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fix switching off and g in 5.5.3
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also fix switch of f and g in 5.5.1 (three cases -> choose f/g)
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abegehr committed Aug 21, 2018
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28 changes: 14 additions & 14 deletions thesis/.texpadtmp/thesis.blg
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[1] Config.pm:302> INFO - This is Biber 2.11
[1] Config.pm:305> INFO - Logfile is '/Users/anton/Documents/Studium/BA/frechet/thesis/.texpadtmp/thesis.blg'
[62] biber-darwin:313> INFO - === Tue Aug 21, 2018, 11:00:43
[95] Biber.pm:371> INFO - Reading '/Users/anton/Documents/Studium/BA/frechet/thesis/.texpadtmp/thesis.bcf'
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[0] Config.pm:305> INFO - Logfile is '/Users/anton/Documents/Studium/BA/frechet/thesis/.texpadtmp/thesis.blg'
[56] biber-darwin:313> INFO - === Tue Aug 21, 2018, 15:22:50
[86] Biber.pm:371> INFO - Reading '/Users/anton/Documents/Studium/BA/frechet/thesis/.texpadtmp/thesis.bcf'
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[235] bibtex.pm:1316> INFO - Found BibTeX data source '/Users/anton/Documents/Studium/BA/frechet/thesis/bibliography.bib'
[259] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'variable = shifted' with 'variable = non-ignorable'
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[265] bbl.pm:720> INFO - Output to /Users/anton/Documents/Studium/BA/frechet/thesis/.texpadtmp/thesis.bbl
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[236] bibtex.pm:1316> INFO - Found BibTeX data source '/Users/anton/Documents/Studium/BA/frechet/thesis/bibliography.bib'
[267] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'variable = shifted' with 'variable = non-ignorable'
[267] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'normalization = NFD' with 'normalization = prenormalized'
[267] Biber.pm:3804> INFO - Sorting list 'nty/global//global/global' of type 'entry' with template 'nty' and locale 'en-US'
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[271] bbl.pm:617> INFO - Writing '/Users/anton/Documents/Studium/BA/frechet/thesis/.texpadtmp/thesis.bbl' with encoding 'UTF-8'
[272] bbl.pm:720> INFO - Output to /Users/anton/Documents/Studium/BA/frechet/thesis/.texpadtmp/thesis.bbl
4 changes: 2 additions & 2 deletions thesis/.texpadtmp/thesis.log
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This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2018) (preloaded format=pdflatex 2018.5.8) 21 AUG 2018 11:14
This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2018) (preloaded format=pdflatex 2018.5.8) 21 AUG 2018 15:24
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Expand Down Expand Up @@ -1012,7 +1012,7 @@ pdfTeX warning (dest): name{Hfootnote.3} has been referenced but does not exist,
pdfTeX warning (dest): name{Hfootnote.2} has been referenced but does not exist, replaced by a fixed one

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Output written on /Users/anton/Documents/Studium/BA/frechet/thesis/.texpadtmp/thesis.pdf (42 pages, 2658682 bytes).
Output written on /Users/anton/Documents/Studium/BA/frechet/thesis/.texpadtmp/thesis.pdf (42 pages, 2658667 bytes).
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10 changes: 5 additions & 5 deletions thesis/3_main.tex
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Expand Up @@ -428,8 +428,8 @@ \subsubsection{Simplifications and First Derivative}\label{sec:first_derivative}
There are three cases to consider:

\begin{enumerate}
\item $\hat{f}'_{\epsilon_0}(\epsilon_0) > \hat{g}'_{\epsilon_0}(\epsilon_0)$: Choose traversal for $f$.
\item $\hat{f}'_{\epsilon_0}(\epsilon_0) < \hat{g}'_{\epsilon_0}(\epsilon_0)$: Choose traversal for $g$.
\item $\hat{f}'_{\epsilon_0}(\epsilon_0) < \hat{g}'_{\epsilon_0}(\epsilon_0)$: Choose traversal for $f$.
\item $\hat{f}'_{\epsilon_0}(\epsilon_0) > \hat{g}'_{\epsilon_0}(\epsilon_0)$: Choose traversal for $g$.
\item $\hat{f}'_{\epsilon_0}(\epsilon_0) = \hat{g}'_{\epsilon_0}(\epsilon_0)$: Cannot decide.
\end{enumerate}

Expand All @@ -453,7 +453,7 @@ \subsubsection{Second Derivative}
[h^{-1}]''(\epsilon) &= - \frac{u^2}{a^2\left(\frac{\epsilon^2 - u^2}{a}\right)^{\frac{3}{2}}}\label{eq:hinvd2}
\end{align}

We can write a simple equation for $[h^{-1}]'(\epsilon_0)$ by using the equations \ref{eq:hinvd2} and \ref{eq:veps0}:
We can write a simple equation for $[h^{-1}]''(\epsilon_0)$ by using the equations \ref{eq:hinvd2} and \ref{eq:veps0}:

\begin{align}
[h^{-1}]''(\epsilon_0) &= - \frac{u^2}{a^2 \left| v \right|^3}\label{eq:hinvd2eps0}
Expand All @@ -476,7 +476,7 @@ \subsubsection{Example: First Derivative}
\end{figure}
\footnotetext{View the example here: \url{https://abegehr.github.io/frechet/?p=(0_1)(4_1)(2_2)&q=(1_0)(1_4)(3_3)}}

In the example depicted in figure \ref{fig:ces_ad_1}, there are two classical critical events of type b at critical height $\epsilon_0 = 3$ to consider: $C_1(1, 4)$ and $C_2(4, 1)$.
In the example depicted in figure \ref{fig:ces_ad_1}, there are two classical critical events of type b at critical height $\epsilon_0 = 3$ to consider: $C_1(4, 1)$ and $C_2(1, 4)$.

The algorithm \ref{alg:gengraph} that generates a traversability graph for multiple critical events at the same height as defined in section \ref{sec:gengraph} returns that there are two possible paths through the critical events $C_1$ and $C_2$. Either the graph is traversed from the startpoint to the endpoint over $C_1$ (traversal path $f$) or over $C_2$ (traversal path $g$).

Expand Down Expand Up @@ -509,7 +509,7 @@ \subsubsection{Example: First Derivative}

$$\hat{f}'_{\epsilon_0}(\epsilon_0) \approx 2.118 < \hat{g}'_{\epsilon_0}(\epsilon_0) \approx 3.236$$

The descent from height $\epsilon_0$ is steeper when taking path $g$ over $C_2(4, 1)$. This can also be seen in figure \ref{fig:ces_ad_1}. The contour lines (lines of equal $\epsilon$) are closer together on the right of $C_2(4, 1)$ than they are above $C_1(1, 4)$.
The descent from height $\epsilon_0$ is steeper when taking path $f$ over $C_1(4, 1)$. This can also be seen in figure \ref{fig:ces_ad_1}. The contour lines (lines of equal $\epsilon$) are closer together on the right of $C_2(4, 1)$ than they are above $C_2(1, 4)$.


\subsubsection{Example: Second Derivative}
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