\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{principle}[theorem]{Principle}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{Truth}[theorem]{Truth}
\newtheorem{AlmostTruth}[theorem]{Almost Truth}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{slogan}[theorem]{Slogan}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{defwarn}[theorem]{Definition and Warning}
\newtheorem{image}[theorem]{Image}
\newtheorem{lemmadefinition}[theorem]{Lemma-Definition}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{question}[theorem]{Question}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{prize}[theorem]{Prize}

\theoremstyle{remark}
\newtheorem{comment}[theorem]{Comment}
\newtheorem{comments}[theorem]{Comments}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{hint}[theorem]{Hint}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{warning}[theorem]{Warning}

\newcommand{\prtag}[1]{\tag*{\llap{$#1$\hskip-\displaywidth}}}

\newcommand{\mathmode}[1]{$#1$}
\newlength{\standardunitlength}
\setlength{\standardunitlength}{0.00083333in}

\newcommand{\eepic}[2]{
  \setlength{\unitlength}{#2\standardunitlength}
  \begin{array}{c}
    {\input #1.eepic } 
  \end{array}
}
\newcommand{\seepic}[2]{
  \setlength{\unitlength}{#2\standardunitlength}
  \begin{array}{c}  \hspace{-1.7mm}
    \raisebox{-2pt}{\input #1.eepic }
    \hspace{-1.9mm}
  \end{array}
}

\newcommand{\fig}[1]{figure~\ref{#1}}
\def\smily{{
  \setlength{\unitlength}{0.4\standardunitlength}
  \begin{array}{c}  \hspace{-1.7mm} \raisebox{-2pt}{
    \begin{picture}(616,629)(0,-10)
    \thicklines
    \put(308.000,344.500){\arc{375.000}{0.6435}{2.4981}}
    \put(308,307){\ellipse{600}{600}}
    \put(195,382){\blacken\ellipse{76}{76}}
    \put(195,382){\ellipse{76}{76}}
    \put(420,382){\blacken\ellipse{76}{76}}
    \put(420,382){\ellipse{76}{76}}
    \end{picture}
  }
  \hspace{-1.9mm}
  \end{array}
}}

\catcode`\@=11
\long\def\@makecaption#1#2{%
    \vskip 10pt
    \setbox\@tempboxa\hbox{%\ifvoid\tinybox\else\box\tinybox\fi
      \small\sf{\bfcaptionfont #1. }\ignorespaces #2}%
    \ifdim \wd\@tempboxa >\captionwidth {%
        \rightskip=\@captionmargin\leftskip=\@captionmargin
        \unhbox\@tempboxa\par}%
      \else
        \hbox to\hsize{\hfil\box\@tempboxa\hfil}%
    \fi}
\font\bfcaptionfont=cmssbx10 scaled \magstephalf
\newdimen\@captionmargin\@captionmargin=2\parindent
\newdimen\captionwidth\captionwidth=\hsize
\catcode`\@=12

\newcommand{\ad}{\operatorname{ad}}
\newcommand{\Ad}{\operatorname{Ad}}
\newcommand{\Alt}{\operatorname{Alt}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\Diff}{\operatorname{Diff}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\gr}{\operatorname{gr}}
\newcommand{\im}{\operatorname{im}}
\newcommand{\mor}{\operatorname{mor}}
\newcommand{\Span}{\operatorname{span}}
\newcommand{\sign}{\operatorname{sign}}
\newcommand{\sym}{\operatorname{sym}}
\newcommand{\tr}{\operatorname{tr}}

\def\qed{{\hfill\text{$\Box$}}}
\def\eqdef{\overset{\text{def}}{=}}

\newlength{\globalparindent}
\setlength{\globalparindent}{\parindent}

\newcommand{\udot}{{\mathaccent\cdot\cup}}

\def\proof{{\par\noindent {\em Proof. } }}

\newenvironment{myitemize}{
        \begin{list}{$\bullet$}{\setlength{\leftmargin}{16pt}
        \setlength{\labelwidth}{12pt}
        \setlength{\labelsep}{4pt}}
}{
        \end{list}
}
\newenvironment{myenumi}{
        \begin{list}{\theenumi}{\usecounter{enumi}
        \setlength{\leftmargin}{16pt}
        \setlength{\labelwidth}{12pt}
        \setlength{\labelsep}{4pt}}
}{
        \end{list}
}

\def\arXiv#1{{\href{http://front.math.ucdavis.edu/#1}{arXiv:#1}}}