\subsubsection{Broadcast time}\label{subsubsec:rect2krtime}

Since in all the experiments we have always reached a coverage of at least
\(99\%\), here we will only discuss the broadcast time needed to reach the 99th
percentile of the coverage.

We have a large unexplained variation (\(12.31\%\)), similar to the cases of the
other scenarios. The reason for this result is the same and it will be discussed
in \chref{ch:starting-node}.

We can see that the most important factor is the broadcast radius that accounts
for the \(65.40\%\) of the variation, followed by the size of the hear window
(\(20.58\%\)). Other factors and their combinations are irrelevant. These
results are aligned with those obtained for the other scenarios, so the same
considerations apply.

Also in this case we had to use a logarithmic transformation of the predicted
variable in order to meet the assumption of finite variance for the residuals.


\chapter{Tentative Methodology}\label{ch3}
In order to create a graph $G = <V,E>$, where set of vertices is $V = <v_1,v_2,... v_n>$ and
set of edges $E = <e_1, e_2,...,e_m>$. Here vertex $v_i = <m_i, c_i>$ where $m_i$ is a tweet and $c_i$
is a concept from dbpedia.
So each node is a pair of tweeter mention $m_i$ and its corresponding concept $c_i$.

To create the nodes, we need to collect tweets. But the tweets donot give adequate data.
Since a single tweet doesnot give us adequate data, we can use a set of tweets discuss about same topic
leading us to some data to disambiguate meaning. In order to work with tweets, we need to
perform some Natural Language Processing. For this we will use GATE Developer ~\cite{url0gate} , a
text analysis or language processing toolkit. It helps us to annotate, gazetize, corefer
 etc in the tweets.

Next we need to collect data from dbpedia because the tweets themselves
donot provide adequate data. After collecting the dbpedia data, we will convert them into $n$-grams.
For each acticles in dbpedia, we will create its corresponding concept vector, $c$. After that, we need to
use machine learning to train how to detect a concept and find relations between them. We also need
to train on matching twitter mentions $m$ with their corresponding concept vectors $c$.

 Thus we will have nodes $<m_i, c_i>$. After that, we need to connect the nodes with edges.
 We will use link detection tecniques as discussed in ~\cite{ref3LinkWikipedia} to
 find possible links and disambiguate links.

\endinput


\subsection{Surveys, catalogs, and population studies}
\label{ssec:surveys-catalogs-and-population-studies}

Sky surveys have a large potential for new source detections, and new phenomena
discovery. They also offer less selection bias to perform source population
studies over a large set of coherently detected and modelled objects. Early
versions of \textit{gammapy} were developed in parallel of the preparation of
the HESS Galactic plane survey catalog~\citep[HGPS][]{2018A&A...612A...1H} and
the associated PWN and SNR populations studies~\citep{2018A&A...612A...2H,
	2018A&A...612A...3H}.

The increase in sensitivity  and resolution provided by new generation of
instruments scale up the number of sources detectable and the complexity of the
models needed to represent them accurately. As an example If we compare the
results of the HGPS to the expectations from the CTA Galactic Plane survey
simulations we jump from 78 sources detected by HESS to about 500 detectable by
CTA~\citep{2021arXiv210903729R}.

Studies performed on simulations not only offer a first glimpse on what could
be the sky seen by CTA (according to our current knowledge on source
populations), but also give us the opportunity to test the software on complex
use cases\footnote{Note that the CTA-GPS simulations were performed with the
	\textit{ctools} package~\citep{2016A&A...593A...1K} and analysed with both
	\textit{ctools} and \textit{gammapy} packages in order to cross-validate
	them.}. So we can  improve performances, optimize our analyses strategies, and
identify the needs in term of parallelisation to process the large datasets
provided by the surveys.

In short the production of catalogs from gamma-ray surveys can be divided in
four main steps : data-reduction; object detection; model fitting and model
selection; associations and classification. The IACTs data-reduction step is
done in the same way than described in the previous sections but scale-up to
few thousand of observations. The object detection step consists a minima in
finding local maxima in the significance, or TS maps,  given by the
ExcessMapEstimator, or TSMapEstimator, respectively.  Further refinements can
include for example  filtering and detection on these maps with techniques from
scikit-image package~\citep{scikit-image}, and outlier detection from
scikit-learn package~\citep{scikit-learn}. Tests on simulations shown that it
reduces the spurious detections at this stage and then speed up the next step
as less objects will have to be fitted simultaneously. During the modelling
step each object is alternatively fitted with different models in order to
determine their optimal parameters, and the best-candidate model. The
subpackage gammapy.modeling.models offers a large variety of choice, and the
possibility to add custom models.  Several spatial models (point-source, disk,
gaussian...), and spectral models (power-law, log-parabola...) may be tested
for each object, so the complexity of the problem increases rapidly in regions
crowded with multiple extended sources. Finally an object is discarded if its
best-fit model is not significantly preferred over the null hypothesis (no
source) comparing the difference in log-Likelihood between these two
hypotheses. For the association and classification step, that is tightly
connected to the population studies, we can a minima compare the fitted models
to the set of gammapy-ray catalogs available in gammapy.catalog. However
further multi-wavelength cross-matches are usually required to characterize the
sources.


%!TEX root = ../Main.tex
In this section a range of tests will be conducted on different NPG structures in order to uncover the effect of chemical modification of the bridges between the Graphene Nano Ribbons (GNRs) in the NPG. From an applied perspective, one of the main motivations is to find out how these bridges can be chemically modified in order to control the current through the material. A recent study\cite{unpub}, submitted for publication in Journal of American Chemical Society, has shown how one could possibly confine current flow to a single GNR channel by modification of bridges between GNRs in NPG utilising \textit{Quantum Interference} (QI) effects (see \cref{studyfig3}). This provides a solution to an important requirement in carbon-based nano circuitry design, namely nano confinement of electron flow. The study was focused on the difference in effect of having \textit{meta} and \textit{para} bridges between GNRs. Meta and para bridges are essentially benzene rings connected in two different ways (see \cref{Parametastructfig,Metaparastructfig}). The meta and para bridges are 'static' cases in the sense that once made through organic synthesis, they are not interchangeable. However, if the bridges are functionalised with oxygen\cite{Soler_2002} they become sensitive to f.ex. hydrogenation in basic/acidic environments, which tends to affect QI. Thus by hydrogenation it should be possible to tune the electrical properties of the material and make it sensitive to external environments. Studies\cite{li_single_2019} show that hydrogenation of specific sites on nano meter scale is possible experimentally. By the use of the functions developed in previous sections, this section will try to uncover what happens when oxygen is bonded to the benzene rings in the meta and para bridges. Subsequently hydrogenation of the oxygen will be tested. Following is a section introducing para and meta NPG in detail.
\begin{figure}[ht]
	\centering
	\begin{subfigure}[b]{.45\textwidth}
		\centering
		\includegraphics[width=.9\textwidth]{Figures/Parametagraphic.eps}
		\caption{}
		\label{Parametastructfig}
	\end{subfigure}
	\quad
	\begin{subfigure}[b]{.45\textwidth}
		\centering
		\includegraphics[width=.9\textwidth]{Figures/Metaparagraphic.eps}
		\caption{}
		\label{Metaparastructfig}
	\end{subfigure}
	\vskip
	\begin{subfigure}[b]{.45\textwidth}
		\centering
		\includegraphics[width=.9\textwidth]{Figures/Fig_3para.eps}
		\caption{}
		\label{studyfig3para}
	\end{subfigure}
	\quad
	\begin{subfigure}[b]{.45\textwidth}
		\centering
		\includegraphics[width=.9\textwidth]{Figures/Fig_3meta.eps}
		\caption{}
		\label{studyfig3}
	\end{subfigure}
	\caption{a) showing the para NPG, b) showing the meta NPG.\newline
		c-d) Figures showing the currents through the GNRs in a large peace of NPG. Note how the current is spread across GNRs with the para bridge a) and confined to a single GNR with meta bridge c).\newline
		Used with permission from Isaac Alcón Rovira.}
	\label{studyfigs}
\end{figure}
\subsection{Bridge effects in para-NPG and meta-NPG}\label{metaparasection}
In broad terms the difference in the \textit{meta} and \textit{para} structures lies in the path an electron will travel through the benzene ring to get across the bridge between GNRs in NPG. In the para bridge, the path across the aromatic ring is symmetric and so the electron will pass above or below with equal phase. Since the para bridge has three bonds in each direction across the ring, the path length in the para bridge is the same on each side (see \cref{Parametastructfig}). This will cause constructive quantum interference of the states once the waves meet on the other side of the ring. For para NPG this causes electronic coupling between the GNR channels. In the meta bridge, the way across the aromatic ring is not symmetric in the sense that there is two bonds across the path below and four bonds across on the path above (see \cref{Metaparastructfig}). This will cause a shift by half a wavelength between the two paths and thus create destructive quantum interference between waves meeting on the other side of the ring. This causes electronic decoupling between GNR channels, allowing for confinement of injected currents in a single GNR channel\cite{unpub} (see \cref{studyfig3}). In \cref{metapara} band plots as well as transmission plots from the study\cite{unpub} are shown for para and meta NPG. In the band plots the two sets of valence/conduction bands around the Fermi level shows band splitting for para and interference for meta NPG. Looking at the transmission plots next to the band plots one can see there is transmission and thus coupling of the states between the GNRs for para. The area between the two peaks at \SI{0.5}{\electronvolt} and  \SI{1.2}{\electronvolt} show transport between the GNRs. In the transmission plot for meta the transmission is suppressed (\cref{metapara} right). The effect of quantum interference can be seen in the transmission plot at \SI{0.7}{\electronvolt}. Here the transmission is close to 0. This means GNRs are electrically decoupled in meta system. To summarise these results qualitatively: Splitting of the bands in the band plots, corresponds to effective coupling of the GNRs. Bands on top of each other, showing QI, corresponds to decoupling of the GNRs and thus electric confinement of injected currents on NPG in single GNR channels (see \cref{studyfig3}). In the following sections, band plots will mainly be used to show results. Therefore one should keep in mind how the band structures, qualitatively, relate to transmission. In \cref{appfigs}, \cref{allbands} a figure obtained from the developed functions with band plots of normal, para and meta NPG can be seen.
\begin{figure}[ht]
	\centering
	\includegraphics[width=.8\textwidth]{Figures/metapararesultdraft.eps}
	\caption{Figure showing band plots and transmission plots for para and meta NPG. In the bottom of each plot is a schematic showing which way the transmission occurs in relation to the NPG. Quantum interference marked with arrows.\newline
		Used with permission from Isaac Alcón Rovira}
	\label{metapara}
\end{figure}
\newpage
\subsection{Tests with oxygen modified meta and para-NPGs}\label{testintroduc}
The tests consist of two kinds of chemical modification. Firstly it will be bonding oxygen sites to the meta and para NPG (addition of Oxygen will be simulated by addition of another carbon site to the structure, effectively adding another pi-electron). Secondly simulations of hydrogenation will be done by bonding hydroxide groups to the bridges. Because of the complex nature of DFT the band plots obtained via DFT makes it hard to interpret what is actually going on in the system chemically. Therefore the aim is to show that using the methods developed, based on tight-binding models, it is possible to reproduce bands structures calculated from DFT. If successful, it will provide a much easier way to understand the effect of the implemented chemical modification compared to the DFT approach. Band plots, as well as some on-site potential maps, calculated with DFT has been provided by supervisor Isaac Alcón Rovira for comparison. The geometries, used for calculations, are provided by supervisor Isaac Alcón Rovira and have been optimised by DFT using the SIESTA package\cite{Soler_2002} and the PBE functional\cite{perdew1996a}.
\subsection{Test 1: Para-O\mathinhead{_4}{_4}-NPG}\label{test1}
The first test is considering the Para-O\(_4\)-NPG. The basic structure is para-NPG where 4 oxygen sites are bonded, two on each benzene ring (see \cref{PS4OOW} ). Starting by looking at the resulting DFT plot in \cref{PS4ODFT} the first thing to notice is that the valence bands show QI and thus decouples the GNRs. So in spite having a para bridge, which normally gives coupling, decoupling occurs when oxygen is bonded to the bridges. Our first assumption to simulate the effect of such oxygen functionalisation is to add those atoms as active \(p_z\)-orbitals in our pi-conjugated system, using the standard parameters (i.e. as for carbon atoms - specifically the on-site potential and hopping value). As a baseline the potential of these atoms are thus the same. Moreover, the hydrogen, included in the provided optimised geometries, will be removed before any calculations take place. In \cref{PS4Odevnomod} one can see the resulting band structure plot, using such an electronic model and the DFT optimised structure. The band plot does not show much resemblance with the one obtained, using DFT. It is pretty much symmetric around the Fermi level and it is hard to make out whether there is coupling or decoupling between GNRs, based on the reasoning discussed in \cref{metaparasection}. Taking a look at the DFT-calculated potential map of \cref{potmapPS4O}, the sites where the oxygen is bonded, have a potential much different from that of the rest of the system (the four dark blue spots). As mentioned, the baseline of the method does not take into account differentiating on-site potentials within a cell. So to compensate for this, the on-site potential of the four bonded oxygen is changed. In \cref{PS4Odevmod} one can see the resulting band plot after the on-site potential of the bonded oxygen has been changed to \SI{-0.50}{\electronvolt}. The resulting band plot resembles the DFT calculations much more. There is now QI in the valence bands which means decoupling of the GNRs.
\begin{figure}[H]
	\centering
	\begin{subfigure}[b]{.8\textwidth}
		\centering
		\includegraphics[width=.8\textwidth]{Figures/para_O4.png}
		\vspace{-1\baselineskip}
		\caption{}
		\label{PS4OOW}
	\end{subfigure}
	\vskip
	\begin{subfigure}[b]{.3\textwidth}
		\centering
		\includegraphics[width=.8\textwidth]{Figures/PS4ODFT.eps}
		\vspace{-1\baselineskip}
		\caption{}
		\label{PS4ODFT}
	\end{subfigure}
	\hspace{-20pt}
	\begin{subfigure}[b]{.3\textwidth}
		\centering
		\includegraphics[width=.8\textwidth]{Figures/PS4Onomod.eps}
		\vspace{-2\baselineskip}
		\caption{}
		\label{PS4Odevnomod}
	\end{subfigure}
	\hspace{-30pt}
	\begin{subfigure}[b]{.3\textwidth}
		\centering
		\includegraphics[width=.8\textwidth]{Figures/PS4Omod.eps}
		\vspace{-2\baselineskip}
		\caption{}
		\label{PS4Odevmod}
	\end{subfigure}
	\vskip
	\begin{subfigure}[b]{.8\textwidth}
		\hspace{50pt}
		\includegraphics[width=.8\textwidth]{Figures/PS4O.eps}
		\vspace{-.5\baselineskip}
		\caption{}
		\label{potmapPS4O}
	\end{subfigure}
	\caption{a) Overview of the system with carbon (grey), oxygen (red) and hydrogen atoms (blue) and highlighting the added oxygen or hydroxide groups as spheres. b) Band structure obtained using DFT. c) Band structures obtained using developed scripts with no on-site potential mods. c) Band structures obtained using developed scripts with the on-site potential changed to \SI{-0.5}{\electronvolt}. d). Potential map of the system. In b and d, quantum interference is highlighted with arrows.}
	\label{PS4O}
\end{figure}
To summarise the results the tests showed significant qualitative agreement with the DFT calculation. This helps to uncover the main effect of functionalisation of para-NPG with oxygen. Knowing that the added \(p_z\) electrons where participating in the pi-conjugated system, those orbitals had to be accounted for in the developed model by addition of extra sites. However, to qualitatively reproduce the DFT results, the on-site potential of those extra sites had to be lowered by \SI{-0.5}{\electronvolt}. This helped to explain why functionalisation with oxygen only affects the valence band of NPG and not the conduction band. In addition the developed method also reproduced the flat bands seen around the Fermi level in the DFT calculation (the developed method showed bands exactly at the Fermi level). In broad terms, these bands are related to benzene bridge in the NPG, but they originate because of the presence of the bonded oxygen. The oxygen do have an dispersive effect on the valence electrons of NPG, but they also seem to be localised in nature, which is why the flat bands occur.
\subsection{Test 2: Para-(OH)\mathinhead{_4}{_4}-NPG}\label{test2}
Next test will be with the Para-(OH)\(_4\)-NPG system. Again the basic structure is para-NPG with bonded oxygen, but here the aim is to simulate hydrogenation, so therefore the oxygen atoms added will become hydroxide groups (see \cref{PS4OHOW}). The DFT plot in \cref{PS4OHDFT} shows band splitting in the valence/conduction bands, so the GNRs have been coupled again. Here the first approach is to use the successful scripts reproducing the DFT results in \cref{test1}: i.e. considering oxygen positions as active \(p_z\) orbitals in our model, but lowering their energy by \SI{-0.5}{\electronvolt}. In \cref{PS4OHmod2} the resulting band plot can be seen. It shows exactly the same as \cref{PS4Odevmod}. The result does therefore not resemble that of the DFT in \cref{PS4OHDFT}, which in turn is very similar to that of the pristine para-NPG (see \cref{metapara} left panel). Looking at the potential map in \cref{potmapPS4OH} it can be seen that the bonded oxygen are shifted by approximately \SI{2}{\electronvolt} down in energy with respect to the GNRs carbon atoms. Using the potential plot in \cref{potmapPS4OH} as a guide, the on-sites of the oxygen are changed to \SI{-2}{\electronvolt}. The resulting plot can be seen in \cref{PS4OHdevmod}. The plot show good agreement with the DFT. The bands have split again and the GNRs are coupled. This means hydrogenation decouples the oxygen from the GNRs pi-conjugated system, so by lowering the on-site potential of one can get effective decoupling of oxygen. As seen this gives a satisfying result. Now that it is known that hydrogenation decouples the oxygen, a last hypothesis is that by simply not considering the oxygen atoms in our model the result will be similar to \cref{PS4OHdevmod}. In \cref{PS4OHremove} the resulting plot can be seen. As hypothesised, the resulting band plot show resemblance with both the DFT and \cref{PS4OHdevmod}. So the decoupling of oxygen by hydrogenation can be simulated simply by removing the hydroxide group entirely. Again there is split in the bands and thus coupling of the GNRs. This means that the developed method can uncover what happens chemically when the NPG is hydrogenated and it can do so in a simple manner.
\begin{figure}[H]
	\centering
	\begin{subfigure}[b]{.8\textwidth}
		\centering
		\includegraphics[width=.8\textwidth]{Figures/para_OH4.png}
		\vspace{-1\baselineskip}
		\caption{}
		\label{PS4OHOW}
	\end{subfigure}
	\vskip
	\begin{subfigure}[b]{.25\textwidth}
		\centering\hspace{-20pt}
		\includegraphics[width=.8\textwidth]{Figures/PS4OHDFT.eps}
		\vspace{-1\baselineskip}
		\caption{}
		\label{PS4OHDFT}
	\end{subfigure}
	\hspace{-20pt}
	\begin{subfigure}[b]{.25\textwidth}
		\centering
		\includegraphics[width=.8\textwidth]{Figures/PS4OHmod2.eps}
		\vspace{-2\baselineskip}
		\caption{}
		\label{PS4OHmod2}
	\end{subfigure}
	\hspace{-20pt}
	\begin{subfigure}[b]{.25\textwidth}
		\centering
		\includegraphics[width=.8\textwidth]{Figures/PS4OHmod.eps}
		\vspace{-2\baselineskip}
		\caption{}
		\label{PS4OHdevmod}
	\end{subfigure}
	\hspace{-20pt}
	\begin{subfigure}[b]{.25\textwidth}
		\centering
		\includegraphics[width=.8\textwidth]{Figures/PS4OHSitesRemoved.eps}
		\vspace{-2\baselineskip}
		\caption{}
		\label{PS4OHremove}
	\end{subfigure}
	\vskip
	\begin{subfigure}[b]{.8\textwidth}
		\centering\hspace{30pt}
		\includegraphics[width=.8\textwidth]{Figures/PS4OH.eps}
		\vspace{-.5\baselineskip}
		\caption{}
		\label{potmapPS4OH}
	\end{subfigure}
	\caption{a) Overview of the system with carbon (grey), oxygen (red) and hydrogen atoms (blue) and highlighting the added oxygen or hydroxide groups as spheres. b) Band structure obtained using DFT. c) Band structures obtained using developed scripts with with on-site potential changed to \SI{-0.5}{\electronvolt} and quantum interference is highlighted with arrow. d) Band structures obtained using developed scripts with on-site potential changed to \SI{-2}{\electronvolt}. e) Band structures obtained using the developed scripts with sites removed. f) Potential map of the system.}
	\label{PS4OH}
\end{figure}
\subsection{Test 3: Meta-O\mathinhead{_2}{_2}-NPG}\label{test3}
Moving on to the meta NPG, the test is going to be exactly the same as \cref{test1}. To see the structure of the system look in \cref{MS2OOW}. Recall from \cref{metaparasection} that the meta NPG gives rise to QI and thus decoupling of the GNRs. However as seen in \cref{test1}, functionalising the bridges with oxygen caused some quantum interference in the para and thus decoupling of the GNRs in para NPG. Therefore one should expect some effect as well for the meta connected NPG. First the DFT plot in \cref{MS2ODFT} show that there is indeed band splitting for the valence states. Here one can also see shifts in the valence and conduction bands. The split is bigger in the valence band compared to the conduction band. So by functionalising with oxygen the effect is most prevalent at the valence band for both para and meta. Following the rationale of \cref{testintroduc}, we first consider the effect of oxygen atoms including them as active \(p_z\) orbitals in our mode, but with the same on-site energy value as the rest of the sites. As seen in \cref{MS2Odevnomod} we may see band splitting for both valence and conduction states. However, the plot is totally symmetric around 0 so it is not entirely accurate. Again the potential map in \cref{potmapM2SO} is used to get the on-site potential of the oxygen, so that they can be modified with the developed method. The result is seen \cref{MS2Odevmod}. Now the band plot shows split of the bands downwards, just like the DFT and the valence bands have a bigger split compared to the conduction bands. So effectively the GNRs are coupled. Now the flat bands  at the Fermi level in \cref{MS2Odevnomod} are associated with the benzene rings and they arise because of the oxygen functionalisation (i.e. they are not present in pristine meta-NPG). When the on-site potential of \SI{-0.2}{\electronvolt} is applied in \cref{MS2Odevmod} the result is in remarkably good agreement with the DFT calculations. The shape of the bands when applying the on-site potential, indicates that they are slightly more dispersive than those of \cref{PS4ODFT,PS4Odevmod}.
\begin{figure}[H]
	\centering
	\begin{subfigure}[b]{.8\textwidth}
		\centering
		\includegraphics[width=.8\textwidth]{Figures/meta_O4.png}
		\vspace{-1\baselineskip}
		\caption{}
		\label{MS2OOW}
	\end{subfigure}
	\vskip
	\begin{subfigure}[b]{.3\textwidth}
		\centering
		\includegraphics[width=.8\textwidth]{Figures/MS2ODFT.eps}
		\vspace{-1\baselineskip}
		\caption{}
		\label{MS2ODFT}
	\end{subfigure}
	\hspace{-20pt}
	\begin{subfigure}[b]{.3\textwidth}
		\centering
		\includegraphics[width=.8\textwidth]{Figures/MS2Onomod.eps}
		\vspace{-2\baselineskip}
		\caption{}
		\label{MS2Odevnomod}
	\end{subfigure}
	\hspace{-30pt}
	\begin{subfigure}[b]{.3\textwidth}
		\centering
		\includegraphics[width=.8\textwidth]{Figures/MS2Omod.eps}
		\vspace{-2\baselineskip}
		\caption{}
		\label{MS2Odevmod}
	\end{subfigure}
	\vskip
	\begin{subfigure}[b]{.8\textwidth}
		\hspace{50pt}
		\includegraphics[width=.8\textwidth]{Figures/MS2Opotmap.eps}
		\vspace{-.5\baselineskip}
		\caption{}
		\label{potmapM2SO}
	\end{subfigure}
	\caption{a) Overview of the system with carbon (grey), oxygen (red) and hydrogen atoms (blue) and highlighting the added oxygen or hydroxide groups as spheres. b) Band structure obtained using DFT. c) Band structures obtained using developed scripts with no on-site potential mods. c) Band structures obtained using developed scripts with on-site potential changed to \SI{-0.2}{\electronvolt}. d) Potential map of the system.}
	\label{MS2O}
\end{figure}
\subsection{Test 4: Meta-(OH)\mathinhead{_2}{_2}-NPG}\label{test4}
The fourth and final test will be with the same approach as \cref{test2} only here the hydrogenation is of the meta-NPG with oxygen bonded. (see \cref{MS2OHOW}). The DFT plot in \cref{MS2OHDFT} shows QI for both valence and conduction bands. It is known from the para-NPG functionalised cases that hydrogen decouples the oxygen atoms from the GNRs pi-conjugated system by lowering their energy with respect to the carbon system. In \cref{potmapMS2OH} one can see that this is indeed the case. The on-site value of the bonded oxygen is much lower than that of the carbon atoms. Therefore we try to model the system by considering the oxygen atoms and lowering their the on-site potentials to \SI{-2.3}{\electronvolt} according to the potential map in \cref{potmapMS2OH}. The resulting band plot can be seen in \cref{MS2OHdevmod}. The plot is in good agreement with DFT, showing effective decoupling of oxygen and thus decoupling of GNRs just like the unmodified meta NPG discussed in \cref{metaparasection}. Finally by removing the hydroxide group entirely, using the developed method afterwards, the resulting band plot in \cref{MS2OHremove} is obtained. The result is also in good agreement with DFT.
\begin{figure}[H]
	\centering
	\begin{subfigure}[b]{.8\textwidth}
		\centering
		\includegraphics[width=.8\textwidth]{Figures/meta_OH4.png}
		\vspace{-1\baselineskip}
		\caption{}
		\label{MS2OHOW}
	\end{subfigure}
	\vskip
	\begin{subfigure}[b]{.3\textwidth}
		\centering
		\includegraphics[width=.8\textwidth]{Figures/MS2OHDFT.eps}
		\vspace{-1\baselineskip}
		\caption{}
		\label{MS2OHDFT}
	\end{subfigure}
	\hspace{-20pt}
	\begin{subfigure}[b]{.3\textwidth}
		\centering
		\includegraphics[width=.8\textwidth]{Figures/MS2OHmod.eps}
		\vspace{-2\baselineskip}
		\caption{}
		\label{MS2OHdevmod}
	\end{subfigure}
	\hspace{-30pt}
	\begin{subfigure}[b]{.3\textwidth}
		\centering
		\includegraphics[width=.8\textwidth]{Figures/MS2OHSitesRemoved.eps}
		\vspace{-2\baselineskip}
		\caption{}
		\label{MS2OHremove}
	\end{subfigure}
	\vskip
	\begin{subfigure}[b]{.8\textwidth}
		\centering\hspace{30pt}
		\includegraphics[width=.8\textwidth]{Figures/MS2OH.eps}
		\vspace{-.5\baselineskip}
		\caption{}
		\label{potmapMS2OH}
	\end{subfigure}
	\caption{a) Overview of the system with carbon (grey), oxygen (red) and hydrogen atoms (blue) and highlighting the added oxygen or hydroxide groups as spheres. b) Band structure obtained using DFT. c) Band structures obtained using developed scripts with on-site potential changed to \SI{-2.3}{\electronvolt}. d) Band structures obtained using the developed scripts with sites removed. e) Potential map of the system. In c and d, quantum interference is highlighted with arrows.}
	\label{MS2OH}
\end{figure}
\subsection{Test summary}
For the para systems, the added oxygen introduced some QI to the system. It was possible to show that the developed program could reproduce the results obtained from DFT. This did require some adjustment to the on-site potential of the added oxygen. By hydrogenation of the oxygen it was shown that the system again returned to a split in the bands. Again by adjusting the on-site potential it was possible to obtain results very close to the DFT calculation. The results are also comparable with the ones mentioned in \cref{metaparasection}. This shows that hydrogenation of oxygen can change the system to resemble regular para-NPG, even if it has oxygen bonded. The same can be said for the meta tests. All in all these results give a better, intuitive understanding of what happens with the studied NPG systems when adding oxygen. The tests have shown that it is possible to tune the electrical properties of chemically modified meta and para NPG by hydrogenation and thus providing a novel approach in designing sensitive nano circuits, for f.ex. pH sensors.


\section{Reconstruction code}

The energy deposited by the particles in the towers produces scintillating light that is propagated with optic fibers through the different layers to APD placed at the base of the cells. The APDs amplify the signal and generate an electronic pulse shape that is stored in the raw data format. From this pulse shape, we extract the signal amplitude and the arrival time. The pulse shape is fitted during the reconstruction via a parametrized function and TMinuit, and those 2 values are extracted.

A particle produces signals in different towers (electromagnetic shower expands more than its Moli\`ere radius which is a cell size). The next step is the formation of clusters of cells that belong to the same particle, although depending on the energy, granularity, clusterization algorithm or event type, those clusters might have contributions from different particles. The default algorithm in pp collisions is a simple aggregation of neighboring cells until there is no more cells above a certain energy threshold (named {\it clusterizer V1}). In case of Pb-Pb collisions environment, where particle showers merge quite often, we apply another algorithm that aggregates cells to the clusters until reaching a cell with more energy than the precedent (named {\it clusterizer V2}). Depending on the analysis type, one might want to use one or the other clusterization type. For this reason, a re-clusterization is also possible at the analysis level. A last clusterizer is implemented, which makes 3x3 clusters. It has been used in jet analysis for instance in order to avoid biasing jet reconstruction where one is interested in the energy flow over a large area without explicit reconstruction of photon showers and where the driving consideration is that the wide clusterizer does not interfere with the jet finder. For $\pi^{0}$, $\eta$, and direct $\gamma$ analyses, {\it V2} is most likely preferable).

Once the cluster is defined, we calculate cluster parameters, shower shape parameters, that will help at the analysis level to identify each cluster as one particle type. Also, we compare the cluster position information with the propagation of tracks measured in the central barrel to the EMCAL surface, to identify the clusters generated by charged particles.

The final analysis objects, ESDs and AODs, contain all the cluster and cell basic informations allowing to redo the clusterization if needed at the analysis level.

%The class AliReconstruction manages this part. This step can
%begin either from the Digits or from the Raw data. We can distinguish
%different steps described in the following sections.

\subsection{Offline data base access}

How to create explained OCDB/OADB section.

\subsubsection{Energy calibration}

\subsubsection{Bad channels - Marie, Alexis}

\subsubsection{Alignment - Marco}

\subsection{Raw data fitting: from ADC sample to digits - David}

As also discussed in Sec.~\ref{sec:simu_raw}, the recorded Raw data consists
of instead a series of so-called time samples with 10-bit ADC counts per channel.
Each time bin is 100 ns wide, corresponding to a 10 MHz readout.
The expected signal (Gamma-2) shape is described e.g. in the AliEMCALRawResponse class,
in the RawResponseFunction method.
The reconstruction from Raw data to Digits is done in the AliEMCALRawUtils class,
Raw2Digits method.
The Raw ADC time samples data is kept in AliCaloBunchInfo objects, which are given
as input to an AliCaloRawAnalyzer object, which returns the signal amplitude and time
information (in the form of an AliCaloFitResults object).
There are several different AloCaloRawAnalyzer versions, which can be selected via
AliEMCALRawUtils::SetFittingAlgorithm(). They are:

\begin{itemize}
 \item kStandard:
   AliCaloRawAnalyzerKStandard, which is a (slower but simple) Gamma-2 fit implementation.
 \item  kFastFit:
   AliCaloRawAnalyzerFastFit, which is a faster Gamma-2 fit implementation from Aleksei Pavlinov.
 \item kNeuralNet:
   AliCaloRawAnalyzerNN, which is a neural network implementation from Paola La Rocca and Franco Riggi.
 \item kPeakFinder:
   AliCaloRawAnalyzerPeakFinder, which is a fast (parameterized vector operations) implementation from Per Thomas Hille.
 \item kCrude:
   AliCaloRawAnalyzerCrude, which is the simplest possible algorithm: just take the maximum ADC value as the signal amplitude.
 \item kFakeAltro:
   AliCaloRawAnalyzerFakeALTRO, which is an algorithm intended for the Trigger/TRU raw data analysis, i.e. not for the regular FEE or cell/tower data.
\end{itemize}

\subsection{Clusterization: From digits to clusters - Adam}

\input{clusterization}

\subsection{Cluster-Track matching - Rongrong, Shingo, Michael}

Even though EMCal is intended to measure the energy of particles that interact with EMCal via electromagnetic showering, e.g. photons and electrons, charged hadrons can also deposit energy in EMCal, most commonly via minimum ionization, but also via nuclear interactions generating hadronic showers. In the analysis where the distinction between hadroic and electromagnetic showers is necessary, cluster-track matching is often used to meet this requirement.

The main method used to extrapolate tracks in the ALICE software framework is:

\begin{DDbox}{\linewidth}
\begin{lstlisting}
 static Bool_t PropagateTrackToBxByBz(AliExternalTrackParam *track, Double_t x, Double_t m,Double_t maxStep, Bool_t rotateTo=kTRUE, Double_t maxSnp=0.8,Int_t sign=0, Bool_t addTimeStep=kFALSE);
\end{lstlisting}
\end{DDbox}
which takes the following arguments: {\it ``track''} stores all the information of the starting point for the extrapolation; {\it ``x''} is the coordinate of the destination plane in the local coordinate system; {\it ``m''} is the mass assumption for the track; {\it ``maxStep''} is the step size used in the extrapolation. This method extrapolates the track trajectory to a destination plane in a magnetic field, taking into account the energy loss of the tracks when going through detector materials. However, the energy loss model is tuned for charged hadrons, so it does not work very well for electrons or positions whose primary energy loss process is bremsstrahlung.

For EMCal, the track-cluster matching is done by default in the reconstruction chain and the code is implemented in:

\begin{DDbox}{\linewidth}
\begin{lstlisting}
 AliEMCALTracker::PropagateBack(AliESDEvent* esd)
\end{lstlisting}
\end{DDbox}

The logic of the matching procedure is the following:
\begin{itemize}
\item Check wheter TPC is available in DAQ/reco. See AliEMCALTracker::LoadClusters(). In case there is no TPC, ITS specific extrapolation will be used.
\item Find all the EMCal clusters in the event. See AliEMCALTracker::LoadClusters().
\item Find all the good tracks in the event. See AliEMCALTracker:: LoadTracks(). Several cuts are applied to select good tracks
	\begin{itemize}
	\item Minimum $p_{\rm{T}}$ cut, which can be set during the reconstruction.
	\item Cut on number of TPC clusters, which can be set during the reconstruction. This specific cut is avoided in case there is only ITS available in reconstruction.
	\item $|\eta|<0.8$ and $20^{\circ} < \varphi < 120^{\circ} $. These fiducial cuts are hard coded since tracks out of this range should never make it to EMCal.
	\end{itemize}
\item For each good track, find the nearest cluster as matched if their residuals fall within the cuts. See AliEMCALTracker::FindMatchedCluster(), which follows the following steps:
	\begin{itemize}
	\item Get the starting point: if the {\it friendTrack} is available, use the last point on the TPC. Otherwise, use the point at the inner wall of the TPC.
	\item If only ITS tracks are available in reconstruction, the propagation will use the track information from the vertex.
	\item Extrapolate tracks to the EMCal surface at 430 cm, and apply fiducial cuts on the extrapolated points: $|\eta|<0.75$ and $70^{\circ} < \varphi < 190^{\circ} $. The step size in the extrapolation can be set in the reconstruction, and the default value is 20 cm.
	\item Extrapolate tracks further, with 5 cm step size, to the positions of all the EMCal clusters which are in the vicinity of the extrapolated points from last step. Then the distance between extrapolated tracks to the clusters are calculated, and the nearest cluster is assigned as matched if the residuals fall within cuts. By fitting the distributions of the residuals using Gaussian functions, we can choose to cut on $N\sigma$ of the residuals.To further improve the matching performance, $p_{\rm{T}}$ and charge dependent cuts can be used.
	\end{itemize}
\end{itemize}

\subsection{How to execute the reconstruction}

Executing the reconstruction is very similar to the simulation case, see the macro TestEMCALReconstruction.C (a bit more detailed than the one in \$ALICE\_ROOT/EMCAL/macros) :

\begin{DDbox}{\linewidth}
\begin{lstlisting}
void TestEMCALReconstruction() {

TString detector=``EMCAL TPC'';//Same function as in Simulation.C
// TString detector=``EMCAL ITS''; if user wants ITS tracking to be used.

AliReconstruction rec; //Create reconstruction object

//Making Tracking
rec.SetRunTracking(detector) ;

//Particle Reconstruction. Make Rec Points
rec.SetRunReconstruction(detector);

//read RAW data. Give directory where raw data is stored
//rec.SetInput(``RawDataDirectory/raw.root");

//Make vertex finder
rec.SetRunVertexFinder(kFALSE) ; // false only if the tracking detectors are not included.

//Fill ESD file with RecPoints information.
rec.SetFillESD(detector) ;

//Run Reconstruction
rec.Run() ;
}
\end{lstlisting}
\end{DDbox}


\section{$\ee \rightarrow \ttbar$ at threshold}
One of the unique capabilities of an $\ee$ linear collider is the ability
to carry out cross section measurements at particle
production thresholds.  The accurately known and readily variable beam
energy of the ILC makes it possible to measure the shape of the cross section
at any pair-production threshold within its range.
Because of the leptonic initial state, it is also  possible to tune
the initial spin state, giving additional options for precision threshold
measurements.
The $t\bar t$ pair production threshold, located at a center of mass energy
energy  $\sqrt{s}\approx 2 m_t$,  allows for precise measurements of
the $t$~quark mass $m_t$ as well as the $t$~quark
total width $\Gamma_t$ and the QCD coupling $\alpha_s$.
 Because the top is a spin-$\frac{1}{2}$ fermion, the
 $t\bar t$ pair is produced in an angular $S$-wave state.  This leads to
a clearly visible rise of the cross section even when folded with the
 ILC luminosity spectrum. Moreover, because the top pair is
produced in a color singlet state, the experimental measurements can be
compared with very accurate and unambiguous
analytic theoretical predictions of the cross section
with negligible hadronization effects. The dependence of the
$t$~quark cross section shape on the $t$~quark mass and interactions
is computable to high precision with full control over the renormalization
scheme dependence of the top mass parameter.  In this section, we will
review the expectations for the theory and ILC measurements of the
$t$~quark threshold cross section shape. The case of the $t$~quark
threshold is not only important in its own right but also serves
 as a prototype case for other particle thresholds that might be accessible at the ILC.

\subsection{ Status of QCD theory}

The calculation of the total top pair production cross section makes use of the
 method of non-relativistic effective
theories. The $t$~quark mass parameter used in this calculation is
defined at the scale of about 10~GeV corresponding to the typical physical
separation of the $t$ and $\bar t$.  This mass parameter can be
converted to the $\msb$ mass in a controlled way. The summation of QCD
 Coulomb singularities treated by a
non-relativistic fixed-order expansion is well known up to
NNLO~\cite{Hoang:2000yr} and has recently been extended accounting
also for NNNLO corrections~\cite{Beneke:2008cr}. Large QCD velocity
logarithms have been determined using renormalization-group-improved
non-relativistic perturbation theory up to NLL order, with a partial
treatment of NNLL effects~\cite{Hoang:2001mm,Hoang:2003ns,Pineda:2006ri}.
Recently the dominant ultrasoft NNLL corrections have been
completed~\cite{Hoang:2006ht,Pineda:2011aw,Hoang:2011gy}. The accuracy
in this calculation is illustrated in Fig.~\ref{fig:tthresh-hs}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centering
%\includegraphics[width=0.7\columnwidth]{ttthresh_hs.pdf}
\includegraphics[width=0.7\columnwidth]{TotCrossCombined.pdf}
\caption{
%Accuracy on the prediction of the top pair production
%cross section at the $t\bar{t}$~threshold at the ILC as achieved by
%recent calculations of QCD corrections (NNLL). For further explanations
%see text. The figure has been taken from~\cite{bib:topws-ms}.
Bands show the theoretical QCD uncertainties of the prediction of the top pair production cross section at the $t\bar t$ threshold
   at the ILC as achieved by recent renormalization-group-improved QCD calculations~\cite{Hoang:2013uda}. Their analysis estimated
   the theoretical uncertainty in the total cross section as $d\sigma_{t\bar t}/\sigma_{t\bar t}\pm 5\%$. For further explanation see text.
}
\label{fig:tthresh-hs}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Since the $t$~quark kinetic energy is of the order of the $t$~quark width,
electro-weak effects, which also include finite-lifetime and interference
contributions, are crucial as well. This makes the cross section dependent
on the experimental prescription concerning the reconstructed final state.
%Results published in ~\cite{Beneke:2010mp,Hoang:2010gu,Penin:2011gg} put approximate NNLL order
%predictions within reach. Recently the NLO resonant calculation of $\ttbar$ in the publications cited before has been extended to NNLO  accuracy~\cite{Jantzen:2013gpa}.
Recently the NLO non-resonant calculation of $\ttbar$ production in~\cite{Beneke:2010mp,Penin:2011gg} has been extended to NNLO accuracy~\cite{Hoang:2010gu,Jantzen:2013gpa}.

Theoretical predictions for differential cross sections such as the
top momentum distribution and forward-backward
asymmetries are only known at the NNLO level
and are thus much less developed.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centering
\includegraphics[width=0.50\columnwidth]{ILC_TopThreshold.pdf}
\includegraphics[width=0.40\columnwidth]{contours-10-ilc.pdf}
\caption{Illustration of a $t$~quark threshold measurement at the ILC.
In the simulation, the $t$~quark mass has been
chosen to be 174.~GeV.  The blue lines show the effect of varying this
mass by 200~MeV. The study is based on full detector simulation and takes
initial state radiation (ISR) and beamstrahlung (BS) and other relevant
machine effects into account: (left) the simulated threshold scan.  (right)
error ellipse for the determination of $m_t$ and $\alpha_s$. The figure is
 taken from~\cite{Seidel:2013sqa}.}
\label{fig:ttthresh-scan}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection{Simulations and measurements}

The most recent experimental study has been carried out by Seidel, Simon, and Tesar, for which the
results are shown in Fig.~\ref{fig:ttthresh-scan}~\cite{Seidel:2013sqa}. That study is based on a full detector simulation
using the ILD detector. It takes the initial state radiation and beamstrahlung of the colliding beams into
account. has been carried out by Martinez and Miquel in~\cite{Martinez:2002st}.
These authors assumed a total integrated luminosity of $100$~fb$^{-1}$, distributed over 10 equidistant energy points
in a $10$~GeV range around the threshold, using the ILC and CLIC beam parameters. To treat the
strong correlation of the input theory parameters, simultaneous fits
were carried out for the $t$~quark mass, the QCD coupling and the $t$~quark width from
measurements of the total cross section that was simulated based on the code TOPPIK with NNLO corrections~\cite{Hoang:1999zc}.
The results are shown in Table~\ref{tab:ThresholdResults} and demonstrate that the statistical precision of the $t$~quark mass in this study is of the order of 30\,MeV.
The results are compatible with the results reported in~\cite{Horiguchi:2013wra}. There the width of the $t$ quark is determined to a statistical precision at the 2\% level.

\begin{table}
\centering
\begin{tabular}{|ccc|}
\hline
\multicolumn{3}{|c|}{1S top mass and $\alpha_s$ combined 2D fit}\\
\hline
Error type & ILC & CLIC \\
\hline
 $m_t$ stat. error &  27~MeV & 34~MeV\\
 $m_t$ theory syst. (1\%/3\%) &  5~MeV / 9~MeV & 5~MeV / 8~MeV\\
 $\alpha_s$ stat. error & 0.0008 & 0.0009\\
 $\alpha_s$ theory syst. (1\%/3\%) & 0.0007 / 0.0022 & 0.0008 / 0.0022 \\
\hline
\end{tabular}

\caption{Results summary for the 2D simultaneous top mass and $\alpha_s$ determination with a threshold scan at CLIC and ILC for 10 points with a total integrated luminosity of 100\,fb$^{-1}$~\cite{Seidel:2013sqa}. \label{tab:ThresholdResults}}
%Event selection and background rejection from CLIC\_ILD is used.\label{tab:ThresholdResultsILC}}
\end{table}

%\begin{table}[ht]
%\begin{footnotesize}
%\begin{minipage}[b]{0.45\linewidth}\centering
%\begin{tabular}{l|c}
%\hline
%\multicolumn{2}{c}{1S top mass and $\alpha_s$ combined 2D fit}\\
%\hline
% $m_t$ stat. error &  34~MeV\\
% $m_t$ theory syst. (1\%/3\%) &  5~MeV / 8~MeV\\
% $\alpha_s$ stat. error & 0.0009\\
% $\alpha_s$ theory syst. (1\%/3\%) & 0.0008 / 0.0022 \\
%\hline
%\end{tabular}

%\caption{Results summary for the 2D simultaneous top mass and $\alpha_s$ determination with a threshold scan at CLIC for 10 points with a total integrated luminosity of 100 fb$^{-1}$. \label{tab:ThresholdResults}}

%\end{minipage}
%\hspace{0.5cm}
%\begin{minipage}[b]{0.45\linewidth}
%\centering
%\begin{tabular}{l|c}
%\hline
%\multicolumn{2}{c}{1S top mass and $\alpha_s$ combined 2D fit}\\
%\hline
% $m_t$ stat. error &  27~MeV\\
% $m_t$ theory syst. (1\%/3\%) &  5~MeV / 9~MeV\\
% $\alpha_s$ stat. error & 0.0008\\
% $\alpha_s$ theory syst. (1\%/3\%) & 0.0007 / 0.0022 \\
%\hline
%\end{tabular}
%\caption{Results summary for the 2D simultaneous top mass and $\alpha_s$ determination with a threshold scan at ILC for 10 points with a total integrated luminosity of 100 fb$^{-1}$. \label{tab:ThresholdResultsILC}}
%Event selection and background rejection from CLIC\_ILD is used.\label{tab:ThresholdResultsILC}}

%\end{minipage}
%\end{footnotesize}
%\end{table}

As shown in an older study by Martinez and Miquel in~\cite{Martinez:2002st} even better precision on the $t$ mass may be achieved by taking
other observables into account.

%The most thorough experimental study of the $t$~quark threshold
%has been carried out by Martinez and Miquel in~\cite{Martinez:2002st}.
%These authors assumed a total integrated luminosity
% of $300$~fb$^{-1}$, distributed over 10 equidistant energy points
%in a $10$~GeV range around the threshold, using the TELSA beam
%parameters.  To treat the
%strong correlation of the input theory parameters, simultaneous fits
%were carried out for the
%$t$~quark mass, the QCD coupling and the $t$~quark width from
%measurments of
%the total cross section, the top momentum distributions and
%the forward-backward asymmetry.
%These were simulated
%based on the code TOPPIK with
%NNLO corrections~\cite{Hoang:1999zc}.

%The study obtained the
%uncertainties $\Delta m_t=19$~MeV, $\Delta \alpha_s(m_Z)=0.0012$
%and $\Delta \Gamma_t=32$~MeV, when  all observables were accounted for
%Using just the total cross section measurements, the results were
%$\Delta m_t=34$~MeV, $\Delta \alpha_s(m_Z)=0.0023$ and
%$\Delta \Gamma_t=42$~MeV.  The
%difference shows the discriminating power of additional observables of the
%threshold region.  The analysis included
%a theory uncertainty in the cross section
%codes of $3\%$, which at this time is only approached for total cross
%section computations. Although the analysis was only based on fixed
%order NNLO predictions, the quoted uncertainties should be realistic.

%The analysis in \cite{Martinez:2002st} did not yet include a complete
%study of
%experimental systematic uncertainties, including, in particular,
%uncertainties in the knowledge of the luminosity spectrum.

%This last point is addressed
%in a more recent study by Seidel, Simon, and Tesar, for which the
%results are shown in Fig.~\ref{fig:ttthresh-scan}~\cite{Seidel:2013sqa}.
%That study was carried with a full detector simulation using the
%ILD detector. It takes the
%initial state radiation and beamstrahlung of the colliding beams into
%account.  The figure underlines the high sensitivity of the threshold region
% to the actual value of the $t$~quark mass. The statistical precision
%obtained on the $t$~quark mass in this  study is of the order
%of 30\,MeV. Due to the QCD corrections relevant for a precise calculation
%of the $t$~quark mass, the threshold scan is sensitive to the
%value of $\alpha_s$. The error ellipse as obtained in a combined
%determination of $\alpha_s$ and $m_t$ is shown in the right-hand panel
%of Fig.~\ref{fig:ttthresh-scan}.

The threshold $t$~quark mass determined in the study of Seidel et al. must still be
converted to the standard $t$~quark $\msb$ mass.  The conversion
formula, to three-loop order, is given in \cite{Hoang:1999zc}.  The
conversion adds an error of about 100~MeV from truncation of the QCD
perturbation series and an error of 70~MeV for each uncertainty of
$0.001$ in the value of $\alphas$. Both sources of uncertainty should be
reduced by the time of the ILC running.  In particular, the study of
event shapes in $\ee\to q\bar q$ at the high energies available at
ILC should resolve current questions concerning the precision determination of
$\alphas$.  We recall that these estimates are the results of a
precision theory of the relation between the threshold mass and the
$t$~quark $\msb$ mass. A comparable theory does not yet exist for the
conversion of the $t$~quark mass measured in hadronic collisions to the
$\msb$ value.

The precise determination of the $t$~quark mass is likely to have
important implications for fundamental theory. A value of the top
quark mass accurate at the level that a linear collider will provide is for example a key input to models of the
vacuum stability of the universe.

%We have given one example at the end of Section~2.1.
%In principle, the contribution of the Higgs exchance potential to the
%5$t\bar t$ threshold makes it possible to measure that Higgs coupling
%to $t\bar t$.   However, the precision of this measurement is strongly
%limited by the fact that the Higgs corrections are
%suppressed by the inverse square of the Higgs mass. For a Higgs mass
%of $m_H=120$~GeV the study in~\cite{Martinez:2002st} found that
%uncertainties of at least several $10\%$ should be expected in a
%measurement of the top quark Higgs Yukawa coupling.  This coupling can
%be measured more accurately from the cross section for $\ee\to t\bar t h$,
%as is explained in Section 2.6 and 2.7 of this report.


\section{VERIFICATION} \label{ch:verification}

In any field of research verification is of the utmost importance.  The goal being to not only guarantee that the work you have done is correct, but to also know the work can be reliably extended.  In this work the verification relies heavily on the individual testing of GLIMMER-CISM and PHAML.  Once the components are sure the mapping between them can be tested, and then further research can be conducted using the combination.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{GLIMMER-CISM}\label{sec:chp5cism}

The discussion about verification for GLIMMER-CISM is really a discussion about how it implements the shallow-ice approximation (SIA), how accurate SIA is overall, compensations that exist for the SIA, the staggered grid type that GLIMMER-CISM uses, and modifications made to the standard SIA in GLIMMER-CISM

Verification issues are compounded by the fact that when working with glaciers inter-model comparisons are essential due to the complex and insolvable nature of the equations that govern the ice flow.  This means making adjustments, approximations, and compromises is a must, and each model must decide which features it deems more important for the goal at hand.  The output must then be compared to other model runs and theoretical expected results for portions of the simulation.  This means that very specific tests must be defined for models to run and compare for a small detailed result that can be theoretically justified.

\subsubsection{The Shallow-Ice Approximation}

Due to the complex nature of non-Newtonian fluids, of which ice is, ice-sheets have historically been simulated with what is known as the shallow-ice approximation (SIA).  This approximation, as its name implies, models an ice sheet as if it were shallow.  This allows the equations to be reduced down to rather simple equations in two directions that need to be solved.  The SIA has been shown to only be effective within a certain ratio of the size of the ice sheet to ice depth and is better over longer periods of time. %with a grid size of whatever based on the depth of the ice.(reference)

%grid size min/max

\subsubsection{Exact Solutions}

The SIA can not be verified by itself, but whether the code implementing it is programmed correctly can be verified-both in the uncoupled \citep{bueler:2005} and thermo-mechanically coupled cases.\citep{bueler:2007}  This method tests the equations by the method of manufactured solutions where a compensatory heating and accumulation are added to the SIA equations in order to balance them so the only variable in the simulation is the application of the equations.  This method should produce the same basic results as the EISMINT tests. This verification technique has not been fully tested with GLIMMER-CISM.  Correctness of the SIA implementation in GLIMMER-CISM is taken mainly from intermodel comparison, specifically the EISMINT experiments.

\subsubsection{EISMINT}

Several models using the SIA have been tested with the EISMINT (European Ice Sheet Modeling INiTiative) tests, which is a framework designed with very specific experiments in order to allow glacial models to compare results, find uncertainties, and enhance the understanding of the simulations.  The EISMINT tests and results are well documented and GLIMMER-CISM's results are also published according to the specifications. \citep{Huybrechts:1996}\citep{glimmerdoc}

Several other intermodel comparison tests exist for ice-sheet models which examine specific aspects of the simulation.  The EISMINT experiments examined an idealized ice-sheet over time with specific parameters, but some experiments such as MISMIP (Marine Ice Sheet Model Intercomparison Project) tests just one aspect of ice sheet dynamics over time.  MISMIP compared grounding line migration of different SIA models to each other and to analytical solutions for very specific situations. \cite{mismip4}

%GLIMMER-CISM has an excellent record with the SIA and it's results are goooooooooooooooooooooooooooood.

\subsubsection{Known Issues}

There are many issues with the SIA, and this is to be expected given the simplification of the equations.  The SIA works very well for large ice sheets, especially in the interior, over long periods of time.  The shortcoming is that it does rather poorly in understanding short-term ice-sheet physics.  Aggravating this issue is the problem of discrete finite difference methods of solving the SIA equations which can lead to many round off errors and inaccuracies that can propagate into larger issues over time.\citep{Bergetal2006b}

%check berg citing

Recently there have been many advances in compensating for some of these shortcomings.  For instance, using a special technique to calculate where the grounding line is between two nodes has been used to drastically increase the accuracy of grounding line migration. \citet{2004JF000202,Pattyn2006JGR}  This technique was further developed to calculate the flux of the ice-sheet at the grounding line and ensure that the ice margin moved approximately the same regardless of the grid resolution used for the finite difference approximation. \citep{Pollard:2009}  The hope though is that many of these types of compensations won't be needed with a better model approximation and more accurate tools.

\subsubsection{Higher Order Model}

Currently, GLIMMER-CISM has a new first-order model that is being tested and used in different simulation scenarios.  This model should fix many of the issues associated with the shallow-ice approximation.  This model should allow for much greater accuracy and more dynamic movement of the ice from ice-shelves, streams, grounding line migration, etc.  A first-order approximation should allow GLIMMER-CISM to explore many new possibilities and test newer hypotheses which were previously impossible with the SIA.  Unfortunately, some of these tests will again need to rely on inter-model comparison which will require other ice-sheet models to catch up.  These advancements come at a cost of complexity-both at a theoretical level and implementation level.

\subsubsection{ISMIP-HOM}

The ISMIP-HOM (Ice Sheet Model Intercomparison Project - Higher Order Models) tests have been completed for the first-order model of GLIMMER-CISM.  The results compare favorably to other higher-order models in the experiment with a bias towards high velocities, but always within a standard deviation of the other models.\citep{Bocek:2009}  These results are encouraging that the model serves as a good basis moving forward.

%Many interesting results could come out of coupling this model with PHAML for certain operations.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{PHAML}\label{sec:chp5phaml}

PHAML is a very sophisticated piece of software that allows for many different approaches to a problem and for specifying when a problem has been solved.  Many of these are covered in chapter \ref{ch:overview}.  Formal verification testing then becomes much more complicated since you're relying on many different libraries as well as implementing a vast array of methods for numerical approximation.

Since PHAML uses hp-adaptivity it can be much harder to analytically understand and verify, but the benefits of hp-FEM are well worth the extra cost.  Using hp-adaptive methods also causes the programming to be much more complicated and makes it more prone to errors meaning verification is even more important when using hp-FEM than simply using standard finite element methods.
\citep{Demkowicz:2007}

There are no formal verification tests published on PHAML, but PHAML has been extensively tested and the techniques it uses as well as the software implementation within PHAML are proven.  Many 2D hp-FEM tests for accuracy are proposed by the author of PHAML, and PHAML holds within the standard margin of error. \citet{mitchell:2d}

\subsubsection{Analytical Solution}

Although no extensive verification testing is being done with PHAML since that is beyond the scope of this thesis, a simple comparison and error analysis have been done on a Laplacian example where an analytical solution is known to exist.  The example is of a unit square from $(0,0)$ to $(1,1)$ where the side boundary $y=1$ is held at a value of one and the three other sides are held at zero giving Dirichlet boundary conditions.  This is shown in figure \ref{fig:laplace_square}.

\addfigurepdf{laplace_square}{0.5}{Laplace Domain: The domain and boundary conditions for the analytical Laplace solution}{The domain and boundary conditions for the analytical Laplace solution.}{fig:laplace_square}

\newpage
Through a rigorous derivation the analytical equation can be found as:

\begin{equation}
T(x,y) = \frac{2}{\pi}\sum_{n=1}\frac{1 - \cos(n \pi)}{n (e^{n\pi} - e^{-n \pi})}\sin(n \pi x)(e^{n \pi y} - e^{-n \pi y})
\label{eq:anlap}
\end{equation}

\cite{sheehan:website}

The sum has no ending point because technically the solution sums from one to infinity over every point that the solution is evaluated at.  For the example here though, once the new terms were negligible the result was used.  Only the first 200 terms were required in the approximation to give ample precision for the comparison.

This example does have one caveat of the boundary conditions.  At the points (0,1) and (1,1) the analytical solution places the point on the outside boundaries giving them values of zero rather than one, while PHAML assumes those two points to be of value one.  These two points dramatically skew the error analysis so the error with and without those two points are analyzed.

\begin{table}[ht]
\caption{PHAML vs Analytical Laplace Differences}
\centering
\begin{center}
    \begin{tabular}{ | c | c | c |}
    %\hline
    %\multicolumn{3}{|c|}{}\\
    \hline
    Value of $(0,1),(1,1)$ & Max Difference & Avg Difference\\ \hline
    0 & 0.0452 & 0.0005 \\ \hline
    1 & 1.0000 & 0.0018 \\ \hline
\end{tabular}
\end{center}
\end{table}

The two solutions are superimposed in figure \ref{fig:phamanal1}.  The only noticeable differences are along the $y=1$ boundary.  As mentioned the difference in the two points $(0,1)$ and $(1,1)$ drastically change the error since two points are off by one.  When the points are at their default value of one the error analysis (figure \ref{fig:phamanalerr1}) is dominated by those two points and it is difficult to see anything else.  With the correction of those two points though (figure \ref{fig:phamanalerr0}), the smaller differences between the two solutions can be viewed.  Even the largest differences in this case though are small.

\addfigure{phamanal1}{1.0}{PHAML/Analytical Laplace Comparison: The resulting solution of PHAML compared to the analytical result}{The resulting PHAML solution to the Laplace PDE with the analytical solution.}{fig:phamanal1}

\addfigure{phamanalerr1}{1}{Error Analysis: The difference between the PHAML and analytical solutions with $(0,1)$ and $(1,1)$ equal to one.}{Error Analysis: The difference between the PHAML and analytical solutions with $(0,1)$ and $(1,1)$ equal to one.}{fig:phamanalerr1}

\addfigure{phamanalerr0}{1}{Error Analysis: The difference between the PHAML and analytical solutions with $(0,1)$ and $(1,1)$ equal to zero}{Error Analysis: The difference between the PHAML and analytical solutions with $(0,1)$ and $(1,1)$ equal to zero.}{fig:phamanalerr0}

The level of accuracy for PHAML's solution against the analytical solution of this Laplace PDE is extremely good.  The border was intentionally held to one by the defined problem, so the small errors on the border are not an issue and expected.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage

\subsection{GLIMMER-CISM/PHAML Together}\label{sec:chp5phamlcism}

Given the acceptability of a solution from PHAML and the ice dynamics in GLIMMER-CISM, the necessary verification deals with assuring that they communicate correctly, input and output data is mapped appropriately, and that the interface is flexible to allow change in both of the applications in future versions.

\subsubsection{Data Mapping}

In order to test the communication and data mapping a simple Laplace PDE and Poisson were run with the two together.  For clarity, and for demonstrating that the outputs are identical, the GLIMMER-CISM NetCDF image and the OpenGL output from PHAML are compared for each.

The Laplacian example is shown in figure \ref{fig:QLapComp} and \ref{fig:QLapCompGL}.  For this example a perfectly round ice-sheet was used as the domain, then one quarter of the ice sheet was set to a value of one, and the rest of the domain was set to zero.  The outer boundary of the quarter was then held to one for the simulation.  The output is identical for both GLIMMER-CISM and PHAML.

\addfigure{QLapComp}{0.8}{Laplace Test: Running a basic Laplace simulation}{A Laplace run with initial conditions of 1 over a quarter and then held at one on the border.  Viewed in NetCDF.}{fig:QLapComp}

\addfigure{QLapCompGL}{0.8}{Laplace Test in GL: Running a basic Laplace simulation}{A Laplace run with initial conditions of 1 over a quarter and then held at one on the border.  Viewed in OpenGL.}{fig:QLapCompGL}

For the Poisson example a source function was set to one on the diagonal where the X and Y coordinates were equal.  The domain was the same as in the Laplacian example and was set to zero.  The source where X was equal to Y was set to one and held to one as a boundary condition.  Both the NetCDF output (\ref{fig:pois}) and the PHAML OpenGL output (\ref{fig:poisgl}) are arranged from the initial condition in the top left, the first timestep in the top right, and then the 20th and 50th timesteps on the bottom.  Again the output is identical between the two datasets, showing that the mapping between the two is working properly.

\addfigure{pois}{1.0}{Poisson Test: Results of a Poisson simulation viewed in NetCDF}{A Poisson simulation with source and boundary at x=y=1 viewed in NetCDF.  The order of steps from top left is initial condition, 1 timestep, 20 timesteps, then 50 timesteps.}{fig:pois}

\addfigure{poisgl}{1.0}{Poisson Test: Results of a Poisson simulation viewed in OpenGL}{A Poisson simulation with source and boundary at x=y=1 viewed in OpenGL.  The order of steps from top left is initial condition, 1 timestep, 20 timesteps, then 50 timesteps.}{fig:poisgl}

\subsubsection{Flexibility}

As previously mentioned the simplicity of how PHAML is called within each module adds some complexity to using the library, but direct access to all of the functionality in PHAML is available.  This makes the modules extremely flexible for any type of problem.  Also, since each module can define its own set of PHAML callbacks, there is no restriction on the problem definition.  Thus, any changes in either CISM or PHAML can be handled in the future.


%---------------------------------------------------------------------
%  Writeup of the improved search of Run II data for single top quark
%  production at DZero.
%  Started: Oct 2006
%  Authors: The Single Top Working Group
%---------------------------------------------------------------------
%

\appendix
\section*{Appendix 2 --- Lepton Isolation Probabilities}
\label{epsilons}

Normalization of the $W$+jets and multijet backgrounds to data is
performed using the matrix method as explained in
Section~\ref{matrix-method}. To determine the separate contributions
from signal-like real-lepton events and background fake-lepton events
in our data, we rely on the measurement of two probabilities,
$\varepsilon_{{\rm real-}\ell}$ and $\varepsilon_{{\rm fake-}\ell}$.

$\varepsilon_{{\rm real-}\ell}$ is the probability for a real lepton
to pass the isolation requirements. It is obtained using a $Z{\rar}
ee$ sample for the electrons and $Z {\rar} \mu\mu$ sample for the
muons. One of the leptons is ``tagged'' as \emph{tight}, then the
$\varepsilon_{{\rm real-}\ell}$ is calculated measuring the
probability of the other ``probe'' lepton to pass the
\emph{tight} selection cut.

The $\pt$ and $\eta$ dependence for the electron case is shown in
Fig.~\ref{cc-eps-signal}. The $\pt$ and jet multiplicity dependence in
the muon case is shown in Fig.~\ref{mu-eps-signal}. The average values
for $\varepsilon_{{\rm fake-}e}$ and $\varepsilon_{{\rm fake-}\mu}$ as
a function of the jet multiplicity are given in
Table~\ref{eps-signal}. The overall uncertainty for $\varepsilon_{{\rm
fake-}\ell}$ is $1\%$ statistical and $2\%$ systematic.

\vspace{-0.1in}
\begin{figure}[!h!tbp]
\includegraphics[width=0.34\textwidth]
{figures/eps_sig_el_pt.eps}
\hspace{0.5in}
\includegraphics[width=0.34\textwidth]
{figures/eps_sig_el_eta.eps}
\vspace{-0.1in}
\caption[ccepssignal]{Transverse momentum and detector
pseudorapidity dependence of $\varepsilon_{{\rm real-}e}$.}
\label{cc-eps-signal}
\end{figure}

\vspace{-0.3in}
\begin{figure}[!h!tbp]
\includegraphics[width=0.36\textwidth]
{figures/eps_sig_mu_pt.eps}
\hspace{0.5in}
\includegraphics[width=0.36\textwidth]
{figures/eps_sig_mu_njets.eps}
\vspace{-0.1in}
\caption[muepssignal]{Transverse momentum and jet
multiplicity dependence of $\varepsilon_{{\rm real-}\mu}$.}
\label{mu-eps-signal}
\end{figure}

\vspace{-0.2in}
\begin{table}[!h!tbp]
\begin{center}
\begin{minipage}{2.5 in}
\begin{ruledtabular}
\begin{tabular}{c||cc}
\multicolumn{3}{c}{\underline{Real-Lepton Probabilities}}
\vspace{0.1in}\\
No. of jets & $\varepsilon_{{\rm real-}e}$
            & $\varepsilon_{{\rm real-}\mu}$ \\
\hline
  1  &  $(87.3 \pm 2.1)\%$  &  $(99.1 \pm 2.2)\%$  \\
  2  &  $(87.4 \pm 2.1)\%$  &  $(98.9 \pm 2.2)\%$  \\
  3  &  $(87.4 \pm 2.1)\%$  &  $(98.7 \pm 2.2)\%$  \\
  4  &  $(87.5 \pm 2.1)\%$  &  $(96.1 \pm 2.1)\%$
\end{tabular}
\end{ruledtabular}
\vspace{-0.1in}
\caption[epssignal]{Average values for $\varepsilon_{{\rm real-}\ell}$
in the electron and muon samples for different jet multiplicities.}
\label{eps-signal}
\end{minipage}
\end{center}
\end{table}

\vspace{-0.2in}
$\varepsilon_{{\rm fake-}\ell}$ is the probability for a fake lepton
to pass the isolation requirements. It is obtained using a data
sample. Assuming that the low $\met$ region ($\met < 10$~GeV) is
dominated by the multijet background, and that $\varepsilon_{{\rm
fake-}\ell}$ is independent of the $\met$ cut, then $\varepsilon_{{\rm
fake-}\ell}$ is determined as the ratio of \emph{tight} over
\emph{loose} events in the low $\met$ region.
We studied $\varepsilon_{{\rm fake-}e}$ as a function of several
parameters, finding no dependence on the lepton $\pt$ or lepton
$\eta$, but dependence on the trigger version (because the isolation
requirement in the trigger changed) and jet multiplicity. For
$\varepsilon_{{\rm fake-}\mu}$, we performed a similar set of studies
and found no need to parametrize as a function of the trigger version
as the muon isolation in the trigger did not change during this
period. However, we did paramterize as a function of the muon
pseudorapidity where there is a small dependence. Although there is
also dependence on muon transverse momentum, parametrizing in this as
well does not change the overall result, so we have not done this.
Figure~\ref{fig:eps-qcd-e} show the calculation of $\varepsilon_{{\rm
fake-}e}$ from the ratio of tight to loose events as a function of
{\met} for each jet multiplicity, with all trigger versions
combined. Figure~\ref{fig:eps-qcd-mu} shows $\varepsilon_{{\rm
fake-}\mu}$ as a function of the muon pseudorapidity. A summary of the
results is presented in Table~\ref{eps-qcd}.

\begin{figure}[!h!tbp]
\includegraphics[width=0.98\textwidth]
{figures/eps_qcd_el_alltrigs_met}
\vspace{-0.1in}
\caption[figepsqcde]{$\varepsilon_{{\rm fake}e}$ calculated in
the $0 < {\met} < 10$ GeV region of the distributions of tight/loose
events, with all trigger versions combined.}
\label{fig:eps-qcd-e}
\end{figure}

\begin{figure}[!h!tbp]
\includegraphics[width=0.98\textwidth]
{figures/eps_qcd_mu_eta}
\vspace{-0.1in}
\caption[figepsqcdmu]{$\varepsilon_{{\rm fake}\mu}$ as a function of
the muon pseudorapidity.}
\label{fig:eps-qcd-mu}
\end{figure}

\begin{table}[!h!tbp]
\begin{center}
\begin{minipage}{6.5 in}
\begin{ruledtabular}
\begin{tabular}{c||ccccc|c}
\multicolumn{7}{c}{\hspace{0.8in}\underline{Fake-Lepton Probabilities}}
\vspace{0.05in}\\
& \multicolumn{5}{c|}{ $\varepsilon_{{\rm fake-}e}$, Trigger Version} & \\
No. of jets & v8--v11   &     v12     &     v13a    &     v13b    &     v14     &     $\varepsilon_{{\rm fake-}\mu}$ \\
\hline
 1 & $(11.2 \pm 0.5)\%$ & $(17.9 \pm 0.6)\%$ & $(18.7 \pm 1.2)\%$ & $(19.1 \pm 0.6)\%$ & $(18.5 \pm 0.6)\%$ & $40.8\%$ \\
 2 & $(12.8 \pm 1.0)\%$ & $(19.2 \pm 1.0)\%$ & $(18.8 \pm 2.2)\%$ & $(19.4 \pm 1.1)\%$ & $(22.0 \pm 1.2)\%$ & $35.8\%$ \\
 3 & $(13.6 \pm 1.5)\%$ & $(19.5 \pm 1.6)\%$ & $(19.8 \pm 3.4)\%$ & $(19.2 \pm 1.6)\%$ & $(19.4 \pm 1.7)\%$ & $34.2\%$ \\
 4 & $(10.0 \pm 2.8)\%$ & $(15.5 \pm 2.9)\%$ & $(20.9 \pm 8.6)\%$ & $(17.7 \pm 3.3)\%$ & $(20.7 \pm 3.7)\%$ & $30.9\%$
\end{tabular}
\end{ruledtabular}
\vspace{-0.1in}
\caption[epsqcd]{$\varepsilon_{{\rm fake-}e}$ as a function of
the trigger version and jet multiplicity, and $\varepsilon_{{\rm
fake-}\mu}$ averaged over $\eta$.}
\label{eps-qcd}
\end{minipage}
\end{center}
\end{table}

\clearpage


A measurement of the cross section times branching fration of \ttH (\Hgg)~production is performed by defining regions of the data which are highly enriched in \ttH events. These regions, called ``signal regions'', are constructed through a set of requirements placed on all candidate events.
The requirements consist of two components:
(1) a ``loose preselection'', which selects events with at least some of the expected decay products of the \ttH system and
(2) a selection based on the output of a binary classification algorithm (called ``BDT-bkg''), trained to separate \ttH from the SM background processes. The loose preselection aims to maintain a high signal efficiency and defines the phase space in which BDT-bkg is trained.
The BDT-bkg algorithm is trained on MC simulation of signal and background, as well as a data-driven description of some backgrounds. After training the BDT-bkg algorithm, signal regions are constructed by placing requirements on the output of BDT-bkg (on top of the preselection requirements).
Within these signal regions, signal and background models are constructed and a measurement of the \ttH cross section is calculated by performing a simultaneous fit to events in all of the signal regions.
%Although the loose preselection maintains an efficiency of X\% (FIXME) on \ttH events, the SM background is still far too large to perform a precise measurement of the \ttH cross section. The output of BDT-bkg is then used to construct signal regions.
%
\subsection{The \Hgg Decay Mode}
The analysis targets \ttH events in which the Higgs boson decays into two photons (\Hgg). As the photon is a massless particle, it does not couple directly to the Higgs boson.
Instead, the dominant process through which the Higgs boson decays to two photons involves a top quark loop, as shown in Figure~\ref{fig:hgg_feynman}.
\begin{figure}[h!]
    \centering
    \includegraphics[width=0.6\textwidth]{figures/feynman_diagrams/hgg}
    \caption{Feynman diagram of \Hgg decay proceeding via a top quark loop.}
    \label{fig:hgg_feynman}
\end{figure}
The \Hgg branching ratio is quite small ($\approx 0.2\%$) in comparison to other commonly studied decay modes, but the diphoton channel presents several key advantages.

First, the CMS ECAL provides excellent energy resolution ($\sigma_E/E$) for reconstructed photons, which ranges from 1-5\% \cite{Chatrchyan:2013dga}.
In general, photons with smaller absolute values of pseudorapidity and higher values of the shower shape variable $R_9$ (defined in Section \ref{sec:evt_photon}) are reconstructed with better energy resolution.
The resulting resolution of the invariant mass of diphoton pairs then ranges from 1-2\% for events considered in this analysis.
%Exact values for the mass resolution in each signal region are given in Section \ref{sec:sig_bkg_models}.
The mass resolution for other Higgs decay modes is typically much worse.
The CMS observation of the $\text{H} \to \text{b}\bar{\text{b}}$ decay mode, for example, achieved a mass resolution of 10-13\% \cite{Hbb_obs}.
The excellent mass resolution of the diphoton channel contributes to its competitive sensitivity -- the SM background follows a steeply falling distribution as a function of increasing diphoton invariant mass, while \Hgg events are clustered around \mH with a resolution of 1-2\%.
The narrow peak around \mH allows for greater discrimination against the SM background processes.

The second advantage of the diphoton decay channel is the relatively small SM background.
At the LHC, final states with photons or leptons are significantly rarer than final states with hadrons.
Each photon or lepton in a final state introduces another factor of the fine structure constant, $\alpha \approx 1/137$, in the cross section times branching ratio for a given process.
A crude, back-of-the-envelope estimate suggests that final states with $N$ leptons plus photons have characteristic cross sections times branching ratios a factor of $\alpha^{-N}$ ($\alpha \approx 137$) times smaller than characteristic cross sections times branching ratios for all-hadronic final states.

Finally, the diphoton decay channel has relatively low systematic uncertainties in comparison to other Higgs decay modes.
This is due in part to the fact that photons are well-measured by the CMS ECAl and in part to the fact that the background is estimated directly from data (described in Sec.~\ref{sec:tth_bkg_models}).
%Recent \ttH cross section measurements in multilepton and $\text{b}\bar{\text{b}}$ final states reported total systematic uncertainties of FIXME and CITE, while this result reports a systematic uncertainty of FIXME.
The fact that the uncertainty on measurements in the \Hgg channel are dominated by statistical uncertainties casts it as the ``golden channel'' for future Higgs studies in Run 3 of the LHC; the statistical uncertainty on measurements scales roughly with the inverse square root of the luminosity, while the systematic uncertainty remains roughly constant as a function of luminosity.

\subsection{Estimates of Expected Sensitvity} \label{sec:tth_za}
In developing the \ttH analysis, the following question frequently arises: will strategy A or strategy B give a better result?
``Strategy'' may refer to a machine learning algorithm, a description of the SM background, etc.
``Better'' is a somewhat subjective matter, but the primary measurement of the \ttH analysis is that of $\mu_{\ttH}$, defined as the ratio of the observed cross section times branching fraction of \ttH (\Hgg) and the predicted cross section times branching fraction:
\begin{equation}
    \mu_{\ttH} = \frac{(\sigma_{\ttH} \times \mathcal B (\Hgg))_{\text{obs}}}{(\sigma_{\ttH} \times \mathcal B (\Hgg))_{\text{SM}}}.
\end{equation}
Therefore, ``better'' is taken to mean ``giving a smaller expected uncertainty on the measurement of $\mu_{\ttH}$''.
The superlative is framed in terms of the expected uncertainty, rather than the observed uncertainty, as the analysis is developed in a blinded fashion to avoid introducing bias.
The full calculation of $\mu_{\ttH}$ involves extensive computing (and human) time, so a simplified measure of the expected uncertainty is used.
For these purposes, we use $Z_A$~\cite{cowan_za}, defined as
\begin{equation}
    Z_A(s,b) = \sqrt{2 \bigg((s+b)\ln(1 + \frac{s}{b})-s\bigg)},
\end{equation}
where $s$ is the number of signal events and $b$ is the number of background events.
In the limit $s<<b$, the Taylor expansion of $Z_A$ gives the familiar $s/\sqrt{b}$:
\begin{equation}
    Z_A(s,b) = \frac{s}{\sqrt{b}}\bigg[1 + \mathcal O \bigg(\frac{s}{b}\bigg) \Bigg].
\end{equation}
The signal yield $s$ is estimated by fitting a Gaussian function to the \mgg distribution of signal MC events and integrating the fitted function over the signal mass window.
The background yield $b$ is estimated by counting the yield of all events in the \mgg sidebands (100 GeV $< \mgg < 120$ GeV, 130 GeV $< \mgg < 180$ GeV) and scaling to the size of the signal mass window.
The signal mass window is taken to be $\mH \pm 1.645 \times \sigma_{\text{eff}}$, with $\sigma_{\text{eff}}$ taken from the fitted Gaussian.
It is not necessary to employ more sophisticated estimates of the signal and background yields (as is done in Sec.~\ref{sec:tth_sig_bkg_models}) because $Z_A$ is only used in comparing the performance of two strategies; the absolute value of $Z_A$ is not relevant.
When quoting an improvement obtained by using a specific method, the value is taken as the percentage difference between the maximum $Z_A$ values obtained by the two methods.


\title{Black Holes in Galaxies}

\section{Basics}

Astronomical evidence now suggests that every galaxy in the Universe
with a mass $M\sim M_\ast$ hosts a supermassive ($\sim
10^7$--$10^{10}$ $M_\odot$) black hole at its center. While early
evidence for the existence of such black holes originally came in the
1960s through their manifestation as extremely luminous quasars,
nowadays there is direct evidence for extremely massive objects at the
centers of galaxies.  There are a number of ways to estimate the
masses of these black holes, using the orbital velocities of stars and
gas surrounding the black hole as a function of distance. These masses
correlate well with global properties of the galaxies, such as the
total luminosity or velocity dispersion of the galaxies; apparently
these correlations are especially good with relation to the properties
of the galactic bulges.

\subsection{Sphere of influence}

Supermassive black hole masses are measured dynamically. However,
these black holes are surrounded by the host galaxy, and therefore we
need to define their {\it sphere of influence} within which the black
hole dominates the dynamics. This region extends to the radius that
the circular velocity around the black hole is comparable to the
velocity dispersion of the galaxy as a whole:
\begin{equation}
r_{\rm infl} = \frac{GM_{\rm BH}}{\sigma^2}
\end{equation}

\subsection{The Milky Way's Black Hole}

At the center of the Milky Way exists a dense stellar cluster as well
as neutral, molecular and hot X-ray-emitting gas. The gas emits radio
continuum emission, and there exists a very compact radio source known
as Sgr A$\ast$. VLBI observations have constrained its size to be
roughly a few light-minutes.  Sgr A$\ast$ is the location of a massive
($\sim 4\times 10^6$ $M_\odot$) black hole.  The study of this black
hole and its environment is reviewed by \citet{genzel10a}.

The first spectroscopic observations of individual stars in the
central region were in the 1990s, with near-infrared speckle imaging
and (later) adaptive optics based imaging and spectroscopy that
definitively characterized the central black hole through study of
individual orbits.  The population of stars within 1 arcsec (or about
0.04 pc) consists primarily of a cluster of old stars ($>1$ Gyr) with
apparently random orbits, and a much smaller group of young massive
stars that appear to be mostly in one disk, with a second distinct
disk possibly existing. The orbital times are 10s of years.

\subsection{Stellar dynamics signatures}

For other galaxies the observations are less detailed. High resolution
imaging and spectroscopy from space have allowed central stellar
velocity dispersion and rotation maps of galaxies within a fraction of
an arcsec (10s of pc for galaxies within 10 Mpc or so). These
observations, showing central spikes in velocity dispersion point to
the common existence of central point masses that we associate with
black holes (e.g. \citealt{kormendy96b}).

\subsection{Gas dynamic signatures}

Ionized gas dynamics near the centers of galaxies also reveal rapid
rotation near the galactic centers for some of these galaxies. For
example, M87 shows these signatures.

In some cases, the gas near the centers of the galaxies are {\it
masers} (the microwave versions of {\it lasers}). Continuum radio
emission from the central AGN provides the pumping mechanism. In at
least one case (NGC 4258), the masing gas is in a convenient disk-like
geometry that allows its angular and Doppler rotational velocity to be
measured, allowing measurement of the mass and distance to the black
hole.

\subsection{Reverberation mapping}

Reverberation mapping provides a third measurement of black hole
mass. In this case we measure the continuum and line emission of
quasars over time. The continuum varies over week-to-month long time
scales. The broad line emission is powered by the continuum but with a
delay of months (depending on the line, since different ionization
stages have different ionization energies) because of the size of the
broad line emitting region. Measurement of the delay determines the
size of the region emitting for any given line, and the Doppler width
of the line then allows the use of the virial theorem to determine the
mass. A dimensionless factor associated with the geometry still must
be determined.

\subsection{$M$--$\sigma$ relation}

Measurements of the black hole masses can be correlated with the
galaxy stellar mass and other properties. $M_{\rm BH}$ and the stellar
mass are generally correlated, with a fair amount of scatter and a
typical black hole to stellar mass ratio of a few tenths of a
percent. A smaller scatter is obtained between $M_{\rm BH}$ as
measured by central stellar dynamics and the bulge velocity dispersion
(the center of the galaxy, but outside the black holes sphere of
influence; \citealt{ferrarese00a, tremaine02a}). There is some
evidence that maser-based black hole mass measurements, which can
probe smaller spheres of influence, find a smaller mass population of
black holes; this finding raises the possibility that some of the
correlation with velocity dispersion is a selection effect
(\citealt{greene16a}).

\section{Commentary}

The significance of the $M$-$\sigma$ relation is unclear. It is not
particularly surprising that the black hole mass correlates with the
galaxy mass, because with with its central location the black hole can
grow as a result of the accretion and merging processes that also
drive the galaxy growth. However, the specific scaling of the black
hole mass must tell us something about how these processes are
linked. Furthermore, the improved correlation with bulge properties
may indicate that the black hole growth is related mostly to the
processes that grow the bulge specifically.

\section{Key References}

\begin{itemize}
  \item {\it Coevolution (Or Not) of Supermassive Black Holes and Host
  Galaxies, \citet{kormendy13a}}
\end{itemize}

\citet{gunn06a}

\section{Order-of-magnitude Exercises}

\begin{enumerate}
\item Calculate the sizes of the spheres of influence for
\item Ionization energies of BLR lines
\end{enumerate}

\section{Analytic Exercises}

\begin{enumerate}
\item Sphere of influence
\end{enumerate}

\section{Numerics and Data Exercises}

\begin{enumerate}
\item Effect of resolution on central velocity dispersion measurements
\end{enumerate}

\bibliographystyle{apj}
\bibliography{exex}


\subsection{SecretBit}
This is the same as the $\sbit$ in the underlying SCALE virtual machine.

\subsubsection{Conversion of Data}
\func{SecretBit::from(a)}
This takes an integer value $a$ and loads it into a value
of type $\SecretBit$.
\begin{lstlisting}
    let sa = SecretBit::from(true);
\end{lstlisting}
You can also load a $\SecretModp$ value into the secret bit,
although it is up to the programmer to ensure that
the value really is a bit. If it is not a bit, then some
information can leak.
\begin{lstlisting}
    let cb = SecretModp::from(0);
    let sa = SecretBit::from(sa);
\end{lstlisting}

\subsubsection{Memory Access}
There is no $\SecretBit$ memory recall!

\subsubsection{Testing Data}
\func{x.test()}
In the SCALE target this takes the $\SecretBit$ value $x$,
applies a reveal operation to it, and then writes the
resulting $\regint$ value into memory.

In the Test target this takes a value stored in the $\regint$
memory saved on the last SCALE invocation, and compares it to
$x$. If the two values are the same it prints the value, and the
line number in the rust file where this was executed.
Otherwise it aborts.

\func{x.test_value(y)}
Test whether the value held in $x$ is the same as the value held in $y$.
This is the same as \verb|x.test()| except the value is compared to
$y$ and not the value emulated in the test environment.


\section{Power and Weight Budget}
\label{sec:PW_Budget}

%We will copy the thing from the provisional application and update it with new values....
%\begin{table}[H}
%\centering
%\label{budget}
%	\begin{tabular}{|c|c|c|c|c|c|}\hline
%	\caption{Power and weight budget for SORA 2.0}
%	Component & Voltage (VDC) & Peak Current (mA) & Duty Cycle (\%) & Power (mW) & Weight (g)\\ \hline
%	30 to 12 V DC/DC Converter & 30 &  & 100 & NA & NA \\ \hline
%	Polulu Driver & 30 &  & 100 & NA & NA \\ \hline
%	Polulu Driver & 30 &  & 100 & NA & NA \\ \hline
%	RESU & 12 & 65 & 100 & 780 & 9 \\ \hline
%	Pump 1 Heater & 12 &  & 10 & NA & NA \\ \hline
%	Pump 2 Heater & 12 &  & 10 & NA & NA \\ \hline
%	Pump 1 & 24 & 270 & 75 & 6480 & 1769.01 \\ \hline
%	Pump 2 & 24 & 270 & 75 & 6480 & 1769.01 \\ \hline
%	Solenoid 1 & 12 & 241.67 & 50 & 2900.04 & 5 \\ \hline
%	Solenoid 2 & 12 & 241.67 & 50 & 2900.04 & 5 \\ \hline
%	Servo Motor & 12 & 7 & 0.1 & 84 & 10 \\ \hline
%	Temperature Sensor 1 & 5 & 0.09 & 100 & 0.45 & 0.02 \\ \hline
%	Temperature Sensor 2 & 5 & 0.09 & 100 & 0.45 & 0.02 \\ \hline
%	Temperature Sensor 3 & 5 & 0.09 & 100 & 0.45 & 0.02 \\ \hline
%	Temperature Sensor 4 & 5 & 0.09 & 100 & 0.45 & 0.02 \\ \hline
%	Temperature Sensor 5 & 5 & 0.09 & 100 & 0.45 & 0.02 \\ \hline
%	Temperature Sensor 6 & 5 & 0.09 & 100 & 0.45 & 0.02 \\ \hline
%	Temperature Sensor 7 & 5 & 0.09 & 100 & 0.45 & 0.02 \\ \hline
%	Pressure \& Altitude Sensor 1 & 3.3 & 1.4 & 100 & 4.62 & 0.005 \\ \hline
%	Real Time Clock (RTC) & 3.3 & 0 & 100 & 0.00198 & 0.005 \\ \hline
%	GPS & 3.3 & 41 & 100 & 135.3 & 0.005 \\ \hline
%	Real Time Clock (RTC) & 3.3 & 0 & 100 & 0.00198 & 0.005 \\ \hline
%	Humidity Sensor & 3.3 & 0.5 & 100 & 1.65 & 0.0025 \\ \hline
%	BNO Multisensor & 3.3 & 0.1 & 100 & 0.33 & 0.005 \\ \hline
%	Total &  & 1138.97 &  & 22839.53 & 3992.24 \\ \hline
%
%	\end{tabular}
%\end{table}

In order to stay within the power constraints, a robust power supply will need to  handle all the components of the payload.  The power supply we will be using is the same PPM-DC-ATX-P by WinSystems INC that we used on our first flight~\cite{SORA}.  During our last flight it operated flawlessly and powered our payload throughout the whole mission.  It offers the desired number of \SI{+5}{\volt} and \SI{+12}{\volt} outputs needed to power the payload's electronics.  This power supply effectively takes \SI{+30}{\volt} and steps it down to two \SI{+12}{\volt} and two \SI{+5 }{\volt} outputs.  One of the \SI{+12}{\volt} outputs goes to the Arduino since it can step down to the appropriate voltages internally while the other goes to a PWM motor for the solenoid.  One of the \SI{+5 }{\volt} outputs powers two analog sensors that will be sent to HASP through the EDAC connection (more on that in the next sections).  The Radiation subsystem will be powered by batteries for the duration of the flight (see Appendix A).  The power supply also has four ground outputs that will be used by each respective component.
\begin{table}[H]
\centering
\caption{Power and weight budget for SORA 2.0}
\label{tab:budget}
\bigskip
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multicolumn{1}{|c|}{\bfseries Component} & \multicolumn{1}{c|}{\bfseries Voltage (VDC)} &  \multicolumn{1}{c|}{\bfseries Current (mA)} & \multicolumn{1}{c|}{\bfseries Duty Cycle (\%)} & \multicolumn{1}{c|}{\bfseries Power (mW)} & \multicolumn{1}{c|}{\bfseries Weight (g)} \\
\hline
    30 to 12 V DC/DC Converter & 30 & 1500 & 100 & 45000 & 10 \\ \hline
	%Polulu Driver & 30 & 1000 & CAL & CAL & 1 \\ \hline
	%Polulu Driver & 30 & 1000 & CAL & CAL & 1 \\ \hline
	RESU and MiniPIX~\ref{tab:Sensors} & 12 & 290 & 100 & 3480 & 15 \\ \hline
	%Pump 1 Heater with driver & 12 & 180 & 40 & 2160 & 2 \\ \hline
	%Pump 2 Heater with driver & 12 & 180 & 40 & 2160 & 2 \\ \hline
	Pump 1 w/ Solenoid & 24 & 670 & 80 & 16080 & 1800 \\ \hline
	Pump 2 w/ Solenoid & 24 & 670 & 80 & 16080 & 1800 \\ \hline
	%Solenoid 1 & 12 & 241.67 & 50 & 2900.04 & 5 \\ \hline
	%Solenoid 2 & 12 & 241.67 & 50 & 2900.04 & 5 \\ \hline
	%Servo Motor & 12 & 7 & 0.1 & 84 & 10 \\ \hline
	%Temperature Sensor 1 & 5 & 0.09 & 100 & 0.45 & 0.02 \\ \hline
	%Temperature Sensor 2 & 5 & 0.09 & 100 & 0.45 & 0.02 \\ \hline
	%Temperature Sensor 3 & 5 & 0.09 & 100 & 0.45 & 0.02 \\ \hline
	%MiniPIX & 5 & 0.09 & 100 & 0.45 & 0.02 \\ \hline
	%Pressure \& Altitude Sensor 1 & 3.3 & 1.4 & 100 & 4.62 & 0.005 \\ \hline
	%Real Time Clock (RTC) & 3.3 & 0 & 100 & 0.00198 & 0.005 \\ \hline
	%GPS & 3.3 & 41 & 100 & 135.3 & 0.005 \\ \hline
	%Real Time Clock (RTC) & 3.3 & 0 & 100 & 0.00198 & 0.005 \\ \hline
	%Humidity Sensor & 3.3 & 0.5 & 100 & 1.65 & 0.0025 \\ \hline
	%BNO 6055 & 3.3 & 0.1 & 100 & 0.33 & 0.005 \\ \hline
	Clean Box & N/A & N/A & N/A & N/A & 10000 \\ \hline
	Structure w/ bolts & N/A & N/A & N/A & N/A & 5000 \\ \hline
	Total & 30 & 1.74 (peak) & 100 & 45000 & 18764 \\ \hline
\end{tabular}
\medskip
\end{table}

% \subsubsection{Powering It All Up}


\section{Appendix}

\subsection{Experimental details}
\label{ch:app-A}

\begin{table}[hbt!]
	\caption{Optimal hyperparameters of the conventional machine learning methods}
	\label{Tab:hyperparameters}
	\begin{tabular}{|p{0.12\textwidth}|p{0.32\textwidth}|p{0.32\textwidth}|p{0.32\textwidth}|}
		\hline
		\textbf{classifier} & \textbf{unbalanced} & \textbf{undersampled} & \textbf{oversampled} \\ \hline
		Decision Tree       & max\_leaf\_nodes=15, min\_samples\_leaf=10 & class\_weight='balanced', criterion='entropy', max\_depth=10, max\_leaf\_nodes=15, min\_samples\_split=40 & class\_weight='balanced', criterion='entropy' \\ \hline
		Random Forest       & criterion='entropy', max\_depth=10, max\_features='log2' & max\_depth=10, max\_features='sqrt' & criterion='entropy', max\_features='sqrt' \\ \hline
		SVM                 & kernel='linear' & kernel='linear' & all default parameters \\ \hline
		Logistic Regression & C=1.2, solver='lbfgs' & C=1.2, solver='saga' & C=1.2, solver='lbfgs' \\ \hline
	\end{tabular}
\end{table}


\chapter{Objects}
\label{Chp:Objects}

The LAGraph library depends on a number of objects to represent
graphs, vectors, and other types associated with graph algorithms in LAGraph.
Other objects are not directly associated with the input and output variables,
but instead modify the behavior of the LAGraph functions.  These are
related to the descriptors in the GraphBLAS.

In this chapter, we need to describe all of the objects a user of LAGraph
needs to understand.  This is also where we describe the types of LAGraph
objects and any constraints on those types.