%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%              tW                     %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Single-top \texorpdfstring{$tW$}{tW} associated production}
\label{subsec:tW}

This section describes the MC samples used for the modelling of single-top $tW$ associated production.
Section~\ref{subsubsec:tW_PP8} describes the \POWPY[8] samples -- both for the diagram removal (DR) set-ups,
which are used for the nominal prediction as well as uncertainties due to additional radiation and PDFs,
and for the diagram subtraction (DS) set-ups, which are used for the uncertainty due to the treatment
of the overlap with \ttbar production.
Section~\ref{subsubsec:tW_PH7} describes the \POWHER[7] samples used for the uncertainty
due to parton showering and hadronisation modelling, and Section~\ref{subsubsec:tW_aMCP8} describes the
\MGNLOPY[8] samples used for the uncertainty due to the choice of matching scheme.

The reference cross-section values are extracted from Ref.~\cite{LHCTopWGsgtopXsec}.

\subsubsection[Powheg+Pythia8]{\POWPY[8]}
\label{subsubsec:tW_PP8}

\paragraph{Samples}
%\label{par:tW_PP8_samples}

Table~\ref{tab:tW_PP8} gives the DSIDs of the $tW$ \POWPY[8] samples, for both the DR and DS schemes.
Single-top and single-anti-top ($tW^-$ and $\bar{t}W^+$) events were generated in different samples.
The dileptonic samples overlap with the inclusive ones.
\begin{table}[htbp]
\begin{center}
\caption{Single-top $tW$ associated production samples produced with \POWPY[8].}
\label{tab:tW_PP8}
\begin{tabular}{ l | l }
\hline
DSID & Description \\
\hline
410646 & $tW^-$ (DR) inclusive \\
410647 & $\bar{t}W^+$ (DR) inclusive \\
410648 & $tW^-$ (DR) dileptonic \\
410649 & $\bar{t}W^+$ (DR) dileptonic \\
\hline
410654 & $tW^-$ (DS) inclusive \\
410655 & $\bar{t}W^+$ (DS) inclusive \\
410656 & $tW^-$ (DS) dileptonic \\
410657 & $\bar{t}W^+$ (DS) dileptonic \\
\hline
\end{tabular}
\end{center}
\end{table}

\paragraph{Short description:}

The associated production of top quarks with $W$ bosons ($tW$) was
modelled by the
\POWHEGBOX[v2]~\cite{Re:2010bp,Nason:2004rx,Frixione:2007vw,Alioli:2010xd}
generator at NLO in QCD using the five-flavour scheme and the
\NNPDF[3.0nlo] set of PDFs~\cite{Ball:2014uwa}.
The diagram removal scheme~\cite{Frixione:2008yi} was used to
remove interference and overlap with \ttbar production.
The related uncertainty was estimated by comparison with an alternative sample
generated using the diagram subtraction scheme~\cite{Frixione:2008yi,ATL-PHYS-PUB-2016-020}.\footnote{Analyses which do not use this approach
should obviously not use this sentence in their description.}
The events were interfaced to \PYTHIA[8.230]~\cite{Sjostrand:2014zea} using the A14
tune~\cite{ATL-PHYS-PUB-2014-021} and the \NNPDF[2.3lo] set of
PDFs~\cite{Ball:2012cx}.

\input{top_isrfsr_powpy_short}

\paragraph{Long description:}

Single-top $tW$ associated production was modelled using the
\POWHEGBOX[v2]~\cite{Re:2010bp,Nason:2004rx,Frixione:2007vw,Alioli:2010xd}
generator, which provided matrix elements at next-to-leading
order~(NLO) in the strong coupling constant \alphas\ in the five-flavour
scheme with the \NNPDF[3.0nlo]~\cite{Ball:2014uwa} parton
distribution function~(PDF) set.  The functional form of the
renormalisation and factorisation scales was set to the default scale,
which is equal to the top-quark mass ($\mtop = 172.5\,\GeV$).
The diagram removal scheme~\cite{Frixione:2008yi} was employed to handle the interference
with \ttbar production~\cite{ATL-PHYS-PUB-2016-020}.  The events were
interfaced with \PYTHIA[8.230]~\cite{Sjostrand:2014zea} using the A14
tune~\cite{ATL-PHYS-PUB-2014-021} and the \NNPDF[2.3lo] PDF set.  The
decays of bottom and charm hadrons were simulated using the
\EVTGEN[1.6.0] program~\cite{Lange:2001uf}.

The inclusive cross-section was corrected to the theory prediction
calculated at NLO in QCD with NNLL soft-gluon
corrections~\cite{Kidonakis:2010ux,Kidonakis:2013zqa}.  For proton--proton
collisions at a centre-of-mass energy of \rts~=~\SI{13}{\TeV}, this
cross-section corresponds to $\sigma(tW)_\text{NLO+NNLL}= 71.7\pm3.8$\,pb,
using a top-quark mass of $\mtop = 172.5\,\GeV$.  The uncertainty in
the cross-section due to the PDF was calculated using the \MSTW[nnlo] 90\%
CL~\cite{Martin:2009iq,Martin:2009bu} PDF set, and was added in
quadrature to the effect of the scale uncertainty.

\input{top_isrfsr_powpy_long}

The nominal \POWPY[8] sample was compared with an alternative
sample generated using the diagram subtraction
scheme~\cite{Frixione:2008yi,ATL-PHYS-PUB-2016-020} to estimate the
uncertainty arising from the interference with \ttbar production.

To evaluate the PDF uncertainties for the nominal PDF, the 100
variations for \NNPDF[3.0nlo] were taken into account.
In addition, the central value of this PDF was compared with the
central values of the \CT[14nnlo]~\cite{Dulat:2015mca} and
\MMHT[nnlo]~\cite{Harland-Lang:2014zoa} PDF sets.

\subsubsection[Powheg+Herwig7]{\POWHER[7]}
\label{subsubsec:tW_PH7}

\paragraph{Samples}
%\label{par:tW_PH7_samples}

Table~\ref{tab:tW_PH7} gives the DSIDs of the $tW$ \POWHER[7] DR samples.
Single-top and single-anti-top ($tW^-$ and $\bar{t}W^+$) events were generated in different samples.
The dileptonic samples overlap with the inclusive ones.
\begin{table}[htbp]
\begin{center}
\caption{Single-top $tW$ associated production samples produced with \POWHER[7].}
\label{tab:tW_PH7}
\begin{tabular}{ l | l }
\hline
DSID & Description \\
\hline
411036 & $tW^-$ (DR) inclusive \\
411037 & $\bar{t}W^+$ (DR) inclusive \\
411038 & $tW^-$ (DR) dileptonic \\
411039 & $\bar{t}W^+$ (DR) dileptonic \\
\hline
\end{tabular}
\end{center}
\end{table}

\paragraph{Short description:}

The uncertainty due to the parton shower and hadronisation model was
evaluated by comparing the nominal sample of events with a sample where
events generated with the
\POWHEGBOX[v2]~\cite{Re:2010bp,Nason:2004rx,Frixione:2007vw,Alioli:2010xd}
generator were interfaced to
\HERWIG[7.04]~\cite{Bahr:2008pv,Bellm:2015jjp}, using the H7UE set
of tuned parameters~\cite{Bellm:2015jjp} and the \MMHT[lo] PDF set~\cite{Harland-Lang:2014zoa}.

\paragraph{Long description:}

The impact of using a different parton shower and hadronisation model was evaluated
by comparing the nominal $tW$ sample with another sample produced with the
\POWHEGBOX[v2]~\cite{Re:2010bp,Nason:2004rx,Frixione:2007vw,Alioli:2010xd}
generator but interfaced with \HERWIG[7.04]~\cite{Bahr:2008pv,Bellm:2015jjp},
using the H7UE set of tuned parameters~\cite{Bellm:2015jjp} and the
\MMHT[lo] PDF set \cite{Harland-Lang:2014zoa}.
\POWHEGBOX provided matrix elements at next-to-leading order~(NLO) in the
strong coupling constant \alphas in the five-flavour scheme with the
\NNPDF[3.0nlo]~\cite{Ball:2014uwa} parton distribution function~(PDF).
The functional form of the renormalisation and factorisation scales was set to
the default scale, which is equal to the top-quark mass.  The diagram removal
scheme~\cite{Frixione:2008yi} was employed to handle the interference
with \ttbar production~\cite{ATL-PHYS-PUB-2016-020}.  The decays of bottom
and charm hadrons are simulated using the \EVTGEN[1.6.0]
program~\cite{Lange:2001uf}.

\subsubsection[MadGraph5\_aMC@NLO+Pythia8]{\MGNLOPY[8]}
\label{subsubsec:tW_aMCP8}

\paragraph{Samples}
%\label{par:tW_aMCP8_samples}

Table~\ref{tab:tW_aMCP8} gives the DSIDs of the $tW$ \MGNLOPY[8] samples.
The dileptonic sample overlaps with the inclusive one.
\begin{table}[htbp]
\begin{center}
\caption{Single-top $tW$ associated production samples produced with \MGNLOPY[8].}
\label{tab:tW_aMCP8}
\begin{tabular}{ l | l }
\hline
DSID & Description \\
\hline
412002 & $tW$ inclusive \\
412003 & $tW$ dileptonic \\
\hline
\end{tabular}
\end{center}
\end{table}

\paragraph{Short description:}

To assess the uncertainty in the matching of NLO matrix elements to the
parton shower, the nominal $tW$ sample was compared with a sample generated
with the \MGNLO[2.6.2] generator at NLO in QCD using the five-flavour
scheme and the \NNPDF[2.3nlo]~\cite{Ball:2014uwa} PDF set. The events were
interfaced with \PYTHIA[8.230]~\cite{Sjostrand:2014zea}, using the A14
set of tuned parameters~\cite{ATL-PHYS-PUB-2014-021} and the \NNPDF[2.3lo]
PDF.

\paragraph{Long description:}

To assess the uncertainty due to the choice of matching scheme, the nominal $tW$ sample was compared with a sample generated
with the \MGNLO[2.6.2] generator, which provided matrix elements at next-to-leading order~(NLO) in the strong coupling constant \alphas
in the five-flavour scheme, using the \NNPDF[2.3nlo]~\cite{Ball:2014uwa} PDF set.
The functional form of the renormalisation and factorisation scale was set to the default scale, which is equal to the top-quark mass.
The parton-shower starting scale had the functional form $\muQ = H_{\mathrm T}/2$~\cite{ATL-PHYS-PUB-2017-007},
where $H_{\mathrm T}$ is defined as the scalar sum of the \pT of all outgoing partons.
The diagram removal scheme~\cite{Frixione:2008yi} was employed to handle the interference with \ttbar production~\cite{ATL-PHYS-PUB-2016-020}.
The events were interfaced with \PYTHIA[8.230]~\cite{Sjostrand:2014zea}, using the A14 set of tuned parameters~\cite{ATL-PHYS-PUB-2014-021}
and the \NNPDF[2.3lo] PDF.
The decays of bottom and charm hadrons were simulated using the \EVTGEN[1.6.0] program~\cite{Lange:2001uf}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%              t-channel              %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Single-top \texorpdfstring{$t$}{t}-channel production}
\label{subsec:tchan}

This section describes the MC samples used for the modelling of single-top $t$-channel production.
Section~\ref{subsubsec:tchan_PP8} describes the \POWPY[8] samples used for the nominal prediction
and for the uncertainty from additional radiation and due to PDFs.
Section~\ref{subsubsec:tchan_PH7} describes the \POWHER[7] samples used for the uncertainty due to the choice of parton shower and hadronisation model,
and Section~\ref{subsubsec:tchan_aMCP8} describes the \MGNLOPY[8] samples used for the uncertainty due to the choice of matching scheme.

The reference cross-section values are extracted from Ref.~\cite{LHCTopWGsgtopXsec}.

\subsubsection[Powheg+Pythia8]{\POWPY[8]}
\label{subsubsec:tchan_PP8}

\paragraph{Samples}
%\label{par:tchan_PP8_samples}

Table~\ref{tab:tchan_PP8} gives the DSIDs of the $t$-channel \POWPY[8] samples.
Single-top and single-anti-top events were generated in distinct samples.
\begin{table}[!htbp]
\begin{center}
\caption{Single-top $t$-channel event samples produced with \POWPY[8].}
\label{tab:tchan_PP8}
\begin{tabular}{ l | l }
\hline
DSID & Description \\
\hline
410658 & $t$-channel $t$ leptonic \\
410659 & $t$-channel $\bar{t}$ leptonic \\
\hline
\end{tabular}
\end{center}
\end{table}

\paragraph{Short description:}

Single-top $t$-channel production was modelled using the
\POWHEGBOX[v2]~\cite{Frederix:2012dh,Nason:2004rx,Frixione:2007vw,Alioli:2010xd}
generator at NLO in QCD using the four-flavour scheme and the
corresponding \NNPDF[3.0nlo] set of PDFs~\cite{Ball:2014uwa}.  The events were
interfaced with \PYTHIA[8.230]~\cite{Sjostrand:2014zea} using the A14
tune~\cite{ATL-PHYS-PUB-2014-021} and the \NNPDF[2.3lo] set of
PDFs~\cite{Ball:2012cx}.

\input{top_isrfsr_powpy_short}

\paragraph{Long description:}

Single-top $t$-channel production was modelled using the
\POWHEGBOX[v2]~\cite{Frederix:2012dh,Nason:2004rx,Frixione:2007vw,Alioli:2010xd}
generator, which provided matrix elements at next-to-leading-order~(NLO)
accuracy in the strong coupling constant \alphas\ in the four-flavour
scheme with the corresponding \NNPDF[3.0nlo]~\cite{Ball:2014uwa} parton
distribution function~(PDF) set.  The functional form of the
renormalisation and factorisation scales was set to
$\sqrt{m_b^2 + p_{\text{T},b}^2}$ following the
recommendation of Ref.~\cite{Frederix:2012dh}. Top quarks were decayed at
LO using \MADSPIN~\cite{Frixione:2007zp,Artoisenet:2012st} to preserve
all spin correlations.  The events were interfaced with
\PYTHIA[8.230]~\cite{Sjostrand:2014zea} using the A14
tune~\cite{ATL-PHYS-PUB-2014-021} and the \NNPDF[2.3lo] PDF set.
The decays of bottom and charm hadrons were simulated using the
\EVTGEN[1.6.0] program~\cite{Lange:2001uf}.

The inclusive cross-section was corrected to the theory prediction calculated at NLO in QCD with
\HATHOR[2.1]~\cite{Aliev:2010zk,Kant:2014oha}.
For proton--proton collisions at a centre-of-mass energy of \rts~=~\SI{13}{\TeV}, this cross-section corresponds to
$\sigma(t,t\text{-chan})_\text{NLO}= 136.02^{+5.40}_{-4.57}$\,pb ($\sigma(\bar{t},t\text{-chan})_\text{NLO}=80.95^{+4.06}_{-3.61}$\,pb)
for single-top (single-anti-top) production, using a top-quark mass of $\mtop = 172.5\,\GeV$.
The uncertainties in the cross-section due to the PDF and \alphas were calculated using the \PDFforLHC prescription~\cite{Butterworth:2015oua}
with the \MSTW[nlo] 68\% CL~\cite{Martin:2009iq,Martin:2009bu}, \CT[10nlo]~\cite{Lai:2010vv}
and \NNPDF[2.3nlo]~\cite{Ball:2012cx} PDF sets,
and were added in quadrature to the effect of the scale uncertainty.

\input{top_isrfsr_powpy_long}

To evaluate the PDF uncertainties for the nominal PDF, the 100 variations for \NNPDF[3.0nlo] were taken into account.
In addition, the central value of this PDF was compared with the central values of the \CT[14nnlo]~\cite{Dulat:2015mca}
and \MMHT[nnlo]~\cite{Harland-Lang:2014zoa} PDF sets.

\subsubsection[Powheg+Herwig7]{\POWHER[7]}
\label{subsubsec:tchan_PH7}

\paragraph{Samples}
%\label{par:tchan_PH7_samples}

Table~\ref{tab:tchan_PH7} gives the DSIDs of the $t$-channel \POWHER[7] samples.
Single-top and single-anti-top events were generated in distinct samples.
\begin{table}[htbp]
\begin{center}
\caption{Single-top $t$-channel event samples produced with \POWHER[7].}
\label{tab:tchan_PH7}
\begin{tabular}{ l | l }
\hline
DSID & Description \\
\hline
411032 & $t$-channel $\bar t$ leptonic \\
411033 & $t$-channel $t$ leptonic \\
\hline
\end{tabular}
\end{center}
\end{table}

\paragraph{Short description:}

The uncertainty due to the parton shower and hadronisation model was
evaluated by comparing the nominal sample of events with a sample where
the events generated with the
\POWHEGBOX[v2]~\cite{Frederix:2012dh,Nason:2004rx,Frixione:2007vw,Alioli:2010xd}
generator were interfaced to
\HERWIG[7.04]~\cite{Bahr:2008pv,Bellm:2015jjp}, using the H7UE set
of tuned parameters~\cite{Bellm:2015jjp} and the \MMHT[lo] PDF set
\cite{Harland-Lang:2014zoa}.

\paragraph{Long description:}

The impact of using a different parton shower and hadronisation model was evaluated by comparing the nominal sample
with another sample produced with the \POWHEGBOX[v2]~\cite{Frederix:2012dh,Nason:2004rx,Frixione:2007vw,Alioli:2010xd}
generator but interfaced with \HERWIG[7.04]~\cite{Bahr:2008pv,Bellm:2015jjp}, using the H7UE set of
tuned parameters~\cite{Bellm:2015jjp} and the \MMHT[lo] PDF set \cite{Harland-Lang:2014zoa}.
\POWHEGBOX provided matrix elements at next-to-leading order~(NLO) in the strong coupling constant \alphas
in the four-flavour scheme with the corresponding \NNPDF[3.0nlo]~\cite{Ball:2014uwa} parton distribution function~(PDF).
The functional form of the renormalisation and factorisation scales was set to $\sqrt{m_b^2 + p_{\text{T},b}^2}$
following the recommendation of Ref.~\cite{Frederix:2012dh}.
Top quarks were decayed at LO using \MADSPIN~\cite{Frixione:2007zp,Artoisenet:2012st} to preserve all spin correlations.
The decays of bottom and charm hadrons were simulated using the \EVTGEN[1.6.0] program~\cite{Lange:2001uf}.

\subsubsection[MadGraph5\_aMC@NLO+Pythia8]{\MGNLOPY[8]}
\label{subsubsec:tchan_aMCP8}

\paragraph{Samples}
%\label{par:tchan_aMCP8_samples}

Table~\ref{tab:tchan_aMCP8} gives the DSIDs of the $t$-channel \MGNLOPY[8] samples.
\begin{table}[htbp]
\begin{center}
\caption{Single-top $t$-channel event samples produced with \MGNLOPY[8].}
\label{tab:tchan_aMCP8}
\begin{tabular}{ l | l }
\hline
DSID & Description \\
\hline
412004 & $t$-channel leptonic \\
\hline
\end{tabular}
\end{center}
\end{table}

\paragraph{Short description:}

To assess the uncertainty in the matching of NLO matrix elements to the
parton shower, the nominal sample was compared with a sample generated
with the \MGNLO[2.6.2] generator at NLO in QCD using the five-flavour
scheme and the \NNPDF[2.3nlo]~\cite{Ball:2014uwa} PDF set. The events were
interfaced with \PYTHIA[8.230]~\cite{Sjostrand:2014zea}, using the A14
set of tuned parameters~\cite{ATL-PHYS-PUB-2014-021} and the \NNPDF[2.3lo] PDF set.

\paragraph{Long description:}

To assess the uncertainty due to the choice of matching scheme, the nominal sample was compared with a sample generated
with the \MGNLO[2.6.2] generator, which provided matrix elements at next-to-leading order~(NLO) in the strong coupling constant \alphas
in the four-flavour scheme, using the corresponding \NNPDF[3.0nlo]~\cite{Ball:2014uwa} PDF set.
The functional form of the renormalisation and factorisation scales was set to $\sqrt{m_b^2 + p_{\text{T},b}^2}$
following the recommendation of Ref.~\cite{Frederix:2012dh}.
The parton-shower starting scale had the functional form $\muQ = H_\text{T}/2$~\cite{ATL-PHYS-PUB-2017-007},
where $H_\text{T}$ is defined as the scalar sum of the \pT of all outgoing partons.
Top quarks were decayed at LO using \MADSPIN~\cite{Frixione:2007zp,Artoisenet:2012st} to preserve all spin correlations.
The events were interfaced with \PYTHIA[8.230]~\cite{Sjostrand:2014zea}, using the A14 set of tuned parameters~\cite{ATL-PHYS-PUB-2014-021}
and the \NNPDF[2.3lo] PDF set.
The decays of bottom and charm hadrons were simulated using the \EVTGEN[1.6.0] program~\cite{Lange:2001uf}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%              s-channel              %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Single-top \texorpdfstring{$s$}{s}-channel production}
\label{subsec:schan}

This section describes the MC samples used for the modelling of single-top $s$-channel production.
Section~\ref{subsubsec:schan_PP8} describes the \POWPY[8] samples used for the nominal prediction
and for the uncertainty from additional radiation and due to PDFs.
Section~\ref{subsubsec:schan_PH7} describes the \POWHER[7] samples used for the uncertainty due to the parton shower and hadronisation model,
and Section~\ref{subsubsec:schan_aMCP8} describes the \MGNLOPY[8] samples used for the uncertainty due to the choice of matching scheme.

The reference cross-section values are extracted from Ref.~\cite{LHCTopWGsgtopXsec}.

\subsubsection[Powheg+Pythia8]{\POWPY[8]}
\label{subsubsec:schan_PP8}

\paragraph{Samples}
%\label{par:schan_PP8_samples}

Table~\ref{tab:schan_PP8} gives the DSIDs of the $s$-channel \POWPY[8] samples.
Single-top and single-anti-top events were generated in distinct samples.
\begin{table}[htbp]
\begin{center}
\caption{Single-top $s$-channel event samples produced with \POWPY[8].}
\label{tab:schan_PP8}
\begin{tabular}{ l | l }
\hline
DSID & Description \\
\hline
410644 & $s$-channel $t$ leptonic \\
410645 & $s$-channel $\bar{t}$ leptonic \\
\hline
\end{tabular}
\end{center}
\end{table}

\paragraph{Short description:}

Single-top $s$-channel production was modelled using the \POWHEGBOX[v2]~\cite{Alioli:2009je,Nason:2004rx,Frixione:2007vw,Alioli:2010xd}
generator at NLO in QCD in the five-flavour scheme with the \NNPDF[3.0nlo]~\cite{Ball:2014uwa} parton distribution function~(PDF) set.
The events were interfaced with \PYTHIA[8.230]~\cite{Sjostrand:2014zea} using the A14 tune~\cite{ATL-PHYS-PUB-2014-021} and the
\NNPDF[2.3lo] PDF set.

\input{top_isrfsr_powpy_short}

\paragraph{Long description:}

Single-top $s$-channel production was modelled using the \POWHEGBOX[v2]~\cite{Alioli:2009je,Nason:2004rx,Frixione:2007vw,Alioli:2010xd}
generator, which provided matrix elements at next-to-leading order~(NLO) in the strong coupling constant \alphas in the
five-flavour scheme with the \NNPDF[3.0nlo]~\cite{Ball:2014uwa} parton distribution function~(PDF) set.
The functional form of the renormalisation and factorisation scales was set to the default scale, which was equal to the top-quark mass.
The events were interfaced with \PYTHIA[8.230]~\cite{Sjostrand:2014zea} using the A14 tune~\cite{ATL-PHYS-PUB-2014-021} and the
\NNPDF[2.3lo] PDF set.
The decays of bottom and charm hadrons were simulated using the \EVTGEN[1.6.0] program~\cite{Lange:2001uf}.

The inclusive cross-section was corrected to the theory prediction calculated at NLO in QCD with
\HATHOR[2.1]~\cite{Aliev:2010zk,Kant:2014oha}.
For proton--proton collisions at a centre-of-mass energy of \rts~=~\SI{13}{\TeV}, this cross-section corresponds to
$\sigma(t,s\text{-chan})_\text{NLO}= 6.35^{+0.23}_{-0.20}$\,pb ($\sigma(\bar{t},s\text{-chan})_\text{NLO}=3.97^{+0.19}_{-0.17}$\,pb)
for single-top (single-anti-top) production, using a top-quark mass of $\mtop = 172.5\,\GeV$.
The uncertainties in the cross-section due to the PDF and \alphas were calculated using the \PDFforLHC prescription~\cite{Butterworth:2015oua}
with the \MSTW[nlo] 68\% CL~\cite{Martin:2009iq,Martin:2009bu}, \CT[10nlo]~\cite{Lai:2010vv} and \NNPDF[2.3nlo]~\cite{Ball:2012cx} PDF sets,
and were added in quadrature to the effect of the scale uncertainty.

\input{top_isrfsr_powpy_long}

To evaluate the PDF uncertainties for the nominal PDF, the 100 variations for \NNPDF[3.0nlo] were taken into account.
In addition, the central value of this PDF was compared with the central values of the
\CT[14nnlo]~\cite{Dulat:2015mca} and \MMHT[nnlo]~\cite{Harland-Lang:2014zoa} PDF sets.

\subsubsection[Powheg+Herwig7]{\POWHER[7]}
\label{subsubsec:schan_PH7}

\paragraph{Samples}
%\label{par:schan_PH7_samples}

Table~\ref{tab:schan_PH7} gives the DSIDs of the $s$-channel \POWHER[7] samples.
Single-top and single-anti-top events were generated in distinct samples.
\begin{table}[htbp]
\begin{center}
\caption{Single-top $s$-channel event samples produced with \POWHER[7].}
\label{tab:schan_PH7}
\begin{tabular}{ l | l }
\hline
DSID & Description \\
\hline
411034 & $s$-channel $t$ leptonic \\
411035 & $s$-channel $\bar{t}$ leptonic \\
\hline
\end{tabular}
\end{center}
\end{table}

\paragraph{Short description:}

The impact of using a different parton shower and hadronisation model was evaluated by comparing the nominal sample
with another sample produced with the \POWHEGBOX[v2]~\cite{Alioli:2009je,Nason:2004rx,Frixione:2007vw,Alioli:2010xd}
generator at NLO in the strong coupling constant \alphas in the five-flavour scheme using the
\NNPDF[3.0nlo]~\cite{Ball:2014uwa} parton distribution function~(PDF).
Events in the latter sample were interfaced with \HERWIG[7.04]~\cite{Bahr:2008pv,Bellm:2015jjp}, using the H7UE set of
tuned parameters~\cite{Bellm:2015jjp} and the \MMHT[lo] PDF set \cite{Harland-Lang:2014zoa}.

\paragraph{Long description:}

The impact of using a different parton shower and hadronisation model was evaluated by comparing the nominal sample
with another sample produced with the \POWHEGBOX[v2]~\cite{Alioli:2009je,Nason:2004rx,Frixione:2007vw,Alioli:2010xd}
generator but interfaced with \HERWIG[7.04]~\cite{Bahr:2008pv,Bellm:2015jjp}, using the H7UE set of
tuned parameters~\cite{Bellm:2015jjp} and the \MMHT[lo] PDF set \cite{Harland-Lang:2014zoa}.
\POWHEGBOX provided matrix elements at next-to-leading order~(NLO) in the strong coupling constant \alphas
in the five-flavour scheme with the \NNPDF[3.0nlo]~\cite{Ball:2014uwa} parton distribution function~(PDF).
The functional form of the renormalisation and factorisation scales was set to the default scale, which is equal to the top-quark mass.
The decays of bottom and charm hadrons were simulated using the \EVTGEN[1.6.0] program~\cite{Lange:2001uf}.

\subsubsection[MadGraph5\_aMC@NLO+Pythia8]{\MGNLOPY[8]}
\label{subsubsec:schan_aMCP8}

\paragraph{Samples}
%\label{par:schan_aMCP8_samples}

Table~\ref{tab:schan_aMCP8} gives the DSIDs of the $s$-channel \MGNLOPY[8] samples.
\begin{table}[htbp]
\begin{center}
\caption{Single-top $s$-channel event samples produced with \MGNLOPY[8].}
\label{tab:schan_aMCP8}
\begin{tabular}{ l | l }
\hline
DSID & Description \\
\hline
412005 & $s$-channel leptonic \\
\hline
\end{tabular}
\end{center}
\end{table}

\paragraph{Short description:}

%% the MG5_aMC@NLO version was doulble checked in the tag collector
To assess the uncertainty due to the choice of matching scheme, the nominal sample was compared with a sample generated
with the \MGNLO[2.6.2] generator at NLO in the strong coupling constant \alphas in the five-flavour scheme,
using the \NNPDF[3.0nlo]~\cite{Ball:2014uwa} PDF set.
The events were interfaced with \PYTHIA[8.230]~\cite{Sjostrand:2014zea}, using the A14 set of tuned parameters~\cite{ATL-PHYS-PUB-2014-021}
and the \NNPDF[2.3lo] PDF set.

\paragraph{Long description:}

To assess the uncertainty due to the choice of matching scheme, the nominal sample was compared with a sample generated
with the \MGNLO[2.6.2] generator, which provided matrix elements at next-to-leading order~(NLO) in the strong coupling constant \alphas
in the five-flavour scheme with the \NNPDF[3.0nlo]~\cite{Ball:2014uwa} parton distribution function~(PDF).
The functional form of the renormalisation and factorisation scales was set to the default scale, which is equal to the top-quark mass.
The parton-shower starting scale had the functional form $\muQ = H_{\mathrm T}/2$~\cite{ATL-PHYS-PUB-2017-007},
where $H_{\mathrm T}$ is defined as the scalar sum of the \pT of all outgoing partons.
Top quarks were decayed at LO using \MADSPIN~\cite{Frixione:2007zp,Artoisenet:2012st} to preserve all spin correlations.
The events were interfaced with \PYTHIA[8.230]~\cite{Sjostrand:2014zea}, using the A14 set of tuned parameters~\cite{ATL-PHYS-PUB-2014-021}
and the \NNPDF[2.3lo] PDF set.
The decays of bottom and charm hadrons were simulated using the \EVTGEN[1.6.0] program~\cite{Lange:2001uf}.


\begin{figure}
\begin{center}
    \includegraphics[width=0.9\textwidth]{figs/abc_slope_AV_starforming.pdf}
    \caption{\label{fig:slope}
    The attenuation-slope relation of star-forming galaxies ($\ssfr >
    10^{-11}yr^{-1}$), using the attenuation curves predicted by our
    \eda~prescription for the median posterior parameter values of SIMBA
    (left), TNG (center) and EAGLE (right).
    For comparison, we include the observed attenuation-slope relation
    from GSWLC2~\citep{salim2020}.
    We use $A_V$ and $S = A(1500\AA)/A_V$ as measurements of attenuation and
    slope, respectively.
    \emph{The \eda~does not predict $A_V < 0.3$ because star-forming galaxies in
    the simulations are intrinsically too lumnious and require significant attenuation to
    match observations.}
    Beyond $A_V > 0.3$, however, there is good agreement between the
    attenuation-slope relation predicted by the \eda~and observations.
    }
\end{center}
\end{figure}

\subsection{Comparison to Dust Observations} \label{sec:reproduce}
In addition to  reproducing observations, the \eda~assigns dust attenuation
curves to each simulated galaxy so we can compare the \eda~attenuation curves
to dust attenuation measured from observations.
We begin with the well-established attenuation-slope relation: star-forming
galaxies with higher dust attenuation have shallower attenuation curves.
This relation is a consequence of dust scattering dominating absorption at
low attenuation while dust absorption dominates at high
attenuation~\citep{gordon1994, witt2000, draine2003, chevallard2013}.
In Figure~\ref{fig:slope}, we present the attenuation-slope relation of
star-forming galaxies with $\ssfr > 10^{-11}yr^{-1}$ using the
dust attenuation curves predicted by the \eda~for the median posteriors of
SIMBA (left), TNG (center) and EAGLE (right).
For comparison, we include the observed attenuation-slope relations of
GSWLC2 galaxies~\citep[grey shaded;][]{salim2020}.
For attenuation, we use $A_V$; for slope, we use the UV-optical slope, $S
= A(1500\AA)/A_V$.
The contours mark the 68 and 95 percentiles.

Most noticably, we find that the \eda~does not predict $A_V < 0.3$ for
star-forming galaxies.
%\chedit{
%    This is not due to the selection function imposed by our forward model.
%    The attenuation-slope relation of star-forming galaxies in GSWLC2 does not
%    change significantly if we impose similar selection cuts as our
%    observational sample.
%    Instead, the lack of star-forming galaxies with $A_V < 0.3$ is a
This is a consequence of SIMBA, TNG, and EAGLE predicting star-forming galaxies
that are intrinsically more luminous than observations.
All of the simulations have star-forming galaxies with intrinsic $M_r <
-21$ and $\gr < 0.5$ (Figure~\ref{fig:obs}).
This is further corroborated by the $\sfr-M*$ relations in
Figure~\ref{fig:smf_msfr}, where the simulations all have star-forming
galaxies with $M_* > 10^{11}M_\odot$, not found in SDSS.
To reproduce the SDSS optical color-magnitude relation these galaxies would
need to be significantly reddened and attenuated so any dust prescription would
need to assign high $A_V$ to star-forming galaxies.
Nevertheless, for $A_V > 0.3$, we find good agreement between the
attenuation-slope relation predicted by the \eda~and observations.
We refrain from further scrutinizing on $A_V$ values, since they can vary {\em
significantly} between different observational measurements --- even for the
same galaxy.
SDSS star-forming galaxies, for instance, have significantly higher
$A_V > 0.3$ according to the \cite{brinchmann2004} measurements
(Appendix~\ref{sec:slab}).

%We note that the difference in the $A_V$ ranges is due to the $M_r$
%completeness limit imposed by our forward model (Section~\ref{sec:fm}).
%The GSWLC2 sample in \cite{salim2020} extends down to $M_* \sim
%10^{8.5}M_\odot$; however, the TNG and EAGLE samples do not extend below
%$M_* \sim10^{10}M_\odot$.
%\emph{The \eda~predicts attenuation-slope relations for TNG and EAGLE that
%are in excellent agreement with observations.}

\begin{figure}
\begin{center}
    \includegraphics[width=0.5\textwidth]{figs/abc_sf_attenuation.pdf}
    \caption{\label{fig:sfatten}
    The normalized attenuation curves of star-forming galaxies predicted by
    the \eda~for median posterior parameter values of SIMBA (orange), TNG
    (blue), and EAGLE (green).
    We classify galaxies with $\log \ssfr > -11~yr^{-1}$ as star-forming.
    The attenuation curves are normalized at $3000\AA$ and we mark the
    68 percentile of the attenuation curves with the shaded region.
    For comparison, we include $A(\lambda)/A(3000\AA)$ measurements from
    the~\cite{narayanan2018} radiative transfer simulation (dashed) and
    \cite{salim2018} observations (dotted).
    %The \cite{calzetti2000} and \cite{battisti2017} attenuation curves are shallower than the \eda~attenuation curves; however, they probe lower $M_*$ galaxies than our forward modeled TNG and EAGLE samples.  For attenuation curve from \cite{salim2018}, which probe a similar $M_*$ range, we find goood agreement.
    {\em The \eda~predict attenuation curves of star-forming galaxies
    are in good agreement with the attenuation curves measured from
    the simulation and observations in the literature.}
    %We also find good agreement with median attenuation curve of star-forming galaxies in the radiative transfer simulations of \cite{narayanan2018}.
    }
\end{center}
\end{figure}

In addition to the attenuation-slope relation, we can also directly compare
the attenuation curves predicted by the \eda~to measurements from
observations for star-forming galaxies.
In Figure~\ref{fig:sfatten}, we present the normalized attenuation curves
of star-forming galaxies predicted by the \eda~for the median posterior
parameter values of SIMBA(orange), TNG (blue), and EAGLE (green).
We again define galaxies with $\ssfr > 10^{-11}{yr}^{-1}$ as star-forming.
The attenuation curves are normalized at $3000\AA$ and we present the
variation in the attenuation curves in the shaded region, 68 percentile.
For comparison, we include $A(\lambda)/A(3000\AA)$ from the
\cite{narayanan2018} radiative transfer simulation (dashed) and
observations~\citep[][dotted]{salim2018}.
The attenuation curve from \cite{salim2018} corresponds to star-forming
galaxies with $M_* > 10^{10.5}M_\odot$, a similar $M_*$ range as our
forward modeled samples.
Since we do not vary the UV bump in our \eda~prescription, we ignore any
discrepancies in the amplitudes of the bump.
\emph{Overall, we find good agreement between the \eda~attenuation curves for
star-forming galaxies and the attenuation curves from observations and
simulations.}

%Again, the fact that we reproduce the detailed dust attenuation curves of star-forming galaxies in observations and simulations with the \eda~without fitting for them, highlights the advantages of a forward modeling approach.

%The \eda~attenuation curves are slightly steeper than the \cite{calzetti2000} and \cite{battisti2017} curves.
%These attenuation curves, however, are derived from $M_* < 10^{9.9}M_\odot$
%star-forming galaxies, which lie below the $M_*$ limit of our forward
%modeled TNG and EAGLE samples.
%Meanwhile, the TNG and EAGLE \eda~attenuation curves are in good agreement
%with the \cite{salim2018} attenuation curve for $M_* > 10^{10.5}M_\odot$ star-forming galaxies.
%They are also consistent with the median curve of \cite{narayanan2018}.

%The \eda~attenuation curves are noticeably steeper than the \cite{calzetti2000} and \cite{battisti2017} curves.  These attenuation curves, however, are derived from $M_* < 10^{9.9}M_\odot$ star-forming galaxies --- below our $M_*$ range.
%Since we find $\mdeltam < 0$ for both the TNG and EAGLE posteriors, the \eda~attenuation curves are consistent with \cite{calzetti2000} and \cite{battisti2017}.

%\chedit{
%    The \eda~predicts higher dust attenuation at lower wavelenghts for
%    star-forming galaxies.
%    Without dust attenuation, both TNG and EAGLE predict star-forming galaxies
%    that are bluer in the optical and UV than observations
%    (Figure~\ref{fig:obs}).
%    To reproduce the SDSS, the \eda~significantly reddens star-forming galaxies.
%}
%In Figure~\ref{fig:raw_atten}, we also find that more massive star-forming
%galaxies have higher attenuation. This is because the simulations overpredict
%luminous blue star-forming galaxies, which must be attenuated to reproduce
%observations.

%At low attenuation, dust scattering dominates absoprtion so the
%attenuation curve steepens because red light scatters isotropically while blue light
%scatters forward~\citep{gordon1994, witt2000, draine2003}. %, which causes more optical-to-IR light to escape the galaxy than UV light
%At high attenuation dust absorption is dominant and the attenuation curve is
%shallower~\citep{chevallard2013}. For the $A_V$ range probed by the DEM, the
%$A_V$--slope relation is in good agreement with GSWLC2 galaxies~\citep[black shaded][]{salim2020}.
%They are also consistent with \cite{leja2017}. We also compare our results to
%theoretical predictions from radiative transfer models, \cite{inoue2005}
%(dotted), the radiative transfer models considered in \cite{chevallard2013}
%(dot dashed), and \cite{trayford2020} (light shaded), which all predict shallower
%attenuation curves than observations. This is also the case for the
%\cite{narayanan2018} attenuation curves (not included).
%\emph{The attenuation curve slopes from the DEM for are in excellent
%agreement with observations and better reproduces the observed
%attenuation--slope relation than radiative transfer models.}


%-------------------------------------------------------------------------------
\section{Event reconstruction}
\label{sec:objects}
%-------------------------------------------------------------------------------

The event reconstruction is affected by multiple $pp$ collisions in a single bunch crossing and by collisions
in neighbouring bunch crossings, referred to as pile-up.
Interaction vertices from the $pp$ collisions are reconstructed from at least two tracks
with transverse momentum ($\pt$) larger than $400~\mev$ that are consistent with originating from the
beam collision region in the $x$--$y$ plane. If more than one primary vertex candidate is found, the
candidate whose associated tracks form the largest sum of squared $\pt$~\cite{ATL-PHYS-PUB-2015-026}
is selected as the hard-scatter primary vertex.

Electron candidates~\cite{ATLAS-CONF-2016-024,Aaboud:2018ugz} are reconstructed from energy
clusters in the EM calorimeter that are matched to reconstructed tracks in the inner detector;
electron candidates in the transition region between the EM barrel and endcap calorimeters
($1.37 < |\eta_{\textrm{cluster}}| < 1.52$) are excluded.
Electron candidates are required to have $|\eta_{\textrm{cluster}}| < 2.47$, and to satisfy tight likelihood-based identification
criteria~\cite{ATLAS-CONF-2016-024} based on calorimeter, tracking and combined variables that provide
separation between electrons and jets.

Muon candidates~\cite{Aad:2016jkr} are reconstructed by matching track segments in %the various
different layers of the muon spectrometer to tracks found in the inner detector;
the resulting muon candidates are re-fitted using the complete track information from both detector systems.
Muon candidates are required to satisfy medium identification criteria~\cite{Aad:2016jkr}.

Electron (muon) candidates are matched to the primary vertex by requiring that the significance of their transverse impact parameter, $d_0$,
satisfies $|d_0/\sigma(d_0)|<5\,(3)$, where $\sigma(d_0)$ is the measured uncertainty in $d_0$,
and by requiring that their longitudinal impact parameter, $z_0$, satisfies $|z_0 \sin\theta|<0.5$~mm. To reduce the impact of the trigger efficiency uncertainty around the threshold, the leptons are required to have \pt 1~GeV above the threshold.
To further reduce the background from non-prompt leptons, photon conversions and hadrons, lepton candidates are also required to be isolated
in the tracker and in the calorimeter.
A track-based lepton isolation criterion is defined by calculating the quantity $I_R = \sum \pt^{\textrm{trk}}$ and $E_R = \sum E^{\textrm{clst}}$, where
the scalar sum includes all tracks and energy deposit (excluding the lepton candidate itself) within the cone defined by $\Delta R<0.2 (0.3)$ around the %axis
direction of the electron (muon). The electron (muon) candidates are required satisfy both $I_R/\pt^\ell < 0.2~(0.3)$ and $E_R/\pt^\ell < 0.2~(0.3)$.

Tight isolation working points are also applied in some channels to reduce fake and non-prompt lepton contributions based a trained isolation boosted decision tree \texttt{PromptLeptonVeto}(PLIV), which is to identify non-prompt light leptons using lifetime information associated with a track jet that matches the selected light lepton~\cite{ATLAS-CONF-2019-045}. These additional reconstructed charged particle tracks inside the jet can be
used to increase the precision of identifying the displaced decay vertex of heavy flavor (b, c) hadrons that produced non-prompt leptons.
The tight working points are used for high $\pt$ lepton.
The scale factors for the efficiencies of the tight \texttt{PLIV} working points are measured using the tag and probe method
with $Z\rightarrow l^+l^-$ events. These scale factors are also checked applicable to the electron or muon from the tau decay
using $Z\rightarrow\tau\tau\rightarrow e\mu$ samples, which is consistent with 1 at 2\% level. To be conservative, we assign
additional $\pm 2\%$ systematic uncertainty for the PLIV efficiency for the tau-lepton in the lepton+$\thad$ channels.
%Additionally, the $\Htautau$ search requires leptons to
%satisfy a calorimeter-based isolation criterion; a requirement is made on the sum of the transverse energy of
%topological clusters within the cone defined by $\Delta R<0.2$ around the lepton, after subtracting the contribution
%from the energy deposit of the lepton itself and correcting for pile-up effects, to be less than a $\pt$-dependent
%percentage of the lepton energy.
%The combined efficiency of track-based and calorimeter-based isolation requirements made by the $\Htautau$
%search is 90\% (99\%) at $\pt=25\;(60)~\gev$.

%a Gradient isolation working points are used for both electrons and muons. The working points are based on cut maps derived from the $Z\to l^+l^-$ Monte Carlo samples. The efficiency maps are simply derived from the sum of transverse energy of the clusters deposited in the calorimeter in the given cone. The efficiency for the leptons with a certain $\pt$ is $(0.1143*\pt[\GeV]+92.14)\%*(0.1143*\pt[\GeV]+92.14)\%$.
%draft 1 version
%Candidate jets are reconstructed with the anti-$k_t$ algorithm~\cite{Cacciari:2008gp,Cacciari:2005hq,Cacciari:2011ma} with a
%radius parameter $R=0.4$, using topological clusters~\cite{Aad:2016upy}
%built from energy deposits in the calorimeters calibrated to the electromagnetic scale.
Candidate jets are reconstructed using the anti-$k_t$ algorithm~\cite{Cacciari:2008gp,Cacciari:2005hq} with a
radius parameter $R=0.4$ applied to topological energy clusters~\cite{Aad:2016upy} and charged-particle tracks, processed using
a particle-flow algorithm~\cite{Aad:2017epj77}. %newref Eur. Phys. J. C 77 (2017) 466
%as implemented in the \fastjet\ package~\cite{Cacciari:2011ma}.
%Jet reconstruction in the calorimeter starts from topological clustering~\cite{Aad:2016upy} of individual calorimeter cells calibrated to the electromagnetic energy scale.
The reconstructed jets are then calibrated to the particle level by the application of a jet energy scale
derived from simulation and in situ corrections based on $\sqrt{s}=13~\tev$ data~\cite{Aaboud:2017jcu}. %(TBD: check PFlow jets description)
The calibrated jets used are required to have $\pt > 25~\gev$ and $|\eta| < 2.5$.
Jet four-momenta are corrected for pile-up effects using the jet-area method~\cite{Cacciari:2008gn}.

Quality criteria are imposed to reject events that contain any jets arising from non-collision sources
or detector noise~\cite{ATLAS-CONF-2015-029}.  To reduce the contamination due to jets originating from pile-up interactions,
additional requirements are imposed on the jet vertex tagger (JVT)~\cite{Aad:2015ina} output for jets with $\pt<60~\gev$ and $|\eta| < 2.4$.

Jets containing $b$-hadrons are identified ($b$-tagged) via the DL1r tagger~\cite{Aad:2019epj79,ATL-PHYS-PUB-2017-013} %newref
%\cite{Aad:2015ydr,ATL-PHYS-PUB-2016-012}
that uses multivariate techniques to combine information about the impact parameters of displaced tracks and the  topological properties
of secondary and tertiary decay vertices reconstructed within the jet. For each jet, a value for the multivariate $b$-tagging discriminant is
calculated. A jet is considered $b$-tagged if this value is above the threshold corresponding to
an average 70\% efficiency to tag a $b$-quark jet, with a light-jet\footnote{Light-jet refers to a jet originating from the hadronisation of a light quark ($u$, $d$, $s$) or a gluon.} rejection factor of about 385 and a charm-jet rejection factor of about 12, as determined for jets with
$\pt >20~\gev$ and $|\eta|<2.5$ in simulated $\ttbar$ events.

Hadronically decaying $\tau$-lepton ($\had$) candidates are reconstructed from energy clusters in the calorimeters and
associated inner-detector tracks~\cite{ATL-PHYS-PUB-2019-033}. %newref
%~\cite{ATL-PHYS-PUB-2015-045}.
Candidates are required to have either one or three associated tracks,
with a total charge of $\pm 1$. Candidates are required to have \pt 5 GeV higher than the trigger threshold, and $|\eta|<2.5$,
excluding the EM calorimeter's transition region.
A Recurrent Neural Network (RNN)~\cite{Graves:2012SCI}
%~\cite{Breiman:1984jka,Friedman:2002we,Freund:1997xna}
%A. Graves, Supervised Sequence Labelling with Recurrent Neural Networks, Studies in Computational Intelligence 385, Springer, 2012.
using calorimeter- and tracking-based variables is used to identify $\had$ candidates and reject jet backgrounds.
Three working points labelled loose, medium and tight are defined, and correspond to different $\had$ identification efficiency values, with the efficiency designed to be independent of $\pt$. The $\Htautau$ search uses the medium
working point for the nominal selection.
%, while the loose working point is used for background estimation.
The medium working point has a combined reconstruction and identification efficiency of 75\% (60\%) for one-prong (three-prong) $\had$
decays, and an expected rejection factor against light-jets of 35 (240)~\cite{ATL-PHYS-PUB-2019-033}.
Electrons that are reconstructed as one-prong $\had$ candidates are removed via a BDT trained to reject electrons.
Events with$\had$ candidate that is $b$-tagged are rejected.
%(TBD: check RNN description and update then numbers)

%Overlaps between candidate objects are removed sequentially, following this order: muons, electrons,
%$\tauhad$ (only for the $\Htautau$ search), and jets. In the $\Hbb$ search, firstly, electron candidates that lie
%within $\Delta R = 0.01$ of a muon candidate are removed to suppress contributions from muon bremsstrahlung.
%Overlaps between electron and jet candidates are resolved next, and finally, overlaps between remaining jet candidates
%and muon candidates are removed. Clusters from identified electrons are not excluded during jet reconstruction.
%In order to avoid double-counting of electrons as jets, the closest jet whose axis is within ${\Delta}R = 0.2$ of an electron
%is discarded. If the electron is within ${\Delta}R = 0.4$ of the axis of any jet after this initial removal, the jet is retained and  the electron is removed.
%The overlap removal procedure between the remaining jet candidates and muon candidates is designed to remove those muons
%that are likely to have arisen in the decay chain of hadrons and to retain the overlapping jet instead.
%Jets and muons may also appear in close proximity when the jet results from high-$\pt$ muon bremsstrahlung,
%and in such cases the jet should be removed and the muon retained. Such jets are characterised by having very
%few matching inner-detector tracks. Selected muons that satisfy $\Delta R(\mu,{\textrm{jet}}) < 0.04+10~\gev/\pt^\mu$ are rejected
%if the jet has at least three tracks originating from the primary vertex; otherwise the jet is removed and the muon is kept.
%In the $\Htautau$ search, a fixed cone size of $\Delta R=0.2$ is used to determine the overlap between
%candidate objects, and only the highest-$\pt$ (leading) or the two leading $\tauhad$ candidates (depending on the
%analysis channel, see Section~\ref{sec:data_presel}) are considered to resolve their overlap with jets.

Overlaps between reconstructed objects are removed sequentially. Firstly, electron candidates that lie
within $\Delta R = 0.01$ of a muon candidate are removed to suppress contributions from muon bremsstrahlung.
Overlaps between electron and jet candidates are resolved next, and finally, overlaps between remaining jet candidates
and muon candidates are removed. Energy clusters from identified electrons are not excluded during jet reconstruction.
In order to avoid double-counting of electrons as jets, the closest jet whose axis is within ${\Delta}R = 0.2$ of an electron
is discarded if the jet is not $b$-tagged, otherwise the electron is removed.
If the electron is within ${\Delta}R = 0.4$ of the axis of any jet after this initial removal, the jet is retained and  the electron is removed.
The overlap removal procedure between the remaining jet candidates and muon candidates is designed to remove those muons
that are likely to have arisen in the decay of hadrons and to retain the overlapping jet instead.
Jets and muons may also appear in close proximity when the jet results from high-$\pt$ muon bremsstrahlung,
and in such cases the jet should be removed and the muon retained. Such jets are characterised by having very
few matching inner-detector tracks. Selected muons that satisfy $\Delta R(\mu,{\textrm{jet}}) < 0.2$ are rejected
if the jet is either $b$-tagged or has at least three tracks originating from the primary vertex; otherwise the jet is removed and the muon is kept.
The $\tauhad$ within a $\Delta R=0.2$ cone of an electron or muon are removed.
In order to avoid double-counting of $\tauhad$ as jets, the closest jet whose axis is
within ${\Delta}R = 0.2$ of a $\tauhad$ is discarded if the jet is not $b$-tagged, otherwise the $\tauhad$ is removed.
% TEXT FROM BOYANG. NEED TO RECHECK WHAT WE HAVE ABOVE IS ACCURATE
%In the $\Htautau$ search, the objects are removed with the following sequence: if two electrons have overlapping second-layer cluster, or shared tracks, the electron with lower $\pt$ is removed; $\tauhad$ within a $\Delta R=0.2$ cone of an electron or muon are removed; if a muon sharing an ID track with an electron and the muon is calo-tagged, the muon is removed, otherwise the electron is removed; jets within a $\Delta R=0.2$ cone of an electron are removed; electrons within a $\Delta R=0.4$ cone of a jet are removed; when a muon ID track is ghost associated to a jet or within a $\Delta R=0.2$ cone of a jet, the jet is removed if it has less than 3 tracks with $\pt>500$ MeV or has a relative small $\pt$ ($\pt^{\mu}>0.5\pt^{\text{jet}} \text{ and } \pt^{\mu}>0.7[\text{the scalar sum of the } \pt \text{'s of the jet tracks with } \pt>500$ MeV]); muons within a $\Delta R=0.4$ cone of a jet are removed; jets within a $\Delta R=0.2$ cone of the leading $\tauhad$ ($\lephad$), or with the two leading $\tauhad$'s ($\hadhad$), are removed.

%firstly, the electron within a ${\Delta}R = 0.2$ cone of muons are excluded, then the $\tau_{had}$'s or jets within a ${\Delta}R = 0.2$ cone of an electron or muon are excluded. Finally, the jets within a ${\Delta}R = 0.2$ cone of the leading $\had$ ($\lephad$), or with the two leading $\had$’s ($\hadhad$) are excluded.

The missing transverse momentum $\mpt$ (with magnitude $\met$) is defined as the negative vector sum of the
$\pt$ of all selected and calibrated objects in the event, including a term to account for momentum from soft particles
in the event which are not associated with any of the selected objects.
This soft term is calculated from inner-detector tracks matched to the selected primary vertex to make it more resilient to
contamination from pile-up interactions~\cite{Aaboud:2018tkc}.


% Footnote with ATLAS coordinate system
\newcommand{\AtlasCoordFootnote}{%
ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP)
in the centre of the detector and the \(z\)-axis along the beam pipe.
The \(x\)-axis points from the IP to the centre of the LHC ring,
and the \(y\)-axis points upwards.
Cylindrical coordinates \((r,\phi)\) are used in the transverse plane,
\(\phi\) being the azimuthal angle around the \(z\)-axis.
The pseudorapidity is defined in terms of the polar angle \(\theta\) as \(\eta = -\ln \tan(\theta/2)\).
Angular distance is measured in units of \(\Delta R \equiv \sqrt{(\Delta\eta)^{2} + (\Delta\phi)^{2}}\).}

%-------------------------------------------------------------------------------
\subsection{Run 2 ATLAS detector example for a Letter}
\label{sec:atlas2a}
%-------------------------------------------------------------------------------

The ATLAS experiment~\cite{PERF-2007-01} at the LHC is a multi-purpose particle detector
with a forward-backward symmetric cylindrical geometry and a near \(4\pi\) coverage in
solid angle.\footnote{\AtlasCoordFootnote}
It consists of an inner tracking detector surrounded by a thin superconducting solenoid
providing a \SI{2}{\tesla} axial magnetic field, electromagnetic and hadron calorimeters, and a muon spectrometer.
The inner tracking detector covers the pseudorapidity range \(|\eta| < 2.5\).
It consists of silicon pixel, silicon microstrip, and transition radiation tracking detectors.
Lead/liquid-argon (LAr) sampling calorimeters provide electromagnetic (EM) energy measurements
with high granularity.
A hadron (steel/scintillator-tile) calorimeter covers the central pseudorapidity range (\(|\eta| < 1.7\)).
The end-cap and forward regions are instrumented with LAr calorimeters
for both EM and hadronic energy measurements up to \(|\eta| = 4.9\).
The muon spectrometer surrounds the calorimeters and is based on
three large air-core toroidal superconducting magnets with eight coils each.
The field integral of the toroids ranges between \num{2.0} and \SI{6.0}{\tesla\metre}
across most of the detector.
The muon spectrometer includes a system of precision tracking chambers and fast detectors for triggering.
A three-level trigger system is used to select events.
The first-level trigger is implemented in hardware and uses a subset of the detector information
to reduce the accepted rate to at most nearly \SI{100}{\kHz}.
This is followed by two software-based trigger levels that
together reduce the accepted event rate to \SI{1}{\kHz} on average
depending on the data-taking conditions.

%-------------------------------------------------------------------------------
\subsection{Run 2 ATLAS detector example for a paper making use of the whole detector}
\label{sec:atlas2b}
%-------------------------------------------------------------------------------

The ATLAS detector~\cite{PERF-2007-01} at the LHC covers nearly the entire solid angle around the collision point.\footnote{\AtlasCoordFootnote}
It consists of an inner tracking detector surrounded by a thin superconducting solenoid, electromagnetic and hadronic calorimeters,
and a muon spectrometer incorporating three large superconducting toroidal magnets.
The inner-detector system (ID) is immersed in a \SI{2}{\tesla} axial magnetic field
and provides charged-particle tracking in the range \(|\eta| < 2.5\).

The high-granularity silicon pixel detector covers the vertex region and typically provides four measurements per track,
the first hit being normally in the insertable B-layer (IBL) installed before Run~2~\cite{ATLAS-TDR-2010-19,PIX-2018-001}.
It is followed by the silicon microstrip tracker (SCT) which usually provides eight measurements per track.
These silicon detectors are complemented by the transition radiation tracker (TRT),
which enables radially extended track reconstruction up to \(|\eta| = 2.0\).
The TRT also provides electron identification information
based on the fraction of hits (typically 30 in total) above a higher energy-deposit threshold corresponding to transition radiation.

The calorimeter system covers the pseudorapidity range \(|\eta| < 4.9\).
Within the region \(|\eta|< 3.2\), electromagnetic calorimetry is provided by barrel and
endcap high-granularity lead/liquid-argon (LAr) calorimeters,
with an additional thin LAr presampler covering \(|\eta| < 1.8\),
to correct for energy loss in material upstream of the calorimeters.
Hadronic calorimetry is provided by the steel/scintillating-tile calorimeter,
segmented into three barrel structures within \(|\eta| < 1.7\), and two copper/LAr hadronic endcap calorimeters.
The solid angle coverage is completed with forward copper/LAr and tungsten/LAr calorimeter modules
optimised for electromagnetic and hadronic measurements respectively.

The muon spectrometer (MS) comprises separate trigger and
high-precision tracking chambers measuring the deflection of muons in a magnetic field generated by superconducting air-core toroids.
The field integral of the toroids ranges between \num{2.0} and \SI{6.0}{\tesla\metre}
across most of the detector.
A set of precision chambers covers the region \(|\eta| < 2.7\) with three layers of monitored drift tubes,
complemented by cathode-strip chambers in the forward region, where the background is highest.
The muon trigger system covers the range \(|\eta| < 2.4\) with resistive-plate chambers in the barrel, and thin-gap chambers in the endcap regions.
Interesting events are selected to be recorded by the first-level trigger system implemented in custom hardware,
followed by selections made by algorithms implemented in software in the high-level trigger~\cite{TRIG-2016-01}.
The first-level trigger reduces the \SI{40}{\MHz} bunch crossing rate to below \SI{100}{\kHz},
which the high-level trigger further reduces in order to record events to disk at about \SI{1}{\kHz}.

%-------------------------------------------------------------------------------
\subsection{Run 1 ATLAS detector example for a Letter}
\label{sec:atlas1a}
%-------------------------------------------------------------------------------

The ATLAS experiment~\cite{PERF-2007-01} at the LHC is a multi-purpose particle detector
with a forward-backward symmetric cylindrical geometry and a near \(4\pi\) coverage in
solid angle.\footnote{\AtlasCoordFootnote}
It consists of an inner tracking detector surrounded by a thin superconducting solenoid
providing a \SI{2}{\tesla} axial magnetic field, electromagnetic and hadron calorimeters, and a muon spectrometer.
The inner tracking detector covers the pseudorapidity range \(|\eta| < 2.5\).
It consists of silicon pixel, silicon microstrip, and transition radiation tracking detectors.
Lead/liquid-argon (LAr) sampling calorimeters provide electromagnetic (EM) energy measurements
with high granularity.
A hadron (steel/scintillator-tile) calorimeter covers the central pseudorapidity range (\(|\eta| < 1.7\)).
The end-cap and forward regions are instrumented with LAr calorimeters
for both EM and hadronic energy measurements up to \(|\eta| = 4.9\).
The muon spectrometer surrounds the calorimeters and is based on
three large air-core toroidal superconducting magnets with eight coils each.
The field integral of the toroids ranges between \num{2.0} and \SI{6.0}{\tesla\metre}
across most of the detector.
The muon spectrometer includes a system of precision tracking chambers and fast detectors for triggering.
A three-level trigger system is used to select events.
The first-level trigger is implemented in hardware and uses a subset of the detector information
to reduce the accepted rate to at most \SI{75}{\kilo\hertz}.
This is followed by two software-based trigger levels that
together reduce the accepted event rate to \SI{400}{\hertz} on average
depending on the data-taking conditions during 2012.

%-------------------------------------------------------------------------------
\subsection{Run 1 ATLAS detector example for a paper making use of the whole detector}
\label{sec:atlas1b}
%-------------------------------------------------------------------------------

The ATLAS detector~\cite{PERF-2007-01} at the LHC covers nearly the entire solid angle around the collision point.\footnote{\AtlasCoordFootnote}
It consists of an inner tracking detector surrounded by a thin superconducting solenoid, electromagnetic and hadronic calorimeters,
and a muon spectrometer incorporating three large superconducting toroidal magnets.
The inner-detector system (ID) is immersed in a \SI{2}{\tesla} axial magnetic field
and provides charged-particle tracking in the range \(|\eta| < 2.5\).

The high-granularity silicon pixel detector covers the vertex region and typically provides three measurements per track,
the first hit being normally in the innermost layer.
It is followed by the silicon microstrip tracker (SCT) which usually provides eight measurements per track.
These silicon detectors are complemented by the transition radiation tracker (TRT),
which enables radially extended track reconstruction up to \(|\eta| = 2.0\).
The TRT also provides electron identification information
based on the fraction of hits (typically 30 in total) above a higher energy deposit threshold corresponding to transition radiation.

The calorimeter system covers the pseudorapidity range \(|\eta| < 4.9\).
Within the region \(|\eta|< 3.2\), electromagnetic calorimetry is provided by barrel and
endcap high-granularity lead/liquid-argon (LAr) calorimeters,
with an additional thin LAr presampler covering \(|\eta| < 1.8\),
to correct for energy loss in material upstream of the calorimeters.
Hadronic calorimetry is provided by the steel/scintillating-tile calorimeter,
segmented into three barrel structures within \(|\eta| < 1.7\), and two copper/LAr hadronic endcap calorimeters.
The solid angle coverage is completed with forward copper/LAr and tungsten/LAr calorimeter modules
optimised for electromagnetic and hadronic measurements respectively.

The muon spectrometer (MS) comprises separate trigger and
high-precision tracking chambers measuring the deflection of muons in a magnetic field generated by superconducting air-core toroids.
The field integral of the toroids ranges between \num{2.0} and \SI{6.0}{\tesla\metre}
across most of the detector.
A set of precision chambers covers the region \(|\eta| < 2.7\) with three layers of monitored drift tubes,
complemented by cathode-strip chambers in the forward region, where the background is highest.
The muon trigger system covers the range \(|\eta| < 2.4\) with resistive plate chambers in the barrel, and thin gap chambers in the endcap regions.
A three-level trigger system is used to select interesting events~\cite{PERF-2011-02}.
The Level-1 trigger is implemented in hardware and uses a subset of detector information
to reduce the event rate to a design value of at most \SI{75}{\kHz}.
This is followed by two software-based trigger levels which together reduce the event rate to about \SI{200}{\Hz}.


\section{Hadron propagation.}

\hspace{1.0em}
So far the hadrons are transported along the straight line paths (cascade
approach), i. e. particle coordinates ${\bf r}$ for time "shift"
between $t_0$ and
$t_1$ are updated according to
\begin{equation}
\label{HP1} {\bf r}(t_1) = {\bf r}(t_0) + \frac{ {\bf p}(t_0)}{E(t_0)},
\end{equation}
where ${\bf p}$ and $E$ denote particles momenta and energies.
The intranuclear nucleons are assumed to be "frozen" before collision,
i. e.  they are allowed to move only with the c. m. nucleus velocity.
However, their Fermi momenta are taken into account to find interaction
cross sections or characteristics of secondary particles produced in the
course of an intranuclear collision.


\hypertarget{performance-metrics-for-parallel-systems}{%
\section{Performance Metrics for Parallel
Systems}\label{performance-metrics-for-parallel-systems}}

\hypertarget{analytical-modeling}{%
\subsection{Analytical Modeling}\label{analytical-modeling}}

\begin{itemize}
\tightlist
\item
  Sequential Runtime

  \begin{itemize}
  \tightlist
  \item
    A sequential algorithm is evaluated by its runtime
  \end{itemize}
\item
  Parallel Runtime

  \begin{itemize}
  \tightlist
  \item
    The parallel runtime of a program depends on

    \begin{itemize}
    \tightlist
    \item
      the input size n,
    \item
      the number of processors p,
    \item
      and the communication parameters of the machine.
    \end{itemize}
  \item
    An algorithm must therefore be analyzed in the context of the
    underlying platform.
  \end{itemize}
\item
  Parallel System

  \begin{itemize}
  \tightlist
  \item
    A parallel system is a combination of a parallel algorithm and an
    underlying parallel platform.
  \end{itemize}
\item
  Wall-clock time

  \begin{itemize}
  \tightlist
  \item
    the time from the start of the first processor to the end of the
    last processor in a parallel ensemble
  \item
    in other words: the operation time of the processor which needs the
    most time
  \end{itemize}
\end{itemize}

\hypertarget{metrics}{%
\subsubsection{Metrics}\label{metrics}}

\begin{itemize}
\tightlist
\item
  $T_s(n)$: Zeitkomplexität des besten sequentiellen Vergleichalgorithmus
\item
  $T_p(p,n)$: Zeitkomplexität des besten parallelen Algorithmus
\item
  n: eindimensionale Inputgrösse
\item
  $p$: Anzahl ``Prozessoren'' (evtl. auch Threads, Cores, etc.)
\item
  Total Parallel Overhead $T_O = pT_P - T_S$
\item
  Speedup $S = T_S / T_P$
\item
  Speedup per processor = Efficiency $E = S / p = T_S / pT_P$
\end{itemize}

\hypertarget{speedup}{%
\subsubsection{Speedup}\label{speedup}}

\begin{itemize}
\tightlist
\item
  The lower bound of the speedup can be 0 (for parallel programs that
  never terminate)
\item
  The upper bound of the speedup should in theory be p, but in practice
  it is often lower than p.
\end{itemize}

\begin{figure}[H]
\centering
\includegraphics[width=0.7\textwidth]{figures/metricExample.png}
\caption{Metric Example}
\end{figure}

\hypertarget{big-o-landau-notation}{%
\subsubsection{Big-O / Landau Notation}\label{big-o-landau-notation}}

\begin{itemize}
\tightlist
\item
  $\mathcal{O}$ (Big O): Obergrenze, maximal so viele Schritte
\item
  $\Omega$ (Omega): Untergrenze, mindestens so viele Schritte
\item
  $\Theta$ (Theta): Obergrenze = Untergrenze, genau so viel
\end{itemize}

\hypertarget{cost-of-a-parallel-system}{%
\subsection{Cost of a Parallel System}\label{cost-of-a-parallel-system}}

\begin{itemize}
\tightlist
\item
  Cost (amount of total work)

  \begin{itemize}
  \tightlist
  \item
    is the product of parallel runtime and the number of processing
    elements used
  \item
    Cost $= p * T_P$
  \end{itemize}
\item
  Cost-Optimal System

  \begin{itemize}
  \tightlist
  \item
    a parallel system is said to be cost-optimal if the cost of solving
    a problem on a parallel computer is asymptotically identical to
    serial cost since $E = T_S / pT_P$, for cost-optimal systems: $E = \mathcal{O}(1)$
  \end{itemize}
\end{itemize}

\hypertarget{impact-of-non-cost-optimality}{%
\subsection{Impact of
Non-Cost-Optimality}\label{impact-of-non-cost-optimality}}

\begin{itemize}
\tightlist
\item
  The efficiency is calculated from the speedup over the number of
  processors (S/P)
\item
  Increasing the processors doesn't always solve the problem. Even if
  the speedup is increasing, the efficiency can shrink
\item
  speedup goes down as the problem size n is increased for a given p
\item
  efficiency doesn't depend on p, but goes down as the problem size is
  increased
\end{itemize}

\hypertarget{effect-of-granularity-on-performance}{%
\subsection{Effect of Granularity on
Performance}\label{effect-of-granularity-on-performance}}

\begin{figure}[H]
\centering
\includegraphics[width=0.7\textwidth]{figures/granularityOnPerformance.png}
\caption{Effect of Granularity on Performance}
\end{figure}

\begin{itemize}
\tightlist
\item
  Scaling-Down a parallel system is using fewer than the maximum
  possible number of processing elements to execute a parallel algorithm
\item
  This often improves the efficiency
\item
  One way of scaling down could be

  \begin{itemize}
  \tightlist
  \item
    think of each processor in the original case as a virtual processor
    and to assign virtual processors equally to scaled down processors
  \item
    since the number of processing elements decreases by a factor of
    $n/p$, the computation at each processing element increases by a
    factor of $n/p$
  \end{itemize}
\end{itemize}

\clearpage


% !TEX root = ./busty_transcription.tex
\section{Discussion and future work}

The study of gene expression is one of the dominant themes of modern biology,
made all the more urgent by the dizzying pace at which genomes are being
sequenced. But there is a troubling Achilles heel buried in all of that genomic
data, which is our inability to find and interpret regulatory sequence. In
many cases, this is not possible even qualitatively, let alone the possibility
of quantitative dissection of the regulatory parts of genomes in a predictive
fashion. Other recent work has tackled the challenge of finding and annotating
the regulatory part of genomes \cite{Belliveau2018, Ireland2020}. Once we have
determined the architecture of the regulatory part of the genome, we are then
faced with the next class of questions which are sharpened by formulating them
in mathematical terms, namely, what are the input-output properties of these
regulatory circuits and what knobs control them?

The present work has tackled that question in the context of the first
regulatory architecture hypothesized in the molecular biology era, namely, the
repressor-operator model of Jacob and Monod~\cite{Jacob1961}. Regulation in that
architecture is the result of a competition between a repressor which inhibits
transcription and RNAP polymerase which undertakes it. Through the labors of
generations of geneticists, molecular biologists and biochemists, an
overwhelming amount of information and insight has been garnered into this
simple regulatory motif, licensing it as what one might call the ``hydrogen
atom'' of regulatory biology. It is from that perspective that the present paper
explores the extent to which some of the different models that have been
articulated to describe that motif allow us to understand both the average level
of gene expression found in a population of cells, the intrinsic cell-to-cell
variability, and the full gene expression distribution found in such a population
as would be reported in a single molecule mRNA Fluorescence \textit{in situ}
Hybridization experiment, for example.
We do so by contrasting two theoretical frameworks to think about the problem.
On the one hand, thermodynamic models are convenient ways to describe the
problem of gene regulation because of the relatively small number of parameters
compared with kinetic models, and the agnostic nature with respect to the full
reaction topology that leads to a transcription event. In addition, either
implicitly or explicitly, many approaches to gene regulation focus strictly on
occupancy and ideas such as $K_d$s, and thus is of great interest to test these
models. This comes with the limitation that thermodynamic models are only able
to make statements about mean gene expression of a quasi-steady state process.
On the other hand, kinetic models break those limitations, being able to make
explicit predictions about higher moments of the gene expression distribution,
and the transient states towards steady state, at the cost of propagating the
number of parameters, and demanding the full set of reactions

Our key insights can be summarized as follows. First, as shown in
Figure~\ref{fig1:means_cartoons}, the mean expression in the simple repression
architecture is captured by a master curve in which the action of repressor and
the details of the RNAP interaction with the promoter appear separately and
additively in an effective free energy. Interestingly, as has been shown
elsewhere in the context of the Monod-Wyman-Changeux model, these kinds of
coarse-graining results are an exact mathematical result and do not constitute
hopeful approximations or biological naivete \cite{Razo-Mejia2018, Chure2019}.
The fact that thermodynamic and kinetic models for the simple repression motif
can not only be reconciled, but are indistinguishable from each other at the
level of mean mRNA count is a consequence of the possible transitions between
promoter microstates. The promoter microstates in all models shown in Figure~1
can explicitly be separated into two groups: 1) promoters with repressor bound,
and 2) all other promoters states. The single path between these two groups of
states guarantees detailed-balance between the groups. We can therefore separate
\textit{expression} related transitions, which by definition must be out of
equilibrium, with \textit{regulation} transitions that can in principle be in
equilibrium. This implies that the probability distribution of the promoter
microstates not involving the repressor take the same functional form, allowing
us to write the fold-change as an effective free energy involving a regulation
term, $\Delta F_R$, and a promoter details term, $\rho$. We offer this argument
as a conjecture, and we suspect that a careful argument using the
graph-theoretic framework based on the Matrix-Tree
Theorem~\cite{Gunawardena2012}, might furnish a ``proof'' not with some
challenges as this theoretical framework applies, at the present, to finite
graphs.

To further dissect the relative merits of the different models, we must appeal
to higher moments of the gene expression probability distribution. To that end,
our second set of insights focus on gene expression noise, where it is seen that
a treatment of the constitutive promoter already reveals that some models have
Fano factors (variance/mean) that are less than one, at odds with any and all
experimental data that we are aware of~\cite{So2011, Jones2014}. This
theoretical result allows us to directly discard a subset of the models (models
1-3 in Figure~\ref{fig2:constit_cartoons}(A)) since they cannot be reconciled
with experimental observations. The two remaining models (models 4 and 5 in
Figure~\ref{fig2:constit_cartoons}) appear to contain enough microscopic realism
to be able to reproduce the data. A previous exploration of model 4 demonstrated
the ``sloppy''~\cite{Transtrum2015} nature of the model in which data on
single-cell mRNA counts alone cannot constrain the value of all parameters
simultaneously \cite{Razo-Mejia2020}. Here we demonstrate that the proposed
one-state bursty promoter model (model 5 in
Figure~\ref{fig2:constit_cartoons}(A)) emerges as a limit of the commonly used
two-state promoter model \cite{Peccoud1995, Shahrezaei2008, So2011, Sanchez2013,
Jones2014}. We put the idea to the test that this level of coarse-graining is
rich enough to reproduce previous experimental observations. In particular we
perform Bayesian inference to determine the two parameters describing the full
steady-state mRNA distribution, finding that the model is able to provide a
quantitative description of a plethora of promoter sequences with different mean
levels of expression and noise.

With the results of the constitutive promoter in hand, we then fix the
parameters associated with this class of promoters and use them as input for
evaluating the noise in gene expression for the simple repression motif itself.
This allows us to provide a single overarching analysis of both the constitutive
and simple repression architectures using one simple model and corresponding set
of self-consistent parameters, demonstrating not only a predictive framework,
but also reconciling the thermodynamic and kinetic views of the same simple
repression constructs. More specifically, we obtained values for the
transcription factor association and dissociation rates by performing Bayesian
inference on the full mRNA distribution for data obtained from simple-repression
promoters with varying number of transcription factors per cell and affinity of
such transcription factors for the binding site. The free energy value obtained
from these kinetic rates --computed as the log ratio of the rates-- agrees with
previous inferences performed only from mean gene expression measurements, that
assumed a thermodynamic rather than a kinetic
framework~\cite{Garcia2011a,Razo-Mejia2018}. Interestingly, to constrain the
binding and unbinding rates inferred for Figure~\ref{fig4:repressed_post_full},
all of the rates need to be simultaneously fit. This is because a single
combination of operator (repressor binding site) and aTc concentration
(repressor copy number) cannot constraint both rates independently, but only
their ratio (See Appendix~\ref{sec:bayesian} for further details).

It is important to be clear on the nature of the assumptions  present in all
models, including model 5 from Figure~\ref{fig2:constit_cartoons}(A). First, as
with many of the models used routinely in the analysis of transcription, the
models considered here imagine the promoter as existing in a series of discrete
states. This assumption is clearly an oversimplification due to effects such as
DNA supercoiling~\cite{Chong2014,Sevier2016} or DNA
looping~\cite{Boedicker2013a} which are both parameterized by  continuous
variables describing DNA shape; although there are formal and very interesting
ways of integrating out such degrees of freedom. Another facet of the models
used here is that they can feature irreversible processes characterized by only
a single rate process with no corresponding return pathway at odds with
thermodynamics. Such models are mathematically convenient and serve only as an
approximation to the more realistic situation in which the forward rates are
much larger than the backward rates. One consequence is that the inferred rates
for the transcription burst rate and burst size  can only be thought of as
effective rate constants. Another idealization of the models considered here is
that when computing the fold-change in gene expression (Section 2) we assume
that the translation efficiency of an mRNA transcript is the same in cells with
and without regulation. There might be some known biological effects that could
limit the applicability of this assumption such as cooperation between the
transcription and translation machinery~\cite{Proshkin2010}, or coupling of mRNA
transcription and degradation~\cite{Chen2015} among other potential effects.
Another implicit assumption in all of the models discussed here is that the
effects due to changes in gene copy number during the cell cycle have no effects
on the expression profile. For moderate growth rates such as those used
in~\cite{Jones2014} from which we obtained the data, at all points during the
cell cycle there are at most two copies of the gene of interest. In a recent
publication, we explored the effects that this change in gene dosage can have in
gene expression, finding that the effects are much more drastic at the protein
level~\cite{Razo-Mejia2020}.

More specifically, for model 5 from Figure~\ref{fig2:constit_cartoons}(A), it is
interesting to speculate what microscopic details are being coarse-grained by
our burst rate and burst size in Figure~\ref{fig2:constit_cartoons}(A), model 5.
Chromosomal locus is one possible influence we have not addressed in this work,
as all the single-molecule mRNA data from~\cite{Jones2014} that we considered
was from a construct integrated at the \textit{galK} locus. The results
of~\cite{Chong2014} indicate that transcription-induced supercoiling contributes
substantially in driving transcriptional bursting, and furthermore, their
Figure~7 suggests that the parameters describing the rate, duration, and size of
bursts vary substantially for transcription from different genomic loci.
Although the authors of~\cite{Englaender2017} do not address noise, they note
enormous differences in mean expression levels when an identical construct is
integrated at different genomic loci. The authors of~\cite{Engl2020} attribute
noise and burstiness in their single-molecule mRNA data to the influence of
different sigma factors, which is a reasonable conclusion from their data. Could
the difference also be due to the different chromosomal locations of the two
operons? What features of different loci are and are not important? Could our
preferred coarse-grained model capture the variability across different loci? If
so, and we were to repeat the parameter inference as done in this work, is there
a simple theoretical model we could build to understand the resulting
parameters?

In summary, this work took up the challenge of exploring the extent to which a
single specific mechanistic model of the simple-repression regulatory
architecture suffices to explain the broad sweep of experimental data for this
system. Pioneering early experimental efforts from the M\"{u}ller-Hill lab
established the simple-repression motif as an arena for the quantitative
dissection of regulatory response in bacteria, with similar work emerging in
examples such as the \textit{ara} and \textit{gal} operons as
well~\cite{Dunn1984b, Oehler1990, Weickert1993, Oehler1994, Schleif2000,
Semsey2002, SwintKruse2009}. In light of a new generation of precision
measurements on these systems, the definition of what it means to understand
them can now be formulated as a rigorous quantitative question. In particular,
we believe understanding of the simple repression motif has advanced
sufficiently that the design of new versions of the architecture is now
possible, based upon predictions about how repressor copy number and DNA binding
site strength control expression. In our view, the next step in the progression
is to first perform similar rigorous analyses of the fundamental ``basis set''
of regulatory architectures. Natural follow-ups to this work are explorations of
motifs such as simple activation that is regulated by a single activator binding
site, and the repressor-activator architecture, mediated by the binding of both
a single activator and a single repressor, and beyond. With the individual
input-output functions in hand, similar quantitative dissections including the
rigorous analysis of their tuning parameters can be undertaken for the ``basis
set'' of full gene-regulatory networks such as switches, feed-forward
architectures and oscillators for example, building upon the recent impressive
efforts from systems biologists and synthetic biologists~\cite{Milo2002,
Alon2007}.


\subsection{Ranged}
\subsubsection{Bow-type}
\begin{longtable}{|C{2cm} L{2cm} L{2cm} L{8cm}|}
\hline
\large{\textbf{Name}} &
\large{\textbf{Cost}} &
\large{\textbf{Handedness}} &
\large{\textbf{Damage}}
\\ \hline
\WeaponRow{Shortbow}{60 m}{Two}{
Depends on the ammunition, $\frac{AmmoDamage}{10}*Strength$
}{Shortbows are simple bows usually made for practice, sportsmanship or hunting rather than serious military use. They have a short range, and don't offer any bonuses to the damage caused by the arrows used. While the caused damage depends on the specific type of arrow used, such bows typically fail at penetrating armour.}
\WeaponRow{Longbow}{120 m}{Two}{
Depends on the ammunition, $\frac{AmmoDamage*1.5}{10}*Strength$
}{Longbows are sturdier bows roughly the height of their users. Typically used as weapons of war, they require quite a bit of strength to use, and thus, longbowmen are usually trained from a rather young age. They are famous for their long range, and their extra strength gives them some armour-penetrating ability, especially when used with bodkin arrows.}
\WeaponRow{Recurve Bow}{160 m}{Two}{
Depends on the ammunition, $\frac{AmmoDamage*1.5}{10}*Strength$
}{Bows where the limbs curve away from the archer when unstrung, typically made from composites of multiple materials, these weapons store and deliver energy more effectively than regular bows, but also require more strength to use. They are effectively shorter bows, but with the effectiveness of a longbow - though, they are also costlier to manufacture.}
\end{longtable}
\subsubsection{Crossbow}
\begin{longtable}{|C{2cm} L{2cm} L{2cm} L{8cm}|}
\hline
\large{\textbf{Name}} &
\large{\textbf{Cost}} &
\large{\textbf{Handedness}} &
\large{\textbf{Damage}}
\\ \hline
\WeaponRow{Stirrup Crossbow}{100 m}{Two}{
Depends on the ammunition
}{The earliest type of crossbow, it has a draw weight low enough to allow crossbowmen to draw the string back with their bare hands, but this also limits the maximum amount of draw weight such a crossbow can have, which in turn puts a limit on the strength of such a crossbow. With mundane ammunition, such crossbows are typically too weak to penetrate plate armour, merely causing dents.}
\WeaponRow{Goatsfoot Crossbow}{150 m}{Two}{
Depends on the ammunition, $AmmoDamage*1.5$
}{A variation of the stirrup crossbow - when draw weight is increased to the point that a crossbowman cannot pull the string back by their bare hands, some utilities are needed to do the job. The first of such utilities was the goat's foot lever, which slightly reduces rate of fire, but greatly increases the maximum draw weight - therefore maximum firepower - a crossbow can potentially have.}
\WeaponRow{Windlass Crossbow}{200 m}{Two}{
Depends on the ammunition, $AmmoDamage*2$
}{A further evolution of the crossbow - at one point, not even the goat's foot could lever could pull back the strings of the increasingly heavier crossbows, requiring a new utility to do the job: the windlass. Greatly increasing draw weight - potential firepower - at the expense of rate of fire, the windlass crossbow is only slighter faster to reload than a muzzle-loaded firearm, but compensates by penetrating platemail.}
\WeaponRow{Repeating Crossbow}{150 m}{Two}{
Depends on the ammunition, $\frac{AmmoDamage}{4}$ per shot
}{Effectively pump-action repeating crossbow, the repeating crossbow has low firepower and short range, but compensates by having quite a high rate of fire, making it effective at crowd-control against larger number of unarmoured opponents.}
\WeaponRow{Hand Crossbow}{80 m}{One}{
Depends on the ammunition, $\frac{AmmoDamage}{3}$
}{A pistol-shaped handheld crossbow small enough to be held in a single hand, it has a low firepower, short range and a rate of fire comparable to that of a stirrup crossbow, but compensates by being a lightweight hanheld weapon that can be held in a single hand.}
\end{longtable}
\subsubsection{Other}
\begin{longtable}{|C{2cm} L{2cm} L{2cm} L{8cm}|}
\hline
\large{\textbf{Name}} &
\large{\textbf{Cost}} &
\large{\textbf{Handedness}} &
\large{\textbf{Damage}}
\\ \hline
\WeaponRow{Sling}{50 m}{One}{
Depends on the ammunition
}{Ancient weapons that are used to hurl projectiles at the enemy - thus causing crushing damage -, short-ranged and inaccurate, they pack quite a punch when they actually hit their intended target.}
\WeaponRow{Javelin}{80 m}{One}{
\textit{Piercing:} $\frac{2d8 \pm modifiers}{5}*Strength$\hfill \break
\textit{Crushing:} $\frac{1d4 \pm modifiers}{5}*Strength$
}{Shorter spears that are made for throwing. Not much else can be said about them.}
\WeaponRow{Blowgun}{50 m}{Two}{
Depends on the ammunition, $\frac{AmmoDamage}{10}*Strength$
}{Primitive weapons where the projectiles are expelled from a narrow tube via the user's exhalation. Typically used with poisonous darts.}
\WeaponRow{Hand Cannon}{300 m}{Two}{
Depends on the ammunition
}{Primitive firearms powered by either gunpowder or a magical substance that fufills a similar function, hand cannons have an abysmally low rate of fire and a pathetically short range - even shorter than that of a shortbow, comparable to the pathetic range of repeating crossbows and hand crossbows - but compensate for all of this by having a devastating amount of firepower, and by making a loud noise that strikes fear into the hearts of uninitiated. At the weapon's effective range, no known armour can offer meaningful resistance to the bullets this weapon fires. The intended target's only hope for salvation is the weapon's infamously comical inaccuracy.}
\end{longtable}


\chapter{Parameterization of Human Knee Joints}
\label{sec:KneeParams}

The proposed and prototyped knee joint requires a specific input of patient parameters to achieve its goal of matching a person's knee movement. Therefore, a system must be developed to non-invasively identify a relationship between the flexion of the patient's knee joint and the linear movement of the shank away from the knee's joint center. There have been several attempts to parameterize the human knee joint using many different methods, including \cite{3DKinKneeJointOldStabby}, \cite{MRIKneeShape_Unloaded}, \cite{MRIKneeShape_Loaded}, and \cite{ModelAnalysisDeepKneeFlexion}. All of these studies focus on the tibiofemoral relationship, i.e. the bone movement within the lower limbs. However, rehabilitation exoskeletons connect to the skin of its user. Other studies have identified a difference between identifying knee kinematics through skin markers and bone pins \cite{reinschmidt1997effect}, suggesting there is in fact a distinct difference between tibiofemoral movement and skin movement around the knee. Therefore, an imaging workflow and software processing program needs to be developed to be able to parameterize a patient's knee movement with consideration to skin movement.

To identify the knee relationship, an imaging workflow as well as an accompanying software analysis tool is proposed and tested with pilot data. This imaging method uses a motion capture system to identify movement of specific markers attached to a person's skin. This will all be processed using a custom software workflow consisting of both Vicon image processing and custom Python scripts that will extract the parameters from the data. To test this workflow, an experiment is proposed using medical MR imaging to identify the kinematics of the knee joint, compare the developed knee joint to a traditional pin joint, and determine the relationship between the knee joint kinematics and the skin movement surrounding the knee.

\section{Proposed Method of Imaging}
\label{sec:ImaginKneeProcedure}

\begin{figure}[ht!]
    \centering
    \includegraphics[width=0.65\textwidth]{Figures/Param/StudyPatientPositioning.jpg}
    \caption{Depicts the patient position during motion capture. The patient will then flex and extend their knee while keeping the heel on the table.}
    \label{fig:ParamPatientPosition}
\end{figure}

The proposed imaging workflow can be split up into two major parts: the imaging session and the software processing. For flexibility and speed, the usage of a motion capture system is proposed, which can develop accurately track the position of motion dots. In this study, the motion capture system used was a 10-camera Vicon Vantage 5 \footnote{See Vicon system here: \url{https://www.vicon.com/hardware/cameras/vantage/}}, but any motion capture system is usable given enough precision.

To measure the knee flexion, a patient is placed on their back, shown in \autoref{fig:ParamPatientPosition}.Motion capture dots are then placed on the patient using double-sided tape to track the positions where the exoskeleton will attach to the skin. Two additional points can be placed on the knee for extra feedback and the final joint position, but are not necessary for the software system. Then, the patient will flex and extend their knee joint while keeping the heel on the table to stabilize the knee and prevent shanking. The placement of the patient should allow for a technician to help the patient bend their knee if their condition does not allow them to do so themselves. Once the data is collected, it can be processed by the parameterization software system.

\section{Parameterization Software}
% TODO: Add software flowchart
The collected data first goes through a Vicon workflow for processing. This will properly format the data, create virtual joints, and determine joint centers. This is done using the built-in tools called SCoRE (Symmetrical Center of Rotation Estimation) and SARA (Symmetrical Axis of Rotation Analysis). Additionally, a built-in tool will determine the angle of the knee joint using the four dots on each section of the patient's leg. This Vicon specific workflow can be exported, shared, and reused for others who are using a similar system. The output of the workflow is a CSV file which can be parsed by a custom script.

To finish the parameterization of the knee, custom software was developed and written in Python. This software utilizes the AiM Vicon Python module \footnote{The AiM Vicon Python module is an open-source project found here: \url{https://github.com/WPI-AIM/AIM_Vicon}} which parses the CSV file from above. The joint angle, SARA, SCoRE, and marker positions are extracted from the data and placed in their specified datatypes. Since the AiM Vicon module did not have all the tools needed for this project, I developed and contributed to the Python module. Additional features added include SARA and SCoRE support, a new 3D visualizer, and stability improvements when importing different workflows.

\begin{figure}[ht!]
    \centering
    \includegraphics[width=\textwidth]{Figures/Param/3D_Marker_Animation.png}
    \caption{An example of a possible output from the Parameterization Software System shown in a visualizer}
    \label{fig:SoftwareSampleData}
\end{figure}

With the data parsed and imported, the final step is to calculate the final joint angles and determine the linear extension between the thigh and shank. The axis of rotation from SCoRE and all marker positions are projected onto a plane, which is calculated to be approximately the same as the plane of the manufactured joint. Then the final distance and angle are calculated by drawing two virtual lines as shown in \autoref{fig:SoftwareSampleData}: 1. from the projected primary thigh marker to the projected SCoRE location and 2. from the projected SCoRE location to the projected primary shank marker. The lengths of these lines then become the calculated linear extension and the angle between these two lines become the flexion. These two metrics are then combined into a dataset, and a best fit quartic curve is selected to become the patient's knee parameters.

\section{Testing with Pilot Data}
Due to time constraints, IRB (Internal Review Board) approval was not obtained to run a full study. However, some data was used to develop the software platform that will analyze the data. The graphs in \autoref{fig:Subj1KneeParams} and \autoref{fig:Subj2KneeParams} show the relationships between the knee joint flexion (angle of the knee) and the linear extension (distance of the shank to the joint center).

\begin{figure}[ht!]
    \centering
    \includegraphics[width=\textwidth]{Figures/Param/Subj1_KneeProfile.png}
    \caption{A visualization of the angular flexion and linear extension of the shank from the joint center of the first dataset with subject 1. Subject bent and extended their knee joint 4 times.}
    \label{fig:Subj1KneeParams}
\end{figure}

\begin{figure}[ht!]
    \centering
    \includegraphics[width=\textwidth]{Figures/Param/Subj2_KneeProfile.png}
    \caption{A visualization of the angular flexion and linear extension of the shank from the joint center of the second dataset with subject 2. Subject bent and extended their knee joint 3 times.}
    \label{fig:Subj2KneeParams}
\end{figure}

The data displayed in \autoref{fig:Subj1KneeParams} has a variation of up to \(7mm\) for any given angle. However, what is most interesting is the difference in the direction of the flexion. When the knee joint angle is increasing (knee is being bent), the markers are closer together, compared to when the knee joint is decreasing. In any given direction of movement, variation is less than \(2mm\), suggesting that this seemingly cyclical trajectory is not due to measurement variations or inaccuracies.

\autoref{fig:Subj2KneeParams}'s data is seemingly less dependent on the knee joint angular velocity direction. All data is within a \(3mm\) variation at any given angle. The notable exception is the \(7mm\) spike that can be seen at roughly \(25^\circ\) mark. This can likely be explained by either an accidental shift in the markers or a slight measurement error in the motion capture system. However, there is not sufficient data to make a conclusion as to the reasoning.

The data analyzed both show that the maximum variation in skin movement through the flexion of the knee never exceeded \(7mm\), even with the outlier event from \autoref{fig:Subj2KneeParams}. In comparison, studies presented in \autoref{sec:KneeModel} describe a tibio\-femoral relationship that varies over \(40mm\) throughout the joint flexion. While these two data points are not statistically significant enough to make a conclusion for all human knees, the preliminary pilot data suggests that the skin movement is different than the tibiofemoral movement. An in-depth study is needed to make a definitive conclusion.

\section{Study Outline}

As a part of this thesis, an study was designed to test the developed parameterization software as well as the knee joint's capability to be customized to a specific person. However, due to time constraints, IRB (Internal Review Board) approval to run the human study was not obtained, and therefore the study was not run. The study's methodology is outlined below, and the IRB proposal documents are attached in \autoref{apx:ParamStudyDocs}.

Up to 6 subjects are to be selected to partake in this study. Each subject should not have any prior severe knee injury, and should be cleared to be imaged with an MRI. The study will start with imaging the knee using the motion capture procedures outlined in \autoref{sec:ImaginKneeProcedure}, and parameters will be selected using the parameterization software developed. These parameters will be used to manufacture an MRI safe personalized knee joint. Additionally, a second non-personalized pin joint will be manufactured and used as a control.

The second stage of the study will compare the fit of the two joints. The subject will undergo MR scans at up to 3 different knee angles, first with no joint and then subsequently with each type of joint. The movement (if any) of the joint in reference to the subjects body will be measured. A successful experiment should demonstrate that the customized knee joint moves less than a non-customized pin joint. Additionally, the tibiofemoral movement can be compared to the measured parameters to determine how much the skin moves relative to the knee joint.


%!TEX root = ../dissertation_vkslm.tex

\chapter{Automatic Handwritten signature verification systems}
\section{Handwritten Signatures}
In this context, the scientific
literature has proposed two ways to synthesize the signa-
tures [7], [8]:

1) Generation of duplicated samples. In this case algo-
rithms create artificial intra-personal variability from reference
specimens. These reference specimens are the seed of these
algorithms. Therefore, they do not create new individuals but
possible signatures executed by the signers. Both in on-line [9],
[10], [11], [12] and off-line [13], [14], [15], [16], [17], [18],
[19], [20], the algorithms are able to generate new samples of
a real user. Duplicated signatures present several advantages
such as training improvement in ASV, statistically meaningful
evaluations, enlarging the number of samples in a database,
etc.

2) Generation of new identities. Algorithms are designed
in this modality by taking into account different rules and
logics in order to simulate realistic effects. In [21] a model
is described to generate dynamic signatures based on visual
characteristics extracted from the time domain. The method
was evaluated from a visual validation. In Galbally et al. [7],
[8] the full generation of on-line flourish-based signatures was
carried out through two algorithms based on spectral analysis
and the kinematic theory of rapid human movements. The
first attempt to generate static flourishes in context of static
signature image is due to [22]. Two follow-up works were
presented in [1], [2] where authors coped with the generation
of Western static and dynamic signatures (text and flourish) as
well as skilled forgeries under a neuromotor approach.

\section{Generation of duplicated samples}

\section{Generation of new identities}