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Location: AENC/resampling_chain/diagram_circle_lemma.tex - annotation
5f02d6bd36ea
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text/x-tex
Add proof of Lemma 12
aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 c2fa910c4916 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 c2fa910c4916 c2fa910c4916 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 c2fa910c4916 c2fa910c4916 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 aad4326b06f9 c2fa910c4916 c2fa910c4916 aad4326b06f9 aad4326b06f9 aad4326b06f9 c2fa910c4916 c2fa910c4916 c2fa910c4916 aad4326b06f9 aad4326b06f9 | \documentclass{standalone}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{parskip}
\usepackage[usenames,dvipsnames]{color}
\usepackage[hidelinks]{hyperref}
\renewcommand*{\familydefault}{\sfdefault}
\usepackage{bbm} %For \mathbbm{1}
%\usepackage{bbold}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[scale=0.8]
\def\r{4};
\def\step{15};
% Line and circles
\foreach \a in {0,...,23} {
\draw[-] ({\a*\step}:\r) -- ({(\a+1)*\step}:\r);
\draw ({\a*\step}:\r) circle (0.05);
}
% Red dots
\foreach \a in {2,3,6,8,9,10,11,14,15,17,20} {
\draw[fill,red] ({\a*\step}:\r) circle (0.09);
}
% Blue squares
\foreach \a in {2,3,20} {
\draw[fill,blue] ({\a*\step}:{\r-0.3})+(-0.1,-0.1) rectangle +(0.1,0.1);
}
% Green triangles
\foreach \a in {6,8,9,10,11,14,15,17} {
\draw[fill,green] ({\a*\step}:{\r-0.3}) +(-0.15,-0.1) -- +(0.15,-0.1) -- +(0,0.15);
}
% Legend
\draw (-1,0)+(-0.5,-1) rectangle +(2.6,1);
\draw[fill,red] (-1.1,0.5) circle (0.09);
\draw[fill,blue] (-1.1,0) +(-0.1,-0.1) rectangle +(0.1,0.1);
\draw[fill,green] (-1.1,-0.5) +(-0.15,-0.1) -- +(0.15,-0.1) -- +(0,0.15);
\draw[anchor=west] (-1,0.5) node {$I$};
\draw[anchor=west] (-1,0) node {$I\cap [{\color{gray}n}{-l},r]_0$};
\draw[anchor=west] (-1,-0.5) node {$I\setminus[{\color{gray}n}{-l},r]_0$};
% Bigger circle
\foreach \a in {-5,...,3} {
\draw[-,dashed] ({\a*\step}:\r+1) -- ({(\a+1)*\step}:\r+1);
%\draw ({\a*\step}:\r+1) circle (0.05);
}
\draw ({-5*\step}:\r+1) +(0.2,-0.2) node {${\color{gray}n}{-l}$};
\draw ({4*\step}:\r+1) +(0.2,0.2) node {$r$};
%
% Arrows with labels
%
% 0
\draw ({0*\step}:\r-0.3) node {$0$};
% 1
\draw ({1*\step}:\r-0.3) node {$1$};
% n/2
\draw ({12*\step}:\r-0.3) node {$\frac{n}{2}$};
% n-1
\draw ({-1*\step}:\r-0.3) +(-0.4,0) node {${\color{gray}n}{-1}$};
% i_*
\draw[->] ({10*\step}:\r-1) -- ({10*\step}:\r-0.2);
\draw ({10*\step}:\r-1) +(0.3,0) node {$i_*$};
% j
\draw[->] ({0*\step}:\r-1.3) -- ({0*\step}:\r-0.8);
\draw ({0*\step}:\r-1) +(-0.4,0) node {$j$};
\end{tikzpicture}
\end{document}
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