From c2327f731ba243a91d47c48ad14ecbcd3b058caf Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Tue, 23 Mar 2021 23:42:48 +0100 Subject: Change title and authors, remove sample --- buch/papers/punktgruppen/Makefile.inc | 12 +++----- buch/papers/punktgruppen/main.tex | 30 ++----------------- buch/papers/punktgruppen/teil0.tex | 22 -------------- buch/papers/punktgruppen/teil1.tex | 55 ----------------------------------- buch/papers/punktgruppen/teil2.tex | 40 ------------------------- buch/papers/punktgruppen/teil3.tex | 40 ------------------------- 6 files changed, 7 insertions(+), 192 deletions(-) delete mode 100644 buch/papers/punktgruppen/teil0.tex delete mode 100644 buch/papers/punktgruppen/teil1.tex delete mode 100644 buch/papers/punktgruppen/teil2.tex delete mode 100644 buch/papers/punktgruppen/teil3.tex (limited to 'buch/papers') diff --git a/buch/papers/punktgruppen/Makefile.inc b/buch/papers/punktgruppen/Makefile.inc index 7c6e70d..629abca 100644 --- a/buch/papers/punktgruppen/Makefile.inc +++ b/buch/papers/punktgruppen/Makefile.inc @@ -3,12 +3,8 @@ # # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -dependencies-punktgruppen = \ - papers/punktgruppen/packages.tex \ - papers/punktgruppen/main.tex \ - papers/punktgruppen/references.bib \ - papers/punktgruppen/teil0.tex \ - papers/punktgruppen/teil1.tex \ - papers/punktgruppen/teil2.tex \ - papers/punktgruppen/teil3.tex +dependencies-punktgruppen = \ + papers/punktgruppen/packages.tex \ + papers/punktgruppen/main.tex \ + papers/punktgruppen/references.bib diff --git a/buch/papers/punktgruppen/main.tex b/buch/papers/punktgruppen/main.tex index fc91913..71e9a92 100644 --- a/buch/papers/punktgruppen/main.tex +++ b/buch/papers/punktgruppen/main.tex @@ -3,34 +3,10 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:punktgruppen}} -\lhead{Thema} +\chapter{Crystal M\rotatebox[origin=c]{180}{a}th\label{chapter:punktgruppen}} +\lhead{Crystal M\rotatebox[origin=c]{180}{a}th} \begin{refsection} -\chapterauthor{Hans Muster} - -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} - -\input{papers/punktgruppen/teil0.tex} -\input{papers/punktgruppen/teil1.tex} -\input{papers/punktgruppen/teil2.tex} -\input{papers/punktgruppen/teil3.tex} +\chapterauthor{Tim T\"onz, Naoki Pross} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/punktgruppen/teil0.tex b/buch/papers/punktgruppen/teil0.tex deleted file mode 100644 index 5a8278e..0000000 --- a/buch/papers/punktgruppen/teil0.tex +++ /dev/null @@ -1,22 +0,0 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{punktgruppen:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{punktgruppen:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - - diff --git a/buch/papers/punktgruppen/teil1.tex b/buch/papers/punktgruppen/teil1.tex deleted file mode 100644 index 228af33..0000000 --- a/buch/papers/punktgruppen/teil1.tex +++ /dev/null @@ -1,55 +0,0 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{punktgruppen:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{punktgruppen:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{punktgruppen:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{punktgruppen:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{punktgruppen:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. - - diff --git a/buch/papers/punktgruppen/teil2.tex b/buch/papers/punktgruppen/teil2.tex deleted file mode 100644 index b48e785..0000000 --- a/buch/papers/punktgruppen/teil2.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{punktgruppen:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{punktgruppen:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/punktgruppen/teil3.tex b/buch/papers/punktgruppen/teil3.tex deleted file mode 100644 index 94abd74..0000000 --- a/buch/papers/punktgruppen/teil3.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{punktgruppen:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{punktgruppen:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - -- cgit v1.2.1 From cb47736dc6b9eba6917914b273230b8a50a8c5da Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Sat, 10 Apr 2021 18:23:01 +0200 Subject: Add outline --- buch/papers/punktgruppen/main.tex | 28 ++++++++++++++++++++++++++++ buch/papers/punktgruppen/packages.tex | 5 +---- 2 files changed, 29 insertions(+), 4 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/punktgruppen/main.tex b/buch/papers/punktgruppen/main.tex index 71e9a92..603f293 100644 --- a/buch/papers/punktgruppen/main.tex +++ b/buch/papers/punktgruppen/main.tex @@ -8,5 +8,33 @@ \begin{refsection} \chapterauthor{Tim T\"onz, Naoki Pross} +%% TODO: remove +%% Some ideas to motivate the topic: +%% - Physics in a crystal lattice structure +%% - Birifrencenge and scattering of light / Xray in Crystals +%% - Electron density function in a lattice +%% - Heat diffusion with lattice model +%% - Ising model for ferromagnetism (?? => H.D. Lang) +%% +%% - Homomorphic encryption (or lattice based cryptography) +%% + Q: Is it possible to edit encrypted data without decrypting it first? + +%% TODO: translated and move into a file {{{ + +\section{Motivation} +% birifrengence + +\section{Math} +% lattice group +% symmetry +% space group + +\section{Physics} +\subsection{Electromagnetic Waves} +\subsection{Crystal Lattice} + + +%% }}} + \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/punktgruppen/packages.tex b/buch/papers/punktgruppen/packages.tex index 971bcfe..9953339 100644 --- a/buch/papers/punktgruppen/packages.tex +++ b/buch/papers/punktgruppen/packages.tex @@ -4,7 +4,4 @@ % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % -% if your paper needs special packages, add package commands as in the -% following example -%\usepackage{packagename} - +\usepackage{tikz-3dplot} -- cgit v1.2.1 From 23326eb7047812366848812919aebf85c04f589e Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Tue, 20 Apr 2021 12:30:50 +0200 Subject: Presentation added --- buch/papers/reedsolomon/RS presentation/RS.aux | 30 + buch/papers/reedsolomon/RS presentation/RS.log | 956 +++++++++++++++++++++ buch/papers/reedsolomon/RS presentation/RS.nav | 9 + buch/papers/reedsolomon/RS presentation/RS.out | 0 buch/papers/reedsolomon/RS presentation/RS.pdf | Bin 0 -> 53965 bytes buch/papers/reedsolomon/RS presentation/RS.snm | 0 .../reedsolomon/RS presentation/RS.synctex.gz | Bin 0 -> 3637 bytes buch/papers/reedsolomon/RS presentation/RS.tex | 25 + buch/papers/reedsolomon/RS presentation/RS.toc | 1 + buch/papers/reedsolomon/RS presentation/Thumbs.db | Bin 0 -> 89088 bytes 10 files changed, 1021 insertions(+) create mode 100644 buch/papers/reedsolomon/RS presentation/RS.aux create mode 100644 buch/papers/reedsolomon/RS presentation/RS.log create mode 100644 buch/papers/reedsolomon/RS presentation/RS.nav create mode 100644 buch/papers/reedsolomon/RS presentation/RS.out create mode 100644 buch/papers/reedsolomon/RS presentation/RS.pdf create mode 100644 buch/papers/reedsolomon/RS presentation/RS.snm create mode 100644 buch/papers/reedsolomon/RS presentation/RS.synctex.gz create mode 100644 buch/papers/reedsolomon/RS presentation/RS.tex create mode 100644 buch/papers/reedsolomon/RS presentation/RS.toc create mode 100644 buch/papers/reedsolomon/RS presentation/Thumbs.db (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/RS presentation/RS.aux b/buch/papers/reedsolomon/RS presentation/RS.aux new file mode 100644 index 0000000..17ce46b --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS.aux @@ -0,0 +1,30 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel 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+\usetheme{Hannover} +\begin{document} + \author{Joshua Bär und Michael Steiner} + \title{Reed-Solomon-Code} + \subtitle{} + \logo{} + \institute{OST Ostschweizer Fachhochschule} + \date{26.04.2021} + \subject{Mathematisches Seminar} + \setbeamercovered{transparent} + \setbeamertemplate{navigation symbols}{} + \begin{frame}[plain] + \maketitle + \end{frame} + + \begin{frame} + \frametitle{Test} + Ich mag Züge. + \end{frame} +\end{document} \ No newline at end of file diff --git a/buch/papers/reedsolomon/RS presentation/RS.toc b/buch/papers/reedsolomon/RS presentation/RS.toc new file mode 100644 index 0000000..4cd1c86 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS.toc @@ -0,0 +1 @@ +\babel@toc {ngerman}{} diff --git a/buch/papers/reedsolomon/RS presentation/Thumbs.db b/buch/papers/reedsolomon/RS presentation/Thumbs.db new file mode 100644 index 0000000..1626e26 Binary files /dev/null and b/buch/papers/reedsolomon/RS presentation/Thumbs.db differ -- cgit v1.2.1 From 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Mon Sep 17 00:00:00 2001 From: JODBaer <55744603+JODBaer@users.noreply.github.com> Date: Wed, 21 Apr 2021 11:31:56 +0200 Subject: Creat gitignor --- buch/papers/reedsolomon/.gitignor | 12 ++++++++++++ 1 file changed, 12 insertions(+) create mode 100644 buch/papers/reedsolomon/.gitignor (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/.gitignor b/buch/papers/reedsolomon/.gitignor new file mode 100644 index 0000000..5f0787b --- /dev/null +++ b/buch/papers/reedsolomon/.gitignor @@ -0,0 +1,12 @@ +RS*.aux +RS*.bbl +RS*.bib +RS*.blg +RS*.idx +RS*.ilg +RS*.ind +RS*.log +RS*.out +RS*.pdf +RS*.run.xml +RS*.toc -- cgit v1.2.1 From b804afc336594a0c0a1ecec56c96d02bb97427f4 Mon Sep 17 00:00:00 2001 From: JODBaer <55744603+JODBaer@users.noreply.github.com> Date: Wed, 21 Apr 2021 12:39:57 +0200 Subject: update gitignor --- buch/papers/reedsolomon/.gitignor | 12 ++++++++++++ 1 file changed, 12 insertions(+) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/.gitignor b/buch/papers/reedsolomon/.gitignor index 5f0787b..466d238 100644 --- a/buch/papers/reedsolomon/.gitignor +++ b/buch/papers/reedsolomon/.gitignor @@ -10,3 +10,15 @@ RS*.out RS*.pdf RS*.run.xml RS*.toc +*.aux +*.lof +*.log +*.lot +*.fls +*.out +*.toc +*.fmt +*.fot +*.cb +*.cb2 +.*.lb -- cgit v1.2.1 From 44b5dcffb75c9f7dc0d28fd5af9794608cd9b395 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Wed, 21 Apr 2021 12:47:00 +0200 Subject: Presentation#1 --- buch/papers/reedsolomon/RS presentation/RS.aux | 29 +- buch/papers/reedsolomon/RS presentation/RS.bbl | 0 buch/papers/reedsolomon/RS presentation/RS.log | 698 +++++++++++-------- buch/papers/reedsolomon/RS presentation/RS.nav | 19 +- buch/papers/reedsolomon/RS presentation/RS.out | 1 + buch/papers/reedsolomon/RS presentation/RS.pdf | Bin 53965 -> 117082 bytes .../reedsolomon/RS presentation/RS.synctex.gz | Bin 3637 -> 6763 bytes buch/papers/reedsolomon/RS presentation/RS.tex | 50 +- buch/papers/reedsolomon/RS presentation/RS.toc | 1 + buch/papers/reedsolomon/RS presentation/Thumbs.db | Bin 89088 -> 0 bytes .../reedsolomon/RS presentation/images/fig1.pdf | Bin 3071 -> 11898 bytes .../RS presentation/images/fig1.pdf_tex | 81 --- .../reedsolomon/RS presentation/images/fig1.png | Bin 27373 -> 0 bytes .../reedsolomon/RS presentation/images/fig2.pdf | Bin 0 -> 13901 bytes .../reedsolomon/RS presentation/images/fig2.png | Bin 30489 -> 0 bytes .../reedsolomon/RS presentation/images/fig3.pdf | Bin 0 -> 13099 bytes .../reedsolomon/RS presentation/images/fig3.png | Bin 16007 -> 0 bytes .../reedsolomon/RS presentation/images/fig4.pdf | Bin 0 -> 14995 bytes .../reedsolomon/RS presentation/images/fig4.png | Bin 27548 -> 0 bytes .../reedsolomon/RS presentation/images/fig5.pdf | Bin 0 -> 13298 bytes .../reedsolomon/RS presentation/images/fig5.png | Bin 30167 -> 0 bytes .../reedsolomon/RS presentation/images/fig6.pdf | Bin 0 -> 13688 bytes .../reedsolomon/RS presentation/images/fig6.png | Bin 22604 -> 0 bytes .../reedsolomon/RS 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buch/papers/reedsolomon/RS presentation/images/fig1.png create mode 100644 buch/papers/reedsolomon/RS presentation/images/fig2.pdf delete mode 100644 buch/papers/reedsolomon/RS presentation/images/fig2.png create mode 100644 buch/papers/reedsolomon/RS presentation/images/fig3.pdf delete mode 100644 buch/papers/reedsolomon/RS presentation/images/fig3.png create mode 100644 buch/papers/reedsolomon/RS presentation/images/fig4.pdf delete mode 100644 buch/papers/reedsolomon/RS presentation/images/fig4.png create mode 100644 buch/papers/reedsolomon/RS presentation/images/fig5.pdf delete mode 100644 buch/papers/reedsolomon/RS presentation/images/fig5.png create mode 100644 buch/papers/reedsolomon/RS presentation/images/fig6.pdf delete mode 100644 buch/papers/reedsolomon/RS presentation/images/fig6.png create mode 100644 buch/papers/reedsolomon/RS presentation/images/fig7.pdf delete mode 100644 buch/papers/reedsolomon/RS presentation/images/fig7.png create mode 100644 buch/papers/reedsolomon/RS presentation/images/polynom1.aux create mode 100644 buch/papers/reedsolomon/RS presentation/images/polynom1.log create mode 100644 buch/papers/reedsolomon/RS presentation/images/polynom1.pdf create mode 100644 buch/papers/reedsolomon/RS presentation/images/polynom1.synctex.gz create mode 100644 buch/papers/reedsolomon/RS presentation/images/polynom1.tex create mode 100644 buch/papers/reedsolomon/RS presentation/images/polynom2.aux create mode 100644 buch/papers/reedsolomon/RS presentation/images/polynom2.log create mode 100644 buch/papers/reedsolomon/RS presentation/images/polynom2.pdf create mode 100644 buch/papers/reedsolomon/RS presentation/images/polynom2.synctex.gz create mode 100644 buch/papers/reedsolomon/RS presentation/images/polynom2.tex (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/RS presentation/RS.aux b/buch/papers/reedsolomon/RS presentation/RS.aux index 17ce46b..fff632d 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.aux +++ b/buch/papers/reedsolomon/RS presentation/RS.aux @@ -14,17 +14,28 @@ \fi} \global\let\hyper@last\relax \gdef\HyperFirstAtBeginDocument#1{#1} -\providecommand\HyField@AuxAddToFields[1]{} -\providecommand\HyField@AuxAddToCoFields[2]{} -\@nameuse{bbl@beforestart} +\providecommand*\HyPL@Entry[1]{} +\bbl@beforestart \catcode `"\active +\HyPL@Entry{0<>} \babel@aux{ngerman}{} \@writefile{nav}{\headcommand {\slideentry {0}{0}{1}{1/1}{}{0}}} \@writefile{nav}{\headcommand {\beamer@framepages {1}{1}}} -\@writefile{nav}{\headcommand {\slideentry {0}{0}{2}{2/2}{}{0}}} +\HyPL@Entry{1<>} +\@writefile{toc}{\beamer@sectionintoc {1}{Introduction}{2}{0}{1}} +\@writefile{nav}{\headcommand {\beamer@sectionpages {1}{1}}} +\@writefile{nav}{\headcommand {\beamer@subsectionpages {1}{1}}} +\@writefile{nav}{\headcommand {\sectionentry {1}{Introduction}{2}{Introduction}{0}}} +\@writefile{nav}{\headcommand {\slideentry {1}{0}{1}{2/2}{}{0}}} \@writefile{nav}{\headcommand {\beamer@framepages 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b/buch/papers/reedsolomon/RS presentation/RS.bbl deleted file mode 100644 index e69de29..0000000 diff --git a/buch/papers/reedsolomon/RS presentation/RS.log b/buch/papers/reedsolomon/RS presentation/RS.log index f7dc931..824b9b5 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.log +++ b/buch/papers/reedsolomon/RS presentation/RS.log @@ -1,10 +1,10 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/W32TeX) (preloaded format=pdflatex 2019.11.30) 20 APR 2021 12:21 +This is XeTeX, Version 3.14159265-2.6-0.999991 (TeX Live 2019/W32TeX) (preloaded format=xelatex 2019.10.25) 21 APR 2021 12:30 entering extended mode restricted \write18 enabled. %&-line parsing enabled. **RS.tex (./RS.tex -LaTeX2e <2019-10-01> patch level 3 +LaTeX2e <2019-10-01> patch level 1 (c:/texlive/2019/texmf-dist/tex/latex/beamer/beamer.cls Document Class: beamer 2019/09/29 v3.57 A class for typesetting presentations (c:/texlive/2019/texmf-dist/tex/latex/beamer/beamerbasemodes.sty @@ -24,12 +24,9 @@ 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Use iftex instead. - -(c:/texlive/2019/texmf-dist/tex/generic/iftex/iftex.sty -Package: iftex 2019/11/07 v1.0c TeX engine tests -)) +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/ifpdf.sty +Package: ifpdf 2018/09/07 v3.3 Provides the ifpdf switch +) \headdp=\dimen102 \footheight=\dimen103 \sidebarheight=\dimen104 @@ -84,11 +81,12 @@ Package: keyval 2014/10/28 v1.15 key=value parser (DPC) (c:/texlive/2019/texmf-dist/tex/latex/geometry/geometry.sty Package: geometry 2018/04/16 v5.8 Page Geometry -(c:/texlive/2019/texmf-dist/tex/generic/iftex/ifvtex.sty -Package: ifvtex 2019/10/25 v1.7 ifvtex legacy package. Use iftex instead. +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/ifvtex.sty +Package: ifvtex 2016/05/16 v1.6 Detect VTeX and its facilities (HO) +Package ifvtex Info: VTeX not detected. ) -(c:/texlive/2019/texmf-dist/tex/generic/iftex/ifxetex.sty -Package: ifxetex 2019/10/25 v0.7 ifxetex legacy package. Use iftex instead. +(c:/texlive/2019/texmf-dist/tex/generic/ifxetex/ifxetex.sty +Package: ifxetex 2010/09/12 v0.6 Provides ifxetex conditional ) \Gm@cnth=\count105 \Gm@cntv=\count106 @@ -104,14 +102,14 @@ Package: ifxetex 2019/10/25 v0.7 ifxetex legacy package. Use iftex instead. \Gm@dimlist=\toks15 ) (c:/texlive/2019/texmf-dist/tex/latex/base/size11.clo -File: size11.clo 2019/10/25 v1.4k Standard LaTeX file (size option) +File: size11.clo 2019/08/27 v1.4j Standard LaTeX file (size option) ) (c:/texlive/2019/texmf-dist/tex/latex/pgf/basiclayer/pgfcore.sty (c:/texlive/2019/texmf-dist/tex/latex/graphics/graphicx.sty Package: graphicx 2017/06/01 v1.1a Enhanced LaTeX Graphics (DPC,SPQR) (c:/texlive/2019/texmf-dist/tex/latex/graphics/graphics.sty -Package: graphics 2019/11/01 v1.3d Standard LaTeX Graphics (DPC,SPQR) +Package: graphics 2019/10/08 v1.3c Standard LaTeX Graphics (DPC,SPQR) (c:/texlive/2019/texmf-dist/tex/latex/graphics/trig.sty Package: trig 2016/01/03 v1.10 sin cos tan (DPC) @@ -119,10 +117,10 @@ Package: trig 2016/01/03 v1.10 sin cos tan (DPC) (c:/texlive/2019/texmf-dist/tex/latex/graphics-cfg/graphics.cfg File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration ) -Package graphics Info: Driver file: pdftex.def on input line 105. +Package graphics Info: Driver file: xetex.def on input line 105. -(c:/texlive/2019/texmf-dist/tex/latex/graphics-def/pdftex.def -File: pdftex.def 2018/01/08 v1.0l Graphics/color driver for pdftex +(c:/texlive/2019/texmf-dist/tex/latex/graphics-def/xetex.def +File: xetex.def 2017/06/24 v5.0h Graphics/color driver for xetex )) \Gin@req@height=\dimen126 \Gin@req@width=\dimen127 @@ -178,18 +176,23 @@ Package: pgfsys 2019/08/03 v3.1.4b (3.1.4b) (c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgf.cfg File: pgf.cfg 2019/08/03 v3.1.4b (3.1.4b) ) -Driver file for pgf: pgfsys-pdftex.def +Driver file for pgf: pgfsys-xetex.def + +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-xetex.def +File: pgfsys-xetex.def 2019/08/03 v3.1.4b (3.1.4b) -(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def -File: pgfsys-pdftex.def 2019/08/03 v3.1.4b (3.1.4b) +(c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-dvipdfmx.def +File: pgfsys-dvipdfmx.def 2019/08/03 v3.1.4b (3.1.4b) (c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-common-pdf.def File: pgfsys-common-pdf.def 2019/08/03 v3.1.4b (3.1.4b) +) +\pgfsys@objnum=\count113 ))) (c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsyssoftpath.code.tex File: pgfsyssoftpath.code.tex 2019/08/03 v3.1.4b (3.1.4b) -\pgfsyssoftpath@smallbuffer@items=\count113 -\pgfsyssoftpath@bigbuffer@items=\count114 +\pgfsyssoftpath@smallbuffer@items=\count114 +\pgfsyssoftpath@bigbuffer@items=\count115 ) (c:/texlive/2019/texmf-dist/tex/generic/pgf/systemlayer/pgfsysprotocol.code.tex File: pgfsysprotocol.code.tex 2019/08/03 v3.1.4b (3.1.4b) @@ -199,9 +202,8 @@ Package: xcolor 2016/05/11 v2.12 LaTeX color extensions (UK) (c:/texlive/2019/texmf-dist/tex/latex/graphics-cfg/color.cfg File: color.cfg 2016/01/02 v1.6 sample color configuration ) -Package xcolor Info: Driver file: pdftex.def on input line 225. +Package xcolor Info: Driver file: xetex.def on input line 225. Package xcolor Info: Model `cmy' substituted by `cmy0' on input line 1348. -Package xcolor Info: Model `hsb' substituted by `rgb' on input line 1352. Package xcolor Info: Model `RGB' extended on input line 1364. Package xcolor Info: Model `HTML' substituted by `rgb' on input line 1366. Package xcolor Info: Model `Hsb' substituted by `hsb' on input line 1367. @@ -218,7 +220,7 @@ Package: pgfcore 2019/08/03 v3.1.4b (3.1.4b) (c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathutil.code.tex) (c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathparser.code.tex \pgfmath@dimen=\dimen140 -\pgfmath@count=\count115 +\pgfmath@count=\count116 \pgfmath@box=\box37 \pgfmath@toks=\toks23 \pgfmath@stack@operand=\toks24 @@ -242,7 +244,7 @@ x) (c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.integerarithm etics.code.tex))) (c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfmathfloat.code.tex -\c@pgfmathroundto@lastzeros=\count116 +\c@pgfmathroundto@lastzeros=\count117 )) (c:/texlive/2019/texmf-dist/tex/generic/pgf/math/pgfint.code.tex) (c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcorepoints.code.tex @@ -279,7 +281,7 @@ File: pgfcorescopes.code.tex 2019/08/03 v3.1.4b (3.1.4b) \pgfpic=\box38 \pgf@hbox=\box39 \pgf@layerbox@main=\box40 -\pgf@picture@serial@count=\count117 +\pgf@picture@serial@count=\count118 ) (c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoregraphicstate.code .tex @@ -310,8 +312,8 @@ File: pgfcorearrows.code.tex 2019/08/03 v3.1.4b (3.1.4b) (c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreshade.code.tex File: pgfcoreshade.code.tex 2019/08/03 v3.1.4b (3.1.4b) \pgf@max=\dimen164 -\pgf@sys@shading@range@num=\count118 -\pgf@shadingcount=\count119 +\pgf@sys@shading@range@num=\count119 +\pgf@shadingcount=\count120 ) (c:/texlive/2019/texmf-dist/tex/generic/pgf/basiclayer/pgfcoreimage.code.tex File: pgfcoreimage.code.tex 2019/08/03 v3.1.4b (3.1.4b) @@ -333,8 +335,8 @@ File: pgfcorepatterns.code.tex 2019/08/03 v3.1.4b (3.1.4b) File: pgfcorerdf.code.tex 2019/08/03 v3.1.4b (3.1.4b) ))) (c:/texlive/2019/texmf-dist/tex/latex/pgf/utilities/xxcolor.sty Package: xxcolor 2003/10/24 ver 0.1 -\XC@nummixins=\count120 -\XC@countmixins=\count121 +\XC@nummixins=\count121 +\XC@countmixins=\count122 ) (c:/texlive/2019/texmf-dist/tex/generic/oberdiek/atbegshi.sty Package: atbegshi 2016/06/09 v1.18 At begin shipout hook (HO) @@ -346,41 +348,46 @@ Package: infwarerr 2016/05/16 v1.4 Providing info/warning/error messages (HO) Package: ltxcmds 2016/05/16 v1.23 LaTeX kernel commands for general use (HO) )) (c:/texlive/2019/texmf-dist/tex/latex/hyperref/hyperref.sty -Package: hyperref 2019/11/10 v7.00c Hypertext links for LaTeX +Package: hyperref 2019/09/28 v7.00a Hypertext links for LaTeX -(c:/texlive/2019/texmf-dist/tex/latex/pdftexcmds/pdftexcmds.sty -Package: pdftexcmds 2019/11/24 v0.31 Utility functions of pdfTeX for LuaTeX (HO -) -Package pdftexcmds Info: \pdf@primitive is available. -Package pdftexcmds Info: \pdf@ifprimitive is available. -Package pdftexcmds Info: \pdfdraftmode found. -) -(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/kvsetkeys.sty -Package: kvsetkeys 2016/05/16 v1.17 Key value parser (HO) +(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/hobsub-hyperref.sty +Package: hobsub-hyperref 2016/05/16 v1.14 Bundle oberdiek, subset hyperref (HO) -(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/etexcmds.sty -Package: etexcmds 2016/05/16 v1.6 Avoid name clashes with e-TeX commands (HO) -(c:/texlive/2019/texmf-dist/tex/generic/iftex/ifluatex.sty -Package: ifluatex 2019/10/25 v1.5 ifluatex legacy package. 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Package: letltxmacro 2016/05/16 v1.5 Let assignment for LaTeX macros (HO) +Package: hopatch 2016/05/16 v1.3 Wrapper for package hooks (HO) +Package: xcolor-patch 2016/05/16 xcolor patch +Package: atveryend 2016/05/16 v1.9 Hooks at the very end of document (HO) +Package hobsub Info: Skipping package `atbegshi' (already loaded). +Package: refcount 2016/05/16 v3.5 Data extraction from label references (HO) +Package: hycolor 2016/05/16 v1.8 Color options for hyperref/bookmark (HO) ) (c:/texlive/2019/texmf-dist/tex/latex/oberdiek/auxhook.sty Package: auxhook 2016/05/16 v1.4 Hooks for auxiliary files (HO) @@ -389,85 +396,79 @@ Package: auxhook 2016/05/16 v1.4 Hooks for auxiliary files (HO) Package: kvoptions 2016/05/16 v3.12 Key value format for package options (HO) ) \@linkdim=\dimen165 -\Hy@linkcounter=\count122 -\Hy@pagecounter=\count123 +\Hy@linkcounter=\count123 +\Hy@pagecounter=\count124 (c:/texlive/2019/texmf-dist/tex/latex/hyperref/pd1enc.def -File: pd1enc.def 2019/11/10 v7.00c Hyperref: PDFDocEncoding definition (HO) -Now handling font encoding PD1 ... -... no UTF-8 mapping file for font encoding PD1 -) -(c:/texlive/2019/texmf-dist/tex/generic/oberdiek/intcalc.sty -Package: intcalc 2016/05/16 v1.2 Expandable calculations with integers (HO) +File: pd1enc.def 2019/09/28 v7.00a Hyperref: PDFDocEncoding definition (HO) ) -\Hy@SavedSpaceFactor=\count124 +\Hy@SavedSpaceFactor=\count125 (c:/texlive/2019/texmf-dist/tex/latex/latexconfig/hyperref.cfg File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive ) -Package hyperref Info: Option `bookmarks' set `true' on input line 4409. -Package hyperref Info: Option `bookmarksopen' set `true' on input line 4409. -Package hyperref Info: Option `implicit' set `false' on input line 4409. -Package hyperref Info: Hyper figures OFF on input line 4535. -Package hyperref Info: Link nesting OFF on input line 4540. -Package hyperref Info: Hyper index ON on input line 4543. -Package hyperref Info: Plain pages OFF on input line 4550. -Package hyperref Info: Backreferencing OFF on input line 4555. +Package hyperref Info: Option `bookmarks' set `true' on input line 4414. +Package hyperref Info: Option `bookmarksopen' set `true' on input line 4414. +Package hyperref Info: Option `implicit' set `false' on input line 4414. +Package hyperref Info: Hyper figures OFF on input line 4540. +Package hyperref Info: Link nesting OFF on input line 4545. +Package hyperref Info: Hyper index ON on input line 4548. +Package hyperref Info: Plain pages OFF on input line 4555. +Package hyperref Info: Backreferencing OFF on input line 4560. 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\section{Introduction} + \begin{frame} + \frametitle{Idee} + + \end{frame} + \begin{frame} - \frametitle{Test} - Ich mag Züge. + \begin{figure} + \only<1>{ + \includegraphics[width=0.9\linewidth]{images/fig1.pdf} + } + \only<2>{ + \includegraphics[width=0.9\linewidth]{images/fig2.pdf} + } + \only<3>{ + \includegraphics[width=0.9\linewidth]{images/fig3.pdf} + } + \only<4>{ + \includegraphics[width=0.9\linewidth]{images/fig4.pdf} + } + \only<5>{ + \includegraphics[width=0.9\linewidth]{images/fig5.pdf} + } + \only<6>{ + \includegraphics[width=0.9\linewidth]{images/fig6.pdf} + } + \only<7>{ + \includegraphics[width=0.9\linewidth]{images/fig7.pdf} + } + \end{figure} \end{frame} + + \begin{frame} + Übertragen von den Zahlen + \textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5} + als $ p(x) = \textcolor{blue}{2}x^2 + \textcolor{blue}{1}x + \textcolor{blue}{5} $.\newline + Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, + \textcolor{green}{15}, \textcolor{green}{26}, + \textcolor{green}{ 41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})$ + \only<1>{ + \includegraphics[]{images/polynom1.pdf}} + \only<2>{ + \includegraphics[]{images/polynom2.pdf}} + \end{frame} + + \end{document} \ No newline at end of file diff --git a/buch/papers/reedsolomon/RS presentation/RS.toc b/buch/papers/reedsolomon/RS presentation/RS.toc index 4cd1c86..32e7e8d 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.toc +++ b/buch/papers/reedsolomon/RS presentation/RS.toc @@ -1 +1,2 @@ \babel@toc {ngerman}{} +\beamer@sectionintoc {1}{Introduction}{2}{0}{1} diff --git a/buch/papers/reedsolomon/RS presentation/Thumbs.db b/buch/papers/reedsolomon/RS presentation/Thumbs.db deleted file mode 100644 index 1626e26..0000000 Binary files a/buch/papers/reedsolomon/RS presentation/Thumbs.db and /dev/null differ diff --git a/buch/papers/reedsolomon/RS presentation/images/fig1.pdf b/buch/papers/reedsolomon/RS presentation/images/fig1.pdf index 5cff7fe..abde60c 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+LaTeX Font Warning: Some font shapes were not available, defaults substituted. + + ) +Here is how much of TeX's memory you used: + 25292 strings out of 492483 + 638772 string characters out of 6132858 + 794987 words of memory out of 5000000 + 29364 multiletter control sequences out of 15000+600000 + 13575 words of font info for 45 fonts, out of 8000000 for 9000 + 1348 hyphenation exceptions out of 8191 + 68i,6n,98p,840b,724s stack positions out of 5000i,500n,10000p,200000b,80000s + +Output written on polynom2.pdf (1 page). diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom2.pdf b/buch/papers/reedsolomon/RS presentation/images/polynom2.pdf new file mode 100644 index 0000000..05f4ba0 Binary files /dev/null and b/buch/papers/reedsolomon/RS presentation/images/polynom2.pdf differ diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom2.synctex.gz b/buch/papers/reedsolomon/RS presentation/images/polynom2.synctex.gz new file mode 100644 index 0000000..24859aa 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\textcolor{green}{83}, \textcolor{green}{110})$ + + + \begin{tikzpicture}[>=latex,thick] + + \draw[color=blue, line width=1.4pt] + plot[domain=0:8, samples=100] + ({\x},{(2*\x^2+1*\x+5)/\teiler}); + \draw[->] (-0.2,0) -- (8,0) coordinate[label={$x$}]; + \draw[->] (0,-0.2) -- (0,150/\teiler) coordinate[label={right:$p(x)$}]; + \def\punkt#1{ + \fill[color=green] #1 circle[radius=0.08]; + \draw #1 circle[radius=0.07]; + } + \punkt{(1,8/\teiler)} + %\punkt{(2,15/\teiler)} + %\punkt{(3,26/\teiler)} + \punkt{(4,41/\teiler)} + \punkt{(5,60/\teiler)} + \punkt{(6,83/\teiler)} + \punkt{(7,110/\teiler)} + \draw[color=gray,line width=1pt,dashed] + plot[domain=0.5:7, samples=100] + ({\x},{(0.1958*\x^2-1.2875*\x+3.0417)}); + \def\erpunkt#1{ + \fill[color=red] #1 circle[radius=0.08]; + \draw #1 circle[radius=0.07]; + } + \erpunkt{(2,50/\teiler)} + \erpunkt{(3,0.9414)} + + + \draw(0,100/\teiler) -- (-0.1,100/\teiler) coordinate[label={left:$100$}]; + \draw(1,0) -- (1,-0.1) coordinate[label={below:$1$}]; + + + + + \end{tikzpicture} +\end{document} -- cgit v1.2.1 From 10f3cdb829c001c341ea31415efb44ff6a2878b8 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Wed, 21 Apr 2021 17:30:50 +0200 Subject: Persentation stand 17:30 --- buch/papers/reedsolomon/RS presentation/RS.aux | 37 +++++- buch/papers/reedsolomon/RS presentation/RS.log | 145 +++++++++++---------- buch/papers/reedsolomon/RS presentation/RS.nav | 26 +++- buch/papers/reedsolomon/RS presentation/RS.out | 4 +- buch/papers/reedsolomon/RS presentation/RS.pdf | Bin 117082 -> 132691 bytes buch/papers/reedsolomon/RS presentation/RS.snm | 1 + .../reedsolomon/RS presentation/RS.synctex.gz | Bin 6763 -> 19501 bytes buch/papers/reedsolomon/RS presentation/RS.tex | 109 ++++++++++++++-- buch/papers/reedsolomon/RS presentation/RS.toc | 4 +- 9 files changed, 235 insertions(+), 91 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/RS presentation/RS.aux b/buch/papers/reedsolomon/RS 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\begin{frame} - Übertragen von den Zahlen - \textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5} - als $ p(x) = \textcolor{blue}{2}x^2 + \textcolor{blue}{1}x + \textcolor{blue}{5} $.\newline - Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, - \textcolor{green}{15}, \textcolor{green}{26}, - \textcolor{green}{ 41}, \textcolor{green}{60}, - \textcolor{green}{83}, \textcolor{green}{110})$ + \uncover<1->{ + Wie ist die Anzahl 0 definiert zum mitgeben? + Indem die Polymereigenschaft genutzt werden. + } + \uncover<2->{ + Wie wird der Fehler lokalisiert? + Indem in einem Endlichen Körper gerechnet wird. + } + + \end{frame} + +\section{Polynom Ansatz} + \begin{frame} + Die Diskrite Fouren Transformation ist so gegeben + \[ + \label{ft_discrete} + \hat{c}_{k} + = \frac{1}{N} \sum_{n=0}^{N-1} + {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} + \]. + + \[ + w = e^{-\frac{2\pi j}{N} k} + \] + Wenn $N$ konstant: + \[ + \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) + \] + \end{frame} + + \begin{frame} + Beispiel 2, 1, 5 Versenden und auf 2 Fehler absichern. + \end{frame} + \begin{frame} + Übertragen von + ${f}_2=$\textcolor{blue}{2}, ${f}_1$\textcolor{blue}{1}, ${f}_0$\textcolor{blue}{5} + als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. + \only<1>{ - \includegraphics[]{images/polynom1.pdf}} + Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, + \textcolor{green}{15}, \textcolor{green}{26}, + \textcolor{green}{ 41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})$ + \includegraphics[scale = 1.2]{images/polynom1.pdf}} \only<2>{ - \includegraphics[]{images/polynom2.pdf}} + Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, + \textcolor{red}{50}, \textcolor{red}{37}, + \textcolor{green}{ 41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})$ + \includegraphics[scale = 1.2]{images/polynom2.pdf} + \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} + \end{frame} + + \begin{frame} + \frametitle{Parameter} + \begin{center} + \begin{tabular}{ c c c } + \hline + "Nutzlast" & Fehler & Versenden \\ + \hline + 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ + 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + &&\\ + k & t & k+2t Werte eines Polynoms vom Grad k-1 \\ + \hline + \end{tabular} + \end{center} + \end{frame} +\section{Diskrete Fourien Transformation} + \begin{frame} + \[ + \begin{pmatrix} + \hat{c}_1 \\\hat{c}_2 \\\hat{c}_3 \\ \vdots \\\hat{c}_n + \end{pmatrix} + = + \begin{pmatrix} + w^0 & w^0 & w^0 & \dots &w^0 \\ + w^0 & w^1 &w^2 & \dots &w^n \\ + w^0 & w^2 &w^4 & \dots &w^{2n} \\ + \vdots & \vdots &\vdots &\ddots &\vdots \\ + w^0 & w^{1n}&w^{2n}& \dots &w^{n} \\ + \end{pmatrix} + \begin{pmatrix} + \textcolor{blue}{5} \\ + \textcolor{blue}{1} \\ + \textcolor{blue}{2} \\ + \vdots \\ + 0 \\ + \end{pmatrix} + \] \end{frame} - \end{document} \ No newline at end of file diff --git a/buch/papers/reedsolomon/RS presentation/RS.toc b/buch/papers/reedsolomon/RS presentation/RS.toc index 32e7e8d..ff200c6 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.toc +++ b/buch/papers/reedsolomon/RS presentation/RS.toc @@ -1,2 +1,4 @@ \babel@toc {ngerman}{} -\beamer@sectionintoc {1}{Introduction}{2}{0}{1} +\beamer@sectionintoc {1}{Einführung}{2}{0}{1} +\beamer@sectionintoc {2}{Polynom Ansatz}{12}{0}{2} +\beamer@sectionintoc {3}{Diskrete Fourien Transformation}{17}{0}{3} -- cgit v1.2.1 From 7c0937851938305c2bb760f3cd4c2084c4493217 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Wed, 21 Apr 2021 18:18:22 +0200 Subject: Presentation neu arangiert --- buch/papers/reedsolomon/RS presentation/RS.tex | 186 +++++++++++++------------ 1 file changed, 96 insertions(+), 90 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 9bdf947..1a1cefd 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -21,12 +21,60 @@ \end{frame} \section{Einführung} \begin{frame} - \frametitle{Idee} + \frametitle{Einführung} \begin{itemize} \item Reed-Solomon-Code beschäftigt sich mit der Übertragung von Daten und deren Fehler Erkennung. - \item Idee Fourier Transformieren und dann senden. - \item Danach Empfangen und Rücktransformieren. + \end{itemize} + \end{frame} +\section{Polynom Ansatz} + \begin{frame} + Beispiel 2, 1, 5 Versenden und auf 2 Fehler absichern. + \end{frame} + \begin{frame} + Übertragen von + ${f}_2=$\textcolor{blue}{2}, ${f}_1$\textcolor{blue}{1}, ${f}_0$\textcolor{blue}{5} + als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. + + \only<1>{ + Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, + \textcolor{green}{15}, \textcolor{green}{26}, + \textcolor{green}{ 41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})$ + \includegraphics[scale = 1.2]{images/polynom1.pdf}} + \only<2>{ + Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, + \textcolor{red}{50}, \textcolor{red}{37}, + \textcolor{green}{ 41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})$ + \includegraphics[scale = 1.2]{images/polynom2.pdf} + \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} + \end{frame} + + \begin{frame} + \frametitle{Parameter} + \begin{center} + \begin{tabular}{ c c c } + \hline + "Nutzlast" & Fehler & Versenden \\ + \hline + 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ + 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + &&\\ + k & t & k+2t Werte eines Polynoms vom Grad k-1 \\ + \hline + \end{tabular} + \end{center} + + Ausserdem können bis zu 2t Fehler erkannt werden! + \end{frame} +\section{Fourier Transformation} + \begin{frame} + \frametitle{Idee} + \begin{itemize} + \item Idee mit Fourier Transformieren und dann senden. + \item Danach Empfangen und Rücktransformieren. \end{itemize} \end{frame} @@ -56,99 +104,57 @@ \end{figure} \end{frame} - +\section{Diskrete Fourier Transformation} \begin{frame} - \uncover<1->{ - Wie ist die Anzahl 0 definiert zum mitgeben? - Indem die Polymereigenschaft genutzt werden. - } - \uncover<2->{ - Wie wird der Fehler lokalisiert? - Indem in einem Endlichen Körper gerechnet wird. - } - + \frametitle{Diskrete Fourier Transformation} + Die Diskrete Fourier Transformation ist so gegeben: + \[ + \label{ft_discrete} + \hat{c}_{k} + = \frac{1}{N} \sum_{n=0}^{N-1} + {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} + \]. + + \[ + w = e^{-\frac{2\pi j}{N} k} + \] + Wenn $N$ konstant: + \[ + \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) + \] \end{frame} -\section{Polynom Ansatz} - \begin{frame} - Die Diskrite Fouren Transformation ist so gegeben - \[ - \label{ft_discrete} - \hat{c}_{k} - = \frac{1}{N} \sum_{n=0}^{N-1} - {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} - \]. - - \[ - w = e^{-\frac{2\pi j}{N} k} - \] - Wenn $N$ konstant: - \[ - \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) - \] - \end{frame} - - \begin{frame} - Beispiel 2, 1, 5 Versenden und auf 2 Fehler absichern. - \end{frame} - \begin{frame} - Übertragen von - ${f}_2=$\textcolor{blue}{2}, ${f}_1$\textcolor{blue}{1}, ${f}_0$\textcolor{blue}{5} - als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. - \only<1>{ - Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, - \textcolor{green}{15}, \textcolor{green}{26}, - \textcolor{green}{ 41}, \textcolor{green}{60}, - \textcolor{green}{83}, \textcolor{green}{110})$ - \includegraphics[scale = 1.2]{images/polynom1.pdf}} - \only<2>{ - Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, - \textcolor{red}{50}, \textcolor{red}{37}, - \textcolor{green}{ 41}, \textcolor{green}{60}, - \textcolor{green}{83}, \textcolor{green}{110})$ - \includegraphics[scale = 1.2]{images/polynom2.pdf} - \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} - \end{frame} - - \begin{frame} - \frametitle{Parameter} - \begin{center} - \begin{tabular}{ c c c } - \hline - "Nutzlast" & Fehler & Versenden \\ - \hline - 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ - 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ - 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ - &&\\ - k & t & k+2t Werte eines Polynoms vom Grad k-1 \\ - \hline - \end{tabular} - \end{center} - \end{frame} -\section{Diskrete Fourien Transformation} \begin{frame} + \frametitle{Diskrete Fourier Transformation} \[ - \begin{pmatrix} - \hat{c}_1 \\\hat{c}_2 \\\hat{c}_3 \\ \vdots \\\hat{c}_n - \end{pmatrix} - = - \begin{pmatrix} - w^0 & w^0 & w^0 & \dots &w^0 \\ - w^0 & w^1 &w^2 & \dots &w^n \\ - w^0 & w^2 &w^4 & \dots &w^{2n} \\ - \vdots & \vdots &\vdots &\ddots &\vdots \\ - w^0 & w^{1n}&w^{2n}& \dots &w^{n} \\ - \end{pmatrix} - \begin{pmatrix} - \textcolor{blue}{5} \\ - \textcolor{blue}{1} \\ - \textcolor{blue}{2} \\ - \vdots \\ - 0 \\ - \end{pmatrix} + \begin{pmatrix} + \hat{c}_1 \\\hat{c}_2 \\\hat{c}_3 \\ \vdots \\\hat{c}_n + \end{pmatrix} + = + \begin{pmatrix} + w^0 & w^0 & w^0 & \dots &w^0 \\ + w^0 & w^1 &w^2 & \dots &w^n \\ + w^0 & w^2 &w^4 & \dots &w^{2n} \\ + \vdots & \vdots &\vdots &\ddots &\vdots \\ + w^0 & w^{1n}&w^{2n}& \dots &w^{n} \\ + \end{pmatrix} + \begin{pmatrix} + \textcolor{blue}{f_0} \\ + \textcolor{blue}{f_1} \\ + \textcolor{blue}{f_2} \\ + \vdots \\ + 0 \\ + \end{pmatrix} \] \end{frame} - +\section{Probleme und Fragen} + \begin{frame} + \frametitle{Probleme und Fragen} + + Wie wird der Fehler lokalisiert? + \only<2>{ + Indem in einem Endlichen Körper gerechnet wird. + } + \end{frame} \end{document} \ No newline at end of file -- cgit v1.2.1 From 264bd585ba37fcf0a8fed6c83b38edfe2495daef Mon Sep 17 00:00:00 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presentation/RS.pdf and b/buch/papers/reedsolomon/RS presentation/RS.pdf differ diff --git a/buch/papers/reedsolomon/RS presentation/RS.snm b/buch/papers/reedsolomon/RS presentation/RS.snm index 8b82641..6607ea8 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.snm +++ b/buch/papers/reedsolomon/RS presentation/RS.snm @@ -1 +1 @@ -\beamer@slide {ft_discrete}{12} +\beamer@slide {ft_discrete}{15} diff --git a/buch/papers/reedsolomon/RS presentation/RS.synctex.gz b/buch/papers/reedsolomon/RS presentation/RS.synctex.gz index 96af4cc..001b5c8 100644 Binary files a/buch/papers/reedsolomon/RS presentation/RS.synctex.gz and b/buch/papers/reedsolomon/RS presentation/RS.synctex.gz differ diff --git a/buch/papers/reedsolomon/RS presentation/RS.toc b/buch/papers/reedsolomon/RS presentation/RS.toc index ff200c6..44c06ab 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.toc +++ b/buch/papers/reedsolomon/RS presentation/RS.toc @@ -1,4 +1,6 @@ \babel@toc {ngerman}{} \beamer@sectionintoc {1}{Einführung}{2}{0}{1} -\beamer@sectionintoc {2}{Polynom Ansatz}{12}{0}{2} -\beamer@sectionintoc {3}{Diskrete Fourien Transformation}{17}{0}{3} +\beamer@sectionintoc {2}{Polynom Ansatz}{3}{0}{2} +\beamer@sectionintoc {3}{Fourier Transformation}{7}{0}{3} +\beamer@sectionintoc {4}{Diskrete Fourier Transformation}{15}{0}{4} +\beamer@sectionintoc {5}{Probleme und Fragen}{17}{0}{5} -- cgit v1.2.1 From 308c797ad63e094b1553d6417d477b4b7e792358 Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Wed, 21 Apr 2021 22:53:22 +0200 Subject: Update RS.tex --- buch/papers/reedsolomon/RS presentation/RS.tex | 708 ++++++++++++++++++++++++- 1 file changed, 707 insertions(+), 1 deletion(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 3d2be8f..400e654 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -17,9 +17,715 @@ \begin{frame}[plain] \maketitle \end{frame} - +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Test} Ich mag Züge. \end{frame} + + \begin{frame} + \frametitle{Reed-Solomon in Endlichen Körpern} + + \begin{itemize} + \item Warum Endliche Körper? + + \qquad bessere Laufzeit + + \vspace{10pt} + + \item Nachricht = Nutzdaten + Fehlerkorrekturteil + + \vspace{10pt} + + \item den Fehlerkorrekturteil brauchen wir im Optimalfall nicht + + \vspace{10pt} + + \item Im Fehlerfall sollen wir aus der Nachricht ein Lokatorpolynom berechnen können, welches die Fehlerhaften Stellen beinhaltet + +% Wir sollten im Fehlerfall in der Lage sein, aus der Nachricht ein Lokatorpolynom zu berechnen, welches die Fehlerhaften Stellen beinhaltet + + \end{itemize} + +% TODO + +% erklärung und einführung der endlichen körper, was wollen wir erreichen? + +% wir versenden im endefekt mehr daten als unsere nachricht umfasst, damit die korrektur sichergestellt werden kann + +% sollten wir fehler bekommen, was uns die korrekturstellen mitgeteilt wird, dann ist es unsere aufgabe ein lokatorpolynom zu finden, welches uns verrät, auf welchen zeilen der Fehler aufgetreten ist + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Definition eines Beispiels} + + \begin{itemize} + + \item Endlicher Körper $q = 11$ + + \only<1->{ist eine Primzahl} + + \only<1->{beinhaltet die Zahlen $\mathbb{Z}_{11} = [0,1,2,3,4,5,6,7,8,9,10]$} + + \vspace{10pt} + + \only<1->{\item Nachrichtenblock $n = q-1$} + + wird an den Empfänger gesendet + + \vspace{10pt} + + \only<1->{\item max. Fehler $z = 2$} + + maximale Anzahl von Fehler, die wir noch korrigieren können + + \vspace{10pt} + + \only<1->{\item Nutzlast $k = n -2t = 6$ Zahlen} + + Fehlerstellen $2t = 4$ Zahlen + + \only<1->{Nachricht $m = [0,0,0,0,4,7,2,5,8,1]$} + + \only<1->{als Polynom $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$} + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Codierung} + + \begin{itemize} + \item Ansatz aus den Komplexen Zahlen mit der Fouriertransformation + + \vspace{10pt} + + \item $\mathrm{e}$ existiert nicht in $\mathbb{Z}_{11}$ + + \vspace{10pt} + + \item wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{Z}_{11}$ abdeckt + + $\mathbb{Z}_{11}\setminus\{0\} = [a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9]$ + + \vspace{10pt} + + \item wir wählen $a = 8$ + + $\mathbb{Z}_{11}\setminus\{0\} = [1,8,9,6,4,10,3,2,5,7]$ + + 8 ist eine Primitive Einheitswurzel + + \vspace{10pt} + + \item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$ + + $\Rightarrow$ \qquad können wir auch als Matrix schreiben + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Codierung} + + \begin{itemize} + \item Übertragungsvektor $V$ + + \item $V = A \cdot m$ + + \end{itemize} + + \[ + V = \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^3& 8^6& 8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $V = [5,3,6,5,2,10,2,7,10,4]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Decodierung ohne Fehler} + + \begin{itemize} + \item Der Empfänger erhält den unveränderten Vektor $V = [5,3,6,5,2,10,2,7,10,4]$ + + \vspace{10pt} + + \item Wir suchen die Inverse der Matrix A + + \end{itemize} + + \begin{columns}[t] + \begin{column}{0.50\textwidth} + + Inverse der Fouriertransformation + \vspace{10pt} + \[ + F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt + \] + \vspace{10pt} + \[ + f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega + \] + + \end{column} + \begin{column}{0.50\textwidth} + + Inverse von a + \vspace{10pt} + \[ + 8^{1} \Rightarrow 8^{-1} + \] + + Inverse finden wir über den Eulkidischen Algorithmus + \vspace{10pt} + \end{column} + \end{columns} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Euklidische Algorithmus} + + \begin{columns}[t] + \begin{column}{0.50\textwidth} + + Recap aus der Vorlesung: + + Gegeben $a \in \mathbb{F}_p$, finde $b = a^{-1} \in \mathbb{F}_p$ + + \begin{tabular}{rcl} + $a b$ &$\equiv$& $1 \mod p$\\ + $a b$ &$=$& $1 + n p$\\ + $a b - n p$ &$=$& $1$\\ + &&\\ + $\operatorname{ggT}(a,p)$&$=$& $1$\\ + $sa + tp$&$=$& $1$\\ + $b$&$=$&$s$\\ + $n$&$=$&$-t$ + \end{tabular} + + \end{column} + \begin{column}{0.50\textwidth} + + \begin{center} + + \begin{tabular}{| c | c c | c | c c |} + \hline + $k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\ + \hline + & & & & $1$& $0$\\ + $0$& $8$& $11$& $0$& $0$& $1$\\ + $1$& $11$& $8$& $1$& $1$& $0$\\ + $2$& $8$& $3$& $2$& $-1$& $1$\\ + $3$& $3$& $2$& $1$& $3$& $-2$\\ + $4$& $2$& $1$& $2$& $-4$& $3$\\ + $5$& $1$& $0$& & $11$& $-8$\\ + \hline + \end{tabular} + + \vspace{10pt} + + \begin{tabular}{rcl} + $-4\cdot 8 + 3 \cdot 11$ &$=$& $1$\\ + $7 \cdot 8 + 3 \cdot 11$ &$=$& $1$\\ + $8^{-1}$ &$=$& $7$ + + \end{tabular} + + \end{center} + + \end{column} + \end{columns} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Decodirung mit Inverser Matrix} + + \begin{itemize} + \item $V = [5,3,6,5,2,10,2,7,10,4]$ + + \item $m = 1/10 \cdot A^{-1} \cdot V$ + + \item $m = 10 \cdot A^{-1} \cdot V$ + + \end{itemize} + + \[ + m = \begin{pmatrix} + 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ + 7^0& 7^1& 7^2& 7^3& 7^4& 7^5& 7^6& 7^7& 7^8& 7^9\\ + 7^0& 7^2& 7^4& 7^6& 7^8& 7^{10}& 7^{12}& 7^{14}& 7^{16}& 7^{18}\\ + 7^0& 7^3& 7^6& 7^9& 7^{12}& 7^{15}& 7^{18}& 7^{21}& 7^{24}& 7^{27}\\ + 7^0& 7^4& 7^8& 7^{12}& 7^{16}& 7^{20}& 7^{24}& 7^{28}& 7^{32}& 7^{36}\\ + 7^0& 7^5& 7^{10}& 7^{15}& 7^{20}& 7^{25}& 7^{30}& 7^{35}& 7^{40}& 7^{45}\\ + 7^0& 7^6& 7^{12}& 7^{18}& 7^{24}& 7^{30}& 7^{36}& 7^{42}& 7^{48}& 7^{54}\\ + 7^0& 7^7& 7^{14}& 7^{21}& 7^{28}& 7^{35}& 7^{42}& 7^{49}& 7^{56}& 7^{63}\\ + 7^0& 7^8& 7^{16}& 7^{24}& 7^{32}& 7^{40}& 7^{48}& 7^{56}& 7^{64}& 7^{72}\\ + 7^0& 7^9& 7^{18}& 7^{27}& 7^{36}& 7^{45}& 7^{54}& 7^{63}& 7^{72}& 7^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 5 \\ 2 \\ 10 \\ 2 \\ 7 \\ 10 \\ 4 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $m = [0,0,0,0,4,7,2,5,8,1]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Decodierung mit Fehler - Ansatz} + + \begin{itemize} + \item Gesendet: $V = [5,3,6,5,2,10,2,7,10,4]$ + + \item Empfangen: $W = [5,3,6,8,2,10,2,7,1,4]$ + + \item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerstellen}5,4,5,7,6,7]$ + \end{itemize} + + Wie finden wir die Fehler? + + \begin{itemize} + \item $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ + + \item $r(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$ + + \item $e(X) = r(X) - m(X)$ + \end{itemize} + + \begin{center} + + \begin{tabular}{c c c c c c c c c c c} + \hline + $i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\ + \hline + $r(a^{i})$& $5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$\\ + $m(a^{i})$& $5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$\\ + $e(a^{i})$& $0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$\\ + \hline + \end{tabular} + + \end{center} + + \begin{itemize} + \item Alle Stellen, die nicht Null sind, sind Fehler + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Nullstellen des Fehlerpolynoms finden} + + \begin{itemize} + \item Satz von Fermat: $f(X) = X^{q-1}-1=0$ + + \vspace{10pt} + + \item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$ + + \vspace{10pt} + + \item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$ + + \vspace{10pt} + + \item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$ + + \vspace{10pt} + + \item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat + + \vspace{10pt} + + $\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9)$ + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Nullstellen des Fehlerpolynoms finden} + + \begin{itemize} + + \item Satz von Fermat: $f(X) = X^{q-1}-1=0$ + + \vspace{10pt} + + \item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$ + + \vspace{10pt} + + \item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$ + + \vspace{10pt} + + \item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$ + + \vspace{10pt} + + \item $\operatorname{kgV}$ gibt uns eine Liste von aller Nullstellen, die wir in $e$ und $d$ zerlegen können + + \vspace{10pt} + + $\operatorname{kgV}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot $ + + \qquad \qquad \qquad \qquad $(X-a^7)(X-a^8)(X-a^9) \cdot q(X)$ + + $= d(X) \cdot e(X)$ + + \vspace{10pt} + + \item Lokatorpolynom $d(X) = (X-a^3)(X-a^8)$ + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{kennen wir $e$?} + + \begin{itemize} + + \item $e$ ist unbekannt auf der Empfängerseite + + \vspace{10pt} + + \item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt? + + \vspace{10pt} + + \item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$ + + In den bekannten Stellen liegt auch die Information, wo es Fehler gegeben hat + + \vspace{10pt} + + \item daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$ + + \vspace{10pt} + + \item $f(X) = X^{10} - 1 = X^{10} + 10$ + + \vspace{10pt} + + \item jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Euklidische Algorithmus (nochmal)} + + $\operatorname{ggT}(f(X),e(X))$ hat den Grad 8 + + \[ + \arraycolsep=1.4pt + \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + X^{10}& & & & & & &+& 10& & & & &:&5X^9&+&7X^8&+& 4X^7&+&10X^6&+&p(X)&=&9X&+&5\\ + X^{10}&+& 8X^9&+& 3X^8&+&2X^7&+& p(X)& & & & & & & & & & & & & & & & \\ \cline{1-9} + && 3X^9&+& 8X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ + && 3X^9&+& 2X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ \cline{3-9} + & & & &6X^8&+&0X^7&+&p(X)& & & & & & & & & & & & \\ + \end{array} + \] + + \[ + \arraycolsep=1.4pt + \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + 5X^9&+& 7X^8&+& 4X^7&+& 10X^6&+& p(X)& & & & &:&6X^8&+&0X^7& & & & & & &=&10X&+&3\\ + 5X^9&+& 0X^8&+& p(X)& & & & & & & & & & & & & & & & & & & & \\ \cline{1-5} + && 7X^8&+& p(X)& & & & & & & & & & & & & & & & \\ + \end{array} + \] + + \vspace{10pt} + + $\operatorname{ggT}(f(X),e(X)) = 6X^8$ + + \vspace{10pt} + + $\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen + + \end{frame} + +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Erweiterte Euklidische Algorithmus} + + \begin{center} + + \begin{tabular}{| c | c | c c |} + \hline + $k$ & $q_i$ & $e_i$ & $f_i$\\ + \hline + & & $0$& $1$\\ + $0$& $9X + 5$& $1$& $0$\\ + $1$& $10X + 3$& $9X+5$& $1$\\ + $2$& & $2X^2 + 0X + 5$& $10X + 3$\\ + \hline + \end{tabular} + + \end{center} + + \vspace{10pt} + + \begin{tabular}{ll} + Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\ + Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\ + Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$ + \end{tabular} + + \vspace{10pt} + + \begin{center} + $a^i = 5 \qquad \Rightarrow \qquad i = 3$ + + $a^i = 6 \qquad \Rightarrow \qquad i = 8$ + \end{center} + + $D = [3,8]$ + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \begin{itemize} + + \item $W = [5,3,6,8,2,10,2,7,1,4]$ + + \item $D = [3,8]$ + + \end{itemize} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 8 \\ 2 \\ 10 \\ 2 \\ 7 \\ 1 \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^3& 8^6& 8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item Fehlerstellen entfernen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item Nullstellen entfernen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \vspace{5pt} + + \begin{itemize} + \item Matrix in eine Quadratische Form bringen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \vspace{5pt} + + \begin{itemize} + \item Matrix Invertieren + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + = + \begin{pmatrix} + 1& 1& 1& 1& 1& 1\\ + 1& 8& 9& 6& 4& 10\\ + 1& 9& 4& 3& 5& 1\\ + 1& 4& 5& 9& 3& 1\\ + 1& 10& 1& 10& 1& 10\\ + 1& 3& 9& 5& 4& 1\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \begin{center} + $\Downarrow$ + \end{center} + \[ + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + = + \begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + \] + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + = + \begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $m = [4,7,2,5,8,1]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- \end{document} \ No newline at end of file -- cgit v1.2.1 From 8473571bc77425cd198b4bba515a3f5fe10c8cd2 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Wed, 21 Apr 2021 22:53:49 +0200 Subject: Style verbessert --- buch/papers/reedsolomon/RS presentation/RS.tex | 17 +++++++++++------ 1 file changed, 11 insertions(+), 6 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 1a1cefd..65f8431 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -64,12 +64,16 @@ &&\\ k & t & k+2t Werte eines Polynoms vom Grad k-1 \\ \hline + &&\\ + &&\\ + &Ausserdem können bis zu 2t Fehler erkannt werden!\\ \end{tabular} \end{center} - Ausserdem können bis zu 2t Fehler erkannt werden! + + \end{frame} -\section{Fourier Transformation} +\section{Diskrete Fourier Transformation} \begin{frame} \frametitle{Idee} \begin{itemize} @@ -104,7 +108,7 @@ \end{figure} \end{frame} -\section{Diskrete Fourier Transformation} + \begin{frame} \frametitle{Diskrete Fourier Transformation} Die Diskrete Fourier Transformation ist so gegeben: @@ -134,10 +138,10 @@ = \begin{pmatrix} w^0 & w^0 & w^0 & \dots &w^0 \\ - w^0 & w^1 &w^2 & \dots &w^n \\ - w^0 & w^2 &w^4 & \dots &w^{2n} \\ + w^0 & w^1 &w^2 & \dots &w^N \\ + w^0 & w^2 &w^4 & \dots &w^{2N} \\ \vdots & \vdots &\vdots &\ddots &\vdots \\ - w^0 & w^{1n}&w^{2n}& \dots &w^{n} \\ + w^0 & w^{1(N-1)}&w^{2(N-1)}& \dots &w^{(N-1)(N-1)} \\ \end{pmatrix} \begin{pmatrix} \textcolor{blue}{f_0} \\ @@ -154,6 +158,7 @@ Wie wird der Fehler lokalisiert? \only<2>{ + \newline Indem in einem Endlichen Körper gerechnet wird. } \end{frame} -- cgit v1.2.1 From 38d0c69842308be5f096375ff070c5233b395c4c Mon Sep 17 00:00:00 2001 From: JODBaer Date: Thu, 22 Apr 2021 16:01:46 +0200 Subject: kleine korrekturen --- buch/papers/reedsolomon/RS presentation/RS.tex | 45 +++++++++++++++----------- 1 file changed, 26 insertions(+), 19 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index eecd66b..618121c 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -19,14 +19,18 @@ \begin{frame}[plain] \maketitle \end{frame} - \section{Einführung} +%------------------------------------------------------------------------------- +\section{Einführung} \begin{frame} \frametitle{Einführung} \begin{itemize} \item Reed-Solomon-Code beschäftigt sich mit der Übertragung von Daten und deren Fehler Erkennung. + \item Wird verwendet in: + \only<2>{CD, QR-Codes, Voyager-Sonde, etc.} \end{itemize} \end{frame} +%------------------------------------------------------------------------------- \section{Polynom Ansatz} \begin{frame} Beispiel 2, 1, 5 Versenden und auf 2 Fehler absichern. @@ -50,7 +54,7 @@ \includegraphics[scale = 1.2]{images/polynom2.pdf} \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} \end{frame} - +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Parameter} \begin{center} @@ -59,20 +63,24 @@ "Nutzlast" & Fehler & Versenden \\ \hline 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ - 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ - 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ +\only<2->{3}& +\only<2->{2}& +\only<2->{7 Werte eines Polynoms vom Grad 2} \\ &&\\ - k & t & k+2t Werte eines Polynoms vom Grad k-1 \\ +\only<3->{k} & +\only<3->{t} & +\only<3->{k+2t Werte eines Polynoms vom Grad k-1} \\ \hline &&\\ &&\\ - &Ausserdem können bis zu 2t Fehler erkannt werden!\\ + \multicolumn{3}{l} { + \only<4>{Ausserdem können bis zu 2t Fehler erkannt werden!} + } \end{tabular} - \end{center} - - - + \end{center} \end{frame} +%------------------------------------------------------------------------------- \section{Diskrete Fourier Transformation} \begin{frame} \frametitle{Idee} @@ -81,7 +89,7 @@ \item Danach Empfangen und Rücktransformieren. \end{itemize} \end{frame} - +%------------------------------------------------------------------------------- \begin{frame} \begin{figure} \only<1>{ @@ -107,8 +115,7 @@ } \end{figure} \end{frame} - - +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Diskrete Fourier Transformation} Die Diskrete Fourier Transformation ist so gegeben: @@ -117,8 +124,8 @@ \hat{c}_{k} = \frac{1}{N} \sum_{n=0}^{N-1} {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} - \]. - + \] + Ersetzten als: \[ w = e^{-\frac{2\pi j}{N} k} \] @@ -128,14 +135,14 @@ \] \end{frame} - +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Diskrete Fourier Transformation} \[ \begin{pmatrix} \hat{c}_1 \\\hat{c}_2 \\\hat{c}_3 \\ \vdots \\\hat{c}_n \end{pmatrix} - = + = \frac{1}{N} \begin{pmatrix} w^0 & w^0 & w^0 & \dots &w^0 \\ w^0 & w^1 &w^2 & \dots &w^N \\ @@ -152,7 +159,7 @@ \end{pmatrix} \] \end{frame} - +%------------------------------------------------------------------------------- \section{Probleme und Fragen} \begin{frame} \frametitle{Probleme und Fragen} @@ -163,7 +170,7 @@ Indem in einem Endlichen Körper gerechnet wird. } \end{frame} - +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Reed-Solomon in Endlichen Körpern} -- cgit v1.2.1 From 9ce4fb55792c297989d1c001a621793303f31689 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Thu, 22 Apr 2021 22:13:29 +0200 Subject: Verbesserungen und anmerkungen umgesetzt --- buch/papers/reedsolomon/RS presentation/RS.tex | 56 ++++++++++++++------------ 1 file changed, 31 insertions(+), 25 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 618121c..9811cf6 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -22,36 +22,38 @@ %------------------------------------------------------------------------------- \section{Einführung} \begin{frame} - \frametitle{Einführung} + \frametitle{Reed-Solomon-Code:} \begin{itemize} - \item Reed-Solomon-Code beschäftigt sich mit der Übertragung von Daten - und deren Fehler Erkennung. - \item Wird verwendet in: - \only<2>{CD, QR-Codes, Voyager-Sonde, etc.} + \item \only<1>{Für Übertragung von Daten} + \item \only<2->{Ermöglicht Korrektur von Übertragungsfehler} + \item \only<3->{Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc.} \end{itemize} \end{frame} %------------------------------------------------------------------------------- \section{Polynom Ansatz} \begin{frame} - Beispiel 2, 1, 5 Versenden und auf 2 Fehler absichern. + \begin{itemize} + \item Beispiel $2, 1, 5$ versenden und auf 2 Fehler absichern + \end{itemize} \end{frame} \begin{frame} Übertragen von - ${f}_2=$\textcolor{blue}{2}, ${f}_1$\textcolor{blue}{1}, ${f}_0$\textcolor{blue}{5} + ${f}_2=\textcolor{blue}{2}$, ${f}_1=\textcolor{blue}{1}$, ${f}_0=\textcolor{blue}{5}$ als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. \only<1>{ - Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, + Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, \textcolor{green}{15}, \textcolor{green}{26}, - \textcolor{green}{ 41}, \textcolor{green}{60}, + \textcolor{green}{41}, \textcolor{green}{60}, \textcolor{green}{83}, \textcolor{green}{110})$ \includegraphics[scale = 1.2]{images/polynom1.pdf}} \only<2>{ - Versende $ (p(1),p(2),...,p(7)) = (\textcolor{green}{8}, + Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, \textcolor{red}{50}, \textcolor{red}{37}, - \textcolor{green}{ 41}, \textcolor{green}{60}, + \textcolor{green}{41}, \textcolor{green}{60}, \textcolor{green}{83}, \textcolor{green}{110})$ \includegraphics[scale = 1.2]{images/polynom2.pdf} + \newline \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} \end{frame} %------------------------------------------------------------------------------- @@ -60,22 +62,22 @@ \begin{center} \begin{tabular}{ c c c } \hline - "Nutzlast" & Fehler & Versenden \\ + ``Nutzlas´´ & Fehler & Versenden \\ \hline 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ \only<2->{3}& -\only<2->{2}& -\only<2->{7 Werte eines Polynoms vom Grad 2} \\ +\only<2->{3}& +\only<3->{9 Werte eines Polynoms vom Grad 2} \\ &&\\ -\only<3->{k} & -\only<3->{t} & -\only<3->{k+2t Werte eines Polynoms vom Grad k-1} \\ +\only<4->{$k$} & +\only<4->{$t$} & +\only<4->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ \hline &&\\ &&\\ \multicolumn{3}{l} { - \only<4>{Ausserdem können bis zu 2t Fehler erkannt werden!} + \only<4>{Ausserdem können bis zu $2t$ Fehler erkannt werden!} } \end{tabular} \end{center} @@ -85,8 +87,9 @@ \begin{frame} \frametitle{Idee} \begin{itemize} - \item Idee mit Fourier Transformieren und dann senden. - \item Danach Empfangen und Rücktransformieren. + \item Fourier-transformieren + \item Übertragung + \item Rücktransformieren \end{itemize} \end{frame} %------------------------------------------------------------------------------- @@ -118,14 +121,16 @@ %------------------------------------------------------------------------------- \begin{frame} \frametitle{Diskrete Fourier Transformation} - Die Diskrete Fourier Transformation ist so gegeben: + \begin{itemize} + \item Diskrete Fourier-Transformation gegeben durch: + \[ \label{ft_discrete} \hat{c}_{k} = \frac{1}{N} \sum_{n=0}^{N-1} {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} \] - Ersetzten als: + \item Ersetzte \[ w = e^{-\frac{2\pi j}{N} k} \] @@ -133,6 +138,7 @@ \[ \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) \] + \end{itemize} \end{frame} %------------------------------------------------------------------------------- @@ -145,8 +151,8 @@ = \frac{1}{N} \begin{pmatrix} w^0 & w^0 & w^0 & \dots &w^0 \\ - w^0 & w^1 &w^2 & \dots &w^N \\ - w^0 & w^2 &w^4 & \dots &w^{2N} \\ + w^0 & w^1 &w^2 & \dots &w^{N-1} \\ + w^0 & w^2 &w^4 & \dots &w^{2(N-1)} \\ \vdots & \vdots &\vdots &\ddots &\vdots \\ w^0 & w^{1(N-1)}&w^{2(N-1)}& \dots &w^{(N-1)(N-1)} \\ \end{pmatrix} @@ -167,7 +173,7 @@ Wie wird der Fehler lokalisiert? \only<2>{ \newline - Indem in einem Endlichen Körper gerechnet wird. + Indem in einem endlichen Körper gerechnet wird. } \end{frame} %------------------------------------------------------------------------------- -- cgit v1.2.1 From 5bca0960f8c9635375d2ca53c93d2bc5a2e37c10 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Thu, 22 Apr 2021 22:59:07 +0200 Subject: Animation verbessert --- buch/papers/reedsolomon/RS presentation/RS.tex | 37 ++++++++++++++------------ 1 file changed, 20 insertions(+), 17 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 9811cf6..732cee5 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -24,9 +24,9 @@ \begin{frame} \frametitle{Reed-Solomon-Code:} \begin{itemize} - \item \only<1>{Für Übertragung von Daten} - \item \only<2->{Ermöglicht Korrektur von Übertragungsfehler} - \item \only<3->{Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc.} + \visible<1->{\item Für Übertragung von Daten} + \visible<2->{\item Ermöglicht Korrektur von Übertragungsfehler} + \visible<3->{\item Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc.} \end{itemize} \end{frame} %------------------------------------------------------------------------------- @@ -37,6 +37,7 @@ \end{itemize} \end{frame} \begin{frame} + \frametitle{Beispiel} Übertragen von ${f}_2=\textcolor{blue}{2}$, ${f}_1=\textcolor{blue}{1}$, ${f}_0=\textcolor{blue}{5}$ als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. @@ -66,18 +67,18 @@ \hline 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ -\only<2->{3}& -\only<2->{3}& -\only<3->{9 Werte eines Polynoms vom Grad 2} \\ +\visible<2->{3}& +\visible<2->{3}& +\visible<3->{9 Werte eines Polynoms vom Grad 2} \\ &&\\ -\only<4->{$k$} & -\only<4->{$t$} & -\only<4->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ +\visible<4->{$k$} & +\visible<4->{$t$} & +\visible<4->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ \hline &&\\ &&\\ \multicolumn{3}{l} { - \only<4>{Ausserdem können bis zu $2t$ Fehler erkannt werden!} + \visible<4>{Ausserdem können bis zu $2t$ Fehler erkannt werden!} } \end{tabular} \end{center} @@ -123,21 +124,23 @@ \frametitle{Diskrete Fourier Transformation} \begin{itemize} \item Diskrete Fourier-Transformation gegeben durch: - + \visible<1->{ \[ \label{ft_discrete} \hat{c}_{k} = \frac{1}{N} \sum_{n=0}^{N-1} {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} - \] + \]} + \visible<2->{ \item Ersetzte \[ w = e^{-\frac{2\pi j}{N} k} - \] - Wenn $N$ konstant: + \]} + \visible<3->{ + \item Wenn $N$ konstant: \[ \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) - \] + \]} \end{itemize} \end{frame} @@ -166,12 +169,12 @@ \] \end{frame} %------------------------------------------------------------------------------- -\section{Probleme und Fragen} + \begin{frame} \frametitle{Probleme und Fragen} Wie wird der Fehler lokalisiert? - \only<2>{ + \visible<2>{ \newline Indem in einem endlichen Körper gerechnet wird. } -- cgit v1.2.1 From 967ff1f33d3faaa1e344ff687aff6c07cde29b77 Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Thu, 22 Apr 2021 23:33:02 +0200 Subject: Update RS.tex --- buch/papers/reedsolomon/RS presentation/RS.tex | 288 ++++++++++++++----------- 1 file changed, 165 insertions(+), 123 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 732cee5..61324f7 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -15,6 +15,7 @@ \date{26.04.2021} \subject{Mathematisches Seminar} \setbeamercovered{transparent} + %\setbeamercovered{invisible} \setbeamertemplate{navigation symbols}{} \begin{frame}[plain] \maketitle @@ -83,7 +84,11 @@ \end{tabular} \end{center} \end{frame} +<<<<<<< Updated upstream %------------------------------------------------------------------------------- +======= + +>>>>>>> Stashed changes \section{Diskrete Fourier Transformation} \begin{frame} \frametitle{Idee} @@ -179,26 +184,38 @@ Indem in einem endlichen Körper gerechnet wird. } \end{frame} +<<<<<<< Updated upstream %------------------------------------------------------------------------------- +======= + +\section{Reed-Solomon in Endlichen Körpern} + +>>>>>>> Stashed changes \begin{frame} \frametitle{Reed-Solomon in Endlichen Körpern} \begin{itemize} - \item Warum Endliche Körper? + \onslide<1->{\item Warum endliche Körper?} - \qquad bessere Laufzeit + \onslide<1->{\qquad konkrete Zahlen $\rightarrow$ keine Rundungsfehler} - \vspace{10pt} + \onslide<1->{\qquad digitale Fehlerkorrektur} - \item Nachricht = Nutzdaten + Fehlerkorrekturteil + \onslide<1->{\qquad bessere Laufzeit} \vspace{10pt} - \item den Fehlerkorrekturteil brauchen wir im Optimalfall nicht + \onslide<1->{\item Nachricht = Nutzdaten + Fehlerkorrekturteil} \vspace{10pt} - \item Im Fehlerfall sollen wir aus der Nachricht ein Lokatorpolynom berechnen können, welches die Fehlerhaften Stellen beinhaltet + \onslide<1->{\item aus Fehlerkorrekturteil die Fehlerstellen finden} + + \onslide<1->{\qquad $\Rightarrow$ gesucht ist ein Lokatorpolynom} + +% \vspace{10pt} + +% \onslide<1->{\item Im Fehlerfall sollen wir aus der Nachricht ein Lokatorpolynom berechnen können, welches die fehlerhaften Stellen beinhaltet} % Wir sollten im Fehlerfall in der Lage sein, aus der Nachricht ein Lokatorpolynom zu berechnen, welches die Fehlerhaften Stellen beinhaltet @@ -212,35 +229,35 @@ % sollten wir fehler bekommen, was uns die korrekturstellen mitgeteilt wird, dann ist es unsere aufgabe ein lokatorpolynom zu finden, welches uns verrät, auf welchen zeilen der Fehler aufgetreten ist \end{frame} -%------------------------------------------------------------------------------- +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Definition eines Beispiels} \begin{itemize} - \item Endlicher Körper $q = 11$ + \only<1->{\item endlicher Körper $q = 11$} \only<1->{ist eine Primzahl} - \only<1->{beinhaltet die Zahlen $\mathbb{Z}_{11} = [0,1,2,3,4,5,6,7,8,9,10]$} + \only<1->{beinhaltet die Zahlen $\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}$} \vspace{10pt} - \only<1->{\item Nachrichtenblock $n = q-1$} + \only<1->{\item Nachrichtenblock $=$ Nutzlast $+$ Fehlerkorrekturstellen - wird an den Empfänger gesendet + $n = q - 1 = 10$ Zahlen} \vspace{10pt} - \only<1->{\item max. Fehler $z = 2$} + \only<1->{\item Max.~Fehler $z = 2$ - maximale Anzahl von Fehler, die wir noch korrigieren können + maximale Anzahl von Fehler, die wir noch korrigieren können} \vspace{10pt} \only<1->{\item Nutzlast $k = n -2t = 6$ Zahlen} - Fehlerstellen $2t = 4$ Zahlen + \only<1->{Fehlerkorrkturstellen $2t = 4$ Zahlen} \only<1->{Nachricht $m = [0,0,0,0,4,7,2,5,8,1]$} @@ -250,52 +267,54 @@ \end{frame} %------------------------------------------------------------------------------- +\section{Codierung eines Beispiels} \begin{frame} \frametitle{Codierung} \begin{itemize} - \item Ansatz aus den Komplexen Zahlen mit der Fouriertransformation + \only<1->{\item Ansatz aus den komplexen Zahlen mit der diskreten Fouriertransformation} \vspace{10pt} - \item $\mathrm{e}$ existiert nicht in $\mathbb{Z}_{11}$ + \only<1->{\item Eulersche Zahl $\mathrm{e}$ existiert nicht in $\mathbb{F}_{11}$} \vspace{10pt} - \item wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{Z}_{11}$ abdeckt + \only<1->{\item Wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken - $\mathbb{Z}_{11}\setminus\{0\} = [a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9]$ + $\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}$} \vspace{10pt} - \item wir wählen $a = 8$ + \only<1->{\item Wir wählen $a = 8$} - $\mathbb{Z}_{11}\setminus\{0\} = [1,8,9,6,4,10,3,2,5,7]$ + \only<1->{$\mathbb{Z}_{11}\setminus\{0\} = \{1,8,9,6,4,10,3,2,5,7\}$} - 8 ist eine Primitive Einheitswurzel + \only<1->{$8$ ist eine primitive Einheitswurzel} \vspace{10pt} - \item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$ + \only<1->{\item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$} - $\Rightarrow$ \qquad können wir auch als Matrix schreiben + \only<1->{$\Rightarrow$ \qquad können wir auch als Matrix schreiben} \end{itemize} \end{frame} -%------------------------------------------------------------------------------- +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Codierung} \begin{itemize} - \item Übertragungsvektor $V$ + \only<1->{\item Übertragungsvektor $v$} - \item $V = A \cdot m$ + \only<1->{\item $v = A \cdot m$} \end{itemize} \[ - V = \begin{pmatrix} + \only<1->{ + v = \begin{pmatrix} 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ @@ -311,29 +330,34 @@ \begin{pmatrix} 1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{pmatrix} + } \] - + \only<1->{ \begin{itemize} - \item $V = [5,3,6,5,2,10,2,7,10,4]$ + \item $v = [5,3,6,5,2,10,2,7,10,4]$ \end{itemize} - + } \end{frame} %------------------------------------------------------------------------------- +\section{Decodierung ohne Fehler} \begin{frame} \frametitle{Decodierung ohne Fehler} \begin{itemize} - \item Der Empfänger erhält den unveränderten Vektor $V = [5,3,6,5,2,10,2,7,10,4]$ + \only<1->{\item Der Empfänger erhält den unveränderten Vektor + $v = [5,3,6,5,2,10,2,7,10,4]$} \vspace{10pt} - \item Wir suchen die Inverse der Matrix A + \only<1->{\item Wir suchen die Inverse der Matrix $A$} + + \vspace{10pt} \end{itemize} \begin{columns}[t] \begin{column}{0.50\textwidth} - + \only<1->{ Inverse der Fouriertransformation \vspace{10pt} \[ @@ -341,25 +365,26 @@ \] \vspace{10pt} \[ - f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega + \mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega \] - + } \end{column} \begin{column}{0.50\textwidth} - - Inverse von a + \only<1->{ + Inverse von $a$} \vspace{10pt} + \only<1->{ \[ 8^{1} \Rightarrow 8^{-1} \] - - Inverse finden wir über den Eulkidischen Algorithmus + } + \only<1->{Inverse finden wir über den Eulkidischen Algorithmus} \vspace{10pt} \end{column} \end{columns} \end{frame} -%------------------------------------------------------------------------------- +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Der Euklidische Algorithmus} @@ -385,8 +410,8 @@ \begin{column}{0.50\textwidth} \begin{center} - - \begin{tabular}{| c | c c | c | c c |} + \only<1->{ + \begin{tabular}{| c | c c | c | r r |} \hline $k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\ \hline @@ -395,17 +420,17 @@ $1$& $11$& $8$& $1$& $1$& $0$\\ $2$& $8$& $3$& $2$& $-1$& $1$\\ $3$& $3$& $2$& $1$& $3$& $-2$\\ - $4$& $2$& $1$& $2$& $-4$& $3$\\ + $4$& $2$& $1$& $2$& \textcolor<3->{blue}{$-4$}& \textcolor<3->{red}{$3$}\\ $5$& $1$& $0$& & $11$& $-8$\\ \hline \end{tabular} - + } \vspace{10pt} \begin{tabular}{rcl} - $-4\cdot 8 + 3 \cdot 11$ &$=$& $1$\\ - $7 \cdot 8 + 3 \cdot 11$ &$=$& $1$\\ - $8^{-1}$ &$=$& $7$ + \only<1->{$\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$}\\ + \only<1->{$7 \cdot 8 + 3 \cdot 11$ &$=$& $1$}\\ + \only<1->{$8^{-1}$ &$=$& $7$} \end{tabular} @@ -417,17 +442,17 @@ \end{frame} %------------------------------------------------------------------------------- \begin{frame} - \frametitle{Decodirung mit Inverser Matrix} + \frametitle{Decodierung mit Inverser Matrix} \begin{itemize} - \item $V = [5,3,6,5,2,10,2,7,10,4]$ + \only<1->{\item $v = [5,3,6,5,2,10,2,7,10,4]$} - \item $m = 1/10 \cdot A^{-1} \cdot V$ + \only<1->{\item $m = 1/10 \cdot A^{-1} \cdot v$} - \item $m = 10 \cdot A^{-1} \cdot V$ + \only<1->{\item $m = 10 \cdot A^{-1} \cdot v$} \end{itemize} - + \only<1->{ \[ m = \begin{pmatrix} 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ @@ -446,85 +471,95 @@ 5 \\ 3 \\ 6 \\ 5 \\ 2 \\ 10 \\ 2 \\ 7 \\ 10 \\ 4 \\ \end{pmatrix} \] - + } + \only<1->{ \begin{itemize} \item $m = [0,0,0,0,4,7,2,5,8,1]$ \end{itemize} - + } \end{frame} %------------------------------------------------------------------------------- +\section{Decodierung mit Fehler} \begin{frame} \frametitle{Decodierung mit Fehler - Ansatz} \begin{itemize} - \item Gesendet: $V = [5,3,6,5,2,10,2,7,10,4]$ + \only<1->{\item Gesendet: $v = [5,3,6,5,2,10,2,7,10,4]$} - \item Empfangen: $W = [5,3,6,8,2,10,2,7,1,4]$ + \only<1->{\item Empfangen: $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$} + + \only<1->{\item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]$} - \item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerstellen}5,4,5,7,6,7]$ \end{itemize} - Wie finden wir die Fehler? + \only<1->{Wie finden wir die Fehler?} + \only<1->{ \begin{itemize} \item $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ \item $r(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$ + %\only<7->{\item $e(X) = r(X) - m(X)$} + \item $e(X) = r(X) - m(X)$ + \end{itemize} - + } + \begin{center} - + \only<1->{ \begin{tabular}{c c c c c c c c c c c} \hline $i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\ \hline - $r(a^{i})$& $5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$\\ - $m(a^{i})$& $5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$\\ - $e(a^{i})$& $0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$\\ + $r(a^{i})$& \only<1->{$5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$}\\ + $m(a^{i})$& \only<1->{$5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$}\\ + $e(a^{i})$& \only<1->{$0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$}\\ \hline \end{tabular} - + } \end{center} - + + \only<1->{ \begin{itemize} \item Alle Stellen, die nicht Null sind, sind Fehler \end{itemize} - + } + \end{frame} -%------------------------------------------------------------------------------- +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Nullstellen des Fehlerpolynoms finden} \begin{itemize} - \item Satz von Fermat: $f(X) = X^{q-1}-1=0$ + \only<1->{\item Satz von Fermat: $f(X) = X^{q-1}-1=0$} \vspace{10pt} - \item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$ + \only<1->{\item $f(X) = X^{10}-1 = 0$ \qquad für $X \in \{1,2,3,4,5,6,7,8,9,10\}$} \vspace{10pt} - \item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + \only<1->{\item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ - \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$ + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$} \vspace{10pt} - \item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + \only<1->{\item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ - \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$ + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$} \vspace{10pt} - \item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat + \only<1->{\item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat} \vspace{10pt} - $\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + \only<1->{$\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ - \qquad \qquad \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9)$ + \qquad \qquad \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9)$} \end{itemize} @@ -574,33 +609,33 @@ \end{frame} %------------------------------------------------------------------------------- \begin{frame} - \frametitle{kennen wir $e$?} + \frametitle{Kennen wir $e(X)$?} \begin{itemize} - \item $e$ ist unbekannt auf der Empfängerseite + \only<1->{\item $e(X)$ ist unbekannt auf der Empfängerseite} \vspace{10pt} - \item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt? + \only<1->{\item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt?} \vspace{10pt} - \item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$ + \only<1->{\item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$ - In den bekannten Stellen liegt auch die Information, wo es Fehler gegeben hat + In den bekannten Stellen liegt auch die Information, wo es Fehler gegeben hat} \vspace{10pt} - \item daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$ + \only<1->{\item Daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$} \vspace{10pt} - \item $f(X) = X^{10} - 1 = X^{10} + 10$ + \only<1->{\item $f(X) = X^{10} - 1 = X^{10} + 10$} \vspace{10pt} - \item jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen + \only<1->{\item Jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen} \end{itemize} \end{frame} @@ -608,8 +643,8 @@ \begin{frame} \frametitle{Der Euklidische Algorithmus (nochmal)} - $\operatorname{ggT}(f(X),e(X))$ hat den Grad 8 - + \only<1->{$\operatorname{ggT}(f(X),e(X))$ hat den Grad $8$} + \only<1->{ \[ \arraycolsep=1.4pt \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} @@ -620,7 +655,8 @@ & & & &6X^8&+&0X^7&+&p(X)& & & & & & & & & & & & \\ \end{array} \] - + } + \only<1->{ \[ \arraycolsep=1.4pt \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} @@ -629,14 +665,14 @@ && 7X^8&+& p(X)& & & & & & & & & & & & & & & & \\ \end{array} \] - + } \vspace{10pt} - $\operatorname{ggT}(f(X),e(X)) = 6X^8$ + \only<1->{$\operatorname{ggT}(f(X),e(X)) = 6X^8$} \vspace{10pt} - $\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen + \only<1->{ $\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen } \end{frame} @@ -653,7 +689,7 @@ & & $0$& $1$\\ $0$& $9X + 5$& $1$& $0$\\ $1$& $10X + 3$& $9X+5$& $1$\\ - $2$& & $2X^2 + 0X + 5$& $10X + 3$\\ + $2$& & \textcolor<2->{blue}{$2X^2 + 0X + 5$}& $10X + 3$\\ \hline \end{tabular} @@ -662,49 +698,54 @@ \vspace{10pt} \begin{tabular}{ll} - Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\ - Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\ - Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$ + \only<1->{Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\} + \only<1->{Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\} + \only<1->{Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$} \end{tabular} \vspace{10pt} - + \only<1->{ \begin{center} $a^i = 5 \qquad \Rightarrow \qquad i = 3$ $a^i = 6 \qquad \Rightarrow \qquad i = 8$ \end{center} - - $D = [3,8]$ + } + \only<1->{$d(X) = (X-a^3)(X-a^8)$} \end{frame} -%------------------------------------------------------------------------------- +%------------------------------------------------------------------------------- +\section{Nachricht Rekonstruieren} \begin{frame} \frametitle{Rekonstruktion der Nachricht} \begin{itemize} - \item $W = [5,3,6,8,2,10,2,7,1,4]$ + \only<1->{\item $w = [5,3,6,8,2,10,2,7,1,4]$} - \item $D = [3,8]$ + \only<1->{\item $d(X) = (X-\textcolor<4->{red}{a^3})(X-\textcolor<4->{red}{a^8})$} \end{itemize} - + \only<1->{ \[ + \textcolor{gray}{ \begin{pmatrix} - 5 \\ 3 \\ 6 \\ 8 \\ 2 \\ 10 \\ 2 \\ 7 \\ 1 \\ 4 \\ + a^0 \\ a^1 \\ a^2 \\ \textcolor<4->{red}{a^3} \\ a^4 \\ a^5 \\ a^6 \\ a^7 \\ \textcolor<4->{red}{a^8} \\ a^9 \\ + \end{pmatrix}} + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ \textcolor<4->{red}{8} \\ 2 \\ 10 \\ 2 \\ 7 \\ \textcolor<4->{red}{1} \\ 4 \\ \end{pmatrix} = \begin{pmatrix} 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ - 8^0& 8^3& 8^6& 8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\ + \textcolor<4->{red}{8^0}& \textcolor<4->{red}{8^3}& \textcolor<4->{red}{8^6}& \textcolor<4->{red}{8^9}& \textcolor<4->{red}{8^{12}}& \textcolor<4->{red}{8^{15}}& \textcolor<4->{red}{8^{18}}& \textcolor<4->{red}{8^{21}}& \textcolor<4->{red}{8^{24}}& \textcolor<4->{red}{8^{27}}\\ 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ - 8^0& 8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\ + \textcolor<4->{red}{8^0}& \textcolor<4->{red}{8^8}& \textcolor<4->{red}{8^{16}}& \textcolor<4->{red}{8^{24}}& \textcolor<4->{red}{8^{32}}& \textcolor<4->{red}{8^{40}}& \textcolor<4->{red}{8^{48}}& \textcolor<4->{red}{8^{56}}& \textcolor<4->{red}{8^{64}}& \textcolor<4->{red}{8^{72}}\\ 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ \end{pmatrix} \cdot @@ -712,13 +753,14 @@ m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ \end{pmatrix} \] - + } + \only<1->{ \begin{itemize} \item Fehlerstellen entfernen \end{itemize} - + } \end{frame} -%------------------------------------------------------------------------------- +%------------------------------------------------------------------------------- \begin{frame} \frametitle{Rekonstruktion der Nachricht} @@ -728,25 +770,25 @@ \end{pmatrix} = \begin{pmatrix} - 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ - 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ - 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ - 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ - 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ - 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ - 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ - 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& \textcolor<3->{green}{8^0}& \textcolor<3->{green}{8^0}& \textcolor<3->{green}{8^0}& \textcolor<3->{green}{8^0}\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& \textcolor<3->{green}{8^6}& \textcolor<3->{green}{8^7}& \textcolor<3->{green}{8^8}& \textcolor<3->{green}{8^9}\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& \textcolor<3->{green}{8^{12}}& \textcolor<3->{green}{8^{14}}& \textcolor<3->{green}{8^{16}}& \textcolor<3->{green}{8^{18}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor<3->{green}{8^{24}}& \textcolor<3->{green}{8^{28}}& \textcolor<3->{green}{8^{32}}& \textcolor<3->{green}{8^{36}}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor<3->{green}{8^{30}}& \textcolor<3->{green}{8^{35}}& \textcolor<3->{green}{8^{40}}& \textcolor<3->{green}{8^{45}}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor<3->{green}{8^{36}}& \textcolor<3->{green}{8^{42}}& \textcolor<3->{green}{8^{48}}& \textcolor<3->{green}{8^{54}}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor<3->{green}{8^{42}}& \textcolor<3->{green}{8^{49}}& \textcolor<3->{green}{8^{56}}& \textcolor<3->{green}{8^{63}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor<3->{green}{8^{54}}& \textcolor<3->{green}{8^{63}}& \textcolor<3->{green}{8^{72}}& \textcolor<3->{green}{8^{81}}\\ \end{pmatrix} \cdot \begin{pmatrix} - m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor<2->{green}{m_6} \\ \textcolor<2->{green}{m_7} \\ \textcolor<2->{green}{m_8} \\ \textcolor<2->{green}{m_9} \\ \end{pmatrix} \] - + \only<1->{ \begin{itemize} \item Nullstellen entfernen \end{itemize} - + } \end{frame} %------------------------------------------------------------------------------- \begin{frame} @@ -754,7 +796,7 @@ \[ \begin{pmatrix} - 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor<2->{red}{7} \\ \textcolor<2->{red}{4} \\ \end{pmatrix} = \begin{pmatrix} @@ -764,8 +806,8 @@ 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ - 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}\\ - 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}\\ + \textcolor<2->{red}{8^0}& \textcolor<2->{red}{8^7}& \textcolor<2->{red}{8^{14}}& \textcolor<2->{red}{8^{21}}& \textcolor<2->{red}{8^{28}}& \textcolor<2->{red}{8^{35}}\\ + \textcolor<2->{red}{8^0}& \textcolor<2->{red}{8^9}& \textcolor<2->{red}{8^{18}}& \textcolor<2->{red}{8^{27}}& \textcolor<2->{red}{8^{36}}& \textcolor<2->{red}{8^{45}}\\ \end{pmatrix} \cdot \begin{pmatrix} @@ -774,11 +816,11 @@ \] \vspace{5pt} - + \only<1->{ \begin{itemize} \item Matrix in eine Quadratische Form bringen \end{itemize} - + } \end{frame} %------------------------------------------------------------------------------- \begin{frame} -- cgit v1.2.1 From 8c6a8e56c125c238dc64c21d1269fcdc7542c5cd Mon Sep 17 00:00:00 2001 From: JODBaer Date: Thu, 22 Apr 2021 23:45:32 +0200 Subject: =?UTF-8?q?merge=20lines=20gel=C3=B6scht?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- buch/papers/reedsolomon/RS presentation/RS.tex | 9 +++------ 1 file changed, 3 insertions(+), 6 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 61324f7..943f2da 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -84,11 +84,9 @@ \end{tabular} \end{center} \end{frame} -<<<<<<< Updated upstream + %------------------------------------------------------------------------------- -======= ->>>>>>> Stashed changes \section{Diskrete Fourier Transformation} \begin{frame} \frametitle{Idee} @@ -184,13 +182,12 @@ Indem in einem endlichen Körper gerechnet wird. } \end{frame} -<<<<<<< Updated upstream + %------------------------------------------------------------------------------- -======= + \section{Reed-Solomon in Endlichen Körpern} ->>>>>>> Stashed changes \begin{frame} \frametitle{Reed-Solomon in Endlichen Körpern} -- cgit v1.2.1 From 179ea16b001b6640e9b720d53ffc06f3e2389ff2 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Fri, 23 Apr 2021 00:30:36 +0200 Subject: appostroph verbessert --- buch/papers/reedsolomon/RS presentation/RS.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 943f2da..d09d77d 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -64,7 +64,7 @@ \begin{center} \begin{tabular}{ c c c } \hline - ``Nutzlas´´ & Fehler & Versenden \\ + ``Nutzlast'' & Fehler & Versenden \\ \hline 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ -- cgit v1.2.1 From ded210e33924d4c078e5a0d899c0585d7f987565 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Fri, 23 Apr 2021 12:58:40 +0200 Subject: Folien Verbesserungen animation --- buch/papers/reedsolomon/RS presentation/RS.aux | 167 +++++++++++----- buch/papers/reedsolomon/RS presentation/RS.log | 212 ++++++++++++++++----- buch/papers/reedsolomon/RS presentation/RS.nav | 117 ++++++++---- buch/papers/reedsolomon/RS presentation/RS.out | 9 +- buch/papers/reedsolomon/RS presentation/RS.pdf | Bin 135643 -> 207741 bytes buch/papers/reedsolomon/RS presentation/RS.snm | 2 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{8}{0}{5}{53/53}{}{0}} +\headcommand {\beamer@framepages {53}{53}} +\headcommand {\slideentry {8}{0}{6}{54/54}{}{0}} +\headcommand {\beamer@framepages {54}{54}} +\headcommand {\beamer@partpages {1}{54}} +\headcommand {\beamer@subsectionpages {43}{54}} +\headcommand {\beamer@sectionpages {43}{54}} +\headcommand {\beamer@documentpages {54}} +\headcommand {\gdef \inserttotalframenumber {29}} diff --git a/buch/papers/reedsolomon/RS presentation/RS.out b/buch/papers/reedsolomon/RS presentation/RS.out index 32b9a2c..364319e 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.out +++ b/buch/papers/reedsolomon/RS presentation/RS.out @@ -1,5 +1,8 @@ \BOOKMARK [2][]{Outline0.1}{Einführung}{}% 1 \BOOKMARK [2][]{Outline0.2}{Polynom\040Ansatz}{}% 2 -\BOOKMARK [2][]{Outline0.3}{Fourier\040Transformation}{}% 3 -\BOOKMARK [2][]{Outline0.4}{Diskrete\040Fourier\040Transformation}{}% 4 -\BOOKMARK [2][]{Outline0.5}{Probleme\040und\040Fragen}{}% 5 +\BOOKMARK [2][]{Outline0.3}{Diskrete\040Fourier\040Transformation}{}% 3 +\BOOKMARK [2][]{Outline0.4}{Reed-Solomon in Endlichen Körpern}{}% 4 +\BOOKMARK [2][]{Outline0.5}{Codierung\040eines\040Beispiels}{}% 5 +\BOOKMARK [2][]{Outline0.6}{Decodierung\040ohne\040Fehler}{}% 6 +\BOOKMARK [2][]{Outline0.7}{Decodierung\040mit\040Fehler}{}% 7 +\BOOKMARK [2][]{Outline0.8}{Nachricht\040Rekonstruieren}{}% 8 diff --git a/buch/papers/reedsolomon/RS presentation/RS.pdf b/buch/papers/reedsolomon/RS presentation/RS.pdf index 913bc42..d9d6693 100644 Binary files a/buch/papers/reedsolomon/RS presentation/RS.pdf and b/buch/papers/reedsolomon/RS presentation/RS.pdf differ diff --git a/buch/papers/reedsolomon/RS presentation/RS.snm b/buch/papers/reedsolomon/RS presentation/RS.snm index 6607ea8..86859c9 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.snm +++ b/buch/papers/reedsolomon/RS presentation/RS.snm @@ -1 +1 @@ -\beamer@slide {ft_discrete}{15} +\beamer@slide {ft_discrete}{21} diff --git a/buch/papers/reedsolomon/RS presentation/RS.synctex.gz b/buch/papers/reedsolomon/RS presentation/RS.synctex.gz index 001b5c8..04bd239 100644 Binary files a/buch/papers/reedsolomon/RS presentation/RS.synctex.gz and b/buch/papers/reedsolomon/RS presentation/RS.synctex.gz differ diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index d09d77d..7b2c4da 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -43,18 +43,18 @@ ${f}_2=\textcolor{blue}{2}$, ${f}_1=\textcolor{blue}{1}$, ${f}_0=\textcolor{blue}{5}$ als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. - \only<1>{ - Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, - \textcolor{green}{15}, \textcolor{green}{26}, - \textcolor{green}{41}, \textcolor{green}{60}, - \textcolor{green}{83}, \textcolor{green}{110})$ - \includegraphics[scale = 1.2]{images/polynom1.pdf}} - \only<2>{ - Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, - \textcolor{red}{50}, \textcolor{red}{37}, - \textcolor{green}{41}, \textcolor{green}{60}, - \textcolor{green}{83}, \textcolor{green}{110})$ - \includegraphics[scale = 1.2]{images/polynom2.pdf} + + Versende $ (p(1),p(2),\dots,p(7))$ + \visible<2->{ = (\textcolor{green}{8},} + \only<2>{\textcolor{green}{15},} + \only<3>{\textcolor{red}{50},} + \only<2>{\textcolor{green}{26},} + \only<3>{\textcolor{red}{37},} + \visible<2->{\textcolor{green}{41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})} + \only<2>{\includegraphics[scale = 1.2]{images/polynom1.pdf}} + \only<3>{\includegraphics[scale = 1.2]{images/polynom2.pdf}} + \visible<3>{ \newline \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} \end{frame} diff --git a/buch/papers/reedsolomon/RS presentation/RS.toc b/buch/papers/reedsolomon/RS presentation/RS.toc index 44c06ab..095b5e6 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.toc +++ b/buch/papers/reedsolomon/RS presentation/RS.toc @@ -1,6 +1,9 @@ \babel@toc {ngerman}{} \beamer@sectionintoc {1}{Einführung}{2}{0}{1} -\beamer@sectionintoc {2}{Polynom Ansatz}{3}{0}{2} -\beamer@sectionintoc {3}{Fourier Transformation}{7}{0}{3} -\beamer@sectionintoc {4}{Diskrete Fourier Transformation}{15}{0}{4} -\beamer@sectionintoc {5}{Probleme und Fragen}{17}{0}{5} +\beamer@sectionintoc {2}{Polynom Ansatz}{5}{0}{2} +\beamer@sectionintoc {3}{Diskrete Fourier Transformation}{13}{0}{3} +\beamer@sectionintoc {4}{Reed-Solomon in Endlichen Körpern}{27}{0}{4} +\beamer@sectionintoc {5}{Codierung eines Beispiels}{29}{0}{5} +\beamer@sectionintoc {6}{Decodierung ohne Fehler}{31}{0}{6} +\beamer@sectionintoc {7}{Decodierung mit Fehler}{36}{0}{7} +\beamer@sectionintoc {8}{Nachricht Rekonstruieren}{43}{0}{8} -- cgit v1.2.1 From 0a80be4477602e2d909e5eda40dae485ec6acd56 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Fri, 23 Apr 2021 13:02:38 +0200 Subject: Read me erstellt --- buch/papers/reedsolomon/RS presentation/README.txt | 1 + 1 file changed, 1 insertion(+) create mode 100644 buch/papers/reedsolomon/RS presentation/README.txt (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/RS presentation/README.txt b/buch/papers/reedsolomon/RS presentation/README.txt new file mode 100644 index 0000000..4d0620f --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/README.txt @@ -0,0 +1 @@ +Dies ist die Presentation des Reed-Solomon-Code \ No newline at end of file -- cgit v1.2.1 From d1b6d92a02d9c44b3860b73d5660c5c6863de0df Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Fri, 23 Apr 2021 21:19:34 +0200 Subject: handout added --- buch/papers/reedsolomon/RS presentation/RS.tex | 290 +++---- .../reedsolomon/RS presentation/RS_handout.tex | 921 +++++++++++++++++++++ 2 files changed, 1069 insertions(+), 142 deletions(-) create mode 100644 buch/papers/reedsolomon/RS presentation/RS_handout.tex (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/RS presentation/RS.tex b/buch/papers/reedsolomon/RS presentation/RS.tex index 943f2da..c215e66 100644 --- a/buch/papers/reedsolomon/RS presentation/RS.tex +++ b/buch/papers/reedsolomon/RS presentation/RS.tex @@ -14,8 +14,8 @@ \institute{OST Ostschweizer Fachhochschule} \date{26.04.2021} \subject{Mathematisches Seminar} - \setbeamercovered{transparent} - %\setbeamercovered{invisible} + %\setbeamercovered{transparent} + \setbeamercovered{invisible} \setbeamertemplate{navigation symbols}{} \begin{frame}[plain] \maketitle @@ -64,22 +64,22 @@ \begin{center} \begin{tabular}{ c c c } \hline - ``Nutzlas´´ & Fehler & Versenden \\ + Nutzlas & Fehler & Versenden \\ \hline 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ -\visible<2->{3}& -\visible<2->{3}& -\visible<3->{9 Werte eines Polynoms vom Grad 2} \\ +\visible<1->{3}& +\visible<1->{3}& +\visible<1->{9 Werte eines Polynoms vom Grad 2} \\ &&\\ -\visible<4->{$k$} & -\visible<4->{$t$} & -\visible<4->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ +\visible<1->{$k$} & +\visible<1->{$t$} & +\visible<1->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ \hline &&\\ &&\\ \multicolumn{3}{l} { - \visible<4>{Ausserdem können bis zu $2t$ Fehler erkannt werden!} + \visible<1>{Ausserdem können bis zu $2t$ Fehler erkannt werden!} } \end{tabular} \end{center} @@ -194,21 +194,21 @@ \begin{itemize} \onslide<1->{\item Warum endliche Körper?} - \onslide<1->{\qquad konkrete Zahlen $\rightarrow$ keine Rundungsfehler} + \onslide<2->{\qquad konkrete Zahlen $\rightarrow$ keine Rundungsfehler} - \onslide<1->{\qquad digitale Fehlerkorrektur} + \onslide<3->{\qquad digitale Fehlerkorrektur} - \onslide<1->{\qquad bessere Laufzeit} + %\onslide<4->{\qquad bessere Laufzeit} \vspace{10pt} - \onslide<1->{\item Nachricht = Nutzdaten + Fehlerkorrekturteil} + \onslide<4->{\item Nachricht = Nutzdaten + Fehlerkorrekturteil} \vspace{10pt} - \onslide<1->{\item aus Fehlerkorrekturteil die Fehlerstellen finden} + \onslide<5->{\item aus Fehlerkorrekturteil die Fehlerstellen finden} - \onslide<1->{\qquad $\Rightarrow$ gesucht ist ein Lokatorpolynom} + \onslide<6->{\qquad $\Rightarrow$ gesucht ist ein Lokatorpolynom} % \vspace{10pt} @@ -232,33 +232,33 @@ \begin{itemize} - \only<1->{\item endlicher Körper $q = 11$} + \onslide<1->{\item endlicher Körper $q = 11$} - \only<1->{ist eine Primzahl} + \onslide<2->{ist eine Primzahl} - \only<1->{beinhaltet die Zahlen $\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}$} + \onslide<3->{beinhaltet die Zahlen $\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}$} \vspace{10pt} - \only<1->{\item Nachrichtenblock $=$ Nutzlast $+$ Fehlerkorrekturstellen + \onslide<4->{\item Nachrichtenblock $=$ Nutzlast $+$ Fehlerkorrekturstellen} - $n = q - 1 = 10$ Zahlen} + \onslide<5->{$n = q - 1 = 10$ Zahlen} \vspace{10pt} - \only<1->{\item Max.~Fehler $z = 2$ + \onslide<6->{\item Max.~Fehler $t = 2$} - maximale Anzahl von Fehler, die wir noch korrigieren können} + \onslide<7->{maximale Anzahl von Fehler, die wir noch korrigieren können} \vspace{10pt} - \only<1->{\item Nutzlast $k = n -2t = 6$ Zahlen} + \onslide<8->{\item Nutzlast $k = n -2t = 6$ Zahlen} - \only<1->{Fehlerkorrkturstellen $2t = 4$ Zahlen} + \onslide<9->{Fehlerkorrkturstellen $2t = 4$ Zahlen} - \only<1->{Nachricht $m = [0,0,0,0,4,7,2,5,8,1]$} + \onslide<10->{Nachricht $m = [0,0,0,0,4,7,2,5,8,1]$} - \only<1->{als Polynom $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$} + \onslide<11->{als Polynom $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$} \end{itemize} @@ -269,31 +269,31 @@ \frametitle{Codierung} \begin{itemize} - \only<1->{\item Ansatz aus den komplexen Zahlen mit der diskreten Fouriertransformation} + \onslide<1->{\item Ansatz aus den komplexen Zahlen mit der diskreten Fouriertransformation} \vspace{10pt} - \only<1->{\item Eulersche Zahl $\mathrm{e}$ existiert nicht in $\mathbb{F}_{11}$} + \onslide<2->{\item Eulersche Zahl $\mathrm{e}$ existiert nicht in $\mathbb{F}_{11}$} \vspace{10pt} - \only<1->{\item Wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken + \onslide<3->{\item Wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken} - $\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}$} + \onslide<4->{$\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}$} \vspace{10pt} - \only<1->{\item Wir wählen $a = 8$} + \onslide<5->{\item Wir wählen $a = 8$} - \only<1->{$\mathbb{Z}_{11}\setminus\{0\} = \{1,8,9,6,4,10,3,2,5,7\}$} + \onslide<6->{$\mathbb{Z}_{11}\setminus\{0\} = \{1,8,9,6,4,10,3,2,5,7\}$} - \only<1->{$8$ ist eine primitive Einheitswurzel} + \onslide<7->{$8$ ist eine primitive Einheitswurzel} \vspace{10pt} - \only<1->{\item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$} + \onslide<8->{\item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$} - \only<1->{$\Rightarrow$ \qquad können wir auch als Matrix schreiben} + \onslide<9->{$\Rightarrow$ \qquad können wir auch als Matrix schreiben} \end{itemize} @@ -303,14 +303,14 @@ \frametitle{Codierung} \begin{itemize} - \only<1->{\item Übertragungsvektor $v$} + \onslide<1->{\item Übertragungsvektor $v$} - \only<1->{\item $v = A \cdot m$} + \onslide<2->{\item $v = A \cdot m$} \end{itemize} \[ - \only<1->{ + \onslide<3->{ v = \begin{pmatrix} 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ @@ -329,11 +329,11 @@ \end{pmatrix} } \] - \only<1->{ + \begin{itemize} - \item $v = [5,3,6,5,2,10,2,7,10,4]$ + \onslide<4->{\item $v = [5,3,6,5,2,10,2,7,10,4]$} \end{itemize} - } + \end{frame} %------------------------------------------------------------------------------- \section{Decodierung ohne Fehler} @@ -341,41 +341,44 @@ \frametitle{Decodierung ohne Fehler} \begin{itemize} - \only<1->{\item Der Empfänger erhält den unveränderten Vektor - $v = [5,3,6,5,2,10,2,7,10,4]$} + \onslide<1->{\item Der Empfänger erhält den unveränderten Vektor $v = [5,3,6,5,2,10,2,7,10,4]$} \vspace{10pt} - \only<1->{\item Wir suchen die Inverse der Matrix $A$} + \onslide<2->{\item Wir suchen die Inverse der Matrix $A$} \vspace{10pt} \end{itemize} \begin{columns}[t] - \begin{column}{0.50\textwidth} - \only<1->{ - Inverse der Fouriertransformation + \begin{column}{0.55\textwidth} + \onslide<3->{ Inverse der Fouriertransformation} \vspace{10pt} + \onslide<4->{ \[ F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt \] + } \vspace{10pt} + \onslide<5->{ \[ \mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega \] } \end{column} - \begin{column}{0.50\textwidth} - \only<1->{ - Inverse von $a$} + \begin{column}{0.45\textwidth} + \onslide<6->{Inverse von $a$} + \vspace{10pt} - \only<1->{ + + \onslide<7->{ \[ 8^{1} \Rightarrow 8^{-1} \] } - \only<1->{Inverse finden wir über den Eulkidischen Algorithmus} + + \onslide<8->{Inverse finden wir über den Eulkidischen Algorithmus} \vspace{10pt} \end{column} \end{columns} @@ -407,7 +410,7 @@ \begin{column}{0.50\textwidth} \begin{center} - \only<1->{ + \onslide<1->{ \begin{tabular}{| c | c c | c | r r |} \hline $k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\ @@ -417,17 +420,18 @@ $1$& $11$& $8$& $1$& $1$& $0$\\ $2$& $8$& $3$& $2$& $-1$& $1$\\ $3$& $3$& $2$& $1$& $3$& $-2$\\ - $4$& $2$& $1$& $2$& \textcolor<3->{blue}{$-4$}& \textcolor<3->{red}{$3$}\\ + $4$& $2$& $1$& $2$& \textcolor<2->{blue}{$-4$}& \textcolor<2->{red}{$3$}\\ $5$& $1$& $0$& & $11$& $-8$\\ \hline \end{tabular} } + \vspace{10pt} \begin{tabular}{rcl} - \only<1->{$\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$}\\ - \only<1->{$7 \cdot 8 + 3 \cdot 11$ &$=$& $1$}\\ - \only<1->{$8^{-1}$ &$=$& $7$} + \onslide<3->{$\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$}\\ + \onslide<4->{$7 \cdot 8 + 3 \cdot 11$ &$=$& $1$}\\ + \onslide<5->{$8^{-1}$ &$=$& $7$} \end{tabular} @@ -442,16 +446,16 @@ \frametitle{Decodierung mit Inverser Matrix} \begin{itemize} - \only<1->{\item $v = [5,3,6,5,2,10,2,7,10,4]$} + \onslide<1->{\item $v = [5,3,6,5,2,10,2,7,10,4]$} - \only<1->{\item $m = 1/10 \cdot A^{-1} \cdot v$} + \onslide<2->{\item $m = 1/10 \cdot A^{-1} \cdot v$} - \only<1->{\item $m = 10 \cdot A^{-1} \cdot v$} + \onslide<3->{\item $m = 10 \cdot A^{-1} \cdot v$} \end{itemize} - \only<1->{ + \onslide<4->{ \[ - m = \begin{pmatrix} + m = 10 \cdot \begin{pmatrix} 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ 7^0& 7^1& 7^2& 7^3& 7^4& 7^5& 7^6& 7^7& 7^8& 7^9\\ 7^0& 7^2& 7^4& 7^6& 7^8& 7^{10}& 7^{12}& 7^{14}& 7^{16}& 7^{18}\\ @@ -469,11 +473,11 @@ \end{pmatrix} \] } - \only<1->{ + \begin{itemize} - \item $m = [0,0,0,0,4,7,2,5,8,1]$ + \onslide<5->{\item $m = [0,0,0,0,4,7,2,5,8,1]$} \end{itemize} - } + \end{frame} %------------------------------------------------------------------------------- \section{Decodierung mit Fehler} @@ -481,48 +485,46 @@ \frametitle{Decodierung mit Fehler - Ansatz} \begin{itemize} - \only<1->{\item Gesendet: $v = [5,3,6,5,2,10,2,7,10,4]$} + \onslide<1->{\item Gesendet: $v = [5,3,6,5,2,10,2,7,10,4]$} - \only<1->{\item Empfangen: $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$} + \onslide<2->{\item Empfangen: $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$} - \only<1->{\item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]$} + \onslide<3->{\item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]$} \end{itemize} - \only<1->{Wie finden wir die Fehler?} + \onslide<4->{Wie finden wir die Fehler?} - \only<1->{ \begin{itemize} - \item $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ + \onslide<5->{\item $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$} - \item $r(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$ + \onslide<6->{\item $r(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$} %\only<7->{\item $e(X) = r(X) - m(X)$} - \item $e(X) = r(X) - m(X)$ + \onslide<7->{\item $e(X) = r(X) - m(X)$} \end{itemize} - } \begin{center} - \only<1->{ + \onslide<8->{ \begin{tabular}{c c c c c c c c c c c} \hline $i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\ \hline - $r(a^{i})$& \only<1->{$5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$}\\ - $m(a^{i})$& \only<1->{$5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$}\\ - $e(a^{i})$& \only<1->{$0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$}\\ + $r(a^{i})$& \onslide<9->{$5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$}\\ + $m(a^{i})$& \onslide<10->{$5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$}\\ + $e(a^{i})$& \onslide<11->{$0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$}\\ \hline \end{tabular} } \end{center} - \only<1->{ + \begin{itemize} - \item Alle Stellen, die nicht Null sind, sind Fehler + \onslide<12->{\item Alle Stellen, die nicht Null sind, sind Fehler} \end{itemize} - } + \end{frame} %------------------------------------------------------------------------------- @@ -530,31 +532,31 @@ \frametitle{Nullstellen des Fehlerpolynoms finden} \begin{itemize} - \only<1->{\item Satz von Fermat: $f(X) = X^{q-1}-1=0$} + \onslide<1->{\item Satz von Fermat: $f(X) = X^{q-1}-1=0$} \vspace{10pt} - \only<1->{\item $f(X) = X^{10}-1 = 0$ \qquad für $X \in \{1,2,3,4,5,6,7,8,9,10\}$} + \onslide<2->{\item $f(X) = X^{10}-1 = 0$ \qquad für $X \in \{1,2,3,4,5,6,7,8,9,10\}$} \vspace{10pt} - \only<1->{\item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + \onslide<3->{\item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$} \vspace{10pt} - \only<1->{\item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + \onslide<4->{\item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$} \vspace{10pt} - \only<1->{\item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat} + \onslide<5->{\item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat} \vspace{10pt} - \only<1->{$\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + \onslide<6->{$\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ \qquad \qquad \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9)$} @@ -567,39 +569,39 @@ \begin{itemize} - \item Satz von Fermat: $f(X) = X^{q-1}-1=0$ + \onslide<1->{\item Satz von Fermat: $f(X) = X^{q-1}-1=0$} \vspace{10pt} - \item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$ + \onslide<1->{\item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$} \vspace{10pt} - \item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + \onslide<1->{\item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ - \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$ + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$} \vspace{10pt} - \item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + \onslide<1->{\item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ - \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$ + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$} \vspace{10pt} - \item $\operatorname{kgV}$ gibt uns eine Liste von aller Nullstellen, die wir in $e$ und $d$ zerlegen können + \onslide<1->{\item $\operatorname{kgV}$ gibt uns eine Liste von aller Nullstellen, die wir in $e$ und $d$ zerlegen können} \vspace{10pt} - $\operatorname{kgV}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot $ + \onslide<2->{$\operatorname{kgV}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot $ - \qquad \qquad \qquad \qquad $(X-a^7)(X-a^8)(X-a^9) \cdot q(X)$ + \qquad \qquad \qquad \qquad $(X-a^7)(X-a^8)(X-a^9) \cdot q(X)$} - $= d(X) \cdot e(X)$ + \onslide<3->{$= d(X) \cdot e(X)$} \vspace{10pt} - \item Lokatorpolynom $d(X) = (X-a^3)(X-a^8)$ + \onslide<4->{\item Lokatorpolynom $d(X) = (X-a^3)(X-a^8)$} \end{itemize} @@ -610,29 +612,29 @@ \begin{itemize} - \only<1->{\item $e(X)$ ist unbekannt auf der Empfängerseite} + \onslide<1->{\item $e(X)$ ist unbekannt auf der Empfängerseite} \vspace{10pt} - \only<1->{\item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt?} + \onslide<2->{\item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt?} \vspace{10pt} - \only<1->{\item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$ + \onslide<3->{\item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$ In den bekannten Stellen liegt auch die Information, wo es Fehler gegeben hat} \vspace{10pt} - \only<1->{\item Daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$} + \onslide<4->{\item Daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$} \vspace{10pt} - \only<1->{\item $f(X) = X^{10} - 1 = X^{10} + 10$} + \onslide<5->{\item $f(X) = X^{10} - 1 = X^{10} + 10$} \vspace{10pt} - \only<1->{\item Jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen} + \onslide<6->{\item Jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen} \end{itemize} \end{frame} @@ -640,8 +642,8 @@ \begin{frame} \frametitle{Der Euklidische Algorithmus (nochmal)} - \only<1->{$\operatorname{ggT}(f(X),e(X))$ hat den Grad $8$} - \only<1->{ + \onslide<1->{$\operatorname{ggT}(f(X),e(X))$ hat den Grad $8$} + \onslide<2->{ \[ \arraycolsep=1.4pt \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} @@ -653,7 +655,7 @@ \end{array} \] } - \only<1->{ + \onslide<3->{ \[ \arraycolsep=1.4pt \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} @@ -665,11 +667,11 @@ } \vspace{10pt} - \only<1->{$\operatorname{ggT}(f(X),e(X)) = 6X^8$} + \onslide<4->{$\operatorname{ggT}(f(X),e(X)) = 6X^8$} \vspace{10pt} - \only<1->{ $\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen } + \onslide<5->{ $\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen } \end{frame} @@ -695,20 +697,22 @@ \vspace{10pt} \begin{tabular}{ll} - \only<1->{Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\} - \only<1->{Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\} - \only<1->{Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$} + \onslide<3->{Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\} + \onslide<4->{Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\} + \onslide<5->{Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$} \end{tabular} \vspace{10pt} - \only<1->{ + + \onslide<6->{ \begin{center} $a^i = 5 \qquad \Rightarrow \qquad i = 3$ $a^i = 6 \qquad \Rightarrow \qquad i = 8$ \end{center} - } - \only<1->{$d(X) = (X-a^3)(X-a^8)$} + } + + \onslide<7->{$d(X) = (X-a^3)(X-a^8)$} \end{frame} %------------------------------------------------------------------------------- @@ -718,12 +722,12 @@ \begin{itemize} - \only<1->{\item $w = [5,3,6,8,2,10,2,7,1,4]$} + \onslide<1->{\item $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$} - \only<1->{\item $d(X) = (X-\textcolor<4->{red}{a^3})(X-\textcolor<4->{red}{a^8})$} + \onslide<2->{\item $d(X) = (X-\textcolor<4->{red}{a^3})(X-\textcolor<4->{red}{a^8})$} \end{itemize} - \only<1->{ + \onslide<3->{ \[ \textcolor{gray}{ \begin{pmatrix} @@ -751,11 +755,11 @@ \end{pmatrix} \] } - \only<1->{ + \begin{itemize} - \item Fehlerstellen entfernen + \onslide<5->{\item Fehlerstellen entfernen} \end{itemize} - } + \end{frame} %------------------------------------------------------------------------------- \begin{frame} @@ -767,25 +771,25 @@ \end{pmatrix} = \begin{pmatrix} - 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& \textcolor<3->{green}{8^0}& \textcolor<3->{green}{8^0}& \textcolor<3->{green}{8^0}& \textcolor<3->{green}{8^0}\\ - 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& \textcolor<3->{green}{8^6}& \textcolor<3->{green}{8^7}& \textcolor<3->{green}{8^8}& \textcolor<3->{green}{8^9}\\ - 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& \textcolor<3->{green}{8^{12}}& \textcolor<3->{green}{8^{14}}& \textcolor<3->{green}{8^{16}}& \textcolor<3->{green}{8^{18}}\\ - 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor<3->{green}{8^{24}}& \textcolor<3->{green}{8^{28}}& \textcolor<3->{green}{8^{32}}& \textcolor<3->{green}{8^{36}}\\ - 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor<3->{green}{8^{30}}& \textcolor<3->{green}{8^{35}}& \textcolor<3->{green}{8^{40}}& \textcolor<3->{green}{8^{45}}\\ - 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor<3->{green}{8^{36}}& \textcolor<3->{green}{8^{42}}& \textcolor<3->{green}{8^{48}}& \textcolor<3->{green}{8^{54}}\\ - 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor<3->{green}{8^{42}}& \textcolor<3->{green}{8^{49}}& \textcolor<3->{green}{8^{56}}& \textcolor<3->{green}{8^{63}}\\ - 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor<3->{green}{8^{54}}& \textcolor<3->{green}{8^{63}}& \textcolor<3->{green}{8^{72}}& \textcolor<3->{green}{8^{81}}\\ + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& \textcolor<4->{green}{8^0}& \textcolor<4->{green}{8^0}& \textcolor<4->{green}{8^0}& \textcolor<4->{green}{8^0}\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& \textcolor<4->{green}{8^6}& \textcolor<4->{green}{8^7}& \textcolor<4->{green}{8^8}& \textcolor<4->{green}{8^9}\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& \textcolor<4->{green}{8^{12}}& \textcolor<4->{green}{8^{14}}& \textcolor<4->{green}{8^{16}}& \textcolor<4->{green}{8^{18}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor<4->{green}{8^{24}}& \textcolor<4->{green}{8^{28}}& \textcolor<4->{green}{8^{32}}& \textcolor<4->{green}{8^{36}}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor<4->{green}{8^{30}}& \textcolor<4->{green}{8^{35}}& \textcolor<4->{green}{8^{40}}& \textcolor<4->{green}{8^{45}}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor<4->{green}{8^{36}}& \textcolor<4->{green}{8^{42}}& \textcolor<4->{green}{8^{48}}& \textcolor<4->{green}{8^{54}}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor<4->{green}{8^{42}}& \textcolor<4->{green}{8^{49}}& \textcolor<4->{green}{8^{56}}& \textcolor<4->{green}{8^{63}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor<4->{green}{8^{54}}& \textcolor<4->{green}{8^{63}}& \textcolor<4->{green}{8^{72}}& \textcolor<4->{green}{8^{81}}\\ \end{pmatrix} \cdot \begin{pmatrix} m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor<2->{green}{m_6} \\ \textcolor<2->{green}{m_7} \\ \textcolor<2->{green}{m_8} \\ \textcolor<2->{green}{m_9} \\ \end{pmatrix} \] - \only<1->{ + \begin{itemize} - \item Nullstellen entfernen + \onslide<3->{\item Nullstellen entfernen} \end{itemize} - } + \end{frame} %------------------------------------------------------------------------------- \begin{frame} @@ -793,7 +797,7 @@ \[ \begin{pmatrix} - 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor<2->{red}{7} \\ \textcolor<2->{red}{4} \\ + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor<3->{red}{7} \\ \textcolor<3->{red}{4} \\ \end{pmatrix} = \begin{pmatrix} @@ -803,8 +807,8 @@ 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ - \textcolor<2->{red}{8^0}& \textcolor<2->{red}{8^7}& \textcolor<2->{red}{8^{14}}& \textcolor<2->{red}{8^{21}}& \textcolor<2->{red}{8^{28}}& \textcolor<2->{red}{8^{35}}\\ - \textcolor<2->{red}{8^0}& \textcolor<2->{red}{8^9}& \textcolor<2->{red}{8^{18}}& \textcolor<2->{red}{8^{27}}& \textcolor<2->{red}{8^{36}}& \textcolor<2->{red}{8^{45}}\\ + \textcolor<3->{red}{8^0}& \textcolor<3->{red}{8^7}& \textcolor<3->{red}{8^{14}}& \textcolor<3->{red}{8^{21}}& \textcolor<3->{red}{8^{28}}& \textcolor<3->{red}{8^{35}}\\ + \textcolor<3->{red}{8^0}& \textcolor<3->{red}{8^9}& \textcolor<3->{red}{8^{18}}& \textcolor<3->{red}{8^{27}}& \textcolor<3->{red}{8^{36}}& \textcolor<3->{red}{8^{45}}\\ \end{pmatrix} \cdot \begin{pmatrix} @@ -813,11 +817,11 @@ \] \vspace{5pt} - \only<1->{ + \begin{itemize} - \item Matrix in eine Quadratische Form bringen + \onslide<2->{\item Matrix in eine Quadratische Form bringen} \end{itemize} - } + \end{frame} %------------------------------------------------------------------------------- \begin{frame} @@ -845,7 +849,7 @@ \vspace{5pt} \begin{itemize} - \item Matrix Invertieren + \onslide<2->{\item Matrix Invertieren} \end{itemize} \end{frame} @@ -873,9 +877,10 @@ \] \begin{center} - $\Downarrow$ + \onslide<2->{$\Downarrow$} \end{center} \[ + \onslide<3->{ \begin{pmatrix} m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \end{pmatrix} @@ -892,6 +897,7 @@ \begin{pmatrix} 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \end{pmatrix} + } \] \end{frame} @@ -919,7 +925,7 @@ \] \begin{itemize} - \item $m = [4,7,2,5,8,1]$ + \onslide<2->{\item $m = [4,7,2,5,8,1]$} \end{itemize} \end{frame} diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.tex b/buch/papers/reedsolomon/RS presentation/RS_handout.tex new file mode 100644 index 0000000..863b3a2 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.tex @@ -0,0 +1,921 @@ +\documentclass[11pt,aspectratio=169]{beamer} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[ngerman]{babel} +\usepackage{tikz} +\usetheme{Hannover} + +\begin{document} + \author{Joshua Bär und Michael Steiner} + \title{Reed-Solomon-Code} + \subtitle{} + \logo{} + \institute{OST Ostschweizer Fachhochschule} + \date{26.04.2021} + \subject{Mathematisches Seminar} + %\setbeamercovered{transparent} + \setbeamercovered{invisible} + \setbeamertemplate{navigation symbols}{} + \begin{frame}[plain] + \maketitle + \end{frame} +%------------------------------------------------------------------------------- +\section{Einführung} + \begin{frame} + \frametitle{Reed-Solomon-Code:} + \begin{itemize} + \visible<1->{\item Für Übertragung von Daten} + \visible<2->{\item Ermöglicht Korrektur von Übertragungsfehler} + \visible<3->{\item Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc.} + \end{itemize} + \end{frame} +%------------------------------------------------------------------------------- +\section{Polynom Ansatz} + \begin{frame} + \begin{itemize} + \item Beispiel $2, 1, 5$ versenden und auf 2 Fehler absichern + \end{itemize} + \end{frame} + \begin{frame} + \frametitle{Beispiel} + Übertragen von + ${f}_2=\textcolor{blue}{2}$, ${f}_1=\textcolor{blue}{1}$, ${f}_0=\textcolor{blue}{5}$ + als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. + + \only<1>{ + Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, + \textcolor{green}{15}, \textcolor{green}{26}, + \textcolor{green}{41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})$ + \includegraphics[scale = 1.2]{images/polynom1.pdf}} + \only<2>{ + Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, + \textcolor{red}{50}, \textcolor{red}{37}, + \textcolor{green}{41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})$ + \includegraphics[scale = 1.2]{images/polynom2.pdf} + \newline + \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Parameter} + \begin{center} + \begin{tabular}{ c c c } + \hline + Nutzlas & Fehler & Versenden \\ + \hline + 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ +\visible<1->{3}& +\visible<1->{3}& +\visible<1->{9 Werte eines Polynoms vom Grad 2} \\ + &&\\ +\visible<1->{$k$} & +\visible<1->{$t$} & +\visible<1->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ + \hline + &&\\ + &&\\ + \multicolumn{3}{l} { + \visible<1>{Ausserdem können bis zu $2t$ Fehler erkannt werden!} + } + \end{tabular} + \end{center} + \end{frame} + +%------------------------------------------------------------------------------- + +\section{Diskrete Fourier Transformation} + \begin{frame} + \frametitle{Idee} + \begin{itemize} + \item Fourier-transformieren + \item Übertragung + \item Rücktransformieren + \end{itemize} + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \begin{figure} + \only<1>{ + \includegraphics[width=0.9\linewidth]{images/fig1.pdf} + } + \only<2>{ + \includegraphics[width=0.9\linewidth]{images/fig2.pdf} + } + \only<3>{ + \includegraphics[width=0.9\linewidth]{images/fig3.pdf} + } + \only<4>{ + \includegraphics[width=0.9\linewidth]{images/fig4.pdf} + } + \only<5>{ + \includegraphics[width=0.9\linewidth]{images/fig5.pdf} + } + \only<6>{ + \includegraphics[width=0.9\linewidth]{images/fig6.pdf} + } + \only<7>{ + \includegraphics[width=0.9\linewidth]{images/fig7.pdf} + } + \end{figure} + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Diskrete Fourier Transformation} + \begin{itemize} + \item Diskrete Fourier-Transformation gegeben durch: + \visible<1->{ + \[ + \label{ft_discrete} + \hat{c}_{k} + = \frac{1}{N} \sum_{n=0}^{N-1} + {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} + \]} + \visible<2->{ + \item Ersetzte + \[ + w = e^{-\frac{2\pi j}{N} k} + \]} + \visible<3->{ + \item Wenn $N$ konstant: + \[ + \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) + \]} + \end{itemize} + \end{frame} + +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Diskrete Fourier Transformation} + \[ + \begin{pmatrix} + \hat{c}_1 \\\hat{c}_2 \\\hat{c}_3 \\ \vdots \\\hat{c}_n + \end{pmatrix} + = \frac{1}{N} + \begin{pmatrix} + w^0 & w^0 & w^0 & \dots &w^0 \\ + w^0 & w^1 &w^2 & \dots &w^{N-1} \\ + w^0 & w^2 &w^4 & \dots &w^{2(N-1)} \\ + \vdots & \vdots &\vdots &\ddots &\vdots \\ + w^0 & w^{1(N-1)}&w^{2(N-1)}& \dots &w^{(N-1)(N-1)} \\ + \end{pmatrix} + \begin{pmatrix} + \textcolor{blue}{f_0} \\ + \textcolor{blue}{f_1} \\ + \textcolor{blue}{f_2} \\ + \vdots \\ + 0 \\ + \end{pmatrix} + \] + \end{frame} +%------------------------------------------------------------------------------- + + \begin{frame} + \frametitle{Probleme und Fragen} + + Wie wird der Fehler lokalisiert? + \visible<2>{ + \newline + Indem in einem endlichen Körper gerechnet wird. + } + \end{frame} + +%------------------------------------------------------------------------------- + + +\section{Reed-Solomon in Endlichen Körpern} + + \begin{frame} + \frametitle{Reed-Solomon in Endlichen Körpern} + + \begin{itemize} + \item Warum endliche Körper? + + \qquad konkrete Zahlen $\rightarrow$ keine Rundungsfehler + + \qquad digitale Fehlerkorrektur + + %\onslide<4->{\qquad bessere Laufzeit} + + \vspace{10pt} + + \item Nachricht = Nutzdaten + Fehlerkorrekturteil + + \vspace{10pt} + + \item aus Fehlerkorrekturteil die Fehlerstellen finden + + \qquad $\Rightarrow$ gesucht ist ein Lokatorpolynom + +% \vspace{10pt} + +% \onslide<1->{\item Im Fehlerfall sollen wir aus der Nachricht ein Lokatorpolynom berechnen können, welches die fehlerhaften Stellen beinhaltet} + +% Wir sollten im Fehlerfall in der Lage sein, aus der Nachricht ein Lokatorpolynom zu berechnen, welches die Fehlerhaften Stellen beinhaltet + + \end{itemize} + +% TODO + +% erklärung und einführung der endlichen körper, was wollen wir erreichen? + +% wir versenden im endefekt mehr daten als unsere nachricht umfasst, damit die korrektur sichergestellt werden kann + +% sollten wir fehler bekommen, was uns die korrekturstellen mitgeteilt wird, dann ist es unsere aufgabe ein lokatorpolynom zu finden, welches uns verrät, auf welchen zeilen der Fehler aufgetreten ist + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Definition eines Beispiels} + + \begin{itemize} + + \item endlicher Körper $q = 11$ + + ist eine Primzahl + + beinhaltet die Zahlen $\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}$ + + \vspace{10pt} + + \item Nachrichtenblock $=$ Nutzlast $+$ Fehlerkorrekturstellen + + $n = q - 1 = 10$ Zahlen + + \vspace{10pt} + + \item Max.~Fehler $t = 2$ + + maximale Anzahl von Fehler, die wir noch korrigieren können + + \vspace{10pt} + + \item Nutzlast $k = n -2t = 6$ Zahlen + + Fehlerkorrkturstellen $2t = 4$ Zahlen + + Nachricht $m = [0,0,0,0,4,7,2,5,8,1]$ + + als Polynom $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Codierung eines Beispiels} + \begin{frame} + \frametitle{Codierung} + + \begin{itemize} + \item Ansatz aus den komplexen Zahlen mit der diskreten Fouriertransformation + + \vspace{10pt} + + \item Eulersche Zahl $\mathrm{e}$ existiert nicht in $\mathbb{F}_{11}$ + + \vspace{10pt} + + \item Wir suchen $a$ so, dass $a^i$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken + + $\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}$ + + \vspace{10pt} + + \item Wir wählen $a = 8$ + + $\mathbb{Z}_{11}\setminus\{0\} = \{1,8,9,6,4,10,3,2,5,7\}$ + + $8$ ist eine primitive Einheitswurzel + + \vspace{10pt} + + \item $m(8^0) = 4\cdot1 + 7\cdot1 + 2\cdot1 + 5\cdot1 + 8\cdot1 + 1 = 5$ + + $\Rightarrow$ \qquad können wir auch als Matrix schreiben + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Codierung} + + \begin{itemize} + \item Übertragungsvektor $v$ + + \item $v = A \cdot m$ + + \end{itemize} + + \[ + v = \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^3& 8^6& 8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $v = [5,3,6,5,2,10,2,7,10,4]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Decodierung ohne Fehler} + \begin{frame} + \frametitle{Decodierung ohne Fehler} + + \begin{itemize} + \item Der Empfänger erhält den unveränderten Vektor $v = [5,3,6,5,2,10,2,7,10,4]$ + + \vspace{10pt} + + \item Wir suchen die Inverse der Matrix $A$ + + \vspace{10pt} + + \end{itemize} + + \begin{columns}[t] + \begin{column}{0.55\textwidth} + Inverse der Fouriertransformation + \vspace{10pt} + + \[ + F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt + \] + + \vspace{10pt} + + \[ + \mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega + \] + + \end{column} + \begin{column}{0.45\textwidth} + Inverse von $a$ + + \vspace{10pt} + + \[ + 8^{1} \Rightarrow 8^{-1} + \] + + Inverse finden wir über den Eulkidischen Algorithmus + \vspace{10pt} + \end{column} + \end{columns} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Euklidische Algorithmus} + + \begin{columns}[t] + \begin{column}{0.50\textwidth} + + Recap aus der Vorlesung: + + Gegeben $a \in \mathbb{F}_p$, finde $b = a^{-1} \in \mathbb{F}_p$ + + \begin{tabular}{rcl} + $a b$ &$\equiv$& $1 \mod p$\\ + $a b$ &$=$& $1 + n p$\\ + $a b - n p$ &$=$& $1$\\ + &&\\ + $\operatorname{ggT}(a,p)$&$=$& $1$\\ + $sa + tp$&$=$& $1$\\ + $b$&$=$&$s$\\ + $n$&$=$&$-t$ + \end{tabular} + + \end{column} + \begin{column}{0.50\textwidth} + + \begin{center} + + \begin{tabular}{| c | c c | c | r r |} + \hline + $k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\ + \hline + & & & & $1$& $0$\\ + $0$& $8$& $11$& $0$& $0$& $1$\\ + $1$& $11$& $8$& $1$& $1$& $0$\\ + $2$& $8$& $3$& $2$& $-1$& $1$\\ + $3$& $3$& $2$& $1$& $3$& $-2$\\ + $4$& $2$& $1$& $2$& \textcolor{blue}{$-4$}& \textcolor{red}{$3$}\\ + $5$& $1$& $0$& & $11$& $-8$\\ + \hline + \end{tabular} + + + \vspace{10pt} + + \begin{tabular}{rcl} + $\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$\\ + $7 \cdot 8 + 3 \cdot 11$ &$=$& $1$\\ + $8^{-1}$ &$=$& $7$ + + \end{tabular} + + \end{center} + + \end{column} + \end{columns} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Decodierung mit Inverser Matrix} + + \begin{itemize} + \item $v = [5,3,6,5,2,10,2,7,10,4]$ + + \item $m = 1/10 \cdot A^{-1} \cdot v$ + + \item $m = 10 \cdot A^{-1} \cdot v$ + + \end{itemize} + + \[ + m = 10 \cdot \begin{pmatrix} + 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ + 7^0& 7^1& 7^2& 7^3& 7^4& 7^5& 7^6& 7^7& 7^8& 7^9\\ + 7^0& 7^2& 7^4& 7^6& 7^8& 7^{10}& 7^{12}& 7^{14}& 7^{16}& 7^{18}\\ + 7^0& 7^3& 7^6& 7^9& 7^{12}& 7^{15}& 7^{18}& 7^{21}& 7^{24}& 7^{27}\\ + 7^0& 7^4& 7^8& 7^{12}& 7^{16}& 7^{20}& 7^{24}& 7^{28}& 7^{32}& 7^{36}\\ + 7^0& 7^5& 7^{10}& 7^{15}& 7^{20}& 7^{25}& 7^{30}& 7^{35}& 7^{40}& 7^{45}\\ + 7^0& 7^6& 7^{12}& 7^{18}& 7^{24}& 7^{30}& 7^{36}& 7^{42}& 7^{48}& 7^{54}\\ + 7^0& 7^7& 7^{14}& 7^{21}& 7^{28}& 7^{35}& 7^{42}& 7^{49}& 7^{56}& 7^{63}\\ + 7^0& 7^8& 7^{16}& 7^{24}& 7^{32}& 7^{40}& 7^{48}& 7^{56}& 7^{64}& 7^{72}\\ + 7^0& 7^9& 7^{18}& 7^{27}& 7^{36}& 7^{45}& 7^{54}& 7^{63}& 7^{72}& 7^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 5 \\ 2 \\ 10 \\ 2 \\ 7 \\ 10 \\ 4 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $m = [0,0,0,0,4,7,2,5,8,1]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- +\section{Decodierung mit Fehler} + \begin{frame} + \frametitle{Decodierung mit Fehler - Ansatz} + + \begin{itemize} + \item Gesendet: $v = [5,3,6,5,2,10,2,7,10,4]$ + + \item Empfangen: $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$ + + \item Rücktransformation: $r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]$ + + \end{itemize} + + Wie finden wir die Fehler? + + \begin{itemize} + \item $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ + + \item $r(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$ + + \item $e(X) = r(X) - m(X)$ + + \end{itemize} + + \begin{center} + + \begin{tabular}{c c c c c c c c c c c} + \hline + $i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\ + \hline + $r(a^{i})$& $5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$\\ + $m(a^{i})$& $5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$\\ + $e(a^{i})$& $0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$\\ + \hline + \end{tabular} + + \end{center} + + \begin{itemize} + \item Alle Stellen, die nicht Null sind, sind Fehler + \end{itemize} + + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Nullstellen des Fehlerpolynoms finden} + + \begin{itemize} + \item Satz von Fermat: $f(X) = X^{q-1}-1=0$ + + \vspace{10pt} + + \item $f(X) = X^{10}-1 = 0$ \qquad für $X \in \{1,2,3,4,5,6,7,8,9,10\}$ + + \vspace{10pt} + + \item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$ + + \vspace{10pt} + + \item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$ + + \vspace{10pt} + + \item $\operatorname{ggT}$ gibt uns eine Liste der Nullstellen, an denen es keine Fehler gegeben hat + + \vspace{10pt} + + $\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9)$ + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Nullstellen des Fehlerpolynoms finden} + + \begin{itemize} + + \item Satz von Fermat: $f(X) = X^{q-1}-1=0$ + + \vspace{10pt} + + \item $f(X) = X^{10}-1 = 0$ \qquad für $X = [1,2,3,4,5,6,7,8,9,10]$ + + \vspace{10pt} + + \item $f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7)(X-a^8)(X-a^9)$ + + \vspace{10pt} + + \item $e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6) \cdot$ + + \qquad \qquad $(X-a^7) \qquad \qquad (X-a^9) \cdot p(x)$ + + \vspace{10pt} + + \item $\operatorname{kgV}$ gibt uns eine Liste von aller Nullstellen, die wir in $e$ und $d$ zerlegen können + + \vspace{10pt} + + $\operatorname{kgV}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6) \cdot $ + + \qquad \qquad \qquad \qquad $(X-a^7)(X-a^8)(X-a^9) \cdot q(X)$ + + $= d(X) \cdot e(X)$ + + \vspace{10pt} + + \item Lokatorpolynom $d(X) = (X-a^3)(X-a^8)$ + + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Kennen wir $e(X)$?} + + \begin{itemize} + + \item $e(X)$ ist unbekannt auf der Empfängerseite + + \vspace{10pt} + + \item $e(X) = r(X) - m(X)$ \qquad $\rightarrow$ \qquad $m(X)$ ist unbekannt? + + \vspace{10pt} + + \item $m$ ist nicht gänzlich unbekannt: $m = [0,0,0,0,?,?,?,?,?,?]$ + + In den bekannten Stellen liegt auch die Information, wo es Fehler gegeben hat + + \vspace{10pt} + + \item Daraus folgt $e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$ + + \vspace{10pt} + + \item $f(X) = X^{10} - 1 = X^{10} + 10$ + + \vspace{10pt} + + \item Jetzt können wir den $\operatorname{ggT}$ von $f(X)$ und $e(X)$ berechnen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Euklidische Algorithmus (nochmal)} + + $\operatorname{ggT}(f(X),e(X))$ hat den Grad $8$ + + \[ + \arraycolsep=1.4pt + \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + X^{10}& & & & & & &+& 10& & & & &:&5X^9&+&7X^8&+& 4X^7&+&10X^6&+&p(X)&=&9X&+&5\\ + X^{10}&+& 8X^9&+& 3X^8&+&2X^7&+& p(X)& & & & & & & & & & & & & & & & \\ \cline{1-9} + && 3X^9&+& 8X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ + && 3X^9&+& 2X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ \cline{3-9} + & & & &6X^8&+&0X^7&+&p(X)& & & & & & & & & & & & \\ + \end{array} + \] + + \[ + \arraycolsep=1.4pt + \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + 5X^9&+& 7X^8&+& 4X^7&+& 10X^6&+& p(X)& & & & &:&6X^8&+&0X^7& & & & & & &=&10X&+&3\\ + 5X^9&+& 0X^8&+& p(X)& & & & & & & & & & & & & & & & & & & & \\ \cline{1-5} + && 7X^8&+& p(X)& & & & & & & & & & & & & & & & \\ + \end{array} + \] + + \vspace{10pt} + + $\operatorname{ggT}(f(X),e(X)) = 6X^8$ + + \vspace{10pt} + + $\operatorname{kgV}$ durch den erweiterten Euklidischen Algorithmus bestimmen + + \end{frame} + +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Der Erweiterte Euklidische Algorithmus} + + \begin{center} + + \begin{tabular}{| c | c | c c |} + \hline + $k$ & $q_i$ & $e_i$ & $f_i$\\ + \hline + & & $0$& $1$\\ + $0$& $9X + 5$& $1$& $0$\\ + $1$& $10X + 3$& $9X+5$& $1$\\ + $2$& & \textcolor{blue}{$2X^2 + 0X + 5$}& $10X + 3$\\ + \hline + \end{tabular} + + \end{center} + + \vspace{10pt} + + \begin{tabular}{ll} + Somit erhalten wir den Faktor& $d(X) = 2X^2 + 5$\\ + Faktorisiert erhalten wir& $d(X) = 2(X-5)(X-6)$\\ + Lokatorpolynom& $d(X) = (X-a^i)(X-a^i)$ + \end{tabular} + + \vspace{10pt} + + \begin{center} + $a^i = 5 \qquad \Rightarrow \qquad i = 3$ + + $a^i = 6 \qquad \Rightarrow \qquad i = 8$ + \end{center} + + + $d(X) = (X-a^3)(X-a^8)$ + + \end{frame} +%------------------------------------------------------------------------------- +\section{Nachricht Rekonstruieren} + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \begin{itemize} + + \item $w = [5,3,6,\textcolor{red}{8},2,10,2,7,\textcolor{red}{1},4]$ + + \item $d(X) = (X-\textcolor{red}{a^3})(X-\textcolor{red}{a^8})$ + + \end{itemize} + + \[ + \textcolor{gray}{ + \begin{pmatrix} + a^0 \\ a^1 \\ a^2 \\ \textcolor{red}{a^3} \\ a^4 \\ a^5 \\ a^6 \\ a^7 \\ \textcolor{red}{a^8} \\ a^9 \\ + \end{pmatrix}} + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ \textcolor{red}{8} \\ 2 \\ 10 \\ 2 \\ 7 \\ \textcolor{red}{1} \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^3}& \textcolor{red}{8^6}& \textcolor{red}{8^9}& \textcolor{red}{8^{12}}& \textcolor{red}{8^{15}}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{27}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^8}& \textcolor{red}{8^{16}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{32}}& \textcolor{red}{8^{40}}& \textcolor{red}{8^{48}}& \textcolor{red}{8^{56}}& \textcolor{red}{8^{64}}& \textcolor{red}{8^{72}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item Fehlerstellen entfernen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& \textcolor{green}{8^0}& \textcolor{green}{8^0}& \textcolor{green}{8^0}& \textcolor{green}{8^0}\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& \textcolor{green}{8^6}& \textcolor{green}{8^7}& \textcolor{green}{8^8}& \textcolor{green}{8^9}\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& \textcolor{green}{8^{12}}& \textcolor{green}{8^{14}}& \textcolor{green}{8^{16}}& \textcolor{green}{8^{18}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor{green}{8^{24}}& \textcolor{green}{8^{28}}& \textcolor{green}{8^{32}}& \textcolor{green}{8^{36}}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor{green}{8^{30}}& \textcolor{green}{8^{35}}& \textcolor{green}{8^{40}}& \textcolor{green}{8^{45}}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor{green}{8^{36}}& \textcolor{green}{8^{42}}& \textcolor{green}{8^{48}}& \textcolor{green}{8^{54}}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor{green}{8^{42}}& \textcolor{green}{8^{49}}& \textcolor{green}{8^{56}}& \textcolor{green}{8^{63}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor{green}{8^{54}}& \textcolor{green}{8^{63}}& \textcolor{green}{8^{72}}& \textcolor{green}{8^{81}}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor{green}{m_6} \\ \textcolor{green}{m_7} \\ \textcolor{green}{m_8} \\ \textcolor{green}{m_9} \\ + \end{pmatrix} + \] + + \begin{itemize} + \item Nullstellen entfernen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor{red}{7} \\ \textcolor{red}{4} \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^7}& \textcolor{red}{8^{14}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{28}}& \textcolor{red}{8^{35}}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^9}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{27}}& \textcolor{red}{8^{36}}& \textcolor{red}{8^{45}}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \vspace{5pt} + + \begin{itemize} + \item Matrix in eine Quadratische Form bringen + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + = + \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \vspace{5pt} + + \begin{itemize} + \item Matrix Invertieren + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + = + \begin{pmatrix} + 1& 1& 1& 1& 1& 1\\ + 1& 8& 9& 6& 4& 10\\ + 1& 9& 4& 3& 5& 1\\ + 1& 4& 5& 9& 3& 1\\ + 1& 10& 1& 10& 1& 10\\ + 1& 3& 9& 5& 4& 1\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + \] + + \begin{center} + $\Downarrow$ + \end{center} + \[ + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + = + \begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + \] + + \end{frame} +%------------------------------------------------------------------------------- + \begin{frame} + \frametitle{Rekonstruktion der Nachricht} + + \[ + \begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ + \end{pmatrix} + = + \begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ + \end{pmatrix} + \cdot + \begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ + \end{pmatrix} + \] + + \begin{itemize} + \item $m = [4,7,2,5,8,1]$ + \end{itemize} + + \end{frame} +%------------------------------------------------------------------------------- + +\end{document} -- cgit v1.2.1 From df810d1315cfb1c4b876d5145846d6ea70753141 Mon Sep 17 00:00:00 2001 From: JODBaer Date: Sat, 24 Apr 2021 15:27:05 +0200 Subject: Handout animation deleted --- .../reedsolomon/RS presentation/RS_handout.aux | 143 +++ 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a/buch/papers/reedsolomon/RS presentation/RS_handout.out b/buch/papers/reedsolomon/RS presentation/RS_handout.out new file mode 100644 index 0000000..364319e --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.out @@ -0,0 +1,8 @@ +\BOOKMARK [2][]{Outline0.1}{Einführung}{}% 1 +\BOOKMARK [2][]{Outline0.2}{Polynom\040Ansatz}{}% 2 +\BOOKMARK [2][]{Outline0.3}{Diskrete\040Fourier\040Transformation}{}% 3 +\BOOKMARK [2][]{Outline0.4}{Reed-Solomon in Endlichen Körpern}{}% 4 +\BOOKMARK [2][]{Outline0.5}{Codierung\040eines\040Beispiels}{}% 5 +\BOOKMARK [2][]{Outline0.6}{Decodierung\040ohne\040Fehler}{}% 6 +\BOOKMARK [2][]{Outline0.7}{Decodierung\040mit\040Fehler}{}% 7 +\BOOKMARK [2][]{Outline0.8}{Nachricht\040Rekonstruieren}{}% 8 diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.pdf b/buch/papers/reedsolomon/RS presentation/RS_handout.pdf new file mode 100644 index 0000000..382049d Binary files /dev/null and b/buch/papers/reedsolomon/RS presentation/RS_handout.pdf differ diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.snm b/buch/papers/reedsolomon/RS presentation/RS_handout.snm new file mode 100644 index 0000000..1796304 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.snm @@ -0,0 +1 @@ +\beamer@slide {ft_discrete}{13} diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.synctex.gz b/buch/papers/reedsolomon/RS presentation/RS_handout.synctex.gz new file mode 100644 index 0000000..c28a28a Binary files /dev/null and b/buch/papers/reedsolomon/RS presentation/RS_handout.synctex.gz differ diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.tex b/buch/papers/reedsolomon/RS presentation/RS_handout.tex index 863b3a2..1cbb6ef 100644 --- a/buch/papers/reedsolomon/RS presentation/RS_handout.tex +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.tex @@ -25,38 +25,29 @@ \begin{frame} \frametitle{Reed-Solomon-Code:} \begin{itemize} - \visible<1->{\item Für Übertragung von Daten} - \visible<2->{\item Ermöglicht Korrektur von Übertragungsfehler} - \visible<3->{\item Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc.} + \item Für Übertragung von Daten + \item Ermöglicht Korrektur von Übertragungsfehler + \item Wird verwendet in: CD, QR-Codes, Voyager-Sonde, etc. \end{itemize} \end{frame} %------------------------------------------------------------------------------- \section{Polynom Ansatz} \begin{frame} \begin{itemize} - \item Beispiel $2, 1, 5$ versenden und auf 2 Fehler absichern + \item $2, 1, 5$ versenden und auf 2 Fehler absichern \end{itemize} - \end{frame} - \begin{frame} \frametitle{Beispiel} Übertragen von ${f}_2=\textcolor{blue}{2}$, ${f}_1=\textcolor{blue}{1}$, ${f}_0=\textcolor{blue}{5}$ als $ p(w) = \textcolor{blue}{2}w^2 + \textcolor{blue}{1}w + \textcolor{blue}{5} $. - - \only<1>{ - Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, - \textcolor{green}{15}, \textcolor{green}{26}, - \textcolor{green}{41}, \textcolor{green}{60}, - \textcolor{green}{83}, \textcolor{green}{110})$ - \includegraphics[scale = 1.2]{images/polynom1.pdf}} - \only<2>{ - Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, - \textcolor{red}{50}, \textcolor{red}{37}, - \textcolor{green}{41}, \textcolor{green}{60}, - \textcolor{green}{83}, \textcolor{green}{110})$ - \includegraphics[scale = 1.2]{images/polynom2.pdf} - \newline - \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern.} + \newline + Versende $ (p(1),p(2),\dots,p(7)) = (\textcolor{green}{8}, + \textcolor{red}{50}, \textcolor{red}{37}, + \textcolor{green}{41}, \textcolor{green}{60}, + \textcolor{green}{83}, \textcolor{green}{110})$ + \includegraphics[scale = 1.2]{images/polynom2.pdf} + \newline + \textcolor{green}{7} Zahlen versenden, um \textcolor{blue}{3} Zahlen gegen \textcolor{red}{2} Fehlern abzusichern. \end{frame} %------------------------------------------------------------------------------- \begin{frame} @@ -68,18 +59,14 @@ \hline 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ -\visible<1->{3}& -\visible<1->{3}& -\visible<1->{9 Werte eines Polynoms vom Grad 2} \\ + 3& 3& 9 Werte eines Polynoms vom Grad 2 \\ &&\\ -\visible<1->{$k$} & -\visible<1->{$t$} & -\visible<1->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ + $k$ & $t$ & $k+2t$ Werte eines Polynoms vom Grad $k-1$ \\ \hline &&\\ &&\\ \multicolumn{3}{l} { - \visible<1>{Ausserdem können bis zu $2t$ Fehler erkannt werden!} + Ausserdem können bis zu $2t$ Fehler erkannt werden! } \end{tabular} \end{center} @@ -127,23 +114,23 @@ \frametitle{Diskrete Fourier Transformation} \begin{itemize} \item Diskrete Fourier-Transformation gegeben durch: - \visible<1->{ + \[ \label{ft_discrete} \hat{c}_{k} = \frac{1}{N} \sum_{n=0}^{N-1} {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} - \]} - \visible<2->{ + \] + \item Ersetzte \[ w = e^{-\frac{2\pi j}{N} k} - \]} - \visible<3->{ + \] + \item Wenn $N$ konstant: \[ \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) - \]} + \] \end{itemize} \end{frame} @@ -177,10 +164,9 @@ \frametitle{Probleme und Fragen} Wie wird der Fehler lokalisiert? - \visible<2>{ \newline Indem in einem endlichen Körper gerechnet wird. - } + \end{frame} %------------------------------------------------------------------------------- diff --git a/buch/papers/reedsolomon/RS presentation/RS_handout.toc b/buch/papers/reedsolomon/RS presentation/RS_handout.toc new file mode 100644 index 0000000..ce1bdc2 --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/RS_handout.toc @@ -0,0 +1,9 @@ +\babel@toc {ngerman}{} +\beamer@sectionintoc {1}{Einführung}{2}{0}{1} +\beamer@sectionintoc {2}{Polynom Ansatz}{3}{0}{2} +\beamer@sectionintoc {3}{Diskrete Fourier Transformation}{5}{0}{3} +\beamer@sectionintoc {4}{Reed-Solomon in Endlichen Körpern}{16}{0}{4} +\beamer@sectionintoc {5}{Codierung eines Beispiels}{18}{0}{5} +\beamer@sectionintoc {6}{Decodierung ohne Fehler}{20}{0}{6} +\beamer@sectionintoc {7}{Decodierung mit Fehler}{23}{0}{7} +\beamer@sectionintoc {8}{Nachricht Rekonstruieren}{29}{0}{8} -- cgit v1.2.1 From 8dc8c7a998d5a2862df90adc8b45d025e692d2d1 Mon Sep 17 00:00:00 2001 From: "User-PC\\User" Date: Wed, 5 May 2021 14:09:44 +0200 Subject: =?UTF-8?q?Arbeiten=20am=20Kapitel,=20zur=20Probe,=20weiteren=20Zu?= =?UTF-8?q?sammenarbeit,=20sodass=20Roy=20Seitz=20es=20einsehen=20k=C3=B6n?= =?UTF-8?q?nte?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- buch/papers/spannung/Einleitung.tex | 91 +++++++++++++++++++++ .../Grafiken/infinitesimalerW\303\274rfel.jpg" | Bin 0 -> 31604 bytes buch/papers/spannung/teil0.tex | 53 +++++++----- buch/papers/spannung/teil1.tex | 77 +++++++---------- buch/papers/spannung/teil2.tex | 48 +++-------- 5 files changed, 164 insertions(+), 105 deletions(-) create mode 100644 buch/papers/spannung/Einleitung.tex create mode 100644 "buch/papers/spannung/Grafiken/infinitesimalerW\303\274rfel.jpg" (limited to 'buch/papers') diff --git a/buch/papers/spannung/Einleitung.tex b/buch/papers/spannung/Einleitung.tex new file mode 100644 index 0000000..17ca1c9 --- /dev/null +++ b/buch/papers/spannung/Einleitung.tex @@ -0,0 +1,91 @@ +\section{Einleitung\label{spannung:section:Einleitung}} +In diesem Kapitel geht es darum die Matrix im dreidimensionalen Spannungszustand genauer zu untersuchen. +In der Geotechnik wendet man solche Matrizen an, um Spannungen im Boden zu berechnen. +Mit diesen Grundlagen dimensioniert man beispielsweise Böschungen, Fundationen, Dämme und Tunnels. +Ebenfalls benötigt man diese Matrix, um aus Versuchen Kennzahlen über den anstehenden Boden zu gewinnen. +Besonderes Augenmerk liegt dabei auf dem Oedometer - Versuch. + +Bei dieser Untersuchung der zugehörigen Berechnungen hat man es mit Vektoren, Matrizen und Tensoren zu tun. +Um die mathematische Untersuchung vorzunehmen, beschäftigt man sich zuerst mit den spezifischen Gegebenheiten und Voraussetzungen. +Ebenfalls gilt es ein paar wichtige Begriffe und deren mathematisches Zeichen einzuführen, +damit sich den Berechnungen schlüssig folgen lässt. + +In diesem Kapitel hat man es insbesondere mit Spannungen und Dehnungen zu tun. +Mit einer Spannung ist hier jedoch keine elektrische Spannung gemeint, +sondern eine Kraft geteilt durch Fläche. + +\section{Einführung wichtige Begriffe\label{spannung:section:Wichtige Begriffe}} +\[ +\l += +Ausgangslänge\enspace[m] +\] +\[ +\Delta l += +Längenänderung\enspacenach\enspaceKraftauftrag\enspace[m] +\] +\[ +\varepsilon += +Dehnung\enspace[-] +\] +\[ +\sigma += +Spannung\enspace[kPa] +\] +\[ +E += +Elastizitätsmodul +\] +\[ +F += +Kraft\enspace[kN] +\] +\[ +A += +Fläche\enspace[m^2] +\] +\[ +t += +Tiefe\enspace[m] +\] +\[ +s += +Setzung,\enspaceAbsenkung\enspace[m] +\] + +Beziehungen +\[ +\varepsilon += +\frac{\Delta l}{l_0} +\] +\[ +\varepsilon_q += +\frac{\Delta b}{l_0} += +\varepsilon_\upsilon +\] +\[ +\sigma += +\frac{N}{A} +\] +\[ +N += +\int_{A} \sigma \dA +\] +\[ +\varepsilon^{\prime} += +\frac{1}{l_0}\] + diff --git "a/buch/papers/spannung/Grafiken/infinitesimalerW\303\274rfel.jpg" "b/buch/papers/spannung/Grafiken/infinitesimalerW\303\274rfel.jpg" new file mode 100644 index 0000000..e3875bb Binary files /dev/null and "b/buch/papers/spannung/Grafiken/infinitesimalerW\303\274rfel.jpg" differ diff --git a/buch/papers/spannung/teil0.tex b/buch/papers/spannung/teil0.tex index cf47a18..ee19778 100644 --- a/buch/papers/spannung/teil0.tex +++ b/buch/papers/spannung/teil0.tex @@ -1,22 +1,37 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{spannung:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{spannung:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. +\section{Spannungsausbreitung\label{spannung:section:Spannungsausbreitung}} +\rhead{Spannungsausbreitung} +Anhand untenstehendem Bild kann ein einfaches Beispiel betrachtet werden. +Es gibt eine Kraft, diese wird auf den Boden abgetragen. +Diese Kraft muss dann vom Boden aufgenommen werden. +Im Boden entsteht eine Spannung. Diese Spannung ist abhängig von $\sigma(x,y,t)$ +Je nach dem, wo man sich im Boden befindet variert die Spannung. +Mit der Tiefe wird die Spannung geringer. +Die Ausbreitung der Spannung im Boden hat die Form einer Zwiebel. +Durch Untersuchung der Spannung an verschiedenen Punkten im Boden, kann man eine Funktion abtragen. +Dasselbe macht man auch mit der Dehnung. Es zeigt sich, dass die Form der beiden Funktionen gleich ist. +Dies erklärt sich dadurch, dass die Spannung und die Dehnung proportional sind zueinander sind. -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. +Anhand eines etwas schwierigeren Beispiels sieht man, +dass die Spannungsausbreitung nicht immer ganz einfach ist. +Man hat hier eine Baugrube mit einem Baugrubenabschluss, wo ein Teil des Bodens abgetragen wurde. +Was aber immer noch gilt ist, dass die Spannung von drei Variablen abhängig ist. $\sigma(x,y,t)$ +Ansätze um die Spannungsausbreitung zu berechnen gibt es je nach Bodentyp verschiedene. + +Die Spannungsausbreitung ist uns jedoch gegeben, es geht nicht darum, dies genauer zu untersuchen. +Durch die Spannungsausbreitung und das Elastizitätsmodul kann man eine Dehnung berechnen. +Anhand dieser Dehnung kann man mit einem Integral wiederum die Setzung berechnen. +\[ +\varepsilon += +\frac{\sigma}{E} +\] +\[ +s += +\int_{\0}^{\infty} \varepsilon \dt +\] +Die Setzung zu bestimmen ist in der Geotechnik sehr wichtig. +Besonders ungleichmässige Setzungen können bei Bauwerken Probleme ergeben. +Es gilt also die Bauwerke so zu dimensionieren, dass es verträgliche Setzungen gibt. diff --git a/buch/papers/spannung/teil1.tex b/buch/papers/spannung/teil1.tex index 95e6f0a..70dbb5a 100644 --- a/buch/papers/spannung/teil1.tex +++ b/buch/papers/spannung/teil1.tex @@ -1,55 +1,34 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{spannung:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx +\section{Proportionalität Spannung-Dehnung\label{spannung:section:Proportionalität Spannung-Dehnung}} +\rhead{Proportionalität Spannung-Dehnung} +Das Hooksche Gesetz beschreibt die elastische Längenänderung von Festkörpern im Zusammenhang mit einer Krafteinwirkung. +Die Längenänderung $\delta l$ ist proportional zur Krafteinwirkung. +$F\sim \Delta l$ +Man kann dies nur im Bereich vom linearen elastischen Materialverhalten anwenden. +Das heisst das alle Verformungen reversibel sind, sobald man die Kraft wegnimmt. +Es findet somit keine dauernde Verformung statt. +Da es sehr praktisch ist die Längenänderung nicht absolut auszudrücken haben wir $\varepsilon$. +$\varepsilon$ beschreibt die relative Längenänderung. +$\varepsilon$ ist wiederum proportional zu der aufgebrachten Spannung. +Im Bauingenieurwesen hat man es oft mit grösseren Teilen oder Grösseren Betrachtungsräumen zu tun. +Da ist es nun natürlich sehr sinnvoll, wenn wir nicht mit absoluten Zahlen rechnen, +sondern unabhängig von der Länge den Zustand mit Epsilon beschreiben können. +Mithilfe vom E-Modul, (steht für Elastizitätsmodul) einer Proportionalitätskonstante, +kann man das in eine Gleichung bringen, wie man hier sieht. Das E-Modul beschreibt, +das Verhältnis von Kraftaufnahme eines Werkstoffes und dessen zusammenhängender Längenveränderung. +\[ +E = -\left[ \frac13 x^3 \right]_a^b +\frac{\Delta\sigma}{\Delta\varepsilon} = -\frac{b^3-a^3}3. -\label{spannung:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. +const. +\] -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? +Aus diesem Verhältnis kann man das E-Modul berechnen. +Je nach Material ist dies verschieden. +Das E-Modul lässt sich nur im linearen-elastischen Materialverhalten anwenden. +Für Bodenmaterial gibt es ein spezielles E-Modul. Dieses wird mit dem Oedometerversuch ermittelt. +Es wird mit $E_{OED}$ ausgedrückt. Dieser Versuch wird später noch beschrieben. +Der Oedometerversuch ist abhängig von den diesem Kapitel zu untersuchenden Matrizen. -\subsection{De finibus bonorum et malorum -\label{spannung:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{spannung:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{spannung:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. diff --git a/buch/papers/spannung/teil2.tex b/buch/papers/spannung/teil2.tex index 37d3242..8eb54cb 100644 --- a/buch/papers/spannung/teil2.tex +++ b/buch/papers/spannung/teil2.tex @@ -1,40 +1,14 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{spannung:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? +\section{Dreiachsiger Spannungszustand\label{spannung:section:Dreiachsiger Spannungszustand}} +\rhead{Proportionalität Spannung-Dehnung} +Wie im Kapitel Spannungsausbreitung beschrieben herrscht in jedem Punkt ein anderer Spannungszustand. +Um die Spannung im Boden genauer untersuchen zu können für man einen infinitesimalen Würfel ein. +\begin{figure} + \includegraphics{C:/Users/User/Documents/SeminarMatrizen/buch/papers/spannung/Grafiken/infinitesimalerWürfel.jpg} + \caption{infinitesimaler Würfel} + \label{fig:infintesimaler-wurfel} +\end{figure} -\subsection{De finibus bonorum et malorum -\label{spannung:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +Sobald eine Kraft von oben wirkt hat man auch Kräfte die seitlich wirken. +Nun alle Kräfte ansehen des Infintesimalen Körpers -- cgit v1.2.1 From 7268e7363fddd5878b35de9169b64090a38a8fc5 Mon Sep 17 00:00:00 2001 From: "User-PC\\User" Date: Thu, 6 May 2021 16:51:10 +0200 Subject: Push --- buch/papers/spannung/Einleitung.tex | 6 ++++++ buch/papers/spannung/teil0.tex | 2 +- buch/papers/spannung/teil2.tex | 39 +++++++++++++++++++++++++++++++++++-- 3 files changed, 44 insertions(+), 3 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/spannung/Einleitung.tex b/buch/papers/spannung/Einleitung.tex index 17ca1c9..efc3809 100644 --- a/buch/papers/spannung/Einleitung.tex +++ b/buch/papers/spannung/Einleitung.tex @@ -89,3 +89,9 @@ N = \frac{1}{l_0}\] +Der Begriff Tensor +Tensoren werden unter anderem in der Elastizitätstheorie gebraucht. +In der Elastizitätstheorie geht es darum viele verschiedene Komponenten zu beschreiben. + + + diff --git a/buch/papers/spannung/teil0.tex b/buch/papers/spannung/teil0.tex index ee19778..67896b8 100644 --- a/buch/papers/spannung/teil0.tex +++ b/buch/papers/spannung/teil0.tex @@ -28,7 +28,7 @@ Anhand dieser Dehnung kann man mit einem Integral wiederum die Setzung berechnen \[ s = -\int_{\0}^{\infty} \varepsilon \dt +\int_{0}^{\infty}\varepsilon\enspace dt \] Die Setzung zu bestimmen ist in der Geotechnik sehr wichtig. Besonders ungleichmässige Setzungen können bei Bauwerken Probleme ergeben. diff --git a/buch/papers/spannung/teil2.tex b/buch/papers/spannung/teil2.tex index 8eb54cb..4aa8204 100644 --- a/buch/papers/spannung/teil2.tex +++ b/buch/papers/spannung/teil2.tex @@ -3,12 +3,47 @@ Wie im Kapitel Spannungsausbreitung beschrieben herrscht in jedem Punkt ein anderer Spannungszustand. Um die Spannung im Boden genauer untersuchen zu können für man einen infinitesimalen Würfel ein. \begin{figure} - \includegraphics{C:/Users/User/Documents/SeminarMatrizen/buch/papers/spannung/Grafiken/infinitesimalerWürfel.jpg} + \centering + \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken\infinitesimalerWürfel.jpg} \caption{infinitesimaler Würfel} \label{fig:infintesimaler-wurfel} \end{figure} Sobald eine Kraft von oben wirkt hat man auch Kräfte die seitlich wirken. -Nun alle Kräfte ansehen des Infintesimalen Körpers +An diesem infinitesimalen Würfel hat man ein räumliches Koordinatensystem, die Achsen (1,2,3). +Jede dieser 6 Flächen dieses Würfels hat damit 3 Pfeile. +Geschrieben werden diese mit $\sigma$ mit jeweils zwei Indizes gibt. +Die Indizes geben uns an, in welche Richtung der Pfeil zeigt. +Zur Notation wird die Voigt`sche Notation benutzt. Das sieht wie folgt aus: + +\[ +\overline{\sigma} += +\left[ \begin{array}{rrr} + \sigma_{11} & \sigma_{12} & \sigma_{13} \\ + \sigma_{21} & \sigma_{22} & \sigma_{23} \\ + \sigma_{31} & \sigma_{32} & \sigma_{33} \\ +\end{array}\right] += +\left[ \begin{array}{rrr} + \sigma_{11} & \sigma_{12} & \sigma_{13} \\ + & \sigma_{22} & \sigma_{23} \\ + sym & & \sigma_{33} \\ +\end{array}\right] +\Rightarrow +\overrightarrow{\sigma} += +\left(\begin{array}{c}\sigma_{11}\\\sigma_{22}\\\sigma_{33}\\\sigma_{23}\\\sigma_{13}\\\sigma_{12}\end{array}\right) +\] + +Voigt`sche Notation besagt, dass man diesen Spannungstensor als Vektor aufschreiben darf. +Die Reihenfolge folgt der Regel von Ecke links oben, diagonal zur Ecke rechts unten. +Danach ist noch $\sigma_{23}$, $\sigma_{13}$ und $\sigma_{12}$ aufzuschreiben. + +Eine weitere Besonderheit ist die Symmetrie der Matrix. + +?????Was könnte man hier noch zu den Pfeilen erklären vom Würfel??????? + + -- cgit v1.2.1 From 51dc9a5ccc1b6a238a94e4520082594c4b3b7d26 Mon Sep 17 00:00:00 2001 From: "User-PC\\User" Date: Wed, 12 May 2021 17:04:05 +0200 Subject: Diverse Anpassungen/Korrekturen --- buch/papers/spannung/Einleitung.tex | 25 ++- .../Grafiken/DiagrammOedometer-Versuch.jpg | Bin 0 -> 34089 bytes .../spannung/Grafiken/infinitesimalerWuerfel.jpg | Bin 0 -> 31604 bytes buch/papers/spannung/main.tex | 21 +- buch/papers/spannung/teil1.tex | 2 +- buch/papers/spannung/teil2.tex | 249 +++++++++++++++++++-- buch/papers/spannung/teil3.tex | 130 +++++++---- buch/papers/spannung/teil4.tex | 68 ++++++ 8 files changed, 414 insertions(+), 81 deletions(-) create mode 100644 buch/papers/spannung/Grafiken/DiagrammOedometer-Versuch.jpg create mode 100644 buch/papers/spannung/Grafiken/infinitesimalerWuerfel.jpg create mode 100644 buch/papers/spannung/teil4.tex (limited to 'buch/papers') diff --git a/buch/papers/spannung/Einleitung.tex b/buch/papers/spannung/Einleitung.tex index efc3809..f1d5d70 100644 --- a/buch/papers/spannung/Einleitung.tex +++ b/buch/papers/spannung/Einleitung.tex @@ -16,39 +16,44 @@ sondern eine Kraft geteilt durch Fläche. \section{Einführung wichtige Begriffe\label{spannung:section:Wichtige Begriffe}} \[ -\l +l = -Ausgangslänge\enspace[m] +\text{Ausgangslänge [\si{\meter}]} \] \[ \Delta l = -Längenänderung\enspacenach\enspaceKraftauftrag\enspace[m] +\text{Längenänderung nach Kraftauftrag [\si{\meter}]} \] \[ \varepsilon = -Dehnung\enspace[-] +\text{Dehnung [$-$]} \] \[ \sigma = -Spannung\enspace[kPa] +\text{Spannung [\si{\kilo\pascal}]} \] \[ E = -Elastizitätsmodul +\text{Elastizitätsmodul} +\] +\[ +\nu += +\text{Querdehnungszahl} \] \[ F = -Kraft\enspace[kN] +\text{Kraft [\si{\kilo\newton}]} \] \[ A = -Fläche\enspace[m^2] +\text{Fläche [\si{\meter\squared}]} \] \[ t @@ -58,7 +63,7 @@ Tiefe\enspace[m] \[ s = -Setzung,\enspaceAbsenkung\enspace[m] +\text{Setzung, Absenkung [m]} \] Beziehungen @@ -82,7 +87,7 @@ Beziehungen \[ N = -\int_{A} \sigma \dA +\int_{A} \sigma dA \] \[ \varepsilon^{\prime} diff --git a/buch/papers/spannung/Grafiken/DiagrammOedometer-Versuch.jpg b/buch/papers/spannung/Grafiken/DiagrammOedometer-Versuch.jpg new file mode 100644 index 0000000..52f1b5c Binary files /dev/null and b/buch/papers/spannung/Grafiken/DiagrammOedometer-Versuch.jpg differ diff --git a/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.jpg b/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.jpg new file mode 100644 index 0000000..e3875bb Binary files /dev/null and b/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.jpg differ diff --git a/buch/papers/spannung/main.tex b/buch/papers/spannung/main.tex index 585a423..60696d4 100644 --- a/buch/papers/spannung/main.tex +++ b/buch/papers/spannung/main.tex @@ -8,29 +8,14 @@ \begin{refsection} \chapterauthor{Adrian Schuler und Thomas Reichlin} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} +% TODO Text +\input{papers/spannung/Einleitung.tex} \input{papers/spannung/teil0.tex} \input{papers/spannung/teil1.tex} \input{papers/spannung/teil2.tex} \input{papers/spannung/teil3.tex} +\input{papers/spannung/teil4.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/spannung/teil1.tex b/buch/papers/spannung/teil1.tex index 70dbb5a..cc55664 100644 --- a/buch/papers/spannung/teil1.tex +++ b/buch/papers/spannung/teil1.tex @@ -1,7 +1,7 @@ \section{Proportionalität Spannung-Dehnung\label{spannung:section:Proportionalität Spannung-Dehnung}} \rhead{Proportionalität Spannung-Dehnung} Das Hooksche Gesetz beschreibt die elastische Längenänderung von Festkörpern im Zusammenhang mit einer Krafteinwirkung. -Die Längenänderung $\delta l$ ist proportional zur Krafteinwirkung. +Die Längenänderung $\Delta l$ ist proportional zur Krafteinwirkung. $F\sim \Delta l$ Man kann dies nur im Bereich vom linearen elastischen Materialverhalten anwenden. Das heisst das alle Verformungen reversibel sind, sobald man die Kraft wegnimmt. diff --git a/buch/papers/spannung/teil2.tex b/buch/papers/spannung/teil2.tex index 4aa8204..d11b3f6 100644 --- a/buch/papers/spannung/teil2.tex +++ b/buch/papers/spannung/teil2.tex @@ -1,49 +1,270 @@ -\section{Dreiachsiger Spannungszustand\label{spannung:section:Dreiachsiger Spannungszustand}} +\section{Dreiachsiger Spannungszustand\label{spannung:section:Dreiachsiger_Spannungszustand}} \rhead{Proportionalität Spannung-Dehnung} Wie im Kapitel Spannungsausbreitung beschrieben herrscht in jedem Punkt ein anderer Spannungszustand. Um die Spannung im Boden genauer untersuchen zu können für man einen infinitesimalen Würfel ein. \begin{figure} \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken\infinitesimalerWürfel.jpg} - \caption{infinitesimaler Würfel} + \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/infinitesimalerWuerfel.jpg} + \caption{Infinitesimaler Würfel} \label{fig:infintesimaler-wurfel} \end{figure} Sobald eine Kraft von oben wirkt hat man auch Kräfte die seitlich wirken. -An diesem infinitesimalen Würfel hat man ein räumliches Koordinatensystem, die Achsen (1,2,3). +An diesem infinitesimalen Würfel hat man ein räumliches Koordinatensystem, die Achsen $(1,2,3)$. Jede dieser 6 Flächen dieses Würfels hat damit 3 Pfeile. Geschrieben werden diese mit $\sigma$ mit jeweils zwei Indizes gibt. Die Indizes geben uns an, in welche Richtung der Pfeil zeigt. -Zur Notation wird die Voigt`sche Notation benutzt. Das sieht wie folgt aus: +Der erste Index ist die Achse auf welcher man sich befindet. +Der zweite Index gibt an, in welche Richtung der Pfeil zeigt. +Zur Notation wird die Voigt'sche Notation benutzt. Das sieht wie folgt aus: \[ \overline{\sigma} = -\left[ \begin{array}{rrr} +\begin{pmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ - \sigma_{31} & \sigma_{32} & \sigma_{33} \\ -\end{array}\right] + \sigma_{31} & \sigma_{32} & \sigma_{33} +\end{pmatrix} = -\left[ \begin{array}{rrr} +\begin{pmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ - & \sigma_{22} & \sigma_{23} \\ + & \sigma_{22} & \sigma_{23} \\ sym & & \sigma_{33} \\ -\end{array}\right] +\end{pmatrix} \Rightarrow \overrightarrow{\sigma} = -\left(\begin{array}{c}\sigma_{11}\\\sigma_{22}\\\sigma_{33}\\\sigma_{23}\\\sigma_{13}\\\sigma_{12}\end{array}\right) +\begin{pmatrix} + \sigma_{11}\\ + \sigma_{22}\\ + \sigma_{33}\\ + \sigma_{23}\\ + \sigma_{13}\\ + \sigma_{12} +\end{pmatrix} \] -Voigt`sche Notation besagt, dass man diesen Spannungstensor als Vektor aufschreiben darf. +Voigt'sche Notation besagt, dass man diesen Spannungstensor als Vektor aufschreiben darf. Die Reihenfolge folgt der Regel von Ecke links oben, diagonal zur Ecke rechts unten. Danach ist noch $\sigma_{23}$, $\sigma_{13}$ und $\sigma_{12}$ aufzuschreiben. Eine weitere Besonderheit ist die Symmetrie der Matrix. +So entspricht $\sigma_{23}$ dem Wert $\sigma_{32}$ oder $\sigma_{13}$ dem Wert $\sigma_{31}$. +Dies ist dadurch bedingt, dass die Kräfte in seitlicher Richtung im Boden die gleichen Werte annehmen. +Man hat in dieser Berechnung ein isotropes Material. +Im infinitesimalen Körper muss ein Gleichgewicht vorherrschen. +Ist kein Gleichgewicht vorhanden, würde sich der Körper zu drehen beginnen. +Es macht somit keinen Unterschied, ob man auf der Achse 2 in Richtung drei geht, +oder auf der Achse 3 in Richtung 2. -?????Was könnte man hier noch zu den Pfeilen erklären vom Würfel??????? +Da die Spannung proportional zur Dehnung ist, kann man die ganze Voigt'sche Notation auch mit der Dehnung ausdrücken. +Auch hier wandelt man das ganze gemäss der Reihenfolge in einen Vektor um. +\[ +\bar{\varepsilon} += +\begin{pmatrix} + \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ + \varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\ + \varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33} +\end{pmatrix} += +\begin{pmatrix} + \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ + & \varepsilon_{22} & \varepsilon_{23} \\ + \text{sym} & & \varepsilon_{33} +\end{pmatrix} +\qquad +\Rightarrow +\qquad +\vec{\varepsilon} += +\begin{pmatrix} + \varepsilon_{11} \\ + \varepsilon_{22} \\ + \varepsilon_{33} \\ + \varepsilon_{23} \\ + \varepsilon_{13} \\ + \varepsilon_{12} +\end{pmatrix} +\] + + +Mit der hergeleiteten Beziehung für die Spannungsgleichung anhand vom E-Modul, +der allgemeinen linearen Spannungsgleichung kann man diese Beziehungen neu aufschreiben. +Man benötigt dazu den zuvor berechneten Dehnungsvektor. +Die Gleichung besagt: +Spannungsvektor $=$ Elastitzitätstensor $\times$ Dehnungsvektor + +\[ +\overrightarrow{\sigma} += +\overline{\overline{C}}\cdot \overrightarrow{\varepsilon} +\] + +Die Vektoren haben je 6 Einträge. Um das ganze auszudrücken braucht es einen 6 x 6 Elastizitätstensor. (Kann man das noch weiter erklären weshalb?????) +Das ganze sieht dann wie folgt aus: + +\[ +\begin{pmatrix} + \sigma_{11} \\ + \sigma_{22} \\ + \sigma_{33} \\ + \sigma_{23} \\ + \sigma_{13} \\ + \sigma_{12} +\end{pmatrix} += +\begin{pmatrix} + C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\ + C_{21} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\ + C_{31} & C_{32} & C_{33} & C_{34} & C_{35} & C_{36} \\ + C_{41} & C_{42} & C_{43} & C_{44} & C_{45} & C_{46} \\ + C_{51} & C_{52} & C_{53} & C_{54} & C_{55} & C_{56} \\ + C_{61} & C_{62} & C_{63} & C_{64} & C_{65} & C_{66} +\end{pmatrix} +\begin{pmatrix} + \varepsilon_{11} \\ + \varepsilon_{22} \\ + \varepsilon_{33} \\ + \varepsilon_{23} \\ + \varepsilon_{13} \\ + \varepsilon_{12} +\end{pmatrix} +\] + +IST DIESE REIHENFOLGE KORREKT???? BEI DEHNUNG + +Die Spannung $\sigma_{11}$ besteht somit aus Anteilen von all diesen sechs Konstanten und den verschiedenen Dehnungen. +Zuvor bei der Voigt'schen Notation hat man jedoch gesehen, dass die Tensoren symmetrisch sind. +Folglich muss auch dieser Elastizitätstensor symmetrisch sein. +Das sind folgendermassen aus: + +\[ +\begin{pmatrix} + \sigma_{11} \\ + \sigma_{22} \\ + \sigma_{33} \\ + \sigma_{23} \\ + \sigma_{13} \\ + \sigma_{12} +\end{pmatrix} += +\begin{pmatrix} + C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\ + & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\ + & & C_{33} & C_{34} & C_{35} & C_{36} \\ + & & & C_{44} & C_{45} & C_{46} \\ + & & & & C_{55} & C_{56} \\ + \text{sym} & & & & & C_{66} +\end{pmatrix} +\begin{pmatrix} + \varepsilon_{11} \\ + \varepsilon_{22} \\ + \varepsilon_{33} \\ + \varepsilon_{23} \\ + \varepsilon_{13} \\ + \varepsilon_{12} +\end{pmatrix} +\] + +Die Konstanten $C$ kann man nun anders ausdrücken. +Und zwar bewerkstelligt man dies mithilfe vom Hook'schen Gesetz. + +\[ +\begin{pmatrix} + \sigma_{11}\\ + \sigma_{22}\\ + \sigma_{33}\\ + \sigma_{23}\\ + \sigma_{13}\\ + \sigma_{12} +\end{pmatrix} += +\frac{E}{(1+\nu)(1-2\nu)} +\begin{pmatrix} + 1- 2\nu & \nu & \nu & 0 & 0 & 0\\ + \nu & 1- 2\nu & \nu & 0 & 0 & 0\\ + \nu & \nu & 1- 2\nu & 0 & 0 & 0\\ + 0 & 0 & 0 & \frac{1}{2} & 0 & 0\\ + 0 & 0 & 0 & 0 & \frac{1}{2} & 0\\ + 0 & 0 & 0 & 0 & 0 & \frac{1}{2} +\end{pmatrix} +\begin{pmatrix} + \varepsilon_{11}\\ + \varepsilon_{22}\\ + \varepsilon_{33}\\ + \varepsilon_{23}\\ + \varepsilon_{13}\\ + \varepsilon_{12} +\end{pmatrix} +\] + +Mithilfe der Poissonzahl, welche uns die Querdehnung angibt, +sprich wie viel sich der Körper in Querrichtung verformt und dem E-Modul kann man alle Konstanten ausdrücken. +Bei einigen fällt auf, dass diese 0 werden. Der Tensor besagt also, +dass diese jeweiligen Konstanten keinen Einfluss auf unsere Spannung haben. +Als Beispiel kann man sich $\sigma_{33}$ anschauen. +Es ist ersichtlich, dass die Konstante $C_{31}$, $C_{32}$, $C_{33}$, $C_{35}$ und $C_{36}$ keinen Einfluss auf $\sigma_{33}$ haben. +Dies kann wie folgt erklärt werden. Auf Achse 3 geht $\sigma_{33}$ in Richtung 3. +Der Einfluss von $C_{31}$, Achse 3 in Richtung 1 hat keinen Einfluss auf $\sigma_{33}$ + +Von $\overline{\overline{C}}$ bildet man nun die Inverse Matrix $\overline{\overline{C}}~^{-1}$ stellt sich die ganze Gleichung um. + +\[ +\vec{\varepsilon} += +\overline{\overline{C}}~^{-1}\cdot \vec{\sigma} +\] + +\[ +\begin{pmatrix} + \varepsilon_{11}\\ + \varepsilon_{22}\\ + \varepsilon_{33}\\ + \varepsilon_{23}\\ + \varepsilon_{13}\\ + \varepsilon_{12} +\end{pmatrix} += +\frac{1}{E} +\begin{pmatrix} + 1 & -\nu & -\nu & 0 & 0 & 0\\ + -\nu & 1 & -\nu & 0 & 0 & 0\\ + -\nu & -\nu & 1 & 0 & 0 & 0\\ + 0 & 0 & 0 & 2+2\nu & 0 & 0\\ + 0 & 0 & 0 & 0 & 2+2\nu & 0\\ + 0 & 0 & 0 & 0 & 0 & 2+2\nu +\end{pmatrix} +\begin{pmatrix} + \sigma_{11}\\ + \sigma_{22}\\ + \sigma_{33}\\ + \sigma_{23}\\ + \sigma_{13}\\ + \sigma_{12} +\end{pmatrix} +\] + +Die zwei Blöcke links unten und rechts oben sind immer noch vorhanden. +Im Vergleich wo wir die Inverse noch nicht gemacht haben hat sich das nicht geändert. +Um die Einflüsse der Parameter zu veranschaulichen schreibt man folgende Gleichung. + +\[ +\varepsilon_{22} += +\frac{1}{E}\sigma_{22} - \frac{\nu}{E}\sigma_{11} - \frac{\nu}{E}\sigma_{33} +\] + +$\varepsilon_{22}$ beschreibt die Dehnung in Achse 2 und in Richtung 2. +In erster Linie hängt $\varepsilon_{22}$ von $\sigma_{22}$ ab. +Wenn die Poisson - Zahl grösser wird oder $\sigma_{11}$ oder $\sigma_{33}$, dann wird dadurch die Dehnung $\varepsilon_{22}$ kleiner. +Das heisst, auf Kosten von Verformung in anderer Richtung als Achse 2 Richtung 2 erfolgt die Verformung an anderer Stelle. +Wiederum hat die Schubspannung auf $\sigma_{11}$ keinen Einfluss. +Nun kennt man die Beziehung der 6 Dehnungen mit den 6 Spannungen. +In der Geotechnik wäre das aufgrund der vielen Komponenten sehr umständlich um damit Berechnungen zu machen. +Es braucht daher eine Vereinfachung mit Invarianten, welche im nächsten Kapitel beschrieben sind. diff --git a/buch/papers/spannung/teil3.tex b/buch/papers/spannung/teil3.tex index ce7d50f..a3b0b7d 100644 --- a/buch/papers/spannung/teil3.tex +++ b/buch/papers/spannung/teil3.tex @@ -1,40 +1,94 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{spannung:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{spannung:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +\section{Spannungsausbreitung\label{spannung:section:Invarianten}} +\rhead{Invarianten} +Trotz der Vereinfachung lässt sich mit den Invarianten die Realität adäquat abbilden. +Als erste Bedingung stellt man folgendes Verhältnis auf: +\[ +\sigma_{22} += +\sigma_{33} +\] +Dies deshalb, da man von einem isotropen Bodenmaterial ausgeht. +In Achse 22, Richtung 22 hat man den gleichen Boden wie in Achse 33 und Richtung 33. +Das Verhalten bezüglich Kraftaufnahme, Dehnung Spannung ist somit dasselbe. + +Man führt die zwei Werte p als hydrostatische Spannung und q als deviatorische Spannung ein. +Die Berechnung von p und q sieht wie folgt aus: + +\[ +p += +\frac{\sigma_{11}+\sigma_{22}+\sigma_{33}}{3} +\] + +oder durch Vereinfachung, da $\sigma_{22}=\sigma_{33}$ : + +\[ +p += +\frac{\sigma_{11}+2\sigma_{33}}{3} +\] + +\[ +q += +\sigma_{11}-\sigma_{33} +\] + +p ist das arithmetische Mittel von der Spannung im infinitesimalen Würfel. +q ist die Differenz zwischen der Spannung in vertikaler Richtung und der Spannung in Richtung 2 und 3. +Man kann p als Druckspannung und q als Schubspannung anschauen. + +Aus der Formel vom vorherigen Kapitel konnten wir die Spannungen berechnen. +Deshalb kann man nun p und q in die Gleichung einsetzen. +Die Dehnungen werden mit neuen Variablen eingeführt. +Die Deviatorische Dehnung kann mit einer Schubdehnung verglichen werden. +Die hydrostatische Dehnung kann mit einer Kompressionsdehnung verglichen werden. + +\[ +\overbrace{\sigma_{11}-\sigma_{33}}^{q} += +\frac{3E}{2(1+\nu)} \overbrace{\frac{2}{3}(\varepsilon_{11} - \varepsilon_{33})}^{\varepsilon_{\nu}} +\] + +\[ +\overbrace{\frac{\sigma_{11}+2\sigma_{33}}{3}}^{p} += +\frac{E}{3(1-2\nu)} \overbrace{(\varepsilon_{11} - 2\varepsilon_{33})}^{\varepsilon_{s}} +\] + +\[ +\varepsilon_{s} += +Hydrostatische Dehnung [-] +\] + +\[ +\varepsilon_{\nu} += +Deviatorische Dehnung [-] +\] + +Diese Komponenten kann man nun in die Vereinfachte Matrix einsetzen. +Man hat dann eine Matrix multipliziert mit einem Vektor und erhält einen Vektor. + +\[ +\begin{pmatrix} + q\\ + p +\end{pmatrix} += +\begin{pmatrix} + \frac{3E}{2(1+\nu)} & 0 \\ + 0 & \frac{E}{3(1-2\nu)} +\end{pmatrix} +\begin{pmatrix} + \varepsilon_{s}\\ + \varepsilon_{\nu} +\end{pmatrix} +\] + +Mit dieser Formel lassen sich verschieden Parameter von Versuchen analysieren und berechnen. +Ein solcher Versuch, den oft in der Geotechnik durchgeführt wird ist der Oedometer-Versuch. +Im nächsten Kapitel wird die Anwendung der Matrix an diesem Versuch beschrieben. diff --git a/buch/papers/spannung/teil4.tex b/buch/papers/spannung/teil4.tex new file mode 100644 index 0000000..f1437b1 --- /dev/null +++ b/buch/papers/spannung/teil4.tex @@ -0,0 +1,68 @@ +\section{Spannungsausbreitung\label{spannung:section:Oedometer - Versuch}} +\rhead{Oedometer - Versuch} +Beim Oedometer - Versucht hat man einen Stahlring mit einer Filterplatte am Boden. +In diesen Stahlring wird eine Bodenprobe eingefüllt. +Anschliessend wir mit einer Platte das Bodenmaterial mit einer ansteigenden Kraft belastet. + +Die Probe wird sich so verdichten. Das Volumen nimmt ab. +Der Stahlring verhindert ein seitliches ausbrechen oder entweichen der Bodenprobe. +Die Dehnung auf der Seite beträgt somit 0. +Mit dem Wert der Kraft und der Fläche lässt sich die Spannung berechnen. +Anhand der Volumenabnahme errechnet man die Dehnung. +Aus diesen Werten lässt sich wiederum das E-Modul bestimmen. +Beim Oedometer Versuch ist das E-Modul als $E_{OED}$ bezeichnet. + +Das $E_{OED}$ hat man speziell in der Geotechnik. +Dies aufgrund der speziellen Situation wo man sich mit dem infinitesimalen Würfel befindet. +Mit dem Stahlring, der verhindert das Material seitlich entweichen kann hat man ganz ähnliche Verhältnisse wie tief im Untergrund. +Auch dort kann das Material bei einer Belastung nicht seitlich entweichen. + +Wichtig ist nochmals zu betonen, dass alle diese beschriebenen Berechnungen ausschliesslich im linear-elastischen Materialverhalten funktionieren. +So ist es auch beim Oedometer - Versuch. +Den Versuch kann man auf einem $\sigma$ und $\varepsilon$ Diagramm abtragen. + +\begin{figure} + \centering + \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/DiagrammOedometer-Versuch.jpg} + \caption{Diagramm Oedometer - Versuch} + \label{fig:Diagramm Oedometer - Versuch} +\end{figure} + +Bei einem Versuch mit anderem Baumaterial wie beispielsweise Holz nimmt die Dehnung im Laufe des Versuchs stärker zu, obwohl weniger Spannung abgetragen wird. +Bei den meisten Böden ist dies anders. Durch die Komprimierung nimmt der Boden mehr Spannung auf, und verformt sich zugleich weniger stark. + +Man kann die Dehnung in unsere vereinfachte Matrix einsetzen. Das E-Modul ersetzt man mit dem $E_{OED}$. + +\[ +\overbrace{\sigma_{11}-\sigma_{33}}^{q} += +\frac{3E}{2(1+\nu)} \overbrace{\frac{2}{3}(\varepsilon_{11} - 0)}^{\varepsilon_{\nu}} +\] + +\[ +\overbrace{\frac{\sigma_{11}+2\sigma_{33}}{3}}^{p} += +\frac{E}{3(1-2\nu)} \overbrace{(\varepsilon_{11} - 2\cdot0)}^{\varepsilon_{s}} +\] + +\[ +\begin{pmatrix} + \sigma_{11}-\sigma_{33} \\ + \sigma_{11}+2\sigma_{33} +\end{pmatrix} += +\begin{bmatrix} + \frac{E_{OED}}{(1+\nu)} & 0 \\ + 0 & \frac{E_{OED}}{(1-2\nu)} +\end{bmatrix} +\begin{pmatrix} + \varepsilon_{11}\\ + \varepsilon_{11} +\end{pmatrix} +\] + +An einem geeigneten Punkt, wo man noch im linear-elastischen Materialverhalten ist, kann man nun das $E_{OED}$ abtragen. +Es wird nur ein Delta betrachtet um $E_{OED}$ zu berechnen. +Man darf die Dehnung nicht über den gesamten Verlauf betrachten um $E_{OED}$ zu berechnen. + +Mit diesem ermittelten E-Modul kann man nun weitere Berechnungen für die Geotechnik durchführen. -- cgit v1.2.1 From 8c0f3f0193804f257bc6646aef8c3be0f9c9166b Mon Sep 17 00:00:00 2001 From: "User-PC\\User" Date: Sat, 15 May 2021 17:29:20 +0200 Subject: =?UTF-8?q?=C3=9Cberarbeitungen?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- buch/papers/spannung/Einleitung.tex | 42 ++-- buch/papers/spannung/Grafiken/Bild3.png | Bin 0 -> 45727 bytes buch/papers/spannung/Grafiken/Bild4.png | Bin 0 -> 72520 bytes buch/papers/spannung/Grafiken/Bild5.png | Bin 0 -> 34721 bytes .../Grafiken/DiagrammOedometer-Versuch.jpg | Bin 34089 -> 0 bytes .../Grafiken/DiagrammOedometer-Versuch.png | Bin 0 -> 23361 bytes .../spannung/Grafiken/infinitesimalerWuerfel.jpg | Bin 31604 -> 0 bytes .../spannung/Grafiken/infinitesimalerWuerfel.png | Bin 0 -> 24852 bytes .../Grafiken/infinitesimalerW\303\274rfel.jpg" | Bin 31604 -> 0 bytes buch/papers/spannung/teil0.tex | 39 +++- buch/papers/spannung/teil1.tex | 35 +-- buch/papers/spannung/teil2.tex | 249 ++++++++++++++++++--- buch/papers/spannung/teil3.tex | 4 +- buch/papers/spannung/teil4.tex | 6 +- 14 files changed, 299 insertions(+), 76 deletions(-) create mode 100644 buch/papers/spannung/Grafiken/Bild3.png create mode 100644 buch/papers/spannung/Grafiken/Bild4.png create mode 100644 buch/papers/spannung/Grafiken/Bild5.png delete mode 100644 buch/papers/spannung/Grafiken/DiagrammOedometer-Versuch.jpg create mode 100644 buch/papers/spannung/Grafiken/DiagrammOedometer-Versuch.png delete mode 100644 buch/papers/spannung/Grafiken/infinitesimalerWuerfel.jpg create mode 100644 buch/papers/spannung/Grafiken/infinitesimalerWuerfel.png delete mode 100644 "buch/papers/spannung/Grafiken/infinitesimalerW\303\274rfel.jpg" (limited to 'buch/papers') diff --git a/buch/papers/spannung/Einleitung.tex b/buch/papers/spannung/Einleitung.tex index f1d5d70..37c2ec2 100644 --- a/buch/papers/spannung/Einleitung.tex +++ b/buch/papers/spannung/Einleitung.tex @@ -7,7 +7,7 @@ Besonderes Augenmerk liegt dabei auf dem Oedometer - Versuch. Bei dieser Untersuchung der zugehörigen Berechnungen hat man es mit Vektoren, Matrizen und Tensoren zu tun. Um die mathematische Untersuchung vorzunehmen, beschäftigt man sich zuerst mit den spezifischen Gegebenheiten und Voraussetzungen. -Ebenfalls gilt es ein paar wichtige Begriffe und deren mathematisches Zeichen einzuführen, +Ebenfalls gilt es ein paar wichtige Begriffe und deren mathematischen Zeichen einzuführen, damit sich den Berechnungen schlüssig folgen lässt. In diesem Kapitel hat man es insbesondere mit Spannungen und Dehnungen zu tun. @@ -16,7 +16,7 @@ sondern eine Kraft geteilt durch Fläche. \section{Einführung wichtige Begriffe\label{spannung:section:Wichtige Begriffe}} \[ -l +l_0 = \text{Ausgangslänge [\si{\meter}]} \] @@ -26,6 +26,11 @@ l \text{Längenänderung nach Kraftauftrag [\si{\meter}]} \] \[ +\Delta b += +\text{Längenänderung in Querrichtung nach Kraftauftrag [\si{\meter}]} +\] +\[ \varepsilon = \text{Dehnung [$-$]} @@ -38,12 +43,12 @@ l \[ E = -\text{Elastizitätsmodul} +\text{Elastizitätsmodul [\si{\kilo\pascal}]} \] \[ \nu = -\text{Querdehnungszahl} +\text{Querdehnungszahl; Poissonzahl [$-$]} \] \[ F @@ -58,7 +63,7 @@ A \[ t = -Tiefe\enspace[m] +\text{Tiefe [\si{\meter}]} \] \[ s @@ -77,7 +82,7 @@ Beziehungen = \frac{\Delta b}{l_0} = -\varepsilon_\upsilon +\varepsilon\cdot\nu \] \[ \sigma @@ -85,18 +90,29 @@ Beziehungen \frac{N}{A} \] \[ -N +F = \int_{A} \sigma dA \] \[ \varepsilon^{\prime} = -\frac{1}{l_0}\] +\frac{1}{l_0} +\] -Der Begriff Tensor -Tensoren werden unter anderem in der Elastizitätstheorie gebraucht. +\section{Einführung wichtige Begriffe\label{spannung:section:Tensoren}} +Tensoren wurden als erstes in der Elastizitätstheorie eingesetzt. (Quelle Herr Müller) In der Elastizitätstheorie geht es darum viele verschiedene Komponenten zu beschreiben. - - - +Mit einer Matrix oder einem Vektor kann man dies nicht mehr bewerkstelligen. +Wenn man den dreidimensionalen Spannungszustand abbilden möchte, müsste man mehrere Vektoren haben. +Deshalb wurden 1840 von Rowan Hamilton Tensoren in die Mathematik eingeführt. +Woldemar Voigt hat den Begriff in die moderne Bedeutung von Skalar, Matrix und Vektor verallgemeinert. +Albert Einstein hat Tensoren zudem in der allgemeinen Relativitätstheorie benutzt. +Tensor sind eine Stufe höher als Matrizen. Matrizen sind 2. Stufe. +Da Tensoren eine Stufe höher sind, kann man auch Matrizen, Vektoren und Skalare als Tensoren bezeichnen. +Der Nachteil von den Tensoren ist, dass man die gewohnten Rechenregeln, die man bei Vektoren oder Matrizen kennt, +nicht darauf anwenden kann. Man ist deshalb bestrebt die Tensoren als Vektoren und Matrizen darzustellen, +damit man die gewohnten Rechenregeln darauf anwenden kann. (Quelle Wikipedia) +In der vorliegenden Arbeit sind bereits alle Tensoren als Matrizen 2. Stufe abgebildet. +Trotzdem kann man diese Matrizen wie vorher beschrieben als Tensor bezeichnen. +Da diese als Matrizen abgebildet sind, dürfen wir die bekannten Rechenregeln auf unsere Tensoren anwenden. \ No newline at end of file diff --git a/buch/papers/spannung/Grafiken/Bild3.png b/buch/papers/spannung/Grafiken/Bild3.png new file mode 100644 index 0000000..8ca72a1 Binary files /dev/null and b/buch/papers/spannung/Grafiken/Bild3.png differ diff --git a/buch/papers/spannung/Grafiken/Bild4.png b/buch/papers/spannung/Grafiken/Bild4.png new file mode 100644 index 0000000..526ee7b Binary files /dev/null and b/buch/papers/spannung/Grafiken/Bild4.png differ diff --git a/buch/papers/spannung/Grafiken/Bild5.png b/buch/papers/spannung/Grafiken/Bild5.png new file mode 100644 index 0000000..6ee004d Binary files /dev/null and b/buch/papers/spannung/Grafiken/Bild5.png differ diff --git a/buch/papers/spannung/Grafiken/DiagrammOedometer-Versuch.jpg b/buch/papers/spannung/Grafiken/DiagrammOedometer-Versuch.jpg deleted file mode 100644 index 52f1b5c..0000000 Binary files a/buch/papers/spannung/Grafiken/DiagrammOedometer-Versuch.jpg and /dev/null differ diff --git a/buch/papers/spannung/Grafiken/DiagrammOedometer-Versuch.png b/buch/papers/spannung/Grafiken/DiagrammOedometer-Versuch.png new file mode 100644 index 0000000..31505bd Binary files /dev/null and b/buch/papers/spannung/Grafiken/DiagrammOedometer-Versuch.png differ diff --git a/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.jpg b/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.jpg deleted file mode 100644 index e3875bb..0000000 Binary files a/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.jpg and /dev/null differ diff --git a/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.png b/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.png new file mode 100644 index 0000000..398529c Binary files /dev/null and b/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.png differ diff --git "a/buch/papers/spannung/Grafiken/infinitesimalerW\303\274rfel.jpg" "b/buch/papers/spannung/Grafiken/infinitesimalerW\303\274rfel.jpg" deleted file mode 100644 index e3875bb..0000000 Binary files "a/buch/papers/spannung/Grafiken/infinitesimalerW\303\274rfel.jpg" and /dev/null differ diff --git a/buch/papers/spannung/teil0.tex b/buch/papers/spannung/teil0.tex index 67896b8..2f4d23b 100644 --- a/buch/papers/spannung/teil0.tex +++ b/buch/papers/spannung/teil0.tex @@ -1,22 +1,43 @@ \section{Spannungsausbreitung\label{spannung:section:Spannungsausbreitung}} \rhead{Spannungsausbreitung} Anhand untenstehendem Bild kann ein einfaches Beispiel betrachtet werden. -Es gibt eine Kraft, diese wird auf den Boden abgetragen. -Diese Kraft muss dann vom Boden aufgenommen werden. -Im Boden entsteht eine Spannung. Diese Spannung ist abhängig von $\sigma(x,y,t)$ +Es gibt eine Flächenlast (Kraft), diese wird auf den Boden abgetragen. +Diese Last muss dann vom Boden aufgenommen werden. +Im Boden entsteht nebst der Eigenspannung eine weitere Spannung durch diese Last (Zusatzspannung). +Diese Zusatzspannung $\sigma$ ist abhängig von $(x,y,t)$. Je nach dem, wo man sich im Boden befindet variert die Spannung. -Mit der Tiefe wird die Spannung geringer. -Die Ausbreitung der Spannung im Boden hat die Form einer Zwiebel. +Mit der Tiefe wird die Zusatzspannung geringer. +Die Ausbreitung der Zusatzspannung im Boden hat die Form einer Zwiebel. Durch Untersuchung der Spannung an verschiedenen Punkten im Boden, kann man eine Funktion abtragen. Dasselbe macht man auch mit der Dehnung. Es zeigt sich, dass die Form der beiden Funktionen gleich ist. -Dies erklärt sich dadurch, dass die Spannung und die Dehnung proportional sind zueinander sind. +Dies erklärt sich dadurch, dass die Spannung und die Dehnung proportional zueinander sind. +\begin{figure} + \centering + \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild4.png} + \caption{Ausbreitung der Spannung im Boden} + \label{fig:Bild4} +\end{figure} + +\begin{figure} + \centering + \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild5.png} + \caption{Funktionen Spannung und Dehnung} + \label{fig:Bild5} +\end{figure} Anhand eines etwas schwierigeren Beispiels sieht man, dass die Spannungsausbreitung nicht immer ganz einfach ist. Man hat hier eine Baugrube mit einem Baugrubenabschluss, wo ein Teil des Bodens abgetragen wurde. -Was aber immer noch gilt ist, dass die Spannung von drei Variablen abhängig ist. $\sigma(x,y,t)$ +Was aber immer noch gilt ist, dass die Spannung $\sigma$ von drei Variablen abhängig ist $(x,y,t)$. Ansätze um die Spannungsausbreitung zu berechnen gibt es je nach Bodentyp verschiedene. +\begin{figure} + \centering + \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild3.png} + \caption{Beispiel Lastauftrag auf Boden} + \label{fig:Bild3} +\end{figure} + Die Spannungsausbreitung ist uns jedoch gegeben, es geht nicht darum, dies genauer zu untersuchen. Durch die Spannungsausbreitung und das Elastizitätsmodul kann man eine Dehnung berechnen. Anhand dieser Dehnung kann man mit einem Integral wiederum die Setzung berechnen. @@ -32,6 +53,4 @@ s \] Die Setzung zu bestimmen ist in der Geotechnik sehr wichtig. Besonders ungleichmässige Setzungen können bei Bauwerken Probleme ergeben. -Es gilt also die Bauwerke so zu dimensionieren, dass es verträgliche Setzungen gibt. - - +Es gilt also die Bauwerke so zu dimensionieren, dass es verträgliche Setzungen gibt. \ No newline at end of file diff --git a/buch/papers/spannung/teil1.tex b/buch/papers/spannung/teil1.tex index cc55664..9467d21 100644 --- a/buch/papers/spannung/teil1.tex +++ b/buch/papers/spannung/teil1.tex @@ -1,20 +1,30 @@ \section{Proportionalität Spannung-Dehnung\label{spannung:section:Proportionalität Spannung-Dehnung}} \rhead{Proportionalität Spannung-Dehnung} -Das Hooksche Gesetz beschreibt die elastische Längenänderung von Festkörpern im Zusammenhang mit einer Krafteinwirkung. -Die Längenänderung $\Delta l$ ist proportional zur Krafteinwirkung. -$F\sim \Delta l$ -Man kann dies nur im Bereich vom linearen elastischen Materialverhalten anwenden. -Das heisst das alle Verformungen reversibel sind, sobald man die Kraft wegnimmt. +Das Hook'sche Gesetz beschreibt die elastische Längenänderung von Festkörpern im Zusammenhang mit einer Krafteinwirkung. +Die Längenänderung $\Delta l$ ist proportional zur Krafteinwirkung $F$. +\[ +F +\sim +\Delta l +\] +Man kann dies nur im Bereich vom linearen-elastischen Materialverhalten anwenden. +Das heisst, dass alle Verformungen reversibel sind, sobald man die Kraft wegnimmt. Es findet somit keine dauernde Verformung statt. Da es sehr praktisch ist die Längenänderung nicht absolut auszudrücken haben wir $\varepsilon$. -$\varepsilon$ beschreibt die relative Längenänderung. -$\varepsilon$ ist wiederum proportional zu der aufgebrachten Spannung. -Im Bauingenieurwesen hat man es oft mit grösseren Teilen oder Grösseren Betrachtungsräumen zu tun. +Die Dehnung $\varepsilon$ beschreibt die relative Längenänderung. +Die Dehnung $\varepsilon$ ist wiederum proportional zu der aufgebrachten Spannung. +Im Bauingenieurwesen hat man es oft mit grösseren Teilen oder grösseren Betrachtungsräumen zu tun. Da ist es nun natürlich sehr sinnvoll, wenn wir nicht mit absoluten Zahlen rechnen, -sondern unabhängig von der Länge den Zustand mit Epsilon beschreiben können. +sondern unabhängig von der Länge den Zustand mit Dehnung $\varepsilon$ beschreiben können. Mithilfe vom E-Modul, (steht für Elastizitätsmodul) einer Proportionalitätskonstante, kann man das in eine Gleichung bringen, wie man hier sieht. Das E-Modul beschreibt, das Verhältnis von Kraftaufnahme eines Werkstoffes und dessen zusammenhängender Längenveränderung. +(Quelle Wikipedia) +\[ +\sigma += +E\cdot\varepsilon +\] \[ E = @@ -26,9 +36,6 @@ const. Aus diesem Verhältnis kann man das E-Modul berechnen. Je nach Material ist dies verschieden. Das E-Modul lässt sich nur im linearen-elastischen Materialverhalten anwenden. -Für Bodenmaterial gibt es ein spezielles E-Modul. Dieses wird mit dem Oedometerversuch ermittelt. +Für Bodenmaterial gibt es ein spezielles E-Modul. Dieses wird mit dem Oedometer-Versuch ermittelt. Es wird mit $E_{OED}$ ausgedrückt. Dieser Versuch wird später noch beschrieben. -Der Oedometerversuch ist abhängig von den diesem Kapitel zu untersuchenden Matrizen. - - - +Der Oedometer-Versuch ist abhängig von den diesem Kapitel zu untersuchenden Matrizen. \ No newline at end of file diff --git a/buch/papers/spannung/teil2.tex b/buch/papers/spannung/teil2.tex index d11b3f6..3db3e26 100644 --- a/buch/papers/spannung/teil2.tex +++ b/buch/papers/spannung/teil2.tex @@ -1,22 +1,197 @@ \section{Dreiachsiger Spannungszustand\label{spannung:section:Dreiachsiger_Spannungszustand}} \rhead{Proportionalität Spannung-Dehnung} Wie im Kapitel Spannungsausbreitung beschrieben herrscht in jedem Punkt ein anderer Spannungszustand. -Um die Spannung im Boden genauer untersuchen zu können für man einen infinitesimalen Würfel ein. +Um die Spannung im Boden genauer untersuchen zu können, führt man einen infinitesimales Bodenteilchen ein. +Das Bodenteilchen ist geometrisch gesehen ein Würfel. +An diesem Bodenteilchen trägt man die Spannungen ein in alle Richtungen. + \begin{figure} \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/infinitesimalerWuerfel.jpg} - \caption{Infinitesimaler Würfel} + \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/infinitesimalerWuerfel.png} + \caption{Infinitesimales Bodenteilchen} \label{fig:infintesimaler-wurfel} \end{figure} -Sobald eine Kraft von oben wirkt hat man auch Kräfte die seitlich wirken. +An diesem infinitesimalen Bodenteilchen hat man ein räumliches Koordinatensystem, die Achsen $(1,2,3)$. +Die Achsen vom Koordinatensystem zeigen aus den 3 ersichtlichen Flächen heraus. +Pro ersichtliche Fläche haben wir eine Normalspannung und zwei Schubspannungen. +Im Gegensatz zum eindimensionalen Zustand entstehen bei einer Belastung des Bodenteilchens eine Vielzahl an Spannungen. +Es entstehen diverse Normal- und Schubspannungen. +Die Schubspannungen befinden sich an der Fläche, sie gehen rechtwinklig von den Achsen weg. +Die Schubspannungen auf einer Fläche stehen im 90 Grad Winkel zueinander. +Geschrieben werden diese mit $\sigma$, mit jeweils zwei Indizes. +Die Indizes geben uns an, in welche Richtung die Spannungen zeigen. +Der erste Index ist die Fläche auf welcher man sich befindet. +Der zweite Index gibt an, in welche Richtung die Spannung zeigt, dabei referenzieren die Indizes auch auf die Achsen $(1,2,3)$. +Bei den Spannungen sind immer positive als auch negative Spannungen möglich. +Es können also Druck- oder Zugspannungen sein. + +Zunächst wird untenstehend der allgemeine Spannungszustand betrachtet. + +Spannungstensor 2. Stufe i,j $\in$ {1,2,3} +\[ +\overline{\sigma} += +\sigma_{ij} += +\begin{pmatrix} + \sigma_{11} & \sigma_{12} & \sigma_{13} \\ + \sigma_{21} & \sigma_{22} & \sigma_{23} \\ + \sigma_{31} & \sigma_{32} & \sigma_{33} +\end{pmatrix} += +\qquad +\Rightarrow +\qquad +\vec{\sigma} += +\begin{pmatrix} + \sigma_{11}\\ + \sigma_{12}\\ + \sigma_{13}\\ + \sigma_{21}\\ + \sigma_{22}\\ + \sigma_{23}\\ + \sigma_{31}\\ + \sigma_{32}\\ + \sigma_{33} +\end{pmatrix} +\] + +Dehnungstensor 2. Stufe k,l $\in$ {1,2,3} + +\[ +\overline{\varepsilon} += +\varepsilon_{kl} += +\begin{pmatrix} + \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ + \varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\ + \varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33} +\end{pmatrix} += +\qquad +\Rightarrow +\qquad +\vec{\varepsilon} += +\begin{pmatrix} + \varepsilon_{11} \\ + \varepsilon_{12} \\ + \varepsilon_{13} \\ + \varepsilon_{21} \\ + \varepsilon_{22} \\ + \varepsilon_{23} \\ + \varepsilon_{31} \\ + \varepsilon_{32} \\ + \varepsilon_{33} +\end{pmatrix} +\] + +Bei diesen zwei obenstehenden Formeln kann man sehen wie Matrizen zu einem Vektor umgewandelt wurden. +Unter dem Kapitel Hadamard-Algebra kann man sehen, dass man dabei Zeile um Zeile in eine Spalte schreiben kann, +sodass es einen Vektor ergibt. + +Elastizitätstensor 4. Stufe i,j,k,l $\in$ {1,2,3} +\[ +\overline\overline{C} += +C_{ijkl} += +\begin{pmatrix} +C_{1111} & C_{1112} & C_{1113} & C_{1121} & C_{1122} & C_{1123} & C_{1131} & C_{1132} & C_{1133} \\ +C_{1211} & C_{1212} & C_{1213} & C_{1221} & C_{1222} & C_{1223} & C_{1231} & C_{1232} & C_{1233} \\ +C_{1311} & C_{1312} & C_{1313} & C_{1321} & C_{1322} & C_{1323} & C_{1331} & C_{1332} & C_{1333} \\ +C_{2111} & C_{2112} & C_{2113} & C_{2121} & C_{2122} & C_{2123} & C_{2131} & C_{2132} & C_{2133} \\ +C_{2211} & C_{2212} & C_{1113} & C_{2221} & C_{2222} & C_{2223} & C_{2231} & C_{2232} & C_{2233} \\ +C_{2311} & C_{2312} & C_{2313} & C_{2321} & C_{2322} & C_{2323} & C_{2331} & C_{2332} & C_{2333} \\ +C_{3111} & C_{3112} & C_{3113} & C_{3121} & C_{3122} & C_{3123} & C_{3131} & C_{3132} & C_{3133} \\ +C_{3211} & C_{3212} & C_{3213} & C_{3221} & C_{3222} & C_{3223} & C_{3231} & C_{3232} & C_{3233} \\ +C_{3311} & C_{3312} & C_{3313} & C_{3321} & C_{3322} & C_{3323} & C_{3331} & C_{3332} & C_{3333} +\end{pmatrix} +\] + +Dieser Elastizitätstensor muss eine quadratische Matrix mit $3^{4}$ Einträgen ergeben, +da die Basis mit den drei Richtungen $1, 2, 3$ und die Potenz mit den 4 Indizes mit je $1, 2, 3$ definiert sind. +Dies gibt daher eine 9 x 9 Matrix, welche zudem symmetrisch ist. + +Folglich gilt: +\[ +\overline{\overline{C}} += +\overline{\overline{C}}~^{T} +\] + +Allgemeine Spannungsgleichung (mit Vektoren und Tensor) +\[ +\vec\sigma += +\overline{\overline{C}}\cdot\vec{\varepsilon} +\] + +\[ +\begin{pmatrix} + \sigma_{11}\\ + \sigma_{12}\\ + \sigma_{13}\\ + \sigma_{21}\\ + \sigma_{22}\\ + \sigma_{23}\\ + \sigma_{31}\\ + \sigma_{32}\\ + \sigma_{33} +\end{pmatrix} += +\frac{E}{(1+\nu)(1-2\nu)} +\begin{pmatrix} + 1-2\nu & 0 & 0 & 0 & \nu & 0 & 0 & 0 & \nu \\ + 0 & frac{1}{4} & 0 & frac{1}{4} & 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & frac{1}{4} & 0 & 0 & 0 & frac{1}{4} & 0 & 0 \\ + 0 & frac{1}{4} & 0 & frac{1}{4} & 0 & 0 & 0 & 0 & 0 \\ + \nu & 0 & 0 & 0 & 1-2\nu & 0 & 0 & 0 & \nu \\ + 0 & 0 & 0 & 0 & 0 & frac{1}{4} & 0 & frac{1}{4} & 0 \\ + 0 & 0 & frac{1}{4} & 0 & 0 & 0 & frac{1}{4} & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 & frac{1}{4} & 0 & frac{1}{4} & 0 \\ + \nu & 0 & 0 & 0 & \nu & 0 & 0 & 0 & 1-2\nu +\end{pmatrix} +\begin{pmatrix} + \varepsilon_{11} \\ + \varepsilon_{12} \\ + \varepsilon_{13} \\ + \varepsilon_{21} \\ + \varepsilon_{22} \\ + \varepsilon_{23} \\ + \varepsilon_{31} \\ + \varepsilon_{32} \\ + \varepsilon_{33} +\end{pmatrix} +\] + +Man kann das zudem auch als Indexnotation aufschreiben. + +\[ +\sigma_{ij} += += +\sum_k=1^3 +\sum_l=1^3 +C_{ijkl}\cdot\varepsilon_{kl} +\] + +Um die Berechnung an einem Beispiel zu veranschaulichen: + +\[ +\sigma_{22} += +\frac{E\cdot\nu}{(1+\nu)(1-2\nu)}\cdot\varepsilon_{11}+\frac{E}{(1+\nu)}\cdot\varepsilon_{22}+\frac{E\cdot\nu}{(1+\nu)(1-2\nu)}\cdot\varepsilon_{33} +\] + +Anhand dem Tensor der allgemeinen Spannungsgleichung kann man zwar eine Symmetrie erkennen. +Die verschiedenen Einträge wechseln sich aber mit einander ab und es gibt keine klaren Blöcke mit nur einem gleichen Eintrag. +Man greift deshalb auf die Voigt'sche Notation zurück. + -An diesem infinitesimalen Würfel hat man ein räumliches Koordinatensystem, die Achsen $(1,2,3)$. -Jede dieser 6 Flächen dieses Würfels hat damit 3 Pfeile. -Geschrieben werden diese mit $\sigma$ mit jeweils zwei Indizes gibt. -Die Indizes geben uns an, in welche Richtung der Pfeil zeigt. -Der erste Index ist die Achse auf welcher man sich befindet. -Der zweite Index gibt an, in welche Richtung der Pfeil zeigt. Zur Notation wird die Voigt'sche Notation benutzt. Das sieht wie folgt aus: \[ @@ -30,14 +205,14 @@ Zur Notation wird die Voigt'sche Notation benutzt. Das sieht wie folgt aus: = \begin{pmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ - & \sigma_{22} & \sigma_{23} \\ - sym & & \sigma_{33} \\ + & \sigma_{22} & \sigma_{23} \\ + sym & & \sigma_{33} \end{pmatrix} \Rightarrow -\overrightarrow{\sigma} +\vec{\sigma} = \begin{pmatrix} - \sigma_{11}\\ + \sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{23}\\ @@ -46,24 +221,23 @@ Zur Notation wird die Voigt'sche Notation benutzt. Das sieht wie folgt aus: \end{pmatrix} \] -Voigt'sche Notation besagt, dass man diesen Spannungstensor als Vektor aufschreiben darf. -Die Reihenfolge folgt der Regel von Ecke links oben, diagonal zur Ecke rechts unten. -Danach ist noch $\sigma_{23}$, $\sigma_{13}$ und $\sigma_{12}$ aufzuschreiben. +In der Voigt'sche Notation hat man die Reihenfolge von der Ecke links oben, diagonal zur Ecke rechts unten. +Danach ist noch $\sigma_{23}$, $\sigma_{13}$ und $\sigma_{12}$ aufzuschreiben um den Vektor zu erhalten. Eine weitere Besonderheit ist die Symmetrie der Matrix. -So entspricht $\sigma_{23}$ dem Wert $\sigma_{32}$ oder $\sigma_{13}$ dem Wert $\sigma_{31}$. +So entspricht $\sigma_{23}$ dem Wert $\sigma_{32}$ und $\sigma_{13}$ dem Wert $\sigma_{31}$. Dies ist dadurch bedingt, dass die Kräfte in seitlicher Richtung im Boden die gleichen Werte annehmen. Man hat in dieser Berechnung ein isotropes Material. Im infinitesimalen Körper muss ein Gleichgewicht vorherrschen. Ist kein Gleichgewicht vorhanden, würde sich der Körper zu drehen beginnen. -Es macht somit keinen Unterschied, ob man auf der Achse 2 in Richtung drei geht, +Es macht somit keinen Unterschied, ob man auf der Achse 2 in Richtung 3 geht, oder auf der Achse 3 in Richtung 2. Da die Spannung proportional zur Dehnung ist, kann man die ganze Voigt'sche Notation auch mit der Dehnung ausdrücken. Auch hier wandelt man das ganze gemäss der Reihenfolge in einen Vektor um. \[ -\bar{\varepsilon} +\overline{\varepsilon} = \begin{pmatrix} \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ @@ -96,15 +270,22 @@ Mit der hergeleiteten Beziehung für die Spannungsgleichung anhand vom E-Modul, der allgemeinen linearen Spannungsgleichung kann man diese Beziehungen neu aufschreiben. Man benötigt dazu den zuvor berechneten Dehnungsvektor. Die Gleichung besagt: -Spannungsvektor $=$ Elastitzitätstensor $\times$ Dehnungsvektor - \[ -\overrightarrow{\sigma} +\text{Spannungsvektor} += +\text{Elastizitätstensor}\cdot\text{Dehnungsvektor} +\] +\[ +\vec{\sigma} = -\overline{\overline{C}}\cdot \overrightarrow{\varepsilon} +\overline{\overline{C}}\cdot\vec{\varepsilon} \] -Die Vektoren haben je 6 Einträge. Um das ganze auszudrücken braucht es einen 6 x 6 Elastizitätstensor. (Kann man das noch weiter erklären weshalb?????) +Die Vektoren haben je 6 Einträge. Um das ganze auszudrücken braucht es einen 6 x 6 Elastizitätstensor. +Der Tensor hat sich also im Vergleich zum 9 x 9 Tensor verkleinert. +Dies ist deshalb der Fall, da man in den Achsen 2 und 3 Symmetrien hat. +Dadurch kann man die Einträge $(\varepsilon_{21}=\varepsilon_{12}; \varepsilon_{31}=\varepsilon_{13}; \varepsilon_{32}=\varepsilon_{23})$ +zusammenfassen und drei Einträge verschwinden, da drei Dehnungen gleich sind. Das ganze sieht dann wie folgt aus: \[ @@ -135,8 +316,6 @@ Das ganze sieht dann wie folgt aus: \end{pmatrix} \] -IST DIESE REIHENFOLGE KORREKT???? BEI DEHNUNG - Die Spannung $\sigma_{11}$ besteht somit aus Anteilen von all diesen sechs Konstanten und den verschiedenen Dehnungen. Zuvor bei der Voigt'schen Notation hat man jedoch gesehen, dass die Tensoren symmetrisch sind. Folglich muss auch dieser Elastizitätstensor symmetrisch sein. @@ -206,10 +385,12 @@ Mithilfe der Poissonzahl, welche uns die Querdehnung angibt, sprich wie viel sich der Körper in Querrichtung verformt und dem E-Modul kann man alle Konstanten ausdrücken. Bei einigen fällt auf, dass diese 0 werden. Der Tensor besagt also, dass diese jeweiligen Konstanten keinen Einfluss auf unsere Spannung haben. +Man sieht nun auch ganz gut, dass sich im Vergleich bei der allgemeinen Darstellung der Spannungsgleichung, +die Einträge verschoben haben. Man hat nun eine sehr vorteilhafte Anordnung der verschiedenen Blöcke im Tensor. Als Beispiel kann man sich $\sigma_{33}$ anschauen. Es ist ersichtlich, dass die Konstante $C_{31}$, $C_{32}$, $C_{33}$, $C_{35}$ und $C_{36}$ keinen Einfluss auf $\sigma_{33}$ haben. Dies kann wie folgt erklärt werden. Auf Achse 3 geht $\sigma_{33}$ in Richtung 3. -Der Einfluss von $C_{31}$, Achse 3 in Richtung 1 hat keinen Einfluss auf $\sigma_{33}$ +Der Einfluss von $C_{31}$, Achse 3 in Richtung 1 hat keinen Einfluss auf $\sigma_{33}$. Von $\overline{\overline{C}}$ bildet man nun die Inverse Matrix $\overline{\overline{C}}~^{-1}$ stellt sich die ganze Gleichung um. @@ -231,12 +412,12 @@ Von $\overline{\overline{C}}$ bildet man nun die Inverse Matrix $\overline{\ove = \frac{1}{E} \begin{pmatrix} - 1 & -\nu & -\nu & 0 & 0 & 0\\ - -\nu & 1 & -\nu & 0 & 0 & 0\\ - -\nu & -\nu & 1 & 0 & 0 & 0\\ - 0 & 0 & 0 & 2+2\nu & 0 & 0\\ - 0 & 0 & 0 & 0 & 2+2\nu & 0\\ - 0 & 0 & 0 & 0 & 0 & 2+2\nu + 1 & -\nu & -\nu & 0 & 0 & 0 \\ + -\nu & 1 & -\nu & 0 & 0 & 0 \\ + -\nu & -\nu & 1 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 2+2\nu & 0 & 0 \\ + 0 & 0 & 0 & 0 & 2+2\nu & 0 \\ + 0 & 0 & 0 & 0 & 0 & 2+2\nu \end{pmatrix} \begin{pmatrix} \sigma_{11}\\ diff --git a/buch/papers/spannung/teil3.tex b/buch/papers/spannung/teil3.tex index a3b0b7d..4054262 100644 --- a/buch/papers/spannung/teil3.tex +++ b/buch/papers/spannung/teil3.tex @@ -80,8 +80,8 @@ Man hat dann eine Matrix multipliziert mit einem Vektor und erhält einen Vektor \end{pmatrix} = \begin{pmatrix} - \frac{3E}{2(1+\nu)} & 0 \\ - 0 & \frac{E}{3(1-2\nu)} + \frac{3E}{2(1+\nu)} & 0 \\ + 0 & \frac{E}{3(1-2\nu)} \end{pmatrix} \begin{pmatrix} \varepsilon_{s}\\ diff --git a/buch/papers/spannung/teil4.tex b/buch/papers/spannung/teil4.tex index f1437b1..85e9b1b 100644 --- a/buch/papers/spannung/teil4.tex +++ b/buch/papers/spannung/teil4.tex @@ -23,7 +23,7 @@ Den Versuch kann man auf einem $\sigma$ und $\varepsilon$ Diagramm abtragen. \begin{figure} \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/DiagrammOedometer-Versuch.jpg} + \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/DiagrammOedometer-Versuch.png} \caption{Diagramm Oedometer - Versuch} \label{fig:Diagramm Oedometer - Versuch} \end{figure} @@ -52,8 +52,8 @@ Man kann die Dehnung in unsere vereinfachte Matrix einsetzen. Das E-Modul ersetz \end{pmatrix} = \begin{bmatrix} - \frac{E_{OED}}{(1+\nu)} & 0 \\ - 0 & \frac{E_{OED}}{(1-2\nu)} + \frac{E_{OED}}{(1+\nu)} & 0 \\ + 0 & \frac{E_{OED}}{(1-2\nu)} \end{bmatrix} \begin{pmatrix} \varepsilon_{11}\\ -- cgit v1.2.1 From d4d4b02c3476fa406e0852ee0eb947d32e3a3d19 Mon Sep 17 00:00:00 2001 From: "User-PC\\User" Date: Sat, 15 May 2021 18:00:28 +0200 Subject: =?UTF-8?q?main.tex=20=C3=BCberarbeitet?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- buch/papers/spannung/main.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) (limited to 'buch/papers') diff --git a/buch/papers/spannung/main.tex b/buch/papers/spannung/main.tex index 60696d4..bbdf730 100644 --- a/buch/papers/spannung/main.tex +++ b/buch/papers/spannung/main.tex @@ -4,7 +4,7 @@ % (c) 2020 Hochschule Rapperswil % \chapter{Thema\label{chapter:spannung}} -\lhead{Thema} +\lhead{Dreiachsiger Spannungszustand} \begin{refsection} \chapterauthor{Adrian Schuler und Thomas Reichlin} -- cgit v1.2.1 From dd7bd6ca3b6517435dfc6b740ab96f51aa15ac2e Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Sun, 16 May 2021 16:03:36 +0200 Subject: edit main.tex add chapters --- buch/papers/reedsolomon/main.tex | 10 +++++++++- 1 file changed, 9 insertions(+), 1 deletion(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/main.tex b/buch/papers/reedsolomon/main.tex index 8219b63..a7485cd 100644 --- a/buch/papers/reedsolomon/main.tex +++ b/buch/papers/reedsolomon/main.tex @@ -3,7 +3,7 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:reedsolomon}} +\chapter{Reed-Solomon-Code\label{chapter:reedsolomon}} \lhead{Thema} \begin{refsection} \chapterauthor{Joshua Bär und Michael Steiner} @@ -27,10 +27,18 @@ Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. \end{itemize} +% Joshua \input{papers/reedsolomon/teil0.tex} \input{papers/reedsolomon/teil1.tex} \input{papers/reedsolomon/teil2.tex} \input{papers/reedsolomon/teil3.tex} +% Michael +\input{papers/reedsolomon/endlichekoerper} +\input{papers/reedsolomon/codebsp} +\input{papers/reedsolomon/decohnefehler} +\input{papers/reedsolomon/decmitfehler} +\input{papers/reedsolomon/rekonstruktion} + \printbibliography[heading=subbibliography] \end{refsection} -- cgit v1.2.1 From 898274b6cb5f825fe710eec58349799cdc5f6bc3 Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Sun, 16 May 2021 16:04:13 +0200 Subject: create endlichekoerper.tex added chapter description --- buch/papers/reedsolomon/endlichekoerper.tex | 23 +++++++++++++++++++++++ 1 file changed, 23 insertions(+) create mode 100644 buch/papers/reedsolomon/endlichekoerper.tex (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/endlichekoerper.tex b/buch/papers/reedsolomon/endlichekoerper.tex new file mode 100644 index 0000000..8ccd918 --- /dev/null +++ b/buch/papers/reedsolomon/endlichekoerper.tex @@ -0,0 +1,23 @@ +% +% teil1.tex -- Beispiel-File für das Paper +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Reed-Solomon in Endlichen Körpern +\label{reedsolomon:section:endlichekoerper}} +\rhead{Problemstellung} + +TODO: + +Das rechnen in endlichen Körpern bietet einige Vorteile: + +\begin{itemize} + \item Konkrete Zahlen: In endlichen Körpern gibt es weder rationale noch komplexe Zahlen. Zudem beschränken sich die möglichen Rechenoperationen auf das Addieren und Multiplizieren. Somit können wir nur ganze Zahlen als Resultat erhalten. + + \item Digitale Fehlerkorrektur: lässt sich nur in endlichen Körpern umsetzen. + +\end{itemize} + +Um jetzt eine Nachricht in den endlichen Körpern zu konstruieren legen wir fest, dass diese Nachricht aus einem Nutzdatenteil und einem Fehlerkorrekturteil bestehen muss. Somit ist die zu übertragende Nachricht immer grösser als die Daten, die wir übertragen wollen. Zudem müssen wir einen Weg finden, den Fehlerkorrekturteil so aus den Nutzdaten zu berechnen, dass wir die Nutzdaten auf der Empfängerseite wieder rekonstruieren können, sollte es zu einer fehlerhaften Übertragung kommen. + +Nun stellt sich die Frage, wie wir eine Fehlerhafte Nachricht korrigieren können, ohne ihren ursprünglichen Inhalt zu kennen. Der Reed-Solomon-Code erzielt dies, indem aus dem Fehlerkorrekturteil ein sogenanntes "Lokatorpolynom" generiert werden kann. Dieses Polynom gibt dem Emfänger an, welche Stellen in der Nachricht feherhaft sind. -- cgit v1.2.1 From 17ada1bd9c8f385c037ab5348292c027380e3cde Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 16 May 2021 20:44:40 +0200 Subject: Korrekturen um build zu garantieren --- buch/papers/spannung/teil2.tex | 14 +++++++------- buch/papers/spannung/teil3.tex | 4 ++-- 2 files changed, 9 insertions(+), 9 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/spannung/teil2.tex b/buch/papers/spannung/teil2.tex index 3db3e26..7dcf65f 100644 --- a/buch/papers/spannung/teil2.tex +++ b/buch/papers/spannung/teil2.tex @@ -95,7 +95,7 @@ sodass es einen Vektor ergibt. Elastizitätstensor 4. Stufe i,j,k,l $\in$ {1,2,3} \[ -\overline\overline{C} +\overline{\overline{C}} = C_{ijkl} = @@ -146,13 +146,13 @@ Allgemeine Spannungsgleichung (mit Vektoren und Tensor) \frac{E}{(1+\nu)(1-2\nu)} \begin{pmatrix} 1-2\nu & 0 & 0 & 0 & \nu & 0 & 0 & 0 & \nu \\ - 0 & frac{1}{4} & 0 & frac{1}{4} & 0 & 0 & 0 & 0 & 0 \\ - 0 & 0 & frac{1}{4} & 0 & 0 & 0 & frac{1}{4} & 0 & 0 \\ - 0 & frac{1}{4} & 0 & frac{1}{4} & 0 & 0 & 0 & 0 & 0 \\ + 0 &\frac{1}{4} & 0 &\frac{1}{4} & 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 &\frac{1}{4} & 0 & 0 & 0 &\frac{1}{4} & 0 & 0 \\ + 0 &\frac{1}{4} & 0 &\frac{1}{4} & 0 & 0 & 0 & 0 & 0 \\ \nu & 0 & 0 & 0 & 1-2\nu & 0 & 0 & 0 & \nu \\ - 0 & 0 & 0 & 0 & 0 & frac{1}{4} & 0 & frac{1}{4} & 0 \\ - 0 & 0 & frac{1}{4} & 0 & 0 & 0 & frac{1}{4} & 0 & 0 \\ - 0 & 0 & 0 & 0 & 0 & frac{1}{4} & 0 & frac{1}{4} & 0 \\ + 0 & 0 & 0 & 0 & 0 &\frac{1}{4} & 0 &\frac{1}{4} & 0 \\ + 0 & 0 &\frac{1}{4} & 0 & 0 & 0 &\frac{1}{4} & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 &\frac{1}{4} & 0 &\frac{1}{4} & 0 \\ \nu & 0 & 0 & 0 & \nu & 0 & 0 & 0 & 1-2\nu \end{pmatrix} \begin{pmatrix} diff --git a/buch/papers/spannung/teil3.tex b/buch/papers/spannung/teil3.tex index 4054262..500c404 100644 --- a/buch/papers/spannung/teil3.tex +++ b/buch/papers/spannung/teil3.tex @@ -61,13 +61,13 @@ Die hydrostatische Dehnung kann mit einer Kompressionsdehnung verglichen werden. \[ \varepsilon_{s} = -Hydrostatische Dehnung [-] +\text{Hydrostatische Dehnung} [-] \] \[ \varepsilon_{\nu} = -Deviatorische Dehnung [-] +\text{Deviatorische Dehnung} [-] \] Diese Komponenten kann man nun in die Vereinfachte Matrix einsetzen. -- cgit v1.2.1 From 46fa4763d730b1312741eefb8a2981c73389ccae Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Mon, 17 May 2021 19:32:32 +0200 Subject: update of codebsp started, restetabelle 1&2 created --- buch/papers/reedsolomon/codebsp.tex | 71 +++++++++++++++++++++++++++++++ buch/papers/reedsolomon/restetabelle1.tex | 24 +++++++++++ buch/papers/reedsolomon/restetabelle2.tex | 24 +++++++++++ 3 files changed, 119 insertions(+) create mode 100644 buch/papers/reedsolomon/codebsp.tex create mode 100644 buch/papers/reedsolomon/restetabelle1.tex create mode 100644 buch/papers/reedsolomon/restetabelle2.tex (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/codebsp.tex b/buch/papers/reedsolomon/codebsp.tex new file mode 100644 index 0000000..e9359f9 --- /dev/null +++ b/buch/papers/reedsolomon/codebsp.tex @@ -0,0 +1,71 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Codierung eines Beispiels +\label{reedsolomon:section:codebsp}} +\rhead{Koerper Festlegen} + +Um die Funktionsweise eines Reed-Solomon-Codes besser zu verstehen werden wir die einzelnen Probleme und ihre Lösungen anhand eines Beispiels betrachten. +Da wir in Endlichen Körpern Rechnen werden wir zuerst solch ein Körper festlegen. Dabei müssen wir die \textcolor{red}{Definition 4.6} berücksichtigen, die besagt, dass nur Primzahlen für endliche Körper in Frage kommen. +Wir legen für unser Beispiel den endlichen Körper $q = 11$ fest. +Alle folgenden Berechnungen wurden mit den beiden Restetabellen \textcolor{red}{xx} und \textcolor{red}{yy} durchgeführt. + +% die beiden Restetabellen von F_11 +%\input{papers/reedsolomon/restetabelle1} +%\input{papers/reedsolomon/restetabelle2} + + + + + +\textbf{DUMP} + +Da Körper laut der \textcolor{red}{Definition 4.6} eine Primzahl sein muss, + + +Dieser Körper sollte jedoch über eine nullteilerfreie Restetabelle verfügen. Somit kommen nur Primzahlen als Körper in frage. + + + Für das Beispiel wählen wir die Zahl $11$. + + uns zu aller erst auf ein sochen Körper festlegen. + +Um die Funktionsweise eines Reed-Solomon-Codes besser zu verstehen werden wir dies anhand eines Beispiels betrachten. + +Um die Nachfolgende Rechenwege besser zu verstehen, werden wir die einzelnen Rechenschritte anhand eines Beispiels betrachten. + + + + +Als erstes muss festgelegt werden, in welchem endlichen Körper gerechnet werden soll. +Da die Restetabelle eines Körpers nullteilerfrei sein soll, kommen so nur Primzahlen in Frage. +Für das Beispiel verwenden wir den Körper $\mathbb{F}_{11}$. So wählen wir + + +$q = 11$ + + +und beinhaltet die Zahlen + + +$Z_{11} = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]$ + +\subsection{De finibus bonorum et malorum +\label{reedsolomon:subsection:malorum}} +At vero eos et accusamus et iusto odio dignissimos ducimus qui +blanditiis praesentium voluptatum deleniti atque corrupti quos +dolores et quas molestias excepturi sint occaecati cupiditate non +provident, similique sunt in culpa qui officia deserunt mollitia +animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis +est et expedita distinctio. Nam libero tempore, cum soluta nobis +est eligendi optio cumque nihil impedit quo minus id quod maxime +placeat facere possimus, omnis voluptas assumenda est, omnis dolor +repellendus. Temporibus autem quibusdam et aut officiis debitis aut +rerum necessitatibus saepe eveniet ut et voluptates repudiandae +sint et molestiae non recusandae. Itaque earum rerum hic tenetur a +sapiente delectus, ut aut reiciendis voluptatibus maiores alias +consequatur aut perferendis doloribus asperiores repellat. + + diff --git a/buch/papers/reedsolomon/restetabelle1.tex b/buch/papers/reedsolomon/restetabelle1.tex new file mode 100644 index 0000000..a5055c0 --- /dev/null +++ b/buch/papers/reedsolomon/restetabelle1.tex @@ -0,0 +1,24 @@ +% created by Michael Steiner +% +% Restetabelle von F_11: Addition +\begin{figure} +\begin{center} +\begin{tabular}{|>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}|} +\hline ++&0&1&2&3&4&5&6&7&8&9&10\\ +\hline +0&0&1&2&3&4&5&6&7&8&9&10\\ +1&1&2&3&4&5&6&7&8&9&10&0\\ +2&2&3&4&5&6&7&8&9&10&0&1\\ +3&3&4&5&6&7&8&9&10&0&1&2\\ +4&4&5&6&7&8&9&10&0&1&2&3\\ +5&5&6&7&8&9&10&0&1&2&3&4\\ +6&6&7&8&9&10&0&1&2&3&4&5\\ +7&7&8&9&10&0&1&2&3&4&5&6\\ +8&8&9&10&0&1&2&3&4&5&6&7\\ +9&9&10&0&1&2&3&4&5&6&7&8\\ +10&10&0&1&2&3&4&5&6&7&8&9\\ +\hline +\end{tabular} +\end{center} +\end{figure} \ No newline at end of file diff --git a/buch/papers/reedsolomon/restetabelle2.tex b/buch/papers/reedsolomon/restetabelle2.tex new file mode 100644 index 0000000..887c981 --- /dev/null +++ b/buch/papers/reedsolomon/restetabelle2.tex @@ -0,0 +1,24 @@ +% created by Michael Steiner +% +% Restetabelle von F_11: Multiplikation +\begin{figure} +\begin{center} +\begin{tabular}{|>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}|} +\hline +\cdot&0&1&2&3&4&5&6&7&8&9&10\\ +\hline +0&0&0&0&0&0&0&0&0&0&0&0\\ +1&0&1&2&3&4&5&6&7&8&9&10\\ +2&0&2&4&6&8&10&1&3&5&7&9\\ +3&0&3&6&9&1&4&7&10&2&5&8\\ +4&0&4&8&1&5&9&2&6&10&3&7\\ +5&0&5&10&4&9&3&8&2&7&1&6\\ +6&0&6&1&7&2&8&3&9&4&10&5\\ +7&0&7&3&10&6&2&9&5&1&8&4\\ +8&0&8&5&2&10&7&4&1&9&6&3\\ +9&0&9&7&5&3&1&10&8&6&4&2\\ +10&0&10&9&8&7&6&5&4&3&2&1\\ +\hline +\end{tabular} +\end{center} +\end{figure} \ No newline at end of file -- cgit v1.2.1 From 55fc006b2133da4f79eb6eb5179d584c130824a2 Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Tue, 18 May 2021 18:29:59 +0200 Subject: updated codebsp.tex, created decohnefehler.tex (with blindtext) --- buch/papers/reedsolomon/codebsp.tex | 174 +++++++++++++++++++++--------- buch/papers/reedsolomon/decohnefehler.tex | 40 +++++++ 2 files changed, 161 insertions(+), 53 deletions(-) create mode 100644 buch/papers/reedsolomon/decohnefehler.tex (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/codebsp.tex b/buch/papers/reedsolomon/codebsp.tex index e9359f9..5b67c43 100644 --- a/buch/papers/reedsolomon/codebsp.tex +++ b/buch/papers/reedsolomon/codebsp.tex @@ -11,61 +11,129 @@ Um die Funktionsweise eines Reed-Solomon-Codes besser zu verstehen werden wir di Da wir in Endlichen Körpern Rechnen werden wir zuerst solch ein Körper festlegen. Dabei müssen wir die \textcolor{red}{Definition 4.6} berücksichtigen, die besagt, dass nur Primzahlen für endliche Körper in Frage kommen. Wir legen für unser Beispiel den endlichen Körper $q = 11$ fest. Alle folgenden Berechnungen wurden mit den beiden Restetabellen \textcolor{red}{xx} und \textcolor{red}{yy} durchgeführt. +Aus den Tabellen folgt auch, dass uns nur die Zahlen \[\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}\] zur Verfügung stehen. % die beiden Restetabellen von F_11 %\input{papers/reedsolomon/restetabelle1} %\input{papers/reedsolomon/restetabelle2} - - - - -\textbf{DUMP} - -Da Körper laut der \textcolor{red}{Definition 4.6} eine Primzahl sein muss, - - -Dieser Körper sollte jedoch über eine nullteilerfreie Restetabelle verfügen. Somit kommen nur Primzahlen als Körper in frage. - - - Für das Beispiel wählen wir die Zahl $11$. - - uns zu aller erst auf ein sochen Körper festlegen. - -Um die Funktionsweise eines Reed-Solomon-Codes besser zu verstehen werden wir dies anhand eines Beispiels betrachten. - -Um die Nachfolgende Rechenwege besser zu verstehen, werden wir die einzelnen Rechenschritte anhand eines Beispiels betrachten. - - - - -Als erstes muss festgelegt werden, in welchem endlichen Körper gerechnet werden soll. -Da die Restetabelle eines Körpers nullteilerfrei sein soll, kommen so nur Primzahlen in Frage. -Für das Beispiel verwenden wir den Körper $\mathbb{F}_{11}$. So wählen wir - - -$q = 11$ - - -und beinhaltet die Zahlen - - -$Z_{11} = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]$ - -\subsection{De finibus bonorum et malorum -\label{reedsolomon:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - +Die grösse des endlichen Körpers legt auch fest, wie gross unsere Nachricht $n$ bestehend aus Nutzdatenteil und Fehlerkorrekturteil sein kann und beträgt in unserem Beispiel +\[ +n = q - 1 = 10 \text{ Zahlen}. +\] + +Im nächsten Schritt bestimmen wir, wie viele Fehler $t$ maximal während der Übertragung auftreten dürfen, damit wir sie noch korrigieren können. +Unser Beispielcode sollte in der Lage sein +\[ +t = 2 +\] +Fehlerstellen korrigieren zu können. + +Die Grösse des Nutzdatenteils hängt von der Grösse der Nachricht sowie der Anzahl der Fehlerkorrekturstellen. Je robuster der Code sein muss, desto weniger Platz für Nutzdaten $k$ bleibt in der Nachricht übrig. +Bei maximal 2 Fehler können wir noch +\[ +k = n - 2t = 6\text{ Zahlen} +\] +übertragen. + +Zusammenfassend haben wir einen Codeblock mit der Länge von 10 Zahlen definiert, der 6 Zahlen als Nutzlast beinhaltet und in der Lage ist aus 2 fehlerhafte Stellen im Block die ursprünglichen Nutzdaten rekonstruieren kann. Zudem werden wir im weiteren feststellen, dass dieser Code maximal 4 Fehlerstellen erkennen, diese aber nicht rekonstruieren kann. + +Wir legen nun die Nachricht +\[ +m = [0,0,0,0,4,7,2,5,8,1] +\] +fest, die wir gerne an einen Empfänger übertragen möchten, wobei die vorderen vier Nullstellen für die Fehlerkorrektur zuständig sind. +Die Nachricht können wir auch als Polynom +\[ +m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1 +\] +darstellen. + +\subsection{Der Ansatz der diskreten Fouriertransformation + \label{reedsolomon:subsection:diskFT}} + +In einem vorherigen Kapitel (???) haben wir schon einmal die diskrete Fouriertransformation zum Codieren einer Nachricht verwendet. In den endlichen Körpern wird dies jedoch nicht gelingen, da die Eulerische Zahl $\mathrm{e}$ in $\mathbb{F}_{11}$ nicht existiert. +Wir suchen also eine Zahl $a^i$, die in endlichen Körpern existiert und den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken kann. +Dazu schreiben wir +\[ +\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\} +\] +um in +\[ +\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}. +\] + +Wenn wir alle möglichen Werte für $a$ einsetzen, also + +%\begin{align} +%a = 0 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{0, 0, 0, 0, 0, 0, 0, 0, 0, 0\} \\ +%a = 1 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 1, 1, 1, 1, 1, 1, 1, 1, 1\} \\ +%a = 2 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 2, 4, 8, 5, 10, 9, 7, 3, 6\} \\ +%a = 3 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 3, 9, 5, 4, 1, 3, 9, 5, 4\} \\ +%a = 4 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 4, 5, 9, 3, 1, 4, 5, 9, 3\} \\ +%a = 5 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 5, 3, 4, 9, 1, 5, 3, 4, 9\} \\ +%a = 6 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 6, 3, 7, 9, 10, 5, 8, 4, 2\} \\ +%a = 7 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 7, 5, 2, 3, 10, 4, 6, 9, 8\} \\ +%a = 8 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 8, 9, 6, 4, 10, 3, 2, 5, 7\} \\ +%a = 9 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 9, 4, 3, 5, 1, 9, 4, 3, 5\} \\ +%a = 10 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 10, 1, 10, 1, 10, 1, 10, 1, 10\} +%\end{align} + +\begin{center} +\begin{tabular}{c r c l} +%$a = 0 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{0, 0, 0, 0, 0, 0, 0, 0, 0, 0\}$ \\ +$a = 1 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 1, 1, 1, 1, 1, 1, 1, 1, 1\}$ \\ +$a = 2 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 2, 4, 8, 5, 10, 9, 7, 3, 6\}$ \\ +$a = 3 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 3, 9, 5, 4, 1, 3, 9, 5, 4\}$ \\ +$a = 4 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 4, 5, 9, 3, 1, 4, 5, 9, 3\}$ \\ +$a = 5 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 5, 3, 4, 9, 1, 5, 3, 4, 9\}$ \\ +$a = 6 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 6, 3, 7, 9, 10, 5, 8, 4, 2\}$ \\ +$a = 7 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 7, 5, 2, 3, 10, 4, 6, 9, 8\}$ \\ +$a = 8 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 8, 9, 6, 4, 10, 3, 2, 5, 7\}$ \\ +$a = 9 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 9, 4, 3, 5, 1, 9, 4, 3, 5\}$ \\ +$a = 10 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 10, 1, 10, 1, 10, 1, 10, 1, 10\}$ +\end{tabular} +\end{center} + +so fällt uns auf, dass die Zahlen $2,6,7,8$ tatsächlich den gesamten Zahlenraum von $\mathbb{F}_{11}$ abbilden. Solche Zahlen werden \em Primitive Einheitswurzel \em genannt. +Für das Beispiel wählen wir die Zahl $a^i = 8$. +Damit wir unsere Nachricht codieren können, müssen wir $8^i$ in $m(X)$ einsetzen. + +\begin{center} + \begin{tabular}{c} + $m(8^0) = 4 \cdot 1 + 7 \cdot 1 + 2 \cdot 1 + 5 \cdot 1 + 8 \cdot 1 + 1 = 5$ \\ + $m(8^1) = 4 \cdot 8 + 7 \cdot 8 + 2 \cdot 8 + 5 \cdot 8 + 8 \cdot 8 + 1 = 3$ \\ + \vdots + \end{tabular} +\end{center} + +Für eine elegantere Formulierung stellen wir das ganze als Matrix dar, wobei $m$ unser Nachrichtenvektor, $A$ die Transformationsmatrix und $v$ unser Übertragungsvektor ist. + +\[ +v = A \cdot m \qquad \Rightarrow \qquad v = \begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^3& 8^6& 8^9& 8^{12}& 8^{15}& 8^{18}& 8^{21}& 8^{24}& 8^{27}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^8& 8^{16}& 8^{24}& 8^{32}& 8^{40}& 8^{48}& 8^{56}& 8^{64}& 8^{72}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + 1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\ +\end{pmatrix} +\] + +Somit bekommen wir für unseren Übertragungsvektor +\[ +v = [5,3,6,5,2,10,2,7,10,4], +\] +den wir jetzt über einen beliebigen Nachrichtenkanal versenden können. + +\textbf{NOTES} + +warum wird 0 weggelassen? diff --git a/buch/papers/reedsolomon/decohnefehler.tex b/buch/papers/reedsolomon/decohnefehler.tex new file mode 100644 index 0000000..832d63f --- /dev/null +++ b/buch/papers/reedsolomon/decohnefehler.tex @@ -0,0 +1,40 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Decodierung ohne Fehler +\label{reedsolomon:section:decohnefehler}} +\rhead{Teil 3} +Sed ut perspiciatis unde omnis iste natus error sit voluptatem +accusantium doloremque laudantium, totam rem aperiam, eaque ipsa +quae ab illo inventore veritatis et quasi architecto beatae vitae +dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit +aspernatur aut odit aut fugit, sed quia consequuntur magni dolores +eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam +est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci +velit, sed quia non numquam eius modi tempora incidunt ut labore +et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima +veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, +nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure +reprehenderit qui in ea voluptate velit esse quam nihil molestiae +consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla +pariatur? + +\subsection{De finibus bonorum et malorum +\label{reedsolomon:subsection:malorum}} +At vero eos et accusamus et iusto odio dignissimos ducimus qui +blanditiis praesentium voluptatum deleniti atque corrupti quos +dolores et quas molestias excepturi sint occaecati cupiditate non +provident, similique sunt in culpa qui officia deserunt mollitia +animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis +est et expedita distinctio. Nam libero tempore, cum soluta nobis +est eligendi optio cumque nihil impedit quo minus id quod maxime +placeat facere possimus, omnis voluptas assumenda est, omnis dolor +repellendus. Temporibus autem quibusdam et aut officiis debitis aut +rerum necessitatibus saepe eveniet ut et voluptates repudiandae +sint et molestiae non recusandae. Itaque earum rerum hic tenetur a +sapiente delectus, ut aut reiciendis voluptatibus maiores alias +consequatur aut perferendis doloribus asperiores repellat. + + -- cgit v1.2.1 From 9c25485518e7f80050a8ee2a12b94abb009c9a58 Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Tue, 18 May 2021 21:14:36 +0200 Subject: finished first final version of decohnefehler.tex --- buch/papers/reedsolomon/decohnefehler.tex | 128 ++++++++++++++++++++++-------- 1 file changed, 97 insertions(+), 31 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/decohnefehler.tex b/buch/papers/reedsolomon/decohnefehler.tex index 832d63f..90f8ba8 100644 --- a/buch/papers/reedsolomon/decohnefehler.tex +++ b/buch/papers/reedsolomon/decohnefehler.tex @@ -5,36 +5,102 @@ % \section{Decodierung ohne Fehler \label{reedsolomon:section:decohnefehler}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{reedsolomon:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +\rhead{fehlerlose rekonstruktion} +Im ersten Teil zur Decodierung des Übertragungsvektor betrachten wir den Übertragungskanal als fehlerfrei. +Wir erhalten also unseren Übertragungsvektor +\[ +v = [5,3,6,5,2,10,2,7,10,4]. +\] +Gesucht ist nun einen Weg, mit dem wir auf unseren Nachrichtenvektor zurückrechnen können. +Ein banaler Ansatz ist das Invertieren der Glechung +\[ +v = A \cdot m \qquad \Rightarrow \qquad m = A^{-1} \cdot v. +\] +Nur stellt sich dann die Frage, wie wir auf die Inverse der Matix $A$ kommen. +Dazu können wir wiederum den Ansatz der Fouriertransformation uns zur Hilfe nehmen, +jedoch betrachten wir jetzt deren Inverse. +Definiert ist sie als +\[ +F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt \qquad \Rightarrow \qquad \mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega. +\] + +In unserem Fall suchen wir also eine inverse für die Primitive Einheitswurzel $a$, also +\[ +8^1 \qquad \Rightarrow \qquad 8^{-1}. +\] + +Im Abschnitt \textcolor{red}{4.1} haben wir den euklidischen Algorithmus kennengelernt, den wir auf unseren Fall anwenden können. + +\subsection{Der Euklidische Algorithmus +\label{reedsolomon:subsection:eukAlgo}} + +Die Funktionsweise des euklidischen Algorithmus ist im Kapitel \textcolor{red}{4.1} ausführlich beschrieben. +Für unsere Anwendung wählen wir die Parameter $a_i = 8$ und $b_i = 11$. +Daraus erhalten wir + +\begin{center} + +\begin{tabular}{| c | c c | c | r r |} + \hline + $k$ & $a_i$ & $b_i$ & $q_i$ & $c_i$ & $d_i$\\ + \hline + & & & & $1$& $0$\\ + $0$& $8$& $11$& $0$& $0$& $1$\\ + $1$& $11$& $8$& $1$& $1$& $0$\\ + $2$& $8$& $3$& $2$& $-1$& $1$\\ + $3$& $3$& $2$& $1$& $3$& $-2$\\ + $4$& $2$& $1$& $2$& \textcolor{blue}{$-4$}& \textcolor{red}{$3$}\\ + $5$& $1$& $0$& & $11$& $-8$\\ + \hline +\end{tabular} + +\end{center} +\begin{center} + +\begin{tabular}{rcl} + $\textcolor{blue}{-4} \cdot 8 + \textcolor{red}{3} \cdot 11$ &$=$& $1$\\ + $7 \cdot 8 + 3 \cdot 11$ &$=$& $1$\\ + $8^{-1}$ &$=$& $7$ + +\end{tabular} + +\end{center} + +als Inverse der Primitiven Einheitswurzel. + +Nun haben wir fast alles für die Rücktransformation beisammen. Wie auch bei der Inversen Fouriertransformation haben wir nun einen Vorfaktor +\[ +m = \textcolor{red}{s} \cdot A^{-1} \cdot v +\] +den wir noch bestimmen müssen. +Glücklicherweise lässt der sich analog wie bei der Inversen Fouriertransformation bestimmen und beträgt +\[ +s = \frac{1}{10}. +\] +Da $\frac{1}{10} = 10^{-1}$ entspricht können wir $s$ ebenfalls mit dem euklidischen Algorithmus bestimmen und stellen fest, dass $10^{-1} = 10$ ergibt. +Somit lässt sich den Nachrichtenvektor einfach bestimmen mit +\[ +m = 10 \cdot A^{-1} \cdot v \qquad \Rightarrow \qquad m = 10 \cdot \begin{pmatrix} + 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ + 7^0& 7^1& 7^2& 7^3& 7^4& 7^5& 7^6& 7^7& 7^8& 7^9\\ + 7^0& 7^2& 7^4& 7^6& 7^8& 7^{10}& 7^{12}& 7^{14}& 7^{16}& 7^{18}\\ + 7^0& 7^3& 7^6& 7^9& 7^{12}& 7^{15}& 7^{18}& 7^{21}& 7^{24}& 7^{27}\\ + 7^0& 7^4& 7^8& 7^{12}& 7^{16}& 7^{20}& 7^{24}& 7^{28}& 7^{32}& 7^{36}\\ + 7^0& 7^5& 7^{10}& 7^{15}& 7^{20}& 7^{25}& 7^{30}& 7^{35}& 7^{40}& 7^{45}\\ + 7^0& 7^6& 7^{12}& 7^{18}& 7^{24}& 7^{30}& 7^{36}& 7^{42}& 7^{48}& 7^{54}\\ + 7^0& 7^7& 7^{14}& 7^{21}& 7^{28}& 7^{35}& 7^{42}& 7^{49}& 7^{56}& 7^{63}\\ + 7^0& 7^8& 7^{16}& 7^{24}& 7^{32}& 7^{40}& 7^{48}& 7^{56}& 7^{64}& 7^{72}\\ + 7^0& 7^9& 7^{18}& 7^{27}& 7^{36}& 7^{45}& 7^{54}& 7^{63}& 7^{72}& 7^{81}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 5 \\ 2 \\ 10 \\ 2 \\ 7 \\ 10 \\ 4 \\ +\end{pmatrix} +\] +und wir erhalten +\[ +m = [0,0,0,0,4,7,2,5,8,1] +\] +als unsere Nachricht zurück. \ No newline at end of file -- cgit v1.2.1 From 6981f2935af17ca2dfce29f0d2d169d9f527b487 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Sat, 22 May 2021 13:58:27 +0200 Subject: Write a bit about symmetry --- buch/papers/punktgruppen/Makefile.inc | 2 + buch/papers/punktgruppen/intro.tex | 1 + buch/papers/punktgruppen/main.tex | 29 +--------- buch/papers/punktgruppen/symmetry.tex | 103 ++++++++++++++++++++++++++++++++++ 4 files changed, 108 insertions(+), 27 deletions(-) create mode 100644 buch/papers/punktgruppen/intro.tex create mode 100644 buch/papers/punktgruppen/symmetry.tex (limited to 'buch/papers') diff --git a/buch/papers/punktgruppen/Makefile.inc b/buch/papers/punktgruppen/Makefile.inc index 629abca..06be362 100644 --- a/buch/papers/punktgruppen/Makefile.inc +++ b/buch/papers/punktgruppen/Makefile.inc @@ -6,5 +6,7 @@ dependencies-punktgruppen = \ papers/punktgruppen/packages.tex \ papers/punktgruppen/main.tex \ + papers/punktgruppen/intro.tex \ + papers/punktgruppen/symmetry.tex \ papers/punktgruppen/references.bib diff --git a/buch/papers/punktgruppen/intro.tex b/buch/papers/punktgruppen/intro.tex new file mode 100644 index 0000000..4a84465 --- /dev/null +++ b/buch/papers/punktgruppen/intro.tex @@ -0,0 +1 @@ +\section{Einleitung} diff --git a/buch/papers/punktgruppen/main.tex b/buch/papers/punktgruppen/main.tex index 603f293..d7690fd 100644 --- a/buch/papers/punktgruppen/main.tex +++ b/buch/papers/punktgruppen/main.tex @@ -8,33 +8,8 @@ \begin{refsection} \chapterauthor{Tim T\"onz, Naoki Pross} -%% TODO: remove -%% Some ideas to motivate the topic: -%% - Physics in a crystal lattice structure -%% - Birifrencenge and scattering of light / Xray in Crystals -%% - Electron density function in a lattice -%% - Heat diffusion with lattice model -%% - Ising model for ferromagnetism (?? => H.D. Lang) -%% -%% - Homomorphic encryption (or lattice based cryptography) -%% + Q: Is it possible to edit encrypted data without decrypting it first? - -%% TODO: translated and move into a file {{{ - -\section{Motivation} -% birifrengence - -\section{Math} -% lattice group -% symmetry -% space group - -\section{Physics} -\subsection{Electromagnetic Waves} -\subsection{Crystal Lattice} - - -%% }}} +\input{papers/punktgruppen/intro} +\input{papers/punktgruppen/symmetry} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/punktgruppen/symmetry.tex b/buch/papers/punktgruppen/symmetry.tex new file mode 100644 index 0000000..9a1a945 --- /dev/null +++ b/buch/papers/punktgruppen/symmetry.tex @@ -0,0 +1,103 @@ +\section{Symmetrie} +Das Wort Symmetrie ist sehr alt und hat sich seltsamerweise von seinem +ursprünglichen griechischen Wort +\(\mathrm{\sigma\nu\mu\mu\varepsilon\tau\rho\iota\alpha}\) +\footnote{\emph{Simmetr\'ia}: ``ein gemeinsames Mass habend, gleichmässig, +verhältnismässig''} fast nicht verändert. In der Alltagssprache mag es ein +locker definierter Begriff sein, aber in der Mathematik hat Symmetrie eine sehr +präzise Bedeutung. +\begin{definition}[Symmetrie] + Ein mathematisches Objekt wird als symmetrisch bezeichnet, wenn es unter einer + bestimmten Operation invariant ist. +\end{definition} + +Wenn der Leser noch nicht mit der Gruppentheorie in Berührung gekommen ist, ist +vielleicht nicht ganz klar, was eine Operation ist, aber die Definition sollte +trotzdem Sinn machen. Die Formalisierung dieser Idee wird bald kommen, aber +zunächst wollen wir etwas Intuition aufbauen. + +\begin{figure}[h] + \centering + \begin{tikzpicture}[ + node distance = 2cm, + shapetheme/.style = { + very thick, draw = black, fill = magenta!20!white, + minimum size = 2cm, + }, + line/.style = {thick, draw = darkgray}, + axis/.style = {line, dashed}, + dot/.style = { + circle, draw = darkgray, fill = darkgray, + minimum size = 1mm, inner sep = 0, outer sep = 0, + }, + ] + + \node[ + shapetheme, + rectangle + ] (R) {}; + \node[dot] at (R) {}; + \draw[axis] (R) ++(-1.5, 0) to ++(3, 0) node[right] {\(\sigma\)}; + + \node[ + shapetheme, + regular polygon, + regular polygon sides = 5, + right = of R, + ] (Ps) {}; + \node[dot] (P) at (Ps) {}; + \draw[line, dotted] (P) to ++(18:1.5); + \draw[line, dotted] (P) to ++(90:1.5); + \draw[line, ->] (P) ++(18:1.2) + arc (18:90:1.2) node[midway, above right] {\(r, 72^\circ\)}; + + \node[ + shapetheme, + circle, right = of P + ] (Cs) {}; + \node[dot] (C) at (Cs) {}; + \draw[line, dotted] (C) to ++(1.5,0); + \draw[line, dotted] (C) to ++(60:1.5); + \draw[line, ->] (C) ++(1.2,0) + arc (0:60:1.2) node[midway, above right] {\(r, \alpha\)}; + + \end{tikzpicture} + \caption{ + Beispiele für geometrisch symmetrische Formen. + \label{fig:punktgruppen:geometry-example} + } +\end{figure} + +Die intuitivsten Beispiele kommen aus der Geometrie, daher werden wir mit +einigen geometrischen Beispielen beginnen. Wie wir jedoch später sehen werden, +ist das Konzept der Symmetrie eigentlich viel allgemeiner. In Abbildung +\ref{fig:punktgruppen:geometry-example} haben wir einige Formen, die +offensichtlich symmetrisch sind. Zum Beispiel hat ein Quadrat viele Achsen, um +die es gedreht werden kann, ohne sein Aussehen zu verändern. Regelmässige +Polygone mit \(n\) Seiten sind gute Beispiele, um eine diskrete +Rotationssymmetrie zu veranschaulichen, was bedeutet, dass eine Drehung um +einen Punkt um einen bestimmten Winkel \(360^\circ/n\) sie unverändert lässt. +Das letzte Beispiel auf der rechten Seite ist eine unendliche +Rotationssymmetrie. Sie wird so genannt, weil es unendlich viele Werte für +\(\alpha \in \mathbb{R}\) gibt, die die Form unverändert lassen. Dies ist +hoffentlich ausreichend, um die Bedeutung hinter der Notation zu verstehen, die +nun eingeführt wird. + +\begin{definition}[Symmetriegruppe] + Sei \(g\) eine Operation, die ein mathematisches Objekt unverändert lässt. + Bei einer anderen Operation \(r\) definieren wir die Komposition \(r\circ g\) + als die Anwendung der Operationen nacheinander. Alle Operationen \(g_i\) + bilden unter Komposition eine Gruppe, die Symmetriegruppe genannt wird. +\end{definition} + +Mit dem oben Gesagten können wir das \(n\)-Gon Beispiel formalisieren. Wenn wir +\(r\) eine Drehung von \(2\pi/n\) sein lassen, gibt es eine wohlbekannte Symmetriegruppe +\[ + C_n = \left\{\mathbf{1}, r, r^2, \ldots, r^{n-1}\right\} +\] +die Zyklische Gruppe heisst. + +\begin{definition}[Gruppenwirkung] +\end{definition} + +% vim:ts=2 sw=2 spell spelllang=de: -- cgit v1.2.1 From 2d2e4369b5d58bc9cd4dcb83ac43e3cda6341f3b Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Sat, 22 May 2021 18:01:15 +0200 Subject: More on symmetry --- buch/papers/punktgruppen/packages.tex | 2 +- buch/papers/punktgruppen/references.bib | 14 ++++++++- buch/papers/punktgruppen/symmetry.tex | 54 +++++++++++++++++++++++++++++---- 3 files changed, 62 insertions(+), 8 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/punktgruppen/packages.tex b/buch/papers/punktgruppen/packages.tex index 9953339..a6efdbf 100644 --- a/buch/papers/punktgruppen/packages.tex +++ b/buch/papers/punktgruppen/packages.tex @@ -4,4 +4,4 @@ % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % -\usepackage{tikz-3dplot} +\usepackage{dsfont} diff --git a/buch/papers/punktgruppen/references.bib b/buch/papers/punktgruppen/references.bib index aa7eb14..0d4e30a 100644 --- a/buch/papers/punktgruppen/references.bib +++ b/buch/papers/punktgruppen/references.bib @@ -4,6 +4,19 @@ % (c) 2020 Autor, Hochschule Rapperswil % +@book{punktgruppen:pinter-algebra, + title = {A Book of Abstract Algebra}, + author = {Charles C. Pinter}, + publisher = {Dover Publications Inc.; 2. Edition}, + year = {2010}, + month = {1}, + day = {10}, + isbn = {978-0486474175}, + inseries = {Dover Books on Mathematics}, + volume = {1} +} + + @online{punktgruppen:bibtex, title = {BibTeX}, url = {https://de.wikipedia.org/wiki/BibTeX}, @@ -32,4 +45,3 @@ pages = {607--627}, url = {https://doi.org/10.1016/j.acha.2017.11.004} } - diff --git a/buch/papers/punktgruppen/symmetry.tex b/buch/papers/punktgruppen/symmetry.tex index 9a1a945..58950da 100644 --- a/buch/papers/punktgruppen/symmetry.tex +++ b/buch/papers/punktgruppen/symmetry.tex @@ -86,18 +86,60 @@ nun eingeführt wird. \begin{definition}[Symmetriegruppe] Sei \(g\) eine Operation, die ein mathematisches Objekt unverändert lässt. Bei einer anderen Operation \(r\) definieren wir die Komposition \(r\circ g\) - als die Anwendung der Operationen nacheinander. Alle Operationen \(g_i\) - bilden unter Komposition eine Gruppe, die Symmetriegruppe genannt wird. + als die Anwendung der Operationen nacheinander. Alle Operationen bilden unter + Komposition eine Gruppe, die Symmetriegruppe genannt wird. \end{definition} Mit dem oben Gesagten können wir das \(n\)-Gon Beispiel formalisieren. Wenn wir \(r\) eine Drehung von \(2\pi/n\) sein lassen, gibt es eine wohlbekannte Symmetriegruppe \[ - C_n = \left\{\mathbf{1}, r, r^2, \ldots, r^{n-1}\right\} + C_n = \langle r \rangle + = \left\{\mathds{1}, r, r^2, \ldots, r^{n-1}\right\} + = \mathbb{Z}/n\mathbb{Z}, \] -die Zyklische Gruppe heisst. - -\begin{definition}[Gruppenwirkung] +die Zyklische Gruppe heisst. Hier die Potenzen von \(r\) sind als wiederholte +Komposition gemeint, d.h. \(r^n = r\circ r \circ \cdots r\circ r\). Die +Schreibweise mit den spitzen Klammern wird als Erzeugendensystem bezeichnet. +Das liegt daran, dass alle Elemente der Symmetriegruppe aus Kombinationen einer +Teilmenge erzeugt werden, die als erzeugende Elemente bezeichnet +werden\cite{punktgruppen:pinter-algebra}. Die Reflexionssymmetriegruppe ist +nicht so interessant, da sie nur +\(\left\{\mathds{1}, \sigma\right\}\) enthält. Kombiniert man sie jedoch mit +der Rotation, erhält man die so genannte Diedergruppe +\[ + D_n = \langle r, \sigma : r^{n-1} = \sigma^2 = (\sigma r)^2 = \mathds{1} \rangle + . +\] +Wir haben nun unseren Operationen Symbole gegeben, mit denen es tatsächlich +möglich ist, eine nicht kommutative Algebra zu erstellen. Die naheliegende +Frage ist dann, könnte es sein, dass wir bereits etwas haben, das dasselbe tut? +Natürlich, ja. Dafür führen wir den Begriff der Darstellung ein. +\begin{definition}[Darstellung einer Gruppe, Gruppenhomomorphismus] + Seien \(G\) und \(H\) Gruppe mit unterschiedlicher Operation \(\diamond\) + bzw. \(\star\). Ein Homomorphismus ist eine Funktion \(f: G \to H\), so dass + für jedes \(a, b \in G\) gilt \(f(a\diamond b) = f(a) \star f(b)\). Man + sagt, dass der Homomorphismus \(f\) \(G\) in \(H\) transformiert, oder dass + \(H\) eine Darstellung von \(G\) ist\cite{punktgruppen:pinter-algebra}. \end{definition} +\begin{beispiel} + Die Elemente \(r^k \in C_n\), wobei \(0 < k < n\), stellen abstrakt eine + Drehung von \(2\pi k/n\) um den Ursprung dar. Die mit der Matrix + \[ + \Phi(r^k) = \begin{pmatrix} + \cos(2\pi k/n) & -\sin(2\pi k/n) \\ + \sin(2\pi k/n) & \cos(2\pi k/n) + \end{pmatrix} + \] + definierte Funktion von \(C_n\) nach \(O(2)\) ist eine Darstellung von + \(C_n\). In diesem Fall ist die erste Gruppenoperation die Komposition und + die zweite die Matrixmultiplikation. Man kann zwar überprüfen, dass + \(\Phi(r^2 \circ r) = \Phi(r^2)\Phi(r)\). +\end{beispiel} +\begin{beispiel} + Die Rotationssymmetrie des Kreises \(C_\infty\), mit einem unendlichen + Kontinuum von Werten \(\alpha \in \mathbb{R}\), entspricht perfekt dem + komplexen Einheitskreis. Der Homomorphismus \(\phi: C_\infty \to \mathbb{C}\) + ist durch die Eulersche Formel \(\phi(r) = e^{i\alpha}\) gegeben. +\end{beispiel} % vim:ts=2 sw=2 spell spelllang=de: -- cgit v1.2.1 From b76677a7d3d40d15c4d9d5bcfa9283c702c4cb02 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Sat, 22 May 2021 18:22:14 +0200 Subject: Create file for crystals --- buch/papers/punktgruppen/Makefile.inc | 1 + buch/papers/punktgruppen/crystals.tex | 1 + buch/papers/punktgruppen/main.tex | 2 ++ buch/papers/punktgruppen/symmetry.tex | 15 ++++++++++----- 4 files changed, 14 insertions(+), 5 deletions(-) create mode 100644 buch/papers/punktgruppen/crystals.tex (limited to 'buch/papers') diff --git a/buch/papers/punktgruppen/Makefile.inc b/buch/papers/punktgruppen/Makefile.inc index 06be362..c7a7d64 100644 --- a/buch/papers/punktgruppen/Makefile.inc +++ b/buch/papers/punktgruppen/Makefile.inc @@ -8,5 +8,6 @@ dependencies-punktgruppen = \ papers/punktgruppen/main.tex \ papers/punktgruppen/intro.tex \ papers/punktgruppen/symmetry.tex \ + papers/punktgruppen/crystals.tex \ papers/punktgruppen/references.bib diff --git a/buch/papers/punktgruppen/crystals.tex b/buch/papers/punktgruppen/crystals.tex new file mode 100644 index 0000000..b104901 --- /dev/null +++ b/buch/papers/punktgruppen/crystals.tex @@ -0,0 +1 @@ +\section{Kristalle} diff --git a/buch/papers/punktgruppen/main.tex b/buch/papers/punktgruppen/main.tex index d7690fd..cbd2af6 100644 --- a/buch/papers/punktgruppen/main.tex +++ b/buch/papers/punktgruppen/main.tex @@ -10,6 +10,8 @@ \input{papers/punktgruppen/intro} \input{papers/punktgruppen/symmetry} +\input{papers/punktgruppen/crystals} +\nocite{punktgruppen:pinter-algebra} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/punktgruppen/symmetry.tex b/buch/papers/punktgruppen/symmetry.tex index 58950da..c0418aa 100644 --- a/buch/papers/punktgruppen/symmetry.tex +++ b/buch/papers/punktgruppen/symmetry.tex @@ -101,15 +101,20 @@ die Zyklische Gruppe heisst. Hier die Potenzen von \(r\) sind als wiederholte Komposition gemeint, d.h. \(r^n = r\circ r \circ \cdots r\circ r\). Die Schreibweise mit den spitzen Klammern wird als Erzeugendensystem bezeichnet. Das liegt daran, dass alle Elemente der Symmetriegruppe aus Kombinationen einer -Teilmenge erzeugt werden, die als erzeugende Elemente bezeichnet -werden\cite{punktgruppen:pinter-algebra}. Die Reflexionssymmetriegruppe ist -nicht so interessant, da sie nur +Teilmenge erzeugt werden, die als erzeugende Elemente bezeichnet werden. Die +Reflexionssymmetriegruppe ist nicht so interessant, da sie nur \(\left\{\mathds{1}, \sigma\right\}\) enthält. Kombiniert man sie jedoch mit der Rotation, erhält man die so genannte Diedergruppe \[ D_n = \langle r, \sigma : r^{n-1} = \sigma^2 = (\sigma r)^2 = \mathds{1} \rangle - . + = \left\{ + \mathds{1}, r, \ldots, r^{n-1}, \sigma, \sigma r, \ldots, \sigma r^{n-1} + \right\}. \] +Diesmal muss die Generator-Notation die Beziehungen zwischen den beiden +Operationen beinhalten. Die ersten beiden sind leicht zu erkennen, für die +letzte empfehlen wir, sie an einem 2D-Quadrat auszuprobieren. + Wir haben nun unseren Operationen Symbole gegeben, mit denen es tatsächlich möglich ist, eine nicht kommutative Algebra zu erstellen. Die naheliegende Frage ist dann, könnte es sein, dass wir bereits etwas haben, das dasselbe tut? @@ -119,7 +124,7 @@ Natürlich, ja. Dafür führen wir den Begriff der Darstellung ein. bzw. \(\star\). Ein Homomorphismus ist eine Funktion \(f: G \to H\), so dass für jedes \(a, b \in G\) gilt \(f(a\diamond b) = f(a) \star f(b)\). Man sagt, dass der Homomorphismus \(f\) \(G\) in \(H\) transformiert, oder dass - \(H\) eine Darstellung von \(G\) ist\cite{punktgruppen:pinter-algebra}. + \(H\) eine Darstellung von \(G\) ist. \end{definition} \begin{beispiel} Die Elemente \(r^k \in C_n\), wobei \(0 < k < n\), stellen abstrakt eine -- cgit v1.2.1 From 3aec8355b975973bfe192dedd216cc0f2b644770 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Sun, 23 May 2021 15:09:31 +0200 Subject: Create file for piezo, update bibliography --- buch/papers/punktgruppen/Makefile.inc | 1 + buch/papers/punktgruppen/main.tex | 4 ++++ buch/papers/punktgruppen/piezo.tex | 1 + buch/papers/punktgruppen/references.bib | 40 ++++++++++++--------------------- 4 files changed, 20 insertions(+), 26 deletions(-) create mode 100644 buch/papers/punktgruppen/piezo.tex (limited to 'buch/papers') diff --git a/buch/papers/punktgruppen/Makefile.inc b/buch/papers/punktgruppen/Makefile.inc index c7a7d64..b6a76c1 100644 --- a/buch/papers/punktgruppen/Makefile.inc +++ b/buch/papers/punktgruppen/Makefile.inc @@ -9,5 +9,6 @@ dependencies-punktgruppen = \ papers/punktgruppen/intro.tex \ papers/punktgruppen/symmetry.tex \ papers/punktgruppen/crystals.tex \ + papers/punktgruppen/piezo.tex \ papers/punktgruppen/references.bib diff --git a/buch/papers/punktgruppen/main.tex b/buch/papers/punktgruppen/main.tex index cbd2af6..d88e221 100644 --- a/buch/papers/punktgruppen/main.tex +++ b/buch/papers/punktgruppen/main.tex @@ -11,7 +11,11 @@ \input{papers/punktgruppen/intro} \input{papers/punktgruppen/symmetry} \input{papers/punktgruppen/crystals} +\input{papers/punktgruppen/piezo} \nocite{punktgruppen:pinter-algebra} +\nocite{punktgruppen:sands-crystal} +\nocite{punktgruppen:lang-elt2} + \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/punktgruppen/piezo.tex b/buch/papers/punktgruppen/piezo.tex new file mode 100644 index 0000000..7ee4174 --- /dev/null +++ b/buch/papers/punktgruppen/piezo.tex @@ -0,0 +1 @@ +\section{Piezoelektrizit\"at} diff --git a/buch/papers/punktgruppen/references.bib b/buch/papers/punktgruppen/references.bib index 0d4e30a..9edb8bd 100644 --- a/buch/papers/punktgruppen/references.bib +++ b/buch/papers/punktgruppen/references.bib @@ -11,37 +11,25 @@ year = {2010}, month = {1}, day = {10}, - isbn = {978-0486474175}, + isbn = {978-0-486-47417-5}, inseries = {Dover Books on Mathematics}, - volume = {1} } +@book{punktgruppen:sands-crystal, + title = {Introduction to Crystallography}, + author = {Donald E. Sands}, + publisher = {Dover Publications Inc.}, + year = {1993}, + isbn = {978-0-486-67839-9}, + inseries = {Dover Books on Science}, +} -@online{punktgruppen:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, +@book{punktgruppen:lang-elt2, + title = {Elektrotechnik 2}, + author = {Hans-Dieter Lang}, + publisher = {Fachhochschule Ostschweiz Rapperswil}, year = {2020}, month = {2}, - day = {6} -} - -@book{punktgruppen:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} + inseries = {Vorlesungsskript zum Modul ELT}, } -@article{punktgruppen:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, - title = { Noncommutative harmonic analysis and image registration }, - journal = { Appl. Comput. Harmon. Anal.}, - year = 2019, - volume = 47, - pages = {607--627}, - url = {https://doi.org/10.1016/j.acha.2017.11.004} -} -- cgit v1.2.1 From b4093cfc873e052d31f644019f0b1134f1db7fbc Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Sun, 23 May 2021 16:24:34 +0200 Subject: On point groups and translational symmetry --- buch/papers/punktgruppen/symmetry.tex | 30 +++++++++++++++++++++++++++++- 1 file changed, 29 insertions(+), 1 deletion(-) (limited to 'buch/papers') diff --git a/buch/papers/punktgruppen/symmetry.tex b/buch/papers/punktgruppen/symmetry.tex index c0418aa..d3ccb4e 100644 --- a/buch/papers/punktgruppen/symmetry.tex +++ b/buch/papers/punktgruppen/symmetry.tex @@ -85,7 +85,7 @@ nun eingeführt wird. \begin{definition}[Symmetriegruppe] Sei \(g\) eine Operation, die ein mathematisches Objekt unverändert lässt. - Bei einer anderen Operation \(r\) definieren wir die Komposition \(r\circ g\) + Bei einer anderen Operation \(h\) definieren wir die Komposition \(h\circ g\) als die Anwendung der Operationen nacheinander. Alle Operationen bilden unter Komposition eine Gruppe, die Symmetriegruppe genannt wird. \end{definition} @@ -147,4 +147,32 @@ Natürlich, ja. Dafür führen wir den Begriff der Darstellung ein. ist durch die Eulersche Formel \(\phi(r) = e^{i\alpha}\) gegeben. \end{beispiel} +Die Symmetrien, die wir bis jetzt besprochen haben, haben immer mindestens +einen Punkt unbesetzt gelassen. Im Fall der Rotation war es der Drehpunkt, bei +der Spiegelung die Achse. Dies ist jedoch keine Voraussetzung für eine +Symmetrie, da es Symmetrien gibt, die jeden Punkt zu einem anderen Punkt +verschieben können. Ein aufmerksamer Leser wird bemerken, dass die +unveränderten Punkte zum Eigenraum\footnote{Zur Erinnerung \(E_\lambda = +\mathrm{null}(\Phi - \lambda I)\)} der Matrixdarstellung der Symmetrieoperation +gehören. Diesen Spezialfall, bei dem mindestens ein Punkt unverändert bleibt, +nennt man Punktsymmetrie. +\begin{definition}[Punktgruppe] + Wenn jede Operation in einer Symmetriegruppe die Eigenschaft hat, mindestens + einen Punkt unverändert zu lassen, sagt man, dass die Symmetriegruppe eine + Punktgruppe ist. +\end{definition} +Um das Konzept zu illustrieren, werden wir den umgekehrten Fall diskutieren: +eine Symmetrie, die keine Punktsymmetrie ist, die aber in der Physik sehr +nützlich ist, nämlich die Translationssymmetrie. Von einem mathematischen +Objekt \(x\) wird gesagt, dass es eine Translationssymmetrie \(Q\) hat, wenn es +die Gleichung +\[ + Q(x) = Q(x + a), +\] +für ein gewisses \(a\), erfüllt. Zum Beispiel besagt das erste Newtonsche +Gesetz, dass ein Objekt, auf das keine Kraft einwirkt, eine +zeitranslationsinvariante Geschwindigkeit hat, d.h. wenn \(\vec{F} = \vec{0}\) +dann \(\vec{v}(t) = \vec{v}(t + \tau)\). + + % vim:ts=2 sw=2 spell spelllang=de: -- cgit v1.2.1 From 5294c40d558e93a034d43846e98176291fb32692 Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Mon, 24 May 2021 14:28:24 +0200 Subject: update decohnefehler.tex, create decmitfehler.tex --- buch/papers/reedsolomon/decmitfehler.tex | 16 ++++++++++++++++ buch/papers/reedsolomon/decohnefehler.tex | 2 +- 2 files changed, 17 insertions(+), 1 deletion(-) create mode 100644 buch/papers/reedsolomon/decmitfehler.tex (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/decmitfehler.tex b/buch/papers/reedsolomon/decmitfehler.tex new file mode 100644 index 0000000..fead10e --- /dev/null +++ b/buch/papers/reedsolomon/decmitfehler.tex @@ -0,0 +1,16 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Decodierung mit Fehler +\label{reedsolomon:section:decmitfehler}} +\rhead{fehlerhafte rekonstruktion} +moin + + +\subsection{Der Satz von Fermat +\label{reedsolomon:subsection:fermat}} +wer ist fermat? + + diff --git a/buch/papers/reedsolomon/decohnefehler.tex b/buch/papers/reedsolomon/decohnefehler.tex index 90f8ba8..6ca577a 100644 --- a/buch/papers/reedsolomon/decohnefehler.tex +++ b/buch/papers/reedsolomon/decohnefehler.tex @@ -80,7 +80,7 @@ Glücklicherweise lässt der sich analog wie bei der Inversen Fouriertransformat s = \frac{1}{10}. \] Da $\frac{1}{10} = 10^{-1}$ entspricht können wir $s$ ebenfalls mit dem euklidischen Algorithmus bestimmen und stellen fest, dass $10^{-1} = 10$ ergibt. -Somit lässt sich den Nachrichtenvektor einfach bestimmen mit +Somit lässt sich der Nachrichtenvektor einfach bestimmen mit \[ m = 10 \cdot A^{-1} \cdot v \qquad \Rightarrow \qquad m = 10 \cdot \begin{pmatrix} 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ -- cgit v1.2.1 From 60bfb41261f51cf20ce65a9242c2624b31d74e75 Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Mon, 24 May 2021 17:17:56 +0200 Subject: decmitfehler.tex updated --- buch/papers/reedsolomon/decmitfehler.tex | 185 ++++++++++++++++++++++++++++++- 1 file changed, 183 insertions(+), 2 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/decmitfehler.tex b/buch/papers/reedsolomon/decmitfehler.tex index fead10e..923c1c5 100644 --- a/buch/papers/reedsolomon/decmitfehler.tex +++ b/buch/papers/reedsolomon/decmitfehler.tex @@ -6,11 +6,192 @@ \section{Decodierung mit Fehler \label{reedsolomon:section:decmitfehler}} \rhead{fehlerhafte rekonstruktion} -moin +Im zweiten Teil zur Decodierung betrachten wir den Fall, dass unser Übertragungskanal nicht fehlerfrei ist. +Wir legen daher den Fehlervektor +\[ +u = [0, 0, 0, 3, 0, 0, 0, 0, 2, 0] +\] +fest, den wir zu unserem Übertragungsvektor als Fehler dazu addieren und somit +\begin{center} + +\begin{tabular}{c | c r } + $v$ & & $[5,3,6,5,2,10,2,7,10,4]$\\ + $u$ & $+$ & $[0,0,0,3,0,0,0,0,2,0]$\\ + \hline + $w$ & & $[5,3,6,8,2,10,2,7,1,4]$\\ +\end{tabular} + +% alternative design +%\begin{tabular}{c | c cccccccccccc } +% $v$ & & $[$&$5,$&$3,$&$6,$&$5,$&$2,$&$10,$&$2,$&$7,$&$10,$&$4$&$]$\\ +% $u$ & $+$ & $[$&$0,$&$0,$&$0,$&$3,$&$0,$&$0,$&$0,$&$0,$&$2,$&$0$&$]$\\ +% \hline +% $w$ & & $[$&$5,$&$3,$&$6,$&$8,$&$2,$&$10,$&$2,$&$7,$&$1,$&$4$&$]$\\ +%\end{tabular} + +\end{center} +als Übertragungsvektor auf der Empfängerseite erhalten. + +Wenn wir den Übertragungsvektor jetzt Rücktransformieren wie im vorherigen Kapitel erhalten wir +\[ +r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]. +\] +Im Vergleich zum vorherigen Kapitel sind die Fehlerkorrekturstellen jetzt $\neq 0$, was bedeutet, dass wir diesen Übertragungsvektor fehlerhaft empfangen haben und sich die Nachricht jetzt nicht mehr so einfach decodieren lässt. + +% warum wir die fehler suchen +Da Reed-Solomon-Codes in der Lage sind, eine Nachricht aus weniger Stellen zu rekonstruieren als wir ursprünglich haben, so müssen wir nur die Fehlerhaften Stellen finden und eliminieren, damit wir unsere Nutzdaten rekonstruieren können. +Damit stellt sich die Frage, wie wir die Fehlerstellen $e$ finden. +Dafür wählen wir einen Primitiven Ansatz mit +\begin{align} + m(X) & = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1 \\ + r(X) & = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7 \\ + e(X) & = r(X) - m(X). +\end{align} +Setzen wir jetzt unsere Einheitswurzel für $X$ ein, so erhalten wir +\begin{center} +\begin{tabular}{c c c c c c c c c c c} + \hline + $i$& $0$& $1$& $2$& $3$& $4$& $5$& $6$& $7$& $8$& $9$\\ + \hline + $r(a^{i})$& $5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$\\ + $m(a^{i})$& $5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$\\ + $e(a^{i})$& $0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$\\ + \hline +\end{tabular} +\end{center} +und damit die Information, dass an allen Stellen, die nicht Null sind, Fehler enthalten. +Um jetzt alle nicht Nullstellen zu finden, wenden wir den Satz von Fermat an. \subsection{Der Satz von Fermat \label{reedsolomon:subsection:fermat}} -wer ist fermat? +Der Satz von Fermat besagt, dass für +\[ +f(X) = X^{q-1} -1 = 0 +\] +gilt, egal was wir für $q$ einsetzen. + +Für unser Beispiel erhalten wir +\[ +f(X) = X^{10}-1 = 0 \qquad \text{für } X = \{1,2,3,4,5,6,7,8,9,10\} +\] +und können $f(X)$ auch umschreiben in +\[ +f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6)(X-a^7)(X-a^8)(X-a^9). +\] +Zur Überprüfung können wir unsere Einheitswurzel in $a$ einsetzen und werden sehen, dass wir für $f(X) = 0$ erhalten werden. +Nach der gleichen Überlegung können wir jetzt auch $e(X)$ darstellen als +\[ +e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6)(X-a^7) \qquad \qquad (X-a^9) \cdot p(x), +\] +wobei $p(X)$ das Restpolynom ist und die Fehlerstellen beinhaltet. +Wenn wir jetzt den grössten gemeinsamen Teiler von $f(X)$ und $e(X)$ berechnen, so erhalten wir mit +\[ +\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6)(X-a^7) \qquad \qquad (X-a^9) +\] +eine Liste von Nullstellen, an denen es keine Fehler gegeben hat. +Da wir uns jedoch für eine Liste mit Nullstellen interessieren, an denen es Fehler gegeben hat berechnen wir stattdessen das kgV von $f(X)$ und $e(X)$ als +\[ +\operatorname{kgV}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6)(X-a^7)(X-a^8)(X-a^9) \cdot q(X). +\] +Wir können das Resultat noch zerlegen in +\[ +\operatorname{kgV}(f(X),e(X)) = d(X) \cdot e(X). +\] +Somit muss $d(X)$ eine Liste von Nullstellen enthalten an denen es Fehler gegeben hat. +\[ +d(X) = (X-a^3)(X-a^8) +\] + + +und ist damit unser gesuchtes Lokatorpolynom. + +Das einzige Problem was jetzt noch bleibt ist, dass wir $e(X)$ berechnet haben aus +\[ +e(X) = r(X) - m(X), +\] +wobei $m(X)$ auf der Empfängerseite unbekannt ist. +Es sieht danach aus, das wir diesen Lösungsansatz nicht verwenden können, da uns ein entscheidender Teil fehlt. +Bei einer näheren Betrachtung von $m(X)$ fällt uns aber auf, dass wir doch etwas über $m(X)$ wissen. +Wir kennen nämlich die ersten vier Stellen, da diese für die Fehlerkorrektur zuständig sind und daher Null sein müssen. +\[ +m = [0,0,0,0,?,?,?,?,?,?] +\] +An genau diesen Stellen liegt auch die Information, wo unsere Fehlerstellen liegen, was uns ermöglicht, den Teil von $e(X)$ zu berechnen, der uns auch interessiert. + +Wir können $e(X)$ also bestimmen als +\[ +e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X) +\] +wobei $p(X)$ wiederum ein unbekanntes Restpolynom ist und +\[ +f(X) = X^{10} - 1 = X^{10} + 10 +\] +ist können wir so in einer ersten Instanz den grössten gemeinsamen Teiler von $f(X)$ und $e(X)$ berechnen. +Dafür nehmen wir uns wiederum den Euklidischen Algorithmus zur Hilfe und berechnen so + +\[ +\arraycolsep=1.4pt +\begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + X^{10}& & & & & & &+& 10& & & & &:&5X^9&+&7X^8&+& 4X^7&+&10X^6&+&p(X)&=&9X&+&5\\ + X^{10}&+& 8X^9&+& 3X^8&+&2X^7&+& p(X)& & & & & & & & & & & & & & & & \\ \cline{1-9} + && 3X^9&+& 8X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ + && 3X^9&+& 2X^8&+& 9X^7&+& p(X)& & & & & & & & & & & & \\ \cline{3-9} + & & & &6X^8&+&0X^7&+&p(X)& & & & & & & & & & & & \\ +\end{array} +\] + +\[ +\arraycolsep=1.4pt +\begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} + 5X^9&+& 7X^8&+& 4X^7&+& 10X^6&+& p(X)& & & & &:&6X^8&+&0X^7& & & & & & &=&10X&+&3\\ + 5X^9&+& 0X^8&+& p(X)& & & & & & & & & & & & & & & & & & & & \\ \cline{1-5} + && 7X^8&+& p(X)& & & & & & & & & & & & & & & & \\ +\end{array} +\] +und erhalten +\[ +\operatorname{ggT}(f(X),e(X)) = 6X^8 +\] +Mit den Resultaten, die wir vom Rechenweg des grössten gemeinsamen Teiler erhalten haben können wir jetzt auch das kleinste Gemeinsame Vielfache berechnen. Eine detailliertere Vorgehensweise findet man in Kapitel ???. +Aus diesem erweiterten Euklidischen Algorithmus erhalten wir +\begin{center} + + \begin{tabular}{| c | c | c c |} + \hline + $k$ & $q_i$ & $e_i$ & $f_i$\\ + \hline + & & $0$& $1$\\ + $0$& $9X + 5$& $1$& $0$\\ + $1$& $10X + 3$& $9X+5$& $1$\\ + $2$& & \textcolor{blue}{$2X^2 + 0X + 5$}& $10X + 3$\\ + \hline + \end{tabular} + +\end{center} +und erhalten auf diesem Weg den Faktor +\[ +d(X) = 2X^2 + 5, +\] +den wir in +\[ +d(X) = 2(X-5)(X-6) +\] +zerlegen können. +Da die unbekannten Stellen im Lokatorpolynom +\[ +d(X) = (X-a^i)(X-a^i) +\] +sind, müssen wir nur noch $i$ berechnen als +\begin{center} + $a^i = 5 \qquad \Rightarrow \qquad i = 3$ + + $a^i = 6 \qquad \Rightarrow \qquad i = 8$. +\end{center} +Somit erhalten wir schliesslich +\[ +d(X) = (X-a^3)(X-a^8) +\] +als unser Lokatorpolynom mit den Fehlerhaften Stellen. \ No newline at end of file -- cgit v1.2.1 From 81527bd39cb20969fa3a84c85a843bca511dcb51 Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Mon, 24 May 2021 17:18:21 +0200 Subject: created rekonstruktion.tex --- buch/papers/reedsolomon/rekonstruktion.tex | 40 ++++++++++++++++++++++++++++++ 1 file changed, 40 insertions(+) create mode 100644 buch/papers/reedsolomon/rekonstruktion.tex (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/rekonstruktion.tex b/buch/papers/reedsolomon/rekonstruktion.tex new file mode 100644 index 0000000..a3edba4 --- /dev/null +++ b/buch/papers/reedsolomon/rekonstruktion.tex @@ -0,0 +1,40 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Nachricht Rekonstruieren +\label{reedsolomon:section:rekonstruktion}} +\rhead{Teil 3} +Sed ut perspiciatis unde omnis iste natus error sit voluptatem +accusantium doloremque laudantium, totam rem aperiam, eaque ipsa +quae ab illo inventore veritatis et quasi architecto beatae vitae +dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit +aspernatur aut odit aut fugit, sed quia consequuntur magni dolores +eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam +est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci +velit, sed quia non numquam eius modi tempora incidunt ut labore +et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima +veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, +nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure +reprehenderit qui in ea voluptate velit esse quam nihil molestiae +consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla +pariatur? + +\subsection{De finibus bonorum et malorum +\label{reedsolomon:subsection:malorum}} +At vero eos et accusamus et iusto odio dignissimos ducimus qui +blanditiis praesentium voluptatum deleniti atque corrupti quos +dolores et quas molestias excepturi sint occaecati cupiditate non +provident, similique sunt in culpa qui officia deserunt mollitia +animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis +est et expedita distinctio. Nam libero tempore, cum soluta nobis +est eligendi optio cumque nihil impedit quo minus id quod maxime +placeat facere possimus, omnis voluptas assumenda est, omnis dolor +repellendus. Temporibus autem quibusdam et aut officiis debitis aut +rerum necessitatibus saepe eveniet ut et voluptates repudiandae +sint et molestiae non recusandae. Itaque earum rerum hic tenetur a +sapiente delectus, ut aut reiciendis voluptatibus maiores alias +consequatur aut perferendis doloribus asperiores repellat. + + -- cgit v1.2.1 From 2cad7439d48a67af39b7b5ec03f8874ec9d9a3c6 Mon Sep 17 00:00:00 2001 From: Pascal Schmid <81317360+paschost@users.noreply.github.com> Date: Mon, 24 May 2021 20:35:18 +0200 Subject: Update section 1 of paper --- buch/papers/verkehr/teil0.tex | 69 ++++++++++++++++++++++++++++++------------- 1 file changed, 49 insertions(+), 20 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/verkehr/teil0.tex b/buch/papers/verkehr/teil0.tex index 5031841..78d9311 100644 --- a/buch/papers/verkehr/teil0.tex +++ b/buch/papers/verkehr/teil0.tex @@ -1,22 +1,51 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{verkehr:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{verkehr:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. +\section{Versuchsreihe} +\label{section:verkehr/versuchsreihe} +Um zwei der vorgestellten Suchalgorithmen zu vergleichen, wurden zwei Versuchsreihen erstellt. Dazu wurden in einem ersten Schritt zufällige Netzwerke generiert und anschliessend der \emph{Dijkstra}-, sowie der \emph{$A^*$}-Algorithmus auf das Netzwerk angewandt. +Dieser Vorgang wurde für die zufällig generierten Netzwerke mit einer Knotenzahl von 10, 20 50, 100, 200, 500 und 1000 je zehnmal repetiert. +Die Anzahl der Knoten im abgesuchten Netzwerk wirkt sich direkt auf die Rechenzeit aus. Der \emph{Dijkstra}-Algorithmus weist eine Zeitkomplexität von $\mathcal{O}(E\log{}V)$ auf, wobei $E$ die Anzahl Kanten (engl. \emph{edges}) und $V$ die Anzahl Knoten (engl. \emph{vertices}) darstellt. +Für den \emph{A*}-Algorithmus ist die Zeitkomplexität einerseits abhängig von der verwendeten Heuristik, andererseits aber auch vom vorliegenden Netzwerk selbst. Aus diesem Grund lässt sich keine defintive Angabe zu $\mathcal{O}$ machen. +Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- und Zielknoten bei der ersten Versuchsreihe im Netzwerk diametral gegenüber liegen. Dadurch gehen viele Knoten verloren, welcher \emph{Dijkstra} als uninformierter Suchalgorithmus absuchen würde. In der zweiten Veruschsreihe werden hingegen Start- un Zielpunkt zufällig im Netzwerk ausgewählt. Es wird deshalb erwwartet, dass die Unterschiede in der Rechenzeit der beiden Algorithmen in der zweiten Versuchsreihe deutlich ausgeprägter sind. + +\subsection{Einfluss der Knotenzahl auf die Rechenzeit} +\label{verkehr:Knotenzahl} + +\begin{wrapfigure}{} +\includegraphics[width=12cm]{figures/chart_Vr1.png} + +\caption{Gemessene Rechenzeiten der ersten Versuchsreihe in Abhängigkeit der Knotenzahl.} +\label{verkehr:Vr1} +\end{wrapfigure} + +In \ref{verkehr:Vr1} ist ersichtlich, dass der Unterschied in der Rechenzeit zwischen \emph{Dijkstra} und \emph{A*} erst aber einer Knotenzahl von ca. $n=500$ merklich ansteigt. Dieses etwas überraschende Resultat ist darauf zurückzuführen, dass bei steigender Knotenzahl die Abweichung des effektiven kürzesten Pfades von der Distanz der Luftlinie abnimmt. +Die Effektivität von \emph{A*} mit euklidischer Heuristik ist wiederum grösser, wenn die Abweichung des kürzesten Pfads von der Luftlinie minimal ist. +Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei die relative Abweichung erstaunlicherweise bei einer Knotenzahl von $n=100$ maximal ist und nach $n=500$ nur noch marginal abnimmt. + +\begin{wrapfigure}{} +\includegraphics[width=12cm]{figures/chart_pathDiff.png} + +\caption{Relative Abweichung des kürzesten Pfads von der Luftlinie.} +\label{verkehr:pathDifference} +\end{wrapfigure} + + +\subsection{Einfluss der Position der Start- und Zielknoten auf die Rechenzeit} + +\begin{wrapfigure}{} +\includegraphics[width=12cm]{figures/chart_Vr2.png}\\ +\caption{Gemessene Rechenzeiten der zweiten Versuchsreihe in Abhängigkeit der Knotenzahl.} +\label{verkehr:Vr2} +\end{wrapfigure} + +Zum Vergleich der Resultate in \ref{verkehr:Knotenzahl} zeigt \ref{verkehr:Vr2} die Rechenzeiten der zweiten Versuchsreihe, in welcher die Start- und Zielknoten zufällig im Netzwerk ausgewählt wurden. Einerseits ist eine reduzierte durchschnittliche Rechenzeit festzustellen, was schlicht daran liegt, dass die zufällige Wahl der Knoten dazu führt, dass diese tendenziell weniger weit auseinander liegen.\\ +Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwischen \emph{Dijkstra} und \emph{A*} deutlich früher abzeichnen. Dieses Phänomen lässt sich leicht durch die zielgerichtete Suche des \emph{A*}-Algorithmus erklären. + +\begin{wrapfigure}{} +\includegraphics[width=6cm]{figures/network_dij.png}\qquad +\includegraphics[width=6cm]{figures/network_aStar.png} +\caption{Suchpfad in grün mit \emph{Dijkstra} (links), und \emph{A*} (rechts). Besuchte Knoten sind in blau, resp. rot markiert.} +\label{verkehr:Comparison} +\end{wrapfigure} + +In \ref{verkehr:Comparison} ist ersichtlich, dass bei einem im Netzwerk liegenden Startknoten die zielgerichtete Suche von \emph{A*} deutlich ausgeprägter zum Zuge kommt, als wenn dieser am Rand des Netzwerks liegen würde. -- cgit v1.2.1 From 1d1a334cce74e76b5ae18701b39d379580e07edb Mon Sep 17 00:00:00 2001 From: Pascal Schmid <81317360+paschost@users.noreply.github.com> Date: Mon, 24 May 2021 20:36:50 +0200 Subject: Update section 2 of paper --- buch/papers/verkehr/teil1.tex | 102 ++++++++++++++++++++---------------------- 1 file changed, 49 insertions(+), 53 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/verkehr/teil1.tex b/buch/papers/verkehr/teil1.tex index 855aef8..78d9311 100644 --- a/buch/papers/verkehr/teil1.tex +++ b/buch/papers/verkehr/teil1.tex @@ -1,55 +1,51 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{verkehr:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{verkehr:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{verkehr:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{verkehr:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{verkehr:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. +\section{Versuchsreihe} +\label{section:verkehr/versuchsreihe} +Um zwei der vorgestellten Suchalgorithmen zu vergleichen, wurden zwei Versuchsreihen erstellt. Dazu wurden in einem ersten Schritt zufällige Netzwerke generiert und anschliessend der \emph{Dijkstra}-, sowie der \emph{$A^*$}-Algorithmus auf das Netzwerk angewandt. +Dieser Vorgang wurde für die zufällig generierten Netzwerke mit einer Knotenzahl von 10, 20 50, 100, 200, 500 und 1000 je zehnmal repetiert. +Die Anzahl der Knoten im abgesuchten Netzwerk wirkt sich direkt auf die Rechenzeit aus. Der \emph{Dijkstra}-Algorithmus weist eine Zeitkomplexität von $\mathcal{O}(E\log{}V)$ auf, wobei $E$ die Anzahl Kanten (engl. \emph{edges}) und $V$ die Anzahl Knoten (engl. \emph{vertices}) darstellt. +Für den \emph{A*}-Algorithmus ist die Zeitkomplexität einerseits abhängig von der verwendeten Heuristik, andererseits aber auch vom vorliegenden Netzwerk selbst. Aus diesem Grund lässt sich keine defintive Angabe zu $\mathcal{O}$ machen. +Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- und Zielknoten bei der ersten Versuchsreihe im Netzwerk diametral gegenüber liegen. Dadurch gehen viele Knoten verloren, welcher \emph{Dijkstra} als uninformierter Suchalgorithmus absuchen würde. In der zweiten Veruschsreihe werden hingegen Start- un Zielpunkt zufällig im Netzwerk ausgewählt. Es wird deshalb erwwartet, dass die Unterschiede in der Rechenzeit der beiden Algorithmen in der zweiten Versuchsreihe deutlich ausgeprägter sind. + +\subsection{Einfluss der Knotenzahl auf die Rechenzeit} +\label{verkehr:Knotenzahl} + +\begin{wrapfigure}{} +\includegraphics[width=12cm]{figures/chart_Vr1.png} + +\caption{Gemessene Rechenzeiten der ersten Versuchsreihe in Abhängigkeit der Knotenzahl.} +\label{verkehr:Vr1} +\end{wrapfigure} + +In \ref{verkehr:Vr1} ist ersichtlich, dass der Unterschied in der Rechenzeit zwischen \emph{Dijkstra} und \emph{A*} erst aber einer Knotenzahl von ca. $n=500$ merklich ansteigt. Dieses etwas überraschende Resultat ist darauf zurückzuführen, dass bei steigender Knotenzahl die Abweichung des effektiven kürzesten Pfades von der Distanz der Luftlinie abnimmt. +Die Effektivität von \emph{A*} mit euklidischer Heuristik ist wiederum grösser, wenn die Abweichung des kürzesten Pfads von der Luftlinie minimal ist. +Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei die relative Abweichung erstaunlicherweise bei einer Knotenzahl von $n=100$ maximal ist und nach $n=500$ nur noch marginal abnimmt. + +\begin{wrapfigure}{} +\includegraphics[width=12cm]{figures/chart_pathDiff.png} + +\caption{Relative Abweichung des kürzesten Pfads von der Luftlinie.} +\label{verkehr:pathDifference} +\end{wrapfigure} + + +\subsection{Einfluss der Position der Start- und Zielknoten auf die Rechenzeit} + +\begin{wrapfigure}{} +\includegraphics[width=12cm]{figures/chart_Vr2.png}\\ +\caption{Gemessene Rechenzeiten der zweiten Versuchsreihe in Abhängigkeit der Knotenzahl.} +\label{verkehr:Vr2} +\end{wrapfigure} + +Zum Vergleich der Resultate in \ref{verkehr:Knotenzahl} zeigt \ref{verkehr:Vr2} die Rechenzeiten der zweiten Versuchsreihe, in welcher die Start- und Zielknoten zufällig im Netzwerk ausgewählt wurden. Einerseits ist eine reduzierte durchschnittliche Rechenzeit festzustellen, was schlicht daran liegt, dass die zufällige Wahl der Knoten dazu führt, dass diese tendenziell weniger weit auseinander liegen.\\ +Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwischen \emph{Dijkstra} und \emph{A*} deutlich früher abzeichnen. Dieses Phänomen lässt sich leicht durch die zielgerichtete Suche des \emph{A*}-Algorithmus erklären. + +\begin{wrapfigure}{} +\includegraphics[width=6cm]{figures/network_dij.png}\qquad +\includegraphics[width=6cm]{figures/network_aStar.png} +\caption{Suchpfad in grün mit \emph{Dijkstra} (links), und \emph{A*} (rechts). Besuchte Knoten sind in blau, resp. rot markiert.} +\label{verkehr:Comparison} +\end{wrapfigure} + +In \ref{verkehr:Comparison} ist ersichtlich, dass bei einem im Netzwerk liegenden Startknoten die zielgerichtete Suche von \emph{A*} deutlich ausgeprägter zum Zuge kommt, als wenn dieser am Rand des Netzwerks liegen würde. -- cgit v1.2.1 From 61c60ad40387ed1401f6685a152529874c07d63d Mon Sep 17 00:00:00 2001 From: Pascal Schmid <81317360+paschost@users.noreply.github.com> Date: Mon, 24 May 2021 20:37:23 +0200 Subject: Update section 3 of paper --- buch/papers/verkehr/teil2.tex | 46 +++++++------------------------------------ 1 file changed, 7 insertions(+), 39 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/verkehr/teil2.tex b/buch/papers/verkehr/teil2.tex index 5170ded..99a0d92 100644 --- a/buch/papers/verkehr/teil2.tex +++ b/buch/papers/verkehr/teil2.tex @@ -1,40 +1,8 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{verkehr:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{verkehr:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - +\section{Ausblick} +\subsection{Optimierungsprobleme bei Graphen} +Das Finden eines kürzesten Pfades, sprich die Minimierung der Summe der Kantengewichte, ist nur eines der Optimierungsprobleme, die sich im Bereich von Grafen aufstellen lassen. Verschiedene, ähnliche Problemstellungen lassen sich teilweise mit denselben Algorithmen lösen.\\ +Im Bereich vom Computernetzwerken könnte zum Beispiel die Minimierung der Knotenzahl zur Datenübbertragung von Interesse sein. Dabei lässt sich dieses Problem einfach dadurch lösen, dass dem \emph{Dijkstra}, oder dem \emph{A*}-Algorithmus anstelle der Graph-Matrix (mit Kantengewichten als Einträgen) die Adjazenz-Matrix als Argument übergeben wird. Der gefundene kürzeste Pfad enstpricht der Anzahl benutzter Kanten, bzw. der Anzahl besuchter Knoten. +\subsection{Wahl der Heuristik} +Ein grundlegendes Problem bei der Anwendung des \emph{A*} oder ähnlicher informierter Suchalgorithmen ist die Wahl der Heurstik. Bei einem physischen Verkehrsnetz kann bspw. die euklidische Distanz problems ermittelt werde. Bei einem regionalen Netzwerk ist die Annahme eines orthogonalen X-Y-Koordinatenetzes absolut ausreichend. Dies gilt z.B. auch für das Vernessungsnetz der Schweiz\footnote{Die aktuelle Schweizer Referenzsystem LV95 benutzt ein E/N-Koordinatennetz, wobei aufgrund zunehmender Abweichung vom Referenzellipsoid bei grosser Entfernung vom Nullpunkt ein Korrekturfaktor für die Höhe angebracht werden muss.} Bei überregionalen Netzwerken (Beispiel: Flugverbindungen) ist hingegen eine Berechnung im dreidimensionalen Raum, oder vereinfacht als Projektion auf das Geoid notwendig. Anonsten ist der Ablauf bei der Ausführung des Algorithmus allerdings identisch.\\ +In nicht-physischen Netzwerken stellt sich jedoch eine zweite Problematik. Da eine physische Distanz entweder nicht ermittelt werden kann, oder aber nicht ausschlaggebend ist, sind andere Netzwerk-Eigenschaften zur Beurteilung beizuziehen. Die Zuverlässigkeit ist dabei aber in den meisten Fällen nicht vergleichbar hoch, wie bei der euklidischen Heuristik. Oftmals werden deshalb bei derartigen Problem auch Algorithmen angewendet, die eine deutlich optimierte Zeitkomplexität aufweisen, dafür aber nicht mit Sicherheit den effizienstesten Pfad finden. -- cgit v1.2.1 From 0cd2f753702cc0ce0c74a281c4b7146ca96ae78f Mon Sep 17 00:00:00 2001 From: Pascal Schmid <81317360+paschost@users.noreply.github.com> Date: Mon, 24 May 2021 20:40:11 +0200 Subject: delete 4th section 4th section not needed --- buch/papers/verkehr/teil3.tex | 40 ---------------------------------------- 1 file changed, 40 deletions(-) delete mode 100644 buch/papers/verkehr/teil3.tex (limited to 'buch/papers') diff --git a/buch/papers/verkehr/teil3.tex b/buch/papers/verkehr/teil3.tex deleted file mode 100644 index 8f79154..0000000 --- a/buch/papers/verkehr/teil3.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{verkehr:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{verkehr:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - -- cgit v1.2.1 From e43c4b251bbed04a843a3b4b13f2c2ae25bc7181 Mon Sep 17 00:00:00 2001 From: Pascal Schmid <81317360+paschost@users.noreply.github.com> Date: Mon, 24 May 2021 20:45:52 +0200 Subject: added /figure folder and figures --- buch/papers/verkehr/figures/chart_Vr1.png | Bin 0 -> 74176 bytes buch/papers/verkehr/figures/chart_Vr2.png | Bin 0 -> 64237 bytes buch/papers/verkehr/figures/chart_pathDiff.png | Bin 0 -> 36673 bytes buch/papers/verkehr/figures/dist_display6.png | Bin 0 -> 399354 bytes buch/papers/verkehr/figures/network_aStar.png | Bin 0 -> 79386 bytes buch/papers/verkehr/figures/network_dij.png | Bin 0 -> 77108 bytes 6 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 buch/papers/verkehr/figures/chart_Vr1.png create mode 100644 buch/papers/verkehr/figures/chart_Vr2.png create mode 100644 buch/papers/verkehr/figures/chart_pathDiff.png create mode 100644 buch/papers/verkehr/figures/dist_display6.png create mode 100644 buch/papers/verkehr/figures/network_aStar.png create mode 100644 buch/papers/verkehr/figures/network_dij.png (limited to 'buch/papers') diff --git a/buch/papers/verkehr/figures/chart_Vr1.png b/buch/papers/verkehr/figures/chart_Vr1.png new file mode 100644 index 0000000..14d4eca Binary files /dev/null and b/buch/papers/verkehr/figures/chart_Vr1.png differ diff --git a/buch/papers/verkehr/figures/chart_Vr2.png b/buch/papers/verkehr/figures/chart_Vr2.png new file mode 100644 index 0000000..2d68681 Binary files /dev/null and b/buch/papers/verkehr/figures/chart_Vr2.png differ diff --git a/buch/papers/verkehr/figures/chart_pathDiff.png b/buch/papers/verkehr/figures/chart_pathDiff.png new file mode 100644 index 0000000..02bded7 Binary files /dev/null and b/buch/papers/verkehr/figures/chart_pathDiff.png differ diff --git a/buch/papers/verkehr/figures/dist_display6.png b/buch/papers/verkehr/figures/dist_display6.png new file mode 100644 index 0000000..3056f43 Binary files /dev/null and b/buch/papers/verkehr/figures/dist_display6.png differ diff --git a/buch/papers/verkehr/figures/network_aStar.png b/buch/papers/verkehr/figures/network_aStar.png new file mode 100644 index 0000000..5a681bd Binary files /dev/null and b/buch/papers/verkehr/figures/network_aStar.png differ diff --git a/buch/papers/verkehr/figures/network_dij.png b/buch/papers/verkehr/figures/network_dij.png new file mode 100644 index 0000000..d9348d7 Binary files /dev/null and b/buch/papers/verkehr/figures/network_dij.png differ -- cgit v1.2.1 From 26ecbb9559558f40e5e05a84ceb8622c5c9bd182 Mon Sep 17 00:00:00 2001 From: Pascal Schmid Date: Mon, 24 May 2021 20:57:36 +0200 Subject: renamed section files --- buch/papers/verkehr/section1.tex | 51 ++++++++++++++++++++++++++++++++++++++++ buch/papers/verkehr/section2.tex | 51 ++++++++++++++++++++++++++++++++++++++++ buch/papers/verkehr/section3.tex | 8 +++++++ buch/papers/verkehr/teil0.tex | 51 ---------------------------------------- buch/papers/verkehr/teil1.tex | 51 ---------------------------------------- buch/papers/verkehr/teil2.tex | 8 ------- 6 files changed, 110 insertions(+), 110 deletions(-) create mode 100644 buch/papers/verkehr/section1.tex create mode 100644 buch/papers/verkehr/section2.tex create mode 100644 buch/papers/verkehr/section3.tex delete mode 100644 buch/papers/verkehr/teil0.tex delete mode 100644 buch/papers/verkehr/teil1.tex delete mode 100644 buch/papers/verkehr/teil2.tex (limited to 'buch/papers') diff --git a/buch/papers/verkehr/section1.tex b/buch/papers/verkehr/section1.tex new file mode 100644 index 0000000..78d9311 --- /dev/null +++ b/buch/papers/verkehr/section1.tex @@ -0,0 +1,51 @@ +\section{Versuchsreihe} +\label{section:verkehr/versuchsreihe} + +Um zwei der vorgestellten Suchalgorithmen zu vergleichen, wurden zwei Versuchsreihen erstellt. Dazu wurden in einem ersten Schritt zufällige Netzwerke generiert und anschliessend der \emph{Dijkstra}-, sowie der \emph{$A^*$}-Algorithmus auf das Netzwerk angewandt. +Dieser Vorgang wurde für die zufällig generierten Netzwerke mit einer Knotenzahl von 10, 20 50, 100, 200, 500 und 1000 je zehnmal repetiert. +Die Anzahl der Knoten im abgesuchten Netzwerk wirkt sich direkt auf die Rechenzeit aus. Der \emph{Dijkstra}-Algorithmus weist eine Zeitkomplexität von $\mathcal{O}(E\log{}V)$ auf, wobei $E$ die Anzahl Kanten (engl. \emph{edges}) und $V$ die Anzahl Knoten (engl. \emph{vertices}) darstellt. +Für den \emph{A*}-Algorithmus ist die Zeitkomplexität einerseits abhängig von der verwendeten Heuristik, andererseits aber auch vom vorliegenden Netzwerk selbst. Aus diesem Grund lässt sich keine defintive Angabe zu $\mathcal{O}$ machen. + +Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- und Zielknoten bei der ersten Versuchsreihe im Netzwerk diametral gegenüber liegen. Dadurch gehen viele Knoten verloren, welcher \emph{Dijkstra} als uninformierter Suchalgorithmus absuchen würde. In der zweiten Veruschsreihe werden hingegen Start- un Zielpunkt zufällig im Netzwerk ausgewählt. Es wird deshalb erwwartet, dass die Unterschiede in der Rechenzeit der beiden Algorithmen in der zweiten Versuchsreihe deutlich ausgeprägter sind. + +\subsection{Einfluss der Knotenzahl auf die Rechenzeit} +\label{verkehr:Knotenzahl} + +\begin{wrapfigure}{} +\includegraphics[width=12cm]{figures/chart_Vr1.png} + +\caption{Gemessene Rechenzeiten der ersten Versuchsreihe in Abhängigkeit der Knotenzahl.} +\label{verkehr:Vr1} +\end{wrapfigure} + +In \ref{verkehr:Vr1} ist ersichtlich, dass der Unterschied in der Rechenzeit zwischen \emph{Dijkstra} und \emph{A*} erst aber einer Knotenzahl von ca. $n=500$ merklich ansteigt. Dieses etwas überraschende Resultat ist darauf zurückzuführen, dass bei steigender Knotenzahl die Abweichung des effektiven kürzesten Pfades von der Distanz der Luftlinie abnimmt. +Die Effektivität von \emph{A*} mit euklidischer Heuristik ist wiederum grösser, wenn die Abweichung des kürzesten Pfads von der Luftlinie minimal ist. +Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei die relative Abweichung erstaunlicherweise bei einer Knotenzahl von $n=100$ maximal ist und nach $n=500$ nur noch marginal abnimmt. + +\begin{wrapfigure}{} +\includegraphics[width=12cm]{figures/chart_pathDiff.png} + +\caption{Relative Abweichung des kürzesten Pfads von der Luftlinie.} +\label{verkehr:pathDifference} +\end{wrapfigure} + + +\subsection{Einfluss der Position der Start- und Zielknoten auf die Rechenzeit} + +\begin{wrapfigure}{} +\includegraphics[width=12cm]{figures/chart_Vr2.png}\\ +\caption{Gemessene Rechenzeiten der zweiten Versuchsreihe in Abhängigkeit der Knotenzahl.} +\label{verkehr:Vr2} +\end{wrapfigure} + +Zum Vergleich der Resultate in \ref{verkehr:Knotenzahl} zeigt \ref{verkehr:Vr2} die Rechenzeiten der zweiten Versuchsreihe, in welcher die Start- und Zielknoten zufällig im Netzwerk ausgewählt wurden. Einerseits ist eine reduzierte durchschnittliche Rechenzeit festzustellen, was schlicht daran liegt, dass die zufällige Wahl der Knoten dazu führt, dass diese tendenziell weniger weit auseinander liegen.\\ +Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwischen \emph{Dijkstra} und \emph{A*} deutlich früher abzeichnen. Dieses Phänomen lässt sich leicht durch die zielgerichtete Suche des \emph{A*}-Algorithmus erklären. + +\begin{wrapfigure}{} +\includegraphics[width=6cm]{figures/network_dij.png}\qquad +\includegraphics[width=6cm]{figures/network_aStar.png} +\caption{Suchpfad in grün mit \emph{Dijkstra} (links), und \emph{A*} (rechts). Besuchte Knoten sind in blau, resp. rot markiert.} +\label{verkehr:Comparison} +\end{wrapfigure} + +In \ref{verkehr:Comparison} ist ersichtlich, dass bei einem im Netzwerk liegenden Startknoten die zielgerichtete Suche von \emph{A*} deutlich ausgeprägter zum Zuge kommt, als wenn dieser am Rand des Netzwerks liegen würde. diff --git a/buch/papers/verkehr/section2.tex b/buch/papers/verkehr/section2.tex new file mode 100644 index 0000000..78d9311 --- /dev/null +++ b/buch/papers/verkehr/section2.tex @@ -0,0 +1,51 @@ +\section{Versuchsreihe} +\label{section:verkehr/versuchsreihe} + +Um zwei der vorgestellten Suchalgorithmen zu vergleichen, wurden zwei Versuchsreihen erstellt. Dazu wurden in einem ersten Schritt zufällige Netzwerke generiert und anschliessend der \emph{Dijkstra}-, sowie der \emph{$A^*$}-Algorithmus auf das Netzwerk angewandt. +Dieser Vorgang wurde für die zufällig generierten Netzwerke mit einer Knotenzahl von 10, 20 50, 100, 200, 500 und 1000 je zehnmal repetiert. +Die Anzahl der Knoten im abgesuchten Netzwerk wirkt sich direkt auf die Rechenzeit aus. Der \emph{Dijkstra}-Algorithmus weist eine Zeitkomplexität von $\mathcal{O}(E\log{}V)$ auf, wobei $E$ die Anzahl Kanten (engl. \emph{edges}) und $V$ die Anzahl Knoten (engl. \emph{vertices}) darstellt. +Für den \emph{A*}-Algorithmus ist die Zeitkomplexität einerseits abhängig von der verwendeten Heuristik, andererseits aber auch vom vorliegenden Netzwerk selbst. Aus diesem Grund lässt sich keine defintive Angabe zu $\mathcal{O}$ machen. + +Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- und Zielknoten bei der ersten Versuchsreihe im Netzwerk diametral gegenüber liegen. Dadurch gehen viele Knoten verloren, welcher \emph{Dijkstra} als uninformierter Suchalgorithmus absuchen würde. In der zweiten Veruschsreihe werden hingegen Start- un Zielpunkt zufällig im Netzwerk ausgewählt. Es wird deshalb erwwartet, dass die Unterschiede in der Rechenzeit der beiden Algorithmen in der zweiten Versuchsreihe deutlich ausgeprägter sind. + +\subsection{Einfluss der Knotenzahl auf die Rechenzeit} +\label{verkehr:Knotenzahl} + +\begin{wrapfigure}{} +\includegraphics[width=12cm]{figures/chart_Vr1.png} + +\caption{Gemessene Rechenzeiten der ersten Versuchsreihe in Abhängigkeit der Knotenzahl.} +\label{verkehr:Vr1} +\end{wrapfigure} + +In \ref{verkehr:Vr1} ist ersichtlich, dass der Unterschied in der Rechenzeit zwischen \emph{Dijkstra} und \emph{A*} erst aber einer Knotenzahl von ca. $n=500$ merklich ansteigt. Dieses etwas überraschende Resultat ist darauf zurückzuführen, dass bei steigender Knotenzahl die Abweichung des effektiven kürzesten Pfades von der Distanz der Luftlinie abnimmt. +Die Effektivität von \emph{A*} mit euklidischer Heuristik ist wiederum grösser, wenn die Abweichung des kürzesten Pfads von der Luftlinie minimal ist. +Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei die relative Abweichung erstaunlicherweise bei einer Knotenzahl von $n=100$ maximal ist und nach $n=500$ nur noch marginal abnimmt. + +\begin{wrapfigure}{} +\includegraphics[width=12cm]{figures/chart_pathDiff.png} + +\caption{Relative Abweichung des kürzesten Pfads von der Luftlinie.} +\label{verkehr:pathDifference} +\end{wrapfigure} + + +\subsection{Einfluss der Position der Start- und Zielknoten auf die Rechenzeit} + +\begin{wrapfigure}{} +\includegraphics[width=12cm]{figures/chart_Vr2.png}\\ +\caption{Gemessene Rechenzeiten der zweiten Versuchsreihe in Abhängigkeit der Knotenzahl.} +\label{verkehr:Vr2} +\end{wrapfigure} + +Zum Vergleich der Resultate in \ref{verkehr:Knotenzahl} zeigt \ref{verkehr:Vr2} die Rechenzeiten der zweiten Versuchsreihe, in welcher die Start- und Zielknoten zufällig im Netzwerk ausgewählt wurden. Einerseits ist eine reduzierte durchschnittliche Rechenzeit festzustellen, was schlicht daran liegt, dass die zufällige Wahl der Knoten dazu führt, dass diese tendenziell weniger weit auseinander liegen.\\ +Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwischen \emph{Dijkstra} und \emph{A*} deutlich früher abzeichnen. Dieses Phänomen lässt sich leicht durch die zielgerichtete Suche des \emph{A*}-Algorithmus erklären. + +\begin{wrapfigure}{} +\includegraphics[width=6cm]{figures/network_dij.png}\qquad +\includegraphics[width=6cm]{figures/network_aStar.png} +\caption{Suchpfad in grün mit \emph{Dijkstra} (links), und \emph{A*} (rechts). Besuchte Knoten sind in blau, resp. rot markiert.} +\label{verkehr:Comparison} +\end{wrapfigure} + +In \ref{verkehr:Comparison} ist ersichtlich, dass bei einem im Netzwerk liegenden Startknoten die zielgerichtete Suche von \emph{A*} deutlich ausgeprägter zum Zuge kommt, als wenn dieser am Rand des Netzwerks liegen würde. diff --git a/buch/papers/verkehr/section3.tex b/buch/papers/verkehr/section3.tex new file mode 100644 index 0000000..99a0d92 --- /dev/null +++ b/buch/papers/verkehr/section3.tex @@ -0,0 +1,8 @@ +\section{Ausblick} +\subsection{Optimierungsprobleme bei Graphen} +Das Finden eines kürzesten Pfades, sprich die Minimierung der Summe der Kantengewichte, ist nur eines der Optimierungsprobleme, die sich im Bereich von Grafen aufstellen lassen. Verschiedene, ähnliche Problemstellungen lassen sich teilweise mit denselben Algorithmen lösen.\\ +Im Bereich vom Computernetzwerken könnte zum Beispiel die Minimierung der Knotenzahl zur Datenübbertragung von Interesse sein. Dabei lässt sich dieses Problem einfach dadurch lösen, dass dem \emph{Dijkstra}, oder dem \emph{A*}-Algorithmus anstelle der Graph-Matrix (mit Kantengewichten als Einträgen) die Adjazenz-Matrix als Argument übergeben wird. Der gefundene kürzeste Pfad enstpricht der Anzahl benutzter Kanten, bzw. der Anzahl besuchter Knoten. + +\subsection{Wahl der Heuristik} +Ein grundlegendes Problem bei der Anwendung des \emph{A*} oder ähnlicher informierter Suchalgorithmen ist die Wahl der Heurstik. Bei einem physischen Verkehrsnetz kann bspw. die euklidische Distanz problems ermittelt werde. Bei einem regionalen Netzwerk ist die Annahme eines orthogonalen X-Y-Koordinatenetzes absolut ausreichend. Dies gilt z.B. auch für das Vernessungsnetz der Schweiz\footnote{Die aktuelle Schweizer Referenzsystem LV95 benutzt ein E/N-Koordinatennetz, wobei aufgrund zunehmender Abweichung vom Referenzellipsoid bei grosser Entfernung vom Nullpunkt ein Korrekturfaktor für die Höhe angebracht werden muss.} Bei überregionalen Netzwerken (Beispiel: Flugverbindungen) ist hingegen eine Berechnung im dreidimensionalen Raum, oder vereinfacht als Projektion auf das Geoid notwendig. Anonsten ist der Ablauf bei der Ausführung des Algorithmus allerdings identisch.\\ +In nicht-physischen Netzwerken stellt sich jedoch eine zweite Problematik. Da eine physische Distanz entweder nicht ermittelt werden kann, oder aber nicht ausschlaggebend ist, sind andere Netzwerk-Eigenschaften zur Beurteilung beizuziehen. Die Zuverlässigkeit ist dabei aber in den meisten Fällen nicht vergleichbar hoch, wie bei der euklidischen Heuristik. Oftmals werden deshalb bei derartigen Problem auch Algorithmen angewendet, die eine deutlich optimierte Zeitkomplexität aufweisen, dafür aber nicht mit Sicherheit den effizienstesten Pfad finden. diff --git a/buch/papers/verkehr/teil0.tex b/buch/papers/verkehr/teil0.tex deleted file mode 100644 index 78d9311..0000000 --- a/buch/papers/verkehr/teil0.tex +++ /dev/null @@ -1,51 +0,0 @@ -\section{Versuchsreihe} -\label{section:verkehr/versuchsreihe} - -Um zwei der vorgestellten Suchalgorithmen zu vergleichen, wurden zwei Versuchsreihen erstellt. Dazu wurden in einem ersten Schritt zufällige Netzwerke generiert und anschliessend der \emph{Dijkstra}-, sowie der \emph{$A^*$}-Algorithmus auf das Netzwerk angewandt. -Dieser Vorgang wurde für die zufällig generierten Netzwerke mit einer Knotenzahl von 10, 20 50, 100, 200, 500 und 1000 je zehnmal repetiert. -Die Anzahl der Knoten im abgesuchten Netzwerk wirkt sich direkt auf die Rechenzeit aus. Der \emph{Dijkstra}-Algorithmus weist eine Zeitkomplexität von $\mathcal{O}(E\log{}V)$ auf, wobei $E$ die Anzahl Kanten (engl. \emph{edges}) und $V$ die Anzahl Knoten (engl. \emph{vertices}) darstellt. -Für den \emph{A*}-Algorithmus ist die Zeitkomplexität einerseits abhängig von der verwendeten Heuristik, andererseits aber auch vom vorliegenden Netzwerk selbst. Aus diesem Grund lässt sich keine defintive Angabe zu $\mathcal{O}$ machen. - -Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- und Zielknoten bei der ersten Versuchsreihe im Netzwerk diametral gegenüber liegen. Dadurch gehen viele Knoten verloren, welcher \emph{Dijkstra} als uninformierter Suchalgorithmus absuchen würde. In der zweiten Veruschsreihe werden hingegen Start- un Zielpunkt zufällig im Netzwerk ausgewählt. Es wird deshalb erwwartet, dass die Unterschiede in der Rechenzeit der beiden Algorithmen in der zweiten Versuchsreihe deutlich ausgeprägter sind. - -\subsection{Einfluss der Knotenzahl auf die Rechenzeit} -\label{verkehr:Knotenzahl} - -\begin{wrapfigure}{} -\includegraphics[width=12cm]{figures/chart_Vr1.png} - -\caption{Gemessene Rechenzeiten der ersten Versuchsreihe in Abhängigkeit der Knotenzahl.} -\label{verkehr:Vr1} -\end{wrapfigure} - -In \ref{verkehr:Vr1} ist ersichtlich, dass der Unterschied in der Rechenzeit zwischen \emph{Dijkstra} und \emph{A*} erst aber einer Knotenzahl von ca. $n=500$ merklich ansteigt. Dieses etwas überraschende Resultat ist darauf zurückzuführen, dass bei steigender Knotenzahl die Abweichung des effektiven kürzesten Pfades von der Distanz der Luftlinie abnimmt. -Die Effektivität von \emph{A*} mit euklidischer Heuristik ist wiederum grösser, wenn die Abweichung des kürzesten Pfads von der Luftlinie minimal ist. -Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei die relative Abweichung erstaunlicherweise bei einer Knotenzahl von $n=100$ maximal ist und nach $n=500$ nur noch marginal abnimmt. - -\begin{wrapfigure}{} -\includegraphics[width=12cm]{figures/chart_pathDiff.png} - -\caption{Relative Abweichung des kürzesten Pfads von der Luftlinie.} -\label{verkehr:pathDifference} -\end{wrapfigure} - - -\subsection{Einfluss der Position der Start- und Zielknoten auf die Rechenzeit} - -\begin{wrapfigure}{} -\includegraphics[width=12cm]{figures/chart_Vr2.png}\\ -\caption{Gemessene Rechenzeiten der zweiten Versuchsreihe in Abhängigkeit der Knotenzahl.} -\label{verkehr:Vr2} -\end{wrapfigure} - -Zum Vergleich der Resultate in \ref{verkehr:Knotenzahl} zeigt \ref{verkehr:Vr2} die Rechenzeiten der zweiten Versuchsreihe, in welcher die Start- und Zielknoten zufällig im Netzwerk ausgewählt wurden. Einerseits ist eine reduzierte durchschnittliche Rechenzeit festzustellen, was schlicht daran liegt, dass die zufällige Wahl der Knoten dazu führt, dass diese tendenziell weniger weit auseinander liegen.\\ -Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwischen \emph{Dijkstra} und \emph{A*} deutlich früher abzeichnen. Dieses Phänomen lässt sich leicht durch die zielgerichtete Suche des \emph{A*}-Algorithmus erklären. - -\begin{wrapfigure}{} -\includegraphics[width=6cm]{figures/network_dij.png}\qquad -\includegraphics[width=6cm]{figures/network_aStar.png} -\caption{Suchpfad in grün mit \emph{Dijkstra} (links), und \emph{A*} (rechts). Besuchte Knoten sind in blau, resp. rot markiert.} -\label{verkehr:Comparison} -\end{wrapfigure} - -In \ref{verkehr:Comparison} ist ersichtlich, dass bei einem im Netzwerk liegenden Startknoten die zielgerichtete Suche von \emph{A*} deutlich ausgeprägter zum Zuge kommt, als wenn dieser am Rand des Netzwerks liegen würde. diff --git a/buch/papers/verkehr/teil1.tex b/buch/papers/verkehr/teil1.tex deleted file mode 100644 index 78d9311..0000000 --- a/buch/papers/verkehr/teil1.tex +++ /dev/null @@ -1,51 +0,0 @@ -\section{Versuchsreihe} -\label{section:verkehr/versuchsreihe} - -Um zwei der vorgestellten Suchalgorithmen zu vergleichen, wurden zwei Versuchsreihen erstellt. Dazu wurden in einem ersten Schritt zufällige Netzwerke generiert und anschliessend der \emph{Dijkstra}-, sowie der \emph{$A^*$}-Algorithmus auf das Netzwerk angewandt. -Dieser Vorgang wurde für die zufällig generierten Netzwerke mit einer Knotenzahl von 10, 20 50, 100, 200, 500 und 1000 je zehnmal repetiert. -Die Anzahl der Knoten im abgesuchten Netzwerk wirkt sich direkt auf die Rechenzeit aus. Der \emph{Dijkstra}-Algorithmus weist eine Zeitkomplexität von $\mathcal{O}(E\log{}V)$ auf, wobei $E$ die Anzahl Kanten (engl. \emph{edges}) und $V$ die Anzahl Knoten (engl. \emph{vertices}) darstellt. -Für den \emph{A*}-Algorithmus ist die Zeitkomplexität einerseits abhängig von der verwendeten Heuristik, andererseits aber auch vom vorliegenden Netzwerk selbst. Aus diesem Grund lässt sich keine defintive Angabe zu $\mathcal{O}$ machen. - -Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- und Zielknoten bei der ersten Versuchsreihe im Netzwerk diametral gegenüber liegen. Dadurch gehen viele Knoten verloren, welcher \emph{Dijkstra} als uninformierter Suchalgorithmus absuchen würde. In der zweiten Veruschsreihe werden hingegen Start- un Zielpunkt zufällig im Netzwerk ausgewählt. Es wird deshalb erwwartet, dass die Unterschiede in der Rechenzeit der beiden Algorithmen in der zweiten Versuchsreihe deutlich ausgeprägter sind. - -\subsection{Einfluss der Knotenzahl auf die Rechenzeit} -\label{verkehr:Knotenzahl} - -\begin{wrapfigure}{} -\includegraphics[width=12cm]{figures/chart_Vr1.png} - -\caption{Gemessene Rechenzeiten der ersten Versuchsreihe in Abhängigkeit der Knotenzahl.} -\label{verkehr:Vr1} -\end{wrapfigure} - -In \ref{verkehr:Vr1} ist ersichtlich, dass der Unterschied in der Rechenzeit zwischen \emph{Dijkstra} und \emph{A*} erst aber einer Knotenzahl von ca. $n=500$ merklich ansteigt. Dieses etwas überraschende Resultat ist darauf zurückzuführen, dass bei steigender Knotenzahl die Abweichung des effektiven kürzesten Pfades von der Distanz der Luftlinie abnimmt. -Die Effektivität von \emph{A*} mit euklidischer Heuristik ist wiederum grösser, wenn die Abweichung des kürzesten Pfads von der Luftlinie minimal ist. -Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei die relative Abweichung erstaunlicherweise bei einer Knotenzahl von $n=100$ maximal ist und nach $n=500$ nur noch marginal abnimmt. - -\begin{wrapfigure}{} -\includegraphics[width=12cm]{figures/chart_pathDiff.png} - -\caption{Relative Abweichung des kürzesten Pfads von der Luftlinie.} -\label{verkehr:pathDifference} -\end{wrapfigure} - - -\subsection{Einfluss der Position der Start- und Zielknoten auf die Rechenzeit} - -\begin{wrapfigure}{} -\includegraphics[width=12cm]{figures/chart_Vr2.png}\\ -\caption{Gemessene Rechenzeiten der zweiten Versuchsreihe in Abhängigkeit der Knotenzahl.} -\label{verkehr:Vr2} -\end{wrapfigure} - -Zum Vergleich der Resultate in \ref{verkehr:Knotenzahl} zeigt \ref{verkehr:Vr2} die Rechenzeiten der zweiten Versuchsreihe, in welcher die Start- und Zielknoten zufällig im Netzwerk ausgewählt wurden. Einerseits ist eine reduzierte durchschnittliche Rechenzeit festzustellen, was schlicht daran liegt, dass die zufällige Wahl der Knoten dazu führt, dass diese tendenziell weniger weit auseinander liegen.\\ -Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwischen \emph{Dijkstra} und \emph{A*} deutlich früher abzeichnen. Dieses Phänomen lässt sich leicht durch die zielgerichtete Suche des \emph{A*}-Algorithmus erklären. - -\begin{wrapfigure}{} -\includegraphics[width=6cm]{figures/network_dij.png}\qquad -\includegraphics[width=6cm]{figures/network_aStar.png} -\caption{Suchpfad in grün mit \emph{Dijkstra} (links), und \emph{A*} (rechts). Besuchte Knoten sind in blau, resp. rot markiert.} -\label{verkehr:Comparison} -\end{wrapfigure} - -In \ref{verkehr:Comparison} ist ersichtlich, dass bei einem im Netzwerk liegenden Startknoten die zielgerichtete Suche von \emph{A*} deutlich ausgeprägter zum Zuge kommt, als wenn dieser am Rand des Netzwerks liegen würde. diff --git a/buch/papers/verkehr/teil2.tex b/buch/papers/verkehr/teil2.tex deleted file mode 100644 index 99a0d92..0000000 --- a/buch/papers/verkehr/teil2.tex +++ /dev/null @@ -1,8 +0,0 @@ -\section{Ausblick} -\subsection{Optimierungsprobleme bei Graphen} -Das Finden eines kürzesten Pfades, sprich die Minimierung der Summe der Kantengewichte, ist nur eines der Optimierungsprobleme, die sich im Bereich von Grafen aufstellen lassen. Verschiedene, ähnliche Problemstellungen lassen sich teilweise mit denselben Algorithmen lösen.\\ -Im Bereich vom Computernetzwerken könnte zum Beispiel die Minimierung der Knotenzahl zur Datenübbertragung von Interesse sein. Dabei lässt sich dieses Problem einfach dadurch lösen, dass dem \emph{Dijkstra}, oder dem \emph{A*}-Algorithmus anstelle der Graph-Matrix (mit Kantengewichten als Einträgen) die Adjazenz-Matrix als Argument übergeben wird. Der gefundene kürzeste Pfad enstpricht der Anzahl benutzter Kanten, bzw. der Anzahl besuchter Knoten. - -\subsection{Wahl der Heuristik} -Ein grundlegendes Problem bei der Anwendung des \emph{A*} oder ähnlicher informierter Suchalgorithmen ist die Wahl der Heurstik. Bei einem physischen Verkehrsnetz kann bspw. die euklidische Distanz problems ermittelt werde. Bei einem regionalen Netzwerk ist die Annahme eines orthogonalen X-Y-Koordinatenetzes absolut ausreichend. Dies gilt z.B. auch für das Vernessungsnetz der Schweiz\footnote{Die aktuelle Schweizer Referenzsystem LV95 benutzt ein E/N-Koordinatennetz, wobei aufgrund zunehmender Abweichung vom Referenzellipsoid bei grosser Entfernung vom Nullpunkt ein Korrekturfaktor für die Höhe angebracht werden muss.} Bei überregionalen Netzwerken (Beispiel: Flugverbindungen) ist hingegen eine Berechnung im dreidimensionalen Raum, oder vereinfacht als Projektion auf das Geoid notwendig. Anonsten ist der Ablauf bei der Ausführung des Algorithmus allerdings identisch.\\ -In nicht-physischen Netzwerken stellt sich jedoch eine zweite Problematik. Da eine physische Distanz entweder nicht ermittelt werden kann, oder aber nicht ausschlaggebend ist, sind andere Netzwerk-Eigenschaften zur Beurteilung beizuziehen. Die Zuverlässigkeit ist dabei aber in den meisten Fällen nicht vergleichbar hoch, wie bei der euklidischen Heuristik. Oftmals werden deshalb bei derartigen Problem auch Algorithmen angewendet, die eine deutlich optimierte Zeitkomplexität aufweisen, dafür aber nicht mit Sicherheit den effizienstesten Pfad finden. -- cgit v1.2.1 From f5ac886bcfec175d61bbcb9fef9dd56c394bbf03 Mon Sep 17 00:00:00 2001 From: Pascal Schmid <81317360+paschost@users.noreply.github.com> Date: Mon, 24 May 2021 20:59:00 +0200 Subject: adjusted input-section names --- buch/papers/verkehr/main.tex | 7 +++---- 1 file changed, 3 insertions(+), 4 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/verkehr/main.tex b/buch/papers/verkehr/main.tex index 332ee7e..01de182 100644 --- a/buch/papers/verkehr/main.tex +++ b/buch/papers/verkehr/main.tex @@ -27,10 +27,9 @@ Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. \end{itemize} -\input{papers/verkehr/teil0.tex} -\input{papers/verkehr/teil1.tex} -\input{papers/verkehr/teil2.tex} -\input{papers/verkehr/teil3.tex} +\input{papers/verkehr/section1.tex} +\input{papers/verkehr/section2.tex} +\input{papers/verkehr/section3.tex} \printbibliography[heading=subbibliography] \end{refsection} -- cgit v1.2.1 From 50995d1062d097f67afc8a11f9d7808539aa1e82 Mon Sep 17 00:00:00 2001 From: Pascal Schmid <81317360+paschost@users.noreply.github.com> Date: Mon, 24 May 2021 21:18:46 +0200 Subject: added chapter title, author --- buch/papers/verkehr/main.tex | 23 ++--------------------- 1 file changed, 2 insertions(+), 21 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/verkehr/main.tex b/buch/papers/verkehr/main.tex index 01de182..6348993 100644 --- a/buch/papers/verkehr/main.tex +++ b/buch/papers/verkehr/main.tex @@ -4,28 +4,9 @@ % (c) 2020 Hochschule Rapperswil % \chapter{Thema\label{chapter:verkehr}} -\lhead{Thema} +\lhead{Verkehrsfluss und Verkehrsnetze} \begin{refsection} -\chapterauthor{Hans Muster} - -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} +\chapterauthor{Pascal Andreas Schmid und Robine Luchsinger} \input{papers/verkehr/section1.tex} \input{papers/verkehr/section2.tex} -- cgit v1.2.1 From c639d259a74ab2fae3a85bc861e0bc5070800b13 Mon Sep 17 00:00:00 2001 From: Pascal Schmid <81317360+paschost@users.noreply.github.com> Date: Mon, 24 May 2021 21:24:56 +0200 Subject: adjusted figures in section 1 --- buch/papers/verkehr/section1.tex | 20 ++++++++++++-------- 1 file changed, 12 insertions(+), 8 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/verkehr/section1.tex b/buch/papers/verkehr/section1.tex index 78d9311..9e40553 100644 --- a/buch/papers/verkehr/section1.tex +++ b/buch/papers/verkehr/section1.tex @@ -11,41 +11,45 @@ Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- \subsection{Einfluss der Knotenzahl auf die Rechenzeit} \label{verkehr:Knotenzahl} -\begin{wrapfigure}{} +\begin{figure} +\centering \includegraphics[width=12cm]{figures/chart_Vr1.png} \caption{Gemessene Rechenzeiten der ersten Versuchsreihe in Abhängigkeit der Knotenzahl.} \label{verkehr:Vr1} -\end{wrapfigure} +\end{figure} In \ref{verkehr:Vr1} ist ersichtlich, dass der Unterschied in der Rechenzeit zwischen \emph{Dijkstra} und \emph{A*} erst aber einer Knotenzahl von ca. $n=500$ merklich ansteigt. Dieses etwas überraschende Resultat ist darauf zurückzuführen, dass bei steigender Knotenzahl die Abweichung des effektiven kürzesten Pfades von der Distanz der Luftlinie abnimmt. Die Effektivität von \emph{A*} mit euklidischer Heuristik ist wiederum grösser, wenn die Abweichung des kürzesten Pfads von der Luftlinie minimal ist. Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei die relative Abweichung erstaunlicherweise bei einer Knotenzahl von $n=100$ maximal ist und nach $n=500$ nur noch marginal abnimmt. -\begin{wrapfigure}{} +\begin{figure} +\centering \includegraphics[width=12cm]{figures/chart_pathDiff.png} \caption{Relative Abweichung des kürzesten Pfads von der Luftlinie.} \label{verkehr:pathDifference} -\end{wrapfigure} +\end{figure} \subsection{Einfluss der Position der Start- und Zielknoten auf die Rechenzeit} -\begin{wrapfigure}{} +\begin{figure} +\centering \includegraphics[width=12cm]{figures/chart_Vr2.png}\\ \caption{Gemessene Rechenzeiten der zweiten Versuchsreihe in Abhängigkeit der Knotenzahl.} \label{verkehr:Vr2} -\end{wrapfigure} +\end{figure} Zum Vergleich der Resultate in \ref{verkehr:Knotenzahl} zeigt \ref{verkehr:Vr2} die Rechenzeiten der zweiten Versuchsreihe, in welcher die Start- und Zielknoten zufällig im Netzwerk ausgewählt wurden. Einerseits ist eine reduzierte durchschnittliche Rechenzeit festzustellen, was schlicht daran liegt, dass die zufällige Wahl der Knoten dazu führt, dass diese tendenziell weniger weit auseinander liegen.\\ Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwischen \emph{Dijkstra} und \emph{A*} deutlich früher abzeichnen. Dieses Phänomen lässt sich leicht durch die zielgerichtete Suche des \emph{A*}-Algorithmus erklären. -\begin{wrapfigure}{} +\begin{figure} +\centering \includegraphics[width=6cm]{figures/network_dij.png}\qquad \includegraphics[width=6cm]{figures/network_aStar.png} \caption{Suchpfad in grün mit \emph{Dijkstra} (links), und \emph{A*} (rechts). Besuchte Knoten sind in blau, resp. rot markiert.} \label{verkehr:Comparison} -\end{wrapfigure} +\end{figure} In \ref{verkehr:Comparison} ist ersichtlich, dass bei einem im Netzwerk liegenden Startknoten die zielgerichtete Suche von \emph{A*} deutlich ausgeprägter zum Zuge kommt, als wenn dieser am Rand des Netzwerks liegen würde. -- cgit v1.2.1 From b5bdeb425d4e8d509ba4d786fab3b167ff48d767 Mon Sep 17 00:00:00 2001 From: Pascal Schmid <81317360+paschost@users.noreply.github.com> Date: Mon, 24 May 2021 21:25:54 +0200 Subject: adjusted figures in section 2 --- buch/papers/verkehr/section2.tex | 20 ++++++++++++-------- 1 file changed, 12 insertions(+), 8 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/verkehr/section2.tex b/buch/papers/verkehr/section2.tex index 78d9311..9e40553 100644 --- a/buch/papers/verkehr/section2.tex +++ b/buch/papers/verkehr/section2.tex @@ -11,41 +11,45 @@ Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- \subsection{Einfluss der Knotenzahl auf die Rechenzeit} \label{verkehr:Knotenzahl} -\begin{wrapfigure}{} +\begin{figure} +\centering \includegraphics[width=12cm]{figures/chart_Vr1.png} \caption{Gemessene Rechenzeiten der ersten Versuchsreihe in Abhängigkeit der Knotenzahl.} \label{verkehr:Vr1} -\end{wrapfigure} +\end{figure} In \ref{verkehr:Vr1} ist ersichtlich, dass der Unterschied in der Rechenzeit zwischen \emph{Dijkstra} und \emph{A*} erst aber einer Knotenzahl von ca. $n=500$ merklich ansteigt. Dieses etwas überraschende Resultat ist darauf zurückzuführen, dass bei steigender Knotenzahl die Abweichung des effektiven kürzesten Pfades von der Distanz der Luftlinie abnimmt. Die Effektivität von \emph{A*} mit euklidischer Heuristik ist wiederum grösser, wenn die Abweichung des kürzesten Pfads von der Luftlinie minimal ist. Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei die relative Abweichung erstaunlicherweise bei einer Knotenzahl von $n=100$ maximal ist und nach $n=500$ nur noch marginal abnimmt. -\begin{wrapfigure}{} +\begin{figure} +\centering \includegraphics[width=12cm]{figures/chart_pathDiff.png} \caption{Relative Abweichung des kürzesten Pfads von der Luftlinie.} \label{verkehr:pathDifference} -\end{wrapfigure} +\end{figure} \subsection{Einfluss der Position der Start- und Zielknoten auf die Rechenzeit} -\begin{wrapfigure}{} +\begin{figure} +\centering \includegraphics[width=12cm]{figures/chart_Vr2.png}\\ \caption{Gemessene Rechenzeiten der zweiten Versuchsreihe in Abhängigkeit der Knotenzahl.} \label{verkehr:Vr2} -\end{wrapfigure} +\end{figure} Zum Vergleich der Resultate in \ref{verkehr:Knotenzahl} zeigt \ref{verkehr:Vr2} die Rechenzeiten der zweiten Versuchsreihe, in welcher die Start- und Zielknoten zufällig im Netzwerk ausgewählt wurden. Einerseits ist eine reduzierte durchschnittliche Rechenzeit festzustellen, was schlicht daran liegt, dass die zufällige Wahl der Knoten dazu führt, dass diese tendenziell weniger weit auseinander liegen.\\ Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwischen \emph{Dijkstra} und \emph{A*} deutlich früher abzeichnen. Dieses Phänomen lässt sich leicht durch die zielgerichtete Suche des \emph{A*}-Algorithmus erklären. -\begin{wrapfigure}{} +\begin{figure} +\centering \includegraphics[width=6cm]{figures/network_dij.png}\qquad \includegraphics[width=6cm]{figures/network_aStar.png} \caption{Suchpfad in grün mit \emph{Dijkstra} (links), und \emph{A*} (rechts). Besuchte Knoten sind in blau, resp. rot markiert.} \label{verkehr:Comparison} -\end{wrapfigure} +\end{figure} In \ref{verkehr:Comparison} ist ersichtlich, dass bei einem im Netzwerk liegenden Startknoten die zielgerichtete Suche von \emph{A*} deutlich ausgeprägter zum Zuge kommt, als wenn dieser am Rand des Netzwerks liegen würde. -- cgit v1.2.1 From 40db89ca6efb1a6b962fe237c5641c939b18dde6 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Tue, 25 May 2021 00:47:51 +0200 Subject: Add book reference, fix typos --- buch/papers/punktgruppen/symmetry.tex | 34 +++++++++++++++++++--------------- 1 file changed, 19 insertions(+), 15 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/punktgruppen/symmetry.tex b/buch/papers/punktgruppen/symmetry.tex index d3ccb4e..23b1411 100644 --- a/buch/papers/punktgruppen/symmetry.tex +++ b/buch/papers/punktgruppen/symmetry.tex @@ -2,8 +2,8 @@ Das Wort Symmetrie ist sehr alt und hat sich seltsamerweise von seinem ursprünglichen griechischen Wort \(\mathrm{\sigma\nu\mu\mu\varepsilon\tau\rho\iota\alpha}\) -\footnote{\emph{Simmetr\'ia}: ``ein gemeinsames Mass habend, gleichmässig, -verhältnismässig''} fast nicht verändert. In der Alltagssprache mag es ein +\footnote{\emph{Simmetr\'ia}: ein gemeinsames Mass habend, gleichmässig, +verhältnismässig} fast nicht verändert. In der Alltagssprache mag es ein locker definierter Begriff sein, aber in der Mathematik hat Symmetrie eine sehr präzise Bedeutung. \begin{definition}[Symmetrie] @@ -109,7 +109,7 @@ der Rotation, erhält man die so genannte Diedergruppe D_n = \langle r, \sigma : r^{n-1} = \sigma^2 = (\sigma r)^2 = \mathds{1} \rangle = \left\{ \mathds{1}, r, \ldots, r^{n-1}, \sigma, \sigma r, \ldots, \sigma r^{n-1} - \right\}. + \right\}. \] Diesmal muss die Generator-Notation die Beziehungen zwischen den beiden Operationen beinhalten. Die ersten beiden sind leicht zu erkennen, für die @@ -121,10 +121,12 @@ Frage ist dann, könnte es sein, dass wir bereits etwas haben, das dasselbe tut? Natürlich, ja. Dafür führen wir den Begriff der Darstellung ein. \begin{definition}[Darstellung einer Gruppe, Gruppenhomomorphismus] Seien \(G\) und \(H\) Gruppe mit unterschiedlicher Operation \(\diamond\) - bzw. \(\star\). Ein Homomorphismus ist eine Funktion \(f: G \to H\), so dass - für jedes \(a, b \in G\) gilt \(f(a\diamond b) = f(a) \star f(b)\). Man - sagt, dass der Homomorphismus \(f\) \(G\) in \(H\) transformiert, oder dass - \(H\) eine Darstellung von \(G\) ist. + bzw. \(\star\). Ein Homomorphismus\footnote{ Für eine ausführlichere + Diskussion siehe \S\ref{buch:grundlagen:subsection:gruppen} im Buch.} ist + eine Funktion \(f: G \to H\), so dass für jedes \(a, b \in G\) gilt + \(f(a\diamond b) = f(a) \star f(b)\). Man sagt, dass der Homomorphismus + \(f\) \(G\) in \(H\) transformiert, oder dass \(H\) eine Darstellung von + \(G\) ist. \end{definition} \begin{beispiel} Die Elemente \(r^k \in C_n\), wobei \(0 < k < n\), stellen abstrakt eine @@ -137,8 +139,8 @@ Natürlich, ja. Dafür führen wir den Begriff der Darstellung ein. \] definierte Funktion von \(C_n\) nach \(O(2)\) ist eine Darstellung von \(C_n\). In diesem Fall ist die erste Gruppenoperation die Komposition und - die zweite die Matrixmultiplikation. Man kann zwar überprüfen, dass - \(\Phi(r^2 \circ r) = \Phi(r^2)\Phi(r)\). + die zweite die Matrixmultiplikation. Man kann überprüfen, dass \(\Phi(r^2 + \circ r) = \Phi(r^2)\Phi(r)\). \end{beispiel} \begin{beispiel} Die Rotationssymmetrie des Kreises \(C_\infty\), mit einem unendlichen @@ -153,9 +155,10 @@ der Spiegelung die Achse. Dies ist jedoch keine Voraussetzung für eine Symmetrie, da es Symmetrien gibt, die jeden Punkt zu einem anderen Punkt verschieben können. Ein aufmerksamer Leser wird bemerken, dass die unveränderten Punkte zum Eigenraum\footnote{Zur Erinnerung \(E_\lambda = -\mathrm{null}(\Phi - \lambda I)\)} der Matrixdarstellung der Symmetrieoperation -gehören. Diesen Spezialfall, bei dem mindestens ein Punkt unverändert bleibt, -nennt man Punktsymmetrie. +\mathrm{null}(\Phi - \lambda I)\), \(\vec{v}\in E_\lambda \implies \Phi \vec{v} += \lambda\vec{v}\)} der Matrixdarstellung der Symmetrieoperation gehören. +Diesen Spezialfall, bei dem mindestens ein Punkt unverändert bleibt, nennt man +Punktsymmetrie. \begin{definition}[Punktgruppe] Wenn jede Operation in einer Symmetriegruppe die Eigenschaft hat, mindestens einen Punkt unverändert zu lassen, sagt man, dass die Symmetriegruppe eine @@ -164,15 +167,16 @@ nennt man Punktsymmetrie. Um das Konzept zu illustrieren, werden wir den umgekehrten Fall diskutieren: eine Symmetrie, die keine Punktsymmetrie ist, die aber in der Physik sehr nützlich ist, nämlich die Translationssymmetrie. Von einem mathematischen -Objekt \(x\) wird gesagt, dass es eine Translationssymmetrie \(Q\) hat, wenn es -die Gleichung +Objekt \(U\) wird gesagt, dass es eine Translationssymmetrie \(Q(x) = x + a\) +hat, wenn es die Gleichung \[ - Q(x) = Q(x + a), + U(x) = U(Q(x)) = U(x + a), \] für ein gewisses \(a\), erfüllt. Zum Beispiel besagt das erste Newtonsche Gesetz, dass ein Objekt, auf das keine Kraft einwirkt, eine zeitranslationsinvariante Geschwindigkeit hat, d.h. wenn \(\vec{F} = \vec{0}\) dann \(\vec{v}(t) = \vec{v}(t + \tau)\). +% \subsection{Sch\"onflies notation} % vim:ts=2 sw=2 spell spelllang=de: -- cgit v1.2.1 From 74e9f65ccf48e09553690d5b9560e1b9c5be84f1 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 25 May 2021 07:45:53 +0200 Subject: fix image paths --- buch/papers/verkehr/Makefile.inc | 12 +++++------- buch/papers/verkehr/section1.tex | 10 +++++----- buch/papers/verkehr/section2.tex | 10 +++++----- 3 files changed, 15 insertions(+), 17 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/verkehr/Makefile.inc b/buch/papers/verkehr/Makefile.inc index 7bd8de1..876d0df 100644 --- a/buch/papers/verkehr/Makefile.inc +++ b/buch/papers/verkehr/Makefile.inc @@ -3,12 +3,10 @@ # # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -dependencies-verkehr = \ +dependencies-verkehr = \ papers/verkehr/packages.tex \ - papers/verkehr/main.tex \ - papers/verkehr/references.bib \ - papers/verkehr/teil0.tex \ - papers/verkehr/teil1.tex \ - papers/verkehr/teil2.tex \ - papers/verkehr/teil3.tex + papers/verkehr/main.tex \ + papers/verkehr/section1.tex \ + papers/verkehr/section2.tex \ + papers/verkehr/references.bib diff --git a/buch/papers/verkehr/section1.tex b/buch/papers/verkehr/section1.tex index 9e40553..638d9dd 100644 --- a/buch/papers/verkehr/section1.tex +++ b/buch/papers/verkehr/section1.tex @@ -13,7 +13,7 @@ Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- \begin{figure} \centering -\includegraphics[width=12cm]{figures/chart_Vr1.png} +\includegraphics[width=12cm]{papers/verkehr/figures/chart_Vr1.png} \caption{Gemessene Rechenzeiten der ersten Versuchsreihe in Abhängigkeit der Knotenzahl.} \label{verkehr:Vr1} @@ -25,7 +25,7 @@ Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei di \begin{figure} \centering -\includegraphics[width=12cm]{figures/chart_pathDiff.png} +\includegraphics[width=12cm]{papers/verkehr/figures/chart_pathDiff.png} \caption{Relative Abweichung des kürzesten Pfads von der Luftlinie.} \label{verkehr:pathDifference} @@ -36,7 +36,7 @@ Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei di \begin{figure} \centering -\includegraphics[width=12cm]{figures/chart_Vr2.png}\\ +\includegraphics[width=12cm]{papers/verkehr/figures/chart_Vr2.png}\\ \caption{Gemessene Rechenzeiten der zweiten Versuchsreihe in Abhängigkeit der Knotenzahl.} \label{verkehr:Vr2} \end{figure} @@ -46,8 +46,8 @@ Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwis \begin{figure} \centering -\includegraphics[width=6cm]{figures/network_dij.png}\qquad -\includegraphics[width=6cm]{figures/network_aStar.png} +\includegraphics[width=6cm]{papers/verkehr/figures/network_dij.png}\qquad +\includegraphics[width=6cm]{papers/verkehr/figures/network_aStar.png} \caption{Suchpfad in grün mit \emph{Dijkstra} (links), und \emph{A*} (rechts). Besuchte Knoten sind in blau, resp. rot markiert.} \label{verkehr:Comparison} \end{figure} diff --git a/buch/papers/verkehr/section2.tex b/buch/papers/verkehr/section2.tex index 9e40553..638d9dd 100644 --- a/buch/papers/verkehr/section2.tex +++ b/buch/papers/verkehr/section2.tex @@ -13,7 +13,7 @@ Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- \begin{figure} \centering -\includegraphics[width=12cm]{figures/chart_Vr1.png} +\includegraphics[width=12cm]{papers/verkehr/figures/chart_Vr1.png} \caption{Gemessene Rechenzeiten der ersten Versuchsreihe in Abhängigkeit der Knotenzahl.} \label{verkehr:Vr1} @@ -25,7 +25,7 @@ Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei di \begin{figure} \centering -\includegraphics[width=12cm]{figures/chart_pathDiff.png} +\includegraphics[width=12cm]{papers/verkehr/figures/chart_pathDiff.png} \caption{Relative Abweichung des kürzesten Pfads von der Luftlinie.} \label{verkehr:pathDifference} @@ -36,7 +36,7 @@ Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei di \begin{figure} \centering -\includegraphics[width=12cm]{figures/chart_Vr2.png}\\ +\includegraphics[width=12cm]{papers/verkehr/figures/chart_Vr2.png}\\ \caption{Gemessene Rechenzeiten der zweiten Versuchsreihe in Abhängigkeit der Knotenzahl.} \label{verkehr:Vr2} \end{figure} @@ -46,8 +46,8 @@ Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwis \begin{figure} \centering -\includegraphics[width=6cm]{figures/network_dij.png}\qquad -\includegraphics[width=6cm]{figures/network_aStar.png} +\includegraphics[width=6cm]{papers/verkehr/figures/network_dij.png}\qquad +\includegraphics[width=6cm]{papers/verkehr/figures/network_aStar.png} \caption{Suchpfad in grün mit \emph{Dijkstra} (links), und \emph{A*} (rechts). Besuchte Knoten sind in blau, resp. rot markiert.} \label{verkehr:Comparison} \end{figure} -- cgit v1.2.1 From 337c10d8861718c88b1c8e4d365a4dd7d678153a Mon Sep 17 00:00:00 2001 From: Alain Date: Wed, 26 May 2021 12:08:46 +0200 Subject: initeur --- buch/papers/ifs/main.log | 6045 +++++++++++++++++++++++++++++++++++++++++++++ buch/papers/ifs/main.tex | 6 +- buch/papers/ifs/teil0.tex | 18 +- buch/papers/ifs/teil1.tex | 11 +- 4 files changed, 6056 insertions(+), 24 deletions(-) create mode 100644 buch/papers/ifs/main.log (limited to 'buch/papers') diff --git a/buch/papers/ifs/main.log b/buch/papers/ifs/main.log new file mode 100644 index 0000000..b818dc7 --- /dev/null +++ b/buch/papers/ifs/main.log @@ -0,0 +1,6045 @@ +This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/W32TeX) (preloaded format=pdflatex 2019.9.25) 27 MAR 2021 11:43 +entering extended mode + restricted \write18 enabled. + %&-line parsing enabled. +**main.tex +(./main.tex +LaTeX2e <2018-12-01> +! 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+Missing character: There is no i in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 21--26 +\/cmr/m/n/10 ^^?o + [] + + +Overfull \hbox (7.50002pt too wide) in paragraph at lines 21--26 +[]\/cmr/m/n/10 A + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 21--26 +\/cmr/m/n/10 ^^?a + [] + +Missing character: There is no B in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no F in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no Z in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (5.55557pt too wide) in paragraph at lines 26--28 +\/cmr/m/n/10 ^^?u + [] + + +Overfull \hbox (7.50002pt too wide) in paragraph at lines 26--28 +[]\/cmr/m/n/10 U + [] + +(./teil0.tex +! Undefined control sequence. +l.6 \section + {Teil 0\label{ifs:section:teil0}} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no 0 in font nullfont! +! Undefined control sequence. +l.7 \rhead + {Teil 0} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no 0 in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no [ in font nullfont! +\citation{ifs:bibtex} +! Undefined control sequence. + ...: Citation `\@citeb ' on page \thepage + \space undefined\on@line . +l.10 erat, sed diam voluptua \cite{ifs:bibtex} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Citation `ifs:bibtex' on page undefined on input line 10. + +Missing character: There is no ] in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! 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+Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! 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+Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no y in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no j in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 15--21 +[] + [] + +) (./teil1.tex +! Undefined control sequence. +l.6 \section + {Teil 1 +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no 1 in font nullfont! +! Undefined control sequence. +l.8 \rhead + {Problemstellung} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no P in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 6--16 +[] + [] + +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <7> on input line 16. +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <5> on input line 16. + +Overfull \hbox (134.08229pt too wide) detected at line 23 +[][] \OML/cmm/m/it/10 x[] dx \/cmr/m/n/10 = [][] = []\OML/cmm/m/it/10 :[] + [] + +Missing character: There is no N in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no U in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no ? in font nullfont! +Missing character: There is no Q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no ? in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 28--33 +[] + [] + +! Undefined control sequence. +l.34 \subsection + {De finibus bonorum et malorum +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! 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Undefined control sequence. +l.40 animi, id est laborum et dolorum fuga \eqref + {000tempmlate:equation1}. +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). 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Undefined control sequence. + ...`ifs:section:loesung' on page \thepage + \space undefined\on@line . +l.43 \ref{ifs:section:loesung} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `ifs:section:loesung' on page undefined on input line + 43. + +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! 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Undefined control sequence. + ...fs:section:folgerung' on page \thepage + \space undefined\on@line . +l.47 \ref{ifs:section:folgerung} + . +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). 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+Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 42--54 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 42--54 +[] + [] + + +Overfull \hbox (10.86105pt too wide) in paragraph at lines 42--54 +[] + [] + +) (./teil2.tex +! Undefined control sequence. +l.6 \section + {Teil 2 +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no 2 in font nullfont! +! Undefined control sequence. +l.8 \rhead + {Teil 2} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no 2 in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no U in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no ? in font nullfont! +Missing character: There is no Q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no ? in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 6--23 +[] + [] + +! Undefined control sequence. +l.24 \subsection + {De finibus bonorum et malorum +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no I in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 24--39 +[] + [] + +) (./teil3.tex +! Undefined control sequence. +l.6 \section + {Teil 3 +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no 3 in font nullfont! +! Undefined control sequence. +l.8 \rhead + {Teil 3} +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no 3 in font nullfont! +Missing character: There is no S in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no U in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no ? in font nullfont! +Missing character: There is no Q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no ? in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 6--23 +[] + [] + +! Undefined control sequence. +l.24 \subsection + {De finibus bonorum et malorum +The control sequence at the end of the top line +of your error message was never \def'ed. If you have +misspelled it (e.g., `\hobx'), type `I' and the correct +spelling (e.g., `I\hbox'). Otherwise just continue, +and I'll forget about whatever was undefined. + +Missing character: There is no D in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no E in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no N in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no x in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no T in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no i in font nullfont! 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LaTeX Error: \begin{document} ended by \end{refsection}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H for immediate help. + ... + +l.36 \end{refsection} + +Your command was ignored. +Type I to replace it with another command, +or to continue without it. + +) +! Emergency stop. +<*> main.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 37 strings out of 492609 + 427 string characters out of 6132818 + 62613 words of memory out of 5000000 + 4043 multiletter control sequences out of 15000+600000 + 4116 words of font info for 16 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 20i,3n,22p,104b,180s stack positions out of 5000i,500n,10000p,200000b,80000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/buch/papers/ifs/main.tex b/buch/papers/ifs/main.tex index 8d70951..48c38f9 100644 --- a/buch/papers/ifs/main.tex +++ b/buch/papers/ifs/main.tex @@ -3,10 +3,10 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:ifs}} -\lhead{Thema} +\chapter{Iterierte Funktionsschemata\label{chapter:ifs}} +\lhead{Iterierte Funktionschemata und ihre Anwendungen} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{Alain Keller} Ein paar Hinweise für die korrekte Formatierung des Textes \begin{itemize} diff --git a/buch/papers/ifs/teil0.tex b/buch/papers/ifs/teil0.tex index b605bfe..7e3d344 100644 --- a/buch/papers/ifs/teil0.tex +++ b/buch/papers/ifs/teil0.tex @@ -4,19 +4,11 @@ % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % \section{Teil 0\label{ifs:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{ifs:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. +\rhead{Was ist ein Iteriertes Funktionsschema} +Mit der Hilfe von Iterierten Funktionsschemata mit nur wenigen Funktionen, komplexe Bilder beschreiben. +In der Regel sind diese Bilder Fraktale. +Wie es dazu kommt, und wie man mit IFS auch Bilder komprimieren kann, wollen wir im folgenden Kapitel untersuchen. -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. +\subsection{Metrische Räume} diff --git a/buch/papers/ifs/teil1.tex b/buch/papers/ifs/teil1.tex index c824cb4..76bc828 100644 --- a/buch/papers/ifs/teil1.tex +++ b/buch/papers/ifs/teil1.tex @@ -3,16 +3,11 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 1 +\section{Fraktale \label{ifs:section:teil1}} \rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt +Bevor wir die IFS genauer ansehen, schauen wir uns Fraktale genauer an. + \begin{equation} \int_a^b x^2\, dx = -- cgit v1.2.1 From 1ffa89d50139ed5061e06ddd3371c5af0d003bd3 Mon Sep 17 00:00:00 2001 From: tim30b Date: Wed, 26 May 2021 15:36:33 +0200 Subject: begin to write kristalls and intro --- buch/papers/punktgruppen/crystals.tex | 15 +++++++++++++++ buch/papers/punktgruppen/intro.tex | 9 +++++++++ buch/papers/punktgruppen/symmetry.tex | 2 +- 3 files changed, 25 insertions(+), 1 deletion(-) (limited to 'buch/papers') diff --git a/buch/papers/punktgruppen/crystals.tex b/buch/papers/punktgruppen/crystals.tex index b104901..6de2bca 100644 --- a/buch/papers/punktgruppen/crystals.tex +++ b/buch/papers/punktgruppen/crystals.tex @@ -1 +1,16 @@ \section{Kristalle} +Unter dem Begriff Kristall sollte sich jeder ein Bild machen können. +Wir werden uns aber nicht auf sein Äusseres fokussieren, sondern was ihn im Inneren ausmacht. +Die Innereien eines Kristalles sind glücklicherweise relativ einfach definiert. +\begin{definition}[Kristall] + Ein Kristall besteht aus Atomen, welche sich in einem Muster arrangieren, welches sich in drei Dimensionen periodisch wiederholt. +\end{definition} + + +Ein Zweidimensionales Beispiel eines solchen Muster ist Abbildung \ref{fig:punktgruppen:lattce-grid}. +Für die Überschaubarkeit haben wir ein simples Muster eines einzelnen XgrauenX Punktes gewählt in nur Zwei Dimensionen. +Die eingezeichneten Vektoren a und b sind die kleinstmöglichen Schritte im Raum bis sich das Kristallgitter wiederholt. +Dadurch können von einem einzelnen XGrauenX Gitterpunkt in \ref{fig:punktgruppen:lattce-grid} können mit einer ganzzahligen Linearkombination von a und b alle anderen Gitterpunkte des Kristalles erreicht werden. +Ein Kristallgitter kann eindeutig mit a und b und deren winkeln beschrieben werden weswegen a und b auch Gitterparameter genannt werden. +Im Dreidimensionalen-Raum können alle Gitterpunkte mit derselben Idee und einem zusätzlichen Vektor also FRMEL FÜR TRANSLATIONSVEKTOR erreicht werden. +Da sich das Ganze Kristallgitter wiederholt, wiederholen sich auch die Eigenschaften eines Gitterpunktes Periodisch mit eiem diff --git a/buch/papers/punktgruppen/intro.tex b/buch/papers/punktgruppen/intro.tex index 4a84465..10dea79 100644 --- a/buch/papers/punktgruppen/intro.tex +++ b/buch/papers/punktgruppen/intro.tex @@ -1 +1,10 @@ \section{Einleitung} +Es gibt viele möglichkeiten sich in Kristallen zu verlieren. +Auch wen man nur die Mathematischen möglichkeiten in betracht zieht, hat man noch viel zu viele Möglichkeiten sich mit kristallen zu beschäftigen. +In diesem Articel ist daher der Fokus "nur" auf die Symmetrie gelegt. +Im Abschitt über Symmetrien werden wir sehen, wie eine Symmetrie eines Objektes weit +2.ter versuch: +Die Kristallographie ist ein grosses Thema, Symmetrien auch. +Für beide bestehen schon bewährte Mathematische Modelle und Definitionen. +Die + diff --git a/buch/papers/punktgruppen/symmetry.tex b/buch/papers/punktgruppen/symmetry.tex index c0418aa..4b179b0 100644 --- a/buch/papers/punktgruppen/symmetry.tex +++ b/buch/papers/punktgruppen/symmetry.tex @@ -14,7 +14,7 @@ präzise Bedeutung. Wenn der Leser noch nicht mit der Gruppentheorie in Berührung gekommen ist, ist vielleicht nicht ganz klar, was eine Operation ist, aber die Definition sollte trotzdem Sinn machen. Die Formalisierung dieser Idee wird bald kommen, aber -zunächst wollen wir etwas Intuition aufbauen. +zunächst wollen wir eine Intuition aufbauen. \begin{figure}[h] \centering -- cgit v1.2.1 From a1a45cd5bd0e487cb69916f8c3e636a5e326c935 Mon Sep 17 00:00:00 2001 From: Alain Date: Wed, 26 May 2021 17:41:38 +0200 Subject: Fraktale Kapitel Fertig --- buch/papers/ifs/images/koch0.eps | 1004 ++ buch/papers/ifs/images/koch1.eps | 1073 ++ buch/papers/ifs/images/koch2.eps | 1085 ++ buch/papers/ifs/images/koch8.eps | 26780 +++++++++++++++++++++++++++++++++++++ buch/papers/ifs/teil0.tex | 2 - buch/papers/ifs/teil1.tex | 116 +- 6 files changed, 30023 insertions(+), 37 deletions(-) create mode 100644 buch/papers/ifs/images/koch0.eps create mode 100644 buch/papers/ifs/images/koch1.eps create mode 100644 buch/papers/ifs/images/koch2.eps create mode 100644 buch/papers/ifs/images/koch8.eps (limited to 'buch/papers') diff --git a/buch/papers/ifs/images/koch0.eps b/buch/papers/ifs/images/koch0.eps new file mode 100644 index 0000000..16e0d6b --- /dev/null +++ b/buch/papers/ifs/images/koch0.eps @@ -0,0 +1,1004 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: (MATLAB, The Mathworks, Inc. 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/equal /greater /question /at +/A /B /C /D /E +/F /G /H /I /J +/K /L /M /N /O +/P /Q /R /S /T +/U /V /W /X /Y +/Z /bracketleft /backslash /bracketright /asciicircum +/underscore /quoteleft /a /b /c +/d /e /f /g /h +/i /j /k /l /m +/n /o /p /q /r +/s /t /u /v /w +/x /y /z /braceleft /bar +/braceright /asciitilde /bullet /Euro /bullet +/quotesinglbase /florin /quotedblbase /ellipsis /dagger +/daggerdbl /circumflex /perthousand /Scaron /guilsinglleft +/OE /bullet /Zcaron /bullet /bullet +/quoteleft /quoteright /quotedblleft /quotedblright /bullet +/endash /emdash /asciitilde /trademark /scaron +/guilsinglright /oe /bullet /zcaron /Ydieresis +/space /exclamdown /cent /sterling /currency +/yen /brokenbar /section /dieresis /copyright +/ordfeminine /guillemotleft /logicalnot /sfthyphen /registered +/macron /degree /plusminus /twosuperior /threesuperior +/acute /mu /paragraph /middot /cedilla +/onesuperior /ordmasculine /guillemotright /onequarter /onehalf +/threequarters /questiondown /Agrave /Aacute /Acircumflex +/Atilde /Adieresis /Aring /AE /Ccedilla +/Egrave /Eacute /Ecircumflex /Edieresis /Igrave +/Iacute /Icircumflex /Idieresis /Eth /Ntilde +/Ograve /Oacute /Ocircumflex /Otilde /Odieresis +/multiply /Oslash /Ugrave /Uacute /Ucircumflex +/Udieresis /Yacute /Thorn /germandbls /agrave +/aacute /acircumflex /atilde /adieresis /aring +/ae /ccedilla /egrave /eacute /ecircumflex +/edieresis /igrave /iacute /icircumflex /idieresis +/eth /ntilde /ograve /oacute /ocircumflex +/otilde /odieresis /divide /oslash /ugrave +/uacute /ucircumflex /udieresis /yacute /thorn +/ydieresis +] def +%%EndResource +%FOPBeginFontReencode +/Courier-Oblique findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Courier-Oblique exch definefont pop +/Courier-BoldOblique findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Courier-BoldOblique exch definefont pop +/Courier-Bold findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Courier-Bold exch definefont pop +/Helvetica findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Helvetica exch definefont pop +/Helvetica-Oblique findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Helvetica-Oblique exch definefont pop +/Helvetica-Bold findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Helvetica-Bold exch definefont pop +/Helvetica-BoldOblique findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Helvetica-BoldOblique exch definefont pop +/Times-Roman 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Wie es dazu kommt, und wie man mit IFS auch Bilder komprimieren kann, wollen wir im folgenden Kapitel untersuchen. -\subsection{Metrische Räume} - diff --git a/buch/papers/ifs/teil1.tex b/buch/papers/ifs/teil1.tex index 76bc828..327a082 100644 --- a/buch/papers/ifs/teil1.tex +++ b/buch/papers/ifs/teil1.tex @@ -8,43 +8,89 @@ \rhead{Problemstellung} Bevor wir die IFS genauer ansehen, schauen wir uns Fraktale genauer an. -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{ifs:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. +\subsection{Was sind Fraktale? +\label{ifs:subsection:finibus}} +Über die genaue Definition von Fraktalen sind sich die Mathematiker noch nicht einig. +In diesem Kapitel orientieren wir uns an den Eigneschaften welche Kenneth Flaconer in seinem Buch Fractal Geometry beschreibt. +Von einem Fraktal $F$ können wir folgende Eigneschaften erwarten: +\begin{enumerate} + \item $F$ hat eine unendlich feine Struktur + \item $F$ kann nicht mit der klassischen Geometrie beschrieben werden. + \item Oftmals haf $F$ eine Form von Selbstähnlichkeit. + \item Die 'fraktale Dimension' ist grösser als die Topologische Dimension + \item Viele Fraktale lassen sich einfach beschrieben +\end{enumerate} +\subsection{Koch Kurve + \label{ifs:subsection:lilkoch}} +Diese Eigenschaften möchten wir nun anhand der Koch Kurve näher anschauen. +In \ref{ifs:kochkurve8} sehen wir die Koch Kurve. Wie man schon erahnen kann, besteht die aus lauter kleineren Kopien von sich selber. +Den Konstruktionvorgang sehen wir in \ref{ifs:kochconst}. +Gestartet wird mit einer einzelnen Strecke der Länge $a$. +Diese wird in ersten Schritt mit vier gleich langen Streckenabschnitte der Länge $\frac{a}{3}$ ersetzt. +In \ref{ifs:kochconstb} ist die Anordnung dieser vier Streckenabschnitte ersichtilich. +Dieser Schritt wird nun für jeden der resultierten Streckenabschnitten wiederholt. +Die Kurve besteht also aus vier kleineren Kopien von der ganzen Kurve, was auch unter Selbstähnlichkeit bekannt ist. -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? -\subsection{De finibus bonorum et malorum -\label{ifs:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. +\begin{figure} + \label{ifs:kochkurve8} + \centering + \includegraphics{papers/ifs/images/koch8} + \caption{Koch Kurve} +\end{figure} -Et harum quidem rerum facilis est et expedita distinctio -\ref{ifs:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{ifs:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. +\begin{figure} + \label{ifs:kochconst} + \centering + \subfigure[]{ + \label{ifs:kochconsta} + \includegraphics[width=0.32\textwidth]{papers/ifs/images/koch0}} + \subfigure[]{ + \label{ifs:kochconstb} + \includegraphics[width=0.32\textwidth]{papers/ifs/images/koch1}} + \subfigure[]{ + \label{kochconstc} + \includegraphics[width=0.32\textwidth]{papers/ifs/images/koch2}} + \caption{(a) Start (b) 1. Iteration (c) 2. Iteration} + \label{fig:foobar} +\end{figure} +Die resultierende Kurve hat ein paar interessante Eigenschaften. +Die Länge der Kurve lasst sich einfach berechnen. +\begin{align*} + l_0 = a ,\quad l_1 = a \frac{4}{3} ,\quad l_2 = a \left( \frac{4}{3}\right)^2 , \quad ... , \quad + l_n = a * \left( \frac{4}{3}\right)^n \quad + \Rightarrow \quad + \lim_{n\to\infty} a \left( \frac{4}{3}\right)^n = \infty +\end{align*} +In jedem Schritt wird die Länge um den Faktor $\frac{4}{3}$ verglängert. Somit divergiert die Länge gegen Unendlich. +Die Fläche unter der Kurve lässt sich folgendermassen berechnen +\begin{align*} + A_0 = 0 , \quad A_1 = \left( \frac{a}{3}\right)^2 \frac{\sqrt{3}}{4} = a^2 \frac{\sqrt{3}}{36}\\ + A_2 = A_1 + 4\left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 \\ + A_3 = A_1 + A_2 + 4^2 \left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 + \left( \frac{4}{9}\right)^2 A_1 +\end{align*} +Wir sehen, dass mit jedem Schritt die neu dazugekommene Fläche um $\frac{4}{9}$ kleiner ist. +Daraus resultiert eine konvergierende Geometrische Rheie. +\begin{align*} + A_n = A_1 \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n = a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n \\ + \lim_{n\to\infty} a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n = \frac{\sqrt{3}}{20} a^2 +\end{align*} +Wie wir sehen ist die Kochkurve ein Konstrukt mit endlicher Fläche, aber unendlichem Umfang. +Zu guter letzt bestimmen wir die Dimension der Kurve. +Es gibt viele verschidene Arten die Dimension zu definieren. Diese können dann auch unterschiedliche Resultate liefern. +Vor allem im Zusammenhang mit Fraktalen findet man in der Literatur viele verschiedene Arten. +In diesem Beispiel werden wir die Ähnlichkeits-Dimension. +\begin{align*} + D = - \frac{log(N)}{log(\epsilon)} +\end{align*} +Mit ihr kann man einfach die Dimension selbstähnlicher Mengen bestimmen. +Als Beispiel nehmen wir ein gleichseitiges Dreieck. Dieses besteht aus $N = 4$ Kopien mit halber ($\epsilon = 1/2$) Kantenlänge. +Somit hat das Dreieck die Dimension $D = 2$. +Die Koch Kurve besteht aus $N = 4$ Kopien mit Kantenlänge $\epsilon = 1/3$. +\begin{align*} + D = - \frac{log(N)}{log(\epsilon)} = - \frac{log(4)}{log(1/3)} \approx 1.2619 +\end{align*} +Wie wir nun sehen besitzt die Kochkurve alle oben beschriebenen Eigenschaften von Fraktalen. +Dies muss jedoch nicht bei allen Fraktalen der Fall. Sonst wäre die Frage nach einer 'richtigen' Definition einfach zu beantworten. -- cgit v1.2.1 From cc0321fcd452279007db89d9d01bf5c970d7fca8 Mon Sep 17 00:00:00 2001 From: Pascal Schmid <81317360+paschost@users.noreply.github.com> Date: Fri, 28 May 2021 11:14:22 +0200 Subject: fixed first section content MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit section was the same as section 2 "Versuchsreihe" instead of "Einführung", fixed that. --- buch/papers/verkehr/section1.tex | 99 +++++++++++++++++++++++----------------- 1 file changed, 58 insertions(+), 41 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/verkehr/section1.tex b/buch/papers/verkehr/section1.tex index 9e40553..1f7c20e 100644 --- a/buch/papers/verkehr/section1.tex +++ b/buch/papers/verkehr/section1.tex @@ -1,55 +1,72 @@ -\section{Versuchsreihe} -\label{section:verkehr/versuchsreihe} +\section{Einführung} +\label{section:verkehr/einfuehrung} -Um zwei der vorgestellten Suchalgorithmen zu vergleichen, wurden zwei Versuchsreihen erstellt. Dazu wurden in einem ersten Schritt zufällige Netzwerke generiert und anschliessend der \emph{Dijkstra}-, sowie der \emph{$A^*$}-Algorithmus auf das Netzwerk angewandt. -Dieser Vorgang wurde für die zufällig generierten Netzwerke mit einer Knotenzahl von 10, 20 50, 100, 200, 500 und 1000 je zehnmal repetiert. -Die Anzahl der Knoten im abgesuchten Netzwerk wirkt sich direkt auf die Rechenzeit aus. Der \emph{Dijkstra}-Algorithmus weist eine Zeitkomplexität von $\mathcal{O}(E\log{}V)$ auf, wobei $E$ die Anzahl Kanten (engl. \emph{edges}) und $V$ die Anzahl Knoten (engl. \emph{vertices}) darstellt. -Für den \emph{A*}-Algorithmus ist die Zeitkomplexität einerseits abhängig von der verwendeten Heuristik, andererseits aber auch vom vorliegenden Netzwerk selbst. Aus diesem Grund lässt sich keine defintive Angabe zu $\mathcal{O}$ machen. +\subsection{Verkehrsnetze} +Das Verkehrsnetz besteht aus allen Anlagen, auf oder unter der Erdoberfläche, auf denen eine räumliche Fortbewegung von Personen oder auch Gütern stattfindet. Verkehrsnetze sind ein Bestandteil der Verkehrsinfrastruktur, die auf topografischen Karten festgehalten werden. Sie umfassen den Schienenverkehr, alle Strassen und Wege, wie auch Flugplätze und alle dazugehörigen Bauwerke. +Aus verkehrsgeografischer Sicht besteht das Verkehrsnetz aus Kanten, Knotenpunkten und dem Hinterland. Die Knotenpunkte werden auch hier durch die Kanten verbunden, die den Verkehrsstrom aufnehmen, wobei das Hinterland durch einzelne Knoten versorgt wird. Die Aufteilung in Kanten und Knotenpunkte ermöglicht eine Vereinfachung komplexer Verkehrsnetze, damit sie mittels der Graphentheorie untersucht werden können. +Grundsätzlich können kurze Wege zwischen den Knotenpunkten das Ziel beim +Aufbau eines Verkehrsnetzes sein. Es kann aber auch versucht werden, die Bau- und Unterhaltskosten des Verkehrsnetzes in einem gewissen Rahmen zu halten. Aus diesen Vorgaben ergibt sich dann, je nach dem was gewünscht wird, eine grob- oder feinmaschige Struktur des Netzes. +Ziel ist aber ein möglichst wirtschaftliches und optimales Verkehrsnetz. -Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- und Zielknoten bei der ersten Versuchsreihe im Netzwerk diametral gegenüber liegen. Dadurch gehen viele Knoten verloren, welcher \emph{Dijkstra} als uninformierter Suchalgorithmus absuchen würde. In der zweiten Veruschsreihe werden hingegen Start- un Zielpunkt zufällig im Netzwerk ausgewählt. Es wird deshalb erwwartet, dass die Unterschiede in der Rechenzeit der beiden Algorithmen in der zweiten Versuchsreihe deutlich ausgeprägter sind. +\subsection{Suchalgorithmen} -\subsection{Einfluss der Knotenzahl auf die Rechenzeit} -\label{verkehr:Knotenzahl} +\subsubsection{Dijkstra-Algorithmus} +Der Algorithmus von Dijkstra ist benannt nach seinem Erfinder dem Mathematik- und Infomratikprofessor Edsger Dijkstra. Den Algorithmus hat er im Jahr 1959 erfunden. +Der Algorithmus von Dijkstra ist ein Greedy-Algorithmus (gieriger Algorithmus), der schrittweise einen Folgezustand auswählt, damit beim Zeitpunkt der Wahl der grösste Gewinn bzw. das beste Ergebnis erzielt werden kann. +Trotz der Schnelligkeit der Greedy-Algorithmen, können viele Probleme nicht optimal gelöst werden. +Vereinfacht wird beim Dijkstra-Algorithmus, ausgehend von einem Startknoten so lange dem kürzesten Pfad gefolgt, bis der Zielknoten erreicht wird. Dabei muss für jeden besuchten Knoten die Kostenfunktion als auch der Pfad dahin (vorheriger Knoten) gespeichert werden. +Dadurch wird hingegen garantiert, dass, wenn der Zielknoten erreicht wird, auch der kürzeste Pfad gefunden wurde. +Grundlegende Voraussetzung für den Dijkstra-Algorithmus ist die strikte Positivität der Kantengewichte. Andernfalls würde ein wiederholtes Ablaufen einer Kante mit negativem Gewicht zu einer stetigen Reduktion der Kostenfunktion führen, was zu einer unendlichen Schlaufe führen würde. -\begin{figure} -\centering -\includegraphics[width=12cm]{figures/chart_Vr1.png} +\subsubsection{A*-Algorithmus} +Suchalgorithmen werden nach einfachen (uninformierte) und heuristischen (informierten) Algorithmen unterschieden. Während einfache Algorithmen den Suchraum intuitiv durchsuchen, beziehen heuristische Algorithmen Wissen über den Suchraum mit ein. +Der A*-Algorithmus geht auf seine Erfinder Peter Hart, Nils Nilsson und Bertram Raphael zurück, die den Algorithmus erstmals im Jahr 1968 beschrieben. +Der A*-Algorithmus ist ein heuristischer Suchalgorithmus, der den kürzesten Pfad zwischen zwei Knoten in einem Graphen mit positiven Kantengewichten berechnet. +Im Gegensatz zu einfachen Suchalgorithmen, wird beim A*-Algorithmus eine Schätzfunktion, die sogenannte Heuristik, verwendet. Dies ermöglicht ein zielgerichtetes Suchen und gleichzeitig wird die Laufzeit verringert. +Ausserdem findet der A*-Algorithmus immer eine optimale Lösung, sofern eine vorhanden ist. +Der A*-Algorithmus wird als Verallgemeinerung gehandhabt und gilt als Erweiterung des Dijkstra-Algorithmus. -\caption{Gemessene Rechenzeiten der ersten Versuchsreihe in Abhängigkeit der Knotenzahl.} -\label{verkehr:Vr1} -\end{figure} +\subsubsection{Floyd-Warshall-Algorithmus} +Der Floyd-Warshall-Algorithmus wurde erstmals im Jahr 1962 von seinen Namensgebern Robert Floyd und Stephen Warshall vorgestellt. +Der Floyd-Warshall-Algorithmus sucht kürzeste Wege innerhalb eines Graphen. Er ermittelt aber nicht nur die Distanz zwischen zwei Knoten, sondern berechnet die kürzesten Wege zwischen allen Knotenpaaren eines gewichteten Graphen. Somit werden die kürzesten , beziehungsweise die optimalsten Wege zwischen allen Paaren von Knoten berechnet, sofern der Graph keinen negativen Kreis (Zyklus) aufweist. +Ein Kreis in einem Graphen ist ein Weg, bei dem Start- und Endpunkt den gleichen Knoten aufweisen. Dieser wird negativ, wenn die Summe der gewichteten Kanten kleiner als Null wird. -In \ref{verkehr:Vr1} ist ersichtlich, dass der Unterschied in der Rechenzeit zwischen \emph{Dijkstra} und \emph{A*} erst aber einer Knotenzahl von ca. $n=500$ merklich ansteigt. Dieses etwas überraschende Resultat ist darauf zurückzuführen, dass bei steigender Knotenzahl die Abweichung des effektiven kürzesten Pfades von der Distanz der Luftlinie abnimmt. -Die Effektivität von \emph{A*} mit euklidischer Heuristik ist wiederum grösser, wenn die Abweichung des kürzesten Pfads von der Luftlinie minimal ist. -Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei die relative Abweichung erstaunlicherweise bei einer Knotenzahl von $n=100$ maximal ist und nach $n=500$ nur noch marginal abnimmt. +\subsubsection{Euklidische Heuristik} +Bei Verkehrsnetzen ist die euklidische Distanz eine gängige und zuverlässige Heurstik. Dabei wird zu den effektiven Reisekosten zum aktuellen Knoten die euklidische Distanz bis zum Zielknoten hinzuaddiert. Dadurch wird die Kostenfunktion konsequent nie überschätzt. Dies stellt eine Voraussetzung an eine zulässige Heuristik dar. +Was bei einem physischen Verkehrsnetz einfach zu bewältigen ist, da Koordinaten von Verkehrsnetzen zur Berechnung der Distanz verwendet werden können, ist bei virtuellen Netzwerken (z.B. Servernetzen) entweder nicht möglich, oder nicht relevant. -\begin{figure} -\centering -\includegraphics[width=12cm]{figures/chart_pathDiff.png} -\caption{Relative Abweichung des kürzesten Pfads von der Luftlinie.} -\label{verkehr:pathDifference} -\end{figure} +\subsection{PageRank-Algorithmus} +Der PageRank-Algorithmus wurde von den Gründern von Google, Larry Page und Sergey Brin im Jahr 1996 entwickelt und zum Patent angemeldet. Zwei Jahre später gründeten sie ihr Unternehmen Google Inc.. +Beim PageRank-Algorithmus handelt es sich um den Algorithmus von Google, aus dem die Google-Matrix abgeleitet wird. +Die Google-Matrix ist eine immens grosse Matrix mit Millionen Zeilen und Spalten, die für die schnelle und vor allem exakte Bestimmung der PageRanks (Gewichtung) eine grosse Bedeutung hat. +Der PageRank-Algorithmus analysiert und gewichtet beispielsweise die Verlinkungsstruktur verschiedener Websites des World Wide Web anhand ihrer Struktur. +Der PageRank wird umso höher, je mehr hochwertige Links auf eine Webseite verweisen und je höher die Gewichtung einer Webseite ist, desto grösser ist der Effekt.\\ +Dabei handelt es sich um einen iterativen Prozess. Ausgegangen wird von der Adjazenz-Matrix $A$, für welche gilt. -\subsection{Einfluss der Position der Start- und Zielknoten auf die Rechenzeit} +%THEORIE... +Grundsätzlich setzt sich der PageRank Algorithmus mit der Fragestellung auseinander, wie eine Suchmaschine wie Google Suchresultate bewertet und somit sortieren soll. Öfters aufgerufene Resultate sollen schliesslich höher gewichtet werden. Dabei wird angenommen, dass eine Website populärer ist, je mehr andere Websites darauf verweisen. -\begin{figure} -\centering -\includegraphics[width=12cm]{figures/chart_Vr2.png}\\ -\caption{Gemessene Rechenzeiten der zweiten Versuchsreihe in Abhängigkeit der Knotenzahl.} -\label{verkehr:Vr2} -\end{figure} +\begin{equation} +A_{i,j}=\left\{ \begin{matrix} +1 & \text{Kante von $j$ nach $i$} \\ 0 & \text{keine Kante von $j$ nach $i$} +\end{matrix} + \right. +\label{verkehr:Adja} +\end{equation} -Zum Vergleich der Resultate in \ref{verkehr:Knotenzahl} zeigt \ref{verkehr:Vr2} die Rechenzeiten der zweiten Versuchsreihe, in welcher die Start- und Zielknoten zufällig im Netzwerk ausgewählt wurden. Einerseits ist eine reduzierte durchschnittliche Rechenzeit festzustellen, was schlicht daran liegt, dass die zufällige Wahl der Knoten dazu führt, dass diese tendenziell weniger weit auseinander liegen.\\ -Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwischen \emph{Dijkstra} und \emph{A*} deutlich früher abzeichnen. Dieses Phänomen lässt sich leicht durch die zielgerichtete Suche des \emph{A*}-Algorithmus erklären. -\begin{figure} -\centering -\includegraphics[width=6cm]{figures/network_dij.png}\qquad -\includegraphics[width=6cm]{figures/network_aStar.png} -\caption{Suchpfad in grün mit \emph{Dijkstra} (links), und \emph{A*} (rechts). Besuchte Knoten sind in blau, resp. rot markiert.} -\label{verkehr:Comparison} -\end{figure} -In \ref{verkehr:Comparison} ist ersichtlich, dass bei einem im Netzwerk liegenden Startknoten die zielgerichtete Suche von \emph{A*} deutlich ausgeprägter zum Zuge kommt, als wenn dieser am Rand des Netzwerks liegen würde. +Für ungerichtete Graphen mit $n$ Knoten gilt \begin{equation}A_{i,j}=A_{j,i}\end{equation} und weiter \begin{equation}A_{i,i}=0\quad\forall i\in \left\{1...n\right\}\end{equation} +Beim PageRank-Algorithmus wird eine abgewandelte Form der Adjazenz-Matrix verwendet. +Dabei werden die Matrix-Einträge spaltenweise durch die jeweilige Spaltensumme geteilt. +\begin{equation} P_{i,j}=\frac{A_{i,j}}{\sum_{i=1}^{n}A_{i,j}} \end{equation} +Anschliessend multipliziert man diese Matrix $P$ mit einem Spaltenvektor $\Vec{r_0}$ mit $n$ Einträgen, für welchen gilt: +\begin{equation} \Vec{r_0}(i) = \frac{1}{n} \quad\forall i\in \left\{1...n\right\} \end{equation} +Dieser Vektor stellt ein neutrales Ranking dar. Alle Knoten werden gleich gewichtet. +Dadurch erhält man wiederum einen $n$-zeiligen Spaltenvektor $\Vec{r_1}$, der das "erste" Ranking darstellt. Durch Multiplikation der ursprünglichen Matrix $P$ mit dem 1. Ranking-Vektor $\Vec{r_1}$ wird auf Basis des ersten Rankings ein zweites erstellt. +\begin{equation} \Vec{r_2} = P\cdot\Vec{r_1} = P\cdot(P\cdot\Vec{r_0}) = P^2\cdot\Vec{r_0}\end{equation} +somit +\begin{equation} \Vec{r_i} = P^i\cdot\Vec{r_0}\end{equation} +Der Vektor $\Vec{r_i}$ konvergiert zu einem Eigenvektor von $P$ und stellt das abschliessende Ranking dar. -- cgit v1.2.1 From 46340ee2972d7f59bf87665fd93298a6a937f797 Mon Sep 17 00:00:00 2001 From: "User-PC\\User" Date: Fri, 28 May 2021 15:06:26 +0200 Subject: =?UTF-8?q?=C3=9Cberarbeitungen=20/=20Verbesserungen?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- buch/papers/spannung/Einleitung.tex | 134 ++++------ buch/papers/spannung/Grafiken/Bild1.png | Bin 0 -> 17190 bytes buch/papers/spannung/Grafiken/Bild2.png | Bin 0 -> 26255 bytes .../spannung/Grafiken/infinitesimalerWuerfel.png | Bin 24852 -> 27082 bytes buch/papers/spannung/teil0.tex | 106 +++++--- buch/papers/spannung/teil1.tex | 58 ++-- buch/papers/spannung/teil2.tex | 296 ++++++++++++--------- buch/papers/spannung/teil3.tex | 12 +- buch/papers/spannung/teil4.tex | 95 +++---- 9 files changed, 372 insertions(+), 329 deletions(-) create mode 100644 buch/papers/spannung/Grafiken/Bild1.png create mode 100644 buch/papers/spannung/Grafiken/Bild2.png (limited to 'buch/papers') diff --git a/buch/papers/spannung/Einleitung.tex b/buch/papers/spannung/Einleitung.tex index 37c2ec2..cf6e916 100644 --- a/buch/papers/spannung/Einleitung.tex +++ b/buch/papers/spannung/Einleitung.tex @@ -1,35 +1,63 @@ \section{Einleitung\label{spannung:section:Einleitung}} -In diesem Kapitel geht es darum die Matrix im dreidimensionalen Spannungszustand genauer zu untersuchen. -In der Geotechnik wendet man solche Matrizen an, um Spannungen im Boden zu berechnen. -Mit diesen Grundlagen dimensioniert man beispielsweise Böschungen, Fundationen, Dämme und Tunnels. -Ebenfalls benötigt man diese Matrix, um aus Versuchen Kennzahlen über den anstehenden Boden zu gewinnen. -Besonderes Augenmerk liegt dabei auf dem Oedometer - Versuch. +In diesem Kapitel geht es darum das Hook'sche Gesetz im Dreidimensionalen zu beschreiben. +Dieses beschreibt die Beziehung von Spannung und Dehnung von linear elastischen Materialien im Eindimensionalen. +Durch variable Krafteinwirkungen entstehen in jedem Punkt des Materials eine Vielzahl an unterschiedlichen Spannungen. +Jeder erdenkliche Punkt im Dreidimensionalen beschreibt daher einen entsprechenden individuellen Spannungszustand. +Um das Hook'sche Gesetz für den 3D Spannungszustand formulieren zu können, reichen Skalare nicht aus. +Darum werden Vektoren, Matrizen und Tensoren zur Hilfe gezogen. +Diese allgemeine Spannungsformel ist Grundlage für Computerprogramme und geotechnische Versuche, wie der Oedometer-Versuch. -Bei dieser Untersuchung der zugehörigen Berechnungen hat man es mit Vektoren, Matrizen und Tensoren zu tun. Um die mathematische Untersuchung vorzunehmen, beschäftigt man sich zuerst mit den spezifischen Gegebenheiten und Voraussetzungen. Ebenfalls gilt es ein paar wichtige Begriffe und deren mathematischen Zeichen einzuführen, damit sich den Berechnungen schlüssig folgen lässt. -In diesem Kapitel hat man es insbesondere mit Spannungen und Dehnungen zu tun. -Mit einer Spannung ist hier jedoch keine elektrische Spannung gemeint, -sondern eine Kraft geteilt durch Fläche. +\section{Spannungsausbreitung\label{spannung:section:Spannungsausbreitung}} +\rhead{Spannungsausbreitung} +Die Geotechnik ist eine Ingenieurdisziplin, bei welcher man Erdbau und den Erdbau tangierende Bauwerke dimensioniert. +Sie beinhaltet aber auch die statische Beurteilung von Boden und Fels. -\section{Einführung wichtige Begriffe\label{spannung:section:Wichtige Begriffe}} +Belastet man den Boden mit einer Spannung \[ -l_0 +\sigma = -\text{Ausgangslänge [\si{\meter}]} +\frac{F}{A} \] +, so wird diese in den Boden geleitet und von diesem kompensiert. +Im Boden entstehen unterschiedlich hohe Zusatzspannung. +Die Zusatzspannung scheint sich räumlich und berechenbar im Boden auszubreiten. +Im Falle einer konstanten Flächenlast $\sigma$ (siehe Abbildung 1.1) breitet sich die Zusatzspannung zwiebelartig aus. +Mit der Tiefe $t$ nimmt diese permanent ab (siehe Abbildung 1.2). +Wie diese Geometrie der Ausbreitung ist wird durch viele Modelle und Ansätze näherungsweise beschrieben. +Diese Zusatzspannung $\sigma$ ist aber sicher abhängig von $(x,y,t)$. + +\begin{figure} + \centering + \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild4.png} + \caption{Ausbreitung der Zusatzspannung im Boden} + \label{fig:Bild4} +\end{figure} + +\begin{figure} + \centering + \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild5.png} + \caption{Funktionen Spannung und Dehnung} + \label{fig:Bild5} +\end{figure} + +Bei jeder dieser Zusatzspannung geht eine entsprechende Zusatzdehnung einher, welche eine Setzung bedeutet. +Im einfachsten Fall kann modellhaft mit \[ -\Delta l +\varepsilon = -\text{Längenänderung nach Kraftauftrag [\si{\meter}]} +\frac{\sigma}{E} \] +die Setzung an einem Punkt an der Bodenoberfläche mit \[ -\Delta b +s = -\text{Längenänderung in Querrichtung nach Kraftauftrag [\si{\meter}]} +\int_{0}^{\infty}\varepsilon\enspace dt \] +berechnet werden mit: \[ \varepsilon = @@ -43,22 +71,7 @@ l_0 \[ E = -\text{Elastizitätsmodul [\si{\kilo\pascal}]} -\] -\[ -\nu -= -\text{Querdehnungszahl; Poissonzahl [$-$]} -\] -\[ -F -= -\text{Kraft [\si{\kilo\newton}]} -\] -\[ -A -= -\text{Fläche [\si{\meter\squared}]} +\text{Elastizitätsmodul; Young-Modul [\si{\kilo\pascal}]} \] \[ t @@ -71,48 +84,17 @@ s \text{Setzung, Absenkung [m]} \] -Beziehungen -\[ -\varepsilon -= -\frac{\Delta l}{l_0} -\] -\[ -\varepsilon_q -= -\frac{\Delta b}{l_0} -= -\varepsilon\cdot\nu -\] -\[ -\sigma -= -\frac{N}{A} -\] -\[ -F -= -\int_{A} \sigma dA -\] -\[ -\varepsilon^{\prime} -= -\frac{1}{l_0} -\] +In der praktischen Geotechnik wird man allerdings weitaus schwierigere Situationen antreffen. +Ein Beispiel wäre eine Baugrube mit einem Baugrubenabschluss, wo ein Teil des Bodens abgetragen ist (siehe Abbildung 1.3). +Die Ausbreitung der Zusatzspannung $\sigma(x,y,t)$ würde hier deutlich komplizierter ausfallen. +Dies bedeutet auch eine komplexere Setzung der Bodenoberfläche infolge einer Flächenlast $\sigma$. +Aus allen zusätzlichen Spannungen müssen die adäquaten Dehnung mit Hilfe einer Spannungsgleichung berechnet werden. +Diese beruht auf Annahmen nach Hooke auf einem linear elastischen Boden. +Generell wird im Ingenieurwesen versucht Phänomene möglichst nach dem Hook'schen Gesetz abbilden zu können. -\section{Einführung wichtige Begriffe\label{spannung:section:Tensoren}} -Tensoren wurden als erstes in der Elastizitätstheorie eingesetzt. (Quelle Herr Müller) -In der Elastizitätstheorie geht es darum viele verschiedene Komponenten zu beschreiben. -Mit einer Matrix oder einem Vektor kann man dies nicht mehr bewerkstelligen. -Wenn man den dreidimensionalen Spannungszustand abbilden möchte, müsste man mehrere Vektoren haben. -Deshalb wurden 1840 von Rowan Hamilton Tensoren in die Mathematik eingeführt. -Woldemar Voigt hat den Begriff in die moderne Bedeutung von Skalar, Matrix und Vektor verallgemeinert. -Albert Einstein hat Tensoren zudem in der allgemeinen Relativitätstheorie benutzt. -Tensor sind eine Stufe höher als Matrizen. Matrizen sind 2. Stufe. -Da Tensoren eine Stufe höher sind, kann man auch Matrizen, Vektoren und Skalare als Tensoren bezeichnen. -Der Nachteil von den Tensoren ist, dass man die gewohnten Rechenregeln, die man bei Vektoren oder Matrizen kennt, -nicht darauf anwenden kann. Man ist deshalb bestrebt die Tensoren als Vektoren und Matrizen darzustellen, -damit man die gewohnten Rechenregeln darauf anwenden kann. (Quelle Wikipedia) -In der vorliegenden Arbeit sind bereits alle Tensoren als Matrizen 2. Stufe abgebildet. -Trotzdem kann man diese Matrizen wie vorher beschrieben als Tensor bezeichnen. -Da diese als Matrizen abgebildet sind, dürfen wir die bekannten Rechenregeln auf unsere Tensoren anwenden. \ No newline at end of file +\begin{figure} + \centering + \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild3.png} + \caption{Beispiel Lastauftrag auf Boden} + \label{fig:Bild3} +\end{figure} \ No newline at end of file diff --git a/buch/papers/spannung/Grafiken/Bild1.png b/buch/papers/spannung/Grafiken/Bild1.png new file mode 100644 index 0000000..32b627e Binary files /dev/null and b/buch/papers/spannung/Grafiken/Bild1.png differ diff --git a/buch/papers/spannung/Grafiken/Bild2.png b/buch/papers/spannung/Grafiken/Bild2.png new file mode 100644 index 0000000..d1321a4 Binary files /dev/null and b/buch/papers/spannung/Grafiken/Bild2.png differ diff --git a/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.png b/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.png index 398529c..2c359e6 100644 Binary files a/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.png and b/buch/papers/spannung/Grafiken/infinitesimalerWuerfel.png differ diff --git a/buch/papers/spannung/teil0.tex b/buch/papers/spannung/teil0.tex index 2f4d23b..be837ac 100644 --- a/buch/papers/spannung/teil0.tex +++ b/buch/papers/spannung/teil0.tex @@ -1,56 +1,84 @@ -\section{Spannungsausbreitung\label{spannung:section:Spannungsausbreitung}} -\rhead{Spannungsausbreitung} -Anhand untenstehendem Bild kann ein einfaches Beispiel betrachtet werden. -Es gibt eine Flächenlast (Kraft), diese wird auf den Boden abgetragen. -Diese Last muss dann vom Boden aufgenommen werden. -Im Boden entsteht nebst der Eigenspannung eine weitere Spannung durch diese Last (Zusatzspannung). -Diese Zusatzspannung $\sigma$ ist abhängig von $(x,y,t)$. -Je nach dem, wo man sich im Boden befindet variert die Spannung. -Mit der Tiefe wird die Zusatzspannung geringer. -Die Ausbreitung der Zusatzspannung im Boden hat die Form einer Zwiebel. -Durch Untersuchung der Spannung an verschiedenen Punkten im Boden, kann man eine Funktion abtragen. -Dasselbe macht man auch mit der Dehnung. Es zeigt sich, dass die Form der beiden Funktionen gleich ist. -Dies erklärt sich dadurch, dass die Spannung und die Dehnung proportional zueinander sind. -\begin{figure} - \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild4.png} - \caption{Ausbreitung der Spannung im Boden} - \label{fig:Bild4} -\end{figure} +\section{Einachsiger Spannungszustand\label{spannung:section:Einachsiger Spannungsustand}} +\rhead{Einachsiger Spannungszustand} +Ein Spannungszustand beschreibt alle Spannungen, welche in einem beliebigen Punkt im Körper wirken (siehe Abbildung 1.4). +Änderungen der äusseren Kräfte verändern die inneren Spannungszustände im Material. +Um alle Spannungen eines Punktes darstellen zu können, wird ein infinitesimales Bodenelement in Form eines Würfels modellhaft vorgestellt. +Man spricht auch von einem Elementarwürfel, da dieser elementar klein ist. \begin{figure} \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild5.png} - \caption{Funktionen Spannung und Dehnung} - \label{fig:Bild5} + \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild2.png} + \caption{Infinitesimales Bodenelement mit den 9 Spannungen} + \label{fig:infintesimaler-wurfel} \end{figure} -Anhand eines etwas schwierigeren Beispiels sieht man, -dass die Spannungsausbreitung nicht immer ganz einfach ist. -Man hat hier eine Baugrube mit einem Baugrubenabschluss, wo ein Teil des Bodens abgetragen wurde. -Was aber immer noch gilt ist, dass die Spannung $\sigma$ von drei Variablen abhängig ist $(x,y,t)$. -Ansätze um die Spannungsausbreitung zu berechnen gibt es je nach Bodentyp verschiedene. +Es werden jeweils drei Seiten dieses Würfels betrachtet, wobei die drei gegenüberliegenden Seiten die selben Spannungen aufweisen. +Das infinitesimale Bodenteilchen hat die Koordinaten $1$, $2$, $3$ muss sich zwingend im Gleichgewicht befinden. +So sind insgesamt 9 verschiedene Spannungen möglich, wobei 3 Normal- und 6 Schubspannungen sind. +Normalspannung wirken normal (mit rechtem Winkel) zur angreifenden Fläche und Schubspannungen parallel zur angreifenden Fläche. +Alle Beträge dieser 9 Spannungen am Elementarwürfel bilden den Spannungszustand. +Daraus können die äquivalenten Dehnungen $\varepsilon$ mit Hilfe des Hook'schen Gesetz berechnet werden. \begin{figure} \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild3.png} - \caption{Beispiel Lastauftrag auf Boden} - \label{fig:Bild3} + \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild1.png} + \caption{1D Spannungszustand aus einer quaderförmigen Bodenprobe} + \label{fig:infintesimaler-wurfel} \end{figure} -Die Spannungsausbreitung ist uns jedoch gegeben, es geht nicht darum, dies genauer zu untersuchen. -Durch die Spannungsausbreitung und das Elastizitätsmodul kann man eine Dehnung berechnen. -Anhand dieser Dehnung kann man mit einem Integral wiederum die Setzung berechnen. +Im einachsigen Spannungszustand herrscht nur die Normalspannung $\sigma_{11}$ (siehe Abbildung). +Das Hook'sche Gesetz beschreibt genau diesen 1D Spannungszustand. +Nach Hooke gilt: +\[ +F +\sim +\Delta l +\] +. +Teilt man beide Seiten mit den Konstanten $A$ und $l_0$ erhält man +\[ +\frac{F}{A} += +\sigma +\sim +\] \[ \varepsilon = -\frac{\sigma}{E} +\frac{\Delta l}{l_0} +\] +und somit +\[ +\sigma +\sim +\varepsilon +\] +. +Mit: +\[ +l_0 += +\text{Länge zu Beginn [\si{\meter}]} +\] +\[ +A += +\text{Fläche [\si{\meter\squared}]} +\] + +Diese Beziehung gilt bei linear elastischen Materialien, welche reversibel sind und nicht dauerhaft verformt werden. +Es ist praktisch die relative Dehnung $\varepsilon$ anzugeben und nicht eine absolute Längenänderung $\Delta l$. +Mithilfe vom Elastizitätsmodul $E$ als Proportionalitätskonstante lässt sich der eindimensionale Fall mit +\[ +\sigma += +E\cdot\varepsilon \] +beschreiben. +Im Falle, dass der E-Modul nicht konstant ist, kann dieser näherungsweise mit \[ -s +E = -\int_{0}^{\infty}\varepsilon\enspace dt +\frac{\Delta\sigma}{\Delta\varepsilon} \] -Die Setzung zu bestimmen ist in der Geotechnik sehr wichtig. -Besonders ungleichmässige Setzungen können bei Bauwerken Probleme ergeben. -Es gilt also die Bauwerke so zu dimensionieren, dass es verträgliche Setzungen gibt. \ No newline at end of file +ausgedrückt werden. \ No newline at end of file diff --git a/buch/papers/spannung/teil1.tex b/buch/papers/spannung/teil1.tex index 9467d21..3b40ee9 100644 --- a/buch/papers/spannung/teil1.tex +++ b/buch/papers/spannung/teil1.tex @@ -1,41 +1,17 @@ -\section{Proportionalität Spannung-Dehnung\label{spannung:section:Proportionalität Spannung-Dehnung}} -\rhead{Proportionalität Spannung-Dehnung} -Das Hook'sche Gesetz beschreibt die elastische Längenänderung von Festkörpern im Zusammenhang mit einer Krafteinwirkung. -Die Längenänderung $\Delta l$ ist proportional zur Krafteinwirkung $F$. -\[ -F -\sim -\Delta l -\] -Man kann dies nur im Bereich vom linearen-elastischen Materialverhalten anwenden. -Das heisst, dass alle Verformungen reversibel sind, sobald man die Kraft wegnimmt. -Es findet somit keine dauernde Verformung statt. -Da es sehr praktisch ist die Längenänderung nicht absolut auszudrücken haben wir $\varepsilon$. -Die Dehnung $\varepsilon$ beschreibt die relative Längenänderung. -Die Dehnung $\varepsilon$ ist wiederum proportional zu der aufgebrachten Spannung. -Im Bauingenieurwesen hat man es oft mit grösseren Teilen oder grösseren Betrachtungsräumen zu tun. -Da ist es nun natürlich sehr sinnvoll, wenn wir nicht mit absoluten Zahlen rechnen, -sondern unabhängig von der Länge den Zustand mit Dehnung $\varepsilon$ beschreiben können. -Mithilfe vom E-Modul, (steht für Elastizitätsmodul) einer Proportionalitätskonstante, -kann man das in eine Gleichung bringen, wie man hier sieht. Das E-Modul beschreibt, -das Verhältnis von Kraftaufnahme eines Werkstoffes und dessen zusammenhängender Längenveränderung. -(Quelle Wikipedia) -\[ -\sigma -= -E\cdot\varepsilon -\] -\[ -E -= -\frac{\Delta\sigma}{\Delta\varepsilon} -= -const. -\] - -Aus diesem Verhältnis kann man das E-Modul berechnen. -Je nach Material ist dies verschieden. -Das E-Modul lässt sich nur im linearen-elastischen Materialverhalten anwenden. -Für Bodenmaterial gibt es ein spezielles E-Modul. Dieses wird mit dem Oedometer-Versuch ermittelt. -Es wird mit $E_{OED}$ ausgedrückt. Dieser Versuch wird später noch beschrieben. -Der Oedometer-Versuch ist abhängig von den diesem Kapitel zu untersuchenden Matrizen. \ No newline at end of file +\section{Skalare, Vektoren, Matrizen und Tensoren\label{spannung:section:Skalare,_Vektoren,_Matrizen_und_Tensoren}} +\rhead{Skalare, Vektoren, Matrizen und Tensoren} +Tensoren wurden als erstes in der Elastizitätstheorie eingesetzt. (Quelle Herr Müller) +In der Elastizitätstheorie geht es darum viele verschiedene Komponenten zu beschreiben. +Mit einer Matrix oder einem Vektor kann man dies nicht mehr bewerkstelligen. +Wenn man den dreidimensionalen Spannungszustand abbilden möchte, müsste man mehrere Vektoren haben. +Deshalb wurden 1840 von Rowan Hamilton Tensoren in die Mathematik eingeführt. +Woldemar Voigt hat den Begriff in die moderne Bedeutung von Skalar, Matrix und Vektor verallgemeinert. +Albert Einstein hat Tensoren zudem in der allgemeinen Relativitätstheorie benutzt. +Tensor sind eine Stufe höher als Matrizen. Matrizen sind 2. Stufe. +Da Tensoren eine Stufe höher sind, kann man auch Matrizen, Vektoren und Skalare als Tensoren bezeichnen. +Der Nachteil von den Tensoren ist, dass man die gewohnten Rechenregeln, die man bei Vektoren oder Matrizen kennt, +nicht darauf anwenden kann. Man ist deshalb bestrebt die Tensoren als Vektoren und Matrizen darzustellen, +damit man die gewohnten Rechenregeln darauf anwenden kann. (Quelle Wikipedia) +In der vorliegenden Arbeit sind bereits alle Tensoren als Matrizen 2. Stufe abgebildet. +Trotzdem kann man diese Matrizen wie vorher beschrieben als Tensor bezeichnen. +Da diese als Matrizen abgebildet sind, dürfen wir die bekannten Rechenregeln auf unsere Tensoren anwenden. \ No newline at end of file diff --git a/buch/papers/spannung/teil2.tex b/buch/papers/spannung/teil2.tex index 7dcf65f..8be0bdc 100644 --- a/buch/papers/spannung/teil2.tex +++ b/buch/papers/spannung/teil2.tex @@ -1,9 +1,47 @@ \section{Dreiachsiger Spannungszustand\label{spannung:section:Dreiachsiger_Spannungszustand}} -\rhead{Proportionalität Spannung-Dehnung} -Wie im Kapitel Spannungsausbreitung beschrieben herrscht in jedem Punkt ein anderer Spannungszustand. -Um die Spannung im Boden genauer untersuchen zu können, führt man einen infinitesimales Bodenteilchen ein. -Das Bodenteilchen ist geometrisch gesehen ein Würfel. -An diesem Bodenteilchen trägt man die Spannungen ein in alle Richtungen. +\rhead{Dreiachsiger Spannungszustand} +Durch komplexe Spannungsausbreitungen im Boden entstehen im 3D Spannungszustand unterschiedliche Normal- und Schubspannungen. +Ein Tensor 0.Stufe, sprich ein Skalar, kann lediglich den 1D Spannungszustand beschreiben. +Um den 3D Spannungszustandes als ein mathematisches Objekt darstellen zu können, wird ein Tensor 2.Stufe, sprich eine Matrix, eingesetzt. +Die Spannungen sind durch die zwei Indizes +\[ +i, j\in\left\{1, 2, 3\right\} +\] + +definiert. +Daher ergeben sich die 9 Spannungen. +Dieser Spannungstensor kann schliesslich mit $3^2$ Einträgen als 3x3 Matrix mit +\[ +\overline{\sigma} += +\sigma_{ij} += +\begin{pmatrix} + \sigma_{11} & \sigma_{12} & \sigma_{13} \\ + \sigma_{21} & \sigma_{22} & \sigma_{23} \\ + \sigma_{31} & \sigma_{32} & \sigma_{33} +\end{pmatrix} +\] +dargestellt werden und beschreibt somit den gesamten Spannungszustand. +Die Dehnungen wirken adäquat zu den Spannungen und sind durch die zwei Indizes +\[ +k, l\in\left\{1, 2, 3\right\} +\] + +definiert. +Der Dehnungstensor ist ebenfalls ein Tensor 2.Stufe und kann somit auch als $3\times3$ Matrix mit +\[ +\overline{\varepsilon} += +\varepsilon_{kl} += +\begin{pmatrix} + \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ + \varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\ + \varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33} +\end{pmatrix} +\] +dargestellt werden und beschreibt den gesamten Dehnungszustand. \begin{figure} \centering @@ -12,23 +50,10 @@ An diesem Bodenteilchen trägt man die Spannungen ein in alle Richtungen. \label{fig:infintesimaler-wurfel} \end{figure} -An diesem infinitesimalen Bodenteilchen hat man ein räumliches Koordinatensystem, die Achsen $(1,2,3)$. -Die Achsen vom Koordinatensystem zeigen aus den 3 ersichtlichen Flächen heraus. -Pro ersichtliche Fläche haben wir eine Normalspannung und zwei Schubspannungen. -Im Gegensatz zum eindimensionalen Zustand entstehen bei einer Belastung des Bodenteilchens eine Vielzahl an Spannungen. -Es entstehen diverse Normal- und Schubspannungen. -Die Schubspannungen befinden sich an der Fläche, sie gehen rechtwinklig von den Achsen weg. -Die Schubspannungen auf einer Fläche stehen im 90 Grad Winkel zueinander. -Geschrieben werden diese mit $\sigma$, mit jeweils zwei Indizes. -Die Indizes geben uns an, in welche Richtung die Spannungen zeigen. -Der erste Index ist die Fläche auf welcher man sich befindet. -Der zweite Index gibt an, in welche Richtung die Spannung zeigt, dabei referenzieren die Indizes auch auf die Achsen $(1,2,3)$. -Bei den Spannungen sind immer positive als auch negative Spannungen möglich. -Es können also Druck- oder Zugspannungen sein. +Der Spannungs- und Dehnungstensor 2.Stufe kann je in einen Tensor 1. Stufe überführt werden, welches ein Spaltenvektor ist. +Gemäss der Hadamard-Algebra dürfen Zeile um Zeile in eine Spalte notiert werden, sodass es einen Spaltenvektor ergibt. +So ergibt sich der Spannungsvektor -Zunächst wird untenstehend der allgemeine Spannungszustand betrachtet. - -Spannungstensor 2. Stufe i,j $\in$ {1,2,3} \[ \overline{\sigma} = @@ -39,7 +64,6 @@ Spannungstensor 2. Stufe i,j $\in$ {1,2,3} \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{pmatrix} -= \qquad \Rightarrow \qquad @@ -57,9 +81,7 @@ Spannungstensor 2. Stufe i,j $\in$ {1,2,3} \sigma_{33} \end{pmatrix} \] - -Dehnungstensor 2. Stufe k,l $\in$ {1,2,3} - +und Dehnungsvektor \[ \overline{\varepsilon} = @@ -70,7 +92,6 @@ Dehnungstensor 2. Stufe k,l $\in$ {1,2,3} \varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\ \varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33} \end{pmatrix} -= \qquad \Rightarrow \qquad @@ -87,13 +108,22 @@ Dehnungstensor 2. Stufe k,l $\in$ {1,2,3} \varepsilon_{32} \\ \varepsilon_{33} \end{pmatrix} -\] +\]. -Bei diesen zwei obenstehenden Formeln kann man sehen wie Matrizen zu einem Vektor umgewandelt wurden. -Unter dem Kapitel Hadamard-Algebra kann man sehen, dass man dabei Zeile um Zeile in eine Spalte schreiben kann, -sodass es einen Vektor ergibt. +Um die Beziehung von Spannung und Dehnung, welche mit Tensoren 2.Stufen ausgedrückt werden, zu beschreiben, wird ein Elastizitätstensor 4.Stufe benötigt. +Dieser ist im 1D Spannungszustand ein Tensor 0.Stufe und somit ein Skalar. +Dieses Skalar ist das Elastizitätsmodul $E$. -Elastizitätstensor 4. Stufe i,j,k,l $\in$ {1,2,3} +Dieser Elastizitätstensor 4.Stufe kann als Tensor 2.Stufe, sprich als Matrix, dargestellt werden. +So wird die Spannungsgleichung stark vereinfacht, da nun ein Vektor mit einer Matrix operiert. +Dieser Tensor muss für eine Spannung jeden Einfluss aus allen 9 Dehnungen mit Konstanten erfassen. +Dies bedeutet um eine von 9 Spannungen berechnen zu können müssen alle 9 Dehnung mit unterschiedlichen Faktoren summiert werden. +Es ergeben sich $9^2$ Einträge, welches mit den 4 Indizes +\[ +i, j, k, l\in\left\{1, 2, 3\right\} +\] +, die zueinander verknüpft werden müssen, zu begründen ist. +Es ergeben sich $3^4$ Einträge, sprich eine $9\times9$ Matrix, welche allgemein mit \[ \overline{\overline{C}} = @@ -104,32 +134,51 @@ C_{1111} & C_{1112} & C_{1113} & C_{1121} & C_{1122} & C_{1123} & C_{1131} & C_{ C_{1211} & C_{1212} & C_{1213} & C_{1221} & C_{1222} & C_{1223} & C_{1231} & C_{1232} & C_{1233} \\ C_{1311} & C_{1312} & C_{1313} & C_{1321} & C_{1322} & C_{1323} & C_{1331} & C_{1332} & C_{1333} \\ C_{2111} & C_{2112} & C_{2113} & C_{2121} & C_{2122} & C_{2123} & C_{2131} & C_{2132} & C_{2133} \\ -C_{2211} & C_{2212} & C_{1113} & C_{2221} & C_{2222} & C_{2223} & C_{2231} & C_{2232} & C_{2233} \\ +C_{2211} & C_{2212} & C_{2213} & C_{2221} & C_{2222} & C_{2223} & C_{2231} & C_{2232} & C_{2233} \\ C_{2311} & C_{2312} & C_{2313} & C_{2321} & C_{2322} & C_{2323} & C_{2331} & C_{2332} & C_{2333} \\ C_{3111} & C_{3112} & C_{3113} & C_{3121} & C_{3122} & C_{3123} & C_{3131} & C_{3132} & C_{3133} \\ C_{3211} & C_{3212} & C_{3213} & C_{3221} & C_{3222} & C_{3223} & C_{3231} & C_{3232} & C_{3233} \\ C_{3311} & C_{3312} & C_{3313} & C_{3321} & C_{3322} & C_{3323} & C_{3331} & C_{3332} & C_{3333} \end{pmatrix} \] - -Dieser Elastizitätstensor muss eine quadratische Matrix mit $3^{4}$ Einträgen ergeben, -da die Basis mit den drei Richtungen $1, 2, 3$ und die Potenz mit den 4 Indizes mit je $1, 2, 3$ definiert sind. -Dies gibt daher eine 9 x 9 Matrix, welche zudem symmetrisch ist. - +ausgedrückt wird. +Dieser Elastizitätstensor muss für isotrope Materialien zwingend symmetrisch sein. Folglich gilt: \[ \overline{\overline{C}} = \overline{\overline{C}}~^{T} -\] +\]. -Allgemeine Spannungsgleichung (mit Vektoren und Tensor) +Die allgemeine Spannungsgleichung lautet nun: \[ \vec\sigma = \overline{\overline{C}}\cdot\vec{\varepsilon} -\] +\]. +Die Konstanten $C$ werden nun nach dem Hook'schen Gesetz mit Hilfe des Elastizitätsmoduls $E$ definiert. +Da dieser Modul durch die eindimensionale Betrachtung definiert ist muss eine weitere Kennzahl eingeführt werden. +Dies ist die Querdehnungszahl $\nu$ (auch Poisson-Zahl), welche mit +\[ +\nu += +\frac{\varepsilon_q}{\varepsilon} += +\frac{\Delta b}{b_0} +\] +und +\[ +\varepsilon += +\text{Längsdehnung [$-$]} +\] +\[ +\varepsilon_q += +\text{Querdehnung [$-$]} +\] +definiert ist. Trägt man die Konstanten in die Matrix ein ergibt sich \[ \begin{pmatrix} \sigma_{11}\\ @@ -168,32 +217,61 @@ Allgemeine Spannungsgleichung (mit Vektoren und Tensor) \end{pmatrix} \] -Man kann das zudem auch als Indexnotation aufschreiben. - +, welche ebenfalls als Indexnotation mit \[ \sigma_{ij} = -= -\sum_k=1^3 -\sum_l=1^3 +\sum_{k=1}^3 +\sum_{l=1}^3 C_{ijkl}\cdot\varepsilon_{kl} \] - -Um die Berechnung an einem Beispiel zu veranschaulichen: +ausgedrückt werden können. +Die Normalspannung $\sigma_{11}$ lässt sich exemplarisch mit \[ \sigma_{22} = \frac{E\cdot\nu}{(1+\nu)(1-2\nu)}\cdot\varepsilon_{11}+\frac{E}{(1+\nu)}\cdot\varepsilon_{22}+\frac{E\cdot\nu}{(1+\nu)(1-2\nu)}\cdot\varepsilon_{33} \] +berechnen. -Anhand dem Tensor der allgemeinen Spannungsgleichung kann man zwar eine Symmetrie erkennen. -Die verschiedenen Einträge wechseln sich aber mit einander ab und es gibt keine klaren Blöcke mit nur einem gleichen Eintrag. -Man greift deshalb auf die Voigt'sche Notation zurück. - - -Zur Notation wird die Voigt'sche Notation benutzt. Das sieht wie folgt aus: +Man betrachte nun die Eigenschaften des Elastizitätstensors. +Dieser ist quadratisch und symmetrisch, die verschiedenen Einträge wechseln sich aber miteinander ab. +Es ergeben sich keine Blöcke mit einheitlichen Einträgen. +Allerdings weiss man, dass im isotropen Boden der Spannungs-, Dehnungs- und daher auch Elastizitätstensor symmetrisch sind. +Wäre dem nicht so, würde sich das Material je nach Richtung unterschiedlich elastisch verhalten. +Diese Symmetrie setzt daher voraus, dass +\[ +\sigma_{12} += +\sigma_{21} +, +\sigma_{13} += +\sigma_{31} +, +\sigma_{23} += +\sigma_{32} +\] +und folglich auch +\[ +\varepsilon_{12} += +\varepsilon_{21} +, +\varepsilon_{13} += +\varepsilon_{31} +, +\varepsilon_{23} += +\varepsilon_{32} +\] +gilt. +Diese Eigenschaft wird durch die Voigt'sche Notation ausgenutzt um die Gleichung vereinfachen zu können. +Durch diese Symmetrie gilt \[ \overline{\sigma} = @@ -208,7 +286,9 @@ Zur Notation wird die Voigt'sche Notation benutzt. Das sieht wie folgt aus: & \sigma_{22} & \sigma_{23} \\ sym & & \sigma_{33} \end{pmatrix} +\qquad \Rightarrow +\qquad \vec{\sigma} = \begin{pmatrix} @@ -220,22 +300,7 @@ Zur Notation wird die Voigt'sche Notation benutzt. Das sieht wie folgt aus: \sigma_{12} \end{pmatrix} \] - -In der Voigt'sche Notation hat man die Reihenfolge von der Ecke links oben, diagonal zur Ecke rechts unten. -Danach ist noch $\sigma_{23}$, $\sigma_{13}$ und $\sigma_{12}$ aufzuschreiben um den Vektor zu erhalten. - -Eine weitere Besonderheit ist die Symmetrie der Matrix. -So entspricht $\sigma_{23}$ dem Wert $\sigma_{32}$ und $\sigma_{13}$ dem Wert $\sigma_{31}$. -Dies ist dadurch bedingt, dass die Kräfte in seitlicher Richtung im Boden die gleichen Werte annehmen. -Man hat in dieser Berechnung ein isotropes Material. -Im infinitesimalen Körper muss ein Gleichgewicht vorherrschen. -Ist kein Gleichgewicht vorhanden, würde sich der Körper zu drehen beginnen. -Es macht somit keinen Unterschied, ob man auf der Achse 2 in Richtung 3 geht, -oder auf der Achse 3 in Richtung 2. - -Da die Spannung proportional zur Dehnung ist, kann man die ganze Voigt'sche Notation auch mit der Dehnung ausdrücken. -Auch hier wandelt man das ganze gemäss der Reihenfolge in einen Vektor um. - +und entsprechend \[ \overline{\varepsilon} = @@ -247,7 +312,7 @@ Auch hier wandelt man das ganze gemäss der Reihenfolge in einen Vektor um. = \begin{pmatrix} \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ - & \varepsilon_{22} & \varepsilon_{23} \\ + & \varepsilon_{22} & \varepsilon_{23} \\ \text{sym} & & \varepsilon_{33} \end{pmatrix} \qquad @@ -263,31 +328,17 @@ Auch hier wandelt man das ganze gemäss der Reihenfolge in einen Vektor um. \varepsilon_{13} \\ \varepsilon_{12} \end{pmatrix} -\] +\]. - -Mit der hergeleiteten Beziehung für die Spannungsgleichung anhand vom E-Modul, -der allgemeinen linearen Spannungsgleichung kann man diese Beziehungen neu aufschreiben. -Man benötigt dazu den zuvor berechneten Dehnungsvektor. -Die Gleichung besagt: -\[ -\text{Spannungsvektor} -= -\text{Elastizitätstensor}\cdot\text{Dehnungsvektor} -\] +Aus den Vereinfachungen der Voigt'schen Notation lassen sich die Spannungs- und Dehnungstensoren als Spaltenvektoren mit je 6 Einträgen darstellen. +Der Elastizitätstensor kann entsprechend auf eine $6\times6$ Matrix reduziert werden. +Es lässt sich nun eine reduzierte allgemeine Spannungsgleichung mit \[ \vec{\sigma} = \overline{\overline{C}}\cdot\vec{\varepsilon} \] - -Die Vektoren haben je 6 Einträge. Um das ganze auszudrücken braucht es einen 6 x 6 Elastizitätstensor. -Der Tensor hat sich also im Vergleich zum 9 x 9 Tensor verkleinert. -Dies ist deshalb der Fall, da man in den Achsen 2 und 3 Symmetrien hat. -Dadurch kann man die Einträge $(\varepsilon_{21}=\varepsilon_{12}; \varepsilon_{31}=\varepsilon_{13}; \varepsilon_{32}=\varepsilon_{23})$ -zusammenfassen und drei Einträge verschwinden, da drei Dehnungen gleich sind. -Das ganze sieht dann wie folgt aus: - +beziehungsweise \[ \begin{pmatrix} \sigma_{11} \\ @@ -315,11 +366,10 @@ Das ganze sieht dann wie folgt aus: \varepsilon_{12} \end{pmatrix} \] - -Die Spannung $\sigma_{11}$ besteht somit aus Anteilen von all diesen sechs Konstanten und den verschiedenen Dehnungen. -Zuvor bei der Voigt'schen Notation hat man jedoch gesehen, dass die Tensoren symmetrisch sind. -Folglich muss auch dieser Elastizitätstensor symmetrisch sein. -Das sind folgendermassen aus: +beschreiben. +Die Spannung $\sigma_{11}$ beispielsweise besteht so aus der Summe aller 6 Produkte der Konstanten $C$ und Dehnungen $\varepsilon$. +Die Symmetrieeigenschaft des Elastizitätstensors bleibt auch hier erhalten. +Nun lässt sich die reduzierte allgemeine Spannungsgleichung mit \[ \begin{pmatrix} @@ -348,9 +398,9 @@ Das sind folgendermassen aus: \varepsilon_{12} \end{pmatrix} \] - -Die Konstanten $C$ kann man nun anders ausdrücken. -Und zwar bewerkstelligt man dies mithilfe vom Hook'schen Gesetz. +beschreiben. +Die Konstanten $C$ und $\nu$ werden wieder nach dem Hook'schen Gesetz definiert. +Dies ergibt die Spannungsgleichung, welche weit möglichst vereinfacht ist: \[ \begin{pmatrix} @@ -379,25 +429,25 @@ Und zwar bewerkstelligt man dies mithilfe vom Hook'schen Gesetz. \varepsilon_{13}\\ \varepsilon_{12} \end{pmatrix} -\] +\]. -Mithilfe der Poissonzahl, welche uns die Querdehnung angibt, -sprich wie viel sich der Körper in Querrichtung verformt und dem E-Modul kann man alle Konstanten ausdrücken. -Bei einigen fällt auf, dass diese 0 werden. Der Tensor besagt also, +Im Elastizitätstensor fallen zwei $3\times3$ Blöcke auf, welche nur Einträge mit $0$ haben. Der Tensor besagt also, dass diese jeweiligen Konstanten keinen Einfluss auf unsere Spannung haben. -Man sieht nun auch ganz gut, dass sich im Vergleich bei der allgemeinen Darstellung der Spannungsgleichung, -die Einträge verschoben haben. Man hat nun eine sehr vorteilhafte Anordnung der verschiedenen Blöcke im Tensor. -Als Beispiel kann man sich $\sigma_{33}$ anschauen. -Es ist ersichtlich, dass die Konstante $C_{31}$, $C_{32}$, $C_{33}$, $C_{35}$ und $C_{36}$ keinen Einfluss auf $\sigma_{33}$ haben. -Dies kann wie folgt erklärt werden. Auf Achse 3 geht $\sigma_{33}$ in Richtung 3. -Der Einfluss von $C_{31}$, Achse 3 in Richtung 1 hat keinen Einfluss auf $\sigma_{33}$. +Man sieht nun auch ganz gut, dass sich im Vergleich zu der allgemeinen Spannungsgleichung, die Einträge verschoben haben. +Da nach Voigt zuerst die Normalspannungen und anschliessend die Schubspannungen notiert worden sind, ergeben sich die $3\times3$ Blöcke. + +Man betrachte als Beispiel die Berechnung von $\sigma_{33}$. +Es ist ersichtlich, dass die Schubdehnungen keinen Einfluss auf $\sigma_{33}$ haben. +Der Einfluss der zu $\sigma_{33}$ äquivalenten Dehnung $\varepsilon_{33}$ hat den grössten Einfluss. +Die anderen Normalspannungen $\sigma_{11}$ und $\sigma_{22}$ haben einen unter anderem mit $\nu$ korrigierten Einfluss. -Von $\overline{\overline{C}}$ bildet man nun die Inverse Matrix $\overline{\overline{C}}~^{-1}$ stellt sich die ganze Gleichung um. +Von $\overline{\overline{C}}$ bildet man noch die inverse Matrix $\overline{\overline{C}}\mathstrut^{-1}$ um die Gleichung umstellen zu können. +Dadurch erhält man die Dehnungsgleichung: \[ \vec{\varepsilon} = -\overline{\overline{C}}~^{-1}\cdot \vec{\sigma} +\overline{\overline{C}}\mathstrut^{-1}\cdot \vec{\sigma} \] \[ @@ -427,25 +477,27 @@ Von $\overline{\overline{C}}$ bildet man nun die Inverse Matrix $\overline{\ove \sigma_{13}\\ \sigma_{12} \end{pmatrix} -\] - -Die zwei Blöcke links unten und rechts oben sind immer noch vorhanden. -Im Vergleich wo wir die Inverse noch nicht gemacht haben hat sich das nicht geändert. -Um die Einflüsse der Parameter zu veranschaulichen schreibt man folgende Gleichung. +\]. +Die zwei $3\times3$ Blöcke links unten und rechts oben sind folglich noch vorhanden. +Um wieder die Einflüsse der Parameter veranschaulichen zu können berechnet man mit \[ \varepsilon_{22} = \frac{1}{E}\sigma_{22} - \frac{\nu}{E}\sigma_{11} - \frac{\nu}{E}\sigma_{33} += +\frac{1}{E}\cdot(\sigma_{22}-\nu\cdot\sigma_{11}-\nu\cdot\sigma_{33}) \] -$\varepsilon_{22}$ beschreibt die Dehnung in Achse 2 und in Richtung 2. -In erster Linie hängt $\varepsilon_{22}$ von $\sigma_{22}$ ab. -Wenn die Poisson - Zahl grösser wird oder $\sigma_{11}$ oder $\sigma_{33}$, dann wird dadurch die Dehnung $\varepsilon_{22}$ kleiner. -Das heisst, auf Kosten von Verformung in anderer Richtung als Achse 2 Richtung 2 erfolgt die Verformung an anderer Stelle. -Wiederum hat die Schubspannung auf $\sigma_{11}$ keinen Einfluss. +die Dehnung $\varepsilon_{22}$. +Diese hängt wieder am meisten von $\sigma_{22}$ ab. +Ist die Querdehnung $\nu$ grösser, so wird die Dehnung $\varepsilon_{22}$ reduziert. +Bei inkompressiblen Medien, bei welchen keine Dehnungen und nur identische Normalspannungen auftreten können, ist folglich +\[ +\nu += +0.5 +\]. + -Nun kennt man die Beziehung der 6 Dehnungen mit den 6 Spannungen. -In der Geotechnik wäre das aufgrund der vielen Komponenten sehr umständlich um damit Berechnungen zu machen. -Es braucht daher eine Vereinfachung mit Invarianten, welche im nächsten Kapitel beschrieben sind. diff --git a/buch/papers/spannung/teil3.tex b/buch/papers/spannung/teil3.tex index 500c404..e5574b8 100644 --- a/buch/papers/spannung/teil3.tex +++ b/buch/papers/spannung/teil3.tex @@ -8,6 +8,7 @@ Als erste Bedingung stellt man folgendes Verhältnis auf: = \sigma_{33} \] +. Dies deshalb, da man von einem isotropen Bodenmaterial ausgeht. In Achse 22, Richtung 22 hat man den gleichen Boden wie in Achse 33 und Richtung 33. @@ -35,6 +36,7 @@ q = \sigma_{11}-\sigma_{33} \] +. p ist das arithmetische Mittel von der Spannung im infinitesimalen Würfel. q ist die Differenz zwischen der Spannung in vertikaler Richtung und der Spannung in Richtung 2 und 3. @@ -44,7 +46,7 @@ Aus der Formel vom vorherigen Kapitel konnten wir die Spannungen berechnen. Deshalb kann man nun p und q in die Gleichung einsetzen. Die Dehnungen werden mit neuen Variablen eingeführt. Die Deviatorische Dehnung kann mit einer Schubdehnung verglichen werden. -Die hydrostatische Dehnung kann mit einer Kompressionsdehnung verglichen werden. +Die hydrostatische Dehnung kann mit einer Kompressionsdehnung verglichen \[ \overbrace{\sigma_{11}-\sigma_{33}}^{q} @@ -70,9 +72,9 @@ Die hydrostatische Dehnung kann mit einer Kompressionsdehnung verglichen werden. \text{Deviatorische Dehnung} [-] \] -Diese Komponenten kann man nun in die Vereinfachte Matrix einsetzen. -Man hat dann eine Matrix multipliziert mit einem Vektor und erhält einen Vektor. +werden. +Diese Komponenten kann man nun in die Vereinfachte Matrix \[ \begin{pmatrix} q\\ @@ -88,7 +90,9 @@ Man hat dann eine Matrix multipliziert mit einem Vektor und erhält einen Vektor \varepsilon_{\nu} \end{pmatrix} \] +einsetzen. +Man hat dann eine Matrix multipliziert mit einem Vektor und erhält einen Vektor. Mit dieser Formel lassen sich verschieden Parameter von Versuchen analysieren und berechnen. Ein solcher Versuch, den oft in der Geotechnik durchgeführt wird ist der Oedometer-Versuch. -Im nächsten Kapitel wird die Anwendung der Matrix an diesem Versuch beschrieben. +Im nächsten Kapitel wird die Anwendung der Matrix an diesem Versuch beschrieben. \ No newline at end of file diff --git a/buch/papers/spannung/teil4.tex b/buch/papers/spannung/teil4.tex index 85e9b1b..60f2518 100644 --- a/buch/papers/spannung/teil4.tex +++ b/buch/papers/spannung/teil4.tex @@ -1,68 +1,69 @@ -\section{Spannungsausbreitung\label{spannung:section:Oedometer - Versuch}} -\rhead{Oedometer - Versuch} -Beim Oedometer - Versucht hat man einen Stahlring mit einer Filterplatte am Boden. -In diesen Stahlring wird eine Bodenprobe eingefüllt. -Anschliessend wir mit einer Platte das Bodenmaterial mit einer ansteigenden Kraft belastet. - -Die Probe wird sich so verdichten. Das Volumen nimmt ab. -Der Stahlring verhindert ein seitliches ausbrechen oder entweichen der Bodenprobe. -Die Dehnung auf der Seite beträgt somit 0. -Mit dem Wert der Kraft und der Fläche lässt sich die Spannung berechnen. -Anhand der Volumenabnahme errechnet man die Dehnung. -Aus diesen Werten lässt sich wiederum das E-Modul bestimmen. -Beim Oedometer Versuch ist das E-Modul als $E_{OED}$ bezeichnet. - -Das $E_{OED}$ hat man speziell in der Geotechnik. -Dies aufgrund der speziellen Situation wo man sich mit dem infinitesimalen Würfel befindet. -Mit dem Stahlring, der verhindert das Material seitlich entweichen kann hat man ganz ähnliche Verhältnisse wie tief im Untergrund. -Auch dort kann das Material bei einer Belastung nicht seitlich entweichen. - -Wichtig ist nochmals zu betonen, dass alle diese beschriebenen Berechnungen ausschliesslich im linear-elastischen Materialverhalten funktionieren. -So ist es auch beim Oedometer - Versuch. -Den Versuch kann man auf einem $\sigma$ und $\varepsilon$ Diagramm abtragen. - -\begin{figure} - \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/DiagrammOedometer-Versuch.png} - \caption{Diagramm Oedometer - Versuch} - \label{fig:Diagramm Oedometer - Versuch} -\end{figure} - -Bei einem Versuch mit anderem Baumaterial wie beispielsweise Holz nimmt die Dehnung im Laufe des Versuchs stärker zu, obwohl weniger Spannung abgetragen wird. -Bei den meisten Böden ist dies anders. Durch die Komprimierung nimmt der Boden mehr Spannung auf, und verformt sich zugleich weniger stark. - -Man kann die Dehnung in unsere vereinfachte Matrix einsetzen. Das E-Modul ersetzt man mit dem $E_{OED}$. +\section{Oedometer-Versuch\label{spannung:section:Oedometer-Versuch}} +\rhead{Oedometer-Versuch} +Mit dem Oedometer-Versuch kann der Oedometrische Elastizitätsmodul $E_{OED}$ bestimmt werden. +Dieser beschreibt ebenfalls das Verhältnis zwischen Spannung und Dehnung, allerdings unter anderen Bedingungen. +Diese Bedingung ist das Verhindern der seitlichen Verformung, sprich der Dehnung in Richtung $1$ und $2$. +Es wird ein Probeelement mit immer grösseren Gewichten belastet, welche gleichmässig auf das Material drücken. +Die seitliche Verschiebung des Materials wird durch einen Stahlring verhindert. +Die Probe wird sich so steig verdichten. +Das Volumen nimmt ab und die Dehnung nimmt immer mehr zu. +Unter diesen Bedingungen wird das Oedometrische E-Modul mit steigender Dehnung zunehmen. +Da im Boden das umgebende Material ähnliche eine seitliche Verformung verhindert, +gibt dieser Oedometrische E-Modul die Realität besser als der gewöhnliche E-Modul wieder. +Durch dieses Verhindern des seitlichen Ausbrechens ist \[ -\overbrace{\sigma_{11}-\sigma_{33}}^{q} +\varepsilon_{22} = -\frac{3E}{2(1+\nu)} \overbrace{\frac{2}{3}(\varepsilon_{11} - 0)}^{\varepsilon_{\nu}} +\varepsilon_{33} += +0 \] - +aber auch \[ -\overbrace{\frac{\sigma_{11}+2\sigma_{33}}{3}}^{p} +\sigma_{22} = -\frac{E}{3(1-2\nu)} \overbrace{(\varepsilon_{11} - 2\cdot0)}^{\varepsilon_{s}} +\sigma_{33} +\neq 0 \] - +Die Spannung $\sigma_{11}$ wird durch durch die aufgebrachte Kraft mit +\[ +\sigma_{11} += +\frac{F}{A} +\] +und die Dehnung $\varepsilon_{11}$ jeweils mit den entsprechenden Setzungen berechnet. +Diese Randbedingen können in die vereinfachte Gleichung eingesetzt. +Diese lautet nun: \[ \begin{pmatrix} \sigma_{11}-\sigma_{33} \\ \sigma_{11}+2\sigma_{33} \end{pmatrix} = -\begin{bmatrix} +\begin{pmatrix} \frac{E_{OED}}{(1+\nu)} & 0 \\ - 0 & \frac{E_{OED}}{(1-2\nu)} -\end{bmatrix} + 0 & \frac{E_{OED}}{(1-2\nu)} +\end{pmatrix} \begin{pmatrix} \varepsilon_{11}\\ \varepsilon_{11} \end{pmatrix} \] +. -An einem geeigneten Punkt, wo man noch im linear-elastischen Materialverhalten ist, kann man nun das $E_{OED}$ abtragen. -Es wird nur ein Delta betrachtet um $E_{OED}$ zu berechnen. -Man darf die Dehnung nicht über den gesamten Verlauf betrachten um $E_{OED}$ zu berechnen. +Daraus lässt sich bei jedem Setzungsgrad das Oedometrische E-Modul $E_{OED}$ und die seitlichen Spannungen $\sigma_{33}$ mit den 2 Gleichungen -Mit diesem ermittelten E-Modul kann man nun weitere Berechnungen für die Geotechnik durchführen. +GLEICHUNGEN... + +berechnen. +Den Versuch kann man auf einem $\sigma$-$\varepsilon$-Diagramm abtragen (siehe Abbildung 1.7). +Durch die Komprimierung nimmt der Boden mehr Spannung auf, und verformt sich zugleich weniger stark. +Mit diesem ermittelten $E_{OED}$ kann man nun weitere Berechnungen für die Geotechnik durchführen. + +\begin{figure} + \centering + \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/DiagrammOedometer-Versuch.png} + \caption{Diagramm Oedometer-Versuch} + \label{fig:Diagramm Oedometer-Versuch} +\end{figure} \ No newline at end of file -- cgit v1.2.1 From e86e0ad0e4415450a9c8b28917024ee6d0d77da5 Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Fri, 28 May 2021 15:23:51 +0200 Subject: text added --- buch/papers/reedsolomon/rekonstruktion.tex | 204 ++++++++++++++++++++++++----- 1 file changed, 174 insertions(+), 30 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/rekonstruktion.tex b/buch/papers/reedsolomon/rekonstruktion.tex index a3edba4..8cb7744 100644 --- a/buch/papers/reedsolomon/rekonstruktion.tex +++ b/buch/papers/reedsolomon/rekonstruktion.tex @@ -5,36 +5,180 @@ % \section{Nachricht Rekonstruieren \label{reedsolomon:section:rekonstruktion}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? +\rhead{Rekonstruktion} +Im letzten Kapitel haben wir eine Möglichkeit gefunden, wie wir die Fehlerhaften Stellen lokalisieren können. +Mit diesen Stellen soll es uns nun möglich sein, aus dem fehlerhaften empfangenen Nachrichtenvektor wieder unsere Nachricht zu rekonstruieren. +Das Lokatorpolynom +\[ +d(X) = (X - a^3)(X-a^8) +\] +markiert dabei diese Fehlerhaften Stellen im Übertragungsvektor +\[ +w = [5,3,6,8,2,10,2,7,1,4]. +\] +Als Ausgangslage verwenden wir die Matrix, mit der wir den Nachrichtenvektor ursprünglich codiert haben. +Unser Ziel ist es wie auch schon im Kapitel X.X (Rekonstuktion ohne Fehler) eine Möglichkeit zu finden, wie wir den Übertragungsvektor decodieren können. +Aufgrund der Fehlerstellen müssen wir aber davon ausgehen, das wir nicht mehr den gleichen Weg verfolgen können wie wir im Kapitel X.X angewendet haben. -\subsection{De finibus bonorum et malorum -\label{reedsolomon:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +Wir stellen also die Matrix auf und markieren gleichzeitig die Fehlerstellen. +\[ +\textcolor{gray}{ + \begin{pmatrix} + a^0 \\ a^1 \\ a^2 \\ \textcolor{red}{a^3} \\ a^4 \\ a^5 \\ a^6 \\ a^7 \\ \textcolor{red}{a^8} \\ a^9 \\ +\end{pmatrix}} +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ \textcolor{red}{8} \\ 2 \\ 10 \\ 2 \\ 7 \\ \textcolor{red}{1} \\ 4 \\ +\end{pmatrix} += +\begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^3}& \textcolor{red}{8^6}& \textcolor{red}{8^9}& \textcolor{red}{8^{12}}& \textcolor{red}{8^{15}}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{27}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^8}& \textcolor{red}{8^{16}}& \textcolor{red}{8^{24}}& \textcolor{red}{8^{32}}& \textcolor{red}{8^{40}}& \textcolor{red}{8^{48}}& \textcolor{red}{8^{56}}& \textcolor{red}{8^{64}}& \textcolor{red}{8^{72}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ +\end{pmatrix} +\] +Die rot markierten Stellen im Übertragungsvektor enthalten Fehler und bringt uns daher kein weiterer Nutzen. +Aus diesem Grund werden diese Stellen aus dem Vektor entfernt, was wir hier ohne Probleme machen können, da dieser Code ja über Fehlerkorrekturstellen verfügt, deren Aufgabe es ist, eine bestimmte Anzahl an Fehler kompensieren zu können. +Die dazugehörigen Zeilen in der Matrix werden ebenfalls entfernt, da die Matrix gleich viele Zeilen wie im Übertragungsvektor aufweisen muss, damit man ihn decodieren kann. +Daraus resultiert +\[ +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ +\end{pmatrix} += +\begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& 8^6& 8^7& 8^8& 8^9\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& 8^{12}& 8^{14}& 8^{16}& 8^{18}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& 8^{24}& 8^{28}& 8^{32}& 8^{36}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& 8^{30}& 8^{35}& 8^{40}& 8^{45}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& 8^{36}& 8^{42}& 8^{48}& 8^{54}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& 8^{42}& 8^{49}& 8^{56}& 8^{63}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& 8^{54}& 8^{63}& 8^{72}& 8^{81}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ +\end{pmatrix} +. +\] +Die Matrix ist jedoch nicht mehr quadratisch, was eine Rekonstruktion durch Inversion ausschliesst. +Um die quadratische Form wieder herzustellen müssen wir zwei Spalten aus der Matrix entfernen. +Wir kennen aber das Resultat aus den letzten vier Spalten, da wir wissen, das die Nachricht aus Nutzdatenteil und Fehlerkorrekturteil besteht, wobei der letzteres bekanntlich aus lauter Nullstellen besteht. +\[ +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ 7 \\ 4 \\ +\end{pmatrix} += +\begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& \textcolor{green}{8^0}& \textcolor{green}{8^0}& \textcolor{green}{8^0}& \textcolor{green}{8^0}\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& \textcolor{green}{8^6}& \textcolor{green}{8^7}& \textcolor{green}{8^8}& \textcolor{green}{8^9}\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& \textcolor{green}{8^{12}}& \textcolor{green}{8^{14}}& \textcolor{green}{8^{16}}& \textcolor{green}{8^{18}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor{green}{8^{24}}& \textcolor{green}{8^{28}}& \textcolor{green}{8^{32}}& \textcolor{green}{8^{36}}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor{green}{8^{30}}& \textcolor{green}{8^{35}}& \textcolor{green}{8^{40}}& \textcolor{green}{8^{45}}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor{green}{8^{36}}& \textcolor{green}{8^{42}}& \textcolor{green}{8^{48}}& \textcolor{green}{8^{54}}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor{green}{8^{42}}& \textcolor{green}{8^{49}}& \textcolor{green}{8^{56}}& \textcolor{green}{8^{63}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor{green}{8^{54}}& \textcolor{green}{8^{63}}& \textcolor{green}{8^{72}}& \textcolor{green}{8^{81}}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor{green}{m_6} \\ \textcolor{green}{m_7} \\ \textcolor{green}{m_8} \\ \textcolor{green}{m_9} \\ +\end{pmatrix} +\] +Wir nehmen die Entsprechenden Spalten aus der Matrix heraus und erhalten so das Überbestimmte Gleichungssystem +\[ +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor{red}{7} \\ \textcolor{red}{4} \\ +\end{pmatrix} += +\begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^7}& \textcolor{red}{8^{14}}& \textcolor{red}{8^{21}}& \textcolor{red}{8^{28}}& \textcolor{red}{8^{35}}\\ + \textcolor{red}{8^0}& \textcolor{red}{8^9}& \textcolor{red}{8^{18}}& \textcolor{red}{8^{27}}& \textcolor{red}{8^{36}}& \textcolor{red}{8^{45}}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ +\end{pmatrix} +. +\] +Die roten Zeilen können wir aufgrund der Überbestimmtheit ebenfalls entfernen und erhalten so die gesuchte quadratische Matrix +\[ +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ +\end{pmatrix} += +\begin{pmatrix} + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ +\end{pmatrix} +. +\] +Nun können wir den Gauss-Algorithmus anwenden um die Matrix zu Invertieren. +\[ +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ +\end{pmatrix} += +\begin{pmatrix} + 1& 1& 1& 1& 1& 1\\ + 1& 8& 9& 6& 4& 10\\ + 1& 9& 4& 3& 5& 1\\ + 1& 4& 5& 9& 3& 1\\ + 1& 10& 1& 10& 1& 10\\ + 1& 3& 9& 5& 4& 1\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ +\end{pmatrix} +\qquad +\Rightarrow +\qquad +\begin{pmatrix} + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ +\end{pmatrix} += +\begin{pmatrix} + 6& 4& 4& 6& 2& 1\\ + 2& 7& 10& 3& 4& 7\\ + 1& 8& 9& 8& 3& 4\\ + 3& 6& 6& 4& 5& 9\\ + 10& 10& 9& 8& 1& 6\\ + 1& 9& 6& 4& 7& 6\\ +\end{pmatrix} +\cdot +\begin{pmatrix} + 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ +\end{pmatrix} +\] +Multiplizieren wir nun aus, erhalten wir unseren Nutzdatenteil +\[ +m = [4,7,2,5,8,1] +\] +zurück, den wir ursprünglich versendet haben. -- cgit v1.2.1 From 98bca60d5b3d77f0396903747f70ea2b3c7ad5bd Mon Sep 17 00:00:00 2001 From: Malarius1999 Date: Sun, 30 May 2021 23:14:28 +0200 Subject: First Version Nicht sicher, ob das Buch kompiliert, weil ich nicht weiss wie man alles zusammen kompiliert. In einem separaten File hat es aber geklappt. Ich bin auch nicht sicher welche Packages wirklich alle notwendig sind. --- buch/papers/clifford/0_ElevatorPitch.tex | 2 + buch/papers/clifford/10_Quaternionen.tex | 60 ++++++++ buch/papers/clifford/1_Vektordarstellung.tex | 71 +++++++++ buch/papers/clifford/2_QuadratVektoren.tex | 110 ++++++++++++++ buch/papers/clifford/3_MultiplikationVektoren.tex | 175 ++++++++++++++++++++++ buch/papers/clifford/4_GeometrischesProdukt.tex | 59 ++++++++ buch/papers/clifford/5_PolareDarstellung.tex | 29 ++++ buch/papers/clifford/6_Dirac-Matrizen.tex | 7 + buch/papers/clifford/7_Reflektion.tex | 32 ++++ buch/papers/clifford/8_Rotation.tex | 99 ++++++++++++ buch/papers/clifford/9_KomplexeZahlen.tex | 27 ++++ buch/papers/clifford/main.tex | 19 ++- buch/papers/clifford/packages.tex | 23 ++- buch/papers/clifford/papers/clifford/teil0.tex | 0 buch/papers/clifford/teil0.tex | 22 --- buch/papers/clifford/teil1.tex | 55 ------- buch/papers/clifford/teil2.tex | 40 ----- buch/papers/clifford/teil3.tex | 40 ----- 18 files changed, 706 insertions(+), 164 deletions(-) create mode 100644 buch/papers/clifford/0_ElevatorPitch.tex create mode 100644 buch/papers/clifford/10_Quaternionen.tex create mode 100644 buch/papers/clifford/1_Vektordarstellung.tex create mode 100644 buch/papers/clifford/2_QuadratVektoren.tex create mode 100644 buch/papers/clifford/3_MultiplikationVektoren.tex create mode 100644 buch/papers/clifford/4_GeometrischesProdukt.tex create mode 100644 buch/papers/clifford/5_PolareDarstellung.tex create mode 100644 buch/papers/clifford/6_Dirac-Matrizen.tex create mode 100644 buch/papers/clifford/7_Reflektion.tex create mode 100644 buch/papers/clifford/8_Rotation.tex create mode 100644 buch/papers/clifford/9_KomplexeZahlen.tex create mode 100644 buch/papers/clifford/papers/clifford/teil0.tex delete mode 100644 buch/papers/clifford/teil0.tex delete mode 100644 buch/papers/clifford/teil1.tex delete mode 100644 buch/papers/clifford/teil2.tex delete mode 100644 buch/papers/clifford/teil3.tex (limited to 'buch/papers') diff --git a/buch/papers/clifford/0_ElevatorPitch.tex b/buch/papers/clifford/0_ElevatorPitch.tex new file mode 100644 index 0000000..a599903 --- /dev/null +++ b/buch/papers/clifford/0_ElevatorPitch.tex @@ -0,0 +1,2 @@ + +GA [Geometric Algebra i.a.W. Clifford Algebra] provides a unified language for the whole of physics and for much of mathematics and its applications that is conceptually and computationally superior to alternative mathematical systems in many application domains. \ No newline at end of file diff --git a/buch/papers/clifford/10_Quaternionen.tex b/buch/papers/clifford/10_Quaternionen.tex new file mode 100644 index 0000000..c987fc3 --- /dev/null +++ b/buch/papers/clifford/10_Quaternionen.tex @@ -0,0 +1,60 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Quaternionen} +Wie die komplexen Zahlen eine Erweiterung der reellen Zahlen sind, sind die Quaternionen eine Erweiterung der komplexen Zahlen für den 3 dimensionalen Raum. Sie haben, wie die komplexen Zahlen, eine dreh-streckende Eigenschaft. +Sie finden beispielsweise in der Computergraphik und in der Robotik Anwendung. +Die Quaternionen werden so definiert. +\begin{align} + q = w + xi + yj + zk; \quad w,x,y,z \in \mathbb{R};\enspace q \in \mathbb{H} +\end{align} +Eine Drehstreckung wird dabei mit dieser Formel erreicht. +\begin{align} \label{QuatRot} + \begin{split} + &v'' = qvq^{-1};\quad q,v,q^{-1} \in \mathbb{H}\\ + &Re(q) = Re(q^{-1});\enspace Im(q) = -Im(q^-1) + \end{split} +\end{align} +Die Quaternionen besitzen im Gegensatz zu dem komplexen Zahlen 3 imaginäre Einheiten $i,j,k$. Wieso 3? Weil es in der dritten Dimension 3 Drehachsen gibt, anstatt nur eine. Nun haben wir ein kleines Problem. Wie sollen wir die Quaternionen darstellen? Wir bräuchten 4 Achsen für die 3 Imaginären Einheiten und die eine reelle Einheit. Ein weiterer Nachteil in visueller Hinsicht entsteht beim Anwenden eines Quaternion auf einen Vektor. Sie befinden sich nicht im gleichen Raum und müssen zuerst ineinander umgewandelt werden, um damit zu rechnen, wie man bei $v$ in der Formel (\ref{QuatRot}) sieht. + +\subsection{geometrischen Algebra} +Die geometrische Algebra besitzt die Fähigkeit beide Probleme zu lösen. Die Quaternionen können, wie schon im 2 dimensionalen Fall durch die gerade Grade $\mathbb{G}_3^+ \cong \mathbb{H}$ dargestellt werden. Da wir uns jetzt aber in $\mathbb{G}_3$ befinden haben wir 3 Basisvektoren $e_1, e_2, e_3$ und können somit 3 Bivektoren bilden $e_{12}, e_{23}, e_{31}$. +\begin{align} + \mathbf{q} = w + x\mathbf{e_{12}} + y\mathbf{e_{23}} + z\mathbf{e_{31}}; \quad w,x,y,z \in \mathbb{R};\enspace q \in \mathbb{G}_3^+ +\end{align} +Die Probleme werden dadurch gelöst, da wir die Bivektoren im Raum nicht durch einzelne Achsen darstellen müssen, sondern sie als eine orientiere Fläche darstellen können. Anstatt die Vektoren in Quaternionen umzurechnen, können wir jetzt die Vektoren separat im gleichen Raum darstellen. +\\BILD VEKTOR, QUATERNION IN G3\\ +Wie schon im 2 dimensionalen Fall beschreibt ein Bivektor, um wie viel der um 90 grad gedrehte orginale Vektor gestreckt wird. Dabei dreht jeder Bivektor den Vektor um eine andere Achse. +\\BILD?\\ +In der Computergraphik und Robotik macht eine Drehstreckung aber nicht viel Sinn. Wieso sollte ein Objekt bei einer Drehung zusätzlich noch grösser werden? Darum verwendet man sogenannte Einheitsquaternion, welche den Betrag $|q|=1$ haben. Sie rotieren die Objekte bzw. Vektoren lediglich. +\begin{align} + \mathbf{q} = \cos(\alpha) + sin(\alpha)(x\mathbf{e_{12}} + y\mathbf{e_{23}} + z\mathbf{e_{31}}) +\end{align} +wobei definiert ist, dass $x^2+y^2+z^2=1$. Somit beträgt der Betrag immer 1. +\begin{align} + |q| = \sqrt{cos(\alpha)^2 + sin(\alpha)^2(x^2+y^2+z^2) } = \sqrt{cos(\alpha)^2 + sin(\alpha)^2} = 1 +\end{align} +Man verwendet um einen Vektor zu drehen wieder die gleiche Formel, wie auch schon im 2 dimensionalen Fall. +\begin{align} \label{QuatRot} + \begin{split} + &v'' = qvq^{-1}\\ + &Re(q) = Re(q^{-1});\enspace Im(q) = -Im(q^-1) + \end{split} +\end{align} +Es ist wichtig bei Quaternionen für eine reine Drehstreckung mit $q$ und $q^{-1}$ beidseitig zu multiplizieren, sonst werden die senkrechten Anteile zu den Bivektorebenen ebenfalls beeinflusst, wie man im Kapitel Rotation bei der Formel (\ref{RotAufPerpPar}) sehen kann + +\subsection{Gimbal-Lock und Interpolation} + +\subsection{Fazit} +andere Darstellungsweise. Besser für Verständnis => komplexe Zahlen erscheinen ähnlicher zu Quaternionen? Eine Sprache für alle Geometrische Probleme + + +\begin{tikzpicture} + \draw[thin,gray!40] (-3,-3) grid (3,3); + \draw[<->] (-3,0)--(3,0) node[right]{$x$}; + \draw[<->] (0,-3)--(0,3) node[above]{$y$}; + \draw[line width=2pt,blue,-stealth](0,0)--(1,1) node[anchor=south west]{$\boldsymbol{u}$}; + \draw[line width=2pt,red,-stealth](0,0)--(-1,-1) node[anchor=north east]{$\boldsymbol{-u}$}; +\end{tikzpicture} \ No newline at end of file diff --git a/buch/papers/clifford/1_Vektordarstellung.tex b/buch/papers/clifford/1_Vektordarstellung.tex new file mode 100644 index 0000000..cb6e7af --- /dev/null +++ b/buch/papers/clifford/1_Vektordarstellung.tex @@ -0,0 +1,71 @@ +\section{Teil 0\label{clifford:section:Vektoroperationen}} +\rhead{Vektoroperationen} +\rhead{Vektordarstellung} +Vektoren können neben der üblichen Darstellung, auch als Linearkombination aus Basisvektoren dargestellt werden +\begin{equation} + \begin{split} + \textbf{a} + &= + \begin{pmatrix} + a_1 \\ a_2 \\ \vdots \\ a_n + \end{pmatrix} + = + a_1 \begin{pmatrix} + 1 \\ 0 \\ \vdots \\ 0 + \end{pmatrix} + + + a_2\begin{pmatrix} + 0 \\ 1 \\ \vdots \\ 0 + \end{pmatrix} + \dots + + + a_n\begin{pmatrix} + 0 \\ 0 \\ \vdots \\ 1 + \end{pmatrix} \\\ + &= + a_1\textbf{e}_1 + + + a_2\textbf{e}_2 + + + \dots + a_n\textbf{e}_n + = + \sum_{i=1}^{n} a_i \textbf{e}_i + \qquad + a_i \in \mathbb{R} + , \textbf{e}_i \in \mathbb{R}^n. + \end{split} +\end{equation} +Diese Basisvektoren sollen orthonormal sein und um die Darstellung zu vereinfachen werden sie durch $\textbf{e}_1 , \textbf{e}_2, ...$ ersetzt. +\begin{beispiel} +Linearkombination von Basisvektoren in $\mathbb{R}^4$ + \begin{equation} + \begin{pmatrix} + 42 \\ 2 \\ 1291 \\ 4 + \end{pmatrix} + = + 42 \begin{pmatrix} + 1 \\ 0 \\ 0 \\ 0 + \end{pmatrix} + + + 2 \begin{pmatrix} + 0 \\ 1 \\ 0 \\ 0 + \end{pmatrix} + + + 1291 + \begin{pmatrix} + 0 \\ 0 \\ 1 \\ 0 + \end{pmatrix} + + + 4 \begin{pmatrix} + 0 \\ 0 \\ 0 \\ 1 + \end{pmatrix} + = + 42\textbf{e}_1 + + + 2\textbf{e}_2 + + + 1291\textbf{e}_3 + + + 4\textbf{e}_4 + \end{equation} +\end{beispiel} +Wobei Beispiel für einen vier dimensionalen Vektor ist, dies kann selbstverständlich für beliebig viele Dimensionen nach demselben Schema erweitert werden. \ No newline at end of file diff --git a/buch/papers/clifford/2_QuadratVektoren.tex b/buch/papers/clifford/2_QuadratVektoren.tex new file mode 100644 index 0000000..cfb05d6 --- /dev/null +++ b/buch/papers/clifford/2_QuadratVektoren.tex @@ -0,0 +1,110 @@ +\subsection{Quadrat von Vektoren} +Was eine Addition von Vektoren bedeutet ist sehr intuitiv und auch leicht geometrisch darzustellen, was allerdings das Produkt von Vektoren ergibt mag anfänglich unintuitiv wirken. +Was soll es schon heissen zwei Vektoren miteinander zu multiplizieren? +\newline +Im Folgenden werden wir versuchen diese Operation ähnlich intuitiv darzustellen. +\newline +Um sinnvoll eine neue Operation zwischen zwei Elementen einer Algebra, in diesem Fall Vektoren, zu definieren, muss man überlegen, was das Ziel dieser Operation ist. +Als grundsätzliches Ziel wird definiert, dass das Quadrat eines Vektor dessen Länge im Quadrat ergibt, da dies auch in vielen anderen Bereichen der Mathematik,zum Beispiel bei komplexen Zahlen, auch so definiert ist. +\newline +Zusätzlich wollen wir auch das Assoziativgesetz und das Kommutativgesetz für Skalare beibehalten. Wobei das Kommutativgesetz leider, oder wie man sehen wird zum Glück, in der geometrischen Algebra im generellen nicht mehr gilt. Das heisst wir dürfen ausklammern \ref{eq:assoziativ} und die Position von Skalaren im Produkt ändern \ref{eq:kommSkalar}, allerdings nicht die Position der Vektoren \ref{eq:kommVector}. +\begin{equation} + \label{eq:assoziativ} + \textbf{e}_i(\textbf{e}_j + \textbf{e}_k) + = + \textbf{e}_i\textbf{e}_j + \textbf{e}_i\textbf{e}_k +\end{equation} +\begin{equation} + \label{eq:kommSkalar} + a\textbf{e}_ib\textbf{e}_j + = + ab\textbf{e}_i\textbf{e}_j +\end{equation} +\begin{equation} + \label{eq:kommVector} + \textbf{e}_i\textbf{e}_j + \neq + \textbf{e}_j\textbf{e}_i +\end{equation} +Betrachten wir nun mit diesen Regeln das Quadrat eines Vektors. +\begin{align} + \textbf{a}^2 &= + \left ( + \sum_{i=1}^{n} a_i \textbf{e}_i + \right ) + \left ( + \sum_{i=1}^{n} a_i \textbf{e}_i + \right ) + \label{eq:quad_a_1} + \\ + &= + \textcolor{red}{\sum_{i=1}^{n} a_i^2\textbf{e}_i^2} + + + \textcolor{blue}{\sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j } + \label{eq:quad_a_2} + \\ + &= \textcolor{cyan}{\sum_{i=1}^{n} a_i^2} + \textcolor{orange}{\sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j}. + \label{eq:quad_a_3} +\end{align} + +\begin{beispiel} +Quadrat eines Vektors in $\mathbb{R}^2$ +\begin{equation} + \begin{split} + \textbf{a}^2 + &= (a_1\textbf{e}_1+a_2\textbf{e}_2)(a_1\textbf{e}_1+a_2\textbf{e}_2) \\\ + &= \textcolor{red}{a_1^2\textbf{e}_1^2 + a_2^2\textbf{e}_2^2} + + \textcolor{blue}{a_1\textbf{e}_1a_2\textbf{e}_2 + a_2\textbf{e}_2a_1\textbf{e}_2} \\\ + & = \textcolor{cyan}{a_1^2 + a_2^2} + \textcolor{orange}{a_1b\textbf{e}_1a_2\textbf{e}_2 + a_2\textbf{e}_2a_1\textbf{e}_2} + \end{split} +\end{equation} + +\end{beispiel} +Der Vektor wird in \ref{eq:quad_a_1} als Linearkombination geschrieben. +Das Quadrat kann, wie in \ref{eq:quad_a_2} gezeigt, in zwei Summen aufteilen werden , wobei die roten Summe die quadrierten Terme und die blaue Summe die Mischterme beinhaltet. +\newline +Da $\textbf{e}_i^2 = 1$ gilt, da zuvor vorausgesetzt wurde, dass man mit orthonormalen Einheitsvektoren arbeitet, wird dies nun eingesetzt ergibt sich \ref{eq:quad_a_3} +\newline +Die hellblaue Teil ist nun bereits Länge im Quadrat eines Vektors, also das Ziel der Multiplikation. +Daher muss der restliche Teil dieser Gleichung null ergeben. +Aus dieser Erkenntnis leiten wir in \ref{eq:Mischterme_Null} weitere Eigenschaften für die Multiplikation her. +\begin{equation} + \label{eq:Mischterme_Null} + \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j = \textcolor{blue}{a_1a_2(\textbf{e}_1\textbf{e}_2 + \textbf{e}_2\textbf{e}_1)} + a_1a_3(\textbf{e}_1\textbf{e}_3 + \textbf{e}_3\textbf{e}_1) + \dots = 0 +\end{equation} +Da dies für beliebige $a_i$ gelten muss werden alle Terme bis auf $a_1$ und $a_2$ gleich null gesetzt. Somit fallen alle Terme bis auf den blauen weg. Wird dies weiter vereinfacht ergibt sich +\begin{equation} +\begin{split} + a_1a_2(\textbf{e}_1\textbf{e}_2 + \textbf{e}_2\textbf{e}_1) &= 0 \\ + a_1a_2\textbf{e}_1\textbf{e}_2 &= -a_1a_2\textbf{e}_2\textbf{e}_1 \\ + \textbf{e}_1\textbf{e}_2 &= -\textbf{e}_2\textbf{e}_1. +\end{split} +\end{equation} +\begin{satz} + Die Multiplikation von Vektoren ist antikommutativ, wenn die multiplizierten Vektoren orthogonal sind. + \begin{equation} + \textbf{e}_i\textbf{e}_j = -\textbf{e}_j\textbf{e}_i \qquad \textbf{e}_i \perp \textbf{e}_j + \end{equation} +\end{satz} +Dieses Wissen reicht nun bereits um alle Produkte der Basisvektoren zu berechnen, was in \ref{tab:multip_vec} gemacht wurde. +\begin{table} +\caption{Multiplikationstabelle für Vektoren} +\label{tab:multip_vec} +\begin{center} +\begin{tabular}{ |c|c|c|c|c|c| } + \hline + & $\textbf{e}_1$ & $\textbf{e}_2$ & $\dots$ & $\textbf{e}_{n-1}$ & $\textbf{e}_{n}$ \\ + \hline + $\textbf{e}_1$ & 1 & $\textbf{e}_1\textbf{e}_2$ & $\dots$ & $\textbf{e}_1\textbf{e}_{n-1}$ & $\textbf{e}_1\textbf{e}_{n}$ \\ + \hline + $\textbf{e}_2$ & $-\textbf{e}_1\textbf{e}_2$ & 1 & $\dots$ & $\textbf{e}_2\textbf{e}_{n-1}$ & $\textbf{e}_2\textbf{e}_{n}$ \\ + \hline + $\vdots$ & $\vdots$ & $\vdots$ & $\ddots$ & $\vdots$ & $\vdots$ \\ + \hline + $\textbf{e}_{n-1}$ & $-\textbf{e}_1\textbf{e}_{n-1}$ & $-\textbf{e}_2\textbf{e}_{n-1}$ & $\dots$ & $1$ & $\textbf{e}_{n-1}\textbf{e}_{n}$ \\ + \hline + $\textbf{e}_{n}$ & $-\textbf{e}_1\textbf{e}_{n}$ & $-\textbf{e}_2\textbf{e}_{n}$ & $\dots$ & $-\textbf{e}_{n-1}\textbf{e}_{n}$ & 1 \\ + \hline +\end{tabular} +\end{center} +\end{table} \ No newline at end of file diff --git a/buch/papers/clifford/3_MultiplikationVektoren.tex b/buch/papers/clifford/3_MultiplikationVektoren.tex new file mode 100644 index 0000000..841dde4 --- /dev/null +++ b/buch/papers/clifford/3_MultiplikationVektoren.tex @@ -0,0 +1,175 @@ +\subsection{Multiplikation von Vektoren} +Was geschieht nun wenn zwei beliebige Vektoren,$u$ und $v$, miteinander multipliziert werden? +\begin{equation} + \textbf{u} = + \sum_{i=1}^{n} u_i \textbf{e}_i + \qquad + \textbf{v} = \sum_{i=1}^{n} v_i \textbf{e}_i +\end{equation} +\begin{equation} + \begin{split} + \textbf{u}\textbf{v} + = + \left ( + \sum_{i=1}^{n} u_i \textbf{e}_i + \right ) + \left ( + \sum_{i=1}^{n} v_i \textbf{e}_i + \right) + = + \sum_{i=1}^n u_iv_i\underbrace{\textbf{e}_i^2}_{1} + + \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j + \end{split} +\end{equation} +\begin{beispiel} + Multiplikation von Vektoren in $\mathbb{R}^2$ +\end{beispiel} +\begin{equation} + \begin{split} + \textbf{u}\textbf{v} + &= + (u_1\textbf{e}_1 + u_2\textbf{e}_2)(v_1\textbf{e}_1 + v_2\textbf{e}_2) + = + u_1v_1\textbf{e}_1^2 + + + u_2v_2\textbf{e}_2^2 + + + u_1v_2\textbf{e}_1\textbf{e}_2 + + + u_2v_1\underbrace{\textbf{e}_2\textbf{e}_1}_{-\textbf{e}_1\textbf{e}_2} + \\\ + &= + \underbrace{(u_1v_1 + u_2v_2)}_{\text{Skalarprodukt}} + + + \underbrace{(u_1v_2 - u_2v_1)\textbf{e}_1\textbf{e}_2}_{\text{Äusseres Produkt}} + \end{split} +\end{equation} +Der linke Teil dieser Multiplikation ergibt das Skalarprodukt der zwei Vektoren, der rechte Term ergibt etwas neues das sich das äussere Produkt der zwei Vektoren nennt. +\subsubsection{Äusseres Produkt} +Das äussere Produkt von zwei Vektoren wird mit einem $\wedge$ dargestellt +\begin{equation} + \textbf{u}\wedge \textbf{v} + = + \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j +\end{equation} +\begin{beispiel} +Äusseres Produkt von zwei Vektoren in $\mathbb{R}^3$ +\end{beispiel} +\begin{equation} + \begin{split} + u \wedge v + &= + u_1v_2\textbf{e}_1\textbf{e}_2 + + + u_1v_3\textbf{e}_1\textbf{e}_3 + + + u_2v_2\textbf{e}_2\textbf{e}_3 + + + u_2v_1\textbf{e}_2\textbf{e}_1 + + + u_3v_1\textbf{e}_3\textbf{e}_1 + + + u_3v_2\textbf{e}_3\textbf{e}_2 \\\ + &= + (u_1v_2 - u_2v_1)\textbf{e}_1\textbf{e}_2 + + + (u_1v_3 - v_3u_1)\textbf{e}_1\textbf{e}_3 + + + (u_2v_3 - u_3v_2)\textbf{e}_2\textbf{e}_3 + \end{split} +\end{equation} +Im letzten Schritt des Beispiels wurden nun, mit Hilfe der antikommutativität des Produkts, die Vektorprodukte, welche die gleichen Einheitsvektoren beinhalten, zusammengefasst. Dieses Vorgehen kann man auch allgemein anwenden, wie in den Gleichungen \ref{eq:u_wedge_v}-\ref{eq:u_wedge_v_5} hergeleitet. +\begin{align} + \textbf{u}\wedge \textbf{v} + &= + \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n + u_iv_j\textbf{e}_i\textbf{e}_j + \label{eq:u_wedge_v} + \\ + \label{eq:u_wedge_v_1} + &= + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j + + + \sum_{\begin{subarray}{l}i,j=1\\j < i\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j + \\ + \label{eq:u_wedge_v_2} + &= + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j + + + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_jv_i\textbf{e}_j\textbf{e}_i + \\ + \label{eq:u_wedge_v_3} + &= + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j + - + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_jv_i\textbf{e}_i\textbf{e}_j + \\ + \label{eq:u_wedge_v_4} + &= + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n (u_iv_j -u_jv_i)\textbf{e}_i\textbf{e}_j + \\ + \label{eq:u_wedge_v_5} + &= + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n \begin{vmatrix} + u_i & v_i \\ + u_j & v_j + \end{vmatrix}\textbf{e}_i\textbf{e}_j +\end{align} +Die Summe aus \ref{eq:u_wedge_v_1} wird in \ref{eq:u_wedge_v} in zwei verschiedene Summen aufgeteilt. +Wobei die linke Summe jeweils den Basisvektor mit dem höheren Index an erster Stelle und die rechte Summe diesen jeweils an zweiter Stelle hat. +\newline +Bei \ref{eq:u_wedge_v_2} werden die Indexe der zweiten Summe vertauscht, damit man nun bei beiden Teilen die gleiche Summe hat. +Danach werden in \ref{eq:u_wedge_v_3}, mit Hilfe der Antikommutativität, die Einheitsvektoren der zweiten Summe vertauscht. +\newline +Nun können die Summen, wie in \ref{eq:u_wedge_v_4} wieder in eine Summe zusammengefasst werden. +\newline +Der Term in der Klammer in \ref{eq:u_wedge_v_4} kann auch als Determinante einer 2x2 Matrix dargestellt werden, was in \ref{eq:u_wedge_v_5} gemacht wird. +\newline +Die Determinante einer Matrix beschreibt welche von den Spaltenvektoren aufgespannt wird, wie in Abbildung \ref{figure:det} dargestellt. +\begin{figure} +\centering +\begin{tikzpicture} + \draw[thin,gray!40] (0,0) grid (4,4); + \draw[<->] (0,0)--(4,0) ; + \draw[<->] (0,0)--(0,4) ; + \draw[line width=0,fill=gray!40] (0,0)--(3,1)--(4,3)--(1,2); + \draw[line width=2pt,blue,-stealth](0,0)--(3,1) node[anchor=north + west]{$\boldsymbol{u}$}; + \draw[line width=2pt,red,-stealth](0,0)--(1,2) node[anchor=south east]{$\boldsymbol{v}$}; + \draw[black] (2,1.5)--(-0.5,2.5) node[anchor = east]{$\begin{vmatrix} + u_i & v_i \\ + u_j & v_j + \end{vmatrix} = u_iv_j - v_iu_j$}; +\end{tikzpicture} +\caption{Geometrische Interpretation der Determinante einer 2x2 Matrix\label{figure:det}} +\end{figure} +\newline +Das äussere Produkt besteht nun also aus der Summe + $\sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n$ + von Flächen + $\begin{vmatrix} + u_i & v_i \\ + u_j & v_j + \end{vmatrix}$, welche in $\textbf{e}_i\textbf{e}_j$ aufgespannt sind, wie man in \ref{eq:u_wedge_v_5} sieht. +Dieses Produkt $\textbf{e}_i\textbf{e}_j$ der Basisvektoren interpretiert man als Umlaufrichtung. +Wobei die gebildete Fläche in Richtung des ersten Vektors umschritten wird. +Dies ist in \ref{figure:wedge} dargestellt, wobei bei diesem Beispiel die Umlaufrichtung im Gegenuhrzeigersinn ist, da die Fläche in Richtung u umschritten wird. +Diese Fläche mit einer Richtung nennt man in der geometrischen Algebra einen Bivektor, da er eine Art zwei dimensionaler Vektor ist. +\begin{figure} +\centering +\begin{tikzpicture} + \draw[thin,gray!40] (0,0) grid (4,4); + \draw[<->] (0,0)--(4,0) node[right]{$x$}; + \draw[<->] (0,0)--(0,4) node[above]{$y$}; + \draw[line width=0,fill=gray!40] (0,0)--(3,1)--(4,3)--(1,2); + \draw[line width=2pt,blue,-stealth](0,0)--(3,1) node[anchor=north + west]{$\boldsymbol{u}$}; + \draw[line width=2pt,red,-stealth](0,0)--(1,2) node[anchor=south east]{$\boldsymbol{v}$}; + \draw[->] (2.15,1.5) arc (0:310:0.3); + \draw[black] (2,1.5)--(-0.5,2.5) node[anchor = east]{$u\wedge v = \begin{vmatrix} + u_i & v_i \\ + u_j & v_j + \end{vmatrix} e_1e_2 = (u_iv_j - v_iu_j)\textbf{e}_1\textbf{e}_2$}; +\end{tikzpicture} +\caption{Geometrische Interpretation des äusseren Produkt in $\mathbb{R}^2$\label{figure:wedge}} +\end{figure} \ No newline at end of file diff --git a/buch/papers/clifford/4_GeometrischesProdukt.tex b/buch/papers/clifford/4_GeometrischesProdukt.tex new file mode 100644 index 0000000..a19e983 --- /dev/null +++ b/buch/papers/clifford/4_GeometrischesProdukt.tex @@ -0,0 +1,59 @@ +\subsection{Geometrisches Produkt} +Die Multiplikation von zwei Vektoren nennt man in der Clifford Algebra das geometrische Produkt, dieses können wir nun als Summe aus dem Skalar- und dem äusseren Produkt darstellen +\begin{equation} + \textbf{u}\textbf{v} = \textbf{u}\cdot \textbf{v} + \textbf{u} \wedge \textbf{v}. +\end{equation} +Dieses Additionszeichen zwischen diesen zwei Produkten mag vielleicht ein wenig eigenartig wirken, da uns das Skalarprodukt ein Skalar und das äussere Produkt einen Bivektor zurück gibt. Was bedeutet es nun also diese beiden Elemente zu addieren? +Man kann sich die Addition wie bei den komplexen Zahlen vorstellen, wobei die imaginäre Einheit auch nicht explizit zu dem reelen Teil addiert werden kann, sondern die zwei Teile zusammen ein Objekt, eine komplexe Zahl bilden. +Dieses Objekt, also die Summe von verschiedenen Elemente der Clifford Algebra, wird Multivektor genannt. +\begin{definition} +Ein Multivektor besteht aus den verschiedenen Bauteilen, wie zum Beispiel Vektoren, Bivektoren oder Trivektoren (Volumen mit einer Richtung), der Clifford Algebra. +\begin{equation} + M = \sum \left ( \prod a_i\textbf{e}_j \right) +\end{equation} +\end{definition} +Besteht eine Clifford Algebra aus n Basisvektoren so hat sie n Dimensionen, dies wird nicht wie in der linearen Algebra mit $\mathbb{R}^n$ sondern mit $\mathbb{G}^n$ beschrieben. +\begin{beispiel} +Allgemeiner Multivektor in $\mathbb{G}^3$ +\begin{equation} + M = a + + + \underbrace{b\textbf{e}_1 + c\textbf{e}_2 + d\textbf{e}_3}_{\text{Vektorteil}} + + + \underbrace{f\textbf{e}_1\textbf{e}_2 + g\textbf{e}_1\textbf{e}_3 + h\textbf{e}_2\textbf{e}_3 }_{\text{Bivektorteil}} + + + \underbrace{k\textbf{e}_1\textbf{e}_2\textbf{e}_3}_{\text{Trivektorteil}} +\end{equation} +\end{beispiel} +\begin{definition} +Um das Produkt von Basisvektoren in Zukunft darzustellen wird folgende Notation definiert + \begin{equation} + e_ie_j = e_{ij} + \end{equation} +\end{definition} +Nun da das geometrische Produkt vollständig definiert wurde können Multiplikationstabellen für verschiedene Dimensionen $\mathbb{G}^n$ erstellt werden. In \ref{tab:multip} ist dies für $\mathbb{G}^3$ gemacht. +\begin{table} + \caption{Multiplikationstabelle für $\mathbb{G^3}$} + \label{tab:multip} + \begin{center} + \begin{tabular}{ |c|c|c|c|c|c|c|c| } + \hline + 1 & $\textbf{e}_1$ & $\textbf{e}_2$ &$\textbf{e}_3$ & $\textbf{e}_{12}$ & $\textbf{e}_{13}$ & $\textbf{e}_{23}$ & $\textbf{e}_{123}$\\ + \hline + $\textbf{e}_1$ & 1 & $\textbf{e}_{12}$ & $\textbf{e}_{12}$ & $\textbf{e}_2$ & $\textbf{e}_3$ & $\textbf{e}_{123}$ & $\textbf{e}_{23}$\\ + \hline + $\textbf{e}_2$ & $-\textbf{e}_{12}$ & 1 & $\textbf{e}_{23}$ & $-\textbf{e}_1$ & $-\textbf{e}_{123}$ & $\textbf{e}_3$ & $-\textbf{e}_{13}$\\ + \hline + $\textbf{e}_3$ & $-\textbf{e}_{13}$ & $-\textbf{e}_{23}$ & 1 & $\textbf{e}_{123}$ & $-\textbf{e}_1$ & $-\textbf{e}_2$ & $\textbf{e}_{12}$\\ + \hline + $\textbf{e}_{12}$ & -$\textbf{e}_2$ & $\textbf{e}_1$& $\textbf{e}_{123}$ & -1 & $-\textbf{e}_{23}$ & $\textbf{e}_{13}$ & $-\textbf{e}_{3}$\\ + \hline + $\textbf{e}_{13}$ & $-\textbf{e}_{3}$ & $-\textbf{e}_{123}$ & $\textbf{e}_{1}$ & $\textbf{e}_{23}$ & -1 & $-\textbf{e}_{12}$ & $\textbf{e}_{2}$\\ + \hline + $\textbf{e}_{23}$ & $\textbf{e}_{123}$ & $-\textbf{e}_{3}$ & $\textbf{e}_{2}$ & $-\textbf{e}_{13}$ & $\textbf{e}_{12}$ & -1 & $-\textbf{e}_{1}$ \\ + \hline + $\textbf{e}_{123}$ & $\textbf{e}_{23}$ & $-\textbf{e}_{13}$ & $\textbf{e}_{12}$ & $-\textbf{e}_{3}$& $\textbf{e}_{2}$ & $-\textbf{e}_{1}$ & -1 \\ + \hline + \end{tabular} + \end{center} +\end{table} diff --git a/buch/papers/clifford/5_PolareDarstellung.tex b/buch/papers/clifford/5_PolareDarstellung.tex new file mode 100644 index 0000000..80fb49f --- /dev/null +++ b/buch/papers/clifford/5_PolareDarstellung.tex @@ -0,0 +1,29 @@ +\subsection{Polare Darstellung des geometrischen Produktes} +Beide Teile des geometrischen Produktes lassen sich durch trigonometrische Terme beschreiben. Das Skalarprodukt kann als +\begin{equation} + \textbf{u}\cdot \textbf{v} = |\textbf{u}||\textbf{v}|\cos{\alpha} +\end{equation} +beschrieben werden. Wobei $\alpha$ den Winkel zwischen den beiden Vektoren beschreibt. +\newline +Beim äusseren Produkt wurde bereits erwähnt, dass es aus dem Produkt der Fläche des von den zwei Vektoren aufgespannten Parallelogram und einer Umlaufrichtung beschrieben wird. Die Fläche eines Parallelograms lässt sich auch mit einen Sinus Term beschreiben +\begin{equation} + \textbf{u} \wedge \textbf{v} + = + \begin{vmatrix} + u_i & v_i \\ + u_j & v_j + \end{vmatrix}\textbf{e}_i\textbf{e}_j + = + \underbrace{|u||v|\sin{\alpha}}_{\text{Fläche}}\textbf{e}_i\textbf{e}_j +\end{equation} +Wobei die Fläche des Parallelogram auf der von $\textbf{e}_i$ und $\textbf{e}_j$ aufgespannten Ebene liegen.\newline +Nun kann man diese Terme wieder zum geometrischen Produkt vereinen +\begin{equation} + \textbf{u}\textbf{v} + = + |\textbf{u}||\textbf{v}|\cos{(\alpha)} + + + |\textbf{u}||\textbf{v}|\sin{(\alpha)} \textbf{e}_i\textbf{e}_j + = + |\textbf{u}||\textbf{v}|(\cos{(\alpha)} + \sin{(\alpha)}\textbf{e}_i\textbf{e}_j) +\end{equation} \ No newline at end of file diff --git a/buch/papers/clifford/6_Dirac-Matrizen.tex b/buch/papers/clifford/6_Dirac-Matrizen.tex new file mode 100644 index 0000000..e68f0f6 --- /dev/null +++ b/buch/papers/clifford/6_Dirac-Matrizen.tex @@ -0,0 +1,7 @@ +% +% einleitung.tex -- Beispiel-File für die Einleitung +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Dirac-Matrizen} + diff --git a/buch/papers/clifford/7_Reflektion.tex b/buch/papers/clifford/7_Reflektion.tex new file mode 100644 index 0000000..dfe86b8 --- /dev/null +++ b/buch/papers/clifford/7_Reflektion.tex @@ -0,0 +1,32 @@ +% +% teil1.tex -- Beispiel-File für das Paper +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Reflektion/ Spiegelung} +Die Spiegelung ist eine grundlegende, geometrische Operation, aus welcher man weitere, wie beispielsweise die später beschriebene Rotation, ableiten kann. Da die Geometrische Algebra für geometrische Anwendungen ausgelegt ist, sollte die Reflektion auch eine einfache, praktische Formulierung besitzen. \\HIER BILD +\subsection{linearen Algebra} +Aus der linearen Algebra ist bekannt, dass man eine Reflektion wie folgt beschreiben kann. +\begin{align} \label{RefLinAlg} + \mathbf{v^{'}} = \mathbf{v} - 2 \cdot \mathbf{v_{\perp u}} +\end{align} +Dabei stellt $\mathbf{u}$ die Spiegelachse dar. +Es scheint für diese Formel aber umständlich zu sein, weitere Reflektionen, mit weiteren Spiegelachsen, anzufügen. Man kann die Abbildung des Vektors auf den Reflektierten Vektor auch als Matrix schreiben, welche aus den Komponenten des zu der Spiegelachse orthonormalen Vektors $\mathbf{\hat{n}}$ besteht. +\\MATRIZEN O(2) und O(3) zeigen\\ +Diese Matrizen gehören der Matrizengruppe $O(n)$ an.... +\subsection{geometrischen Algebra} +Die Geometrische Algebra leitet aus der obigen Formel (\ref{RefLinAlg}) eine einfache und intuitive Form her, welche auch für weitere Operationen einfach erweitert werden kann. +\begin{align} + \mathbf{v'} = \mathbf{uvu^{-1}} +\end{align} +wobei die Inverse eines Vektors so definiert ist, dass multipliziert mit sich selbst das neutrale Element 1 ergibt. +\begin{align} + u^{-1} = \dfrac{u}{|u|^2} \Rightarrow uu^{-1} = 1 +\end{align} +verwendet man für $\mathbf{u}$ nur einen Einheitsvektor $\mathbf{\hat{u}}$, welcher die Länge 1 besitzt, wird somit die Formel reduziert zu einer beidseitigen Multiplikation von $\mathbf{\hat{u}}$. +\begin{align} + \mathbf{v'} = \mathbf{\hat{u}v\hat{u}} +\end{align} +Im Gegensatz zu den Abbildungen in der linearen Algebra, welche in jeder anderen Dimension durch andere Matrizen beschrieben werden müssen, ist es in der geometrischen Algebra immer der gleiche Vorgehensweise. +Zudem ist diese kompakte Schreibweise in der linearen Algebra nicht möglich, da keine Multiplikation von Vektoren definiert ist. +\\BEISPIEL? \ No newline at end of file diff --git a/buch/papers/clifford/8_Rotation.tex b/buch/papers/clifford/8_Rotation.tex new file mode 100644 index 0000000..ebd278c --- /dev/null +++ b/buch/papers/clifford/8_Rotation.tex @@ -0,0 +1,99 @@ +% +% teil2.tex -- Beispiel-File für teil2 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Rotation} +Eine Rotation kann man aus zwei, aufeinanderfolgende Reflektionen bilden. Das war für mich zuerst eine verwirrende Aussage, da man aus den vorherig gezeigten Formeln annehmen könnte, dass die Reflektion schon für eine Drehung ausreicht. Obwohl sich die Längen, Winkel und Volumen sich bei einer Reflektion, wie bei einer Rotation, nicht ändert, sind sie doch verschieden, da die Orientierung bei der Reflektion invertiert wird. Stellt man sich beispielsweise ein Objekt in 3D vor und spiegelt dieses an einer Fläche, dann ist es unmöglich nur durch eine Rotation (egal an welchem Punkt) das ursprüngliche Objekt deckungsgleich auf das Gespiegelte zu drehen. Hingegen ist es wiederum möglich ein zweifach gespiegeltes Objekt durch eine Drehung zu erreichen. Das liegt daran, da die Orientierung zwei mal invertiert wurde. +\\BILD + +\subsection{linearen Algebra} +In der linearen Algebra haben wir Drehungen durch die Matrizen der Gruppe $SO(n)$ beschrieben. Die SO(2) werden beispielsweise auf diese Weise gebildet. +\begin{align} + D = + \begin{pmatrix} + cos(\alpha) & sin(\alpha) \\ + -sin(\alpha) & cos(\alpha) + \end{pmatrix} +\end{align} + +\subsection{geometrischen Algebra} +Da wir jetzt aus der Geometrie wissen, dass eine Rotation durch zwei Reflektionen gebildet werden kann, können wir die Rotation einfach herleiten. +\begin{align} \label{rotGA} + v'' = wv'w^{-1} = w(uvu^{-1})w^{-1} +\end{align} +Die Vektoren $\mathbf{w}$ und $\mathbf{u}$ bilden hier wiederum die Spiegelachsen. Diese versuchen wir jetzt noch zu verbessern. Dazu leiten wir zuerst die bekannte Polarform her. (Anmerkung: Hier wird eine Rotation auf der $\mathbf{e_{12}}$ Ebene hergeleitet. Weitere Drehungen können in höheren Dimensionen durch Linearkombinationen von Drehungen in den $\mathbf{e_{ij}}, i\not=j$ Ebenen erreicht werden) +\begin{align} + \mathbf{w} = |w| \left[\cos(\theta_w) e_1 + \sin(\theta_w) e_2\right] +\end{align} +Dabei können wir ausnützen, dass $e_1^2 = 1$ ist. Was nichts ändert wenn wir es einfügen. Zudem klammern wir dann $e_1$ aus. +\begin{align} + \mathbf{w} = |w| \left[\cos(\theta_w) e_1 + \sin(\theta_w) e_1e_1e_2\right] +\end{align} +\begin{align} \label{e1ausklammern} + \mathbf{w} = |w|e_1\left[\cos(\theta_w)+ \sin(\theta_w) e_{12}\right] +\end{align} +Durch die Reihenentwicklung ist es uns jetzt möglich den Term in eckigen Klammern mit der e-Funktion zu schreiben. +\begin{align} + \mathbf{w} = |w|\mathbf{e_1} e^{\theta_w \mathbf{e_{12}}} +\end{align} +Man kann es so interpretieren, dass der Einheitsvektor $e_1$ um die Länge w gestreckt und um $theta_w$ gedreht wird. +Nun werden wir den Effekt von zwei aneinandergereihten Vektoren $(wu)$ betrachten. +\begin{align} + \mathbf{wu} = |w|\mathbf{e_1} e^{\theta_w \mathbf{e_{12}}}||u||\mathbf{e_1} e^{\theta_u \mathbf{e_{12}}} +\end{align} +Um die beiden $\mathbf{e_1}$ zu kürzen, können wir die Reihenfolge des exponential Terms mit $\mathbf{e_1}$ wechseln, indem man bei der Gleichung (\ref{e1ausklammern}), anstatt mit $\mathbf{e_1e_1e_2}$ mit $\mathbf{e_2e_1e_1}$ erweitert. +\begin{align} + \mathbf{w} = |w|\left[\cos(\theta_w)+ \sin(\theta_w) \mathbf{e_2e_1}\right]\mathbf{e_1} +\end{align} +Da $\mathbf{e_2e_1 = -e_{12}}$ können wir einfach den Winkel negieren. +Jetzt können wir wieder $e_1e_1 = 1$ kürzen. Die Längen können als Skalare beliebig verschoben werden und die exponential Terme zusammengefasst werden. +\begin{align} + \mathbf{wu} = |w||u|e^{-\theta_w \mathbf{e_{12}}}\mathbf{e_1}\mathbf{e_1} e^{\theta_u \mathbf{e_{12}}} +\end{align} +\begin{align} + \mathbf{wu} = |w||u|e^{(\theta_u-\theta_w) \mathbf{e_{12}}} +\end{align} +der Term $\mathbf{u^{-1}w^{-1}}$ kann durch die selbe Methode zusammengefasst werden. +\begin{align} + \mathbf{u^{-1}w^{-1}} = \dfrac{1}{|w||u|}e^{(\theta_w-\theta_u) \mathbf{e_{12}}} +\end{align} +Dabei definieren wir den Winkel zwischen den Vektoren $\mathbf{w}$ und $\mathbf{u}$ als $\theta = \theta_w - \theta_u$. Setzten wir nun unsere neuen Erkenntnisse in die Gleichung (\ref{rotGA}) ein. +\begin{align} + \mathbf{v''} = |w||u|e^{-\theta \mathbf{e_{12}}} v \dfrac{1}{|w||u|}e^{\theta \mathbf{e_{12}}} +\end{align} +HIER DEFINITION/IST WICHTIGE FORMEL +\begin{align} + \mathbf{v''} = e^{-\theta \mathbf{e_{12}}} v e^{\theta \mathbf{e_{12}}} +\end{align} +Wir wissen nun, dass das diese beidseitige Multiplikation die Länge von $\mathbf{v}$ nicht verändert, da sich die Längen von $\mathbf{w}$ und $\mathbf{u}$ kürzen. Betrachten wir nun den Effekt der Exponentialterme auf $\mathbf{v}$. Dabei Teilen wir den Vektor $\mathbf{v}$ auf in einen Anteil $\mathbf{v_\parallel}$, welcher auf der Ebene $\mathbf{e_{12}}$ liegt, und einen Anteil $\mathbf{v_\perp}$, welcher senkrecht zu der Ebene steht. +\begin{align} \label{RotAufPerpPar} + \mathbf{v''} = e^{-\theta \mathbf{e_{12}}} (\mathbf{v_\perp + v_\parallel}) e^{\theta \mathbf{e_{12}}} +\end{align} +\begin{align} + \mathbf{v''} = e^{-\theta \mathbf{e_{12}}} \mathbf{v_\perp} e^{\theta \mathbf{e_{12}}} + e^{-\theta \mathbf{e_{12}}} \mathbf{v_\parallel} e^{\theta \mathbf{e_{12}}} +\end{align} +Auf eine allgemeine Herleitung wird hier zwar verzichtet, aber man kann zeigen, dass die Reihenfolge so vertauscht werden kann. Der Winkel wird dabei beim parallelen Term negiert. +\begin{align} + \mathbf{v''} = \mathbf{v_\perp} e^{-\theta \mathbf{e_{12}}} e^{\theta \mathbf{e_{12}}} + \mathbf{v_\parallel} e^{-(-\theta) \mathbf{e_{12}}} e^{\theta \mathbf{e_{12}}} +\end{align} +\begin{align} + \mathbf{v''} = \mathbf{v_\perp} + \mathbf{v_\parallel} e^{2\theta \mathbf{e_{12}}} +\end{align} +Man kann an dieser Gleichung sehen, dass nur der parallele Anteil des Vektors $\mathbf{v}$ auf der Ebene $\mathbf{e_{12}}$ um $2\theta$ gedreht wird. Der senkrechte Anteil bleibt gleich. Wichtig dabei zu sehen ist, dass nur der Winkel zwischen den Vektoren $\mathbf{w}$ und $\mathbf{u}$ von Bedeutung ist. Die Länge und Richtung der einzelnen Vektoren spielt keine Rolle. +\\BEISPIEL +\begin{align} + \begin{split} + &\mathbf{v} = 1\mathbf{e_1} + 2\mathbf{e_2} + 3\mathbf{e_3}\quad\Rightarrow\quad \mathbf{v_\parallel} = 1\mathbf{e_1} + 2\mathbf{e_2}; \quad \mathbf{v_\perp} = 3\mathbf{e_3}\\ &\mathbf{wu} = 1e^{(-\pi/2) \mathbf{e_{12}}} = 1[\cos(-\pi/2)\mathbf{e_1}+\sin(-\pi/2)\mathbf{e_2}] = -\mathbf{e_2}; \\ &\mathbf{u^{-1}w^{-1}} = 1e^{(\pi/2) \mathbf{e_{12}}} = \mathbf{e_2} + \end{split} +\end{align} +\begin{align} + \begin{split} + \mathbf{v''} = &\mathbf{(wu)v(u^{-1}w^{-1})} \\ + &-\mathbf{e_2} (1\mathbf{e_1} + 2\mathbf{e_2} + 3\mathbf{e_3}) \mathbf{e_2} \\ + & -1\mathbf{e_2e_1e_2} - 2\mathbf{e_2e_2e_2} - 3\mathbf{e_2e_3e_2} \\ + & 1\mathbf{e_2e_2e_1} - 2\mathbf{e_2} + 3\mathbf{e_2e_2e_3} \\ + & 1\mathbf{e_1} - 2\mathbf{e_2} + 3\mathbf{e_3} + \end{split} +\end{align} +Man sieht, dass sich der Vektor $\mathbf{v_\parallel}$ sich um $2\cdot90^\circ$ gedreht hat und der Vektor $\mathbf{v_\perp}$ unverändert blieb. \ No newline at end of file diff --git a/buch/papers/clifford/9_KomplexeZahlen.tex b/buch/papers/clifford/9_KomplexeZahlen.tex new file mode 100644 index 0000000..735eead --- /dev/null +++ b/buch/papers/clifford/9_KomplexeZahlen.tex @@ -0,0 +1,27 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{komplexe Zahlen} +Die komplexen Zahlen finden eine Vielzahl von Anwendungsgebiete in den Ingenieurwissenschaften. Das liegt daran, weil die komplexen Zahlen Rotationen und Schwingungen gut beschreiben können. Nachdem vorherigen Kapitel überrascht es wahrscheinlich nicht viele, dass es möglich ist Komplexe Zahlen in der geometrischen Algebra darzustellen. Sie können durch die geraden Grade der 2 Dimensionalen geometrischen Algebra vollständig beschrieben werden: $\mathbb{G}_2^+ \cong \mathbb{C}$. Das bedeutet eine komplexe Zahl kann durch ein Skalar (Grade 0) und einem Bivektor (Grade 2) dargestellt werden. Als Abkürzung nehme ich die Bezeichnung $g_n \in \mathbb{G}_2^+$. +\begin{align} + a_0 + a_1 j \cong a_0 + a_1 e_{12} = g_n;\quad a_0, a_1 \in \mathbb{R} +\end{align} +oder in Polarform. +\begin{align} + |r|e^{\theta j} \cong |r|e^{\theta e_{12}} = g_n; \quad r, \theta \in \mathbb{R} +\end{align} +Man beachte, dass wenn wir, wie bei den komplexen Zahlen, Elemente von $\mathbb{G}_2^+$ miteinander Multiplizieren, ist es nicht, wie im Kapitel Rotation bei der Formel (\ref{rotGA})beschrieben, eine Multiplikation von zwei $g_n$ mit einem Vektor. Im 2 dimensionalen bewirken beide Multiplikationen grundsätzlich das Gleiche (eine Drehstreckung), aber die Multiplikation von mehreren $g_n$ ist kommutativ, wie wir es von den komplexen zahlen kennen. +\begin{align} + \begin{split} + &(a + b \mathbf{e_{12}})(c + d \mathbf{e_{12}}) = (c + d \mathbf{e_{12}})(a + b \mathbf{e_{12}})\\ + &(a + b \mathbf{e_{12}})(x\mathbf{e_1}+y\mathbf{e_2})(c + d \mathbf{e_{12}}) \not= (a + b \mathbf{e_{12}})(c + d \mathbf{e_{12}})(x\mathbf{e_1}+y\mathbf{e_2}) + \end{split} +\end{align} +Um später die Auswirkung der Quaternionen besser zu verstehen, möchte ich kurz darauf eingehen, was ein $g_n$ für eine Auswirkung auf einen Vektor hat. +Wir kennen diesen Effekt schon von den komplexen Zahlen. Wenn eine komplexe Zahl $c_1=a+bj$ mit einer zweiten $c_2=c+dj$ multipliziert wird, dann kann man diese so aufteilen. +\begin{align} + c = (a + bj)(c + dj) = c\cdot(a+bj) + dj\cdot(a+bj) +\end{align} +Wobei $c\cdot(a+bj)$ die jetzige komplexe Zahl $c_1$ um den Faktor $c$ steckt und $dj\cdot(a+bj)$ die um 90° im gegenuhrzeigersinn gedrehte Zahl $c_1$ um den Faktor $d$ streckt. Diese Anteile addiert ergeben, dann den um $c_2$ drehgestreckten Vektor $c_1$. Die wirklichen Vorteile der geometrischen Algebra werden sich aber erst bei den Quaternionen zeigen. diff --git a/buch/papers/clifford/main.tex b/buch/papers/clifford/main.tex index 5533c55..d94e065 100644 --- a/buch/papers/clifford/main.tex +++ b/buch/papers/clifford/main.tex @@ -4,9 +4,9 @@ % (c) 2020 Hochschule Rapperswil % \chapter{Thema\label{chapter:clifford}} -\lhead{Thema} +\lhead{Clifford Algebra} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{Thierry Schwaller, Marius Baumann} Ein paar Hinweise für die korrekte Formatierung des Textes \begin{itemize} @@ -27,10 +27,17 @@ Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. \end{itemize} -\input{papers/clifford/teil0.tex} -\input{papers/clifford/teil1.tex} -\input{papers/clifford/teil2.tex} -\input{papers/clifford/teil3.tex} +\input{0_ElevatorPitch} +\input{1_Vektordarstellung} +\input{2_QuadratVektoren} +\input{3_MultiplikationVektoren} +\input{4_GeometrischesProdukt} +\input{5_PolareDarstellung} +\input{6_Dirac-Matrizen} +\input{7_Reflektion} +\input{8_Rotation} +\input{9_KomplexeZahlen} +\input{10_Quaternionen} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/clifford/packages.tex b/buch/papers/clifford/packages.tex index 8abcef1..f6e94e0 100644 --- a/buch/papers/clifford/packages.tex +++ b/buch/papers/clifford/packages.tex @@ -7,4 +7,25 @@ % if your paper needs special packages, add package commands as in the % following example %\usepackage{packagename} - +\usepackage[utf8]{inputenc} +\usepackage{a4wide} +\usepackage{ngerman} +\usepackage{tikz} +\usepackage{mathdots} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{amsthm} +\newtheorem{definition}{Definition}[chapter] +\newtheorem{beispiel}[definition]{Beispiel} +\newtheorem{bemerkung}[definition]{Bemerkung} +\newtheorem{lemma}[definition]{Lemma} +\newtheorem{satz}[definition]{Satz} +\newtheorem{hauptsatz}[definition]{Hauptsatz} +\newtheorem{corollar}[definition]{Korollar} +\usepackage[german]{babel} +\usepackage[T1]{fontenc} +\usepackage{fullpage} +\usepackage{graphicx} +\usepackage{float} +\usepackage{colortbl} +\usepackage{multirow} \ No newline at end of file diff --git a/buch/papers/clifford/papers/clifford/teil0.tex b/buch/papers/clifford/papers/clifford/teil0.tex new file mode 100644 index 0000000..e69de29 diff --git a/buch/papers/clifford/teil0.tex b/buch/papers/clifford/teil0.tex deleted file mode 100644 index ac943f4..0000000 --- a/buch/papers/clifford/teil0.tex +++ /dev/null @@ -1,22 +0,0 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{clifford:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{clifford:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - - diff --git a/buch/papers/clifford/teil1.tex b/buch/papers/clifford/teil1.tex deleted file mode 100644 index 0674afb..0000000 --- a/buch/papers/clifford/teil1.tex +++ /dev/null @@ -1,55 +0,0 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{clifford:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{clifford:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{clifford:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{clifford:section:loesung}. -Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil -impedit quo minus id quod maxime placeat facere possimus, omnis -voluptas assumenda est, omnis dolor repellendus -\ref{clifford:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. - - diff --git a/buch/papers/clifford/teil2.tex b/buch/papers/clifford/teil2.tex deleted file mode 100644 index bbcefb0..0000000 --- a/buch/papers/clifford/teil2.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{clifford:section:teil2}} -\rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{clifford:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - diff --git a/buch/papers/clifford/teil3.tex b/buch/papers/clifford/teil3.tex deleted file mode 100644 index f50d42d..0000000 --- a/buch/papers/clifford/teil3.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{clifford:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? - -\subsection{De finibus bonorum et malorum -\label{clifford:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. - - -- cgit v1.2.1 From 401325ee8d395ec4de27f4dcede73e860f3e28a8 Mon Sep 17 00:00:00 2001 From: "User-PC\\User" Date: Mon, 31 May 2021 10:47:48 +0200 Subject: =?UTF-8?q?=C3=9Cberarbeitung=20und=20Verbesserung=20der=20Kapitel?= =?UTF-8?q?=20Bearbeitung=20Literaturverzeichnis=20(im=20Literaturverzeich?= =?UTF-8?q?nis=20noch=20nicht=20alles=20korrekt)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- buch/papers/spannung/Einleitung.tex | 81 ++++++++----------- buch/papers/spannung/references.bib | 49 +++++++---- buch/papers/spannung/teil0.tex | 70 ++++++++-------- buch/papers/spannung/teil1.tex | 37 +++++---- buch/papers/spannung/teil2.tex | 156 +++++++++++++++++------------------- buch/papers/spannung/teil3.tex | 107 +++++++++++++------------ buch/papers/spannung/teil4.tex | 44 ++++++---- 7 files changed, 281 insertions(+), 263 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/spannung/Einleitung.tex b/buch/papers/spannung/Einleitung.tex index cf6e916..c80db64 100644 --- a/buch/papers/spannung/Einleitung.tex +++ b/buch/papers/spannung/Einleitung.tex @@ -1,15 +1,18 @@ \section{Einleitung\label{spannung:section:Einleitung}} +\rhead{Einleitung} +Das Hook'sche Gesetz beschreibt die Beziehung von Spannung und Dehnung von linear-elastischen Materialien im Eindimensionalen. In diesem Kapitel geht es darum das Hook'sche Gesetz im Dreidimensionalen zu beschreiben. -Dieses beschreibt die Beziehung von Spannung und Dehnung von linear elastischen Materialien im Eindimensionalen. Durch variable Krafteinwirkungen entstehen in jedem Punkt des Materials eine Vielzahl an unterschiedlichen Spannungen. -Jeder erdenkliche Punkt im Dreidimensionalen beschreibt daher einen entsprechenden individuellen Spannungszustand. +In jedem erdenklichen Punkt im Dreidimensionalen herrscht daher ein entsprechender individueller Spannungszustand. Um das Hook'sche Gesetz für den 3D Spannungszustand formulieren zu können, reichen Skalare nicht aus. Darum werden Vektoren, Matrizen und Tensoren zur Hilfe gezogen. -Diese allgemeine Spannungsformel ist Grundlage für Computerprogramme und geotechnische Versuche, wie der Oedometer-Versuch. +Mit diesen lässt sich eine Spannungsformel für den 3D Spannungszustand bilden. +Diese Spannungsformel ist Grundlage für Computerprogramme und geotechnische Versuche, wie der Oedometer-Versuch. Um die mathematische Untersuchung vorzunehmen, beschäftigt man sich zuerst mit den spezifischen Gegebenheiten und Voraussetzungen. -Ebenfalls gilt es ein paar wichtige Begriffe und deren mathematischen Zeichen einzuführen, -damit sich den Berechnungen schlüssig folgen lässt. +Ebenfalls gilt es ein paar wichtige Begriffe und deren mathematischen Zeichen einzuführen. +In diesem Kapitel gehen wir auch auf die Zusammenhänge von Spannung, Dehnungen und Verformungen an elastischen Materialien ein, +wie sie in gängigen Lehrbüchern der Mechanik oder der Geotechnik behandelt werden. z. B. [\cite{spannung:Grundlagen der Geotechnik}] \section{Spannungsausbreitung\label{spannung:section:Spannungsausbreitung}} \rhead{Spannungsausbreitung} @@ -21,30 +24,34 @@ Belastet man den Boden mit einer Spannung \sigma = \frac{F}{A} +, \] -, so wird diese in den Boden geleitet und von diesem kompensiert. -Im Boden entstehen unterschiedlich hohe Zusatzspannung. -Die Zusatzspannung scheint sich räumlich und berechenbar im Boden auszubreiten. +so wird diese in den Boden geleitet und von diesem kompensiert. +Im Boden entstehen unterschiedlich hohe Zusatzspannungen. +Diese Zusatzspannung breitet sich räumlich im Boden aus. Im Falle einer konstanten Flächenlast $\sigma$ (siehe Abbildung 1.1) breitet sich die Zusatzspannung zwiebelartig aus. -Mit der Tiefe $t$ nimmt diese permanent ab (siehe Abbildung 1.2). -Wie diese Geometrie der Ausbreitung ist wird durch viele Modelle und Ansätze näherungsweise beschrieben. -Diese Zusatzspannung $\sigma$ ist aber sicher abhängig von $(x,y,t)$. \begin{figure} \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild4.png} - \caption{Ausbreitung der Zusatzspannung im Boden} + \includegraphics[width=0.4\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild4.png} + \caption{Ausbreitung der Zusatzspannung im Boden infolge einfacher Flächenlast} \label{fig:Bild4} \end{figure} +Mit der Tiefe $t$ nimmt diese permanent ab (siehe Abbildung 1.2). +Wie diese Geometrie der Ausbreitung ist, kann durch viele Modelle und Ansätze näherungsweise beschrieben werden. +Diese Zusatzspannung $\sigma$ ist im Wesentlichen abhängig von $(x,y,t)$. +Je nach Modell werden noch andere Parameter berücksichtigt. +Das können beispielsweise jenste Bodenkennwerte oder auch der Wassergehalt sein. + \begin{figure} \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild5.png} - \caption{Funktionen Spannung und Dehnung} + \includegraphics[width=0.35\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild5.png} + \caption{Funktionen der Spannung und Dehnung im Zusammenhang mit der Tiefe} \label{fig:Bild5} \end{figure} -Bei jeder dieser Zusatzspannung geht eine entsprechende Zusatzdehnung einher, welche eine Setzung bedeutet. +Bei jeder dieser Zusatzspannung geht eine entsprechende Zusatzdehnung des Bodens einher, welche eine Setzung bedeutet. Im einfachsten Fall kann modellhaft mit \[ \varepsilon @@ -58,43 +65,25 @@ s \int_{0}^{\infty}\varepsilon\enspace dt \] berechnet werden mit: -\[ -\varepsilon -= -\text{Dehnung [$-$]} -\] -\[ -\sigma -= -\text{Spannung [\si{\kilo\pascal}]} -\] -\[ -E -= -\text{Elastizitätsmodul; Young-Modul [\si{\kilo\pascal}]} -\] -\[ -t -= -\text{Tiefe [\si{\meter}]} -\] -\[ -s -= -\text{Setzung, Absenkung [m]} -\] - +\begin{align*} + \varepsilon &= \text{Dehnung [$-$]} \\ + \sigma &= \text{Spannung [\si{\kilo\pascal}]} \\ + E &= \text{Elastizitätsmodul; Young-Modul [\si{\kilo\pascal}]}\\ + t &= \text{Tiefe [\si{\meter}]} \\ + s &= \text{Setzung, Absenkung [m].} +\end{align*} +Diese Zusammenhänge sind wie erwähnt unter anderem im Lehrbuch [\cite{spannung:Grundlagen der Geotechnik}] beschrieben. In der praktischen Geotechnik wird man allerdings weitaus schwierigere Situationen antreffen. Ein Beispiel wäre eine Baugrube mit einem Baugrubenabschluss, wo ein Teil des Bodens abgetragen ist (siehe Abbildung 1.3). Die Ausbreitung der Zusatzspannung $\sigma(x,y,t)$ würde hier deutlich komplizierter ausfallen. Dies bedeutet auch eine komplexere Setzung der Bodenoberfläche infolge einer Flächenlast $\sigma$. -Aus allen zusätzlichen Spannungen müssen die adäquaten Dehnung mit Hilfe einer Spannungsgleichung berechnet werden. -Diese beruht auf Annahmen nach Hooke auf einem linear elastischen Boden. +Aus allen zusätzlichen Spannungen müssen die adäquaten Dehnungen mit Hilfe einer Spannungsgleichung berechnet werden. +Diese beruht auf Annahmen nach Hooke auf einem linear-elastischen Boden. Generell wird im Ingenieurwesen versucht Phänomene möglichst nach dem Hook'schen Gesetz abbilden zu können. \begin{figure} \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild3.png} - \caption{Beispiel Lastauftrag auf Boden} + \includegraphics[width=0.45\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild3.png} + \caption{Beispiel eines Lastauftrags auf den Boden bei einer komplexeren Situation, welches kompliziertere Spannungsausbreitung zur Folge hat} \label{fig:Bild3} \end{figure} \ No newline at end of file diff --git a/buch/papers/spannung/references.bib b/buch/papers/spannung/references.bib index ed5703c..090e3c3 100644 --- a/buch/papers/spannung/references.bib +++ b/buch/papers/spannung/references.bib @@ -4,27 +4,46 @@ % (c) 2020 Autor, Hochschule Rapperswil % -@online{spannung:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, - year = {2020}, - month = {2}, +@online{spannung:Tensor, + title = {Tensor}, + url = {https://de.wikipedia.org/wiki/Tensor}, + date = {2021-05-29}, + year = {2021}, + month = {5}, day = {6} } -@book{spannung:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} +@online{spannung:Voigtsche Notation, + title = {Voigtsche Notation}, + url = {https://de.wikipedia.org/wiki/Voigtsche_Notation}, + date = {2021-05-29}, + year = {2021}, + month = {5}, + day = {6} +} + +@book{spannung:Grundlagen der Geotechnik, + title = {Grundlagen der Geotechnik}, + author = {Hans-Henning Schmidt and Roland F. Buchmaier and Carola Vogt-Breyer}, + publisher = {Springer Fachmedien Wiesbaden GmbH}, + year = {2017}, + isbn = {978-3-658-14930-7}, + inseries = {Geotechnik nach Eurocode}, + volume = {5} +} + +@book{spannung:Stoffgesetze und numerische Modellierung in der Geotechnik, + title = {Stoffgesetze und numerische Modellierung in der Geotechnik}, + author = {Carlo Rabaiotti and Alessio Höttges}, + publisher = {Hochschule Rapperswil}, + year = {2021}, + isbn = {}, + inseries = {}, + volume = {} } @article{spannung:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, + author = { Tabea Méndez and Andreas Müller }, title = { Noncommutative harmonic analysis and image registration }, journal = { Appl. Comput. Harmon. Anal.}, year = 2019, diff --git a/buch/papers/spannung/teil0.tex b/buch/papers/spannung/teil0.tex index be837ac..ffc9009 100644 --- a/buch/papers/spannung/teil0.tex +++ b/buch/papers/spannung/teil0.tex @@ -1,48 +1,47 @@ -\section{Einachsiger Spannungszustand\label{spannung:section:Einachsiger Spannungsustand}} -\rhead{Einachsiger Spannungszustand} -Ein Spannungszustand beschreibt alle Spannungen, welche in einem beliebigen Punkt im Körper wirken (siehe Abbildung 1.4). +\section{Der Spannungszustand\label{spannung:section:Der Spannungsustand}} +\rhead{Der Spannungszustand} +Ein Spannungszustand ist durch alle Spannungen, welche in einem beliebigen Punkt im Körper wirken, definiert (siehe Abbildung 1.4). Änderungen der äusseren Kräfte verändern die inneren Spannungszustände im Material. Um alle Spannungen eines Punktes darstellen zu können, wird ein infinitesimales Bodenelement in Form eines Würfels modellhaft vorgestellt. Man spricht auch von einem Elementarwürfel, da dieser elementar klein ist. \begin{figure} \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild2.png} + \includegraphics[width=0.4\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild2.png} \caption{Infinitesimales Bodenelement mit den 9 Spannungen} - \label{fig:infintesimaler-wurfel} + \label{fig:Bild2} \end{figure} -Es werden jeweils drei Seiten dieses Würfels betrachtet, wobei die drei gegenüberliegenden Seiten die selben Spannungen aufweisen. -Das infinitesimale Bodenteilchen hat die Koordinaten $1$, $2$, $3$ muss sich zwingend im Gleichgewicht befinden. -So sind insgesamt 9 verschiedene Spannungen möglich, wobei 3 Normal- und 6 Schubspannungen sind. -Normalspannung wirken normal (mit rechtem Winkel) zur angreifenden Fläche und Schubspannungen parallel zur angreifenden Fläche. -Alle Beträge dieser 9 Spannungen am Elementarwürfel bilden den Spannungszustand. +Es werden jeweils drei Seiten dieses Würfels betrachtet, wobei die drei gegenüberliegenden Seiten im Betrag die selben Spannungen aufweisen, +sodass der Elementarwürfel im Gleichgewicht ist. +Wäre dieses Gleichgewicht nicht vorhanden, käme es zu Verschiebungen und Drehungen. +Das infinitesimale Bodenteilchen hat die Koordinaten $1$, $2$, $3$. +Veränderungen der Normalspannungen können durch Schubspannungen kompensiert werden und umgekehrt. +So sind insgesamt neun verschiedene Spannungen möglich, wobei drei Normal- und sechs Schubspannungen sind. +Normalspannungen wirken normal (mit rechtem Winkel) zur angreifenden Fläche und Schubspannungen parallel zur angreifenden Fläche. +Alle Beträge dieser neun Spannungen am Elementarwürfel bilden den Spannungszustand. Daraus können die äquivalenten Dehnungen $\varepsilon$ mit Hilfe des Hook'schen Gesetz berechnet werden. +Daher gibt es auch den entsprechenden Dehnungszustand. -\begin{figure} - \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild1.png} - \caption{1D Spannungszustand aus einer quaderförmigen Bodenprobe} - \label{fig:infintesimaler-wurfel} -\end{figure} -Im einachsigen Spannungszustand herrscht nur die Normalspannung $\sigma_{11}$ (siehe Abbildung). +\section{Spannungszustand\label{spannung:section:Spannungsustand}} +\rhead{Spannungszustand} + +Im einachsigen Spannungszustand herrscht nur die Normalspannung $\sigma_{11}$ (siehe Abbildung 1.5). Das Hook'sche Gesetz beschreibt genau diesen 1D Spannungszustand. Nach Hooke gilt: \[ F \sim \Delta l -\] . -Teilt man beide Seiten mit den Konstanten $A$ und $l_0$ erhält man +\] +Teilt man beide Seiten durch die Konstanten $A$ und $l_0$, erhält man \[ \frac{F}{A} = \sigma \sim -\] -\[ \varepsilon = \frac{\Delta l}{l_0} @@ -52,22 +51,21 @@ und somit \sigma \sim \varepsilon +, \] -. -Mit: -\[ -l_0 -= -\text{Länge zu Beginn [\si{\meter}]} -\] -\[ -A -= -\text{Fläche [\si{\meter\squared}]} -\] - -Diese Beziehung gilt bei linear elastischen Materialien, welche reversibel sind und nicht dauerhaft verformt werden. +mit +\begin{align*} + l_0 &= \text{Länge zu Beginn [\si{\meter}]} \\ + A &= \text{Fläche [\si{\meter\squared}].} +\end{align*} +Diese Beziehung gilt bei linear-elastischen Materialien, welche reversible Verformungen zulassen. Es ist praktisch die relative Dehnung $\varepsilon$ anzugeben und nicht eine absolute Längenänderung $\Delta l$. +\begin{figure} + \centering + \includegraphics[width=0.35\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild1.png} + \caption{1D Spannungszustand aus einer quaderförmigen Bodenprobe} + \label{fig:Bild1} +\end{figure} Mithilfe vom Elastizitätsmodul $E$ als Proportionalitätskonstante lässt sich der eindimensionale Fall mit \[ \sigma @@ -75,7 +73,7 @@ Mithilfe vom Elastizitätsmodul $E$ als Proportionalitätskonstante lässt sich E\cdot\varepsilon \] beschreiben. -Im Falle, dass der E-Modul nicht konstant ist, kann dieser näherungsweise mit +Im Falle, dass $E$ nicht konstant ist, kann dieser näherungsweise durch \[ E = diff --git a/buch/papers/spannung/teil1.tex b/buch/papers/spannung/teil1.tex index 3b40ee9..2db244e 100644 --- a/buch/papers/spannung/teil1.tex +++ b/buch/papers/spannung/teil1.tex @@ -1,17 +1,24 @@ \section{Skalare, Vektoren, Matrizen und Tensoren\label{spannung:section:Skalare,_Vektoren,_Matrizen_und_Tensoren}} \rhead{Skalare, Vektoren, Matrizen und Tensoren} -Tensoren wurden als erstes in der Elastizitätstheorie eingesetzt. (Quelle Herr Müller) -In der Elastizitätstheorie geht es darum viele verschiedene Komponenten zu beschreiben. -Mit einer Matrix oder einem Vektor kann man dies nicht mehr bewerkstelligen. -Wenn man den dreidimensionalen Spannungszustand abbilden möchte, müsste man mehrere Vektoren haben. -Deshalb wurden 1840 von Rowan Hamilton Tensoren in die Mathematik eingeführt. -Woldemar Voigt hat den Begriff in die moderne Bedeutung von Skalar, Matrix und Vektor verallgemeinert. -Albert Einstein hat Tensoren zudem in der allgemeinen Relativitätstheorie benutzt. -Tensor sind eine Stufe höher als Matrizen. Matrizen sind 2. Stufe. -Da Tensoren eine Stufe höher sind, kann man auch Matrizen, Vektoren und Skalare als Tensoren bezeichnen. -Der Nachteil von den Tensoren ist, dass man die gewohnten Rechenregeln, die man bei Vektoren oder Matrizen kennt, -nicht darauf anwenden kann. Man ist deshalb bestrebt die Tensoren als Vektoren und Matrizen darzustellen, -damit man die gewohnten Rechenregeln darauf anwenden kann. (Quelle Wikipedia) -In der vorliegenden Arbeit sind bereits alle Tensoren als Matrizen 2. Stufe abgebildet. -Trotzdem kann man diese Matrizen wie vorher beschrieben als Tensor bezeichnen. -Da diese als Matrizen abgebildet sind, dürfen wir die bekannten Rechenregeln auf unsere Tensoren anwenden. \ No newline at end of file +Der Begriff Tensor kann als Überbegriff, der mathematischen Objekte Skalar, Vektor und Matrix, betrachtet werden. +Allerdings sind noch höhere Stufen dieser Objekte beinhaltet. +Ein Skalar, ein Vektor oder eine Matrix ist daher auch ein Tensor. +Ein Skalar ist ein Tensor 0. Stufe. +Mit einem Vektor können mehrere Skalare auf einmal beschrieben werden. +Ein Vektor hat daher die Stufe 1 und ist höherstufig als ein Skalar. +Mit einer Matrix können wiederum mehrere Vektoren auf einmal beschrieben werden. +Eine Matrix hat daher die Stufe 2 und ist noch höherstufig als ein Vektor. +Versteht man diese Stufen, so versteht man den Sinn des Begriffs Tensor. + +Jede Stufe von Tensoren verlangt andere Rechenregeln. +So zeigt sich auch der Nachteil von Tensoren mit Stufen höher als 2. +Man ist also bestrebt höherstufige Tensoren mit Skalaren, Vektoren oder Matrizen zu beschreiben. + +Der Begriff Tensor wurde 1840 von Rowan Hamilton in die Mathematik eingeführt. +James Clerk Maxwell hat bereits mit Tensoren operiert, ohne den Begriff Tensor gekannt zu haben. +Erst Woldemar Voigt hat den Begriff in die moderne Bedeutung von Skalar, Matrix und Vektor verallgemeinert. +Er hat in der Elastizitätstheorie als erstes Tensoren eingesetzt und beschrieben. +Auch Albert Einstein hat solche Tensoren eingesetzt, +um in der Relativitätstheorie die Änderung der 4D Raumzeit beschreiben zu können. +\cite{spannung:Tensor} +\cite{spannung:Voigtsche Notation} \ No newline at end of file diff --git a/buch/papers/spannung/teil2.tex b/buch/papers/spannung/teil2.tex index 8be0bdc..afd2c21 100644 --- a/buch/papers/spannung/teil2.tex +++ b/buch/papers/spannung/teil2.tex @@ -1,16 +1,22 @@ \section{Dreiachsiger Spannungszustand\label{spannung:section:Dreiachsiger_Spannungszustand}} \rhead{Dreiachsiger Spannungszustand} Durch komplexe Spannungsausbreitungen im Boden entstehen im 3D Spannungszustand unterschiedliche Normal- und Schubspannungen. -Ein Tensor 0.Stufe, sprich ein Skalar, kann lediglich den 1D Spannungszustand beschreiben. -Um den 3D Spannungszustandes als ein mathematisches Objekt darstellen zu können, wird ein Tensor 2.Stufe, sprich eine Matrix, eingesetzt. +\begin{figure} + \centering + \includegraphics[width=0.4\linewidth,keepaspectratio]{papers/spannung/Grafiken/infinitesimalerWuerfel.png} + \caption{Beispiel eines Spannungszustandes; Vergrösserung eines infinitesimalen Bodenteilchen} + \label{fig:infinitesimalerWuerfel} +\end{figure} +Ein Tensor 0. Stufe, sprich ein Skalar, kann lediglich den 1D Spannungszustand beschreiben. +Um den 3D Spannungszustandes als ein mathematisches Objekt darstellen zu können, wird ein Tensor 2. Stufe, sprich eine Matrix, eingesetzt. Die Spannungen sind durch die zwei Indizes \[ i, j\in\left\{1, 2, 3\right\} \] - definiert. -Daher ergeben sich die 9 Spannungen. -Dieser Spannungstensor kann schliesslich mit $3^2$ Einträgen als 3x3 Matrix mit +Daher ergeben sich die neun Spannungen. +Die nachfolgenden Zusammenhänge sind in \cite{spannung:Voigtsche Notation} beschrieben. +Dieser Spannungstensor kann schliesslich mit $3^2$ Einträgen als $3\times3$ Matrix mit \[ \overline{\sigma} = @@ -23,13 +29,12 @@ Dieser Spannungstensor kann schliesslich mit $3^2$ Einträgen als 3x3 Matrix mit \end{pmatrix} \] dargestellt werden und beschreibt somit den gesamten Spannungszustand. -Die Dehnungen wirken adäquat zu den Spannungen und sind durch die zwei Indizes +Die Dehnungen wirken in die gleichen Richtungen wie die korrespondierenden Spannungen und sind durch die zwei Indizes \[ k, l\in\left\{1, 2, 3\right\} \] - definiert. -Der Dehnungstensor ist ebenfalls ein Tensor 2.Stufe und kann somit auch als $3\times3$ Matrix mit +Der Dehnungstensor ist ebenfalls ein Tensor 2. Stufe und kann somit auch als $3\times3$ Matrix mit \[ \overline{\varepsilon} = @@ -43,14 +48,7 @@ Der Dehnungstensor ist ebenfalls ein Tensor 2.Stufe und kann somit auch als $3\t \] dargestellt werden und beschreibt den gesamten Dehnungszustand. -\begin{figure} - \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/infinitesimalerWuerfel.png} - \caption{Infinitesimales Bodenteilchen} - \label{fig:infintesimaler-wurfel} -\end{figure} - -Der Spannungs- und Dehnungstensor 2.Stufe kann je in einen Tensor 1. Stufe überführt werden, welches ein Spaltenvektor ist. +Der Spannungs- und Dehnungstensor 2. Stufe kann je in einen Tensor 1. Stufe überführt werden, welches ein Spaltenvektor ist. Gemäss der Hadamard-Algebra dürfen Zeile um Zeile in eine Spalte notiert werden, sodass es einen Spaltenvektor ergibt. So ergibt sich der Spannungsvektor @@ -108,22 +106,22 @@ und Dehnungsvektor \varepsilon_{32} \\ \varepsilon_{33} \end{pmatrix} -\]. - -Um die Beziehung von Spannung und Dehnung, welche mit Tensoren 2.Stufen ausgedrückt werden, zu beschreiben, wird ein Elastizitätstensor 4.Stufe benötigt. -Dieser ist im 1D Spannungszustand ein Tensor 0.Stufe und somit ein Skalar. -Dieses Skalar ist das Elastizitätsmodul $E$. +. +\] +Um die Beziehung von Spannung und Dehnung, welche mit Tensoren 2. Stufe ausgedrückt werden, zu beschreiben, wird ein Elastizitätstensor 4. Stufe benötigt. +Dieser ist im 1D Spannungszustand ein Tensor 0. Stufe und somit ein Skalar, der Elastizitätsmodul $E$. -Dieser Elastizitätstensor 4.Stufe kann als Tensor 2.Stufe, sprich als Matrix, dargestellt werden. -So wird die Spannungsgleichung stark vereinfacht, da nun ein Vektor mit einer Matrix operiert. +Dieser Elastizitätstensor 4. Stufe kann als Tensor 2. Stufe, sprich als Matrix, dargestellt werden. +So wird die Spannungsgleichung stark vereinfacht, da nun eine Matrix auf einen Vektor operiert. Dieser Tensor muss für eine Spannung jeden Einfluss aus allen 9 Dehnungen mit Konstanten erfassen. Dies bedeutet um eine von 9 Spannungen berechnen zu können müssen alle 9 Dehnung mit unterschiedlichen Faktoren summiert werden. Es ergeben sich $9^2$ Einträge, welches mit den 4 Indizes \[ i, j, k, l\in\left\{1, 2, 3\right\} +, \] -, die zueinander verknüpft werden müssen, zu begründen ist. -Es ergeben sich $3^4$ Einträge, sprich eine $9\times9$ Matrix, welche allgemein mit +die zueinander verknüpft werden müssen, zu begründen ist. +Es ergeben sich $3^4$ Einträge, sprich eine $9\times9$ Matrix, welche allgemein \[ \overline{\overline{C}} = @@ -141,25 +139,26 @@ C_{3211} & C_{3212} & C_{3213} & C_{3221} & C_{3222} & C_{3223} & C_{3231} & C_{ C_{3311} & C_{3312} & C_{3313} & C_{3321} & C_{3322} & C_{3323} & C_{3331} & C_{3332} & C_{3333} \end{pmatrix} \] -ausgedrückt wird. +geschrieben werden kann. Dieser Elastizitätstensor muss für isotrope Materialien zwingend symmetrisch sein. Folglich gilt: \[ \overline{\overline{C}} = \overline{\overline{C}}~^{T} -\]. - +. +\] Die allgemeine Spannungsgleichung lautet nun: \[ \vec\sigma = \overline{\overline{C}}\cdot\vec{\varepsilon} -\]. - +. +\] Die Konstanten $C$ werden nun nach dem Hook'schen Gesetz mit Hilfe des Elastizitätsmoduls $E$ definiert. -Da dieser Modul durch die eindimensionale Betrachtung definiert ist muss eine weitere Kennzahl eingeführt werden. -Dies ist die Querdehnungszahl $\nu$ (auch Poisson-Zahl), welche mit +Da dieser Modul durch die eindimensionale Betrachtung definiert ist, +muss für die dreidimensionale Betrachtung eine weitere Kennzahl eingeführt werden. +Dies ist die Querdehnungszahl $\nu$ (auch Poisson-Zahl), welche durch \[ \nu = @@ -168,17 +167,11 @@ Dies ist die Querdehnungszahl $\nu$ (auch Poisson-Zahl), welche mit \frac{\Delta b}{b_0} \] und -\[ -\varepsilon -= -\text{Längsdehnung [$-$]} -\] -\[ -\varepsilon_q -= -\text{Querdehnung [$-$]} -\] -definiert ist. Trägt man die Konstanten in die Matrix ein ergibt sich +\begin{align*} + \varepsilon &= \text{Längsdehnung [$-$]} \\ + \varepsilon_q &= \text{Querdehnung [$-$]} +\end{align*} +definiert ist. Trägt man die Konstanten in die Matrix ein, ergibt sich \[ \begin{pmatrix} \sigma_{11}\\ @@ -215,9 +208,9 @@ definiert ist. Trägt man die Konstanten in die Matrix ein ergibt sich \varepsilon_{32} \\ \varepsilon_{33} \end{pmatrix} +, \] - -, welche ebenfalls als Indexnotation mit +welche ebenfalls als Indexnotation mit \[ \sigma_{ij} = @@ -225,9 +218,8 @@ definiert ist. Trägt man die Konstanten in die Matrix ein ergibt sich \sum_{l=1}^3 C_{ijkl}\cdot\varepsilon_{kl} \] -ausgedrückt werden können. -Die Normalspannung $\sigma_{11}$ lässt sich exemplarisch mit - +ausgedrückt werden kann. +Die Normalspannung $\sigma_{22}$ lässt sich exemplarisch als \[ \sigma_{22} = @@ -247,10 +239,12 @@ Diese Symmetrie setzt daher voraus, dass = \sigma_{21} , +\qquad \sigma_{13} = \sigma_{31} , +\qquad \sigma_{23} = \sigma_{32} @@ -261,16 +255,18 @@ und folglich auch = \varepsilon_{21} , +\qquad \varepsilon_{13} = \varepsilon_{31} , +\qquad \varepsilon_{23} = \varepsilon_{32} \] gilt. -Diese Eigenschaft wird durch die Voigt'sche Notation ausgenutzt um die Gleichung vereinfachen zu können. +Diese Eigenschaft wird durch die Voigt'sche Notation \cite{spannung:Voigtsche Notation} ausgenutzt, um die Gleichung vereinfachen zu können. Durch diese Symmetrie gilt \[ \overline{\sigma} @@ -284,7 +280,7 @@ Durch diese Symmetrie gilt \begin{pmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ & \sigma_{22} & \sigma_{23} \\ - sym & & \sigma_{33} + \text{sym} & & \sigma_{33} \end{pmatrix} \qquad \Rightarrow @@ -328,9 +324,10 @@ und entsprechend \varepsilon_{13} \\ \varepsilon_{12} \end{pmatrix} -\]. +. +\] -Aus den Vereinfachungen der Voigt'schen Notation lassen sich die Spannungs- und Dehnungstensoren als Spaltenvektoren mit je 6 Einträgen darstellen. +Aus den Vereinfachungen der Voigt'schen Notation lassen sich die Spannungs- und Dehnungstensoren als Spaltenvektoren mit je sechs Einträgen darstellen. Der Elastizitätstensor kann entsprechend auf eine $6\times6$ Matrix reduziert werden. Es lässt sich nun eine reduzierte allgemeine Spannungsgleichung mit \[ @@ -350,12 +347,12 @@ beziehungsweise \end{pmatrix} = \begin{pmatrix} - C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\ - C_{21} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\ - C_{31} & C_{32} & C_{33} & C_{34} & C_{35} & C_{36} \\ - C_{41} & C_{42} & C_{43} & C_{44} & C_{45} & C_{46} \\ - C_{51} & C_{52} & C_{53} & C_{54} & C_{55} & C_{56} \\ - C_{61} & C_{62} & C_{63} & C_{64} & C_{65} & C_{66} + C_{1111} & C_{1122} & C_{1133} & C_{1123} & C_{1113} & C_{1112} \\ + C_{2211} & C_{2222} & C_{2233} & C_{2223} & C_{2213} & C_{2212} \\ + C_{3311} & C_{3322} & C_{3333} & C_{3323} & C_{3313} & C_{3312} \\ + C_{2311} & C_{2322} & C_{2333} & C_{2323} & C_{2313} & C_{2312} \\ + C_{1311} & C_{1322} & C_{1333} & C_{1323} & C_{1313} & C_{1312} \\ + C_{1211} & C_{1222} & C_{1233} & C_{1223} & C_{1213} & C_{1212} \end{pmatrix} \begin{pmatrix} \varepsilon_{11} \\ @@ -367,9 +364,9 @@ beziehungsweise \end{pmatrix} \] beschreiben. -Die Spannung $\sigma_{11}$ beispielsweise besteht so aus der Summe aller 6 Produkte der Konstanten $C$ und Dehnungen $\varepsilon$. +Die Spannung $\sigma_{11}$ beispielsweise erhält man, wenn man die sechs Produkte aus den Konstanten $C$ und Dehnungen $\varepsilon$ summiert. Die Symmetrieeigenschaft des Elastizitätstensors bleibt auch hier erhalten. -Nun lässt sich die reduzierte allgemeine Spannungsgleichung mit +Somit lässt sich die reduzierte allgemeine Spannungsgleichung mit \[ \begin{pmatrix} @@ -382,12 +379,12 @@ Nun lässt sich die reduzierte allgemeine Spannungsgleichung mit \end{pmatrix} = \begin{pmatrix} - C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\ - & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\ - & & C_{33} & C_{34} & C_{35} & C_{36} \\ - & & & C_{44} & C_{45} & C_{46} \\ - & & & & C_{55} & C_{56} \\ - \text{sym} & & & & & C_{66} + C_{1111} & C_{1122} & C_{1133} & C_{1123} & C_{1113} & C_{1112} \\ + & C_{2222} & C_{2233} & C_{2223} & C_{2213} & C_{2212} \\ + & & C_{3333} & C_{3323} & C_{3313} & C_{3312} \\ + & & & C_{2323} & C_{2313} & C_{2312} \\ + & & & & C_{1313} & C_{1312} \\ + \text{sym} & & & & & C_{1212} \end{pmatrix} \begin{pmatrix} \varepsilon_{11} \\ @@ -399,9 +396,8 @@ Nun lässt sich die reduzierte allgemeine Spannungsgleichung mit \end{pmatrix} \] beschreiben. -Die Konstanten $C$ und $\nu$ werden wieder nach dem Hook'schen Gesetz definiert. +Die Konstanten $C$ werden wieder nach dem Hook'schen Gesetz definiert. Dies ergibt die Spannungsgleichung, welche weit möglichst vereinfacht ist: - \[ \begin{pmatrix} \sigma_{11}\\ @@ -429,10 +425,11 @@ Dies ergibt die Spannungsgleichung, welche weit möglichst vereinfacht ist: \varepsilon_{13}\\ \varepsilon_{12} \end{pmatrix} -\]. +. +\] Im Elastizitätstensor fallen zwei $3\times3$ Blöcke auf, welche nur Einträge mit $0$ haben. Der Tensor besagt also, -dass diese jeweiligen Konstanten keinen Einfluss auf unsere Spannung haben. +dass diese jeweiligen Dehnungen keinen Einfluss auf unsere Spannung haben. Man sieht nun auch ganz gut, dass sich im Vergleich zu der allgemeinen Spannungsgleichung, die Einträge verschoben haben. Da nach Voigt zuerst die Normalspannungen und anschliessend die Schubspannungen notiert worden sind, ergeben sich die $3\times3$ Blöcke. @@ -477,27 +474,18 @@ Dadurch erhält man die Dehnungsgleichung: \sigma_{13}\\ \sigma_{12} \end{pmatrix} -\]. - +. +\] Die zwei $3\times3$ Blöcke links unten und rechts oben sind folglich noch vorhanden. -Um wieder die Einflüsse der Parameter veranschaulichen zu können berechnet man mit +Um wieder die Einflüsse der Parameter veranschaulichen zu können berechnet man die Dehnung \[ \varepsilon_{22} = \frac{1}{E}\sigma_{22} - \frac{\nu}{E}\sigma_{11} - \frac{\nu}{E}\sigma_{33} = \frac{1}{E}\cdot(\sigma_{22}-\nu\cdot\sigma_{11}-\nu\cdot\sigma_{33}) +. \] - -die Dehnung $\varepsilon_{22}$. Diese hängt wieder am meisten von $\sigma_{22}$ ab. Ist die Querdehnung $\nu$ grösser, so wird die Dehnung $\varepsilon_{22}$ reduziert. -Bei inkompressiblen Medien, bei welchen keine Dehnungen und nur identische Normalspannungen auftreten können, ist folglich -\[ -\nu -= -0.5 -\]. - - - +Bei inkompressiblen Medien, bei welchen keine Dehnungen und nur identische Normalspannungen auftreten können, ist folglich $\nu=0.5$. \ No newline at end of file diff --git a/buch/papers/spannung/teil3.tex b/buch/papers/spannung/teil3.tex index e5574b8..438ac31 100644 --- a/buch/papers/spannung/teil3.tex +++ b/buch/papers/spannung/teil3.tex @@ -1,80 +1,86 @@ -\section{Spannungsausbreitung\label{spannung:section:Invarianten}} -\rhead{Invarianten} -Trotz der Vereinfachung lässt sich mit den Invarianten die Realität adäquat abbilden. -Als erste Bedingung stellt man folgendes Verhältnis auf: +\section{Die geotechnischen Invarianten\label{spannung:section:Die geotechnischen Invarianten}} +\rhead{Die geotechnischen Invarianten} +In vielen Fällen in der Geotechnik und auch in Versuchen hat man gleichmässige Belastungen über eine grössere Fläche. +Durch eine solche Belastung auf den Boden, entstehen gleichermassen Spannungen in Richtung $2$ und $3$, +wenn man von einem isotropen Bodenmaterial ausgeht. +Folglich gilt: \[ \sigma_{22} = \sigma_{33} -\] . - -Dies deshalb, da man von einem isotropen Bodenmaterial ausgeht. -In Achse 22, Richtung 22 hat man den gleichen Boden wie in Achse 33 und Richtung 33. -Das Verhalten bezüglich Kraftaufnahme, Dehnung Spannung ist somit dasselbe. - -Man führt die zwei Werte p als hydrostatische Spannung und q als deviatorische Spannung ein. -Die Berechnung von p und q sieht wie folgt aus: - +\] +Dadurch wird der Spannungszustand vereinfacht. +Diesen vereinfachten Spannungszustand kann man mit den zwei geotechnischen Invarianten abbilden. +Die erste Invariante ist die volumetrische Spannung \[ p = \frac{\sigma_{11}+\sigma_{22}+\sigma_{33}}{3} +, \] - -oder durch Vereinfachung, da $\sigma_{22}=\sigma_{33}$ : - +welche als arithmetisches Mittel aller Normalspannungen im infinitesimalen Würfel definiert ist. +Die zweite Invariante ist die deviatorische Spannung +\[ +q += +\sqrt{\frac{(\sigma_{11}-\sigma_{22})^{2}+(\sigma_{11}-\sigma_{33})^{2}+(\sigma_{22}-\sigma_{33})^{2}}{2}} +. +\] +Diese Zusammenhänge werden im Skript [\cite{spannung:Stoffgesetze und numerische Modellierung in der Geotechnik}] aufgezeigt. +Die hydrostatische Spannung $p$ kann gemäss Gleichung (Nr) als \[ p = \frac{\sigma_{11}+2\sigma_{33}}{3} \] - +vereinfacht werden. +Die deviatorische Spannung $q$ wird gemäss Gleichung (Nr) als \[ q = \sigma_{11}-\sigma_{33} \] -. - -p ist das arithmetische Mittel von der Spannung im infinitesimalen Würfel. -q ist die Differenz zwischen der Spannung in vertikaler Richtung und der Spannung in Richtung 2 und 3. -Man kann p als Druckspannung und q als Schubspannung anschauen. - -Aus der Formel vom vorherigen Kapitel konnten wir die Spannungen berechnen. -Deshalb kann man nun p und q in die Gleichung einsetzen. -Die Dehnungen werden mit neuen Variablen eingeführt. -Die Deviatorische Dehnung kann mit einer Schubdehnung verglichen werden. -Die hydrostatische Dehnung kann mit einer Kompressionsdehnung verglichen - -\[ -\overbrace{\sigma_{11}-\sigma_{33}}^{q} -= -\frac{3E}{2(1+\nu)} \overbrace{\frac{2}{3}(\varepsilon_{11} - \varepsilon_{33})}^{\varepsilon_{\nu}} -\] +vereinfacht. Man kann $p$ als Isotrop und $q$ als Schub betrachten. +Die Invarianten können mit der Spannungsformel (Nr..xxx) berechnet werden. +Durch geschickte Umformung dieser Gleichung, lassen sich die Module als Faktor separieren. +Dabei entstehen spezielle Faktoren mit den Dehnungskomponenten. +So ergibt sich \[ \overbrace{\frac{\sigma_{11}+2\sigma_{33}}{3}}^{p} = -\frac{E}{3(1-2\nu)} \overbrace{(\varepsilon_{11} - 2\varepsilon_{33})}^{\varepsilon_{s}} +\frac{E}{3(1-2\nu)} \overbrace{(\varepsilon_{11} - 2\varepsilon_{33})}^{\varepsilon_{v}} \] - +und \[ -\varepsilon_{s} +\overbrace{\sigma_{11}-\sigma_{33}}^{q} = -\text{Hydrostatische Dehnung} [-] +\frac{3E}{2(1+\nu)} \overbrace{\frac{2}{3}(\varepsilon_{11} - \varepsilon_{33})}^{\varepsilon_{s}} +. \] - +Die Faktoren mit den Dehnungskomponenten können so mit \[ -\varepsilon_{\nu} +\varepsilon_{v} = -\text{Deviatorische Dehnung} [-] +(\varepsilon_{11} - 2\varepsilon_{33}) +\qquad +\text{und} +\qquad +\varepsilon_{s} += +\frac{2}{3}(\varepsilon_{11} - \varepsilon_{33}) \] - -werden. - -Diese Komponenten kann man nun in die Vereinfachte Matrix +eingeführt werden, mit +\begin{align*} + \varepsilon_{v} &= \text{Hydrostatische Dehnung [-]} \\ + \varepsilon_{s} &= \text{Deviatorische Dehnung [-].} +\end{align*} +Die hydrostatische Dehnung $\varepsilon_{v}$ kann mit einer Kompression verglichen werden. +Die deviatorische Dehnung $\varepsilon_{s}$ kann mit einer Verzerrung verglichen werden. + +Diese zwei Gleichungen kann man durch die Matrixschreibweise \[ \begin{pmatrix} q\\ @@ -87,12 +93,13 @@ Diese Komponenten kann man nun in die Vereinfachte Matrix \end{pmatrix} \begin{pmatrix} \varepsilon_{s}\\ - \varepsilon_{\nu} + \varepsilon_{v} \end{pmatrix} \] -einsetzen. -Man hat dann eine Matrix multipliziert mit einem Vektor und erhält einen Vektor. +(sollte nummeriert sein) vereinfachen. +Man hat so eine Matrix multipliziert mit einem Vektor und erhält einen Vektor. +Änderungen des Spannungszustandes können mit dieser Gleichung vollumfänglich erfasst werden. -Mit dieser Formel lassen sich verschieden Parameter von Versuchen analysieren und berechnen. -Ein solcher Versuch, den oft in der Geotechnik durchgeführt wird ist der Oedometer-Versuch. +Mit dieser Formel lassen sich verschieden Ergebnisse von Versuchen analysieren und berechnen. +Ein solcher Versuch, den oft in der Geotechnik durchgeführt wird, ist der Oedometer-Versuch. Im nächsten Kapitel wird die Anwendung der Matrix an diesem Versuch beschrieben. \ No newline at end of file diff --git a/buch/papers/spannung/teil4.tex b/buch/papers/spannung/teil4.tex index 60f2518..d524f13 100644 --- a/buch/papers/spannung/teil4.tex +++ b/buch/papers/spannung/teil4.tex @@ -1,16 +1,16 @@ \section{Oedometer-Versuch\label{spannung:section:Oedometer-Versuch}} \rhead{Oedometer-Versuch} -Mit dem Oedometer-Versuch kann der Oedometrische Elastizitätsmodul $E_{OED}$ bestimmt werden. +Mit dem Oedometer-Versuch kann der oedometrische Elastizitätsmodul $E_{OED}$ bestimmt werden. Dieser beschreibt ebenfalls das Verhältnis zwischen Spannung und Dehnung, allerdings unter anderen Bedingungen. Diese Bedingung ist das Verhindern der seitlichen Verformung, sprich der Dehnung in Richtung $1$ und $2$. Es wird ein Probeelement mit immer grösseren Gewichten belastet, welche gleichmässig auf das Material drücken. Die seitliche Verschiebung des Materials wird durch einen Stahlring verhindert. -Die Probe wird sich so steig verdichten. +Die Probe wird sich so stetig verdichten. Das Volumen nimmt ab und die Dehnung nimmt immer mehr zu. -Unter diesen Bedingungen wird das Oedometrische E-Modul mit steigender Dehnung zunehmen. +Unter diesen Bedingungen wird der oedometrische Elastizitätsmodul mit steigender Dehnung zunehmen. -Da im Boden das umgebende Material ähnliche eine seitliche Verformung verhindert, -gibt dieser Oedometrische E-Modul die Realität besser als der gewöhnliche E-Modul wieder. +Da im Boden das umgebende Material ähnlich eine seitliche Verformung verhindert, +bildet dieser oedometrische Elastizitätsmodul die Realität besser ab, als der gewöhnliche Elastizitätsmodul. Durch dieses Verhindern des seitlichen Ausbrechens ist \[ \varepsilon_{22} @@ -25,15 +25,16 @@ aber auch = \sigma_{33} \neq 0 +. \] -Die Spannung $\sigma_{11}$ wird durch durch die aufgebrachte Kraft mit +Die Spannung $\sigma_{11}$ wird durch die aufgebrachte Kraft mit \[ \sigma_{11} = \frac{F}{A} \] und die Dehnung $\varepsilon_{11}$ jeweils mit den entsprechenden Setzungen berechnet. -Diese Randbedingen können in die vereinfachte Gleichung eingesetzt. +Diese Randbedingungen können in die vereinfachte Gleichung (Nrxxx) eingesetzt werden. Diese lautet nun: \[ \begin{pmatrix} @@ -42,21 +43,30 @@ Diese lautet nun: \end{pmatrix} = \begin{pmatrix} - \frac{E_{OED}}{(1+\nu)} & 0 \\ - 0 & \frac{E_{OED}}{(1-2\nu)} + \frac{E_{OED}}{(1+\nu)} & 0 \\ + 0 & \frac{E_{OED}}{3(1-2\nu)} \end{pmatrix} \begin{pmatrix} \varepsilon_{11}\\ \varepsilon_{11} \end{pmatrix} -\] . - -Daraus lässt sich bei jedem Setzungsgrad das Oedometrische E-Modul $E_{OED}$ und die seitlichen Spannungen $\sigma_{33}$ mit den 2 Gleichungen - -GLEICHUNGEN... - +\] +Daraus lässt sich bei jedem Setzungsgrad der oedometrische Elastitzitätsmodul $E_{OED}$ und die seitlichen Spannungen $\sigma_{33}$ mit den 2 Gleichungen +\[ +\sigma_{11}-\sigma_{33} += +\frac{E_{OED}}{(1+\nu)}\cdot\varepsilon_{11} +\] +und +\[ +\sigma_{11}+2\sigma_{33} += +\frac{E_{OED}}{3(1-2\nu)}\cdot\varepsilon_{11} +\] berechnen. +Mit diesen Gleichungen hat man das Gleichungssystem um $E_{OED}$ und $\sigma_{33}$ zu berechnen. +Die Poisson-Zahl muss als Kennwert gemäss der Bodenklasse gewählt werden. Den Versuch kann man auf einem $\sigma$-$\varepsilon$-Diagramm abtragen (siehe Abbildung 1.7). Durch die Komprimierung nimmt der Boden mehr Spannung auf, und verformt sich zugleich weniger stark. Mit diesem ermittelten $E_{OED}$ kann man nun weitere Berechnungen für die Geotechnik durchführen. @@ -64,6 +74,6 @@ Mit diesem ermittelten $E_{OED}$ kann man nun weitere Berechnungen für die Geot \begin{figure} \centering \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/DiagrammOedometer-Versuch.png} - \caption{Diagramm Oedometer-Versuch} - \label{fig:Diagramm Oedometer-Versuch} + \caption{Diagramm Charakteristik verschiedener Elastizitätsmodule bei gleichem Material} + \label{fig:DiagrammOedometer-Versuch} \end{figure} \ No newline at end of file -- cgit v1.2.1 From e52cb985f06bec15524ae4029b65dc537716384d Mon Sep 17 00:00:00 2001 From: Malarius1999 Date: Tue, 1 Jun 2021 14:31:06 +0200 Subject: fixed imports and rheads --- buch/papers/clifford/0_ElevatorPitch.tex | 2 +- buch/papers/clifford/10_Quaternionen.tex | 1 + buch/papers/clifford/1_Vektordarstellung.tex | 4 +-- buch/papers/clifford/6_Dirac-Matrizen.tex | 2 +- buch/papers/clifford/7_Reflektion.tex | 1 + buch/papers/clifford/8_Rotation.tex | 1 + buch/papers/clifford/9_KomplexeZahlen.tex | 1 + buch/papers/clifford/main.tex | 42 ++++++++-------------------- buch/papers/clifford/packages.tex | 22 --------------- 9 files changed, 20 insertions(+), 56 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/clifford/0_ElevatorPitch.tex b/buch/papers/clifford/0_ElevatorPitch.tex index a599903..0db5617 100644 --- a/buch/papers/clifford/0_ElevatorPitch.tex +++ b/buch/papers/clifford/0_ElevatorPitch.tex @@ -1,2 +1,2 @@ - +TODO... GA [Geometric Algebra i.a.W. Clifford Algebra] provides a unified language for the whole of physics and for much of mathematics and its applications that is conceptually and computationally superior to alternative mathematical systems in many application domains. \ No newline at end of file diff --git a/buch/papers/clifford/10_Quaternionen.tex b/buch/papers/clifford/10_Quaternionen.tex index c987fc3..8945ba8 100644 --- a/buch/papers/clifford/10_Quaternionen.tex +++ b/buch/papers/clifford/10_Quaternionen.tex @@ -4,6 +4,7 @@ % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % \section{Quaternionen} +\rhead{Quaternionen} Wie die komplexen Zahlen eine Erweiterung der reellen Zahlen sind, sind die Quaternionen eine Erweiterung der komplexen Zahlen für den 3 dimensionalen Raum. Sie haben, wie die komplexen Zahlen, eine dreh-streckende Eigenschaft. Sie finden beispielsweise in der Computergraphik und in der Robotik Anwendung. Die Quaternionen werden so definiert. diff --git a/buch/papers/clifford/1_Vektordarstellung.tex b/buch/papers/clifford/1_Vektordarstellung.tex index cb6e7af..88a5789 100644 --- a/buch/papers/clifford/1_Vektordarstellung.tex +++ b/buch/papers/clifford/1_Vektordarstellung.tex @@ -1,6 +1,6 @@ -\section{Teil 0\label{clifford:section:Vektoroperationen}} +\section{Vektoroperationen\label{clifford:section:Vektoroperationen}} \rhead{Vektoroperationen} -\rhead{Vektordarstellung} +\subsection{Vektordarstellung\label{clifford:section:Vektordarstellung}} Vektoren können neben der üblichen Darstellung, auch als Linearkombination aus Basisvektoren dargestellt werden \begin{equation} \begin{split} diff --git a/buch/papers/clifford/6_Dirac-Matrizen.tex b/buch/papers/clifford/6_Dirac-Matrizen.tex index e68f0f6..6417bb3 100644 --- a/buch/papers/clifford/6_Dirac-Matrizen.tex +++ b/buch/papers/clifford/6_Dirac-Matrizen.tex @@ -4,4 +4,4 @@ % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % \section{Dirac-Matrizen} - +\rhead{Dirac-Matrizen} diff --git a/buch/papers/clifford/7_Reflektion.tex b/buch/papers/clifford/7_Reflektion.tex index dfe86b8..d4942e0 100644 --- a/buch/papers/clifford/7_Reflektion.tex +++ b/buch/papers/clifford/7_Reflektion.tex @@ -4,6 +4,7 @@ % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % \section{Reflektion/ Spiegelung} +\rhead{Reflektion/ Spiegelung} Die Spiegelung ist eine grundlegende, geometrische Operation, aus welcher man weitere, wie beispielsweise die später beschriebene Rotation, ableiten kann. Da die Geometrische Algebra für geometrische Anwendungen ausgelegt ist, sollte die Reflektion auch eine einfache, praktische Formulierung besitzen. \\HIER BILD \subsection{linearen Algebra} Aus der linearen Algebra ist bekannt, dass man eine Reflektion wie folgt beschreiben kann. diff --git a/buch/papers/clifford/8_Rotation.tex b/buch/papers/clifford/8_Rotation.tex index ebd278c..c2928bf 100644 --- a/buch/papers/clifford/8_Rotation.tex +++ b/buch/papers/clifford/8_Rotation.tex @@ -4,6 +4,7 @@ % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % \section{Rotation} +\rhead{Rotation} Eine Rotation kann man aus zwei, aufeinanderfolgende Reflektionen bilden. Das war für mich zuerst eine verwirrende Aussage, da man aus den vorherig gezeigten Formeln annehmen könnte, dass die Reflektion schon für eine Drehung ausreicht. Obwohl sich die Längen, Winkel und Volumen sich bei einer Reflektion, wie bei einer Rotation, nicht ändert, sind sie doch verschieden, da die Orientierung bei der Reflektion invertiert wird. Stellt man sich beispielsweise ein Objekt in 3D vor und spiegelt dieses an einer Fläche, dann ist es unmöglich nur durch eine Rotation (egal an welchem Punkt) das ursprüngliche Objekt deckungsgleich auf das Gespiegelte zu drehen. Hingegen ist es wiederum möglich ein zweifach gespiegeltes Objekt durch eine Drehung zu erreichen. Das liegt daran, da die Orientierung zwei mal invertiert wurde. \\BILD diff --git a/buch/papers/clifford/9_KomplexeZahlen.tex b/buch/papers/clifford/9_KomplexeZahlen.tex index 735eead..4dbab2c 100644 --- a/buch/papers/clifford/9_KomplexeZahlen.tex +++ b/buch/papers/clifford/9_KomplexeZahlen.tex @@ -4,6 +4,7 @@ % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % \section{komplexe Zahlen} +\rhead{komplexe Zahlen} Die komplexen Zahlen finden eine Vielzahl von Anwendungsgebiete in den Ingenieurwissenschaften. Das liegt daran, weil die komplexen Zahlen Rotationen und Schwingungen gut beschreiben können. Nachdem vorherigen Kapitel überrascht es wahrscheinlich nicht viele, dass es möglich ist Komplexe Zahlen in der geometrischen Algebra darzustellen. Sie können durch die geraden Grade der 2 Dimensionalen geometrischen Algebra vollständig beschrieben werden: $\mathbb{G}_2^+ \cong \mathbb{C}$. Das bedeutet eine komplexe Zahl kann durch ein Skalar (Grade 0) und einem Bivektor (Grade 2) dargestellt werden. Als Abkürzung nehme ich die Bezeichnung $g_n \in \mathbb{G}_2^+$. \begin{align} a_0 + a_1 j \cong a_0 + a_1 e_{12} = g_n;\quad a_0, a_1 \in \mathbb{R} diff --git a/buch/papers/clifford/main.tex b/buch/papers/clifford/main.tex index d94e065..46d04bd 100644 --- a/buch/papers/clifford/main.tex +++ b/buch/papers/clifford/main.tex @@ -3,41 +3,23 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:clifford}} +\chapter{Clifford Algebra\label{chapter:clifford}} \lhead{Clifford Algebra} \begin{refsection} \chapterauthor{Thierry Schwaller, Marius Baumann} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} -\input{0_ElevatorPitch} -\input{1_Vektordarstellung} -\input{2_QuadratVektoren} -\input{3_MultiplikationVektoren} -\input{4_GeometrischesProdukt} -\input{5_PolareDarstellung} -\input{6_Dirac-Matrizen} -\input{7_Reflektion} -\input{8_Rotation} -\input{9_KomplexeZahlen} -\input{10_Quaternionen} +\input{papers/clifford/0_ElevatorPitch.tex} +\input{papers/clifford/1_Vektordarstellung.tex} +\input{papers/clifford/2_QuadratVektoren.tex} +\input{papers/clifford/3_MultiplikationVektoren.tex} +\input{papers/clifford/4_GeometrischesProdukt.tex} +\input{papers/clifford/5_PolareDarstellung.tex} +\input{papers/clifford/6_Dirac-Matrizen.tex} +\input{papers/clifford/7_Reflektion.tex} +\input{papers/clifford/8_Rotation.tex} +\input{papers/clifford/9_KomplexeZahlen.tex} +\input{papers/clifford/10_Quaternionen.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/clifford/packages.tex b/buch/papers/clifford/packages.tex index f6e94e0..8fb4bd9 100644 --- a/buch/papers/clifford/packages.tex +++ b/buch/papers/clifford/packages.tex @@ -7,25 +7,3 @@ % if your paper needs special packages, add package commands as in the % following example %\usepackage{packagename} -\usepackage[utf8]{inputenc} -\usepackage{a4wide} -\usepackage{ngerman} -\usepackage{tikz} -\usepackage{mathdots} -\usepackage{amssymb} -\usepackage{amsmath} -\usepackage{amsthm} -\newtheorem{definition}{Definition}[chapter] -\newtheorem{beispiel}[definition]{Beispiel} -\newtheorem{bemerkung}[definition]{Bemerkung} -\newtheorem{lemma}[definition]{Lemma} -\newtheorem{satz}[definition]{Satz} -\newtheorem{hauptsatz}[definition]{Hauptsatz} -\newtheorem{corollar}[definition]{Korollar} -\usepackage[german]{babel} -\usepackage[T1]{fontenc} -\usepackage{fullpage} -\usepackage{graphicx} -\usepackage{float} -\usepackage{colortbl} -\usepackage{multirow} \ No newline at end of file -- cgit v1.2.1 From b70156cbf2d76d1850ddd1fc6f58e79bdc5c5203 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Wed, 2 Jun 2021 07:53:42 +0200 Subject: Makefile in clifford, references in spannung --- buch/papers/clifford/Makefile.inc | 20 +++++++++++++------- buch/papers/spannung/Einleitung.tex | 6 +++--- buch/papers/spannung/references.bib | 6 +++--- buch/papers/spannung/teil1.tex | 2 +- buch/papers/spannung/teil2.tex | 6 +++--- buch/papers/spannung/teil3.tex | 4 ++-- 6 files changed, 25 insertions(+), 19 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/clifford/Makefile.inc b/buch/papers/clifford/Makefile.inc index 7b941b3..8cdd02e 100644 --- a/buch/papers/clifford/Makefile.inc +++ b/buch/papers/clifford/Makefile.inc @@ -3,12 +3,18 @@ # # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -dependencies-clifford = \ +dependencies-clifford = \ papers/clifford/packages.tex \ papers/clifford/main.tex \ - papers/clifford/references.bib \ - papers/clifford/teil0.tex \ - papers/clifford/teil1.tex \ - papers/clifford/teil2.tex \ - papers/clifford/teil3.tex - + papers/clifford/references.bib \ + papers/clifford/0_ElevatorPitch.tex \ + papers/clifford/1_Vektordarstellung.tex \ + papers/clifford/2_QuadratVektoren.tex \ + papers/clifford/3_MultiplikationVektoren.tex \ + papers/clifford/4_GeometrischesProdukt.tex \ + papers/clifford/5_PolareDarstellung.tex \ + papers/clifford/6_Dirac-Matrizen.tex \ + papers/clifford/7_Reflektion.tex \ + papers/clifford/8_Rotation.tex \ + papers/clifford/9_KomplexeZahlen.tex \ + papers/clifford/10_Quaternionen.tex diff --git a/buch/papers/spannung/Einleitung.tex b/buch/papers/spannung/Einleitung.tex index c80db64..0cb1433 100644 --- a/buch/papers/spannung/Einleitung.tex +++ b/buch/papers/spannung/Einleitung.tex @@ -12,7 +12,7 @@ Diese Spannungsformel ist Grundlage für Computerprogramme und geotechnische Ver Um die mathematische Untersuchung vorzunehmen, beschäftigt man sich zuerst mit den spezifischen Gegebenheiten und Voraussetzungen. Ebenfalls gilt es ein paar wichtige Begriffe und deren mathematischen Zeichen einzuführen. In diesem Kapitel gehen wir auch auf die Zusammenhänge von Spannung, Dehnungen und Verformungen an elastischen Materialien ein, -wie sie in gängigen Lehrbüchern der Mechanik oder der Geotechnik behandelt werden. z. B. [\cite{spannung:Grundlagen der Geotechnik}] +wie sie in gängigen Lehrbüchern der Mechanik oder der Geotechnik behandelt werden, z.~B.~\cite{spannung:Grundlagen-der-Geotechnik}. \section{Spannungsausbreitung\label{spannung:section:Spannungsausbreitung}} \rhead{Spannungsausbreitung} @@ -72,7 +72,7 @@ berechnet werden mit: t &= \text{Tiefe [\si{\meter}]} \\ s &= \text{Setzung, Absenkung [m].} \end{align*} -Diese Zusammenhänge sind wie erwähnt unter anderem im Lehrbuch [\cite{spannung:Grundlagen der Geotechnik}] beschrieben. +Diese Zusammenhänge sind wie erwähnt unter anderem im Lehrbuch [\cite{spannung:Grundlagen-der-Geotechnik}] beschrieben. In der praktischen Geotechnik wird man allerdings weitaus schwierigere Situationen antreffen. Ein Beispiel wäre eine Baugrube mit einem Baugrubenabschluss, wo ein Teil des Bodens abgetragen ist (siehe Abbildung 1.3). Die Ausbreitung der Zusatzspannung $\sigma(x,y,t)$ würde hier deutlich komplizierter ausfallen. @@ -86,4 +86,4 @@ Generell wird im Ingenieurwesen versucht Phänomene möglichst nach dem Hook'sch \includegraphics[width=0.45\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild3.png} \caption{Beispiel eines Lastauftrags auf den Boden bei einer komplexeren Situation, welches kompliziertere Spannungsausbreitung zur Folge hat} \label{fig:Bild3} -\end{figure} \ No newline at end of file +\end{figure} diff --git a/buch/papers/spannung/references.bib b/buch/papers/spannung/references.bib index 090e3c3..02f8d09 100644 --- a/buch/papers/spannung/references.bib +++ b/buch/papers/spannung/references.bib @@ -13,7 +13,7 @@ day = {6} } -@online{spannung:Voigtsche Notation, +@online{spannung:Voigtsche-Notation, title = {Voigtsche Notation}, url = {https://de.wikipedia.org/wiki/Voigtsche_Notation}, date = {2021-05-29}, @@ -22,7 +22,7 @@ day = {6} } -@book{spannung:Grundlagen der Geotechnik, +@book{spannung:Grundlagen-der-Geotechnik, title = {Grundlagen der Geotechnik}, author = {Hans-Henning Schmidt and Roland F. Buchmaier and Carola Vogt-Breyer}, publisher = {Springer Fachmedien Wiesbaden GmbH}, @@ -32,7 +32,7 @@ volume = {5} } -@book{spannung:Stoffgesetze und numerische Modellierung in der Geotechnik, +@book{spannung:Stoffgesetze-und-numerische-Modellierung-in-der-Geotechnik, title = {Stoffgesetze und numerische Modellierung in der Geotechnik}, author = {Carlo Rabaiotti and Alessio Höttges}, publisher = {Hochschule Rapperswil}, diff --git a/buch/papers/spannung/teil1.tex b/buch/papers/spannung/teil1.tex index 2db244e..74516c1 100644 --- a/buch/papers/spannung/teil1.tex +++ b/buch/papers/spannung/teil1.tex @@ -21,4 +21,4 @@ Er hat in der Elastizitätstheorie als erstes Tensoren eingesetzt und beschriebe Auch Albert Einstein hat solche Tensoren eingesetzt, um in der Relativitätstheorie die Änderung der 4D Raumzeit beschreiben zu können. \cite{spannung:Tensor} -\cite{spannung:Voigtsche Notation} \ No newline at end of file +\cite{spannung:Voigtsche-Notation} diff --git a/buch/papers/spannung/teil2.tex b/buch/papers/spannung/teil2.tex index afd2c21..921d2b8 100644 --- a/buch/papers/spannung/teil2.tex +++ b/buch/papers/spannung/teil2.tex @@ -15,7 +15,7 @@ i, j\in\left\{1, 2, 3\right\} \] definiert. Daher ergeben sich die neun Spannungen. -Die nachfolgenden Zusammenhänge sind in \cite{spannung:Voigtsche Notation} beschrieben. +Die nachfolgenden Zusammenhänge sind in \cite{spannung:Voigtsche-Notation} beschrieben. Dieser Spannungstensor kann schliesslich mit $3^2$ Einträgen als $3\times3$ Matrix mit \[ \overline{\sigma} @@ -266,7 +266,7 @@ und folglich auch \varepsilon_{32} \] gilt. -Diese Eigenschaft wird durch die Voigt'sche Notation \cite{spannung:Voigtsche Notation} ausgenutzt, um die Gleichung vereinfachen zu können. +Diese Eigenschaft wird durch die Voigt'sche Notation \cite{spannung:Voigtsche-Notation} ausgenutzt, um die Gleichung vereinfachen zu können. Durch diese Symmetrie gilt \[ \overline{\sigma} @@ -488,4 +488,4 @@ Um wieder die Einflüsse der Parameter veranschaulichen zu können berechnet man \] Diese hängt wieder am meisten von $\sigma_{22}$ ab. Ist die Querdehnung $\nu$ grösser, so wird die Dehnung $\varepsilon_{22}$ reduziert. -Bei inkompressiblen Medien, bei welchen keine Dehnungen und nur identische Normalspannungen auftreten können, ist folglich $\nu=0.5$. \ No newline at end of file +Bei inkompressiblen Medien, bei welchen keine Dehnungen und nur identische Normalspannungen auftreten können, ist folglich $\nu=0.5$. diff --git a/buch/papers/spannung/teil3.tex b/buch/papers/spannung/teil3.tex index 438ac31..8d99733 100644 --- a/buch/papers/spannung/teil3.tex +++ b/buch/papers/spannung/teil3.tex @@ -28,7 +28,7 @@ q \sqrt{\frac{(\sigma_{11}-\sigma_{22})^{2}+(\sigma_{11}-\sigma_{33})^{2}+(\sigma_{22}-\sigma_{33})^{2}}{2}} . \] -Diese Zusammenhänge werden im Skript [\cite{spannung:Stoffgesetze und numerische Modellierung in der Geotechnik}] aufgezeigt. +Diese Zusammenhänge werden im Skript [\cite{spannung:Stoffgesetze-und-numerische-Modellierung-in-der-Geotechnik}] aufgezeigt. Die hydrostatische Spannung $p$ kann gemäss Gleichung (Nr) als \[ p @@ -102,4 +102,4 @@ Man hat so eine Matrix multipliziert mit einem Vektor und erhält einen Vektor. Mit dieser Formel lassen sich verschieden Ergebnisse von Versuchen analysieren und berechnen. Ein solcher Versuch, den oft in der Geotechnik durchgeführt wird, ist der Oedometer-Versuch. -Im nächsten Kapitel wird die Anwendung der Matrix an diesem Versuch beschrieben. \ No newline at end of file +Im nächsten Kapitel wird die Anwendung der Matrix an diesem Versuch beschrieben. -- cgit v1.2.1 From dfb9b5075e428e41f02cdf2d758a02899eea7e1e Mon Sep 17 00:00:00 2001 From: Alain Date: Fri, 4 Jun 2021 18:55:37 +0200 Subject: New Chapter IFS --- buch/papers/ifs/images/koch0-eps-converted-to.pdf | Bin 0 -> 5087 bytes buch/papers/ifs/images/koch1-eps-converted-to.pdf | Bin 0 -> 5141 bytes buch/papers/ifs/images/koch2-eps-converted-to.pdf | Bin 0 -> 5210 bytes buch/papers/ifs/images/koch8-eps-converted-to.pdf | Bin 0 -> 103521 bytes buch/papers/ifs/images/sierpinski.PNG | Bin 0 -> 293448 bytes buch/papers/ifs/images/sierpinski1.PNG | Bin 0 -> 11571 bytes buch/papers/ifs/images/sierpinski2.PNG | Bin 0 -> 12811 bytes buch/papers/ifs/images/sierpinski3.PNG | Bin 0 -> 14204 bytes buch/papers/ifs/images/sierpinski6.PNG | Bin 0 -> 30626 bytes buch/papers/ifs/main.tex | 19 ---- buch/papers/ifs/teil2.tex | 128 +++++++++++++++++----- buch/papers/ifs/teil3.tex | 46 +++----- 12 files changed, 114 insertions(+), 79 deletions(-) create mode 100644 buch/papers/ifs/images/koch0-eps-converted-to.pdf create mode 100644 buch/papers/ifs/images/koch1-eps-converted-to.pdf create mode 100644 buch/papers/ifs/images/koch2-eps-converted-to.pdf create mode 100644 buch/papers/ifs/images/koch8-eps-converted-to.pdf create mode 100644 buch/papers/ifs/images/sierpinski.PNG create mode 100644 buch/papers/ifs/images/sierpinski1.PNG create mode 100644 buch/papers/ifs/images/sierpinski2.PNG create mode 100644 buch/papers/ifs/images/sierpinski3.PNG create mode 100644 buch/papers/ifs/images/sierpinski6.PNG (limited to 'buch/papers') diff --git a/buch/papers/ifs/images/koch0-eps-converted-to.pdf b/buch/papers/ifs/images/koch0-eps-converted-to.pdf new file mode 100644 index 0000000..078c399 Binary files /dev/null and b/buch/papers/ifs/images/koch0-eps-converted-to.pdf differ diff --git a/buch/papers/ifs/images/koch1-eps-converted-to.pdf b/buch/papers/ifs/images/koch1-eps-converted-to.pdf new file mode 100644 index 0000000..81dcf18 Binary files /dev/null and b/buch/papers/ifs/images/koch1-eps-converted-to.pdf differ diff --git a/buch/papers/ifs/images/koch2-eps-converted-to.pdf b/buch/papers/ifs/images/koch2-eps-converted-to.pdf new file mode 100644 index 0000000..b7c7de7 Binary files /dev/null and b/buch/papers/ifs/images/koch2-eps-converted-to.pdf differ diff --git a/buch/papers/ifs/images/koch8-eps-converted-to.pdf b/buch/papers/ifs/images/koch8-eps-converted-to.pdf new file mode 100644 index 0000000..0bafd03 Binary files /dev/null and b/buch/papers/ifs/images/koch8-eps-converted-to.pdf differ diff --git a/buch/papers/ifs/images/sierpinski.PNG b/buch/papers/ifs/images/sierpinski.PNG new file mode 100644 index 0000000..1e57bf1 Binary files /dev/null and b/buch/papers/ifs/images/sierpinski.PNG differ diff --git a/buch/papers/ifs/images/sierpinski1.PNG b/buch/papers/ifs/images/sierpinski1.PNG new file mode 100644 index 0000000..91195f9 Binary files /dev/null and b/buch/papers/ifs/images/sierpinski1.PNG differ diff --git a/buch/papers/ifs/images/sierpinski2.PNG b/buch/papers/ifs/images/sierpinski2.PNG new file mode 100644 index 0000000..df57c13 Binary files /dev/null and b/buch/papers/ifs/images/sierpinski2.PNG differ diff --git a/buch/papers/ifs/images/sierpinski3.PNG b/buch/papers/ifs/images/sierpinski3.PNG new file mode 100644 index 0000000..055818f Binary files /dev/null and b/buch/papers/ifs/images/sierpinski3.PNG differ diff --git a/buch/papers/ifs/images/sierpinski6.PNG b/buch/papers/ifs/images/sierpinski6.PNG new file mode 100644 index 0000000..7990497 Binary files /dev/null and b/buch/papers/ifs/images/sierpinski6.PNG differ diff --git a/buch/papers/ifs/main.tex b/buch/papers/ifs/main.tex index 48c38f9..8ae0fad 100644 --- a/buch/papers/ifs/main.tex +++ b/buch/papers/ifs/main.tex @@ -8,25 +8,6 @@ \begin{refsection} \chapterauthor{Alain Keller} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} - \input{papers/ifs/teil0.tex} \input{papers/ifs/teil1.tex} \input{papers/ifs/teil2.tex} diff --git a/buch/papers/ifs/teil2.tex b/buch/papers/ifs/teil2.tex index bfd1684..a3d5ee1 100644 --- a/buch/papers/ifs/teil2.tex +++ b/buch/papers/ifs/teil2.tex @@ -3,38 +3,106 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 2 +\section{Fraktale mit IFS \label{ifs:section:teil2}} \rhead{Teil 2} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? +Wollen wir nun eine bestimmte Art anschauen, wie man Fraktale machen kann. +Zur veranschaulichung dieser Methode nehmen wir das Sierpinski Dreieck. +\begin{figure} + \label{ifs:sierpinski10} + \centering + \includegraphics[width=0.5\textwidth]{papers/ifs/images/sierpinski} + \caption{Sierpinski-Dreieck} +\end{figure} +Wenn man das Dreieck genau anschaut, erkennt man schnell, dass es aus drei kleineren Kopien seiner selbst besteht. +Es ist also ein Selbstähnliches Konstrukt. +Diese Eigenschaft wollen wir uns zunutze machen. -\subsection{De finibus bonorum et malorum -\label{ifs:subsection:bonorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +Wir definieren das Dreieck mit kantenlänge 1 als Menge $X$. +Ausserdem bestimmen wir drei Funktionen, welche die gesamte Menge auf eine ihrer kleineren Kopien abbildet +\begin{align*} + f_1(x,y) + = + \begin{pmatrix} + \frac{1}{2} & 0 \\ + 0 & \frac{1}{2} \\ + \end{pmatrix} + \begin{pmatrix} + x\\ + y\\ + \end{pmatrix} + ,\quad + f_2(x,y) + = + \begin{pmatrix} + \frac{1}{2} & 0 \\ + 0 & \frac{1}{2} \\ + \end{pmatrix} + \begin{pmatrix} + x\\ + y\\ + \end{pmatrix} + + + \begin{pmatrix} + \frac{1}{2} \\ + 0 + \end{pmatrix} + , \quad + f_3(x,y) + = + \begin{pmatrix} + \frac{1}{2} & 0 \\ + 0 & \frac{1}{2} \\ + \end{pmatrix} + \begin{pmatrix} + x\\ + y\\ + \end{pmatrix} + + + \begin{pmatrix} + \frac{1}{4} \\ + \frac{1}{2} + \end{pmatrix}\\ +\end{align*} +$f_1$ bildet das Dreieck auf das Teilstück unten links ab, $f_2$ auf das Teilstück unten rechts und $f_3$ auf das obere Teilstück. +Wendet man alle drei Funktionen auf das Sierpinski-Dreieck an, entsteht also wieder ein Sierpinski-Dreieck. +\begin{align*} + X = \bigcup\limits_{i = 1}^{3} f_i(X) +\end{align*} +Man kann sogar noch einen Schritt weiter gehen, und sagen: Wenn wir die Funktionen auf eine beliebige Startmenge anwenden, konvergeiert die Menge gegen das Sierpinski-Dreieck. +\begin{figure} + \label{ifs:sierpconst} + \centering + \subfigure[]{ + \label{ifs:sierpconsta} + \includegraphics[width=0.25\textwidth]{papers/ifs/images/sierpinski1}} + \subfigure[]{ + \label{ifs:sierpconstb} + \includegraphics[width=0.25\textwidth]{papers/ifs/images/sierpinski2}} + \subfigure[]{ + \label{ifs:sierpconstc} + \includegraphics[width=0.25\textwidth]{papers/ifs/images/sierpinski3}} + \subfigure[]{ + \label{ifs:sierpconstd} + \includegraphics[width=0.25\textwidth]{papers/ifs/images/sierpinski6}} + \caption{Konstruktion eines Sierpinski-Dreiecks mit einem Schwarzen Quadrat als Start\\ + (a) 1. Iteration (b) 2. Iteration (c) 3. Iteration (d) 5. Iteration} +\end{figure} +Im Beispiel der Abbildung \ref{ifs:sierpconst} sehen wir, wie das Bild nach jeder Iteration dem Sierpinski-Dreieck ähnlicher wird. +Der Abstand zum Original wird immer kleiner, und konvergiert bei unendlich Iterationen gegen null. + +\subsection{Iterierte Funktionensysteme +\label{ifs:subsection:bonorum}} +In diesem Unterkapitel wollen wir die Erkenntniss, wie wir aus einer beliebigen Menge ein Sierpinski-Dreieck genereieren können, verallgemeinern. +TODO TEXT +$S_1_...,S_n$ sind Kontraktionen auf die Menge $D \subset \mathbb{R}^n$. Es gilt +\begin{align} + |S_i(x) - S_i(y)| \leq c_i|x - y| +\end{align} +für jedes i mit einem $c_i < 1$. Dann existiert eine eindeutige kompakte Menge $F$ für die gilt +\begin{equation} + F = \bigcup\limits_{i = 1}^{m} S_i(F) +\end{equation} +TODO Text diff --git a/buch/papers/ifs/teil3.tex b/buch/papers/ifs/teil3.tex index 23fabbc..bba6e32 100644 --- a/buch/papers/ifs/teil3.tex +++ b/buch/papers/ifs/teil3.tex @@ -3,38 +3,24 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 3 +\section{Fraktale Bildkomprimierung \label{ifs:section:teil3}} -\rhead{Teil 3} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? +\rhead{Fraktale Bildkomprimierung} +Mit dem Prinzip dieser IFS ist es auch möglich Bilder zu Komprimieren. +Diese Idee hatte der Mathematiker Michael Barnsley, welcher mit seinem Buch Fractals Everywhere einen wichtigen beitrag zum verständnis von Fraktalen geiefert hat. +Das Ziel ist es ein IFS zu finden, welches das Bild als Attraktor hat. +In diesem Unterkapitel wollen wir eine Methode dafür anschauen. -\subsection{De finibus bonorum et malorum +\subsection{Titel \label{ifs:subsection:malorum}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis -est et expedita distinctio. Nam libero tempore, cum soluta nobis -est eligendi optio cumque nihil impedit quo minus id quod maxime -placeat facere possimus, omnis voluptas assumenda est, omnis dolor -repellendus. Temporibus autem quibusdam et aut officiis debitis aut -rerum necessitatibus saepe eveniet ut et voluptates repudiandae -sint et molestiae non recusandae. Itaque earum rerum hic tenetur a -sapiente delectus, ut aut reiciendis voluptatibus maiores alias -consequatur aut perferendis doloribus asperiores repellat. +Bis jetzt wurde in Zusammenhnag mit IFS immer erwähnt, dass die Transformationen auf die ganze Menge angewendet werden. +Dies muss jedoch nicht so sein. +Es gibt auch einen Attraktor, wenn die Transformationen nur Teile der Menge auf die ganze Menge abbilden. +Diese Eigenschaft wollen wir uns in der Fraktalen Bildkompression zunutze machen. +Sie ermöglicht uns Ähnlichkeiten zwischen kleineren Teilen des Bildes zunutze machen. +Es ist wohl nicht Falsch zu sagen, dass Ähnlichkeiten zur gesamten Menge, wie wir sie zum Beispiel beim Barnsley Fern gesehen haben, bei Bilder aus dem Alltag eher selten anzutreffen sind. +Doch wie Finden wir die richtigen Affinen Transformationen, welche als IFS das Bild als Attraktor haben. + + -- cgit v1.2.1 From 1bfb8ee184dad8fec1aee19cd7d57f62374f9c2a Mon Sep 17 00:00:00 2001 From: Alain Date: Sat, 5 Jun 2021 14:00:27 +0200 Subject: chap3 a bit --- buch/papers/ifs/teil3.tex | 68 ++++++++++++++++++++++++++++++++++++++++++++--- 1 file changed, 65 insertions(+), 3 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/ifs/teil3.tex b/buch/papers/ifs/teil3.tex index bba6e32..d31eee7 100644 --- a/buch/papers/ifs/teil3.tex +++ b/buch/papers/ifs/teil3.tex @@ -11,16 +11,78 @@ Diese Idee hatte der Mathematiker Michael Barnsley, welcher mit seinem Buch Frac Das Ziel ist es ein IFS zu finden, welches das Bild als Attraktor hat. In diesem Unterkapitel wollen wir eine Methode dafür anschauen. -\subsection{Titel -\label{ifs:subsection:malorum}} + Bis jetzt wurde in Zusammenhnag mit IFS immer erwähnt, dass die Transformationen auf die ganze Menge angewendet werden. Dies muss jedoch nicht so sein. Es gibt auch einen Attraktor, wenn die Transformationen nur Teile der Menge auf die ganze Menge abbilden. Diese Eigenschaft wollen wir uns in der Fraktalen Bildkompression zunutze machen. Sie ermöglicht uns Ähnlichkeiten zwischen kleineren Teilen des Bildes zunutze machen. Es ist wohl nicht Falsch zu sagen, dass Ähnlichkeiten zur gesamten Menge, wie wir sie zum Beispiel beim Barnsley Fern gesehen haben, bei Bilder aus dem Alltag eher selten anzutreffen sind. -Doch wie Finden wir die richtigen Affinen Transformationen, welche als IFS das Bild als Attraktor haben. +Doch wie Finden wir die richtigen Affinen Transformationen, welche als IFS das Bild als Attraktor haben? + +\subsection{Titel +\label{ifs:subsection:malorum}} +In der Beschreibung des Verfahrens wird sich auf Graustufenbilder bezogen. Wie das Verfahren für Farbbilder verwendet werden kann, wird später erläutert. + +In einem ersten Schritt teilen wir das Bild in disjunkte benachbarte $b \times b$ Pixel-Quadrate auf. Diese Blöcke nennen wir Range-Blöcke der Menge $R=\{R_0,R_1,...R_m\}$ +Im nächesten Schritt teilen wir das Bild in alle möglichen $2b \times 2b$ Pixel-Quadrate auf. Diese sind die Domain-Blöcke der Menge $D = \{D_0,D_1,...D_n\}$. +Im dritten und letzten Schritt wird für jeden Range-Block $R_i$ ein Domain-Block $D_j$ gesucht, welcher ihm am ähnlichsten ist. + +\subsubsection{Finden des ähnlichsten $D_j$} +Zuerst braucen wir die Transformation um ein Element aus $D$ auf ein Element von $R$ Abzubilden. +\begin{align*} + T(x,y,z) = + \begin{pmatrix} + a & b & 0 \\ + c & d & 0 \\ + 0 & 0 & s + \end{pmatrix} + \begin{pmatrix} + x \\ + y \\ + z + \end{pmatrix} + + + \begin{pmatrix} + \alpha \\ + \beta \\ + g + \end{pmatrix} +\end{align*} +Diese Transformation bildet den Pixel $P$ auf Koordinate $(x,y)$ und Graustufe $z$ auf den Pixel $P'$ ab. +Da wir mit Pixeln arbeiten, sind die Transformationen in der Ebene Beschränkt. +Diese wird durch die Paramenter $a,b,c$ und $d$ bestimmt. +Mögliche Transfomrationen sind auf folgende Liste Beschränkt: +\begin{itemize} + \item Identische Transformation, keine änderung + \item Drehung um 90, 180 oder 270 Grad. + \item Spiegelung an der vertikalen, horizontalen und den Diagonalachsen. +\end{itemize} +$\alpha$ und $\beta$ verschieben den Pixel an die richtige Stelle. +Da wir ein $2b \times 2b$ Feld auf ein $b \times b$ Feld abbilden möcheen, müssen wir zuerst $G_j$ um $1/2$ skalieren. +Dies erreichen wir, indem wir alle disjunkten $2 \times 2$ px Blöcke mit einem Pixel des Grautones deren Mittelwertes ersetzen. +Skaliert und transformiert erhalten wir $\tilde{D_j}$ +Die Parameter $s$ und $g$ beschreiben die Änderung des Grautones. $s$ verändert den Kontrast und $g$ verschiebt die Töne auf die richtige Helligkeit. +$s$ und $g$ werden mit der linearen Regression ermittelt. +\begin{align*} + z' = sz + g \\ + f(\tilde{D_j}) \text{, Funktion um Grauton von Pixel zu erhalten} \\ + s = \frac{cov(f(R_i), f(\tilde{D_j}))}{var(\tilde{D_j})} \\ + g = E(f(R_i)) - s E(f(\tilde{D_j})) +\end{align*} +Mit diesen Parameteren haben wir nun die Transformation vollständig bestimmt. +Um zu beurteilen ob der Domain-Block $D_j$ mit der gefundenen Transfromation $T$ dem Range-Block $R_i$ genügend ähnlich ist, berechnet man den quadratischen Abstand $e$. +\begin{align*} + e = d(f(R_i), f(T(D_j))) +\end{align*} +Dieser Abstand sollte so klein wie möglich sein. +Die beste Kombination von $D_j$ und $T_i$ ist also diese, welche den kleinsten Abstand zum Block $R_i$ hat, und somit am ähnlichsten ist. +Am Ende des Verfahrens haben wir also für jeden $R_i$ einen passenden $D_i$ mit der zugehörigen Abbildung $T_i$ gefunden. +\subsubsection{Rekonstruktion des Bildes} +Mit den Gefundenen Abbildungen lässt sich das Bild generieren. +Wir beginnen wie schon im letzten Kapitel mit einer beliebigen Startmenge. +In unserem Fall ist dieses ein Bild derselben Grösse. -- cgit v1.2.1 From 668b065f377691fde6727ba10fc979a82c1e5c7b Mon Sep 17 00:00:00 2001 From: Alain Date: Sat, 5 Jun 2021 15:15:57 +0200 Subject: La Reconstruction Text. --- buch/papers/ifs/teil3.tex | 11 +++++++++-- 1 file changed, 9 insertions(+), 2 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/ifs/teil3.tex b/buch/papers/ifs/teil3.tex index d31eee7..bc848bc 100644 --- a/buch/papers/ifs/teil3.tex +++ b/buch/papers/ifs/teil3.tex @@ -68,7 +68,7 @@ Die Parameter $s$ und $g$ beschreiben die Änderung des Grautones. $s$ veränder $s$ und $g$ werden mit der linearen Regression ermittelt. \begin{align*} z' = sz + g \\ - f(\tilde{D_j}) \text{, Funktion um Grauton von Pixel zu erhalten} \\ + f(\tilde{D_j}) \text{, Funktion um das Bild eins Blockes zu erhalten} \\ s = \frac{cov(f(R_i), f(\tilde{D_j}))}{var(\tilde{D_j})} \\ g = E(f(R_i)) - s E(f(\tilde{D_j})) \end{align*} @@ -85,4 +85,11 @@ Am Ende des Verfahrens haben wir also für jeden $R_i$ einen passenden $D_i$ mit \subsubsection{Rekonstruktion des Bildes} Mit den Gefundenen Abbildungen lässt sich das Bild generieren. Wir beginnen wie schon im letzten Kapitel mit einer beliebigen Startmenge. -In unserem Fall ist dieses ein Bild derselben Grösse. +In unserem Fall ist dieses ein Bild $f_0$ derselben Grösse. +Nun ersetzen wir jedes $R_i$ mit der Transformierten des zugehörigen Domain-Blocks $T(G_j)$. +Dies wird verkürzt als Operator $W$ geschrieben. +So erhalten wir ein neues Bild $f_1 = W(f_0)$. +Dieses Vorgehen führen wir iteriert aus bis wir von $f_n = W(f_{n-1})$ zu $f_{n-1}$ kaum mehr einen unterschied fesstellen. Die Iteration hat nun ihren Fixpunkt, das Bild, erreicht. + +TODO Bilder Beispiel +TODO Performance und Kompressonsverhältnis -- cgit v1.2.1 From 74bbee4492a76486091554e24625767440018056 Mon Sep 17 00:00:00 2001 From: Alain Date: Sun, 6 Jun 2021 14:03:33 +0200 Subject: typos --- buch/papers/ifs/teil1.tex | 16 ++++++++-------- buch/papers/ifs/teil2.tex | 10 +++++----- buch/papers/ifs/teil3.tex | 22 +++++++++++----------- 3 files changed, 24 insertions(+), 24 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/ifs/teil1.tex b/buch/papers/ifs/teil1.tex index 327a082..f02aff6 100644 --- a/buch/papers/ifs/teil1.tex +++ b/buch/papers/ifs/teil1.tex @@ -11,8 +11,8 @@ Bevor wir die IFS genauer ansehen, schauen wir uns Fraktale genauer an. \subsection{Was sind Fraktale? \label{ifs:subsection:finibus}} Über die genaue Definition von Fraktalen sind sich die Mathematiker noch nicht einig. -In diesem Kapitel orientieren wir uns an den Eigneschaften welche Kenneth Flaconer in seinem Buch Fractal Geometry beschreibt. -Von einem Fraktal $F$ können wir folgende Eigneschaften erwarten: +In diesem Kapitel orientieren wir uns an den Eigenschaften welche Kenneth Falconer in seinem Buch Fractal Geometry beschreibt. +Von einem Fraktal $F$ können wir folgende Eigenschaften erwarten: \begin{enumerate} \item $F$ hat eine unendlich feine Struktur \item $F$ kann nicht mit der klassischen Geometrie beschrieben werden. @@ -24,10 +24,10 @@ Von einem Fraktal $F$ können wir folgende Eigneschaften erwarten: \label{ifs:subsection:lilkoch}} Diese Eigenschaften möchten wir nun anhand der Koch Kurve näher anschauen. In \ref{ifs:kochkurve8} sehen wir die Koch Kurve. Wie man schon erahnen kann, besteht die aus lauter kleineren Kopien von sich selber. -Den Konstruktionvorgang sehen wir in \ref{ifs:kochconst}. +Den Konstruktionsvorgang sehen wir in \ref{ifs:kochconst}. Gestartet wird mit einer einzelnen Strecke der Länge $a$. Diese wird in ersten Schritt mit vier gleich langen Streckenabschnitte der Länge $\frac{a}{3}$ ersetzt. -In \ref{ifs:kochconstb} ist die Anordnung dieser vier Streckenabschnitte ersichtilich. +In \ref{ifs:kochconstb} ist die Anordnung dieser vier Streckenabschnitte ersichtlich. Dieser Schritt wird nun für jeden der resultierten Streckenabschnitten wiederholt. Die Kurve besteht also aus vier kleineren Kopien von der ganzen Kurve, was auch unter Selbstähnlichkeit bekannt ist. @@ -63,7 +63,7 @@ Die Länge der Kurve lasst sich einfach berechnen. \Rightarrow \quad \lim_{n\to\infty} a \left( \frac{4}{3}\right)^n = \infty \end{align*} -In jedem Schritt wird die Länge um den Faktor $\frac{4}{3}$ verglängert. Somit divergiert die Länge gegen Unendlich. +In jedem Schritt wird die Länge um den Faktor $\frac{4}{3}$ verlängert. Somit divergiert die Länge gegen Unendlich. Die Fläche unter der Kurve lässt sich folgendermassen berechnen \begin{align*} A_0 = 0 , \quad A_1 = \left( \frac{a}{3}\right)^2 \frac{\sqrt{3}}{4} = a^2 \frac{\sqrt{3}}{36}\\ @@ -71,14 +71,14 @@ Die Fläche unter der Kurve lässt sich folgendermassen berechnen A_3 = A_1 + A_2 + 4^2 \left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 + \left( \frac{4}{9}\right)^2 A_1 \end{align*} Wir sehen, dass mit jedem Schritt die neu dazugekommene Fläche um $\frac{4}{9}$ kleiner ist. -Daraus resultiert eine konvergierende Geometrische Rheie. +Daraus resultiert eine konvergierende Geometrische Reihe. \begin{align*} A_n = A_1 \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n = a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n \\ \lim_{n\to\infty} a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n = \frac{\sqrt{3}}{20} a^2 \end{align*} Wie wir sehen ist die Kochkurve ein Konstrukt mit endlicher Fläche, aber unendlichem Umfang. -Zu guter letzt bestimmen wir die Dimension der Kurve. -Es gibt viele verschidene Arten die Dimension zu definieren. Diese können dann auch unterschiedliche Resultate liefern. +Zu guter Letzt bestimmen wir die Dimension der Kurve. +Es gibt viele verschiedene Arten die Dimension zu definieren. Diese können dann auch unterschiedliche Resultate liefern. Vor allem im Zusammenhang mit Fraktalen findet man in der Literatur viele verschiedene Arten. In diesem Beispiel werden wir die Ähnlichkeits-Dimension. \begin{align*} diff --git a/buch/papers/ifs/teil2.tex b/buch/papers/ifs/teil2.tex index a3d5ee1..a728340 100644 --- a/buch/papers/ifs/teil2.tex +++ b/buch/papers/ifs/teil2.tex @@ -7,7 +7,7 @@ \label{ifs:section:teil2}} \rhead{Teil 2} Wollen wir nun eine bestimmte Art anschauen, wie man Fraktale machen kann. -Zur veranschaulichung dieser Methode nehmen wir das Sierpinski Dreieck. +Zur Veranschaulichung dieser Methode nehmen wir das Sierpinski Dreieck. \begin{figure} \label{ifs:sierpinski10} \centering @@ -19,7 +19,7 @@ Es ist also ein Selbstähnliches Konstrukt. Diese Eigenschaft wollen wir uns zunutze machen. -Wir definieren das Dreieck mit kantenlänge 1 als Menge $X$. +Wir definieren das Dreieck mit Kantenlänge 1 als Menge $X$. Ausserdem bestimmen wir drei Funktionen, welche die gesamte Menge auf eine ihrer kleineren Kopien abbildet \begin{align*} f_1(x,y) @@ -70,7 +70,7 @@ Wendet man alle drei Funktionen auf das Sierpinski-Dreieck an, entsteht also wie \begin{align*} X = \bigcup\limits_{i = 1}^{3} f_i(X) \end{align*} -Man kann sogar noch einen Schritt weiter gehen, und sagen: Wenn wir die Funktionen auf eine beliebige Startmenge anwenden, konvergeiert die Menge gegen das Sierpinski-Dreieck. +Man kann sogar noch einen Schritt weiter gehen, und sagen: Wenn wir die Funktionen auf eine beliebige Startmenge anwenden, konvergiert die Menge gegen das Sierpinski-Dreieck. \begin{figure} \label{ifs:sierpconst} \centering @@ -94,10 +94,10 @@ Der Abstand zum Original wird immer kleiner, und konvergiert bei unendlich Itera \subsection{Iterierte Funktionensysteme \label{ifs:subsection:bonorum}} -In diesem Unterkapitel wollen wir die Erkenntniss, wie wir aus einer beliebigen Menge ein Sierpinski-Dreieck genereieren können, verallgemeinern. +In diesem Unterkapitel wollen wir die Erkenntnis, wie wir aus einer beliebigen Menge ein Sierpinski-Dreieck generieren können, verallgemeinern. TODO TEXT -$S_1_...,S_n$ sind Kontraktionen auf die Menge $D \subset \mathbb{R}^n$. Es gilt +$S_1,...,S_n$ sind Kontraktionen auf die Menge $D \subset \mathbb{R}^n$. Es gilt \begin{align} |S_i(x) - S_i(y)| \leq c_i|x - y| \end{align} diff --git a/buch/papers/ifs/teil3.tex b/buch/papers/ifs/teil3.tex index bc848bc..c3e8a65 100644 --- a/buch/papers/ifs/teil3.tex +++ b/buch/papers/ifs/teil3.tex @@ -7,12 +7,12 @@ \label{ifs:section:teil3}} \rhead{Fraktale Bildkomprimierung} Mit dem Prinzip dieser IFS ist es auch möglich Bilder zu Komprimieren. -Diese Idee hatte der Mathematiker Michael Barnsley, welcher mit seinem Buch Fractals Everywhere einen wichtigen beitrag zum verständnis von Fraktalen geiefert hat. +Diese Idee hatte der Mathematiker Michael Barnsley, welcher mit seinem Buch Fractals Everywhere einen wichtigen Beitrag zum Verständnis von Fraktalen geliefert hat. Das Ziel ist es ein IFS zu finden, welches das Bild als Attraktor hat. In diesem Unterkapitel wollen wir eine Methode dafür anschauen. -Bis jetzt wurde in Zusammenhnag mit IFS immer erwähnt, dass die Transformationen auf die ganze Menge angewendet werden. +Bis jetzt wurde in Zusammenhang mit IFS immer erwähnt, dass die Transformationen auf die ganze Menge angewendet werden. Dies muss jedoch nicht so sein. Es gibt auch einen Attraktor, wenn die Transformationen nur Teile der Menge auf die ganze Menge abbilden. Diese Eigenschaft wollen wir uns in der Fraktalen Bildkompression zunutze machen. @@ -25,11 +25,11 @@ Doch wie Finden wir die richtigen Affinen Transformationen, welche als IFS das B In der Beschreibung des Verfahrens wird sich auf Graustufenbilder bezogen. Wie das Verfahren für Farbbilder verwendet werden kann, wird später erläutert. In einem ersten Schritt teilen wir das Bild in disjunkte benachbarte $b \times b$ Pixel-Quadrate auf. Diese Blöcke nennen wir Range-Blöcke der Menge $R=\{R_0,R_1,...R_m\}$ -Im nächesten Schritt teilen wir das Bild in alle möglichen $2b \times 2b$ Pixel-Quadrate auf. Diese sind die Domain-Blöcke der Menge $D = \{D_0,D_1,...D_n\}$. +Im nächsten Schritt teilen wir das Bild in alle möglichen $2b \times 2b$ Pixel-Quadrate auf. Diese sind die Domain-Blöcke der Menge $D = \{D_0,D_1,...D_n\}$. Im dritten und letzten Schritt wird für jeden Range-Block $R_i$ ein Domain-Block $D_j$ gesucht, welcher ihm am ähnlichsten ist. \subsubsection{Finden des ähnlichsten $D_j$} -Zuerst braucen wir die Transformation um ein Element aus $D$ auf ein Element von $R$ Abzubilden. +Zuerst brauchen wir die Transformation um ein Element aus $D$ auf ein Element von $R$ Abzubilden. \begin{align*} T(x,y,z) = \begin{pmatrix} @@ -52,15 +52,15 @@ Zuerst braucen wir die Transformation um ein Element aus $D$ auf ein Element von Diese Transformation bildet den Pixel $P$ auf Koordinate $(x,y)$ und Graustufe $z$ auf den Pixel $P'$ ab. Da wir mit Pixeln arbeiten, sind die Transformationen in der Ebene Beschränkt. -Diese wird durch die Paramenter $a,b,c$ und $d$ bestimmt. -Mögliche Transfomrationen sind auf folgende Liste Beschränkt: +Diese wird durch die Parameter $a,b,c$ und $d$ bestimmt. +Mögliche Transformationen sind auf folgende Liste Beschränkt: \begin{itemize} - \item Identische Transformation, keine änderung + \item Identische Transformation, keine Änderung \item Drehung um 90, 180 oder 270 Grad. \item Spiegelung an der vertikalen, horizontalen und den Diagonalachsen. \end{itemize} $\alpha$ und $\beta$ verschieben den Pixel an die richtige Stelle. -Da wir ein $2b \times 2b$ Feld auf ein $b \times b$ Feld abbilden möcheen, müssen wir zuerst $G_j$ um $1/2$ skalieren. +Da wir ein $2b \times 2b$ Feld auf ein $b \times b$ Feld abbilden möchten, müssen wir zuerst $G_j$ um $1/2$ skalieren. Dies erreichen wir, indem wir alle disjunkten $2 \times 2$ px Blöcke mit einem Pixel des Grautones deren Mittelwertes ersetzen. Skaliert und transformiert erhalten wir $\tilde{D_j}$ @@ -72,8 +72,8 @@ $s$ und $g$ werden mit der linearen Regression ermittelt. s = \frac{cov(f(R_i), f(\tilde{D_j}))}{var(\tilde{D_j})} \\ g = E(f(R_i)) - s E(f(\tilde{D_j})) \end{align*} -Mit diesen Parameteren haben wir nun die Transformation vollständig bestimmt. -Um zu beurteilen ob der Domain-Block $D_j$ mit der gefundenen Transfromation $T$ dem Range-Block $R_i$ genügend ähnlich ist, berechnet man den quadratischen Abstand $e$. +Mit diesen Parametern haben wir nun die Transformation vollständig bestimmt. +Um zu beurteilen ob der Domain-Block $D_j$ mit der gefundenen Transformation $T$ dem Range-Block $R_i$ genügend ähnlich ist, berechnet man den quadratischen Abstand $e$. \begin{align*} e = d(f(R_i), f(T(D_j))) \end{align*} @@ -89,7 +89,7 @@ In unserem Fall ist dieses ein Bild $f_0$ derselben Grösse. Nun ersetzen wir jedes $R_i$ mit der Transformierten des zugehörigen Domain-Blocks $T(G_j)$. Dies wird verkürzt als Operator $W$ geschrieben. So erhalten wir ein neues Bild $f_1 = W(f_0)$. -Dieses Vorgehen führen wir iteriert aus bis wir von $f_n = W(f_{n-1})$ zu $f_{n-1}$ kaum mehr einen unterschied fesstellen. Die Iteration hat nun ihren Fixpunkt, das Bild, erreicht. +Dieses Vorgehen führen wir iteriert aus bis wir von $f_n = W(f_{n-1})$ zu $f_{n-1}$ kaum mehr einen unterschied feststellen. Die Iteration hat nun ihren Fixpunkt, das Bild, erreicht. TODO Bilder Beispiel TODO Performance und Kompressonsverhältnis -- cgit v1.2.1 From 021d83730d896b7cef1050fbdd4c4c766992a9b0 Mon Sep 17 00:00:00 2001 From: Alain Date: Sun, 6 Jun 2021 17:36:05 +0200 Subject: ifs work --- buch/papers/ifs/images/farn.eps | 2372 ++++++++++++++++++++++++++++++ buch/papers/ifs/images/farncolor.eps | 2666 ++++++++++++++++++++++++++++++++++ buch/papers/ifs/teil2.tex | 26 +- 3 files changed, 5063 insertions(+), 1 deletion(-) create mode 100644 buch/papers/ifs/images/farn.eps create mode 100644 buch/papers/ifs/images/farncolor.eps (limited to 'buch/papers') diff --git a/buch/papers/ifs/images/farn.eps b/buch/papers/ifs/images/farn.eps new file mode 100644 index 0000000..597745b --- /dev/null +++ b/buch/papers/ifs/images/farn.eps @@ -0,0 +1,2372 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: (MATLAB, The Mathworks, Inc. Version 9.7.0.1434023 \(R2019b\) Update 6. 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+DIF$m8W>c?0Oh)?;m[dC'hEBbpTD!!jG?/A&C$2pABB9V#X*k0prZ;Kei_0~> + +%AXGEndBitmap +GR +GR +GS +[0.48 0 0 0.48 921.60007 0.39996] CT +[1 0 0 1 0 0] CT +N +-1920 0 M +480 0 L +480 1170 L +-1920 1170 L +-1920 0 L +cp +clip +GS +0 0 translate +480 234 scale +%AXGBeginBitmap: java.awt.image.BufferedImage +{{ +/RawData currentfile /ASCII85Decode filter def +/Data RawData /FlateDecode filter def +/DeviceRGB setcolorspace +<< + /ImageType 1 + /Decode [0 1 0 1 0 1] + /DataSource Data + /Height 234 + /ImageMatrix [480 0 0 234 0 0] + /Width 480 + /BitsPerComponent 8 +>> image +} stopped {handleerror} if + RawData flushfile +} exec +Gb"0;0`_7S!5bE.WTRt%TE"rlzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz +zzzzzzzzzzzzzzzzzzzzzzzzz!!!#W#$rcmOo~> + +%AXGEndBitmap +GR +GR +%%Trailer +%%Pages: 1 +%%EOF diff --git a/buch/papers/ifs/teil2.tex b/buch/papers/ifs/teil2.tex index a728340..8a7f76f 100644 --- a/buch/papers/ifs/teil2.tex +++ b/buch/papers/ifs/teil2.tex @@ -105,4 +105,28 @@ für jedes i mit einem $c_i < 1$. Dann existiert eine eindeutige kompakte Menge \begin{equation} F = \bigcup\limits_{i = 1}^{m} S_i(F) \end{equation} -TODO Text +Weiter definieren wir die Transformation S auf kompakte Mengen ohne die leere Menge. +\begin{equation} + S(E) = \bigcup\limits_{i = 1}^m S_i(E) +\end{equation} +Wird diese Transformation Iterativ ausgeführt, das heisst $S^0(E) = E, S^k(E) = S(S^{k-1}(E))$, und für jedes $i$ $S_i(E) \subset E$, gilt +\begin{equation} + F = \bigcap\limits_{k = 1}^{\infty} S^k(E). +\end{equation} +In Worte gefasst bedeutet das, dass jede Gruppe von Kontraktionen iterativ ausgeführt, gegen eine eindeutige Menge konvergiert. +Dies für jede Startmenge, solange diese ihre Transformierten wieder beinhaltet. +Auf den Beweis wird verzichtet. +\subsection{Beispiel: Barnsley-Farn} +\begin{figure} + \label{ifs:farn} + \centering + \makebox[\textwidth][c]{ + \includegraphics[width=1.4\textwidth]{papers/ifs/images/farn}} + \caption{Barnsley-Farn} +\end{figure} +\begin{figure} + \label{ifs:farncolor} + \centering + \includegraphics[width=0.7\textwidth]{papers/ifs/images/farncolor} + \caption{Vier Transformationen des Barnsley-Farn} +\end{figure} -- cgit v1.2.1 From f0006b3ae7eb70a1fc33b26f482308a43445969e Mon Sep 17 00:00:00 2001 From: Alain Date: Mon, 7 Jun 2021 17:26:10 +0200 Subject: Farn und Compression --- buch/papers/ifs/teil2.tex | 61 +++++++++++++++++++++++++++++++++++++++++++++++ buch/papers/ifs/teil3.tex | 12 ++++++++++ 2 files changed, 73 insertions(+) (limited to 'buch/papers') diff --git a/buch/papers/ifs/teil2.tex b/buch/papers/ifs/teil2.tex index 8a7f76f..5e36f97 100644 --- a/buch/papers/ifs/teil2.tex +++ b/buch/papers/ifs/teil2.tex @@ -117,6 +117,67 @@ In Worte gefasst bedeutet das, dass jede Gruppe von Kontraktionen iterativ ausge Dies für jede Startmenge, solange diese ihre Transformierten wieder beinhaltet. Auf den Beweis wird verzichtet. \subsection{Beispiel: Barnsley-Farn} +Der Barnsley-Farn, Abbildung \ref{ifs:farn}, ist ein weiteres Fraktal, welches mit einem IFS generiert werden kann. +Wie man schnell erkennen kann, besteht der Farn aus Blättern, welche eine grosse Ähnlichkeit zum ganzen Farn haben. +\begin{align*} + {S_1(x,y)} + = + \begin{pmatrix} + 0 & 0 \\ + 0 & 0.16 \\ + \end{pmatrix} + \begin{pmatrix} + x\\ + y\\ + \end{pmatrix}, \quad + {S_2(x,y)} + = + \begin{pmatrix} + 0.85 & 0.04 \\ + -0.04 & 0.85 \\ + \end{pmatrix} + \begin{pmatrix} + x\\ + y\\ + \end{pmatrix} + + + \begin{pmatrix} + 0 \\ + 1.6 + \end{pmatrix}\\ + {S_3(x,y)} + = + \begin{pmatrix} + 0.2 & -0.26 \\ + 0.23 & 0.22 \\ + \end{pmatrix} + \begin{pmatrix} + x\\ + y\\ + \end{pmatrix} + + + \begin{pmatrix} + 0 \\ + 1.6 + \end{pmatrix}, \quad + {S_4(x,y)} + = + \begin{pmatrix} + -0.15 & 0.28 \\ + 0.26 & 0.24 \\ + \end{pmatrix} + \begin{pmatrix} + x\\ + y\\ + \end{pmatrix} + + + \begin{pmatrix} + 0 \\ + 0.44 + \end{pmatrix}\\ +\end{align*} +In der Abbildung \ref{ifs:farncolor} sehen wir die vier Transformationen farblich dargestellt. +$S_1$ \begin{figure} \label{ifs:farn} \centering diff --git a/buch/papers/ifs/teil3.tex b/buch/papers/ifs/teil3.tex index c3e8a65..fa4130b 100644 --- a/buch/papers/ifs/teil3.tex +++ b/buch/papers/ifs/teil3.tex @@ -91,5 +91,17 @@ Dies wird verkürzt als Operator $W$ geschrieben. So erhalten wir ein neues Bild $f_1 = W(f_0)$. Dieses Vorgehen führen wir iteriert aus bis wir von $f_n = W(f_{n-1})$ zu $f_{n-1}$ kaum mehr einen unterschied feststellen. Die Iteration hat nun ihren Fixpunkt, das Bild, erreicht. +\subsubsection{Farbbilder} +Dieses Verfahren mit Graustufenbilder lässt sich ganz einfach auf Farbbilder erweitern. +Jeder Pixel eines Farbbildes besteht aus einem Rot, Grün und Blauwert (RGB). +Teilt man ein Bild in die drei Farbkanäle auf, das heisst, es wird nur noch ein Farbwert benutzt, erhält man drei Bilder, welche wie ein Graustufenbild sind. +Nun wendet man auf jeden dieser Farbkanalbilder den Algorithmus an, und fügt nach der Rekonstruktion die Kanäle wieder zusammen. + +\subsubsection{Performance des Verfahren} +Dieser Grundalgorithmus der Fraktalen Bildkompression ist offensichtlich recht langsam und skaliert auch schlecht mit grösseren Bilder. +Man kann die Laufzeit zwar verbessern indem man die Domain-Blöcke auch disjunkt macht, und für weniger detailreiche Bilder ein grösseres $b$ wählt, jedoch wird er auch so nie so schnell wie zum Beispiel das jpeg verfahren. + +\subsection{Beispiel} +Kommen wir nun zu einem Beispiel TODO Bilder Beispiel TODO Performance und Kompressonsverhältnis -- cgit v1.2.1 From 6b86c10028987f4e08ca3e25ac13291f256375fa Mon Sep 17 00:00:00 2001 From: Alain Date: Tue, 8 Jun 2021 14:53:07 +0200 Subject: Barnsley Farn & Kompression bsp --- buch/papers/ifs/images/faroe.png | Bin 0 -> 987 bytes buch/papers/ifs/images/faroe0.PNG | Bin 0 -> 80239 bytes buch/papers/ifs/images/faroe1.PNG | Bin 0 -> 104146 bytes buch/papers/ifs/images/faroe5.PNG | Bin 0 -> 73790 bytes buch/papers/ifs/images/original.png | Bin 0 -> 138885 bytes buch/papers/ifs/images/rapperswil.png | Bin 0 -> 851 bytes buch/papers/ifs/images/rapperswil0.PNG | Bin 0 -> 66375 bytes buch/papers/ifs/images/rapperswil001.PNG | Bin 0 -> 93116 bytes buch/papers/ifs/images/rapperswil01.PNG | Bin 0 -> 81696 bytes buch/papers/ifs/images/rapperswil04.PNG | Bin 0 -> 60921 bytes buch/papers/ifs/images/rapperswil1.PNG | Bin 0 -> 82594 bytes buch/papers/ifs/images/rapperswil4.PNG | Bin 0 -> 60837 bytes 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/dev/null and b/buch/papers/ifs/images/rapperswil04.PNG differ diff --git a/buch/papers/ifs/images/rapperswil1.PNG b/buch/papers/ifs/images/rapperswil1.PNG new file mode 100644 index 0000000..6c085db Binary files /dev/null and b/buch/papers/ifs/images/rapperswil1.PNG differ diff --git a/buch/papers/ifs/images/rapperswil4.PNG b/buch/papers/ifs/images/rapperswil4.PNG new file mode 100644 index 0000000..56d1331 Binary files /dev/null and b/buch/papers/ifs/images/rapperswil4.PNG differ diff --git a/buch/papers/ifs/images/zurich.png b/buch/papers/ifs/images/zurich.png new file mode 100644 index 0000000..bb70f7d Binary files /dev/null and b/buch/papers/ifs/images/zurich.png differ diff --git a/buch/papers/ifs/teil2.tex b/buch/papers/ifs/teil2.tex index 5e36f97..d25004f 100644 --- a/buch/papers/ifs/teil2.tex +++ b/buch/papers/ifs/teil2.tex @@ -177,7 +177,24 @@ Wie man schnell erkennen kann, besteht der Farn aus Blättern, welche eine gross \end{pmatrix}\\ \end{align*} In der Abbildung \ref{ifs:farncolor} sehen wir die vier Transformationen farblich dargestellt. -$S_1$ + +$S_1$ erstellt den Stiel des Farnblattes (rot). +Die Transformation bildet das Gesamte Blatt auf die Y-Achse ab. +$S_2$ (grün) erstellt den Hauptteil des Farnes. +Sie verkleinert und dreht das gesamte Bild und stellt es auf das Ende des Stiels aus $S_1$. +$S_3$ bildet das gesamte Blatt auf das blaue Teilblatt unten Links ab. +$S_4$ Spiegelt das Blatt und bildet es auf das magentafarbene Teilblatt ab. + +Wir führen im Zusammenhang mit dem Barnsley-Farn noch eine weitere Methode ein, um IFS auszuführen. +Bis jetzt wurde immer davon gesprochen, die Transformationen auf die gesamte Menge anzuwenden. +Bei komplizierteren IFS welche viele Iterationen brauchen, bis man den Attraktor erkennen kann, ist diese Methode ziemlich rechenintensiv. +Eine Alternative ist das Chaos-Game. +Bei dieser Methode werden die Transformationen nicht auf die Menge angewendet, sondern nur auf einen einzelnen Punkt. +Der Startpunkt kann dabei ein beliebiger Punkt in $E$ sein. +Es wird bei jedem Iterationsschritt nur eine Transformation, welche zufällig gewählt wurde, angewendet. +Da, wie wir beim Barnsley-Farn gut sehen, dass nicht jede Transformation gleich viel des Bildes ausmacht, werden diese beim Chaos-Game gewichtet. +Die Gewichtung erfolgt über den Anteil der Gesamtmasse. +Im Fall des Barnsley-Fern wird $S_1$ in $1\%$, $S_2$ in $85\%$ und $S_3 \& S_4$ in $7\%$ der Iterationen ausgeführt. \begin{figure} \label{ifs:farn} \centering diff --git a/buch/papers/ifs/teil3.tex b/buch/papers/ifs/teil3.tex index fa4130b..515fd81 100644 --- a/buch/papers/ifs/teil3.tex +++ b/buch/papers/ifs/teil3.tex @@ -102,6 +102,42 @@ Dieser Grundalgorithmus der Fraktalen Bildkompression ist offensichtlich recht l Man kann die Laufzeit zwar verbessern indem man die Domain-Blöcke auch disjunkt macht, und für weniger detailreiche Bilder ein grösseres $b$ wählt, jedoch wird er auch so nie so schnell wie zum Beispiel das jpeg verfahren. \subsection{Beispiel} -Kommen wir nun zu einem Beispiel -TODO Bilder Beispiel -TODO Performance und Kompressonsverhältnis +Kommen wir nun zu einem Beispiel. +Wir Verwenden dafür den oben beschriebenen Algorithmus. +Die Range-Blöcke wurden $4\times4$ gewählt und die Dommain dementsprechend $8\times8$. +Um etwas Zeit bei der Komprimierung zu ersparen, wurden nur disjunkte Domain-Blöcke gebraucht. +Als erstes Beispiel wählen wir das 360x360px Bild von Rapperswil in Abbildung \ref{ifs:original}. +Der Algorithmus liefert uns für jeden Range-Block die benötigten Parameter. +Mit diesen lässt sich das Bild im Anschluss wieder Rekonstruieren. + +Als Startbild wird ein mittelgraues 360x360px Bild gewählt, Abbildung \ref{ifs:bild0}. +Nun lassen wir das IFS laufen. +Wie wir in Abbildung \ref{ifs:rappirecoa} sehen, ist schon nach der ersten Iteration das Bild schon erkennbar. +Nach der fünften Iteration , Abbildung \ref{ifs:rappirecoc} gibt es fast keinen Unterschied mehr zur letzten Iteration, wir können die Rekonstruktion beenden. +\begin{figure} + \label{ifs:original} + \centering + \includegraphics[width=0.4\textwidth]{papers/ifs/images/original} + \caption{Original Bild von Rapperswil} +\end{figure} +\begin{figure} + \label{ifs:bild0} + \centering + \includegraphics[width=0.4\textwidth]{papers/ifs/images/rapperswil} + \caption{Startbild} +\end{figure} + +\begin{figure} + \label{ifs:rappireco} + \centering + \subfigure[]{ + \label{ifs:rappirecoa} + \includegraphics[width=0.32\textwidth]{papers/ifs/images/rapperswil01}} + \subfigure[]{ + \label{ifs:rappirecob} + \includegraphics[width=0.32\textwidth]{papers/ifs/images/rapperswil001}} + \subfigure[]{ + \label{ifs:rappirecoc} + \includegraphics[width=0.32\textwidth]{papers/ifs/images/rapperswil04}} + \caption{(a) 1. Iteration (b) 2. Iteration (c) 5. Iteration} +\end{figure} -- cgit v1.2.1 From 72c6e0954eb2acd262a7db6701ed1d04bb8943c5 Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Tue, 8 Jun 2021 15:34:22 +0200 Subject: created Hilfstabellen.tex, reworked codebsp.tex --- buch/papers/reedsolomon/codebsp.tex | 94 ++++++++++++++++++++++--------- buch/papers/reedsolomon/hilfstabellen.tex | 21 +++++++ 2 files changed, 87 insertions(+), 28 deletions(-) create mode 100644 buch/papers/reedsolomon/hilfstabellen.tex (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/codebsp.tex b/buch/papers/reedsolomon/codebsp.tex index 5b67c43..818078e 100644 --- a/buch/papers/reedsolomon/codebsp.tex +++ b/buch/papers/reedsolomon/codebsp.tex @@ -8,19 +8,35 @@ \rhead{Koerper Festlegen} Um die Funktionsweise eines Reed-Solomon-Codes besser zu verstehen werden wir die einzelnen Probleme und ihre Lösungen anhand eines Beispiels betrachten. -Da wir in Endlichen Körpern Rechnen werden wir zuerst solch ein Körper festlegen. Dabei müssen wir die \textcolor{red}{Definition 4.6} berücksichtigen, die besagt, dass nur Primzahlen für endliche Körper in Frage kommen. -Wir legen für unser Beispiel den endlichen Körper $q = 11$ fest. -Alle folgenden Berechnungen wurden mit den beiden Restetabellen \textcolor{red}{xx} und \textcolor{red}{yy} durchgeführt. -Aus den Tabellen folgt auch, dass uns nur die Zahlen \[\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}\] zur Verfügung stehen. +Da wir in Endlichen Körpern Rechnen werden wir zuerst solch einen Körper festlegen. Dabei müssen wir die \textcolor{red}{Definition 4.6 (wie verweist man auf eine definition?)} berücksichtigen, die besagt, dass nur Primzahlen für endliche Körper in Frage kommen. +Wir legen für unser Beispiel den endlichen Körper mit $q = 11$ fest. +Zur Hilfestellung können dazu die beiden Tabellen \ref{reedsolomon:subsection:adtab} und +\ref{reedsolomon:subsection:mptab} hinzugezogen werden. Diese Tabellen enthalten sämtliche Resultate aller gültigen Operationen \textcolor{red}{(Notiz: nach meinem Wissen gibt es ja nur addition und multiplikation als gültige operationen)}, die in diesem Körper durchgeführt werden können. +Aus der Definition der Endlichen Körper (ersichtlich auch in den Tabellen) folgt, dass uns nur die Zahlen \[\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}\] zur verfügung stehen und somit $11 = 0$ gelten muss. + +% OLD TEXT +%Alle folgenden Berechnungen wurden mit den beiden Restetabellen \ref{reedsolomon:subsection:adtab} und \ref{reedsolomon:subsection:mptab} durchgeführt. +%Aus den Tabellen folgt auch, dass uns nur die Zahlen \[\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}\] zur Verfügung stehen. % die beiden Restetabellen von F_11 %\input{papers/reedsolomon/restetabelle1} %\input{papers/reedsolomon/restetabelle2} -Die grösse des endlichen Körpers legt auch fest, wie gross unsere Nachricht $n$ bestehend aus Nutzdatenteil und Fehlerkorrekturteil sein kann und beträgt in unserem Beispiel +Anhand der Menge uns zur Verfügung stehenden Zahlen wird auch festgelegt, wie viele Zahlen ein Nachrichtenblock $n$, bestehend aus Nutzdatenteil und Fehlerkorrekturteil, umfassen kann. +Der Nachrichtenblock im Beispiel besteht aus \[ -n = q - 1 = 10 \text{ Zahlen}. +n = q - 1 = 10 \text{ Zahlen}, \] +wobei die null weggelassen wird. Wenn wir versuchen würden, mit der null zu codieren, so stellen wir fest, dass wir wieder null an der gleichen Stelle erhalten und somit wäre die Codierung nicht eindeutig. + +% Notes +%Da bei allen Codes, die codiert werden wird an der gleichen Stelle eine Nullstelle auftreten. + +% Old Text +%Die grösse des endlichen Körpers legt auch fest, wie gross unsere Nachricht $n$ bestehend aus Nutzdatenteil und Fehlerkorrekturteil sein kann und beträgt in unserem Beispiel +%\[ +%n = q - 1 = 10 \text{ Zahlen}. +%\] Im nächsten Schritt bestimmen wir, wie viele Fehler $t$ maximal während der Übertragung auftreten dürfen, damit wir sie noch korrigieren können. Unser Beispielcode sollte in der Lage sein @@ -29,41 +45,63 @@ t = 2 \] Fehlerstellen korrigieren zu können. -Die Grösse des Nutzdatenteils hängt von der Grösse der Nachricht sowie der Anzahl der Fehlerkorrekturstellen. Je robuster der Code sein muss, desto weniger Platz für Nutzdaten $k$ bleibt in der Nachricht übrig. +Die Grösse des Nutzdatenteils hängt von der Grösse des Nachrichtenblocks sowie der Anzahl der Fehlerkorrekturstellen ab. Je robuster der Code sein muss, desto weniger Platz für Nutzdaten $k$ bleibt in der Nachricht übrig. Bei maximal 2 Fehler können wir noch \[ k = n - 2t = 6\text{ Zahlen} \] übertragen. -Zusammenfassend haben wir einen Codeblock mit der Länge von 10 Zahlen definiert, der 6 Zahlen als Nutzlast beinhaltet und in der Lage ist aus 2 fehlerhafte Stellen im Block die ursprünglichen Nutzdaten rekonstruieren kann. Zudem werden wir im weiteren feststellen, dass dieser Code maximal 4 Fehlerstellen erkennen, diese aber nicht rekonstruieren kann. +Zusammenfassend haben wir einen Nachrichtenblock mit der Länge von 10 Zahlen definiert, der 6 Zahlen als Nutzlast beinhaltet und in der Lage ist aus 2 fehlerhafte Stellen im Block die ursprünglichen Nutzdaten zu rekonstruieren. Zudem werden wir im weiteren feststellen, dass dieser Code maximal vier Fehlerstellen erkennen, diese aber nicht rekonstruieren kann. Wir legen nun die Nachricht \[ m = [0,0,0,0,4,7,2,5,8,1] \] -fest, die wir gerne an einen Empfänger übertragen möchten, wobei die vorderen vier Nullstellen für die Fehlerkorrektur zuständig sind. -Die Nachricht können wir auch als Polynom +fest, die wir gerne an einen Empfänger übertragen möchten, wobei die vorderen vier Stellen für die Fehlerkorrektur zuständig sind. +Solange diese Stellen vor dem Codieren und nach dem Decodieren den Wert null haben, so ist die Nachricht Fehlerfrei übertragen worden. + +Da wir in den folgenden Abschnitten mit Polynomen arbeiten, stellen wir die Nachicht auch noch als Polynom \[ m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1 \] -darstellen. +dar. + +% Old Text +%Die Nachricht können wir auch als Polynom +%\[ +%m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1 +%\] +%darstellen. \subsection{Der Ansatz der diskreten Fouriertransformation \label{reedsolomon:subsection:diskFT}} -In einem vorherigen Kapitel (???) haben wir schon einmal die diskrete Fouriertransformation zum Codieren einer Nachricht verwendet. In den endlichen Körpern wird dies jedoch nicht gelingen, da die Eulerische Zahl $\mathrm{e}$ in $\mathbb{F}_{11}$ nicht existiert. -Wir suchen also eine Zahl $a^i$, die in endlichen Körpern existiert und den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken kann. -Dazu schreiben wir +In einem vorherigen Kapitel \textcolor{red}{(???)} haben wir schon einmal die diskrete Fouriertransformation zum Codieren einer Nachricht verwendet. In den endlichen Körpern wird dies jedoch nicht gelingen, da die Eulerische Zahl $e$ in endlichen Körpern nicht existiert. +Wir legen deshalb die Zahl $a$ fest. Diese Zahl soll die gleichen aufgaben haben, wie $e^{\frac{j}{2 \pi}}$ in der Diskreten Fouriertransformation, nur mit dem Unterschied, dass $a$ in $\mathbb{F}_{11}$ existiert. Dazu soll $a$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken, um \[ \mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\} \] -um in +in \[ \mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}. \] - -Wenn wir alle möglichen Werte für $a$ einsetzen, also +umzuschreiben. + +Wenn wir jetzt sämtliche Zahlen von $\mathbb{F}_{11}$ in $a$ einsetzen + +% Old Text +%Wir suchen also eine Zahl $a$, die in endlichen Körpern existiert und den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken kann. +%Dazu schreiben wir +%\[ +%\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\} +%\] +%um in +%\[ +%\mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}. +%\] +% +%Wenn wir alle möglichen Werte für $a$ einsetzen, also %\begin{align} %a = 0 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{0, 0, 0, 0, 0, 0, 0, 0, 0, 0\} \\ @@ -94,21 +132,26 @@ $a = 9 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 9, 4, 3, 5, 1, 9, $a = 10 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 10, 1, 10, 1, 10, 1, 10, 1, 10\}$ \end{tabular} \end{center} +so fällt uns auf, dass für $a$ die Zahlen $2,6,7,8$ erhalten, die tatsächlich den gesamten Zahlenraum von $\mathbb{F}_{11}$ abbilden. Solche Zahlen werden \em Primitive Einheitswurzel \em genannt. +Wenden wir diese Vorgehensweise auch für andere Endliche Körper an, so werden wir sehen, dass wir immer mindestens zwei solcher Einheitswurzel finden werden. Somit ist es uns überlassen, eine dieser Einheitswurzeln auszuwählen, mit der wir weiter rechnen wollen. -so fällt uns auf, dass die Zahlen $2,6,7,8$ tatsächlich den gesamten Zahlenraum von $\mathbb{F}_{11}$ abbilden. Solche Zahlen werden \em Primitive Einheitswurzel \em genannt. Für das Beispiel wählen wir die Zahl $a^i = 8$. Damit wir unsere Nachricht codieren können, müssen wir $8^i$ in $m(X)$ einsetzen. \begin{center} \begin{tabular}{c} - $m(8^0) = 4 \cdot 1 + 7 \cdot 1 + 2 \cdot 1 + 5 \cdot 1 + 8 \cdot 1 + 1 = 5$ \\ - $m(8^1) = 4 \cdot 8 + 7 \cdot 8 + 2 \cdot 8 + 5 \cdot 8 + 8 \cdot 8 + 1 = 3$ \\ - \vdots + $m(8^0) = 4 \cdot 1^5 + 7 \cdot 1^4 + 2 \cdot 1^3 + 5 \cdot 1^2 + 8 \cdot 1^1 + 1 = 5$ \\ + $m(8^1) = 4 \cdot 8^5 + 7 \cdot 8^4 + 2 \cdot 8^3 + 5 \cdot 8^2 + 8 \cdot 8^1 + 1 = 3$ \\ + \vdots \\ + $m(8^9) = 4 \cdot 7^5 + 7 \cdot 7^4 + 2 \cdot 7^3 + 5 \cdot 7^2 + 8 \cdot 7^1 + 1 = 4$ \end{tabular} \end{center} -Für eine elegantere Formulierung stellen wir das ganze als Matrix dar, wobei $m$ unser Nachrichtenvektor, $A$ die Transformationsmatrix und $v$ unser Übertragungsvektor ist. - + +\subsection{Allgemeine Codierung + \label{reedsolomon:subsection:algCod}} + +Für eine elegantere Formulierung stellen wir das ganze als Matrix dar, wobei $m$ unsere Nachricht, $A$ die Transformationsmatrix und $v$ unser Übertragungsvektor ist. \[ v = A \cdot m \qquad \Rightarrow \qquad v = \begin{pmatrix} 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ @@ -127,13 +170,8 @@ v = A \cdot m \qquad \Rightarrow \qquad v = \begin{pmatrix} 1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{pmatrix} \] - Somit bekommen wir für unseren Übertragungsvektor \[ v = [5,3,6,5,2,10,2,7,10,4], \] den wir jetzt über einen beliebigen Nachrichtenkanal versenden können. - -\textbf{NOTES} - -warum wird 0 weggelassen? diff --git a/buch/papers/reedsolomon/hilfstabellen.tex b/buch/papers/reedsolomon/hilfstabellen.tex new file mode 100644 index 0000000..10e4fd1 --- /dev/null +++ b/buch/papers/reedsolomon/hilfstabellen.tex @@ -0,0 +1,21 @@ +% +% hilfstabellen.tex +% Autor: Michael Steiner +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{$\mathbb{F}_{11}$ Hilfstabellen + \label{reedsolomon:section:hilfstabellen}} +\rhead{Hilfstabellen} + +\textbf{TODO}: gibt es eine besser darstellungsart der tabellen? (\& platzierung der subsections) + +Um das rechnen zu erleichtern findet man in diesem Abschnitt die Resultate, die bei der Addition und der Multiplikation in $\mathbb{F}_{11}$ resultieren. + +\subsection{Additionstabelle + \label{reedsolomon:subsection:adtab}} +\input{papers/reedsolomon/restetabelle1.tex} + +\subsection{Multiplikationstabelle + \label{reedsolomon:subsection:mptab}} +\input{papers/reedsolomon/restetabelle2.tex} \ No newline at end of file -- cgit v1.2.1 From d408309e04a27315a2ce8788872095334dbea183 Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Tue, 8 Jun 2021 17:33:56 +0200 Subject: updated codebsp.tex and decohnefehler.tex --- buch/papers/reedsolomon/codebsp.tex | 24 ++++++++------ buch/papers/reedsolomon/decohnefehler.tex | 54 ++++++++++++++++++------------- 2 files changed, 46 insertions(+), 32 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/codebsp.tex b/buch/papers/reedsolomon/codebsp.tex index 818078e..262297e 100644 --- a/buch/papers/reedsolomon/codebsp.tex +++ b/buch/papers/reedsolomon/codebsp.tex @@ -87,9 +87,6 @@ in \mathbb{Z}_{11}\setminus\{0\} = \{a^0, a^1, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9\}. \] umzuschreiben. - -Wenn wir jetzt sämtliche Zahlen von $\mathbb{F}_{11}$ in $a$ einsetzen - % Old Text %Wir suchen also eine Zahl $a$, die in endlichen Körpern existiert und den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken kann. %Dazu schreiben wir @@ -102,7 +99,6 @@ Wenn wir jetzt sämtliche Zahlen von $\mathbb{F}_{11}$ in $a$ einsetzen %\] % %Wenn wir alle möglichen Werte für $a$ einsetzen, also - %\begin{align} %a = 0 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{0, 0, 0, 0, 0, 0, 0, 0, 0, 0\} \\ %a = 1 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 1, 1, 1, 1, 1, 1, 1, 1, 1\} \\ @@ -117,6 +113,10 @@ Wenn wir jetzt sämtliche Zahlen von $\mathbb{F}_{11}$ in $a$ einsetzen %a = 10 : \qquad \mathbb{Z}_{11}\setminus\{0\} = \{1, 10, 1, 10, 1, 10, 1, 10, 1, 10\} %\end{align} +\subsubsection{Die primitiven Einheitswurzeln + \label{reedsolomon:subsection:primsqrt}} + +Wenn wir jetzt sämtliche Zahlen von $\mathbb{F}_{11}$ in $a$ einsetzen \begin{center} \begin{tabular}{c r c l} %$a = 0 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{0, 0, 0, 0, 0, 0, 0, 0, 0, 0\}$ \\ @@ -133,11 +133,15 @@ $a = 10 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 10, 1, 10, 1, 10, \end{tabular} \end{center} so fällt uns auf, dass für $a$ die Zahlen $2,6,7,8$ erhalten, die tatsächlich den gesamten Zahlenraum von $\mathbb{F}_{11}$ abbilden. Solche Zahlen werden \em Primitive Einheitswurzel \em genannt. -Wenden wir diese Vorgehensweise auch für andere Endliche Körper an, so werden wir sehen, dass wir immer mindestens zwei solcher Einheitswurzel finden werden. Somit ist es uns überlassen, eine dieser Einheitswurzeln auszuwählen, mit der wir weiter rechnen wollen. +Wenden wir diese Vorgehensweise auch für andere Endliche Körper an, so werden wir sehen, dass wir immer mindestens zwei solcher Einheitswurzel finden werden. Somit ist es uns überlassen, eine dieser Einheitswurzeln auszuwählen, mit der wir weiter rechnen wollen. Für das Beispiel wählen wir die Zahl $a^i = 8$. -Für das Beispiel wählen wir die Zahl $a^i = 8$. -Damit wir unsere Nachricht codieren können, müssen wir $8^i$ in $m(X)$ einsetzen. +\subsubsection{Bildung einer Transformationsmatrix + \label{reedsolomon:subsection:transMat}} +Mit der Wahl einer Einheitswurzel ist es uns jetzt möglich, unsere Nachricht zu Codieren. Daraus sollen wir dann einen Übertragungsvektor $v$ erhalten, den wir an den Empfänger schicken können. Für die Codierung müssen wir alle $a^i$ in das Polynom $m(X)$ einsetzen. Da wir $a^i = 8^i$ gewählt haben ergibt sich daraus +% +%Damit wir unsere Nachricht codieren können, müssen wir $8^i$ in $m(X)$ einsetzen. +% \begin{center} \begin{tabular}{c} $m(8^0) = 4 \cdot 1^5 + 7 \cdot 1^4 + 2 \cdot 1^3 + 5 \cdot 1^2 + 8 \cdot 1^1 + 1 = 5$ \\ @@ -146,12 +150,12 @@ Damit wir unsere Nachricht codieren können, müssen wir $8^i$ in $m(X)$ einsetz $m(8^9) = 4 \cdot 7^5 + 7 \cdot 7^4 + 2 \cdot 7^3 + 5 \cdot 7^2 + 8 \cdot 7^1 + 1 = 4$ \end{tabular} \end{center} - +unser Übertragungsvektor. Um das ganze noch ein wenig übersichtlicher zu gestalten können wir die Polynome zu einer Matrix zusammenfassen und bildet so unsere Transformationsmatrix $A$. \subsection{Allgemeine Codierung \label{reedsolomon:subsection:algCod}} -Für eine elegantere Formulierung stellen wir das ganze als Matrix dar, wobei $m$ unsere Nachricht, $A$ die Transformationsmatrix und $v$ unser Übertragungsvektor ist. +Für die Codierung benötigen wir die Nachricht $m$, die Codiert werden soll sowie die Transformationsmatrix $A$. Daraus erhalten wir den Übertragungsvektor $v$. Setzen wir die Zahlen aus dem Beispiel ein erhalten wir folgende Darstellung. \[ v = A \cdot m \qquad \Rightarrow \qquad v = \begin{pmatrix} 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ @@ -170,7 +174,7 @@ v = A \cdot m \qquad \Rightarrow \qquad v = \begin{pmatrix} 1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{pmatrix} \] -Somit bekommen wir für unseren Übertragungsvektor +Für unseren Übertragungsvektor resultiert \[ v = [5,3,6,5,2,10,2,7,10,4], \] diff --git a/buch/papers/reedsolomon/decohnefehler.tex b/buch/papers/reedsolomon/decohnefehler.tex index 6ca577a..3b709f3 100644 --- a/buch/papers/reedsolomon/decohnefehler.tex +++ b/buch/papers/reedsolomon/decohnefehler.tex @@ -3,41 +3,50 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Decodierung ohne Fehler +\section{Decodierung: Ansatz ohne Fehler \label{reedsolomon:section:decohnefehler}} \rhead{fehlerlose rekonstruktion} -Im ersten Teil zur Decodierung des Übertragungsvektor betrachten wir den Übertragungskanal als fehlerfrei. -Wir erhalten also unseren Übertragungsvektor + +In diesem Abschnitt betrachten wie die Überlegung, wie wir auf der Empfängerseite die Nachricht aus dem empfangenen Übertragungsvektor erhalten. Nach einer einfachen Überlegung müssen wir den Übertragungsvektor decodieren, was auf den ersten Blick nicht allzu kompliziert sein sollte, solange wir davon ausgehen können, dass es während der Übertragung keine Fehler gegeben hat. Wir betrachten deshalb den Übertragungskanal als fehlerfrei. + +Der Übertragungsvektor empfangen wir also als \[ v = [5,3,6,5,2,10,2,7,10,4]. \] - -Gesucht ist nun einen Weg, mit dem wir auf unseren Nachrichtenvektor zurückrechnen können. -Ein banaler Ansatz ist das Invertieren der Glechung +% Old Text +%Im ersten Teil zur Decodierung des Übertragungsvektor betrachten wir den Übertragungskanal als fehlerfrei. +%Wir erhalten also unseren Übertragungsvektor +%\[ +%v = [5,3,6,5,2,10,2,7,10,4]. +%\] +Nach einem banalen Ansatz ist die Decodierung die Inverse der Codierung. Dank der Matrixschreibweise lässt sich dies relativ einfach umsetzen. +% Old Text +%Gesucht ist nun einen Weg, mit dem wir auf unseren Nachrichtenvektor zurückrechnen können. +%Ein banaler Ansatz ist das Invertieren der Glechung \[ -v = A \cdot m \qquad \Rightarrow \qquad m = A^{-1} \cdot v. +v = A \cdot m \qquad \Rightarrow \qquad m = A^{-1} \cdot v \] - -Nur stellt sich dann die Frage, wie wir auf die Inverse der Matix $A$ kommen. +Nur stellt sich jetzt die Frage, wie wir die Inverse von $A$ berechnen. Dazu können wir wiederum den Ansatz der Fouriertransformation uns zur Hilfe nehmen, jedoch betrachten wir jetzt deren Inverse. Definiert ist sie als \[ F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt \qquad \Rightarrow \qquad \mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega. \] - -In unserem Fall suchen wir also eine inverse für die Primitive Einheitswurzel $a$, also +Damit beschäftigen wir uns im Abschnitt \ref{reedsolomon:subsection:algdec} weiter, konkret suchen wir momentan aber eine Inverse für unsere primitive Einheitswurzel $a$. \[ -8^1 \qquad \Rightarrow \qquad 8^{-1}. +8^1 \qquad \rightarrow \qquad 8^{-1} \] +Mit einem solchen Problem haben wir uns bereits in Abschnitt \ref{buch:section:euklid} befasst und so den euklidischen Algorithmus kennengelernt, den wir auf unseren Fall anwenden können. -Im Abschnitt \textcolor{red}{4.1} haben wir den euklidischen Algorithmus kennengelernt, den wir auf unseren Fall anwenden können. +% Old Text +%Im Abschnitt \textcolor{red}{4.1} haben wir den euklidischen Algorithmus kennengelernt, den wir auf unseren Fall anwenden können. -\subsection{Der Euklidische Algorithmus -\label{reedsolomon:subsection:eukAlgo}} +\subsection{Inverse der primitiven Einheitswurzel +\label{reedsolomon:subsection:invEinh}} -Die Funktionsweise des euklidischen Algorithmus ist im Kapitel \textcolor{red}{4.1} ausführlich beschrieben. -Für unsere Anwendung wählen wir die Parameter $a_i = 8$ und $b_i = 11$. +Die Funktionsweise des euklidischen Algorithmus ist im Kapitel \ref{buch:section:euklid} ausführlich beschrieben. +Für unsere Anwendung wählen wir die Parameter $a = 8$ und $b = 11$ ($\mathbb{F}_{11}$). Daraus erhalten wir \begin{center} @@ -67,20 +76,21 @@ Daraus erhalten wir \end{tabular} \end{center} +als Inverse der primitiven Einheitswurzel. Die inverse Transformationsmatrix $A^{-1}$ bilden wir indem wir jetzt die inverse primitive Einheitswurzel anstelle der primitiven Einheitswurzel in die Matrix einsetzen. -als Inverse der Primitiven Einheitswurzel. +\subsection{Allgemeine Decodierung + \label{reedsolomon:subsection:algdec}} -Nun haben wir fast alles für die Rücktransformation beisammen. Wie auch bei der Inversen Fouriertransformation haben wir nun einen Vorfaktor +Wir haben jetzt fast alles für eine erfolgreiche Rücktransformation beisammen. Wir haben aber noch nicht alle Aspekte der inversen diskreten Fouriertransformation befolgt, so fehlt uns noch einen Vorfaktor \[ m = \textcolor{red}{s} \cdot A^{-1} \cdot v \] den wir noch bestimmen müssen. -Glücklicherweise lässt der sich analog wie bei der Inversen Fouriertransformation bestimmen und beträgt +Glücklicherweise lässt der sich analog wie bei der inversen diskreten Fouriertransformation bestimmen und beträgt \[ s = \frac{1}{10}. \] -Da $\frac{1}{10} = 10^{-1}$ entspricht können wir $s$ ebenfalls mit dem euklidischen Algorithmus bestimmen und stellen fest, dass $10^{-1} = 10$ ergibt. -Somit lässt sich der Nachrichtenvektor einfach bestimmen mit +Da $\frac{1}{10} = 10^{-1}$ entspricht können wir $s$ ebenfalls mit dem euklidischen Algorithmus bestimmen und stellen fest, dass $10^{-1} = 10$ in $\mathbb{F}_{11}$ ergibt. Somit lässt sich der Nachrichtenvektor einfach bestimmen mit \[ m = 10 \cdot A^{-1} \cdot v \qquad \Rightarrow \qquad m = 10 \cdot \begin{pmatrix} 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ -- cgit v1.2.1 From 73d5c3d4df0f73e96c1bac2ae1ce3b4dfcdc9d90 Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Thu, 10 Jun 2021 12:23:57 +0200 Subject: updated a lot --- buch/papers/reedsolomon/decmitfehler.tex | 292 +++++++++++++++++++--------- buch/papers/reedsolomon/endlichekoerper.tex | 6 +- buch/papers/reedsolomon/main.tex | 7 + buch/papers/reedsolomon/references.bib | 69 ++++--- buch/papers/reedsolomon/rekonstruktion.tex | 33 ++-- 5 files changed, 275 insertions(+), 132 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/decmitfehler.tex b/buch/papers/reedsolomon/decmitfehler.tex index 923c1c5..db6e586 100644 --- a/buch/papers/reedsolomon/decmitfehler.tex +++ b/buch/papers/reedsolomon/decmitfehler.tex @@ -3,52 +3,109 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Decodierung mit Fehler +\section{Decodierung: Ansatz mit Fehlerkorrektur \label{reedsolomon:section:decmitfehler}} \rhead{fehlerhafte rekonstruktion} -Im zweiten Teil zur Decodierung betrachten wir den Fall, dass unser Übertragungskanal nicht fehlerfrei ist. -Wir legen daher den Fehlervektor +Bisher haben wir die Decodierung unter der Bedingung durchgeführt, dass der Übertragungsvektor fehlerlos versendet und empfangen wurde. +In der realen Welt müssen wir uns jedoch damit abfinden, dass kein Übertragungskanal garantiert fehlerfrei ist und das wir früher oder später mit Fehlern rechnen müssen. +Genau für dieses Problem wurden Fehler korrigierende Codes, wie der Reed-Solomon-Code, entwickelt. +In diesem Abschnitt betrachten wir somit die Idee der Fehlerkorrektur und wie wir diese auf unser Beispiel anwenden können. +Der Übertragungskanal im Beispiel weisst jetzt den Fehlervektor \[ u = [0, 0, 0, 3, 0, 0, 0, 0, 2, 0] \] -fest, den wir zu unserem Übertragungsvektor als Fehler dazu addieren und somit +auf. +Senden wir jetzt unser Übertragungsvektor $v$ durch diesen Kanal addiert sich der Fehlervektor $u$ auf unsere Übertragung und wir erhalten \begin{center} - -\begin{tabular}{c | c r } - $v$ & & $[5,3,6,5,2,10,2,7,10,4]$\\ - $u$ & $+$ & $[0,0,0,3,0,0,0,0,2,0]$\\ - \hline - $w$ & & $[5,3,6,8,2,10,2,7,1,4]$\\ -\end{tabular} - -% alternative design -%\begin{tabular}{c | c cccccccccccc } -% $v$ & & $[$&$5,$&$3,$&$6,$&$5,$&$2,$&$10,$&$2,$&$7,$&$10,$&$4$&$]$\\ -% $u$ & $+$ & $[$&$0,$&$0,$&$0,$&$3,$&$0,$&$0,$&$0,$&$0,$&$2,$&$0$&$]$\\ + + \begin{tabular}{c | c r } + $v$ & & $[5,3,6,5,2,10,2,7,10,4]$\\ + $u$ & $+$ & $[0,0,0,3,0,0,0,0,2,0]$\\ + \hline + $w$ & & $[5,3,6,8,2,10,2,7,1,4]$\\ + \end{tabular} + + % alternative design + %\begin{tabular}{c | c cccccccccccc } + % $v$ & & $[$&$5,$&$3,$&$6,$&$5,$&$2,$&$10,$&$2,$&$7,$&$10,$&$4$&$]$\\ + % $u$ & $+$ & $[$&$0,$&$0,$&$0,$&$3,$&$0,$&$0,$&$0,$&$0,$&$2,$&$0$&$]$\\ + % \hline + % $w$ & & $[$&$5,$&$3,$&$6,$&$8,$&$2,$&$10,$&$2,$&$7,$&$1,$&$4$&$]$\\ + %\end{tabular} + +\end{center} +als neuen, fehlerbehafteten Übertragungsvektor $w$ auf der Empfängerseite. +% Old Text +%In diesem Abschnitt gehen wir genauer darauf ein, wie der Reed-Solomon-Code eine solche Feherkorrektur vornimt. +% +%In diesem Abschnitt betrachten wir das Problem, dass während der Übertragung des Übertragungsvektors von unserem Beispiel +% +% +%Zu diesem Zweck wurden Fehler korrigierende Codes entwickelt. +% +%Dieser Optimalfall kann jedoch mit keinem Übertragungskanal garantiert werden +% +% +%Im zweiten Teil zur Decodierung betrachten wir den Fall, dass unser Übertragungskanal nicht fehlerfrei ist. +%Wir legen daher den Fehlervektor +%\[ +%u = [0, 0, 0, 3, 0, 0, 0, 0, 2, 0] +%\] +%fest, den wir zu unserem Übertragungsvektor als Fehler dazu addieren und somit +% +%\begin{center} +% +%\begin{tabular}{c | c r } +% $v$ & & $[5,3,6,5,2,10,2,7,10,4]$\\ +% $u$ & $+$ & $[0,0,0,3,0,0,0,0,2,0]$\\ % \hline -% $w$ & & $[$&$5,$&$3,$&$6,$&$8,$&$2,$&$10,$&$2,$&$7,$&$1,$&$4$&$]$\\ +% $w$ & & $[5,3,6,8,2,10,2,7,1,4]$\\ %\end{tabular} - -\end{center} -als Übertragungsvektor auf der Empfängerseite erhalten. - -Wenn wir den Übertragungsvektor jetzt Rücktransformieren wie im vorherigen Kapitel erhalten wir +% +%% alternative design +%%\begin{tabular}{c | c cccccccccccc } +%% $v$ & & $[$&$5,$&$3,$&$6,$&$5,$&$2,$&$10,$&$2,$&$7,$&$10,$&$4$&$]$\\ +%% $u$ & $+$ & $[$&$0,$&$0,$&$0,$&$3,$&$0,$&$0,$&$0,$&$0,$&$2,$&$0$&$]$\\ +%% \hline +%% $w$ & & $[$&$5,$&$3,$&$6,$&$8,$&$2,$&$10,$&$2,$&$7,$&$1,$&$4$&$]$\\ +%%\end{tabular} +% +%\end{center} +%als Übertragungsvektor auf der Empfängerseite erhalten. +Wir jetzt als Empfänger wissen jedoch nicht, dass der erhaltene Übertragungsvektor jetzt fehlerbehaftet ist und werden dementsprechend den Ansatz aus Abschnitt \ref{reedsolomon:section:decohnefehler} anwenden. +Wir stellen jedoch recht schnell fest, dass am decodierten Nachrichtenblock \[ -r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]. +r = [\underbrace{5,7,4,10,}_{\text{Syndrom}}5,4,5,7,6,7]. \] -Im Vergleich zum vorherigen Kapitel sind die Fehlerkorrekturstellen jetzt $\neq 0$, was bedeutet, dass wir diesen Übertragungsvektor fehlerhaft empfangen haben und sich die Nachricht jetzt nicht mehr so einfach decodieren lässt. +etwas nicht in Ordnung ist, denn die vorderen vier Fehlerkorrekturstellen haben nicht mehr den Wert null. +Der Nachrichtenblock weisst jetzt ein \em Syndrom \em auf, welches anzeigt, dass der Übertragungsvektor fehlerhaft empfangen wurde. +% Old Text +%Wenn wir den Übertragungsvektor jetzt Rücktransformieren wie im vorherigen Kapitel erhalten wir +%\[ +%r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]. +%\] +Jetzt stellt sich natürlich die Frage, wie wir daraus den ursprünglich gesendeten Nachrichtenvektor zurückerhalten sollen. Laut der Definition über die Funktionsweise eines Reed-Solomon-Codes können wir aus den Fehlerkorrekturstellen ein ``Lokatorpolynom'' berechnen, welches die Information enthält, welche stellen innerhalb des empfangenen Übertragungsvektors fehlerhaft sind. -% warum wir die fehler suchen -Da Reed-Solomon-Codes in der Lage sind, eine Nachricht aus weniger Stellen zu rekonstruieren als wir ursprünglich haben, so müssen wir nur die Fehlerhaften Stellen finden und eliminieren, damit wir unsere Nutzdaten rekonstruieren können. -Damit stellt sich die Frage, wie wir die Fehlerstellen $e$ finden. -Dafür wählen wir einen Primitiven Ansatz mit -\begin{align} - m(X) & = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1 \\ - r(X) & = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7 \\ - e(X) & = r(X) - m(X). -\end{align} -Setzen wir jetzt unsere Einheitswurzel für $X$ ein, so erhalten wir +\subsection{Das Fehlerstellenpolynom $d(X)$ + \label{reedsolomon:subsection:fehlerpolynom}} +Bevor wir unser Lokatorpolynom berechnen können, müssen wir zuerst eine Möglichkeit finden, die Fehlerhaften von den Korrekten Stellen im Übertragungsvektor unterscheiden zu können. In einem ersten Versuch könnten wir $d$ berechnen mit +\begin{center} +\begin{tabular}{r c l} + $m(X)$ & $=$ & $4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ \\ + $r(X)$ & $=$ & $5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$ \\ + $d(X)$ & $=$ & $r(X) - m(X)$. +\end{tabular} +\end{center} +TODO (rewrite sentence): Dies wird uns zwar andere sorgen wegen $m(X)$ bereiten, \textcolor{red}{die werden wir jedoch zu einem späteren Zeitpunkt betrachten (todo: verweis auf kapitel?)}. +Setzen wir jetzt noch unsere Einheitswurzel aus dem Beispiel ein so erhalten wir +% Old Text +%\begin{align} +% m(X) & = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1 \\ +% r(X) & = 5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7 \\ +% e(X) & = r(X) - m(X). +%\end{align} +%Setzen wir jetzt unsere Einheitswurzel für $X$ ein, so erhalten wir \begin{center} \begin{tabular}{c c c c c c c c c c c} \hline @@ -56,80 +113,137 @@ Setzen wir jetzt unsere Einheitswurzel für $X$ ein, so erhalten wir \hline $r(a^{i})$& $5$& $3$& $6$& $8$& $2$& $10$& $2$& $7$& $1$& $4$\\ $m(a^{i})$& $5$& $3$& $6$& $5$& $2$& $10$& $2$& $7$& $10$& $4$\\ - $e(a^{i})$& $0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$\\ + $d(a^{i})$& $0$& $0$& $0$& $3$& $0$& $0$& $0$& $0$& $2$& $0$\\ \hline \end{tabular} \end{center} -und damit die Information, dass an allen Stellen, die nicht Null sind, Fehler enthalten. -Um jetzt alle nicht Nullstellen zu finden, wenden wir den Satz von Fermat an. +und damit die Information, dass allen Stellen, die nicht Null sind, Fehler enthalten. +Aus der Tabelle lesen wir, das in unserem Beispiel die Fehler an der Stelle drei und acht zu finden sind. + +Für das einfache Bestimmen von Hand mag dies ja noch ausreichen, jedoch können wir mit diesen Stellen nicht das Lokatorpolynom bestimmen, denn dafür bräuchten wir alle Nullstellen, an denen es Fehler gegeben hat (also sozusagen genau das umgekehrte). Um dies zu erreichen wenden wir eine andere Herangehensweise und nehmen uns den Satz von Fermat sowie den kleinsten gemeinsamen Teiler zur Hilfe. -\subsection{Der Satz von Fermat -\label{reedsolomon:subsection:fermat}} -Der Satz von Fermat besagt, dass für +\subsection{Mit dem grössten gemeinsamen Teiler auf Nullstellenjagd +\label{reedsolomon:subsection:ggT}} + +Zuerst betrachten wir mal den Satz von Fermat deren Funktionsweise wir in Abschnitt \ref{buch:section:galoiskoerper} kennengelernt haben. Der besagt, dass für \[ f(X) = X^{q-1} -1 = 0 \] -gilt, egal was wir für $q$ einsetzen. - -Für unser Beispiel erhalten wir +wobei dies für jedes $q$ gilt. Setzen wir also das $q$ von unserem Beispiel ein \[ f(X) = X^{10}-1 = 0 \qquad \text{für } X = \{1,2,3,4,5,6,7,8,9,10\} \] -und können $f(X)$ auch umschreiben in +und stellen dies als Nullstellenform (\textcolor{red}{richtiger name für die Schreibweise?}) dar. So ergibt sich die Darstellung \[ f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6)(X-a^7)(X-a^8)(X-a^9). \] Zur Überprüfung können wir unsere Einheitswurzel in $a$ einsetzen und werden sehen, dass wir für $f(X) = 0$ erhalten werden. -Nach der gleichen Überlegung können wir jetzt auch $e(X)$ darstellen als + +Wir können jetzt auch $d(X)$ nach der gleichen Überlegung darstellen als \[ -e(X) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6)(X-a^7) \qquad \qquad (X-a^9) \cdot p(x), +d(X) = (X-a^0)(X-a^1)(X-a^2)\textcolor{gray!40}{(X-a^3)}(X-a^4)(X-a^5)(X-a^6)(X-a^7)\textcolor{gray!40}{(X-a^8)}(X-a^9) \cdot p(x), \] -wobei $p(X)$ das Restpolynom ist und die Fehlerstellen beinhaltet. -Wenn wir jetzt den grössten gemeinsamen Teiler von $f(X)$ und $e(X)$ berechnen, so erhalten wir mit +wobei diese Darstellung nicht mehr alle Nullstellen umfasst wie es noch in $f(X)$ der Fall war. +Dies liegt daran, dass wir ja zwei Fehlerstellen (grau markiert) haben, die nicht Null sind. Diese fassen wir zum Restpolynom $p(X)$ (\textcolor{red}{eventuell farblich kennzeichnen?}) zusammen. +Wenn wir jetzt den grössten gemeinsamen Teiler von $f(X)$ und $d(X)$ berechnen, so erhalten wir mit \[ -\operatorname{ggT}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2) \qquad \qquad (X-a^4)(X-a^5)(X-a^6)(X-a^7) \qquad \qquad (X-a^9) +\operatorname{ggT}(f(X),d(X)) = (X-a^0)(X-a^1)(X-a^2)\textcolor{gray!40}{(X-a^3)}(X-a^4)(X-a^5)(X-a^6)(X-a^7)\textcolor{gray!40}{(X-a^8)}(X-a^9) \] eine Liste von Nullstellen, an denen es keine Fehler gegeben hat. -Da wir uns jedoch für eine Liste mit Nullstellen interessieren, an denen es Fehler gegeben hat berechnen wir stattdessen das kgV von $f(X)$ und $e(X)$ als +Dies scheint zuerst nicht sehr hilfreich zu sein, da wir für das Lokatorpolynom ja eine Liste der Nullstellen suchen, an denen es Fehler gegeben hat. Aus diesem Grund berechnen wir im nächsten Schritt das kleinste gemeinsame Vielfache von $f(X)$ und $d(X)$. + +%Wir werden auch feststellen, das unsere Bemühungen bisher nicht umsonst waren. + +\subsection{Mit dem kgV fehlerhafte Nullstellen finden + \label{reedsolomon:subsection:kgV}} + +Das kgV hat nämlich die Eigenschaft sämtliche Nullstellen zu finden, also nicht nur die fehlerhaften sondern auch die korrekten, was in \[ -\operatorname{kgV}(f(X),e(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6)(X-a^7)(X-a^8)(X-a^9) \cdot q(X). +\operatorname{kgV}(f(X),d(X)) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6)(X-a^7)(X-a^8)(X-a^9) \cdot q(X). \] -Wir können das Resultat noch zerlegen in +ersichtlich ist. +Aus dem vorherigen Abschnitt wissen wir auch, dass $d(X)$ alle korrekten Nullstellen beinhaltet. Teilen wir das kgV jetzt auf in \[ -\operatorname{kgV}(f(X),e(X)) = d(X) \cdot e(X). +\operatorname{kgV}(f(X),d(X)) = d(X) \cdot l(X) \] -Somit muss $d(X)$ eine Liste von Nullstellen enthalten an denen es Fehler gegeben hat. +sollten wir für $l(X)$ eine Liste mit allen fehlerhaften Nullstellen erhalten. +Somit ist \[ -d(X) = (X-a^3)(X-a^8) +l(X) = (X-a^3)(X-a^8) \] +unser gesuchtes Lokatorpolynom. +Es scheint so als müssten wir nur noch an den besagten Stellen den Übertragungsvektor korrigieren und wir währen fertig mit der Fehlerkorrektur. +Jedoch haben wir noch ein grundlegendes Problem, dass zu beginn aufgetaucht ist, wir aber beiseite geschoben haben. Die Rede ist natürlich vom Nachrichtenvektor $m(X)$, mit dem wir in erster Linie das wichtige Fehlerstellenpolynom $d(X)$ berechnet haben. +\subsection{Der problematische Nachrichtenvektor $m(X)$ + \label{reedsolomon:subsection:nachrichtenvektor}} -und ist damit unser gesuchtes Lokatorpolynom. - -Das einzige Problem was jetzt noch bleibt ist, dass wir $e(X)$ berechnet haben aus +In Abschnitt \ref{reedsolomon:section:decmitfehler} haben wir \[ -e(X) = r(X) - m(X), +d(X) = r(X) - m(X) \] -wobei $m(X)$ auf der Empfängerseite unbekannt ist. -Es sieht danach aus, das wir diesen Lösungsansatz nicht verwenden können, da uns ein entscheidender Teil fehlt. -Bei einer näheren Betrachtung von $m(X)$ fällt uns aber auf, dass wir doch etwas über $m(X)$ wissen. -Wir kennen nämlich die ersten vier Stellen, da diese für die Fehlerkorrektur zuständig sind und daher Null sein müssen. +in Abhängigkeit von $m(X)$ berechnet. +Jedoch haben wir ausser acht gelassen, dass $m(X)$ auf der Empfängerseite nicht existiert und somit gänzlich unbekannt ist. +Es scheint so als würde dieser Lösungsansatz, den wir bisher verfolgt haben, nicht funktioniert. +Wir könnten uns höchstens noch fragen, ob wir tatsächlich nichts über den Nachrichtenvektor im Beispiel wissen. Wenn wir noch einmal den Vektor betrachten als \[ -m = [0,0,0,0,?,?,?,?,?,?] +m = [0,0,0,0,4,7,2,5,8,1] \] -An genau diesen Stellen liegt auch die Information, wo unsere Fehlerstellen liegen, was uns ermöglicht, den Teil von $e(X)$ zu berechnen, der uns auch interessiert. - -Wir können $e(X)$ also bestimmen als +fällt uns aber auf, dass wir doch etwas über diesen Vektor wissen, nämlich den Wert der ersten 2t (im Beispiel vier) stellen. +Im Normalfall sollen diese nämlich den Wert null betragen und somit sind nur die letzten k stellen (im Beispiel sechs) für uns unbekannt, dargestellt als \[ -e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X) +m = [0,0,0,0,?,?,?,?,?,?]. \] -wobei $p(X)$ wiederum ein unbekanntes Restpolynom ist und +Wie der Zufall es so will liegt an diesen vier Stellen auch die Information, wo die Fehlerstellen liegen. Daher reicht es auch aus +% darum werden die stellen auch als fehlerkorrekturstellen bezeichnet \[ -f(X) = X^{10} - 1 = X^{10} + 10 +d(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X) \] -ist können wir so in einer ersten Instanz den grössten gemeinsamen Teiler von $f(X)$ und $e(X)$ berechnen. -Dafür nehmen wir uns wiederum den Euklidischen Algorithmus zur Hilfe und berechnen so +so zu berechnen, dass wir die wichtigen vier Stellen kennen, der Rest des Polynoms jedoch im unbekannten Restpolynom $p(X)$ enthalten ist. + +\textcolor{red}{ist das wechseln zwischen 2t,k aus dem allgemeinfall und vier,sechs aus dem beispiel zu verwirrend?} + +\subsection{Die Berechnung der Fehlerstellen + \label{reedsolomon:subsection:nachrichtenvektor}} + +Um die Fehlerstellen zu berechnen wenden wir die gleiche Vorgehensweise wie zuvor an, also zuerst den ggT, danach berechnen wir das kgV um am Ende das Lokatorpolynom zu erhalten. + +\subsubsection{Schritt 1: ggT} +Wir berechnen den ggT von $f(X)$ und $d(X)$ mit +\begin{center} +\begin{tabular}{r c l} + $f(X)$ & $=$ & $X^{10} - 1 = X^{10} + 10$ \\ + $d(X)$ & $=$ & $5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X)$ +\end{tabular} +\end{center} +% +% +% +%Das einzige Problem was jetzt noch bleibt ist, dass wir $e(X)$ berechnet haben aus +%\[ +%e(X) = r(X) - m(X), +%\] +%wobei $m(X)$ auf der Empfängerseite unbekannt ist. +%Es sieht danach aus, das wir diesen Lösungsansatz nicht verwenden können, da uns ein entscheidender Teil fehlt. +%Bei einer näheren Betrachtung von $m(X)$ fällt uns aber auf, dass wir doch etwas über $m(X)$ wissen. +%Wir kennen nämlich die ersten vier Stellen, da diese für die Fehlerkorrektur zuständig sind und daher Null sein müssen. +%\[ +%m = [0,0,0,0,?,?,?,?,?,?] +%\] +%An genau diesen Stellen liegt auch die Information, wo unsere Fehlerstellen liegen, was uns ermöglicht, den Teil von $e(X)$ zu berechnen, der uns auch interessiert. +% +%Wir können $e(X)$ also bestimmen als +%\[ +%e(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X) +%\] +%wobei $p(X)$ wiederum ein unbekanntes Restpolynom ist und +%\[ +%f(X) = X^{10} - 1 = X^{10} + 10 +%\] +%ist können wir so in einer ersten Instanz den grössten gemeinsamen Teiler von $f(X)$ und $e(X)$ berechnen. +%Dafür nehmen wir uns wiederum den Euklidischen Algorithmus zur Hilfe und berechnen so +% \[ \arraycolsep=1.4pt \begin{array}{rcrcrcrcccrcrcrcrcrcrcrcrcr} @@ -151,11 +265,16 @@ Dafür nehmen wir uns wiederum den Euklidischen Algorithmus zur Hilfe und berech \] und erhalten \[ -\operatorname{ggT}(f(X),e(X)) = 6X^8 +\operatorname{ggT}(f(X),e(X)) = 6X^8. \] -Mit den Resultaten, die wir vom Rechenweg des grössten gemeinsamen Teiler erhalten haben können wir jetzt auch das kleinste Gemeinsame Vielfache berechnen. Eine detailliertere Vorgehensweise findet man in Kapitel ???. -Aus diesem erweiterten Euklidischen Algorithmus erhalten wir +\subsubsection{Schritt 2: kgV} + +Mit dem Resultat das wir vom ggT erhalten haben können wir jetzt das kgV berechnen. Dazu können wir jetzt den erweiterten Euklidischen Algorithmus verwenden, den wir in Abschnitt \ref{buch:subsection:daskgv} kennengelernt haben. +% +%Mit den Resultaten, die wir vom Rechenweg des grössten gemeinsamen Teiler erhalten haben können wir jetzt auch das kleinste Gemeinsame Vielfache berechnen. Eine detailliertere Vorgehensweise findet man in Kapitel ???. +% +%Aus diesem erweiterten Euklidischen Algorithmus erhalten wir \begin{center} \begin{tabular}{| c | c | c c |} @@ -170,28 +289,23 @@ Aus diesem erweiterten Euklidischen Algorithmus erhalten wir \end{tabular} \end{center} -und erhalten auf diesem Weg den Faktor +Daraus erhalten wir die Faktoren \[ -d(X) = 2X^2 + 5, +l(X) = 2X^2 + 5 \qquad \rightarrow \qquad l(X) = 2(X-5)(X-6). \] -den wir in +Unser gesuchtes Lokatorpolynom hat also die Form \[ -d(X) = 2(X-5)(X-6) +l(X) = (X-a^i)(X-a^j). \] -zerlegen können. -Da die unbekannten Stellen im Lokatorpolynom -\[ -d(X) = (X-a^i)(X-a^i) -\] -sind, müssen wir nur noch $i$ berechnen als +Also brauchen wir nur noch $i$ und $j$ zu berechnen und wir haben unsere gesuchten Fehlerstellen. +Diese bekommen wir recht einfach mit \begin{center} $a^i = 5 \qquad \Rightarrow \qquad i = 3$ - $a^i = 6 \qquad \Rightarrow \qquad i = 8$. + $a^j = 6 \qquad \Rightarrow \qquad j = 8$. \end{center} - -Somit erhalten wir schliesslich +Schlussendlich erhalten wir \[ d(X) = (X-a^3)(X-a^8) \] -als unser Lokatorpolynom mit den Fehlerhaften Stellen. \ No newline at end of file +als unser Lokatorpolynom mit den fehlerhaften Stellen. diff --git a/buch/papers/reedsolomon/endlichekoerper.tex b/buch/papers/reedsolomon/endlichekoerper.tex index 8ccd918..146067a 100644 --- a/buch/papers/reedsolomon/endlichekoerper.tex +++ b/buch/papers/reedsolomon/endlichekoerper.tex @@ -7,9 +7,9 @@ \label{reedsolomon:section:endlichekoerper}} \rhead{Problemstellung} -TODO: +\textcolor{red}{TODO: (warten auf den 1. Teil)} -Das rechnen in endlichen Körpern bietet einige Vorteile: +Das Rechnen in endlichen Körpern bietet einige Vorteile: \begin{itemize} \item Konkrete Zahlen: In endlichen Körpern gibt es weder rationale noch komplexe Zahlen. Zudem beschränken sich die möglichen Rechenoperationen auf das Addieren und Multiplizieren. Somit können wir nur ganze Zahlen als Resultat erhalten. @@ -20,4 +20,4 @@ Das rechnen in endlichen Körpern bietet einige Vorteile: Um jetzt eine Nachricht in den endlichen Körpern zu konstruieren legen wir fest, dass diese Nachricht aus einem Nutzdatenteil und einem Fehlerkorrekturteil bestehen muss. Somit ist die zu übertragende Nachricht immer grösser als die Daten, die wir übertragen wollen. Zudem müssen wir einen Weg finden, den Fehlerkorrekturteil so aus den Nutzdaten zu berechnen, dass wir die Nutzdaten auf der Empfängerseite wieder rekonstruieren können, sollte es zu einer fehlerhaften Übertragung kommen. -Nun stellt sich die Frage, wie wir eine Fehlerhafte Nachricht korrigieren können, ohne ihren ursprünglichen Inhalt zu kennen. Der Reed-Solomon-Code erzielt dies, indem aus dem Fehlerkorrekturteil ein sogenanntes "Lokatorpolynom" generiert werden kann. Dieses Polynom gibt dem Emfänger an, welche Stellen in der Nachricht feherhaft sind. +Nun stellt sich die Frage, wie wir eine fehlerhafte Nachricht korrigieren können, ohne ihren ursprünglichen Inhalt zu kennen. Der Reed-Solomon-Code erzielt dies, indem aus dem Fehlerkorrekturteil ein sogenanntes ``Lokatorpolynom'' generiert werden kann. Dieses Polynom gibt dem Emfänger an, welche Stellen in der Nachricht feherhaft sind. diff --git a/buch/papers/reedsolomon/main.tex b/buch/papers/reedsolomon/main.tex index a7485cd..9822d25 100644 --- a/buch/papers/reedsolomon/main.tex +++ b/buch/papers/reedsolomon/main.tex @@ -39,6 +39,13 @@ Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren \input{papers/reedsolomon/decohnefehler} \input{papers/reedsolomon/decmitfehler} \input{papers/reedsolomon/rekonstruktion} +\input{papers/reedsolomon/hilfstabellen} +%\input{papers/reedsolomon/glossar} -> geplant zur besseren orientierung +%\input{papers/reedsolomon/anwendungen} -> geplant + +\nocite{reedsolomon:weitz} +\nocite{reedsolomon:informationkommunikation} +%\nocite{reedsolomon:mendezmueller} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/reedsolomon/references.bib b/buch/papers/reedsolomon/references.bib index 38613bd..4c1d17a 100644 --- a/buch/papers/reedsolomon/references.bib +++ b/buch/papers/reedsolomon/references.bib @@ -4,32 +4,53 @@ % (c) 2020 Autor, Hochschule Rapperswil % -@online{reedsolomon:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, - year = {2020}, - month = {2}, - day = {6} +@online{reedsolomon:weitz, + title = {Fehlerkorrektur mit Reed-Solomon-Codes}, + url = {https://youtu.be/uOLW43OIZJ0}, + date = {2021-06-10}, + year = {2021}, + month = {6}, + day = {10} } -@book{reedsolomon:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} -} +% https://link.springer.com/chapter/10.1007%2F978-3-8351-9077-1_9 -@article{reedsolomon:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, - title = { Noncommutative harmonic analysis and image registration }, - journal = { Appl. Comput. Harmon. Anal.}, - year = 2019, - volume = 47, - pages = {607--627}, - url = {https://doi.org/10.1016/j.acha.2017.11.004} +@book{reedsolomon:informationkommunikation, + title = {Information und Kommunikation}, + author = {Markus Hufschmid}, + publisher = {Teubner}, + year = {2007}, + isbn = {978-3-8351-0122-7}, + inseries = {}, + volume = {1} } +% Beispiele +%@online{reedsolomon:bibtex, +% title = {BibTeX}, +% url = {https://de.wikipedia.org/wiki/BibTeX}, +% date = {2020-02-06}, +% year = {2020}, +% month = {2}, +% day = {6} +%} +% +%@book{reedsolomon:numerical-analysis, +% title = {Numerical Analysis}, +% author = {David Kincaid and Ward Cheney}, +% publisher = {American Mathematical Society}, +% year = {2002}, +% isbn = {978-8-8218-4788-6}, +% inseries = {Pure and applied undegraduate texts}, +% volume = {2} +%} +% +%@article{reedsolomon:mendezmueller, +% author = { Tabea Méndez and Andreas Müller }, +% title = { Noncommutative harmonic analysis and image registration }, +% journal = { Appl. Comput. Harmon. Anal.}, +% year = 2019, +% volume = 47, +% pages = {607--627}, +% url = {https://doi.org/10.1016/j.acha.2017.11.004} +%} \ No newline at end of file diff --git a/buch/papers/reedsolomon/rekonstruktion.tex b/buch/papers/reedsolomon/rekonstruktion.tex index 8cb7744..89a700f 100644 --- a/buch/papers/reedsolomon/rekonstruktion.tex +++ b/buch/papers/reedsolomon/rekonstruktion.tex @@ -1,24 +1,25 @@ % -% teil3.tex -- Beispiel-File für Teil 3 +% rekonstruktion.tex +% Autor: Michael Steiner % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % \section{Nachricht Rekonstruieren \label{reedsolomon:section:rekonstruktion}} \rhead{Rekonstruktion} -Im letzten Kapitel haben wir eine Möglichkeit gefunden, wie wir die Fehlerhaften Stellen lokalisieren können. +Im letzten Kapitel haben wir eine Möglichkeit gefunden, wie wir die fehlerhaften Stellen lokalisieren können. Mit diesen Stellen soll es uns nun möglich sein, aus dem fehlerhaften empfangenen Nachrichtenvektor wieder unsere Nachricht zu rekonstruieren. Das Lokatorpolynom \[ -d(X) = (X - a^3)(X-a^8) +l(X) = (X - a^3)(X-a^8) \] -markiert dabei diese Fehlerhaften Stellen im Übertragungsvektor +markiert dabei diese fehlerhaften Stellen im Übertragungsvektor \[ w = [5,3,6,8,2,10,2,7,1,4]. \] Als Ausgangslage verwenden wir die Matrix, mit der wir den Nachrichtenvektor ursprünglich codiert haben. -Unser Ziel ist es wie auch schon im Kapitel X.X (Rekonstuktion ohne Fehler) eine Möglichkeit zu finden, wie wir den Übertragungsvektor decodieren können. -Aufgrund der Fehlerstellen müssen wir aber davon ausgehen, das wir nicht mehr den gleichen Weg verfolgen können wie wir im Kapitel X.X angewendet haben. +Unser Ziel ist es wie auch schon im Abschnitt \ref{reedsolomon:section:decohnefehler} eine Möglichkeit zu finden, wie wir den Übertragungsvektor decodieren können. +Aufgrund der Fehlerstellen müssen wir aber davon ausgehen, das wir nicht mehr den gleichen Weg verfolgen können wie wir im Abschnitt \ref{reedsolomon:section:decohnefehler} angewendet haben. Wir stellen also die Matrix auf und markieren gleichzeitig die Fehlerstellen. \[ @@ -82,21 +83,21 @@ Wir kennen aber das Resultat aus den letzten vier Spalten, da wir wissen, das di \end{pmatrix} = \begin{pmatrix} - 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& \textcolor{green}{8^0}& \textcolor{green}{8^0}& \textcolor{green}{8^0}& \textcolor{green}{8^0}\\ - 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& \textcolor{green}{8^6}& \textcolor{green}{8^7}& \textcolor{green}{8^8}& \textcolor{green}{8^9}\\ - 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& \textcolor{green}{8^{12}}& \textcolor{green}{8^{14}}& \textcolor{green}{8^{16}}& \textcolor{green}{8^{18}}\\ - 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor{green}{8^{24}}& \textcolor{green}{8^{28}}& \textcolor{green}{8^{32}}& \textcolor{green}{8^{36}}\\ - 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor{green}{8^{30}}& \textcolor{green}{8^{35}}& \textcolor{green}{8^{40}}& \textcolor{green}{8^{45}}\\ - 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor{green}{8^{36}}& \textcolor{green}{8^{42}}& \textcolor{green}{8^{48}}& \textcolor{green}{8^{54}}\\ - 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor{green}{8^{42}}& \textcolor{green}{8^{49}}& \textcolor{green}{8^{56}}& \textcolor{green}{8^{63}}\\ - 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor{green}{8^{54}}& \textcolor{green}{8^{63}}& \textcolor{green}{8^{72}}& \textcolor{green}{8^{81}}\\ + 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& \textcolor{darkgreen}{8^0}& \textcolor{darkgreen}{8^0}& \textcolor{darkgreen}{8^0}& \textcolor{darkgreen}{8^0}\\ + 8^0& 8^1& 8^2& 8^3& 8^4& 8^5& \textcolor{darkgreen}{8^6}& \textcolor{darkgreen}{8^7}& \textcolor{darkgreen}{8^8}& \textcolor{darkgreen}{8^9}\\ + 8^0& 8^2& 8^4& 8^6& 8^8& 8^{10}& \textcolor{darkgreen}{8^{12}}& \textcolor{darkgreen}{8^{14}}& \textcolor{darkgreen}{8^{16}}& \textcolor{darkgreen}{8^{18}}\\ + 8^0& 8^4& 8^8& 8^{12}& 8^{16}& 8^{20}& \textcolor{darkgreen}{8^{24}}& \textcolor{darkgreen}{8^{28}}& \textcolor{darkgreen}{8^{32}}& \textcolor{darkgreen}{8^{36}}\\ + 8^0& 8^5& 8^{10}& 8^{15}& 8^{20}& 8^{25}& \textcolor{darkgreen}{8^{30}}& \textcolor{darkgreen}{8^{35}}& \textcolor{darkgreen}{8^{40}}& \textcolor{darkgreen}{8^{45}}\\ + 8^0& 8^6& 8^{12}& 8^{18}& 8^{24}& 8^{30}& \textcolor{darkgreen}{8^{36}}& \textcolor{darkgreen}{8^{42}}& \textcolor{darkgreen}{8^{48}}& \textcolor{darkgreen}{8^{54}}\\ + 8^0& 8^7& 8^{14}& 8^{21}& 8^{28}& 8^{35}& \textcolor{darkgreen}{8^{42}}& \textcolor{darkgreen}{8^{49}}& \textcolor{darkgreen}{8^{56}}& \textcolor{darkgreen}{8^{63}}\\ + 8^0& 8^9& 8^{18}& 8^{27}& 8^{36}& 8^{45}& \textcolor{darkgreen}{8^{54}}& \textcolor{darkgreen}{8^{63}}& \textcolor{darkgreen}{8^{72}}& \textcolor{darkgreen}{8^{81}}\\ \end{pmatrix} \cdot \begin{pmatrix} - m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor{green}{m_6} \\ \textcolor{green}{m_7} \\ \textcolor{green}{m_8} \\ \textcolor{green}{m_9} \\ + m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ \textcolor{darkgreen}{m_6} \\ \textcolor{darkgreen}{m_7} \\ \textcolor{darkgreen}{m_8} \\ \textcolor{darkgreen}{m_9} \\ \end{pmatrix} \] -Wir nehmen die Entsprechenden Spalten aus der Matrix heraus und erhalten so das Überbestimmte Gleichungssystem +Wir nehmen die entsprechenden Spalten aus der Matrix heraus und erhalten so das Überbestimmte Gleichungssystem \[ \begin{pmatrix} 5 \\ 3 \\ 6 \\ 2 \\ 10 \\ 2 \\ \textcolor{red}{7} \\ \textcolor{red}{4} \\ -- cgit v1.2.1 From 82672c8b82f0d082daa05cfc212a1b05a7f79650 Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Thu, 10 Jun 2021 15:22:44 +0200 Subject: hilfstabellen updated --- buch/papers/reedsolomon/hilfstabellen.tex | 2 - buch/papers/reedsolomon/restetabelle1.tex | 190 ++++++++++++++++++++++++++--- buch/papers/reedsolomon/restetabelle2.tex | 192 ++++++++++++++++++++++++++---- 3 files changed, 343 insertions(+), 41 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/hilfstabellen.tex b/buch/papers/reedsolomon/hilfstabellen.tex index 10e4fd1..4e39de5 100644 --- a/buch/papers/reedsolomon/hilfstabellen.tex +++ b/buch/papers/reedsolomon/hilfstabellen.tex @@ -8,8 +8,6 @@ \label{reedsolomon:section:hilfstabellen}} \rhead{Hilfstabellen} -\textbf{TODO}: gibt es eine besser darstellungsart der tabellen? (\& platzierung der subsections) - Um das rechnen zu erleichtern findet man in diesem Abschnitt die Resultate, die bei der Addition und der Multiplikation in $\mathbb{F}_{11}$ resultieren. \subsection{Additionstabelle diff --git a/buch/papers/reedsolomon/restetabelle1.tex b/buch/papers/reedsolomon/restetabelle1.tex index a5055c0..3969ef2 100644 --- a/buch/papers/reedsolomon/restetabelle1.tex +++ b/buch/papers/reedsolomon/restetabelle1.tex @@ -1,24 +1,176 @@ % created by Michael Steiner % % Restetabelle von F_11: Addition -\begin{figure} + +% alternatives design +%\begin{figure} +%\begin{center} +%\begin{tabular}{|>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}|} +%\hline +%+&0&1&2&3&4&5&6&7&8&9&10\\ +%\hline +%0&0&1&2&3&4&5&6&7&8&9&10\\ +%1&1&2&3&4&5&6&7&8&9&10&0\\ +%2&2&3&4&5&6&7&8&9&10&0&1\\ +%3&3&4&5&6&7&8&9&10&0&1&2\\ +%4&4&5&6&7&8&9&10&0&1&2&3\\ +%5&5&6&7&8&9&10&0&1&2&3&4\\ +%6&6&7&8&9&10&0&1&2&3&4&5\\ +%7&7&8&9&10&0&1&2&3&4&5&6\\ +%8&8&9&10&0&1&2&3&4&5&6&7\\ +%9&9&10&0&1&2&3&4&5&6&7&8\\ +%10&10&0&1&2&3&4&5&6&7&8&9\\ +%\hline +%\end{tabular} +%\end{center} +%\end{figure} + \begin{center} -\begin{tabular}{|>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}|} -\hline -+&0&1&2&3&4&5&6&7&8&9&10\\ -\hline -0&0&1&2&3&4&5&6&7&8&9&10\\ -1&1&2&3&4&5&6&7&8&9&10&0\\ -2&2&3&4&5&6&7&8&9&10&0&1\\ -3&3&4&5&6&7&8&9&10&0&1&2\\ -4&4&5&6&7&8&9&10&0&1&2&3\\ -5&5&6&7&8&9&10&0&1&2&3&4\\ -6&6&7&8&9&10&0&1&2&3&4&5\\ -7&7&8&9&10&0&1&2&3&4&5&6\\ -8&8&9&10&0&1&2&3&4&5&6&7\\ -9&9&10&0&1&2&3&4&5&6&7&8\\ -10&10&0&1&2&3&4&5&6&7&8&9\\ -\hline -\end{tabular} + +\begin{tikzpicture}[>=latex,thick,scale=0.45] +\fill[color=gray!40] (0,0) rectangle (18,-1.5); +\fill[color=gray!40] (0,0) rectangle (1.5,-18); +\draw[step = 1.5, gray,very thin] (0,0) grid (18,-18); +\draw[very thick] (0,0) rectangle (18,-18); +\draw[very thick] (0,-1.5) -- (18,-1.5); +\draw[very thick] (1.5,0) -- (1.5,-18); +\node at (0.75,-0.75) {$+$}; +\foreach \x in {0,...,10} + \node at (2.25+\x*1.5,-0.75) {$\x$}; +\foreach \y in {0,...,10} + \node at (0.75,-2.25+\y*-1.5) {$\y$}; +% Row 0 +\node at ( 2.25,-2.25) {$0$}; +\node at ( 3.75,-2.25) {$1$}; +\node at ( 5.25,-2.25) {$2$}; +\node at ( 6.75,-2.25) {$3$}; +\node at ( 8.25,-2.25) {$4$}; +\node at ( 9.75,-2.25) {$5$}; +\node at (11.25,-2.25) {$6$}; +\node at (12.75,-2.25) {$7$}; +\node at (14.25,-2.25) {$8$}; +\node at (15.75,-2.25) {$9$}; +\node at (17.25,-2.25) {$10$}; +% Row 1 +\node at ( 2.25,-3.75) {$1$}; +\node at ( 3.75,-3.75) {$2$}; +\node at ( 5.25,-3.75) {$3$}; +\node at ( 6.75,-3.75) {$4$}; +\node at ( 8.25,-3.75) {$5$}; +\node at ( 9.75,-3.75) {$6$}; +\node at (11.25,-3.75) {$7$}; +\node at (12.75,-3.75) {$8$}; +\node at (14.25,-3.75) {$9$}; +\node at (15.75,-3.75) {$10$}; +\node at (17.25,-3.75) {$0$}; +% Row 2 +\node at ( 2.25,-5.25) {$2$}; +\node at ( 3.75,-5.25) {$3$}; +\node at ( 5.25,-5.25) {$4$}; +\node at ( 6.75,-5.25) {$5$}; +\node at ( 8.25,-5.25) {$6$}; +\node at ( 9.75,-5.25) {$7$}; +\node at (11.25,-5.25) {$8$}; +\node at (12.75,-5.25) {$9$}; +\node at (14.25,-5.25) {$10$}; +\node at (15.75,-5.25) {$0$}; +\node at (17.25,-5.25) {$1$}; +% Row 3 +\node at ( 2.25,-6.75) {$3$}; +\node at ( 3.75,-6.75) {$4$}; +\node at ( 5.25,-6.75) {$5$}; +\node at ( 6.75,-6.75) {$6$}; +\node at ( 8.25,-6.75) {$7$}; +\node at ( 9.75,-6.75) {$8$}; +\node at (11.25,-6.75) {$9$}; +\node at (12.75,-6.75) {$10$}; +\node at (14.25,-6.75) {$0$}; +\node at (15.75,-6.75) {$1$}; +\node at (17.25,-6.75) {$2$}; +% Row 4 +\node at ( 2.25,-8.25) {$4$}; +\node at ( 3.75,-8.25) {$5$}; +\node at ( 5.25,-8.25) {$6$}; +\node at ( 6.75,-8.25) {$7$}; +\node at ( 8.25,-8.25) {$8$}; +\node at ( 9.75,-8.25) {$9$}; +\node at (11.25,-8.25) {$10$}; +\node at (12.75,-8.25) {$0$}; +\node at (14.25,-8.25) {$1$}; +\node at (15.75,-8.25) {$2$}; +\node at (17.25,-8.25) {$3$}; +% Row 5 +\node at ( 2.25,-9.75) {$5$}; +\node at ( 3.75,-9.75) {$6$}; +\node at ( 5.25,-9.75) {$7$}; +\node at ( 6.75,-9.75) {$8$}; +\node at ( 8.25,-9.75) {$9$}; +\node at ( 9.75,-9.75) {$10$}; +\node at (11.25,-9.75) {$0$}; +\node at (12.75,-9.75) {$1$}; +\node at (14.25,-9.75) {$2$}; +\node at (15.75,-9.75) {$3$}; +\node at (17.25,-9.75) {$4$}; +% Row 6 +\node at ( 2.25,-11.25) {$6$}; +\node at ( 3.75,-11.25) {$7$}; +\node at ( 5.25,-11.25) {$8$}; +\node at ( 6.75,-11.25) {$9$}; +\node at ( 8.25,-11.25) {$10$}; +\node at ( 9.75,-11.25) {$0$}; +\node at (11.25,-11.25) {$1$}; +\node at (12.75,-11.25) {$2$}; +\node at (14.25,-11.25) {$3$}; +\node at (15.75,-11.25) {$4$}; +\node at (17.25,-11.25) {$5$}; +% Row 7 +\node at ( 2.25,-12.75) {$7$}; +\node at ( 3.75,-12.75) {$8$}; +\node at ( 5.25,-12.75) {$9$}; +\node at ( 6.75,-12.75) {$10$}; +\node at ( 8.25,-12.75) {$0$}; +\node at ( 9.75,-12.75) {$1$}; +\node at (11.25,-12.75) {$2$}; +\node at (12.75,-12.75) {$3$}; +\node at (14.25,-12.75) {$4$}; +\node at (15.75,-12.75) {$5$}; +\node at (17.25,-12.75) {$6$}; +% Row 8 +\node at ( 2.25,-14.25) {$8$}; +\node at ( 3.75,-14.25) {$9$}; +\node at ( 5.25,-14.25) {$10$}; +\node at ( 6.75,-14.25) {$0$}; +\node at ( 8.25,-14.25) {$1$}; +\node at ( 9.75,-14.25) {$2$}; +\node at (11.25,-14.25) {$3$}; +\node at (12.75,-14.25) {$4$}; +\node at (14.25,-14.25) {$5$}; +\node at (15.75,-14.25) {$6$}; +\node at (17.25,-14.25) {$7$}; +% Row 9 +\node at ( 2.25,-15.75) {$9$}; +\node at ( 3.75,-15.75) {$10$}; +\node at ( 5.25,-15.75) {$0$}; +\node at ( 6.75,-15.75) {$1$}; +\node at ( 8.25,-15.75) {$2$}; +\node at ( 9.75,-15.75) {$3$}; +\node at (11.25,-15.75) {$4$}; +\node at (12.75,-15.75) {$5$}; +\node at (14.25,-15.75) {$6$}; +\node at (15.75,-15.75) {$7$}; +\node at (17.25,-15.75) {$8$}; +% Row 10 +\node at ( 2.25,-17.25) {$10$}; +\node at ( 3.75,-17.25) {$0$}; +\node at ( 5.25,-17.25) {$1$}; +\node at ( 6.75,-17.25) {$2$}; +\node at ( 8.25,-17.25) {$3$}; +\node at ( 9.75,-17.25) {$4$}; +\node at (11.25,-17.25) {$5$}; +\node at (12.75,-17.25) {$6$}; +\node at (14.25,-17.25) {$7$}; +\node at (15.75,-17.25) {$8$}; +\node at (17.25,-17.25) {$9$}; +\end{tikzpicture} + \end{center} -\end{figure} \ No newline at end of file diff --git a/buch/papers/reedsolomon/restetabelle2.tex b/buch/papers/reedsolomon/restetabelle2.tex index 887c981..1a9815c 100644 --- a/buch/papers/reedsolomon/restetabelle2.tex +++ b/buch/papers/reedsolomon/restetabelle2.tex @@ -1,24 +1,176 @@ % created by Michael Steiner % % Restetabelle von F_11: Multiplikation -\begin{figure} + +% alternatives design +%\begin{figure} +%\begin{center} +%\begin{tabular}{|>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}|} +%\hline +%\cdot&0&1&2&3&4&5&6&7&8&9&10\\ +%\hline +%0&0&0&0&0&0&0&0&0&0&0&0\\ +%1&0&1&2&3&4&5&6&7&8&9&10\\ +%2&0&2&4&6&8&10&1&3&5&7&9\\ +%3&0&3&6&9&1&4&7&10&2&5&8\\ +%4&0&4&8&1&5&9&2&6&10&3&7\\ +%5&0&5&10&4&9&3&8&2&7&1&6\\ +%6&0&6&1&7&2&8&3&9&4&10&5\\ +%7&0&7&3&10&6&2&9&5&1&8&4\\ +%8&0&8&5&2&10&7&4&1&9&6&3\\ +%9&0&9&7&5&3&1&10&8&6&4&2\\ +%10&0&10&9&8&7&6&5&4&3&2&1\\ +%\hline +%\end{tabular} +%\end{center} +%\end{figure} + \begin{center} -\begin{tabular}{|>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}|} -\hline -\cdot&0&1&2&3&4&5&6&7&8&9&10\\ -\hline -0&0&0&0&0&0&0&0&0&0&0&0\\ -1&0&1&2&3&4&5&6&7&8&9&10\\ -2&0&2&4&6&8&10&1&3&5&7&9\\ -3&0&3&6&9&1&4&7&10&2&5&8\\ -4&0&4&8&1&5&9&2&6&10&3&7\\ -5&0&5&10&4&9&3&8&2&7&1&6\\ -6&0&6&1&7&2&8&3&9&4&10&5\\ -7&0&7&3&10&6&2&9&5&1&8&4\\ -8&0&8&5&2&10&7&4&1&9&6&3\\ -9&0&9&7&5&3&1&10&8&6&4&2\\ -10&0&10&9&8&7&6&5&4&3&2&1\\ -\hline -\end{tabular} -\end{center} -\end{figure} \ No newline at end of file + + \begin{tikzpicture}[>=latex,thick,scale=0.45] + \fill[color=gray!40] (0,0) rectangle (18,-1.5); + \fill[color=gray!40] (0,0) rectangle (1.5,-18); + \draw[step = 1.5, gray,very thin] (0,0) grid (18,-18); + \draw[very thick] (0,0) rectangle (18,-18); + \draw[very thick] (0,-1.5) -- (18,-1.5); + \draw[very thick] (1.5,0) -- (1.5,-18); + \node at (0.75,-0.75) {$\cdot$}; + \foreach \x in {0,...,10} + \node at (2.25+\x*1.5,-0.75) {$\x$}; + \foreach \y in {0,...,10} + \node at (0.75,-2.25+\y*-1.5) {$\y$}; + % Row 0 + \node at ( 2.25,-2.25) {$0$}; + \node at ( 3.75,-2.25) {$0$}; + \node at ( 5.25,-2.25) {$0$}; + \node at ( 6.75,-2.25) {$0$}; + \node at ( 8.25,-2.25) {$0$}; + \node at ( 9.75,-2.25) {$0$}; + \node at (11.25,-2.25) {$0$}; + \node at (12.75,-2.25) {$0$}; + \node at (14.25,-2.25) {$0$}; + \node at (15.75,-2.25) {$0$}; + \node at (17.25,-2.25) {$0$}; + % Row 1 + \node at ( 2.25,-3.75) {$0$}; + \node at ( 3.75,-3.75) {$1$}; + \node at ( 5.25,-3.75) {$2$}; + \node at ( 6.75,-3.75) {$3$}; + \node at ( 8.25,-3.75) {$4$}; + \node at ( 9.75,-3.75) {$5$}; + \node at (11.25,-3.75) {$6$}; + \node at (12.75,-3.75) {$7$}; + \node at (14.25,-3.75) {$8$}; + \node at (15.75,-3.75) {$9$}; + \node at (17.25,-3.75) {$10$}; + % Row 2 + \node at ( 2.25,-5.25) {$0$}; + \node at ( 3.75,-5.25) {$2$}; + \node at ( 5.25,-5.25) {$4$}; + \node at ( 6.75,-5.25) {$6$}; + \node at ( 8.25,-5.25) {$8$}; + \node at ( 9.75,-5.25) {$10$}; + \node at (11.25,-5.25) {$1$}; + \node at (12.75,-5.25) {$3$}; + \node at (14.25,-5.25) {$5$}; + \node at (15.75,-5.25) {$7$}; + \node at (17.25,-5.25) {$9$}; + % Row 3 + \node at ( 2.25,-6.75) {$0$}; + \node at ( 3.75,-6.75) {$3$}; + \node at ( 5.25,-6.75) {$6$}; + \node at ( 6.75,-6.75) {$9$}; + \node at ( 8.25,-6.75) {$1$}; + \node at ( 9.75,-6.75) {$4$}; + \node at (11.25,-6.75) {$7$}; + \node at (12.75,-6.75) {$10$}; + \node at (14.25,-6.75) {$2$}; + \node at (15.75,-6.75) {$5$}; + \node at (17.25,-6.75) {$8$}; + % Row 4 + \node at ( 2.25,-8.25) {$0$}; + \node at ( 3.75,-8.25) {$4$}; + \node at ( 5.25,-8.25) {$8$}; + \node at ( 6.75,-8.25) {$1$}; + \node at ( 8.25,-8.25) {$5$}; + \node at ( 9.75,-8.25) {$9$}; + \node at (11.25,-8.25) {$2$}; + \node at (12.75,-8.25) {$6$}; + \node at (14.25,-8.25) {$10$}; + \node at (15.75,-8.25) {$3$}; + \node at (17.25,-8.25) {$7$}; + % Row 5 + \node at ( 2.25,-9.75) {$0$}; + \node at ( 3.75,-9.75) {$5$}; + \node at ( 5.25,-9.75) {$10$}; + \node at ( 6.75,-9.75) {$4$}; + \node at ( 8.25,-9.75) {$9$}; + \node at ( 9.75,-9.75) {$3$}; + \node at (11.25,-9.75) {$8$}; + \node at (12.75,-9.75) {$2$}; + \node at (14.25,-9.75) {$7$}; + \node at (15.75,-9.75) {$1$}; + \node at (17.25,-9.75) {$6$}; + % Row 6 + \node at ( 2.25,-11.25) {$0$}; + \node at ( 3.75,-11.25) {$6$}; + \node at ( 5.25,-11.25) {$1$}; + \node at ( 6.75,-11.25) {$7$}; + \node at ( 8.25,-11.25) {$2$}; + \node at ( 9.75,-11.25) {$8$}; + \node at (11.25,-11.25) {$3$}; + \node at (12.75,-11.25) {$9$}; + \node at (14.25,-11.25) {$4$}; + \node at (15.75,-11.25) {$10$}; + \node at (17.25,-11.25) {$5$}; + % Row 7 + \node at ( 2.25,-12.75) {$0$}; + \node at ( 3.75,-12.75) {$7$}; + \node at ( 5.25,-12.75) {$3$}; + \node at ( 6.75,-12.75) {$10$}; + \node at ( 8.25,-12.75) {$6$}; + \node at ( 9.75,-12.75) {$2$}; + \node at (11.25,-12.75) {$9$}; + \node at (12.75,-12.75) {$5$}; + \node at (14.25,-12.75) {$1$}; + \node at (15.75,-12.75) {$8$}; + \node at (17.25,-12.75) {$4$}; + % Row 8 + \node at ( 2.25,-14.25) {$0$}; + \node at ( 3.75,-14.25) {$8$}; + \node at ( 5.25,-14.25) {$5$}; + \node at ( 6.75,-14.25) {$2$}; + \node at ( 8.25,-14.25) {$10$}; + \node at ( 9.75,-14.25) {$7$}; + \node at (11.25,-14.25) {$4$}; + \node at (12.75,-14.25) {$1$}; + \node at (14.25,-14.25) {$9$}; + \node at (15.75,-14.25) {$6$}; + \node at (17.25,-14.25) {$3$}; + % Row 9 + \node at ( 2.25,-15.75) {$0$}; + \node at ( 3.75,-15.75) {$9$}; + \node at ( 5.25,-15.75) {$7$}; + \node at ( 6.75,-15.75) {$5$}; + \node at ( 8.25,-15.75) {$3$}; + \node at ( 9.75,-15.75) {$1$}; + \node at (11.25,-15.75) {$10$}; + \node at (12.75,-15.75) {$8$}; + \node at (14.25,-15.75) {$6$}; + \node at (15.75,-15.75) {$4$}; + \node at (17.25,-15.75) {$2$}; + % Row 10 + \node at ( 2.25,-17.25) {$0$}; + \node at ( 3.75,-17.25) {$10$}; + \node at ( 5.25,-17.25) {$9$}; + \node at ( 6.75,-17.25) {$8$}; + \node at ( 8.25,-17.25) {$7$}; + \node at ( 9.75,-17.25) {$6$}; + \node at (11.25,-17.25) {$5$}; + \node at (12.75,-17.25) {$4$}; + \node at (14.25,-17.25) {$3$}; + \node at (15.75,-17.25) {$2$}; + \node at (17.25,-17.25) {$1$}; + \end{tikzpicture} + +\end{center} \ No newline at end of file -- cgit v1.2.1 From d15eaa234f3f1622289e2486db54fe0ce7309b8f Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Thu, 10 Jun 2021 18:22:35 +0200 Subject: nachschlagewerk created --- buch/papers/reedsolomon/decmitfehler.tex | 3 ++- buch/papers/reedsolomon/main.tex | 2 +- buch/papers/reedsolomon/nachschlagewerk.tex | 4 ++++ 3 files changed, 7 insertions(+), 2 deletions(-) create mode 100644 buch/papers/reedsolomon/nachschlagewerk.tex (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/decmitfehler.tex b/buch/papers/reedsolomon/decmitfehler.tex index db6e586..feaa027 100644 --- a/buch/papers/reedsolomon/decmitfehler.tex +++ b/buch/papers/reedsolomon/decmitfehler.tex @@ -97,7 +97,8 @@ Bevor wir unser Lokatorpolynom berechnen können, müssen wir zuerst eine Mögli $d(X)$ & $=$ & $r(X) - m(X)$. \end{tabular} \end{center} -TODO (rewrite sentence): Dies wird uns zwar andere sorgen wegen $m(X)$ bereiten, \textcolor{red}{die werden wir jedoch zu einem späteren Zeitpunkt betrachten (todo: verweis auf kapitel?)}. +Dies wird uns zwar andere sorgen wegen $m(X)$ bereiten, wir werden werden deshalb erst in Abschnitt \ref{reedsolomon:subsection:nachrichtenvektor} darauf zurückkommen. + Setzen wir jetzt noch unsere Einheitswurzel aus dem Beispiel ein so erhalten wir % Old Text %\begin{align} diff --git a/buch/papers/reedsolomon/main.tex b/buch/papers/reedsolomon/main.tex index 9822d25..fa20936 100644 --- a/buch/papers/reedsolomon/main.tex +++ b/buch/papers/reedsolomon/main.tex @@ -39,8 +39,8 @@ Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren \input{papers/reedsolomon/decohnefehler} \input{papers/reedsolomon/decmitfehler} \input{papers/reedsolomon/rekonstruktion} +\input{papers/reedsolomon/nachschlagewerk} \input{papers/reedsolomon/hilfstabellen} -%\input{papers/reedsolomon/glossar} -> geplant zur besseren orientierung %\input{papers/reedsolomon/anwendungen} -> geplant \nocite{reedsolomon:weitz} diff --git a/buch/papers/reedsolomon/nachschlagewerk.tex b/buch/papers/reedsolomon/nachschlagewerk.tex new file mode 100644 index 0000000..60b857e --- /dev/null +++ b/buch/papers/reedsolomon/nachschlagewerk.tex @@ -0,0 +1,4 @@ +\section{Nachschlagewerk + \label{reedsolomon:section:nachschlagen}} +\rhead{nachschlagewerk} +todo: auflistung von z.b nachrichtenvektor, übertragungsvektor usw. inklusiver erklärung was es ist falls man beim lesen den faden verliert \ No newline at end of file -- cgit v1.2.1 From 09ca369b5a078dae6d55cc21e85452ac04a4a939 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 11 Jun 2021 08:30:04 +0200 Subject: Fix references.bib --- buch/papers/reedsolomon/references.bib | 31 ------------------------------- 1 file changed, 31 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/references.bib b/buch/papers/reedsolomon/references.bib index 4c1d17a..731bd35 100644 --- a/buch/papers/reedsolomon/references.bib +++ b/buch/papers/reedsolomon/references.bib @@ -13,8 +13,6 @@ day = {10} } -% https://link.springer.com/chapter/10.1007%2F978-3-8351-9077-1_9 - @book{reedsolomon:informationkommunikation, title = {Information und Kommunikation}, author = {Markus Hufschmid}, @@ -25,32 +23,3 @@ volume = {1} } -% Beispiele -%@online{reedsolomon:bibtex, -% title = {BibTeX}, -% url = {https://de.wikipedia.org/wiki/BibTeX}, -% date = {2020-02-06}, -% year = {2020}, -% month = {2}, -% day = {6} -%} -% -%@book{reedsolomon:numerical-analysis, -% title = {Numerical Analysis}, -% author = {David Kincaid and Ward Cheney}, -% publisher = {American Mathematical Society}, -% year = {2002}, -% isbn = {978-8-8218-4788-6}, -% inseries = {Pure and applied undegraduate texts}, -% volume = {2} -%} -% -%@article{reedsolomon:mendezmueller, -% author = { Tabea Méndez and Andreas Müller }, -% title = { Noncommutative harmonic analysis and image registration }, -% journal = { Appl. Comput. Harmon. Anal.}, -% year = 2019, -% volume = 47, -% pages = {607--627}, -% url = {https://doi.org/10.1016/j.acha.2017.11.004} -%} \ No newline at end of file -- cgit v1.2.1 From 99d2ddf90c75e83fc8ee82f5d0145a17db9a6338 Mon Sep 17 00:00:00 2001 From: Alain Date: Sun, 13 Jun 2021 15:59:24 +0200 Subject: minor changes, refernezen --- buch/papers/ifs/main.tex | 1 + buch/papers/ifs/references.bib | 48 +++++++++++++++++++++++++++++++++++------- buch/papers/ifs/teil0.tex | 2 +- buch/papers/ifs/teil1.tex | 18 +++++++--------- buch/papers/ifs/teil2.tex | 20 +++++++++--------- buch/papers/ifs/teil3.tex | 14 ++++++------ 6 files changed, 67 insertions(+), 36 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/ifs/main.tex b/buch/papers/ifs/main.tex index 8ae0fad..cceaf87 100644 --- a/buch/papers/ifs/main.tex +++ b/buch/papers/ifs/main.tex @@ -13,5 +13,6 @@ \input{papers/ifs/teil2.tex} \input{papers/ifs/teil3.tex} + \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/ifs/references.bib b/buch/papers/ifs/references.bib index 716857f..790c15c 100644 --- a/buch/papers/ifs/references.bib +++ b/buch/papers/ifs/references.bib @@ -13,14 +13,29 @@ day = {6} } -@book{ifs:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} +@online{ifs:chaos, + title = {Chaosspiel}, + url = {https://de.wikipedia.org/wiki/Iteriertes_Funktionensystem#Chaosspiel}, + date = {20201-06-13}, + year = {2021}, + month = {6}, + day = {13} +} + +@online{ifs:barnsleyfern, + title = {Barnsley fern}, + url = {https://en.wikipedia.org/wiki/Barnsley_fern}, + date = {20201-06-13}, + year = {2021}, + month = {6}, + day = {13} +} +@book{ifs:fractal-geometry, + title = {Fractal Geometry}, + author = {Kenneth Falconer}, + publisher = {John Wiley & Sons}, + year = {1900}, + isbn = {0-471-92287-0}, } @article{ifs:mendezmueller, @@ -33,3 +48,20 @@ url = {https://doi.org/10.1016/j.acha.2017.11.004} } +@Inbook{ifs:Rousseau2012, + author= {Rousseau, Christiane + and Saint-Aubin, Yvan + and Stern, Manfred}, + title={Bildkompression: Iterierte Funktionensysteme}, + bookTitle={Mathematik und Technologie}, + year={2012}, + publisher={Springer Berlin Heidelberg}, + address={Berlin, Heidelberg}, + pages={341--386}, + abstract={Dieses Kapitel kann in ein bis zwei Wochen Vorlesungen behandelt werden. Steht nur eine Woche zur Verfugung, dann konnen Sie kurz die Einfuhrung behandeln (Abschnitt 11.1) und anschlie{\ss}end ausf{\"u}hrlich den Begriff des Attraktors eines iterierten Funktionensystems betrachten (Abschnitt 11.3), wobei Sie sich auf das Sierpi{\'{n}}ski- Dreieck (Beispiel 11.5) konzentrieren. Beweisen Sie den Satz {\"u}ber die Konstruktion von affinen Transformationen, die drei Punkte der Ebene auf drei Punkte der Ebene abbilden und diskutieren Sie die speziellen affinen Transformationen, die h{\"a}ufig bei iterierten Funktionensystemen verwendet werden (Abschnitt 11.2).}, + isbn={978-3-642-30092-9}, + doi={10.1007/978-3-642-30092-9_11}, + url={https://doi.org/10.1007/978-3-642-30092-9_11} +} + + diff --git a/buch/papers/ifs/teil0.tex b/buch/papers/ifs/teil0.tex index d61c013..7cb218f 100644 --- a/buch/papers/ifs/teil0.tex +++ b/buch/papers/ifs/teil0.tex @@ -7,6 +7,6 @@ \rhead{Was ist ein Iteriertes Funktionsschema} Mit der Hilfe von Iterierten Funktionsschemata mit nur wenigen Funktionen, komplexe Bilder beschreiben. In der Regel sind diese Bilder Fraktale. -Wie es dazu kommt, und wie man mit IFS auch Bilder komprimieren kann, wollen wir im folgenden Kapitel untersuchen. +Wie es dazu kommt, und wie man mit IFS auch Bilder komprimieren kann, wollen wir in diesem Kapitel untersuchen. diff --git a/buch/papers/ifs/teil1.tex b/buch/papers/ifs/teil1.tex index f02aff6..54089ec 100644 --- a/buch/papers/ifs/teil1.tex +++ b/buch/papers/ifs/teil1.tex @@ -8,10 +8,9 @@ \rhead{Problemstellung} Bevor wir die IFS genauer ansehen, schauen wir uns Fraktale genauer an. -\subsection{Was sind Fraktale? -\label{ifs:subsection:finibus}} -Über die genaue Definition von Fraktalen sind sich die Mathematiker noch nicht einig. -In diesem Kapitel orientieren wir uns an den Eigenschaften welche Kenneth Falconer in seinem Buch Fractal Geometry beschreibt. + +Über die genaue Definition von Fraktalen sind sich die Mathematiker nicht einig. +In diesem Kapitel orientieren wir uns an den Eigenschaften welche Kenneth Falconer in seinem Buch Fractal Geometry \cite{ifs:fractal-geometry} beschreibt. Von einem Fraktal $F$ können wir folgende Eigenschaften erwarten: \begin{enumerate} \item $F$ hat eine unendlich feine Struktur @@ -23,8 +22,8 @@ Von einem Fraktal $F$ können wir folgende Eigenschaften erwarten: \subsection{Koch Kurve \label{ifs:subsection:lilkoch}} Diese Eigenschaften möchten wir nun anhand der Koch Kurve näher anschauen. -In \ref{ifs:kochkurve8} sehen wir die Koch Kurve. Wie man schon erahnen kann, besteht die aus lauter kleineren Kopien von sich selber. -Den Konstruktionsvorgang sehen wir in \ref{ifs:kochconst}. +In \ref{ifs:kochkurve8} sehen wir die Koch Kurve. Wie man schon erahnen kann, besteht sie aus lauter kleineren Kopien von sich selber. +Den Konstruktionsvorgang ist in Abbildung \ref{ifs:kochconst} dargestellt. Gestartet wird mit einer einzelnen Strecke der Länge $a$. Diese wird in ersten Schritt mit vier gleich langen Streckenabschnitte der Länge $\frac{a}{3}$ ersetzt. In \ref{ifs:kochconstb} ist die Anordnung dieser vier Streckenabschnitte ersichtlich. @@ -33,14 +32,13 @@ Die Kurve besteht also aus vier kleineren Kopien von der ganzen Kurve, was auch \begin{figure} - \label{ifs:kochkurve8} \centering \includegraphics{papers/ifs/images/koch8} \caption{Koch Kurve} + \label{ifs:kochkurve8} \end{figure} \begin{figure} - \label{ifs:kochconst} \centering \subfigure[]{ \label{ifs:kochconsta} @@ -52,7 +50,7 @@ Die Kurve besteht also aus vier kleineren Kopien von der ganzen Kurve, was auch \label{kochconstc} \includegraphics[width=0.32\textwidth]{papers/ifs/images/koch2}} \caption{(a) Start (b) 1. Iteration (c) 2. Iteration} - \label{fig:foobar} + \label{ifs:kochconst} \end{figure} Die resultierende Kurve hat ein paar interessante Eigenschaften. @@ -80,7 +78,7 @@ Wie wir sehen ist die Kochkurve ein Konstrukt mit endlicher Fläche, aber unendl Zu guter Letzt bestimmen wir die Dimension der Kurve. Es gibt viele verschiedene Arten die Dimension zu definieren. Diese können dann auch unterschiedliche Resultate liefern. Vor allem im Zusammenhang mit Fraktalen findet man in der Literatur viele verschiedene Arten. -In diesem Beispiel werden wir die Ähnlichkeits-Dimension. +In diesem Beispiel werden wir die Ähnlichkeits-Dimension \cite{ifs:fractal-geometry}. \begin{align*} D = - \frac{log(N)}{log(\epsilon)} \end{align*} diff --git a/buch/papers/ifs/teil2.tex b/buch/papers/ifs/teil2.tex index d25004f..143317a 100644 --- a/buch/papers/ifs/teil2.tex +++ b/buch/papers/ifs/teil2.tex @@ -9,10 +9,10 @@ Wollen wir nun eine bestimmte Art anschauen, wie man Fraktale machen kann. Zur Veranschaulichung dieser Methode nehmen wir das Sierpinski Dreieck. \begin{figure} - \label{ifs:sierpinski10} \centering \includegraphics[width=0.5\textwidth]{papers/ifs/images/sierpinski} \caption{Sierpinski-Dreieck} + \label{ifs:sierpinski10} \end{figure} Wenn man das Dreieck genau anschaut, erkennt man schnell, dass es aus drei kleineren Kopien seiner selbst besteht. Es ist also ein Selbstähnliches Konstrukt. @@ -71,8 +71,7 @@ Wendet man alle drei Funktionen auf das Sierpinski-Dreieck an, entsteht also wie X = \bigcup\limits_{i = 1}^{3} f_i(X) \end{align*} Man kann sogar noch einen Schritt weiter gehen, und sagen: Wenn wir die Funktionen auf eine beliebige Startmenge anwenden, konvergiert die Menge gegen das Sierpinski-Dreieck. -\begin{figure} - \label{ifs:sierpconst} +\begin{figure} \centering \subfigure[]{ \label{ifs:sierpconsta} @@ -88,6 +87,7 @@ Man kann sogar noch einen Schritt weiter gehen, und sagen: Wenn wir die Funktion \includegraphics[width=0.25\textwidth]{papers/ifs/images/sierpinski6}} \caption{Konstruktion eines Sierpinski-Dreiecks mit einem Schwarzen Quadrat als Start\\ (a) 1. Iteration (b) 2. Iteration (c) 3. Iteration (d) 5. Iteration} + \label{ifs:sierpconst} \end{figure} Im Beispiel der Abbildung \ref{ifs:sierpconst} sehen wir, wie das Bild nach jeder Iteration dem Sierpinski-Dreieck ähnlicher wird. Der Abstand zum Original wird immer kleiner, und konvergiert bei unendlich Iterationen gegen null. @@ -95,7 +95,7 @@ Der Abstand zum Original wird immer kleiner, und konvergiert bei unendlich Itera \subsection{Iterierte Funktionensysteme \label{ifs:subsection:bonorum}} In diesem Unterkapitel wollen wir die Erkenntnis, wie wir aus einer beliebigen Menge ein Sierpinski-Dreieck generieren können, verallgemeinern. -TODO TEXT + $S_1,...,S_n$ sind Kontraktionen auf die Menge $D \subset \mathbb{R}^n$. Es gilt \begin{align} @@ -185,26 +185,26 @@ Sie verkleinert und dreht das gesamte Bild und stellt es auf das Ende des Stiels $S_3$ bildet das gesamte Blatt auf das blaue Teilblatt unten Links ab. $S_4$ Spiegelt das Blatt und bildet es auf das magentafarbene Teilblatt ab. -Wir führen im Zusammenhang mit dem Barnsley-Farn noch eine weitere Methode ein, um IFS auszuführen. +Wir führen im Zusammenhang mit dem Barnsley-Farn \cite{ifs:barnsleyfern} noch eine weitere Methode ein, um IFS auszuführen. Bis jetzt wurde immer davon gesprochen, die Transformationen auf die gesamte Menge anzuwenden. Bei komplizierteren IFS welche viele Iterationen brauchen, bis man den Attraktor erkennen kann, ist diese Methode ziemlich rechenintensiv. -Eine Alternative ist das Chaos-Game. +Eine Alternative ist das Chaosspiel \cite{ifs:chaos}. Bei dieser Methode werden die Transformationen nicht auf die Menge angewendet, sondern nur auf einen einzelnen Punkt. Der Startpunkt kann dabei ein beliebiger Punkt in $E$ sein. Es wird bei jedem Iterationsschritt nur eine Transformation, welche zufällig gewählt wurde, angewendet. -Da, wie wir beim Barnsley-Farn gut sehen, dass nicht jede Transformation gleich viel des Bildes ausmacht, werden diese beim Chaos-Game gewichtet. +Da, wie wir beim Barnsley-Farn gut sehen, dass nicht jede Transformation gleich viel des Bildes ausmacht, werden diese beim Chaosspiel gewichtet. Die Gewichtung erfolgt über den Anteil der Gesamtmasse. Im Fall des Barnsley-Fern wird $S_1$ in $1\%$, $S_2$ in $85\%$ und $S_3 \& S_4$ in $7\%$ der Iterationen ausgeführt. -\begin{figure} - \label{ifs:farn} +\begin{figure} \centering \makebox[\textwidth][c]{ \includegraphics[width=1.4\textwidth]{papers/ifs/images/farn}} \caption{Barnsley-Farn} + \label{ifs:farn} \end{figure} \begin{figure} - \label{ifs:farncolor} \centering \includegraphics[width=0.7\textwidth]{papers/ifs/images/farncolor} \caption{Vier Transformationen des Barnsley-Farn} + \label{ifs:farncolor} \end{figure} diff --git a/buch/papers/ifs/teil3.tex b/buch/papers/ifs/teil3.tex index 515fd81..24f0751 100644 --- a/buch/papers/ifs/teil3.tex +++ b/buch/papers/ifs/teil3.tex @@ -9,7 +9,7 @@ Mit dem Prinzip dieser IFS ist es auch möglich Bilder zu Komprimieren. Diese Idee hatte der Mathematiker Michael Barnsley, welcher mit seinem Buch Fractals Everywhere einen wichtigen Beitrag zum Verständnis von Fraktalen geliefert hat. Das Ziel ist es ein IFS zu finden, welches das Bild als Attraktor hat. -In diesem Unterkapitel wollen wir eine Methode dafür anschauen. +In diesem Unterkapitel wollen wir eine Methode dafür anschauen.\cite{ifs:Rousseau2012} Bis jetzt wurde in Zusammenhang mit IFS immer erwähnt, dass die Transformationen auf die ganze Menge angewendet werden. @@ -17,10 +17,10 @@ Dies muss jedoch nicht so sein. Es gibt auch einen Attraktor, wenn die Transformationen nur Teile der Menge auf die ganze Menge abbilden. Diese Eigenschaft wollen wir uns in der Fraktalen Bildkompression zunutze machen. Sie ermöglicht uns Ähnlichkeiten zwischen kleineren Teilen des Bildes zunutze machen. -Es ist wohl nicht Falsch zu sagen, dass Ähnlichkeiten zur gesamten Menge, wie wir sie zum Beispiel beim Barnsley Fern gesehen haben, bei Bilder aus dem Alltag eher selten anzutreffen sind. +Es ist wohl nicht falsch zu sagen, dass Ähnlichkeiten zur gesamten Menge, wie wir sie zum Beispiel beim Barnsley Farn gesehen haben, bei Bilder aus dem Alltag eher selten anzutreffen sind. Doch wie Finden wir die richtigen Affinen Transformationen, welche als IFS das Bild als Attraktor haben? -\subsection{Titel +\subsection{das Kompressionsverfahren \label{ifs:subsection:malorum}} In der Beschreibung des Verfahrens wird sich auf Graustufenbilder bezogen. Wie das Verfahren für Farbbilder verwendet werden kann, wird später erläutert. @@ -114,21 +114,20 @@ Als Startbild wird ein mittelgraues 360x360px Bild gewählt, Abbildung \ref{ifs: Nun lassen wir das IFS laufen. Wie wir in Abbildung \ref{ifs:rappirecoa} sehen, ist schon nach der ersten Iteration das Bild schon erkennbar. Nach der fünften Iteration , Abbildung \ref{ifs:rappirecoc} gibt es fast keinen Unterschied mehr zur letzten Iteration, wir können die Rekonstruktion beenden. -\begin{figure} - \label{ifs:original} +\begin{figure} \centering \includegraphics[width=0.4\textwidth]{papers/ifs/images/original} \caption{Original Bild von Rapperswil} + \label{ifs:original} \end{figure} \begin{figure} - \label{ifs:bild0} \centering \includegraphics[width=0.4\textwidth]{papers/ifs/images/rapperswil} \caption{Startbild} + \label{ifs:bild0} \end{figure} \begin{figure} - \label{ifs:rappireco} \centering \subfigure[]{ \label{ifs:rappirecoa} @@ -140,4 +139,5 @@ Nach der fünften Iteration , Abbildung \ref{ifs:rappirecoc} gibt es fast keinen \label{ifs:rappirecoc} \includegraphics[width=0.32\textwidth]{papers/ifs/images/rapperswil04}} \caption{(a) 1. Iteration (b) 2. Iteration (c) 5. Iteration} + \label{ifs:rappireco} \end{figure} -- cgit v1.2.1 From e6f890beb3ad6030abc3f7082a7cd3ce0a8dabd8 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 14 Jun 2021 07:41:27 +0200 Subject: fix paper/ifs/references.bib --- buch/papers/ifs/references.bib | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/ifs/references.bib b/buch/papers/ifs/references.bib index 790c15c..fbf75f4 100644 --- a/buch/papers/ifs/references.bib +++ b/buch/papers/ifs/references.bib @@ -33,7 +33,7 @@ @book{ifs:fractal-geometry, title = {Fractal Geometry}, author = {Kenneth Falconer}, - publisher = {John Wiley & Sons}, + publisher = {John Wiley \& Sons}, year = {1900}, isbn = {0-471-92287-0}, } @@ -58,7 +58,7 @@ publisher={Springer Berlin Heidelberg}, address={Berlin, Heidelberg}, pages={341--386}, - abstract={Dieses Kapitel kann in ein bis zwei Wochen Vorlesungen behandelt werden. Steht nur eine Woche zur Verfugung, dann konnen Sie kurz die Einfuhrung behandeln (Abschnitt 11.1) und anschlie{\ss}end ausf{\"u}hrlich den Begriff des Attraktors eines iterierten Funktionensystems betrachten (Abschnitt 11.3), wobei Sie sich auf das Sierpi{\'{n}}ski- Dreieck (Beispiel 11.5) konzentrieren. Beweisen Sie den Satz {\"u}ber die Konstruktion von affinen Transformationen, die drei Punkte der Ebene auf drei Punkte der Ebene abbilden und diskutieren Sie die speziellen affinen Transformationen, die h{\"a}ufig bei iterierten Funktionensystemen verwendet werden (Abschnitt 11.2).}, + abstract={Dieses Kapitel kann in ein bis zwei Wochen Vorlesungen behandelt werden. Steht nur eine Woche zur Verfügung, dann können Sie kurz die Einführung behandeln (Abschnitt 11.1) und anschlie{\ss}end ausf{\"u}hrlich den Begriff des Attraktors eines iterierten Funktionensystems betrachten (Abschnitt 11.3), wobei Sie sich auf das Sierpi{\'{n}}ski- Dreieck (Beispiel 11.5) konzentrieren. Beweisen Sie den Satz {\"u}ber die Konstruktion von affinen Transformationen, die drei Punkte der Ebene auf drei Punkte der Ebene abbilden und diskutieren Sie die speziellen affinen Transformationen, die h{\"a}ufig bei iterierten Funktionensystemen verwendet werden (Abschnitt 11.2).}, isbn={978-3-642-30092-9}, doi={10.1007/978-3-642-30092-9_11}, url={https://doi.org/10.1007/978-3-642-30092-9_11} -- cgit v1.2.1 From 81d11a125976ab6c877b934cdeb79806a1105bca Mon Sep 17 00:00:00 2001 From: Alain Date: Fri, 18 Jun 2021 10:47:53 +0200 Subject: reworks --- buch/papers/ifs/images/farnnotweight.eps | 2179 ++++++++++++++++++++++++++++++ buch/papers/ifs/teil0.tex | 2 +- buch/papers/ifs/teil1.tex | 80 +- buch/papers/ifs/teil2.tex | 34 +- buch/papers/ifs/teil3.tex | 109 +- 5 files changed, 2331 insertions(+), 73 deletions(-) create mode 100644 buch/papers/ifs/images/farnnotweight.eps (limited to 'buch/papers') diff --git a/buch/papers/ifs/images/farnnotweight.eps b/buch/papers/ifs/images/farnnotweight.eps new file mode 100644 index 0000000..975c384 --- /dev/null +++ b/buch/papers/ifs/images/farnnotweight.eps @@ -0,0 +1,2179 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: (MATLAB, The Mathworks, Inc. 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b/buch/papers/ifs/teil0.tex index 7cb218f..b8a678d 100644 --- a/buch/papers/ifs/teil0.tex +++ b/buch/papers/ifs/teil0.tex @@ -5,7 +5,7 @@ % \section{Teil 0\label{ifs:section:teil0}} \rhead{Was ist ein Iteriertes Funktionsschema} -Mit der Hilfe von Iterierten Funktionsschemata mit nur wenigen Funktionen, komplexe Bilder beschreiben. +Mit der Hilfe von Iterierten Funktionsschemata (IFS) kann mit nur wenigen affinen Funktionen, komplexe Bilder beschreiben werden. In der Regel sind diese Bilder Fraktale. Wie es dazu kommt, und wie man mit IFS auch Bilder komprimieren kann, wollen wir in diesem Kapitel untersuchen. diff --git a/buch/papers/ifs/teil1.tex b/buch/papers/ifs/teil1.tex index 54089ec..68e2e44 100644 --- a/buch/papers/ifs/teil1.tex +++ b/buch/papers/ifs/teil1.tex @@ -15,20 +15,20 @@ Von einem Fraktal $F$ können wir folgende Eigenschaften erwarten: \begin{enumerate} \item $F$ hat eine unendlich feine Struktur \item $F$ kann nicht mit der klassischen Geometrie beschrieben werden. - \item Oftmals haf $F$ eine Form von Selbstähnlichkeit. - \item Die 'fraktale Dimension' ist grösser als die Topologische Dimension + \item Oftmals hat $F$ eine Form von Selbstähnlichkeit. + \item Die 'fraktale Dimension' ist grösser als die topologische Dimension \item Viele Fraktale lassen sich einfach beschrieben \end{enumerate} \subsection{Koch Kurve \label{ifs:subsection:lilkoch}} -Diese Eigenschaften möchten wir nun anhand der Koch Kurve näher anschauen. -In \ref{ifs:kochkurve8} sehen wir die Koch Kurve. Wie man schon erahnen kann, besteht sie aus lauter kleineren Kopien von sich selber. +Diese Eigenschaften möchten wir nun am Beispiel der Koch Kurve näher anschauen. +In Abbildung \ref{ifs:kochkurve8} sehen wir die Koch Kurve. Sie besteht aus lauter kleineren Kopien von sich selber. Den Konstruktionsvorgang ist in Abbildung \ref{ifs:kochconst} dargestellt. Gestartet wird mit einer einzelnen Strecke der Länge $a$. -Diese wird in ersten Schritt mit vier gleich langen Streckenabschnitte der Länge $\frac{a}{3}$ ersetzt. +Diese wird in ersten Schritt durch vier gleich langen Streckenabschnitte der Länge $\frac{a}{3}$ ersetzt. In \ref{ifs:kochconstb} ist die Anordnung dieser vier Streckenabschnitte ersichtlich. Dieser Schritt wird nun für jeden der resultierten Streckenabschnitten wiederholt. -Die Kurve besteht also aus vier kleineren Kopien von der ganzen Kurve, was auch unter Selbstähnlichkeit bekannt ist. +Die Kurve besteht also aus vier kleineren Kopien der ganzen Kurve, was auch unter Selbstähnlichkeit bekannt ist. \begin{figure} @@ -54,41 +54,79 @@ Die Kurve besteht also aus vier kleineren Kopien von der ganzen Kurve, was auch \end{figure} Die resultierende Kurve hat ein paar interessante Eigenschaften. -Die Länge der Kurve lasst sich einfach berechnen. +Die Länge der Kurve der jeweiligen Iteration lässt sich mit \begin{align*} - l_0 = a ,\quad l_1 = a \frac{4}{3} ,\quad l_2 = a \left( \frac{4}{3}\right)^2 , \quad ... , \quad - l_n = a * \left( \frac{4}{3}\right)^n \quad + l_0 = a ,\quad l_1 = a \frac{4}{3} ,\quad l_2 = a \left( \frac{4}{3}\right)^2 , \quad \cdots , \quad + l_n = a \cdot \left( \frac{4}{3}\right)^n \quad \Rightarrow \quad \lim_{n\to\infty} a \left( \frac{4}{3}\right)^n = \infty \end{align*} -In jedem Schritt wird die Länge um den Faktor $\frac{4}{3}$ verlängert. Somit divergiert die Länge gegen Unendlich. +beschreiben. +In jedem Schritt wird die Länge um den Faktor $\frac{4}{3}$ verlängert. Daraus resultiert, dass die Länge gegen $\infty$ divergiert. + + Die Fläche unter der Kurve lässt sich folgendermassen berechnen \begin{align*} - A_0 = 0 , \quad A_1 = \left( \frac{a}{3}\right)^2 \frac{\sqrt{3}}{4} = a^2 \frac{\sqrt{3}}{36}\\ + A_0 = 0 \\ + A_1 = \left( \frac{a}{3}\right)^2 \frac{\sqrt{3}}{4} = a^2 \frac{\sqrt{3}}{36}\\ A_2 = A_1 + 4\left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 \\ A_3 = A_1 + A_2 + 4^2 \left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 + \left( \frac{4}{9}\right)^2 A_1 \end{align*} Wir sehen, dass mit jedem Schritt die neu dazugekommene Fläche um $\frac{4}{9}$ kleiner ist. -Daraus resultiert eine konvergierende Geometrische Reihe. +Die Gesamtfläche ist daher gegeben durch die geometrische Reihe, \begin{align*} A_n = A_1 \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n = a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n \\ - \lim_{n\to\infty} a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n = \frac{\sqrt{3}}{20} a^2 \end{align*} -Wie wir sehen ist die Kochkurve ein Konstrukt mit endlicher Fläche, aber unendlichem Umfang. +mit dem Grenzwert +\begin{align*} + \lim_{n\to\infty} a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n = \frac{\sqrt{3}}{20} a^2. +\end{align*} +Wie wir sehen ist die Koch-Kurve eine Kurve mit endlicher Fläche, aber unendlicher Umfang. + + Zu guter Letzt bestimmen wir die Dimension der Kurve. -Es gibt viele verschiedene Arten die Dimension zu definieren. Diese können dann auch unterschiedliche Resultate liefern. -Vor allem im Zusammenhang mit Fraktalen findet man in der Literatur viele verschiedene Arten. +Es gibt viele verschiedene Methoden die Dimension zu definieren. Diese können dann auch unterschiedliche Resultate liefern. +Vor allem im Zusammenhang mit Fraktalen findet man in der Literatur unterschiedliche Arten. In diesem Beispiel werden wir die Ähnlichkeits-Dimension \cite{ifs:fractal-geometry}. +Die Ähnlichkeits-Dimension ist das Verhältnis der Logarithmen der Anzahl Kopien $N$ des Originales und deren Skalierungsfaktor $\epsilon$ + \begin{align*} - D = - \frac{log(N)}{log(\epsilon)} + D = - \frac{\log N}{\log \epsilon }. \end{align*} Mit ihr kann man einfach die Dimension selbstähnlicher Mengen bestimmen. -Als Beispiel nehmen wir ein gleichseitiges Dreieck. Dieses besteht aus $N = 4$ Kopien mit halber ($\epsilon = 1/2$) Kantenlänge. +Als Beispiel nehmen wir ein gleichseitiges Dreieck. Dieses besteht aus $N = 4$ Kopien mit halber ($\epsilon = 1/2$) Kantenlänge $l$, Abbildung \ref{ifs:trinagle}. Somit hat das Dreieck die Dimension $D = 2$. -Die Koch Kurve besteht aus $N = 4$ Kopien mit Kantenlänge $\epsilon = 1/3$. +Die Koch Kurve besteht aus $N = 4$ Kopien mit Kantenlänge $\epsilon =l \cdot 1/3$. \begin{align*} - D = - \frac{log(N)}{log(\epsilon)} = - \frac{log(4)}{log(1/3)} \approx 1.2619 + D = - \frac{\log N }{\log \epsilon } = - \frac{\log 4 }{\log 1/3 } \approx 1.2619 \end{align*} -Wie wir nun sehen besitzt die Kochkurve alle oben beschriebenen Eigenschaften von Fraktalen. +Wie wir nun sehen besitzt die Koch-Kurve alle oben beschriebenen Eigenschaften von Fraktalen. Dies muss jedoch nicht bei allen Fraktalen der Fall. Sonst wäre die Frage nach einer 'richtigen' Definition einfach zu beantworten. +\begin{figure} + \centering + \begin{tikzpicture} + + % draw the background + \draw [line width=1.5pt, fill=gray!2] (0,0) -- (60:4) -- (4,0) -- cycle; + + \coordinate[label=left:$A$] (A) at (0,0); + \coordinate[label=right:$B$] (B) at (4,0); + \coordinate[label=above:$C$] (C) at (2,3.464); + + \coordinate[label=below:$l$](c) at ($ (A)!.5!(B) $); + \coordinate[label=left:$l$] (b) at ($ (A)!.5!(C) $); + \coordinate[label=right:$l$](a) at ($ (B)!.5!(C) $); + + \coordinate[label=below:$l/2$](d) at ($ (b)!.5!(a)$); + + % the triangle + \draw [line width=1.5pt] (A) -- (B) -- (C) -- cycle; + \draw [line width=0.5pt] (a) -- (b); + \draw [line width=0.5pt] (a) -- (c); + \draw [line width=0.5pt] (c) -- (b); + + \end{tikzpicture} + \caption{Selbstähnlichkeit eines gleichseitigen Dreiecks} + \label{ifs:trinagle} +\end{figure} diff --git a/buch/papers/ifs/teil2.tex b/buch/papers/ifs/teil2.tex index 143317a..5de3d4b 100644 --- a/buch/papers/ifs/teil2.tex +++ b/buch/papers/ifs/teil2.tex @@ -14,13 +14,13 @@ Zur Veranschaulichung dieser Methode nehmen wir das Sierpinski Dreieck. \caption{Sierpinski-Dreieck} \label{ifs:sierpinski10} \end{figure} -Wenn man das Dreieck genau anschaut, erkennt man schnell, dass es aus drei kleineren Kopien seiner selbst besteht. -Es ist also ein Selbstähnliches Konstrukt. +Es besteht aus drei kleineren Kopien von sich selbst. +Es ist also ein Selbstähnliches Gebilde. Diese Eigenschaft wollen wir uns zunutze machen. Wir definieren das Dreieck mit Kantenlänge 1 als Menge $X$. -Ausserdem bestimmen wir drei Funktionen, welche die gesamte Menge auf eine ihrer kleineren Kopien abbildet +Ausserdem bestimmen wir drei Funktionen \begin{align*} f_1(x,y) = @@ -63,13 +63,15 @@ Ausserdem bestimmen wir drei Funktionen, welche die gesamte Menge auf eine ihrer \begin{pmatrix} \frac{1}{4} \\ \frac{1}{2} - \end{pmatrix}\\ + \end{pmatrix}, \end{align*} +welche die gesamte Menge auf eine ihrer kleineren Kopien abbildet $f_1$ bildet das Dreieck auf das Teilstück unten links ab, $f_2$ auf das Teilstück unten rechts und $f_3$ auf das obere Teilstück. -Wendet man alle drei Funktionen auf das Sierpinski-Dreieck an, entsteht also wieder ein Sierpinski-Dreieck. +Wendet man alle drei Funktionen auf das Sierpinski-Dreieck an \begin{align*} - X = \bigcup\limits_{i = 1}^{3} f_i(X) + X = \bigcup\limits_{i = 1}^{3} f_i(X), \end{align*} +entsteht also wieder ein Sierpinski-Dreieck. Man kann sogar noch einen Schritt weiter gehen, und sagen: Wenn wir die Funktionen auf eine beliebige Startmenge anwenden, konvergiert die Menge gegen das Sierpinski-Dreieck. \begin{figure} \centering @@ -90,11 +92,11 @@ Man kann sogar noch einen Schritt weiter gehen, und sagen: Wenn wir die Funktion \label{ifs:sierpconst} \end{figure} Im Beispiel der Abbildung \ref{ifs:sierpconst} sehen wir, wie das Bild nach jeder Iteration dem Sierpinski-Dreieck ähnlicher wird. -Der Abstand zum Original wird immer kleiner, und konvergiert bei unendlich Iterationen gegen null. +Der Abstand zum Original wird immer kleiner, und konvergiert gegen null. \subsection{Iterierte Funktionensysteme \label{ifs:subsection:bonorum}} -In diesem Unterkapitel wollen wir die Erkenntnis, wie wir aus einer beliebigen Menge ein Sierpinski-Dreieck generieren können, verallgemeinern. +In diesem Abschnitt wollen wir die Erkenntnis, wie wir aus einer beliebigen Menge ein Sierpinski-Dreieck generieren können, verallgemeinern. $S_1,...,S_n$ sind Kontraktionen auf die Menge $D \subset \mathbb{R}^n$. Es gilt @@ -114,10 +116,11 @@ Wird diese Transformation Iterativ ausgeführt, das heisst $S^0(E) = E, S^k(E) = F = \bigcap\limits_{k = 1}^{\infty} S^k(E). \end{equation} In Worte gefasst bedeutet das, dass jede Gruppe von Kontraktionen iterativ ausgeführt, gegen eine eindeutige Menge konvergiert. +Diese Menge ist auch als Attraktor des IFS bekannt. Dies für jede Startmenge, solange diese ihre Transformierten wieder beinhaltet. Auf den Beweis wird verzichtet. \subsection{Beispiel: Barnsley-Farn} -Der Barnsley-Farn, Abbildung \ref{ifs:farn}, ist ein weiteres Fraktal, welches mit einem IFS generiert werden kann. +Der Barnsley-Farn, Abbildung \ref{ifs:farn}, ist ein Beispiel eines Fraktal, welches mit einem IFS generiert werden kann. Wie man schnell erkennen kann, besteht der Farn aus Blättern, welche eine grosse Ähnlichkeit zum ganzen Farn haben. \begin{align*} {S_1(x,y)} @@ -183,9 +186,9 @@ Die Transformation bildet das Gesamte Blatt auf die Y-Achse ab. $S_2$ (grün) erstellt den Hauptteil des Farnes. Sie verkleinert und dreht das gesamte Bild und stellt es auf das Ende des Stiels aus $S_1$. $S_3$ bildet das gesamte Blatt auf das blaue Teilblatt unten Links ab. -$S_4$ Spiegelt das Blatt und bildet es auf das magentafarbene Teilblatt ab. - -Wir führen im Zusammenhang mit dem Barnsley-Farn \cite{ifs:barnsleyfern} noch eine weitere Methode ein, um IFS auszuführen. +$S_4$ spiegelt das Blatt und bildet es auf das magentafarbene Teilblatt ab. +\subsection{Chaosspiel} +Wir führen im Zusammenhang mit dem Barnsley-Farn \cite{ifs:barnsleyfern} noch eine weitere Methode ein, um ein IFS zu zeichnen. Bis jetzt wurde immer davon gesprochen, die Transformationen auf die gesamte Menge anzuwenden. Bei komplizierteren IFS welche viele Iterationen brauchen, bis man den Attraktor erkennen kann, ist diese Methode ziemlich rechenintensiv. Eine Alternative ist das Chaosspiel \cite{ifs:chaos}. @@ -208,3 +211,10 @@ Im Fall des Barnsley-Fern wird $S_1$ in $1\%$, $S_2$ in $85\%$ und $S_3 \& S_4$ \caption{Vier Transformationen des Barnsley-Farn} \label{ifs:farncolor} \end{figure} +\begin{figure} + \centering + \makebox[\textwidth][c]{ + \includegraphics[width=1.4\textwidth]{papers/ifs/images/farnnotweight}} + \caption{Chaosspiel ohne Gewichtung} + \label{ifs:farnNoWeight} +\end{figure} diff --git a/buch/papers/ifs/teil3.tex b/buch/papers/ifs/teil3.tex index 24f0751..39a808f 100644 --- a/buch/papers/ifs/teil3.tex +++ b/buch/papers/ifs/teil3.tex @@ -12,30 +12,35 @@ Das Ziel ist es ein IFS zu finden, welches das Bild als Attraktor hat. In diesem Unterkapitel wollen wir eine Methode dafür anschauen.\cite{ifs:Rousseau2012} -Bis jetzt wurde in Zusammenhang mit IFS immer erwähnt, dass die Transformationen auf die ganze Menge angewendet werden. +Bis jetzt wurde in Zusammenhang mit IFS immer erwähnt, dass die Transformationen, welche das IFS bilden, auf die gesamte Menge. Dies muss jedoch nicht so sein. Es gibt auch einen Attraktor, wenn die Transformationen nur Teile der Menge auf die ganze Menge abbilden. Diese Eigenschaft wollen wir uns in der Fraktalen Bildkompression zunutze machen. Sie ermöglicht uns Ähnlichkeiten zwischen kleineren Teilen des Bildes zunutze machen. Es ist wohl nicht falsch zu sagen, dass Ähnlichkeiten zur gesamten Menge, wie wir sie zum Beispiel beim Barnsley Farn gesehen haben, bei Bilder aus dem Alltag eher selten anzutreffen sind. -Doch wie Finden wir die richtigen Affinen Transformationen, welche als IFS das Bild als Attraktor haben? +Doch wie finden wir die richtigen affinen Transformationen, welche als IFS das Bild als Attraktor haben? \subsection{das Kompressionsverfahren \label{ifs:subsection:malorum}} -In der Beschreibung des Verfahrens wird sich auf Graustufenbilder bezogen. Wie das Verfahren für Farbbilder verwendet werden kann, wird später erläutert. - +Wir beschränken das Verfahren für Graustufenbilder. Wie das Verfahren für Farbbilder verwendet werden kann, wird später erläutert. +Ein Graustufenbild kann man als Pixelraster mit einer x und y Achse verstehen. +Jedem dieser Pixel wird ein Grauwert zugeordnet. +Ein Bild ist also eine Funktion, die jedem Pixel einen Grauwert $z$ zuweist +\begin{align*} + z = f(x,y). +\end{align*} In einem ersten Schritt teilen wir das Bild in disjunkte benachbarte $b \times b$ Pixel-Quadrate auf. Diese Blöcke nennen wir Range-Blöcke der Menge $R=\{R_0,R_1,...R_m\}$ Im nächsten Schritt teilen wir das Bild in alle möglichen $2b \times 2b$ Pixel-Quadrate auf. Diese sind die Domain-Blöcke der Menge $D = \{D_0,D_1,...D_n\}$. Im dritten und letzten Schritt wird für jeden Range-Block $R_i$ ein Domain-Block $D_j$ gesucht, welcher ihm am ähnlichsten ist. \subsubsection{Finden des ähnlichsten $D_j$} -Zuerst brauchen wir die Transformation um ein Element aus $D$ auf ein Element von $R$ Abzubilden. +Zuerst brauchen wir die Transformation \begin{align*} - T(x,y,z) = + T_i(x,y,z) = \begin{pmatrix} - a & b & 0 \\ - c & d & 0 \\ - 0 & 0 & s + a_i & b_i & 0 \\ + c_i & d_i & 0 \\ + 0 & 0 & s_i \end{pmatrix} \begin{pmatrix} x \\ @@ -44,52 +49,80 @@ Zuerst brauchen wir die Transformation um ein Element aus $D$ auf ein Element vo \end{pmatrix} + \begin{pmatrix} - \alpha \\ - \beta \\ - g + \alpha_i \\ + \beta_i \\ + g_i \end{pmatrix} \end{align*} -Diese Transformation bildet den Pixel $P$ auf Koordinate $(x,y)$ und Graustufe $z$ auf den Pixel $P'$ ab. - -Da wir mit Pixeln arbeiten, sind die Transformationen in der Ebene Beschränkt. -Diese wird durch die Parameter $a,b,c$ und $d$ bestimmt. -Mögliche Transformationen sind auf folgende Liste Beschränkt: +um ein Element aus $D$ auf ein Element von $R$ Abzubilden. +Wenn wir die Grauwerte ausser acht lassen, haben wir die affine Abbildung +\begin{align} + t_i(x,y) = + \begin{pmatrix} + a_i & b_i \\ + c_i & d_i + \end{pmatrix} + \begin{pmatrix} + x \\ + y + \end{pmatrix} + + + \begin{pmatrix} + \alpha_i \\ + \beta_i + \end{pmatrix}. +\label{ifs:affTrans} +\end{align} +Da wir mit Pixeln arbeiten, ist die Auswahl der möglichen Abbildungen begrenzt. +Wir sind auf folgende acht Abbildungen beschränkt: \begin{itemize} \item Identische Transformation, keine Änderung \item Drehung um 90, 180 oder 270 Grad. \item Spiegelung an der vertikalen, horizontalen und den Diagonalachsen. \end{itemize} -$\alpha$ und $\beta$ verschieben den Pixel an die richtige Stelle. Da wir ein $2b \times 2b$ Feld auf ein $b \times b$ Feld abbilden möchten, müssen wir zuerst $G_j$ um $1/2$ skalieren. Dies erreichen wir, indem wir alle disjunkten $2 \times 2$ px Blöcke mit einem Pixel des Grautones deren Mittelwertes ersetzen. -Skaliert und transformiert erhalten wir $\tilde{D_j}$ -Die Parameter $s$ und $g$ beschreiben die Änderung des Grautones. $s$ verändert den Kontrast und $g$ verschiebt die Töne auf die richtige Helligkeit. -$s$ und $g$ werden mit der linearen Regression ermittelt. + +Die Parameter $s_i$ und $g_i$ beschreiben die Änderung des Grautones. $s$ verändert den Kontrast und $g$ verschiebt die Töne auf die richtige Helligkeit, sie bilden die lineare Funktion +\begin{align*} + z' = s_i z + g_i. +\end{align*} +Für die Bestimmung dieser Parameter führen wir zuerst die Bildfunktionen $f_{R_i}$ und $\tilde{f_{R_i}}$ ein. +$f_{R_i}$ ist die Bildfunktion des Range-Blockes $R_i$ und $\tilde{f_{R_i}}$ ist die Bildfunktion des zuerst Skalierten und dann mit \ref{ifs:affTrans} transformierten Domain-Blocks $D_j$. +$s$ und $g$ werden mit der einfachen linearen Regression ermittelt. +Wir suchen $s_i$ und $g_i$ so das \begin{align*} - z' = sz + g \\ - f(\tilde{D_j}) \text{, Funktion um das Bild eins Blockes zu erhalten} \\ - s = \frac{cov(f(R_i), f(\tilde{D_j}))}{var(\tilde{D_j})} \\ - g = E(f(R_i)) - s E(f(\tilde{D_j})) + f_{R_i} = s_i \tilde{f_{R_i}} + g_i = \bar{f_{R_i}}. \end{align*} +Die Parameter lassen sich mit +\begin{align*} + s = \frac{\operatorname{cov}(f_{R_i}), f(\tilde{f_{R_i}}))}{\operatorname{var}(\tilde{f_{R_i}})} \\ + g = E(f_{R_i}) - s E(f(\tilde{f_{R_i}})) +\end{align*} +berechnen. Mit diesen Parametern haben wir nun die Transformation vollständig bestimmt. -Um zu beurteilen ob der Domain-Block $D_j$ mit der gefundenen Transformation $T$ dem Range-Block $R_i$ genügend ähnlich ist, berechnet man den quadratischen Abstand $e$. +Um zu beurteilen wie ähnlich der Domain-Block $D_j$ mit der gefundenen Transformation $T$ dem Range-Block ist, berechnet man den quadratischen Abstand \begin{align*} - e = d(f(R_i), f(T(D_j))) + e = d(f_{R_i}, \bar{f_{R_i}}). \end{align*} Dieser Abstand sollte so klein wie möglich sein. -Die beste Kombination von $D_j$ und $T_i$ ist also diese, welche den kleinsten Abstand zum Block $R_i$ hat, und somit am ähnlichsten ist. -Am Ende des Verfahrens haben wir also für jeden $R_i$ einen passenden $D_i$ mit der zugehörigen Abbildung $T_i$ gefunden. +Wir bestimmen die Parameter $s$ und $g$ für jede der acht möglichen affinen Abbildungen und das mit jedem Domain-Block. +Die Kombination von $D_j$ und $T_i$, welche den kleinsten Abstand $e$ hat, ist die beste. + +Diese Schritte führen wir für jeden Range-Block $R_i$ aus. +Am Ende des Algorithmus haben wir für jeden Range-Block den zugehörigen Domain-Block und Transformation gefunden. + \subsubsection{Rekonstruktion des Bildes} -Mit den Gefundenen Abbildungen lässt sich das Bild generieren. +Mit den gefundenen Abbildungen lässt sich das Bild generieren. Wir beginnen wie schon im letzten Kapitel mit einer beliebigen Startmenge. In unserem Fall ist dieses ein Bild $f_0$ derselben Grösse. Nun ersetzen wir jedes $R_i$ mit der Transformierten des zugehörigen Domain-Blocks $T(G_j)$. Dies wird verkürzt als Operator $W$ geschrieben. So erhalten wir ein neues Bild $f_1 = W(f_0)$. -Dieses Vorgehen führen wir iteriert aus bis wir von $f_n = W(f_{n-1})$ zu $f_{n-1}$ kaum mehr einen unterschied feststellen. Die Iteration hat nun ihren Fixpunkt, das Bild, erreicht. +Dieses Vorgehen führen wir iteriert aus bis wir von $f_n = W(f_{n-1})$ zu $f_{n-1}$ kaum mehr einen Unterschied feststellen. Die Iteration hat nun ihren Attraktor, das Bild, erreicht. \subsubsection{Farbbilder} Dieses Verfahren mit Graustufenbilder lässt sich ganz einfach auf Farbbilder erweitern. @@ -98,19 +131,17 @@ Teilt man ein Bild in die drei Farbkanäle auf, das heisst, es wird nur noch ein Nun wendet man auf jeden dieser Farbkanalbilder den Algorithmus an, und fügt nach der Rekonstruktion die Kanäle wieder zusammen. \subsubsection{Performance des Verfahren} -Dieser Grundalgorithmus der Fraktalen Bildkompression ist offensichtlich recht langsam und skaliert auch schlecht mit grösseren Bilder. -Man kann die Laufzeit zwar verbessern indem man die Domain-Blöcke auch disjunkt macht, und für weniger detailreiche Bilder ein grösseres $b$ wählt, jedoch wird er auch so nie so schnell wie zum Beispiel das jpeg verfahren. +Dieser Grundalgorithmus der fraktalen Bildkompression ist recht langsam und skaliert auch schlecht für grössere Bilder. +Man kann die Laufzeit zwar verbessern indem man die Domain-Blöcke auch disjunkt macht, und für weniger detailreiche Bilder ein grösseres $b$ wählt, jedoch wird er auch so nie so schnell wie zum Beispiel das JPEG-Verfahren. \subsection{Beispiel} -Kommen wir nun zu einem Beispiel. -Wir Verwenden dafür den oben beschriebenen Algorithmus. +Wir Verwenden dafür den oben beschriebenen Algorithmus, welcher uns für jeden Range-Block die benötigten Parameter liefert. +Mit diesen lässt sich das Bild im Anschluss wieder Rekonstruieren. Die Range-Blöcke wurden $4\times4$ gewählt und die Dommain dementsprechend $8\times8$. Um etwas Zeit bei der Komprimierung zu ersparen, wurden nur disjunkte Domain-Blöcke gebraucht. Als erstes Beispiel wählen wir das 360x360px Bild von Rapperswil in Abbildung \ref{ifs:original}. -Der Algorithmus liefert uns für jeden Range-Block die benötigten Parameter. -Mit diesen lässt sich das Bild im Anschluss wieder Rekonstruieren. - -Als Startbild wird ein mittelgraues 360x360px Bild gewählt, Abbildung \ref{ifs:bild0}. +Das Startbild ist ein mittelgraues 360x360px Bild, Abbildung \ref{ifs:bild0}. +Es kann jedoch ein beliebiges Startbild Nun lassen wir das IFS laufen. Wie wir in Abbildung \ref{ifs:rappirecoa} sehen, ist schon nach der ersten Iteration das Bild schon erkennbar. Nach der fünften Iteration , Abbildung \ref{ifs:rappirecoc} gibt es fast keinen Unterschied mehr zur letzten Iteration, wir können die Rekonstruktion beenden. -- cgit v1.2.1 From 180789bb3f452a49dca3f3769630e0899357208e Mon Sep 17 00:00:00 2001 From: Alain Date: Sat, 19 Jun 2021 18:17:20 +0200 Subject: imporvements --- buch/papers/ifs/images/farncolor2.eps | 1137 +++++++++++++++++++++++++++++++++ buch/papers/ifs/images/faroe.png | Bin 987 -> 0 bytes buch/papers/ifs/images/faroe0.PNG | Bin 80239 -> 0 bytes buch/papers/ifs/images/faroe1.PNG | Bin 104146 -> 0 bytes buch/papers/ifs/images/faroe5.PNG | Bin 73790 -> 0 bytes buch/papers/ifs/teil1.tex | 15 +- buch/papers/ifs/teil2.tex | 77 ++- buch/papers/ifs/teil3.tex | 6 +- 8 files changed, 1199 insertions(+), 36 deletions(-) create mode 100644 buch/papers/ifs/images/farncolor2.eps delete mode 100644 buch/papers/ifs/images/faroe.png delete mode 100644 buch/papers/ifs/images/faroe0.PNG delete mode 100644 buch/papers/ifs/images/faroe1.PNG delete mode 100644 buch/papers/ifs/images/faroe5.PNG (limited to 'buch/papers') diff --git a/buch/papers/ifs/images/farncolor2.eps b/buch/papers/ifs/images/farncolor2.eps new file mode 100644 index 0000000..5d7d1d7 --- /dev/null +++ b/buch/papers/ifs/images/farncolor2.eps @@ -0,0 +1,1137 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: (MATLAB, The Mathworks, Inc. 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diff --git a/buch/papers/ifs/images/faroe.png b/buch/papers/ifs/images/faroe.png deleted file mode 100644 index 183702a..0000000 Binary files a/buch/papers/ifs/images/faroe.png and /dev/null differ diff --git a/buch/papers/ifs/images/faroe0.PNG b/buch/papers/ifs/images/faroe0.PNG deleted file mode 100644 index 7396ed3..0000000 Binary files a/buch/papers/ifs/images/faroe0.PNG and /dev/null differ diff --git a/buch/papers/ifs/images/faroe1.PNG b/buch/papers/ifs/images/faroe1.PNG deleted file mode 100644 index b2aff49..0000000 Binary files a/buch/papers/ifs/images/faroe1.PNG and /dev/null differ diff --git a/buch/papers/ifs/images/faroe5.PNG b/buch/papers/ifs/images/faroe5.PNG deleted file mode 100644 index 50f9564..0000000 Binary files a/buch/papers/ifs/images/faroe5.PNG and /dev/null differ diff --git a/buch/papers/ifs/teil1.tex b/buch/papers/ifs/teil1.tex index 68e2e44..385abcf 100644 --- a/buch/papers/ifs/teil1.tex +++ b/buch/papers/ifs/teil1.tex @@ -17,7 +17,7 @@ Von einem Fraktal $F$ können wir folgende Eigenschaften erwarten: \item $F$ kann nicht mit der klassischen Geometrie beschrieben werden. \item Oftmals hat $F$ eine Form von Selbstähnlichkeit. \item Die 'fraktale Dimension' ist grösser als die topologische Dimension - \item Viele Fraktale lassen sich einfach beschrieben + \item Viele Fraktale lassen sich einfach beschrieben TODO \end{enumerate} \subsection{Koch Kurve \label{ifs:subsection:lilkoch}} @@ -29,6 +29,7 @@ Diese wird in ersten Schritt durch vier gleich langen Streckenabschnitte der Lä In \ref{ifs:kochconstb} ist die Anordnung dieser vier Streckenabschnitte ersichtlich. Dieser Schritt wird nun für jeden der resultierten Streckenabschnitten wiederholt. Die Kurve besteht also aus vier kleineren Kopien der ganzen Kurve, was auch unter Selbstähnlichkeit bekannt ist. +Man spricht von einer selbstähnlichen Menge, wenn sich diese Menge überdecken lässt mit echten Teilmengen, die zur ganzen Menge ähnlich sind. \begin{figure} @@ -61,16 +62,16 @@ Die Länge der Kurve der jeweiligen Iteration lässt sich mit \Rightarrow \quad \lim_{n\to\infty} a \left( \frac{4}{3}\right)^n = \infty \end{align*} -beschreiben. +berechnen. In jedem Schritt wird die Länge um den Faktor $\frac{4}{3}$ verlängert. Daraus resultiert, dass die Länge gegen $\infty$ divergiert. Die Fläche unter der Kurve lässt sich folgendermassen berechnen \begin{align*} - A_0 = 0 \\ - A_1 = \left( \frac{a}{3}\right)^2 \frac{\sqrt{3}}{4} = a^2 \frac{\sqrt{3}}{36}\\ - A_2 = A_1 + 4\left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 \\ - A_3 = A_1 + A_2 + 4^2 \left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 + \left( \frac{4}{9}\right)^2 A_1 + A_0 &= 0 \\ + A_1 &= \left( \frac{a}{3}\right)^2 \frac{\sqrt{3}}{4} = a^2 \frac{\sqrt{3}}{36}\\ + A_2 &= A_1 + 4\left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 \\ + A_3 &= A_1 + A_2 + 4^2 \left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 + \left( \frac{4}{9}\right)^2 A_1. \end{align*} Wir sehen, dass mit jedem Schritt die neu dazugekommene Fläche um $\frac{4}{9}$ kleiner ist. Die Gesamtfläche ist daher gegeben durch die geometrische Reihe, @@ -81,7 +82,7 @@ mit dem Grenzwert \begin{align*} \lim_{n\to\infty} a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n = \frac{\sqrt{3}}{20} a^2. \end{align*} -Wie wir sehen ist die Koch-Kurve eine Kurve mit endlicher Fläche, aber unendlicher Umfang. +Wie wir sehen ist die Koch-Kurve ein Objekt mit endlicher Fläche, aber unendlichem Umfang. Zu guter Letzt bestimmen wir die Dimension der Kurve. diff --git a/buch/papers/ifs/teil2.tex b/buch/papers/ifs/teil2.tex index 5de3d4b..be3d354 100644 --- a/buch/papers/ifs/teil2.tex +++ b/buch/papers/ifs/teil2.tex @@ -65,7 +65,7 @@ Ausserdem bestimmen wir drei Funktionen \frac{1}{2} \end{pmatrix}, \end{align*} -welche die gesamte Menge auf eine ihrer kleineren Kopien abbildet +welche die gesamte Menge auf eine ihrer kleineren Kopien abbildet. $f_1$ bildet das Dreieck auf das Teilstück unten links ab, $f_2$ auf das Teilstück unten rechts und $f_3$ auf das obere Teilstück. Wendet man alle drei Funktionen auf das Sierpinski-Dreieck an \begin{align*} @@ -99,31 +99,36 @@ Der Abstand zum Original wird immer kleiner, und konvergiert gegen null. In diesem Abschnitt wollen wir die Erkenntnis, wie wir aus einer beliebigen Menge ein Sierpinski-Dreieck generieren können, verallgemeinern. -$S_1,...,S_n$ sind Kontraktionen auf die Menge $D \subset \mathbb{R}^n$. Es gilt +$S_1,\dots,S_n$ sind Kontraktionen auf die Menge $D \subset \mathbb{R}^n$. Es gilt \begin{align} |S_i(x) - S_i(y)| \leq c_i|x - y| \end{align} -für jedes i mit einem $c_i < 1$. Dann existiert eine eindeutige kompakte Menge $F$ für die gilt +für jedes i mit einem $c_i < 1$. +Der Banachsche Fixpunktsatz besagt, dass für solche Kontraktionen ein Eindeutiges $A$ existiert, für das $S(A) = A$ gilt. +Den Beweis kann man in \cite{ifs:Rousseau2012} nachlesen. +Hat man nicht nur eine sondern mehrere Kontraktionen, dann existiert eine eindeutige kompakte Menge $F$ für die gilt \begin{equation} - F = \bigcup\limits_{i = 1}^{m} S_i(F) + F = \bigcup\limits_{i = 1}^{m} S_i(F). \end{equation} -Weiter definieren wir die Transformation S auf kompakte Mengen ohne die leere Menge. +Weiter definieren wir die Transformation S auf kompakte Mengen $E$ ohne die leere Menge. \begin{equation} S(E) = \bigcup\limits_{i = 1}^m S_i(E) \end{equation} Wird diese Transformation Iterativ ausgeführt, das heisst $S^0(E) = E, S^k(E) = S(S^{k-1}(E))$, und für jedes $i$ $S_i(E) \subset E$, gilt \begin{equation} F = \bigcap\limits_{k = 1}^{\infty} S^k(E). + \label{ifs:ifsForm} \end{equation} In Worte gefasst bedeutet das, dass jede Gruppe von Kontraktionen iterativ ausgeführt, gegen eine eindeutige Menge konvergiert. Diese Menge ist auch als Attraktor des IFS bekannt. -Dies für jede Startmenge, solange diese ihre Transformierten wieder beinhaltet. -Auf den Beweis wird verzichtet. +Der Beweis für die Existenz eines eindeutigen Attraktors ist in \cite{ifs:fractal-geometry} beschrieben. +Aus diesem Beweis folgt, dass die Startmenge $E$, anders als in \ref{ifs:ifsForm} beschrieben ist, beliebig sein kann, \subsection{Beispiel: Barnsley-Farn} Der Barnsley-Farn, Abbildung \ref{ifs:farn}, ist ein Beispiel eines Fraktal, welches mit einem IFS generiert werden kann. Wie man schnell erkennen kann, besteht der Farn aus Blättern, welche eine grosse Ähnlichkeit zum ganzen Farn haben. -\begin{align*} - {S_1(x,y)} +Die vier affinen Transformationen +\begin{align} + & {S_1(x,y)} = \begin{pmatrix} 0 & 0 \\ @@ -132,9 +137,9 @@ Wie man schnell erkennen kann, besteht der Farn aus Blättern, welche eine gross \begin{pmatrix} x\\ y\\ - \end{pmatrix}, \quad + \end{pmatrix}, \quad & {S_2(x,y)} - = + &= \begin{pmatrix} 0.85 & 0.04 \\ -0.04 & 0.85 \\ @@ -148,7 +153,7 @@ Wie man schnell erkennen kann, besteht der Farn aus Blättern, welche eine gross 0 \\ 1.6 \end{pmatrix}\\ - {S_3(x,y)} + & {S_3(x,y)} = \begin{pmatrix} 0.2 & -0.26 \\ @@ -162,9 +167,9 @@ Wie man schnell erkennen kann, besteht der Farn aus Blättern, welche eine gross \begin{pmatrix} 0 \\ 1.6 - \end{pmatrix}, \quad + \end{pmatrix}, \quad & {S_4(x,y)} - = + &= \begin{pmatrix} -0.15 & 0.28 \\ 0.26 & 0.24 \\ @@ -178,26 +183,44 @@ Wie man schnell erkennen kann, besteht der Farn aus Blättern, welche eine gross 0 \\ 0.44 \end{pmatrix}\\ -\end{align*} -In der Abbildung \ref{ifs:farncolor} sehen wir die vier Transformationen farblich dargestellt. - + \label{ifs:farnFormel} +\end{align} +, welche für die konstruktion des Farns benötigt werden sind in der Abbildung \ref{ifs:farncolor} farblich dargestellt. +Das gesamte Farnblatt ist in der schwarzen Box. +Auf diese werden die Transformationen angewendet $S_1$ erstellt den Stiel des Farnblattes (rot). Die Transformation bildet das Gesamte Blatt auf die Y-Achse ab. $S_2$ (grün) erstellt den Hauptteil des Farnes. Sie verkleinert und dreht das gesamte Bild und stellt es auf das Ende des Stiels aus $S_1$. $S_3$ bildet das gesamte Blatt auf das blaue Teilblatt unten Links ab. $S_4$ spiegelt das Blatt und bildet es auf das magentafarbene Teilblatt ab. -\subsection{Chaosspiel} -Wir führen im Zusammenhang mit dem Barnsley-Farn \cite{ifs:barnsleyfern} noch eine weitere Methode ein, um ein IFS zu zeichnen. +\subsection{Erzeugung eines Bildes mit einem IFS} +Es gibt zwei verschiedene Methoden um ein Bild mit einem IFS zu erzeugen. +Die erste Methode ist wahrscheinlich die intuitivste. +Wir beginnen mit einm Startbild, zum Beispiel ein Schwarzes Quadrat, und bilden dieses mit den affinen Transformationen des IFS ab. +Das neue Bild, dass entsteht, ist die nächste Iterierte. +Dieses wird wieder mit den Transformationen abgebildet. +Wir wiederholen den letzten schritt, bis wir zufrieden mit der neusten Iterierten sind. +Diesen Vorgang haben wir beim Sierpinski-Dreieck in Abbildung \ref{ifs:sierpconst} gebraucht. + + +Die zweite Methode ist das Chaosspiel \cite{ifs:chaos}. Bis jetzt wurde immer davon gesprochen, die Transformationen auf die gesamte Menge anzuwenden. -Bei komplizierteren IFS welche viele Iterationen brauchen, bis man den Attraktor erkennen kann, ist diese Methode ziemlich rechenintensiv. -Eine Alternative ist das Chaosspiel \cite{ifs:chaos}. -Bei dieser Methode werden die Transformationen nicht auf die Menge angewendet, sondern nur auf einen einzelnen Punkt. +Bei komplizierteren IFS welche viele Iterationen brauchen, bis man den Attraktor erkennen kann, ist die erste Methode ziemlich rechenintensiv. +Beim Chaosspiel werden die Transformationen nicht auf die Menge angewendet, sondern nur auf einen einzelnen Punkt. Der Startpunkt kann dabei ein beliebiger Punkt in $E$ sein. Es wird bei jedem Iterationsschritt nur eine Transformation, welche zufällig gewählt wurde, angewendet. -Da, wie wir beim Barnsley-Farn gut sehen, dass nicht jede Transformation gleich viel des Bildes ausmacht, werden diese beim Chaosspiel gewichtet. -Die Gewichtung erfolgt über den Anteil der Gesamtmasse. -Im Fall des Barnsley-Fern wird $S_1$ in $1\%$, $S_2$ in $85\%$ und $S_3 \& S_4$ in $7\%$ der Iterationen ausgeführt. +Da, wie wir beim Barnsley-Farn gut sehen, nicht jede Transformation gleich viel des Bildes ausmacht, werden diese beim Chaosspiel gewichtet. +Je mehr eine Transformation kontrahiert, desto weniger Punkte braucht es um die resultierende Teilabbildung darzustellen. +Im Fall des Barnsley-Fern wird $S_1$ in $1\%$, $S_2$ in $85\%$ und $S_3 \& S_4$ in $7\%$ der Iterationen ausgeführt. +Wir sehen auch in Abbildung \ref{ifs:farncolor} gut, dass der rote Stiel, $S_1$, einiges weniger Punkte braucht als der grüne Hauptteil des Blattes, $S_2$. + +In Abbildung \ref{ifs:farnNoWeight} wurden die vier gleich stark gewichtet. +Man sieht, dass trotzt gleich vieler Iterationen wie in Abbildung \ref{ifs:farn}, der Farn kaum nicht so gut abgebildet ist. + + + + \begin{figure} \centering \makebox[\textwidth][c]{ @@ -207,8 +230,8 @@ Im Fall des Barnsley-Fern wird $S_1$ in $1\%$, $S_2$ in $85\%$ und $S_3 \& S_4$ \end{figure} \begin{figure} \centering - \includegraphics[width=0.7\textwidth]{papers/ifs/images/farncolor} - \caption{Vier Transformationen des Barnsley-Farn} + \includegraphics[width=\textwidth]{papers/ifs/images/farncolor2} + \caption{Vier Transformationen des Barnsley-Farn in unterschiedlichen Farben} \label{ifs:farncolor} \end{figure} \begin{figure} diff --git a/buch/papers/ifs/teil3.tex b/buch/papers/ifs/teil3.tex index 39a808f..b3dff85 100644 --- a/buch/papers/ifs/teil3.tex +++ b/buch/papers/ifs/teil3.tex @@ -9,7 +9,7 @@ Mit dem Prinzip dieser IFS ist es auch möglich Bilder zu Komprimieren. Diese Idee hatte der Mathematiker Michael Barnsley, welcher mit seinem Buch Fractals Everywhere einen wichtigen Beitrag zum Verständnis von Fraktalen geliefert hat. Das Ziel ist es ein IFS zu finden, welches das Bild als Attraktor hat. -In diesem Unterkapitel wollen wir eine Methode dafür anschauen.\cite{ifs:Rousseau2012} +In diesem Unterkapitel wollen wir eine Methode dafür anschauen, wie sie in \cite{ifs:Rousseau2012} beschrieben ist. Bis jetzt wurde in Zusammenhang mit IFS immer erwähnt, dass die Transformationen, welche das IFS bilden, auf die gesamte Menge. @@ -132,7 +132,9 @@ Nun wendet man auf jeden dieser Farbkanalbilder den Algorithmus an, und fügt na \subsubsection{Performance des Verfahren} Dieser Grundalgorithmus der fraktalen Bildkompression ist recht langsam und skaliert auch schlecht für grössere Bilder. -Man kann die Laufzeit zwar verbessern indem man die Domain-Blöcke auch disjunkt macht, und für weniger detailreiche Bilder ein grösseres $b$ wählt, jedoch wird er auch so nie so schnell wie zum Beispiel das JPEG-Verfahren. +Dies resultiert aus eigenen Experimenten. +Man kann die Laufzeit zwar verbessern indem man die Domain-Blöcke auch disjunkt macht, und für weniger detailreiche Bilder ein grösseres $b$ wählt, jedoch wird er auch so nicht so schnell wie zum Beispiel das JPEG-Verfahren. +Es wurden bessere Algorithmen der fraktalen Bildkompression entwickelt, doch auch diese können, vor allem in der Laufzeit, noch nicht mit herkömmlichen Komprimierungsverfahren mithalten. \subsection{Beispiel} Wir Verwenden dafür den oben beschriebenen Algorithmus, welcher uns für jeden Range-Block die benötigten Parameter liefert. -- cgit v1.2.1 From 8cb994306345986d642fd46759c92e7adee4e4ef Mon Sep 17 00:00:00 2001 From: Alain Date: Sun, 20 Jun 2021 22:09:47 +0200 Subject: Changes --- buch/papers/ifs/images/FIC.pdf | 2003 +++++++++++++++++++++++++++++ buch/papers/ifs/images/farnrightwight.eps | 1027 +++++++++++++++ buch/papers/ifs/images/rapperswil0.PNG | Bin 66375 -> 0 bytes buch/papers/ifs/images/rapperswil1.PNG | Bin 82594 -> 0 bytes buch/papers/ifs/images/rapperswil4.PNG | Bin 60837 -> 0 bytes buch/papers/ifs/images/zurich.png | Bin 71780 -> 0 bytes buch/papers/ifs/references.bib | 33 +- buch/papers/ifs/teil0.tex | 2 +- buch/papers/ifs/teil1.tex | 10 +- buch/papers/ifs/teil2.tex | 36 +- buch/papers/ifs/teil3.tex | 25 +- 11 files changed, 3092 insertions(+), 44 deletions(-) create mode 100644 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/dev/null differ diff --git a/buch/papers/ifs/images/zurich.png b/buch/papers/ifs/images/zurich.png deleted file mode 100644 index bb70f7d..0000000 Binary files a/buch/papers/ifs/images/zurich.png and /dev/null differ diff --git a/buch/papers/ifs/references.bib b/buch/papers/ifs/references.bib index fbf75f4..817c5a4 100644 --- a/buch/papers/ifs/references.bib +++ b/buch/papers/ifs/references.bib @@ -4,15 +4,6 @@ % (c) 2020 Autor, Hochschule Rapperswil % -@online{ifs:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, - year = {2020}, - month = {2}, - day = {6} -} - @online{ifs:chaos, title = {Chaosspiel}, url = {https://de.wikipedia.org/wiki/Iteriertes_Funktionensystem#Chaosspiel}, @@ -38,16 +29,6 @@ isbn = {0-471-92287-0}, } -@article{ifs:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, - title = { Noncommutative harmonic analysis and image registration }, - journal = { Appl. Comput. Harmon. Anal.}, - year = 2019, - volume = 47, - pages = {607--627}, - url = {https://doi.org/10.1016/j.acha.2017.11.004} -} - @Inbook{ifs:Rousseau2012, author= {Rousseau, Christiane and Saint-Aubin, Yvan @@ -64,4 +45,18 @@ url={https://doi.org/10.1007/978-3-642-30092-9_11} } +@article{ifs:pifs, + title = {Applications of Partitioned Iterated Function Systems in Image and Video Compression}, + journal = {Journal of Visual Communication and Image Representation}, + volume = 7, + number = {2}, + pages = {144-154}, + year = 1996, + issn = {1047-3203}, + doi = {https://doi.org/10.1006/jvci.1996.0014}, + url = {https://www.sciencedirect.com/science/article/pii/S1047320396900140}, + author = {Guojun Lu and Toon Lin Yew}, + abstract = {Iterated function systems (IFS) have been used to compress image data. Because of difficulty in finding IFS in natural images, a technique based on partitioned IFS (PIFS) has been proposed for image compression. In this technique, an image to be compressed is divided into nonoverlapping blocks. For each block an affine transformation is found in the image. This set of affine transformations (called PIFS) corresponds to a unique image. In the simplest case, images are partitioned into fixed size blocks. In this paper, we investigate image and video compression techniques using variable block sizes based on the quadtree partition. One property of images generated using PIFS is scalability: they have fine detail in any scale. We exploit this property to reduce required compression time and improve compression performance. There are large amounts of temporal redundancy between fames of a video sequence. We describe a method to remove temporal redundancies effectively using a quadtree partitioning technique. We have implemented the above schemes to compress image and video sequences and will report our experimental results.} +} + diff --git a/buch/papers/ifs/teil0.tex b/buch/papers/ifs/teil0.tex index b8a678d..833748c 100644 --- a/buch/papers/ifs/teil0.tex +++ b/buch/papers/ifs/teil0.tex @@ -3,7 +3,7 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 0\label{ifs:section:teil0}} +\section{Einleitung \label{ifs:section:teil0}} \rhead{Was ist ein Iteriertes Funktionsschema} Mit der Hilfe von Iterierten Funktionsschemata (IFS) kann mit nur wenigen affinen Funktionen, komplexe Bilder beschreiben werden. In der Regel sind diese Bilder Fraktale. diff --git a/buch/papers/ifs/teil1.tex b/buch/papers/ifs/teil1.tex index 385abcf..70b0b1b 100644 --- a/buch/papers/ifs/teil1.tex +++ b/buch/papers/ifs/teil1.tex @@ -6,18 +6,18 @@ \section{Fraktale \label{ifs:section:teil1}} \rhead{Problemstellung} -Bevor wir die IFS genauer ansehen, schauen wir uns Fraktale genauer an. +Bevor wir die IFS ansehen, schauen wir uns Fraktale genauer an. Über die genaue Definition von Fraktalen sind sich die Mathematiker nicht einig. In diesem Kapitel orientieren wir uns an den Eigenschaften welche Kenneth Falconer in seinem Buch Fractal Geometry \cite{ifs:fractal-geometry} beschreibt. -Von einem Fraktal $F$ können wir folgende Eigenschaften erwarten: +Von einem Fraktal $F$ können wir folgende Eigenschaften erwarten: \begin{enumerate} \item $F$ hat eine unendlich feine Struktur \item $F$ kann nicht mit der klassischen Geometrie beschrieben werden. \item Oftmals hat $F$ eine Form von Selbstähnlichkeit. \item Die 'fraktale Dimension' ist grösser als die topologische Dimension - \item Viele Fraktale lassen sich einfach beschrieben TODO + \item Viele Fraktale lassen sich auf eine simple Art definieren. Es genügen zum Beispiel nur wenige Funktionen, welche rekursiv ausgeführt werden, um ein Fraktal zu definieren. \end{enumerate} \subsection{Koch Kurve \label{ifs:subsection:lilkoch}} @@ -74,7 +74,7 @@ Die Fläche unter der Kurve lässt sich folgendermassen berechnen A_3 &= A_1 + A_2 + 4^2 \left( \frac{a}{3^2}\right)^2 \frac{\sqrt{3}}{4} = A_1 + \frac{4}{9} A_1 + \left( \frac{4}{9}\right)^2 A_1. \end{align*} Wir sehen, dass mit jedem Schritt die neu dazugekommene Fläche um $\frac{4}{9}$ kleiner ist. -Die Gesamtfläche ist daher gegeben durch die geometrische Reihe, +Die Gesamtfläche ist daher gegeben durch die konvergierende geometrische Reihe, \begin{align*} A_n = A_1 \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n = a^2 \frac{\sqrt{3}}{36} \sum_{i = 0}^{n-1} \left( \frac{4}{9}\right)^n \\ \end{align*} @@ -89,7 +89,7 @@ Zu guter Letzt bestimmen wir die Dimension der Kurve. Es gibt viele verschiedene Methoden die Dimension zu definieren. Diese können dann auch unterschiedliche Resultate liefern. Vor allem im Zusammenhang mit Fraktalen findet man in der Literatur unterschiedliche Arten. In diesem Beispiel werden wir die Ähnlichkeits-Dimension \cite{ifs:fractal-geometry}. -Die Ähnlichkeits-Dimension ist das Verhältnis der Logarithmen der Anzahl Kopien $N$ des Originales und deren Skalierungsfaktor $\epsilon$ +Die Ähnlichkeits-Dimension $D$ ist das Verhältnis der Logarithmen der Anzahl Kopien $N$ des Originales und deren Skalierungsfaktor $\epsilon$ \begin{align*} D = - \frac{\log N}{\log \epsilon }. diff --git a/buch/papers/ifs/teil2.tex b/buch/papers/ifs/teil2.tex index be3d354..0c957d6 100644 --- a/buch/papers/ifs/teil2.tex +++ b/buch/papers/ifs/teil2.tex @@ -95,7 +95,7 @@ Im Beispiel der Abbildung \ref{ifs:sierpconst} sehen wir, wie das Bild nach jede Der Abstand zum Original wird immer kleiner, und konvergiert gegen null. \subsection{Iterierte Funktionensysteme -\label{ifs:subsection:bonorum}} +\label{ifs:subsection:IteratedFunktionensysteme}} In diesem Abschnitt wollen wir die Erkenntnis, wie wir aus einer beliebigen Menge ein Sierpinski-Dreieck generieren können, verallgemeinern. @@ -110,9 +110,10 @@ Hat man nicht nur eine sondern mehrere Kontraktionen, dann existiert eine eindeu \begin{equation} F = \bigcup\limits_{i = 1}^{m} S_i(F). \end{equation} -Weiter definieren wir die Transformation S auf kompakte Mengen $E$ ohne die leere Menge. +Weiter definieren wir die Transformation S auf kompakte Mengen $E$ ohne die leere Menge \begin{equation} - S(E) = \bigcup\limits_{i = 1}^m S_i(E) + S(E) = \bigcup\limits_{i = 1}^m S_i(E). + \label{ifs:transformation} \end{equation} Wird diese Transformation Iterativ ausgeführt, das heisst $S^0(E) = E, S^k(E) = S(S^{k-1}(E))$, und für jedes $i$ $S_i(E) \subset E$, gilt \begin{equation} @@ -122,7 +123,8 @@ Wird diese Transformation Iterativ ausgeführt, das heisst $S^0(E) = E, S^k(E) = In Worte gefasst bedeutet das, dass jede Gruppe von Kontraktionen iterativ ausgeführt, gegen eine eindeutige Menge konvergiert. Diese Menge ist auch als Attraktor des IFS bekannt. Der Beweis für die Existenz eines eindeutigen Attraktors ist in \cite{ifs:fractal-geometry} beschrieben. -Aus diesem Beweis folgt, dass die Startmenge $E$, anders als in \ref{ifs:ifsForm} beschrieben ist, beliebig sein kann, +Aus diesem Beweis folgt, dass die Startmenge $E$, anders als in \ref{ifs:ifsForm} beschrieben ist, beliebig sein kann. + \subsection{Beispiel: Barnsley-Farn} Der Barnsley-Farn, Abbildung \ref{ifs:farn}, ist ein Beispiel eines Fraktal, welches mit einem IFS generiert werden kann. Wie man schnell erkennen kann, besteht der Farn aus Blättern, welche eine grosse Ähnlichkeit zum ganzen Farn haben. @@ -194,14 +196,17 @@ $S_2$ (grün) erstellt den Hauptteil des Farnes. Sie verkleinert und dreht das gesamte Bild und stellt es auf das Ende des Stiels aus $S_1$. $S_3$ bildet das gesamte Blatt auf das blaue Teilblatt unten Links ab. $S_4$ spiegelt das Blatt und bildet es auf das magentafarbene Teilblatt ab. -\subsection{Erzeugung eines Bildes mit einem IFS} -Es gibt zwei verschiedene Methoden um ein Bild mit einem IFS zu erzeugen. +\subsection{Erzeugung eines Bildes zu einem IFS} +Es gibt zwei verschiedene Methoden um das Bild zu einem IFS zu erzeugen. Die erste Methode ist wahrscheinlich die intuitivste. Wir beginnen mit einm Startbild, zum Beispiel ein Schwarzes Quadrat, und bilden dieses mit den affinen Transformationen des IFS ab. Das neue Bild, dass entsteht, ist die nächste Iterierte. Dieses wird wieder mit den Transformationen abgebildet. Wir wiederholen den letzten schritt, bis wir zufrieden mit der neusten Iterierten sind. + Diesen Vorgang haben wir beim Sierpinski-Dreieck in Abbildung \ref{ifs:sierpconst} gebraucht. +In Abbildung \ref{ifs:sierpinski10} ist die zehnte Iterierte zu sehen. +Weitere Iterationen hätten in dieser Darstellungsgrösse kaum mehr einen Unterschied gemacht. Die zweite Methode ist das Chaosspiel \cite{ifs:chaos}. @@ -216,8 +221,12 @@ Im Fall des Barnsley-Fern wird $S_1$ in $1\%$, $S_2$ in $85\%$ und $S_3 \& S_4$ Wir sehen auch in Abbildung \ref{ifs:farncolor} gut, dass der rote Stiel, $S_1$, einiges weniger Punkte braucht als der grüne Hauptteil des Blattes, $S_2$. In Abbildung \ref{ifs:farnNoWeight} wurden die vier gleich stark gewichtet. -Man sieht, dass trotzt gleich vieler Iterationen wie in Abbildung \ref{ifs:farn}, der Farn kaum nicht so gut abgebildet ist. +Man sieht, dass trotzt gleich vieler Iterationen wie in Abbildung \ref{ifs:farn}, der Farn nicht so gut abgebildet wird. +Am besten sieht man den Effekt einer schlechten Gewichtung in Abbildung \ref{ifs:farnrightWeight}. +Hier wurde $S_4$, welches für das rechte untere Teilblatt zuständig ist, mit nur $1\%$ statt $7\%$ gewichtet. +Man sieht, wie sich der Mangel an Punkten auf die anderen Abbildungen das Farnblattes auswirkt. +In jeder Kopie des ganzen Farns fehlen die Punkte für dieses rechte untere Teilblatt. @@ -234,10 +243,15 @@ Man sieht, dass trotzt gleich vieler Iterationen wie in Abbildung \ref{ifs:farn} \caption{Vier Transformationen des Barnsley-Farn in unterschiedlichen Farben} \label{ifs:farncolor} \end{figure} + \begin{figure} \centering - \makebox[\textwidth][c]{ - \includegraphics[width=1.4\textwidth]{papers/ifs/images/farnnotweight}} - \caption{Chaosspiel ohne Gewichtung} - \label{ifs:farnNoWeight} + \subfigure[]{ + \label{ifs:farnNoWeight} + \includegraphics[width=0.45\textwidth]{papers/ifs/images/farnnotweight}} + \subfigure[]{ + \label{ifs:farnrightWeight} + \includegraphics[width=0.45\textwidth]{papers/ifs/images/farnrightwight}} + \caption{(a) Chaosspiel ohne Gewichtung (b) $S_4$ zu wenig gewichtet} + \label{ifs:farnweight} \end{figure} diff --git a/buch/papers/ifs/teil3.tex b/buch/papers/ifs/teil3.tex index b3dff85..ebae0fb 100644 --- a/buch/papers/ifs/teil3.tex +++ b/buch/papers/ifs/teil3.tex @@ -11,14 +11,14 @@ Diese Idee hatte der Mathematiker Michael Barnsley, welcher mit seinem Buch Frac Das Ziel ist es ein IFS zu finden, welches das Bild als Attraktor hat. In diesem Unterkapitel wollen wir eine Methode dafür anschauen, wie sie in \cite{ifs:Rousseau2012} beschrieben ist. - -Bis jetzt wurde in Zusammenhang mit IFS immer erwähnt, dass die Transformationen, welche das IFS bilden, auf die gesamte Menge. -Dies muss jedoch nicht so sein. -Es gibt auch einen Attraktor, wenn die Transformationen nur Teile der Menge auf die ganze Menge abbilden. -Diese Eigenschaft wollen wir uns in der Fraktalen Bildkompression zunutze machen. -Sie ermöglicht uns Ähnlichkeiten zwischen kleineren Teilen des Bildes zunutze machen. Es ist wohl nicht falsch zu sagen, dass Ähnlichkeiten zur gesamten Menge, wie wir sie zum Beispiel beim Barnsley Farn gesehen haben, bei Bilder aus dem Alltag eher selten anzutreffen sind. -Doch wie finden wir die richtigen affinen Transformationen, welche als IFS das Bild als Attraktor haben? +Ein IFS, wie wir es in \ref{ifs:subsection:IteratedFunktionensysteme} definiert haben, wird uns also nicht weiter helfen. +Die Lösung dazu sind Partitionierte IFS (PIFS) \cite{ifs:pifs}. +In \ref{ifs:transformation} wurde definiert, dass die Kontraktionen $S_i$ bei IFS auf die gesamte Menge $E$ angewendet werden. +Bei einem PIFS wird der Attraktor in disjunkte Teilmengen aufgeteilt. +Für jede dieser Teilmengen $R_i$ braucht es dann eine grössere Teilmenge, welche mit einer affinen Transformation eine zu $R_i$ ähnliche Menge bildet. +Wir müssen nicht mehr Ähnlichkeiten zum ganzen Bild finden, sondern zwischen Teilen des Bildes. +Doch wie finden wir das PIFS, welches das Bild als Attraktor hat? \subsection{das Kompressionsverfahren \label{ifs:subsection:malorum}} @@ -29,9 +29,12 @@ Ein Bild ist also eine Funktion, die jedem Pixel einen Grauwert $z$ zuweist \begin{align*} z = f(x,y). \end{align*} + +Wir suchen ein PIFS welches das zu komprimierende Bild als Attraktor hat. In einem ersten Schritt teilen wir das Bild in disjunkte benachbarte $b \times b$ Pixel-Quadrate auf. Diese Blöcke nennen wir Range-Blöcke der Menge $R=\{R_0,R_1,...R_m\}$ Im nächsten Schritt teilen wir das Bild in alle möglichen $2b \times 2b$ Pixel-Quadrate auf. Diese sind die Domain-Blöcke der Menge $D = \{D_0,D_1,...D_n\}$. Im dritten und letzten Schritt wird für jeden Range-Block $R_i$ ein Domain-Block $D_j$ gesucht, welcher ihm am ähnlichsten ist. +Zwei Beispiele wie solche Domain-, und Range-Block Paare aussehen können, sehen wir in Abbildung \ref{ifs:FIC} \subsubsection{Finden des ähnlichsten $D_j$} Zuerst brauchen wir die Transformation @@ -114,6 +117,12 @@ Die Kombination von $D_j$ und $T_i$, welche den kleinsten Abstand $e$ hat, ist d Diese Schritte führen wir für jeden Range-Block $R_i$ aus. Am Ende des Algorithmus haben wir für jeden Range-Block den zugehörigen Domain-Block und Transformation gefunden. +\begin{figure} + \centering + \includegraphics[width=\textwidth]{papers/ifs/images/FIC} + \caption{Domain-, und Range-Block Paare in Grün und Rot} + \label{ifs:FIC} +\end{figure} \subsubsection{Rekonstruktion des Bildes} Mit den gefundenen Abbildungen lässt sich das Bild generieren. @@ -144,7 +153,7 @@ Um etwas Zeit bei der Komprimierung zu ersparen, wurden nur disjunkte Domain-Bl Als erstes Beispiel wählen wir das 360x360px Bild von Rapperswil in Abbildung \ref{ifs:original}. Das Startbild ist ein mittelgraues 360x360px Bild, Abbildung \ref{ifs:bild0}. Es kann jedoch ein beliebiges Startbild -Nun lassen wir das IFS laufen. +Nun lassen wir das PIFS laufen. Wie wir in Abbildung \ref{ifs:rappirecoa} sehen, ist schon nach der ersten Iteration das Bild schon erkennbar. Nach der fünften Iteration , Abbildung \ref{ifs:rappirecoc} gibt es fast keinen Unterschied mehr zur letzten Iteration, wir können die Rekonstruktion beenden. \begin{figure} -- cgit v1.2.1 From cceb539b3b83de6cf4296e6062c8d2f6e31aec72 Mon Sep 17 00:00:00 2001 From: Alain Date: Tue, 22 Jun 2021 17:28:15 +0200 Subject: minor changes --- buch/papers/ifs/teil1.tex | 3 ++- buch/papers/ifs/teil2.tex | 5 ++--- buch/papers/ifs/teil3.tex | 4 ++-- 3 files changed, 6 insertions(+), 6 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/ifs/teil1.tex b/buch/papers/ifs/teil1.tex index 70b0b1b..a75b529 100644 --- a/buch/papers/ifs/teil1.tex +++ b/buch/papers/ifs/teil1.tex @@ -98,8 +98,9 @@ Mit ihr kann man einfach die Dimension selbstähnlicher Mengen bestimmen. Als Beispiel nehmen wir ein gleichseitiges Dreieck. Dieses besteht aus $N = 4$ Kopien mit halber ($\epsilon = 1/2$) Kantenlänge $l$, Abbildung \ref{ifs:trinagle}. Somit hat das Dreieck die Dimension $D = 2$. Die Koch Kurve besteht aus $N = 4$ Kopien mit Kantenlänge $\epsilon =l \cdot 1/3$. +Ihre Ähnlichkeits-Dimension ist somit \begin{align*} - D = - \frac{\log N }{\log \epsilon } = - \frac{\log 4 }{\log 1/3 } \approx 1.2619 + D = - \frac{\log N }{\log \epsilon } = - \frac{\log 4 }{\log 1/3 } \approx 1.2619. \end{align*} Wie wir nun sehen besitzt die Koch-Kurve alle oben beschriebenen Eigenschaften von Fraktalen. Dies muss jedoch nicht bei allen Fraktalen der Fall. Sonst wäre die Frage nach einer 'richtigen' Definition einfach zu beantworten. diff --git a/buch/papers/ifs/teil2.tex b/buch/papers/ifs/teil2.tex index 0c957d6..fd10634 100644 --- a/buch/papers/ifs/teil2.tex +++ b/buch/papers/ifs/teil2.tex @@ -115,15 +115,14 @@ Weiter definieren wir die Transformation S auf kompakte Mengen $E$ ohne die leer S(E) = \bigcup\limits_{i = 1}^m S_i(E). \label{ifs:transformation} \end{equation} -Wird diese Transformation Iterativ ausgeführt, das heisst $S^0(E) = E, S^k(E) = S(S^{k-1}(E))$, und für jedes $i$ $S_i(E) \subset E$, gilt +Wird diese Transformation Iterativ ausgeführt, das heisst $S^0(E) = E, S^k(E) = S(S^{k-1}(E))$, gilt \begin{equation} F = \bigcap\limits_{k = 1}^{\infty} S^k(E). \label{ifs:ifsForm} \end{equation} In Worte gefasst bedeutet das, dass jede Gruppe von Kontraktionen iterativ ausgeführt, gegen eine eindeutige Menge konvergiert. -Diese Menge ist auch als Attraktor des IFS bekannt. +Diese Menge ist auch als Attraktor eines IFS bekannt. Der Beweis für die Existenz eines eindeutigen Attraktors ist in \cite{ifs:fractal-geometry} beschrieben. -Aus diesem Beweis folgt, dass die Startmenge $E$, anders als in \ref{ifs:ifsForm} beschrieben ist, beliebig sein kann. \subsection{Beispiel: Barnsley-Farn} Der Barnsley-Farn, Abbildung \ref{ifs:farn}, ist ein Beispiel eines Fraktal, welches mit einem IFS generiert werden kann. diff --git a/buch/papers/ifs/teil3.tex b/buch/papers/ifs/teil3.tex index ebae0fb..78fb935 100644 --- a/buch/papers/ifs/teil3.tex +++ b/buch/papers/ifs/teil3.tex @@ -87,13 +87,13 @@ Da wir ein $2b \times 2b$ Feld auf ein $b \times b$ Feld abbilden möchten, müs Dies erreichen wir, indem wir alle disjunkten $2 \times 2$ px Blöcke mit einem Pixel des Grautones deren Mittelwertes ersetzen. -Die Parameter $s_i$ und $g_i$ beschreiben die Änderung des Grautones. $s$ verändert den Kontrast und $g$ verschiebt die Töne auf die richtige Helligkeit, sie bilden die lineare Funktion +Die Parameter $s_i$ und $g_i$ beschreiben die Änderung des Grautones. $s$ verändert den Kontrast und $g$ verschiebt die Grautöne auf die richtige Helligkeit, sie bilden die lineare Funktion \begin{align*} z' = s_i z + g_i. \end{align*} Für die Bestimmung dieser Parameter führen wir zuerst die Bildfunktionen $f_{R_i}$ und $\tilde{f_{R_i}}$ ein. $f_{R_i}$ ist die Bildfunktion des Range-Blockes $R_i$ und $\tilde{f_{R_i}}$ ist die Bildfunktion des zuerst Skalierten und dann mit \ref{ifs:affTrans} transformierten Domain-Blocks $D_j$. -$s$ und $g$ werden mit der einfachen linearen Regression ermittelt. + Wir suchen $s_i$ und $g_i$ so das \begin{align*} f_{R_i} = s_i \tilde{f_{R_i}} + g_i = \bar{f_{R_i}}. -- cgit v1.2.1 From f04279543c41d828b0684fe603e09cfb4f9ed8b1 Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Wed, 23 Jun 2021 20:00:21 +0200 Subject: several changes --- buch/papers/reedsolomon/codebsp.tex | 72 ++++++++++++++++++----------- buch/papers/reedsolomon/decmitfehler.tex | 2 +- buch/papers/reedsolomon/endlichekoerper.tex | 8 ++-- buch/papers/reedsolomon/main.tex | 2 +- buch/papers/reedsolomon/rekonstruktion.tex | 2 +- 5 files changed, 51 insertions(+), 35 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/codebsp.tex b/buch/papers/reedsolomon/codebsp.tex index 262297e..6ab792a 100644 --- a/buch/papers/reedsolomon/codebsp.tex +++ b/buch/papers/reedsolomon/codebsp.tex @@ -5,14 +5,14 @@ % \section{Codierung eines Beispiels \label{reedsolomon:section:codebsp}} -\rhead{Koerper Festlegen} +\rhead{Codierung eines Beispiels} Um die Funktionsweise eines Reed-Solomon-Codes besser zu verstehen werden wir die einzelnen Probleme und ihre Lösungen anhand eines Beispiels betrachten. -Da wir in Endlichen Körpern Rechnen werden wir zuerst solch einen Körper festlegen. Dabei müssen wir die \textcolor{red}{Definition 4.6 (wie verweist man auf eine definition?)} berücksichtigen, die besagt, dass nur Primzahlen für endliche Körper in Frage kommen. -Wir legen für unser Beispiel den endlichen Körper mit $q = 11$ fest. +Da wir in endlichen Körpern rechnen, werden wir zuerst solch einen Körper festlegen. Dabei müssen wir die \textcolor{red}{Definition 4.6 (verweis auf eine Definition im Buch ohne label)} berücksichtigen, die besagt, dass nur Primzahlen für endliche Körper in Frage kommen. +Wir legen für unser Beispiel den endlichen Körper $\mathbb{F}_{q}$ mit $q = 11$ fest. Zur Hilfestellung können dazu die beiden Tabellen \ref{reedsolomon:subsection:adtab} und -\ref{reedsolomon:subsection:mptab} hinzugezogen werden. Diese Tabellen enthalten sämtliche Resultate aller gültigen Operationen \textcolor{red}{(Notiz: nach meinem Wissen gibt es ja nur addition und multiplikation als gültige operationen)}, die in diesem Körper durchgeführt werden können. -Aus der Definition der Endlichen Körper (ersichtlich auch in den Tabellen) folgt, dass uns nur die Zahlen \[\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}\] zur verfügung stehen und somit $11 = 0$ gelten muss. +\ref{reedsolomon:subsection:mptab} hinzugezogen werden. Diese Tabellen enthalten die Resultate der arithmetischen Operationen im Körper $\mathbb{F}_{11}$, die durchgeführt werden können. +Aus der Definition der endlichen Körper (ersichtlich auch in den Tabellen) folgt, dass uns nur die Zahlen \[\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}\] zur Verfügung stehen und somit $11 = 0$ gelten muss. % OLD TEXT %Alle folgenden Berechnungen wurden mit den beiden Restetabellen \ref{reedsolomon:subsection:adtab} und \ref{reedsolomon:subsection:mptab} durchgeführt. @@ -22,7 +22,7 @@ Aus der Definition der Endlichen Körper (ersichtlich auch in den Tabellen) folg %\input{papers/reedsolomon/restetabelle1} %\input{papers/reedsolomon/restetabelle2} -Anhand der Menge uns zur Verfügung stehenden Zahlen wird auch festgelegt, wie viele Zahlen ein Nachrichtenblock $n$, bestehend aus Nutzdatenteil und Fehlerkorrekturteil, umfassen kann. +Die Menge uns zur Verfügung stehender Zahlen legt auch fest, wie viele Zahlen ein Nachrichtenblock $n$, bestehend aus Nutzdatenteil und Fehlerkorrekturteil, umfassen kann. Der Nachrichtenblock im Beispiel besteht aus \[ n = q - 1 = 10 \text{ Zahlen}, @@ -52,16 +52,16 @@ k = n - 2t = 6\text{ Zahlen} \] übertragen. -Zusammenfassend haben wir einen Nachrichtenblock mit der Länge von 10 Zahlen definiert, der 6 Zahlen als Nutzlast beinhaltet und in der Lage ist aus 2 fehlerhafte Stellen im Block die ursprünglichen Nutzdaten zu rekonstruieren. Zudem werden wir im weiteren feststellen, dass dieser Code maximal vier Fehlerstellen erkennen, diese aber nicht rekonstruieren kann. +Zusammenfassend haben wir einen Nachrichtenblock mit der Länge von 10 Zahlen definiert, der 6 Zahlen als Nutzlast beinhaltet und in der Lage ist, aus 2 fehlerhafte Stellen im Block die ursprünglichen Nutzdaten zu rekonstruieren. Zudem werden wir im weiteren feststellen, dass dieser Code maximal vier Fehlerstellen erkennen, diese aber nicht rekonstruieren kann. -Wir legen nun die Nachricht +Wir legen nun für das Beispiel die Nachricht \[ m = [0,0,0,0,4,7,2,5,8,1] \] fest, die wir gerne an einen Empfänger übertragen möchten, wobei die vorderen vier Stellen für die Fehlerkorrektur zuständig sind. -Solange diese Stellen vor dem Codieren und nach dem Decodieren den Wert null haben, so ist die Nachricht Fehlerfrei übertragen worden. +Solange diese Stellen vor dem Codieren und nach dem Decodieren den Wert null haben, so ist die Nachricht fehlerfrei übertragen worden. -Da wir in den folgenden Abschnitten mit Polynomen arbeiten, stellen wir die Nachicht auch noch als Polynom +Da wir in den folgenden Abschnitten mit Polynomen arbeiten, stellen wir die Nachricht auch noch als Polynom \[ m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1 \] @@ -77,8 +77,8 @@ dar. \subsection{Der Ansatz der diskreten Fouriertransformation \label{reedsolomon:subsection:diskFT}} -In einem vorherigen Kapitel \textcolor{red}{(???)} haben wir schon einmal die diskrete Fouriertransformation zum Codieren einer Nachricht verwendet. In den endlichen Körpern wird dies jedoch nicht gelingen, da die Eulerische Zahl $e$ in endlichen Körpern nicht existiert. -Wir legen deshalb die Zahl $a$ fest. Diese Zahl soll die gleichen aufgaben haben, wie $e^{\frac{j}{2 \pi}}$ in der Diskreten Fouriertransformation, nur mit dem Unterschied, dass $a$ in $\mathbb{F}_{11}$ existiert. Dazu soll $a$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken, um +In einem vorherigen Abschnitt \textcolor{red}{(???)} haben wir schon einmal die diskrete Fouriertransformation zum Codieren einer Nachricht verwendet. In den endlichen Körpern wird dies jedoch nicht gelingen, da die Eulerische Zahl $e$ in endlichen Körpern nicht existiert. +Wir wählen deshalb eine Zahl $a$, die die gleichen Aufgaben haben soll wie $e^{\frac{j}{2 \pi}}$ in der diskreten Fouriertransformation, nur mit dem Unterschied, dass $a$ in $\mathbb{F}_{11}$ ist. Dazu soll die Potenz von $a$ den gesamten Zahlenbereich von $\mathbb{F}_{11}$ abdecken, um \[ \mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\} \] @@ -118,22 +118,36 @@ umzuschreiben. Wenn wir jetzt sämtliche Zahlen von $\mathbb{F}_{11}$ in $a$ einsetzen \begin{center} -\begin{tabular}{c r c l} -%$a = 0 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{0, 0, 0, 0, 0, 0, 0, 0, 0, 0\}$ \\ -$a = 1 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 1, 1, 1, 1, 1, 1, 1, 1, 1\}$ \\ -$a = 2 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 2, 4, 8, 5, 10, 9, 7, 3, 6\}$ \\ -$a = 3 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 3, 9, 5, 4, 1, 3, 9, 5, 4\}$ \\ -$a = 4 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 4, 5, 9, 3, 1, 4, 5, 9, 3\}$ \\ -$a = 5 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 5, 3, 4, 9, 1, 5, 3, 4, 9\}$ \\ -$a = 6 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 6, 3, 7, 9, 10, 5, 8, 4, 2\}$ \\ -$a = 7 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 7, 5, 2, 3, 10, 4, 6, 9, 8\}$ \\ -$a = 8 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 8, 9, 6, 4, 10, 3, 2, 5, 7\}$ \\ -$a = 9 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 9, 4, 3, 5, 1, 9, 4, 3, 5\}$ \\ -$a = 10 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 10, 1, 10, 1, 10, 1, 10, 1, 10\}$ +\begin{tabular}{c c c c c c c} +$a = 1$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 1, 1, 1, 1, 1, 1, 1, 1, 1\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 2$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 2, 4, 8, 5, 10, 9, 7, 3, 6\}$ & $ = $ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 3$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 3, 9, 5, 4, 1, 3, 9, 5, 4\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 4$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 4, 5, 9, 3, 1, 4, 5, 9, 3\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 5$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 5, 3, 4, 9, 1, 5, 3, 4, 9\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 6$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 6, 3, 7, 9, 10, 5, 8, 4, 2\}$ & $ = $ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 7$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 7, 5, 2, 3, 10, 4, 6, 9, 8\}$ & $ = $ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 8$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 8, 9, 6, 4, 10, 3, 2, 5, 7\}$ & $ = $ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 9$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 9, 4, 3, 5, 1, 9, 4, 3, 5\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\ +$a = 10$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 10, 1, 10, 1, 10, 1, 10, 1, 10\}$ & $\neq$ & $\mathbb{F}_{11}\setminus\{0\}$ \\ \end{tabular} \end{center} -so fällt uns auf, dass für $a$ die Zahlen $2,6,7,8$ erhalten, die tatsächlich den gesamten Zahlenraum von $\mathbb{F}_{11}$ abbilden. Solche Zahlen werden \em Primitive Einheitswurzel \em genannt. -Wenden wir diese Vorgehensweise auch für andere Endliche Körper an, so werden wir sehen, dass wir immer mindestens zwei solcher Einheitswurzel finden werden. Somit ist es uns überlassen, eine dieser Einheitswurzeln auszuwählen, mit der wir weiter rechnen wollen. Für das Beispiel wählen wir die Zahl $a^i = 8$. +%\begin{center} +%\begin{tabular}{c r c l} +%%$a = 0 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{0, 0, 0, 0, 0, 0, 0, 0, 0, 0\}$ \\ +%$a = 1 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 1, 1, 1, 1, 1, 1, 1, 1, 1\}$ \\ +%$a = 2 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 2, 4, 8, 5, 10, 9, 7, 3, 6\}$ \\ +%$a = 3 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 3, 9, 5, 4, 1, 3, 9, 5, 4\}$ \\ +%$a = 4 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 4, 5, 9, 3, 1, 4, 5, 9, 3\}$ \\ +%$a = 5 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 5, 3, 4, 9, 1, 5, 3, 4, 9\}$ \\ +%$a = 6 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 6, 3, 7, 9, 10, 5, 8, 4, 2\}$ \\ +%$a = 7 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 7, 5, 2, 3, 10, 4, 6, 9, 8\}$ \\ +%$a = 8 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 8, 9, 6, 4, 10, 3, 2, 5, 7\}$ \\ +%$a = 9 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 9, 4, 3, 5, 1, 9, 4, 3, 5\}$ \\ +%$a = 10 :$& $\qquad \mathbb{Z}_{11}\setminus\{0\}$ &$=$& $\{1, 10, 1, 10, 1, 10, 1, 10, 1, 10\}$ +%\end{tabular} +%\end{center} +so fällt uns auf, dass für $a$ die Zahlen $2,6,7,8$ erhalten, die tatsächlich den gesamten Zahlenraum von $\mathbb{F}_{11}$ abbilden. Solche Zahlen werden \em primitive Einheitswurzel \em genannt. +Wenden wir diese Vorgehensweise auch für andere endliche Körper an, so werden wir sehen, dass wir immer mindestens zwei solcher Einheitswurzel finden werden. Somit ist es uns überlassen, eine dieser Einheitswurzeln auszuwählen, mit der wir weiter rechnen wollen. Für das Beispiel wählen wir die Zahl $a = 8$. \subsubsection{Bildung einer Transformationsmatrix \label{reedsolomon:subsection:transMat}} @@ -150,12 +164,13 @@ Mit der Wahl einer Einheitswurzel ist es uns jetzt möglich, unsere Nachricht zu $m(8^9) = 4 \cdot 7^5 + 7 \cdot 7^4 + 2 \cdot 7^3 + 5 \cdot 7^2 + 8 \cdot 7^1 + 1 = 4$ \end{tabular} \end{center} -unser Übertragungsvektor. Um das ganze noch ein wenig übersichtlicher zu gestalten können wir die Polynome zu einer Matrix zusammenfassen und bildet so unsere Transformationsmatrix $A$. +unser Übertragungsvektor. \subsection{Allgemeine Codierung \label{reedsolomon:subsection:algCod}} +Um das Ganze noch ein wenig übersichtlicher zu gestalten können wir die Polynome zu einer Matrix zusammenfassen, die unsere Transformationsmatrix $A$ bildet. -Für die Codierung benötigen wir die Nachricht $m$, die Codiert werden soll sowie die Transformationsmatrix $A$. Daraus erhalten wir den Übertragungsvektor $v$. Setzen wir die Zahlen aus dem Beispiel ein erhalten wir folgende Darstellung. +Für die allgemeine Codierung benötigen wir die Nachricht $m$, die codiert werden soll, sowie die Transformationsmatrix $A$. Daraus erhalten wir den Übertragungsvektor $v$. Setzen wir die Zahlen aus dem Beispiel ein erhalten wir folgende Darstellung: \[ v = A \cdot m \qquad \Rightarrow \qquad v = \begin{pmatrix} 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0& 8^0\\ @@ -173,6 +188,7 @@ v = A \cdot m \qquad \Rightarrow \qquad v = \begin{pmatrix} \begin{pmatrix} 1 \\ 8 \\ 5 \\ 2 \\ 7 \\ 4 \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{pmatrix} +. \] Für unseren Übertragungsvektor resultiert \[ diff --git a/buch/papers/reedsolomon/decmitfehler.tex b/buch/papers/reedsolomon/decmitfehler.tex index feaa027..1f195e9 100644 --- a/buch/papers/reedsolomon/decmitfehler.tex +++ b/buch/papers/reedsolomon/decmitfehler.tex @@ -5,7 +5,7 @@ % \section{Decodierung: Ansatz mit Fehlerkorrektur \label{reedsolomon:section:decmitfehler}} -\rhead{fehlerhafte rekonstruktion} +\rhead{Decodierung mit Fehler} Bisher haben wir die Decodierung unter der Bedingung durchgeführt, dass der Übertragungsvektor fehlerlos versendet und empfangen wurde. In der realen Welt müssen wir uns jedoch damit abfinden, dass kein Übertragungskanal garantiert fehlerfrei ist und das wir früher oder später mit Fehlern rechnen müssen. Genau für dieses Problem wurden Fehler korrigierende Codes, wie der Reed-Solomon-Code, entwickelt. diff --git a/buch/papers/reedsolomon/endlichekoerper.tex b/buch/papers/reedsolomon/endlichekoerper.tex index 146067a..19e5dd4 100644 --- a/buch/papers/reedsolomon/endlichekoerper.tex +++ b/buch/papers/reedsolomon/endlichekoerper.tex @@ -5,10 +5,10 @@ % \section{Reed-Solomon in Endlichen Körpern \label{reedsolomon:section:endlichekoerper}} -\rhead{Problemstellung} - -\textcolor{red}{TODO: (warten auf den 1. Teil)} - +\rhead{Reed-Solomon in endlichen Körpern} +\[ +\textcolor{red}{\text{TODO: (warten auf den 1. Teil)}} +\] Das Rechnen in endlichen Körpern bietet einige Vorteile: \begin{itemize} diff --git a/buch/papers/reedsolomon/main.tex b/buch/papers/reedsolomon/main.tex index fa20936..b4899cb 100644 --- a/buch/papers/reedsolomon/main.tex +++ b/buch/papers/reedsolomon/main.tex @@ -4,7 +4,7 @@ % (c) 2020 Hochschule Rapperswil % \chapter{Reed-Solomon-Code\label{chapter:reedsolomon}} -\lhead{Thema} +\lhead{Reed-Solomon-Code} \begin{refsection} \chapterauthor{Joshua Bär und Michael Steiner} diff --git a/buch/papers/reedsolomon/rekonstruktion.tex b/buch/papers/reedsolomon/rekonstruktion.tex index 89a700f..40919d7 100644 --- a/buch/papers/reedsolomon/rekonstruktion.tex +++ b/buch/papers/reedsolomon/rekonstruktion.tex @@ -6,7 +6,7 @@ % \section{Nachricht Rekonstruieren \label{reedsolomon:section:rekonstruktion}} -\rhead{Rekonstruktion} +\rhead{Rekonstruktion der Nachricht} Im letzten Kapitel haben wir eine Möglichkeit gefunden, wie wir die fehlerhaften Stellen lokalisieren können. Mit diesen Stellen soll es uns nun möglich sein, aus dem fehlerhaften empfangenen Nachrichtenvektor wieder unsere Nachricht zu rekonstruieren. Das Lokatorpolynom -- cgit v1.2.1 From 0359a35136adf760ebaea4d4719e7801532b7e71 Mon Sep 17 00:00:00 2001 From: "User-PC\\User" Date: Thu, 24 Jun 2021 10:37:48 +0200 Subject: Diverse Anpassungen, Nummerierung und Referenzierung auf Formeln --- buch/papers/spannung/Einleitung.tex | 6 +++--- buch/papers/spannung/teil0.tex | 4 ++-- buch/papers/spannung/teil2.tex | 31 +++++++++++++++++-------------- buch/papers/spannung/teil3.tex | 25 ++++++++++++++----------- buch/papers/spannung/teil4.tex | 4 ++-- 5 files changed, 38 insertions(+), 32 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/spannung/Einleitung.tex b/buch/papers/spannung/Einleitung.tex index 0cb1433..b1588ff 100644 --- a/buch/papers/spannung/Einleitung.tex +++ b/buch/papers/spannung/Einleitung.tex @@ -29,7 +29,7 @@ Belastet man den Boden mit einer Spannung so wird diese in den Boden geleitet und von diesem kompensiert. Im Boden entstehen unterschiedlich hohe Zusatzspannungen. Diese Zusatzspannung breitet sich räumlich im Boden aus. -Im Falle einer konstanten Flächenlast $\sigma$ (siehe Abbildung 1.1) breitet sich die Zusatzspannung zwiebelartig aus. +Im Falle einer konstanten Flächenlast $\sigma$ siehe Abbildung~\ref{spannung:Bild4} breitet sich die Zusatzspannung zwiebelartig aus. \begin{figure} \centering @@ -38,7 +38,7 @@ Im Falle einer konstanten Flächenlast $\sigma$ (siehe Abbildung 1.1) breitet si \label{fig:Bild4} \end{figure} -Mit der Tiefe $t$ nimmt diese permanent ab (siehe Abbildung 1.2). +Mit der Tiefe $t$ nimmt diese permanent ab (siehe Abbildung~\ref{spannung:Bild5}). Wie diese Geometrie der Ausbreitung ist, kann durch viele Modelle und Ansätze näherungsweise beschrieben werden. Diese Zusatzspannung $\sigma$ ist im Wesentlichen abhängig von $(x,y,t)$. Je nach Modell werden noch andere Parameter berücksichtigt. @@ -74,7 +74,7 @@ berechnet werden mit: \end{align*} Diese Zusammenhänge sind wie erwähnt unter anderem im Lehrbuch [\cite{spannung:Grundlagen-der-Geotechnik}] beschrieben. In der praktischen Geotechnik wird man allerdings weitaus schwierigere Situationen antreffen. -Ein Beispiel wäre eine Baugrube mit einem Baugrubenabschluss, wo ein Teil des Bodens abgetragen ist (siehe Abbildung 1.3). +Ein Beispiel wäre eine Baugrube mit einem Baugrubenabschluss, wo ein Teil des Bodens abgetragen ist (siehe Abbildung~\ref{spannung:Bild3}). Die Ausbreitung der Zusatzspannung $\sigma(x,y,t)$ würde hier deutlich komplizierter ausfallen. Dies bedeutet auch eine komplexere Setzung der Bodenoberfläche infolge einer Flächenlast $\sigma$. Aus allen zusätzlichen Spannungen müssen die adäquaten Dehnungen mit Hilfe einer Spannungsgleichung berechnet werden. diff --git a/buch/papers/spannung/teil0.tex b/buch/papers/spannung/teil0.tex index ffc9009..7647252 100644 --- a/buch/papers/spannung/teil0.tex +++ b/buch/papers/spannung/teil0.tex @@ -1,6 +1,6 @@ \section{Der Spannungszustand\label{spannung:section:Der Spannungsustand}} \rhead{Der Spannungszustand} -Ein Spannungszustand ist durch alle Spannungen, welche in einem beliebigen Punkt im Körper wirken, definiert (siehe Abbildung 1.4). +Ein Spannungszustand ist durch alle Spannungen, welche in einem beliebigen Punkt im Körper wirken, definiert (siehe Abbildung~\ref{spannung:Bild2}). Änderungen der äusseren Kräfte verändern die inneren Spannungszustände im Material. Um alle Spannungen eines Punktes darstellen zu können, wird ein infinitesimales Bodenelement in Form eines Würfels modellhaft vorgestellt. Man spricht auch von einem Elementarwürfel, da dieser elementar klein ist. @@ -27,7 +27,7 @@ Daher gibt es auch den entsprechenden Dehnungszustand. \section{Spannungszustand\label{spannung:section:Spannungsustand}} \rhead{Spannungszustand} -Im einachsigen Spannungszustand herrscht nur die Normalspannung $\sigma_{11}$ (siehe Abbildung 1.5). +Im einachsigen Spannungszustand herrscht nur die Normalspannung $\sigma_{11}$ (siehe Abbildung~\ref{spannung:Bild1}). Das Hook'sche Gesetz beschreibt genau diesen 1D Spannungszustand. Nach Hooke gilt: \[ diff --git a/buch/papers/spannung/teil2.tex b/buch/papers/spannung/teil2.tex index 921d2b8..6326eab 100644 --- a/buch/papers/spannung/teil2.tex +++ b/buch/papers/spannung/teil2.tex @@ -155,6 +155,17 @@ Die allgemeine Spannungsgleichung lautet nun: \overline{\overline{C}}\cdot\vec{\varepsilon} . \] + +Als Indexnotation +\[ +\sigma_{ij} += +\sum_{k=1}^3 +\sum_{l=1}^3 +C_{ijkl}\cdot\varepsilon_{kl} +\] +kann dies ebenfalls geschrieben werden. + Die Konstanten $C$ werden nun nach dem Hook'schen Gesetz mit Hilfe des Elastizitätsmoduls $E$ definiert. Da dieser Modul durch die eindimensionale Betrachtung definiert ist, muss für die dreidimensionale Betrachtung eine weitere Kennzahl eingeführt werden. @@ -208,17 +219,8 @@ definiert ist. Trägt man die Konstanten in die Matrix ein, ergibt sich \varepsilon_{32} \\ \varepsilon_{33} \end{pmatrix} -, -\] -welche ebenfalls als Indexnotation mit -\[ -\sigma_{ij} -= -\sum_{k=1}^3 -\sum_{l=1}^3 -C_{ijkl}\cdot\varepsilon_{kl} +. \] -ausgedrückt werden kann. Die Normalspannung $\sigma_{22}$ lässt sich exemplarisch als \[ \sigma_{22} @@ -308,7 +310,7 @@ und entsprechend = \begin{pmatrix} \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\ - & \varepsilon_{22} & \varepsilon_{23} \\ + & \varepsilon_{22} & \varepsilon_{23} \\ \text{sym} & & \varepsilon_{33} \end{pmatrix} \qquad @@ -397,8 +399,8 @@ Somit lässt sich die reduzierte allgemeine Spannungsgleichung mit \] beschreiben. Die Konstanten $C$ werden wieder nach dem Hook'schen Gesetz definiert. -Dies ergibt die Spannungsgleichung, welche weit möglichst vereinfacht ist: -\[ +Dies ergibt die Spannungsformel, welche weit möglichst vereinfacht ist: +\begin{equation} \begin{pmatrix} \sigma_{11}\\ \sigma_{22}\\ @@ -426,7 +428,8 @@ Dies ergibt die Spannungsgleichung, welche weit möglichst vereinfacht ist: \varepsilon_{12} \end{pmatrix} . -\] +\label{spannung:Spannungsgleichung} +\end{equation} Im Elastizitätstensor fallen zwei $3\times3$ Blöcke auf, welche nur Einträge mit $0$ haben. Der Tensor besagt also, dass diese jeweiligen Dehnungen keinen Einfluss auf unsere Spannung haben. diff --git a/buch/papers/spannung/teil3.tex b/buch/papers/spannung/teil3.tex index 8d99733..3e456c3 100644 --- a/buch/papers/spannung/teil3.tex +++ b/buch/papers/spannung/teil3.tex @@ -14,29 +14,31 @@ Folglich gilt: Dadurch wird der Spannungszustand vereinfacht. Diesen vereinfachten Spannungszustand kann man mit den zwei geotechnischen Invarianten abbilden. Die erste Invariante ist die volumetrische Spannung -\[ +\begin{equation} p = \frac{\sigma_{11}+\sigma_{22}+\sigma_{33}}{3} +\label{spannung:Invariante_p} , -\] +\end{equation} welche als arithmetisches Mittel aller Normalspannungen im infinitesimalen Würfel definiert ist. Die zweite Invariante ist die deviatorische Spannung -\[ +\begin{equation} q = \sqrt{\frac{(\sigma_{11}-\sigma_{22})^{2}+(\sigma_{11}-\sigma_{33})^{2}+(\sigma_{22}-\sigma_{33})^{2}}{2}} +\label{spannung:Invariante_q} . -\] +\end{equation} Diese Zusammenhänge werden im Skript [\cite{spannung:Stoffgesetze-und-numerische-Modellierung-in-der-Geotechnik}] aufgezeigt. -Die hydrostatische Spannung $p$ kann gemäss Gleichung (Nr) als +Die hydrostatische Spannung $p$ kann gemäss Gleichung \eqref{spannung:Invariante_p} als \[ p = \frac{\sigma_{11}+2\sigma_{33}}{3} \] vereinfacht werden. -Die deviatorische Spannung $q$ wird gemäss Gleichung (Nr) als +Die deviatorische Spannung $q$ wird gemäss Gleichung \eqref{spannung:Invariante_q}als \[ q = @@ -44,7 +46,7 @@ q \] vereinfacht. Man kann $p$ als Isotrop und $q$ als Schub betrachten. -Die Invarianten können mit der Spannungsformel (Nr..xxx) berechnet werden. +Die Invarianten können mit der Spannungsformel \eqref{spannung:Spannungsgleichung} berechnet werden. Durch geschickte Umformung dieser Gleichung, lassen sich die Module als Faktor separieren. Dabei entstehen spezielle Faktoren mit den Dehnungskomponenten. So ergibt sich @@ -81,7 +83,7 @@ Die hydrostatische Dehnung $\varepsilon_{v}$ kann mit einer Kompression verglich Die deviatorische Dehnung $\varepsilon_{s}$ kann mit einer Verzerrung verglichen werden. Diese zwei Gleichungen kann man durch die Matrixschreibweise -\[ +\begin{equation} \begin{pmatrix} q\\ p @@ -95,11 +97,12 @@ Diese zwei Gleichungen kann man durch die Matrixschreibweise \varepsilon_{s}\\ \varepsilon_{v} \end{pmatrix} -\] -(sollte nummeriert sein) vereinfachen. +\label{spannung:Matrixschreibweise} +\end{equation} +vereinfachen. Man hat so eine Matrix multipliziert mit einem Vektor und erhält einen Vektor. Änderungen des Spannungszustandes können mit dieser Gleichung vollumfänglich erfasst werden. -Mit dieser Formel lassen sich verschieden Ergebnisse von Versuchen analysieren und berechnen. +Mit dieser Formel \eqref{spannung:Matrixschreibweise} lassen sich verschieden Ergebnisse von Versuchen analysieren und berechnen. Ein solcher Versuch, den oft in der Geotechnik durchgeführt wird, ist der Oedometer-Versuch. Im nächsten Kapitel wird die Anwendung der Matrix an diesem Versuch beschrieben. diff --git a/buch/papers/spannung/teil4.tex b/buch/papers/spannung/teil4.tex index d524f13..2f2e4ce 100644 --- a/buch/papers/spannung/teil4.tex +++ b/buch/papers/spannung/teil4.tex @@ -34,7 +34,7 @@ Die Spannung $\sigma_{11}$ wird durch die aufgebrachte Kraft mit \frac{F}{A} \] und die Dehnung $\varepsilon_{11}$ jeweils mit den entsprechenden Setzungen berechnet. -Diese Randbedingungen können in die vereinfachte Gleichung (Nrxxx) eingesetzt werden. +Diese Randbedingungen können in die vereinfachte Gleichung \eqref{spannung:Matrixschreibweise} eingesetzt werden. Diese lautet nun: \[ \begin{pmatrix} @@ -67,7 +67,7 @@ und berechnen. Mit diesen Gleichungen hat man das Gleichungssystem um $E_{OED}$ und $\sigma_{33}$ zu berechnen. Die Poisson-Zahl muss als Kennwert gemäss der Bodenklasse gewählt werden. -Den Versuch kann man auf einem $\sigma$-$\varepsilon$-Diagramm abtragen (siehe Abbildung 1.7). +Den Versuch kann man auf einem $\sigma$-$\varepsilon$-Diagramm abtragen (siehe Abbildung~\ref{spannung:DiagrammOedometer-Versuch}). Durch die Komprimierung nimmt der Boden mehr Spannung auf, und verformt sich zugleich weniger stark. Mit diesem ermittelten $E_{OED}$ kann man nun weitere Berechnungen für die Geotechnik durchführen. -- cgit v1.2.1 From fa0d3a4ead1df4b8587035b8b62b42375f970ba9 Mon Sep 17 00:00:00 2001 From: michael-OST <75078383+michael-OST@users.noreply.github.com> Date: Thu, 24 Jun 2021 18:47:13 +0200 Subject: all files updated and corrected --- buch/papers/reedsolomon/codebsp.tex | 6 +- buch/papers/reedsolomon/decmitfehler.tex | 43 ++++++----- buch/papers/reedsolomon/decohnefehler.tex | 115 +++++++++++++++++++++++++--- buch/papers/reedsolomon/hilfstabellen.tex | 2 +- buch/papers/reedsolomon/main.tex | 6 +- buch/papers/reedsolomon/nachschlagewerk.tex | 4 - buch/papers/reedsolomon/rekonstruktion.tex | 9 ++- buch/papers/reedsolomon/zusammenfassung.tex | 15 ++++ 8 files changed, 156 insertions(+), 44 deletions(-) delete mode 100644 buch/papers/reedsolomon/nachschlagewerk.tex create mode 100644 buch/papers/reedsolomon/zusammenfassung.tex (limited to 'buch/papers') diff --git a/buch/papers/reedsolomon/codebsp.tex b/buch/papers/reedsolomon/codebsp.tex index 6ab792a..0339d9c 100644 --- a/buch/papers/reedsolomon/codebsp.tex +++ b/buch/papers/reedsolomon/codebsp.tex @@ -147,12 +147,12 @@ $a = 10$ & $\Rightarrow$ & $\{a^i | 0 \le i \le 10\}$ & $=$ & $\{1, 10, 1, 10, 1 %\end{tabular} %\end{center} so fällt uns auf, dass für $a$ die Zahlen $2,6,7,8$ erhalten, die tatsächlich den gesamten Zahlenraum von $\mathbb{F}_{11}$ abbilden. Solche Zahlen werden \em primitive Einheitswurzel \em genannt. -Wenden wir diese Vorgehensweise auch für andere endliche Körper an, so werden wir sehen, dass wir immer mindestens zwei solcher Einheitswurzel finden werden. Somit ist es uns überlassen, eine dieser Einheitswurzeln auszuwählen, mit der wir weiter rechnen wollen. Für das Beispiel wählen wir die Zahl $a = 8$. +Wenden wir diese Vorgehensweise auch für andere endliche Körper an, so werden wir sehen, dass wir immer mindestens zwei solcher Einheitswurzel finden werden. Somit ist es uns überlassen, eine dieser Einheitswurzel auszuwählen, mit der wir weiter rechnen wollen. Für das Beispiel wählen wir die Zahl $a = 8$. \subsubsection{Bildung einer Transformationsmatrix \label{reedsolomon:subsection:transMat}} -Mit der Wahl einer Einheitswurzel ist es uns jetzt möglich, unsere Nachricht zu Codieren. Daraus sollen wir dann einen Übertragungsvektor $v$ erhalten, den wir an den Empfänger schicken können. Für die Codierung müssen wir alle $a^i$ in das Polynom $m(X)$ einsetzen. Da wir $a^i = 8^i$ gewählt haben ergibt sich daraus +Mit der Wahl einer Einheitswurzel ist es uns jetzt möglich, unsere Nachricht zu Codieren. Daraus sollen wir dann einen Übertragungsvektor $v$ erhalten, den wir an den Empfänger schicken können. Für die Codierung müssen wir alle $a^i$ in das Polynom $m(X)$ einsetzen. Da wir $a^i = 8^i$ gewählt haben, ergibt sich daraus % %Damit wir unsere Nachricht codieren können, müssen wir $8^i$ in $m(X)$ einsetzen. % @@ -164,7 +164,7 @@ Mit der Wahl einer Einheitswurzel ist es uns jetzt möglich, unsere Nachricht zu $m(8^9) = 4 \cdot 7^5 + 7 \cdot 7^4 + 2 \cdot 7^3 + 5 \cdot 7^2 + 8 \cdot 7^1 + 1 = 4$ \end{tabular} \end{center} -unser Übertragungsvektor. +als unser Übertragungsvektor. \subsection{Allgemeine Codierung \label{reedsolomon:subsection:algCod}} diff --git a/buch/papers/reedsolomon/decmitfehler.tex b/buch/papers/reedsolomon/decmitfehler.tex index 1f195e9..a46d7da 100644 --- a/buch/papers/reedsolomon/decmitfehler.tex +++ b/buch/papers/reedsolomon/decmitfehler.tex @@ -10,12 +10,12 @@ Bisher haben wir die Decodierung unter der Bedingung durchgeführt, dass der Üb In der realen Welt müssen wir uns jedoch damit abfinden, dass kein Übertragungskanal garantiert fehlerfrei ist und das wir früher oder später mit Fehlern rechnen müssen. Genau für dieses Problem wurden Fehler korrigierende Codes, wie der Reed-Solomon-Code, entwickelt. In diesem Abschnitt betrachten wir somit die Idee der Fehlerkorrektur und wie wir diese auf unser Beispiel anwenden können. + Der Übertragungskanal im Beispiel weisst jetzt den Fehlervektor \[ u = [0, 0, 0, 3, 0, 0, 0, 0, 2, 0] \] auf. - Senden wir jetzt unser Übertragungsvektor $v$ durch diesen Kanal addiert sich der Fehlervektor $u$ auf unsere Übertragung und wir erhalten \begin{center} @@ -73,10 +73,10 @@ als neuen, fehlerbehafteten Übertragungsvektor $w$ auf der Empfängerseite. % %\end{center} %als Übertragungsvektor auf der Empfängerseite erhalten. -Wir jetzt als Empfänger wissen jedoch nicht, dass der erhaltene Übertragungsvektor jetzt fehlerbehaftet ist und werden dementsprechend den Ansatz aus Abschnitt \ref{reedsolomon:section:decohnefehler} anwenden. +Als Empfänger wissen wir jedoch nicht, dass der erhaltene Übertragungsvektor jetzt fehlerbehaftet ist und werden dementsprechend den Ansatz aus Abschnitt \ref{reedsolomon:section:decohnefehler} anwenden. Wir stellen jedoch recht schnell fest, dass am decodierten Nachrichtenblock \[ -r = [\underbrace{5,7,4,10,}_{\text{Syndrom}}5,4,5,7,6,7]. +r = [\underbrace{5,7,4,10,}_{\text{Syndrom}}5,4,5,7,6,7] \] etwas nicht in Ordnung ist, denn die vorderen vier Fehlerkorrekturstellen haben nicht mehr den Wert null. Der Nachrichtenblock weisst jetzt ein \em Syndrom \em auf, welches anzeigt, dass der Übertragungsvektor fehlerhaft empfangen wurde. @@ -85,21 +85,29 @@ Der Nachrichtenblock weisst jetzt ein \em Syndrom \em auf, welches anzeigt, dass %\[ %r = [\underbrace{5,7,4,10,}_{Fehlerinfo}5,4,5,7,6,7]. %\] -Jetzt stellt sich natürlich die Frage, wie wir daraus den ursprünglich gesendeten Nachrichtenvektor zurückerhalten sollen. Laut der Definition über die Funktionsweise eines Reed-Solomon-Codes können wir aus den Fehlerkorrekturstellen ein ``Lokatorpolynom'' berechnen, welches die Information enthält, welche stellen innerhalb des empfangenen Übertragungsvektors fehlerhaft sind. +Jetzt stellt sich natürlich die Frage, wie wir daraus den ursprünglich gesendeten Nachrichtenvektor zurückerhalten sollen. Laut der Definition über die Funktionsweise eines Reed-Solomon-Codes können wir aus den Fehlerkorrekturstellen ein ``Lokatorpolynom'' berechnen, welches die Information enthält, welche Stellen innerhalb des empfangenen Übertragungsvektors fehlerhaft sind. \subsection{Das Fehlerstellenpolynom $d(X)$ \label{reedsolomon:subsection:fehlerpolynom}} -Bevor wir unser Lokatorpolynom berechnen können, müssen wir zuerst eine Möglichkeit finden, die Fehlerhaften von den Korrekten Stellen im Übertragungsvektor unterscheiden zu können. In einem ersten Versuch könnten wir $d$ berechnen mit +Bevor wir unser Lokatorpolynom berechnen können, müssen wir zuerst eine Möglichkeit finden, die fehlerhaften von den korrekten Stellen im Übertragungsvektor unterscheiden zu können. +In einem ersten Versuch berechnen wir die Differenz $d$ des empfangenen und dem gesendeten Übertragungsvektor mit +%Alle Stellen in $d$, die nicht null sind sind demnach fehler. +% +%In einem ersten Versuch könnten wir $d$ berechnen mit \begin{center} \begin{tabular}{r c l} $m(X)$ & $=$ & $4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$ \\ $r(X)$ & $=$ & $5X^9 + 7X^8 + 4X^7 + 10X^6 + 5X^5 + 4X^4 + 5X^3 + 7X^2 + 6X + 7$ \\ - $d(X)$ & $=$ & $r(X) - m(X)$. + $d(X)$ & $=$ & $r(X) - m(X)$ \end{tabular} \end{center} -Dies wird uns zwar andere sorgen wegen $m(X)$ bereiten, wir werden werden deshalb erst in Abschnitt \ref{reedsolomon:subsection:nachrichtenvektor} darauf zurückkommen. +und nennen $d(X)$ als unseres Fehlerstellenpolynom. Dieses Polynom soll uns sagen, welche Stellen korrekt und welche fehlerhaft sind. -Setzen wir jetzt noch unsere Einheitswurzel aus dem Beispiel ein so erhalten wir +Durch das verwenden von $m(X)$ stossen wir auf weitere Probleme, da wir den Nachrichtenvektor auf der Empfängerseite nicht kennen (unser Ziel ist es ja genau diesen zu finden). Dieses Problem betrachten wir im Abschnitt \ref{reedsolomon:subsection:nachrichtenvektor} genauer. Um die Überlegungen in den folgenden Abschnitten besser zu verstehen sei $m(X)$ bekannt auf der Empfängerseite. + +%Dies wird uns zwar andere sorgen wegen $m(X)$ bereiten, wir werden werden deshalb erst in Abschnitt \ref{reedsolomon:subsection:nachrichtenvektor} darauf zurückkommen. + +Setzen wir jetzt unsere Einheitswurzel aus dem Beispiel ein so erhalten wir % Old Text %\begin{align} % m(X) & = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1 \\ @@ -126,15 +134,15 @@ Für das einfache Bestimmen von Hand mag dies ja noch ausreichen, jedoch können \subsection{Mit dem grössten gemeinsamen Teiler auf Nullstellenjagd \label{reedsolomon:subsection:ggT}} -Zuerst betrachten wir mal den Satz von Fermat deren Funktionsweise wir in Abschnitt \ref{buch:section:galoiskoerper} kennengelernt haben. Der besagt, dass für +Zuerst betrachten wir den Satz von Fermat, dessen Funktionsweise wir in Abschnitt \ref{buch:section:galoiskoerper} kennengelernt haben. Der besagt, dass \[ f(X) = X^{q-1} -1 = 0 \] -wobei dies für jedes $q$ gilt. Setzen wir also das $q$ von unserem Beispiel ein +gilt für jedes $X$. Setzen wir das $q$ von unserem Beispiel ein \[ f(X) = X^{10}-1 = 0 \qquad \text{für } X = \{1,2,3,4,5,6,7,8,9,10\} \] -und stellen dies als Nullstellenform (\textcolor{red}{richtiger name für die Schreibweise?}) dar. So ergibt sich die Darstellung +und stellen dies als Faktorisierung dar. So ergibt sich die Darstellung \[ f(X) = (X-a^0)(X-a^1)(X-a^2)(X-a^3)(X-a^4)(X-a^5)(X-a^6)(X-a^7)(X-a^8)(X-a^9). \] @@ -145,7 +153,7 @@ Wir können jetzt auch $d(X)$ nach der gleichen Überlegung darstellen als d(X) = (X-a^0)(X-a^1)(X-a^2)\textcolor{gray!40}{(X-a^3)}(X-a^4)(X-a^5)(X-a^6)(X-a^7)\textcolor{gray!40}{(X-a^8)}(X-a^9) \cdot p(x), \] wobei diese Darstellung nicht mehr alle Nullstellen umfasst wie es noch in $f(X)$ der Fall war. -Dies liegt daran, dass wir ja zwei Fehlerstellen (grau markiert) haben, die nicht Null sind. Diese fassen wir zum Restpolynom $p(X)$ (\textcolor{red}{eventuell farblich kennzeichnen?}) zusammen. +Dies liegt daran, dass wir ja zwei Fehlerstellen (grau markiert) haben, die nicht Null sind. Diese fassen wir zum Restpolynom $p(X)$ zusammen. Wenn wir jetzt den grössten gemeinsamen Teiler von $f(X)$ und $d(X)$ berechnen, so erhalten wir mit \[ \operatorname{ggT}(f(X),d(X)) = (X-a^0)(X-a^1)(X-a^2)\textcolor{gray!40}{(X-a^3)}(X-a^4)(X-a^5)(X-a^6)(X-a^7)\textcolor{gray!40}{(X-a^8)}(X-a^9) @@ -174,7 +182,7 @@ l(X) = (X-a^3)(X-a^8) \] unser gesuchtes Lokatorpolynom. Es scheint so als müssten wir nur noch an den besagten Stellen den Übertragungsvektor korrigieren und wir währen fertig mit der Fehlerkorrektur. -Jedoch haben wir noch ein grundlegendes Problem, dass zu beginn aufgetaucht ist, wir aber beiseite geschoben haben. Die Rede ist natürlich vom Nachrichtenvektor $m(X)$, mit dem wir in erster Linie das wichtige Fehlerstellenpolynom $d(X)$ berechnet haben. +Jedoch haben wir noch ein grundlegendes Problem, dass zu Beginn aufgetaucht ist, wir aber beiseite geschoben haben. Die Rede ist natürlich vom Nachrichtenvektor $m(X)$, mit dem wir in erster Linie das wichtige Fehlerstellenpolynom $d(X)$ berechnet haben, auf der Empfängerseite aber nicht kennen. \subsection{Der problematische Nachrichtenvektor $m(X)$ \label{reedsolomon:subsection:nachrichtenvektor}} @@ -190,20 +198,18 @@ Wir könnten uns höchstens noch fragen, ob wir tatsächlich nichts über den Na \[ m = [0,0,0,0,4,7,2,5,8,1] \] -fällt uns aber auf, dass wir doch etwas über diesen Vektor wissen, nämlich den Wert der ersten 2t (im Beispiel vier) stellen. -Im Normalfall sollen diese nämlich den Wert null betragen und somit sind nur die letzten k stellen (im Beispiel sechs) für uns unbekannt, dargestellt als +fällt uns aber auf, dass wir doch etwas über diesen Vektor wissen, nämlich den Wert der ersten $2t$ (im Beispiel vier) stellen. +Im Normalfall sollen diese nämlich den Wert null betragen und somit sind nur die letzten $k$ stellen (im Beispiel sechs) für uns unbekannt, dargestellt als \[ m = [0,0,0,0,?,?,?,?,?,?]. \] -Wie der Zufall es so will liegt an diesen vier Stellen auch die Information, wo die Fehlerstellen liegen. Daher reicht es auch aus +Nach der Definition des Reed-Solomon-Codes soll an genau diesen vier Stellen auch die Information befinden, wo die Fehlerstellen liegen. Daher reicht es auch aus % darum werden die stellen auch als fehlerkorrekturstellen bezeichnet \[ d(X) = 5X^9 + 7X^8 + 4X^7 + 10X^6 + p(X) \] so zu berechnen, dass wir die wichtigen vier Stellen kennen, der Rest des Polynoms jedoch im unbekannten Restpolynom $p(X)$ enthalten ist. -\textcolor{red}{ist das wechseln zwischen 2t,k aus dem allgemeinfall und vier,sechs aus dem beispiel zu verwirrend?} - \subsection{Die Berechnung der Fehlerstellen \label{reedsolomon:subsection:nachrichtenvektor}} @@ -294,6 +300,7 @@ Daraus erhalten wir die Faktoren \[ l(X) = 2X^2 + 5 \qquad \rightarrow \qquad l(X) = 2(X-5)(X-6). \] +\subsubsection{Schritt 3: Fehlerstellen bestimmen} Unser gesuchtes Lokatorpolynom hat also die Form \[ l(X) = (X-a^i)(X-a^j). diff --git a/buch/papers/reedsolomon/decohnefehler.tex b/buch/papers/reedsolomon/decohnefehler.tex index 3b709f3..0470db0 100644 --- a/buch/papers/reedsolomon/decohnefehler.tex +++ b/buch/papers/reedsolomon/decohnefehler.tex @@ -5,7 +5,7 @@ % \section{Decodierung: Ansatz ohne Fehler \label{reedsolomon:section:decohnefehler}} -\rhead{fehlerlose rekonstruktion} +\rhead{Decodierung ohne Fehler} In diesem Abschnitt betrachten wie die Überlegung, wie wir auf der Empfängerseite die Nachricht aus dem empfangenen Übertragungsvektor erhalten. Nach einer einfachen Überlegung müssen wir den Übertragungsvektor decodieren, was auf den ersten Blick nicht allzu kompliziert sein sollte, solange wir davon ausgehen können, dass es während der Übertragung keine Fehler gegeben hat. Wir betrachten deshalb den Übertragungskanal als fehlerfrei. @@ -33,7 +33,7 @@ Definiert ist sie als \[ F(\omega) = \int_{-\infty}^{\infty} f(t) \mathrm{e}^{-j\omega t} dt \qquad \Rightarrow \qquad \mathfrak{F}^{-1}(F(\omega)) = f(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) \mathrm{e}^{j \omega t} d\omega. \] -Damit beschäftigen wir uns im Abschnitt \ref{reedsolomon:subsection:algdec} weiter, konkret suchen wir momentan aber eine Inverse für unsere primitive Einheitswurzel $a$. +Damit beschäftigen wir uns im Abschnitt \ref{reedsolomon:subsection:sfaktor} weiter, konkret suchen wir momentan aber eine Inverse für unsere primitive Einheitswurzel $a$. \[ 8^1 \qquad \rightarrow \qquad 8^{-1} \] @@ -45,7 +45,7 @@ Mit einem solchen Problem haben wir uns bereits in Abschnitt \ref{buch:section:e \subsection{Inverse der primitiven Einheitswurzel \label{reedsolomon:subsection:invEinh}} -Die Funktionsweise des euklidischen Algorithmus ist im Kapitel \ref{buch:section:euklid} ausführlich beschrieben. +Die Funktionsweise des euklidischen Algorithmus ist im Abschnitt \ref{buch:section:euklid} ausführlich beschrieben. Für unsere Anwendung wählen wir die Parameter $a = 8$ und $b = 11$ ($\mathbb{F}_{11}$). Daraus erhalten wir @@ -76,21 +76,112 @@ Daraus erhalten wir \end{tabular} \end{center} -als Inverse der primitiven Einheitswurzel. Die inverse Transformationsmatrix $A^{-1}$ bilden wir indem wir jetzt die inverse primitive Einheitswurzel anstelle der primitiven Einheitswurzel in die Matrix einsetzen. +als Inverse der primitiven Einheitswurzel. Die inverse Transformationsmatrix $A^{-1}$ bilden wir, indem wir jetzt die inverse primitive Einheitswurzel anstelle der primitiven Einheitswurzel in die Matrix einsetzen: +\[ +\begin{pmatrix} + 8^0 & 8^0 & 8^0 & 8^0 & \dots & 8^0 \\ + 8^0 & 8^{-1} & 8^{-2} & 8^{-3} & \dots & 8^{-9} \\ + 8^0 & 8^{-2} & 8^{-4} & 8^{-6} & \dots & 8^{-18} \\ + 8^0 & 8^{-3} & 8^{-6} & 8^{-9} & \dots & 8^{-27} \\ + \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ + 8^0 & 8^{-9} & 8^{-18} & 8^{-27} & \dots & 8^{-81} \\ +\end{pmatrix} +\qquad +\Rightarrow +\qquad +\begin{pmatrix} + 7^0 & 7^0 & 7^0 & 7^0 & \dots & 7^0 \\ + 7^0 & 7^{1} & 7^{2} & 7^{3} & \dots & 7^{9} \\ + 7^0 & 7^{2} & 7^{4} & 7^{6} & \dots & 7^{18} \\ + 7^0 & 7^{3} & 7^{6} & 7^{9} & \dots & 7^{27} \\ + \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ + 7^0 & 7^{9} & 7^{18} & 7^{27} & \dots & 7^{81} \\ +\end{pmatrix} +\] -\subsection{Allgemeine Decodierung - \label{reedsolomon:subsection:algdec}} +\subsection{Der Faktor $s$ + \label{reedsolomon:subsection:sfaktor}} +Die diskrete Fouriertransformation benötigt für die Inverse einen Vorfaktor von $\frac{1}{2\pi}$. +Primitiv nehmen wir an, dass wir für die Inverse Transformationsmatrix ebenfalls einen benötigen. +Nur stellt sich jetzt die Frage, wie wir diesen Vorfaktor in unserem Fall ermitteln können. +Dafür betrachten wir eine Regel aus der Linearen Algebra, nämlich dass -Wir haben jetzt fast alles für eine erfolgreiche Rücktransformation beisammen. Wir haben aber noch nicht alle Aspekte der inversen diskreten Fouriertransformation befolgt, so fehlt uns noch einen Vorfaktor \[ -m = \textcolor{red}{s} \cdot A^{-1} \cdot v +A \cdot A^{-1} = E +\] +entsprechen muss. +Ist dies nicht der Fall, so benötigt $A^{-1}$ eben genau diesen Korrekturfaktor und ändert die Gleichung so zu +\begin{equation} + A \cdot s \cdot A^{-1} = E. + \label{reedsolomon:equation:sfaktor} +\end{equation} +%\[ +%A \cdot s \cdot A^{-1} = E. +%\] +Somit sollte es für uns ein leichtes Spiel sein, $s$ für unser Beispiel zu ermitteln: +\[ +\begin{pmatrix} + 8^0 & 8^0 & 8^0 & \dots & 8^0 \\ + 8^0 & 8^1 & 8^2 & \dots & 8^9 \\ + 8^0 & 8^2 & 8^4 & \dots & 8^{18} \\ + \vdots & \vdots & \vdots & \ddots & \vdots \\ + 8^0 & 8^9 & 8^{18} & \dots & 8^{81} \\ +\end{pmatrix} +\cdot +\begin{pmatrix} + 7^0 & 7^0 & 7^0 & \dots & 7^0 \\ + 7^0 & 7^{1} & 7^{2} & \dots & 7^{9} \\ + 7^0 & 7^{2} & 7^{4} & \dots & 7^{18} \\ + \vdots & \vdots & \vdots & \ddots & \vdots \\ + 7^0 & 7^{9} & 7^{18} & \dots & 7^{81} \\ +\end{pmatrix} += +\begin{pmatrix} + 10 & 0 & 0 & \dots & 0 \\ + 0 & 10 & 0 & \dots & 0 \\ + 0 & 0 & 10 & \dots & 0 \\ + \vdots & \vdots & \vdots & \ddots & \vdots \\ + 0 & 0 & 0 & \dots & 10 \\ +\end{pmatrix} \] -den wir noch bestimmen müssen. -Glücklicherweise lässt der sich analog wie bei der inversen diskreten Fouriertransformation bestimmen und beträgt +Aus der letzten Matrix folgt, dass wir \[ -s = \frac{1}{10}. +s = \dfrac{1}{10} \] -Da $\frac{1}{10} = 10^{-1}$ entspricht können wir $s$ ebenfalls mit dem euklidischen Algorithmus bestimmen und stellen fest, dass $10^{-1} = 10$ in $\mathbb{F}_{11}$ ergibt. Somit lässt sich der Nachrichtenvektor einfach bestimmen mit +als unseren Vorfaktor setzen müssen um die Gleichung \ref{reedsolomon:equation:sfaktor} zu erfüllen. Da wir in $\mathbb{F}_{11}$ nur mit ganzen Zahlen arbeiten schreiben wir $\frac{1}{10}$ in $10^{-1}$ um und bestimmen diese Inverse erneut mit dem euklidischen Algorithmus und erhalten für $10^{-1} = 10$ als unseren Vorfaktor in $\mathbb{F}_{11}$. +% +%erfüllt wird. Wir schreiben den Bruch um in $\frac{1}{10} = 10^{-1}$ und wenden darauf erneut den euklidischen Algorithmus an und erhalten somit den Vorfaktor $10^{-1} = 10 = s$ in $\mathbb{F}_{11}$. +% +%Um $s$ eindeutig zu bestimmen müssen wir $\frac{1}{10}$ nur noch in den Bereich von $\mathbb{F}_{11}$ verschieben. Wie sich herausstellt können wir das recht einfach bewerkstelligen, da $\frac{1}{10} = 10^{-1}$ entspricht. Daraus können wir $s$ mit dem euklidischen Algorithmus bestimmen und stellen fest, dass $10^{-1} = 10$ in $\mathbb{F}_{11}$ ergibt. +% +%Da $s$ jetzt ein Bruch ist brauchen wir ihn nur noch in $\mathbb{F}_{11}$ zu schieben. Praktischerweise können wir $\frac{1}{10} = 10^{-1}$ darstellen +% +%Da $\frac{1}{10} = 10^{-1}$ entspricht können wir $s$ ebenfalls mit dem euklidischen Algorithmus bestimmen und stellen fest, dass $10^{-1} = 10$ in $\mathbb{F}_{11}$ ergibt. +% +%Daher nehmen wir an, dass wir für die Inverse Transformationsmatrix ebenfalls ein solcher Vorfaktor benötigen. Dieser Faktor hat seinen Ursprung in der Gleichung +%\[ +%A \cdot A^{-1} = E. +%\] +%Sollte diese Gleichung nicht aufgehen, so muss die Inverse mit +\subsection{Allgemeine Decodierung + \label{reedsolomon:subsection:algdec}} + +Wir haben jetzt alles für eine erfolgreiche Rücktransformation vom empfangenen Nachrichtenvektor beisammen. Die allgemeine Gleichung für die Rücktransformation lautet +\[ +m = s \cdot A^{-1} \cdot v. +\] +Setzen wir nun die Werte ein in +% +%Wir haben aber noch nicht alle Aspekte der inversen diskreten Fouriertransformation befolgt, so fehlt uns noch einen Vorfaktor +%\[ +%m = \textcolor{red}{s} \cdot A^{-1} \cdot v +%\] +%den wir noch bestimmen müssen. +%Glücklicherweise lässt der sich analog wie bei der inversen diskreten Fouriertransformation bestimmen und beträgt +%\[ +%s = \frac{1}{10}. +%\] +%Da $\frac{1}{10} = 10^{-1}$ entspricht können wir $s$ ebenfalls mit dem euklidischen Algorithmus bestimmen und stellen fest, dass $10^{-1} = 10$ in $\mathbb{F}_{11}$ ergibt. Somit lässt sich der Nachrichtenvektor einfach bestimmen mit \[ m = 10 \cdot A^{-1} \cdot v \qquad \Rightarrow \qquad m = 10 \cdot \begin{pmatrix} 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0& 7^0\\ diff --git a/buch/papers/reedsolomon/hilfstabellen.tex b/buch/papers/reedsolomon/hilfstabellen.tex index 4e39de5..b006f21 100644 --- a/buch/papers/reedsolomon/hilfstabellen.tex +++ b/buch/papers/reedsolomon/hilfstabellen.tex @@ -4,7 +4,7 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{$\mathbb{F}_{11}$ Hilfstabellen +\section{Hilfstabellen für $\mathbb{F}_{11}$ \label{reedsolomon:section:hilfstabellen}} \rhead{Hilfstabellen} diff --git a/buch/papers/reedsolomon/main.tex b/buch/papers/reedsolomon/main.tex index b4899cb..4e2fd60 100644 --- a/buch/papers/reedsolomon/main.tex +++ b/buch/papers/reedsolomon/main.tex @@ -1,7 +1,7 @@ % % main.tex -- Paper zum Thema % -% (c) 2020 Hochschule Rapperswil +% (c) 2021 Joshua Bär und Michael Steiner, Hochschule Rapperswil % \chapter{Reed-Solomon-Code\label{chapter:reedsolomon}} \lhead{Reed-Solomon-Code} @@ -39,9 +39,9 @@ Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren \input{papers/reedsolomon/decohnefehler} \input{papers/reedsolomon/decmitfehler} \input{papers/reedsolomon/rekonstruktion} -\input{papers/reedsolomon/nachschlagewerk} -\input{papers/reedsolomon/hilfstabellen} +\input{papers/reedsolomon/zusammenfassung} %\input{papers/reedsolomon/anwendungen} -> geplant +\input{papers/reedsolomon/hilfstabellen} \nocite{reedsolomon:weitz} \nocite{reedsolomon:informationkommunikation} diff --git a/buch/papers/reedsolomon/nachschlagewerk.tex b/buch/papers/reedsolomon/nachschlagewerk.tex deleted file mode 100644 index 60b857e..0000000 --- a/buch/papers/reedsolomon/nachschlagewerk.tex +++ /dev/null @@ -1,4 +0,0 @@ -\section{Nachschlagewerk - \label{reedsolomon:section:nachschlagen}} -\rhead{nachschlagewerk} -todo: auflistung von z.b nachrichtenvektor, übertragungsvektor usw. inklusiver erklärung was es ist falls man beim lesen den faden verliert \ No newline at end of file diff --git a/buch/papers/reedsolomon/rekonstruktion.tex b/buch/papers/reedsolomon/rekonstruktion.tex index 40919d7..04e748c 100644 --- a/buch/papers/reedsolomon/rekonstruktion.tex +++ b/buch/papers/reedsolomon/rekonstruktion.tex @@ -7,7 +7,7 @@ \section{Nachricht Rekonstruieren \label{reedsolomon:section:rekonstruktion}} \rhead{Rekonstruktion der Nachricht} -Im letzten Kapitel haben wir eine Möglichkeit gefunden, wie wir die fehlerhaften Stellen lokalisieren können. +Im letzten Abschnitt haben wir eine Möglichkeit gefunden, wie wir die fehlerhaften Stellen lokalisieren können. Mit diesen Stellen soll es uns nun möglich sein, aus dem fehlerhaften empfangenen Nachrichtenvektor wieder unsere Nachricht zu rekonstruieren. Das Lokatorpolynom \[ @@ -21,7 +21,7 @@ Als Ausgangslage verwenden wir die Matrix, mit der wir den Nachrichtenvektor urs Unser Ziel ist es wie auch schon im Abschnitt \ref{reedsolomon:section:decohnefehler} eine Möglichkeit zu finden, wie wir den Übertragungsvektor decodieren können. Aufgrund der Fehlerstellen müssen wir aber davon ausgehen, das wir nicht mehr den gleichen Weg verfolgen können wie wir im Abschnitt \ref{reedsolomon:section:decohnefehler} angewendet haben. -Wir stellen also die Matrix auf und markieren gleichzeitig die Fehlerstellen. +Wir stellen also die Matrix auf und markieren gleichzeitig die Fehlerstellen: \[ \textcolor{gray}{ \begin{pmatrix} @@ -47,8 +47,9 @@ Wir stellen also die Matrix auf und markieren gleichzeitig die Fehlerstellen. \begin{pmatrix} m_0 \\ m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \\ m_6 \\ m_7 \\ m_8 \\ m_9 \\ \end{pmatrix} +. \] -Die rot markierten Stellen im Übertragungsvektor enthalten Fehler und bringt uns daher kein weiterer Nutzen. +Die rot markierten Stellen im Übertragungsvektor enthalten Fehler und bringt uns daher keinen weiterer Nutzen. Aus diesem Grund werden diese Stellen aus dem Vektor entfernt, was wir hier ohne Probleme machen können, da dieser Code ja über Fehlerkorrekturstellen verfügt, deren Aufgabe es ist, eine bestimmte Anzahl an Fehler kompensieren zu können. Die dazugehörigen Zeilen in der Matrix werden ebenfalls entfernt, da die Matrix gleich viele Zeilen wie im Übertragungsvektor aufweisen muss, damit man ihn decodieren kann. @@ -183,3 +184,5 @@ m = [4,7,2,5,8,1] \] zurück, den wir ursprünglich versendet haben. +Wir möchten noch anmerken, dass es mehrere Wege für die Rekonstruktion des Nutzdatenteils gibt, diese aber alle auf dem Lokatorpolynom basieren. + diff --git a/buch/papers/reedsolomon/zusammenfassung.tex b/buch/papers/reedsolomon/zusammenfassung.tex new file mode 100644 index 0000000..568356f --- /dev/null +++ b/buch/papers/reedsolomon/zusammenfassung.tex @@ -0,0 +1,15 @@ +\section{Zusammenfassung + \label{reedsolomon:section:zf}} +\rhead{Zusammenfassung} +Dieser Abschnitt beinhaltet eine Übersicht über die Funktionsweise eines Reed-Solomon-Codes für beliebige endliche Körper. + +TODO: + +\subsubsection{Schritt 1: primitives Element} + +\subsubsection{Schritt 2: Codierung} + +\subsubsection{Schritt 3: Decodierung ohne Fehler} + +\subsubsection{Schritt 4: Decodierung mit Fehler} + -- cgit v1.2.1