diff --git a/thesis/.texpadtmp/thesis.blg b/thesis/.texpadtmp/thesis.blg index cf92ab4..72081f1 100644 --- a/thesis/.texpadtmp/thesis.blg +++ b/thesis/.texpadtmp/thesis.blg @@ -1,15 +1,15 @@ -[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' -[198] Biber.pm:857> INFO - Found 2 citekeys in bib section 0 +[0] Config.pm:302> INFO - This is Biber 2.11 +[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' +[193] Biber.pm:857> INFO - Found 2 citekeys in bib section 0 [214] Biber.pm:3976> INFO - Processing section 0 -[229] Biber.pm:4136> INFO - Looking for bibtex format file '/Users/anton/Documents/Studium/BA/frechet/thesis/bibliography.bib' for section 0 -[230] bibtex.pm:1500> INFO - LaTeX decoding ... -[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' -[260] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'normalization = NFD' with 'normalization = prenormalized' -[260] Biber.pm:3804> INFO - Sorting list 'nty/global//global/global' of type 'entry' with template 'nty' and locale 'en-US' -[260] Biber.pm:3810> INFO - No sort tailoring available for locale 'en-US' -[263] bbl.pm:617> INFO - Writing '/Users/anton/Documents/Studium/BA/frechet/thesis/.texpadtmp/thesis.bbl' with encoding 'UTF-8' -[265] bbl.pm:720> INFO - Output to /Users/anton/Documents/Studium/BA/frechet/thesis/.texpadtmp/thesis.bbl +[230] Biber.pm:4136> INFO - Looking for bibtex format file '/Users/anton/Documents/Studium/BA/frechet/thesis/bibliography.bib' for section 0 +[231] bibtex.pm:1500> INFO - LaTeX decoding ... +[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' +[267] Biber.pm:3810> INFO - No sort tailoring available for locale 'en-US' +[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 diff --git a/thesis/.texpadtmp/thesis.log b/thesis/.texpadtmp/thesis.log index 13ba38b..0afe7a4 100644 --- a/thesis/.texpadtmp/thesis.log +++ b/thesis/.texpadtmp/thesis.log @@ -1,4 +1,4 @@ -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 entering extended mode restricted \write18 enabled. file:line:error style messages enabled. @@ -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 -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). PDF statistics: 843 PDF objects out of 1000 (max. 8388607) 698 compressed objects within 7 object streams diff --git a/thesis/.texpadtmp/thesis.synctex.gz b/thesis/.texpadtmp/thesis.synctex.gz index 94d3506..6a68a00 100644 Binary files a/thesis/.texpadtmp/thesis.synctex.gz and b/thesis/.texpadtmp/thesis.synctex.gz differ diff --git a/thesis/3_main.tex b/thesis/3_main.tex index d41f90e..6403f9d 100644 --- a/thesis/3_main.tex +++ b/thesis/3_main.tex @@ -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} @@ -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} @@ -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 $g$). @@ -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} diff --git a/thesis/thesis.pdf b/thesis/thesis.pdf index da48824..a6f5bf1 100644 Binary files a/thesis/thesis.pdf and b/thesis/thesis.pdf differ diff --git a/thesis/title.pdf b/thesis/title.pdf new file mode 100644 index 0000000..dd6a699 Binary files /dev/null and b/thesis/title.pdf differ