Three new figures

diff --git a/fig/eta_r.tex b/fig/eta_r.tex
new file mode 100644 (file)
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--- /dev/null
+++ b/fig/eta_r.tex
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+\centering
+\begin{subfigure}[t]{0.4\textwidth}
+    \centering
+    \begin{tikzpicture}
+        \draw[->] (0,0) -- (4,0) node[right] {$t$};
+        \draw[->] (0,0) -- (0,1.5) node[above] {};
+        \draw[line,domain=0:1] plot ({\x},{0});
+        \draw[line,domain=1:2] plot ({\x},{\x-1});
+        %\draw[line,domain=2:4,dashed] plot ({\x},{2*\x-3});
+        \draw[line,domain=2:4] plot ({\x},{1});
+
+        \draw (1,2pt) -- (1,-2pt) node[
+            font=\small,
+            anchor=north
+        ]
+        {$s\vphantom{+}$};
+
+        \draw (2,2pt) -- (2,-2pt) node[
+            font=\small,
+            anchor=north
+        ]
+        {$s+r$};
+
+        \draw (2pt,1) -- (-2pt,1) node[
+            font=\small,
+            anchor=east
+        ]
+        {$1$};
+
+    \end{tikzpicture}
+    \caption{$\eta_r(t)$}
+    \label{fig:eta_r}
+\end{subfigure}
+~
+\begin{subfigure}[t]{0.4\textwidth}
+    \centering
+    \begin{tikzpicture}
+        \draw[->] (0,0) -- (4,0) node[right] {$t$};
+        \draw[->] (0,0) -- (0,1.5) node[above] {};
+        \draw[line,domain=0:1] plot ({\x},{0});
+        \draw[line,domain=1:2] plot ({\x},{1});
+        %\draw[line,domain=2:4,dashed] plot ({\x},{2*\x-3});
+        \draw[line,domain=2:4] plot ({\x},{0});
+
+        \draw (1,2pt) -- (1,-2pt) node[
+            font=\small,
+            anchor=north
+        ]
+        {$s\vphantom{+}$};
+
+        \draw (2,2pt) -- (2,-2pt) node[
+            font=\small,
+            anchor=north
+        ]
+        {$s+r$};
+
+    \end{tikzpicture}
+    \caption{$\eta_r'(t)$}
+    \label{fig:eta_r_diff}
+\end{subfigure}
+\caption{
+    Visualization of the cut-off function $\eta_r(t)$ and its derivative.
+}
+\label{fig:eta_r_both}

diff --git a/fig/line_param.tex b/fig/line_param.tex
new file mode 100644 (file)
index 0000000..b163b72
--- /dev/null
+++ b/fig/line_param.tex
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+\centering
+\begin{tikzpicture}
+
+    \newcommand{\pgfextractangle}[3]{%
+        \pgfmathanglebetweenpoints{\pgfpointanchor{#2}{center}}
+        {\pgfpointanchor{#3}{center}}
+        \global\let#1\pgfmathresult  
+    }
+
+    \draw[->] (-0.5,0) -- (5,0) node[right] {$x$};
+    \draw[->] (0,-0.5) -- (0,4) node[above] {};
+
+    \node[] (a) at (-1, 4) {};
+    \node[] (b) at (5, -1) {};
+    \node[] (c) at ($(a)!(0,0)!(b)$) {};
+    \node[] (o) at (0,0) {};
+    \draw[line] (a) -- (b);
+    \draw[line,thin,black] ($(a)!(0,0)!(b)$) -- node[above] {$\rho$} (0,0);
+
+    \pgfextractangle{\angle}{o}{c}
+    \draw (0,0) +(0:1cm) arc (0:\angle:1cm);
+    \path (0,0) +(25:1.3cm) node {$\phi$};
+
+\end{tikzpicture}
+\caption{
+    The blue line is parametrized by the angle angle $\phi$ and the
+    distance from the origin to the line $\rho$.
+}
+\label{fig:line_param}

diff --git a/fig/lower_semicont.tex b/fig/lower_semicont.tex
new file mode 100644 (file)
index 0000000..32850ee
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+++ b/fig/lower_semicont.tex
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+\centering
+\begin{tikzpicture}
+    \draw[->] (0,0) -- (6,0) node[right] {$x$};
+    \draw[line,domain=0:3] plot ({\x},{(\x-3)*(\x-3)/9 + 0.5});
+    \draw[line,domain=3:6] plot ({\x},{3 - (5 - \x)*(5 - \x) / 5});
+
+    \node[
+        circle,
+        fill=blue,
+        draw=blue,
+        minimum size=7pt,
+        inner sep=0pt,
+        font=\footnotesize
+    ] (s) at (3, 0.52) {};
+    \node[
+        circle,
+        fill=white,
+        draw=blue,
+        minimum size=7pt,
+        inner sep=0pt,
+        font=\footnotesize
+    ] (s) at (3, 2.19) {};
+
+\end{tikzpicture}
+\caption{
+    A lower semicontinuous function $f : \mathbb{R} \to \mathbb{R}$ can
+    have discontinuities, but for a convergent sequence $x_k \to x$ we
+    always have $f(x) \leq \liminf_{k \to \infty} f(x_k)$.
+}
+\label{fig:lower_semicont}