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[2][-]{subsection.2.2.2}{Orthognormalbasis }{section.2.2}% 17 +\BOOKMARK [2][-]{subsection.2.2.3}{Symmetrische und selbstadjungierte Abbilungen }{section.2.2}% 18 +\BOOKMARK [2][-]{subsection.2.2.4}{Orthogonale und unit\344re Matrizen }{section.2.2}% 19 +\BOOKMARK [2][-]{subsection.2.2.5}{Orthogonale Unterr\344ume }{section.2.2}% 20 +\BOOKMARK [2][-]{subsection.2.2.6}{Andere Normen auf Vektorr\344umen }{section.2.2}% 21 +\BOOKMARK [1][-]{section.2.3}{Algebraische Strukturen }{chapter.2}% 22 +\BOOKMARK [2][-]{subsection.2.3.1}{Gruppen }{section.2.3}% 23 +\BOOKMARK [2][-]{subsection.2.3.2}{Ringe und Moduln }{section.2.3}% 24 +\BOOKMARK [2][-]{subsection.2.3.3}{Algebren }{section.2.3}% 25 +\BOOKMARK [2][-]{subsection.2.3.4}{K\366rper }{section.2.3}% 26 +\BOOKMARK [1][-]{section.2.4}{Hadamard-Algebra }{chapter.2}% 27 +\BOOKMARK [2][-]{subsection.2.4.1}{Hadamard-Produkt }{section.2.4}% 28 +\BOOKMARK [2][-]{subsection.2.4.2}{Hadamard-Produkt und Matrizenalgebra }{section.2.4}% 29 +\BOOKMARK 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{subsubsection}{Addition}{10}{section*.6}% +\contentsline {subsubsection}{Multiplikation}{10}{section*.7}% +\contentsline {subsubsection}{Rechenregeln}{10}{section*.8}% +\contentsline {subsubsection}{Teilbarkeit}{11}{section*.9}% +\contentsline {subsubsection}{Konstruktion der nat\IeC {\"u}rlichen Zahlen aus der Mengenlehre}{11}{section*.10}% +\contentsline {subsubsection}{Nat\IeC {\"u}rliche Zahlen als \IeC {\"A}quivalenzklassen}{12}{section*.11}% +\contentsline {section}{\numberline {1.2}Ganze Zahlen }{12}{section.1.2}% +\contentsline {subsubsection}{Paare von nat\IeC {\"u}rlichen Zahlen}{12}{section*.12}% +\contentsline {subsubsection}{\IeC {\"A}quivalenzrelation}{13}{section*.13}% +\contentsline {subsubsection}{Entgegengesetzter Wert}{13}{section*.14}% +\contentsline {subsubsection}{L\IeC {\"o}sung von Gleichungen}{13}{section*.15}% +\contentsline {subsubsection}{Ring}{14}{section*.16}% +\contentsline {section}{\numberline {1.3}Rationale Zahlen }{14}{section.1.3}% +\contentsline {subsubsection}{Br\IeC {\"u}che}{14}{section*.17}% +\contentsline {subsubsection}{K\IeC {\"u}rzen}{15}{section*.18}% +\contentsline {subsubsection}{Kehrwert}{15}{section*.19}% +\contentsline {subsubsection}{L\IeC {\"o}sung von linearen Gleichungen}{15}{section*.20}% +\contentsline {subsubsection}{K\IeC {\"o}rper}{15}{section*.21}% +\contentsline {section}{\numberline {1.4}Reelle Zahlen }{16}{section.1.4}% +\contentsline {section}{\numberline {1.5}Komplexe Zahlen }{16}{section.1.5}% +\contentsline {subsubsection}{Imagin\IeC {\"a}re und komplexe Zahlen}{17}{section*.22}% +\contentsline {subsubsection}{Real- und Imagin\IeC {\"a}rteil}{17}{section*.23}% +\contentsline {subsubsection}{Komplexe Konjugation}{17}{section*.24}% +\contentsline {subsubsection}{Betrag}{18}{section*.25}% +\contentsline {subsubsection}{Division}{18}{section*.26}% +\contentsline {subsubsection}{Gausssche Zahlenebene}{18}{section*.27}% +\contentsline {subsubsection}{Geometrische Interpretation der Rechenoperationen}{19}{section*.29}% +\contentsline {subsubsection}{Algebraische Vollst\IeC {\"a}ndigkeit}{19}{section*.30}% +\contentsline {subsubsection}{Quaternionen und Octonionen}{20}{section*.31}% +\contentsline {chapter}{\numberline {2}Vektoren und Matrizen }{23}{chapter.2}% +\contentsline {section}{\numberline {2.1}Lineare Algebra }{23}{section.2.1}% +\contentsline {subsection}{\numberline {2.1.1}Vektoren }{23}{subsection.2.1.1}% +\contentsline {subsubsection}{Zeilen- und Spaltenvektoren}{23}{section*.32}% +\contentsline {subsubsection}{Standardbasisvektoren}{24}{section*.33}% +\contentsline {subsubsection}{Vektorraum}{24}{section*.34}% +\contentsline {subsubsection}{Gleichungssysteme in Vektorform}{25}{section*.35}% +\contentsline {subsubsection}{Lineare Abh\IeC {\"a}ngigkeit}{26}{section*.36}% +\contentsline {subsubsection}{Basis}{26}{section*.37}% +\contentsline {subsubsection}{Unterr\IeC {\"a}ume}{26}{section*.38}% +\contentsline {subsection}{\numberline {2.1.2}Matrizen }{27}{subsection.2.1.2}% +\contentsline {subsubsection}{Definition einer Matrix}{27}{section*.39}% +\contentsline {subsubsection}{Addition und Multiplikation mit Skalaren}{27}{section*.40}% +\contentsline {subsubsection}{Multiplikation}{28}{section*.41}% +\contentsline {subsubsection}{Einheitsmatrix}{28}{section*.42}% +\contentsline {subsection}{\numberline {2.1.3}Gleichungssysteme }{28}{subsection.2.1.3}% +\contentsline {subsubsection}{Eindeutige L\IeC {\"o}sung}{28}{section*.43}% +\contentsline {subsubsection}{Inhomogene und homogene Gleichungssysteme}{29}{section*.44}% +\contentsline {subsubsection}{Gauss-Algorithmus}{29}{section*.45}% +\contentsline {subsubsection}{L\IeC {\"o}sungsmenge}{31}{section*.47}% +\contentsline {subsubsection}{Inverse Matrix}{31}{section*.48}% +\contentsline {subsubsection}{Determinante}{32}{section*.49}% +\contentsline {subsection}{\numberline {2.1.4}Lineare Abbildungen }{32}{subsection.2.1.4}% +\contentsline {subsubsection}{Definition}{32}{section*.50}% +\contentsline {subsubsection}{Matrix}{33}{section*.51}% +\contentsline {subsubsection}{Basiswechsel}{33}{section*.52}% +\contentsline {subsubsection}{Umkehrabbbildung}{34}{section*.53}% +\contentsline {subsubsection}{Kern und Bild}{34}{section*.54}% +\contentsline {subsubsection}{Rang und Defekt}{35}{section*.55}% +\contentsline {subsubsection}{Quotient}{35}{section*.56}% +\contentsline {section}{\numberline {2.2}Skalarprodukt }{35}{section.2.2}% +\contentsline {subsection}{\numberline {2.2.1}Bilinearformen und Skalarprodukte }{36}{subsection.2.2.1}% +\contentsline {subsubsection}{Symmetrische bilineare Funktionen}{36}{section*.57}% +\contentsline {subsubsection}{Positiv definite Bilinearformen und Skalarprodukt}{36}{section*.58}% +\contentsline {subsubsection}{Dreiecksungleichung}{37}{section*.59}% +\contentsline {subsubsection}{Polarformel}{38}{section*.60}% +\contentsline {subsubsection}{Komplexe Vektorr\IeC {\"a}ume und Sesquilinearformen}{38}{section*.61}% +\contentsline {subsection}{\numberline {2.2.2}Orthognormalbasis }{39}{subsection.2.2.2}% +\contentsline {subsubsection}{Gram-Matrix}{39}{section*.62}% +\contentsline {subsubsection}{Orthonormalbasis}{39}{section*.63}% +\contentsline {subsubsection}{Gram-Schmidt-Orthonormalisierung}{39}{section*.64}% +\contentsline {subsubsection}{Orthogonalisierung}{40}{section*.65}% +\contentsline {subsubsection}{Orthonormalbasen in komplexen Vektorr\IeC {\"a}umen}{40}{section*.66}% +\contentsline {subsection}{\numberline {2.2.3}Symmetrische und selbstadjungierte Abbilungen }{40}{subsection.2.2.3}% +\contentsline {subsubsection}{Symmetrische Abbildungen}{41}{section*.67}% +\contentsline {subsubsection}{Selbstadjungierte Abbildungen}{41}{section*.68}% +\contentsline {subsubsection}{Die Adjungierte}{41}{section*.69}% +\contentsline {subsection}{\numberline {2.2.4}Orthogonale und unit\IeC {\"a}re Matrizen }{42}{subsection.2.2.4}% +\contentsline {subsection}{\numberline {2.2.5}Orthogonale Unterr\IeC {\"a}ume }{42}{subsection.2.2.5}% +\contentsline {subsection}{\numberline {2.2.6}Andere Normen auf Vektorr\IeC {\"a}umen }{42}{subsection.2.2.6}% +\contentsline {subsubsection}{$l^1$-Norm}{42}{section*.70}% +\contentsline {subsubsection}{$l^\infty $-Norm}{43}{section*.71}% +\contentsline {subsubsection}{Operatornorm}{43}{section*.72}% +\contentsline {subsubsection}{Normen auf Funktionenr\IeC {\"a}umen}{44}{section*.73}% +\contentsline {section}{\numberline {2.3}Algebraische Strukturen }{44}{section.2.3}% +\contentsline {subsection}{\numberline {2.3.1}Gruppen }{44}{subsection.2.3.1}% +\contentsline {subsubsection}{Beispiele von Gruppen}{46}{section*.75}% +\contentsline {subsubsection}{Einige einfache Rechenregeln in Gruppen}{47}{section*.76}% +\contentsline {subsubsection}{Homomorphismen}{47}{section*.77}% +\contentsline {subsubsection}{Normalteiler}{48}{section*.78}% +\contentsline {subsubsection}{Faktorgruppen}{48}{section*.79}% +\contentsline {subsubsection}{Darstellungen}{49}{section*.80}% +\contentsline {subsection}{\numberline {2.3.2}Ringe und Moduln }{49}{subsection.2.3.2}% +\contentsline {subsubsection}{Definition eines Rings}{49}{section*.81}% +\contentsline {subsubsection}{Beispiele von Ringen}{50}{section*.82}% +\contentsline {subsubsection}{Einheiten}{51}{section*.84}% +\contentsline {subsubsection}{Nullteiler}{51}{section*.85}% +\contentsline {subsubsection}{Homomorphismus}{52}{section*.86}% +\contentsline {subsubsection}{Ideale}{52}{section*.87}% +\contentsline {subsection}{\numberline {2.3.3}Algebren }{53}{subsection.2.3.3}% +\contentsline {subsubsection}{Die Algebra der Funktionen $\Bbbk ^X$}{54}{section*.89}% +\contentsline {subsubsection}{Die Algebra der stetigen Funktionen $C([a,b])$}{54}{section*.90}% +\contentsline {subsection}{\numberline {2.3.4}K\IeC {\"o}rper }{54}{subsection.2.3.4}% +\contentsline {section}{\numberline {2.4}Hadamard-Algebra }{55}{section.2.4}% +\contentsline {subsection}{\numberline {2.4.1}Hadamard-Produkt }{55}{subsection.2.4.1}% +\contentsline {subsection}{\numberline {2.4.2}Hadamard-Produkt und Matrizenalgebra }{56}{subsection.2.4.2}% +\contentsline {subsubsection}{Unvertr\IeC {\"a}glichkeit von Hadamard- und Matrizen-Produkt}{56}{section*.91}% +\contentsline {subsubsection}{Einbettung der Hadamard-Algebra ein eine Matrizenalgebra}{56}{section*.92}% +\contentsline {subsubsection}{Beispiel: Faltung und Fourier-Theorie}{57}{section*.93}% +\contentsline {subsection}{\numberline {2.4.3}Weitere Verkn\IeC {\"u}pfungen }{57}{subsection.2.4.3}% +\contentsline {subsubsection}{Transposition}{57}{section*.94}% +\contentsline {subsubsection}{Frobeniusnorm}{57}{section*.95}% +\contentsline {subsubsection}{Skalarprodukt}{58}{section*.96}% +\contentsline {chapter}{\numberline {3}Polynome }{61}{chapter.3}% +\contentsline {section}{\numberline {3.1}Definitionen }{62}{section.3.1}% +\contentsline {subsection}{\numberline {3.1.1}Skalare }{62}{subsection.3.1.1}% +\contentsline {subsection}{\numberline {3.1.2}Der Polynomring }{63}{subsection.3.1.2}% +\contentsline {subsection}{\numberline {3.1.3}Grad }{64}{subsection.3.1.3}% +\contentsline {subsection}{\numberline {3.1.4}Teilbarkeit }{66}{subsection.3.1.4}% +\contentsline {subsubsection}{Polynomdivision}{66}{section*.98}% +\contentsline {subsubsection}{Euklidische Ringe und Faktorzerlegung}{67}{section*.99}% +\contentsline {subsubsection}{Irreduzible Polynome}{68}{section*.100}% +\contentsline {subsubsection}{Faktorisierung in einem Polynomring}{68}{section*.101}% +\contentsline {subsection}{\numberline {3.1.5}Formale Potenzreihen }{68}{subsection.3.1.5}% +\contentsline {section}{\numberline {3.2}Polynome als Vektoren }{68}{section.3.2}% +\contentsline {subsection}{\numberline {3.2.1}Polynome beliebigen Grades }{69}{subsection.3.2.1}% +\contentsline {subsection}{\numberline {3.2.2}Multiplikative Struktur }{70}{subsection.3.2.2}% +\contentsline {section}{\numberline {3.3}Polynommultiplikation mit Matrizen }{70}{section.3.3}% +\contentsline {section}{\numberline {3.4}Minimalpolynom }{70}{section.3.4}% +\contentsline {chapter}{\numberline {4}Endliche K\IeC {\"o}rper }{71}{chapter.4}% +\contentsline {section}{\numberline {4.1}Der euklidische Algorithmus }{71}{section.4.1}% +\contentsline {subsection}{\numberline {4.1.1}Ganze Zahlen}{71}{subsection.4.1.1}% +\contentsline {subsection}{\numberline {4.1.2}Matrixschreibweise }{73}{subsection.4.1.2}% +\contentsline {subsection}{\numberline {4.1.3}Vereinfachte Durchf\IeC {\"u}hrung }{74}{subsection.4.1.3}% +\contentsline {subsection}{\numberline {4.1.4}Polynome}{76}{subsection.4.1.4}% +\contentsline {section}{\numberline {4.2}Galois-K\IeC {\"o}rper }{77}{section.4.2}% +\contentsline {subsection}{\numberline {4.2.1}Arithmetik modulo $p$ }{78}{subsection.4.2.1}% +\contentsline {subsubsection}{Restklassenring}{78}{section*.103}% +\contentsline {subsubsection}{Division in $\mathbb {Z}/n\mathbb {Z}$}{79}{section*.104}% +\contentsline {subsubsection}{Der kleine Satz von Fermat}{80}{section*.105}% +\contentsline {subsubsection}{Der Satz von Wilson}{81}{section*.106}% +\contentsline {subsection}{\numberline {4.2.2}Charakteristik }{81}{subsection.4.2.2}% +\contentsline {subsubsection}{Primk\IeC {\"o}rper}{82}{section*.107}% +\contentsline {subsubsection}{Teilbarkeit von Binomialkoeffizienten}{82}{section*.108}% +\contentsline {subsubsection}{Frobenius-Automorphismus}{84}{section*.111}% +\contentsline {section}{\numberline {4.3}Wurzeln }{85}{section.4.3}% +\contentsline {subsection}{\numberline {4.3.1}Irreduzible Polynome }{85}{subsection.4.3.1}% +\contentsline {subsection}{\numberline {4.3.2}K\IeC {\"o}rpererweiterungen }{87}{subsection.4.3.2}% +\contentsline {subsubsection}{Erweiterung mit einem irreduziblen Polynom}{87}{section*.112}% +\contentsline {subsubsection}{Matrixrealisierung der Multiplikation mit $\alpha $}{87}{section*.113}% +\contentsline {subsubsection}{Inverse}{88}{section*.114}% +\contentsline {subsubsection}{Algebraische Konstruktion}{91}{section*.116}% +\contentsline {subsubsection}{Reduktion modulo $m$}{91}{section*.117}% +\contentsline {subsubsection}{Multiplikative Inverse}{92}{section*.118}% +\contentsline {subsection}{\numberline {4.3.3}Zerf\IeC {\"a}llungsk\IeC {\"o}rper }{94}{subsection.4.3.3}% +\contentsline {chapter}{\numberline {5}Eigenwerte und Eigenvektoren }{101}{chapter.5}% +\contentsline {section}{\numberline {5.1}Grundlagen }{101}{section.5.1}% +\contentsline {subsection}{\numberline {5.1.1}Kern und Bild von Matrixpotenzen }{101}{subsection.5.1.1}% +\contentsline {subsection}{\numberline {5.1.2}Invariante Unterr\IeC {\"a}ume }{103}{subsection.5.1.2}% +\contentsline {subsection}{\numberline {5.1.3}Nilpotente Matrizen }{104}{subsection.5.1.3}% +\contentsline {subsection}{\numberline {5.1.4}Eigenwerte und Eigenvektoren }{105}{subsection.5.1.4}% +\contentsline {subsection}{\numberline {5.1.5}Verallgemeinerte Eigenr\IeC {\"a}ume }{107}{subsection.5.1.5}% +\contentsline {subsection}{\numberline {5.1.6}Zerlegung in invariante Unterr\IeC {\"a}ume }{108}{subsection.5.1.6}% +\contentsline {subsection}{\numberline {5.1.7}Das charakteristische Polynom }{109}{subsection.5.1.7}% +\contentsline {section}{\numberline {5.2}Normalformen }{111}{section.5.2}% +\contentsline {subsection}{\numberline {5.2.1}Diagonalform}{111}{subsection.5.2.1}% +\contentsline {subsection}{\numberline {5.2.2}Jordan-Normalform }{111}{subsection.5.2.2}% +\contentsline {subsection}{\numberline {5.2.3}Reelle Normalform }{114}{subsection.5.2.3}% +\contentsline {subsection}{\numberline {5.2.4}Obere Hessenberg-Form }{114}{subsection.5.2.4}% +\contentsline {section}{\numberline {5.3}Funktionen einer Matrix }{114}{section.5.3}% +\contentsline {subsection}{\numberline {5.3.1}Polynom-Funktionen }{114}{subsection.5.3.1}% +\contentsline {subsection}{\numberline {5.3.2}Approximation von $f(A)$ }{117}{subsection.5.3.2}% +\contentsline {subsection}{\numberline {5.3.3}Potenzreihen }{118}{subsection.5.3.3}% +\contentsline {subsection}{\numberline {5.3.4}Gelfand-Radius und Eigenwerte }{118}{subsection.5.3.4}% +\contentsline {subsubsection}{Spezialfall: Diagonalisierbare Matrizen}{118}{section*.121}% +\contentsline {subsubsection}{Blockmatrizen}{119}{section*.122}% +\contentsline {subsubsection}{Jordan-Bl\IeC {\"o}cke}{120}{section*.123}% +\contentsline {subsubsection}{Iterationsfolgen}{120}{section*.124}% +\contentsline {subsubsection}{Der Satz von Gelfand}{121}{section*.125}% +\contentsline {section}{\numberline {5.4}Numerische Verfahren zur Eigenwertbestimmung }{122}{section.5.4}% +\contentsline {section}{\numberline {5.5}Spektraltheorie }{122}{section.5.5}% +\contentsline {chapter}{\numberline {6}Permutationen }{125}{chapter.6}% +\contentsline {section}{\numberline {6.1}Permutationen einer endlichen Menge }{125}{section.6.1}% +\contentsline {subsection}{\numberline {6.1.1}Permutationen als $2\times n$-Matrizen}{125}{subsection.6.1.1}% +\contentsline {subsection}{\numberline {6.1.2}Zyklenzerlegung }{126}{subsection.6.1.2}% +\contentsline {subsection}{\numberline {6.1.3}Konjugierte Elemente in $S_n$}{127}{subsection.6.1.3}% +\contentsline {section}{\numberline {6.2}Permutationen und Transpositionen }{127}{section.6.2}% +\contentsline {subsection}{\numberline {6.2.1}Zyklus und Permutationen aus Transpositionen}{128}{subsection.6.2.1}% +\contentsline {subsection}{\numberline {6.2.2}Signum einer Permutation}{128}{subsection.6.2.2}% +\contentsline {section}{\numberline {6.3}Permutationsmatrizen }{129}{section.6.3}% +\contentsline {subsection}{\numberline {6.3.1}Matrizen}{130}{subsection.6.3.1}% +\contentsline {subsection}{\numberline {6.3.2}Transpositionen}{130}{subsection.6.3.2}% +\contentsline {subsection}{\numberline {6.3.3}Determinante und Vorzeichen}{131}{subsection.6.3.3}% +\contentsline {section}{\numberline {6.4}Determinante }{132}{section.6.4}% +\contentsline {chapter}{\numberline {7}Matrizengruppen }{133}{chapter.7}% +\contentsline {section}{\numberline {7.1}Symmetrien }{133}{section.7.1}% +\contentsline {section}{\numberline {7.2}Lie-Gruppen }{133}{section.7.2}% +\contentsline {section}{\numberline {7.3}Lie-Algebren }{133}{section.7.3}% +\contentsline {section}{\numberline {7.4}Homogene R\IeC {\"a}ume }{133}{section.7.4}% +\contentsline {chapter}{\numberline {8}Graphen }{135}{chapter.8}% +\contentsline {section}{\numberline {8.1}Beschreibung von Graphen mit Matrizen }{135}{section.8.1}% +\contentsline {subsection}{\numberline {8.1.1}Definition von Graphen }{136}{subsection.8.1.1}% +\contentsline {subsubsection}{Ungerichtete Graphen}{136}{section*.129}% +\contentsline {subsubsection}{Gerichtete Graphen}{136}{section*.130}% +\contentsline {subsubsection}{Adjazenzmatrix}{137}{section*.131}% +\contentsline {subsubsection}{Adjazenzmatrix und die Anzahl der Pfade}{138}{section*.134}% +\contentsline {subsubsection}{Beschriftete Graphen}{140}{section*.136}% +\contentsline {subsection}{\numberline {8.1.2}Inzidenzmatrix}{140}{subsection.8.1.2}% +\contentsline {subsubsection}{Beschriftete Graphen}{140}{section*.137}% +\contentsline {subsubsection}{Inzidenzmatrix und Adjazenzmatrix}{141}{section*.138}% +\contentsline {subsubsection}{Gradmatrix}{141}{section*.139}% +\contentsline {subsubsection}{Gerichtete Graphen}{141}{section*.140}% +\contentsline {subsubsection}{Anwendung: Netlist}{141}{section*.141}% +\contentsline {subsection}{\numberline {8.1.3}Die Adjazenzmatrix und Laplace-Matrix }{141}{subsection.8.1.3}% +\contentsline {section}{\numberline {8.2}Spektrale Graphentheorie }{142}{section.8.2}% +\contentsline {section}{\numberline {8.3}Wavelets auf Graphen }{142}{section.8.3}% +\contentsline {subsection}{\numberline {8.3.1}Funktionen auf einem Graphen und die Laplace-Matrix}{142}{subsection.8.3.1}% +\contentsline {subsection}{\numberline {8.3.2}Standardbasis und Eigenbasis }{143}{subsection.8.3.2}% +\contentsline {subsection}{\numberline {8.3.3}Wavelet-Basen }{144}{subsection.8.3.3}% +\contentsline {chapter}{\numberline {9}Wahrscheinlichkeitsmatrizen }{145}{chapter.9}% +\contentsline {section}{\numberline {9.1}Google-Matrix }{145}{section.9.1}% +\contentsline {subsection}{\numberline {9.1.1}Ein Modell f\IeC {\"u}r Webseitenbesucher }{146}{subsection.9.1.1}% +\contentsline {subsection}{\numberline {9.1.2}Wahrscheinlichkeitsinterpretation }{146}{subsection.9.1.2}% +\contentsline {subsubsection}{Ereignisse und Wahrscheinlichkeiten}{146}{section*.144}% +\contentsline {subsubsection}{Bedingte Wahrscheinlichkeit}{147}{section*.145}% +\contentsline {subsubsection}{Totale Wahrscheinlichkeit}{147}{section*.146}% +\contentsline {subsection}{\numberline {9.1.3}``Freier Wille'' }{148}{subsection.9.1.3}% +\contentsline {subsubsection}{Erweiterung der Link-Matrix}{148}{section*.147}% +\contentsline {subsubsection}{Die Google-Matrix}{149}{section*.148}% +\contentsline {subsection}{\numberline {9.1.4}Wahrscheinlichkeitsverteilung }{150}{subsection.9.1.4}% +\contentsline {subsubsection}{Station\IeC {\"a}re Verteilung}{150}{section*.149}% +\contentsline {subsubsection}{Potenzverfahren}{151}{section*.150}% +\contentsline {section}{\numberline {9.2}Diskrete Markov-Ketten und Wahrscheinlichkeitsmatrizen }{152}{section.9.2}% +\contentsline {subsection}{\numberline {9.2.1}Markov-Eigenschaft}{152}{subsection.9.2.1}% +\contentsline {subsubsection}{Ged\IeC {\"a}chtnislosigkeit}{152}{section*.151}% +\contentsline {subsubsection}{Die Chapman-Kolmogorov-Gleichung}{153}{section*.152}% +\contentsline {subsection}{\numberline {9.2.2}Diskrete Markov-Kette}{153}{subsection.9.2.2}% +\contentsline {subsubsection}{Zustandswahrscheinlichkeiten}{155}{section*.154}% +\contentsline {subsubsection}{Zeitunabh\IeC {\"a}ngige \IeC {\"U}bergangswahrscheinlichkeiten}{155}{section*.155}% +\contentsline {subsubsection}{Station\IeC {\"a}re Verteilung}{155}{section*.156}% +\contentsline {subsubsection}{Irreduzible Markov-Ketten}{156}{section*.157}% +\contentsline {subsubsection}{Die konvexe Menge der station\IeC {\"a}ren Verteilungen}{158}{section*.159}% +\contentsline {subsubsection}{Grenzverteilung}{158}{section*.161}% +\contentsline {subsubsection}{Erwartungswert und Varianz}{159}{section*.162}% +\contentsline {subsubsection}{Erwartungswert von Werten auf \IeC {\"U}berg\IeC {\"a}ngen}{159}{section*.163}% +\contentsline {subsection}{\numberline {9.2.3}Absorbierende Zust\IeC {\"a}nde}{160}{subsection.9.2.3}% +\contentsline {subsubsection}{Absorbtionszeit}{161}{section*.165}% +\contentsline {subsubsection}{Wartezeit}{162}{section*.166}% +\contentsline {section}{\numberline {9.3}Positive Vektoren und Matrizen }{162}{section.9.3}% +\contentsline {subsection}{\numberline {9.3.1}Elementare Eigenschaften }{163}{subsection.9.3.1}% +\contentsline {subsection}{\numberline {9.3.2}Die verallgemeinerte Dreiecksungleichung }{166}{subsection.9.3.2}% +\contentsline {subsection}{\numberline {9.3.3}Der Satz von Perron-Frobenius }{168}{subsection.9.3.3}% +\contentsline {section}{\numberline {9.4}Das Paradoxon von Parrondo }{172}{section.9.4}% +\contentsline {subsection}{\numberline {9.4.1}Die beiden Teilspiele }{172}{subsection.9.4.1}% +\contentsline {subsubsection}{Das Spiel $A$}{172}{section*.172}% +\contentsline {subsubsection}{Das Spiel $B$}{172}{section*.173}% +\contentsline {subsubsection}{\IeC {\"U}bergangsmatrix im Spiel $B$}{172}{section*.174}% +\contentsline {subsubsection}{Gewinnerwartung in einem Einzelspiel $B$}{173}{section*.176}% +\contentsline {subsubsection}{Das wiederholte Spiel $B$}{174}{section*.177}% +\contentsline {subsubsection}{Das modifizierte Spiel $\mathaccentV {tilde}07E{B}$}{175}{section*.178}% +\contentsline {subsubsection}{Gewinnerwartung im modifizierten Einzelspiel}{176}{section*.180}% +\contentsline {subsubsection}{Iteration des modifizierten Spiels}{177}{section*.181}% +\contentsline {subsection}{\numberline {9.4.2}Kombination der Spiele }{177}{subsection.9.4.2}% +\contentsline {subsubsection}{Das Spiel $C$}{178}{section*.182}% +\contentsline {subsubsection}{Das iterierte Spiel $C$}{178}{section*.183}% +\contentsline {chapter}{\numberline {10}Anwendungen in Kryptographie und Codierungstheorie }{179}{chapter.10}% +\contentsline {section}{\numberline {10.1}Arithmetik f\IeC {\"u}r die Kryptographie }{179}{section.10.1}% +\contentsline {subsection}{\numberline {10.1.1}Potenzieren }{179}{subsection.10.1.1}% +\contentsline {subsection}{\numberline {10.1.2}Rechenoperationen in $\mathbb {F}_p$ }{179}{subsection.10.1.2}% +\contentsline {subsection}{\numberline {10.1.3}Rechenoperationen in $\mathbb {F}_{2^l}$ }{179}{subsection.10.1.3}% +\contentsline {section}{\numberline {10.2}Kryptographie und endliche K\IeC {\"o}rper }{179}{section.10.2}% +\contentsline {subsection}{\numberline {10.2.1}Potenzen in $\mathbb {F}_p$ und diskreter Logarithmus }{179}{subsection.10.2.1}% +\contentsline {subsection}{\numberline {10.2.2}Diffie-Hellman-Schl\IeC {\"u}sseltausch }{181}{subsection.10.2.2}% +\contentsline {subsection}{\numberline {10.2.3}Elliptische Kurven }{181}{subsection.10.2.3}% +\contentsline {subsubsection}{Elliptische Kurven}{182}{section*.185}% +\contentsline {subsubsection}{Geometrische Definition der Gruppenoperation}{183}{section*.187}% +\contentsline {subsubsection}{Gruppenoperation, algebraische Konstruktion}{184}{section*.188}% +\contentsline {subsubsection}{Beispiele}{187}{section*.189}% +\contentsline {subsubsection}{Diffie-Hellman in einer elliptischen Kurve}{187}{section*.190}% +\contentsline {section}{\numberline {10.3}Advanced Encryption Standard -- AES }{187}{section.10.3}% +\contentsline {section}{\numberline {10.4}Fehlerkorrigierende Codes nach Reed-Solomon }{187}{section.10.4}% +\contentsline {subsection}{\numberline {10.4.1}Was ist ein Code? }{188}{subsection.10.4.1}% +\contentsline {subsection}{\numberline {10.4.2}Reed-Solomon-Code }{188}{subsection.10.4.2}% +\contentsline {subsection}{\numberline {10.4.3}Decodierung }{188}{subsection.10.4.3}% +\contentsline {chapter}{\numberline {11}Homologie }{189}{chapter.11}% +\contentsline {section}{\numberline {11.1}Simplexe und simpliziale Komplexe }{189}{section.11.1}% +\contentsline {subsection}{\numberline {11.1.1}Simplexe und Rand }{189}{subsection.11.1.1}% +\contentsline {subsubsection}{Rand eines Dreiecks}{189}{section*.193}% +\contentsline {subsubsection}{Simplizes}{190}{section*.194}% +\contentsline {subsubsection}{Rechnen mit Simplizes}{191}{section*.195}% +\contentsline {subsubsection}{Rand eines Simplex}{191}{section*.196}% +\contentsline {subsection}{\numberline {11.1.2}Triangulation }{192}{subsection.11.1.2}% +\contentsline {section}{\numberline {11.2}Kettenkomplexe }{192}{section.11.2}% +\contentsline {subsection}{\numberline {11.2.1}Randoperator von Simplexen }{192}{subsection.11.2.1}% +\contentsline {subsection}{\numberline {11.2.2}Kettenkomplexe und Morphismen }{192}{subsection.11.2.2}% +\contentsline {section}{\numberline {11.3}Homologie }{192}{section.11.3}% +\contentsline {subsection}{\numberline {11.3.1}Homologie eines Kettenkomplexes }{192}{subsection.11.3.1}% +\contentsline {subsection}{\numberline {11.3.2}Induzierte Abbildung }{192}{subsection.11.3.2}% +\contentsline {subsection}{\numberline {11.3.3}Homologie eines simplizialen Komplexes }{192}{subsection.11.3.3}% +\contentsline {section}{\numberline {11.4}Exaktheit und die Mayer-Vietoris-Folge }{192}{section.11.4}% +\contentsline {subsection}{\numberline {11.4.1}Kurze exakte Folgen von Kettenkomplexen }{192}{subsection.11.4.1}% +\contentsline {subsection}{\numberline {11.4.2}Schlangenlemma und lange exakte Folgen }{192}{subsection.11.4.2}% +\contentsline {subsection}{\numberline {11.4.3}Mayer-Vietoris-Folge }{192}{subsection.11.4.3}% +\contentsline {section}{\numberline {11.5}Fixpunkte }{192}{section.11.5}% +\contentsline {subsection}{\numberline {11.5.1}Lefshetz-Spurformel }{192}{subsection.11.5.1}% +\contentsline {subsection}{\numberline {11.5.2}Brower-Fixpunktsatz }{192}{subsection.11.5.2}% +\contentsline {part}{II\hspace {1em}Anwendungen und weiterf\IeC {\"u}hrende Themen}{193}{part.2}% +\contentsline {chapter}{\numberline {12}Thema}{197}{chapter.12}% +\contentsline {section}{\numberline {12.1}Teil 0}{197}{section.12.1}% +\contentsline {section}{\numberline {12.2}Teil 1 }{197}{section.12.2}% +\contentsline {subsection}{\numberline {12.2.1}De finibus bonorum et malorum }{198}{subsection.12.2.1}% +\contentsline {section}{\numberline {12.3}Teil 2 }{198}{section.12.3}% +\contentsline {subsection}{\numberline {12.3.1}De finibus bonorum et malorum }{198}{subsection.12.3.1}% +\contentsline {section}{\numberline {12.4}Teil 3 }{199}{section.12.4}% +\contentsline {subsection}{\numberline {12.4.1}De finibus bonorum et malorum }{199}{subsection.12.4.1}% +\contentsline {chapter}{\numberline {13}Thema}{201}{chapter.13}% +\contentsline {section}{\numberline {13.1}Teil 0}{201}{section.13.1}% +\contentsline {section}{\numberline {13.2}Teil 1 }{201}{section.13.2}% +\contentsline {subsection}{\numberline {13.2.1}De finibus bonorum et malorum }{202}{subsection.13.2.1}% +\contentsline {section}{\numberline {13.3}Teil 2 }{202}{section.13.3}% +\contentsline {subsection}{\numberline {13.3.1}De finibus bonorum et malorum }{202}{subsection.13.3.1}% +\contentsline {section}{\numberline {13.4}Teil 3 }{203}{section.13.4}% +\contentsline {subsection}{\numberline {13.4.1}De finibus bonorum et malorum }{203}{subsection.13.4.1}% +\contentsline {chapter}{\numberline {14}Thema}{205}{chapter.14}% +\contentsline {section}{\numberline {14.1}Teil 0}{205}{section.14.1}% +\contentsline {section}{\numberline {14.2}Teil 1 }{205}{section.14.2}% +\contentsline {subsection}{\numberline {14.2.1}De finibus bonorum et malorum }{206}{subsection.14.2.1}% +\contentsline {section}{\numberline {14.3}Teil 2 }{206}{section.14.3}% +\contentsline {subsection}{\numberline {14.3.1}De finibus bonorum et malorum }{206}{subsection.14.3.1}% +\contentsline {section}{\numberline {14.4}Teil 3 }{207}{section.14.4}% +\contentsline {subsection}{\numberline {14.4.1}De finibus bonorum et malorum }{207}{subsection.14.4.1}% +\contentsline {chapter}{\numberline {15}Thema}{209}{chapter.15}% +\contentsline {section}{\numberline {15.1}Teil 0}{209}{section.15.1}% +\contentsline {section}{\numberline {15.2}Teil 1 }{209}{section.15.2}% +\contentsline {subsection}{\numberline {15.2.1}De finibus bonorum et malorum }{210}{subsection.15.2.1}% +\contentsline {section}{\numberline {15.3}Teil 2 }{210}{section.15.3}% +\contentsline {subsection}{\numberline {15.3.1}De finibus bonorum et malorum }{210}{subsection.15.3.1}% +\contentsline {section}{\numberline {15.4}Teil 3 }{211}{section.15.4}% +\contentsline {subsection}{\numberline {15.4.1}De finibus bonorum et malorum }{211}{subsection.15.4.1}% +\contentsline {chapter}{\numberline {16}Iterierte Funktionsschemata}{213}{chapter.16}% +\contentsline {section}{\numberline {16.1}Teil 0}{213}{section.16.1}% +\contentsline {section}{\numberline {16.2}Teil 1 }{213}{section.16.2}% +\contentsline {subsection}{\numberline {16.2.1}De finibus bonorum et malorum }{214}{subsection.16.2.1}% +\contentsline {section}{\numberline {16.3}Teil 2 }{214}{section.16.3}% +\contentsline {subsection}{\numberline {16.3.1}De finibus bonorum et malorum }{214}{subsection.16.3.1}% +\contentsline {section}{\numberline {16.4}Teil 3 }{215}{section.16.4}% +\contentsline {subsection}{\numberline {16.4.1}De finibus bonorum et malorum }{215}{subsection.16.4.1}% +\contentsline {chapter}{\numberline {17}McEliece-Kryptosystem}{217}{chapter.17}% +\contentsline {section}{\numberline {17.1}Teil 0}{217}{section.17.1}% +\contentsline {section}{\numberline {17.2}Teil 1 }{217}{section.17.2}% +\contentsline {subsection}{\numberline {17.2.1}De finibus bonorum et malorum }{218}{subsection.17.2.1}% +\contentsline {section}{\numberline {17.3}Teil 2 }{218}{section.17.3}% +\contentsline {subsection}{\numberline {17.3.1}De finibus bonorum 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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! 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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! 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+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 6--14 +[] + [] + + +Overfull \hbox (5.43053pt too wide) in paragraph at lines 6--14 +[] + [] + +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! +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! 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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'). 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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 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 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! 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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 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') 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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License terms: http://www.apache.org/licenses/LICENSE-2.0) +/bd{bind def}bind def +/ld{load def}bd +/GR/grestore ld +/GS/gsave ld +/RM/rmoveto ld +/C/curveto ld +/t/show ld +/L/lineto ld +/ML/setmiterlimit ld +/CT/concat ld +/f/fill ld +/N/newpath ld +/S/stroke ld +/CC/setcmykcolor ld +/A/ashow ld +/cp/closepath ld +/RC/setrgbcolor ld +/LJ/setlinejoin ld +/GC/setgray ld +/LW/setlinewidth ld +/M/moveto ld +/re {4 2 roll M +1 index 0 rlineto +0 exch rlineto +neg 0 rlineto +cp } bd +/_ctm matrix def +/_tm matrix def +/BT { _ctm currentmatrix pop matrix _tm copy pop 0 0 moveto } bd +/ET { _ctm setmatrix } bd +/iTm { _ctm setmatrix _tm concat } bd +/Tm { _tm astore pop iTm 0 0 moveto } bd +/ux 0.0 def +/uy 0.0 def +/F { + /Tp exch def + /Tf exch def + Tf findfont Tp scalefont setfont + /cf Tf def /cs Tp def +} bd +/ULS {currentpoint /uy exch def /ux exch def} bd +/ULE { + /Tcx currentpoint pop def + gsave + newpath + cf findfont cs scalefont dup + /FontMatrix get 0 get /Ts exch def /FontInfo get dup + /UnderlinePosition get Ts mul /To exch def + /UnderlineThickness get Ts mul /Tt exch def + ux uy To add moveto Tcx uy To add lineto + Tt setlinewidth stroke + grestore +} bd +/OLE { + /Tcx currentpoint pop def + gsave + newpath + cf findfont cs scalefont dup + /FontMatrix get 0 get /Ts exch def /FontInfo get dup + /UnderlinePosition get Ts mul /To exch def + /UnderlineThickness get Ts mul /Tt exch def + ux uy To add cs add moveto Tcx uy To add cs add lineto + Tt setlinewidth stroke + grestore +} bd +/SOE { + /Tcx currentpoint pop def + gsave + newpath + cf findfont cs scalefont dup + /FontMatrix get 0 get /Ts exch def /FontInfo get dup + /UnderlinePosition get Ts mul /To exch def + /UnderlineThickness get Ts mul /Tt exch def + ux uy To add cs 10 mul 26 idiv add moveto Tcx uy To add cs 10 mul 26 idiv add lineto + Tt setlinewidth stroke + grestore +} bd +/QT { +/Y22 exch store +/X22 exch store +/Y21 exch store +/X21 exch store +currentpoint +/Y21 load 2 mul add 3 div exch +/X21 load 2 mul add 3 div exch +/X21 load 2 mul /X22 load add 3 div +/Y21 load 2 mul /Y22 load add 3 div +/X22 load /Y22 load curveto +} bd +/SSPD { +dup length /d exch dict def +{ +/v exch def +/k exch def +currentpagedevice k known { +/cpdv currentpagedevice k get def +v cpdv ne { +/upd false def +/nullv v type /nulltype eq def +/nullcpdv cpdv type /nulltype eq def +nullv nullcpdv or +{ +/upd true def +} { +/sametype v type cpdv type eq def +sametype { +v type /arraytype eq { +/vlen v length def +/cpdvlen cpdv length def +vlen cpdvlen eq { +0 1 vlen 1 sub { +/i exch def +/obj v i get def +/cpdobj cpdv i get def +obj cpdobj ne { +/upd true def +exit +} if +} for +} { +/upd true def +} ifelse +} { +v type /dicttype eq { +v { +/dv exch def +/dk exch def +/cpddv cpdv dk get def +dv cpddv ne { +/upd true def +exit +} if +} forall +} { +/upd true def +} ifelse +} ifelse +} if +} ifelse +upd true eq { +d k v put +} if +} if +} if +} forall +d length 0 gt { +d setpagedevice +} if +} bd +/RE { % /NewFontName [NewEncodingArray] /FontName RE - + findfont dup length dict begin + { + 1 index /FID ne + {def} {pop pop} ifelse + } forall + /Encoding exch def + /FontName 1 index def + currentdict definefont pop + end +} bind def +%%EndResource +%%BeginResource: procset (Apache XML Graphics EPS ProcSet) 1.0 0 +%%Version: 1.0 0 +%%Copyright: (Copyright 2002-2003 The Apache Software Foundation. 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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 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') 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') 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') 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') 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') 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') 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 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') 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}} -Ãœber 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. + +Ãœber 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 77ba89eb398fbb3d47a55280fbc0983f42629a15 Mon Sep 17 00:00:00 2001 From: Alain Date: Sun, 13 Jun 2021 22:35:03 +0200 Subject: removed build-files --- buch/.gitignore | 16 + buch/buch-blx.bib | 11 - buch/buch.aux | 956 -------------- buch/buch.bbl | 20 - buch/buch.blg | 74 -- buch/buch.idx | 167 --- buch/buch.log | 3498 -------------------------------------------------- buch/buch.out | 234 ---- buch/buch.pdf | Bin 1412544 -> 2646094 bytes buch/buch.run.xml | 521 -------- buch/buch.synctex.gz | Bin 2189601 -> 0 bytes buch/buch.toc | 394 ------ buch/buch1-blx.aux | 15 - buch/buch10-blx.aux | 13 - buch/buch11-blx.aux | 13 - buch/buch2-blx.aux | 13 - buch/buch3-blx.aux | 13 - buch/buch4-blx.aux | 13 - buch/buch5-blx.aux | 13 - buch/buch6-blx.aux | 12 - buch/buch7-blx.aux | 13 - buch/buch8-blx.aux | 13 - buch/buch9-blx.aux | 13 - buch/test1.tex | 93 -- 24 files changed, 16 insertions(+), 6112 deletions(-) create mode 100644 buch/.gitignore delete mode 100644 buch/buch-blx.bib delete mode 100644 buch/buch.aux delete mode 100644 buch/buch.bbl delete mode 100644 buch/buch.blg delete mode 100644 buch/buch.idx delete mode 100644 buch/buch.log delete mode 100644 buch/buch.out delete mode 100644 buch/buch.run.xml delete mode 100644 buch/buch.synctex.gz delete mode 100644 buch/buch.toc delete mode 100644 buch/buch1-blx.aux delete mode 100644 buch/buch10-blx.aux delete mode 100644 buch/buch11-blx.aux delete mode 100644 buch/buch2-blx.aux delete mode 100644 buch/buch3-blx.aux delete mode 100644 buch/buch4-blx.aux delete mode 100644 buch/buch5-blx.aux delete mode 100644 buch/buch6-blx.aux delete mode 100644 buch/buch7-blx.aux delete mode 100644 buch/buch8-blx.aux delete mode 100644 buch/buch9-blx.aux delete mode 100644 buch/test1.tex (limited to 'buch') diff --git a/buch/.gitignore b/buch/.gitignore new file mode 100644 index 0000000..3c3ac3d --- /dev/null +++ b/buch/.gitignore @@ -0,0 +1,16 @@ +*.aux +*.bbl +*.bib +*.blg +*.idx +*.ilg +*.ind +*.log +*.out +*.rpt +buch*.pdf +*.run.xml +*.toc +.build/ +*.synctex.gz +*.DS_Store \ No newline at end of file diff --git a/buch/buch-blx.bib b/buch/buch-blx.bib deleted file mode 100644 index 21bae92..0000000 --- a/buch/buch-blx.bib +++ /dev/null @@ -1,11 +0,0 @@ -@Comment{$ biblatex control file $} -@Comment{$ biblatex bcf format version 3.6 $} -% Do not modify this file! -% -% This is an auxiliary file used by the 'biblatex' package. -% This file may safely be deleted. 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-\BOOKMARK [0][-]{chapter*.3}{Einleitung}{part.1}% 2 -\BOOKMARK [0][-]{chapter.1}{Zahlen }{part.1}% 3 -\BOOKMARK [1][-]{section.1.1}{Nat\374rliche Zahlen }{chapter.1}% 4 -\BOOKMARK [1][-]{section.1.2}{Ganze Zahlen }{chapter.1}% 5 -\BOOKMARK [1][-]{section.1.3}{Rationale Zahlen }{chapter.1}% 6 -\BOOKMARK [1][-]{section.1.4}{Reelle Zahlen }{chapter.1}% 7 -\BOOKMARK [1][-]{section.1.5}{Komplexe Zahlen }{chapter.1}% 8 -\BOOKMARK [0][-]{chapter.2}{Vektoren und Matrizen }{part.1}% 9 -\BOOKMARK [1][-]{section.2.1}{Lineare Algebra }{chapter.2}% 10 -\BOOKMARK [2][-]{subsection.2.1.1}{Vektoren }{section.2.1}% 11 -\BOOKMARK [2][-]{subsection.2.1.2}{Matrizen }{section.2.1}% 12 -\BOOKMARK [2][-]{subsection.2.1.3}{Gleichungssysteme }{section.2.1}% 13 -\BOOKMARK [2][-]{subsection.2.1.4}{Lineare Abbildungen }{section.2.1}% 14 -\BOOKMARK [1][-]{section.2.2}{Skalarprodukt }{chapter.2}% 15 -\BOOKMARK [2][-]{subsection.2.2.1}{Bilinearformen und Skalarprodukte }{section.2.2}% 16 -\BOOKMARK [2][-]{subsection.2.2.2}{Orthognormalbasis }{section.2.2}% 17 -\BOOKMARK [2][-]{subsection.2.2.3}{Symmetrische und selbstadjungierte Abbilungen }{section.2.2}% 18 -\BOOKMARK [2][-]{subsection.2.2.4}{Orthogonale und unit\344re Matrizen }{section.2.2}% 19 -\BOOKMARK [2][-]{subsection.2.2.5}{Orthogonale Unterr\344ume }{section.2.2}% 20 -\BOOKMARK [2][-]{subsection.2.2.6}{Andere Normen auf Vektorr\344umen }{section.2.2}% 21 -\BOOKMARK [1][-]{section.2.3}{Algebraische Strukturen }{chapter.2}% 22 -\BOOKMARK [2][-]{subsection.2.3.1}{Gruppen }{section.2.3}% 23 -\BOOKMARK [2][-]{subsection.2.3.2}{Ringe und Moduln }{section.2.3}% 24 -\BOOKMARK [2][-]{subsection.2.3.3}{Algebren }{section.2.3}% 25 -\BOOKMARK [2][-]{subsection.2.3.4}{K\366rper }{section.2.3}% 26 -\BOOKMARK [1][-]{section.2.4}{Hadamard-Algebra }{chapter.2}% 27 -\BOOKMARK [2][-]{subsection.2.4.1}{Hadamard-Produkt }{section.2.4}% 28 -\BOOKMARK [2][-]{subsection.2.4.2}{Hadamard-Produkt und Matrizenalgebra }{section.2.4}% 29 -\BOOKMARK [2][-]{subsection.2.4.3}{Weitere Verkn\374pfungen }{section.2.4}% 30 -\BOOKMARK [0][-]{chapter.3}{Polynome }{part.1}% 31 -\BOOKMARK [1][-]{section.3.1}{Definitionen }{chapter.3}% 32 -\BOOKMARK [2][-]{subsection.3.1.1}{Skalare }{section.3.1}% 33 -\BOOKMARK [2][-]{subsection.3.1.2}{Der Polynomring }{section.3.1}% 34 -\BOOKMARK [2][-]{subsection.3.1.3}{Grad }{section.3.1}% 35 -\BOOKMARK [2][-]{subsection.3.1.4}{Teilbarkeit }{section.3.1}% 36 -\BOOKMARK [2][-]{subsection.3.1.5}{Formale Potenzreihen }{section.3.1}% 37 -\BOOKMARK [1][-]{section.3.2}{Polynome als Vektoren }{chapter.3}% 38 -\BOOKMARK [2][-]{subsection.3.2.1}{Polynome beliebigen Grades }{section.3.2}% 39 -\BOOKMARK [2][-]{subsection.3.2.2}{Multiplikative Struktur }{section.3.2}% 40 -\BOOKMARK [1][-]{section.3.3}{Polynommultiplikation mit Matrizen }{chapter.3}% 41 -\BOOKMARK [1][-]{section.3.4}{Minimalpolynom }{chapter.3}% 42 -\BOOKMARK [0][-]{chapter.4}{Endliche K\366rper }{part.1}% 43 -\BOOKMARK [1][-]{section.4.1}{Der euklidische Algorithmus }{chapter.4}% 44 -\BOOKMARK [2][-]{subsection.4.1.1}{Ganze Zahlen}{section.4.1}% 45 -\BOOKMARK [2][-]{subsection.4.1.2}{Matrixschreibweise }{section.4.1}% 46 -\BOOKMARK [2][-]{subsection.4.1.3}{Vereinfachte Durchf\374hrung }{section.4.1}% 47 -\BOOKMARK [2][-]{subsection.4.1.4}{Polynome}{section.4.1}% 48 -\BOOKMARK [1][-]{section.4.2}{Galois-K\366rper }{chapter.4}% 49 -\BOOKMARK [2][-]{subsection.4.2.1}{Arithmetik modulo p }{section.4.2}% 50 -\BOOKMARK [2][-]{subsection.4.2.2}{Charakteristik }{section.4.2}% 51 -\BOOKMARK [1][-]{section.4.3}{Wurzeln }{chapter.4}% 52 -\BOOKMARK [2][-]{subsection.4.3.1}{Irreduzible Polynome }{section.4.3}% 53 -\BOOKMARK [2][-]{subsection.4.3.2}{K\366rpererweiterungen }{section.4.3}% 54 -\BOOKMARK [2][-]{subsection.4.3.3}{Zerf\344llungsk\366rper }{section.4.3}% 55 -\BOOKMARK [0][-]{chapter.5}{Eigenwerte und Eigenvektoren }{part.1}% 56 -\BOOKMARK [1][-]{section.5.1}{Grundlagen }{chapter.5}% 57 -\BOOKMARK [2][-]{subsection.5.1.1}{Kern und Bild von Matrixpotenzen }{section.5.1}% 58 -\BOOKMARK [2][-]{subsection.5.1.2}{Invariante Unterr\344ume }{section.5.1}% 59 -\BOOKMARK [2][-]{subsection.5.1.3}{Nilpotente Matrizen }{section.5.1}% 60 -\BOOKMARK [2][-]{subsection.5.1.4}{Eigenwerte und Eigenvektoren }{section.5.1}% 61 -\BOOKMARK [2][-]{subsection.5.1.5}{Verallgemeinerte Eigenr\344ume }{section.5.1}% 62 -\BOOKMARK [2][-]{subsection.5.1.6}{Zerlegung in invariante Unterr\344ume }{section.5.1}% 63 -\BOOKMARK [2][-]{subsection.5.1.7}{Das charakteristische Polynom }{section.5.1}% 64 -\BOOKMARK [1][-]{section.5.2}{Normalformen }{chapter.5}% 65 -\BOOKMARK [2][-]{subsection.5.2.1}{Diagonalform}{section.5.2}% 66 -\BOOKMARK [2][-]{subsection.5.2.2}{Jordan-Normalform }{section.5.2}% 67 -\BOOKMARK [2][-]{subsection.5.2.3}{Reelle Normalform }{section.5.2}% 68 -\BOOKMARK [2][-]{subsection.5.2.4}{Obere Hessenberg-Form }{section.5.2}% 69 -\BOOKMARK [1][-]{section.5.3}{Funktionen einer Matrix }{chapter.5}% 70 -\BOOKMARK [2][-]{subsection.5.3.1}{Polynom-Funktionen }{section.5.3}% 71 -\BOOKMARK [2][-]{subsection.5.3.2}{Approximation von f\(A\) }{section.5.3}% 72 -\BOOKMARK [2][-]{subsection.5.3.3}{Potenzreihen }{section.5.3}% 73 -\BOOKMARK [2][-]{subsection.5.3.4}{Gelfand-Radius und Eigenwerte }{section.5.3}% 74 -\BOOKMARK [1][-]{section.5.4}{Numerische Verfahren zur Eigenwertbestimmung }{chapter.5}% 75 -\BOOKMARK [1][-]{section.5.5}{Spektraltheorie }{chapter.5}% 76 -\BOOKMARK [0][-]{chapter.6}{Permutationen }{part.1}% 77 -\BOOKMARK [1][-]{section.6.1}{Permutationen einer endlichen Menge }{chapter.6}% 78 -\BOOKMARK [2][-]{subsection.6.1.1}{Permutationen als 2n-Matrizen}{section.6.1}% 79 -\BOOKMARK [2][-]{subsection.6.1.2}{Zyklenzerlegung }{section.6.1}% 80 -\BOOKMARK [2][-]{subsection.6.1.3}{Konjugierte Elemente in Sn}{section.6.1}% 81 -\BOOKMARK [1][-]{section.6.2}{Permutationen und Transpositionen }{chapter.6}% 82 -\BOOKMARK [2][-]{subsection.6.2.1}{Zyklus und Permutationen aus Transpositionen}{section.6.2}% 83 -\BOOKMARK [2][-]{subsection.6.2.2}{Signum einer Permutation}{section.6.2}% 84 -\BOOKMARK [1][-]{section.6.3}{Permutationsmatrizen }{chapter.6}% 85 -\BOOKMARK [2][-]{subsection.6.3.1}{Matrizen}{section.6.3}% 86 -\BOOKMARK [2][-]{subsection.6.3.2}{Transpositionen}{section.6.3}% 87 -\BOOKMARK [2][-]{subsection.6.3.3}{Determinante und Vorzeichen}{section.6.3}% 88 -\BOOKMARK [1][-]{section.6.4}{Determinante }{chapter.6}% 89 -\BOOKMARK [0][-]{chapter.7}{Matrizengruppen }{part.1}% 90 -\BOOKMARK [1][-]{section.7.1}{Symmetrien }{chapter.7}% 91 -\BOOKMARK [1][-]{section.7.2}{Lie-Gruppen }{chapter.7}% 92 -\BOOKMARK [1][-]{section.7.3}{Lie-Algebren }{chapter.7}% 93 -\BOOKMARK [1][-]{section.7.4}{Homogene R\344ume }{chapter.7}% 94 -\BOOKMARK [0][-]{chapter.8}{Graphen }{part.1}% 95 -\BOOKMARK [1][-]{section.8.1}{Beschreibung von Graphen mit Matrizen }{chapter.8}% 96 -\BOOKMARK [2][-]{subsection.8.1.1}{Definition von Graphen }{section.8.1}% 97 -\BOOKMARK [2][-]{subsection.8.1.2}{Inzidenzmatrix}{section.8.1}% 98 -\BOOKMARK [2][-]{subsection.8.1.3}{Die Adjazenzmatrix und Laplace-Matrix }{section.8.1}% 99 -\BOOKMARK [1][-]{section.8.2}{Spektrale Graphentheorie }{chapter.8}% 100 -\BOOKMARK [1][-]{section.8.3}{Wavelets auf Graphen }{chapter.8}% 101 -\BOOKMARK [2][-]{subsection.8.3.1}{Funktionen auf einem Graphen und die Laplace-Matrix}{section.8.3}% 102 -\BOOKMARK [2][-]{subsection.8.3.2}{Standardbasis und Eigenbasis }{section.8.3}% 103 -\BOOKMARK [2][-]{subsection.8.3.3}{Wavelet-Basen }{section.8.3}% 104 -\BOOKMARK [0][-]{chapter.9}{Wahrscheinlichkeitsmatrizen }{part.1}% 105 -\BOOKMARK [1][-]{section.9.1}{Google-Matrix }{chapter.9}% 106 -\BOOKMARK [2][-]{subsection.9.1.1}{Ein Modell f\374r Webseitenbesucher }{section.9.1}% 107 -\BOOKMARK [2][-]{subsection.9.1.2}{Wahrscheinlichkeitsinterpretation }{section.9.1}% 108 -\BOOKMARK [2][-]{subsection.9.1.3}{``Freier Wille'' }{section.9.1}% 109 -\BOOKMARK [2][-]{subsection.9.1.4}{Wahrscheinlichkeitsverteilung }{section.9.1}% 110 -\BOOKMARK [1][-]{section.9.2}{Diskrete Markov-Ketten und Wahrscheinlichkeitsmatrizen }{chapter.9}% 111 -\BOOKMARK [2][-]{subsection.9.2.1}{Markov-Eigenschaft}{section.9.2}% 112 -\BOOKMARK [2][-]{subsection.9.2.2}{Diskrete Markov-Kette}{section.9.2}% 113 -\BOOKMARK [2][-]{subsection.9.2.3}{Absorbierende Zust\344nde}{section.9.2}% 114 -\BOOKMARK [1][-]{section.9.3}{Positive Vektoren und Matrizen }{chapter.9}% 115 -\BOOKMARK [2][-]{subsection.9.3.1}{Elementare Eigenschaften }{section.9.3}% 116 -\BOOKMARK [2][-]{subsection.9.3.2}{Die verallgemeinerte Dreiecksungleichung }{section.9.3}% 117 -\BOOKMARK [2][-]{subsection.9.3.3}{Der Satz von Perron-Frobenius }{section.9.3}% 118 -\BOOKMARK [1][-]{section.9.4}{Das Paradoxon von Parrondo }{chapter.9}% 119 -\BOOKMARK [2][-]{subsection.9.4.1}{Die beiden Teilspiele }{section.9.4}% 120 -\BOOKMARK [2][-]{subsection.9.4.2}{Kombination der Spiele }{section.9.4}% 121 -\BOOKMARK [0][-]{chapter.10}{Anwendungen in Kryptographie und Codierungstheorie }{part.1}% 122 -\BOOKMARK [1][-]{section.10.1}{Arithmetik f\374r die Kryptographie }{chapter.10}% 123 -\BOOKMARK [2][-]{subsection.10.1.1}{Potenzieren }{section.10.1}% 124 -\BOOKMARK [2][-]{subsection.10.1.2}{Rechenoperationen in Fp }{section.10.1}% 125 -\BOOKMARK [2][-]{subsection.10.1.3}{Rechenoperationen in F2l }{section.10.1}% 126 -\BOOKMARK [1][-]{section.10.2}{Kryptographie und endliche K\366rper }{chapter.10}% 127 -\BOOKMARK [2][-]{subsection.10.2.1}{Potenzen in Fp und diskreter Logarithmus }{section.10.2}% 128 -\BOOKMARK [2][-]{subsection.10.2.2}{Diffie-Hellman-Schl\374sseltausch }{section.10.2}% 129 -\BOOKMARK [2][-]{subsection.10.2.3}{Elliptische Kurven }{section.10.2}% 130 -\BOOKMARK [1][-]{section.10.3}{Advanced Encryption Standard \205 AES }{chapter.10}% 131 -\BOOKMARK [1][-]{section.10.4}{Fehlerkorrigierende Codes nach Reed-Solomon }{chapter.10}% 132 -\BOOKMARK [2][-]{subsection.10.4.1}{Was ist ein Code? }{section.10.4}% 133 -\BOOKMARK [2][-]{subsection.10.4.2}{Reed-Solomon-Code }{section.10.4}% 134 -\BOOKMARK [2][-]{subsection.10.4.3}{Decodierung }{section.10.4}% 135 -\BOOKMARK [0][-]{chapter.11}{Homologie }{part.1}% 136 -\BOOKMARK [1][-]{section.11.1}{Simplexe und simpliziale Komplexe }{chapter.11}% 137 -\BOOKMARK [2][-]{subsection.11.1.1}{Simplexe und Rand }{section.11.1}% 138 -\BOOKMARK [2][-]{subsection.11.1.2}{Triangulation }{section.11.1}% 139 -\BOOKMARK [1][-]{section.11.2}{Kettenkomplexe }{chapter.11}% 140 -\BOOKMARK [2][-]{subsection.11.2.1}{Randoperator von Simplexen }{section.11.2}% 141 -\BOOKMARK [2][-]{subsection.11.2.2}{Kettenkomplexe und Morphismen }{section.11.2}% 142 -\BOOKMARK [1][-]{section.11.3}{Homologie }{chapter.11}% 143 -\BOOKMARK [2][-]{subsection.11.3.1}{Homologie eines Kettenkomplexes }{section.11.3}% 144 -\BOOKMARK [2][-]{subsection.11.3.2}{Induzierte Abbildung }{section.11.3}% 145 -\BOOKMARK [2][-]{subsection.11.3.3}{Homologie eines simplizialen Komplexes }{section.11.3}% 146 -\BOOKMARK [1][-]{section.11.4}{Exaktheit und die Mayer-Vietoris-Folge }{chapter.11}% 147 -\BOOKMARK [2][-]{subsection.11.4.1}{Kurze exakte Folgen von Kettenkomplexen }{section.11.4}% 148 -\BOOKMARK [2][-]{subsection.11.4.2}{Schlangenlemma und lange exakte Folgen }{section.11.4}% 149 -\BOOKMARK [2][-]{subsection.11.4.3}{Mayer-Vietoris-Folge }{section.11.4}% 150 -\BOOKMARK [1][-]{section.11.5}{Fixpunkte }{chapter.11}% 151 -\BOOKMARK [2][-]{subsection.11.5.1}{Lefshetz-Spurformel }{section.11.5}% 152 -\BOOKMARK [2][-]{subsection.11.5.2}{Brower-Fixpunktsatz }{section.11.5}% 153 -\BOOKMARK [-1][-]{part.2}{II Anwendungen und weiterf\374hrende Themen}{}% 154 -\BOOKMARK [0][-]{chapter.12}{Thema}{part.2}% 155 -\BOOKMARK [1][-]{section.12.1}{Teil 0}{chapter.12}% 156 -\BOOKMARK [1][-]{section.12.2}{Teil 1 }{chapter.12}% 157 -\BOOKMARK [2][-]{subsection.12.2.1}{De finibus bonorum et malorum }{section.12.2}% 158 -\BOOKMARK [1][-]{section.12.3}{Teil 2 }{chapter.12}% 159 -\BOOKMARK [2][-]{subsection.12.3.1}{De finibus bonorum et malorum }{section.12.3}% 160 -\BOOKMARK [1][-]{section.12.4}{Teil 3 }{chapter.12}% 161 -\BOOKMARK [2][-]{subsection.12.4.1}{De finibus bonorum et malorum }{section.12.4}% 162 -\BOOKMARK [0][-]{chapter.13}{Thema}{part.2}% 163 -\BOOKMARK [1][-]{section.13.1}{Teil 0}{chapter.13}% 164 -\BOOKMARK [1][-]{section.13.2}{Teil 1 }{chapter.13}% 165 -\BOOKMARK [2][-]{subsection.13.2.1}{De finibus bonorum et malorum }{section.13.2}% 166 -\BOOKMARK [1][-]{section.13.3}{Teil 2 }{chapter.13}% 167 -\BOOKMARK [2][-]{subsection.13.3.1}{De finibus bonorum et malorum }{section.13.3}% 168 -\BOOKMARK [1][-]{section.13.4}{Teil 3 }{chapter.13}% 169 -\BOOKMARK [2][-]{subsection.13.4.1}{De finibus bonorum et malorum }{section.13.4}% 170 -\BOOKMARK [0][-]{chapter.14}{Thema}{part.2}% 171 -\BOOKMARK [1][-]{section.14.1}{Teil 0}{chapter.14}% 172 -\BOOKMARK [1][-]{section.14.2}{Teil 1 }{chapter.14}% 173 -\BOOKMARK [2][-]{subsection.14.2.1}{De finibus 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{subsubsection}{Addition}{10}{section*.6}% -\contentsline {subsubsection}{Multiplikation}{10}{section*.7}% -\contentsline {subsubsection}{Rechenregeln}{10}{section*.8}% -\contentsline {subsubsection}{Teilbarkeit}{11}{section*.9}% -\contentsline {subsubsection}{Konstruktion der nat\IeC {\"u}rlichen Zahlen aus der Mengenlehre}{11}{section*.10}% -\contentsline {subsubsection}{Nat\IeC {\"u}rliche Zahlen als \IeC {\"A}quivalenzklassen}{12}{section*.11}% -\contentsline {section}{\numberline {1.2}Ganze Zahlen }{12}{section.1.2}% -\contentsline {subsubsection}{Paare von nat\IeC {\"u}rlichen Zahlen}{12}{section*.12}% -\contentsline {subsubsection}{\IeC {\"A}quivalenzrelation}{13}{section*.13}% -\contentsline {subsubsection}{Entgegengesetzter Wert}{13}{section*.14}% -\contentsline {subsubsection}{L\IeC {\"o}sung von Gleichungen}{13}{section*.15}% -\contentsline {subsubsection}{Ring}{14}{section*.16}% -\contentsline {section}{\numberline {1.3}Rationale Zahlen }{14}{section.1.3}% -\contentsline {subsubsection}{Br\IeC {\"u}che}{14}{section*.17}% -\contentsline {subsubsection}{K\IeC {\"u}rzen}{15}{section*.18}% -\contentsline {subsubsection}{Kehrwert}{15}{section*.19}% -\contentsline {subsubsection}{L\IeC {\"o}sung von linearen Gleichungen}{15}{section*.20}% -\contentsline {subsubsection}{K\IeC {\"o}rper}{15}{section*.21}% -\contentsline {section}{\numberline {1.4}Reelle Zahlen }{16}{section.1.4}% -\contentsline {section}{\numberline {1.5}Komplexe Zahlen }{16}{section.1.5}% -\contentsline {subsubsection}{Imagin\IeC {\"a}re und komplexe Zahlen}{17}{section*.22}% -\contentsline {subsubsection}{Real- und Imagin\IeC {\"a}rteil}{17}{section*.23}% -\contentsline {subsubsection}{Komplexe Konjugation}{17}{section*.24}% -\contentsline {subsubsection}{Betrag}{18}{section*.25}% -\contentsline {subsubsection}{Division}{18}{section*.26}% -\contentsline {subsubsection}{Gausssche Zahlenebene}{18}{section*.27}% -\contentsline {subsubsection}{Geometrische Interpretation der Rechenoperationen}{19}{section*.29}% -\contentsline {subsubsection}{Algebraische Vollst\IeC {\"a}ndigkeit}{19}{section*.30}% -\contentsline {subsubsection}{Quaternionen und Octonionen}{20}{section*.31}% -\contentsline {chapter}{\numberline {2}Vektoren und Matrizen }{23}{chapter.2}% -\contentsline {section}{\numberline {2.1}Lineare Algebra }{23}{section.2.1}% -\contentsline {subsection}{\numberline {2.1.1}Vektoren }{23}{subsection.2.1.1}% -\contentsline {subsubsection}{Zeilen- und Spaltenvektoren}{23}{section*.32}% -\contentsline {subsubsection}{Standardbasisvektoren}{24}{section*.33}% -\contentsline {subsubsection}{Vektorraum}{24}{section*.34}% -\contentsline {subsubsection}{Gleichungssysteme in Vektorform}{25}{section*.35}% -\contentsline {subsubsection}{Lineare Abh\IeC {\"a}ngigkeit}{26}{section*.36}% -\contentsline {subsubsection}{Basis}{26}{section*.37}% -\contentsline {subsubsection}{Unterr\IeC {\"a}ume}{26}{section*.38}% -\contentsline {subsection}{\numberline {2.1.2}Matrizen }{27}{subsection.2.1.2}% -\contentsline {subsubsection}{Definition einer Matrix}{27}{section*.39}% -\contentsline {subsubsection}{Addition und Multiplikation mit Skalaren}{27}{section*.40}% -\contentsline {subsubsection}{Multiplikation}{28}{section*.41}% -\contentsline {subsubsection}{Einheitsmatrix}{28}{section*.42}% -\contentsline {subsection}{\numberline {2.1.3}Gleichungssysteme }{28}{subsection.2.1.3}% -\contentsline {subsubsection}{Eindeutige L\IeC {\"o}sung}{28}{section*.43}% -\contentsline {subsubsection}{Inhomogene und homogene Gleichungssysteme}{29}{section*.44}% -\contentsline {subsubsection}{Gauss-Algorithmus}{29}{section*.45}% -\contentsline {subsubsection}{L\IeC {\"o}sungsmenge}{31}{section*.47}% -\contentsline {subsubsection}{Inverse Matrix}{31}{section*.48}% -\contentsline {subsubsection}{Determinante}{32}{section*.49}% -\contentsline {subsection}{\numberline {2.1.4}Lineare Abbildungen }{32}{subsection.2.1.4}% -\contentsline {subsubsection}{Definition}{32}{section*.50}% -\contentsline {subsubsection}{Matrix}{33}{section*.51}% -\contentsline {subsubsection}{Basiswechsel}{33}{section*.52}% -\contentsline {subsubsection}{Umkehrabbbildung}{34}{section*.53}% -\contentsline {subsubsection}{Kern und Bild}{34}{section*.54}% -\contentsline {subsubsection}{Rang und Defekt}{35}{section*.55}% -\contentsline {subsubsection}{Quotient}{35}{section*.56}% -\contentsline {section}{\numberline {2.2}Skalarprodukt }{35}{section.2.2}% -\contentsline {subsection}{\numberline {2.2.1}Bilinearformen und Skalarprodukte }{36}{subsection.2.2.1}% -\contentsline {subsubsection}{Symmetrische bilineare Funktionen}{36}{section*.57}% -\contentsline {subsubsection}{Positiv definite Bilinearformen und Skalarprodukt}{36}{section*.58}% -\contentsline {subsubsection}{Dreiecksungleichung}{37}{section*.59}% -\contentsline {subsubsection}{Polarformel}{38}{section*.60}% -\contentsline {subsubsection}{Komplexe Vektorr\IeC {\"a}ume und Sesquilinearformen}{38}{section*.61}% -\contentsline {subsection}{\numberline {2.2.2}Orthognormalbasis }{39}{subsection.2.2.2}% -\contentsline {subsubsection}{Gram-Matrix}{39}{section*.62}% -\contentsline {subsubsection}{Orthonormalbasis}{39}{section*.63}% -\contentsline {subsubsection}{Gram-Schmidt-Orthonormalisierung}{39}{section*.64}% -\contentsline {subsubsection}{Orthogonalisierung}{40}{section*.65}% -\contentsline {subsubsection}{Orthonormalbasen in komplexen Vektorr\IeC {\"a}umen}{40}{section*.66}% -\contentsline {subsection}{\numberline {2.2.3}Symmetrische und selbstadjungierte Abbilungen }{40}{subsection.2.2.3}% -\contentsline {subsubsection}{Symmetrische Abbildungen}{41}{section*.67}% -\contentsline {subsubsection}{Selbstadjungierte Abbildungen}{41}{section*.68}% -\contentsline {subsubsection}{Die Adjungierte}{41}{section*.69}% -\contentsline {subsection}{\numberline {2.2.4}Orthogonale und unit\IeC {\"a}re Matrizen }{42}{subsection.2.2.4}% -\contentsline {subsection}{\numberline {2.2.5}Orthogonale Unterr\IeC {\"a}ume }{42}{subsection.2.2.5}% -\contentsline {subsection}{\numberline {2.2.6}Andere Normen auf Vektorr\IeC {\"a}umen }{42}{subsection.2.2.6}% -\contentsline {subsubsection}{$l^1$-Norm}{42}{section*.70}% -\contentsline {subsubsection}{$l^\infty $-Norm}{43}{section*.71}% -\contentsline {subsubsection}{Operatornorm}{43}{section*.72}% -\contentsline {subsubsection}{Normen auf Funktionenr\IeC {\"a}umen}{44}{section*.73}% -\contentsline {section}{\numberline {2.3}Algebraische Strukturen }{44}{section.2.3}% -\contentsline {subsection}{\numberline {2.3.1}Gruppen }{44}{subsection.2.3.1}% -\contentsline {subsubsection}{Beispiele von Gruppen}{46}{section*.75}% -\contentsline {subsubsection}{Einige einfache Rechenregeln in Gruppen}{47}{section*.76}% -\contentsline {subsubsection}{Homomorphismen}{47}{section*.77}% -\contentsline {subsubsection}{Normalteiler}{48}{section*.78}% -\contentsline {subsubsection}{Faktorgruppen}{48}{section*.79}% -\contentsline {subsubsection}{Darstellungen}{49}{section*.80}% -\contentsline {subsection}{\numberline {2.3.2}Ringe und Moduln }{49}{subsection.2.3.2}% -\contentsline {subsubsection}{Definition eines Rings}{49}{section*.81}% -\contentsline {subsubsection}{Beispiele von Ringen}{50}{section*.82}% -\contentsline {subsubsection}{Einheiten}{51}{section*.84}% -\contentsline {subsubsection}{Nullteiler}{51}{section*.85}% -\contentsline {subsubsection}{Homomorphismus}{52}{section*.86}% -\contentsline {subsubsection}{Ideale}{52}{section*.87}% -\contentsline {subsection}{\numberline {2.3.3}Algebren }{53}{subsection.2.3.3}% -\contentsline {subsubsection}{Die Algebra der Funktionen $\Bbbk ^X$}{54}{section*.89}% -\contentsline {subsubsection}{Die Algebra der stetigen Funktionen $C([a,b])$}{54}{section*.90}% -\contentsline {subsection}{\numberline {2.3.4}K\IeC {\"o}rper }{54}{subsection.2.3.4}% -\contentsline {section}{\numberline {2.4}Hadamard-Algebra }{55}{section.2.4}% -\contentsline {subsection}{\numberline {2.4.1}Hadamard-Produkt }{55}{subsection.2.4.1}% -\contentsline {subsection}{\numberline {2.4.2}Hadamard-Produkt und Matrizenalgebra }{56}{subsection.2.4.2}% -\contentsline {subsubsection}{Unvertr\IeC {\"a}glichkeit von Hadamard- und Matrizen-Produkt}{56}{section*.91}% -\contentsline {subsubsection}{Einbettung der Hadamard-Algebra ein eine Matrizenalgebra}{56}{section*.92}% -\contentsline {subsubsection}{Beispiel: Faltung und Fourier-Theorie}{57}{section*.93}% -\contentsline {subsection}{\numberline {2.4.3}Weitere Verkn\IeC {\"u}pfungen }{57}{subsection.2.4.3}% -\contentsline {subsubsection}{Transposition}{57}{section*.94}% -\contentsline {subsubsection}{Frobeniusnorm}{57}{section*.95}% -\contentsline {subsubsection}{Skalarprodukt}{58}{section*.96}% -\contentsline {chapter}{\numberline {3}Polynome }{61}{chapter.3}% -\contentsline {section}{\numberline {3.1}Definitionen }{62}{section.3.1}% -\contentsline {subsection}{\numberline {3.1.1}Skalare }{62}{subsection.3.1.1}% -\contentsline {subsection}{\numberline {3.1.2}Der Polynomring }{63}{subsection.3.1.2}% -\contentsline {subsection}{\numberline {3.1.3}Grad }{64}{subsection.3.1.3}% -\contentsline {subsection}{\numberline {3.1.4}Teilbarkeit }{66}{subsection.3.1.4}% -\contentsline {subsubsection}{Polynomdivision}{66}{section*.98}% -\contentsline {subsubsection}{Euklidische Ringe und Faktorzerlegung}{67}{section*.99}% -\contentsline {subsubsection}{Irreduzible Polynome}{68}{section*.100}% -\contentsline {subsubsection}{Faktorisierung in einem Polynomring}{68}{section*.101}% -\contentsline {subsection}{\numberline {3.1.5}Formale Potenzreihen }{68}{subsection.3.1.5}% -\contentsline {section}{\numberline {3.2}Polynome als Vektoren }{68}{section.3.2}% -\contentsline {subsection}{\numberline {3.2.1}Polynome beliebigen Grades }{69}{subsection.3.2.1}% -\contentsline {subsection}{\numberline {3.2.2}Multiplikative Struktur }{70}{subsection.3.2.2}% -\contentsline {section}{\numberline {3.3}Polynommultiplikation mit Matrizen }{70}{section.3.3}% -\contentsline {section}{\numberline {3.4}Minimalpolynom }{70}{section.3.4}% -\contentsline {chapter}{\numberline {4}Endliche K\IeC {\"o}rper }{71}{chapter.4}% -\contentsline {section}{\numberline {4.1}Der euklidische Algorithmus }{71}{section.4.1}% -\contentsline {subsection}{\numberline {4.1.1}Ganze Zahlen}{71}{subsection.4.1.1}% -\contentsline {subsection}{\numberline {4.1.2}Matrixschreibweise }{73}{subsection.4.1.2}% -\contentsline {subsection}{\numberline {4.1.3}Vereinfachte Durchf\IeC {\"u}hrung }{74}{subsection.4.1.3}% -\contentsline {subsection}{\numberline {4.1.4}Polynome}{76}{subsection.4.1.4}% -\contentsline {section}{\numberline {4.2}Galois-K\IeC {\"o}rper }{77}{section.4.2}% -\contentsline {subsection}{\numberline {4.2.1}Arithmetik modulo $p$ }{78}{subsection.4.2.1}% -\contentsline {subsubsection}{Restklassenring}{78}{section*.103}% -\contentsline {subsubsection}{Division in $\mathbb {Z}/n\mathbb {Z}$}{79}{section*.104}% -\contentsline {subsubsection}{Der kleine Satz von Fermat}{80}{section*.105}% -\contentsline {subsubsection}{Der Satz von Wilson}{81}{section*.106}% -\contentsline {subsection}{\numberline {4.2.2}Charakteristik }{81}{subsection.4.2.2}% -\contentsline {subsubsection}{Primk\IeC {\"o}rper}{82}{section*.107}% -\contentsline {subsubsection}{Teilbarkeit von Binomialkoeffizienten}{82}{section*.108}% -\contentsline {subsubsection}{Frobenius-Automorphismus}{84}{section*.111}% -\contentsline {section}{\numberline {4.3}Wurzeln }{85}{section.4.3}% -\contentsline {subsection}{\numberline {4.3.1}Irreduzible Polynome }{85}{subsection.4.3.1}% -\contentsline {subsection}{\numberline {4.3.2}K\IeC {\"o}rpererweiterungen }{87}{subsection.4.3.2}% -\contentsline {subsubsection}{Erweiterung mit einem irreduziblen Polynom}{87}{section*.112}% -\contentsline {subsubsection}{Matrixrealisierung der Multiplikation mit $\alpha $}{87}{section*.113}% -\contentsline {subsubsection}{Inverse}{88}{section*.114}% -\contentsline {subsubsection}{Algebraische Konstruktion}{91}{section*.116}% -\contentsline {subsubsection}{Reduktion modulo $m$}{91}{section*.117}% -\contentsline {subsubsection}{Multiplikative Inverse}{92}{section*.118}% -\contentsline {subsection}{\numberline {4.3.3}Zerf\IeC {\"a}llungsk\IeC {\"o}rper }{94}{subsection.4.3.3}% -\contentsline {chapter}{\numberline {5}Eigenwerte und Eigenvektoren }{101}{chapter.5}% -\contentsline {section}{\numberline {5.1}Grundlagen }{101}{section.5.1}% -\contentsline {subsection}{\numberline {5.1.1}Kern und Bild von Matrixpotenzen }{101}{subsection.5.1.1}% -\contentsline {subsection}{\numberline {5.1.2}Invariante Unterr\IeC {\"a}ume }{103}{subsection.5.1.2}% -\contentsline {subsection}{\numberline {5.1.3}Nilpotente Matrizen }{104}{subsection.5.1.3}% -\contentsline {subsection}{\numberline {5.1.4}Eigenwerte und Eigenvektoren }{105}{subsection.5.1.4}% -\contentsline {subsection}{\numberline {5.1.5}Verallgemeinerte Eigenr\IeC {\"a}ume }{107}{subsection.5.1.5}% -\contentsline {subsection}{\numberline {5.1.6}Zerlegung in invariante Unterr\IeC {\"a}ume }{108}{subsection.5.1.6}% -\contentsline {subsection}{\numberline {5.1.7}Das charakteristische Polynom }{109}{subsection.5.1.7}% -\contentsline {section}{\numberline {5.2}Normalformen }{111}{section.5.2}% -\contentsline {subsection}{\numberline {5.2.1}Diagonalform}{111}{subsection.5.2.1}% -\contentsline {subsection}{\numberline {5.2.2}Jordan-Normalform }{111}{subsection.5.2.2}% -\contentsline {subsection}{\numberline {5.2.3}Reelle Normalform }{114}{subsection.5.2.3}% -\contentsline {subsection}{\numberline {5.2.4}Obere Hessenberg-Form }{114}{subsection.5.2.4}% -\contentsline {section}{\numberline {5.3}Funktionen einer Matrix }{114}{section.5.3}% -\contentsline {subsection}{\numberline {5.3.1}Polynom-Funktionen }{114}{subsection.5.3.1}% -\contentsline {subsection}{\numberline {5.3.2}Approximation von $f(A)$ }{117}{subsection.5.3.2}% -\contentsline {subsection}{\numberline {5.3.3}Potenzreihen }{118}{subsection.5.3.3}% -\contentsline {subsection}{\numberline {5.3.4}Gelfand-Radius und Eigenwerte }{118}{subsection.5.3.4}% -\contentsline {subsubsection}{Spezialfall: Diagonalisierbare Matrizen}{118}{section*.121}% -\contentsline {subsubsection}{Blockmatrizen}{119}{section*.122}% -\contentsline {subsubsection}{Jordan-Bl\IeC {\"o}cke}{120}{section*.123}% -\contentsline {subsubsection}{Iterationsfolgen}{120}{section*.124}% -\contentsline {subsubsection}{Der Satz von Gelfand}{121}{section*.125}% -\contentsline {section}{\numberline {5.4}Numerische Verfahren zur Eigenwertbestimmung }{122}{section.5.4}% -\contentsline {section}{\numberline {5.5}Spektraltheorie }{122}{section.5.5}% -\contentsline {chapter}{\numberline {6}Permutationen }{125}{chapter.6}% -\contentsline {section}{\numberline {6.1}Permutationen einer endlichen Menge }{125}{section.6.1}% -\contentsline {subsection}{\numberline {6.1.1}Permutationen als $2\times n$-Matrizen}{125}{subsection.6.1.1}% -\contentsline {subsection}{\numberline {6.1.2}Zyklenzerlegung }{126}{subsection.6.1.2}% -\contentsline {subsection}{\numberline {6.1.3}Konjugierte Elemente in $S_n$}{127}{subsection.6.1.3}% -\contentsline {section}{\numberline {6.2}Permutationen und Transpositionen }{127}{section.6.2}% -\contentsline {subsection}{\numberline {6.2.1}Zyklus und Permutationen aus Transpositionen}{128}{subsection.6.2.1}% -\contentsline {subsection}{\numberline {6.2.2}Signum einer Permutation}{128}{subsection.6.2.2}% -\contentsline {section}{\numberline {6.3}Permutationsmatrizen }{129}{section.6.3}% -\contentsline {subsection}{\numberline {6.3.1}Matrizen}{130}{subsection.6.3.1}% -\contentsline {subsection}{\numberline {6.3.2}Transpositionen}{130}{subsection.6.3.2}% -\contentsline {subsection}{\numberline {6.3.3}Determinante und Vorzeichen}{131}{subsection.6.3.3}% -\contentsline {section}{\numberline {6.4}Determinante }{132}{section.6.4}% -\contentsline {chapter}{\numberline {7}Matrizengruppen }{133}{chapter.7}% -\contentsline {section}{\numberline {7.1}Symmetrien }{133}{section.7.1}% -\contentsline {section}{\numberline {7.2}Lie-Gruppen }{133}{section.7.2}% -\contentsline {section}{\numberline {7.3}Lie-Algebren }{133}{section.7.3}% -\contentsline {section}{\numberline {7.4}Homogene R\IeC {\"a}ume }{133}{section.7.4}% -\contentsline {chapter}{\numberline {8}Graphen }{135}{chapter.8}% -\contentsline {section}{\numberline {8.1}Beschreibung von Graphen mit Matrizen }{135}{section.8.1}% -\contentsline {subsection}{\numberline {8.1.1}Definition von Graphen }{136}{subsection.8.1.1}% -\contentsline {subsubsection}{Ungerichtete Graphen}{136}{section*.129}% -\contentsline {subsubsection}{Gerichtete Graphen}{136}{section*.130}% -\contentsline {subsubsection}{Adjazenzmatrix}{137}{section*.131}% -\contentsline {subsubsection}{Adjazenzmatrix und die Anzahl der Pfade}{138}{section*.134}% -\contentsline {subsubsection}{Beschriftete Graphen}{140}{section*.136}% -\contentsline {subsection}{\numberline {8.1.2}Inzidenzmatrix}{140}{subsection.8.1.2}% -\contentsline {subsubsection}{Beschriftete Graphen}{140}{section*.137}% -\contentsline {subsubsection}{Inzidenzmatrix und Adjazenzmatrix}{141}{section*.138}% -\contentsline {subsubsection}{Gradmatrix}{141}{section*.139}% -\contentsline {subsubsection}{Gerichtete Graphen}{141}{section*.140}% -\contentsline {subsubsection}{Anwendung: Netlist}{141}{section*.141}% -\contentsline {subsection}{\numberline {8.1.3}Die Adjazenzmatrix und Laplace-Matrix }{141}{subsection.8.1.3}% -\contentsline {section}{\numberline {8.2}Spektrale Graphentheorie }{142}{section.8.2}% -\contentsline {section}{\numberline {8.3}Wavelets auf Graphen }{142}{section.8.3}% -\contentsline {subsection}{\numberline {8.3.1}Funktionen auf einem Graphen und die Laplace-Matrix}{142}{subsection.8.3.1}% -\contentsline {subsection}{\numberline {8.3.2}Standardbasis und Eigenbasis }{143}{subsection.8.3.2}% -\contentsline {subsection}{\numberline {8.3.3}Wavelet-Basen }{144}{subsection.8.3.3}% -\contentsline {chapter}{\numberline {9}Wahrscheinlichkeitsmatrizen }{145}{chapter.9}% -\contentsline {section}{\numberline {9.1}Google-Matrix }{145}{section.9.1}% -\contentsline {subsection}{\numberline {9.1.1}Ein Modell f\IeC {\"u}r Webseitenbesucher }{146}{subsection.9.1.1}% -\contentsline {subsection}{\numberline {9.1.2}Wahrscheinlichkeitsinterpretation }{146}{subsection.9.1.2}% -\contentsline {subsubsection}{Ereignisse und Wahrscheinlichkeiten}{146}{section*.144}% -\contentsline {subsubsection}{Bedingte Wahrscheinlichkeit}{147}{section*.145}% -\contentsline {subsubsection}{Totale Wahrscheinlichkeit}{147}{section*.146}% -\contentsline {subsection}{\numberline {9.1.3}``Freier Wille'' }{148}{subsection.9.1.3}% -\contentsline {subsubsection}{Erweiterung der Link-Matrix}{148}{section*.147}% -\contentsline {subsubsection}{Die Google-Matrix}{149}{section*.148}% -\contentsline {subsection}{\numberline {9.1.4}Wahrscheinlichkeitsverteilung }{150}{subsection.9.1.4}% -\contentsline {subsubsection}{Station\IeC {\"a}re Verteilung}{150}{section*.149}% -\contentsline {subsubsection}{Potenzverfahren}{151}{section*.150}% -\contentsline {section}{\numberline {9.2}Diskrete Markov-Ketten und Wahrscheinlichkeitsmatrizen }{152}{section.9.2}% -\contentsline {subsection}{\numberline {9.2.1}Markov-Eigenschaft}{152}{subsection.9.2.1}% -\contentsline {subsubsection}{Ged\IeC {\"a}chtnislosigkeit}{152}{section*.151}% -\contentsline {subsubsection}{Die Chapman-Kolmogorov-Gleichung}{153}{section*.152}% -\contentsline {subsection}{\numberline {9.2.2}Diskrete Markov-Kette}{153}{subsection.9.2.2}% -\contentsline {subsubsection}{Zustandswahrscheinlichkeiten}{155}{section*.154}% -\contentsline {subsubsection}{Zeitunabh\IeC {\"a}ngige \IeC {\"U}bergangswahrscheinlichkeiten}{155}{section*.155}% -\contentsline {subsubsection}{Station\IeC {\"a}re Verteilung}{155}{section*.156}% -\contentsline {subsubsection}{Irreduzible Markov-Ketten}{156}{section*.157}% -\contentsline {subsubsection}{Die konvexe Menge der station\IeC {\"a}ren Verteilungen}{158}{section*.159}% -\contentsline {subsubsection}{Grenzverteilung}{158}{section*.161}% -\contentsline {subsubsection}{Erwartungswert und Varianz}{159}{section*.162}% -\contentsline {subsubsection}{Erwartungswert von Werten auf \IeC {\"U}berg\IeC {\"a}ngen}{159}{section*.163}% -\contentsline {subsection}{\numberline {9.2.3}Absorbierende Zust\IeC {\"a}nde}{160}{subsection.9.2.3}% -\contentsline {subsubsection}{Absorbtionszeit}{161}{section*.165}% -\contentsline {subsubsection}{Wartezeit}{162}{section*.166}% -\contentsline {section}{\numberline {9.3}Positive Vektoren und Matrizen }{162}{section.9.3}% -\contentsline {subsection}{\numberline {9.3.1}Elementare Eigenschaften }{163}{subsection.9.3.1}% -\contentsline {subsection}{\numberline {9.3.2}Die verallgemeinerte Dreiecksungleichung }{166}{subsection.9.3.2}% -\contentsline {subsection}{\numberline {9.3.3}Der Satz von Perron-Frobenius }{168}{subsection.9.3.3}% -\contentsline {section}{\numberline {9.4}Das Paradoxon von Parrondo }{172}{section.9.4}% -\contentsline {subsection}{\numberline {9.4.1}Die beiden Teilspiele }{172}{subsection.9.4.1}% -\contentsline {subsubsection}{Das Spiel $A$}{172}{section*.172}% -\contentsline {subsubsection}{Das Spiel $B$}{172}{section*.173}% -\contentsline {subsubsection}{\IeC {\"U}bergangsmatrix im Spiel $B$}{172}{section*.174}% -\contentsline {subsubsection}{Gewinnerwartung in einem Einzelspiel $B$}{173}{section*.176}% -\contentsline {subsubsection}{Das wiederholte Spiel $B$}{174}{section*.177}% -\contentsline {subsubsection}{Das modifizierte Spiel $\mathaccentV {tilde}07E{B}$}{175}{section*.178}% -\contentsline {subsubsection}{Gewinnerwartung im modifizierten Einzelspiel}{176}{section*.180}% -\contentsline {subsubsection}{Iteration des modifizierten Spiels}{177}{section*.181}% -\contentsline {subsection}{\numberline {9.4.2}Kombination der Spiele }{177}{subsection.9.4.2}% -\contentsline {subsubsection}{Das Spiel $C$}{178}{section*.182}% -\contentsline {subsubsection}{Das iterierte Spiel $C$}{178}{section*.183}% -\contentsline {chapter}{\numberline {10}Anwendungen in Kryptographie und Codierungstheorie }{179}{chapter.10}% -\contentsline {section}{\numberline {10.1}Arithmetik f\IeC {\"u}r die Kryptographie }{179}{section.10.1}% -\contentsline {subsection}{\numberline {10.1.1}Potenzieren }{179}{subsection.10.1.1}% -\contentsline {subsection}{\numberline {10.1.2}Rechenoperationen in $\mathbb {F}_p$ }{179}{subsection.10.1.2}% -\contentsline {subsection}{\numberline {10.1.3}Rechenoperationen in $\mathbb {F}_{2^l}$ }{179}{subsection.10.1.3}% -\contentsline {section}{\numberline {10.2}Kryptographie und endliche K\IeC {\"o}rper }{179}{section.10.2}% -\contentsline {subsection}{\numberline {10.2.1}Potenzen in $\mathbb {F}_p$ und diskreter Logarithmus }{179}{subsection.10.2.1}% -\contentsline {subsection}{\numberline {10.2.2}Diffie-Hellman-Schl\IeC {\"u}sseltausch }{181}{subsection.10.2.2}% -\contentsline {subsection}{\numberline {10.2.3}Elliptische Kurven }{181}{subsection.10.2.3}% -\contentsline {subsubsection}{Elliptische Kurven}{182}{section*.185}% -\contentsline {subsubsection}{Geometrische Definition der Gruppenoperation}{183}{section*.187}% -\contentsline {subsubsection}{Gruppenoperation, algebraische Konstruktion}{184}{section*.188}% -\contentsline {subsubsection}{Beispiele}{187}{section*.189}% -\contentsline {subsubsection}{Diffie-Hellman in einer elliptischen Kurve}{187}{section*.190}% -\contentsline {section}{\numberline {10.3}Advanced Encryption Standard -- AES }{187}{section.10.3}% -\contentsline {section}{\numberline {10.4}Fehlerkorrigierende Codes nach Reed-Solomon }{187}{section.10.4}% -\contentsline {subsection}{\numberline {10.4.1}Was ist ein Code? }{188}{subsection.10.4.1}% -\contentsline {subsection}{\numberline {10.4.2}Reed-Solomon-Code }{188}{subsection.10.4.2}% -\contentsline {subsection}{\numberline {10.4.3}Decodierung }{188}{subsection.10.4.3}% -\contentsline {chapter}{\numberline {11}Homologie }{189}{chapter.11}% -\contentsline {section}{\numberline {11.1}Simplexe und simpliziale Komplexe }{189}{section.11.1}% -\contentsline {subsection}{\numberline {11.1.1}Simplexe und Rand }{189}{subsection.11.1.1}% -\contentsline {subsubsection}{Rand eines Dreiecks}{189}{section*.193}% -\contentsline {subsubsection}{Simplizes}{190}{section*.194}% -\contentsline {subsubsection}{Rechnen mit Simplizes}{191}{section*.195}% -\contentsline {subsubsection}{Rand eines Simplex}{191}{section*.196}% -\contentsline {subsection}{\numberline {11.1.2}Triangulation }{192}{subsection.11.1.2}% -\contentsline {section}{\numberline {11.2}Kettenkomplexe }{192}{section.11.2}% -\contentsline {subsection}{\numberline {11.2.1}Randoperator von Simplexen }{192}{subsection.11.2.1}% -\contentsline {subsection}{\numberline {11.2.2}Kettenkomplexe und Morphismen }{192}{subsection.11.2.2}% -\contentsline {section}{\numberline {11.3}Homologie }{192}{section.11.3}% -\contentsline {subsection}{\numberline {11.3.1}Homologie eines Kettenkomplexes }{192}{subsection.11.3.1}% -\contentsline {subsection}{\numberline {11.3.2}Induzierte Abbildung }{192}{subsection.11.3.2}% -\contentsline {subsection}{\numberline {11.3.3}Homologie eines simplizialen Komplexes }{192}{subsection.11.3.3}% -\contentsline {section}{\numberline {11.4}Exaktheit und die Mayer-Vietoris-Folge }{192}{section.11.4}% -\contentsline {subsection}{\numberline {11.4.1}Kurze exakte Folgen von Kettenkomplexen }{192}{subsection.11.4.1}% -\contentsline {subsection}{\numberline {11.4.2}Schlangenlemma und lange exakte Folgen }{192}{subsection.11.4.2}% -\contentsline {subsection}{\numberline {11.4.3}Mayer-Vietoris-Folge }{192}{subsection.11.4.3}% -\contentsline {section}{\numberline {11.5}Fixpunkte }{192}{section.11.5}% -\contentsline {subsection}{\numberline {11.5.1}Lefshetz-Spurformel }{192}{subsection.11.5.1}% -\contentsline {subsection}{\numberline {11.5.2}Brower-Fixpunktsatz }{192}{subsection.11.5.2}% -\contentsline {part}{II\hspace {1em}Anwendungen und weiterf\IeC {\"u}hrende Themen}{193}{part.2}% -\contentsline {chapter}{\numberline {12}Thema}{197}{chapter.12}% -\contentsline {section}{\numberline {12.1}Teil 0}{197}{section.12.1}% -\contentsline {section}{\numberline {12.2}Teil 1 }{197}{section.12.2}% -\contentsline {subsection}{\numberline {12.2.1}De finibus bonorum et malorum }{198}{subsection.12.2.1}% -\contentsline {section}{\numberline {12.3}Teil 2 }{198}{section.12.3}% -\contentsline {subsection}{\numberline {12.3.1}De finibus bonorum et malorum }{198}{subsection.12.3.1}% -\contentsline {section}{\numberline {12.4}Teil 3 }{199}{section.12.4}% -\contentsline {subsection}{\numberline {12.4.1}De finibus bonorum et malorum }{199}{subsection.12.4.1}% -\contentsline {chapter}{\numberline {13}Thema}{201}{chapter.13}% -\contentsline {section}{\numberline {13.1}Teil 0}{201}{section.13.1}% -\contentsline {section}{\numberline {13.2}Teil 1 }{201}{section.13.2}% -\contentsline {subsection}{\numberline {13.2.1}De finibus bonorum et malorum }{202}{subsection.13.2.1}% -\contentsline {section}{\numberline {13.3}Teil 2 }{202}{section.13.3}% -\contentsline {subsection}{\numberline {13.3.1}De finibus bonorum et malorum }{202}{subsection.13.3.1}% -\contentsline {section}{\numberline {13.4}Teil 3 }{203}{section.13.4}% -\contentsline {subsection}{\numberline {13.4.1}De finibus bonorum et malorum }{203}{subsection.13.4.1}% -\contentsline {chapter}{\numberline {14}Thema}{205}{chapter.14}% -\contentsline {section}{\numberline {14.1}Teil 0}{205}{section.14.1}% -\contentsline {section}{\numberline {14.2}Teil 1 }{205}{section.14.2}% -\contentsline {subsection}{\numberline {14.2.1}De finibus bonorum et malorum }{206}{subsection.14.2.1}% -\contentsline {section}{\numberline {14.3}Teil 2 }{206}{section.14.3}% -\contentsline {subsection}{\numberline {14.3.1}De finibus bonorum et malorum }{206}{subsection.14.3.1}% -\contentsline {section}{\numberline {14.4}Teil 3 }{207}{section.14.4}% -\contentsline {subsection}{\numberline {14.4.1}De finibus bonorum et malorum }{207}{subsection.14.4.1}% -\contentsline {chapter}{\numberline {15}Thema}{209}{chapter.15}% -\contentsline {section}{\numberline {15.1}Teil 0}{209}{section.15.1}% -\contentsline {section}{\numberline {15.2}Teil 1 }{209}{section.15.2}% -\contentsline {subsection}{\numberline {15.2.1}De finibus bonorum et malorum }{210}{subsection.15.2.1}% -\contentsline {section}{\numberline {15.3}Teil 2 }{210}{section.15.3}% -\contentsline {subsection}{\numberline {15.3.1}De finibus bonorum et malorum }{210}{subsection.15.3.1}% -\contentsline {section}{\numberline {15.4}Teil 3 }{211}{section.15.4}% -\contentsline {subsection}{\numberline {15.4.1}De finibus bonorum et malorum }{211}{subsection.15.4.1}% -\contentsline {chapter}{\numberline {16}Iterierte Funktionsschemata}{213}{chapter.16}% -\contentsline {section}{\numberline {16.1}Teil 0}{213}{section.16.1}% -\contentsline {section}{\numberline {16.2}Teil 1 }{213}{section.16.2}% -\contentsline {subsection}{\numberline {16.2.1}De finibus bonorum et malorum }{214}{subsection.16.2.1}% -\contentsline {section}{\numberline {16.3}Teil 2 }{214}{section.16.3}% -\contentsline {subsection}{\numberline {16.3.1}De finibus bonorum et malorum }{214}{subsection.16.3.1}% -\contentsline {section}{\numberline {16.4}Teil 3 }{215}{section.16.4}% -\contentsline {subsection}{\numberline {16.4.1}De finibus bonorum et malorum }{215}{subsection.16.4.1}% -\contentsline {chapter}{\numberline {17}McEliece-Kryptosystem}{217}{chapter.17}% -\contentsline {section}{\numberline {17.1}Teil 0}{217}{section.17.1}% -\contentsline {section}{\numberline {17.2}Teil 1 }{217}{section.17.2}% -\contentsline {subsection}{\numberline {17.2.1}De finibus bonorum et malorum }{218}{subsection.17.2.1}% -\contentsline {section}{\numberline {17.3}Teil 2 }{218}{section.17.3}% -\contentsline {subsection}{\numberline {17.3.1}De finibus bonorum et malorum }{218}{subsection.17.3.1}% -\contentsline {section}{\numberline {17.4}Teil 3 }{219}{section.17.4}% -\contentsline {subsection}{\numberline {17.4.1}De finibus bonorum et malorum }{219}{subsection.17.4.1}% -\contentsline {chapter}{\numberline {18}Thema}{221}{chapter.18}% -\contentsline {section}{\numberline {18.1}Teil 0}{221}{section.18.1}% -\contentsline {section}{\numberline {18.2}Teil 1 }{221}{section.18.2}% -\contentsline {subsection}{\numberline {18.2.1}De finibus bonorum et malorum }{222}{subsection.18.2.1}% -\contentsline {section}{\numberline {18.3}Teil 2 }{222}{section.18.3}% -\contentsline {subsection}{\numberline {18.3.1}De finibus bonorum et malorum }{222}{subsection.18.3.1}% -\contentsline {section}{\numberline {18.4}Teil 3 }{223}{section.18.4}% -\contentsline {subsection}{\numberline {18.4.1}De finibus bonorum et malorum }{223}{subsection.18.4.1}% -\contentsline {chapter}{\numberline {19}Thema}{225}{chapter.19}% -\contentsline {section}{\numberline {19.1}Teil 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It will be recreated as -% required. -% -\relax -\bibstyle{biblatex} -\bibdata{buch-blx,chapters/references,papers/verkehr/references,papers/multiplikation/references,papers/punktgruppen/references,papers/reedsolomon/references,papers/ifs/references,papers/mceliece/references,papers/clifford/references,papers/spannung/references,papers/erdbeben/references} -\citation{biblatex-control} -\citation{clifford:bibtex} diff --git a/buch/buch9-blx.aux b/buch/buch9-blx.aux deleted file mode 100644 index 20f2950..0000000 --- a/buch/buch9-blx.aux +++ /dev/null @@ -1,13 +0,0 @@ -% $ biblatex auxiliary file $ -% $ biblatex bbl format version 3.1 $ -% Do not modify this file! -% -% This is an auxiliary file used by the 'biblatex' package. -% This file may safely be deleted. It will be recreated as -% required. -% -\relax -\bibstyle{biblatex} -\bibdata{buch-blx,chapters/references,papers/verkehr/references,papers/multiplikation/references,papers/punktgruppen/references,papers/reedsolomon/references,papers/ifs/references,papers/mceliece/references,papers/clifford/references,papers/spannung/references,papers/erdbeben/references} -\citation{biblatex-control} -\citation{spannung:bibtex} diff --git a/buch/test1.tex b/buch/test1.tex deleted file mode 100644 index 8345230..0000000 --- a/buch/test1.tex +++ /dev/null @@ -1,93 +0,0 @@ -% -% test1.tex -- Test 1 -% -% (c) 2012 Prof. Dr. Andreas Mueller, HSR -% -%\documentclass[a4paper,12pt]{book} -\documentclass[a4paper,12pt]{article} -\usepackage{geometry} -\geometry{papersize={210mm,297mm},total={165mm,260mm}} -\usepackage{ngerman} -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{times} -\usepackage{amsmath} -\usepackage{amssymb} -\usepackage{amsfonts} -\usepackage{amsthm} -\usepackage{graphicx} -\usepackage{fancyhdr} -\usepackage{textcomp} -\usepackage[all]{xy} -\usepackage{txfonts} -\usepackage{alltt} -\usepackage{verbatim} -\usepackage{paralist} -\usepackage{makeidx} -\usepackage{array} -\usepackage{hyperref} -\usepackage{caption} -\usepackage{subcaption} -\usepackage{standalone} -\usepackage{environ} -\usepackage{tikz} -\input{../common/linsys.tex} -\newcounter{beispiel} -\newenvironment{beispiele}{ -\bgroup\smallskip\parindent0pt\bf Beispiele\egroup - -\begin{list}{\arabic{beispiel}.} - {\usecounter{beispiel} - \setlength{\labelsep}{5mm} - \setlength{\rightmargin}{0pt} -}}{\end{list}} -\newcounter{uebungsaufgabe} -% environment fuer uebungsaufgaben -\newenvironment{uebungsaufgaben}{ -\begin{list}{\arabic{uebungsaufgabe}.} - {\usecounter{uebungsaufgabe} - \setlength{\labelwidth}{2cm} - \setlength{\leftmargin}{0pt} - \setlength{\labelsep}{5mm} - \setlength{\rightmargin}{0pt} - \setlength{\itemindent}{0pt} -}}{\end{list}\vfill\pagebreak} -\newenvironment{teilaufgaben}{ -\begin{enumerate} -\renewcommand{\labelenumi}{\alph{enumi})} -}{\end{enumerate}} -% Loesung -\NewEnviron{loesung}{% -\begin{proof}[L"osung]% -\renewcommand{\qedsymbol}{$\bigcirc$} -\BODY -\end{proof}} -\NewEnviron{bewertung}{\relax} -\NewEnviron{diskussion}{ -\BODY -} -\RenewEnviron{loesung}{\relax} -\RenewEnviron{diskussion}{\relax} -\newenvironment{hinweis}{% -\renewcommand{\qedsymbol}{} -\begin{proof}[Hinweis]}{\end{proof}} - -\begin{document} -{\parindent0pt\hbox to\hsize{% -Name: \hbox to7cm{\dotfill} Vorname: \dotfill}} -\vspace{0.5cm} - -\section*{Kurztest 1} - -\begin{uebungsaufgaben} - -\item -\input chapters/30-endlichekoerper/uebungsaufgaben/3003.tex -\item -\input chapters/30-endlichekoerper/uebungsaufgaben/3004.tex -\item -\input chapters/30-endlichekoerper/uebungsaufgaben/3005.tex - -\end{uebungsaufgaben} - -\end{document} -- cgit v1.2.1 From ba58a06b206d92a30af1b21448a2104b12a9dbbe Mon Sep 17 00:00:00 2001 From: LordMcFungus Date: Sun, 13 Jun 2021 22:37:05 +0200 Subject: Delete buch.pdf --- buch/buch.pdf | Bin 2646094 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 buch/buch.pdf (limited to 'buch') diff --git a/buch/buch.pdf b/buch/buch.pdf deleted file mode 100644 index 2ffffe0..0000000 Binary files a/buch/buch.pdf and /dev/null differ -- cgit v1.2.1