\newcommand{\spp}{\vspace{1.5ex}}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bdm}{\begin{displaymath}}
\newcommand{\edm}{\end{displaymath}}

\newlength\arraycolsepsave
\setlength\arraycolsepsave{.5\arraycolsep}\arraycolsep\arraycolsepsave
\newcommand{\bea}{\arraycolsep.13em\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}\arraycolsep\arraycolsepsave}
\newcommand{\beas}{\arraycolsep.13em\begin{eqnarray*}}
\newcommand{\eeas}{\end{eqnarray*}\arraycolsep\arraycolsepsave}

\def\bmat#1{\left[\begin{array}{#1}}
\def\emat{\end{array}\right]}
\def\bvec{\left[\begin{array}{c}}
\let\evec=\emat

\newcommand{\sbea}{\nopagebreak[3]\samepage\begin{eqnarray}}
\newcommand{\seea}{\end{eqnarray}\pagebreak[0]}
\newcommand{\sbeas}{\nopagebreak[3]\samepage\begin{eqnarray*}}
\newcommand{\seeas}{\end{eqnarray*}\pagebreak[0]}

\def\intii{\int_{-\infty}^\infty}
\def\intzi{\int_0^\infty}
\def\dint{\mathop{\int\!\!\int}\limits}

\def\lb{\label} % WARNING don't use \let here;
%                 it interferes with dlabels.sty
\newcommand{\er}[1]{{\rm(\ref{#1})}}
\newcommand{\sler}[1]{{\sl(\ref{#1})}}
\newcommand{\nr}[1]{{\rm\ref{#1}}}
\newcommand{\slnr}[1]{{\sl\ref{#1}}}
\newcommand{\rmbox}[1]{\mbox{\rm #1}}

\let\la=\langle
\let\ra=\rangle

\def\qaq{\quad\mbox{\rm and}\quad}
\gdef\dlabel@box{\relax}
\gdef\dlabel@code{\relax}
%\def\ebox{\raisebox{-.24ex}{\mbox{\large$\Box$}}}

% Replace \ebox with IEEEtran.cls definition of \QEDopen
% to avoid using amssymb package:
%\def\eboxIEEE{{\setlength{\fboxsep}{0pt}\setlength{\fboxrule}{0.2pt}\fbox{\rule[0pt]{0pt}{1.3ex}\rule[0pt]{1.3ex}{0pt}}}}
\def\ebox{{\setlength{\fboxsep}{0pt}\setlength{\fboxrule}{0.45pt}\fbox{\rule[.0pt]{0pt}{1.5ex}\rule[0pt]{1.5ex}{0pt}}}}
\def\QED{\hspace*{\fill}~\ebox}
\def\eQED{\gdef\@eqnnum{\ebox}}
\def\myeqno#1{\gdef\@eqnnum{$#1$}}
\def\fix{\gdef\@eqnnum{{\rm (\theequation)\dlabel@box}\dlabel@code}\addtocounter{equation}{-1}}
\def\eqsymbol#1{\gdef\@eqnnum{#1}}

%\def\eboxme{\raisebox{-.24ex}{\mbox{\large$\square$}}} \def\mycircs{\raisebox{.123ex}{\hspace*{.05em}$\scriptstyle\bigcirc$}}
% \def\nstwepts{{\mbox{\scriptsize$\not\mathrel{\kern-.9pt\mycircs}$}}}

% \def\mycircd{\raisebox{.08ex}{\kern-.02em\footnotesize$\bigcirc$}}
% \def\nstweptd{{\mbox{$\not\mathrel{\mycircd}$}}}

% \def\nstwept{{\mathchoice\nstweptd\nstweptd\nstwepts\nstwepts}}

%\def\nstenpt{{\mbox{$\not\mathrel{\raisebox{.2pt}{\kern-.5pt\mycircd}}$}}}
% \let\nstenpt=\nstwept

% \let\ns=\nstwept
\def\ns{\mbox{\large$\varnothing$}}  % Requires amssymb package if macro is used

% tenpt footnotesize \ns
\def\nsft{\mbox{\scriptsize$\not\mathrel{\kern-.9pt\mycircs}$}}

\def\bgz{\makebox[0pt][c]{\huge 0}}

\def\conv{\protect\mathop{\kern.1em\raisebox{-.15ex}{$\ast$}\kern-.62em{\scriptstyle\bigcirc}}}

\def\indep{\mathop{\bot\kern-.6em\bot}}

\def\DIC{{\hbox{\rm\kern.2em\raise.45ex%
\hbox{$\scriptscriptstyle |$}\kern-.35em C}}}
%\def\DIC{{\hbox{\rm\kern.2em\raise.36ex% OLD
%\hbox{$\scriptstyle |$}\kern-.4em C}}}
\def\SIC{{\hbox{\scriptsize\rm\kern.2em\raise.325ex%
\hbox{$\scriptscriptstyle|$}\kern-.375em C}}}
%\def\SIC{{\hbox{\scriptsize\rm\kern.2em\raise.4ex% OLD
%\hbox{$\scriptscriptstyle |$}\kern-.4em C}}}
\def\IC{{\mathchoice\DIC\DIC\SIC\SIC}}

%\def\DIQ{{
%
%\hbox{\rm\kern.2em\raise.35ex\hbox{$\scriptstyle |$}\kern-.37em Q}
%
%}}
\def\DIQ{{\hbox{\rm\kern.2em\raise.33ex%
\hbox{$\scriptstyle |$}\kern-.365em Q}}}
%\def\DIQ{{\hbox{\rm\kern.2em\raise.4ex%
%\hbox{$\scriptstyle |$}\kern-.4em Q}}}
\def\SIQ{{\hbox{\scriptsize\rm\kern.2em\raise.4ex%
\hbox{$\scriptscriptstyle |$}\kern-.4em Q}}}
%\def\SIQ{{\hbox{\scriptsize\rm\kern.2em\raise.4ex%
%\hbox{$\scriptscriptstyle |$}\kern-.4em Q}}}
\def\IQ{{\mathchoice\DIQ\DIQ\SIQ\SIQ}}

\def\IR{{\rm I\!R}}
\def\IP{{\rm I\!P}}
\def\IE{{\rm I\!E}}
\def\IN{{\rm I\!N}}
\def\Ind{{\bf 1}}

\def\DZZ{{\hbox{\sf Z\kern-.41em Z}}}
\def\SZZ{{\hbox{\scriptsize\sf Z\kern-.41em Z}}}
\def\ZZ{{\mathchoice\DZZ\DZZ\SZZ\SZZ}}
\let\IZ=\ZZ

\def\bE{{\sf E}}
\let\bEf=\bE
% \def\bEf{\hbox{\footnotesize\sf E}}
\def\bP{{\sf P}}
\let\bPf=\bP
% \def\bPf{\hbox{\footnotesize\sf P}}
%\def\sP{\hbox{\kern-.15em\raise.5ex\hbox{\large$\wp$}}}
\def\sP{\hbox{\kern-.05em\raise.5ex\hbox{\large$\wp$}}}
\def\sPf%
{\hbox{\kern-.05em\raise.5ex\hbox{\normalsize$\wp$}}} % \footnotesize \sP
\def\Pr{{\rm Pr}}
\def\sB{{\cal B}}
\def\sF{{\cal F}}
\def\sG{{\cal G}}

%\def\de{  \buildrel {\rm \bigtriangleup} \over = }
\def\de{\buildrel{\scriptscriptstyle\bigtriangleup}\over=}
\def\cde{\mathrel{:=}}
\def\dec{\mathrel{=:}}
\let\oliminf=\liminf
\let\olimsup=\limsup
\def\lim@sup{\mathop{\overline{\rm lim}}}
\def\lim@inf{\mathop{\underline{\rm lim}}}
\let\limsup=\lim@sup
\let\liminf=\lim@inf
\def\argmin{\mathop{\rm argmin}}
\def\argmax{\mathop{\rm argmax}}
\def\mspan{\mathop{\rm span}\nolimits}
\def\range{\mathop{\rm range}\nolimits}
\def\tr{\mathop{\rm tr}\nolimits}
\def\diag{\mathop{\rm diag}\nolimits}
\def\re{\mathop{\rm Re}\nolimits}
\def\im{\mathop{\rm Im}\nolimits}
\def\sgn{\mathop{\rm sgn}\nolimits}
\def\sinc{\mathop{\rm sinc}\nolimits}
\def\Si{\mathop{\rm Si}\nolimits}
\def\Cin{\mathop{\rm Cin}\nolimits}
\def\Ein{\mathop{\rm Ein}\nolimits}
\def\mod{\mathop{\rm mod}\nolimits}
\def\supp{\mathop{\rm supp}\nolimits}
\def\co{\mathop{\rm co}\nolimits}
\def\aff{\mathop{\rm aff}\nolimits}
\def\cl{\mathop{\rm cl}\nolimits}
\def\clco{\overline{\co}}
\def\var{\mathop{\sf var}\nolimits}
\def\cov{\mathop{\sf cov}\nolimits}
\let\downto=\downarrow
\def\hg#1#2{{\,}_{#1}\kern-.2ex F_{#2}} % Hypergeometric

\def\Dbt{{\raisebox{-.8ex}{\mbox{\LARGE\sf X}}}}
\def\Dpbt{{\raisebox{-.4ex}{\mbox{\large\sf X}}}}
\def\Sbt{{\raisebox{-.4ex}{\mbox{\sf X}}}}
\def\bigtimes{\mathop{\mathchoice\Dbt\Dpbt\Sbt\Sbt}}

\def\ds{\displaystyle}
\def\ts{\textstyle}
%\def\lrt{\mathop{\raisebox{-.5ex}{$\buildrel{\ds >}\over<$}}}
%\def\LRT{\ds {{{H_1} \atop {\ds >}} \atop {{\ds <} \atop {H_0}}}}
%\def\LRT#1#2{\mathop
%{\;{{{#1} \atop >} \atop {< \atop {#2}}}}
%}
%\def\LRT#1#2{\mathop{
%\raisebox{.15ex}{$
%{{\scriptstyle\;#1}\atop{\scriptstyle\gtrless}}
%\atop 
%{\raisebox{-.9ex}{$\scriptstyle\;#2$}}
%$}
%}}
\def\LRT#1#2{\mathop{
\raisebox{.2ex}{$
{{\scriptstyle\;#1}\atop{\ds\gtrless}}
\atop 
{\raisebox{-1.25ex}{$\scriptstyle\;#2$}}
$}
}}


\def\mx{{\rm max}}
\def\mn{{\rm min}}
\def\crs{\times}
\def\half{{\textstyle{ 1 \over 2}}}
\def\ninv{(1/n)}

\def\same{\rule{2.3em}{0.3pt}\,, }

\def\f@left#1{\makebox[0pt][l]{$\displaystyle#1$}}
% \def\f@oleft[#1]#2{\makebox[0pt][l]{\hspace*{-#1}$\displaystyle#2$}}
\def\f@oleft[#1]#2{\makebox[0pt][l]{$\displaystyle#2$}\hspace*{#1}}
\def\lefteqn{\@ifnextchar[{\f@oleft}{\f@left}}

\def\eqbox#1#2{\makebox[#1]{$\displaystyle#2$}}

\let\refsize=\small
\def\call#1{}
\def\singlespace{}

\outer\def\fl{\@ifnextchar[{\@opfl}{\@fl}}
\def\@opfl[#1]#2{\fl@{#1}{#2}}
\def\@fl#1{\fl@{-2.5ex}{#1}}
\def\fl@#1{}
\def\UC{}

%--------------------------------------------------------------------
%
% Define my version of \mathpalette.  Example:  \def\bx{\pal{\bf x}}
%
\def\pal#1{{\mathchoice{#1}{#1}{\hbox{\scriptsize#1}}{\hbox{\tiny#1}}}}
%
% If font does not exist in \tiny size, use \pals instead.
% Example: \def\sX{\pals{\sf X}}
%
\def\pals#1{{\mathchoice{#1}{#1}{\hbox{\scriptsize#1}}{\hbox{\scriptsize#1}
}}}
%
% To define a text symbol for math mode use:
%
\def\palt#1{{\mathchoice{\hbox{#1}}{\hbox{#1}}{\hbox{\scriptsize#1}}{\hbox{\tiny#1}}}}
%
% Example: \def\bx{\palt{\bfseries\itshape x}}
%
%--------------------------------------------------------------------
\iftrue
\def\optlist{}

\def\list#1#2{\ifnum \@listdepth >5\relax \@toodeep
     \else \global\advance\@listdepth\@ne \fi
  \rightmargin \z@ \listparindent\z@ \itemindent\z@
  \csname @list\romannumeral\the\@listdepth\endcsname
  \def\@itemlabel{#1}\let\makelabel\@mklab \@nmbrlistfalse #2\optlist\relax
  \@trivlist
  \parskip\parsep \parindent\listparindent
  \advance\linewidth -\rightmargin \advance\linewidth -\leftmargin
  \advance\@totalleftmargin \leftmargin
  \parshape \@ne \@totalleftmargin \linewidth
  \ignorespaces}


\def\mylabel#1{\def\@itemlabel{#1}}
\fi
%--------------------------------------------------------------------