\newcommand{\SEC}{\section}
\newcommand{\SUBSEC}{\subsection}

\newcommand{\I}{{\bf i}}
\newcommand{\J}{{\bf j}}
\newcommand{\K}{{\bf k}}
\newcommand{\R}{{\bf R}}

\newcommand{\Z}{{Z}}
\newcommand{\N}{{N}}
%
%% \newcommand{\Z}{{\Bbb Z}}
%% \newcommand{\R}{{\Bbb R}}
%% \newcommand{\N}{{\Bbb N}}

\newcommand{\SP}{\vspace*{1em}}
%
\newcommand{\T}{\sf T}
\newcommand{\ol}[1]{\overline{#1}}
%
\newcommand{\ignore}[1]{\mbox{$\!\!\!$}}
\newcommand{\ig}[1]{\mbox{$\!\!\!$}}
%
\newcommand{\EQ}{\begin{equation}}
\newcommand{\EN}{\end{equation}}
%
\newcommand{\MAT}[4]{           % 2-by-2 matrix
\left(
\!\!
\begin{array}{cc}
#1 & #2 \\
#3 & #4
\end{array}
\!\!
\right)
% \!
}
%
\newcommand{\DETthree}[9]{           % 3-by-3 determinant
\left|
\!\!
\begin{array}{ccc}
#1 & #2 & #3 \\
#4 & #5 & #6 \\
#7 & #8 & #9 \\
\end{array}
\!\!
\right|
% \!
}
\newcommand{\MATthree}[9]{           % 3-by-3 matrix
\left(
\!\!
\begin{array}{ccc}
#1 & #2 & #3 \\
#4 & #5 & #6 \\
#7 & #8 & #9 \\
\end{array}
\!\!
\right)
% \!
}
%
\newcommand{\COMB}[2]{           % 2-by-1 matrix = bin coef
\left(
\!\!
\begin{array}{c}
#1 \\ #2
\end{array}
\!\!
\right)
% \!
}
%
\newcommand{\COMBthree}[3]{           % 3-by-1 matrix
\left(
\!\!
\begin{array}{c}
#1 \\ #2 \\ #3
\end{array}
\!\!
\right)
% \!
}
%
\newcommand{\bu}{\bullet}
\newcommand{\V}{\vfill}
\newcommand{\BC}{\begin{center}}
\newcommand{\EC}{\end{center}}
\newcommand{\UL}[1]{\underline{#1}}
\newcommand{\AND}{\wedge}
\newcommand{\OR}{\vee}
\newcommand{\impl}{\Rightarrow}
\newcommand{\implby}{\Leftarrow}
\newcommand{\quant}[4]{(#1 #2 \ |\ #3 \ : \ #4)}
\newcommand{\set}[3]{(#1 \ |\ #2 \ : \ #3)}