%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Useful notations
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\zero}{\ensuremath{\mathbb{O}}}
\newcommand{\Z}{\ensuremath{\mathbb{Z}}}
\newcommand{\R}{\ensuremath{\mathbb{R}}}
\newcommand{\N}{\ensuremath{\mathbb{N}}}
\newcommand{\mathH}{\ensuremath{\mathcal{H}}}
\newcommand{\E}{\ensuremath{\mathrm{E}}}
\newcommand{\T}{\ensuremath{\mathsf{T}}}
\newcommand{\e}{\ensuremath{\mathbf{e}}}
\newcommand{\thenif}{\ensuremath{\;\mbox{ если }\;}}

%\newcommand{\vecI}{\ensuremath{\overrightarrow{\mathrm{I}}}}
\newcommand{\vecI}{\ensuremath{\mathcal{I}}}
\newcommand{\edge}[1]{\ensuremath{\mathbf{\mathsf{#1}}}}

\newcommand{\ppe}{\xrightarrow{\odot,\,\odot^{-\!1}\!}}
%\newcommand{\ppn}[1]{\xrightarrow[#1]{\odot,\,\odot^{-\!1}\!}}
%\newcommand{\ppt}{\xrightarrow[*]{\odot,\,\odot^{-\!1}\!}}
%\newcommand{\pp}{\xrightarrow{\odot,\,\odot^{-\!1}\!}}
%\newcommand{\pe}{\xrightarrow{}}
%\newcommand{\pn}[1]{\xrightarrow[#1]{}}
%\newcommand{\pt}{\xrightarrow[*]{}}
%\newcommand{\p}{\xrightarrow{}}
\newcommand{\diag}{\ensuremath{\mathop{\mathrm{diag}}}}
\newcommand{\sign}{\ensuremath{\mathop{\mathrm{sign}}}}
\renewcommand{\gcd}{\ensuremath{\mathop{\mbox{НОД}}}}
\newcommand{\integ}{\ensuremath{\mathop{\mathrm{int}}}}
\newcommand{\fract}{\ensuremath{\mathop{\mathrm{frac}}}}
\newcommand{\occ}{\ensuremath{\mathop{\mathrm{occ}}}}
\newcommand{\rhs}{\ensuremath{\mathop{\mathrm{rhs}}}}
%\newcommand{\Tc}[1]{\ensuremath{\mathop{T}\limits^\circ{}_{\!\!#1}}}
%\newcommand{\Fc}[1]{\ensuremath{\mathop{F}\limits^\circ{}_{\!\!#1}}}
\newcommand{\Tc}[1]{\ensuremath{T^{\mathrm{c}}_{#1}}}
\newcommand{\Fc}[1]{\ensuremath{F^{\mathrm{c}}_{#1}}}


% expectation
\DeclareMathOperator{\avg}{\mbox{\sffamily\bfseries E}}
% probability
\DeclareMathOperator{\prob}{\mbox{\sffamily\bfseries P}}
% covariance
\DeclareMathOperator{\cov}{cov}
% variance
\DeclareMathOperator{\var}{\mbox{\sffamily\bfseries D}}
% indicator
\DeclareMathOperator{\ind}{\mbox{\sffamily\bfseries 1}}