new slides

15 files changed, 558 insertions, 0 deletions

diff --git a/vorlesungen/slides/4/galois/aufloesbarkeit.tex b/vorlesungen/slides/4/galois/aufloesbarkeit.tex
new file mode 100644
index 0000000..3215689
--- /dev/null
+++ b/vorlesungen/slides/4/galois/aufloesbarkeit.tex
@@ -0,0 +1,14 @@
+%
+% aufloesbarkeit.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Auflösbarkeit}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\end{column}
+\begin{column}{0.48\textwidth}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/galois/automorphismus.tex b/vorlesungen/slides/4/galois/automorphismus.tex
new file mode 100644
index 0000000..ab666cf
--- /dev/null
+++ b/vorlesungen/slides/4/galois/automorphismus.tex
@@ -0,0 +1,14 @@
+%
+% automorphismus.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Automorphismen}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\end{column}
+\begin{column}{0.48\textwidth}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/galois/erweiterung.tex b/vorlesungen/slides/4/galois/erweiterung.tex
new file mode 100644
index 0000000..1cf0bec
--- /dev/null
+++ b/vorlesungen/slides/4/galois/erweiterung.tex
@@ -0,0 +1,14 @@
+%
+% erweiterung.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Körpererweiterungen}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\end{column}
+\begin{column}{0.48\textwidth}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/galois/images/Makefile b/vorlesungen/slides/4/galois/images/Makefile
new file mode 100644
index 0000000..444944e
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/Makefile
@@ -0,0 +1,12 @@
+#
+# Makefile
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	wuerfel2.png wuerfel.png
+
+wuerfel.png:	wuerfel.pov common.inc
+	povray +A0.1 -W1080 -H1080 -Owuerfel.png wuerfel.pov
+
+wuerfel2.png:	wuerfel2.pov common.inc
+	povray +A0.1 -W1080 -H1080 -Owuerfel2.png wuerfel2.pov
diff --git a/vorlesungen/slides/4/galois/images/common.inc b/vorlesungen/slides/4/galois/images/common.inc
new file mode 100644
index 0000000..6cfcabe
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/common.inc
@@ -0,0 +1,89 @@
+//
+// common.inc
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+#include "textures.inc"
+#include "stones.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.133;
+#declare O = <0, 0, 0>;
+#declare E = <1, 1, 1>;
+#declare a = pow(2, 1/3);
+#declare at = 0.02;
+
+camera {
+	location <3, 2, 12>
+	look_at E * (a / 2) * 0.93
+	right x * imagescale
+	up y * imagescale
+}
+
+light_source {
+	<11, 20, 16> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+sky_sphere {
+	pigment {
+		color rgb<1,1,1>
+	}
+}
+
+#macro wuerfelgitter(A, AT)
+	cylinder { O, <A, 0, 0>, AT }
+	cylinder { O, <0, A, 0>, AT }
+	cylinder { O, <0, 0, A>, AT }
+	cylinder { <A, 0, 0>, <A, A, 0>, AT }
+	cylinder { <A, 0, 0>, <A, 0, A>, AT }
+	cylinder { <0, A, 0>, <A, A, 0>, AT }
+	cylinder { <0, A, 0>, <0, A, A>, AT }
+	cylinder { <0, 0, A>, <A, 0, A>, AT }
+	cylinder { <0, 0, A>, <0, A, A>, AT }
+	cylinder { <A, A, 0>, <A, A, A>, AT }
+	cylinder { <A, 0, A>, <A, A, A>, AT }
+	cylinder { <0, A, A>, <A, A, A>, AT }
+	sphere { <0, 0, 0>, AT }
+	sphere { <A, 0, 0>, AT }
+	sphere { <0, A, 0>, AT }
+	sphere { <0, 0, A>, AT }
+	sphere { <A, A, 0>, AT }
+	sphere { <A, 0, A>, AT }
+	sphere { <0, A, A>, AT }
+	sphere { <A, A, A>, AT }
+#end
+
+#macro wuerfel()
+	union {
+		box { O, E }
+		wuerfelgitter(1, 0.5*at)
+		texture {
+			T_Grnt24
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro wuerfel2()
+	union {
+		wuerfelgitter(a, at)
+		pigment {
+			color rgb<0.8,0.4,0.4>
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
diff --git a/vorlesungen/slides/4/galois/images/wuerfel.png b/vorlesungen/slides/4/galois/images/wuerfel.png
new file mode 100644
index 0000000..ff6fc14
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/wuerfel.png
diff --git a/vorlesungen/slides/4/galois/images/wuerfel.pov b/vorlesungen/slides/4/galois/images/wuerfel.pov
new file mode 100644
index 0000000..a5db465
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/wuerfel.pov
@@ -0,0 +1,9 @@
+//
+// wuerfel.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "common.inc"
+
+wuerfel()
+
diff --git a/vorlesungen/slides/4/galois/images/wuerfel2.png b/vorlesungen/slides/4/galois/images/wuerfel2.png
new file mode 100644
index 0000000..68919cc
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/wuerfel2.png
diff --git a/vorlesungen/slides/4/galois/images/wuerfel2.pov b/vorlesungen/slides/4/galois/images/wuerfel2.pov
new file mode 100644
index 0000000..ac32b2f
--- /dev/null
+++ b/vorlesungen/slides/4/galois/images/wuerfel2.pov
@@ -0,0 +1,9 @@
+//
+// wuerfel.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "common.inc"
+
+wuerfel()
+wuerfel2()
diff --git a/vorlesungen/slides/4/galois/konstruktion.tex b/vorlesungen/slides/4/galois/konstruktion.tex
new file mode 100644
index 0000000..6afa359
--- /dev/null
+++ b/vorlesungen/slides/4/galois/konstruktion.tex
@@ -0,0 +1,145 @@
+%
+% konstruktion.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Konstruktion mit Zirkel und Lineal}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Strahlensatz}
+\uncover<6->{%
+Jedes beliebige rationale Streckenverhältnis $\frac{p}{q}$
+kann mit Zirkel und Lineal konstruiert werden.}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{Kreis--Gerade}
+Aus $c$ und $a$ konstruiere $b=\sqrt{c^2-a^2}$
+\uncover<13->{%
+$\Rightarrow$ jede beliebige Quadratwurzel kann konstruiert werden}
+\end{block}}
+\end{column}
+\end{columns}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\s{0.5}
+\def\t{0.45}
+
+\coordinate (A) at (0,0);
+\coordinate (B) at ({10*\t},0);
+
+\uncover<2->{
+	\draw (0,0) -- (30:{10.5*\s});
+}
+
+\uncover<3->{
+	\foreach \x in {0,...,10}{
+		\fill (30:{\x*\s}) circle[radius=0.03];
+	}
+	\foreach \x in {0,1,2,3,4,7,8,9}{
+		\node at (30:{\x*\s}) [above] {\tiny $\x$};
+	}
+	\node at (30:{10*\s}) [above right] {$q=10$};
+}
+
+\uncover<4->{
+	\foreach \x in {1,...,10}{
+		\fill (0:{\x*\t}) circle[radius=0.03];
+		\draw[->,line width=0.2pt] (30:{\x*\s}) -- (0:{\x*\t});
+	}
+}
+
+\draw (A) -- (0:{10.5*\t});
+\node at (A) [below left] {$A$};
+\node at (B) [below right] {$B$};
+\fill (A) circle[radius=0.05];
+\fill (B) circle[radius=0.05];
+
+\uncover<5->{
+	\node at (30:{6*\s}) [above left] {$p=6$};
+	\draw[line width=0.2pt] (0,0) -- (0,-0.4);
+	\draw[line width=0.2pt] ({6*\t},0) -- ({6*\t},-0.4);
+	\draw[<->] (0,-0.3) -- ({6*\t},-0.3);
+	\node at ({3*\t},-0.4) [below]
+		{$\displaystyle\frac{p}{q}\cdot\overline{AB}$};
+}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+%\foreach \x in {8,...,14}{
+%	\only<\x>{\node at (4,4) {$\x$};}
+%}
+
+\def\r{4}
+\def\a{50}
+
+\coordinate (A) at ({\r*cos(\a)},0);
+
+\uncover<10->{
+	\fill[color=gray] (\r,0) -- (\r,0.3) arc (90:180:0.3) -- cycle;
+	\fill[color=gray]
+		(95:\r) -- ($(95:\r)+(185:0.3)$) arc (185:275:0.3) -- cycle;
+}
+
+\draw[->] (0,0) -- (95:\r);
+\node at (95:{0.5*\r}) [left] {$c$};
+
+\begin{scope}
+	\clip (-1,-0.3) rectangle (4.5,4.1);
+	\uncover<10->{
+		\draw (-1,0) -- (5,0);
+		\draw[->] (0,0) -- (\r,0);
+		\draw (0,0) circle[radius=\r];
+		\draw ({\r*cos(\a)},-1) -- ({\r*cos(\a)},5);
+	}
+\end{scope}
+
+\uncover<11->{
+	\fill[color=blue!20] (0,0) -- (A) -- (\a:\r) -- cycle;
+}
+
+\uncover<9->{
+	\fill[color=gray!80] (A) -- ($(A)+(0,0.5)$) arc (90:180:0.5) -- cycle;
+	\fill[color=gray!120] ($(A)+(-0.2,0.2)$) circle[radius=0.07];
+	\draw ({\r*cos(\a)},-0.3) -- ({\r*cos(\a)},4.1);
+}
+
+\uncover<11->{
+	\draw[color=blue,line width=1.4pt] (0,0) -- (\a:\r);
+	\node[color=blue] at (\a:{0.5*\r}) [above left] {$c$};
+}
+
+\draw[color=blue,line width=1.4pt] (0,0) -- ({\r*cos(\a)},0);
+\fill[color=blue] (0,0) circle[radius=0.04];
+\fill[color=blue] (A) circle[radius=0.04];
+\node[color=blue] at ({0.5*\r*cos(\a)},0) [below] {$a$};
+
+\uncover<12->{
+	\fill[color=white,opacity=0.8]
+		({\r*cos(\a)+0.1},{0.5*\r*sin(\a)-0.25})
+		rectangle
+		({\r*cos(\a)+2},{0.5*\r*sin(\a)+0.25});
+
+	\node[color=red] at ({\r*cos(\a)},{0.5*\r*sin(\a)}) [right]
+		{$b=\sqrt{c^2-a^2}$};
+	\draw[color=red,line width=1.4pt] ({\r*cos(\a)},0) -- (\a:\r);
+	\fill[color=red] (\a:\r) circle[radius=0.05];
+	\fill[color=red] (A) circle[radius=0.05];
+}
+
+\end{tikzpicture}
+\end{center}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/galois/quadratur.tex b/vorlesungen/slides/4/galois/quadratur.tex
new file mode 100644
index 0000000..f5763b9
--- /dev/null
+++ b/vorlesungen/slides/4/galois/quadratur.tex
@@ -0,0 +1,66 @@
+%
+% quadratur.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Quadratur des Kreises}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.44\textwidth}
+\begin{center}
+\uncover<2->{%
+\begin{tikzpicture}[>=latex,thick]
+
+\def\r{2.8}
+\pgfmathparse{sqrt(3.14159)*\r/2}
+\xdef\s{\pgfmathresult}
+
+\fill[color=blue!20] (-\s,-\s) rectangle (\s,\s);
+\fill[color=red!40,opacity=0.5] (0,0) circle[radius=\r];
+
+\uncover<3->{
+	\draw[->,color=red] (0,0) -- (50:\r);
+	\fill[color=red] (0,0) circle[radius=0.04];
+	\node[color=red] at (50:{0.5*\r}) [below right] {$r$};
+}
+
+\uncover<4->{
+	\draw[line width=0.3pt] (-\s,-\s) -- (-\s,{-\s-0.7});
+	\draw[line width=0.3pt] (\s,-\s) -- (\s,{-\s-0.7});
+	\draw[<->,color=blue] (-\s,{-\s-0.6}) -- (\s,{-\s-0.6});
+	\node[color=blue] at (0,{-\s-0.6}) [below] {$l$};
+}
+
+\uncover<5->{
+	\node at (0,{-\s/2}) {${\color{red}\pi r^2}={\color{blue}l^2}
+		\;\Rightarrow\;
+		{\color{blue}l}={\color{red}\sqrt{\pi}r}$};
+}
+
+\end{tikzpicture}}
+\end{center}
+\end{column}
+\begin{column}{0.52\textwidth}
+\begin{block}{Aufgabe}
+Konstruiere ein zu einem Kreis flächengleiches Quadrat
+\end{block}
+\uncover<6->{%
+\begin{block}{Modifizierte Aufgabe}
+Konstruiere eine Strecke, deren Länge Lösung der Gleichung
+$x^2-\pi=0$ ist.
+\end{block}}
+\uncover<7->{%
+\begin{proof}[Unmöglichkeitsbeweis mit Widerspruch]
+\begin{itemize}
+\item<8-> Lösung in einem Erweiterungskörper
+\item<9-> Lösung ist Nullstelle eines Polynoms 
+\item<10-> Lösung ist algebraisch
+\item<11-> $\pi$ ist {\bf nicht} algebraisch
+\uncover<12->{(Lindemann 1882\only<13>{, Weierstrass 1885})}
+\qedhere
+\end{itemize}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/galois/radikale.tex b/vorlesungen/slides/4/galois/radikale.tex
new file mode 100644
index 0000000..52fc4b9
--- /dev/null
+++ b/vorlesungen/slides/4/galois/radikale.tex
@@ -0,0 +1,14 @@
+%
+% radikale.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Radikale}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\end{column}
+\begin{column}{0.48\textwidth}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/galois/sn.tex b/vorlesungen/slides/4/galois/sn.tex
new file mode 100644
index 0000000..0e3ebe2
--- /dev/null
+++ b/vorlesungen/slides/4/galois/sn.tex
@@ -0,0 +1,14 @@
+%
+% sn.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Auflösbarkeit von $S_n$}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\end{column}
+\begin{column}{0.48\textwidth}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/galois/winkeldreiteilung.tex b/vorlesungen/slides/4/galois/winkeldreiteilung.tex
new file mode 100644
index 0000000..54b941b
--- /dev/null
+++ b/vorlesungen/slides/4/galois/winkeldreiteilung.tex
@@ -0,0 +1,94 @@
+%
+% winkeldreiteilung.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Winkeldreiteilung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.43\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\r{5}
+\def\a{25}
+
+\uncover<3->{
+	\draw[line width=0.7pt] (\r,0) arc (0:90:\r);
+}
+
+\fill[color=blue!20] (0,0) -- (\r,0) arc(0:{3*\a}:\r) -- cycle;
+\node[color=blue] at ({1.5*\a}:{1.05*\r}) {$\alpha$};
+
+\draw[color=blue,line width=1.3pt] (\r,0) arc (0:{3*\a}:\r);
+
+\uncover<2->{
+	\fill[color=red!40,opacity=0.5] (0,0) -- (\r,0) arc(0:\a:\r) -- cycle;
+	\draw[color=red,line width=1.4pt] (\r,0) arc (0:\a:\r);
+	\node[color=red] at ({0.5*\a}:{0.7*\r})
+		{$\displaystyle\frac{\alpha}{3}$};
+}
+
+\uncover<3->{
+	\fill[color=blue] ({3*\a}:\r) circle[radius=0.05];
+	\draw[color=blue] ({3*\a}:\r) -- ({\r*cos(3*\a)},-0.1);
+
+	\fill[color=red] ({\a}:\r) circle[radius=0.05];
+	\draw[color=red] ({\a}:\r) -- ({\r*cos(\a)},-0.1);
+
+	\draw[->] (-0.1,0) -- ({\r+0.4},0) coordinate[label={$x$}];
+	\draw[->] (0,-0.1) -- (0,{\r+0.4}) coordinate[label={right:$y$}];
+}
+
+
+\uncover<4->{
+\node at ({0.5*\r},-0.5) [below] {$\displaystyle
+\cos{\color{blue}\alpha}
+=
+4\cos^3{\color{red}\frac{\alpha}3} -3 \cos {\color{red}\frac{\alpha}3}
+$};
+}
+
+\uncover<5->{
+	\node[color=blue] at ({\r*cos(3*\a)},0) [below] {$a\mathstrut$};
+	\node[color=red] at ({\r*cos(\a)},0) [below] {$x\mathstrut$};
+}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.53\textwidth}
+\begin{block}{Aufgabe}
+Teile einen Winkel in drei gleiche Teile
+\end{block}
+\vspace{-2pt}
+\uncover<6->{%
+\begin{block}{Algebraisierte Aufgabe}
+Konstruiere $x$ aus $a$ derart, dass
+\[
+p(x)
+=
+x^3-\frac34 x -a = 0
+\]
+\uncover<7->{%
+$a=0$:}
+\uncover<8->{$p(x) = x(x^2-\frac{3}{4})\uncover<9->{\Rightarrow x = \frac{\sqrt{3}}2}$}
+\end{block}}
+\vspace{-2pt}
+\uncover<10->{%
+\begin{proof}[Unmöglichkeitsbeweis]
+\begin{itemize}
+\item<11->
+$a\ne 0$ $\Rightarrow$  $p(x)$ irreduzibel
+\item<12->
+$p(x)$ definiert eine Körpererweiterung vom Grad $3$
+\item<13->
+Konstruierbar sind nur Körpererweiterungen vom Grad $2^l$
+\qedhere
+\end{itemize}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/galois/wuerfel.tex b/vorlesungen/slides/4/galois/wuerfel.tex
new file mode 100644
index 0000000..ada6079
--- /dev/null
+++ b/vorlesungen/slides/4/galois/wuerfel.tex
@@ -0,0 +1,64 @@
+%
+% wuerfel.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Würfelverdoppelung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel.png}};
+\uncover<2->{
+\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel2.png}};
+}
+
+\uncover<3->{
+	\draw[<->,color=blue] (-1.25,-2.4) -- (2.55,-2.25);
+	\node[color=blue] at (0.75,-2.3) [above] {$a$};
+}
+
+\uncover<4->{
+	\begin{scope}[yshift=0.03cm]
+		\draw[color=red] (-2.13,-2.89) -- (-2.13,-3.19);
+		\draw[color=red] (2.85,-2.7) -- (2.85,-3.0);
+		\draw[<->,color=red] (-2.13,-3.09) -- (2.85,-2.9);
+	\end{scope}
+	\node[color=red] at (0.36,-2.9) [below] {$b$};
+}
+
+\uncover<5->{
+\node at (0,-4) {$
+	2{\color{blue}a}^3={\color{red}b}^3
+	\uncover<6->{\;\Rightarrow\;
+	\frac{b}{a} = \sqrt[3]{2}}$};
+}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.52\textwidth}
+\begin{block}{Aufgabe}
+Konstruiere einen Würfel mit doppeltem Volumen
+\end{block}
+\uncover<7->{%
+\begin{block}{Algebraisierte Aufgabe}
+Konstruiere eine Nullstelle von $p(x)=x^3-2$
+\end{block}}
+\uncover<8->{%
+\begin{proof}[Unmöglichkeitsbeweis]
+\begin{itemize}
+\item<9->
+$p(x)$ irreduzibel
+\item<10->
+$p(x)$ definiert eine Körpererweiterung vom Grad $3$
+\item<11->
+Nur Körpererweiterungen vom Grad $2^l$ sind konstruierbar
+\qedhere
+\end{itemize}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}