adjusted chapter

2 files changed, 53 insertions, 654 deletions

diff --git a/buch/papers/lambertw/main.log b/buch/papers/lambertw/main.log
index 4b0af4d..754563d 100644
--- a/buch/papers/lambertw/main.log
+++ b/buch/papers/lambertw/main.log
@@ -1,14 +1,12 @@
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-            {Thema\label{chapter:lambertw}}
+            {Verfolgungskurven\label{chapter:lambertw}}
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@@ -22,16 +20,28 @@ See the LaTeX manual or LaTeX Companion for explanation.
 Type  H <return>  for immediate help.
  ...                                              
                                                   
-l.6 \chapter{T
-              hema\label{chapter:lambertw}}
+l.6 \chapter{V
+              erfolgungskurven\label{chapter:lambertw}}
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-                  {Hans Muster}
+                  {David Hugentobler und Yanik Kuster}
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+[][]
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@@ -734,16 +124,16 @@ Enter file name:
 <read *> 
          
 l.30 \input{papers/lambertw/teil0.tex}
-                                      ^^M
+                                      
 *** (cannot \read from terminal in nonstop modes)
 
  
 Here is how much of TeX's memory you used:
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- 593 string characters out of 5852527
- 296836 words of memory out of 5000000
- 18242 multiletter control sequences out of 15000+600000
- 403598 words of font info for 28 fonts, out of 8000000 for 9000
+ 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
- 23i,1n,32p,120b,183s stack positions out of 5000i,500n,10000p,200000b,80000s
+ 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/teil1.tex b/buch/papers/lambertw/teil1.tex
index 3415c45..2f71f43 100644
--- a/buch/papers/lambertw/teil1.tex
+++ b/buch/papers/lambertw/teil1.tex
@@ -25,7 +25,7 @@ Wir verwenden die Hergeleiteten Gleichungen
     \frac{1}{4}\left(\left(y_0+r_0\right)\eta+\left(r_0-y_0\right)ln\left(\eta\right)-r_0+3y_0\right) \\
     \chi
     &=
-    \frac{r_0+y_0}{r_0-y_0}; \cdot\chi \\
+    \frac{r_0+y_0}{r_0-y_0}\\
     \eta
     &=
     \left(\frac{x}{x_0}\right)^2 
@@ -37,13 +37,13 @@ Wir verwenden die Hergeleiteten Gleichungen
 Wir definieren einen Treffer wenn die Koordinaten des Verfolgers mit denen des Ziels übereinstimmen bei einem diskreten Zeitpunkt $t_1$. Aus dem vorangegangenem Beispiel, sind die Gleichungen zu den x- und y-Koordinaten des Verfolgers bekannt. Die Des Ziels sind
 
 \begin{equation}
-    \overrightarrow{Z}(t)
+    \vec{Z}(t)
     =
     \left( \begin{array}{c} 0 \\ v \cdot t \end{array} \right)
     =
     \left( \begin{array}{c} 0 \\ t \end{array} \right)
     ;\quad
-    \overrightarrow{V}(t)
+    \vec{V}(t)
     =
     \left( \begin{array}{c} x(t) \\ y(t) \end{array} \right)
     \label{lambertw:Anfangspunkte}
@@ -52,7 +52,7 @@ Wir definieren einen Treffer wenn die Koordinaten des Verfolgers mit denen des Z
 Somit gilt es
 
 \begin{equation*}
-    \overrightarrow{Z}(t_1)=\overrightarrow{V}(t_1)
+    \vec{Z}(t_1)=\vec{V}(t_1)
 \end{equation*}
 
 zu lösen. Da die $y(t)$ 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.
@@ -72,7 +72,10 @@ zu lösen. Da die $y(t)$ viel komplexer ist als $x(t)$ wird das Problem in zwei
     \\
 \end{align*}
 
-Zuerst wird die Bedingung der x-Koordinate betrachtet. Diese kann durch quadrieren und anschliessendes multiplizieren von $\chi$ vereinfacht werden.
+Zuerst wird die Bedingung der x-Koordinate betrachtet.
+Diese kann durch quadrieren und anschliessendes multiplizieren von $\chi$ vereinfacht werden.
+Es ist zu beachten, dass $W(x)$ die Lambert W-Funktion ist, welche im Kapitel \eqref{buch:section:lambertw} behandelt wurde.
+Die Gleichung
 
 \begin{equation}
     0
@@ -80,7 +83,8 @@ Zuerst wird die Bedingung der x-Koordinate betrachtet. Diese kann durch quadrier
     W\left(\chi\cdot e^{\chi-\frac{4t}{r_0-y_0}}\right)
 \end{equation}
 
-Dies entspricht genau den Nullstellen der Lambert W-Funktion. Da die Lambert W-Funktion genau eine Nullstelle bei
+
+entspricht genau den Nullstellen der Lambert W-Funktion. Da die Lambert W-Funktion genau eine Nullstelle bei
 
 \begin{equation*}
     W(0)=0
@@ -100,5 +104,10 @@ Dies nützt nicht viel, da unendlich viel Zeit vergehen müsste damit ein Treffe
 Somit kann nach den Gestellten Bedingungen das Ziel nie getroffen werden.
 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.
+Dies kann mathematisch mit 
+
+\begin{equation}
+    |\vec{V}-\vec{Z]|<a_min \quad a_min\in\mathbb{R}>0
+\end{equation}