\def\paragraph#1{\textit{#1}.}
\def\C{{\mathbb C}}
  
\def\bigO{\mathcal{O}}

\newtheorem{tm}{Theorem}[section]
\newtheorem{pp}[tm]{Proposition}
\newtheorem{lm}[tm]{Lemma}
\newtheorem{cl}[tm]{Corollary}

\def\myurl#1{\textsf{#1}}
\def\mytilde{\textasciitilde}

\def\Vv{\mathcal{V}}
\def\Ww{\mathcal{W}}
\def\Ll{\mathcal{L}}
\def\Mm{\mathcal{M}}
\def\Cc{\mathcal{C}}


\def\Na{N^\alpha}
\def\Ng{N}
\def\Nb{M}
\def\Ka{K^\alpha}
\def\Kb{K}

\def\N{\mathbb{N}}
\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
\def\C{\mathbb{C}}
\def\P{\mathbb{P}}
\def\R{\mathbb{R}}
\def\E{\mathbb{E}}
\def\F{\mathbb{F}}
\def\G{\mathbb{G}}

\def\Id{\mathrm{Id}}
\def\Pdiffn#1#2#3{\frac{\partial^{#3}}{\partial{#2}^{#3}}#1}
\def\Tau{\mathcal{T}}
\def\Union{\bigcup}
\def\abs#1{\left[ {#1} \right]}
\def\binomial#1#2{{{#1}\choose{#2}}}
\def\card#1{\##1}
\def\charact{\mathrm{char}}
\def\comp{\circ}
\def\defl#1{\widetilde{#1}}
\def\deg{\mathrm{deg}}
\def\dim{\mathrm{dim}}
\def\for{\mathrm{for}}
\def\funto{\rightarrow}
\def\fun{\rightarrow}
\def\grad{\nabla}
\def\inter{\cap}
\def\lmon{\mathrm{lm}}
\def\minus{\backslash}
\def\min{\mathrm{min}}
\def\mod{\mathrm{mod}}
\def\monomials#1{\mathcal{M}}
\def\multinomial#1#2{\binomial{#1}{#2}}
\def\mul{\mathrm{mul}}
\def\Norm#1{\left|\left| {#1} \right|\right|}
\def\norm#1{|| {#1} ||}
\def\module#1{| {#1} |}
\def\Module#1{\left| {#1} \right|}
\def\rank{\mathrm{rank}}
\def\supp{\mathrm{supp}}
\def\tens{\otimes}
\def\union{\cup}
\def\val{\mathrm{val}}
\def\conv#1{\mathrm{Conv}(#1)}

\def\coeff#1#2{\frac{D^{#2}#1}{{#2}!}}

\def\mygamma#1#2{\sup_{k\ge 2}\norm{D#1(#2)^{-1}\coeff{#1(#2)}{k}}^{\frac{1}{k-1}}}
\def\gammad{\gamma^\dag}
\def\gammab{\overline\gamma}
\def\betab{\overline\beta}
\def\lambdab{\overline\lambda}
\def\alphab{\overline\alpha}
\def\gammah{\hat\gamma}
\def\betah{\hat\beta}
\def\sigmah{\hat\sigma}
\def\alphah{\hat\alpha}
\def\rhob{\overline\rho}
\def\mub{\overline\mu}
\def\eb{\overline{e}}
\def\cb{\overline{c}}
\def\sigmab{\overline{\sigma}}
\def\deltab{\overline{\delta}}
\def\Sigmab{\overline{\Sigma}}
\def\phib{\overline{\phi}}
\def\Phib{\overline{\Phi}}
\def\fb{\overline{f}}

\def\pdiff#1#2{\frac{\partial #1}{\partial #2}}
\def\Pdiff#1#2{\frac{\partial}{\partial #2} #1}

\def\pdiffn#1#2#3{\frac{\partial^{#3}#1}{\partial{#2}^{#3}}}
\def\spdiffn#1#2#3{\frac{\partial^{#3}#1}{\partial{#2}}}
\def\Pdiffn#1#2#3{\frac{\partial^{#3}}{\partial{#2}^{#3}}#1}

\def\diff#1#2{\frac{d#1}{d#2}}
\def\Diff#1#2{\frac{d}{d#2}#1}

\def\diffn#1#2#3{\frac{d^{#3}#1}{d#2^{#3}}}
\def\Diffn#1#2#3{\frac{d^{#3}}{d#2^{#3}}#1}

\def\defl#1{\widetilde{#1}}

\def\interior#1{\overset{\circ}{#1}}
%\def\adherence#1{\overline{#1}}
\def\adherence#1{\Bar{#1}}
\def\Bb{\Bar{B}}
\def\Bb{\mathcal{B}}
\def\gf{\mathfrak{g}}

\def\trunc#1#2#3{\left[ #1 \right]_{#2}^{#3}}
\def\truncl#1#2{\left\lfloor #1 \right\rfloor_{#2}}
\def\trunch#1#2{\left\lceil #1 \right\rceil^{#2}}
\def\function#1{\texttt{#1}}


\newcounter{equationbak}
\newcounter{equationR}
\setcounter{equationR}{0}

\newenvironment{equationE}{\stepcounter{equationR}\renewcommand{\theequation}{$E_{\arabic{equationR}}$}\begin{equation}}{\end{equation}\renewcommand{\theequation}{$\arabic{equation}$}}


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