acknowledgements to end, remove n, python capitalization

latex/document.tex

@@ -21,8 +21,6 @@ captions=nooneline,captions=tableabove,english,DIV=16,numbers=noenddot,final]{sc
 \clearpage
 \input{./tex/abstract.tex}
 \clearpage
-\input{./tex/thanks.tex}
-\clearpage
 \thispagestyle{empty}
 \mbox{}
 \clearpage
@@ -62,4 +60,7 @@ captions=nooneline,captions=tableabove,english,DIV=16,numbers=noenddot,final]{sc
 \printbibliography
 \clearpage
 \input{./tex/erklaerung.tex}
+\clearpage
+\input{./tex/thanks.tex}
+\clearpage
 \end{document}

latex/tex/introduction.tex

@@ -1,4 +1,4 @@
-n\chapter{Introduction}%
+\chapter{Introduction}%
 \label{chap:intro}
 
 Monte Carlo (MC) methods have been and still are one of the most
@@ -52,15 +52,11 @@ Throughout, natural units with
 otherwise. The fine structure constant's value \(\alpha = 1/137.036\)
 is configured in \sherpa\ and used in analytic calculations.
 
-The compatibility of histograms is tested as discussed in
-\cref{sec:comphist} and the respective \(P\) and \(T\) values are
-being included in the ratio plots.
-
 \section{Source Code}%
 \label{sec:source}
 
-The (literate) python code, used to generate most of the results and
-figures can be found under
+The (literate) \texttt{Python} code, used to generate most of the
+results and figures can be found under
 \url{https://github.com/vale981/bachelor_thesis/} and more
 specifically in the subdirectory \texttt{prog/python/qqgg}.
 
@@ -69,8 +65,8 @@ algorithm related functionality as a module. The file
 \texttt{analytical\_xs.org} contains a literate computation notebook
 that generates all the results of \cref{chap:mc}. The file
 \texttt{parton\_density\_function\_stuff.org} contains all the
-computations for \cref{chap:pdf}. The python code makes heavy use of
-\href{https://www.scipy.org/}{scipy}~\cite{2020Virtanen:Sc} and
+computations for \cref{chap:pdf}. The \texttt{Python} code makes heavy
+use of \href{https://www.scipy.org/}{scipy}~\cite{2020Virtanen:Sc} and
 its component \href{https://numpy.org/}{numpy}).
 
 \section{Software Versions}%

latex/tex/monte-carlo.tex

@@ -10,11 +10,11 @@ Monte Carlo methods for multidimensional integration and sampling of
 probability distributions are central tools of modern particle
 physics. Therefore some simple methods and algorithms are being
 studied and implemented from scratch here and will be applied to the
-results from \cref{chap:qqgg}. The \verb|python| code for the
+results from \cref{chap:qqgg}. The \texttt{Python} code for the
 implementation can be found as described in \cref{sec:source}. The
 sampling and integration intervals, as well as other parameters have
-been chosen as in \cref{sec:compsher} been chosen, so that
-\result{xs/python/eta} and \result{xs/python/ecm}. This chapter is
-based on the appendix~\cite{buckley:2011ge} and supplants that source
-with some derivations and more methods, for instance the
+been chosen as in \cref{sec:compsher}, so that \result{xs/python/eta}
+and \result{xs/python/ecm}. This chapter is based on the
+appendix of~\cite{buckley:2011ge} and supplants that source with some
+derivations and more methods, for instance the
 \vegas~\cite{Lepage:19781an} algorithm.

latex/tex/monte-carlo/integration.tex

@@ -68,13 +68,13 @@ variable \(\vb{x}\), and the integration volume
 \(\Omega\). Accordingly, some ways variance reductions can be
 accomplished are choosing a suitable \(\rho\) (importance sampling),
 by transforming the integral onto another variable or by subdividing
-integration volume into several sub-volumes of different size while
-keeping the sample size constant in all sub-volumes (a special case of
-stratified sampling).\footnote{There are of course still other methods
-  like the multi-channel method.}\footnote{Of course, combinations of
-  these methods can be applied as well.}  Combining ideas from
-importance sampling and stratified sampling leads to the \vegas\
-algorithm~\cite{Lepage:19781an} that approximates the optimal
+the integration volume into several sub-volumes of different size
+while keeping the sample size constant in all sub-volumes (a special
+case of stratified sampling).\footnote{There are of course still other
+  methods like the multi-channel method.}\footnote{Of course,
+  combinations of these methods can be applied as well.}  Combining
+ideas from importance sampling and stratified sampling leads to the
+\vegas\ algorithm~\cite{Lepage:19781an} that approximates the optimal
 distribution of importance sampling by adaptive subdivision of the
 integration volume into a grid.
 
@@ -83,7 +83,7 @@ dimensionality of the integration volume as opposed to many other
 numerical integration algorithms (trapezoid rule, Simpsons rule) that
 usually converge like \(N^{-\frac{k}{n}}\) with
 \(k\in\mathbb{N}_{>0}\) and \(n\) being the dimensionality.  MC
-integration in it's simplest form converges like \(N^{-\frac{1}{2}}\)
+integration in its simplest form converges like \(N^{-\frac{1}{2}}\)
 in \emph{all} dimensions. Because phase space integrals in particle
 physics usually have a high dimensionality, MC integration is a
 suitable approach for those problems. When implementing MC methods,
@@ -171,13 +171,15 @@ equivalent (see \ref{sec:equap}).
 \subsection{Integration with \vegas}
 \label{sec:mcintvegas}
 
-Stratified sampling gives optimal results, when the variance in every
-sub-volume is the same~\cite{Lepage:19781an}. In importance sampling,
-the optimal probability distribution is given
-by \cref{eq:optimalrho}, where \(f(\Omega) \geq 0\) is presumed
-without loss of generality. When applying \vegas\ to multi dimensional
-integrals,~\cref{eq:optimalrho} is usually modified to factorize into
-distributions for each variable to simplify calculations.
+Subdividing the integration volume into sub-volumes and taking the same
+number of samples in each volume (stratified sampling), gives optimal
+results, when the variance in every sub-volume is the
+same~\cite{Lepage:19781an}. In importance sampling, the optimal
+probability distribution is given by \cref{eq:optimalrho}, where
+\(f(\Omega) \geq 0\) is presumed without loss of generality. When
+applying \vegas\ to multi dimensional integrals,~\cref{eq:optimalrho}
+is usually modified to factorize into distributions for each variable
+to simplify calculations.
 %
 \begin{equation}
   \label{eq:optimalrho}
@@ -188,21 +190,23 @@ The idea behind \vegas\ is to subdivide \(\Omega\) into hypercubes
 (create a grid), define \(\rho\) as step-function with constant value
 on those hypercubes and iteratively approximating
 \cref{eq:optimalrho}, instead of trying to minimize the variance
-directly. In the end, the samples are concentrated where \(f\) takes
-on the highest values and changes most rapidly. This is done by
-subdividing the hypercubes into smaller chunks, based on their
-contribution to the integral, which is calculated through MC sampling,
-and then varying the hypercube borders until all hypercubes contain
-the same number of chunks.  So if the contribution of a hypercube is
-large, it will be divided into more chunks than others. When the
-hypercube borders are then shifted, this hypercube will shrink, while
-others will grow by consuming the remaining chunks.  Repeating this
-step in so called \vegas\ iterations will converge the contribution of
-each hypercube to the same value. More details about the algorithm can
-be found in~\cite{Lepage:19781an}.  Note that no knowledge about the
-integrand is required. The probability density used by \vegas\ is
-given in \cref{eq:vegasrho} with \(K\) being the number of hypercubes
-and \(\Omega_i\) being the hypercubes themselves.
+directly. As in stratified sampling, the sample number is kept the
+same in each hypercube to realize \cref{eq:optimalrho}. In the end,
+the samples are concentrated where \(f\) takes on the highest values
+and changes most rapidly. This is done by subdividing the hypercubes
+into smaller chunks, based on their contribution to the integral,
+which is calculated through MC sampling, and then varying the
+hypercube borders until all hypercubes contain the same number of
+chunks.  So if the contribution of a hypercube is large, it will be
+divided into more chunks than others. When the hypercube borders are
+then shifted, this hypercube will shrink, while others will grow by
+consuming the remaining chunks.  Repeating this step in so called
+\vegas\ iterations will converge the contribution of each hypercube to
+the same value. More details about the algorithm can be found
+in~\cite{Lepage:19781an}.  Note that no knowledge about the integrand
+is required. The probability density used by \vegas\ is given in
+\cref{eq:vegasrho} with \(K\) being the number of hypercubes and
+\(\Omega_i\) being the hypercubes themselves.
 %
 \begin{equation}
   \label{eq:vegasrho}
@@ -222,7 +226,7 @@ and \(\Omega_i\) being the hypercubes themselves.
     deviations of the distributions with matching color.}
 \end{figure}
 %
-This algorithm has been implemented in python and applied to
+This algorithm has been implemented in \texttt{Python} and applied to
 \cref{eq:crossec}.  In one dimension the hypercubes become simple
 interval increments and applying \vegas\ to \cref{eq:crossec} with
 \result{xs/python/xs_mc_θ_vegas_K} increments yields

latex/tex/monte-carlo/sampling.tex

@@ -243,9 +243,8 @@ distribution \(f\) is required.
 Yet another approach is to subdivide the sampling volume \(\Omega\)
 into \(K\) sub-volumes \(\Omega_i\subset\Omega\) and then take a
 number of samples from each volume proportional to the integral of the
-function \(f\) in that volume. In the context of sampling of
-distributions, this is known as \emph{stratified
-  sampling}. The advantage of this method is, that it is now possible
+function \(f\) in that volume. This is a method of stratified
+\emph{stratified sampling}, with the advantage that it is now possible
 to optimize the sampling in each sub-volume independently.
 
 Let \(N\) be the total sample count (\(N\gg 1\)),
@@ -340,9 +339,12 @@ in \cref{sec:simpdiphotriv}.
     \caption{\label{fig:histpt} Histogram of the transverse momentum
       (\(\pt\)) distribution.}
   \end{subfigure}
-  \caption{\label{fig:histos} Histograms of observables, generated
-    from a sample of 4-momenta and normalized to unity. The plots
-    include histograms generated by \sherpa\ and \rivet.}
+  \caption[Histograms of observables, generated from a sample of
+  4-momenta and normalized to unity.]{\label{fig:histos} Histograms of
+    observables, generated from a sample of 4-momenta and normalized
+    to unity. The plots include histograms generated by \sherpa\ and
+    \rivet. The \(P\) and \(T\) values are a measure of consistency of
+  the two histograms as is described in \cref{sec:comphist}.}
 \end{figure}
 %
 Where \cref{fig:histeta} shows clear resemblance of
@@ -354,10 +356,14 @@ to \(\pt\) it can be seen in \cref{fig:diff-xs-pt} that there really
 is a singularity at \(\pt = \ecm\), due to a term
 \(1/\sqrt{1-(2\cdot \pt/\ecm)^2}\) stemming from the Jacobian
 determinant. This singularity will vanish once considering a more
-realistic process (see \cref{chap:pdf}). Furthermore the histograms
-\cref{fig:histeta,fig:histpt} have a \(P\)-value (see
-\cref{sec:comphist}) tested for consistency with their
-\rivet-generated counterparts and are therefore considered valid.
+realistic process (see \cref{chap:pdf}).
+
+The compatibility of histograms is tested as discussed in
+\cref{sec:comphist} and the respective \(P\) and \(T\) values are
+being included in the ratio plots. The histograms
+\cref{fig:histeta,fig:histpt} are (see \cref{sec:comphist}) tested for
+consistency with their \rivet-generated counterparts. They have a
+\(P\)-value greater than \(.6\) and are therefore considered valid.
 
 %%% Local Variables:
 %%% mode: latex

latex/tex/pdf/results.tex

@@ -96,7 +96,7 @@ implementation used here can be found as described in
 \cref{sec:source} and employs stratified sampling (as discussed in
 \cref{sec:stratsamp-real}) and the hit-or-miss method. The matrix
 element (ME) and cuts are implemented using
-\texttt{cython}~\cite{behnel2011:cy} to obtain better performance as
+\texttt{Cython}~\cite{behnel2011:cy} to obtain better performance as
 these are evaluated very often. The ME and the cuts are then convolved
 with the PDF (as in \cref{eq:weighteddist}) and wrapped into a simple
 function with a generic interface and plugged into the \vegas\
@@ -123,9 +123,9 @@ one sample is obtained. Taking more than one sample can improve
 performance, but introduces bias, as hypercubes with low weight may be
 oversampled. At various points, the \texttt{numba}~\cite{lam2015:po}
 package has been used to just-in-time compile code to increase
-performance. The python \texttt{multiprocessing} module is used to
-parallelize the sampling and exploit all CPU cores. Although the
-\vegas\ step is very (\emph{very}) time intensive, but the actual
+performance. The \texttt{Python} \texttt{multiprocessing} module is
+used to parallelize the sampling and exploit all CPU cores. Although
+the \vegas\ step is very (\emph{very}) time intensive, but the actual
 sampling performance is in the same order of magnitude as \sherpa, but
 some parameters have to be manually tuned.
 

latex/tex/thanks.tex

@@ -1,3 +1,4 @@
+\thispagestyle{empty}
 \section*{Acknowledgements}
 
 I want to thank Frank Siegert, my thesis advisor, for his patience