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The relationship between the graph map, point evaluations, and constant graph maps. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (YX) at (-2,0) {$Y^X$};
\node (XYX) at (2,0) {$(X \otimes Y)^X$};
\node (XY) at (2,-2) {$X \otimes Y$};
\node (Y) at (-2,-2) {$Y$};
\draw[->, above] (YX) to node {$\Gamma_{\cdot}$} (XYX);
\draw[->,right] (XYX) to node {$\hat{ev}_{x}$} (XY);
\draw[->,below,left,dashed] (YX) to node [xshift=-3pt] {$\langle \overline{x}, ev_{x} \rangle$} (XY);
\draw[->, left] (YX) to node {$ev_x$} (Y);
\draw[->,below] (Y) to node {$\Gamma_{\overline{x}}$} (XY);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The generic Bayesian model. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (1) at (0,2) {$1$};
\node (H) at (-2,0) {$H$};
\node (D) at (2,0) {$D$};
\draw[->,above left] (1) to node {$P_H$} (H);
\draw[->, above] ([yshift=2pt] H.east) to node {$\mathcal{S}$} ([yshift=2pt] D.west);
\draw[->, below,dashed] ([yshift=-2pt] D.west) to node {$\mathcal I$} ([yshift=-2pt] H.east);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The commutativity of these diagrams, together with the measurability of the constant functions and constant graph functions, implies the projection maps _X and _Y are measurable. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (X) at (2,0) {$X$};
\node (Y) at (5,-3) {$Y$};
\node (XY) at (2,-3) {$X \otimes Y$};
\draw[->,right] (X) to node {$\overline{y}$} (Y);
\draw[->,right] (X) to node {$\Gamma_{\overline{y}}$} (XY);
\draw[->,below] (XY) to node {$\pi_Y$} (Y);
\node (Y2) at (-2,0) {$Y$};
\node (X2) at (-5,-3) {$X$};
\node (XY2) at (-2,-3) {$X \otimes Y$};
\draw[->,left] (Y2) to node {$\overline{x}$} (X2);
\draw[->,right] (Y2) to node {$\Gamma_{\overline{x}}$} (XY2);
\draw[->,below] (XY2) to node {$\pi_X$} (X2);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The additional conditions required for a symmetric monoidal category. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (XY) at (-1,0) {$(X \square Y) \square Z$};
\node (YW) at (-1,-3) {$X \square (Y \square Z)$};
\node (YW2) at (-1,-6) {$Y \square (Z \square X)$};
\node (four) at (4,0) {$(Y \square X) \square Z$};
\node (five) at (4,-3) {$Y \square (X \square Z)$};
\node (six) at (4,-6) {$Y \square (Z \square X)$};
\draw[->, above] (XY) to node {$s_{X,Y} \square Id_Z$} (four);
\draw[->,above] (YW2) to node {$a_{Y,Z,X}$} (six);
\draw[->,right] (four) to node {$a_{Y,X, Z}$} (five);
\draw[->,left] (XY) to node {$a_{X ,Y,Z}$} (YW);
\draw[->,left] (YW) to node {$s_{X,Y \square Z}$}(YW2);
\draw[->,right] (five) to node {$Id_Y \otimes s_{X,Z}$} (six);
\node (AI) at (7.5,0) {$X \square I$};
\node (IA) at (9.5,0) {$I \square X$};
\node (A) at (9.5,-2) {$X$};
\draw[->,above] (AI) to node {$s_{X,1}$} (IA);
\draw[->,below,left] (AI) to node {$r_{X}$} (A);
\draw[->,right] (IA) to node {$l_X$} (A);
\node (BI) at (7.5,-4) {$X \square Y$};
\node (IB) at (9.5,-4) {$Y \square X$};
\node (B) at (9.5,-6) {$X \square Y$};
\draw[->,above] (BI) to node {$s_{X,Y}$} (IB);
\draw[->,below,left] (BI) to node {$s_{Y,X}$} (B);
\draw[->,right] (IB) to node {$Id_X$} (B);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The commutativity of both triangles, the measurability of f and ev_x, and the induced -algebra of X Y^X implies the measurability of ev. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (X) at (-3,0) {$X$};
\node (YX) at (3,0) {$Y^X$};
\node (XY) at (0,0) {$ X \otimes Y^X$};
\node (Y) at (0,-3) {$Y$};
\draw[->,above] (X) to node {$\Gamma_{\overline{f}}$} (XY);
\draw[->,left] (X) to node [xshift=-3pt] {$f$} (Y);
\draw[->, above] (YX) to node {$\Gamma_{\overline{x}}$} (XY);
\draw[->,right] (YX) to node [xshift=5pt,yshift=0pt] {$ev_x$} (Y);
\draw[->,right] (XY) to node {$ev_{X,Y}$} (Y);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The defining characteristic property of the evaluation function ev for tensor products of conditionals in P. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (X) at (0,-3) {$X \otimes Z$};
\node (XY) at (0,0) {$ X \otimes Y^X$};
\node (Y) at (3,0) {$Y$};
\node (1) at (-4,-3) {$Z$};
\node (YX) at (-4,0) {$Y^X$};
\draw[->, left] (X) to node {$1_X \otimes P$} (XY);
\draw[->,below, right] (X) to node [xshift=0pt,yshift=0pt] {$\overline{P}$} (Y);
\draw[->,above] (XY) to node {$\delta_{ev}$} (Y);
\draw[->,left] (1) to node {$P$} (YX);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The defining property of a Gaussian Process is the commutativity of a P diagram. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\xv}{\mathbf{x}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (1) at (-1,0) {$1$};
\node (YX) at (2,0) {$Y^X$};
\node (YX0) at (2,-2) {$Y^{X_0}$};
\draw[->,above] (1) to node {$P$} (YX);
\draw[->, right] (YX) to node {$\delta_{Y^{\iota}}$} (YX0);
\draw[->,left ] (1) to node [xshift = -5pt]{$P \iota^{-1}$} (YX0);
\node (12) at (5,0) {$1$};
\node (YX2) at (8,0) {$Y^X$};
\node (YX02) at (8,-2) {$Y$};
\draw[->,above] (12) to node {$P$} (YX2);
\draw[->, right] (YX2) to node {$\delta_{ev_{\xv}}$} (YX02);
\draw[->,left ] (12) to node [xshift = -5pt]{$P ev_{\xv}^{-1}$} (YX02);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The tensor product of conditional 1_X and P in P. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (XZ) at (0,0) {$X \otimes Z$};
\node (X) at (-3,0) {$X$};
\node (Z) at (3,0) {$Z$};
\node (XY) at (0,-3) {$X \otimes Y$};
\node (X2) at (-3,-3) {$X$};
\node (Y) at (3,-3) {$Y$};
\draw[->, right] (XZ) to node {$1_{X} \otimes P$} (XY);
\draw[->,above] (XZ) to node {$\delta_{\pi_X}$} (X);
\draw[->,above] (XZ) to node {$\delta_{\pi_Z}$} (Z);
\draw[->,below] (XY) to node {$\delta_{\pi_X}$} (X2);
\draw[->,below] (XY) to node {$\delta_{\pi_Y}$} (Y);
\draw[->,right] (X) to node {$1_X$} (X2);
\draw[->,left] (Z) to node {$P$} (Y);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The evaluation function ev sets up a bijective correspondence between the two measurable maps f and f. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (X) at (0,-3) {$X \otimes Z$};
\node (XY) at (0,0) {$ X \otimes Y^X$};
\node (Y) at (3,0) {$Y$};
\node (1) at (-4,-3) {$Z$};
\node (YX) at (-4,0) {$Y^X$};
\draw[->, left] (X) to node {$Id_X \otimes \tilde{f}$} (XY);
\draw[->,below, right] (X) to node [xshift=0pt,yshift=0pt] {$f$} (Y);
\draw[->,above] (XY) to node {$ev_{X,Y}$} (Y);
\draw[->,left] (1) to node {$\tilde{f}$} (YX);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The defining characteristic property of the evaluation function ev for graphs. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (X) at (0,-3) {$X \cong X \otimes 1$};
\node (XY) at (0,0) {$ X \otimes Y^X$};
\node (Y) at (3,0) {$Y$};
\node (1) at (-4,-3) {$1$};
\node (YX) at (-4,0) {$Y^X$};
\draw[->, left] (X) to node {$\Gamma_{\overline{f}}\cong Id_X \otimes \ulcorner f \urcorner$} (XY);
\draw[->,below, right] (X) to node [xshift=0pt,yshift=0pt] {$f$} (Y);
\draw[->,above] (XY) to node {$ev_{X,Y}$} (Y);
\draw[->,left] (1) to node {$\ulcorner f \urcorner$} (YX);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The noise free sampling distribution S_nf. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\mcS}{\mathcal{S}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (XYX2) at (0,0) {$X \otimes Y^X$};
\node (XYg) at (4,0) {$X \otimes (X \otimes Y)^X$};
\node (Y) at (8,0) {$X \otimes Y$};
\draw[->,above] (XYX2) to node {$1 \otimes \delta_{\Gamma}$} (XYg);
\draw[->,above] (XYg) to node {$\delta_{ev}$} (Y);
\draw[->,below,out=-45,in=225,looseness=.3] (XYX2) to node {$\mcS_{nf}$ = composite} (Y);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The sampling distribution model in P with additive Gaussian noise. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\mcS}{\mathcal{S}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (XYX) at (1,0) {$X \otimes Y^X$};
\node (XYX2) at (4.5,0) {$X \otimes Y^X$};
\node (XYg) at (8.5,0) {$X \otimes (X \otimes Y)^X$};
\node (Y) at (12,0) {$X \otimes Y$};
\draw[->,below] (XYX) to node {$1_X \otimes N$} (XYX2);
\draw[->,below] (XYX2) to node {$1_X \otimes \delta_{\Gamma_{\cdot}}$} (XYg);
\draw[->,below] (XYg) to node {$\delta_{ev}$} (Y);
\draw[->,below,out=-45,in=225,looseness=.3] (XYX) to node {$\mcS_n$ = composite} (Y);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The categorical characterization of a joint normal distribution. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\NN}{\mathcal{N}}
\newcommand{\sa}{\Sigma}
\newcommand{\mcS}{\mathcal{S}}
\newcommand{\mcI}{\mathcal{I}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (1) at (0,0) {$1$};
\node (XY) at (0,-1.5) {$X \times Y$};
\node (X1) at (-3,-3) {$X$};
\node (X2) at (3,-3) {$Y$};
\draw[->,right] (1) to node {$J$} (XY);
\draw[->,above,left] (XY) to node [yshift=3pt] {$\delta_{\pi_X}$} (X1);
\draw[->,above,right] (XY) to node [yshift=3pt] {$\delta_{\pi_Y}$} (X2);
\draw[->,left,out=180,in=90,looseness=1] (1) to node {$P_1 \sim \NN(\mu_1,\sa_{11})$} (X1);
\draw[->,right,out=0,in=90,looseness=1] (1) to node {$P_2 \sim \NN(\mu_2,\sa_{22})$} (X2);
\draw[->, above] ([yshift=2pt] X1.east) to node [yshift=3pt] {$\overline{\mcS}$} ([yshift=2pt] X2.west);
\draw[->, below] ([yshift=-2pt] X2.west) to node [yshift=-3pt] {$\overline{\mcI}$} ([yshift=-2pt] X1.east);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The defining characteristic property of the evaluation function ev for graphs. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\mcS}{\mathcal{S}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (X1) at (0,-3) {$X \otimes 1$};
\node (XY) at (0,0) {$ X \otimes Y^{X}$};
\node (X2) at (3,0) {$Y$};
\node (1) at (-4,-3) {$1$};
\node (YX) at (-4,0) {$Y^{X}$};
\draw[->, left] (X1) to node {$\Gamma_{\mcS}$} (XY);
\draw[->,below, right] (X1) to node [xshift=0pt,yshift=0pt] {$\overline{\mcS}$} (X2);
\draw[->,above] (XY) to node {$ev_{X,Y}$} (X2);
\draw[->,left] (1) to node {$\mcS$} (YX);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The substitution/evaluation relation. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\xv}{\mathbf{x}}
\newcommand{\fn}{\ulcorner f \urcorner}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (1) at (3,0) {$1$};
\node (X02) at (6,0) {$X_0$};
\node (X2) at (9,0) {$X$};
\node (Y2) at (7.5,-2) {$Y$};
\draw[->,above] (X02) to node {$\iota$} (X2);
\draw[->,above] (1) to node {$\xv$} (X02);
\draw[->, right] (X2) to node {$f$} (Y2);
\draw[->,left, dashed ] (1) to node [yshift = -3pt] {$f(\xv) = f \circ \iota \circ \xv = ev_{\xv}(\fn)$} (Y2);
\draw[->,right] (X02) to node {$f|_{X_0}$} (Y2);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The generic nonparametric Bayesian model for stochastic processes. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\xv}{\mathbf{x}}
\newcommand{\mcS}{\mathcal{S}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (X) at (0,2) {$1$};
\node (H) at (-2,0) {$X \otimes Y^X$};
\node (D) at (2,0) {$X \otimes Y$};
\node (d) at (5,1) {$d$ is measurement data};
\draw[->, left] (X) to node [xshift=-3pt] {$\Gamma_P(\cdot \mid \xv)$} (H);
\draw[->,right,dashed] (X) to node [xshift=2pt] {$d$} (D);
\draw[->, above] ([yshift=2pt] H.east) to node [yshift=3pt] {$\mcS$} ([yshift=2pt] D.west);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The additive Gaussian noise measurement model. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\NN}{\mathcal{N}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (1) at (0,0) {$1$};
\node (Y) at (4,0) {$Y$};
\draw[->,above] (1) to node {$M_y \sim \NN(y,\sigma^2)$} (Y);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The generic parametric Bayesian model. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\Rp}{\mathbb{R}^{^p}}
\newcommand{\xv}{\mathbf{x}}
\newcommand{\mcS}{\mathcal{S}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (X) at (1,2) {$1$};
\node (H) at (-2,0) {$X \otimes \Rp$};
\node (YX) at ( 1,0) {$X \otimes Y^X$};
\node (D) at (4,0) {$X \otimes Y$};
\draw[->,left] (X) to node [xshift=-3pt] {$\Gamma_P(\cdot \mid \xv)$} (H);
\draw[->,below] (H) to node {$1_X \otimes \delta_i$} (YX);
\draw[->, right,dashed] (X) to node [xshift=3pt] {$d$} (D);
\draw[->, below] ([yshift=2pt] YX.east) to node {$\mcS_n$} ([yshift=2pt] D.west);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The noise free sampling distributions S^x given the prior _x P with the dirac measure on the X component. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\mcSS}{\mathcal{S}^{\mathbf{x}}}
\newcommand{\xv}{\mathbf{x}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (YX) at (0,0) {$Y^X$};
\node (Y) at (4,0) {$Y$};
\draw[->,above] (YX) to node {$\mcSS = \delta_{ev_{\xv}}$} (Y);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
Construction of the generic Markov kernel N for modeling the Gaussian additive measurement noise. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\GP}{\mathcal{G}\mathcal{P}}
\newcommand{\xv}{\mathbf{x}}
\newcommand{\NN}{\mathcal{N}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (1) at (0,-1.5) {$1$};
\node (YX) at (4,0) {$Y^X$};
\node (YX2) at (8,0) {$Y^X$};
\node (Y) at (4,-3) {$Y$};
\node (Y2) at (8,-3) {$Y$};
\draw[->,above,left] (1) to node[xshift=-2pt,yshift=10pt] {$P \sim \GP(f,k)$} (YX);
\draw[->,above] (YX) to node {$N$} (YX2);
\draw[->,below,left] (1) to node [xshift=-5pt,yshift=-4pt] {$Pev_{\xv}^{-1} \sim \NN(f(\xv),k(\xv,\xv))$} (Y);
\draw[->,above] (Y) to node {$M$} (Y2);
\draw[->,right] (YX) to node {$\delta_{ev_{\xv}}$} (Y);
\draw[->,right] (YX2) to node {$\delta_{ev_{\xv}}$} (Y2);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The distribution P GP(m,k) can be evaluated on rectangles U=A B by projecting onto the given x coordinate. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\GP}{\mathcal{G}\mathcal{P}}
\newcommand{\xv}{\mathbf{x}}
\newcommand{\NN}{\mathcal{N}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (1) at (-1.5,0) {$1$};
\node (YX) at (3,0) {$Y^X$};
\node (XYX) at (6,0) {$(X \otimes Y)^X$};
\node (XY) at (9,0) {$X \otimes Y$};
\node (Y) at (6,-2) {$Y$};
\draw[->,above] (1) to node {$P \sim \GP(m,k)$} (YX);
\draw[->,above] (YX) to node {$\Gamma_{\cdot}$} (XYX);
\draw[->,above] (XYX) to node {$\hat{ev}_{\xv}$} (XY);
\draw[->,right] (XY) to node {$\delta_{\pi_Y}$} (Y);
\draw[->,above,right,dashed] (YX) to node [xshift=3pt,yshift=4pt] {$\delta_{ev_{\xv}}$} (Y);
\draw[->,below, left] (1) to node [xshift = -5pt,yshift=-5pt] {$P ev_{\xv}^{-1} \sim \NN(m(\xv),k(\xv,\xv))$} (Y);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The parametric model sampling distribution as a composite of four components. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\Rp}{\mathbb{R}^{^p}}
\newcommand{\xv}{\mathbf{x}}
\newcommand{\mcS}{\mathcal{S}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (1) at (1.8,3) {$1$};
\node (Rn) at (-4,0) {$X \otimes \Rp$};
\node (XYX) at (-1,0) {$X \otimes Y^X$};
\node (XYX2) at (2,0) {$X \otimes Y^X$};
\node (XYg) at (6,0) {$X \otimes (X \otimes Y)^X$};
\node (Y) at (9,0) {$X \otimes Y$};
\draw[->,left,above] (1) to node [xshift=-16pt,yshift=3pt] {$\Gamma_P(\cdot \mid \xv)$} (Rn);
\draw[->,above] (Rn) to node {$1_X \otimes \delta_{i}$} (XYX);
\draw[->,above] (XYX) to node {$1_X \otimes N$} (XYX2);
\draw[->,above] (XYX2) to node {$1_X \otimes \delta_{\Gamma}$} (XYg);
\draw[->,above] (XYg) to node {$\delta_{ev}$} (Y);
\draw[->,below,out=-45,in=225,looseness=.3] (Rn) to node {$\mcS$} (Y);
\draw[->,above,out=45,in=135,looseness=.5] (XYX) to node {$\mcS_n$} (Y);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The composite of the prior and noise measurement model is the graph of a GP at x. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\xv}{\mathbf{x}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (X) at (0,0) {$1$};
\node (X1) at (-3,-3) {$X$};
\node (X2) at (-3,-6) {$X$};
\node (YX) at (3,-3) {$Y^X$};
\node (YX2) at (3,-6) {$Y^X$};
\node (XYX) at (0,-3) {$X \otimes Y^X$};
\node (XYX2) at (0,-6) {$X \otimes Y^X$};
\node (X3) at (6,0) {$1$};
\node (XYX3) at (6, -6) {$X \otimes Y^X$};
\draw[->,above,left] (X) to node {$\delta_{\xv}$} (X1);
\draw[->,left] (X1) to node {$1_X$} (X2);
\draw[->,right] (X) to node {$P$} (YX);
\draw[->,right] (YX) to node {$N$} (YX2);
\draw[->,right] (X) to node [yshift=-6pt] {$\Gamma_P(\cdot \mid \xv)$} (XYX);
\draw[->,right] (XYX) to node {$1_X \otimes N$} (XYX2);
\draw[->,above] (XYX) to node {$\delta_{\pi_{X}}$} (X1);
\draw[->,above] (XYX2) to node {$\delta_{\pi_{X}}$} (X2);
\draw[->,above] (XYX) to node {$\delta_{\pi_{Y^X}}$} (YX);
\draw[->,above] (XYX2) to node {$\delta_{\pi_{Y^X}}$} (YX2);
\draw[->,right] (X3) to node {$\Gamma_{N \circ P}(\cdot \mid \xv)$} (XYX3);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The GP N_f can be evaluated on rectangles U=A B by projecting onto the given x coordinate. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\GP}{\mathcal{G}\mathcal{P}}
\newcommand{\xv}{\mathbf{x}}
\newcommand{\NN}{\mathcal{N}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (1) at (-1.5,0) {$1$};
\node (YX) at (3,0) {$Y^X$};
\node (XYX) at (6,0) {$(X \otimes Y)^X$};
\node (XY) at (9,0) {$X \otimes Y$};
\node (Y) at (6,-2) {$Y$};
\draw[->,above] (1) to node {$N_f \sim \GP(f,k_N)$} (YX);
\draw[->,above] (YX) to node {$\delta_{\Gamma_{\cdot}}$} (XYX);
\draw[->,above] (XYX) to node {$\delta_{\hat{ev}_{\xv}}$} (XY);
\draw[->,right] (XY) to node {$\delta_{\pi_Y}$} (Y);
\draw[->,above,right,dashed] (YX) to node [xshift=3pt,yshift=4pt] {$\delta_{ev_{\xv}}$} (Y);
\draw[->,below, left] (1) to node [xshift = -5pt,yshift=-5pt] {$N_f \left( ev_{\xv}^{-1}(\cdot) \right) \sim \NN(f(\xv),\sigma^2)$} (Y);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
Each GP P_Y^X^1, which is parameterized by a measurement (x,y) X Y, determines a conditional P_Y^X^1. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (1) at (-.2,0) {$X \otimes Y$};
\node (YX) at (2.7,0) {$Y^X$};
\node (X) at (-.2,-1) {$X \otimes (X \otimes Y)$};
\node (Y) at (2.7,-1) {$Y$};
\node (pt1) at (-1,-.5) {$$};
\node (pt2) at (3.5,-.5) {$$};
\node (pt3) at (4.1,.25) {$$};
\node (pt4) at (4.1,-1.5) {$$};
\draw[->,above] (1) to node {$P_{Y^X}^1$} (YX);
\draw[->,below] (X) to node {$\overline{P_{Y^X}^1}$} (Y);
\draw[-] (pt1) to node {$$} (pt2);
\pgfsetlinewidth{.5ex}
\draw[<->] (pt3) to node {$$} (pt4);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The Gaussian additive noise parametric sampling distributions S^x_p viewed as a family of sampling distributions, one for each x X. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\Rp}{\mathbb{R}^{^p}}
\newcommand{\xv}{\mathbf{x}}
\newcommand{\mcSS}{\mathcal{S}^{\mathbf{x}}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (Rp) at (-3,0) {$\Rp$};
\node (YX) at (0,0) {$Y^X$};
\node (YX2) at (3,0) {$Y^X$};
\node (Y) at (6,0) {$Y$};
\draw[->,above] (Rp) to node {$\delta_{i}$} (YX);
\draw[->,above] (YX) to node {$N$} (YX2);
\draw[->,above] (YX2) to node {$\delta_{ev_{\xv}}$} (Y);
\draw[->,below,out=-45,in=225,looseness=.5] (Rp) to node {$\mcSS_p$} (Y);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The composite of the prior distribution P GP(m,k) and the sampling distribution S^x give the coordinate projections as priors on Y. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\GP}{\mathcal{G}\mathcal{P}}
\newcommand{\mcSS}{\mathcal{S}^{\mathbf{x}}}
\newcommand{\xv}{\mathbf{x}}
\newcommand{\mcII}{\mathcal{I}^{\mathbf{x}}}
\newcommand{\NN}{\mathcal{N}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (1) at (0,0) {$1$};
\node (YX) at (-2,-2) {$Y^X$};
\node (Y) at (2,-2) {$Y$};
\draw[->,above,left] (1) to node [xshift = -5pt,yshift=5pt] {$P \sim \GP(m,k)$} (YX);
\draw[->, above] ([yshift=2pt] YX.east) to node [yshift=3pt] {$\mcSS =\delta_{ev_{\xv}}$} ([yshift=2pt] Y.west);
\draw[->, below,dashed] ([yshift=-2pt] Y.west) to node [yshift=-3pt] {$\mcII$} ([yshift=-2pt] YX.east);
\draw[->,above, right] (1) to node [xshift = 5pt,yshift=5pt] {$P ev_{\xv}^{-1} \sim \NN(m(\xv),k(\xv,\xv))$} (Y);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
A Markov transformation as the image of a P valued Functor. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\F}{\mathcal{F}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (x1) at (0,0) {$\F(t_1)$};
\node (x2) at (3,0) {$\F(t_2)$};
\node (x3) at (6,0) {$\F(t_3)$};
\node (dots) at (9,0) {$\ldots$};
\draw[->, above] (x1) to node {$\F_{t_1,t_2}$} (x2);
\draw[->, above] (x2) to node {$\F_{t_2,t_3}$} (x3);
\draw[->, above] (x3) to node {$\F_{t_3,t_4}$} (dots);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
A Markov model as the image of a stochastic process. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\F}{\mathcal{F}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (1) at (4.5, 3) {$1$};
\node (x1) at (0,0) {$\F(t_1)$};
\node (x2) at (3,0) {$\F(t_2)$};
\node (x3) at (6,0) {$\F(t_3)$};
\node (dots) at (9,0) {$\ldots$};
\draw[->, above] (x1) to node {$\F_{t_1,t_2}$} (x2);
\draw[->, above] (x2) to node {$\F_{t_2,t_3}$} (x3);
\draw[->, above] (x3) to node {$\F_{t_3,t_4}$} (dots);
\draw[->,left] (1) to node {$\mu_{t_1}$} (x1);
\draw[->,left] (1) to node {$\mu_{t_2}$} (x2);
\draw[->,left] (1) to node {$\mu_{t_3}$} (x3);
\draw[->,left] (1) to node {$\mu_{t_4}$} (dots);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
Proving the joint distribution _ev_x ev_z_Y^ P = P(ev_x^-1() ev_z^-1())) is a normal distribution N(u, ). | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\xv}{\mathbf{x}}
\newcommand{\zv}{\mathbf{z}}
\newcommand{\NN}{\mathcal{N}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (1) at (0,0) {$1$};
\node (YX) at (3,0) {$Y^X$};
\node (Y1) at (9,2.6) {$Y_{\xv}$};
\node (Y2) at (9,-2.6) {$Y_{\zv}$};
\node (YY) at (6,0) {$Y^{X_0}$};
\node (YxY) at (9,0) {$Y_{\xv} \times Y_{\zv}$};
\draw[->,above] (1) to node {$P$} (YX);
\draw[->,above,above] (YX) to node {$\delta_{Y^{\iota}}$} (YY);
\draw[->,above, left] (YX) to node [xshift=-2pt, yshift=3pt] {$\delta_{ev_{\xv}}$} (Y1);
\draw[->,below, left] (YX) to node {$\delta_{ev_{\zv}}$} (Y2);
\draw[->, right] (YxY) to node {$\delta_{\pi_{Y_{\xv}}}$} (Y1);
\draw[->, right] (YxY) to node {$\delta_{\pi_{Y_{\zv}}}$} (Y2);
\draw[->, right] (YY) to node {$\delta_{ev_{\xv}}$} (Y1);
\draw[->, right] (YY) to node {$\delta_{ev_{\zv}}$} (Y2);
\draw[->,above] (YY) to node {$\delta_{ev_{\xv} \times ev_{\zv}}$} (YxY);
\draw[->,out=45,in=180,left] (1) to node [xshift=-5pt,yshift=3pt] {$\NN(m(\xv),k(\xv,\xv))$} (Y1);
\draw[->,out=-45,in=180,left] (1) to node [xshift=-5pt,yshift=-3pt] {$\NN(m(\zv),k(\zv,\zv))$} (Y2);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The restriction of Pi^-1. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\Rn}{\mathbb{R}^{^n}}
\newcommand{\xv}{\mathbf{x}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (1) at (0,2) {$1$};
\node (Rp) at (-3,0) {$\Rn$};
\node (YX) at (0,0) {$Y^X$};
\node (YX0) at (0,-2) {$Y^{X_0}$};
\node (Yprod) at (5,-2) {$\prod_{\xv \in X_0} Y_{\xv}$};
\draw[->,right] (1) to node {$Pi^{-1}$} (YX);
\draw[->,left,above] (1) to node {$P$} (Rp);
\draw[->,above] (Rp) to node {$\delta_{i}$} (YX);
\draw[->,right] (YX) to node {$\delta_{Y^{\iota}}$} (YX0);
\draw[->,left, below] (Rp) to node {$\delta_{Y^{\iota} \circ i}$} (YX0);
\draw[->,above] (YX0) to node {$\delta_{ev_{\xv_1} \times \ldots \times ev_{\xv_{n'}}}$} (Yprod);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
Splitting the Gaussian additive noise Bayesian model (top diagram) into two separate Bayesian models (bottom two diagrams) and composing the inference maps for these two simple Bayesian models gives the inference map for the original Gaussian additive Bayesian model. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\NN}{\mathcal{N}}
\newcommand{\xv}{\mathbf{x}}
\newcommand{\mcSS}{\mathcal{S}^{\mathbf{x}}}
\newcommand{\mcI}{\mathcal{I}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (1) at (3,2) {$1$};
\node (YX) at (0,0) {$Y^X$};
\node (YX2) at (3,0) {$Y^X$};
\node (Y) at (6,0) {$Y$};
\draw[->,above] (1) to node [xshift = -25pt] {$P \sim \NN(m,k)$} (YX);
\draw[->,above] (YX) to node {$N$} (YX2);
\draw[->,right,dashed] (1) to node {$N \circ P$} (YX2);
\draw[->,right,dashed] (1) to node [xshift = 3pt] {$\delta_{\xv} \circ N \circ P \sim \NN(m(\xv),\underbrace{k(\xv,\xv) + k_N(\xv,\xv)}_{=\kappa(\xv,\xv)})$} (Y);
\draw[->,above] (YX2) to node {$\delta_{ev_{\xv}}$} (Y);
\draw[->,below,out=-45,in=225,looseness=.5] (YX) to node {$\mcSS$} (Y);
\node (com) at (6,-2) {$\Downarrow$ Decomposition};
\node (13) at (3,-3) {$1$};
\node (YX3) at (0,-5) {$Y^X$};
\node (YX23) at (3,-5) {$Y^X$};
\draw[->,above] (13) to node [xshift = -25pt] {$P \sim \NN(m,k)$} (YX3);
\draw[->,above] ([yshift=2pt] YX3.east) to node {$N$} ([yshift=2pt] YX23.west);
\draw[->,right,dashed] (13) to node {$N \circ P$} (YX23);
\draw[->,below,dashed] ([yshift=-2pt] YX23.west) to node {$\mcI_*$} ([yshift=-2pt] YX3.east);
\node (14) at (7,-3) {$1$};
\node (YX4) at (7,-5) {$Y^X$};
\node (Y4) at (10,-5) {$Y$};
\draw[->,above] (14) to node [xshift = -25pt] {$N \circ P$} (YX4);
\draw[->,above] ([yshift=2pt] YX4.east) to node {$\delta_{\xv}$} ([yshift=2pt] Y4.west);
\draw[->,right,dashed] (14) to node {$\delta_{\xv} \circ N \circ P$} (Y4);
\draw[->,below,dashed] ([yshift=-2pt] Y4.west) to node {$\mcI_{nf}$} ([yshift=-2pt] YX4.east);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
An arrow in P^T is a natural transformation. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\F}{\mathcal{F}}
\newcommand{\G}{\mathcal{G}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (x1) at (0,0) {$\F(t_1)$};
\node (x2) at (0,-3) {$\F(t_2)$};
\node (y1) at (3,0) {$\G(t_1)$};
\node (y2) at (3,-3) {$\G(t_2)$};
\draw[->, left] (x1) to node {$\F_{t_1,t_2}$} (x2);
\draw[->, above] (x1) to node {$\eta_{t_1}$} (y1);
\draw[->, above] (x2) to node {$\eta_{t_2}$} (y2);
\draw[->, right] (y1) to node {$\G_{t_1,t_2}$} (y2);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The inference map for the parametric model is a composite of two inference maps. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\Rn}{\mathbb{R}^{^n}}
\newcommand{\mcSS}{\mathcal{S}^{\mathbf{x}}}
\newcommand{\mcI}{\mathcal{I}}
\newcommand{\mcII}{\mathcal{I}^{\mathbf{x}}}
\newcommand{\xv}{\mathbf{x}}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\node (1) at (0,2) {$1$};
\node (Rp) at (-3,0) {$\Rn$};
\node (YX) at (0,0) {$Y^X$};
\node (Y) at (3,0) {$Y$};
\node (com) at (6,0) {$\mcSS_{p} = \mcSS_n \circ \delta_{i}$};
\draw[->,right] (1) to node {$Pi^{-1}$} (YX);
\draw[->,left,above] (1) to node {$P$} (Rp);
\draw[->,above] ([yshift=2pt] Rp.east) to node {$\delta_{i}$} ([yshift=2pt] YX.west);
\draw[->,below] ([yshift=-2pt] YX.west) to node {$\mcI_{\star}$} ([yshift=-2pt] Rp.east);
\draw[->,above] ([yshift=2pt] YX.east) to node {$\mcSS_n$} ([yshift=2pt] Y.west);
\draw[->,below] ([yshift=-2pt] Y.west) to node {$\mcII_n$} ([yshift=-2pt] YX.east);
\draw[->,above,right] (1) to node [yshift=4pt] {$\mcSS_n \circ Pi^{-1}$} (Y);
\draw[->,out=225,in=-45,looseness=.5,below] (Y) to node {$\mathcal{I}^{\xv}_{p}$} (Rp);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
The hidden Markov model viewed in P. | \documentclass[12pt]{article}
\usepackage{amsfonts, amssymb, amsmath, amsthm}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usepackage{color,hyperref}
\newcommand{\F}{\mathcal{F}}
\newcommand{\mcS}{\mathcal{S}}
\newcommand{\mcI}{\mathcal{I}}
\begin{document}
\begin{tikzpicture}
\node (1) at (0,2) {$1$};
\node (X1) at (-2,0) {$\F(t_1)$};
\node (X2) at (2,0) {$\F(t_2)$};
\node (Y1) at (-2,-3) {$Y_{t_1}$};
\node (Y2) at (2,-3) {$Y_{t_2}$};
\draw[->, left,above] (1) to node [xshift=-2pt] {$\mu_{t_1}$} (X1);
\draw[->, above] (X1) to node {$\F_{t_1,t_2}$} (X2);
\draw[->, left] ([xshift=-2pt] X1.south) to node [yshift=8pt] {$\mcS_{t_1}$} ([xshift=-2pt] Y1.north);
\draw[->, right] ([xshift=2pt] Y1.north) to node [yshift=8pt] {$\mcI_{t_1}$} ([xshift=2pt] X1.south);
\draw[->,right,dashed] (1) to node [yshift = -9pt] {$d_{t_1}$} (Y1);
\draw[->,above,dashed,out = -180,in=90,looseness=1] (1) to node [xshift=-25pt] {$\hat{\mu}_{t_1} = \mcI \circ d_{t_1}$} (X1);
\draw[->, right] (1) to node {$\F_{t_1,t_2} \circ \hat{\mu}_{t_1}$} (X2);
\draw[->, left] ([xshift=-2pt] X2.south) to node [yshift=8pt] {$\mcS_{t_2}$} ([xshift=-2pt] Y2.north);
\draw[->, right] ([xshift=2pt] Y2.north) to node [yshift=8pt] {$\mcI_{t_2}$} ([xshift=2pt] X2.south);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1312.1445 | arxiv | 2013-12-06T02:03:56 |
|
Five unit squares are completely contained in a right triangle of sides 7 and 73, corresponding to a = 7 and b = 3. In other words 5 = V(7,3)(see Definition def:functionW). | \documentclass[reqno,a4paper]{amsart}
\usepackage{mathrsfs,amsmath,amssymb,graphicx}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[scale=0.6]
\draw[very thick] (0,0) -- (7,0) -- (0,2.333) -- (0,0);
\foreach \x in {0,...,3}
\draw[gray,xshift=\x cm, yshift=0 cm] (0,0) -- (1,0) -- (1,1) -- (0,1) -- (0,0) -- (1,1) -- (1,0) -- (0,1);
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\end{document} | https://arxiv.org/abs/1310.8088 | arxiv | 2015-04-22T02:08:25 |
|
The three possible forms for a level-2 Coxeter graph with 2 cycles. | \documentclass[a4paper,10pt,twoside]{amsart}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz}
\usetikzlibrary{arrows,positioning,decorations.text,calc}
\begin{document}
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\draw (-4,1.2) circle (.8);
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\draw (4,0) circle (2);
\draw (2,0)--(6,0);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1310.8608 | arxiv | 2014-09-05T02:03:38 |
|
Simple roots, fundamental weights, and positive roots of depth 6 of a geometric Coxeter system of rank~3 seen in the affine space spanned by the simple roots. The Coxeter graph is shown in the upper-left corner. | \documentclass[a4paper,10pt,twoside]{amsart}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz}
\usetikzlibrary{arrows,positioning,decorations.text,calc}
\begin{document}
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[scale=1.5,
q/.style={black,line join=round,thin},
racine/.style={red},
poid/.style={blue},
racinesimple/.style={black},
racinedih/.style={blue},
sommet/.style={inner sep=2pt,circle,draw=black,fill=blue,thick,anchor=base},
rotate=0]
\def\grosseur{0.0125}
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%% La courbe Q
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(3.3,1.44) --
(3.29,1.48) --
(3.28,1.52) --
(3.27,1.55) --
(3.26,1.58) --
(3.24,1.63) --
(3.22,1.68) --
(3.21,1.7) --
(3.2,1.72) --
(3.19,1.74) --
(3.18,1.76) --
(3.17,1.78) --
(3.14,1.83) --
(3.08,1.92) --
(3.05,1.96) --
(3.01,2.01) --
(2.84,2.18) --
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(2.68,2.3) --
(2.58,2.36) --
(2.56,2.37) --
(2.54,2.38) --
(2.5,2.4) --
(2.48,2.41) --
(2.41,2.44) --
(2.33,2.47) --
(2.3,2.48) --
(2.23,2.5) --
(2.19,2.51) --
(2.15,2.52) --
(2.1,2.53) --
(2.03,2.54) --
(1.94,2.55) --
(1.93,2.55) --
(1.72,2.55) --
(1.71,2.55) --
(1.62,2.54) --
(1.56,2.53) --
(1.39,2.49) --
(1.36,2.48) --
(1.33,2.47) --
(1.28,2.45) --
(1.23,2.43) --
(1.21,2.42) --
(1.19,2.41) --
(1.17,2.4) --
(1.15,2.39) --
(1.13,2.38) --
(1.08,2.35) --
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(0.88,2.2) --
(0.87,2.19) --
(0.86,2.18) --
(0.85,2.17) --
(0.84,2.16) --
(0.83,2.15) --
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(0.7,1.99) --
(0.68,1.96) --
(0.65,1.91) --
(0.59,1.79) --
(0.57,1.74) --
(0.56,1.71) --
(0.55,1.68) --
(0.54,1.65) --
(0.53,1.61) --
(0.51,1.51) --
(0.5,1.43) --
(0.5,1.42) --
(0.5,1.24) --
(0.5,1.23) --
(0.51,1.15) --
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(0.72,0.61) --
(0.77,0.54) --
(0.81,0.49) --
(0.82,0.48) --
(0.99,0.31) --
(1.0,0.3) --
(1.05,0.26) --
(1.12,0.21) --
(1.15,0.19) --
(1.2,0.16) --
(1.25,0.13) --
(1.27,0.12) --
(1.29,0.11) --
(1.31,0.1) --
(1.33,0.09) --
(1.35,0.08) --
(1.42,0.05) --
(1.6,-0.01) --
(1.64,-0.02) --
(1.69,-0.03) --
(1.74,-0.04) --
(1.8,-0.05) --
(1.81,-0.05) --
(1.9,-0.06) --
(1.92,-0.06) --
(2.09,-0.06) --
(2.1,-0.06) --
(2.11,-0.06) --
(2.2,-0.05) --
(2.27,-0.04) --
(2.32,-0.03) --
(2.36,-0.02) --
(2.4,-0.01) --
(2.52,0.03) --
(2.57,0.05) --
(2.73,0.13) --
(2.78,0.16) --
(2.81,0.18) --
(2.85,0.21) --
(2.89,0.24) --
(2.9,0.25) --
(3.06,0.41) --
(3.07,0.42) --
(3.1,0.46) --
(3.16,0.55) --
(3.19,0.6) --
(3.2,0.62) --
(3.21,0.64) --
(3.22,0.66) --
(3.23,0.68) --
(3.25,0.73) --
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(3.3,0.89) --
(3.31,0.93) --
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(3.33,1.08) --
(3.33,1.09) --
(3.33,1.23) --
(3.33,1.24) --
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cycle;
% Roots of deepness= 1
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\fill[racine] (3.28016235456987,0.811177913876863) circle (\grosseur);
\fill[racine] (3.28596333420256,0.830351548449951) circle (\grosseur);
\fill[racine] (3.28741092636580,0.831969509046017) circle (\grosseur);
\fill[racine] (3.29082086766775,0.846406866255772) circle (\grosseur);
\fill[racine] (3.29194088642621,0.851148915888199) circle (\grosseur);
\fill[racine] (3.31069979214095,0.930572283824989) circle (\grosseur);
\fill[racine] (3.31218697829716,0.936868884227573) circle (\grosseur);
\fill[racine] (3.31555986427533,0.957119690076840) circle (\grosseur);
\fill[racine] (3.31707317073171,0.957556544021818) circle (\grosseur);
\fill[racine] (3.31877200287694,0.976405368751778) circle (\grosseur);
\fill[racine] (3.32023121387283,0.985166470894668) circle (\grosseur);
\fill[racine] (3.32056194125160,0.988057931946486) circle (\grosseur);
\fill[racine] (3.32953249714937,1.06648510386225) circle (\grosseur);
\fill[racine] (3.33014354066986,1.07182729399797) circle (\grosseur);
\fill[racine] (3.33084592329806,1.08442547350144) circle (\grosseur);
\fill[racine] (3.33226810192583,1.10993416480320) circle (\grosseur);
\fill[racine] (3.33365901319003,1.15413644432044) circle (\grosseursimple);
\fill[racine] (3.33463796477495,1.15244085043722) circle (\grosseursimple);
\fill[racine] (3.33858267716535,1.14560840815579) circle (\grosseursimple);
\fill[racine] (3.35483870967742,1.11745213391540) circle (\grosseursimple);
\fill[racine] (3.42857142857143,0.989743318610787) circle (\grosseursimple);
\node[label=right :{$\omega_s$}] at (3.33806146572104, 1.88355406969665) {};
\node[label=left :{$\omega_t$}] at (0.172972972972973, 2.17208533705935) {};
\node[label=below :{$\omega_r$}] at (2.00898472596586, -0.130720815665576) {};
\fill[poid] (3.33806146572104, 1.88355406969665) circle (\grosseursimple);
\fill[poid] (0.172972972972973, 2.17208533705935) circle (\grosseursimple);
\fill[poid] (2.00898472596586, -0.130720815665576) circle (\grosseursimple);
\draw[thin,blue] (3.33806146572104, 1.88355406969665) -- (3.33333333333333, 1.15470053837925);
\draw[thin,blue] (3.33806146572104, 1.88355406969665) -- (2.66666666666667, 2.30940107675850);
\draw[thin,blue] (0.172972972972973, 2.17208533705935) -- (0.552786404500042, 0.957454138327394);
\draw[thin,blue] (0.172972972972973, 2.17208533705935) -- (1.44721359549996, 2.50664747681036);
\draw[thin,blue] (2.00898472596586, -0.130720815665576) -- (2.43643578047198, 0.000000000000000);
\draw[thin,blue] (2.00898472596586, -0.130720815665576) -- (1.56356421952802, 0.000000000000000);
\coordinate (ancre) at (-1.5,2.6);
\node[sommet,label=below left:$s$] (alpha) at (ancre) {};
\node[sommet,label=below right :$t$] (beta) at ($(ancre)+(0.5,0)$) {} edge[thick,dotted] node[auto] {$-1.1$} (alpha);
\node[sommet,label=above:$r$] (gamma) at ($(ancre)+(0.25,0.43)$) {} edge[thick,dotted] node[auto,swap] {$-1.5$} (alpha) edge[thick,dotted] node[auto] {$-1.25$} (beta);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1310.8608 | arxiv | 2014-09-05T02:03:38 |
|
Configuración inicial del sistema. | \documentclass[a4paper,twocolumn,12pt]{article}
\usepackage[latin1]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{tikz,fp,ifthen,fullpage}
\usetikzlibrary{arrows,snakes,backgrounds}
\usepackage{pgfplots}
\usetikzlibrary{shapes,trees}
\usetikzlibrary{calc,through,backgrounds,decorations}
\usetikzlibrary{decorations.shapes,decorations.text,decorations.pathmorphing,backgrounds,fit,calc,through,decorations.fractals}
\usetikzlibrary{fadings,intersections}
\usetikzlibrary{patterns}
\usetikzlibrary{mindmap}
\begin{document}
\begin{tikzpicture}[xscale=1.0,yscale=1.0]
%----------------------------------------------------------
%----------- tren A ----------------
\fill[gray!15](0,0) rectangle (1,0.6);
\draw[color=black] (0,0) rectangle (1,0.6);
\node[] at (0.5,0.3) {\small{$A$}};
%--------------tren B ---------------
\fill[gray!15] (6,0) rectangle (7,0.6);
\draw[color=black] (6,0) rectangle (7,0.6);
\node[] at (6.5,0.3) {\small{$B$}};
%----------------eje x -----------------
\draw[-latex, black] (0cm,0cm)--(8cm,0cm);%--eje
\draw[black] (1cm,-0.2cm)--(1cm,0.2cm);%-origen eje
\node[] at (1.0,-0.4) {\small{$0$}};
\node[] at (8.2,0.0) {$x$};
%----- mosca ------------------------------
\fill[black] (1.05cm,0.5cm) circle(0.05cm); %---mosca
%------- distancia entre trenes ------------------------
\draw[-latex, black] (3.3cm,-0.6cm)--(1.0cm,-0.6cm);
\draw[-latex, black] (4.0cm,-0.6cm)--(6.0cm,-0.6cm);
\node[] at (3.7,-0.6) {$L_{o}$};
%----------------------------------------------------------
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1208.1772 | arxiv | 2012-11-30T02:00:55 |
|
Posición de la mosca y los dos trenes justo cuando la mosca toca el tren B por primera vez. | \documentclass[a4paper,twocolumn,12pt]{article}
\usepackage[latin1]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{tikz,fp,ifthen,fullpage}
\usetikzlibrary{arrows,snakes,backgrounds}
\usepackage{pgfplots}
\usetikzlibrary{shapes,trees}
\usetikzlibrary{calc,through,backgrounds,decorations}
\usetikzlibrary{decorations.shapes,decorations.text,decorations.pathmorphing,backgrounds,fit,calc,through,decorations.fractals}
\usetikzlibrary{fadings,intersections}
\usetikzlibrary{patterns}
\usetikzlibrary{mindmap}
\begin{document}
\begin{tikzpicture}[xscale=1.0,yscale=1.0]
%----------------------------------------------------------
%----------- tren A ----------------
\draw[dashed,color=black] (0,0) rectangle (1,0.6);
%----------- tren A ya movido ----------------
\fill[gray!15](2,0) rectangle (3,0.6);
\draw[color=black] (2,0) rectangle (3,0.6);
\node[] at (2.5,0.3) {\small{$A$}};
%--------------tren B ---------------
\draw[dashed, color=black] (6,0) rectangle (7,0.6);
%--------------tren B ya movido ---------------
\fill[gray!15] (4.5,0) rectangle (5.5,0.6);
\draw[color=black] (4.5,0) rectangle (5.5,0.6);
\node[] at (5.0,0.3) {\small{$B$}};
%----------------eje x -----------------
\draw[-latex, black] (0cm,0cm)--(8cm,0cm);%--eje
\draw[black] (1cm,-0.2cm)--(1cm,0.2cm);%-origen eje
\node[] at (1.0,-0.4) {\small{$0$}};
\node[] at (8.2,0.0) {$x$};
%----- mosca ------------------------------
\fill[black] (4.45cm,0.5cm) circle(0.05cm); %---mosca
%------- distancia origen a tren A movido ------------------------
\draw[-latex, black] (1.7cm,-0.6cm)--(1.0cm,-0.6cm);
\draw[-latex, black] (2.3cm,-0.6cm)--(3.0cm,-0.6cm);
\node[] at (2.05,-0.6) {$d_{A1}$};
\draw[dashed, gray] (3cm,-0.7cm)--(3cm,0.0cm);%--linea limite
%------- distancia tren A movido a tren B movido ------------------------
\draw[-latex, black] (3.5cm,-0.6cm)--(3.0cm,-0.6cm);
\draw[-latex, black] (4.0cm,-0.6cm)--(4.5cm,-0.6cm);
\node[] at (3.7,-0.6) {$L_{1}$};
\draw[dashed, gray] (4.5cm,-0.7cm)--(4.5cm,0.0cm);%--linea limite
%------- distancia tren B movido a tren B ------------------------
\draw[-latex, black] (4.9cm,-0.6cm)--(4.5cm,-0.6cm);
\draw[-latex, black] (5.5cm,-0.6cm)--(6.0cm,-0.6cm);
\node[] at (5.2,-0.6) {$d_{B1}$};
\draw[dashed, gray] (6cm,-0.7cm)--(6cm,0.0cm);%--linea limite
%--
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1208.1772 | arxiv | 2012-11-30T02:00:55 |
|
Configuración inicial después de que la mosca regresa al tren A por primera vez. | \documentclass[a4paper,twocolumn,12pt]{article}
\usepackage[latin1]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{tikz,fp,ifthen,fullpage}
\usetikzlibrary{arrows,snakes,backgrounds}
\usepackage{pgfplots}
\usetikzlibrary{shapes,trees}
\usetikzlibrary{calc,through,backgrounds,decorations}
\usetikzlibrary{decorations.shapes,decorations.text,decorations.pathmorphing,backgrounds,fit,calc,through,decorations.fractals}
\usetikzlibrary{fadings,intersections}
\usetikzlibrary{patterns}
\usetikzlibrary{mindmap}
\begin{document}
\begin{tikzpicture}[xscale=1.0,yscale=1.0]
%----------------------------------------------------------
%----------- tren A ----------------
%----------- tren A ya movido ----------------
\fill[gray!15](2,0) rectangle (3,0.6);
\draw[color=black] (2,0) rectangle (3,0.6);
\node[] at (2.5,0.3) {\small{$A$}};
%--------------tren B ya movido ---------------
\fill[gray!15] (4.5,0) rectangle (5.5,0.6);
\draw[color=black] (4.5,0) rectangle (5.5,0.6);
\node[] at (5.0,0.3) {\small{$B$}};
%----------------eje x -----------------
\draw[-latex, black] (1.5cm,0cm)--(6.5cm,0cm);%--eje
\node[] at (3.0,-0.2) {\small{$0$}};
\node[] at (6.7,0.0) {$x$};
%----- mosca ------------------------------
\fill[black] (3.07cm,0.5cm) circle(0.05cm); %---mosca
\draw[dashed, gray] (3cm,-0.7cm)--(3cm,-0.4cm);%--linea limite Aqui
%------- distancia tren A movido a tren B movido ------------------------
\draw[-latex, black] (3.5cm,-0.6cm)--(3.0cm,-0.6cm);
\draw[-latex, black] (4.0cm,-0.6cm)--(4.5cm,-0.6cm);
\node[] at (3.7,-0.6) {$L_{2}$};
\draw[dashed, gray] (4.5cm,-0.7cm)--(4.5cm,0.0cm);%--linea limite
%--
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1208.1772 | arxiv | 2012-11-30T02:00:55 |
|
Nueva configuración inicial. | \documentclass[a4paper,twocolumn,12pt]{article}
\usepackage[latin1]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{tikz,fp,ifthen,fullpage}
\usetikzlibrary{arrows,snakes,backgrounds}
\usepackage{pgfplots}
\usetikzlibrary{shapes,trees}
\usetikzlibrary{calc,through,backgrounds,decorations}
\usetikzlibrary{decorations.shapes,decorations.text,decorations.pathmorphing,backgrounds,fit,calc,through,decorations.fractals}
\usetikzlibrary{fadings,intersections}
\usetikzlibrary{patterns}
\usetikzlibrary{mindmap}
\begin{document}
\begin{tikzpicture}[xscale=1.0,yscale=1.0]
%----------------------------------------------------------
%----------- tren A ----------------
%----------- tren A ya movido ----------------
\fill[gray!15](2,0) rectangle (3,0.6);
\draw[color=black] (2,0) rectangle (3,0.6);
\node[] at (2.5,0.3) {\small{$A$}};
%--------------tren B ya movido ---------------
\fill[gray!15] (4.5,0) rectangle (5.5,0.6);
\draw[color=black] (4.5,0) rectangle (5.5,0.6);
\node[] at (5.0,0.3) {\small{$B$}};
%----------------eje x -----------------
\draw[-latex, black] (1.5cm,0cm)--(6.5cm,0cm);%--eje
\node[] at (3.0,-0.2) {\small{$0$}};
\node[] at (6.7,0.0) {$x$};
%----- mosca ------------------------------
\fill[black] (4.45cm,0.5cm) circle(0.05cm); %---mosca
\draw[dashed, gray] (3cm,-0.7cm)--(3cm,-0.4cm);%--linea limite Aqui
%------- distancia tren A movido a tren B movido ------------------------
\draw[-latex, black] (3.5cm,-0.6cm)--(3.0cm,-0.6cm);
\draw[-latex, black] (4.0cm,-0.6cm)--(4.5cm,-0.6cm);
\node[] at (3.7,-0.6) {$L_{1}$};
\draw[dashed, gray] (4.5cm,-0.7cm)--(4.5cm,0.0cm);%--linea limite
%--
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1208.1772 | arxiv | 2012-11-30T02:00:55 |
|
Combining two cycles, and rotating a path. | \documentclass[11pt]{article}
\usepackage{amsmath, amsthm, amssymb}
\usepackage{tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}
%%
%% Second
%%
\def\xendtwo{8 cm}
\def\yshifttwo{2.2 cm}
\draw [color=white] (0,0) circle(0.1mm);
\foreach \x in {0.0,0.8,...,8.8} {
\draw [fill=black] (\x cm, \yshifttwo) circle (0.5mm);
}
\foreach \x in {0.0,0.8} {
\draw [-latex] (\x cm,\yshifttwo) -- (\x cm + 0.7cm,\yshifttwo);
}
\draw [-latex, dotted] (1.6 cm,\yshifttwo) -- (2.3cm,\yshifttwo);
\foreach \x in {2.4,3.2,...,4.0} {
\draw [-latex] (\x cm,\yshifttwo) -- (\x cm + 0.7cm,\yshifttwo);
}
\draw [-latex, dotted] (4.8 cm,\yshifttwo) -- (5.5cm,\yshifttwo);
\foreach \x in {5.6,6.4,7.2} {
\draw [-latex] (\x cm,\yshifttwo) -- (\x cm + 0.7cm,\yshifttwo);
}
\draw[-latex] (\xendtwo, \yshifttwo) .. controls (6cm,\yshifttwo + 1.1cm)
and (4.4cm,\yshifttwo + 1.1cm) .. (2.4cm, \yshifttwo + 0.1cm);
\draw[-latex] (1.6cm, \yshifttwo) ..controls(3cm, \yshifttwo + 0.7cm)
and (4.2cm,\yshifttwo + 0.7cm) .. (5.6cm, \yshifttwo + 0.1cm);
\draw (2.4cm, \yshifttwo - 0.4cm) node {$v_{i+1}$};
\draw (1.6cm, \yshifttwo - 0.4cm) node {$v_{i}$};
\draw (5.6cm, \yshifttwo - 0.4cm) node {$v_{j}$};
\draw (4.8cm, \yshifttwo - 0.4cm) node {$v_{j-1}$};
\draw (0.0cm, \yshifttwo - 0.4cm) node {$v_{0}$};
\draw (8.0cm, \yshifttwo - 0.4cm) node {$v_{\ell}$};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1103.5522 | arxiv | 2012-03-30T02:03:52 |
|
A forbidden structure | \documentclass{article}
\usepackage{graphicx,float,color,fancybox,shapepar,setspace,hyperref}
\usepackage{pgf,tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\clip(-0.6,0.62) rectangle (2.2,4.74);
\draw (0.82,3.62)-- (-0.4,2.58);
\draw (0.82,3.62)-- (2,2.56);
\draw (-0.4,2.58)-- (0,1.24);
\draw (0,1.24)-- (1.68,1.24);
\draw (2,2.56)-- (1.68,1.24);
\draw (0.82,3.62)-- (0.82,4.64);
\draw (0.82,3.62)-- (0.82,2.66);
\draw (0.32,3.86) node[anchor=north west] {$v$};
\draw (0.32,3.16) node[anchor=north west] {$e_1$};
\draw (0.3,4.6) node[anchor=north west] {$e_2$};
\draw (-0.3,3.56) node[anchor=north west] {$e_3$};
\draw (1.44,3.48) node[anchor=north west] {$e_4$};
\draw (0.62,1.24) node[anchor=north west] {$C$};
\begin{scriptsize}
\fill [color=black] (0.82,3.62) circle (1.5pt);
\fill [color=black] (-0.4,2.58) circle (1.5pt);
\fill [color=black] (2,2.56) circle (1.5pt);
\fill [color=black] (0,1.24) circle (1.5pt);
\fill [color=black] (1.68,1.24) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1304.6466 | arxiv | 2013-04-25T02:00:40 |
|
Operation (B). | \documentclass{article}
\usepackage{graphicx,float,color,fancybox,shapepar,setspace,hyperref}
\usepackage{pgf,tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\clip(-2.3,2.64) rectangle (4.24,5.24);
\draw (-2,5)-- (-1,4);
\draw (-1,4)-- (0,3);
\draw (-1,4)-- (-2,3);
\draw (-1,4)-- (0,5);
\draw (2.5,4)-- (3.5,3.98);
\draw (2.5,4)-- (2,5);
\draw (2.5,4)-- (2,3);
\draw (3.5,3.98)-- (4,5);
\draw (3.5,3.98)-- (4,3);
\draw (-1.14,5.2) node[anchor=north west] {$f$};
\draw (-1.1,3.4) node[anchor=north west] {$g$};
\draw (2.9,5.14) node[anchor=north west] {$f$};
\draw (2.88,3.44) node[anchor=north west] {$g$};
\draw (-0.86,4.2) node[anchor=north west] {$v$};
\draw (1.88,4.2) node[anchor=north west] {$v_1$};
\draw (3.48,4.2) node[anchor=north west] {$v_2$};
\draw (0.74,4.2) node[anchor=north west] {$\Longrightarrow$};
\begin{scriptsize}
\fill [color=black] (-2,5) circle (1.5pt);
\fill [color=black] (-1,4) circle (1.5pt);
\fill [color=black] (0,3) circle (1.5pt);
\fill [color=black] (-2,3) circle (1.5pt);
\fill [color=black] (0,5) circle (1.5pt);
\fill [color=black] (2.5,4) circle (1.5pt);
\fill [color=black] (3.5,3.98) circle (1.5pt);
\fill [color=black] (2,5) circle (1.5pt);
\fill [color=black] (2,3) circle (1.5pt);
\fill [color=black] (4,5) circle (1.5pt);
\fill [color=black] (4,3) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1304.6466 | arxiv | 2013-04-25T02:00:40 |
|
Operation (C). | \documentclass{article}
\usepackage{graphicx,float,color,fancybox,shapepar,setspace,hyperref}
\usepackage{pgf,tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\clip(-5.24,1.8) rectangle (6.16,4.24);
\draw [dash pattern=on 2pt off 2pt] (-5,4)-- (-5,2);
\draw (-5,4)-- (-4,4);
\draw (-4,4)-- (-3,4);
\draw (-3,4)-- (-3,2);
\draw (-5,2)-- (-4,2);
\draw (-4,2)-- (-3,2);
\draw (-3,4)-- (-2,4);
\draw (-2,4)-- (-1,4);
\draw [dash pattern=on 2pt off 2pt] (-1,4)-- (-1,2);
\draw (-1,2)-- (-2,2);
\draw (-2,2)-- (-3,2);
\draw (2,4)-- (3,4);
\draw [dash pattern=on 2pt off 2pt] (2,4)-- (2,2);
\draw (3,4)-- (4,4);
\draw (4,4)-- (4,2);
\draw (2,2)-- (3,2);
\draw (3,2)-- (4,2);
\draw (4,2)-- (5,2);
\draw (4,4)-- (5,4);
\draw (5,4)-- (6,4);
\draw [dash pattern=on 2pt off 2pt] (6,4)-- (6,2);
\draw (5,2)-- (6,2);
\draw (4,3.38)-- (2,2);
\draw (4,2.6)-- (6,4);
\draw (0.2,3.44) node[anchor=north west] {$\Longrightarrow$};
\begin{scriptsize}
\fill [color=black] (-5,4) circle (1.5pt);
\fill [color=black] (-5,2) circle (1.5pt);
\fill [color=black] (-4,4) circle (1.5pt);
\fill [color=black] (-3,4) circle (1.5pt);
\fill [color=black] (-3,2) circle (1.5pt);
\fill [color=black] (-4,2) circle (1.5pt);
\fill [color=black] (-2,4) circle (1.5pt);
\fill [color=black] (-1,4) circle (1.5pt);
\fill [color=black] (-1,2) circle (1.5pt);
\fill [color=black] (-2,2) circle (1.5pt);
\fill [color=black] (2,4) circle (1.5pt);
\fill [color=black] (3,4) circle (1.5pt);
\fill [color=black] (2,2) circle (1.5pt);
\fill [color=black] (4,4) circle (1.5pt);
\fill [color=black] (4,2) circle (1.5pt);
\fill [color=black] (3,2) circle (1.5pt);
\fill [color=black] (5,2) circle (1.5pt);
\fill [color=black] (5,4) circle (1.5pt);
\fill [color=black] (6,4) circle (1.5pt);
\fill [color=black] (6,2) circle (1.5pt);
\fill [color=black] (4,3.38) circle (1.5pt);
\fill [color=black] (4,2.6) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1304.6466 | arxiv | 2013-04-25T02:00:40 |
|
Re-embedding of G | \documentclass{article}
\usepackage{graphicx,float,color,fancybox,shapepar,setspace,hyperref}
\usepackage{pgf,tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=0.8cm,y=0.8cm]
\clip(-0.86,0.3) rectangle (8.08,4.56);
\draw (0.82,3.62)-- (-0.4,2.58);
\draw (0.82,3.62)-- (2,2.56);
\draw (-0.4,2.58)-- (0,1.24);
\draw (0,1.24)-- (1.68,1.24);
\draw (2,2.56)-- (1.68,1.24);
\draw (0.82,3.62)-- (1.04,2.62);
\draw (0.82,3.62)-- (0.58,2.62);
\draw (0.62,1.14) node[anchor=north west] {$C$};
\draw (0.58,2.62)-- (1.04,2.62);
\draw (0.58,2.62)-- (0.18,2.48);
\draw (0.58,2.62)-- (0.5,2.2);
\draw (1.04,2.62)-- (1.24,2.24);
\draw (1.04,2.62)-- (1.46,2.48);
\draw (-0.4,2.58)-- (-0.72,2.76);
\draw (-0.4,2.58)-- (-0.72,2.76);
\draw (-0.4,2.58)-- (-0.76,2.44);
\draw (0,1.24)-- (-0.38,1.12);
\draw (0,1.24)-- (-0.16,0.84);
\draw (1.68,1.24)-- (2,0.78);
\draw (1.68,1.24)-- (2.26,1.16);
\draw (2,2.56)-- (2.46,2.84);
\draw (2,2.56)-- (2.54,2.46);
\draw (1.84,3.66) node[anchor=north west] {$ f $};
\draw (3.2,2.76) node[anchor=north west] {$ \Longrightarrow $};
\draw (6.2,3.38)-- (4.94,2.3);
\draw (6.2,3.38)-- (7.52,2.4);
\draw (4.94,2.3)-- (5.44,0.96);
\draw (7.52,2.4)-- (7.14,0.96);
\draw (5.44,0.96)-- (7.14,0.96);
\draw (6.2,3.38)-- (5.9,4.04);
\draw (6.2,3.38)-- (6.4,4.04);
\draw (5.9,4.04)-- (6.4,4.04);
\draw (6.4,4.04)-- (6.56,4.46);
\draw (6.4,4.04)-- (6.76,4.12);
\draw (5.9,4.04)-- (5.58,4.4);
\draw (5.9,4.04)-- (5.5,4.08);
\draw (4.94,2.3)-- (4.6,2.56);
\draw (4.94,2.3)-- (4.54,2.16);
\draw (7.52,2.4)-- (8.02,2.62);
\draw (7.52,2.4)-- (8.04,2.24);
\draw (7.14,0.96)-- (7.46,0.52);
\draw (7.14,0.96)-- (7.68,0.88);
\draw (5.44,0.96)-- (5.02,0.64);
\draw (5.44,0.96)-- (5.4,0.42);
\draw (7.26,3.56) node[anchor=north west] {$f$};
\draw (-0.38,3.68) node[anchor=north west] {$f$};
\draw (4.98,3.56) node[anchor=north west] {$f$};
\draw (6.22,1.04) node[anchor=north west] {$C$};
\begin{scriptsize}
\fill [color=black] (0.82,3.62) circle (1.5pt);
\fill [color=black] (-0.4,2.58) circle (1.5pt);
\fill [color=black] (2,2.56) circle (1.5pt);
\fill [color=black] (0,1.24) circle (1.5pt);
\fill [color=black] (1.68,1.24) circle (1.5pt);
\fill [color=black] (1.04,2.62) circle (1.5pt);
\fill [color=black] (0.58,2.62) circle (1.5pt);
\fill [color=black] (6.2,3.38) circle (1.5pt);
\fill [color=black] (4.94,2.3) circle (1.5pt);
\fill [color=black] (7.52,2.4) circle (1.5pt);
\fill [color=black] (5.44,0.96) circle (1.5pt);
\fill [color=black] (7.14,0.96) circle (1.5pt);
\fill [color=black] (5.9,4.04) circle (1.5pt);
\fill [color=black] (6.4,4.04) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1304.6466 | arxiv | 2013-04-25T02:00:40 |
|
Operation (A) | \documentclass{article}
\usepackage{graphicx,float,color,fancybox,shapepar,setspace,hyperref}
\usepackage{pgf,tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\clip(-3.24,0.96) rectangle (7.5,4.06);
\draw (1.68,2.78) node[anchor=north west] {$\Longrightarrow$};
\draw (-1.42,3.54)-- (0.24,3.56);
\draw [dash pattern=on 2pt off 2pt] (-1.42,3.54)-- (-1.92,3.04);
\draw (-1.92,3.04)-- (-2.38,2.52);
\draw (-2.38,2.52)-- (-1.44,1.64);
\draw (-1.44,1.64)-- (0.32,1.64);
\draw (0.24,3.56)-- (1.12,2.62);
\draw (1.12,2.62)-- (0.32,1.64);
\draw (4.16,3.9)-- (5.64,3.9);
\draw (5.64,3.9)-- (6.54,2.82);
\draw (6.54,2.82)-- (6.58,1.96);
\draw (6.58,1.96)-- (5.7,1.32);
\draw (5.7,1.32)-- (4.26,1.3);
\draw (4.26,1.3)-- (3.28,1.86);
\draw (3.28,1.86)-- (3.28,2.72);
\draw (3.28,2.72)-- (3.7,3.3);
\draw [dash pattern=on 2pt off 2pt] (3.7,3.3)-- (4.16,3.9);
\draw (4.94,2.64)-- (3.28,2.72);
\draw (4.94,2.64)-- (3.28,1.86);
\draw (4.94,2.64)-- (6.54,2.82);
\draw (4.94,2.64)-- (6.58,1.96);
\draw [dash pattern=on 2pt off 2pt] (1.12,2.62)-- (1.52,3.06);
\draw (2.7,2.87) node[anchor=north west] {$u_1$};
\draw (2.72,2.35) node[anchor=north west] {$u_2$};
\draw (6.5,3) node[anchor=north west] {$v_1$};
\draw (6.5,2.37) node[anchor=north west] {$v_2$};
\draw (0.6,2.84) node[anchor=north west] {$v$};
\draw (-2.3,2.8) node[anchor=north west] {$u$};
\draw [dash pattern=on 2pt off 2pt] (1.12,2.62)-- (1.5,2.22);
\draw [dash pattern=on 2pt off 2pt] (-2.38,2.52)-- (-2.9,3);
\draw [dash pattern=on 2pt off 2pt] (-2.38,2.52)-- (-2.86,2);
\draw [dash pattern=on 2pt off 2pt] (3.28,2.72)-- (2.9,3.08);
\draw [dash pattern=on 2pt off 2pt] (3.28,1.86)-- (2.9,1.58);
\draw [dash pattern=on 2pt off 2pt] (6.54,2.82)-- (7.08,3.16);
\draw [dash pattern=on 2pt off 2pt] (6.58,1.96)-- (7.06,1.62);
\begin{scriptsize}
\fill [color=black] (-1.42,3.54) circle (1.5pt);
\fill [color=black] (0.24,3.56) circle (1.5pt);
\fill [color=black] (-1.92,3.04) circle (1.5pt);
\fill [color=black] (-2.38,2.52) circle (1.5pt);
\fill [color=black] (-1.44,1.64) circle (1.5pt);
\fill [color=black] (0.32,1.64) circle (1.5pt);
\fill [color=black] (1.12,2.62) circle (1.5pt);
\fill [color=black] (4.16,3.9) circle (1.5pt);
\fill [color=black] (5.64,3.9) circle (1.5pt);
\fill [color=black] (6.54,2.82) circle (1.5pt);
\fill [color=black] (6.58,1.96) circle (1.5pt);
\fill [color=black] (5.7,1.32) circle (1.5pt);
\fill [color=black] (4.26,1.3) circle (1.5pt);
\fill [color=black] (3.28,1.86) circle (1.5pt);
\fill [color=black] (3.28,2.72) circle (1.5pt);
\fill [color=black] (3.7,3.3) circle (1.5pt);
\fill [color=white] (4.94,2.64) circle (1.5pt);
\draw(4.94,2.64) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1304.6466 | arxiv | 2013-04-25T02:00:40 |
|
(G) 5 | \documentclass{article}
\usepackage{graphicx,float,color,fancybox,shapepar,setspace,hyperref}
\usepackage{pgf,tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\clip(-1.38,0.86) rectangle (3.5,4.12);
\draw (-0.94,2.46)-- (-0.22,3.2);
\draw (-0.22,3.2)-- (1.02,3.2);
\draw (1.02,3.2)-- (1.02,1.74);
\draw (1.02,1.74)-- (-0.24,1.74);
\draw (-0.24,1.74)-- (-0.94,2.46);
\draw (1.02,3.2)-- (2.2,3.2);
\draw (2.2,3.2)-- (2.92,2.44);
\draw (2.92,2.44)-- (2.26,1.74);
\draw (2.26,1.74)-- (1.02,1.74);
\draw (1.02,1.74)-- (1.04,0.96);
\draw (1.02,3.2)-- (1.02,4.04);
\draw (0.95,2.32) node[anchor=north west] {$u$};
\draw (0.95,3.2) node[anchor=north west] {$v$};
\draw (-0.38,3.2) node[anchor=north west] {$x_1$};
\draw (1.8,3.2) node[anchor=north west] {$x_2$};
\draw (-0.38,2.32) node[anchor=north west] {$x_3$};
\draw (1.8,2.32) node[anchor=north west] {$x_4$};
\draw (0.28,3.9) node[anchor=north west] {$f_1$};
\draw (1.48,3.9) node[anchor=north west] {$f_2$};
\draw (0.24,1.84) node[anchor=north west] {$f_3$};
\draw (1.44,1.78) node[anchor=north west] {$f_4$};
\draw (-0.22,3.2)-- (-0.34,3.62);
\draw (-0.22,3.2)-- (-0.6,3.36);
\draw (-0.94,2.46)-- (-1.24,2.62);
\draw (-0.94,2.46)-- (-1.24,2.28);
\draw (-0.24,1.74)-- (-0.6,1.54);
\draw (-0.24,1.74)-- (-0.34,1.32);
\draw (2.26,1.74)-- (2.4,1.26);
\draw (2.26,1.74)-- (2.6,1.52);
\draw (2.92,2.44)-- (3.34,2.12);
\draw (2.92,2.44)-- (3.38,2.54);
\draw (2.2,3.2)-- (2.72,3.28);
\draw (2.2,3.2)-- (2.48,3.7);
\begin{scriptsize}
\fill [color=black] (-0.94,2.46) circle (1.5pt);
\fill [color=black] (-0.22,3.2) circle (1.5pt);
\fill [color=black] (1.02,3.2) circle (1.5pt);
\fill [color=black] (1.02,1.74) circle (1.5pt);
\fill [color=black] (-0.24,1.74) circle (1.5pt);
\fill [color=black] (2.2,3.2) circle (1.5pt);
\fill [color=black] (2.92,2.44) circle (1.5pt);
\fill [color=black] (2.26,1.74) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1304.6466 | arxiv | 2013-04-25T02:00:40 |
|
Another re-embedding of G | \documentclass{article}
\usepackage{graphicx,float,color,fancybox,shapepar,setspace,hyperref}
\usepackage{pgf,tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\clip(-0.86,0.3) rectangle (8.16,4.56);
\draw (0.9,3.6)-- (-0.4,2.58);
\draw (0.9,3.6)-- (2.26,2.56);
\draw (-0.4,2.58)-- (-0.04,0.96);
\draw (-0.04,0.96)-- (1.76,0.94);
\draw (2.26,2.56)-- (1.76,0.94);
\draw (0.9,3.6)-- (1.34,2.96);
\draw (0.9,3.6)-- (1.04,2.8);
\draw (0.62,0.94) node[anchor=north west] {$C$};
\draw (1.04,2.8)-- (1.34,2.96);
\draw (1.04,2.8)-- (0.96,2.44);
\draw (1.04,2.8)-- (1.14,2.46);
\draw (1.34,2.96)-- (1.42,2.5);
\draw (1.34,2.96)-- (1.62,2.62);
\draw (-0.4,2.58)-- (0.52,2.06);
\draw (-0.4,2.58)-- (0.52,2.06);
\draw (-0.4,2.58)-- (0.28,1.72);
\draw (-0.04,0.96)-- (-0.38,1.12);
\draw (-0.04,0.96)-- (-0.28,0.7);
\draw (1.76,0.94)-- (2.06,0.74);
\draw (1.76,0.94)-- (2.12,1.08);
\draw (2.26,2.56)-- (2.68,2.76);
\draw (2.26,2.56)-- (2.68,2.42);
\draw (1.92,3.82) node[anchor=north west] {$ f $};
\draw (3.26,1.98) node[anchor=north west] {$ \Longrightarrow $};
\draw (6.2,3.38)-- (5.02,2.38);
\draw (6.2,3.38)-- (7.52,2.4);
\draw (5.02,2.38)-- (5.44,0.96);
\draw (7.52,2.4)-- (7.14,0.96);
\draw (5.44,0.96)-- (7.14,0.96);
\draw (6.2,3.38)-- (5.9,4.04);
\draw (6.2,3.38)-- (6.4,4.04);
\draw (5.9,4.04)-- (6.4,4.04);
\draw (6.4,4.04)-- (6.56,4.46);
\draw (6.4,4.04)-- (6.76,4.12);
\draw (5.9,4.04)-- (5.58,4.4);
\draw (5.9,4.04)-- (5.5,4.08);
\draw (5.02,2.38)-- (4.44,3.1);
\draw (5.02,2.38)-- (4.18,2.74);
\draw (7.52,2.4)-- (8.02,2.62);
\draw (7.52,2.4)-- (8.04,2.24);
\draw (7.14,0.96)-- (7.46,0.52);
\draw (7.14,0.96)-- (7.68,0.88);
\draw (5.44,0.96)-- (5.02,0.64);
\draw (5.44,0.96)-- (5.4,0.42);
\draw (7.26,3.56) node[anchor=north west] {$f$};
\draw (-0.38,3.68) node[anchor=north west] {$f$};
\draw (4.98,3.56) node[anchor=north west] {$f$};
\draw (6,0.94) node[anchor=north west] {$C$};
\draw (0.28,1.72)-- (0.52,2.06);
\draw (0.52,2.06)-- (0.84,1.84);
\draw (0.52,2.06)-- (0.9,2.12);
\draw (0.28,1.72)-- (0.46,1.38);
\draw (0.28,1.72)-- (0.64,1.58);
\draw (4.22,2.72)-- (4.44,3.1);
\draw (4.44,3.1)-- (4.4,3.48);
\draw (4.44,3.1)-- (4.7,3.38);
\draw (4.22,2.72)-- (3.88,2.9);
\draw (4.22,2.72)-- (3.9,2.56);
\draw (-0.92,2.22) node[anchor=north west] {$f$};
\draw (4.72,2.08) node[anchor=north west] {$f$};
\draw (5.22,2.52) node[anchor=north west] {$u$};
\draw (6,3.16) node[anchor=north west] {$v$};
\draw (-0.94,2.85) node[anchor=north west] {$u$};
\draw (0.8,4.06) node[anchor=north west] {$v$};
\begin{scriptsize}
\fill [color=black] (0.9,3.6) circle (1.5pt);
\fill [color=black] (-0.4,2.58) circle (1.5pt);
\fill [color=black] (2.26,2.56) circle (1.5pt);
\fill [color=black] (-0.04,0.96) circle (1.5pt);
\fill [color=black] (1.76,0.94) circle (1.5pt);
\fill [color=black] (1.34,2.96) circle (1.5pt);
\fill [color=black] (1.04,2.8) circle (1.5pt);
\fill [color=black] (0.28,1.72) circle (1.5pt);
\fill [color=black] (6.2,3.38) circle (1.5pt);
\fill [color=black] (5.02,2.38) circle (1.5pt);
\fill [color=black] (7.52,2.4) circle (1.5pt);
\fill [color=black] (5.44,0.96) circle (1.5pt);
\fill [color=black] (7.14,0.96) circle (1.5pt);
\fill [color=black] (5.9,4.04) circle (1.5pt);
\fill [color=black] (6.4,4.04) circle (1.5pt);
\fill [color=black] (4.44,3.1) circle (1.5pt);
\fill [color=black] (4.18,2.74) circle (1.5pt);
\fill [color=black] (0.52,2.06) circle (1.5pt);
\fill [color=black] (0.28,1.72) circle (1.5pt);
\fill [color=black] (4.44,3.1) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1304.6466 | arxiv | 2013-04-25T02:00:40 |
|
An forbidden structure of G. | \documentclass{article}
\usepackage{graphicx,float,color,fancybox,shapepar,setspace,hyperref}
\usepackage{pgf,tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
\clip(-1.06,0.18) rectangle (2.88,4.14);
\draw (0.9,3.6)-- (-0.4,2.58);
\draw (0.9,3.6)-- (2.26,2.56);
\draw (-0.4,2.58)-- (-0.04,0.96);
\draw (-0.04,0.96)-- (1.76,0.94);
\draw (2.26,2.56)-- (1.76,0.94);
\draw (0.9,3.6)-- (1.02,2.82);
\draw (0.9,3.6)-- (0.6,2.86);
\draw (0.68,0.94) node[anchor=north west] {$C$};
\draw (0.6,2.86)-- (1.02,2.82);
\draw (1.76,0.94)-- (2.06,0.74);
\draw (1.76,0.94)-- (2.12,1.08);
\draw (2.26,2.56)-- (2.68,2.76);
\draw (2.26,2.56)-- (2.68,2.42);
\draw (1.62,2.36) node[anchor=north west] {$ f $};
\draw (-0.4,0.9) node[anchor=north west] {$u$};
\draw (0.75,4.06) node[anchor=north west] {$v$};
\draw (-0.04,0.96)-- (0.22,1.72);
\draw (-0.03,0.99)-- (0.56,1.44);
\draw (0.22,1.72)-- (0.56,1.44);
\draw (-0.4,2.58)-- (-0.64,2.92);
\draw (-0.4,2.58)-- (-0.74,2.58);
\draw (1.02,2.82)--(0.56,1.44);
\draw (1.5,0.9) node[anchor=north west] {$u_1$};
\draw (2.16,3.32) node[anchor=north west] {$u_2$};
\draw (1.06,2.94) node[anchor=north west] {$u_3$};
\draw (0.54,1.7) node[anchor=north west] {$u_4$};
\draw (-0.22,2.68) node[anchor=north west] {$g$};
\draw (0.04,2.2) node[anchor=north west] {$u_5$};
\draw (0.3,2.86) node[anchor=north west] {$u_6$};
\draw (-0.86,2.52) node[anchor=north west] {$w$};
\begin{scriptsize}
\fill [color=black] (0.9,3.6) circle (1.5pt);
\fill [color=black] (-0.4,2.58) circle (1.5pt);
\fill [color=black] (2.26,2.56) circle (1.5pt);
\fill [color=black] (-0.04,0.96) circle (1.5pt);
\fill [color=black] (1.76,0.94) circle (1.5pt);
\fill [color=black] (1.02,2.82) circle (1.5pt);
\fill [color=black] (0.6,2.86) circle (1.5pt);
\fill [color=black] (0.22,1.72) circle (1.5pt);
\fill [color=black] (-0.03,0.99) circle (1.5pt);
\fill [color=black] (0.56,1.44) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1304.6466 | arxiv | 2013-04-25T02:00:40 |
|
The square with whiskers. | \documentclass[10pt,a4paper]{article}
\usepackage{amsmath,amsxtra,amssymb,latexsym,amscd,amsfonts,multicol,enumerate,ifthen,indentfirst,amsthm,amstext}
\usepackage{multicol,color}
\usepackage{tikz}
\usepackage[T1]{fontenc}
\begin{document}
\begin{tikzpicture}
\draw (0,0) rectangle (2,2) ;
\draw (2,1) -- (3,1);
\draw (0,1) -- (-1,1);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1306.3397 | arxiv | 2015-02-11T02:17:34 |
|
The angle, an example of non-local convexity. | \documentclass[10pt,a4paper]{article}
\usepackage{amsmath,amsxtra,amssymb,latexsym,amscd,amsfonts,multicol,enumerate,ifthen,indentfirst,amsthm,amstext}
\usepackage{multicol,color}
\usepackage{tikz}
\usepackage[T1]{fontenc}
\begin{document}
\begin{tikzpicture}
\draw[thick] (0,0) -- (3,0) node[right]{$S_1$};
\draw[thick] (0,0) -- (2,2) node[right]{$S_2$};
\draw (-0.3,0) arc (180:45:0.3);
\draw (-0.2,0.4) node{$\beta$};
\draw[dashed] (-2,0)--(0,0);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1306.3397 | arxiv | 2015-02-11T02:17:34 |
|
Two tangent edges. | \documentclass[10pt,a4paper]{article}
\usepackage{amsmath,amsxtra,amssymb,latexsym,amscd,amsfonts,multicol,enumerate,ifthen,indentfirst,amsthm,amstext}
\usepackage{multicol,color}
\usepackage{tikz}
\usepackage[T1]{fontenc}
\begin{document}
\begin{tikzpicture}
\draw[thick] (0,0) arc (-90:0:2);
\draw[thick] (0,0)--(3,0);
\draw (0,0.2) arc (-90:0:1.8);
\draw (0,-0.2) arc (-90:0:2.2);
\draw (0,0.2)--(3,0.2);
\draw (0,-0.2)--(3,-0.2);
\draw (0,0.2) arc (90:270:0.2);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1306.3397 | arxiv | 2015-02-11T02:17:34 |
|
Convex, concave binary and concave ternary points, respectively. | \documentclass[10pt,a4paper]{article}
\usepackage{amsmath,amsxtra,amssymb,latexsym,amscd,amsfonts,multicol,enumerate,ifthen,indentfirst,amsthm,amstext}
\usepackage{multicol,color}
\usepackage{tikz}
\usepackage[T1]{fontenc}
\begin{document}
\begin{tikzpicture}
\fill[fill=black!35!white] (-2.5,0)--(-1.5,1)--(-2.5,1)--(-2.5,0);
\draw[thick] (-2.5,0)--(-1.5,1)--(-2.5,1);
\fill[fill=black!35!white] (1,0)--(0,0)--(1,1)--(0.8,1)--(-0.6,-0.2)--(1,-0.2)--(1,0);
\draw[thick] (1,0)--(0,0)--(1,1);
\draw[dashed] (-1,0)--(0,0);
\draw (-0.2,0) arc (180:45:0.2);
\draw (-0.35,0.25) node{$\beta$};
\fill[fill=black!35!white] (4,0)--(4,1)--(3,1)--(3,0)--(4,0);
\draw[thick] (4,0)--(4,1)--(3,1);
\draw[thick] (4,1)--(4.6,1.6);
\draw(4.3,0) node{$E_1$};
\draw(3,1.2) node{$E_2$};
\draw(4.8,1.8) node{$E_3$};
\draw[dashed] (4,1)--(4,1.6);
\draw (4,1.2) arc (90:45:0.2);
\draw (4.2,1.5) node{$\beta_1$};
\draw[dashed] (4,1)--(3,0);
\draw (3.8,1) arc (-180:-135:0.2);
\draw (3.5,0.8) node{$\beta_2$};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1306.3397 | arxiv | 2015-02-11T02:17:34 |
|
Non-convex polygon with concave binary irregular point. | \documentclass[10pt,a4paper]{article}
\usepackage{amsmath,amsxtra,amssymb,latexsym,amscd,amsfonts,multicol,enumerate,ifthen,indentfirst,amsthm,amstext}
\usepackage{multicol,color}
\usepackage{tikz}
\usepackage[T1]{fontenc}
\begin{document}
\begin{tikzpicture}[scale=0.8]
\draw (1.5,0)--(0,3) --(0,0)--(3,0)--(0,1.5);
\draw (0.3,1.8) node{$S_1$};
\draw (0.5,0.5) node{$S_2$};
\draw (1.8,0.3) node{$S_3$};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1306.3397 | arxiv | 2015-02-11T02:17:34 |
|
Decomposition of S at a binary concave point on the exterior boundary. | \documentclass[10pt,a4paper]{article}
\usepackage{amsmath,amsxtra,amssymb,latexsym,amscd,amsfonts,multicol,enumerate,ifthen,indentfirst,amsthm,amstext}
\usepackage{multicol,color}
\usepackage{tikz}
\usepackage[T1]{fontenc}
\begin{document}
\begin{tikzpicture}[scale=1.5]
\draw (0,0)--(2,0) arc(0:90:1) arc(0:90:1)--(0,0);
\draw[dashed, rounded corners=4pt] (1,1)--(0.75,1) arc (0:180:0.25)--(0,1);
\draw[dashed] (1,0)--(1,1);
\draw (0.5,1) circle (0.15cm);
\draw (0.5,1.5) node{$S_1$};
\draw (1.5,0.5) node{$S_3$};
\draw (0.5, 0.5) node{$S_2$};
\draw (1.2,1.2) node{$P$};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1306.3397 | arxiv | 2015-02-11T02:17:34 |
|
Intersection of -neighborhood sets. | \documentclass[10pt,a4paper]{article}
\usepackage{amsmath,amsxtra,amssymb,latexsym,amscd,amsfonts,multicol,enumerate,ifthen,indentfirst,amsthm,amstext}
\usepackage{multicol,color}
\usepackage{tikz}
\usepackage[T1]{fontenc}
\begin{document}
\begin{tikzpicture}
\fill[fill=black!20!white] (2,0)--(0,0)--(-1.5,1.5)--(-1.5,-0.5)--(2,-0.5)--(2,0);
\draw[thick] (2,0)--(0,0)--(-1.5,1.5);
\draw[dashed] (0.5,0) arc (0:180:0.5);
\draw[dashed] (0.5,0) arc (0:-180:0.5);
\draw[dashed] (-1,0.5)--(2,0.5);
\draw[dashed] (-1,-0.5)--(2,-0.5);
\draw[dashed] (0,-0.5)--(0,0.5);
\draw (-1.5,0)--(0,0);
\draw[dashed, rotate=45] (-0.5,0)--(0.5,0);
\draw[dashed, rotate=45] (0.5,-1)--(0.5,2);
\draw[dashed, rotate=45] (-0.5,-1)--(-0.5,2);
\draw (0,0)--(1,-1);
\draw(-1.4,0.3) node{$S_1$};
\draw (-1,-0.3)node{$S_2$};
\draw(1.5,-0.3) node{$S_3$};
\draw (-0.6,0.2) node{$\beta$};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1306.3397 | arxiv | 2015-02-11T02:17:34 |
|
L-shape. | \documentclass[10pt,a4paper]{article}
\usepackage{amsmath,amsxtra,amssymb,latexsym,amscd,amsfonts,multicol,enumerate,ifthen,indentfirst,amsthm,amstext}
\usepackage{multicol,color}
\usepackage{tikz}
\usepackage[T1]{fontenc}
\begin{document}
\begin{tikzpicture}[scale=0.6]
\draw (0,0)--(2,0)--(2,1)--(1,1)--(1,3)--(0,3)--(0,0);
\draw (2,0)--(2.6,0.4)--(2.6,1.4)--(1.6,1.4)--(1.6,3.4)--(0.6,3.4)--(0,3);
\draw (1,1)--(1.6,1.4);
\draw (1,3)--(1.6,3.4);
\draw (2,1)--(2.6,1.4);
\draw[dashed] (0,0)--(0.6,0.4)--(0.6,3.4);
\draw[dashed] (0.6,0.4)--(2.6,0.4);
\draw[dotted] (1,0)--(1.6,0.4)--(1.6,1.4)-- (0.6,1.4)--(0,1)--(1,1)--(1,0);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1306.3397 | arxiv | 2015-02-11T02:17:34 |
|
Example of a non-convex trihedral. | \documentclass[10pt,a4paper]{article}
\usepackage{amsmath,amsxtra,amssymb,latexsym,amscd,amsfonts,multicol,enumerate,ifthen,indentfirst,amsthm,amstext}
\usepackage{multicol,color}
\usepackage{tikz}
\usepackage[T1]{fontenc}
\begin{document}
\begin{tikzpicture}[scale=0.8]
\draw (0,0)--(2,0)--(3,1)--(3,3)--(1,3)--(0,2)--(0,0);
\draw (2,0)--(2,0.4)--(0.6,0.4)--(0.6,2)--(1.4,2.8)--(2.8,2.8)--(2.8,1.2)--(2,0.4);
\draw (2.8,1.2)--(1.4,1.2)--(1.4,2.8);
\draw (0.6,0.4)--(1.4,1.2);
\draw (0,2)--(0.6,2);
\draw (2.8,2.8)--(3,3);
\draw[dashed] (0,0)--(1,1)--(1,3);
\draw[dashed] (1,1)--(3,1);
\draw[dotted] (0.8,0.8)--(0.8,2.8)--(1.4,2.8)--(1.6,3)--(1.6,1);
\draw[dotted] (0,0.4)--(0.6,0.4)--(0.6,0);
\draw[dotted] (0.8,2.8)--(1.4,2.8)--(1.6,3);
\draw[dotted] (1.4,1.2)--(0.8,1.2)--(1,1.4)--(1.6,1.4)--(1.4,1.2)--(1.4,0.8);
\draw[dotted] (0.8,0.8)--(2.8,0.8)--(2.8,1.2)--(3,1.4)--(1.6,1.4);
\draw[dotted] (0.6,0)--(1.6,1);
\draw[dotted] (0,0.4)--(0.8,1.2);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1306.3397 | arxiv | 2015-02-11T02:17:34 |
|
Decomposition at a concave ternary point. | \documentclass[10pt,a4paper]{article}
\usepackage{amsmath,amsxtra,amssymb,latexsym,amscd,amsfonts,multicol,enumerate,ifthen,indentfirst,amsthm,amstext}
\usepackage{multicol,color}
\usepackage{tikz}
\usepackage[T1]{fontenc}
\begin{document}
\begin{tikzpicture}
\draw (0,0) rectangle (2,2) ;
\draw (2,1) -- (3,1);
\draw (1.2,1.2) node{$S_3$};
\draw (2.2,1.2) node{$S_2$};
\draw (3.1,1.2) node{$S_1$};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1306.3397 | arxiv | 2015-02-11T02:17:34 |
|
Decomposition at a concave point on the interior boundary. | \documentclass[10pt,a4paper]{article}
\usepackage{amsmath,amsxtra,amssymb,latexsym,amscd,amsfonts,multicol,enumerate,ifthen,indentfirst,amsthm,amstext}
\usepackage{multicol,color}
\usepackage{tikz}
\usepackage[T1]{fontenc}
\begin{document}
\begin{tikzpicture}
\draw (0,0)--(3,0)--(3,3)--(0,3)--(0,0);
\draw (2,1)--(1,1)--(1,2)--(2,2)--(2,1);
\draw[dashed] (1,0)--(1,1)--(0,1);
\draw[dashed] (1.5,2)--(1.5,3);
\draw[dashed] (2,1.5)--(3,1.5);
\draw (0.5,2.5) node{$S_1$};
\draw (2.5,0.5) node{$S_3$};
\draw (0.5, 0.5) node{$S_2$};
\draw (2.5,2.5) node{$S_4$};
\draw (1.2,1.2) node{$P$};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1306.3397 | arxiv | 2015-02-11T02:17:34 |
|
Example of a dihedral. | \documentclass[10pt,a4paper]{article}
\usepackage{amsmath,amsxtra,amssymb,latexsym,amscd,amsfonts,multicol,enumerate,ifthen,indentfirst,amsthm,amstext}
\usepackage{multicol,color}
\usepackage{tikz}
\usepackage[T1]{fontenc}
\begin{document}
\begin{tikzpicture}
\draw (0,0)--(2,0)--(3,1)--(1,1)--(1,3)--(0,2)--(0,0);
\draw (0,0)--(1,1);
\draw (0.5,0.5)--(0.8,0.5) arc (0:90:0.3)--(0.5,0.5);
\draw (0.7,0.88) node{$\alpha$};
\draw (1.7,0.5) node{$S_1$};
\draw (0.5, 1.7) node{$S_2$};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1306.3397 | arxiv | 2015-02-11T02:17:34 |
|
At an angle point. | \documentclass[10pt,a4paper]{article}
\usepackage{amsmath,amsxtra,amssymb,latexsym,amscd,amsfonts,multicol,enumerate,ifthen,indentfirst,amsthm,amstext}
\usepackage{multicol,color}
\usepackage{tikz}
\usepackage[T1]{fontenc}
\begin{document}
\begin{tikzpicture}
\draw[thick] (0,0) -- (3,0) node[right]{$S_1$};
\draw[thick] (0,0) -- (2,2) node[right]{$S_3$};
\draw (-0.2, 0.1) node{$S_2$};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1306.3397 | arxiv | 2015-02-11T02:17:34 |
|
Trajectories in string theory and point particle physics | \documentclass{article}
\usepackage{tikz}
\usetikzlibrary{decorations.pathmorphing}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{shapes.multipart}
\usetikzlibrary{intersections}
\usetikzlibrary{calc}
\usetikzlibrary{positioning}
\begin{document}
\begin{tikzpicture}[yscale=0.5, every text node part/.style={align=center},node distance=0.6cm]
\draw[->] (-5,-3) -- +(0,6) node[left] {time};
\foreach \x/\wann in {-3cm/1,0.5cm/2,1.5cm/2}
{
\begin{scope}[xshift=\x]
\draw (0,-3) .. controls (0.5,-1) and (-0.5,1) .. (0,3);
\end{scope}
}
\draw (1,3) ellipse (0.5cm and 0.15cm);
\draw[dotted] (1.5,-3) arc (0:180:0.5cm and 0.15cm);
\draw (0.5,-3) arc (180:360:0.5cm and 0.15cm);
\node (p) at (-3,-5) {Particle trajectory};
\node (s) at (1,-5) {Closed string};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1207.0340 | arxiv | 2012-07-03T02:07:42 |
|
The holonomy from the north pole to Africa, Indonesia and back to the north pole is a rotation by a right angle | \documentclass{article}
\usepackage{tikz}
\usetikzlibrary{decorations.pathmorphing}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{shapes.multipart}
\usetikzlibrary{intersections}
\usetikzlibrary{calc}
\usetikzlibrary{positioning}
\begin{document}
\begin{tikzpicture}[vector/.style={->,thick,>=stealth,blue}]
\draw [name path = ell1] (0,0) ellipse (1.7cm and 1.9cm);
\draw [name path = ell2] (0,0) ellipse (0.5cm and 2cm);
\draw [name path = ell3] (0,0) ellipse (2cm and 0.5cm);
\path [name intersections={of=ell1 and ell2}];
\draw[vector] (intersection-2) -- node[above=1pt] {$v$} +(-0.4,-0.2);
\node[rectangle, xslant=1, yslant=-0.01, minimum height=0.8cm, minimum width=1.5cm, draw] at (intersection-2) {};
\draw[vector] (intersection-2) -- +(0.4,-0.2);
\node[blue] at ($(intersection-2)+(0.56,0.2)$) {$v'''$};
\draw[red, very thick, ->] ($(intersection-2)+(-0.2,-0.1)$) arc (220:310:0.3cm and 0.15cm);%
\draw[dashed] (intersection-2) -- (intersection-4);
\path [name intersections={of=ell2 and ell3}];
\draw[vector] (intersection-3) -- node[left=1pt] {$v'$} +(-0.3,-0.3);
\node[rectangle, xslant=-0.01, yslant=-0.05, minimum size=1.5cm, draw] at (intersection-3) {};
\draw[dashed] (intersection-1) -- (intersection-3);
\path [name intersections={of=ell1 and ell3}];
\draw[vector] (intersection-4) -- node[right=-2pt] {$v''$} +(-0.2,-0.4);
\node[rectangle, xslant=0.1, yslant=0.6, minimum height=1.3cm, minimum width=0.7cm, draw] at (intersection-4) {};
\draw[dashed] (intersection-2) -- (intersection-4);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1207.0340 | arxiv | 2012-07-03T02:07:42 |
|
Information Chain in Trust Reasoning. | \documentclass[11pt,a4paper,twosides]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{hhline,colortbl}
\usepackage{tikz}
\usetikzlibrary{arrows,decorations.pathreplacing,shapes,fit,backgrounds,shadows}
\usepackage{xcolor}
\begin{document}
\begin{tikzpicture}[
line/.style={thick,>=stealth'},
oval/.style={shape=rectangle,rounded corners=2pt,very thick,draw=black!50,fill=white,drop shadow},
]
\node[oval] (obs) {observations (\emph{obs})};
\node[oval,below of=obs] (eval) {evaluations (\emph{eval})};
\node[oval,below of=eval] (tb) {trust beliefs (\emph{tb})};
\node[oval,below of=tb] (ti) {trust intentions (\emph{ti})};
\node[oval,below of=ti] (ta) {trust acts (\emph{ta})};
\draw[line,->] (obs) -- (eval);
\draw[line,->] (eval) -- (tb);
\draw[line,->] (tb) -- (ti);
\draw[line,->] (ti) -- (ta);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1207.0405 | arxiv | 2012-07-03T02:09:14 |
|
An edge and a point chain | \documentclass{amsart}
\usepackage{amsmath, amssymb, amsthm}
\usepackage{tikz}
\usetikzlibrary{calc}
\newcommand{\Mbar}{\overline M}
\begin{document}
\begin{tikzpicture}
\fill (-2.5, 0) circle (2pt) node[left] {$0$} node[right] {$+1$}
(-2.5, 1) circle (2pt) node[left] {$\infty$} node[right] {$-1$};
\draw (-2.5, -0) -- (-2.5, 0.5) node[left] {$\mathbb P^1$}
node[right] {$\zeta$}-- (-2.5, 1);
\draw (0, 0) node[left] {$\Mbar_{g(v), \mathbf w(v)}$} \foreach \i
in {1,...,5} {-- ($\i*(0.4, 0) + (-0.2, 0.5)$) node {$d_{e_\i}$}
-- ($\i*(0.4, 0) + (0, 0.5) + 0.5*pow(-1,\i+1)*(0,1)$)
node[label=-90+180*\i:$d_{v_\i}$] {} } -- (2.2,0.5) node
{$d_{e_6}$} -- (2.4,0) node[right] {$\Mbar_{g(w), \mathbf w(w)}$};
\begin{scope}[xshift=6cm]
\draw (0, 1) node[left] {$\Mbar_{g(v), \mathbf w(v)}$} \foreach
\i in {1,...,4} {-- ($\i*(0.4, 0) + (-0.2, 0.5)$) node
{$d_{e_\i}$} -- ($\i*(0.4, 0) + (0, 0.5) -
0.5*pow(-1,\i+1)*(0,1)$) node[label=90+180*\i:$d_{v_\i}$] {} }
-- (1.8,0.5) node {$d_{e_5}$} -- (2,0) node[right] {point};
\end{scope}
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1306.6580 | arxiv | 2013-06-28T02:02:54 |
|
The triangles T and T'. | \documentclass[psamsfonts]{amsart}
\usepackage{amssymb,amsfonts}
\usepackage{pgf}
\usepackage{pgfkeys}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{calc}
\pgfmathsetmacro{\octagonradius}{3}
\pgfmathsetmacro{\octagonbigradius}{\octagonradius/sin(67.5)}
\tikzset{nomorepostaction/.code={\let\tikz@postactions\pgfutil@empty}}
\begin{document}
\begin{tikzpicture}[scale=1.15]
\draw (0,0) -- node[midway, left]{2} (1.5,1.8) -- node[midway, right]{1} (3,0) -- node[midway, below]{3} (0,0);
\draw (.3,0) arc (0:30:5.6mm);
\draw (.5,.22) node{$\frac{a}{b}\pi$};
\draw (2.75,.3) arc (150:180:5.6mm);
\draw (2.46,.24) node{$\frac{a}{b}\pi$};
\draw (5,0) -- node[midway, left]{2} (5,1.8) -- node[midway, right]{1} (6.5,0) -- node[midway, below]{3} (5,0);
\draw (5,.2) -- (5.2,.2) -- (5.2,0);
\draw (6.25,.3) arc (150:180:5.6mm);
\draw (5.96,.24) node{$\frac{a}{b}\pi$};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1306.6702 | arxiv | 2013-07-02T02:02:36 |
|
The surfaces R(T) and R(T') for a base angle of 13. The arrows indicate edge orientations. Parallel edges are identified iff they share the same orientation. Labels are not shown. | \documentclass[psamsfonts]{amsart}
\usepackage{amssymb,amsfonts}
\usepackage{pgf}
\usepackage{pgfkeys}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{calc}
\pgfmathsetmacro{\octagonradius}{3}
\pgfmathsetmacro{\octagonbigradius}{\octagonradius/sin(67.5)}
\tikzset{nomorepostaction/.code={\let\tikz@postactions\pgfutil@empty}}
\begin{document}
\begin{tikzpicture}
[scale=2,midarrow/.style={thick,decoration={markings,mark=at position 0.5 with {\arrow{>}}},postaction={decorate}}]
% edges of R(T)
\draw[midarrow] (1,0) -- (.5,.866);
\draw[midarrow] (-.5,.866) -- (.5,.866);
\draw[midarrow] (-.5,.866) -- (-1,0);
\draw[midarrow] (-.5,-.866) -- (-1,0);
\draw[midarrow] (-.5,-.866) -- (.5,-.866);
\draw[midarrow] (1,0) -- (.5,-.866);
% internal lines in R(T)
\draw (-1,0) -- (1,0);
\draw (-.5,-.866) -- (.5,.866);
\draw (.5,-.866) -- (-.5,.866);
% edges of R(T')
\draw[midarrow] (3.5+.75,.433) -- (3.5+1,0);
\draw[midarrow] (3.5+.75,.433) -- (3.5+.5,.866);
\draw[midarrow] (3.5+0,.866) -- (3.5+.5,.866);
\draw[midarrow] (3.5+0,.866) -- (3.5+-.5,.866);
\draw[midarrow] (3.5+-.75,.433) -- (3.5+-.5,.866);
\draw[midarrow] (3.5+-.75,.433) -- (3.5+-1,0);
\draw[midarrow] (3.5+.75,-.433) -- (3.5+1,0);
\draw[midarrow] (3.5+.75,-.433) -- (3.5+.5,-.866);
\draw[midarrow] (3.5+0,-.866) -- (3.5+.5,-.866);
\draw[midarrow] (3.5+0,-.866) -- (3.5+-.5,-.866);
\draw[midarrow] (3.5+-.75,-.433) -- (3.5+-.5,-.866);
\draw[midarrow] (3.5+-.75,-.433) -- (3.5+-1,0);
% internal lines in R(T')
\draw (3.5+-1,0) -- (3.5+1,0);
\draw (3.5+-.5,-.866) -- (3.5+.5,.866);
\draw (3.5+.5,-.866) -- (3.5+-.5,.866);
\draw (3.5+0,-.866) -- (3.5+0,.866);
\draw (3.5+.75,-.433) -- (3.5+-.75,.433);
\draw (3.5+-.75,-.433) -- (3.5+.75,.433);
% pi
\draw[->] (1.25,0) to node[above] {$\pi$} (2.25,0);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1306.6702 | arxiv | 2013-07-02T02:02:36 |
|
A simple sensor network comprised of 4 nodes and 3 links in 2 dimensions. Nodes 1 to 3 are sensors; node 4 is an anchor (N_S=3, N_A=1, K_S=2, K_A=1). Eq.~eq:Xi shows the matrix for this network. | \documentclass[10pt,journal]{IEEEtran}
\usepackage[cmex10]{amsmath}
\usepackage{amssymb,bm,cite,tikz,commath}
\usetikzlibrary{patterns,calc}
\begin{document}
\begin{tikzpicture}[every rectangle node/.style={inner sep = 2pt, black}, every circle node/.style={inner sep = 1.6pt, red}, every path/.style={blue}]
\draw (1,0) node[circle,draw,label=right:1] {} -- (0,0) node[circle,draw,label=left:2] {} -- (0,1) node[circle,draw,label=left:3] {} -- (1,1) node[rectangle,draw,label=right:4] {};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1305.7272 | arxiv | 2014-03-17T01:08:58 |
|
Geometry of one sensor node and three anchor nodes. | \documentclass[10pt,journal]{IEEEtran}
\usepackage[cmex10]{amsmath}
\usepackage{amssymb,bm,cite,tikz,commath}
\usetikzlibrary{patterns,calc}
\begin{document}
\begin{tikzpicture}[scale=1,>=stealth,every rectangle node/.style={thick,black,inner sep=2},every circle node/.style={draw,thick,red,inner sep=1.7}]\small
\node [draw,rectangle] at (220:1.5) (A1) {};
\node [draw,rectangle] at (320:1.4) (A2) {};
\node [draw,rectangle] at (70:2) (A3) {};
\node [circle] at (0,0) (S1) {};
\draw [thick,blue] (A1) -- (S1);
\draw [thick,blue] (A2) -- (S1);
\draw [thick,blue] (A3) -- (S1);
\draw [dashed] (S1) -- (3,0);
\draw [->] (1.5,0) arc[start angle=0,delta angle=70,radius=1.5];
\draw [->] (1,0) arc[start angle=0,delta angle=220,radius=1];
\draw [->] (0.5,0) arc[start angle=0,delta angle=320,radius=0.5];
\node at (27:1.75) {$\theta_1$};
\node at (130:1.3) {$\theta_2$};
\node at (270:0.75) {$\theta_3$};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1305.7272 | arxiv | 2014-03-17T01:08:58 |
|
Homotopy collapse of an edge into an adjacent 2-simplex | \documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[>=latex]
%filling the triangle
\draw[fill, blue!30!white] (0,0) -- (2,0) -- (0,2) -- (0,0);
%drawing the edges
\draw[thick] (0,0) -- (2,0) -- (0,2) -- (0,0);
%drawing the nodes
\draw[fill] circle(0.1) (2,0) circle(0.1) (0,2) circle(0.1);
%showing the paring
\draw[red,->] (1,1) -- (0.4, 0.4);
%label the edge e
\draw (1,1) node[anchor=south west]{$e$};
%arrow in between
\draw[->,very thick] (2,1) -- (4,1);
%drawing the collapsed triangle
%drawing the edges
\draw[thick,xshift=4.1cm] (0,0) -- (2,0); \draw[thick,xshift=4.1cm] (0,2) -- (0,0);
%drawing the nodes
\draw[fill,xshift=4.1cm] circle(0.1) (2,0) circle(0.1) (0,2) circle(0.1);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1302.3982 | arxiv | 2013-02-19T02:01:29 |
|
%(Top:) An isolated ``domino'' subgraph of a square lattice. Dashed lines indicate missing edges incident to the subgraph. A domino subgraph in a 3D lattice may also occur with the two cycles meeting at a right angle. (Bottom:) Illustration of the three independent paths between the central qubits of a domino subgraph. If the constraints acting on b do so with different tensor factors , ', '': ^2 and similarly for the constraints , ', '': ^2 acting on e, and the path-constraints are all non-zero, then these form an infeasible system of constraints on the states of b and e. Similar remarks apply for any pair of qubits connected by more than two independent paths. | \documentclass[a4paper,UKenglish]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{times,amsmath,amssymb,amsthm,bm,bbm,mathtools}
\usepackage{tikz}
\usetikzlibrary{calc,shapes}
\begin{document}
\begin{tikzpicture}[scale=0.83,
outer sep=-2pt,
every node/.style={circle,fill=black}
]
\coordinate (d) at (1,0);
\coordinate (e) at (2,0);
\coordinate (f) at (3,0);
\coordinate (a) at (1,1);
\coordinate (b) at (2,1);
\coordinate (c) at (3,1);
\foreach \x in {1,2,3} {
\foreach \y in {0,1} {
\foreach \dx/\dy in {0.75/0,-0.75/0,0/0.75,0/-0.75}
\draw [line width=1.5pt, densely dashed, gray] (\x,\y) -- ($(\x,\y) + (\dx,\dy)$);
}}
\draw [line width=2pt, blue!85!white] (b) -- (e) -- (d) -- (a) -- (b) -- (c) -- (f) -- (e);
\node [label=below left:$a$] at (a) {};
\node [label=below left:$b$] at (b) {};
\node [label=below left:$c$] at (c) {};
\node [label=below left:$d$] at (d) {};
\node [label=below left:$e$] at (e) {};
\node [label=below left:$f$] at (f) {};
% ==========================
\coordinate (b) at (2,-2);
\coordinate (b') at (1.5,-2);
\coordinate (b'') at (2.5,-2);
\coordinate (a) at (1,-2.6);
\coordinate (c) at (3,-2.6);
\coordinate (d) at (1,-3.4);
\coordinate (e) at (2,-4);
\coordinate (f) at (3,-3.4);
\coordinate (e') at (1.5,-4);
\coordinate (e'') at (2.5,-4);
\draw [line width=3pt, blue!85!white] (b) -- (e);
\draw [line width=3pt, blue!85!white] (b') -- (a) -- (d) -- (e');
\draw [line width=3pt, blue!85!white] (b'') -- (c) -- (f) -- (e'');
\node [label=above:$b$] at (b) {};
\node [label=above left:$b$] at (b') {};
\node [label=above right:$b$] at (b'') {};
\node [label=left:$a$] at (a) {};
\node [label=right:$c$] at (c) {};
\node [label=left:$d$] at (d) {};
\node [label=below:$e$] at (e) {};
\node [label=below left:$e$] at (e') {};
\node [label=below right:$e$] at (e'') {};
\node [label=right:$f$] at (f) {};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1403.1588 | arxiv | 2014-07-02T02:11:43 |
|
% Example of a ``figure eight'' graph on 2-1 vertices, for = 8. By Eqn.~eqn:conditionalFrustrBicycleIndptFactors, the probability of such a graph describing a frustrated figure-eight subsystem scales as O(Q_2^2). | \documentclass[a4paper,UKenglish]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{times,amsmath,amssymb,amsthm,bm,bbm,mathtools}
\usepackage{tikz}
\usetikzlibrary{calc,shapes}
\begin{document}
\begin{tikzpicture}[scale=0.78,
every node/.style={circle,fill=black,outer sep=-0.5ex}
]
\def\r{2}
\foreach \o/\j in {-\r/b,\r/a} {
\foreach \l/\d in {0/0,1/45,2/90,3/135,4/180,5/225,6/270,7/315} {
\coordinate (\j\l) at ($(-\o,0) + (\d:\r)$);
}
\foreach \a/\b in {0/1,1/2,2/3,3/4,4/5,5/6,6/7,7/0} {
\draw [line width=2pt, blue!85!white] (\j\a) -- (\j\b);
}
}
\foreach \o/\j in {-\r/b,\r/a} {
\foreach \l/\d in {0/0,1/45,2/90,3/135,4/180,5/225,6/270,7/315} {
\node (\j\l) at ($(-\o,0) + (\d:\r)$) {};
}
}
\node at (a0) [label=left:${x_0 = x_\ell\,}$] {};
\node at (a0) [label=right:${\, = x_{2\ell}}$] {};
\node at (a1) [label=225:$x_1$] {};
\node at (a2) [label=south:$\cdots$] {};
\node at (a6) [label=north:$\ldots$] {};
\node at (a7) [label=135:$x_{\ell-1}$] {};
\node at (b3) [label=305:$x_{\ell+1}$] {};
\node at (b2) [label=south:$\cdots$] {};
\node at (b6) [label=north:$\ldots$] {};
\node at (b5) [label=45:$x_{2\ell-1}$] {};
\node at (a4) [outer sep=1.5em, label=right:{\Large$X_1$}] {};
\node at (b0) [outer sep=1.5em, label=left:{\Large$X_2$}] {};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1403.1588 | arxiv | 2014-07-02T02:11:43 |
|
Architecture of Cyber-Physical Systems. | \documentclass[10pt,final,twocolumn]{IEEEtran}
\usepackage{latexsym, amsmath, color, amsfonts, amssymb,graphicx}
\usepackage{amsmath}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[scale=0.7]
\draw[fill=lightgray, very thick, rotate around={60:(3.464,-2)}] (3.464,-2) ellipse (2 and 0.9);
\node[align=center, rotate=60] at (3.464,-2){\textbf{Control}};
\draw[fill=lightgray,very thick, rotate around={-60:(-3.464,-2)}] (-3.464,-2) ellipse (2 and 0.9);
\node[align=center, rotate=-60] at (-3.464,-2){\textbf{Communication}};
\draw[fill=lightgray,very thick] (0,4) ellipse (2 and 0.9);
\node[align=center] at (0,4){\textbf{Computation}};
\draw[ultra thick] (0,2.8)--(-2.425,-1.4)--(2.425,-1.4)--(0,2.8);
\node[align=center,rotate=60] at (-1.732,1){\textbf{Cyber}};
\node[align=center,rotate=-60] at (1.732,1){\textbf{Physical}};
\node[align=center] at (0,-2){\textbf{System}};
\node[align=center] at (0,0){\textbf{Information}};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1506.07955 | arxiv | 2015-06-29T02:03:03 |
|
The purely geometric quad graph not embedded in any lattice. | \documentclass[a4paper]{article}
\usepackage{cite, amssymb}
\usepackage{amsmath}
\usepackage{color}
\usepackage{tikz}
\usetikzlibrary{patterns}
\begin{document}
\begin{tikzpicture}
\node (x1) at (0,0) [circle,fill,label=-135:$x$] {};
\node (x4) at (0,2.5) [circle,fill,label=135:$x_{1}$] {};
\node (x2) at (2.5,0) [circle,fill,label=-45:$x_{2}$] {};
\node (x3) at (2.5,2.5) [circle,fill,label=45:$x_{12}$] {};
\draw [thick] (x2) to node[below] {$\alpha_{1}$} (x1);
\draw [thick] (x4) to node[above] {$\alpha_{1}$} (x3);
\draw [thick] (x3) to node[right] {$\alpha_{2}$} (x2);
\draw [thick] (x1) to node[left] {$\alpha_{2}$} (x4);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1705.00298 | arxiv | 2017-05-02T02:06:07 |
|
Principal growth directions. | \documentclass[a4paper]{article}
\usepackage{cite, amssymb}
\usepackage{amsmath}
\usepackage{color}
\usepackage{tikz}
\usetikzlibrary{patterns}
\begin{document}
\begin{tikzpicture}[scale=0.5]
\draw[style=help lines,dashed] (0,0) grid[step=1cm] (14,14);
\draw[thick] (0,9)--(0,10)--(1,10)--(1,11)--(2,11)--(2,12)--(3,12)--(3,13)--(4,13)--(4,14)--(5,14);
\node[above right] at (3+1/2,11-1/2) {$(-,-)$};
\draw[thick,->] (2+1/2,12-1/2)--(3+1/2,11-1/2);
\draw[thick,red] (0,7)--(0,6)--(1,6)--(1,5)--(2,5)--(2,4)--(3,4)--(3,3)--(4,3)--(4,2)--(5,2)--(5,1)--(6,1)--(6,0)--(7,0);
\node[above,red] at (5-1/2,4+1/2) {$(-,+)$};
\draw[thick,->,red] (4-1/2,3+1/2)--(5-1/2,4+1/2);
\draw[thick,blue] (8,0)--(9,0)--(9,1)--(10,1)--(10,2)--(11,2)--(11,3)--(12,3)--(12,4)--(13,4)--(13,5)--(14,5)--(14,6);
\node[above,blue] at (11-1/2,4+1/2) {$(+,+)$};
\draw[thick,->,blue] (12-1/2,3+1/2)--(11-1/2,4+1/2);
\draw[thick,teal] (14,7)--(14,8)--(13,8)--(13,9)--(12,9)--(12,10)--(11,10)--(11,11)--(10,11)--(10,12)--(9,12)--(9,13)--(8,13)--(8,14)--(7,14);
\node[below,teal] at (11-1/2-1,11-1/2-1) {$(+,-)$};
\draw[thick,->,teal] (11-1/2,11-1/2)--(11-1/2-1,11-1/2-1);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1705.00298 | arxiv | 2017-05-02T02:06:07 |
|
The ``four stripe'' lattice. | \documentclass[a4paper]{article}
\usepackage{cite, amssymb}
\usepackage{amsmath}
\usepackage{color}
\usepackage{tikz}
\usetikzlibrary{patterns}
\begin{document}
\begin{tikzpicture}[scale=2.5]
\draw [pattern=north west lines,thick] (0,0) rectangle (1,1);
\draw [pattern=north east lines,thick] (1,1) rectangle (2,2);
\draw [pattern=horizontal lines,thick] (1,0) rectangle (2,1);
\draw [pattern=vertical lines,thick] (0,1) rectangle (1,2);
\foreach \x in {0,...,2}{% Two indices running over each
\foreach \y in {0,...,2}{% node on the grid we have drawn
\node[draw,circle,inner sep=2pt,fill] at (\x,\y) {};
}
}
\node[below left] at (0,0) {$x$};
\node[below] at (1,0) {$x_1$};
\node[below right] at (2,0) {$x$};
\node[left] at (0,1) {$x_2$};
\node[below left] at (1,1) {$x_{12}$};
\node[right] at (2,1) {$x_2$};
\node[above left] at (0,2) {$x$};
\node[above] at (1,2) {$x_1$};
\node[above right] at (2,2) {$x$};
\node[] at (1/2,1/2) {$Q$};
\node[] at (1/2+1,1/2) {$|Q$};
\node[] at (1/2,1/2+1) {$\underline{Q}$};
\node[] at (1/2+1,1/2+1) {$|\underline{Q}$};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1705.00298 | arxiv | 2017-05-02T02:06:07 |
|
Defining relation of (A_n). | \documentclass[11pt,reqno]{amsart}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{arrows,shapes,positioning}
\begin{document}
\begin{tikzpicture}[scale=.65]
\draw (2,0) to [bend right=90] (4,0);
\draw (2,0) to [bend left=90] (4,0);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1509.01241 | arxiv | 2016-09-07T02:01:15 |
|
Equations on a Cube | \documentclass[a4paper]{article}
\usepackage{cite, amssymb}
\usepackage{amsmath}
\usepackage{color}
\usepackage{tikz}
\usetikzlibrary{patterns}
\begin{document}
\begin{tikzpicture}[auto,scale=0.8]
\node (x) at (0,0) [circle,fill,label=-135:$x$] {};
\node (x1) at (4,0) [circle,fill,label=-45:$x_{1}$] {};
\node (x2) at (1.5,1.5) [circle,fill,label=-45:$x_{2}$] {};
\node (x3) at (0,4) [circle,fill,label=135:$x_{3}$] {};
\node (x12) at (5.5,1.5) [circle,fill,label=-45:$x_{12}$] {};
\node (x13) at (4,4) [circle,fill,label=-45:$x_{13}$] {};
\node (x23) at (1.5,5.5) [circle,fill,label=135:$x_{23}$] {};
\node (x123) at (5.5,5.5) [circle,fill,label=45:$x_{123}$] {};
\node (A) at (2.75,0.75) {$A$};
\node (Aq) at (2.75,4.75) {$\bar A$};
\node (B) at (0.75,2.75) {$B$};
\node (Bq) at (4.75,2.75) {$\bar B$};
\node (C) at (2,2) {$C$};
\node (Cq) at (3.5,3.5) {$\bar C$};
\draw (x) -- node[below]{$\alpha_{1}$} (x1)
-- node[right] {$\alpha_{2}$} (x12) -- node[right] {$\alpha_{3}$} (x123)
-- node[above] {$\alpha_{1}$} (x23) -- node[left] {$\alpha_{2}$} (x3) -- node[left] {$\alpha_{3}$} (x);
\draw (x3) -- node[above]{$\alpha_{1}$} (x13) -- node[right]{$\alpha_{3}$} (x1);
\draw (x13) -- node[right]{$\alpha_{2}$} (x123);
\draw [dashed] (x) --node[left]{$\alpha_{2}$} (x2)
-- node[above] {$\alpha_{1}$} (x12);
\draw [dashed] (x2) --node[left] {$\alpha_{3}$} (x23);
\draw [dotted,thick] (A) to (Aq);
\draw [dotted,thick] (B) to (Bq);
\draw [dotted,thick] (C) to (Cq);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1705.00298 | arxiv | 2017-05-02T02:06:07 |
|
Interval representation of the graph G_1 in Figure ~fig:ex5. | \documentclass[11pt]{article}
\usepackage{amsmath, amssymb, array}
\usepackage{tikz}
\usetikzlibrary{plotmarks}
\begin{document}
\begin{tikzpicture}
\draw[-][draw=black,very thick] (1,0) -- (4,0);
\draw[-][draw=black,thick] (2,-0.3) -- (5,-0.3);
\draw[-][draw=black,very thick] (3,-.6) -- (7,-.6);
\draw[-][draw=black,thick] (3.5,-.9) -- (8,-.9);
\draw[-][draw=black,very thick] (4.5,-1.2) -- (6,-1.2);
\draw[-][draw=black,thick] (5.8,-1.5) -- (10,-1.5);
\draw[-][draw=black,very thick] (6.5,-1.8) -- (11,-1.8);
\node [left] at (.8,0) {\tiny{$p_{2}$}};
\node [left] at (.8,-.3) {\tiny{$n_{1}$}};
\node [left] at (.8,-.6) {\tiny{$p_{1}$}};
\node [left] at (.8,-.9) {\tiny{$n_{3}$}};
\node [left] at (.8,-1.2) {\tiny{$p_{3}$}};
\node [left] at (.8,-1.5) {\tiny{$n_{2}$}};
\node [left] at (.8,-1.8) {\tiny{$p_{4}$}};
\node at (1,0) {\pgfuseplotmark{square*}};
\node at (4,0) {\pgfuseplotmark{square*}};
\node at (3,-.6) {\pgfuseplotmark{square*}};
\node at (7,-.6) {\pgfuseplotmark{square*}};
\node at (4.5,-1.2) {\pgfuseplotmark{square*}};
\node at (6,-1.2) {\pgfuseplotmark{square*}};
\node at (11,-1.8) {\pgfuseplotmark{square*}};
\node at (6.5,-1.8) {\pgfuseplotmark{square*}};
\node at (1,.3) {\tiny{$1$}};
\node at (1.3,.3) {\tiny{$2$}};
\node at (1.6,.3) {\tiny{$3$}};
\node at (1.9,.3) {\tiny{$4$}};
\node at (2.1,.3) {\tiny{$5$}};
\node at (2.4,.3) {\tiny{$6$}};
\node at (2.7,.3) {\tiny{$7$}};
\node at (3,.3) {\tiny{$8$}};
\node at (3.3,.3) {\tiny{$9$}};
\draw[densely dotted][draw=black,thick] (3.6,0.3) -- (10.8,.3);
\node at (11.1,.3) {\tiny{$32$}};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1607.02922 | arxiv | 2016-07-12T02:13:01 |
|
The graph G_1. | \documentclass[11pt]{article}
\usepackage{amsmath, amssymb, array}
\usepackage{tikz}
\usetikzlibrary{plotmarks}
\begin{document}
\begin{tikzpicture}
\draw[-][draw=black,very thick] (0,0) -- (0,1);
\draw[-][draw=black,very thick] (0,0) -- (1,.5);
\draw[-][draw=black,very thick] (0,0) -- (-1,.5);
\draw[-][draw=black,very thick] (0,0) -- (1,-.5);
\draw[-][draw=black,very thick] (0,0) -- (-1,-.5);
\draw[-][draw=black,very thick] (0,0) -- (0,-1);
\draw[-][draw=black,very thick] (0,1) -- (-1,.5);
\draw[-][draw=black,very thick] (0,1) -- (1,.5);
\draw[-][draw=black,very thick] (-1,.5) -- (-1,-.5);
\draw[-][draw=black,very thick] (-1,-.5) -- (0,-1);
\draw[-][draw=black,very thick] (0,-1) -- (1,-.5);
\draw[-][draw=black,very thick] (1,-.5) -- (1,.5);
\draw[-][draw=black,very thick] (-1,0.5) to [out=90,in=90] (1,-.5);
\draw [fill=black] (0,0) circle [radius=0.09];
\draw [fill=black] (1,0.5) circle [radius=0.09];
\draw [fill=black] (-1,0.5) circle [radius=0.09];
\draw [fill=black] (0,-1) circle [radius=0.09];
\draw [fill=white] (0,1) circle [radius=0.09];
\draw [fill=white] (-1,-.5) circle [radius=0.09];
\draw [fill=white] (1,-.5) circle [radius=0.09];
\node [above] at (0,1) {{$n_{2}$}};
\node [right] at (1,.5) {{$p_{4}$}};
\node [right] at (0,0) {{$p_{1}$}};
\node [right] at (1,-.5) {{$n_{3}$}};
\node [below] at (0,-1) {{$p_{2}$}};
\node [left] at (-1,-.5) {{$n_{1}$}};
\node [left] at (-1,.5) {{$p_{3}$}};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1607.02922 | arxiv | 2016-07-12T02:13:01 |
|
Coxeter graph of type A_n. | \documentclass[11pt,reqno]{amsart}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{arrows,shapes,positioning}
\begin{document}
\begin{tikzpicture}[scale=1,every circle node/.style={draw, circle, fill,inner sep=1pt}, node distance=1.5cm]
\node [circle, label=below:$s_1$] (s1) at (0,0){};
\node [circle, label=below:$s_2$, right of= s1] (s2){};
\node [circle, label=below:$s_3$, right of= s2] (s3){};
\node (dots)[right of=s3, node distance=.75cm]{$\cdots$};
\node [circle, label=below:$s_{n-1}$, right of=s3] (sn-1){};
\node [circle, label=below:$s_{n}$, right of=sn-1] (sn){};
\path[-] (s1) edge (s2);
\path[-] (s2) edge (s3);
\path[-] (sn-1) edge (sn);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1509.01241 | arxiv | 2016-09-07T02:01:15 |
|
Standard k-box. | \documentclass[11pt,reqno]{amsart}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{arrows,shapes,positioning}
\begin{document}
\begin{tikzpicture}[scale=.75]
\draw[gray,thick] (0,0) rectangle (8,2);
\foreach \x in {1,2,7} \filldraw (\x,0) circle (1pt);
\foreach \x in {1,2,7} \filldraw (\x,2) circle (1pt);
\draw (1,2) node[above]{\scriptsize $1$};
\draw (2,2) node[above]{\scriptsize $2$};
\draw (7,2) node[above]{\scriptsize $k$};
\draw (1,0) node[below]{\scriptsize $1'$};
\draw (2,0) node[below]{\scriptsize $2'$};
\draw (7,0) node[below]{\scriptsize $k'$};
\draw (4.25,2) node[above]{$\cdots$};
\draw (4.25,0) node[below]{$\cdots$};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1509.01241 | arxiv | 2016-09-07T02:01:15 |
|
Unique loop-free diagram having only propagating edges. | \documentclass[11pt,reqno]{amsart}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{arrows,shapes,positioning}
\begin{document}
\begin{tikzpicture}[scale=.75]
\draw[gray,thick] (0,0) rectangle (5,2);
\foreach \x in {1,2,4} \filldraw (\x,0) circle (1pt);
\foreach \x in {1,2,4} \filldraw (\x,2) circle (1pt);
\draw (3,1) node{$\cdots$};
\foreach \x in {1,2,4} \draw (\x,0) to (\x,2);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1509.01241 | arxiv | 2016-09-07T02:01:15 |
|
Choosing _c. | \documentclass[a4]{article}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}[>=latex',scale=1.6]
\draw[very thick,->] (0,0) -- (5,0);
\draw[->] (0,0) -- (20:4.95);
\draw[very thick,->] (0,0) -- (40:4.95);
\draw[dashed] (40:5) -- (3.83,0);
\draw[dashed] (3.53,0) -- (3.53,0.3) -- (3.83,0.3);
\draw[<->] (1.5,0) arc[radius=1.5,start angle=0,end angle=40];
\draw[<->] (20:2.5) arc[radius=2.5,start angle=20,end angle=40];
\draw (-5:5) arc[radius=5,start angle=-5,end angle=45];
\node[anchor=north] at (3.83,0) {$\frac{1}{c}$};
\node[anchor=south east] at (40:4.8) {1};
\node[anchor=west] at (12:1.5) {$\arccos \frac{1}{c}$};
\node[anchor=south west] at (36:2.5)
{$\varphi_c=\frac{1}{2}\arccos \frac{1}{c}$};
\draw[black,thick,fill=white] (0,0) circle[radius=0.05];
\draw[black,thick,fill=white] (20:5) circle[radius=0.05];
\draw[black,thick,fill=white] (40:5) circle[radius=0.05];
\draw[black,thick,fill=white] (3.83,0) circle[radius=0.05];
\node[anchor=east] at (0,0) {$Q$};
\node[anchor=south west] at (40:5) {$P$};
\node[anchor=south west] at (20:5) {$S$};
\node[anchor=south west] at (3.83,0) {$R$};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1611.07303 | arxiv | 2016-11-23T02:05:41 |
|
The dynamical system for \{T_0,\,T_1\} | \documentclass[a4paper,12pt]{article}
\usepackage[centertags]{amsmath}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{tikz, subfigure}
\usepackage[latin1]{inputenc}
\usepackage{tikz}
\usepackage{amsfonts,graphicx,amsmath,amssymb,hyperref,color}
\begin{document}
\begin{tikzpicture}[scale=5]
\draw(0,0)node[below]{\scriptsize 0}--(.382,0)node[below]{\scriptsize$\frac{1}{\beta}$}--(.618,0)node[below]{\scriptsize$\frac{1}{\beta(\beta-1)}$}--(1,0)node[below]{\scriptsize$\frac{1}{\beta-1}$}--(1,1)--(0,1)node[left]{\scriptsize$\frac{1}{\beta-1}$}--(0,.5)--(0,0);
\draw[dotted](.382,0)--(.382,1)(0.618,0)--(0.618,1);
\draw[thick](0,0)--(0.618,1)(.382,0)--(1,1);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1406.3263 | arxiv | 2014-07-01T02:12:47 |
|
The Stark fly-by geometry. | \documentclass[handcarry]{aiaa-tc}
\usepackage{tikz}
\usetikzlibrary{calc}
\usepackage{amssymb}
\usepackage{amsmath}
\begin{document}
\begin{tikzpicture}[scale=3]
% Hyperbola parameters
\pgfmathsetmacro{\e}{1.15} % eccentricity incomin
\pgfmathsetmacro{\a}{1}
\pgfmathsetmacro{\b}{(\a*sqrt((\e)^2-1)}
\pgfmathsetmacro{\ee}{1.4} % eccentricity
\pgfmathsetmacro{\aa}{(\a*\e) / \ee}
\pgfmathsetmacro{\bb}{(\aa*sqrt((\ee)^2-1)}
% Plot the circles
\path [draw=none,fill=gray, fill opacity = 0.05] (0,0) circle (0.7);
\path [draw=none,fill=gray, fill opacity = 0.05] (0,0) circle (0.9);
% Plots the axis
\draw[->] (0,0) -- (0.3,0) node[right] {$\mathbf i_{x}$};
\draw[->] (0,0) -- (0,0.3) node[left] {$\mathbf i_{y}$};
% Plot the hyperbola
\draw[red] plot[domain=-0.94:0] ({\b*sinh(\x)}, {\a*cosh(\x) - \e*\a});
\draw[red, ->] plot[domain=0:0.93] ({\bb*sinh(\x)}, {\aa*cosh(\x) + (\a-\aa) - \e*\a});
\draw[] plot[domain=-1.5:-0.94] ({\b*sinh(\x)}, {\a*cosh(\x) - \e*\a});
\draw[] plot[domain=0.93:1.5] ({\bb*sinh(\x)}, {\aa*cosh(\x) + (\a-\aa) - \e*\a});
% Start of thrust
\path [draw=none,fill=black, fill opacity = 0.5] ({\b*sinh(-0.94)}, {\a*cosh(-0.94) - \e*\a}) circle (0.02);
% Plot the asymptotes
%left
\pgfmathsetmacro{\A}{-1.65}
\pgfmathsetmacro{\B}{-0.9}
\draw[dashed] plot[domain=-1.4:0] ({\x},{\A*\x+\B});
%right
\pgfmathsetmacro{\C}{0.89}
\pgfmathsetmacro{\D}{-0.65}
\draw[dashed] plot[domain=-0.2:1.7] ({\x},{\C*\x+\D});
% Plots the in and out velocity vectors
\pgfmathsetmacro{\xv}{ - (\B-\D) / (\A-\C)}
\pgfmathsetmacro{\yv}{\A*\xv+\B}
\draw[thick,->] (\xv,\yv) -- (\xv+0.1,\yv+\A*0.1) node[left] {$\mathbf v^-_\infty$};
\draw[thick,->] (\xv,\yv) -- (\xv+0.2,\yv+\C*0.2) node[right] {$\mathbf v^+_\infty$};
% Plots the x0, y0 and rm
\draw[<->] (-0.615,0.4) -- (0,0.4) node[midway, above] {$-x_0$} ;
\draw[<->] (-0.68,0.32) -- (-0.68,0) node[midway, left] {$y_0$} ;
\draw[<->] (0,-0.01) -- (0,-0.14) node[midway, left] {$r_m$} ;
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1601.04963 | arxiv | 2016-01-21T02:05:38 |
|
The radial fly-by geometry. In the range rr_i, r_o, the spacecraft maintains an additional radial acceleration and is thus not flying along a hyperbola | \documentclass[handcarry]{aiaa-tc}
\usepackage{tikz}
\usetikzlibrary{calc}
\usepackage{amssymb}
\usepackage{amsmath}
\begin{document}
\begin{tikzpicture}[scale=5]
% Hyperbola parameters
\pgfmathsetmacro{\e}{1.1} % eccentricity
\pgfmathsetmacro{\a}{1}
\pgfmathsetmacro{\b}{(\a*sqrt((\e)^2-1)}
\def\offset{0.00} %makes the figure cleaner violating the foci definition
% Plot the circle
\path [draw=none,fill=gray, fill opacity = 0.1] (-\e*\a,0) circle (0.7);
\path [draw=none,fill=white] (-\e*\a,0) circle (0.4);
% Plots the axis
\draw[->] (-\e*\a-\offset,0,0) -- (0.25-\e*\a-\offset,0,0) node[right] {$\mathbf i_{r_m}$};
\draw[->] (-\e*\a-\offset,0,0) -- (-\e*\a-\offset,0.25,0) node[left] {$\mathbf i_{p_m}$};
\draw[] (-\e*\a-\offset,0) -- ({-\a*cosh(-1)},{\b*sinh(-1)}) node[midway, above, fill=white] {$r_o$};
\draw[] (-\e*\a-\offset,0) -- ({-\a*cosh(-0.725)},{\b*sinh(-0.725)}) node[midway, right, fill=white] {$r_i$};
\draw[arrows=<->](-\e*\a-\offset,-0.02)--(0.08-\e*\a-\offset,-0.02) node[midway, below] {$r_m$};
% Plot the hyperbola
\draw[red] plot[domain=-1:-0.725] ({-\a*cosh(\x)},{\b*sinh(\x)});
\draw[red] plot[domain=0.725:1] ({-\a*cosh(\x)},{\b*sinh(\x)});
\draw[->] plot[domain=-1.3:-1] ({-\a*cosh(\x)},{\b*sinh(\x)});
\draw plot[domain=1:1.3] ({-\a*cosh(\x)},{\b*sinh(\x)});
\draw[->] plot[domain=-0.725:0.725] ({-\a*cosh(\x)},{\b*sinh(\x)});
% Plot the asymptotes
\draw[dashed] plot[domain=-1.2:0] ({1.65*\x},{0.85*\x+0.22});
\draw[dashed] plot[domain=-1.2:0] ({1.65*\x},{-0.85*\x-0.22});
% Plots the in and out velocity vectors
\def\xs1{-0.22/0.85}
\def\xf1{-0.158}
\draw[thick,->] (1.65*\xs1,0.85*\xs1+0.22) -- (1.65*\xf1,0.85*\xf1+0.22) node[above] {$\mathbf v^-_\infty$};
\def\xs2{-0.22/0.85}
\def\xf2{-0.36}
\draw[thick,->] (1.65*\xs2,-0.85*\xs2-0.22) -- (1.65*\xf2,-0.85*\xf2-0.22) node[above] {$\mathbf v^+_\infty$};
% Plots the angle delta
\draw [->,domain=30:150] plot ({-1.65*0.22/0.85+0.1*cos(\x)}, {0.1*sin(\x)}) node[right] {$2\delta$};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1601.04963 | arxiv | 2016-01-21T02:05:38 |
|
The high-energy physics software stack of an LHC experiment. | \documentclass[11pt,conference,letterpaper]{IEEEtran}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{tikz}
\usetikzlibrary{arrows,positioning,shapes,topaths,calc,fit,backgrounds,matrix,shadows,automata,patterns,decorations.pathmorphing,decorations.pathreplacing,decorations.text,circuits.logic.US,trees,mindmap}
\begin{document}
\begin{tikzpicture}
\tikzset{
block/.style={thick,rounded corners,draw,align=center, fill=white, drop shadow, minimum width=4cm, minimum height=1.5cm, top color=white},
}
\colorlet{colexternal}{red!50!black!20}
\colorlet{colhep}{green!50!black!20}
\colorlet{coluser}{blue!50!black!20}
\node[label={[rotate=20,font=\bf,red!75, label distance=2mm]right:20 MLOC}, block, bottom color=colexternal] (os) at (0,0) {Grid Libraries\\System Libraries\\OS Kernel};
\node[label={[rotate=20,font=\bf,green!75!black!75, label distance=2mm]right:5 MLOC}, above=4mm of os.north,block,bottom color=colhep,double copy shadow] (root) {High Energy Physics\\Libraries};
\node[label={[rotate=20,font=\bf,green!75!black!75, label distance=2mm]right:4 MLOC}, above=4mm of root.north,block,bottom color=colhep,double copy shadow] (framework) {Experiment\\Software Framework};
\node[label={[rotate=20,font=\bf,blue!75!black!75, label distance=2mm]right:0.1 MLOC}, above=4mm of framework.north,block,bottom color=coluser,double copy shadow] (analyse) {Individual \\Analysis Code};
\draw[very thick,->] (5.5,-0.5) -- node[very near start,rotate=90,red,yshift=-3mm] {stable} node[very near end,rotate=90,blue,yshift=-3mm] {changing} (5.5,6);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1407.3063 | arxiv | 2014-07-14T02:07:01 |
|
The q-ary channel. | \documentclass[10pt,journal,twocolumn,twoside]{IEEEtran}
\usepackage{graphicx,cite,amssymb,amsmath,bm}
\usepackage{amsmath}
\usepackage[utf8]{inputenc}
\usepackage{tikz}
\usetikzlibrary{fadings}
\usetikzlibrary{shadows.blur}
\usetikzlibrary{shapes,arrows}
\usetikzlibrary{calc,shapes.misc}
\usetikzlibrary{decorations.pathreplacing}
\usetikzlibrary{positioning}
\usepackage{pgfplots,relsize}
\usetikzlibrary{plotmarks}
\begin{document}
\begin{tikzpicture}[>=latex']
\tikzset{root/.style={circle, draw, thick, scale=1.3, minimum size=0.1mm, fill=gray}}
% Nodes
\node[root] (in1) at (-3.9,1) {};
\node[root] (out1) at (3.9,1) {};
\node[root] (in2) at (-3.9,-1.5) {};
\node[root] (out2) at (3.9,-1.5) {};
\node[root] (in3) at (-3.9,-6) {};
\node[root] (out3) at (3.9,-6) {};
\node at ($(in2)!.4!(in3)$) {\scalebox{3}\vdots};
\node at ($(out2)!.4!(out3)$) {\scalebox{3}\vdots};
\node (in1l) at (-4.7,1) {};
\node (out1l) at (4.7,1) {};
\node (in2l) at (-4.7,-1.5) {};
\node (out2l) at (4.7,-1.5) {};
\node (in3l) at (-4.9,-6) {};
\node (out3l) at (4.9,-6) {};
\node (ps00) at (0,1.25) {};
\node (ps11) at (0,-1.25) {};
\node (psqq) at (0,-5.7) {};
\node (ps01) at (-1.6,0.6) {};
\node (ps0q) at (-1.7,-0.13) {};
\node (ps10) at (-3,-0.9) {};
\node (ps1q) at (-3,-2.6) {};
\node (psq0) at (-3.15,-4.55) {};
\node (psq1) at (-1,-5) {};
% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %
\draw [-] (in1) -- (out1);
\draw [-] (in2) -- (out2);
\draw [-] (in1) -- (out2);
\draw [-] (in1) -- (out3);
\draw [-] (in2) -- (out1);
\draw [-] (in3) -- (out3);
\draw [-] (in3) -- (out2);
\draw [-] (in3) -- (out1);
\draw [-] (in2) -- (out3);
\node[] at (in1l) {\scalebox{1.7}{$0$}};
\node[] at (in2l) {\scalebox{1.7}{$1$}};
\node[] at (in3l) {\scalebox{1.7}{$q-1$}};
\node[] at (out1l) {\scalebox{1.7}{$0$}};
\node[] at (out2l) {\scalebox{1.7}{$1$}};
\node[] at (out3l) {\scalebox{1.7}{$q-1$}};
\node[] at (ps00) {\scalebox{1.7}{$p_{0,0}$}};
\node[] at (ps01) {\scalebox{1.7}{$p_{0,1}$}};
\node[] at (ps0q) {\scalebox{1.7}{$p_{0,q-1}$}};
\node[] at (ps11) {\scalebox{1.7}{$p_{1,1}$}};
\node[] at (ps10) {\scalebox{1.7}{$p_{1,0}$}};
\node[] at (ps1q) {\scalebox{1.7}{$p_{1,q-1}$}};
\node[] at (psq0) {\scalebox{1.7}{$p_{q-1,0}$}};
\node[] at (psq1) {\scalebox{1.7}{$p_{q-1,1}$}};
\node[] at (psqq) {\scalebox{1.7}{$p_{q-1,q-1}$}};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1705.07011 | arxiv | 2017-05-22T02:07:04 |
|
Example Gauge Transformation | \documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{color}
\usepackage{tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}
\node (8) at (0,1) {8};
\node (7) at (0,2) {7};
\node (6) at (0,3) {6};
\node (5) at (0,4) {5};
\node (1) at (1,5) {1};
\node (2) at (2,5) {2};
\node (3) at (3,5) {3};
\node (4) at (4,5) {4};
\draw[fill=gray] (0.5,0.5) rectangle (4.5,4.5);
\draw[ultra thick,draw=red] (0.2,4) -- (5,4);
\draw[ultra thick,draw=red] (0.2,3) -- (5,3);
\draw[ultra thick,draw=red] (0.2,2) -- (5,2);
\draw[ultra thick,draw=red] (0.2,1) -- (5,1);
\draw[ultra thick,draw=blue](1,4.8) -- (1,0.2);
\draw[ultra thick,draw=blue](2,4.8) -- (2,0.2);
\draw[ultra thick,draw=blue](3,4.8) -- (3,0.2);
\draw[ultra thick,draw=blue](4,4.8) -- (4,0.2);
\draw[fill=black] (4,1) circle (1mm);
\draw[fill=black] (4,2) circle (1mm);
\draw[fill=black] (4,3) circle (1mm);
\draw[fill=black] (4,4) circle (1mm);
\draw[fill=black] (3,1) circle (1mm);
\draw[fill=black] (3,2) circle (1mm);
\draw[fill=black] (3,3) circle (1mm);
\draw[fill=black] (3,4) circle (1mm);
\draw[fill=black] (2,1) circle (1mm);
\draw[fill=black] (2,2) circle (1mm);
\draw[fill=black] (2,3) circle (1mm);
\draw[fill=black] (2,4) circle (1mm);
\draw[fill=black] (1,1) circle (1mm);
\draw[fill=black] (1,2) circle (1mm);
\draw[fill=black] (1,3) circle (1mm);
\draw[fill=black] (1,4) circle (1mm);
\node (16) at (6,1) {16};
\node (15) at (6,2) {15};
\node (14) at (6,3) {14};
\node (13) at (6,4) {13};
\node (9) at (7,5) {9};
\node (10) at (8,5) {10};
\node (11) at (9,5) {11};
\node (12) at (10,5) {12};
\draw[fill=gray] (6.5,0.5) rectangle (10.5,4.5);
\draw[ultra thick,draw=blue] (6.3,4) -- (11,4);
\draw[ultra thick,draw=blue] (6.3,3) -- (11,3);
\draw[ultra thick,draw=blue] (6.3,2) -- (11,2);
\draw[ultra thick,draw=blue] (6.3,1) -- (11,1);
\draw[ultra thick,draw=red](7,4.8) -- (7,0.2);
\draw[ultra thick,draw=red](8,4.8) -- (8,0.2);
\draw[ultra thick,draw=red](9,4.8) -- (9,0.2);
\draw[ultra thick,draw=red](10,4.8) -- (10,0.2);
\draw[fill=black] (7,1) circle (1mm);
\draw[fill=black] (7,2) circle (1mm);
\draw[fill=black] (7,3) circle (1mm);
\draw[fill=black] (7,4) circle (1mm);
\draw[fill=black] (8,1) circle (1mm);
\draw[fill=black] (8,2) circle (1mm);
\draw[fill=black] (8,3) circle (1mm);
\draw[fill=black] (8,4) circle (1mm);
\draw[fill=black] (9,1) circle (1mm);
\draw[fill=black] (9,2) circle (1mm);
\draw[fill=black] (9,3) circle (1mm);
\draw[fill=black] (9,4) circle (1mm);
\draw[fill=black] (10,1) circle (1mm);
\draw[fill=black] (10,2) circle (1mm);
\draw[fill=black] (10,3) circle (1mm);
\draw[fill=black] (10,4) circle (1mm);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1504.08011 | arxiv | 2015-12-18T02:10:50 |
|
The 2-ary de Bruijn graph of order 3, or B(2,3). | \documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{color}
\usepackage{tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}[-,>=stealth',auto,node distance=2cm,
thick,main node/.style={circle,draw,font=\sffamily\bfseries,scale=0.75},new node/.style={circle,fill=black,text=white,draw,font=\sffamily\bfseries,scale=0.75}]
\node[main node] (0) {000};
\node[main node] (1) [above right of=0] {001};
\node[main node] (2) [below right of=1] {010};
\node[main node] (4) [below right of=0] {100};
\node[main node] (5) [right of=2] {101};
\node[main node] (6) [below right of=5] {110};
\node[main node] (3) [above right of=5] {011};
\node[main node] (7) [below right of=3] {111};
\path[every node/.style={font=\sffamily\small}]
(0) edge node [left] {} (1)
(1) edge node [left] {} (3)
edge node [right] {} (2)
(2) edge node{} (5)
edge node [right] {} (4)
(3) edge node [right] {} (6)
edge node [right] {} (7)
(4) edge node [left] {} (0)
edge node [right] {} (1)
(5) edge node [right] {} (3)
(6) edge node [right] {} (5)
edge node [right] {} (4)
(7) edge node [right] {} (6);
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1504.08011 | arxiv | 2015-12-18T02:10:50 |
|
Embedding 3-OR onto the Chimera Graph | \documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{color}
\usepackage{tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}[-,>=stealth',auto,node distance=2cm,
thick,main node/.style={circle,draw,font=\sffamily\bfseries},new node/.style={font=\sffamily\bfseries},bend angle = 15]
\node[main node] (1) {$S_1$};
\node[new node,node distance=7mm] (1L) [left of=1] {$+1$};
\node[new node,node distance=10mm] (12L) [below right of=1,anchor=east] {$-2$};
\node[main node] (2) [right of=1] {$S_2$};
\node[new node,node distance=10mm] (22L) [below left of=2,anchor=west] {$-2$};
\node[new node,node distance=7mm] (2L) [right of=2] {$+1$};
\node[main node] (z1) [below of=1] {$z_1$};
\node[new node,node distance=9mm] (z1L) [left of=z1] {$-1.5$};
\node[main node] (z2) [right of=z1] {$z_1$};
\node[new node,node distance=7mm] (z2L) [right of=z2] {$-1.5$};
\node[main node] (3) [below of=z2] {$S_3$};
\node[new node,node distance=7mm] (3L) [right of=3] {$-1$};
\path[every node/.style={font=\sffamily\footnotesize}]
(1) edge node [above] {+1} (2)
(1) edge node [above left,distance=2mm] {} (z2)
(2) edge node [above right] {} (z1)
(z1) edge node [above] {$\textsf{-J}_{\textsf{FM}}$} (z2)
(z1) edge node [above] {+1} (3)
;
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1504.08011 | arxiv | 2015-12-18T02:10:50 |
|
Mapping from OR-clauses to Ising Models | \documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{color}
\usepackage{tikz}
\usetikzlibrary{arrows}
\begin{document}
\begin{tikzpicture}[-,>=stealth',auto,node distance=2cm,
thick,main node/.style={circle,draw,font=\sffamily\bfseries},new node/.style={font=\sffamily\bfseries},bend angle = 15]
\node[main node] (21) {$S_1$};
\node[new node,node distance=6mm] (21L) [above of=21] {$-1$};
\node[main node] (22) [right of=21] {$S_2$};
\node[new node,node distance=6mm] (22L) [above of=22] {$-1$};
\node[new node] (2) [above right of=21] {2-OR};
\node[main node] (31) [right of=22] {$S_1$};
\node[new node,node distance=6mm] (31L) [above of=31] {$+1$};
\node[main node] (32) [right of=31] {$S_2$};
\node[new node,node distance=6mm] (32L) [above of=32] {$+1$};
\node[main node,node distance=15mm] (3z1)[below right of=31] {$z_1$};
\node[new node,node distance=7mm] (3z1L) [left of=3z1] {$-3$};
\node[main node] (33) [right of=3z1] {$S_3$};
\node[new node,node distance=6mm] (33L) [above of=33] {$-1$};
\node[new node] (3) [above right of=31] {3-OR};
\node[new node] (4) [below of=22] {4-OR};
\node[main node] (41) [below left of=4] {$S_1$};
\node[new node,node distance=6mm] (41L) [above of=41] {$+1$};
\node[main node] (42) [right of=41] {$S_2$};
\node[new node,node distance=6mm] (42L) [above of=42] {$+1$};
\node[main node,node distance=15mm] (4z1)[below left of=42] {$z_1$};
\node[new node,node distance=7mm] (4z1L) [left of=4z1] {$-3$};
\node[main node] (43) [right of=42] {$S_3$};
\node[new node,node distance=6mm] (43L) [above of=43] {$+1$};
\node[main node] (44) [right of=43] {$S_4$};
\node[new node,node distance=6mm] (44L) [above of=44] {$+1$};
\node[main node,node distance=15mm] (4z2)[below right of=43] {$z_2$};
\node[new node,node distance=6mm] (4z2L) [right of=4z2] {$-3$};
\node[new node] (5) [below of=4z2] {5-OR};
\node[main node,node distance=15mm] (52) [below left of=5] {$S_2$};
\node[new node,node distance=6mm] (52L) [above of=52] {$+1$};
\node[main node,node distance=15mm] (53) [below right of=5] {$S_3$};
\node[new node,node distance=6mm] (53L) [above of=53] {$+1$};
\node[main node] (51) [left of=52] {$S_1$};
\node[new node,node distance=6mm] (51L) [above of=51] {$+1$};
\node[main node] (54) [right of=53] {$S_4$};
\node[new node,node distance=6mm] (54L) [above of=54] {$+1$};
\node[main node,node distance=15mm] (5z1)[below left of=52] {$z_1$};
\node[new node,node distance=7mm] (5z1L) [left of=5z1] {$-1$};
\node[main node,node distance=15mm] (5z2)[below right of=53]{$z_2$};
\node[new node,node distance=6mm] (5z2L) [right of=5z2] {$-1$};
\node[main node,node distance=30mm] (5z3)[below right of=5z1]{$z_3$};
\node[new node,node distance=7mm] (5z3L) [left of=5z3] {$-3$};
\node[main node] (55) [right of=5z3] {$S_5$};
\node[new node,node distance=6mm] (55L) [above of=55] {$-1$};
\node[new node] (6) [below of=5z3] {6-OR};
\node[main node,node distance=15mm] (62) [below left of=6] {$S_2$};
\node[new node,node distance=6mm] (62L) [above of=62] {$+1$};
\node[main node,node distance=15mm] (63) [below right of=6] {$S_3$};
\node[new node,node distance=6mm] (63L) [above of=63] {$+1$};
\node[main node] (61) [left of=62] {$S_1$};
\node[new node,node distance=6mm] (61L) [above of=61] {$+1$};
\node[main node] (64) [right of=63] {$S_4$};
\node[new node,node distance=6mm] (64L) [above of=64] {$+1$};
\node[main node,node distance=15mm] (6z1) [below left of=62] {$z_1$};
\node[new node,node distance=7mm] (6z1L) [left of=6z1] {$-1$};
\node[main node,node distance=15mm] (6z2) [below right of=63] {$z_2$};
\node[new node,node distance=6mm] (6z2L) [right of=6z2] {$-1$};
\node[main node,node distance=30mm] (6z3) [below right of=6z1]{$z_3$};
\node[new node,node distance=7mm] (6z3L) [left of=6z3] {$-3$};
\node[main node,node distance=45mm] (6z4) [right of=6z3] {$z_4$};
\node[new node,node distance=6mm] (6z4L) [right of=6z4] {$-3$};
\node[main node,node distance=15mm] (65) [above left of=6z4] {$S_5$};
\node[new node,node distance=6mm] (65L) [below of=65] {$+1$};
\node[main node,node distance=15mm] (66) [above right of=6z4]{$S_6$};
\node[new node,node distance=6mm] (66L) [below of=66] {$+1$};
\path[every node/.style={font=\sffamily\footnotesize}]
(21) edge node [above] {+1} (22)
(31)edge node [above] {+1} (32)
edge node [left] {-2} (3z1)
(32)edge node [right] {-2} (3z1)
(33)edge node [below] {+1} (3z1)
(41) edge node [above] {+1} (42)
(41) edge node [left] {-2} (4z1)
(4z1) edge node [right] {-2} (42)
(4z1) edge node [below] {+1} (4z2)
(4z2) edge node [left] {-2} (43)
(4z2) edge node [right] {-2} (44)
(43) edge node [above] {+1} (44)
(51) edge node [above] {+1} (52)
(51) edge node [left] {-2} (5z1)
(52) edge node [right] {-2} (5z1)
(5z1) edge node [above] {+1} (5z2)
(5z1) edge node [left] {-2} (5z3)
(5z3) edge node [above] {+1} (55)
(5z3) edge node [right] {-2} (5z2)
(5z2) edge node [left] {-2} (53)
(5z2) edge node [right] {-2} (54)
(53) edge node [above] {+1} (54)
(61) edge node [above] {+1} (62)
(61) edge node [left] {-2} (6z1)
(62) edge node [right] {-2} (6z1)
(63) edge node [above] {+1} (64)
(63) edge node [left] {-2} (6z2)
(64) edge node [right] {-2} (6z2)
(6z1) edge node [above] {+1} (6z2)
(6z1) edge node [left] {-2} (6z3)
(6z2) edge node [right] {-2} (6z3)
(6z3) edge node [above] {+1} (6z4)
(66) edge node [left] {-2} (6z4)
(65) edge node [above] {+1} (66)
(65) edge node [right] {-2} (6z4)
;
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1504.08011 | arxiv | 2015-12-18T02:10:50 |
|
Left panel: A bichromatic word with sum 1, prefix maximum 2, and prefix minimum -1. The number of blocks is 5. Right panel: The word after swapping the second and the third block, indicated by blue and red box, respectively. Note that this swapping operation satisfies the prerequisites of Claim~cl:swap, and consequently the prefix maximum of the word does not increase. | \documentclass[11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{amstext,amsfonts,amsthm,amsmath,amssymb}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[scale=0.68]
\tikzstyle{vertex}=[circle,fill=black,minimum size=0.1cm,inner sep=0pt]
\begin{scope}[shift={(-5.5,0)}]
\draw[gray, dashed] (-7.5,1.6) -- (1.5,1.6);
\node[vertex] (p1) at (-5.7,1.6) {};
\node[red,thick] at (-5.4,0.5) {$+$};
\node[vertex] (p2) at (-5.1,2.2) {};
\draw[red,thick] (p1) -- (p2);
\node[red,thick] at (-4.8,0.5) {$-$};
\node[vertex] (p3) at (-4.5,1.6) {};
\draw[red,thick] (p2) -- (p3);
\node[blue,thick] at (-4.2,0.5) {$+$};
\node[vertex] (p4) at (-3.9,2.2) {};
\draw[blue,thick] (p3) -- (p4);
\node[blue,thick] at (-3.6,0.5) {$+$};
\node[vertex] (p5) at (-3.3,2.8) {};
\draw[blue,thick] (p4) -- (p5);
\node[blue,thick] at (-3.0,0.5) {$-$};
\node[vertex] (p6) at (-2.7,2.2) {};
\draw[blue,thick] (p5) -- (p6);
\node[red,thick] at (-2.4,0.5) {$-$};
\node[vertex] (p7) at (-2.1,1.6) {};
\draw[red,thick] (p6) -- (p7);
\node[blue,thick] at (-1.8,0.5) {$-$};
\node[vertex] (p8) at (-1.5,1.0) {};
\draw[blue,thick] (p7) -- (p8);
\node[red,thick] at (-1.2,0.5) {$+$};
\node[vertex] (p9) at (-0.9,1.6) {};
\draw[red,thick] (p8) -- (p9);
\node[red,thick] at (-0.6,0.5) {$+$};
\node[vertex] (p10) at (-0.3,2.2) {};
\draw[red,thick] (p9) -- (p10);
\draw[blue] (-4.45,0.8) rectangle (-2.75,0.2);
\draw[red] (-2.65,0.8) rectangle (-2.15,0.2);
\end{scope}
\begin{scope}[shift={(5.5,0)}]
\draw[gray, dashed] (-7.5,1.6) -- (1.5,1.6);
\node[vertex] (p1) at (-5.7,1.6) {};
\node[red,thick] at (-5.4,0.5) {$+$};
\node[vertex] (p2) at (-5.1,2.2) {};
\draw[red,thick] (p1) -- (p2);
\node[red,thick] at (-4.8,0.5) {$-$};
\node[vertex] (p3) at (-4.5,1.6) {};
\draw[red,thick] (p2) -- (p3);
\node[red,thick] at (-4.2,0.5) {$-$};
\node[vertex] (p4) at (-3.9,1.0) {};
\draw[red,thick] (p3) -- (p4);
\node[blue,thick] at (-3.6,0.5) {$+$};
\node[vertex] (p5) at (-3.3,1.6) {};
\draw[blue,thick] (p4) -- (p5);
\node[blue,thick] at (-3.0,0.5) {$+$};
\node[vertex] (p6) at (-2.7,2.2) {};
\draw[blue,thick] (p5) -- (p6);
\node[blue,thick] at (-2.4,0.5) {$-$};
\node[vertex] (p7) at (-2.1,1.6) {};
\draw[blue,thick] (p6) -- (p7);
\node[blue,thick] at (-1.8,0.5) {$-$};
\node[vertex] (p8) at (-1.5,1.0) {};
\draw[blue,thick] (p7) -- (p8);
\node[red,thick] at (-1.2,0.5) {$+$};
\node[vertex] (p9) at (-0.9,1.6) {};
\draw[red,thick] (p8) -- (p9);
\node[red,thick] at (-0.6,0.5) {$+$};
\node[vertex] (p10) at (-0.3,2.2) {};
\draw[red,thick] (p9) -- (p10);
\draw[red] (-4.45,0.8) rectangle (-3.95,0.2);
\draw[blue] (-3.85,0.8) rectangle (-2.15,0.2);
\end{scope}
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1701.06937 | arxiv | 2017-01-25T02:07:21 |
|
This is how $T_{\overline{U},\hat{\varphi}}$ and $\pi^{-1}\left(T_{\overline{U},\hat{\varphi}}\right)$ must behave: The \textcolor{red}{red area} is the complement between $S_2$ and $S_1$. The rational tentacle \textcolor{green}{$T_{\overline{U},\hat{\varphi}}$} is contained in $\mathbb{R}^n\backslash S_2$ and \textcolor{green}{$\pi^{-1}\left(T_{\overline{U},\hat{\varphi}}\right)$} is contained in $S_1$. | \documentclass[12pt,abstracton]{scrreprt}
\usepackage[usenames,dvipsnames,x11names]{xcolor}
\usepackage{tikz}
\usetikzlibrary{arrows,decorations.pathmorphing}
\usetikzlibrary{intersections,shapes.arrows}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{color}
\usepackage{enumitem, color, amssymb}
\tikzset{
LL/.style={
draw=black,decorate,
decoration={snake, segment length=3mm, amplitude=1mm,post length=2mm}
}
}
\begin{document}
\begin{tikzpicture}
% the bottom left border of the surface
\path[name path=border1] (0,0) to[out=-10,in=150] (6,-2);
% the upper right border of the surface
\path[name path=border2] (12,1) to[out=150,in=-10] (5.5,3.2);
% a path for a line crossing both borders
\path[name path=redline] (0,-0.4) -- (12,1.5);
\path[name path=redlinex] (0,-0.4) -- (10,3);
% intersections between the borders and the lines
\path[name intersections={of=border1 and redline,by={a}}];
\path[name intersections={of=border2 and redline,by={b}}];
% we draw the surface
\shade[left color=gray!10,right color=gray!80]
(0,0) to[out=-10,in=150] (6,-2) -- (12,1)to[out=150,in=-10] (5.5,3.7) -- cycle;
% Encircler1
\draw[dashed,line width=2.8pt,shorten >= 3pt,shorten <= 3pt, opacity=0.3]
(0,0) to[bend left] (-1,4) to[bend right] (-0.3,3.5);
\draw[dashed,line width=2.8pt,shorten >= 3pt,shorten <= 3pt, opacity=0.3]
(-1,4) to[bend right] (6,7) to [bend left] (5.4,3.7);
\draw[dashed,line width=2.8pt,shorten >= 3pt,shorten <= 3pt, opacity=0.3]
(6,7) to[bend left] (9,7.5);
% Tentacles 2nd circle
\shade [ball color=white] (5,2.5) circle [radius=1cm];
\draw[blue ,line width=1.8pt,shorten >= 3pt,shorten <= 3pt]
(5.4,1.5) to[bend right] (8.1,4.6);
\draw[blue ,line width=1.8pt,shorten >= 3pt,shorten <= 3pt]
(5,3.5) to[bend right] (7.7,5.1);
\draw[dashed, blue ,line width=1.8pt,shorten >= 3pt,shorten <= 3pt]
(5,3.5) to[bend right] (5.4,1.5);
\draw[blue ,line width=1.8pt,shorten >= 3pt,shorten <= 3pt]
(7.55,4.15) to[bend left] (5.7,3);
\draw [fill opacity=0.3,fill=blue!80!blue]
(5,3.5) to[bend right] (5.4,1.5) to[bend right] (8.1,4.5) to[bend left] (7.5,4) to [bend left] (7.7,5) to[bend left] (5,3.5);
\draw [fill opacity=0.3,fill=blue!80!blue]
(8.1,4.5) to[bend left] (7.5,4) to [bend left] (7.7,5) to[bend left] (8.1,4.5);
% Red part
\shade[left color=red!20, right color=red!60]
(-0.3,3.5) to[out=-10,in=225] (10,7.5) to[bend right] (9,7.5) to[out=225,in=-10] (1,4);
\shade[left color=red!0, right color=red!50]
(b) to[out=172,in=-10] (a) to[out=10,in=150] (b);
% Top line
\draw[dashed,line width=2.8pt,shorten >= 3pt,shorten <= 3pt]
(-0.3,3.5) to[out=-10,in=225]
coordinate[pos=0.27] (aux1)
coordinate[pos=0.52] (aux2)
coordinate[pos=0.75] (aux3) (10,7.5);
% Red line for S(q)
\draw[dashed,red,line width=2.8pt,shorten >= 3pt,shorten <= 3pt]
(1,4) to[out=-10,in=225]
coordinate[pos=0.1] (auxx1)
coordinate[pos=0.6] (auxx2)
coordinate[pos=0.9] (auxx3) (9,7.5);
% we draw the back line
\draw[dashed,line width=2.8pt,shorten >= 3pt,shorten <= 3pt]
(b) to[out=172,in=-10]
coordinate[pos=0.8] (bux1)
coordinate[pos=0.5] (bux2)
coordinate[pos=0.2] (bux3) (a);
% Red line for S(q)
\draw[dashed,red,line width=2.8pt,shorten >= 3pt,shorten <= 3pt]
(b) to[out=150,in=10]
coordinate[pos=0.9] (buxx1)
coordinate[pos=0.4] (buxx2)
coordinate[pos=0.15] (buxx3) (a);
% Connection lines
% Fill upper part
% Red version
\foreach \coor in {1,2,3}
\draw[dashed,red,line width=1.8pt,shorten >= 3pt,shorten <= 3pt] (auxx\coor)to[bend right] (buxx\coor);
\foreach \coor in {1,2,3}
\draw[dashed,line width=1.8pt,shorten >= 3pt,shorten <= 3pt] (aux\coor)to[bend left] (bux\coor);
%Draw area under the curve
\shade[left color=black!0, right color=black!80]
(a) to[out=-10,in=150] (6,-2) -- (12,1)to[out=150,in=-10] (b) to[out=172,in=-10] (a);
% Fill up
\draw [fill opacity=0.2,fill=red!80!red]
(b) to[out=172,in=-10] (a) to[bend left] (-0.3,3.5) to[out=-10,in=225] (10,7.5) to[bend left] (b);
\draw [fill opacity=0.2,fill=red!80!red]
(b) to[out=150,in=10] (a) to[bend left] (1,4) to[out=-10,in=225] (9,7.5) to[bend left] (b);
%Draw circles
\shade [ball color=white] (7,0.5) circle [radius=1cm];
% Tentacles
\draw[blue ,line width=1.8pt,shorten >= 3pt,shorten <= 3pt]
(7.4,-0.4) to[bend left] (12.1,1);
\draw[blue ,line width=1.8pt,shorten >= 3pt,shorten <= 3pt]
(6.7,1.35) to[bend left] (12.1,2);
\draw[dashed, blue ,line width=1.8pt,shorten >= 3pt,shorten <= 3pt]
(7.4,-0.4) to[bend left] (6.7,1.35);
\draw[blue ,line width=1.8pt,shorten >= 3pt,shorten <= 3pt]
(11.9,1.5) to[bend right] (7.9,0.5);
\draw [fill opacity=0.3,fill=blue!80!blue]
(7.4,-0.4) to[bend left] (6.7,1.35) to[bend left] (12,2) to[bend right] (11.8,1.5) to[bend right] (12,1) to[bend right] (7.4,-0.4);
\draw [fill opacity=0.3,fill=blue!80!blue]
(12,2) to[bend right] (11.8,1.5) to[bend right] (12,1) to[bend right] (12,2);
%Encircler 2
\draw[dashed,line width=2.8pt,shorten >= 3pt,shorten <= 3pt, opacity=0.3]
(-0.3,3.5) to[bend right] (7,4);
\draw[dashed,line width=2.8pt,shorten >= 3pt,shorten <= 3pt, opacity=0.3]
(6,-2) to[bend left] (7,4);
\draw[dashed,line width=2.8pt,shorten >= 3pt,shorten <= 3pt, opacity=0.3]
(7,4) to[bend right] (11.5,6.5);
\draw[dashed,line width=2.8pt,shorten >= 3pt,shorten <= 3pt, opacity=0.3]
(12,1) to[bend right] (11.5,6.5);
\draw[dashed,line width=2.8pt,shorten >= 3pt,shorten <= 3pt, opacity=0.3]
(11.5,6.5) to[bend right] (10,7.5);
% we add some labels
\node[rotate=20] at (4.5,0.4) {\color{red}{$S_2\backslash S_1$}};
\node[rotate=30] at (7.7,0.8) {\color{green}{$\pi^{-1}\left(T_{\overline{U},\hat{\varphi}}\right)$}};
\node[rotate=30] at (5.5,2.5) {\color{green}{$T_{\overline{U},\hat{\varphi}}$}};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1701.07013 | arxiv | 2017-01-26T02:00:09 |
|
At each of the L repetitions we query the closest positions. Since the projected distance X to our target point y is distributed as Bin(k,dist(q,y)/d), we find y with constant probability by setting L=O(X a^-1). | \documentclass[11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage{tikz}
\usetikzlibrary{shapes,decorations.pathreplacing}
\begin{document}
\begin{tikzpicture}
\draw (-4,0) -- ++(30:1.5) node[pos=.5,above] {$a$};
\draw (-4,0) -- (-4,3) node[pos=.7,left] {$X$};
\draw (-4,0) circle (1.5);
\draw (0,0) circle (1.5);
\draw (4,0) circle (1.5);
\draw[fill=black] (-4,0) circle (.05) node[below] {$\sigma_{1,1}h_{1,k}(q)$};
\draw[fill=black] (0,0) circle (.05) node[below] {$\sigma_{2,1}h_{2,k}(q)$};
\draw[fill=black] (.5,.7) circle (.05) node[below] {$\sigma_{2,i}h_{2,k}(q)$};
\draw[fill=black] (4,0) circle (.05) node[below] {$\sigma_{3,1}h_{3,k}(q)$};
\draw[fill=black] (-4,3) circle (.05) node[above] {$h_{1,k}(y)$};
\draw[fill=black] (0,1.5) circle (.05) node[above] {$h_{2,k}(y)$};
\draw[fill=black] (4,2) circle (.05) node[above] {$h_{3,k}(y)$};
\node at (6.5,0) {$\dots$};
\draw [decorate,decoration={brace,amplitude=10}] (6.5,-1.5) -- (-5.5,-1.5)
node[midway,anchor=north,yshift=-10] {$L$};
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1605.02673 | arxiv | 2016-07-21T02:07:53 |
|
Optimal schedule for the indicated pre-fixed absences. | \documentclass[12pt]{article}
\usepackage[ansinew]{inputenc}%ß als Eingabe statt "s u.s.w.
\usepackage[T1]{fontenc}%this is needed for correct output of umlauts in pdf
\usepackage{tikz}
\usepackage{tkz-graph}
\usepackage{color}
\usepackage{colortbl}
\usepackage{amsmath}
\usepackage{amssymb}%
\begin{document}
\begin{tikzpicture}
[transform canvas={scale=1.35}, every node/.style={circle, scale=0.5}]
\node[draw,fill=green!25,scale=1.1] (a) at (0,1) {\(\!\bf3\!\)};
\node[scale=1.3] (a+) at (-0.32,1.07) {\(\bf A\)};
\node[draw,fill=green!25,scale=1.1] (b) at (-0.868,-0.5) {\(\!\bf3\!\)};
\node[scale=1.3] (b+) at (-1.186,-0.57) {\(\bf B\)};
\node[draw,fill=brown!45,scale=1.1] (c) at (0.868,-0.5) {\(\!\bf4\!\)};
\node[scale=1.3] (c+) at (1.186,-0.57) {\(\bf C\)};
\SetUpEdge[lw = 1.8pt,
labelcolor = white,
labeltext = red,
labelstyle = {font=\sffamily\small,scale=0.8,font=\bf,sloped}]
\Edge[color=red!80, labeltext=red, label=Round\,1](a)(c)
\Edge[color=blue!90, labeltext=blue, label=Round\,2](b)(c)
\Edge[color=brown, labeltext=brown, label=Round\,4](a)(b)
\end{tikzpicture}
\end{document} | https://arxiv.org/abs/1509.00488 | arxiv | 2016-05-23T02:12:02 |