diff options
24 files changed, 613 insertions, 1 deletions
diff --git a/vorlesungen/04_msepolynome/slides.tex b/vorlesungen/04_msepolynome/slides.tex index f8a7ba2..46ba8bf 100644 --- a/vorlesungen/04_msepolynome/slides.tex +++ b/vorlesungen/04_msepolynome/slides.tex @@ -15,6 +15,7 @@ \folie{3/division2.tex} \folie{3/teilbarkeit.tex} \folie{3/ideal.tex} +\folie{3/nichthauptideal.tex} \folie{3/idealverband.tex} \folie{3/faktorisierung.tex} \folie{3/faktorzerlegung.tex} diff --git a/vorlesungen/slides/3/Makefile.inc b/vorlesungen/slides/3/Makefile.inc index a70f73b..d6ce6b7 100644 --- a/vorlesungen/slides/3/Makefile.inc +++ b/vorlesungen/slides/3/Makefile.inc @@ -13,6 +13,8 @@ chapter3 = \ ../slides/3/ringstruktur.tex \ ../slides/3/teilbarkeit.tex \ ../slides/3/ideal.tex \ + ../slides/3/nichthauptideal.tex \ + ../slides/3/nichthauptideal2.tex \ ../slides/3/idealverband.tex \ ../slides/3/maximalideal.tex \ ../slides/3/quotientenring.tex \ diff --git a/vorlesungen/slides/3/chapter.tex b/vorlesungen/slides/3/chapter.tex index ea2718d..5287ffe 100644 --- a/vorlesungen/slides/3/chapter.tex +++ b/vorlesungen/slides/3/chapter.tex @@ -11,6 +11,8 @@ \folie{3/ringstruktur.tex} \folie{3/teilbarkeit.tex} \folie{3/ideal.tex} +\folie{3/nichthauptideal.tex} +\folie{3/nichthauptideal2.tex} \folie{3/maximalideal.tex} \folie{3/idealverband.tex} \folie{3/quotientenring.tex} diff --git a/vorlesungen/slides/3/images/Makefile b/vorlesungen/slides/3/images/Makefile new file mode 100644 index 0000000..e338fcf --- /dev/null +++ b/vorlesungen/slides/3/images/Makefile @@ -0,0 +1,55 @@ +# +# Makefile -- build images +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: hauptideal.jpg nichthauptideal.jpg hauptideal2.jpg hauptidealX.jpg \ + hauptidealR.jpg hauptidealXR.jpg ring.jpg + +ring.png: ring.pov common.inc + povray +A0.1 +W1920 +H1080 -Oring.png ring.pov +ring.jpg: ring.png + convert ring.png -density 300 -units PixelsPerInch ring.jpg + +hauptideal.png: hauptideal.pov common.inc + povray +A0.1 +W1920 +H1080 -Ohauptideal.png hauptideal.pov +hauptideal.jpg: hauptideal.png + convert hauptideal.png -density 300 -units PixelsPerInch \ + hauptideal.jpg + +hauptidealR.png: hauptidealR.pov common.inc + povray +A0.1 +W1920 +H1080 -OhauptidealR.png hauptidealR.pov +hauptidealR.jpg: hauptidealR.png + convert hauptidealR.png -density 300 -units PixelsPerInch \ + hauptidealR.jpg + +hauptideal2.png: hauptideal2.pov common.inc + povray +A0.1 +W1920 +H1080 -Ohauptideal2.png hauptideal2.pov +hauptideal2.jpg: hauptideal2.png + convert hauptideal2.png -density 300 -units PixelsPerInch \ + hauptideal2.jpg + +hauptidealX.png: hauptidealX.pov common.inc + povray +A0.1 +W1920 +H1080 -OhauptidealX.png hauptidealX.pov +hauptidealX.jpg: hauptidealX.png + convert hauptidealX.png -density 300 -units PixelsPerInch \ + hauptidealX.jpg + +hauptidealXR.png: hauptidealXR.pov common.inc + povray +A0.1 +W1920 +H1080 -OhauptidealXR.png hauptidealXR.pov +hauptidealXR.jpg: hauptidealXR.png + convert hauptidealXR.png -density 300 -units PixelsPerInch \ + hauptidealXR.jpg + +nichthauptideal.png: nichthauptideal.pov common.inc + povray +A0.1 +W1920 +H1080 -Onichthauptideal.png nichthauptideal.pov +nichthauptideal.jpg: nichthauptideal.png + convert nichthauptideal.png -density 300 -units PixelsPerInch \ + nichthauptideal.jpg + +ideal: ideal.pov ideal.ini common.inc + rm -rf ideal + mkdir ideal + povray +A0.1 +W1920 +H1080 -Oideal/i.png ideal.ini + + diff --git a/vorlesungen/slides/3/images/common.inc b/vorlesungen/slides/3/images/common.inc new file mode 100644 index 0000000..36c4e6b --- /dev/null +++ b/vorlesungen/slides/3/images/common.inc @@ -0,0 +1,277 @@ +// +// common.inc +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.40; +#declare O = <0, 0, 0>; +#declare at = 0.10; + +#declare Xunten = -10; +#declare Xoben = 10; +#declare Yunten = -8; +#declare Yoben = 8; +#declare Zunten = 0; +#declare Zoben = 20; + +#declare phi0 = 2 * pi * 290 / 360; + +camera { + location <60 * cos(2*pi*T+phi0), 15, 60 * sin(2*pi*T+phi0) + 10> + look_at <0, -2, 10> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <-14, 20, -50> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +light_source { + <41, 20, -50> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.1 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + +arrow(< -12.0, 0.0, 0 >, < 12.0, 0.0, 0.0 >, at, Gray) +arrow(< 0.0, 0.0, -2.0>, < 0.0, 0.0, 22.0 >, at, Gray) +arrow(< 0.0, -10.0, 0 >, < 0.0, 10.0, 0.0 >, at, Gray) + +#macro kasten() + box { <-10.5,-8.5,-0.5>, <10.5,8.5,20.5> } +#end + +#declare gruen = rgb<0.2,0.4,0.2>; +#declare blau = rgb<0.0,0.4,0.8>; +#declare rot = rgb<1.0,0.4,0.0>; + +#declare r = 0.4; + +#macro Zring() + union { + #declare X = Xunten; + #while (X <= Xoben + 0.5) + #declare Y = Yunten; + #while (Y <= Yoben + 0.5) + #declare Z = Zunten; + #while (Z <= Zoben + 0.5) + sphere { <X, Y, Z>, r } + + #declare Z = Z + 1; + #end + #declare Y = Y + 1; + #end + #declare X = X + 1; + #end + pigment { + color rot + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro Hauptideal() + union { + #declare A = Xunten; + #while (A <= Xoben + 0.5) + #declare B = Zunten; + #while (B <= Zoben + 0.5) + #declare Y = A + B; + #if ((Y >= Yunten - 0.5) & (Y <= Yoben + 0.5)) + sphere { <A, Y, B>, r } + #end + #declare B = B + 1; + #end + #declare A = A + 1; + #end + pigment { + color blau + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro HauptidealR() + intersection { + kasten() + #declare n = vnormalize(< 1, -2, 1 >); + plane { n, 0.1 } + plane { -n, 0.1 } + pigment { + color blau + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro Ideal2() + union { + #declare X = Xunten; + #while (X <= Xoben + 0.5) + #declare Y = Yunten; + #while (Y <= Yoben + 0.5) + #declare Z = Zunten; + #while (Z <= Zoben + 0.5) + sphere { <X, Y, Z>, r } + #declare Z = Z + 2; + #end + #declare Y = Y + 2; + #end + #declare X = X + 2; + #end + pigment { + color gruen + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro IdealX() + union { + #declare Y = Yunten; + #while (Y <= Yoben + 0.5) + #declare Z = Zunten; + #while (Z <= Zoben + 0.5) + sphere { <0, Y, Z>, r } + #declare Z = Z + 1; + #end + #declare Y = Y + 1; + #end + pigment { + color gruen + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro IdealXR() + intersection { + kasten() + plane { <1,0,0>, 0.1 } + plane { <-1,0,0>, 0.1 } + pigment { + color gruen + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro Nichthauptideal() + union { + #declare X = Xunten/2; + #while (X <= Xoben/2 + 0.5) + #declare Y = Yunten; + #while (Y <= Yoben + 0.5) + #declare Z = 0; + #while (Z <= Zoben + 0.5) + sphere { <2*X,Y,Z>, r } + #declare Z = Z + 1; + #end + #declare Y = Y + 1; + #end + #declare X = X + 1; + #end + pigment { + color gruen + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro NichthauptidealKomplement() + union { + #declare X = Xunten + 1; + #while (X <= Xoben + 0.5) + #declare Y = Yunten; + #while (Y <= Yoben + 0.5) + #declare Z = Zunten; + #while (Z <= Zoben + 0.5) + sphere { <X,Y,Z>, r } + #declare Z = Z + 1; + #end + #declare Y = Y + 1; + #end + #declare X = X + 2; + #end + pigment { + color rot + } + finish { + specular 0.9 + metallic + } + } +#end + + + + + + + diff --git a/vorlesungen/slides/3/images/hauptideal.jpg b/vorlesungen/slides/3/images/hauptideal.jpg Binary files differnew file mode 100644 index 0000000..769f53c --- /dev/null +++ b/vorlesungen/slides/3/images/hauptideal.jpg diff --git a/vorlesungen/slides/3/images/hauptideal.pov b/vorlesungen/slides/3/images/hauptideal.pov new file mode 100644 index 0000000..a934e57 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptideal.pov @@ -0,0 +1,10 @@ +// +// hauptideal.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +Hauptideal() + diff --git a/vorlesungen/slides/3/images/hauptideal2.jpg b/vorlesungen/slides/3/images/hauptideal2.jpg Binary files differnew file mode 100644 index 0000000..51823f3 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptideal2.jpg diff --git a/vorlesungen/slides/3/images/hauptideal2.pov b/vorlesungen/slides/3/images/hauptideal2.pov new file mode 100644 index 0000000..9da5a1a --- /dev/null +++ b/vorlesungen/slides/3/images/hauptideal2.pov @@ -0,0 +1,10 @@ +// +// hauptideal2.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +Ideal2() + diff --git a/vorlesungen/slides/3/images/hauptidealR.jpg b/vorlesungen/slides/3/images/hauptidealR.jpg Binary files differnew file mode 100644 index 0000000..fae5840 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealR.jpg diff --git a/vorlesungen/slides/3/images/hauptidealR.pov b/vorlesungen/slides/3/images/hauptidealR.pov new file mode 100644 index 0000000..330e523 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealR.pov @@ -0,0 +1,10 @@ +// +// hauptidealR.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +HauptidealR() + diff --git a/vorlesungen/slides/3/images/hauptidealX.jpg b/vorlesungen/slides/3/images/hauptidealX.jpg Binary files differnew file mode 100644 index 0000000..f9b4540 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealX.jpg diff --git a/vorlesungen/slides/3/images/hauptidealX.pov b/vorlesungen/slides/3/images/hauptidealX.pov new file mode 100644 index 0000000..d0045f9 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealX.pov @@ -0,0 +1,10 @@ +// +// hauptidealX.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +IdealX() + diff --git a/vorlesungen/slides/3/images/hauptidealXR.jpg b/vorlesungen/slides/3/images/hauptidealXR.jpg Binary files differnew file mode 100644 index 0000000..d8906c8 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealXR.jpg diff --git a/vorlesungen/slides/3/images/hauptidealXR.pov b/vorlesungen/slides/3/images/hauptidealXR.pov new file mode 100644 index 0000000..5daa3e6 --- /dev/null +++ b/vorlesungen/slides/3/images/hauptidealXR.pov @@ -0,0 +1,10 @@ +// +// hauptidealXR.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +IdealXR() + diff --git a/vorlesungen/slides/3/images/ideal.ini b/vorlesungen/slides/3/images/ideal.ini new file mode 100644 index 0000000..66aa191 --- /dev/null +++ b/vorlesungen/slides/3/images/ideal.ini @@ -0,0 +1,7 @@ +Input_File_Name=ideal.pov +Initial_Frame=0 +Final_Frame=2500 +Initial_Clock=0 +Final_Clock=5 +Cyclic_Animation=off +Pause_when_Done=off diff --git a/vorlesungen/slides/3/images/ideal.pov b/vorlesungen/slides/3/images/ideal.pov new file mode 100644 index 0000000..88afaf7 --- /dev/null +++ b/vorlesungen/slides/3/images/ideal.pov @@ -0,0 +1,26 @@ +// +// ideal.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = clock; +#include "common.inc" + +#if (T < 1) +Zring() +#else + #if (T < 2) + Hauptideal() + #else + #if (T < 3) + Ideal2() + #else + #if (T < 4) + IdealX() + #else + Nichthauptideal() + NichthauptidealKomplement() + #end + #end + #end +#end diff --git a/vorlesungen/slides/3/images/nichthauptideal.jpg b/vorlesungen/slides/3/images/nichthauptideal.jpg Binary files differnew file mode 100644 index 0000000..55858d0 --- /dev/null +++ b/vorlesungen/slides/3/images/nichthauptideal.jpg diff --git a/vorlesungen/slides/3/images/nichthauptideal.pov b/vorlesungen/slides/3/images/nichthauptideal.pov new file mode 100644 index 0000000..72a6330 --- /dev/null +++ b/vorlesungen/slides/3/images/nichthauptideal.pov @@ -0,0 +1,10 @@ +// +// hauptideal.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +Nichthauptideal() +NichthauptidealKomplement() diff --git a/vorlesungen/slides/3/images/ring.jpg b/vorlesungen/slides/3/images/ring.jpg Binary files differnew file mode 100644 index 0000000..27721b1 --- /dev/null +++ b/vorlesungen/slides/3/images/ring.jpg diff --git a/vorlesungen/slides/3/images/ring.pov b/vorlesungen/slides/3/images/ring.pov new file mode 100644 index 0000000..f854335 --- /dev/null +++ b/vorlesungen/slides/3/images/ring.pov @@ -0,0 +1,10 @@ +// +// ring.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#declare T = 0; +#include "common.inc" + +Zring() + diff --git a/vorlesungen/slides/3/nichthauptideal.tex b/vorlesungen/slides/3/nichthauptideal.tex new file mode 100644 index 0000000..46074b9 --- /dev/null +++ b/vorlesungen/slides/3/nichthauptideal.tex @@ -0,0 +1,78 @@ +% +% nichthauptideal.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Nicht-Hauptideal in $\mathbb{Z}[X]$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Hauptideal\uncover<2->{ = ``Gerade''}} +\vspace{-10pt} +\begin{align*} +\langle X+1\rangle&=(X+1) = {\color{red}(X+1)\cdot\mathbb{Z}[X]} +\end{align*} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.4] +\draw[->] (-6.3,0) -- (6.8,0) coordinate[label={$\mathbb{Z}$}]; +\draw[->] (0,-6.2) -- (0,6.6) coordinate[label={right:$\mathbb{Z}X$}]; +\foreach \x in {-6,...,6}{ + \fill[color=red] (\x,\x) circle[radius=0.12]; +} +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Ideal mit zwei Erzeugenden} +\vspace{-10pt} +\begin{align*} +\uncover<6->{ +{\color{darkgreen} +\langle 2,X\rangle +} +&=} +\uncover<5->{ +{\color{red}2\cdot\mathbb{Z}[X]} +} +\uncover<6->{+} +\uncover<4->{ +{\color{blue}X\cdot\mathbb{Z}[X]} +} +\end{align*} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=0.4] +\draw[->] (-6.3,0) -- (6.9,0) coordinate[label={$\mathbb{Z}$}]; +\draw[->] (0,-6.2) -- (0,7.0) coordinate[label={right:$\mathbb{Z}X$}]; +\uncover<6->{ + \foreach \x in {-6,-4,...,6}{ + \foreach \y in {-6,...,6}{ + \fill[color=darkgreen] (\x,\y) circle[radius=0.20]; + } + } +} +\uncover<5->{ + \foreach \x in {-6,-4,...,6}{ + \foreach \y in {-6,-4,...,6}{ + \fill[color=red] (\x,\y) circle[radius=0.16]; + } + } +} +\uncover<4->{ + \foreach \y in {-6,...,6}{ + \fill[color=blue] (0,\y) circle[radius=0.12]; + } +} +\end{tikzpicture} +\end{center} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/3/nichthauptideal2.tex b/vorlesungen/slides/3/nichthauptideal2.tex new file mode 100644 index 0000000..f362a4a --- /dev/null +++ b/vorlesungen/slides/3/nichthauptideal2.tex @@ -0,0 +1,92 @@ +% +% nichthauptideal2.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\frametitle{Das Ideal $\langle 2,X\rangle \subset \mathbb{Z}[X]$} +\vspace{-12pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\c{\clip (-2.8,-2.0) rectangle (2.8,2.0);} + +\def\labels{ + \fill[color=white,opacity=0.5] (1.5,-0.1) circle[radius=0.2]; + \node at (1.5,-0.1) {$1$}; + \fill[color=white,opacity=0.5] (-0.9,1.7) circle[radius=0.2]; + \node at (-0.9,1.7) {$X$}; + \fill[color=white,opacity=0.5] (0.8,0.7) circle[radius=0.2]; + \node at (0.8,0.7) {$X^2$}; +} + +\only<-3>{ +\begin{scope}[xshift=3.0cm,yshift=1.9cm] + \begin{scope} + \c + \node at (0,0) + {\includegraphics[width=7cm]{../slides/3/images/ring.jpg}}; + \end{scope} + \node[color=orange] at (1.9,0.1) [right] {$\mathbb{Z}[X]$}; + \fill[color=white,opacity=0.5] (1.5,-0.6) circle[radius=0.2]; + \node at (1.5,-0.6) {$1$}; + \fill[color=white,opacity=0.5] (-0.4,1.7) circle[radius=0.2]; + \node at (-0.4,1.7) {$X$}; +\end{scope} +} + +\uncover<2->{ +\begin{scope}[xshift=-3.0cm,yshift=1.9cm] + \begin{scope} + \c + \node at (0,0) + {\includegraphics[width=7cm]{../slides/3/images/hauptideal.jpg}}; + \end{scope} + \node[color=blue] at (-0.2,-1.2) {$(X+1)\cdot\mathbb{Z}[X]$}; + \labels +\end{scope} +} + +\uncover<3->{ +\begin{scope}[xshift=-3.0cm,yshift=-1.9cm] + \begin{scope} + \c + \node at (0,0) + {\includegraphics[width=7cm]{../slides/3/images/hauptideal2.jpg}}; + \end{scope} + \node[color=darkgreen] at (-3.0,-0.8) {$2\cdot\mathbb{Z}[X]$}; +\end{scope} + +\begin{scope}[xshift=3.0cm,yshift=-1.9cm] + \begin{scope} + \c + \node at (0,0) + {\includegraphics[width=7cm]{../slides/3/images/hauptidealX.jpg}}; + \end{scope} + \node[color=darkgreen] at (2.5,-0.8) {$X\cdot\mathbb{Z}[X]$}; + \labels +\end{scope} +} + +\uncover<4->{ +\begin{scope}[xshift=3.0cm,yshift=1.9cm] + \begin{scope} + \c + \node at (0,0) + {\includegraphics[width=7cm]{../slides/3/images/nichthauptideal.jpg}}; + \end{scope} + \node[color=orange] at (1.9,0.1) [right] {$\mathbb{Z}[X]$}; + \node[color=darkgreen] at (1.9,-0.4) [right] {$\langle 2,X\rangle$}; +\end{scope} +} + +\draw[color=gray!50] (-6.6,0) -- (6.4,0); +\draw[color=gray!50] (0,-3.8) -- (0,3.8); + +\end{tikzpicture} +\end{center} + +\end{frame} +\egroup diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 9402625..276b978 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -3,4 +3,6 @@ % % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % -\folie{3/wurzel2.tex} +%\folie{3/wurzel2.tex} +\folie{3/nichthauptideal.tex} +\folie{3/nichthauptideal2.tex} |
