\titleformat{\section}
{\titlerule
\vspace{.8ex}%
\normalfont\scshape}
{\thesection.}{.5em}{}

\titleformat{\subsection}
{\normalfont\scshape}
{\thesubsection.}{.5em}{}

\titleformat{\subsubsection}
{\normalfont\scshape}
{\thesubsubsection.}{.5em}{}
%
\makeatletter
\renewcommand\section{\@startsection {section}{1}{\z@}%
                                   {.1ex \@plus -1ex \@minus -.2ex}%
                                   {1ex \@plus.1ex}%
                                   {\centering\large\scshape}}
\makeatother
\newcommand{\R}{\mathbb{R}}


