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Diffstat (limited to 'buch/papers/lambertw')
19 files changed, 12503 insertions, 117 deletions
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text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="253" y="1288">–0.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="253" y="1288">–0.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="253" y="1288">–0.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" 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font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="356" y="1288">–0.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="356" y="1288">–0.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="463" y="1288">0.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="463" y="1288">0.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="463" y="1288">0.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="515" y="1288">0.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="515" y="1288">0.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="515" y="1288">0.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="566" y="1288">0.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="566" y="1288">0.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="566" y="1288">0.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="618" y="1288">0.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="618" y="1288">0.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="618" y="1288">0.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="670" y="1288">0.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="670" y="1288">0.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="670" y="1288">0.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="721" y="1288">0.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="721" y="1288">0.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="721" y="1288">0.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="773" y="1288">0.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="773" y="1288">0.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="773" y="1288">0.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="825" y="1288">0.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="825" y="1288">0.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="825" y="1288">0.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="876" y="1288">0.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="876" y="1288">0.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="876" y="1288">0.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="934" y="1288">1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="934" y="1288">1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="934" y="1288">1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="980" y="1288">1.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="980" y="1288">1.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="980" y="1288">1.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1032" y="1288">1.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1032" y="1288">1.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1032" y="1288">1.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1083" y="1288">1.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1083" y="1288">1.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1083" y="1288">1.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1135" y="1288">1.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1135" y="1288">1.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1135" y="1288">1.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1187" y="1288">1.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1187" y="1288">1.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1187" y="1288">1.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1238" y="1288">1.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1238" y="1288">1.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1238" y="1288">1.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1290" y="1288">1.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1290" y="1288">1.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1290" y="1288">1.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1342" y="1288">1.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1342" y="1288">1.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1342" y="1288">1.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1393" y="1288">1.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1393" y="1288">1.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1393" y="1288">1.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1451" y="1288">2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1451" y="1288">2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1451" y="1288">2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1497" y="1288">2.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1497" y="1288">2.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1497" y="1288">2.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1549" y="1288">2.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1549" y="1288">2.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1549" y="1288">2.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1600" y="1288">2.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1600" y="1288">2.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1600" y="1288">2.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1652" y="1288">2.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1652" y="1288">2.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1652" y="1288">2.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1704" y="1288">2.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1704" y="1288">2.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1704" y="1288">2.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1755" y="1288">2.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1755" y="1288">2.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1755" y="1288">2.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1807" y="1288">2.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1807" y="1288">2.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1807" y="1288">2.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1859" y="1288">2.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1859" y="1288">2.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1859" y="1288">2.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1910" y="1288">2.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1910" y="1288">2.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1910" y="1288">2.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1968" y="1288">3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="1968" y="1288">3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="1968" y="1288">3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="2014" y="1288">3.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="2014" y="1288">3.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="2014" y="1288">3.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="2066" y="1288">3.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="2066" y="1288">3.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="2066" y="1288">3.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="2117" y="1288">3.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="2117" y="1288">3.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="2117" y="1288">3.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1226">0.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="1226">0.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1226">0.1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1174">0.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="1174">0.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1174">0.2</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1122">0.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="1122">0.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1122">0.3</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1071">0.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="1071">0.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1071">0.4</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1019">0.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="1019">0.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="1019">0.5</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="967">0.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="967">0.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="967">0.6</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="916">0.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="916">0.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="916">0.7</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="864">0.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="864">0.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="864">0.8</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="812">0.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="395" y="812">0.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="395" y="812">0.9</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="405" y="760">1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="none" stroke="rgb(255, 255, 255)" stroke-linejoin="bevel" stroke-miterlimit="10" stroke-opacity="1" stroke-width="3" text-anchor="start" x="405" y="760">1</text><text font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" dominant-baseline="alphabetic" fill="rgb(0, 0, 0)" fill-opacity="1" stroke="none" text-anchor="start" x="405" y="760">1</text><text 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Undefined control sequence. +l.7 \lhead + {Thema} +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 h 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! + +! LaTeX Error: Environment refsection undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.8 \begin{refsection} + +Your command was ignored. +Type I <command> <return> to replace it with another command, +or <return> to continue without it. + +! Undefined control sequence. +l.9 \chapterauthor + {David Hugentobler und Yanik Kuster} +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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LaTeX Error: File `papers/lambertw/teil0.tex' not found. + +Type X to quit or <RETURN> to proceed, +or enter new name. (Default extension: tex) + +Enter file name: +! Emergency stop. +<read *> + +l.30 \input{papers/lambertw/teil0.tex} + +*** (cannot \read from terminal in nonstop modes) + + +Here is how much of TeX's memory you used: + 22 strings out of 478927 + 609 string characters out of 2852535 + 290175 words of memory out of 3000000 + 17980 multiletter control sequences out of 15000+600000 + 403430 words of font info for 27 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 16i,0n,26p,94b,28s stack positions out of 5000i,500n,10000p,200000b,80000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/buch/papers/lambertw/main.tex b/buch/papers/lambertw/main.tex index c125c33..9e6d04f 100644 --- a/buch/papers/lambertw/main.tex +++ b/buch/papers/lambertw/main.tex @@ -3,34 +3,35 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:lambertw}} -\lhead{Thema} +\chapter{Verfolgungskurven\label{chapter:lambertw}} +\lhead{Verfolgungskurven} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{David Hugentobler und Yanik Kuster} -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} +%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/lambertw/teil0.tex} +%\input{papers/lambertw/teil2.tex} +%\input{papers/lambertw/teil3.tex} +\input{papers/lambertw/teil4.tex} \input{papers/lambertw/teil1.tex} -\input{papers/lambertw/teil2.tex} -\input{papers/lambertw/teil3.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/lambertw/packages.tex b/buch/papers/lambertw/packages.tex index 6581a5a..366de78 100644 --- a/buch/papers/lambertw/packages.tex +++ b/buch/papers/lambertw/packages.tex @@ -8,3 +8,5 @@ % following example %\usepackage{packagename} +\usepackage{graphicx} +\usepackage{float}
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+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 a in font nullfont! +Missing character: There is no l 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 L in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no u 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 e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 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 G 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 t 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 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 t 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 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 f 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 g 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 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 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 g 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 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 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 f 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 g 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 . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 10--11 +[] + [] + + +Overfull \hbox (10.00003pt too wide) in paragraph at lines 10--11 +\/cmr/m/n/10 a + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 10--11 +\/cmr/m/n/10 a + [] + +! Undefined control sequence. +l.13 \subsection + {Verfolger und Verfolgungsstrategie +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 Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.13 \subsection{V + erfolger und Verfolgungsstrategie +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + +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 f 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 g 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 d 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 f 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 g 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 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 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 g 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 W 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 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 e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no h 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 w 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 d 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 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 f 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 g 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 d 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 c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no 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 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 o in font nullfont! +Missing character: There is no l 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 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 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 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 g 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 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 e in font nullfont! +Missing character: There is no r 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 W 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 n 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 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 a 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 d 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 s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no 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 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 f 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 g 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 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 n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 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 f 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 g 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 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 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 g 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 h in font nullfont! +Missing character: There is no l 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 D in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i 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 b 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 v 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 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 g 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 h 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 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 e 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 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 d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 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 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 f 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 g 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 w in font nullfont! +Missing character: There is no h 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 k 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 t 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 D in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no g 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 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 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 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 g 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 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 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 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 u 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 F 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 e 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 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 i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e 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 a 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 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 e 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 d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 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 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 f 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 g 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 k 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 t 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 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 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 k 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 n in font nullfont! +Missing character: There is no . 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 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 f 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 g 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 h 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 n 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 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 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 r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no 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 i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no f 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 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 u in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n 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 G 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 h in font nullfont! +Missing character: There is no w 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 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 k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o 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 M in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s 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 k 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 n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no n 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 e 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 i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no 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 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 B 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 r 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 a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no 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 A in font nullfont! +Missing character: There is no b 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 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 z in font nullfont! +Missing character: There is no w 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 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 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 f 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 g 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 d in font nullfont! +Missing character: There is no Z 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 l in font nullfont! +Missing character: There is no k 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 t 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 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 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 . in font nullfont! +Missing character: There is no W in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 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 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 d 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 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 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 e in font nullfont! +Missing character: There is no r 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 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 d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 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 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 g 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 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 e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no , in font nullfont! +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 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 d 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 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 c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no m 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 r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e 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 D 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 a 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 f 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 g 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 d 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 e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 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 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 g 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 z in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 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 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 d 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 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 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 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 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 e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no 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 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 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 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 r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l 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 r in font nullfont! +Missing character: There is no k 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 p 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 t in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h 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 b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 13--16 +[] + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 13--16 +\/cmr/m/n/10 a + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 13--16 +\/cmr/m/n/10 a + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 13--16 +\/cmr/m/n/10 o + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 13--16 +\/cmr/m/n/10 a + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 13--16 +\/cmr/m/n/10 o + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 13--16 +\/cmr/m/n/10 o + [] + + +! LaTeX Error: Environment table undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.17 \begin{table} + +Your command was ignored. +Type I <command> <return> to replace it with another command, +or <return> to continue without it. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 \begin{tabular} + {|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <7> on input line 18. +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <5> on input line 18. + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +! LaTeX Error: Illegal character in array arg. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.18 ...|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + +! Undefined control sequence. +<recently read> \text + +l.20 \text + {}&\text{Geschwindigkeit}&\text{Abstand}&\text{Richtung}\\ +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. + +! Undefined control sequence. +l.20 \text{}&\text + {Geschwindigkeit}&\text{Abstand}&\text{Richtung}\\ +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 G 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 h in font nullfont! +Missing character: There is no w 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 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 k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +! Undefined control sequence. +l.20 \text{}&\text{Geschwindigkeit}&\text + {Abstand}&\text{Richtung}\\ +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 A in font nullfont! +Missing character: There is no b 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 a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +! Undefined control sequence. +l.20 ...text{Geschwindigkeit}&\text{Abstand}&\text + {Richtung}\\ +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 R in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +! Undefined control sequence. +<recently read> \text + +l.22 \text + {Strategie 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 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 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 g 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 1 in font nullfont! +! Undefined control sequence. +l.23 & \text + {konstant} & \text{-} & \text{direkt auf Ziel hinzu}\\ +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 k 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 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 t in font nullfont! +! Undefined control sequence. +l.23 & \text{konstant} & \text + {-} & \text{direkt auf Ziel hinzu}\\ +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 - in font nullfont! +! Undefined control sequence. +l.23 & \text{konstant} & \text{-} & \text + {direkt auf Ziel hinzu}\\ +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 i 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 k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no Z 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 l 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 n in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no u in font nullfont! +! Undefined control sequence. +l.25 \text + {Strategie 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 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 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 g 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 2 in font nullfont! +! Undefined control sequence. +l.26 & \text + {-} & \text{konstant} & \text{direkt auf Ziel hinzu}\\ +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 - in font nullfont! +! Undefined control sequence. +l.26 & \text{-} & \text + {konstant} & \text{direkt auf Ziel hinzu}\\ +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 k 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 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 t in font nullfont! +! Undefined control sequence. +l.26 & \text{-} & \text{konstant} & \text + {direkt auf Ziel hinzu}\\ +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 i 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 k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no Z 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 l 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 n in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no u in font nullfont! +! Undefined control sequence. +l.28 \text + {Strategie 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 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 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 g 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 3 in font nullfont! +! Undefined control sequence. +l.29 & \text + {konstant} & \text{-} & \text{etwas voraus Zielen}\\ +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 k 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 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 t in font nullfont! +! Undefined control sequence. +l.29 & \text{konstant} & \text + {-} & \text{etwas voraus Zielen}\\ +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 - in font nullfont! +! Undefined control sequence. +l.29 & \text{konstant} & \text{-} & \text + {etwas voraus Zielen}\\ +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 e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no a 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 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 s in font nullfont! +Missing character: There is no Z 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 l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! + +! LaTeX Error: \begin{document} ended by \end{table}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.33 \end{table} + +Your command was ignored. +Type I <command> <return> to replace it with another command, +or <return> to continue without it. + + +Overfull \hbox (20.00006pt too wide) in paragraph at lines 18--34 +[][] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.42 I + n der Tabelle \eqref{lambertw:Strategien} sind drei mögliche Strategi... + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + +Missing character: There is no I 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 T in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +! Undefined control sequence. +l.42 In der Tabelle \eqref + {lambertw:Strategien} sind drei mögliche Strategi... +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 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 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 t in font nullfont! +Missing character: There is no w 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 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 e 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 e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n 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 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 m in font nullfont! +Missing character: There is no g 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 h 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 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 e 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 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 u in font nullfont! +Missing character: There is no f 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 z in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no l 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 F 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 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 w 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 d 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 r 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 c 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 u in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 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 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 g 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 1 in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no a 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 n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no B in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 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 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 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 g 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 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 d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no G 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 h in font nullfont! +Missing character: There is no w 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 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 k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no k 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 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 t 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 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 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 f 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 g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no i 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 e in font nullfont! +Missing character: There is no r 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 r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no Z 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 l 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 n in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no . 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 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 G 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 f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no k in font nullfont! +! Undefined control sequence. +l.45 In der Grafik \eqref + {lambertw:pursuerDGL2} ist das Problem dargestellt. +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 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 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 t in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no : 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 r 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 e in font nullfont! +Missing character: There is no r 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 L in font nullfont! +Missing character: There is no 2 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 d 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 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 d 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 g 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 l 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 W 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 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 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 t 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 e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r 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 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 f 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 g 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 , 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 O in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r 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 Z 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 l 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 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 G 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 h in font nullfont! +Missing character: There is no w 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 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 k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r 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 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 f 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 g in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no 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 D in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k 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 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 t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s 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 w 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 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 k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no k 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 n 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 n 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 t 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 G in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 42--48 +[] + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 42--48 +\/cmr/m/n/10 o + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 42--48 +\/cmr/m/n/10 a + [] + + +Overfull \hbox (10.00023pt too wide) in paragraph at lines 42--48 +[]$ + [] + + +Overfull \hbox (10.00023pt too wide) in paragraph at lines 42--48 +[]$ + [] + + +Overfull \hbox (10.00023pt too wide) in paragraph at lines 42--48 +[]$ + [] + +! Undefined control sequence. +l.51 \quad|A\in\mathbb + {R}>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. + + +Overfull \hbox (122.89459pt too wide) detected at line 52 +\OMS/cmsy/m/n/10 j[]j \/cmr/m/n/10 = \OML/cmm/m/it/10 konst \/cmr/m/n/10 = \OML +/cmm/m/it/10 A \OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 A \OMS/cmsy/m/n/10 2 \OML/cmm +/m/it/10 R > \/cmr/m/n/10 0 + [] + +Missing character: There is no d 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 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 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 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 G 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 h in font nullfont! +Missing character: There is no w 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 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 k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no w 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 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 u in font nullfont! +Missing character: There is no m 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 n in font nullfont! +Missing character: There is no n 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 t 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 G in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g in font nullfont! + +Overfull \hbox (81.8452pt too wide) detected at line 58 +[] \OMS/cmsy/m/n/10 j[]j \/cmr/m/n/10 = [] + [] + +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h 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 e 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 n in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no D 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 c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no S 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 t 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 k 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 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 O in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no 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 n 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 n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t 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 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 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 k 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 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 v 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 P 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 k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g 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 D 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 e in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no g 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 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 s 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 k 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 s 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 l 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 b 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 s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n 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 n 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 w 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 d 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 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 D 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 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 o in font nullfont! +Missing character: There is no n 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 t 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 B 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 r 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 , in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L 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 a in font nullfont! +Missing character: There is no u 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 i 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 f 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 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 e in font nullfont! +Missing character: There is no g 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 A 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 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 r in font nullfont! +Missing character: There is no f in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l 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 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 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 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 a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no 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 h in font nullfont! +Missing character: There is no , 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 s 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 i 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 n in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e 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 n in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no 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 r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t 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 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 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 s 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 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 D 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 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 o 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 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 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 u in font nullfont! +Missing character: There is no s 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 s 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 l 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 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 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 W in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no 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 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 n in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e 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 t 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 z 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 a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no 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 r in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t 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 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 , 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 d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no L 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 g 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 i 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 a in font nullfont! +Missing character: There is no l 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 u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no w 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 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 e in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no 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 m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no 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 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 h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 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 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 k 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 a 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 u in font nullfont! +Missing character: There is no l 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 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 z 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 r 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 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 a in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no 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 s in font nullfont! +Missing character: There is no y 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 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 n in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i 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 f in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no G in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no 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 z 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 d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no z 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 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! + +! LaTeX Error: Environment align undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.65 \begin{align} + +Your command was ignored. +Type I <command> <return> to replace it with another command, +or <return> to continue without it. + +Missing character: There is no - in font nullfont! +! Missing $ inserted. +<inserted text> + $ +l.67 ...}{|\overrightarrow{Z}-\overrightarrow{V}|} + \cdot +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +! Extra }, or forgotten $. +\frac #1#2->{\begingroup #1\endgroup \over #2} + +l.67 ...}{|\overrightarrow{Z}-\overrightarrow{V}|} + \cdot +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Misplaced alignment tab character &. +l.69 & + = +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + +! Misplaced alignment tab character &. +l.73 & + = +I can't figure out why you would want to use a tab mark +here. If you just want an ampersand, the remedy is +simple: Just type `I\&' now. But if some right brace +up above has ended a previous alignment prematurely, +you're probably due for more error messages, and you +might try typing `S' now just to see what is salvageable. + + +! LaTeX Error: \begin{document} ended by \end{align}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.75 \end{align} + +Your command was ignored. +Type I <command> <return> to replace it with another command, +or <return> to continue without it. + +! Missing $ inserted. +<inserted text> + $ +l.75 \end{align} + +I've inserted something that you may have forgotten. +(See the <inserted text> above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +! Missing } inserted. +<inserted text> + } +l.75 \end{align} + +I've inserted something that you may have forgotten. +(See the <inserted text> above.) +With luck, this will get me unwedged. But if you +really didn't forget anything, try typing `2' now; then +my insertion and my current dilemma will both disappear. + +Missing character: There is no D in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 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 G 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 t 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 K 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 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 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 f 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 g 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 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 , 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 s 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 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 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 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 f 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 g 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 d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 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 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 g 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 1 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 w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no . in font nullfont! + +Overfull \hbox (10.00023pt too wide) in paragraph at lines 58--77 + $[]$ + [] + + +Overfull \hbox (10.00023pt too wide) in paragraph at lines 58--77 +[]$ + [] + + +Overfull \hbox (8.05556pt too wide) in paragraph at lines 58--77 +\OML/cmm/m/it/10 V$ + [] + + +Overfull \hbox (7.54167pt too wide) in paragraph at lines 58--77 +\OML/cmm/m/it/10 Z$ + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 58--77 +\/cmr/m/n/10 a + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 58--77 +\/cmr/m/n/10 a + [] + + +Overfull \hbox (8.05556pt too wide) in paragraph at lines 58--77 +\OML/cmm/m/it/10 V$ + [] + + +Overfull \hbox (7.54167pt too wide) in paragraph at lines 58--77 +\OML/cmm/m/it/10 Z$ + [] + + +Overfull \hbox (8.05556pt too wide) in paragraph at lines 58--77 +\OML/cmm/m/it/10 V$ + [] + + +Overfull \hbox (7.54167pt too wide) in paragraph at lines 58--77 +\OML/cmm/m/it/10 Z$ + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 58--77 +\/cmr/m/n/10 o + [] + + +Overfull \hbox (152.45233pt too wide) in paragraph at lines 58--77 +[][]$[]$ + [] + +! Undefined control sequence. +l.79 \subsection + {Ziel +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 Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.79 \subsection{Z + iel +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + +Missing character: There is no Z 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 l 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 s 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 h 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 s 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 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 w 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 u in font nullfont! +Missing character: There is no f 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 s in font nullfont! +Missing character: There is no Z 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 l in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no . in font nullfont! +Missing character: There is no W 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 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 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 f 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 g 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 w 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 d in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no c in font nullfont! 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+Missing character: There is no a 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 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 F 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 c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no 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 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 g 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 h 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 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 , in font nullfont! +Missing character: There is no w in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l 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 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 n 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 f 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 g 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 b in font nullfont! +Missing character: There is no e 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 n 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 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 D in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no 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 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 e 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 e 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 n 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 l 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 a in font nullfont! +Missing character: There is no r in font nullfont! 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+Missing character: There is no B in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no 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 e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no k 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 t 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 i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no Z in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! 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+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 g in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 79--85 +[] + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 79--85 +\/cmr/m/n/10 a + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 79--85 +\/cmr/m/n/10 o + [] + + +Overfull \hbox (5.55557pt too wide) in paragraph at lines 79--85 +\/cmr/m/n/10 u + [] + + +! LaTeX Error: Environment Bmatrix undefined. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.88 \begin{Bmatrix} + +Your command was ignored. +Type I <command> <return> to replace it with another command, +or <return> to continue without it. + + +! LaTeX Error: \begin{equation} on input line 85 ended by \end{Bmatrix}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.91 \end{Bmatrix} + +Your command was ignored. +Type I <command> <return> to replace it with another command, +or <return> to continue without it. + + +Overfull \hbox (38.12263pt too wide) detected at line 92 +[]\/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 ) = 0[] \OML/cmm/m/it/10 t + [] + +Missing character: There is no b in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no h 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 e 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 n in font nullfont! 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Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. +\section{Was sind Verfolgungskurven? +\label{lambertw:section:Was_sind_Verfolgungskurven}} +\rhead{Was sind Verfolgungskurven?} + +Verfolgungskurven tauchen oft auf bei Fragen wie welchen Pfad begeht ein Hund während er einer Katze nachrennt. +Ein solches Problem hat im Kern immer ein Verfolger und sein Ziel. +Der Verfolger verfolgt sein Ziel, das versucht zu entkommen. +Der Pfad, der der Verfolger während der Verfolgung begeht, wird Verfolgungskurve genannt. +Um diese Kurve zu bestimmen, kann das Verfolgungsproblem als Differentialgleichung formuliert werden. +Diese Differentialgleichung entspringt der Verfolgungsstrategie des Verfolgers. + + +\subsection{Verfolger und Verfolgungsstrategie +\label{lambertw:subsection:Verfolger}} +Wie bereits erwähnt, wird der Verfolger durch seine Verfolgungsstrategie definiert. +Wir nehmen an, dass sich der Verfolger stur an eine Verfolgungsstrategie hält. +Dabei gibt es viele mögliche Strategien, die der Verfolger wählen könnte. +Die möglichen Strategien entstehen durch Festlegung einzelner Parameter, die der Verfolger kontrollieren kann. +Der Verfolger hat nur einen direkten Einfluss auf seinen Geschwindigkeitsvektor. +Mit diesem kann er neben Richtung und Betrag auch den Abstand zwischen Verfolger und Ziel kontrollieren. +Wenn zwei dieser drei Parameter durch die Strategie definiert werden, ist der dritte nicht mehr frei. +Daraus folgt, dass eine Strategie zwei dieser drei Parameter festlegen muss, um den Verfolger komplett zu beschreiben. + +\begin{table} + \centering + \begin{tabular}{|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|} + \hline + \text{}&\text{Geschwindigkeit}&\text{Abstand}&\text{Richtung}\\ + \hline + \text{Strategie 1} + & \text{konstant} & \text{-} & \text{direkt auf Ziel hinzu}\\ + + \text{Strategie 2} + & \text{-} & \text{konstant} & \text{direkt auf Ziel hinzu}\\ + + \text{Strategie 3} + & \text{konstant} & \text{-} & \text{etwas voraus Zielen}\\ + \hline + \end{tabular} + \caption{mögliche Verfolgungsstrategien} + \label{lambertw:table:Strategien} +\end{table} + +\begin{figure} + \centering + \includegraphics[scale=0.1]{./papers/lambertw/Bilder/pursuerDGL2.pdf} + \caption{Vektordarstellung Strategie 1} + \label{lambertw:grafic:pursuerDGL2} +\end{figure} + +In der Tabelle \eqref{lambertw:table:Strategien} sind drei mögliche Strategien aufgezählt. +Im Folgenden wird nur noch auf die Strategie 1 eingegangen. +Bei dieser Strategie ist die Geschwindigkeit konstant und der Verfolger bewegt sich immer direkt auf sein Ziel zu. +In der Abbildung \eqref{lambertw:grafic:pursuerDGL2} ist das Problem dargestellt, +wobei $\vec{V}$ der Ortsvektor des Verfolgers, $\vec{Z}$ der Ortsvektor des Ziels und $\dot{\vec{V}}$ der Geschwindigkeitsvektor des Verfolgers ist. +Die konstante Geschwindigkeit kann man mit der Gleichung +\begin{equation} + |\dot{\vec{V}}| + = \operatorname{const} = A + \quad A\in\mathbb{R}>0 +\end{equation} +darstellen. Der Geschwindigkeitsvektor wiederum kann mit der Gleichung +\begin{equation} + \frac{\vec{Z}-\vec{V}}{|\vec{Z}-\vec{V}|}\cdot|\dot{\vec{V}}| + = + \dot{\vec{V}} +\end{equation} +beschrieben werden. +Die Differenz der Ortsvektoren $\vec{V}$ und $\vec{Z}$ ist ein Vektor der vom Punkt $V$ auf $Z$ zeigt. +Da die Länge dieses Vektors beliebig sein kann, wird durch Division durch den Betrag, die Länge auf eins festgelegt. +Aus dem Verfolgungsproblem ist auch ersichtlich, dass die Punkte $V$ und $Z$ nicht am gleichen Ort starten und so eine Division durch Null ausgeschlossen ist. +Wenn die Punkte $V$ und $Z$ trotzdem am gleichen Ort starten, ist die Lösung trivial. +Nun wird die Gleichung mit $\dot{\vec{V}}$ skalar multipliziert, um das Gleichungssystem von zwei auf eine Gleichung zu reduzieren. Somit ergeben sich +\begin{align} + \frac{\vec{Z}-\vec{V}}{|\vec{Z}-\vec{V}|}\cdot|\dot{\vec{V}}|\cdot\dot{\vec{V}} + &= + |\dot{\vec{V}}|^2 + \\ + \label{lambertw:pursuerDGL} + \frac{\vec{Z}-\vec{V}}{|\vec{Z}-\vec{V}|}\cdot \frac{\dot{\vec{V}}}{|\dot{\vec{V}}|} + &= + 1 \text{.} +\end{align} +Die Lösungen dieser Differentialgleichung sind die gesuchten Verfolgungskurven, insofern der Verfolger die Strategie 1 verwendet. + +\subsection{Ziel +\label{lambertw:subsection:Ziel}} +Als nächstes gehen wir auf das Ziel ein. +Wie der Verfolger wird auch unser Ziel sich strikt an eine Fluchtstrategie halten, welche von Anfang an bekannt ist. +Diese Strategie kann als Parameterdarstellung der Position nach der Zeit beschrieben werden. +Zum Beispiel könnte ein Ziel auf einer Geraden flüchten, welches auf einer Ebene mit der Parametrisierung + +\begin{equation} + \vec{Z}(t) + = + \left( \begin{array}{c} 0 \\ t \end{array} \right) +\end{equation} + +beschrieben werden könnte. +Mit dieser Gleichung ist das Ziel auch schon vollumfänglich definiert. +Die Fluchtkurve kann eine beliebige Form haben, jedoch wird die zu lösende Differentialgleichung für die Verfolgungskurve immer komplexer. + + diff --git a/buch/papers/lambertw/teil1.log b/buch/papers/lambertw/teil1.log new file mode 100644 index 0000000..d2eb5c6 --- /dev/null +++ b/buch/papers/lambertw/teil1.log @@ -0,0 +1,3259 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.23 (MiKTeX 21.8) (preloaded format=pdflatex 2021.9.21) 5 APR 2022 23:32 +entering extended mode +**./teil1.tex +(teil1.tex +LaTeX2e <2021-06-01> patch level 1 +L3 programming layer <2021-08-27> +! 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+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 h in font nullfont! +Missing character: There is no w 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 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 k in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no o in font nullfont! 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+Missing character: There is no r in font nullfont! +Missing character: There is no Z in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no k in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no f 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 s in font nullfont! +Missing character: There is no Z 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 l in font nullfont! +Missing character: There is no z in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no g 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 20--24 +[] + [] + + +Overfull \hbox (10.55559pt too wide) in paragraph at lines 20--24 +\/cmr/m/n/10 u + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 20--24 +\/cmr/m/n/10 a + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.25 U + m die DGL dieses Problems herzuleiten wird der Sachverhalt in der Graf... + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + +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 i 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 G in font nullfont! +Missing character: There is no L in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i 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 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 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 h 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 z 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 e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no w 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 d 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 S 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 h 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 h 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 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 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 G 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 f in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no k in font nullfont! +! Undefined control sequence. +l.25 ... wird der Sachverhalt in der Grafik \eqref + {pursuer_grafik1} aufgezeigt. +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 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 u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +! Missing $ inserted. +<inserted text> + $ +l.25 ... Sachverhalt in der Grafik \eqref{pursuer_ + grafik1} aufgezeigt. +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <7> on input line 25. +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <5> on input line 25. +! Extra }, or forgotten $. +l.25 ...halt in der Grafik \eqref{pursuer_grafik1} + aufgezeigt. +I've deleted a group-closing symbol because it seems to be +spurious, as in `$x}$'. But perhaps the } is legitimate and +you forgot something else, as in `\hbox{$x}'. In such cases +the way to recover is to insert both the forgotten and the +deleted material, e.g., by typing `I$}'. + +! Missing $ inserted. +<inserted text> + $ +l.27 + +I've inserted a begin-math/end-math symbol since I think +you left one out. Proceed, with fingers crossed. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 25--27 +[] + [] + + +Overfull \hbox (332.97824pt too wide) in paragraph at lines 25--27 +[]\OML/cmm/m/it/10 rafik\/cmr/m/n/10 1\OML/cmm/m/it/10 aufgezeigt:DerPunktPistd +erVerfolgerundderPunktAistseinZiel:$ + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.28 U + m dies mathematisch beschreiben zu können, wird der Richtungsvektor +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + +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 i 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 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 h 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 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 c in font nullfont! +Missing character: There is no h in font nullfont! 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+Missing character: There is no r in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 28--29 +[] + [] + + +Overfull \hbox (5.00002pt too wide) in paragraph at lines 28--29 +\/cmr/m/n/10 o + [] + + +Overfull \hbox (64.58458pt too wide) detected at line 33 +[] \/cmr/m/n/10 = [] + [] + +Missing character: There is no b 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 g 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 D 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 c in font nullfont! +Missing character: There is no h in font nullfont! +Missing character: There is no d in font nullfont! 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Undefined control sequence. +l.49 \subsection + {Beispiel} +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 Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.49 \subsection{B + eispiel} +You're in trouble here. 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Undefined control sequence. +l.50 ...ieses Problem wurde bereits die DGL \eqref + {eq:PursuerDGL} hergeleitet. +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 e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no : 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 r 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 e in font nullfont! +Missing character: There is no r 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 L 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 r 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 e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no 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 D in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no A in font nullfont! +Missing character: There is no n 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 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 b 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 n 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 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 n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no d in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 49--52 +[] + [] + + +Overfull \hbox (5.55557pt too wide) in paragraph at lines 49--52 +\/cmr/m/n/10 u + [] + + +Overfull \hbox (5.55557pt too wide) in paragraph at lines 49--52 +\/cmr/m/n/10 u + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.53 \begin{equation} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 53--53 +[] + [] + + +Overfull \hbox (135.67946pt too wide) detected at line 57 +[]\/cmr/m/n/10 (\OML/cmm/m/it/10 t\/cmr/m/n/10 )[][] = [][][] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.59 \begin{equation} + +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + + +Overfull \hbox (20.0pt too wide) in paragraph at lines 59--59 +[] + [] + + +Overfull \hbox (61.57726pt too wide) detected at line 65 +[] \OMS/cmsy/m/n/10 [] \/cmr/m/n/10 = 1 = \OML/cmm/m/it/10 v[] + [] + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.79 S + ed ut perspiciatis unde omnis iste natus error sit voluptatem +You're in trouble here. Try typing <return> to proceed. +If that doesn't work, type X <return> to quit. + +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! 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+Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! + +Overfull \hbox (20.0pt too wide) in paragraph at lines 79--86 +[] + [] + + +Overfull \hbox (134.08229pt too wide) detected at line 93 +[][] \OML/cmm/m/it/10 x[] dx \/cmr/m/n/10 = [][] = []\OML/cmm/m/it/10 :[] + [] + +Missing character: There is no N in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no , in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no s in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no c in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no b in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no d in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no g in font nullfont! +Missing character: There is no n in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no i in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no q in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no r in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no v in font nullfont! +Missing character: There is no o in font nullfont! +Missing character: There is no l in font nullfont! +Missing character: There is no u in font nullfont! +Missing character: There is no p in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no a in font nullfont! +Missing character: There is no t in font nullfont! +Missing character: There is no e in font nullfont! +Missing character: There is no m in font nullfont! +Missing character: There is no . in font nullfont! + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.98 U + t enim ad minima veniam, quis nostrum exercitationem ullam corporis +You're in trouble here. 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Undefined control sequence. +l.104 \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. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.104 \subsection{D + e finibus bonorum et malorum +You're in trouble here. 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Undefined control sequence. +l.110 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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LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.112 E + t harum quidem rerum facilis est et expedita distinctio +You're in trouble here. 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Undefined control sequence. +<write> ...ertw:section:loesung' on page \thepage + \space undefined\on@line . +l.113 \ref{lambertw: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'). 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Undefined control sequence. +<write> ...tw:section:folgerung' on page \thepage + \space undefined\on@line . +l.117 \ref{lambertw: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'). Otherwise just continue, +and I'll forget about whatever was undefined. + + +LaTeX Warning: Reference `lambertw:section:folgerung' on page undefined on inp +ut line 117. + +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! 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Emergency stop. +<*> ./teil1.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 35 strings out of 478927 + 720 string characters out of 2852535 + 294175 words of memory out of 3000000 + 17993 multiletter control sequences out of 15000+600000 + 403738 words of font info for 28 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 23i,12n,32p,307b,98s stack positions out of 5000i,500n,10000p,200000b,80000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/buch/papers/lambertw/teil1.tex b/buch/papers/lambertw/teil1.tex index 7b545c3..fa7deb1 100644 --- a/buch/papers/lambertw/teil1.tex +++ b/buch/papers/lambertw/teil1.tex @@ -3,53 +3,167 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 1 -\label{lambertw:section:teil1}} -\rhead{Problemstellung} -Sed ut perspiciatis unde omnis iste natus error sit voluptatem -accusantium doloremque laudantium, totam rem aperiam, eaque ipsa -quae ab illo inventore veritatis et quasi architecto beatae vitae -dicta sunt explicabo. -Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit -aut fugit, sed quia consequuntur magni dolores eos qui ratione -voluptatem sequi nesciunt -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{lambertw:equation1} -\end{equation} -Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, -consectetur, adipisci velit, sed quia non numquam eius modi tempora -incidunt ut labore et dolore magnam aliquam quaerat voluptatem. - -Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis -suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? -Quis autem vel eum iure reprehenderit qui in ea voluptate velit -esse quam nihil molestiae consequatur, vel illum qui dolorem eum -fugiat quo voluptas nulla pariatur? - -\subsection{De finibus bonorum et malorum -\label{lambertw:subsection:finibus}} -At vero eos et accusamus et iusto odio dignissimos ducimus qui -blanditiis praesentium voluptatum deleniti atque corrupti quos -dolores et quas molestias excepturi sint occaecati cupiditate non -provident, similique sunt in culpa qui officia deserunt mollitia -animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{lambertw: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{lambertw:section:folgerung}. -Temporibus autem quibusdam et aut officiis debitis aut rerum -necessitatibus saepe eveniet ut et voluptates repudiandae sint et -molestiae non recusandae. -Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis -voluptatibus maiores alias consequatur aut perferendis doloribus -asperiores repellat. +\section{Wird das Ziel erreicht? +\label{lambertw:section:Wird_das_Ziel_erreicht}} +\rhead{Wird das Ziel erreicht?} + +Sehr oft kommt es vor, dass bei Verfolgungsproblemen die Frage auftaucht, ob das Ziel überhaupt erreicht wird. +Wenn zum Beispiel die Geschwindigkeit des Verfolgers kleiner ist als diejenige des Ziels, gibt es Anfangsbedingungen bei denen das Ziel nie erreicht wird. +Im Anschluss dieser Frage stellt sich meist die nächste Frage, wie lange es dauert bis das Ziel erreicht wird. +Diese beiden Fragen werden in diesem Kapitel behandelt und an einem Beispiel betrachtet. +% +%\subsection{Ziel erreichen (überarbeiten) +%\label{lambertw:subsection:ZielErreichen}} +Für diese Betrachtung wird das Beispiel aus \eqref{lambertw:section:teil4} zur Hilfe genommen. +Wir verwenden die hergeleiteten Gleichungen \eqref{lambertw:eqFunkXNachT} für Startbedingung im ersten Quadranten +\begin{align*} + x\left(t\right) + &= + x_0\cdot\sqrt{\frac{W\left(\chi\cdot e^{\chi-\frac{4t}{r_0-y_0}}\right)}{\chi}} \\ + y(t) + &= + \frac{1}{4}\left(\left(y_0+r_0\right)\left(\frac{x(t)}{x_0}\right)^2+\left(r_0-y_0\right)\operatorname{ln}\left(\left(\frac{x(t)}{x_0}\right)^2\right)-r_0+3y_0\right)\\ + \chi + &= + \frac{r_0+y_0}{r_0-y_0}\\ + \eta + &= + \left(\frac{x}{x_0}\right)^2\\ + r_0 + &= + \sqrt{x_0^2+y_0^2} \text{.}\\ +\end{align*} +% +Das Ziel wird erreicht, wenn die Koordinaten des Verfolgers mit denen des Ziels bei einem diskreten Zeitpunkt $t_1$ übereinstimmen. +Somit gilt es + +\begin{equation*} + \vec{Z}(t_1)=\vec{V}(t_1) +\end{equation*} +% +zu lösen. +Aus dem vorangegangenem Beispiel, ist die Parametrisierung des Verfolgers und des Ziels bekannt. +Das Ziel wird parametrisiert durch + +\begin{equation} + \vec{Z}(t) + = + \left( \begin{array}{c} 0 \\ t \end{array} \right) +\end{equation} +% +und der Verfolger durch + +\begin{equation} + \vec{V}(t) + = + \left( \begin{array}{c} x(t) \\ y(t) \end{array} \right) + \text{.} +\end{equation} +% + Da $y(t)$ viel komplexer ist als $x(t)$ wird das Problem in zwei einzelne Teilprobleme zerlegt. Wobei die Bedingung der x- und y-Koordinaten einzeln überprüft werden. Es entstehen daher folgende Bedingungen + +\begin{align*} + 0 + &= + x(t) + = + x_0\sqrt{\frac{W\left(\chi\cdot e^{\chi-\frac{4t}{r_0-y_0}}\right)}{\chi}} + \\ + t + &= + y(t) + = + \frac{1}{4}\left(\left(y_0+r_0\right)\left(\frac{x(t)}{x_0}\right)^2+\left(r_0-y_0\right)\operatorname{ln}\left(\left(\frac{x(t)}{x_0}\right)^2\right)-r_0+3y_0\right) + \\ +\end{align*} +% +, welche Beide gleichzeitig erfüllt sein müssen, damit das Ziel erreicht wurde. +Zuerst wird die Bedingung der x-Koordinate betrachtet. +Diese kann durch dividieren durch $x_0$, anschliessendes quadrieren und multiplizieren von $\chi$ vereinfacht werden. Daraus folgt +\begin{equation} + 0 + = + W\left(\chi\cdot e^{\chi-\frac{4t}{r_0-y_0}}\right) + \text{.} +\end{equation} +% +Es ist zu beachten, dass $W(x)$ die Lambert W-Funktion ist, welche im Kapitel \eqref{buch:section:lambertw} behandelt wurde. +Diese Gleichung entspricht genau den Nullstellen der Lambert W-Funktion. Da die Lambert W-Funktion genau eine Nullstelle bei + +\begin{equation*} + W(0)=0 +\end{equation*} +% +besitzt, kann die Bedingung weiter vereinfacht werden zu + +\begin{equation} + 0 + = + \chi\cdot e^{\chi-\frac{4t}{r_0-y_0}} + \text{.} +\end{equation} +% +Da $\chi\neq0$ und die Exponentialfunktion nie null sein kann, ist diese Bedingung unmöglich zu erfüllen. +Beim Grenzwert für $t\rightarrow\infty$ geht die Exponentialfunktion gegen null. +Dies nützt nicht viel, da unendlich viel Zeit vergehen müsste damit ein Einholen möglich wäre. +Somit kann nach den Gestellten Bedingungen das Ziel nie erreicht werden. +Aus der Symmetrie des Problems an der y-Achse können auch alle Anfangspunkte im zweiten Quadranten die Bedingungen nicht erfüllen. +Bei allen Anfangspunkten mit $y_0<0$ ist ein Einholen unmöglich, da die Geschwindigkeit des Verfolgers und Ziels übereinstimmen und der Verfolger dem Ziel bereits am Anfang nachgeht. +Wenn die Wertemenge der Anfangsbedingung um die positive y-Achse erweitert wird, kann das Ziel wiederum erreicht werden. +Sobald der Verfolger auf der positiven y-Achse startet, bewegen sich Verfolger und Ziel aufeinander zu, da der Geschwindigkeitsvektor des Verfolgers auf das Ziel zeigt und der Verfolger sich auf der Fluchtgeraden befindet. +Dies führt zwingend dazu, dass der Verfolger das Ziel erreichen wird. +Die Verfolgungskurve kann in diesem Fall mit + +\begin{equation} + \vec{V}(t) + = + \left( \begin{array}{c} 0 \\ y_0-t \end{array} \right) +\end{equation} +% +parametrisiert werden. +Nun kann der Abstand zwischen Verfolger und Ziel leicht bestimmt und nach 0 aufgelöst werden. +Daraus folgt + +\begin{equation} + 0 + = + |\vec{V}(t_1)-\vec{Z}(t_1)| + = + y_0-2t_1 +\end{equation} +% +, was aufgelöst zu + +\begin{equation} + t_1 + = + \frac{y_0}{2} +\end{equation} +% +führt. +Nun ist klar, dass lediglich Anfangspunkte auf der positiven y-Achse oder direkt auf dem Ziel dazu führen, dass der Verfolger das Ziel bei $t_1$ einholt. +Bei allen anderen Anfangspunkten wird der Verfolger das Ziel nie erreichen. +Dieses Resultat ist aber eher akademischer Natur, weil der Verfolger und das Ziel als Punkt betrachtet wurden. +Wobei aber in Realität nicht von Punkten sondern von Objekten mit einer räumlichen Ausdehnung gesprochen werden kann. +Somit wird in einer nächsten Betrachtung untersucht, ob der Verfolger dem Ziel näher kommt als ein definierter Trefferradius. +Falls dies stattfinden sollte, wird dies als Treffer interpretiert. +Mathematisch kann dies mit + +\begin{equation} + |\vec{V}-\vec{Z}|<a_{min} \quad a_{min}\in\mathbb{R}>0 +\end{equation} +% +beschrieben werden, wobei $a_{min}$ dem Trefferradius entspricht. +Durch quadrieren verschwindet die Wurzel des Betrages, womit + +\begin{equation} + |\vec{V}-\vec{Z}|^2<a_{min}^2 \quad a_{min}\in \mathbb{R} > 0 +\end{equation} +% +die neue Bedingung ist. +Da sowohl der Betrag als auch $a_{min}$ grösser null sind, bleibt die Aussage unverändert. + + + diff --git a/buch/papers/lambertw/teil2.log b/buch/papers/lambertw/teil2.log new file mode 100644 index 0000000..00ab7e3 --- /dev/null +++ b/buch/papers/lambertw/teil2.log @@ -0,0 +1,1580 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.23 (MiKTeX 21.8) (preloaded format=pdflatex 2021.9.21) 18 JUL 2022 16:36 +entering extended mode +**./teil2.tex +(teil2.tex +LaTeX2e <2021-06-01> patch level 1 +L3 programming layer <2021-08-27> +! 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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. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.24 \subsection{D + e finibus bonorum et malorum +You're in trouble here. 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+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! 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Nemo enim ipsam voluptatem quia voluptas sit -aspernatur aut odit aut fugit, sed quia consequuntur magni dolores -eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam -est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci -velit, sed quia non numquam eius modi tempora incidunt ut labore -et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima -veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, -nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure -reprehenderit qui in ea voluptate velit esse quam nihil molestiae -consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla -pariatur? -\subsection{De finibus bonorum et malorum -\label{lambertw: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. +\subsection{Strategie 1 +\label{lambertw:subsection:Strategie1}} + + +\subsection{Strategie 2 +\label{lambertw:subsection:Strategie2}} diff --git a/buch/papers/lambertw/teil3.log b/buch/papers/lambertw/teil3.log new file mode 100644 index 0000000..018dcba --- /dev/null +++ b/buch/papers/lambertw/teil3.log @@ -0,0 +1,1580 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.23 (MiKTeX 21.8) (preloaded format=pdflatex 2021.9.21) 5 APR 2022 21:53 +entering extended mode +**./teil3.tex +(teil3.tex +LaTeX2e <2021-06-01> patch level 1 +L3 programming layer <2021-08-27> +! 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'). 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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. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.24 \subsection{D + e finibus bonorum et malorum +You're in trouble here. 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Undefined control sequence. +l.6 \section + {Beispiel Verfolgungskurve +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 Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.6 \section{B + eispiel Verfolgungskurve +You're in trouble here. 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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. + + +! LaTeX Error: Missing \begin{document}. + +See the LaTeX manual or LaTeX Companion for explanation. +Type H <return> for immediate help. + ... + +l.24 \subsection{D + e finibus bonorum et malorum +You're in trouble here. 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Emergency stop. +<*> teil4.tex + +*** (job aborted, no legal \end found) + + +Here is how much of TeX's memory you used: + 16 strings out of 478371 + 382 string characters out of 5852527 + 296836 words of memory out of 5000000 + 18226 multiletter control sequences out of 15000+600000 + 403430 words of font info for 27 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 13i,0n,12p,83b,18s stack positions out of 5000i,500n,10000p,200000b,80000s +! ==> Fatal error occurred, no output PDF file produced! diff --git a/buch/papers/lambertw/teil4.tex b/buch/papers/lambertw/teil4.tex new file mode 100644 index 0000000..84a0ec7 --- /dev/null +++ b/buch/papers/lambertw/teil4.tex @@ -0,0 +1,368 @@ +% +% teil3.tex -- Beispiel-File für Teil 3 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Beispiel einer Verfolgungskurve +\label{lambertw:section:teil4}} +\rhead{Beispiel einer Verfolgungskurve} +In diesem Abschnitt wird rechnerisch das Beispiel einer Verfolgungskurve mit der Verfolgungsstrategie 1 beschreiben. Dafür werden zuerst Bewegungsraum, Anfangspositionen und Bewegungsverhalten definiert, in einem nächsten Schritt soll eine Differentialgleichung dafür aufgestellt werden und anschliessend gelöst werden. + +\subsection{Anfangsbedingungen definieren und einsetzen + \label{lambertw:subsection:Anfangsbedingungen}} +Das zu verfolgende Ziel \(\vec{Z}\) bewegt sich entlang der \(y\)-Achse mit konstanter Geschwindigkeit \(v = 1\), beginnend beim Ursprung des Kartesischen Koordinatensystems. Der Verfolger \(\vec{V}\) startet auf einem beliebigen Punkt im ersten Quadranten und bewegt sich auch mit konstanter Geschwindigkeit \(|\dot{V}| = 1\) in Richtung Ziel. Diese Anfangspunkte oder Anfangsbedingungen können wie folgt formuliert werden: +\begin{equation} + \vec{Z} + = + \left( \begin{array}{c} 0 \\ v \cdot t \end{array} \right) + = + \left( \begin{array}{c} 0 \\ t \end{array} \right) + ,\: + \vec{V} + = + \left( \begin{array}{c} x \\ y \end{array} \right) + \:\text{und}\:\: + \bigl| \dot{V} \bigl| + = + 1. + \label{lambertw:Anfangsbed} +\end{equation} +Wir haben nun die Anfangsbedingungen definiert, jetzt fehlt nur noch eine DGL, welche die fortlaufende Änderung der Position und Bewegungsrichtung des Verfolgers beschreibt. +Diese DGL haben wir bereits in Kapitel \ref{lambertw:subsection:Verfolger} definiert, und zwar Gleichung \eqref{lambertw:pursuerDGL}. Wenn man die Startpunkte einfügt ergibt sich folgender Ausdruck: +\begin{equation} + \frac{\left( \begin{array}{c} 0-x \\ t-y \end{array} \right)}{\sqrt{x^2 + (t-y)^2}} + \cdot + \left(\begin{array}{c} \dot{x} \\ \dot{y} \end{array}\right) + = + 1. + \label{lambertw:eqMitAnfangsbed} +\end{equation} + +\subsection{DGL vereinfachen + \label{lambertw:subsection:DGLvereinfach}} +Nun haben wir eine Gleichung, es stellt sich aber die Frage ob es überhaupt eine geschlossene Lösung dafür gibt. Eine Funktion welche die Beziehung \(y(x)\) beschreibt oder sogar \(x(t)\) und \(y(t)\) liefert. Zum jetzigen Zeitpunkt mag es nicht trivial scheinen, aber mit den gewählten Anfangsbedingungen \eqref{lambertw:Anfangsbed} ist es möglich eine geschlossene Lösung für die Gleichung \eqref{lambertw:eqMitAnfangsbed} zu finden. +Auf dem Weg dahin muss die definierte DGL zuerst wesentlich vereinfacht werden, sei es mittels algebraische Umformungen oder mit den Tools aus der Analysis. Also legen wir los! + +Zuerst müssen wir den Bruch in \eqref{lambertw:eqMitAnfangsbed} los werden, der sieht so nicht handlich aus. Dafür multiplizieren wir beidseitig mit dem Nenner: +\begin{equation} + \left( \begin{array}{c} 0-x \\ t-y \end{array} \right) + \cdot + \left(\begin{array}{c} \dot{x} \\ \dot{y} \end{array}\right) + = \sqrt{x^2 + (t-y)^2}. + \label{lambertw:eqOhneBruch} +\end{equation} +In einem weiteren Schritt, lösen wir das Skalarprodukt auf und erhalten folgende Gleichung \eqref{lambertw:eqOhneSkalarprod} ohne vektorielle Grössen: +\begin{equation} + -x \cdot \dot{x} + (t-y) \cdot \dot{y} + = \sqrt{x^2 + (t-y)^2}. + \label{lambertw:eqOhneSkalarprod} +\end{equation} +Im letzten Schritt, fällt die Nützlichkeit des Skalarproduktes in der Verfolgungsgleichung \eqref{lambertw:pursuerDGL} markant auf. Meiner Meinung ziemlich elegant und nicht selbstverständlich in der Lage zu sein, das Problem auf eine einzige Gleichung reduzieren zu können. + +Die nächsten Schritte sind sehr algebralastig und würden das lesen dieses Papers einfach nur mühsam machen, also werde ich diese auslassen. Hingegen werden ich die algebraische Hauptschritte erwähnen, die notwendig wären falls man es trotzdem selber ausprobieren möchte: +\begin{itemize} + \item + Quadrieren und erweitern. + \item + Gruppieren. + \item + Substitution von einzelnen Thermen mittels der Beziehung \(\dot{x}^2 + \dot{y}^2 = 1\). + \item + Und das erkennen des Musters einer Binomischen Formel. +\end{itemize} +Das Resultat aller dieser Vereinfachungen führen zu folgender Gleichung \eqref{lambertw:eqAlgVerinfacht}, die viel handhabbarer ist als zuvor: +\begin{equation} + (x \dot{y} + (t-y) \dot{x})^2 + = 0. + \label{lambertw:eqAlgVerinfacht} +\end{equation} +Da der linke Term gleich Null ist, muss auch der Inhalt des Quadrates gleich Null sein, somit folgt eine weitere Vereinfachung, welche zu einer im Vergleich zu \eqref{lambertw:eqOhneSkalarprod} wesentlich einfachere DGL führt: +\begin{equation} + x \dot{y} + (t-y) \dot{x} + = 0. + \label{lambertw:eqGanzVerinfacht} +\end{equation} +Kompakt, ohne Wurzelterme und Quadrate, nur elementare Operationen und Ableitungen. Nun stellt sich die Frage wie es weiter gehen soll, bei der Gleichung \eqref{lambertw:eqGanzVerinfacht} scheinen keine weiteren Vereinfachungen möglich zu sein. Wir brauchen einen neuen Ansatz um unser Ziel einer möglichen Lösung zu verfolgen. + +\subsection{Zeitabhängigkeit loswerden + \label{lambertw:subsection:ZeitabhLoswerden}} +Der nächste logischer Schritt schient irgendwie die Zeitabhängigkeit in der Gleichung \eqref{lambertw:eqGanzVerinfacht} loszuwerden, aber wieso? Nun, wie am Anfang von Abschnitt \ref{lambertw:subsection:DGLvereinfach} beschrieben, suchen wir eine Lösung der Art \(y(x)\), dies ist natürlich erst möglich wenn wir die Abhängigkeit nach \(t\) eliminieren können. + +Der erste Schritt auf dem Weg dahin, ist es die zeitlichen Ableitung los zu werden, dafür wird \eqref{lambertw:eqGanzVerinfacht} beidseitig mit \(\dot{x}\) dividiert, was erlaubt ist, weil diese Änderung ungleich Null ist: +\begin{equation} + x \frac{\dot{y}}{\dot{x}} + (t-y) \frac{\dot{x}}{\dot{x}} + = 0. + \label{lambertw:eqVorKeineZeitAbleit} +\end{equation} +Der Grund dafür ist, dass +\begin{equation} + \frac{\displaystyle\dot{y}}{\displaystyle\dot{x}} + = \frac{\displaystyle\frac{dy}{dt}}{\displaystyle\frac{dx}{dt}} + = \frac{dy}{dx} + = y^{\prime}, + \label{lambertw:eqQuotZeitAbleit} +\end{equation} +und somit kann der Quotient dieser zeitlichen Ableitungen in eine Ableitung nach \(x\) umgewandelt werden. +Nach dem diese Eigenschaft \eqref{lambertw:eqQuotZeitAbleit} in \eqref{lambertw:eqVorKeineZeitAbleit} eingesetzt wird und vereinfacht wurde, entsteht folgende neue Gleichung: +\begin{equation} + x y^{\prime} + t - y + = 0. + \label{lambertw:DGLmitT} +\end{equation} +Hier wäre es natürlich passend wenn man die Abhängigkeit nach \(t\) komplett wegbringen könnte. Um dies zu erreichen muss man auf die Definition der Bogenlänge aus der Analysis zurückgreifen, wobei die Strecke \(s\) folgendem entspricht: +\begin{equation} + s + = + v \cdot t + = + 1 \cdot t + = + t + = + \int_{\displaystyle x_0}^{\displaystyle x_{\text{end}}}\sqrt{1+y^{\prime\, 2}} \: dx. + \label{lambertw:eqZuBogenlaenge} +\end{equation} +Nicht gerade auffällig ist die Richtung in welche hier integriert wird. Wenn der Verfolger sich wie vorgesehen am Anfang im ersten Quadranten befindet, dann muss sich dieser nach links bewegen, was nicht der üblichen Integrationsrichtung entspricht. Um eine Integration wie üblich von links nach rechts ausführen zu können, müssen die Integrationsgenerzen vertauscht werden, was in einem Vorzeichenwechsel resultiert. Wenn man nun \eqref{lambertw:eqZuBogenlaenge} in die DGL \eqref{lambertw:DGLmitT} einfügt, dann ergibt sich folgender Ausdruck: +\begin{equation} + x y^{\prime} - \int\sqrt{1+y^{\prime\, 2}} \: dx - y + = 0. + \label{lambertw:DGLohneT} +\end{equation} +Um das Integral los zu werden, leitet man den vorherigen Ausdruck \eqref{lambertw:DGLohneT} nach \(x\) ab und erhaltet folgende DGL \eqref{lambertw:DGLohneInt}: +\begin{align} + y^{\prime}+ xy^{\prime\prime} - \sqrt{1+y^{\prime\, 2}} - y^{\prime} + &= 0, \\ + xy^{\prime\prime} - \sqrt{1+y^{\prime\, 2}} + &= 0. + \label{lambertw:DGLohneInt} +\end{align} +Nun sind wir unserem Ziel einen weiteren Schritt näher. Die Gleichung \eqref{lambertw:DGLohneInt} mag auf den ersten Blick nicht gerade einfach sein, aber im Nächsten Abschnitt werden wir sehen, dass sie relativ einfach zu lösen ist. + +\subsection{DGL lösen + \label{lambertw:subsection:DGLloes}} +Die Gleichung \eqref{lambertw:DGLohneInt} ist eine DGL zweiter Ordnung und kann +mittels der Substitution \(y^{\prime} = u\) in eine DGL erster Ordnung umgewandelt werden: +\begin{equation} + xu^{\prime} - \sqrt{1+u^2} + = 0. + \label{lambertw:DGLmitU} +\end{equation} +Diese \eqref{lambertw:DGLmitU} zu lösen ist ziemlich einfach da sie separierbar ist, aus diesem Grund werde ich direkt zur Lösung \eqref{lambertw:loesDGLmitU} übergehen: +\begin{align} + \operatorname{arsinh}(u) + &= + \operatorname{ln}(x) + C, \\ + u + &= + \operatorname{sinh}(\operatorname{ln}(x) + C). + \label{lambertw:loesDGLmitU} +\end{align} +Indem man die Substitution rückgängig macht, erhält man eine weitere DGL erster Ordnung die bereits separiert ist und erhält folgende Gleichung: +\begin{equation} + y^{\prime} + = + \operatorname{sinh}(\operatorname{ln}(x) + C). + \label{lambertw:loesDGLmitY} +\end{equation} +Diese \eqref{lambertw:loesDGLmitY} kann mit den selben Methoden gelöst werden wie \eqref{lambertw:DGLmitU}, diesmal aber in Kombination mit der exponentiellen Definition der \(\operatorname{sinh}\)-Funktion: +\begin{equation} + y + = + C_1 + C_2 x^2 - \frac{\operatorname{ln}(x)}{8 \cdot C_2}. +\end{equation} +Nun haben wir eine Lösung, aber wie es immer mit Lösungen ist, stellt sich die Frage ob sie überhaupt plausibel ist. Dieser Frage werden wir in nächsten Abschnitt \ref{lambertw:subsection:LoesAnalys} nachgehen. + +\subsection{Lösung analysieren + \label{lambertw:subsection:LoesAnalys}} + +\begin{figure} + \centering + \includegraphics{papers/lambertw/Bilder/VerfolgungskurveBsp.png} + \caption[Graph der Verfolgungskurve]{Graph der Verfolgungskurve wobei, ({\color{red}rot}) die Funktion \ensuremath{y(x)} ist, ({\color{darkgreen}grün}) der quadratische Teil und ({\color{blue}blau}) dem \ensuremath{ln(x)}-Teil entspricht. + \label{lambertw:BildFunkLoes} + } +\end{figure} + +Das Resultat, wie ersichtlich, ist folgende Funktion \eqref{lambertw:funkLoes} welche mittels Anfangsbedingungen parametrisiert werden kann: +\begin{equation} + {\color{red}{y(x)}} + = + C_1 + C_2 {\color{darkgreen}{x^2}} {\color{blue}{-}} \frac{\color{blue}{\operatorname{ln}(x)}}{8 \cdot C_2}. + \label{lambertw:funkLoes} +\end{equation} +Für die Koeffizienten \(C_1\) und \(C_2\) ergibt sich ein Anfangswertproblem, welches für deren Bestimmung gelöst werden muss. Zuerst soll aber eine qualitative Intuition, oder Idee für das Aussehen der Funktion \(y(x)\) geschaffen werden: +\begin{itemize} + \item + Für grosse \(x\)-Werte, welche in der Regel in der Nähe von \(x_0\) sein sollten, ist der quadratisch Term in der Funktion \eqref{lambertw:funkLoes} dominant. + \item + Für immer kleiner werdende \(x\) geht der Verfolger in Richtung \(y\)-Achse, wobei seine Steigung stetig sinkt, was Sinn macht wenn der Verfolgte entlang der \(y\)-Achse steigt. Irgendwann werden Verfolger und Ziel auf gleicher Höhe sein. + \item + Für \(x\)-Werte in der Nähe von \(0\) ist das asymptotische Verhalten des Logarithmus dominant, dies macht auch Sinn da sich der Verfolgte auf der \(y\)-Achse bewegt und der Verfolger im nachgeht. + \item + Aufgrund des Monotoniewechsels in der Kurve \eqref{lambertw:funkLoes} muss diese auch ein Minimum aufweisen. Es stellt sich nun die Frage: Wo befindet sich dieser Punkt? + + Eine Abschätzung darüber kann getroffen werden und zwar, dass dieser dann entsteht, wenn \(A\) und \(P\) die gleiche \(y\)-Koordinaten besitzen. In diesem Moment ändert die Richtung der \(y\)-Komponente der Geschwindigkeit des Verfolgers, somit auch sein Vorzeichen und dadurch entsteht auch das Minimum. +\end{itemize} +Alle diese Eigenschafte stimmen mit dem überein, was man von einer Kurve dieser Art erwarten würde, welche durch die Grafik \ref{lambertw:BildFunkLoes} repräsentiert wurde. Nun stellt sich die Frage wie die Kurve wirklich aussieht. Dies wird im folgenden Abschnitt \ref{lambertw:subsection:AllgLoes} behandelt. + +\subsection{Anfangswertproblem + \label{lambertw:subsection:AllgLoes}} +Wie üblich bei der Suche nach einer exakten Lösung, kommt ein Anfangswertproblem vor. Um dieses zu lösen, müssen wir zuerst die Anfangswerte definieren. Da wir das Problem allgemein lösen wollen, ergeben sich folgende zwei Anfangswerte: +\begin{equation} + y(x)\big \vert_{t=0} + = + y(x_0) + = + y_0 + \label{lambertw:eq1Anfangswert} +\end{equation} +und +\begin{equation} + \frac{dy}{dx}\bigg \vert_{t=0} + = + y^{\prime}(x_0) + = + \frac{y_0}{x_0}. + \label{lambertw:eq2Anfangswert} +\end{equation} +Der zweite Anfangswert \eqref{lambertw:eq2Anfangswert} mag nicht grade offensichtlich sein. Die Erklärung dafür ist aber simpel: Der Verfolger wird sich zum Zeitpunkt \(t=0\) in Richtung Koordinatenursprung bewegen wollen, wo sich das Ziel befindet. Somit entsteht das Steigungsdreieck mit \(\Delta x = x_0\) und \(\Delta y = y_0\). + +Das Lösen des Anfangswertproblems ist ein Problem aus der Algebra, auf welches ich nicht unbedingt eingehen möchte. Zur Vollständigkeit und Nachvollziehbarkeit, werde ich aber das Gleichungssystem \eqref{lambertw:eqGleichungssystem} präsentieren, welches notwendig ist um das Anfangswertproblem zu lösen, sowie auch die allgemeine Lösung \eqref{lambertw:eqAllgLoes} die sich nach dem einsetzen der Koeffizienten \(C_1\) und \(C_2\) in die Funktion \eqref{lambertw:funkLoes} ergibt. + +\begin{itemize} + \item + Gleichungssystem: + \begin{subequations} + \begin{align} + y_0 + &= + C_1 + C_2 x^2_0 - \frac{\operatorname{ln}(x_0)}{8 \cdot C_2}, \\ + \frac{y_0}{x_0} + &= + 2 \cdot C_2 x_0 - \frac{1}{8 \cdot C_2 \cdot x_0}. + \end{align} + \label{lambertw:eqGleichungssystem} + \end{subequations} + \item + Die allgemeine Funktion: + \begin{equation} + y(x) + = + \frac{1}{4}\left(\left(y_0+r_0\right)\eta+\left(r_0-y_0\right)\operatorname{ln}\left(\eta\right)-r_0+3y_0\right) + \label{lambertw:eqAllgLoes} + \end{equation} + Damit die Funkion \eqref{lambertw:eqAllgLoes} trotzdem noch übersichtlich bleibt, wurden \(\eta\) und \(r_0\) wie folgt definiert: + \begin{equation} + \eta + = + \left(\frac{x}{x_0}\right)^2 + \:\:\text{und}\:\: + r_0 + = + \sqrt{x_0^2+y_0^2}. + \end{equation} +\end{itemize} +Diese neue allgemein Funktion \eqref{lambertw:eqAllgLoes} weist immer noch die selbe Struktur wie die vorherig hergeleitete Funktion \eqref{lambertw:funkLoes} auf, einerseits einen quadratischen Teil der in \(\eta\) enthalten ist, anderseits den \(\operatorname{ln}\)-Teil. Aus dieser Ähnlichkeit kann geschlossen werden, dass sich \eqref{lambertw:eqAllgLoes} auf eine ähnliche Art verhalten wird. + +Nun sind wir soweit, dass wir eine \(y(x)\)-Beziehung für beliebige Anfangswerte darstellen können, unser erstes Ziel wurde erreicht. Ist das alles? Nein, wir können einen Schritt weiter gehen und uns Fragen: Ist es analytisch möglich herauszufinden, wo sich Verfolger und Ziel zu jedem Zeitpunkt befinden? Dieser Frage werden wir im nächsten Abschnitt nachgehen. + +\subsection{Funktion nach der Zeit + \label{lambertw:subsection:FunkNachT}} +Lieber Leser sei mir nicht böse, aber in diesem Abschnitt werde ich ein wenig mehr bei den algebraischen Umformungen ins Detail gehen. Dies hat auch einen bestimmten Grund, ich möchte den Einsatz einer speziellen Funktion aufzeigen, sowie auch wann und wieso diese vorkommt. Welche spezielle Funktion? Fragst du dich wahrscheinlich in diesem Moment. Nun, um diese Frage zu kurz zu beantworten, es ist "YouTube's favorite special function" laut dem Mathematiker Michael Penn, die Lambert-W-Funktion \(W(x)\) welche übrigens im Kapitel \ref{buch:section:lambertw} bereits beschrieben wurde. + +Also fangen wir an. Der erste Schritt ist es herauszufinden, wie die Zeitabhängigkeit wieder hinein gebracht werden kann. Dafür greifen wir auf die letzte Gleichung zu, in welcher \(t\) noch enthalten war, und zwar DGL \eqref{lambertw:DGLmitT}, welche zur Übersichtlichkeit hier nochmals aufgeführt wird: +\begin{equation} + x y^{\prime} + t - y + = 0. + \label{lambertw:eqDGLmitTnochmals} +\end{equation} +Wie in \eqref{lambertw:eqDGLmitTnochmals} zu sehen ist, werden \(y\) und deren Ableitung \(y^{\prime}\) benötigt, diese sind: +\begin{subequations} + \begin{align} + y + &= + \frac{1}{4}\left(\left(y_0+r_0\right)\eta+\left(r_0-y_0\right)\operatorname{ln}\left(\eta\right)-r_0+3y_0\right), \\ + \label{lambertw:eqFunkUndAbleit1} + y^\prime + &= + \frac{1}{2}\left(\left(y_0+r_0\right)\frac{x}{x_0^2}+\left(r_0-y_0\right)\frac{1}{x}\right). + \end{align} + \label{lambertw:eqFunkUndAbleit} +\end{subequations} +Wenn man diese Gleichungen \ref{lambertw:eqFunkUndAbleit} in die DGL \label{lambertw:eqDGLmitTnochmals} einfügt, vereinfacht und nach \(t\) auflöst, dann ergibt sich folgenden Ausdruck: +\begin{equation} + -4t + = + \left(y_0+r_0\right)\left(\eta-1\right)+\left(r_0-y_0\right)\operatorname{ln}\left(\eta\right). + \label{lambertw:eqFunkUndAbleitEingefuegt} +\end{equation} +In einem nächsten Schritt wird alles mit \(x\) auf die eine Seite gebracht, der Rest auf die andere Seite und anschliessend beidseitig exponentiert, was wie folgt aussieht: +\begin{align} + -4t+\left(y_0+r_0\right) + &= + \left(y_0+r_0\right)\eta+\left(r_0-y_0\right)\operatorname{ln}\left(\eta\right), \\ + e^{\displaystyle -4t+\left(y_0+r_0\right)} + &= + e^{\displaystyle \left(y_0+r_0\right)\eta}\cdot\eta^{\displaystyle \left(r_0-y_0\right)}. + \label{lambertw:eqMitExp} +\end{align} +Auf dem rechten Term von \eqref{lambertw:eqMitExp} beginnen wir langsam eine ähnliche Struktur wie \(\eta e^\eta\) zu erkennen, dies schreit nach der Struktur die benötigt wird um \(\eta\) mittels der Lambert-W-Funktion \(W(x)\) zu erhalten. Dies macht durchaus Sinn, wenn wir die Funktion \(x(t)\) finden wollen und \(W(x)\) die Umkehrfunktion von \(x e^x\) ist. + +Die erste Sache die uns in \eqref{lambertw:eqMitExp} stört ist, dass \(\eta\) als Potenz da steht. Dieses Problem können wir loswerden, indem wir beidseitig mit \(\:\displaystyle \frac{1}{r_0-y_0}\:\) potenzieren: +\begin{equation} + e^{\displaystyle \frac{-4t}{r_0-y_0}+\frac{y_0+r_0}{r_0-y_0}} + = + \eta\cdot e^{\displaystyle \frac{y_0+r_0}{r_0-y_0}\eta} . + \label{lambertw:eqOhnePotenz} +\end{equation} +Das nächste Problem auf welches wir in \eqref{lambertw:eqOhnePotenz} treffen ist, dass \(\eta\) nicht alleine im Exponent steht. Dies kann elegant mit folgender Substitution gelöst werden: +\begin{equation} + \chi + = + \frac{y_0+r_0}{r_0-y_0}. + \label{lambertw:eqChiSubst} +\end{equation} +Es gäbe natürlich andere Substitutionen wie z.B. +\[\displaystyle \chi=\frac{y_0+r_0}{r_0-y_0}\cdot\eta,\] +die auf das selbe Ergebnis führen würden, aber \eqref{lambertw:eqChiSubst} liefert in einem Schritt die kompakteste Lösung. Also fahren wir mit der Substitution \eqref{lambertw:eqChiSubst} weiter, setzen diese in die Gleichung \eqref{lambertw:eqOhnePotenz} ein und multiplizieren beidseitig mit \(\chi\). Daraus erhalten wir folgende Gleichung: +\begin{equation} + \chi\cdot e^{\displaystyle \chi-\frac{4t}{r_0-y_0}} + = + \chi\eta\cdot e^{\displaystyle \chi\eta}. + \label{lambertw:eqNachSubst} +\end{equation} +Schön oder? Nun sind wir endlich soweit, dass wir die angedeutete Lambert-W-Funktion \(W(x)\)einsetzen können. Wenn wir beidseitig \(W(x)\) anwenden, dann erhalten wir folgenden Ausdruck: +\begin{equation} + W\left(\chi\cdot e^{\displaystyle \chi-\frac{4t}{r_0-y_0}}\right) + = + \chi\eta. +\end{equation} +Nach dem Auflösen nach \(x\) welches in \(\eta\) enthalten ist, erhalten wir die gesuchte \(x(t)\)-Funktion \eqref{lambertw:eqFunkXNachT}. Dieses \(x(t)\) in Kombination mit \eqref{lambertw:eqFunkUndAbleit1} liefert die Position des Verfolgers zu jedem Zeitpunkt. Das Gleichungspaar \eqref{lambertw:eqFunktionenNachT}, besteht aus folgenden Gleichungen: +\begin{subequations} + \begin{align} + \label{lambertw:eqFunkXNachT} + x(t) + &= + x_0\cdot\sqrt{\frac{W\left(\chi\cdot e^{\displaystyle \chi-\frac{4t}{r_0-y_0}}\right)}{\chi}}, \\ + \label{lambertw:eqFunkYNachT} + y(x(t)) + = + y(t) + &= + \frac{1}{4}\left(\left(y_0+r_0\right)\left(\frac{x(t)}{x_0}\right)^2+\left(r_0-y_0\right)\operatorname{ln}\left(\left(\frac{x(t)}{x_0}\right)^2\right)-r_0+3y_0\right). + \end{align} + \label{lambertw:eqFunktionenNachT} +\end{subequations} +Nun haben wir unser letztes Ziel erreicht und sind in der Lage eine Verfolgung rechnerisch sowie graphisch zu repräsentieren. + +Wir sind aber noch nicht ganz fertig, ich muss gestehen, dass ich in diesem Abschnitt einen wichtigen Teil verschwiegen habe. Und zwar wieso, dass ich schon bei der Gleichung \eqref{lambertw:eqFunkUndAbleitEingefuegt} wusste, dass man nach einigen Umformungen die Lambert-W-Funktion eingesetzt werden kann. +Der Grund dafür ist die Struktur +\begin{equation} + y + = + p(x) +\operatorname{ln}(x), + \label{lambertw:eqEinsatzLambW} +\end{equation} +bei welcher \(p(x)\) eine beliebige Potenz von \(x\) darstellt. + +Jedes mal wenn \(x\) gesucht ist und in einer Struktur der Art \eqref{lambertw:eqEinsatzLambW} vorkommt, dann kann mit ein paar Umformungen die Struktur \(f(x)e^{f(x)}\) erzielt werden. Wie bereits in diesem Abschnitt \ref{lambertw:subsection:FunkNachT} gezeigt wurde, kann \(x\) nun mittels der \(W(x)\)-Funktion aufgelöst werden. Erstaunlicherweise ist \eqref{lambertw:eqEinsatzLambW} eine Struktur die oftmals vorkommt, was die Lambert-W-Funktion so wichtig macht.
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