymbol#1{\ensuremath{\ifcase#1\or *\or **\or *{**}\or \dagger\or 
   \ddagger\or \mathsection\or \mathparagraph\or \dagger\dagger
   \or \ddagger\ddagger \else\@ctrerr\fi}}

\newcount\currentpageno%

\newcommand\maketitle{\par
\currentpageno\c@page%
  \thispagestyle{myheadings}
  \begingroup
    \renewcommand\thefootnote{\@@fnsymbol\c@footnote}%
    \def\@makefnmark{\rlap{\@textsuperscript{\normalfont\@thefnmark}}}%
    \long\def\@makefntext##1{\parindent 1em\noindent
            \hb@xt@1.8em{%
%%%
                \hss\@textsuperscript{\normalfont\@thefnmark}}%
                \@tempdima\columnwidth%
                \advance\@tempdima-1.8em%
                \parbox[t]{\@tempdima}{##1}
    }
%%%
      \newpage
      \global\@topnum\z@   % Prevents figures from going at top of page.
      \@maketitle
  \@thanks
  \endgroup
  \setcounter{footnote}{0}%
  \global\let\thanks\relax
  \global\let\maketitle\relax
  \global\let\@maketitle\relax
  \global\let\@thanks\@empty
  \global\let\@author\@empty
  \global\let\@date\@empty
  \global\let\@title\@empty
  \global\let\@dedicatory\@empty
  \global\let\@support\@empty
  \global\let\title\relax
  \global\let\dedicatory\relax
  \global\let\support\relax
  \global\let\author\relax
  \global\let\@keywords\@empty
  \global\let\keywords\relax
}

\def\@maketitle{%
  \newpage
  \null
  \vskip 2pt%%%
  \begin{center}%%%
  \let \footnote \thanks
    {\Large\bfseries \@title \par}%
    \@ifundefined{@dedicatory}%
      {}{\vskip \baselineskip {\normalsize\itshape \@dedicatory \par}}%
    \vskip 20pt%%%
    {\footnotesize By \par}
    \vskip 10pt%%%
    {\normalsize    \@author \par}
  \end{center}
  %\footnote[0]{Communicated by \@comm}
   \footnote[0]{\@comm}
  \footnote[0]{2020 Mathematics Subject Classification(s): \@class}
  \@ifundefined{@keywords}%
      {}{\footnotetext[0]{\@keywords}}%
  \@ifundefined{@support}%
      {}{\footnotetext[0]{\@support}}%
  \@ifundefined{@email}%
      {}{\footnotetext[0]{\@email}}%
  }

\newenvironment{abstract}
{	\vspace*{4pt}
	\begingroup
	\begin{center}
	\centerline{\normalsize\bfseries Abstract}
	\end{center}
	\par \vspace{-12pt} \small
}{%
	\par
	\endgroup
	\vspace{13.5pt}%
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%from latex.ltx
%%%\def\@seccntformat#1{\csname the#1\endcsname\quad}
%%%this command cause blank space (\quad) before title,
%%%even if no chapter mark is setted.
\def\@seccntformat#1{\csname the#1\endcsname}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\renewcommand \thepart {\@Roman\c@part}
\renewcommand \thesection {\@arabic\c@section}
\renewcommand \thesubsection   {\thesection.\@arabic\c@subsection}
\renewcommand \thesubsubsection{\thesubsection.\@arabic\c@subsubsection}
\renewcommand \theparagraph    {\thesubsubsection.\@arabic\c@paragraph}
\renewcommand \thesubparagraph {\theparagraph.\@arabic\c@subparagraph.}


\newcommand\part{\par
   \addvspace{4ex}%
   \@afterindentfalse
   \secdef\@part\@spart
}

\def\@part[#1]#2{%
    \ifnum \c@secnumdepth >\m@ne
      \refstepcounter{part}%
      \addcontentsline{toc}{part}{\thepart\hspace{1em}#1}%
    \else
      \addcontentsline{toc}{part}{#1}%
    \fi
    {\parindent \z@ \raggedright
     \interlinepenalty \@M
     \normalfont
     \ifnum \c@secnumdepth >\m@ne
       \Large\bfseries \partname~\thepart
       \par\nobreak
     \fi
     \huge \bfseries #2%
     \markboth{}{}\par}%
    \nobreak
    \vskip 3ex
    \@afterheading}
\def\@spart#1{%
    {\parindent \z@ \raggedright
     \interlinepenalty \@M
     \normalfont
     \huge \bfseries #1\par}%
     \nobreak
     \vskip 3ex
     \@afterheading}

\def\section{%\par
   \addvspace{2mm}%%%{\baselineskip}
   \@afterindenttrue
   \secdef\@section\@ssection}
\def\@section[#1]#2{%
    \ifnum \c@secnumdepth >\m@ne
      \refstepcounter{section}%
      \addcontentsline{toc}{section}{\S\,\thesection.\hspace{1em}#1}%
    \else
      \addcontentsline{toc}{section}{\hspace{1em}#1}%
    \fi
%%%
    \if@nobreak%
    	\vspace{-2mm}
    \fi
%%%
    {\parindent \z@ 
     \interlinepenalty \@M
     \normalfont
    \par \bigskip \centering{
     \ifnum \c@secnumdepth >\m@ne
       \bfseries \S\,\thesection.\quad%
     \fi%
     \bfseries #2
      }\par\nobreak\bigskip \nobreak}%
    \nobreak
%    \vskip \baselineskip
    \@afterheading}%
\def\@ssection#1{%
%%%
    \if@nobreak%
    	\vspace{-2mm}
    \fi
%%%
    {\parindent \z@ 
     \interlinepenalty \@M
     \normalfont
     \begin{center}
     \bfseries #1
      \end{center}}%
    \nobreak
%    \vskip \baselineskip
    \@afterheading}%

\def\subsection{%\par
   \addvspace{2mm}%%%{\baselineskip}
   \@afterindenttrue
   \secdef\@subsection\@ssubsection}
\def\@subsection[#1]#2{%
    \ifnum \c@secnumdepth >\m@ne
      \refstepcounter{subsection}%
      \addcontentsline{toc}{subsection}{\S\,\thesubsection.\hspace{1em}#1}%
    \else
      \addcontentsline{toc}{subsection}{#1}%
    \fi
%%%
    \if@nobreak%
    	\vspace{-2mm}
    \fi
%%%
    {\parindent \z@ 
     \interlinepenalty \@M
     \normalfont
       \par \medskip \centering{
     \ifnum \c@secnumdepth >\m@ne
       \bfseries \sectionname \thesubsection.\quad%
     \fi%
     \bfseries #2
       }\par\nobreak\medskip \nobreak}%
%    \vskip \baselineskip
    \nobreak
    \@afterheading}%
\def\@ssubsection#1{%
    {\parindent \z@ 
     \interlinepenalty \@M
     \normalfont
    \par \medskip \centering{
     \bfseries #1
       }\par\nobreak\medskip \nobreak}%
    \nobreak
%    \vskip \baselineskip
    \@afterheading}%
%%%%
\def\subsubsection{%
   \addvspace{\baselineskip}%%%{2mm}
   \@afterindenttrue
   \secdef\@subsubsection\@ssubsubsection}
\def\@subsubsection[#1]#2{%
    \ifnum \c@secnumdepth >\m@ne
      \refstepcounter{subsubsection}%
      \addcontentsline{toc}{subsubsection}{\S\,\thesubsubsection.\hspace{1em}#1}%
    \else
      \addcontentsline{toc}{subsubsection}{#1}%
    \fi
%%%
    \if@nobreak%
    	\vspace{-4mm}
    \fi
%%%
    {\parindent \z@ 
     \interlinepenalty \@M
     \normalfont
     \ifnum \c@secnumdepth >\m@ne
       \bfseries \thesubsubsection.\quad%
     \fi%
     \bfseries #2
     }%
    \@afterheading
     }%
\def\@ssubsubsection#1{%
%%%
    \if@nobreak%
    	\vspace{-2mm}
    \fi
%%%
    {\parindent \z@ 
     \interlinepenalty \@M
     \normalfont
     \bfseries #1
     }%
    \@afterheading
     }%
%%%%
%%%%%%%%%%%%%
\def\@seccntformat#1{\csname the#1\endcsname\quad}
%%%%%%%%%%%%%


\newcommand\paragraph{\@startsection{paragraph}{4}{\z@}%
                                    {3.25ex \@plus1ex \@minus.2ex}%
                                    {-1em}%
                                    {\normalfont\normalsize\bfseries}}
\newcommand\subparagraph{\@startsection{subparagraph}{5}{\parindent}%
                                       {3.25ex \@plus1ex \@minus .2ex}%
                                       {-1em}%
                                      {\normalfont\normalsize\bfseries}}


%%%\setlength\leftmargini  {2.5em}
\setlength\leftmargini  {1.8em}%%% If itemlabel is wider, increase this.
\leftmargin  \leftmargini
\setlength\leftmarginii  {2.2em}
\setlength\leftmarginiii {1.87em}
\setlength\leftmarginiv  {1.7em}
\setlength\leftmarginv  {1em}
\setlength\leftmarginvi {1em}
\setlength  \labelsep  {.5em}
\setlength  \labelwidth{\leftmargini}
\addtolength\labelwidth{-\labelsep}
\@beginparpenalty -\@lowpenalty
\@endparpenalty   -\@lowpenalty
\@itempenalty     -\@lowpenalty
\renewcommand\theenumi{\@arabic\c@enumi}
\renewcommand\theenumii{\@alph\c@enumii}
\renewcommand\theenumiii{\@roman\c@enumiii}
\renewcommand\theenumiv{\@Alph\c@enumiv}
\newcommand\labelenumi{\theenumi.}
\newcommand\labelenumii{(\theenumii)}
\newcommand\labelenumiii{\theenumiii.}
\newcommand\labelenumiv{\theenumiv.}
\renewcommand\p@enumii{\theenumi}
\renewcommand\p@enumiii{\theenumi(\theenumii)}
\renewcommand\p@enumiv{\p@enumiii\theenumiii}
\newcommand\labelitemi{\textbullet}
\newcommand\labelitemii{\normalfont\bfseries \textendash}
\newcommand\labelitemiii{\textasteriskcentered}
\newcommand\labelitemiv{\textperiodcentered}

\newenvironment{description}
               {\list{}{\labelwidth\z@ \itemindent-\leftmargin
                        \let\makelabel\descriptionlabel}}
               {\endlist}
\newcommand*\descriptionlabel[1]{\hspace\labelsep
                                \normalfont\bfseries #1}

\newenvironment{verse}
               {\let\\\@centercr
                \list{}{\itemsep      \z@
                        \itemindent   -1.5em%
                        \listparindent\itemindent
                        \rightmargin  \leftmargin
                        \advance\leftmargin 1.5em}%
                \item\relax}
               {\endlist}
\newenvironment{quotation}
               {\list{}{\listparindent 1.5em%
                        \itemindent    \listparindent
                        \rightmargin   \leftmargin
                        \parsep        \z@ \@plus\p@}%
                \item\relax}
               {\endlist}
\newenvironment{quote}
               {\list{}{\rightmargin\leftmargin}%
                \item\relax}
               {\endlist}



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


\newcommand\appendix{\par
  \setcounter{section}{0}%
  \setcounter{subsection}{0}%
  \renewcommand\thesection{Appendix \@Alph\c@section}
  \renewcommand\sectionname{}
%%%  \addcontentsline{toc}{section}{\thesection.\hspace{1em}#1}%
}%



\setlength\arraycolsep{1\p@}
\setlength\tabcolsep{6\p@}
\setlength\arrayrulewidth{.4\p@}
\setlength\doublerulesep{2\p@}
\setlength\tabbingsep{\labelsep}
\skip\@mpfootins = \skip\footins
\setlength\fboxsep{3\p@}
\setlength\fboxrule{.4\p@}


\newcounter{figure}
\renewcommand \thefigure {\@arabic\c@figure}
\def\fps@figure{tbp}
\def\ftype@figure{1}
\def\ext@figure{lof}
\def\fnum@figure{\figurename~\thefigure}
\newenvironment{figure}
               {\@float{figure}}
               {\end@float}
\newenvironment{figure*}
               {\@dblfloat{figure}}
               {\end@dblfloat}

\newcounter{table}
\renewcommand\thetable{\@arabic\c@table}
\def\fps@table{tbp}
\def\ftype@table{2}
\def\ext@table{lot}
\def\fnum@table{\tablename~\thetable}
\newenvironment{table}
               {\@float{table}}
               {\end@float}
\newenvironment{table*}
               {\@dblfloat{table}}
               {\end@dblfloat}

\newlength\abovecaptionskip
\newlength\belowcaptionskip
\setlength\abovecaptionskip{10\p@}
\setlength\belowcaptionskip{0\p@}
\long\def\@makecaption#1#2{%
  \vskip\abovecaptionskip
  \sbox\@tempboxa{#1. #2}%
  \ifdim \wd\@tempboxa >\hsize
    #1. #2\par
  \else
    \global \@minipagefalse
    \hb@xt@\hsize{\hfil\box\@tempboxa\hfil}%
  \fi
  \vskip\belowcaptionskip}

\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}
\DeclareRobustCommand*\cal{\@fontswitch\relax\mathcal}
\DeclareRobustCommand*\mit{\@fontswitch\relax\mathnormal}

\newcommand\@pnumwidth{1.55em}
\newcommand\@tocrmarg{2.55em}
\newcommand\@dotsep{4.5}
\setcounter{tocdepth}{3}
\newcommand\tableofcontents{%
    \vspace*{-6.75pt}
    \section*{\contentsname}%
    \@starttoc{toc}%
    \vspace*{13.5pt}
    }
\newcommand*\l@part[2]{%
  \ifnum \c@tocdepth >-2\relax
    \addpenalty\@secpenalty
    \addvspace{2.25em \@plus\p@}%
    \begingroup
      \parindent \z@ \rightskip \@pnumwidth
      \parfillskip -\@pnumwidth
      {\leavevmode
       \large \bfseries #1\hfil \hb@xt@\@pnumwidth{\hss #2}}\par
       \nobreak
       \if@compatibility
         \global\@nobreaktrue
         \everypar{\global\@nobreakfalse\everypar{}}%
      \fi
    \endgroup
  \fi}
\newcommand*\l@section[2]{% 
  \ifnum \c@tocdepth >\z@
    \addpenalty\@secpenalty
    \addvspace{0.3em \@plus\p@}%%%% changed 20/06/2000
    \setlength\@tempdima{1.5em}%
    \begingroup
      \parindent \z@ \rightskip \@pnumwidth
      \parfillskip -\@pnumwidth
      \leavevmode %\bfseries
      \advance\leftskip\@tempdima
      \hskip -\leftskip
      %#1\nobreak\hfil \nobreak\hb@xt@\@pnumwidth{\hss #2}\par
      #1
      \hfill\mbox{}
      \par
    \endgroup
  \fi}

\newcommand\@tocline[5]{%
  \ifnum #1>\c@tocdepth \else
    \vskip \z@ \@plus.2\p@
    {\leftskip #2\relax \rightskip \@tocrmarg \parfillskip -\rightskip
     \parindent #2\relax\@afterindenttrue
     \interlinepenalty\@M
     \leavevmode
     \@tempdima #3\relax
     \advance\leftskip \@tempdima \null\nobreak\hskip -\leftskip
     {#4}
     \hfill\mbox{}%
     \par}%
  \fi}

\newcommand*\l@subsection{\@tocline{2}{1.5em}{2.3em}}
\newcommand*\l@subsubsection{\@tocline{3}{3.8em}{3.2em}}
\newcommand*\l@paragraph{\@tocline{4}{7.0em}{4.1em}}
\newcommand*\l@subparagraph{\@tocline{5}{10em}{5em}}
\newcommand\listoffigures{%
    \section*{\listfigurename
      \@mkboth{\MakeUppercase\listfigurename}%
              {\MakeUppercase\listfigurename}}%
    \@starttoc{lof}%
    }
\newcommand*\l@figure{\@dottedtocline{1}{1.5em}{2.3em}}
\newcommand\listoftables{%
    \section*{\listtablename
      \@mkboth{%
          \MakeUppercase\listtablename}{\MakeUppercase\listtablename}}%
    \@starttoc{lot}%
    }
\let\l@table\l@figure





\renewcommand\footnoterule{
  \kern-3\p@
  \hrule\@width.4\columnwidth
  \kern2.6\p@}
%%%\newcommand\@makefntext[1]{
%%%    \parindent 1em%
%%%    \noindent
%%%    \hb@xt@1.8em{\hss\@makefnmark}#1}
\long\def\@makefntext#1{%
    \parindent 1em%
    \noindent
    \hb@xt@1.8em{\hss\@makefnmark}%
    \@tempdima\columnwidth%
    \advance\@tempdima-1.8em%
    \parbox[t]{\@tempdima}{#1}
    }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%\@ifpackageloaded{amsgen}{}{\input{amsgen.sty}}
%%%        from amsgen.sty
\let\@xp=\expandafter
\let\@nx=\noexpand
\def\@oparg#1[#2]{\@ifnextchar[{#1}{#1[#2]}}
\long\def\@ifempty#1{\@xifempty#1@@..\@nil}
\long\def\@xifempty#1#2@#3#4#5\@nil{%
  \ifx#3#4\@xp\@firstoftwo\else\@xp\@secondoftwo\fi}
\long\def\@ifnotempty#1{\@ifempty{#1}{}}
\def\@addpunct#1{\ifnum\spacefactor>\@m \else#1\fi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% from  amsthm.sty [1996/10/24 v1.2d]
\newcommand{\theoremstyle}[1]{%
  \@ifundefined{th@#1}{%
    \PackageWarning{amsthm}{Unknown theoremstyle `#1'}%
    \thm@style{plain}%
  }{%
    \thm@style{#1}%
  }%
}
\newtoks\thm@style
\thm@style{plain}
\newtoks\thm@bodyfont
\thm@bodyfont{\itshape}
\newtoks\thm@headfont
\thm@headfont{\bfseries}
\newtoks\thm@notefont
\thm@notefont{}
\newtoks\thm@headpunct
\thm@headpunct{.}
\newtoks\thm@numberfont
\thm@numberfont{\bfseries}
%%%\newskip\thm@preskip \thm@preskip\topsep
\newskip\thm@preskip \thm@preskip\abovedisplayskip
%%%\newskip\thm@postskip \thm@postskip\topsep
\newskip\thm@postskip \thm@postskip \belowdisplayskip
\renewcommand{\newtheorem}{\@ifstar{\@xnthm *}{\@xnthm \relax}}
\def\@xnthm#1#2{%
  \let\@tempa\relax
  \@xp\@ifdefinable\csname #2\endcsname{%
    \global\@xp\let\csname end#2\endcsname\@endtheorem
    \ifx *#1% unnumbered, need to get one more mandatory arg
      \edef\@tempa##1{%
        \gdef\@xp\@nx\csname#2\endcsname{%
          \@nx\@thm{\@xp\@nx\csname th@\the\thm@style\endcsname}%
            {}{##1}}}%
    \else % numbered theorem, need to check for optional arg
      \def\@tempa{\@oparg{\@ynthm{#2}}[]}%
    \fi
  }%
  \@tempa
}
\def\@ynthm#1[#2]#3{%
  \ifx\relax#2\relax
    \def\@tempa{\@oparg{\@xthm{#1}{#3}}[]}%
  \else
    \@ifundefined{c@#2}{%
      \def\@tempa{\@nocounterr{#2}}%
    }{%
      \@xp\xdef\csname the#1\endcsname{\@xp\@nx\csname the#2\endcsname}%
      \toks@{#3}%
      \@xp\xdef\csname#1\endcsname{%
        \@nx\@thm{%
          \let\@nx\thm@swap
            \if S\thm@swap\@nx\@firstoftwo\else\@nx\@gobble\fi
          \@xp\@nx\csname th@\the\thm@style\endcsname}%
            {#2}{\the\toks@}}%
      \let\@tempa\relax
    }%
  \fi
  \@tempa
}
\def\@xthm#1#2[#3]{%
  \ifx\relax#3\relax
    \newcounter{#1}%
  \else
    \newcounter{#1}[#3]%
    \@xp\xdef\csname the#1\endcsname{\@xp\@nx\csname the#3\endcsname
      \@thmcountersep\@thmcounter{#1}}%
  \fi
  \toks@{#2}%
  \@xp\xdef\csname#1\endcsname{%
    \@nx\@thm{%
      \let\@nx\thm@swap
        \if S\thm@swap\@nx\@firstoftwo\else\@nx\@gobble\fi
      \@xp\@nx\csname th@\the\thm@style\endcsname}%
      {#1}{\the\toks@}}%
}
\def\@thm#1#2#3{\normalfont
  \trivlist
  \edef\@restorelabelsep{\labelsep\the\labelsep}%
  \labelsep.5em\relax \let\thmheadnl\relax
%%%  \let\thm@indent\noindent % no indent
  \let\thm@indent\indent % no indent
  \let\thm@swap\@gobble
  \thm@headfont{\bfseries}% heading font bold
  \thm@numberfont{\bfseries}% heading font bold
  \thm@headpunct{.}% add period after heading
%%%  \thm@preskip\topsep
  \thm@preskip \abovedisplayskip
  \thm@postskip\thm@preskip
  #1% style overrides
  \@topsep \thm@preskip               % used by first \item
  \@topsepadd \thm@postskip           % used by \@endparenv
  \def\@tempa{#2}\ifx\@empty\@tempa
    \def\@tempa{\@oparg{\@begintheorem{#3}{}}[]}%
  \else
    \refstepcounter{#2}%
    \def\@tempa{\@oparg{\@begintheorem{#3}{\csname the#2\endcsname}}[]}%
  \fi
  \@tempa
}
\let\@ythm\relax
\let\thmname\@iden \let\thmnote\@iden \let\thmnumber\@iden
\providecommand\@upn{\textup}
\def\thmhead@plain#1#2#3{%
%%%  \thmname{#1}\thmnumber{\@ifnotempty{#1}{ }#2}%
  \thmname{#1}\thmnumber{\@ifnotempty{#1}{ }#2}%
%%%  \thmnote{ {\the\thm@notefont(#3)}}}
\thmnote{ {\the\thm@notefont $($#3)}}}
\let\thmhead\thmhead@plain
\def\swappedhead#1#2#3{%
  \thmnumber{#2}\thmname{\@ifnotempty{#2}{. }#1}%
%%%  \thmnote{ {\the\thm@notefont(#3)}}}
  \thmnote{ {\the\thm@notefont #3}}}
\let\thmheadnl\relax
\def\@begintheorem#1#2[#3]{%
  \item[\normalfont % reset in case body font is abnormal
  \hskip\labelsep
  \the\thm@headfont
  \thm@indent
  \@ifempty{#1}{\let\thmname\@gobble}{\let\thmname\@iden}%
  \@ifempty{#2}{\let\thmnumber\@gobble}{\let\thmnumber\@iden}%
  \@ifempty{#3}{\let\thmnote\@gobble}{\let\thmnote\@iden}%
%%%  \thm@swap\swappedhead\thmhead{#1}{#2}{#3}%
  \thm@swap\swappedhead\thmhead{#1}
  {\the\thm@numberfont{#2}}
  {\rm{#3}}%
%%%  \the\thm@headpunct]%
  \the\thm@headpunct] \quad%
  \@restorelabelsep
  \thmheadnl % possibly a newline.
  \ignorespaces}
\def\nonslanted{\relax
  \@xp\let\@xp\@tempa\csname\f@shape shape\endcsname
  \ifx\@tempa\itshape\upshape
  \else\ifx\@tempa\slshape\upshape\fi\fi}
\def\swapnumbers{\edef\thm@swap{\if S\thm@swap N\else S\fi}}
\def\thm@swap{N}%
\let\@opargbegintheorem\relax
%%%
%%% plain %%%%%%%%%%%%%%%%%%%
\def\th@plain{%
%%  \let\thm@indent\noindent % no indent
%%  \thm@headfont{\bfseries}% heading font is bold
%%  \thm@numberfont{\bfseries}% number font is bold
%%  \thm@notefont{}% same as heading font
%%  \thm@headpunct{.}% add period after heading
%%  \let\thm@swap\@gobble
%%  \thm@preskip\topsep
%%  \thm@postskip\theorempreskipamount
  \itshape % body font
}
%%%
%%% definition %%%%%%%%%%%%%%
\def\th@definition{%
  \normalfont % body font
}
%%%
%%% remark %%%%%%%%%%%%%%%%%%
\def\th@remark{%
  \thm@headfont{\itshape}%
%%%  \thm@numberfont{\itshape}%
  \thm@numberfont{\rm}%
  \normalfont % body font
%%%  \thm@preskip\topsep
  \thm@preskip \abovedisplayskip
%%%  \divide\thm@preskip\tw@
  \thm@postskip\thm@preskip
}
%%%
\def\@endtheorem{\endtrivlist\@endpefalse }
\newcommand{\newtheoremstyle}[9]{%
  \@ifempty{#5}{\dimen@\z@skip}{\dimen@#5\relax}%
  \ifdim\dimen@=\z@
%%%    \toks@{#4\let\thm@indent\noindent}%
    \toks@{#4\let\thm@indent\indent}%
  \else
%%%    \toks@{#4\def\thm@indent{\noindent\hbox to#5{}}}%
    \toks@{#4\def\thm@indent{\indent\hbox to#5{}}}%
  \fi
  \def\@tempa{#8}\ifx\space\@tempa
    \toks@\@xp{\the\toks@ \labelsep\fontdimen\tw@\font\relax}%
  \else
    \def\@tempb{\newline}%
    \ifx\@tempb\@tempa
      \toks@\@xp{\the\toks@ \labelsep\z@skip
        \def\thmheadnl{%
          \@noskipsectrue \global\@nobreaktrue
          \everypar{\global\@minipagefalse \global\@newlistfalse
            \global\@inlabelfalse \global\@nobreakfalse
            {\setbox\z@\lastbox}\box\@labels\par
            \nobreak\vskip-\parskip
            \everypar{}\noindent}}%
      }%
    \else
      \toks@\@xp{\the\toks@ \labelsep#8\relax}%
    \fi
  \fi
  \begingroup \th@plain % to set \thm@preskip and postskip
  \@defaultunits\@tempskipa#2\thm@preskip\relax\@nnil
  \@defaultunits\@tempskipb#3\thm@postskip\relax\@nnil
  \xdef\@gtempa{\thm@preskip\the\@tempskipa
    \thm@postskip\the\@tempskipb\relax}%
  \endgroup
  \@temptokena\@xp{\@gtempa
    \thm@headfont{#6}\thm@headpunct{#7}%
  }%
  \@ifempty{#9}{%
    \let\thmhead\thmhead@plain
  }{%
    \@namedef{thmhead@#1}##1##2##3{#9}%
    \@temptokena\@xp{\the\@temptokena
      \@xp\let\@xp\thmhead\csname thmhead@#1\endcsname}%
  }%
  \@xp\xdef\csname th@#1\endcsname{\the\toks@ \the\@temptokena}%
}
%%% Proof %%%%%%%%%%%%%%%%%%%%
\DeclareRobustCommand{\qed}{%
  \ifmmode % if math mode, assume display: omit penalty etc.
  \else \leavevmode\unskip\penalty9999 \hbox{}\nobreak\hfill
  \fi
  \quad\hbox{\qedsymbol}}
\newcommand{\openbox}{\leavevmode
  \hbox to.77778em{%
  \hfil\vrule
  \vbox to.675em{\hrule width.6em\vfil\hrule}%
  \vrule\hfil}}
\newcommand{\qedsymbol}{\openbox}
\newenvironment{proof}[1][\proofname]{\par
%%%  \vskip \abovedisplayskip%
  \vskip \parsep%
  \normalfont
%%%  \topsep6\p@\@plus6\p@ \trivlist
%%%  \item[\hskip\labelsep\itshape
%%%    #1\@addpunct{.}]\ignorespaces
  {\itshape
  #1\@addpunct{.}}\quad\mdseries \ignorespaces
}{%
  \qed%\endtrivlist
  {\vskip \belowdisplayskip}
}
\newcommand{\proofname}{Proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% end  amsthm.sty [1996/10/24 v1.2d]  by  HIK NACOS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





\newenvironment{thebibliography}[1]
     {\vspace{20pt}
     \section*{\refname
        \@mkboth{\MakeUppercase\refname}{\MakeUppercase\refname}%
        \addcontentsline{toc}{section}{References}}%
      \small%%%
      \list{\@biblabel{\@arabic\c@enumiv}}%
           {%
            \setlength{\parsep}{0pt}%%%
            \setlength{\itemsep}{0pt}%%%
            \settowidth\labelwidth{\@biblabel{#1}}%
            \leftmargin\labelwidth
            \advance\leftmargin\labelsep
            \usecounter{enumiv}%
            \let\p@enumiv\@empty
            \renewcommand\theenumiv{\@arabic\c@enumiv}}%
      \sloppy
      \clubpenalty4000
      \@clubpenalty \clubpenalty
      \widowpenalty4000%
      \sfcode`\.\@m}
     {\def\@noitemerr
       {\@latex@warning{Empty `thebibliography' environment}}%
      \endlist}
\newcommand\newblock{\hskip .11em\@plus.33em\@minus.07em}

\def\@lbibitem[#1]#2{\item[\hfill\@biblabel{#1}]\if@filesw
      {\let\protect\noexpand
       \immediate
       \write\@auxout{\string\bibcite{#2}{#1}}}\fi\ignorespaces}

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}



\newenvironment{theindex}
               {\if@twocolumn
                  \@restonecolfalse
                \else
                  \@restonecoltrue
                \fi
                \columnseprule \z@
                \columnsep 35\p@
                \twocolumn[\section*{\indexname}]%
                \@mkboth{\MakeUppercase\indexname}%
                        {\MakeUppercase\indexname}%
                \thispagestyle{plain}\parindent\z@
                \parskip\z@ \@plus .3\p@\relax
                \let\item\@idxitem}
               {\if@restonecol\onecolumn\else\clearpage\fi}
\newcommand\@idxitem{\par\hangindent 40\p@}
\newcommand\subitem{\@idxitem \hspace*{20\p@}}
\newcommand\subsubitem{\@idxitem \hspace*{30\p@}}
\newcommand\indexspace{\par \vskip 10\p@ \@plus5\p@ \@minus3\p@\relax}



\newcommand\contentsname{Contents}
\newcommand\listfigurename{List of Figures}
\newcommand\listtablename{List of Tables}
\newcommand\refname{References}
\newcommand\indexname{Index}
\newcommand\figurename{Figure}
\newcommand\tablename{Table}
\newcommand\partname{Part}
\newcommand\sectionname{\S\,}
\newcommand\appendixname{Appendix}


\newcommand\dash{%
  \raise .65ex\hbox{\vrule width 36pt height 0.4pt depth 0pt}}


\def\today{\ifcase\month\or
  January\or February\or March\or April\or May\or June\or
  July\or August\or September\or October\or November\or December\fi
  \space\number\day, \number\year}
\setlength\columnsep{10\p@}
\setlength\columnseprule{0\p@}
\pagestyle{headings}
\pagenumbering{arabic}
\normalfont
\normalsize

\newcommand{\bm}[1]{\hbox{\boldmath$#1$}}%
\newcommand{\sbm}[1]{\hbox{\scriptsize\boldmath$#1$}}%

\newenvironment{Enu}[1][\rm 1.]{\begin{enumerate}[#1] \parskip=-0ex \topsep=0ex 
\parsep =0pt \itemsep=0ex}{\end{enumerate}}

\endinput



%%
%% End of file `article.cls'.