\newcommand{\nwc}{\newcommand}
\newcommand{\nwt}{\newtheorem}
\newcommand{\bC}{\hbox{{\rm I}
                   \kern-.8em\hbox{{\rm C}}}} 
\nwc{\ba}{\begin{array}}
\nwc{\be}{\begin{equation}}
\nwc{\beq}{\begin{eqnarray}}
\nwc{\beqn}{\begin{eqnarray*}}
\nwc{\beqast}{\begin{eqnarray*}}
\nwc{\bm}{\boldmath}

\nwc{\m}{\mbox}
\nwc{\ubm}{\unboldmath}

\nwc{\bma}{\m{\bm $a$ \ubm}}
\nwc{\bmma}{\m{\bm $a$\ubm}}
\nwc{\bmb}{\m{\bm $b$ \ubm}}
\nwc{\bmmb}{\m{\bm $b$\ubm}}
\nwc{\bmc}{\m{\bm $c$ \ubm}}
\nwc{\bmmc}{\m{\bm $c$\ubm}}
\nwc{\bmd}{\m{\bm $d$ \ubm}}
\nwc{\bmmd}{\m{\bm $d$\ubm}}
\nwc{\bme}{\m{\bm $e$ \ubm}}
\nwc{\bmme}{\m{\bm $e$\ubm}}
\nwc{\bmf}{\m{\bm $f$ \ubm}}
\nwc{\bmmf}{\m{\bm $f$\ubm}}
\nwc{\bmg}{\m{\bm $g$ \ubm}}
\nwc{\bmmg}{\m{\bm $g$\ubm}}
\nwc{\bmh}{\m{\bm $h$ \ubm}}
\nwc{\bmmh}{\m{\bm $h$\ubm}}
\nwc{\bmi}{\m{\bm $i$ \ubm}}
\nwc{\bmmi}{\m{\bm $i$\ubm}}
\nwc{\bmj}{\m{\bm $j$ \ubm}}
\nwc{\bmmj}{\m{\bm $j$\ubm}}
\nwc{\bmk}{\m{\bm $k$ \ubm}}
\nwc{\bmmk}{\m{\bm $k$\ubm}}
\nwc{\bml}{\m{\bm $l$ \ubm}}
\nwc{\bmml}{\m{\bm $l$\ubm}}
\nwc{\bmm}{\m{\bm $m$ \ubm}}
\nwc{\bmmm}{\m{\bm $m$\ubm}}
\nwc{\bmn}{\m{\bm $n$ \ubm}}
\nwc{\bmmn}{\m{\bm $n$\ubm}}
\nwc{\bmo}{\m{\bm $o$ \ubm}}
\nwc{\bmmo}{\m{\bm $o$\ubm}}
\nwc{\bmp}{\m{\bm $p$ \ubm}}
\nwc{\bmmp}{\m{\bm $p$\ubm}}
\nwc{\bmq}{\m{\bm $q$ \ubm}}
\nwc{\bmmq}{\m{\bm $q$\ubm}}
\nwc{\bmr}{\m{\bm $r$ \ubm}}
\nwc{\bmmr}{\m{\bm $r$\ubm}}
\nwc{\bms}{\m{\bm $s$ \ubm}}
\nwc{\bmms}{\m{\bm $s$\ubm}}
\nwc{\bmt}{\m{\bm $t$ \ubm}}
\nwc{\bmmt}{\m{\bm $t$\ubm}}
\nwc{\bmu}{\m{\bm $u$ \ubm}}
%\nwc{\bmmu}{\m{\bm $u$\ubm}}
\nwc{\bmv}{\m{\bm $v$ \ubm}}
\nwc{\bmmv}{\m{\bm $v$\ubm}}
\nwc{\bmw}{\m{\bm $w$ \ubm}}
\nwc{\bmmw}{\m{\bm $w$\ubm}}
\nwc{\bmx}{\m{\bm $x$ \ubm}}
\nwc{\bmmx}{\m{\bm $x$\ubm}}
\nwc{\bmy}{\m{\bm $y$ \ubm}}
\nwc{\bmmy}{\m{\bm $y$\ubm}}
\nwc{\bmz}{\m{\bm $z$ \ubm}}
\nwc{\bmmz}{\m{\bm $z$\ubm}}

\nwc{\bmA}{\m{\bm $A$\ubm}}
\nwc{\bmB}{\m{\bm $B$\ubm}}
\nwc{\bmD}{\m{\bm $D$\ubm}}
\nwc{\bmF}{\m{\bm $F$\ubm}}
\nwc{\bmI}{\m{\bm $I$\ubm}}
\nwc{\bmJ}{\m{\bm $J$\ubm}}
\nwc{\bmM}{\m{\bm $M$\ubm}}
\nwc{\bmP}{\m{\bm $P$\ubm}}
\nwc{\bmR}{\m{\bm $R$\ubm}}
\nwc{\bmU}{\m{\bm $U$\ubm}}
\nwc{\bmX}{\m{\bm $X$\ubm}}


\nwc{\bmalpha}{\m{\bm  $\alpha$  \ubm}}
\nwc{\bmeta}{\m{\bm    $\eta$    \ubm}}
\nwc{\bmgamma}{\m{\bm  $\gamma$  \ubm}}
\nwc{\bmkappa}{\m{\bm  $\kappa$  \ubm}}
\nwc{\bmlambda}{\m{\bm $\lambda$ \ubm}}
\nwc{\bmmu}{\m{\bm $\mu$\ubm}}
\nwc{\bmnabla}{\m{\bm  $\nabla$  \ubm}}
\nwc{\bmnu}{\m{\bm     $\nu$     \ubm}}
\nwc{\bmomega}{\m{\bm  $\omega$  \ubm}}
\nwc{\bmphi}{\m{\bm    $\phi$    \ubm}}
\nwc{\bmpsi}{\m{\bm    $\psi$    \ubm}}
\nwc{\bmsigma}{\m{\bm  $\sigma$  \ubm}}
\nwc{\bmtau}{\m{\bm    $\tau$    \ubm}}
\nwc{\bmtheta}{\m{\bm  $\theta$  \ubm}}
\nwc{\bmzeta}{\m{\bm  $\zeta$  \ubm}}
 \nwc{\bmxi}{\m{\bm    $\xi$     \ubm}}

\nwc{\bmGamma}{\m{\bm $\Gamma$\ubm}}
\nwc{\bmLambda}{\m{\bm $\Lambda$\ubm}}
\nwc{\bmPhi}{\m{\bm $\Phi$\ubm}}

\nwc{\ca}{{\cal A}}
\nwc{\cao}{{\cal A}^{-1}}
\nwc{\cb}{{\cal B}}
\nwc{\cc}{{\cal C}}
\nwc{\cd}{{\cal D}}
\nwc{\ce}{{\cal E}}
\nwc{\cf}{{\cal F}}
\nwc{\cg}{{\cal G}}
\nwc{\ch}{{\cal H}}
\nwc{\ci}{{\cal I}}
\nwc{\cj}{{\cal J}}
\nwc{\ck}{{\cal K}}
\nwc{\cl}{{\cal L}}
\nwc{\clu}{{\cal L}{\cal U}}
\nwc{\cm}{{\cal M}}
\nwc{\cn}{{\cal N}}
\nwc{\co}{{\cal O}}
\nwc{\cp}{{\cal P}}
\nwc{\cq}{{\cal Q}}
\nwc{\calr}{{\cal R}}
\nwc{\cs}{{\cal S}}
\nwc{\ct}{{\cal T}}
\nwc{\cu}{{\cal U}}
\nwc{\cv}{{\cal V}}
\nwc{\cw}{{\cal W}}
\nwc{\cx}{{\cal X}}
\nwc{\cy}{{\cal Y}}
\nwc{\cz}{{\cal Z}}


\nwc{\ea}{\end{array}}
\nwc{\ee}{\end{equation}}
\nwc{\eeq}{\end{eqnarray}}
\nwc{\eeqn}{\end{eqnarray*}}
\nwc{\eeqast}{\end{eqnarray*}}

\nwc{\half}{\frac{1}{2}}

\nwc{\noi}{\noindent}
\nwc{\non}{\nonumber}
\nwc{\onehalf}{\frac{1}{2}}
\nwc{\onethird}{\frac{1}{3}}

\nwc{\p}{\partial}

\nwc{\uP}{{\em \bf Proof: }}
\nwc{\uT}{\underline{Theorem:}}
\newcommand{\pfend}{{\vrule height4pt width3pt depth2pt}%
   \vspace{1pc}}