\usepackage{amsmath,amsthm,amssymb,graphicx}

\usepackage{mathrsfs}

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\newtheorem{thm}[equation]{Theorem}
\newtheorem{theorem}[equation]{Theorem}
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\newtheorem{Conj}[equation]{Conjecture}
\newtheorem{conj}[equation]{Conjecture}
\newtheorem{conj*}{Conjecture}
\newtheorem{Cor}[equation]{Corollary}
\newtheorem{cor}[equation]{Corollary}
\newtheorem{hyp}[equation]{Hypothesis}
\newtheorem{corollary}[equation]{Corollary}
\newtheorem{Prop}[equation]{Proposition}
\newtheorem{prop}[equation]{Proposition}
\newtheorem{proposition}[equation]{Proposition}
\newtheorem{property}[equation]{Property}
\newtheorem{question}[equation]{Question}
\newtheorem{assumption}[equation]{Assumption}

\newtheorem{Lem}[equation]{Lemma}
\newtheorem{lem}[equation]{Lemma}
\newtheorem{claim}[equation]{Claim}
\newtheorem{lemma}[equation]{Lemma}

\newtheorem{statement}[equation]{Statement}

\theoremstyle{definition}

\newtheorem{defi}{Definition}[section]
%\newtheorem{def}{Definition}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{Def}[equation]{Definition}
%\renewtheorem{def}[equation]{Definition}

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\newtheorem{Rmk}[equation]{Remark}
\newtheorem{rmk}[equation]{Remark}
\newtheorem{remark}[equation]{Remark}
%\newtheorem{aside}[equation]{Aside}
\newtheorem{rem}[equation]{Remark}

%\renewtheorem{def}[equation]{Definition}




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\newcommand{\bbG}{{\mathbb G}}

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\newcommand{\fc}{\frak{c}}
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\newcommand{\fe}{\frak{e}}
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%\newcommand{\fg}{\frak{g}}
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\newcommand{\fz}{\frak{z}}



\newcommand{\fA}{\frak{A}}
\newcommand{\fB}{\frak{B}}
\newcommand{\fC}{\frak{C}}
\newcommand{\fD}{\frak{D}}
\newcommand{\fE}{\frak{E}}
\newcommand{\fF}{\frak{F}} 
\newcommand{\fG}{\frak{G}} 
\newcommand{\fH}{\frak{H}}
\newcommand{\fI}{\frak{I}}
\newcommand{\fJ}{\frak{J}}
\newcommand{\fK}{\frak{K}}
\newcommand{\fL}{\frak{L}}
\newcommand{\fM}{\frak{M}}
\newcommand{\fN}{\frak{N}}
\newcommand{\fO}{\frak{O}}
\newcommand{\fP}{\frak{P}}
\newcommand{\fQ}{\frak{Q}}
\newcommand{\fR}{\frak{R}}
\newcommand{\fS}{\frak{S}}
\newcommand{\fT}{\frak{T}}
\newcommand{\fU}{\frak{U}}
\newcommand{\fV}{\frak{V}}
\newcommand{\fW}{\frak{W}}
\newcommand{\fX}{\frak{X}}
\newcommand{\fY}{\frak{Y}}
\newcommand{\fZ}{\frak{Z}}



\newcommand{\sA}{\mathscr{A}}
\newcommand{\sB}{\mathscr{B}}
\newcommand{\sC}{\mathscr{C}}
\newcommand{\sD}{\mathscr{D}}
\newcommand{\sE}{\mathscr{E}}
\newcommand{\sF}{\mathscr{F}}
\newcommand{\sG}{\mathscr{G}}
\newcommand{\sH}{\mathscr{H}}
\newcommand{\sI}{\mathscr{I}}
\newcommand{\sJ}{\mathscr{J}}
\newcommand{\sK}{\mathscr{K}}
\newcommand{\sL}{\mathscr{L}}
\newcommand{\sM}{\mathscr{M}}
\newcommand{\sN}{\mathscr{N}}
\newcommand{\sO}{\mathscr{O}}
\newcommand{\sP}{\mathscr{P}}
\newcommand{\sQ}{\mathscr{Q}}
\newcommand{\sR}{\mathscr{R}}
\newcommand{\sS}{\mathscr{S}}
\newcommand{\sT}{\mathscr{T}}
\newcommand{\sU}{\mathscr{U}}
\newcommand{\sV}{\mathscr{V}}
\newcommand{\sW}{\mathscr{W}}
\newcommand{\sX}{\mathscr{X}}
\newcommand{\sY}{\mathscr{Y}}
\newcommand{\sZ}{\mathscr{Z}}
 
%\newcommand{\cv}{\mathcal{v}} 
%\newcommand{\cb}{\mathcal{b}}
\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cE}{\mathcal{E}}
\newcommand{\cF}{\mathcal{F}} 
\newcommand{\cG}{\mathcal{G}} 
\newcommand{\cH}{\mathcal{H}}
\newcommand{\cI}{\mathcal{I}}
\newcommand{\cJ}{\mathcal{J}}
\newcommand{\cK}{\mathcal{K}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cN}{\mathcal{N}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cp}{\mathcal{p}}
\newcommand{\cQ}{\mathcal{Q}}
\newcommand{\cR}{\mathcal{R}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cT}{\mathcal{T}}
\newcommand{\cU}{\mathcal{U}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\cW}{\mathcal{W}}
\newcommand{\cX}{\mathcal{X}}
\newcommand{\cY}{\mathcal{Y}}
\newcommand{\cZ}{\mathcal{Z}}

\newcommand{\Or}{\ensuremath{{\mathcal O}}}
\newcommand{\I}{\cI}
\newcommand{\n}{\N}

\newcommand{\vp}{\varpi}     %lowercase alpha

\newcommand{\ha}{\aleph}     %lowercase alpha
\newcommand{\hb}{\beth}     %lowercase alpha
\newcommand{\hg}{\gimel}     %lowercase alpha
\newcommand{\hd}{\daleth}     %lowercase alpha

\newcommand{\ga}{\alpha}     %lowercase alpha
\newcommand{\gb}{\beta}      %lowercase beta
\newcommand{\gd}{\delta}     %lowercase delta
\newcommand{\gD}{\Delta}     %uppercase delta
\newcommand{\sq}{\square}    %square
\newcommand{\gn}{\eta}       %lowercase eta
\newcommand{\gep}{\epsilon}  %lowercase epsilon
\newcommand{\vep}{\varepsilon} %epsilon
\newcommand{\G}{\Gamma}      %Capital Gamma
\newcommand{\Gi}{\Gamma_{\infty}}      %Capital Gamma sub infty
\newcommand{\g}{\gamma}      %lowercase gamma
\newcommand{\gL}{\Lambda}    %capital lamda
\newcommand{\gi}{\iota}    %lowercase iota
\newcommand{\gl}{\lambda}    %lowercase lambda
\newcommand{\gk}{\kappa}    %lowercase kappa
\newcommand{\gs}{\sigma}     %lowercase sigma
\newcommand{\gS}{\Sigma}     %uppercase sigma
\newcommand{\gt}{\theta}     %lowercase theta
\newcommand{\gT}{\Theta}     %lowercase theta
\newcommand{\gz}{\zeta}      %lowercase zeta
\newcommand{\gU}{\Upsilon}      %uppercase upsilon
\newcommand{\gu}{\upsilon}      %uppercase upsilon
\newcommand{\gw}{\omega}      %lowercase omega
\newcommand{\gW}{\Omega}      %lowercase omega

\newcommand{\vf}{\varphi}      

\newcommand{\foh}{\frac{1}{2}}  %onehalf

\newcommand{\vectwo}[2]
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                        #2
                          \end{array}\right) }

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{\left(\begin{array}{c}
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                        #2 \\
                        #3
                          \end{array}\right) }

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{\left(\begin{array}{cc}
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                        #3 &  #4
                          \end{array}\right) }

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{\left(\begin{array}{ccc}
                        #1  & #2 & #3  \\
                        #4  & #5 & #6  \\
                        #7 &  #8 & #9
                          \end{array}\right) }


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\newcommand{\Proj}{\operatorname{Proj}}

\newcommand{\Disc}{\operatorname{Disc}}
\newcommand{\Area}{\operatorname{Area}}


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\renewcommand{\Im}{{\mathfrak{Im}}}

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\renewcommand{\>}{\right\rangle}
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\newcommand{\ra}{\rangle}
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\newcommand{\ba}{\backslash}
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\newcommand{\GibkH}{\Gi\bk\boldH}
\newcommand{\GqbkH}{\G_0(q)\bk\boldH}

\newcommand{\dd}{\partial}

\newcommand{\ddxx}{{\partial^2\over\partial x^2}}
\newcommand{\ddyy}{{\partial^2\over\partial y^2}}
\newcommand{\ddx}{{\partial\over\partial x}}
\newcommand{\ddy}{{\partial\over\partial y}}


\newcommand{\op}{\operatorname}
\newcommand{\e}{\gep}
\newcommand{\Li}{\operatorname{Li}}
\newcommand{\iL}{\operatorname{iL}}

\newcommand{\lcm}{\operatorname{lcm}}