



%\some user defined macros
\def\q{\quad}
\def\qq{\qquad}
\def\qqq{\qquad\qquad}
\def\qqqq{\qquad\qquad\qquad}
\def\2;{\;\;}
\def\noi{\noindent}
%
\def\eps{\epsilon}
\def\Pr{{\hbox{P}_{\hskip -1mm r}}}
\def\pr{{\prime}}
\def\qpr{{\;\prime}}
%
\def\IntN{{\mathbb Z}}
\def\NatN{{\mathbb N}}
\def\NatNo{{\mathbb N}_0}
\def\RealN{{\mathbb R}}
\def\RatN{{\mathbb Q}}
\def\IrrN{{\mathbb I}}
\def\ComplexN{{\mathbb C}}
\def\mathB{{\mathbb B}}
\def\mathD{{\mathbb D}}
\def\mathE{{\mathbb E}}
\def\mathF{{\mathbb F}}
\def\mathH{{\mathbb H}}
\def\mathJ{{\mathbb J}}
\def\mathL{{\mathbb L}}
\def\mathK{{\mathbb K}}
\def\mathP{{\mathbb P}}
\def\mathS{{\mathbb S}}
\def\mathi{\hbox{$\mathit{i}$}}
\def\smathi{\hbox{\fns $\mathit{i}$}}
\def\Li{{\C{L}i_2}}
\def\half{{\frac{1}{2}}}
\def\shalf{{\sfrac{1}{2}}}
\def\sshalf{{\ssfrac{1}{2}}}
\def\Im{{\MF{I}m}\,}
\def\Re{{\MF{R}e}\,}
\newcommand{\Res}{{\MF{R}es}}
\newcommand\marg{\hbox{\minefont\fns$\MF{a}\MF{r}\MF{g}\,$}}
%\newcommand\arg{\hbox{\minefont$\MF{a}\MF{r}\MF{g}\,$}}
\newcommand\Arg{\hbox{\minefont$\MF{A}\MF{r}\MF{g}\,$}}
\def\mmI{\hbox{\minefont\textit{\fns i}}}
\def\mMI{\hbox{\minefont\textit{\small i}}}
\def\mI{\hbox{\minefont\textit{i}}}
\def\MI{\hbox{\minefont\textit{\large i}}}


% Some user defined functions
\def\a#1{{a_#1}}
\def\e#1{{\vec{e}}_#1}
\def\b#1{{\vec{b}}_#1}
\def\t#1{{\vec{t}}_#1}
\def\cc#1{{c_#1}}
\def\i#1{{i_#1}}
\def\j#1{{j_#1}}
\def\k#1{{k_#1}}
\def\n#1{{n_#1}}
\def\m#1{{m_#1}}
\def\vs#1{{v_#1}}
\def\u#1{{u_#1}}
\def\o#1{\overline{#1}}
\def\w#1{\widetilde{#1}}
\def\Ref#1{(\ref{#1})}
\def\B#1{{\bf #1}}
\def\C#1{{\mathcal #1}}
\def\MF#1{{\mathfrak #1}}
\def\SC#1{{\hbox{\textsc{#1}}}}
\def\U#1{\underline{#1}}
\def\x#1{{x_#1}}
\def\y#1{{y_#1}}
\def\z#1{{z_#1}}
\def\xpr#1{{x^\pr_#1}}
\def\ypr#1{{y^\pr_#1}}
\def\zpr#1{{z^\pr_#1}}
\def\xsq#1{{x^2_#1}}
\def\ysq#1{{y^2_#1}}
\def\zsq#1{{z^2_#1}}
\def\h#1{{\mathbf{\widehat{\hbox{$#1$}}}}}
\def\mi{{\vec{e}_1}}
\def\mj{{\vec{e}_2}}
\def\mk{{\vec{e}_3}}

\def\H#1{\hbox{\Large$#1$}}


\def\alphas#1{{\alpha_#1}}
\def\betas#1{{\beta_#1}}
\def\gammas#1{{\gamma_#1}}
\def\lambdas#1{{\lambda_#1}}
\def\chis#1{{\chi_#1}}
\def\sigmas#1{{\sigma_#1}}
\def\etas#1{{\eta_#1}}
\def\taus#1{{\tau_#1}}
\def\zetas#1{{\zeta_#1}}
\def\omegas#1{{\omega_#1}}
\def\phis#1{{\phi_#1}}


\def\binom#1#2{{{#1}\choose{#2}}}
\def\Bi#1#2{{\binom{#1}{#2}}}
\def\fBi#1#2{\hbox{\large $\displaystyle{\binom{#1}{#2}}$}}
\def\sBi#1#2{\hbox{\scriptsize ${\binom{#1}{#2}}$}}
\def\SBi#1#2{\hbox{\footnotesize ${\binom{#1}{#2}}$}}
%\def\qBi#1#2{\left[ {\begin{array}{*{20}c } #1 \\ #2 \\ \end{array}} \right]}
\def\qBi#1#2{{\begin{bmatrix} #1 \\ #2 \end{bmatrix}}}
\def\sqBi#1#2{\hbox{\scriptsize ${\begin{bmatrix} #1 \\ #2 \end{bmatrix}}$}}
\def\ssqBi#1#2{\hbox{\beginpicture \put { } at 0 2 \put { } at 8 11 
                                                                          \plot 2 0 0 0 0 12 2 12 / \plot 6 0 8 0 8 12 6 12 / 
                                                                          \put {{\tiny $#2$}} at 4 3 \put {{\tiny $#1$}} at 4 9 \endpicture}}
\def\fqBi#1#2{\hbox{\large ${\begin{bmatrix} #1 \\ #2 \end{bmatrix}}$}}
\def\Tri#1#2{\binom{#1}{#2}_{\hspace{-1mm}2}}
\def\ceil#1#2{{\lcl #1 #2 \rcl}}
\def\floor#1#2{{\lfl #1 #2 \rfl}}
\def\SSfrac#1#2{\hbox{\Large $\frac{#1}{#2}$}}
\def\Sfrac#1#2{\hbox{\large $\frac{#1}{#2}$}}
\def\sfrac#1#2{\hbox{\nor $\frac{#1}{#2}$}}
\def\ssfrac#1#2{\hbox{\fns $\frac{#1}{#2}$}}
\def\sstack#1#2{\substack{\hbox{\tiny$#1$} \\ \hbox{\tiny$#2$}}}

% Brackets
\def\LB{\left(}         \def\RB{\right)}
\def\LV{\left|}        \def\RV{\right|}
\def\LA{\left\langle}        \def\RA{\right\rangle}
\def\LVV{\left\|}      \def\RVV{\right\|}
\def\lcl{\left\lceil}  \def\rcl{\right\rceil}
\def\lfl{\!\left\lfloor} \def\rfl{\right\rfloor\!}
\def\LC{\left\{}       \def\RC{\right\}}
\def\LH{\left[}        \def\RH{\right]}
\def\largeleft{\hbox{\Large(}}  \def\largeright{\hbox{\Large)}}
\def\hugeleft{\hbox{\huge(}}  \def\hugeright{\hbox{\huge)}}

% Arrows
\def\lrarrow{\leftrightarrow}
\def\LRarrow{\Leftrightarrow}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[theorem]
\newtheorem{lemma}[theorem]{Lemma}
%User defined environment Algorithm
\newcommand{\algorithm}{\vspace{2mm}\noindent{{\scshape \bf Algorithm:} }} 
%User defined environment CLAIM
\newcommand{\Claim}{\vspace{2mm}\noindent{{\scshape \nor Claim:} }} 
\newcommand{\Proof}{{\scshape \noindent \nor Proof of claim: }}
\newcommand{\cqed}{\hskip 2mm $\triangle$ \vskip 2mm}
%\newcommand{\qed}{\hfill {\footnotesize$\square$} \vskip 2mm}
\newcommand{\tqed}{\eqno\hbox{\footnotesize$\square$}}

% Some size
\def\nor{\normalsize}
\def\fns{\scriptsize}

% Alignment
\def\fls{\flushleft}

%Miscalaneous
\def\alev{\hbox{ae}}
\def\etal{\hbox{\textit{et al{.}}}}
\def\lvv{\hbox{\LARGE$|$}}
\def\vv{{\;\hbox{\Large $|$}\;}}
\def\svv{{\;\hbox{$|$}\;}}
\def\sleq{{\leq}}
\def\sgeq{{\geq}}
\def\vars{{\hbox{\Large$\varsigma$}}}
\def\svars{{\hbox{$\varsigma$}}}
\def\indic{{\hbox{$\mathbbm{1}$}}}
\def\indicj{{\hbox{\rm{I}}}}
%\def\edge#1#2{{#1{\sim}#2}}
%\def\dedge#1#2{{#1{\leadsto}#2}}
%\def\edge#1#2{{\langle #1\hspace{1pt} {\sim} \hspace{1pt} #2 \rangle}}
\def\edge#1#2{{\langle #1 \hspace{0.85pt}{\sim}\hspace{0.85pt} #2 \rangle}}
\def\dedge#1#2{{\langle #1\hspace{1.25pt} {\leadsto}\hspace{0.50pt} #2\rangle}}
\def\path#1#2{{#1\hspace{0.75pt} {\leftrightsquigarrow}\hspace{0.75pt} #2}}
\def\dpath#1#2{{#1\hspace{0.75pt} {\rightsquigarrow}\hspace{0.75pt} #2}}
\def\ppr{{\prime\prime}}
\def\deg#1{\hbox{deg}\,#1}
\def\maxdeg#1{\hbox{maxdeg}\,#1}
\def\bB{{\mathbf B}}
\def\bQ{{\mathbf Q}}
\def\bI{{\mathbf I}}
\def\bS{{\mathbf S}}
\def\det{{\hbox{det}}\,}
\def\ande{\:\hbox{and}\;}

% Add and minus inline
\def\thin{ {\hspace{0.75pt}} }
\def\plus{{\hspace{0.85pt}{+}\hspace{0.85pt}}}
\def\minus{{\hspace{0.85pt}{-}\hspace{0.85pt}}}
\def\equals{{\hspace{0.75pt}{=}\hspace{0.75pt}}}
\def\gteqs{{\hspace{0.75pt}{\geq}\hspace{0.75pt}}}
\def\lteqs{{\hspace{0.75pt}{\leq}\hspace{0.75pt}}}
\def\geqs{{\hspace{0.75pt}{>}\hspace{0.75pt}}}
\def\leqs{{\hspace{0.75pt}{<}\hspace{0.75pt}}}
\def\csum{{\hspace{0.75pt} {\#}\hspace{0.75pt}}}

%In different fonts
\def\fy{\hbox{\textit{y}}}
\def\fx{\hbox{\textit{x}}}

%WHITE vertical space
\def\wvs{\hbox{\textcolor{white}{\large |}}}

%hyphenation
\hyphenation{ter-min-olo-gy}
\hyphenation{Metro-polis}
