
\newcommand{\bc}{\begin{center}}
\newcommand{\ec}{\end{center}}
\newcommand{\vs}{\vspace}
\newcommand{\mb}{\mbox{\ }}
\newcommand{\ds}{\displaystyle}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}
%\numberwithin{equation}{section}

\newtheorem{athm}{Theorem}
\renewcommand{\theathm}{\Alph{athm}}


\newtheorem{theorem}{Theorem}[section]
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}



\newcommand{\br}{\mathbb R}
\newcommand{\bv}{\mathbb V}
\newcommand{\bbc}{\mathbb C}
\newcommand{\bw}{\mathbb W}
\newcommand{\bh}{\mathbb H}
\newcommand{\bz}{\mathbb Z}
\newcommand{\bbf}{\mathbb F}
\newcommand{\bp}{\mathbb P}
\newcommand{\bk}{\mathbb K}
\newcommand{\bd}{\mathbb D}
\newcommand{\bbe}{\mathbb E}
\newcommand{\bi}{\mathbb I}
\newcommand{\cb}{{\mathcal B}}
\newcommand{\ce}{{\mathcal E}}
\newcommand{\cc}{{\mathcal C}}
\newcommand{\cf}{{\mathcal F}}
\newcommand{\cg}{{\mathcal G}}
\newcommand{\ch}{{\mathcal H}}
\newcommand{\cq}{{\mathcal Q}}
\newcommand{\cd}{{\mathcal D}}
\newcommand{\cx}{{\mathcal X}}
\newcommand{\cv}{{\mathcal V}}
\newcommand{\cs}{{\mathcal S}}
\newcommand{\ct}{{\mathcal T}}
\newcommand{\up}{\Upsilon}
\newcommand{\ca}{{\mathcal A}}
\newcommand{\cl}{{\mathcal L}}
\newcommand{\vep}{{\varepsilon}}
\newcommand{\ep}{{\epsilon}}
\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\ga}{\gamma}
\newcommand{\de}{\delta}
\newcommand{\et}{\eta}
\newcommand{\io}{\iota}
\newcommand{\ka}{\kappa}
\newcommand{\la}{\lambda}
\newcommand{\om}{\omega}
\renewcommand{\phi}{\varphi}
\newcommand{\ph}{\varphi}
\newcommand{\ps}{\psi}
\newcommand{\si}{\sigma}
\renewcommand{\th}{\theta}
\newcommand{\ta}{\tau}
\newcommand{\ze}{\zeta}
\newcommand{\De}{\Delta}
\newcommand{\Ga}{\Gamma}
\newcommand{\La}{\Lambda}
\newcommand{\Om}{\Omega}
\newcommand{\Ph}{\Phi}
\newcommand{\Ps}{\Psi}
\newcommand{\Si}{\Sigma}
\newcommand{\Th}{\Theta}
\newcommand{\fg}{\mathfrak g}
\newcommand{\fh}{\mathfrak h}
\newcommand{\fp}{\mathfrak p}
\newcommand{\fn}{\mathfrak n}
\newcommand{\fs}{\mathfrak s}
