\breakpagebeforenextheadingtrue
\section{Implementation and Numerical Results}
\label{sec:73results}

\minitoc[0mm]{69mm}{8}

\parbox{1em}{}
\vspace{-3em}

\disableornamentsfornextheadingtrue
\subsection{Implementation}
\label{sec:731implementation}

\paragraph{Parameters, implementation, and geometry}

Details about implementational aspects of the model can be found in
\multicite{Sprenger15Continuum,Roehrle16Two,Valentin18Gradient},
for instance, values for the material parameters.
%
The constitutive law has been implemented in the CMISS software package
(an interactive computer program for Continuum Mechanics,
Image analysis, Signal processing and System identification%
\footnote{%
  \url{https://www.cmiss.org/}%
}).
The emerging PDEs are discretized using quadratic finite element basis
functions and the resulting linearized system is solved with CMISS.
%
The geometry of the human upper limb model is based on
the Visible Human Male's dataset \cite{Spitzer96Visible}.
Again, we refer to \multicite{Sprenger15Continuum,Roehrle16Two} for details
about the geometry.

\subsection{Reference and Sparse Grid Solution}
\label{sec:732solutionTypes}

\paragraph{Reference solution}

Since the model is only two-dimensional, we can compute a reference solution
on a full grid.
To this end, we evaluate the exerted muscle forces $\forceT$ and $\forceB$ on
the full grid
\begin{equation}
  \{\ang{10}, \ang{11}, \dotsc, \ang{150}\} \times \{0, 0.1, \dotsc, 1\}
  \ni (\elbang, \actX),\quad
  X \in \{\mathrm{T}, \mathrm{B}\}.
\end{equation}
The resulting \num{1551} grid points
are interpolated with bicubic full grid splines%
\footnote{%
  Computed with the Geometric Tools Engine \cite{Schneider03Geometric},
  see \url{https://www.geometrictools.com/}.%
}
to obtain \term{reference solutions}
$\forceTref, \forceBref\colon
\clint{\ang{10}, \ang{150}} \times \clint{0, 1} \to \real$,
which are shown in \cref{fig:biomech2ReferenceForce}.
Due to the high resolution of the full grid,
we may assume that the reference solutions are accurate enough
to ensure $\forceTref \approx \forceT$ and $\forceBref \approx \forceB$.
We refer to the resulting equilibrium elbow angle
with $\equielbangref{\forceL}$.
It is displayed in
\cref{fig:biomech2ReferenceEquilibriumAngle}
for the loads of $\forceL = \SI{22}{\newton}$,
$\SI{-60}{\newton}$, and $\SI{180}{\newton}$.

\begin{figure}
  \includegraphics{biomech2ReferenceForce_1}%
  \;\;%
  \includegraphics{biomech2ReferenceForce_2}%
  \hfill%
  \rlap{\raisebox{53mm}{\;$\forceXref$ [\si{\kilo\newton}]}}%
  \includegraphics{biomech2ReferenceForce_3}%
  \caption[Reference triceps and biceps forces]{%
    Reference triceps and biceps forces $\forceXref$
    ($X \in \{\mathrm{T}, \mathrm{B}\}$).%
  }%
  \label{fig:biomech2ReferenceForce}%
\end{figure}

\begin{figure}
  \includegraphics{biomech2ReferenceEquilibriumAngle_4}%
  \\[2mm]%
  \subcaptionbox{%
    $\forceL = \SI{22}{\newton}$%
  }[49mm]{%
    \includegraphics{biomech2ReferenceEquilibriumAngle_1}%
  }%
  \hfill%
  \subcaptionbox{%
    $\forceL = \SI{-60}{\newton}$%
  }[49mm]{%
    \includegraphics{biomech2ReferenceEquilibriumAngle_2}%
  }%
  \hfill%
  \subcaptionbox{%
    $\forceL = \SI{180}{\newton}$%
  }[49mm]{%
    \includegraphics{biomech2ReferenceEquilibriumAngle_3}%
  }%
  \caption[Reference equilibrium elbow angle]{%
    Reference equilibrium elbow angle $\equielbangref{\forceL}$
    for different loads $\forceL$.
    The empty areas correspond to activation pairs
    at which $\equielbangref{\forceL}$ is not well-defined
    (see \cref{eq:equilibriumAngle}).%
  }%
  \label{fig:biomech2ReferenceEquilibriumAngle}%
\end{figure}

\paragraph{Sparse grid solution}

Additionally, we evaluate $\forceT$ and $\forceB$ at the $\ngp = 49$
grid points
\begin{equation}
  \{(\elbang^{(k,\mathrm{unif})}, \actX^{(k,\mathrm{unif})}) \mid
  k = 1, \dotsc, \ngp\}
  \subset \clint{\ang{10}, \ang{150}} \times \clint{0, 1},\quad
  X \in \{\mathrm{T}, \mathrm{B}\},
\end{equation}
of the uniform regular sparse grid $\interiorregsgset{n}{d}$ of
level $n = 5$ in $d = 2$ dimensions
without boundary points (to reduce the number of samples)
and at the sparse Clenshaw--Curtis grid
\begin{equation}
  \{(\elbang^{(k,\cc)}, \actX^{(k,\cc)}) \mid
  k = 1, \dotsc, \ngp\}
  \subset \clint{\ang{10}, \ang{150}} \times \clint{0, 1},\quad
  X \in \{\mathrm{T}, \mathrm{B}\},
\end{equation}
of the same size and level.%
\footnote{%
  The domain $\clint{\ang{10}, \ang{150}} \times \clint{0, 1}$
  is assumed to be implicitly normalized to the unit square
  $\clint{\*0, \*1}$.%
}
These values are interpolated using three
different hierarchical B-spline bases of degree $p = 1$, $3$, and $5$:
modified hierarchical uniform B-splines
$\bspl[\modified]{\*l,\*i}{p}$
(see \cref{sec:313modification}),
modified hierarchical Clenshaw--Curtis B-splines
$\bspl[\cc,\modified]{\*l,\*i}{p}$
(see \cref{sec:314nonUniform}), and
modified hierarchical uniform not-a-knot B-splines
$\bspl[\nak,\modified]{\*l,\*i}{p}$
(see \cref{sec:323modifiedNAKBSplines}).
The implementation was done using the sparse grid toolbox
\sgpp{} \cite{Pflueger10Spatially}.%
\footnote{%
  \url{http://sgpp.sparsegrids.org/}%
}
The corresponding interpolants and resulting quantities
are denoted with the superscripts
``$\sparse,\!p$'', ``$\sparse,\!p,\!\cc$'', or ``$\sparse,\!p,\!\nak$'',
respectively.
A superscript of ``$\sparse$'' without any further specification
means one of the three hierarchical B-spline bases in general.
Note that the equilibrium elbow angle is \emph{not} interpolated
(neither in the full grid nor in the sparse grid case),
but rather obtained by inserting the interpolated muscle forces
into \eqref{eq:totalMomentSurrogate} and \eqref{eq:equilibriumAngleSurrogate}.

\subsection{Errors of Muscle Forces and Equilibrium Angle}
\label{sec:733errors}

\paragraph{Quality of reference interpolants}

Before we turn to the sparse grid interpolants,
we assess the quality of the reference interpolants on the full grid.
For this purpose, we evaluate the full grid interpolants
$\forceTintp, \forceBintp$
at the sparse grid points $(\elbang^{(k)}, \actX^{(k)})$
(which are not a subset of the full grid points!)
and compare the resulting values with the known exact values
$\forceT(\elbang^{(k)}, \actT^{(k)})$ and
$\forceB(\elbang^{(k)}, \actB^{(k)})$
of the muscle forces $\forceT, \forceB$.
We also incorporate the known values at the sparse
Clenshaw--Curtis grid points.
In particular, let $G$ be the union of
$\{(\elbang^{(k,\mathrm{unif})}, \actX^{(k,\mathrm{unif})}) \mid
k = 1, \dotsc, \ngp\}$ and
$\{(\elbang^{(k,\cc)}, \actX^{(k,\cc)}) \mid k = 1, \dotsc, \ngp\}$.
We then approximate the relative $\Ltwo$ interpolation error
of the reference interpolants by
\begin{equation}
  \frac{\normLtwo{\forceX - \forceXref}}{\normLtwo{\forceX}}
  \approx
  \frac{
    \setsize{G}^{-1/2}
    \norm[2]{
      (\forceX(\elbang, \actX) - \forceXref(\elbang, \actX))_
      {(\elbang, \actX) \in G}
    }
  }{
    \setsize{G}^{-1/2}
    \norm[2]{(\forceX(\elbang, \actX))_{(\elbang, \actX) \in G}}
  },\quad
  X \in \{\mathrm{T}, \mathrm{B}\},
\end{equation}
where $\norm[2]{\cdot}$ is the Euclidean norm.%
\footnote{%
  We have $\setsize{G} = 2\ngp - 1$, since sparse grids of
  uniform and Clenshaw--Curtis type only
  share the center point $(\elbang, \actX) = (\ang{80}, 0.5)$,
  if there are no boundary points.%
}
After inserting the known values $\forceX(\elbang, \actX)$ and
$\forceXref(\elbang, \actX)$ ($(\elbang, \actX) \in G$)
on the right-hand side, %\rhs
we obtain
\begin{equation}
  \frac{\normLtwo{\forceT - \forceTref}}{\normLtwo{\forceT}}
  \approx \SI{2.19}{\permille},\qquad
  \frac{\normLtwo{\forceB - \forceBref}}{\normLtwo{\forceB}}
  \approx \SI{2.06}{\permille}.
\end{equation}
These errors are very small, which justifies our assumption of
$\forceTref \approx \forceT$ and $\forceBref \approx \forceB$.

\paragraph{Error of sparse grid muscle forces}

\Cref{tbl:biomech2ErrorL2_1} contains the relative $\Ltwo$
interpolation errors
$\normLtwo{\forceXref - \forceXintp}/\normLtwo{\forceXref}$
($X \in \{\mathrm{T}, \mathrm{B}\}$) of the sparse grid interpolants
for all hierarchical bases and degrees $p = 1, 3, 5$.
All reported errors are relatively small
due to the smoothness of the original functions
(cf.\ $\forceXref$ in \cref{fig:biomech2ReferenceForce}).
All in all, the modified Clenshaw--Curtis B-splines perform best,
achieving relative $\Ltwo$ errors of below \SI{3.6}{\permille}
in the cubic case.
Surprisingly, the not-a-knot B-splines are the worst choice in our
comparison.
Their corresponding errors exceed \SI{1}{\percent} for the triceps
and $p > 1$.
The possible reasons are two-fold:
First, there might be slight noise in the given muscle force data,
which is visible in \cref{fig:biomech2ReferenceForce},
as there seems to be a kink in $\forceBref$ at $\elbang \approx \ang{25}$.
Second, the employed regular sparse grids might be too coarse
as the higher convergence order of not-a-knot B-splines
only pays off in the asymptotic range (see \cref{sec:541interpolation}).
The same observations hold for the degree $p$,
for which $p = 3$ seems to be the best choice,
as the errors increase again for $p = 5$.

\begin{table}
  \newcommand*{\bi}{$\bspl[\modified]{l,i}{p}$}
  \newcommand*{\bii}{$\bspl[\cc,\modified]{l,i}{p}$}
  \newcommand*{\biii}{$\bspl[\nak,\modified]{l,i}{p}$}
  \subcaptionbox{%
    $\normLtwo{\forceXref - \forceXintp}/\normLtwo{\forceXref}$
    [\si{\permille}] given as triceps/biceps pairs
    ($X \in \{\mathrm{T}, \mathrm{B}\}$).%
    \label{tbl:biomech2ErrorL2_1}%
  }[85.2mm]{%
    \setnumberoftableheaderrows{1}%
    \begin{tabular}{%
      >{\kern\tabcolsep}=l<{\kern2mm}%
      +c<{\kern-1mm}+c<{\kern-1mm}+c<{\kern\tabcolsep}%
    }
      \toprulec
      \headerrow
      $p$&   $1$&                  $3$&                  $5$\\
      \midrulec
      \bi&   $3.60,7.12$&          $3.05,7.00$&          $\mathbf{2.98},7.90$\\
      \bii&  $\mathbf{3.28},4.35$& $3.31,\mathbf{3.56}$& $3.35,3.64$\\
      \biii& $3.60,7.12$&          $3.09,10.0$&          $7.13,24.6$\\
      \bottomrulec
    \end{tabular}%
  }%
  \hfill%
  \subcaptionbox{%
    $\normLtwo{\equielbangref{\forceL} - \equielbangintp{\forceL}}/
    \normLtwo{\equielbangref{\forceL}}$
    [\si{\permille}] for $\forceL = \SI{22}{\newton}$.%
    \label{tbl:biomech2ErrorL2_2}%
  }[59mm]{%
    \setnumberoftableheaderrows{1}%
    \begin{tabular}{%
      >{\kern\tabcolsep}=l<{\kern2mm}%
      +c<{\kern-1mm}+c<{\kern-1mm}+c<{\kern\tabcolsep}%
    }
      \toprulec
      \headerrow
      $p$&   $1$&    $3$&             $5$\\
      \midrulec
      \bi&   $4.15$& $3.74$&          $3.72$\\
      \bii&  $3.42$& $\mathbf{2.83}$& $2.86$\\
      \biii& $4.15$& $4.06$&          $8.28$\\
      \bottomrulec
    \end{tabular}%
  }%
  \caption[Relative $L^2$ errors of forces and equilibrium elbow angle]{%
    Relative $\Ltwo$ errors of triceps/biceps force \emph{(left)} and
    equilibrium elbow angle \emph{(right)}
    for different hierarchical bases $\basis{\*l,\*i}$ and
    B-spline degrees $p$.
    Highlighted entries are the best among those with
    the same hierarchical basis or the same degree
    (similar to Nash equilibria).%
  }%
  \label{tbl:biomech2ErrorL2}%
\end{table}

\vspace{\fill}

\Cref{fig:biomech2ErrorForce} shows the pointwise absolute error
$\abs{\forceXref(\elbang, \actX) - \forceXintp(\elbang, \actX)}$
for the modified B-splines $\bspl[\modified]{l,i}{p}$ and
$\bspl[\cc,\modified]{l,i}{p}$
on uniform and Clenshaw--Curtis grids in the cubic case $p = 3$.
Note that in contrast to usual interpolation settings,
the absolute errors $\abs{\forceXref - \forceXintp}$
shown in \cref{fig:biomech2ErrorForce} do not vanish at the
sparse grid points $(\elbang^{(k)}, \actX^{(k)})$
($X \in \{\mathrm{T}, \mathrm{B}\}$, $k = 1, \dotsc, \ngp$),
since $\forceXintp$ does not interpolate $\forceXref$
at these points.%
\footnote{%
  It would have been possible to construct $\forceXintp$
  as a sparse grid interpolant of $\forceXref$.
  However, building a spline surrogate ($\forceXintp$)
  of another spline surrogate ($\forceXref$) would skew the results.%
}
As it is typical for (modified) sparse grid interpolants,
the error is the largest near the boundary of the domain.
However, the Clenshaw--Curtis points help to decrease the error
due to the higher density of grid points near the boundary.
In the Clenshaw--Curtis case, the maximal errors are
\begin{equation}
  \normLinfty{\forceTref - \forceTintp[p,\cc]}
  \approx \SI{10.6}{\newton},\qquad
  \normLinfty{\forceBref - \forceBintp[p,\cc]}
  \approx \SI{9.51}{\newton},
\end{equation}
where $\normLinfty{\forceXref - \forceXintp[p,\cc]}
\ceq \max_{(\elbang, \actX)}
\abs{\forceXref(\elbang, \actX) - \forceXintp[p,\cc](\elbang, \actX)}$
(since the functions are continuous).
If we restrict the domain to
$\clint{\ang{31}, \ang{129}} \times \clint{0.15, 0.85}$
by omitting \SI{15}{\percent} on each side of the original domain,
then the maximal absolute errors drop to only
\SI{6.73}{\newton} (triceps) and \SI{0.967}{\newton} (biceps),
which is small compared to maximal possible forces of
around \SI{1}{\kilo\newton}.

\begin{figure}
  \includegraphics{biomech2ErrorForce_7}%
  \\[2mm]%
  \subcaptionbox{%
    $\abs{\forceXref - \forceXintp[p]}$ for
    $X = \mathrm{T}$ \emph{(left)} and
    $X = \mathrm{B}$ \emph{(right).}%
  }[74mm]{%
    \includegraphics{biomech2ErrorForce_1}%
    \hfill%
    \includegraphics{biomech2ErrorForce_2}%
  }%
  \hfill%
  \subcaptionbox{%
    $\abs{\forceXref - \forceXintp[p,\cc]}$ for
    $X = \mathrm{T}$ \emph{(left)} and
    $X = \mathrm{B}$ \emph{(right).}%
  }[74mm]{%
    \includegraphics{biomech2ErrorForce_3}%
    \hfill%
    \includegraphics{biomech2ErrorForce_4}%
  }%
  \caption[Absolute error of muscle forces]{%
    Absolute error of muscle forces $\forceT, \forceB$ for
    modified cubic B-splines ($p = 3$)
    on sparse grids of uniform type \emph{(left two plots)} and
    of Clenshaw--Curtis type \emph{(right two plots)}
    together with the points of the sparse grid \emph{(dots).}%
  }%
  \label{fig:biomech2ErrorForce}%
\end{figure}

\pagebreak

\paragraph{Error of the equilibrium elbow angle}

The relative $\Ltwo$ errors
$\normLtwo{\equielbangref{\forceL} - \equielbangintp{\forceL}}/
\normLtwo{\equielbangref{\forceL}}$
of the equilibrium elbow angle function are shown in
\cref{tbl:biomech2ErrorL2_1} for the load of
$\forceL = \SI{22}{\newton}$.
Modified cubic Clenshaw--Curtis B-splines achieve the best results.
Therefore, we use this type of hierarchical basis
for the remainder of this chapter.
Pointwise plots of the absolute error
$\abs{\equielbangref{\forceL} - \equielbangintp[p,\cc]{\forceL}}$
are presented in \cref{fig:biomech2ErrorEquilibriumAngle}.
Again, the maximal error is comparatively small:
For $\forceL = \SI{22}{\newton}$, it is only \ang{0.886}.
If we restrict the domain to $\clint{0.15, 0.85}^2$,
then this maximal error drops to \ang{0.103} (or \ang{;6.18;}),
as the areas near the boundary of $\clint{\*0, \*1}$
contribute the most to the error.

\begin{figure}
  \includegraphics{biomech2ErrorEquilibriumAngle_5}%
  \\[2mm]%
  \subcaptionbox{%
    $\forceL = \SI{22}{\newton}$%
  }[49mm]{%
    \includegraphics{biomech2ErrorEquilibriumAngle_1}%
  }%
  \hfill%
  \subcaptionbox{%
    $\forceL = \SI{-60}{\newton}$%
  }[49mm]{%
    \includegraphics{biomech2ErrorEquilibriumAngle_2}%
  }%
  \hfill%
  \subcaptionbox{%
    $\forceL = \SI{180}{\newton}$%
  }[49mm]{%
    \includegraphics{biomech2ErrorEquilibriumAngle_3}%
  }%
  \caption[Absolute error of the equilibrium elbow angle]{%
    Absolute error
    $\abs{\equielbangref{\forceL} - \equielbangintp[p,\cc]{\forceL}}$
    of the equilibrium elbow angle for
    modified hierarchical cubic Clenshaw--Curtis B-splines ($p = 3$)
    for different loads $\forceL$.
    In the empty areas, at least one of
    $\equielbangref{\forceL}$ and $\equielbangintp[p,\cc]{\forceL}$
    is not well-defined (see \cref{eq:equilibriumAngle}).%
  }%
  \label{fig:biomech2ErrorEquilibriumAngle}%
\end{figure}

\subsection{Test Scenario}
\label{sec:734scenario}

\paragraph{Definition of the test scenario}

In the following, we want to assess the performance
of the sparse grid interpolants for the optimization problems
\ref{item:biomech2MinSum} and \ref{item:biomech2MinDist}.
For this goal, we create a test scenario \cite{Valentin18Gradient}
that simulates a pseudo-dynamic sequence of motions
by varying the load force and/or the target elbow angle
in discrete time steps $t$ as seen in \cref{fig:biomech2ScenarioA_1}.
The test scenario is as follows:
%
\begin{figure}
  \includegraphics{biomech2ScenarioA_5}%
  \\[2mm]%
  \subcaptionbox{%
    Load $\forceL$ and target elbow angle $\tarelbang$.%
    \label{fig:biomech2ScenarioA_1}%
  }[73.3mm]{%
    \hspace*{4.5mm}%
    \includegraphics{biomech2ScenarioA_1}%
    \hspace*{2.4mm}%
  }%
  \hfill%
  \subcaptionbox{%
    Optimal activation parameters $\actT$ and $\actB$.%
    \label{fig:biomech2ScenarioA_2}%
  }[73.3mm]{%
    \hspace*{0.0mm}%
    \includegraphics{biomech2ScenarioA_2}%
    \hspace*{9.8mm}%
  }%
  \\[1mm]%
  \subcaptionbox{%
    Deviation $\abs{\equielbang{\forceL} - \tarelbang}$
    of attained elbow angle to target and
    deviation \smash{$|\momentref|$} of the moment from equilibrium.%
    \label{fig:biomech2ScenarioA_3}%
  }[73.3mm]{%
    \includegraphics{biomech2ScenarioA_3}%
  }%
  \hfill%
  \subcaptionbox{%
    Number of evaluations of $\equielbang{\forceL}$
    and number of Newton iterations per evaluation
    of $\equielbang{\forceL}$.%
    \label{fig:biomech2ScenarioA_4}%
  }[73.3mm]{%
    \includegraphics{biomech2ScenarioA_4}%
  }%
  \caption[Settings and results of the test scenario]{%
    Setting (a) of the test scenario and corresponding results (b, c, d).%
  }%
  \label{fig:biomech2ScenarioA}%
\end{figure}
%
\begin{enumerate}
  \item
  Find a feasible initial solution for problem \ref{item:biomech2MinSum}
  with $\forceL(t_0) \ceq \SI{22}{\newton}$ and
  $\tarelbang(t_0) \ceq \ang{75}$.

  \item
  Apply \ref{item:biomech2MinSum} with $\forceL(t_1) \ceq \SI{22}{\newton}$ and
  $\tarelbang(t_1) \ceq \ang{75}$.

  \item
  Apply \ref{item:biomech2MinDist} with $\forceL(t_2) \ceq \SI{22}{\newton}$ and
  $\tarelbang(t_2) \ceq \ang{60}$ (changed target angle).

  \item
  Apply \ref{item:biomech2MinDist} with $\forceL(t_3) \ceq \SI{30}{\newton}$ and
  $\tarelbang(t_3) \ceq \ang{60}$ (changed load).

  \item
  Apply \ref{item:biomech2MinDist} with $\forceL(t_4) \ceq \SI{40}{\newton}$ and
  $\tarelbang(t_4) \ceq \ang{50}$ (changed load and target angle).
\end{enumerate}
%
For each of the steps 2 to 5, the activation levels $\actT, \actB$ obtained
in the previous step (i.e., either the feasible initial solution
of step 1 or the optimal solution of steps 2 to 4) are used
as the input of the optimization problem
\ref{item:biomech2MinSum} or \ref{item:biomech2MinDist}.
The feasible initial solution in step 1 is determined as explained
in \cref{sec:513gradientBasedConstrained}.

\paragraph{Solutions of problem \ref{item:biomech2MinSum}}

We note that independently of $\forceL$ and $\tarelbang$,
every solution $(\actT, \actB)$ of problem \ref{item:biomech2MinSum} will be
on the boundary part of the domain $\clint{\*0, \*1}$,
on which at least one activation parameter vanishes, i.e.,
\begin{equation}
  \{(\actT, \actB) \in \clint{\*0, \*1} \mid
  (\actT = 0) \lor (\actB = 0)\}.
\end{equation}
The reason is that the two muscles triceps and biceps are antagonistic
(see \cref{sec:711models}), meaning that they work against each other.
If both $\actT > 0$ and $\actB > 0$, then the body will waste energy,
as the same target elbow angle can be attained by reducing both
$\actT$ and $\actB$ simultaneously, thus requiring less energy.
A visual example for this is \cref{fig:biomech2ReferenceEquilibriumAngle},
where the contour lines generally go from the bottom left
(small $\actT, \actB$) to the top right (large $\actT, \actB$).
This issue may be prevented by either
more complicated musculoskeletal models with more
than two muscles or different optimization problems
such as problem \ref{item:biomech2MinDist},
where the objective function differs.

\paragraph{Plots of optimization results}

\Cref{fig:biomech2ScenarioA_2,fig:biomech2ScenarioA_3,fig:biomech2ScenarioA_4}
show the results of the test scenario using the muscle forces
$\forceXintp[p,\cc]$ obtained by interpolating with
modified hierarchical cubic Clenshaw--Curtis B-splines (solid lines, $p = 3$).
As comparison, we repeat the solution process
with the forces obtained by interpolating with the
corresponding hierarchical piecewise linear basis (dashed lines, $p = 1$) and
with the reference forces $\forceXref$ (dotted lines).
For the piecewise linear basis,
we use exactly the same method as for the cubic case
(Newton method for $\equielbangintp{\forceL}$,
Augmented Lagrangian with adaptive gradient descent for the
solution of problems \ref{item:biomech2MinSum} and \ref{item:biomech2MinDist}),
although the derivatives of the muscle forces are discontinuous.
For the reference forces, we use the fact that the reference surrogates
are full grid spline interpolants, which can be explicitly differentiated.
Without the full grid interpolants,
we would have to approximate the derivatives with finite differences.

\vspace*{\fill}
\pagebreak

\paragraph{Equilibrium elbow angle}

In \cref{fig:biomech2ScenarioA_2}, we see that the activation levels
of all three methods are more or less the same.
However, \cref{fig:biomech2ScenarioA_3} reveals that even these small
differences lead to deviations of the resulting equilibrium elbow angle
to the target angle that differ by up to two orders of magnitude.
The two green lines with filled markers at the bottom of
\cref{fig:biomech2ScenarioA_3} show the error of
the equilibrium elbow angle $\equielbangintp{\forceL}$
using sparse grid interpolation to the desired target angle $\tarelbang$.
Unsurprisingly, this error is very small as
it is minimized by the optimizer as part of the constraint.
The true error, which is obtained by
using the reference equilibrium elbow angle $\equielbangref{\forceL}$,
is in general much larger
(top two green lines in \cref{fig:biomech2ScenarioA_3}
with hollow markers).
We see that the cubic B-splines decrease the error
by up to two orders of magnitude compared to the
piecewise linear basis.
There are two reasons for this:
First, the error of $\equielbang{\forceL}$ is generally smaller
when using higher-order B-splines as we have seen above.
Second, higher-order B-splines are continuously differentiable,
which makes them suitable for gradient-based optimization.
In contrast, the surrogates obtained by piecewise linear interpolation
have kinks, which may complicate finding optimal points
in the augmented Lagrangian and Newton methods.

\paragraph{Number of evaluations and Newton iterations}

This is supported by \cref{fig:biomech2ScenarioA_4},
which shows the number of evaluations of $\equielbang{\forceL}$
during the optimization and the average number of Newton iterations
per evaluation.
While the number of total evaluations is similar for all three methods,
the number of required Newton iterations to achieve convergence
is in general around \SI{50}{\percent} larger for the piecewise linear
basis functions.

\subsection{Spatial Adaptivity}
\label{sec:735adaptivity}

\paragraph{Generation of a spatially adaptive sparse grid}

As mentioned in \cite{Valentin18Gradient}, spatial adaptivity
may be employed to reduce the number of necessary muscle force samples
even further,
especially for more complicated musculoskeletal systems with
more parameters.
To verify this statement, we remove all grid points
$(\elbang^{(k,\cc)}, \actX^{(k,\cc)})$
from the regular sparse Clenshaw--Curtis grid that satisfy
\begin{equation}
  \frac{
    \abs{\alpha_\mathrm{T}^{(k,p,\cc)}}
  }{
    \max_{k'} \abs{\alpha_\mathrm{T}^{(k',p,\cc)}}
  } < \SI{1}{\percent}
  \quad\text{and}\quad
  \frac{
    \abs{\alpha_\mathrm{B}^{(k,p,\cc)}}
  }{
    \max_{k'} \abs{\alpha_\mathrm{B}^{(k',p,\cc)}}
  } < \SI{1}{\percent},
\end{equation}
where $\alpha_X^{(k,p,\cc)}$ ($X \in \{\mathrm{T}, \mathrm{B}\}$)
is the hierarchical surplus of the basis function $\bspl[\cc,\modified]{k}{p}$
corresponding to $(\elbang^{(k,\cc)}, \actX^{(k,\cc)})$.
For higher-dimensional models,
one would of course not sample muscle data on a regular sparse grid
and then coarsen the data by removing points,
but rather use an a posteriori adaptivity criterion to
decide which grid points to refine iteratively.

\paragraph{Comparison with the regular case}

For the cubic case $p = 3$,
the resulting force interpolants $\forceXintp[p,\cc,\mathrm{adap}]$ together
with the spatially adaptive sparse grid
(which has been coarsened from 49 to 28 points) and
equilibrium elbow angle $\equielbangintp[p,\cc,\mathrm{adap}]{\forceL}$
for $\forceL = \SI{22}{\newton}$ are shown in
\cref{fig:biomech2SpatiallyAdaptive}.
The sparse grid is almost dimensionally adaptive,
as $\forceXref$ seems to be almost linear in the $\actX$ direction
for both $X = \mathrm{T}$ and $X = \mathrm{B}$.
The errors increase slightly:
The relative $\Ltwo$ force errors for $(\mathrm{T}, \mathrm{B})$ increase
from $(\SI{3.31}{\permille}, \SI{3.56}{\permille})$
to $(\SI{3.36}{\permille}, \SI{4.43}{\permille})$,
and the absolute $\Linfty$ errors increase
from $(\SI{10.6}{\newton}, \SI{9.51}{\newton})$
to $(\SI{12.3}{\newton}, \SI{9.57}{\newton})$.
In addition, the relative $\Ltwo$ and absolute $\Linfty$ errors
for $\equielbang{\forceL}$ increase
from \SI{2.83}{\permille} and \ang{0.886}
to \SI{4.12}{\permille} and \ang{1.09}, respectively.
While all these errors are somewhat larger than for the regular sparse grid,
they are still at an acceptable level,
but the number of necessary muscle force evaluations is halved compared
to the regular case.
Additionally, the solution of the test scenario doesn't change significantly
due to the similar errors of $\forceX$ and $\equielbang{\forceL}$.

\begin{figure}
  \hspace*{2mm}%
  \raisebox{0.2mm}{\includegraphics{biomech2ErrorForce_8}}%
  \hspace*{12mm}%
  \includegraphics{biomech2ErrorEquilibriumAngle_6}%
  \\[2mm]%
  \subcaptionbox{%
    $\abs{\forceTref - \forceTintp[p,\cc,\mathrm{adap}]}$%
  }[49mm]{%
    \raisebox{1.02mm}{\includegraphics{biomech2ErrorForce_5}}%
  }%
  \hfill%
  \subcaptionbox{%
    $\abs{\forceBref - \forceBintp[p,\cc,\mathrm{adap}]}$%
  }[49mm]{%
    \raisebox{1.02mm}{\includegraphics{biomech2ErrorForce_6}}%
  }%
  \hfill%
  \subcaptionbox{%
    $\abs{
      \equielbangref{\forceL} - \equielbangintp[p,\cc,\mathrm{adap}]{\forceL}
    }$
    for $\forceL = \SI{22}{\newton}$%
  }[49mm]{%
    \includegraphics{biomech2ErrorEquilibriumAngle_4}%
  }%
  \caption[%
    Errors of muscle forces and equilibrium angle
    for the spatially adaptive case%
  ]{%
    Errors of muscle forces and equilibrium elbow angle
    for the spatially adaptive case
    (modified hierarchical cubic Clenshaw--Curtis B-splines,
    i.e., $p = 3$) together with the points of the
    spatially adaptive sparse grid \emph{(dots).}%
  }%
  \label{fig:biomech2SpatiallyAdaptive}%
\end{figure}


\section{Quantum Phases of Gapped Hamiltonians}
Revision.
There are two definitions of phase.

The first one is this:
$H_1$, $H_2$ are in the same quantum ($T=0$) phase of matter
if and only if
we have a continuous parameter $\lambda$ and a path connecting them such that
$H(\lambda)$ stays open.

The second definition is the traditional thermodynamics definition.
$H_1$ and $H_2$ are in the same phase if and only if
there is a path $H(\lambda)$ such that
\begin{align}
    \lim_{N\to\infty} \frac{E(\lambda)}{N}
\end{align}
is smooth.

The third definition is this:
$H_1$ and $H_2$ are the same phase if and only if
there is a path $H(\lambda)$
with ground states $\ket{\psi_i(\lambda)}$
such that the ground state expectation value
${\langle\mathcal{O}\rangle}_{\lambda}$
is a smooth function of $\lambda$ for every local operator $\mathcal{O}$.

Some remarks:
\begin{itemize}
    \item We will use (1)
    \item (2) is the usual thermodynamic definition in terms of free energy
        density.
    \item (2) and (3) apply for gapless systems.
\end{itemize}

There is a conjecture.
\begin{conjecture}
    If $H(\lambda)$ remains gapped,
    then ${\langle\mathcal{O}\rangle}_{\lambda}$
    is a smooth function of $\lambda$.
\end{conjecture}
This implies that if two phases are the same according to (1)
then they are also different according to (3).

\section{Constant Depth Circuits}
You can have two states in the same phase
but they might not be related by a constant depth local circuit.

We have a bunch of local gates that act on our qubits.
Local quantum circuit is a circuit where each layer has gates with finite
support locally.
It is a composition of these layers.
Constant depth means that the number of layers is less than a constant as the
size goes to infinity.

Let $U$ be a constant depth local unitary circuit.
If $\ket{\psi}$ is the ground state of some Hamiltonian $H$,
then $U\ket{\psi}$ is in the same phase.

However, it is not true that if I have two states $\ket{\psi_1}$ and
$\ket{\psi_2}$
then $\ket{\psi_1} = U\ket{\psi_2}$.

An example of this is the following.
Suppose $\ket{\psi_1}$ is a product state.
The correlations must be exactly zero beyond a certain range,
because it cannot create correlations beyond a certain distance $\xi$ in space
that has to do with the gates and range.

If you have one state which is a trivial phase of matter
but correlations have exponentially decaying tails,
you cannot go to one with $U$.

You can fix this by allowing the notaries have tails,
but then it's tricky
and I've never seen anyone made this work.
It's not that useful.
If only useful to show two states are in the same state.

If you allow logarithmic depth,
you can connect states that are in different phases
and that's too much.

One hope would be to give up the fact you want it to be exactly
$\ket{\psi_1} = U\ket{\psi_2}$
and make it constant error instead.

In particular,
if you have a gapped ground state $\ket{\psi_1}$
and this unitary circuit $U$,
then there should be an adiabatic path between the Hamiltonians,
and you should be able to construct that unitary path
given this circuit.

One more point is that you can add symmetry to the problem.
If you assume the problem has symmetry,
then you must make sure that whole path respects the symmetry.
For example,
quantum hall phases have paths that are sometimes incompatible with
time-reversal symmetry.

When you add symmetry,
you end up splitting into many phases that cannot be connected by
symmetry-respecting paths.

\section{Topology in Gapped Ground States}
Now let's connect this to the mathematical structure of TQFT.

Here is some gapped phase of matter,
but where is the topology?

The properties robust to arbitrary perturbations.

Universal properties that distinguish different phases are usually characterised
by TQFTs.
On the one hand,
I can think of gapped quantum phases of matter.
The idea is that as far as we know, although there's no proof,
these are in one-to-one correspondence with deformation classes of TQFTs.

This statement as I said it is not quite right,
depending on how you define TQFT.
We believe this is true in $1+1$ and $2+1$ spacetime dimensions,
but in higher dimensions,
you could have more complicated things happening.

Topological liquids are interesting.
There is the space of all gapped phases of matter.
Then inside this space,
there are topological liquids
which have one-to-one correspondence with deformation classes of TQFTs.
Whereas gapped phases for $D\ge 3 + 1$,
we're not really sure,
but we thin they are in one-to-one correspondence with
deformation classes of TQFTs with \emph{defect networks}.

The mathematical structure TQFT has a log of topology.
They essentially classify gapped phases of matter with some caveats.
TQFTs may come with continuous parameters.
You have to be really careful what these objects are.

\begin{question}
    What are deformation classes?
\end{question}
A TQFT $\mathcal{T}_\lambda$ can be split up into spaces and points.
You could have some theory $\mathcal{T}^1_\lambda$ for some continuous parameter
$\lambda$,
you could have another one $\mathcal{T}^2_\lambda$
and then you have some which are just points $\mathcal{T}^3_\lambda$, etc.

That there are TQFTs without lattice descriptions,
some TQFTs do not have an exact lattice description.
Every TQFT useful for gapped phases of matter
can arise from some microscopic description on a lattice.
It might not be exactly solvable.

The rough intuition is that if you want to define TQFTs,
you want to make sure all quantities that come up are
topological invariants.
If you use triangulation of the space you're describing,
it should be independent of the triangulation.
Some gapped phases of matter require you to specify some microscopic geometry,
and you can't avoid it.

There's nothing outside of topological liquids outside of 1 and 2 spatial
dimensions,
but in 3 dimensions it's weird and clever.

\section{Topological Quantum Field Theory Basics}
I want to give you the plain vanilla version.
The discussion is abstract and mathematical,
but it's worth doing before moving into examples.

I want to tell you some basics of TQFTs.
I want to talk about ``plain'' TQFT,
but there are additives like oriented and unitary.
I will tell you the plain TQFT and then soup it up.

TQFT is this really amazing and intricate math structure that \emph{emerges} out
of some messy quantum many-body system with an energy gap.
It's not obvious why it should emerge from a gapped quantum many-body system.

To me, it's an absolutely amazing and profound thing.
You can have something so mathematically rich just emerge from some quantum
many-body system.
What I'm going to discuss are the \emph{Atiyah's axioms}.
Atiyah is one of the foremost mathematicians of the century
who died recently.
When discussing with Witten,
Atiyah saw what Witten was doing and axiomised it.
So it listed some axioms TQFTs should satisfy.
So it is an axiomatic framework that describes TQFTs.
One interesting aspect is that it is a way of defining a restricted class of
QFTs in a rigorous fashion.
In QFT, you have problems with lattice discretisation and perturbation theory.
But TQFT can be mathematically rigorously defined and used in physics,
so it's beautiful common ground.

These Atiyah axioms are related to some other axioms proposed at around the same
time, including those by Segal,
who was trying to define 2D \emph{conformal field theory} axioms,
which has a lot more structure that TQFTs.

These are useful for eventually giving a rigorous definition of quantum field
theory.
A few months ago there was an article in Quanta magazine about the mathematical
problem of defining QFT.\@
How do you define a QFT,
no one really knows.
TQFT might be useful.

\begin{question}
    What's the problem with QFT?\@
\end{question}
We know how to rigorously define free field theory.
We know how to rigorously define perturbation theory of free field theory.
There was this whole program of axiomatic quantum field theory in the late 20th
century that never became useful and cannot capture what we know.
We can't understand 2D CFT from the ideas that we had.
You need a whole new set of tools,
like operators algebras, a complicated story.
You can rigorously define some aspects of simple QFTs
but people are after something more broad
that can explain all the things we know about all the QFTs we know.

I'm not going to say anything mathematically complicated.
Feel free to say if you don't understand some mathematical terminology,
don't be shy.

There's an abstract definition of plain vanilla TQFT.
\begin{definition}
    A $(d+1)$-dimensional TQFT is is a symmetric monoidal functor
    between two categories $\mathcal{F}: \mathrm{Cob}_{d + 1}\to \mathrm{Vec}$.
\end{definition}

$\Cob_{d + 1}$ is a tensor category of $(d + 1)$-dimensional cobordisms.

A \emph{category} is a mathematical structure that consists of objects
and \emph{morphisms} between objects.
At a naive level,
it's a set of objects and arrows between objects called morphisms.
And this is a 1-category.
In our case,
objects of $\Cob_{d + 1}$ are $d$-dimensional closed manifolds
and a morphism between two objects,
which are $d$-dimensional manifolds,
say from $\Sigma_1^d\to\Sigma_2^d$,
is a $(d+1)$-dimensional manifold $M^{d+1}$
such that its boundaries are $\Sigma_1^d$ and $\Sigma_2^d$,
i.e.
$\partial M^{d + 1} = \Sigma_1^d\sqcup \bar{\Sigma}_2^d$.

Let's do an example $\Sigma_2 = S^1$, which is a circle
and we have a cobordism to $\Sigma_1 = S^1\sqcup S^1$.
A morphism looks like this
[picture of pair of pants]
The manifold $M^{d+1}$ is called a (co)-bordism
from $\Sigma_1^d$ to $\Sigma_2^d$.
Bord means boundary in French.
The point is that if you hae two manifolds,
and if you can go from one another in a highr-dimensional manifold,
then that's a bordism, sometimes called a co-bordism.

\begin{question}
    So there's nothing differnt betwen cobordism and bodirsm?
\end{question}
Two words for the same hting,
but in some cnotexts,
they'll call maps from this into $U(1)$ as a cobordism.
Mahtematicians do yuse oth interchangably.

\begin{question}
    Can you go other that such that clause?
    What is $\sqcup$?
\end{question}
Objects of this category are $d$-dimensiona closed manifiolds.
A morphism is a way to gor from one objcet toa nother.
A cobordism is a $d+1$-dimeisional manifold with boundaries that are the
objects.
It's disjoint union.

\begin{question}
    Category of sets, category of topological sets.
    Weird notation?
\end{question}
Objects of Vec are vector spaces and the morphisms are maps betwene vector
spaces, so it is a bit weird.

\begin{question}
    Are there some topological properties?
\end{question}
If two manifolds are bordred with each other,
it just means they're boundaries of some higher-dimesional manifold,
but that's about it.

\begin{question}
    Are two manifolds always connected by a $(d+1)$-dimensional manifolds?
\end{question}
No, there's a cobordism group notion,
group by equivalence classes based on whether there is a cobordism betwen them.

Every 2D manifold is related to every 2D manifold,
so that's trivial,
but it's not true in higher-dimensional space.

\begin{question}
    What are the points in between?
\end{question}
A morphism here is this higher dimensional manifold.

For every object, there may or may not be a morphism between them.

It's more than a map.
The TQFT gives a map, but $\mathrm{Cob}_{d+1}$ is the whole manifold.

\begin{question}
    What is a tensor category?
\end{question}
I'll come back to that.

\begin{question}
    What is they're not circles but spheres?
    What does that look like?
\end{question}
You want me to draw in 4D?\@

You have $\Sigma_1^2 = S^2 \sqcup S^2$ and
$\Sigma_2^2=S^2$.
Then the bordism is
$M^3 = B^3\sqcup (S^2\times I)$
and the boundary is
$\partial M^3 = S^2 \sqcup (S^2 \times S^2)$

\begin{question}
    Does the cobordism have to be connected?
\end{question}
No, it doesn't have to connected.
There are many possible morphisms between objects.

\begin{question}
    Can you give an example of what these objects are physically?
\end{question}
To some extent I can,
but it's related to why a miracle TQFT emerges out of a system.
II haven't gotten there yet.
I can define stuff on all kinds of spacetime manifolds.
I'm going to define stuff for eveyr $(d+1)$-dimensional spacetime manifodls.
But Condensed matter systmes don't have spacetime,
only time and space the system lives on.
It's hard to give examples which use all this technology irght now.
But a simple example is this.
Actually, hold that question until I get further.
Let me define a few things about it and then I'll try to address your question.

The next thing is the tensor category.
Tensor means there's a notion of a tensor product here.
There is a tensor product betwene objects
$\Sigma_1\otimes\Sigma_2$.
I need to give a meaning for tensor product,
which I say is just the disjoint union.
$\Sigma_1 \otimes \Sigma_2 := \Sigma_1 \sqcup \Sigma_2$.
Now I need to define $\mathrm{Vec}$, which is a tensor category.
Objects are finite-dimensional vector spaces of $\mathbb{C}$.
Morphisms are linear maps between vector spaces.

In our case,
because we're doing \emph{unitary} TQFT,
we want these linear maps to be unitary maps.
Actually, no unitary for now, we'll add it later.
The tensor aspect should be obvious.
The tensor product is just the tensor product of vector spaces.

A functor is just a map between categories.
For every object on the left side,
I have an object on the right side.
And for every morphism on the left I have on on the right.
Monoidal means it respects the tensor product on the left and right side.
Symmetric means it applies the same on each side.

What the TQFT is telling is is that
for every closed $d$-manifold $\Sigma_d$,
we have a finite-dimensional vector space $V(\Sigma^d)$.
For every $M^{d+1}$ cobordism between
$\Sigma_1^d$ and $\Sigma_2^d$,
we have an unitary linear map
$Z(M^{d+1}):V(\Sigma_1^d)\to V(\Sigma_2^d)$.

This is the outline of the more abstract definitoin.
Now I'm going to be more specific about exactly what data the axioms satisfy.
It's all hidden in this very abstract construction of a symmetric monoidal
uniatry functor between two categories.

\begin{question}
    You can only have unitary vector spaces of the same size?
\end{question}
Actually, it should be a linear map,
only unitary if they are the same dimension.
That's why you want the manifold to be closed.

\begin{question}
    Is this always defined for two objects?
\end{question}
The tensor product $\Sigma_1\otimes\Sigma_2$ defines for me a single object.
Then if I have another one
$\Sigma_1\sqcup\Sigma_2$.
[pair of pants picture]

Again, let me emphasize,
it's totally not obvious why something like this should come out of a quantum
many-body system.
I have linear maps associated with every $(d+1)$-dimensional bordisms.
Why should this structure arise?
It's not obvious.

\begin{question}
    $V(\Sigma^d)$
\end{question}
Every manifold is associated with a vector space.
You can think of $\Sigma^d$ as an argument.
For every $\Sigma^d$ you have a different vector space.

\begin{question}
    Does the dimension of the vector space have something to do with the
    dimension of the manifold?
\end{question}
No.

\begin{question}
    What is monoidal and symmetric?
\end{question}
I can write it down.

Monoidal means that
$\mathcal{F}(\Sigma_1^d\otimes \Sigma_2^d)
\simeq
V(\Sigma_1^d)\otimes V(\Sigma_2^d)$.
Here $\simeq$ means isomorphic.

Symmetric means that
$\mathcal{F}(\Sigma_1^d\otimes\Sigma_2^d)\simeq
\matcal{F}(\Sigma_2^d\otimes\Sigma_1^2)$.

\subsection{Defining data}
Let's define what's the data.
$V(\Sigma^d)$ is a finite-dimensional vector space.
There is a path integral or partition function $Z(M^{d+1})$.
When $M^{d+1}$ is closed,
then $Z(M^{d+1})\in\mathbb{C}$.
When $M^{d+1}$ has boundary, then
$Z(M^{d+1})\in V(\partial M^{d=1})$.

This is an alternative way of thinking about it.
When $M^{d+1}$ is closed, you have a point,
but when it has a boundary, you have a state.

\begin{question}
    If I have a 5D TQFT,
    can we define a state on $\mathbb{CP}^2$.
\end{question}
No path integral will define this.
But you still have a vector space.

\section{Atiyah's Axioms}
You could think of this as the data of a TQFT
and these are the axioms.

\begin{axiom}
    All manifolds have orientation.
\end{axiom}

\begin{axiom}
    $Z$ and $V$ are \emph{functorial} with respect to orientation-preserving
    diffeomorphisms of $M^{d+1}$ and $\Sigma^d$.
\end{axiom}
Let me try to explain what this means.

$M^{d+1}$ had two boundaries,
and $Z(M^{d+1})$ was thought of as a map.

Actually, we're out of time,
I'll tell you what this axiom means next time.

\begin{question}
    What does $V$ mean physically?
\end{question}
$V$ is the ground state degeneracy on that manifold.


\section{The WORG Method}
\label{method}

In order to describe the WORG method, first it is is useful to define
notation for demand curves and their parameterization.  Call $t$ the time
[years] up to some maximal time horizon $T$ (e.g. 50 years) over which time
the demand curve is known.  Then call $f(t)$ the demand curve in the natural
units of the facility type (such as [GWe] for reactors).
$f(t)$ may be any function that is desired, including non-differential
functions. Moreover, $f(t)$ need not return a simple scalar.  In a
multi-objective case, this function would return a vector of independent
objective values.
For example, though, the demand curve for a 1\% growth rate
starting at 90 [GWe] has the following form:
\begin{equation}
\label{f-1}
f(t) = 90\times 1.01^t
\end{equation}
Additionally, call $\Theta$ the deployment schedule for the facilities that
may be constructed to meet the demand.
$\Theta$ is a sequence of $P$ parameters, indexed by $p$, as seen in
Equation \ref{Theta}.
\begin{equation}
\label{Theta}
\Theta = \left\{\theta_1, \theta_2, \ldots, \theta_P\right\}
\end{equation}
Each $\theta_p$ represents that number of facilities to deploy on its
time step. In simple cases where there is only one type of facility
to deploy $P == T$.  However, when the deployment schedules of multiple
facility types are needed to meet the same demand curve, $P > T$.  The usual
example for $P > T$ is for transition scenarios which necessarily require
multiple kinds of reactors.

Now denote $M$ as the sequence for the minimum number of facilities deployable
for each deployment parameter. Also, call $N$ the sequence of the maximum number
of facilities deployable. The deployment parameters are thus each defined
on the range $\theta_p \in [M_p, N_p]$. Furthermore, because only whole
numbers of facilities may be deployed $\theta_p \in \N$.  It is also typical,
but not required, for $M = \mathbf{0}$. Zero is also the lower bound
for all possible $\theta_p$ as facilities may not be forcibly retired via the
deployment schedule.

From here, call $g(t, \Theta)$ the production as a function of time for a
given deployment schedule. This has the same units as the demand curve.
Thus for power demand and reactor deployments, $g$ is in units of [GWe]. The
optimization problem can now be posed as an attempt to find a $\Theta$
that minimizes the difference between $f$ and $g$.

\input{dtw}
\input{gp}
\input{algo}


\documentclass[11pt]{amsart}
\usepackage{geometry}                % See geometry.pdf to learn the layout options. There are lots.
\geometry{letterpaper}                   % ... or a4paper or a5paper or ...
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\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{epstopdf}
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\title{Brief Article}
\author{The Author}
%\date{}                                           % Activate to display a given date or no date

\begin{document}
%\maketitle
%\section{}
%\subsection{}
Let $(X, \eta)$ and $(Y, \rho)$ be two Polish spaces.
$C(X, Y)$ is the set of all continuous mappings $f: X\mapsto Y$.
For $f, g\in C(X, Y)$, we define
$$d (f, g) = \sup_{x\in X} \rho(f(x),  g(x)).$$
\begin{enumerate}
\item
Prove that $(C(X, Y), d)$ is a Polish space.
\item If $K \subset Y$ are compact, is $C(X, K)$ compact in $C(X, Y)$?
\iffalse
\item Let $F \subset C(X, K)$ be a equicontinuous. This means, $\forall \epsilon >0$, there $\exists \delta>0$, such that
\begin{itemize}
\item
if $\eta(x, y) < \delta$ and $f\in F$, then $\rho(f(x), f(y))<\epsilon$.
\end{itemize}
Is $F$ compact in $C(X, Y)$?
\fi
\end{enumerate}

\end{document}


\subsection{Reiterate self-energy requirements}

\begin{frame}
  \frametitle{Reiterate self-energy requirements}
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  \begin{block}{Rules for using self-energies}

    Coupling a \emph{bulk} electrode to a device requires(!) coupling region to behave
    \emph{bulk} as well.

    \vspace{4pt}

    \begin{center}
      \begin{tikzpicture}

        \foreach \x in {0,1,2,3,4,5,6,7,8} {
            \def\tmpcol{black}
            \ifnum\x>2
            \def\tmpcol{red!70!black}
            \fi
            \ifnum\x>3
            \def\tmpcol{green!70!black}
            \fi
            \ifnum\x>4
            \def\tmpcol{red!70!black}
            \fi
            \ifnum\x>5
            \def\tmpcol{black}
            \fi
            \node[block,gray] at ({(\x-0.5)*\bsize},3*\bsize) {};

            \expandafter\fill\expandafter[\tmpcol] ({(\x-0.75)*\bsize},3*\bsize)
            circle (3pt);
            \expandafter\fill\expandafter[\tmpcol] ({(\x-0.25)*\bsize},3*\bsize)
            circle (3pt);

        }

        \node[block dd] at (-1.5*\bsize,0.5*\bsize) {$\SE_{-}$};
        \foreach \x in {0,1,2,3,4,5,6,7,8} {
            \ifnum\x<3
            \def\tmpnum{0}
            \fi
            \ifnum\x>2
            \pgfmathparse{int(\x-2)}
            \edef\tmpnum{\pgfmathresult}
            \fi
            \ifnum\x>5
            \pgfmathparse{int(4)}
            \edef\tmpnum{\pgfmathresult}
            \fi
            \node[block] (A\x) at ({(\x-0.5)*\bsize},0.5*\bsize) {$\HH_\tmpnum$};
            \ifnum\x>0
            \pgfmathparse{int(\x-1)}
            \edef\xp{\pgfmathresult}
            \ifnum\x>6
            \pgfmathparse{int(5)}
            \edef\tmpnum{\pgfmathresult}
            \fi
            \draw[->,dd] (A\xp) to[out=75,in=105] node[above] {$\VV_\tmpnum$} (A\x);
            \draw[<-,dd] (A\xp) to[out=-75,in=-105] node[below] {$\VV^\dagger_\tmpnum$} (A\x);
            \fi
        }
        \node[block dd] at (8.5*\bsize,0.5*\bsize) {$\SE_{+}$};
      \end{tikzpicture}

    \end{center}

    \vspace{-12pt}

    \begin{itemize}
      \item<+-> Remember that $\SE_{-/+}$ is a correction to the Hamiltonian (i.e.
      $\HH' = \HH + \SE$)

      \item \emph{Extremely} important in TranSiesta, electrostatics are long-range!
    \end{itemize}
  \end{block}

  % \begin{block}<2->{Use symmetries whenever you can}

  %   \begin{itemize}
  %     \item If you have transverse periodic electrodes you should apply Bloch's theorem
  %     using the flag \texttt{Bloch}
  %   \end{itemize}

  % \end{block}

\end{frame}

%%% Local Variables:
%%% mode: latex
%%% TeX-master: "talk"
%%% End:


\twocolumn

\chapter*{Appendix}

The table below contains an initial segment of the fine spectrum for each of the classes in this survey. The classes are ordered in lexicographically decreasing order of their fine spectrum sequence and, if available, the sequence is followed by a link to the \url{oeis.org} entry for this sequence.

{\small
%\begin{multicols}{2}
\begin{tabular}{|l|l|l|}\hline
Name& Fine spectrum& \href{oeis.org}{OEIS}\\\hline
\hyperlink{PoMag}{PoMag}& 1, 16, 4051 &No\\
\hyperlink{PoImpA}{PoImpA}& 1, 16, 3981 &No\\
\hyperlink{PoSgrp}{PoSgrp}& 1, 11, 173, 4753, 198838,...&No\\
\hyperlink{Mag}{Mag}& 1, 10, 3330, 178981952,...&\href{http://oeis.org/A001329}{A001329}\\
\hyperlink{Srng}{Srng}& 1, 10, 132, 2341 &No\\
\hyperlink{CPoSgrp}{CPoSgrp}& 1, 7, 83, 1468, 37248,... &No\\
\hyperlink{MedMag}{MedMag}& 1, 7, 75, 3969 &No\\
\hyperlink{IdPoSgrp}{IdPoSgrp}& 1, 7, 69, 1035 &No\\
\hyperlink{MMag}{MMag}& 1, 6, 280&\\
\hyperlink{JImpA}{JImpA}& 1, 6, 245&\\
\hyperlink{MImpA}{MImpA}& 1, 6, 220&\\
\hyperlink{JMag}{JMag}& 1, 6, 220&\\
\hyperlink{ToMag}{ToMag}& 1, 6, 175&\\
\hyperlink{ToImpA}{ToImpA}& 1, 6, 175&\\
\hyperlink{MultLat}{MultLat}& 1, 6, 175&\\
\hyperlink{DLMag}{DLMag}& 1, 6, 175&\\
\hyperlink{DLImpA}{DLImpA}& 1, 6, 175&\\
\hyperlink{LMag}{LMag}& 1, 6, 175&\\
\hyperlink{LImpA}{LImpA}& 1, 6, 175&\\
\hyperlink{DivPos}{DivPos}& 1, 6, 123 &\\
\hyperlink{LrPoMag}{LrPoMag}& 1, 6, 110&\\
\hyperlink{MSgrp}{MSgrp}& 1, 6, 70, 1437 &No\\
\hyperlink{JSgrp}{JSgrp}& 1, 6, 61, 866 &No\\
\hyperlink{CDivPos}{CDivPos}& 1, 6, 55, 1434 &No\\
\hyperlink{DLSgrp}{DLSgrp}& 1, 6, 44, 479 &No\\
\hyperlink{LSgrp}{LSgrp}& 1, 6, 44, 479 &No\\
\hyperlink{ToSgrp}{ToSgrp}& 1, 6, 44, 386 &\href{http://oeis.org/A084965}{A084965}\\
\hyperlink{PoUn}{PoUn}& 1, 6, 43, 452 &No\\
\hyperlink{PoNUn}{PoNUn}& 1, 6, 39, 386, 5203 &No\\
\hyperlink{BMag}{BMag}& 1, 6, 0, 1176, 0, 0, 0 &No\\
\hyperlink{BImpA}{BImpA}& 1, 6, 0, 1176, 0, 0, 0 &No\\
\hyperlink{BSgrp}{BSgrp}& 1, 6, 0, 93, 0, 0, 0 &No\\
\hyperlink{LrPoSgrp}{LrPoSgrp}& 1, 5, 28, 273, 3788 &No\\
\hyperlink{Sgrp}{Sgrp}& 1, 5, 24, 188, 1915, 28634,...&\href{http://oeis.org/A027851}{A027851}\\
\hyperlink{DivJslat}{DivJslat}& 1, 4, 281 &No\\
\hyperlink{DivMslat}{DivMslat}& 1, 4, 216 &\\
\hyperlink{DivLat}{DivLat}& 1, 4, 216&\\
\hyperlink{ToDivLat}{ToDivLat}& 1, 4, 216&\\
\hyperlink{DDivLat}{DDivLat}& 1, 4, 216&\\
\hyperlink{CnjMag}{CnjMag}& 1, 4, 215&\\
\end{tabular}

\begin{tabular}{|l|l|l|}
\hyperlink{CMag}{CMag}& 1, 4, 129, 43968, 254429900,... &\href{http://oeis.org/A001425}{A001425}\\
\hyperlink{CDivJslat}{CDivJslat}& 1, 4, 79, 7545 &No\\
\hyperlink{CDivMslat}{CDivMslat}& 1, 4, 64, 6208 &No\\
\hyperlink{CDivLat}{CDivLat}& 1, 4, 64, 6208 &No\\
\hyperlink{PoMon}{PoMon}& 1, 4, 37, 549 &No\\
\hyperlink{CMSgrp}{CMSgrp}& 1, 4, 32, 432 &\href{http://oeis.org/A009668}{??}\\
\hyperlink{CJSgrp}{CJSgrp}& 1, 4, 29, 289 &No\\
\hyperlink{IdMSgrp}{IdMSgrp}& 1, 4, 28, 308, 4694 &No\\
\hyperlink{CPoMon}{CPoMon}& 1, 4, 27, 301, 4887 &No\\
\hyperlink{IdJSgrp}{IdJSgrp}& 1, 4, 23, 166, 1379 &No\\
\hyperlink{Srng$_0$}{Srng$_0$}& 1, 4, 22, 283 &No\\
\hyperlink{Srng$_1$}{Srng$_1$}& 1, 4, 22, 169, 1819 &No\\
\hyperlink{CDLSgrp}{CDLSgrp}& 1, 4, 20, 149, 1106 &No\\
\hyperlink{CLSgrp}{CLSgrp}& 1, 4, 20, 149, 1427 &No\\
\hyperlink{CToSgrp}{CToSgrp}& 1, 4, 20, 114, 710, 4726,... &\href{http://oeis.org/A346414}{A346414}\\
\hyperlink{IdLSgrp}{IdLSgrp}& 1, 4, 17, 100, 674 &No\\
\hyperlink{DIdLSgrp}{DIdLSgrp}& 1, 4, 17, 100, 576 &No\\
\hyperlink{IdToSgrp}{IdToSgrp}& 1, 4, 17, 82, 422 &\href{http://oeis.org/A181517}{??}\\
\hyperlink{RPoUn}{RPoUn}& 1, 4, 16, 87, 562 &No\\
\hyperlink{GalPos}{GalPos}& 1, 4, 15, 83, 539 &No\\
\hyperlink{InPoMag}{InPoMag}& 1, 4, 12, 77, 498 &No\\
\hyperlink{CyInPoMag}{CyInPoMag}& 1, 4, 12, 76, 481 &No\\
\hyperlink{CInPoMag}{CInPoMag}& 1, 4, 12, 69, 354, 3632 &No\\
\hyperlink{InPoSgrp}{InPoSgrp}& 1, 4, 10, 50, 210, 1721 &No\\
\hyperlink{CyInPoSgrp}{CyInPoSgrp}& 1, 4, 10, 50, 196, 1397 &No\\
\hyperlink{CInPoSgrp}{CInPoSgrp}& 1, 4, 10, 50, 194, 1356 &No\\
\hyperlink{BCSgrp}{BCSgrp}& 1, 4, 0, 35, 0, 0, 0, 1237, 0 &No\\
\hyperlink{BIdSgrp}{BIdSgrp}& 1, 4, 0, 18, 0, 0, 0, 88, 0, 0 &No\\
\hyperlink{LrMMag}{LrMMag}& 1, 3, 52, 4827 &No\\
\hyperlink{LrJMag}{LrJMag}& 1, 3, 52, 4827 &No\\
\hyperlink{LrLMag}{LrLMag}& 1, 3, 50, 4441 &No\\
\hyperlink{DLrLMag}{DLrLMag}& 1, 3, 50, 4441 &No\\
\hyperlink{LrToMag}{LrToMag}& 1, 3, 50, 4116 &\href{http://oeis.org/A071094}{??}\\
\hyperlink{RtQgrp}{RtQgrp}& 1, 3, 44, 14022 &\href{http://oeis.org/A193623}{??}\\
\hyperlink{RPoMag}{RPoMag}& 1, 3, 28, 1200 &No\\
\hyperlink{IdPoMon}{IdPoMon}& 1, 3, 23, 238, 3356 &No\\
\hyperlink{CDDivLat}{CDDivLat}& 1, 3, 20, 364 &\href{http://oeis.org/A003150}{??}\\
\hyperlink{CToDivLat}{CToDivLat}& 1, 3, 20, 294 &No\\
\hyperlink{LrMSgrp}{LrMSgrp}& 1, 3, 19, 199, 2946 &No\\
\hyperlink{LrJSgrp}{LrJSgrp}& 1, 3, 19, 192 &No\\
\hyperlink{CIdPoSgrp}{CIdPoSgrp}& 1, 3, 19, 171, 2069 &No\\
\hyperlink{LrLSgrp}{LrLSgrp}& 1, 3, 18, 183, 2500 &No\\
\hyperlink{DLrLSgrp}{DLrLSgrp}& 1, 3, 18, 183, 1968 &No\\
\hyperlink{LrToSgrp}{LrToSgrp}& 1, 3, 18, 144, 1370 &No\\
\hyperlink{MUn}{MUn}& 1, 3, 17, 138, 1555 &No\\
\hyperlink{QtMag}{QtMag}& 1, 3, 16, 218 &\href{http://oeis.org/A000273}{??}\\
\hyperlink{CRPoMag}{CRPoMag}& 1, 3, 16, 180, 4761 &No\\
\hyperlink{RPoSgrp}{RPoSgrp}& 1, 3, 16, 154, 2100 &No\\
\hyperlink{JUn}{JUn}& 1, 3, 16, 104, 822 &No\\
\hyperlink{MNUn}{MNUn}& 1, 3, 15, 113, 1167 &No\\
\hyperlink{JNUn}{JNUn}& 1, 3, 15, 113, 1167 &No\\
\hyperlink{CIdPoMon}{CIdPoMon}& 1, 3, 13, 86, 759 &No\\
\hyperlink{CRPoSgrp}{CRPoSgrp}& 1, 3, 12, 76, 670 &No\\
\hyperlink{IdLrPoSgrp}{IdLrPoSgrp}& 1, 3, 12, 71, 524 &No\\
\hyperlink{CSgrp}{CSgrp}& 1, 3, 12, 58, 325, 2143, 17291,... &\href{http://oeis.org/A001426}{A001426}\\
\hyperlink{pPos}{pPos}& 1, 3, 11, 47, 243 &No\\
\end{tabular}

\begin{tabular}{|l|l|l|}
\hyperlink{LNUn}{LNUn}& 1, 3, 10, 56, 457 &No\\
\hyperlink{DLNUn}{DLNUn}& 1, 3, 10, 56, 276 &No\\
\hyperlink{LUn}{LUn}& 1, 3, 10, 50, 313 &No\\
\hyperlink{DLUn}{DLUn}& 1, 3, 10, 50, 226 &No\\
\hyperlink{Bnd}{Bnd}& 1, 3, 10, 46, 251, 1682, 13213 &\href{http://oeis.org/A058112}{A058112}\\
\hyperlink{ToNUn}{ToNUn}& 1, 3, 10, 35, 126, 462 &\\
\hyperlink{ToUn}{ToUn}& 1, 3, 10, 35, 126, 462 &\\
\hyperlink{RegSgrp}{RegSgrp}& 1, 3, 9, 42, 206, 1352, 10168,... &\href{http://oeis.org/A001427}{A001427}\\
\hyperlink{NBnd}{NBnd}& 1, 3, 8, 30, 114, 536 &No\\
\hyperlink{InPoMon}{InPoMon}& 1, 3, 5, 20, 39, 179, 500 &No\\
\hyperlink{CyInPoMon}{CyInPoMon}& 1, 3, 5, 20, 39, 176, 493 &No\\
\hyperlink{CInPoMon}{CInPoMon}& 1, 3, 5, 20, 39, 174, 488 &No\\
\hyperlink{InPos}{InPos}& 1, 3, 5, 16, 30, 108 &No\\
\hyperlink{NRng}{NRng}& 1, 3, 5, 35, 10, 99, 24, 3856,...&\href{http://oeis.org/A305858}{A305858}\\
\hyperlink{BDivLat}{BDivLat}& 1, 3, 0, 325 &No\\
\hyperlink{BLrMag}{BLrMag}& 1, 3, 0, 325, 0, 0, 0 &No\\
\hyperlink{BCDivLat}{BCDivLat}& 1, 3, 0, 70, 0, 0, 0 &No\\
\hyperlink{BLrSgrp}{BLrSgrp}& 1, 3, 0, 39, 0 &No\\
\hyperlink{BUn}{BUn}& 1, 3, 0, 15, 0, 0, 0, 147, 0 &No\\
\hyperlink{BNUn}{BNUn}& 1, 3, 0, 15, 0, 0, 0, 147, 0 &No\\
\hyperlink{Shell}{Shell}& 1, 2, 243 &\\
\hyperlink{RMMag}{RMMag}& 1, 2, 20, 1116 &No\\
\hyperlink{RJMag}{RJMag}& 1, 2, 20, 1116 &No\\
\hyperlink{RLMag}{RLMag}& 1, 2, 20, 1116 &No\\
\hyperlink{DRLMag}{DRLMag}& 1, 2, 20, 1116 &No\\
\hyperlink{RToMag}{RToMag}& 1, 2, 20, 980 &\href{http://oeis.org/A008793}{??}\\
\hyperlink{MMon}{MMon}& 1, 2, 14, 168, 3488 &No\\
\hyperlink{RMSgrp}{RMSgrp}& 1, 2, 12, 129, 1852 &No\\
\hyperlink{RJSgrp}{RJSgrp}& 1, 2, 12, 129, 1852 &No\\
\hyperlink{RLSgrp}{RLSgrp}& 1, 2, 12, 129, 1852 &No\\
\hyperlink{IdSrng$_0$}{IdSrng$_0$}& 1, 2, 12, 129, 1852 &No\\
\hyperlink{DRLSgrp}{DRLSgrp}& 1, 2, 12, 129, 1437 &No\\
\hyperlink{RToSgrp}{RToSgrp}& 1, 2, 12, 101, 1003 &No\\
\hyperlink{Sgrp$_0$}{Sgrp$_0$}& 1, 2, 12, 90, 960 &No\\
\hyperlink{JMon}{JMon}& 1, 2, 11, 73, 703 &No\\
\hyperlink{CRMMag}{CRMMag}& 1, 2, 10, 148, 4398 &No\\
\hyperlink{CRJMag}{CRJMag}& 1, 2, 10, 148, 4398 &No\\
\hyperlink{CRLMag}{CRLMag}& 1, 2, 10, 148, 4398 &No\\
\hyperlink{CDRLMag}{CDRLMag}& 1, 2, 10, 148, 3554 &No\\
\hyperlink{CRToMag}{CRToMag}& 1, 2, 10, 112, 2772 &\href{http://oeis.org/A049505}{??}\\
\hyperlink{CMMon}{CMMon}& 1, 2, 10, 92, 1322 &No\\
\hyperlink{IdMMon}{IdMMon}& 1, 2, 10, 81, 950 &No\\
\hyperlink{FL}{FL}& 1, 2, 9, 79, 737 &No\\
\hyperlink{FL$_e$}{FL$_e$}& 1, 2, 9, 63, 492 &No\\
\hyperlink{CJMon}{CJMon}& 1, 2, 9, 55, 437 &No\\
\hyperlink{GalJslat}{GalJslat}& 1, 2, 9, 52, 361, 2947 &No\\
\hyperlink{CRMSgrp}{CRMSgrp}& 1, 2, 8, 57, 550 &No\\
\hyperlink{CRJSgrp}{CRJSgrp}& 1, 2, 8, 57, 550 &No\\
\hyperlink{CRLSgrp}{CRLSgrp}& 1, 2, 8, 57, 550 &No\\
\hyperlink{CRSlSgrp}{CRSlSgrp}& 1, 2, 8, 57, 392 &No\\
\hyperlink{CDRLSgrp}{CDRLSgrp}& 1, 2, 8, 57, 392 &No\\
\hyperlink{CIdMSgrp}{CIdMSgrp}& 1, 2, 8, 53, 498 &No\\
\hyperlink{IdLrMSgrp}{IdLrMSgrp}& 1, 2, 8, 46, 345, 3180 &No\\
\hyperlink{LMon}{LMon}& 1, 2, 8, 45, 347 &No\\
\hyperlink{IdLrJSgrp}{IdLrJSgrp}& 1, 2, 8, 45, 304 &No\\
\hyperlink{DLMon}{DLMon}& 1, 2, 8, 45, 279 &No\\
\hyperlink{CRToSgrp}{CRToSgrp}& 1, 2, 8, 41, 241 &No\\
\hyperlink{ToMon}{ToMon}& 1, 2, 8, 34, 184, 1218,... &\href{http://oeis.org/A346413}{A346413}\\
\hyperlink{IdLrLSgrp}{IdLrLSgrp}& 1, 2, 7, 40, 273 &No\\
\hyperlink{DIdLrLSgrp}{DIdLrLSgrp}& 1, 2, 7, 40, 213 &No\\
\hyperlink{Mon}{Mon}& 1, 2, 7, 35, 228, 2237, 31559 &\href{http://oeis.org/A058129}{A058129}\\
\hyperlink{CIdJSgrp}{CIdJSgrp}& 1, 2, 7, 33, 185 &\\
\hyperlink{IdLrToSgrp}{IdLrToSgrp}& 1, 2, 7, 30, 144, 740 &No\\
\end{tabular}

\begin{tabular}{|l|l|l|}
\hyperlink{IdJMon}{IdJMon}& 1, 2, 7, 29, 136 &\href{http://oeis.org/A307389}{??}\\
\hyperlink{OrdA}{OrdA}& 1, 2, 7, 36, 251 &No\\
\hyperlink{Srng$_{01}$}{Srng$_{01}$}& 1, 2, 6, 40, 295, 3246 &No\\
\hyperlink{FL$_c$}{FL$_c$}& 1, 2, 6, 39, 279 &No\\
\hyperlink{LrPoMon}{LrPoMon}& 1, 2, 6, 32, 234, 2493 &No\\
\hyperlink{CIdMMon}{CIdMMon}& 1, 2, 6, 31, 228, 2205 &No\\
\hyperlink{CLMon}{CLMon}& 1, 2, 6, 31, 199 &No\\
\hyperlink{FL$_{ec}$}{FL$_{ec}$}& 1, 2, 6, 31, 199 &No\\
\hyperlink{CDLMon}{CDLMon}& 1, 2, 6, 31, 149 &No\\
\hyperlink{GalMslat}{GalMslat}& 1, 2, 6, 30, 184, 1373 &No\\
\hyperlink{GalLat}{GalLat}& 1, 2, 6, 30, 184 &No\\
\hyperlink{DGalLat}{DGalLat}& 1, 2, 6, 30, 126 &No\\
\hyperlink{IdLMon}{IdLMon}& 1, 2, 6, 22, 93, 439 &No\\
\hyperlink{CToMon}{CToMon}& 1, 2, 6, 22, 92, 426 &\\
\hyperlink{DIdLMon}{DIdLMon}& 1, 2, 6, 22, 75, 274 &No\\
\hyperlink{GalToLat}{GalToLat}& 1, 2, 6, 20, 70, 252, 924 &\\
\hyperlink{IdToMon}{IdToMon}& 1, 2, 6, 16, 44, 120 &\\
\hyperlink{InLMag}{InLMag}& 1, 2, 5, 42, 342 &No\\
\hyperlink{CyInLMag}{CyInLMag}& 1, 2, 5, 42, 328 &No\\
\hyperlink{DInLMag}{DInLMag}& 1, 2, 5, 42, 164 &No\\
\hyperlink{CyDInLMag}{CyDInLMag}& 1, 2, 5, 42, 156 &No\\
\hyperlink{CInLMag}{CInLMag}& 1, 2, 5, 38, 238, 2722 &No\\
\hyperlink{CDInLMag}{CDInLMag}& 1, 2, 5, 38, 90, 858 &No\\
\hyperlink{InLSgrp}{InLSgrp}& 1, 2, 5, 29, 146, 1308 &No\\
\hyperlink{CyInLSgrp}{CyInLSgrp}& 1, 2, 5, 29, 132, 1018 &No\\
\hyperlink{CInLSgrp}{CInLSgrp}& 1, 2, 5, 29, 130, 984 &No\\
\hyperlink{DInLSgrp}{DInLSgrp}& 1, 2, 5, 29, 63, 454 &No\\
\hyperlink{CyDInLSgrp}{CyDInLSgrp}& 1, 2, 5, 29, 55, 353 &No\\
\hyperlink{CDInLSgrp}{CDInLSgrp}& 1, 2, 5, 29, 53, 330 &No\\
\hyperlink{CInSlSgrp}{CInSlSgrp}& 1, 2, 5, 29, 53, 330 &No\\
\hyperlink{RPoMon}{RPoMon}& 1, 2, 5, 28, 186 &No\\
\hyperlink{CRPoMon}{CRPoMon}& 1, 2, 5, 24, 131, 1001 &No\\
\hyperlink{InToMag}{InToMag}& 1, 2, 5, 22, 142 &No\\
\hyperlink{CyInToMag}{CyInToMag}& 1, 2, 5, 22, 138 &\href{http://oeis.org/A001437}{??}\\
\hyperlink{BCI}{BCI}& 1, 2, 5, 22, 118, 974 &No\\
\hyperlink{CIdLSgrp}{CIdLSgrp}& 1, 2, 5, 19, 86, 462 &No\\
\hyperlink{CMon}{CMon}& 1, 2, 5, 19, 78, 421, 2637 &\href{http://oeis.org/A058131}{A058131}\\
\hyperlink{CDIdLSgrp}{CDIdLSgrp}& 1, 2, 5, 19, 68 &No\\
\hyperlink{CInToMag}{CInToMag}& 1, 2, 5, 18, 72, 384 &No\\
\hyperlink{CIdJMon}{CIdJMon}& 1, 2, 5, 17, 66, 288 &No\\
\hyperlink{Pos}{Pos}& 1, 2, 5, 16, 63, 318, 2045, 16999,... &\href{http://oeis.org/A000112}{A000112}\\
\hyperlink{pMslat}{pMslat}& 1, 2, 5, 16, 60, 262, 1315 &No\\
\hyperlink{pJslat}{pJslat}& 1, 2, 5, 16, 60, 262, 1315 &No\\
\hyperlink{InvSgrp}{InvSgrp}& 1, 2, 5, 16, 52, 208, 911, 4637,... &\href{http://oeis.org/A001428}{A001428}\\
\hyperlink{CInvSgrp}{CInvSgrp}& 1, 2, 5, 16, 51, 201,...&\href{http://oeis.org/A234843}{A234843}\\
\hyperlink{InToSgrp}{InToSgrp}& 1, 2, 5, 14, 43, 147, 578 &\href{http://oeis.org/A137555}{??}\\
\hyperlink{CIdToSgrp}{CIdToSgrp}& 1, 2, 5, 14, 42, 132 &\\
\hyperlink{CyInToSgrp}{CyInToSgrp}& 1, 2, 5, 14, 39, 119 &No\\
\hyperlink{CInToSgrp}{CInToSgrp}& 1, 2, 5, 14, 37, 107 &No\\
\hyperlink{CIdLMon}{CIdLMon}& 1, 2, 4, 12, 41, 159 &No\\
\hyperlink{CDIdLMon}{CDIdLMon}& 1, 2, 4, 12, 31, 90, 241 &No\\
\hyperlink{CIdToMon}{CIdToMon}& 1, 2, 4, 8, 16, 32, 64 &\\
\hyperlink{pLat}{pLat}& 1, 2, 3, 7, 21, 75, 315 &No\\
\hyperlink{pDLat}{pDLat}& 1, 2, 3, 7, 13, 27, 50 &No\\
\hyperlink{pToLat}{pToLat}& 1, 2, 3, 4, 5, 6,... &\href{http://oeis.org/A000027}{A000027}\\
\hyperlink{DivRng}{DivRng}& 1, 2, 3, 3, 5, 0, 7, 4 &No\\
\hyperlink{Rng}{Rng}& 1, 2, 2, 11, 2, 4 &\href{http://oeis.org/A027623}{A027623}\\
\hyperlink{CRng}{CRng}& 1, 2, 2, 9, 2, 4 &\href{http://oeis.org/A037289}{A037289}\\
\hyperlink{LtCanSgrp}{LtCanSgrp}& 1, 2, 2, 4, 2, 5, 2, 9 &No\\
\hyperlink{RecBnd}{RecBnd}& 1, 2, 2, 3, 2, 4, 2, 4, 3, 4 &\\
\end{tabular}

\begin{tabular}{|l|l|l|}
\hyperlink{Sfld}{Sfld}& 1, 2, 1, 1, 1, 0 &\\
\hyperlink{BRMag}{BRMag}& 1, 2, 0, 136, 0 &No\\
\hyperlink{BCRMag}{BCRMag}& 1, 2, 0, 36, 0, 0 &No\\
\hyperlink{BRSgrp}{BRSgrp}& 1, 2, 0, 28, 0, 0 &No\\
\hyperlink{BInMag}{BInMag}& 1, 2, 0, 20, 0 &No\\
\hyperlink{BCyInMag}{BCyInMag}& 1, 2, 0, 20, 0 &No\\
\hyperlink{BCInMag}{BCInMag}& 1, 2, 0, 20, 0 &No\\
\hyperlink{BCRSgrp}{BCRSgrp}& 1, 2, 0, 16, 0, 0 &\\
\hyperlink{BInSgrp}{BInSgrp}& 1, 2, 0, 15, 0, 0 &No\\
\hyperlink{BCyInSgrp}{BCyInSgrp}& 1, 2, 0, 15, 0, 0 &No\\
\hyperlink{BCInSgrp}{BCInSgrp}& 1, 2, 0, 15, 0, 0 &No\\
\hyperlink{BMon}{BMon}& 1, 2, 0, 11, 0, 0, 0, 383 &No\\
\hyperlink{BRUn}{BRUn}& 1, 2, 0, 10, 0, 0, 0, 104 &No\\
\hyperlink{BIdLrSgrp}{BIdLrSgrp}& 1, 2, 0, 10, 0, 0 &No\\
\hyperlink{BGalLat}{BGalLat}& 1, 2, 0, 10, 0, 0 &No\\
\hyperlink{BCMon}{BCMon}& 1, 2, 0, 9, 0, 0, 0 &No\\
\hyperlink{BIdMon}{BIdMon}& 1, 2, 0, 6, 0, 0, 0, 24 &No\\
\hyperlink{BCIdSgrp}{BCIdSgrp}& 1, 2, 0, 5, 0, 0, 0, 13 &No\\
\hyperlink{BCIdMon}{BCIdMon}& 1, 2, 0, 4, 0, 0, 0, 9 &No\\
\hyperlink{pBA}{pBA}& 1, 2, 0, 3, 0, 0, 0, 1, 0, 0 &\\
\hyperlink{Qgrp}{Qgrp}& 1, 1, 5, 35, 1411,... &\href{http://oeis.org/A057991}{A057991}\\
\hyperlink{MouQgrp}{MouQgrp}& 1, 1, 5, 29, 1351 &No\\
\hyperlink{LrMMon}{LrMMon}& 1, 1, 4, 24, 195, 2146 &No\\
\hyperlink{IdRMSgrp}{IdRMSgrp}& 1, 1, 4, 24, 169, 1404 &No\\
\hyperlink{IdRJSgrp}{IdRJSgrp}& 1, 1, 4, 24, 169, 1404 &No\\
\hyperlink{IdRPoSgrp}{IdRPoSgrp}& 1, 1, 4, 24, 169 &No\\
\hyperlink{IdRLSgrp}{IdRLSgrp}& 1, 1, 4, 24, 169 &No\\
\hyperlink{DIdRLSgrp}{DIdRLSgrp}& 1, 1, 4, 24, 124 &No\\
\hyperlink{LrLMon}{LrLMon}& 1, 1, 4, 23, 169, 1635 &No\\
\hyperlink{LrJMon}{LrJMon}& 1, 1, 4, 23, 169, 1635 &No\\
\hyperlink{DLrLMon}{DLrLMon}& 1, 1, 4, 23, 130, 976 &No\\
\hyperlink{LrToMon}{LrToMon}& 1, 1, 4, 17, 92, 609 &No\\
\hyperlink{IdRToSgrp}{IdRToSgrp}& 1, 1, 4, 17, 82 &No\\
\hyperlink{RL}{RL}& 1, 1, 3, 20, 149, 1488, 18554,... &No??\\
\hyperlink{IdSrng$_{01}$}{IdSrng$_{01}$}& 1, 1, 3, 20, 149, 1488, 18554,... &No\\
\hyperlink{bRL}{bRL}& 1, 1, 3, 20, 149, 1488 &No\\
\hyperlink{RMMon}{RMMon}& 1, 1, 3, 20, 149, 1488 &No\\
\hyperlink{RJMon}{RJMon}& 1, 1, 3, 20, 149, 1488 &No\\
\hyperlink{KA}{KA}& 1, 1, 3, 20, 149, 1488 &No\\
\hyperlink{KLat}{KLat}& 1, 1, 3, 16, 149, 1488 &No\\
\hyperlink{ActLat}{ActLat}& 1, 1, 3, 16, 149, 1488 &No\\
\hyperlink{DRL}{DRL}& 1, 1, 3, 20, 115, 899, 7782,... &No\\
\hyperlink{CRL}{CRL}& 1, 1, 3, 16, 100, 794, 7493,... &No\\
\hyperlink{CRMMon}{CRMMon}& 1, 1, 3, 16, 100, 794 &No\\
\hyperlink{CRJMon}{CRJMon}& 1, 1, 3, 16, 100, 794 &No\\
\hyperlink{CDRL}{CDRL}& 1, 1, 3, 16, 70, 399 &No\\
\hyperlink{RToMon}{RToMon}& 1, 1, 3, 15, 84, 575 &No\\
\hyperlink{BCKJslat}{BCKJslat}& 1, 1, 3, 14, 87, 745 &No\\
\hyperlink{IdLrPoMon}{IdLrPoMon}& 1, 1, 3, 12, 59, 350 &No\\
\hyperlink{IdLrMMon}{IdLrMMon}& 1, 1, 3, 12, 59, 348, 2372 &No\\
\hyperlink{CRSlMon}{CRSlMon}& 1, 1, 3, 12, 47, 220 &No\\
\hyperlink{IdLrJMon}{IdLrJMon}& 1, 1, 3, 11, 46, 215, 1114 &No\\
\hyperlink{IdLrLMon}{IdLrLMon}& 1, 1, 3, 11, 46, 215 &No\\
\hyperlink{CRToMon}{CRToMon}& 1, 1, 3, 11, 46, 213 &\\
\hyperlink{DIdLrLMon}{DIdLrLMon}& 1, 1, 3, 11, 37, 134 &No\\
\hyperlink{IdLrToMon}{IdLrToMon}& 1, 1, 3, 8, 22, 60, 164 &\href{http://oeis.org/A155020}{??}\\
\hyperlink{Qnd}{Qnd}& 1, 1, 3, 7, 22, 73, 298, 1581,... &\href{http://oeis.org/A181769}{A181769}\\
\hyperlink{IPoMon}{IPoMon}& 1, 1, 2, 11, 102, 1609 &No\\
\hyperlink{IMMon}{IMMon}& 1, 1, 2, 11, 102, 1569 &No\\
\hyperlink{CIPoMon}{CIPoMon}& 1, 1, 2, 9, 60, 590 &No\\
\hyperlink{CIMMon}{CIMMon}& 1, 1, 2, 9, 60, 572 &No\\
\hyperlink{Polrim}{Polrim}& 1, 1, 2, 9, 51, 409 &No\\
\end{tabular}

\begin{tabular}{|l|l|l|}
\hyperlink{ILrMMon}{ILrMMon}& 1, 1, 2, 9, 51, 408 &No\\
\hyperlink{Porim}{Porim}& 1, 1, 2, 9, 49, 365 &No\\
\hyperlink{IRJMon}{IRJMon}& 1, 1, 2, 9, 49, 364, 3335 &No\\
\hyperlink{IRMMon}{IRMMon}& 1, 1, 2, 9, 49, 364 &No\\
\hyperlink{IJMon}{IJMon}& 1, 1, 2, 9, 49, 364 &No\\
\hyperlink{IRL}{IRL}& 1, 1, 2, 9, 49, 364 &No\\
\hyperlink{ILrJMon}{ILrJMon}& 1, 1, 2, 9, 49, 364 &No\\
\hyperlink{ILrLMon}{ILrLMon}& 1, 1, 2, 9, 49, 364 &No\\
\hyperlink{ILMon}{ILMon}& 1, 1, 2, 9, 49, 364 &No\\
\hyperlink{DIRL}{DIRL}& 1, 1, 2, 9, 49, 359 &No\\
\hyperlink{DILrLMon}{DILrLMon}& 1, 1, 2, 9, 49, 359 &No\\
\hyperlink{DILMon}{DILMon}& 1, 1, 2, 9, 49, 359 &No\\
\hyperlink{InFL}{InFL}& 1, 1, 2, 9, 21, 101, 284, 1464 &No\\
\hyperlink{CyInFL}{CyInFL}& 1, 1, 2, 9, 21, 101, 279, 1433 &No\\
\hyperlink{CInFL}{CInFL}& 1, 1, 2, 9, 21, 100, 276, 1392 &No\\
\hyperlink{DInFL}{DInFL}& 1, 1, 2, 9, 8, 43, 49 &No\\
\hyperlink{CyDInFL}{CyDInFL}& 1, 1, 2, 9, 8, 43, 48 &No\\
\hyperlink{CDInFL}{CDInFL}& 1, 1, 2, 9, 8, 42, 46 &No\\
\hyperlink{IToMon}{IToMon}& 1, 1, 2, 8, 44, 308, 2641,... &\href{http://oeis.org/A253950}{A253950}\\
\hyperlink{IRToMon}{IRToMon}& 1, 1, 2, 8, 44, 308 &\\
\hyperlink{ILrToMon}{ILrToMon}& 1, 1, 2, 8, 44, 308 &\\
\hyperlink{BCKMslat}{BCKMslat}& 1, 1, 2, 8, 38, 265 &No\\
\hyperlink{CIdRPoSgrp}{CIdRPoSgrp}& 1, 1, 2, 8, 36, 203 &No\\
\hyperlink{CIdRMSgrp}{CIdRMSgrp}& 1, 1, 2, 8, 36, 202 &No\\
\hyperlink{CIdRJSgrp}{CIdRJSgrp}& 1, 1, 2, 8, 36, 202 &No\\
\hyperlink{CIdRLSgrp}{CIdRLSgrp}& 1, 1, 2, 8, 36, 202 &No\\
\hyperlink{IdRPoMon}{IdRPoMon}& 1, 1, 2, 8, 32, 148 &No\\
\hyperlink{IdRJMon}{IdRJMon}& 1, 1, 2, 8, 32, 147, 759 &No \\
\hyperlink{IdRMMon}{IdRMMon}& 1, 1, 2, 8, 32, 147 &No\\
\hyperlink{IdRL}{IdRL}& 1, 1, 2, 8, 32, 147 &No\\
\hyperlink{DIdRL}{DIdRL}& 1, 1, 2, 8, 27, 96 &No\\
\hyperlink{CIdRSlSgrp}{CIdRSlSgrp}& 1, 1, 2, 8, 25, 97 &No\\
\hyperlink{CDIdRLSgrp}{CDIdRLSgrp}& 1, 1, 2, 8, 25, 97 &No\\
\hyperlink{RtHp}{RtHp}& 1, 1, 2, 8, 24, 91 &No\\
\hyperlink{Dtoid}{Dtoid}& 1, 1, 2, 7, 61 &No\\
\hyperlink{CIRMMon}{CIRMMon}& 1, 1, 2, 7, 26, 129, 723 &No\\
\hyperlink{CIRL}{CIRL}& 1, 1, 2, 7, 26, 129, 723 &No\\
\hyperlink{CIRJMon}{CIRJMon}& 1, 1, 2, 7, 26, 129, 723 &No\\
\hyperlink{FL$_{ew}$}{FL$_{ew}$}& 1, 1, 2, 7, 26, 129, 723 &No\\
\hyperlink{FL$_w$}{FL$_w$}& 1, 1, 2, 7, 26, 129, 723 &No\\
\hyperlink{Pocrim}{Pocrim}& 1, 1, 2, 7, 26, 129 &No\\
\hyperlink{CIJMon}{CIJMon}& 1, 1, 2, 7, 26, 129 &No\\
\hyperlink{CILMon}{CILMon}& 1, 1, 2, 7, 26, 129 &No\\
\hyperlink{BCKLat}{BCKLat}& 1, 1, 2, 7, 26, 129 &No\\
\hyperlink{CDIRL}{CDIRL}& 1, 1, 2, 7, 26, 124, 645 &No\\
\hyperlink{CDILMon}{CDILMon}& 1, 1, 2, 7, 26, 124, 645 &No\\
\hyperlink{CIRSlMon}{CIRSlMon}& 1, 1, 2, 7, 23, 99, 464 &No\\
\hyperlink{CIToMon}{CIToMon}& 1, 1, 2, 6, 22, 94, 451 &\href{http://oeis.org/A030453}{A030453}\\
\hyperlink{CIRToMon}{CIRToMon}& 1, 1, 2, 6, 22, 94, 451 &Same as above??\\
\hyperlink{CIdRPoMon}{CIdRPoMon}& 1, 1, 2, 6, 20, 78 &\\
\hyperlink{CIdRJMon}{CIdRJMon}& 1, 1, 2, 6, 20, 77, 333 &No\\
\hyperlink{CIdRMMon}{CIdRMMon}& 1, 1, 2, 6, 20, 77 &\\
\hyperlink{CIdRL}{CIdRL}& 1, 1, 2, 6, 20, 77 &\\
\hyperlink{IdRToMon}{IdRToMon}& 1, 1, 2, 6, 16, 44, 120 &No\\
\hyperlink{CDIdRL}{CDIdRL}& 1, 1, 2, 6, 15, 44, 115 &No\\
\hyperlink{Mslat}{Mslat}& 1, 1, 2, 5, 15, 53, 222, 1078,... &\href{http://oeis.org/A006966}{A006966}\\
\hyperlink{Jslat}{Jslat}& 1, 1, 2, 5, 15, 53, 222, 1078,... &\href{http://oeis.org/A006966}{A006966}\\
\hyperlink{ubJslat}{ubJslat}& 1, 1, 2, 5, 15, 53, 222, 1078,... &\href{http://oeis.org/A006966}{A006966}\\
\hyperlink{CIdRToSgrp}{CIdRToSgrp}& 1, 1, 2, 5, 14, 42 &\\
\hyperlink{GBL}{GBL}& 1, 1, 2, 5, 10, 23, 49, 111 &No\\
\hyperlink{BLA}{BLA}& 1, 1, 2, 5, 10, 23, 49, 111 &No\\
\hyperlink{Hp}{Hp}& 1, 1, 2, 5, 10, 23, 49 &No\\
\hyperlink{CIdRSlMon}{CIdRSlMon}& 1, 1, 2, 5, 9, 20, 38 &No\\
\hyperlink{CInSlMon}{CInSlMon}& 1, 1, 2, 5, 8, 20, 36, 90 &No\\
\hyperlink{InToMon}{InToMon}& 1, 1, 2, 4, 8, 17, 38 &\href{http://oeis.org/A229202}{??}\\
\end{tabular}

\begin{tabular}{|l|l|l|}
\hyperlink{CyInToMon}{CyInToMon}& 1, 1, 2, 4, 8, 17, 38, 91 &\\
\hyperlink{CInToMon}{CInToMon}& 1, 1, 2, 4, 8, 17, 36, 81 &No\\
\hyperlink{CIdRToMon}{CIdRToMon}& 1, 1, 2, 4, 8, 16, 32 &\\
\hyperlink{sqMV}{sqMV}& 1, 1, 2, 2, 5, 5, 8 &\\
\hyperlink{qMV}{qMV}& 1, 1, 1, 9, 9, 467 &No\\
\hyperlink{Rng$_1$}{Rng$_1$}& 1, 1, 1, 4, 1, 1, 1, 11, 4, 1 &\href{http://oeis.org/A037291}{A037291}\\
\hyperlink{CRng$_1$}{CRng$_1$}& 1, 1, 1, 4, 1, 1, 1, 10, 4, 1 &\href{http://oeis.org/A127707}{A127707}\\
\hyperlink{HilA}{HilA}& 1, 1, 1, 3, 8, 27, 113 &No\\
\hyperlink{InLat}{InLat}& 1, 1, 1, 3, 5, 14, 27 &No\\
\hyperlink{InPorim}{InPorim}& 1, 1, 1, 3, 3, 13, 17, 84 &No\\
\hyperlink{IInFL}{IInFL}& 1, 1, 1, 3, 3, 12, 17, 78 &No\\
\hyperlink{CyInPorim}{CyInPorim}& 1, 1, 1, 3, 3, 12, 15, 79 &No\\
\hyperlink{CyIInFL}{CyIInFL}& 1, 1, 1, 3, 3, 12, 15, 75 &No\\
\hyperlink{InPocrim}{InPocrim}& 1, 1, 1, 3, 3, 12, 15, 73, 116 &No\\
\hyperlink{CIInFL}{CIInFL}& 1, 1, 1, 3, 3, 12, 15, 70, 112 &No\\
\hyperlink{DIInFL}{DIInFL}& 1, 1, 1, 3, 3, 12, 13, 66 &No\\
\hyperlink{CyDIInFL}{CyDIInFL}& 1, 1, 1, 3, 3, 12, 12, 65 &No\\
\hyperlink{CDIInFL}{CDIInFL}& 1, 1, 1, 3, 3, 12, 12, 60, 73 &No\\
\hyperlink{MZrd}{MZrd}& 1, 1, 1, 3, 3, 8, 12, 35 &No\\
\hyperlink{IMTL}{IMTL}& 1, 1, 1, 3, 3, 8, 12, 35 &No\\
\hyperlink{DInLat}{DInLat}& 1, 1, 1, 3, 1, 4, 3, 11 &No\\
\hyperlink{DmA}{DmA}& 1, 1, 1, 3, 1, 4, 2, 9, 5, 14 &No\\
\hyperlink{Lp}{Lp}& 1, 1, 1, 2, 6, 109, 23746,... &\href{http://oeis.org/A057771}{A057771}\\
\hyperlink{Lat}{Lat}& 1, 1, 1, 2, 5, 15, 53, 222, 1078,... &\href{http://oeis.org/A006966}{A006966}\\
\hyperlink{lbJslat}{lbJslat}& 1, 1, 1, 2, 5, 15, 53 &\href{http://oeis.org/A006966}{A006966}\\
\hyperlink{bLat}{bLat}& 1, 1, 1, 2, 5, 15, 53 &\href{http://oeis.org/A006966}{A006966}\\
\hyperlink{MsdLat}{MsdLat}& 1, 1, 1, 2, 4, 9, 23, 65, 197, 636 &No\\
\hyperlink{JsdLat}{JsdLat}& 1, 1, 1, 2, 4, 9, 23, 65, 197, 636 &No\\
\hyperlink{SdLat}{SdLat}& 1, 1, 1, 2, 4, 9, 22, 60, 174, 534 &\href{http://oeis.org/A292790}{A292790}\\
\hyperlink{ModLat}{ModLat}& 1, 1, 1, 2, 4, 8, 16, 34, 72, 157 &\href{http://oeis.org/A006981}{A006981}\\
\hyperlink{AdLat}{AdLat}& 1, 1, 1, 2, 4 &\\
\hyperlink{IInToMon}{IInToMon}& 1, 1, 1, 2, 3, 7, 12, 35 &\\
\hyperlink{CyIInToMon}{CyIInToMon}& 1, 1, 1, 2, 3, 7, 12, 35 &\\
\hyperlink{IMTLChn}{IMTLChn}& 1, 1, 1, 2, 3, 7, 12, 31, 59 &\href{http://oeis.org/A034786}{A034786}\\
\hyperlink{HA}{HA}& 1, 1, 1, 2, 3, 5, 8, 15, 26, 47 &\href{http://oeis.org/A006982}{A006982}\\
\hyperlink{DLat}{DLat}& 1, 1, 1, 2, 3, 5, 8, 15, 26, 47 &\href{http://oeis.org/A006982}{A006982}\\
\hyperlink{BrSlat}{BrSlat}& 1, 1, 1, 2, 3, 5, 8, 15, 26, 47 &\href{http://oeis.org/A006982}{A006982}\\
\hyperlink{BrA}{BrA}& 1, 1, 1, 2, 3, 5, 8, 15, 26, 47 &\href{http://oeis.org/A006982}{A006982}\\
\hyperlink{bDLat}{bDLat}& 1, 1, 1, 2, 3, 5, 8, 15, 26, 47 &\href{http://oeis.org/A006982}{A006982}\\
\hyperlink{StAlg}{StAlg}& 1, 1, 1, 2, 2, 4, 5, 10, 16, 28 &No\\
\hyperlink{CIdInFL}{CIdInFL}& 1, 1, 1, 2, 2, 4, 4, 9, 10, 21 &No\\
\hyperlink{KLA}{KLA}& 1, 1, 1, 2, 1, 3, 2, 6, 4, 10 &No\\
\hyperlink{PoGrp}{PoGrp}& 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1 &\href{http://oeis.org/A000001}{A000001}\\
\hyperlink{Grp}{Grp}& 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1 &\href{http://oeis.org/A000001}{A000001}\\
\hyperlink{CanSgrp}{CanSgrp}& 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1 &\href{http://oeis.org/A000001}{A000001}\\
\hyperlink{psMV}{psMV}& 1, 1, 1, 2, 1, 2, 1, 3, 2, 2 &\\
\hyperlink{GödA}{GödA}& 1, 1, 1, 2, 1, 2, 1, 3, 1, 2 &\\
\hyperlink{MV}{MV}& 1, 1, 1, 2, 1, 2, 1, 3 &\\
\hyperlink{CanMon}{CanMon}& 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1 &\href{http://oeis.org/A000001}{A000001}\\
\hyperlink{AbPoGrp}{AbPoGrp}& 1, 1, 1, 2, 1, 1, 1, 3, 2, 1 &\href{http://oeis.org/A000688}{A000688}\\
\hyperlink{AbGrp}{AbGrp}& 1, 1, 1, 2, 1, 1, 1, 3, 2, 1 &\href{http://oeis.org/A000688}{A000688}\\
\hyperlink{CanCSgrp}{CanCSgrp}& 1, 1, 1, 2, 1, 1, 1 &\href{http://oeis.org/A000688}{A000688}\\
\hyperlink{CanCMon}{CanCMon}& 1, 1, 1, 2, 1, 1, 1 &\href{http://oeis.org/A000688}{A000688}\\
\hyperlink{InToLat}{InToLat}& 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 &\\
\hyperlink{ToLat}{ToLat}& 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 &\href{http://oeis.org/A000012}{A000012}\\
\hyperlink{Set}{Set}& 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 &\href{http://oeis.org/A000012}{A000012}\\
\hyperlink{UFDom}{UFDom}& 1, 1, 1, 1, 1, 0 &\\
\end{tabular}

\begin{tabular}{|l|l|l|}
\hyperlink{PIDom}{PIDom}& 1, 1, 1, 1, 1, 0 &\\
\hyperlink{IntDom}{IntDom}& 1, 1, 1, 1, 1, 0 &\\
\hyperlink{EucDom}{EucDom}& 1, 1, 1, 1, 1, 0 &\\
\hyperlink{NRng$_1$}{NRng$_1$}& 1, 1, 1, 6, 1, 1, 1, 53, 11, 1 &No\\
\hyperlink{MouLp}{MouLp}& 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1 &\\
\hyperlink{LRng}{LRng}& 1, 1, 1, 2, 3, 5, 8 &\\
\hyperlink{BIdRSgrp}{BIdRSgrp}& 1, 1, 0, 7, 0, 0, 0, 26 &No\\
\hyperlink{BLrMon}{BLrMon}& 1, 1, 0, 6, 0, 0, 0, 90 &\href{http://oeis.org/A305324}{??}\\
\hyperlink{BSlat}{BSlat}& 1, 1, 0, 5, 0, 0, 0 &\\
\hyperlink{BInFL}{BInFL}& 1, 1, 0, 5, 0, 0, 0, 25 &No\\
\hyperlink{BCyInFL}{BCyInFL}& 1, 1, 0, 5, 0, 0, 0 &(Stopped)\\
\hyperlink{BCInFL}{BCInFL}& 1, 1, 0, 5, 0, 0, 0&\\
\hyperlink{BRL}{BRL}& 1, 1, 0, 5, 0, 0&\\
\hyperlink{BCRL}{BCRL}& 1, 1, 0, 5, 0&\\
\hyperlink{RA}{RA}& 1, 1, 0, 3, 0, 0&\\
\hyperlink{BIdLrMon}{BIdLrMon}& 1, 1, 0, 3, 0, 0&\\
\hyperlink{BCIdRSgrp}{BCIdRSgrp}& 1, 1, 0, 3, 0, 0&\\
\hyperlink{IRA}{IRA}& 1, 1, 0, 2, 0, 0, 0, 10, 102, 4412&\\
\hyperlink{BInLat}{BInLat}& 1, 1, 0, 2, 0, 0&\\
\hyperlink{BIdRL}{BIdRL}& 1, 1, 0, 2, 0, 0&\\
\hyperlink{BCIdRL}{BCIdRL}& 1, 1, 0, 2, 0, 0&\\
\hyperlink{CplmLat}{CplmLat}& 1, 1, 0, 1, 2&\\
\hyperlink{CdMLat}{CdMLat}& 1, 1, 0, 1, 1&\\
\hyperlink{OLat}{OLat}& 1, 1, 0, 1, 0, 2, 0, 5, 0, 15&\\
\hyperlink{OMLat}{OMLat}& 1, 1, 0, 1, 0, 1, 0, 2&\\
\hyperlink{BA}{BA}& 1, 1, 0, 1, 0, 0, 0, 1, 0, 1&\\
\hyperlink{BCIInFL}{BCIInFL}& 1, 1, 0, 1, 0, 0, 0, 1, 0&\\
\hyperlink{BIInFL}{BIInFL}& 1, 1, 0, 1, 0, 0, 0, 1&\\
\hyperlink{BGrp}{BGrp}& 1, 1, 0, 1, 0, 0, 0, 1&\\
\hyperlink{BCyIInFL}{BCyIInFL}& 1, 1, 0, 1, 0, 0, 0, 1&\\
\hyperlink{GBA}{GBA}& 1, 1, 0, 1, 0, 0&\\
\hyperlink{BIRL}{BIRL}& 1, 1, 0, 1, 0, 0&\\
\hyperlink{BCIRL}{BCIRL}& 1, 1, 0, 1, 0, 0&\\
\hyperlink{BCIMon}{BCIMon}& 1, 1, 0, 1, 0, 0&\\
\hyperlink{BIMon}{BIMon}& 1, 1, 0, 1, 0&\\
\hyperlink{BILrMon}{BILrMon}& 1, 1, 0, 1, 0&\\
\hyperlink{Bilat}{Bilat}& 1, 0, 0, 1, 3, 32, 284&\\
\hyperlink{RepLGrp}{RepLGrp}& 1, 0, 0, 0, 0, 0&\\
\hyperlink{AbLGrp}{AbLGrp}& 1, 0, 0, 0, 0, 0&\\
\hyperlink{LGrp}{LGrp}& 1, 0, 0, 0, 0, 0&\\
\hyperlink{TrivA}{TrivA}& 1, 0, 0&\\
\hyperlink{ToGrp}{ToGrp}& 1, 0, 0&\\
\hyperlink{CanRL}{CanRL}& 1, 0, 0&\\
\hyperlink{AbToGrp}{AbToGrp}& 1, 0, 0&\\
\hyperlink{Fld}{Fld}& 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1 &\href{https://oeis.org/A069513}{A069513}\\
\hyperlink{pcDLat}{pcDLat}& &\\
\hyperlink{pGrp}{pGrp}& &\\
\hyperlink{WaHp}{WaHp}& &\\
\hyperlink{Unar}{Unar}& &\\
\hyperlink{ToRng}{ToRng}& &\\
\hyperlink{ToFld}{ToFld}& &\\
\hyperlink{TA}{TA}& &\\
\hyperlink{SkLat}{SkLat}& &\\
\hyperlink{SeqA}{SeqA}& &\\
\hyperlink{RegRng}{RegRng}& &\\
\hyperlink{RMod}{RMod}& &\\
\hyperlink{OreDom}{OreDom}& &\\
\hyperlink{OckA}{OckA}& &\\
\hyperlink{NlGrp}{NlGrp}& &\\
\hyperlink{Neofld}{Neofld}& &\\
\hyperlink{NdLat}{NdLat}& &\\
\hyperlink{NaA}{NaA}& &\\
\end{tabular}

\begin{tabular}{|l|l|l|}
\hyperlink{NVLGrp}{NVLGrp}& &\\
\hyperlink{NFld}{NFld}& &\\
\hyperlink{NA}{NA}& &\\
\hyperlink{Mset}{Mset}& &\\
\hyperlink{MonA}{MonA}& &\\
\hyperlink{MTLA}{MTLA}& &\\
\hyperlink{ModOLat}{ModOLat}& &\\
\hyperlink{MALLA}{MALLA}& &\\
\hyperlink{MA}{MA}& &\\
\hyperlink{LieA}{LieA}& &\\
\hyperlink{LNeofld}{LNeofld}& &\\
\hyperlink{LLA}{LLA}& &\\
\hyperlink{LA$_n$}{LA$_n$}& &\\
\hyperlink{JorA}{JorA}& &\\
\hyperlink{ImpLat}{ImpLat}& &\\
\hyperlink{ILLA}{ILLA}& &\\
\hyperlink{Gset}{Gset}& &\\
\hyperlink{GMV}{GMV}& &\\
\hyperlink{FVec}{FVec}& &\\
\hyperlink{FRng}{FRng}& &\\
\hyperlink{DunnMon}{DunnMon}& &\\
\hyperlink{DpAlg}{DpAlg}& &\\
\hyperlink{DdpAlg}{DdpAlg}& &\\
\hyperlink{DblStAlg}{DblStAlg}& &\\
\hyperlink{DmMon}{DmMon}& &\\
\hyperlink{DDblpAlg}{DDblpAlg}& &\\
\hyperlink{CliffSgrp}{CliffSgrp}& &\\
\hyperlink{CToRng}{CToRng}& &\\
\hyperlink{CRegRng}{CRegRng}& &\\
\hyperlink{CA$_2$}{CA$_2$}& &\\
\hyperlink{CLRng}{CLRng}& &\\
\hyperlink{BoolLat}{BoolLat}& &\\
\hyperlink{BilinA}{BilinA}& &\\
\hyperlink{BRMod}{BRMod}& &\\
\hyperlink{BCK}{BCK}& &\\
\hyperlink{AbpGrp}{AbpGrp}& &\\
\hyperlink{AAlg}{AAlg}& \\\hline
\end{tabular}
%\end{multicols}
}

\onecolumn


\documentclass{article}
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\usepackage{adjustbox} % For vertically-aligned side-by-side minipages
\usepackage{array, amsmath,  mhchem}
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\begin{document}
\large

\section*{Problem 1}

Ultimately we would like to calculate the rate of product formation, which from the law of mass action is given by:

\[ \frac{d\left[ P \right]}{dt} = k_2 \left[ C_2 \right] - k_{-2} \left[ P \right] \left[ E \right] \]

Assume that $[C_2]$ is at steady state:

\[ 0 = \frac{d \left[ C_2 \right]}{dt} =  k_r \left[ C_1 \right] + k_{-2} \left[ P \right] \left[ E \right] - k_2 \left[ C_2 \right] = k_r \left[ C_1 \right]  -  \frac{d\left[ P \right]}{dt}   \implies  \frac{d\left[ P \right]}{dt}  = k_r \left[ C_1 \right] \]

Assume that $[C_1]$ is also at steady state:

\[ 0 = \frac{d \left[ C_1 \right]}{dt} =  k_1 \left[ S \right] \left[ E \right] -\left(  k_{-1} + k_r \right)  \left[ C_1 \right] \implies \left[ C_1 \right]  = \frac{k_1 \left[ S \right] \left[ E \right]}{k_{-1} + k_r} \]

Given this expression and the above result:

\begin{eqnarray}
 \frac{d\left[ P \right]}{dt}  = \frac{k_1 k_r \left[ S \right] \left[ E \right]}{k_{-1} + k_r} \label{eqn:prob1a}
 \end{eqnarray}

The desired rate law expression is equivalent to this statement, but does not contain the variable $[E]$, suggesting we must substitute an equivalent expression for $[E]$. Noting that the enzyme moiety is conserved, we can see that:

\begin{eqnarray*}
\left[ E_{\textrm{tot}} \right]  & = &  \left[ E\right]  + \left[ C_1 \right] + \left[ C_2 \right]\\
 \left[ E\right] & = & \left[ E_{\textrm{tot}} \right] - \left[ C_1 \right] - \left[ C_2 \right]\\
 & = & \left[ E_{\textrm{tot}} \right] - \frac{k_1}{k_{-1} + k_r} \left[ S \right] \left[ E \right] - \left[ C_2 \right]
\end{eqnarray*}

To simplify further, we must find an expression for $[C_2]$. The statement that the product readily rebinds free enzyme suggests that it may be appropriate to assume a rapid equilibrium assumption for the right-most reversible reaction:

\begin{eqnarray*}
k_2 \left[ C_2 \right] & = & k_{-2} \left[ P \right] \left[ E \right]\\
\left[ C_2 \right] & = & \frac{k_{-2} }{k_2} \left[ P \right] \left[ E \right]
\end{eqnarray*}

where we have again defined $K_p$ for convenience. Plugging this expression for $[C_2]$ into our expression for $[E]$ derived from moiety conservation and rearranging, we get:

\begin{eqnarray*}
 \left[ E\right] & = & \left[ E_{\textrm{tot}} \right] - \frac{k_1}{k_{-1} + k_r} \left[ S \right] \left[ E \right] - K_p \left[ P \right] \left[ E \right]\\
 \left(1 + \frac{k_1 \left[ S \right]}{k_{-1} + k_r} + \frac{k_{-2} }{k_2}  \left[ P \right] \right) \left[ E \right]   & = & \left[ E_{\textrm{tot}} \right]\\
  \left(\frac{k_{-1}+k_r}{k_1} + \left[ S \right]+ \frac{k_{-2} \left(k_{-1}+k_r \right)}{k_1k_2}  \left[ P \right] \right) \left[ E \right]   & = & \frac{k_{-1}+k_r}{k_1} \left[ E_{\textrm{tot}} \right]\\
   \left( K_m + \left[ S \right] + \frac{K_m}{K_p} \left[ P \right] \right) \left[ E \right] & = & K_m \left[ E_{\textrm{tot}} \right]\\
 \left[ E \right] & = & \frac{K_m \left[ E_{\textrm{tot}} \right]}{\left[ S \right] + K_m \left(1 + \frac{\left[ P \right]}{K_p} \right)}
\end{eqnarray*}

where we have defined $K_m$ and $K_p$ for convenience. This expression for $[E]$ can be plugged into equation \ref{eqn:prob1a} to get:

\begin{eqnarray*}
 \frac{d\left[ P \right]}{dt}  & = & \frac{k_r \left[ S \right]}{K_m} \left( \frac{K_m \left[ E_{\textrm{tot}} \right]}{\left[ S \right] + K_m \left(1 + \frac{\left[ P \right]}{K_p}  \right)} \right)  = \frac{V_{\textrm{max}} \left[ S \right]}{\left[ S \right] + K_m \left(1 + \frac{\left[ P \right]}{K_p} \right)}
 \end{eqnarray*}

 \section*{Problem 2}

We illustrate two approaches: the first uses rapid equilibrium assumptions for binding, and the second uses quasi-steady-state assumptions as the problem statement hints.

\subsection*{Rapid equilibrium assumptions}

Enzyme moiety conservation gives the expression:

\[ \left[ E_{\textrm{tot}} \right] = \left[ E \right] + \left[ ES \right] + \left[ EI \right] + \left[ ESI \right] \]

To simplify this, we will assume that all substrate and inhibitor binding events are in rapid equilibrium:

\begin{eqnarray*}
k_1 \left[ E \right] \left[ S \right] = k_{-1} \left[ ES \right] & \implies & \left[ E \right]  = \frac{k_{-1} \left[ ES \right]}{k_1 \left[ S \right]}\\
k_3 \left[ E \right] \left[ I \right] = k_{-3} \left[ EI \right] & \implies & \left[ EI \right]  = \frac{k_3 \left[ I \right]}{k_{-3}} \left( \frac{k_{-1} \left[ ES \right]}{k_1 \left[ S \right]} \right) = \frac{k_{-1}k_3 \left[ I \right] \left[ ES \right]}{k_{-3}k_1 \left[ S \right]}\\
k_3 \left[ ES \right] \left[ I \right] = k_{-3} \left[ ESI \right] & \implies & \left[ ESI \right] = \frac{k_3 \left[ ES \right] \left[ I \right]}{k_{-3}}
\end{eqnarray*}

Plugging these equations into the moiety conservation statement allows us to find an expression for [ES]:

\begin{eqnarray*}
\left[ E_{\textrm{tot}} \right] & = & \left( \frac{k_{-1}}{k_1 \left[ S \right]} + 1 + \frac{k_{-1}k_3 \left[ I \right]}{k_1 k_{-3} \left[ S \right]} + \frac{k_3 \left[ I \right]}{k_{-3}}  \right) \left[ ES \right]\\
& = & \left( 1 + \frac{k_3 \left[ I \right]}{k_{-3}} \right)\left( 1 + \frac{k_{-1}}{k_{1}\left[ S \right]} \right) \left[ ES \right]\\
\left[ ES \right] & = & \frac{\left[ E_{\textrm{tot}} \right]}{\left( 1 + \frac{k_3 \left[ I \right]}{k_{-3}} \right)\left( 1 + \frac{k_{-1}}{k_{1}\left[ S \right]} \right)} = \frac{\left[ E_{\textrm{tot}} \right] \left[ S \right]}{\left( 1 + \frac{\left[ I \right]}{K_i} \right)\left( \left[ S \right] + K_m\right)}
\end{eqnarray*}

Notice that the definition of $K_m$ here is a bit unexpected: this is due to our use of rapid equilibrium rather than steady-state assumptions. Plugging this into the equation for rate of product formation, we get:
\begin{eqnarray*}
\frac{d \left[ P \right]}{dt} = k_2 \left[ ES \right] = \left( \frac{k_2 \left[ E_{\textrm{tot}} \right]}{1 + \frac{\left[ I \right]}{K_i}} \right) \frac{\left[S\right]}{\left[ S \right] + K_m} = \left( \frac{V_{\textrm{max}}}{1 + \frac{\left[ I \right]}{K_i}} \right) \frac{\left[S\right]}{\left[ S \right] + K_m}
\end{eqnarray*}

\subsection*{Quasi-steady-state assumptions}

The quasi-steady-state assumptions give three expressions:

\begin{eqnarray*}
\frac{d\left[ EI \right]}{dt} & = & k_{3} \left[ E\right] \left[ I\right]  + k_{-1} \left[ ESI \right] - \left( k_1 \left[ S \right] + k_{-3} \right) \left[ EI \right] = 0 \\
\frac{d\left[ ES \right]}{dt} & = & k_{1} \left[ E\right] \left[ S \right]  + k_{-3} \left[ ESI \right] - \left( k_3 \left[ I \right] + k_{-1} + k_2 \right) \left[ ES \right] = 0\\
\frac{d\left[ ESI \right]}{dt} & = & k_{1} \left[ EI \right] \left[ S \right]  + k_{3} \left[ ES \right] \left[ I \right] - \left( k_{-3} + k_{-1} \right) \left[ ESI \right] = 0 \\
\end{eqnarray*}

We can eliminate [E] from these expressions using moiety conservation, i.e.

\[ \left[ E \right] = \left[ E_{\textrm{tot}} \right] - \left[ ES \right] - \left[ EI \right] - \left[ ESI \right] \]

Plugging in and rearranging, we get:

\begin{eqnarray*}
\frac{d\left[ EI \right]}{dt} & = & k_{3} \left[ E_{\textrm{tot}} \right] \left[ I\right]  - k_{3} \left[ I\right]  \left[ ES \right]  + \left(k_{-1} - k_3 \left[ I \right] \right) \left[ ESI \right] - \left( k_1 \left[ S \right] + k_{-3} + k_3 \left[ I \right] \right) \left[ EI \right] = 0 \\
\frac{d\left[ ES \right]}{dt} & = & k_{1} \left[ E_{\textrm{tot}} \right] \left[ S \right]  - k_{1} \left[ S \right] \left[ EI \right]  + \left( k_{-3} - k_1 \left[S\right] \right) \left[ ESI \right] - \left( k_3 \left[ I \right] + k_{-1} + k_2 + k_1 \left[S\right] \right) \left[ ES \right] = 0\\
\frac{d\left[ ESI \right]}{dt} & = & k_{1} \left[ EI \right] \left[ S \right]  + k_{3} \left[ ES \right] \left[ I \right] - \left( k_{-3} + k_{-1} \right) \left[ ESI \right] = 0 \\
\end{eqnarray*}

The third quasi-steady-state assumption gives us that:

\[ \left[ ESI \right] = \frac{k_{1}}{k_{-3} + k_{-1}} \left[ EI \right] \left[ S \right]  + \frac{k_{3}}{k_{-3} + k_{-1}} \left[ ES \right] \left[ I \right] \]

Plugging this in to the remaining two expressions, we get:

\begin{eqnarray*}
\frac{d\left[ EI \right]}{dt} & = & k_{3} \left[ E_{\textrm{tot}} \right] \left[ I\right]  - k_{3} \left[ I\right]  \left( \frac{k_{-3} + k_3 \left[ I \right]}{k_{-3} + k_{-1} } \right) \left[ ES \right]  - \left( k_1 \left[ S \right] \left( \frac{k_{-3} + k_3 \left[ I \right]}{k_{-3} + k_{-1} } \right) + k_{-3} + k_3 \left[ I \right] \right) \left[ EI \right] = 0 \\
\frac{d\left[ ES \right]}{dt} & = & k_{1} \left[ E_{\textrm{tot}} \right] \left[ S \right]
- k_{1} \left[ S \right]  \left( \frac{k_{-1} + k_1 \left[ S \right]}{k_{-3} + k_{-1} } \right) \left[ EI \right] - \left( k_3 \left[ I \right] \left( \frac{k_{-1} + k_1 \left[ S \right]}{k_{-3} + k_{-1} } \right) + k_{-1} + k_2 + k_1 \left[S\right] \right) \left[ ES \right] = 0
\end{eqnarray*}

We can solve the first equation to find an expression for [EI], then plug this into the second equation and solve to find [ES]:

\begin{eqnarray*}
\left[ ES \right] & = & \frac{}{k_3 \left[ I \right] \left( \frac{k_{-1} + k_1 \left[ S \right]}{k_{-3} + k_{-1} } \right) + k_{-1} + k_2 + k_1 \left[S\right]}

\end{eqnarray*}

\begin{lstlisting}
function []  = mutualrepression()
    % Pick some parameter values for plotting
    global k n
    k = 0.5; n=3;

    [x, y] = meshgrid(0:0.05:1.5, 0:0.05:1.5);
    dx = k ./(k+y.^n) - x;
    dy = k ./(k+x.^n) - y;
    r = (dx.^2 + dy.^2).^0.5;
    dx = dx ./ r;
    dy = dy ./ r;

    quiver(x,y,dx,dy); hold on;
    xlabel('[X]')
    ylabel('[Y]')
    axis([0,1.5,0,1.5])
    [t, c] = ode45(@updater, [0 50], [0.7, 0.8]);
    plot(c(:,1),c(:,2),'-r', 'LineWidth', 3)
    plot(0.7, 0.8, 'or');
    [t, c] = ode45(@updater, [0 50], [0.6,0.5]);
    plot(c(:,1),c(:,2),'-g', 'LineWidth', 3);
    plot(0.6, 0.5, 'og');
    [t, c] = ode45(@updater, [0 50], [1.4,0.2]);
    plot(c(:,1),c(:,2),'-b', 'LineWidth', 3);
    plot(1.4, 0.2, 'ob');
    [t, c] = ode45(@updater, [0 50], [0.3,1.2]);
    plot(c(:,1),c(:,2),'-k', 'LineWidth', 3);
    plot(0.3, 1.2, 'ok');
    [t, c] = ode45(@updater, [0 50], [0.3,0.3]);
    plot(c(:,1),c(:,2),'-m', 'LineWidth', 3);
    plot(0.3, 0.3, 'om');

end

function dc = updater(t, c)
    x = c(1);
    y = c(2);
    global k n
    dx = k/(k+y^n) - x;
    dy = k/(k+x^n) - y;
    dc = [dx; dy];
end
\end{lstlisting}

\end{document}

problem set keys/ps2/tex/ps2 answer key.tex


\documentclass[twoside]{MATH77}
\usepackage{multicol}
\usepackage[fleqn,reqno,centertags]{amsmath}
\begin{document}
\begmath 18.2  Sorting Data of Arbitrary Structure in Memory

\silentfootnote{$^\copyright$1997 Calif. Inst. of Technology, \thisyear \ Math \`a la Carte, Inc.}

\subsection{Purpose}

Sort data having an organization or structure not supported by one of the
subprograms in Chapter~18.1, for example, data having more than one key to
determine the sorted order. The subprogram INSORT in Chapter~18.3 has
similar functionality to GSORTP and is more efficient if the data are
initially partly ordered, or the ordering criterion is expensive.

\subsection{Usage}

\subsubsection{Program Prototype}

\begin{description}

\item[INTEGER] \ {\bf N, IP}($\geq |\text{N}|$){\bf , COMPAR}

\item[EXTERNAL] \ {\bf COMPAR}

\end{description}

Assign values to N and data elements indexed by~1 through N. Require
N $\geq $ 1.
$$
\fbox{\bf CALL GSORTP (COMPAR, N, IP)}
$$
Following the call to GSORTP the contents of IP(1) through IP(N) are such
that the $\text{J}^{th}$ element of the sorted sequence is the IP(J$)^{th}$ element
of the original sequence.

\subsubsection{Argument Definitions}

\begin{description}

\item[COMPAR] \ [in] An INTEGER FUNCTION subprogram that defines the relative
order of elements of the data. COMPAR is invoked as COMPAR(I, J), and is
expected to return $-$1 (or any negative integer) if the $\text{I}^{th}$
element of the original data is to precede the $\text{J}^{th}$ element in
the sorted sequence, +1 (or any positive integer)
if the $\text{I}^{th}$ element is to follow the $\text{J}^{th}$ element,
and zero if the order is immaterial. GSORTP does not have access to the
data. It is the caller's responsibility to make the data known to COMPAR.
Since COMPAR is a dummy procedure, it may have any name. Its name must
appear in an EXTERNAL statement in the calling program unit.

\item[N] \ [in] $|$N$|$ is the number of elements to sort, and the upper
 bound of subscripts to use to access IP.  If N $>$ 0 then IP(I) is
 initialized to I, for $1 \leq I \leq N$.  Actual arguments for COMPAR
 are always elements of IP.

\item[IP()] \ [out] An array to contain the definition of the sorted
sequence. IP(1:N) are set so the $\text{J}^{th}$ element of the sorted sequence
is the IP(J$)^{th}$ element of the original sequence.
\end{description}

\subsection{Examples and Remarks}

Program DRGSORTP illustrates the use of GSORTP to sort 1000 randomly
generated real numbers. The output should consist of the single line

\hspace{.2in}GSORTP succeeded

\subparagraph{Stability}

A sorting method is said to be $stable$ if the original relative order of
equal elements is preserved. This subroutine uses the quicksort algorithm,
which is not inherently stable. To impose stability, return COMPAR =
I $-$ J if the $\text{I}^{th}$ and $\text{J}^{th}$ elements are equal.

\subsection{Functional Description}

See Section~18.1.D.

\subsection{Error Procedures and Restrictions}

See Section~18.1.E.

\subsection{Supporting Information}

The source language for these subroutines is ANSI Fortran 77.

\begin{tabular}{@{\bf}l@{\hspace{5pt}}l}
\bf Entry & \hspace{.2in} {\bf Required Files}\vspace{2pt}\\
GSORTP & \hspace{.35in} GSORTP\\\end{tabular}

Designed and coded by W. V. Snyder, JPL~1988.

\begcodenp
\medskip\

\lstset{language=[77]Fortran,showstringspaces=false}
\lstset{xleftmargin=.8in}

\centerline{\bf \large DRGSORTP}\vspace{10pt}
\lstinputlisting{\codeloc{gsortp}}
\end{document}