 % footnote mark should be (1), etc.
 \renewcommand{\thefootnote}{(\arabic{footnote})}

\newcommand{\qed}
 {\ifhmode\unskip\nobreak\fi\quad\ensuremath{\mathrm{Q.E.D.}}}

\newcommand{\binom}[2]{\renewcommand{\arraycolsep}{0pt}
\left(\begin{array}{c}#1\\#2\end{array}\right)}
\newcommand{\mod}{\quad\mathop{\rm mod}}

\newcommand{\Abar}{\overline{A}}
\newcommand{\abar}{\overline{a}}
\newcommand{\bbar}{\overline{b}}
\newcommand{\Cbar}{\overline{C}}
\newcommand{\dbar}{\overline{d}}
\newcommand{\Dbar}{\overline{D}}
\newcommand{\Ebar}{\overline{E}}
\newcommand{\Fbar}{\overline{F}}
\newcommand{\Gbar}{\overline{G}}
\newcommand{\gbar}{\overline{g}}
\newcommand{\Kbar}{\overline{K}}
\newcommand{\Lbar}{\overline{L}}
\newcommand{\Mbar}{\overline{M}}
\newcommand{\Obar}{\overline{O}}
\newcommand{\Pbar}{\overline{P}}
\newcommand{\Vbar}{\overline{V}}
\newcommand{\nbar}{\overline{n}}
\newcommand{\pbar}{\overline{p}}
\newcommand{\rbar}{\overline{r}}
\newcommand{\Fiebar}{\overline{\Phi}}
\newcommand{\fiebar}{\overline{\varphi}}
\newcommand{\albar}{\overline{\alpha}}
\newcommand{\Gabar}{\overline{\Gamma}}
\newcommand{\debar}{\overline{\delta}}
\newcommand{\pibar}{\overline{\pi}}

\newtheorem{Corollary}{Corollary}
\newtheorem{Theorem}{Theorem}
\newtheorem{Lemma}{Lemma}
\newenvironment{proc}{\begin{quote}\em}{\end{quote}}
\newenvironment{NOTE}{\paragraph{Note}}{\par\smallskip}
\newenvironment{EXERCISE}{\paragraph{EXERCISE}}{\par\smallskip}
\newenvironment{EXAMPLES}{\paragraph{EXAMPLES}}{\par\smallskip}
\newenvironment{Questions}{\paragraph{QUESTIONS}}{\par\smallskip}
\newenvironment{Remark}{\paragraph{Remark}}{\par\smallskip}
\newenvironment{Notice}{\paragraph{NOTE}}{\par\smallskip}

% short Greeks
\newcommand\al{\alpha}
\newcommand\be{\beta}
\newcommand\ga{\gamma}
\newcommand\de{\delta}
\newcommand\ep{\varepsilon}
\newcommand\fie{\varphi}
% \newcommand\io{\iota}
% \newcommand\ka{\kappa}
\newcommand\la{\lambda}
\newcommand\si{\sigma}
% \newcommand\th{\theta}
\newcommand\ze{\zeta}
 \newcommand\om{\omega}
 \newcommand\Fie{\Phi}
 \newcommand\Ga{\Gamma}
 \newcommand\De{\Delta}
 \newcommand\La{\Lambda}
 \newcommand\Si{\Sigma}
% \newcommand\Th{\Theta}
 \newcommand\Om{\Omega}