


\def\eqalignno#1{\displ@y \tabskip=\@centering
    \halign to\displaywidth{\hfil$\@lign\displaystyle{##}$
    \tabskip=0pt 
&$\@lign\displaystyle{{}##}$\hfil\tabskip=\@centering
&\llap{$\@lign##$}\tabskip=0pt\crcr#1\crcr}}

\newcommand{\eq}[1]{\refstepcounter{equation} \label{#1}(\theequation)}

\def\proj{{\rm proj}}
\def\conv{{\rm conv}}
\def\cone{{\rm cone}}
\def\aff{{\rm aff}}
\def\det{{\rm det}}
\def\adj{{\rm adj}}     

\def\eqnarrayz#1{\displ@y
\tabskip=0pt \halign to \displaywidth{\hfil$\@lign\displaystyle{##{}}$
\tabskip=\@centering &$\@lign\displaystyle{##{}}$
\tabskip=0pt &$\@lign\displaystyle{##{}}$ 
\tabskip=0pt &$\@lign\displaystyle{##{}}$  
\tabskip=0pt &\hfil$\@lign\displaystyle{{}##}$ 
\tabskip=0pt &\hfil$\@lign\displaystyle{##}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=\@centering &\llap{$\@lign##$}\tabskip=0pt\crcr#1\crcr}}


\def\eqnarrayx#1{\displ@y
\tabskip=0pt \halign to \displaywidth{\hfil$\@lign\displaystyle{##{}}$
\tabskip=\@centering &$\@lign\displaystyle{##{}}$
\tabskip=0pt &\hfil$\@lign\displaystyle{{}##}$ 
\tabskip=0pt &\hfil$\@lign\displaystyle{##}$ \hfil 
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=\@centering &\llap{$\@lign##$}\tabskip=0pt\crcr#1\crcr}}


\def\eqnarrayzb#1{\displ@y
\tabskip=0pt \halign to \displaywidth{\hfil$\@lign\displaystyle{##{}}$
\tabskip=\@centering &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil 
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=\@centering &\llap{$\@lign##$}\tabskip=0pt\crcr#1\crcr}}

\def\eqnarrayzc#1{\displ@y
\tabskip=0pt \halign to \displaywidth{\hfil$\@lign\displaystyle{{}##}$
\tabskip=\@centering &$\@lign\displaystyle{{}##}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil 
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=0pt &$\@lign\displaystyle{##{}}$ \hfil
\tabskip=\@centering &\llap{$\@lign##$}\tabskip=0pt\crcr#1\crcr}}


\def\gcd{{\rm gcd \,}}
\def\bfx{{\bf x}}





