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\chapter{Evaluation of labeled ribbon graphs}
\label{Section:Appendix}
\Correct[noline,caption={DONE Propagator}]{Change propagator to admissible propagator or symmetric propagator. Let's leave it as a propagator. It is the propagator in our theory, hence, we can have some definitions.}
In this appendix, we consider an algebraic propagator $\Prpg$ and the graph pairing $\langle \cdot, \cdot \rangle^{\Prpg}_\Gamma$ (Definition~\ref{Def:EvalRibGraph}), which encapsulates the contribution of a ribbon graph~$\Gamma$ to the map $f_{klg}: (\DBCyc V)^{\otimes k} \rightarrow (\DBCyc V)^{\otimes l}$ defined as a sum of contributions of ribbon graphs (Proposition~\ref{Prop:GraphPairing}). Such maps were already defined in \cite[Section 11]{Cieliebak2015} using coordinates; here we use an invariant framework inspired by~\cite{Mnev2017}. As an example, we work out in details expressions for the canonical $\dIBL$-operations $\OPQ_{210}$ and~$\OPQ_{120}$ (Example~\ref{Ex:Canon}). We also explain the technicality of identifying symmetric maps with maps on symmetric powers (Remark~\ref{Rem:SymMaps}).

Next, we show that the matrix $(T^{ij})$ from Definition~\ref{Def:CanonicaldIBL} corresponds to the algebraic Schwartz kernel (Definition~\ref{Def:LinSchw}) of the identity $\Id$ up to a sign. Assuming that the Hodge propagator $\Prpg$ from Definition~\ref{Def:GreenKernel} is algebraic, we deduce natural candidates for the signs in Definition~\ref{Def:PushforwardMCdeRham} based on the formula from \cite[Remark 12.10]{Cieliebak2015} for the genuine pushforward Maurer-Cartan element~$\PMC$ in the finite-dimensional case. Establishing the formal analogy between the de Rham case and the finite-dimensional case is our main application of the invariant framework. Finally, we sketch how to obtain signs for the Fr\'echet $\dIBL$-structure on $\DR(M)$ (Remark~\ref{Rem:Frechet}).

Throughout this appendix, we will use Notation~\ref{Def:Notation} without further remarks.
%
%Let $V$ be a finite dimensional graded vector space. We will use: the notation from Sec.~\ref{Section:Proof2} for graphs and their labellings; and the algebraic setting from Sec.~\ref{Section:TheoryA}.

\section{Finite dimensional case}

\begin{Def}[Propagator \& graph pairing]\label{Def:EvalRibGraph}
Let $V$ be a graded vector space. We will call a tensor $\Prpg\in V[1]^{\otimes 2}$ an \emph{algebraic propagator}. We call it \emph{admissible} if the symmetry condition
\begin{equation}\label{Eq:SymmetryCondition}
 \tau(\Prpg) = (-1)^{\Abs{\Prpg}} \Prpg
\end{equation}
is satisfied, where $\tau: V[1]^{\otimes 2} \rightarrow V[1]^{\otimes 2}$ is the twist map defined by $\tau(v_1 \otimes v_2) = (-1)^{\Abs{v_1}\Abs{v_2}} v_2 \otimes v_1$ for all $v_1$, $v_2\in V[1]$.

For a ribbon graph $\Gamma \in \RRG_{klg}$ and its labeling $L$, consider the permutation~$\sigma_L$ from Definition~\ref{Def:EdgeVertex}. It acts on tensor powers according to Definition~\ref{Def:Permutations} and thus defines the map
\begin{equation*} \sigma_L: (V[1]^{\otimes 2})^{\otimes e} \otimes V[1]^{\otimes s_1} \otimes \dotsb \otimes V[1]^{\otimes s_l} \longrightarrow V[1]^{\otimes d_1} \otimes \dotsb \otimes V[1]^{\otimes d_k},
\end{equation*}
where $d_i$ and $s_i$ are the valencies of internal vertices $1$, $\dotsc$, $k$ and boundary components $1$, $\dotsc$, $l$, respectively, and $e$ is the number of internal edges. We extend $\sigma_L$ by $0$ to other combinations of tensor powers.

The \emph{graph pairing}
\[ \langle \cdot, \cdot \rangle_{\Gamma}^{\Prpg}\ :\ (\DBCyc V)^{\otimes k} \otimes (\BCyc V)^{\otimes l} \longrightarrow \R  \]
is defined for all $\psi_1$, $\dotsc$, $\psi_{k} \in \DBCyc V$ and generating words $w_i = v_{i1} \dots v_{i m_i}$ with $v_{ij} \in V[1]$ for $m_{i} \in \N$ and $i=1$, $\dotsc$, $l$ by the following formula:
\begin{align*}
&\langle \psi_1\otimes \dotsb \otimes \psi_{k}, w_1 \otimes \dotsb \otimes w_l \rangle_{\Gamma}^{\Prpg} \\
 &\quad \coloneqq \begin{multlined}[t]\smash{\sum_{L_1,\,L_3^b}}\vphantom{\sum_{L}} (\psi_1 \otimes \dotsb \otimes \psi_k)\bigl(\sigma_L(\Prpg^{\otimes e}\otimes  (v_{11}\otimes \dotsb \otimes v_{1 m_1}) \otimes \dotsb \\ \otimes (v_{l1}\otimes \dotsb \otimes v_{l m_{l}})\bigr), \end{multlined}
\end{align*}
where we use the pairing from Definition~\ref{Def:Pairings} and in every summand an $L_2$ compatible with $L_1$ and an $L_{3}^v$ are chosen arbitrarily to get a full labeling $L$ of~$\Gamma$. The graph pairing extends to $\langle \cdot, \cdot\rangle_\Gamma^{\Prpg}\ :\ \RTen \DBCyc V \otimes \RTen \BCyc V \rightarrow \R$, where $\RTen W = \bigoplus_{k=1}^\infty W^{\otimes k}$ is the reduced tensor product.
\end{Def}

\begin{Proposition}\label{Prop:GraphPairing}
In the setting of Definition~\ref{Def:EvalRibGraph}, we denote $w = w_1 \otimes \dotsb \otimes w_l$ and $\psi = \psi_1 \otimes \dotsb\otimes \psi_k$. If $\Prpg$ is admissible, then the following holds:
\begin{ClaimList}
\item The number $\psi(\sigma_L(\Prpg^{\otimes e}\otimes w))$ does not depend on the choice of $L_{3}^v$ and an $L_2$ compatible with $L_1$. Moreover, $\langle\cdot, \cdot \rangle_\Gamma^{\Prpg}$ does not depend on the representative of $[\Gamma]\in \RRG_{klg}$.
\item If $V$ is finite-dimensional, then for every $k$, $l \ge 1$, $g \ge 0$ there is a unique linear map
\[ f_{klg}: (\DBCyc V)^{\otimes k} \longrightarrow (\DBCyc V)^{\otimes l} \] such that
\begin{equation*}
\begin{aligned}
&f_{klg}(\psi_1\otimes \dotsb \otimes \psi_k)(w_1 \otimes \dotsb \otimes w_l) \\
 &\qquad\qquad =  \frac{1}{l!} \sum_{[\Gamma]\in \RRG_{klg}} \frac{1}{\Abs{\Aut(\Gamma)}} \langle \psi_1\otimes \dotsb \otimes \psi_k, w_1 \otimes \dotsb \otimes w_l\rangle^{\Prpg}_{\Gamma}. \end{aligned}
\end{equation*}
\item The following holds for the map $f_{klg}$ from (b):
\begin{itemize}
\item It is homogenous of degree
\begin{equation} \label{Eq:DegreeForm} \Abs{f_{klg}} = -\Abs{\Prpg}(k+l-2+2g). \end{equation}
\item The filtration degree satisfies
\begin{equation} \label{Eq:FiltrDegreeForm}
 \Norm{f_{klg}} \ge - 2 (k+l-2+2g).
\end{equation}
\item For all $\eta\in\Perm_l$ and $\mu\in\Perm_k$, we have
\begin{equation} \label{Eq:SymmetryForm} \eta \circ f_{klg} \circ \mu = (-1)^{\Abs{\Prpg}(\eta + \mu)}f_{klg}.
\end{equation}
\end{itemize}
\end{ClaimList}
\end{Proposition}

\begin{proof}
\begin{ProofList}
\item Let us denote by $\bar{i}$ and $ij$ the operations on $L_2$ given by $\mathrm{e}_i \mapsto -\mathrm{e}_i$ and $\mathrm{e}_i \leftrightarrow \mathrm{e}_j$, respectively. An even number of these operations does not change the orientation of the complex~\eqref{Eq:OrientationComplex}. Their effect in $\sigma_L$ acting on $\Prpg^{\otimes e}\otimes w$ is
\begin{equation*}
%\label{Eq:PPPP}
\bar{i}: \Prpg_i\mapsto \tau(\Prpg_i) = (-1)^{\Abs{\Prpg}} \Prpg_i\quad\text{and}\quad ij: \Prpg_i \dots \Prpg_j\mapsto (-1)^{\Abs{\Prpg}} \Prpg_j \dots \Prpg_i.
\end{equation*}
Therefore, an even number of them does not change $\sigma_L(\Prpg^{\otimes e}\otimes w)$. This proves the independence of the choice of a compatible $L_2$. The independence of the choice of $L_{3}^v$ is clear since $\psi_i$ are cyclic symmetric.

An isomorphism of ribbon graphs $\eta: \Gamma \rightarrow \Gamma'$ induces the map of compatible labelings $L \mapsto L' = \eta_* L$ such that $\sigma_{L} = \sigma_{L'}$. The independence of the choice of a representative of $[\Gamma]$ follows.

\item Suppose that $\psi = \psi_1 \otimes \dotsb \otimes \psi_k$ with $\psi_i \in (\DBCyc V)_{r_i}^{c_i}$, where $r_i\in \N$ and $c_i \in \Z$ for $i=1$, $\dotsc$, $k$. A~general element of $(\DBCyc V)^{\otimes k}$ is then a finite linear combination of such~$\psi$'s.

First of all, let us argue that the sum $\sum_{\RRG_{klg}}$ is finite. The number of internal edges~$e$ is fixed from the Euler formula \eqref{Eq:EulerFormula}. Therefore, the number of contributing graphs $(V_{\mathrm{int}}, E_{\mathrm{int}})$ is finite. In order to bound the number of external vertices, we notice that $d_1 = r_1$, $\dotsc$, $d_k = r_k$ must hold for $\psi(\sigma_L(\Prpg^{\otimes e}\otimes w))$ to be non-zero. Therefore, the sum is finite.

We now have the linear functional
\[ f_{klg}(\psi) \coloneqq \frac{1}{l!}\sum_{[\Gamma]\in \RRG_{klg} } \frac{1}{\Abs{\Aut(\Gamma)}} \langle \psi \mid \cdot \rangle_\Gamma^{\Prpg}: (\BCyc V)^{\otimes l} \longrightarrow \R \]
and need to show that $f_{klg}(\psi)\in (\DBCyc V)^{\otimes l} \subset (\BCyc V)^{\otimes l*}$. Because $V$ has finite dimension, the weight-filtration of $\BCyc V$ satisfies (WG1) \& (WG2) (see \eqref{Eq:WGs} and Proposition~\ref{Prop:Compl}), and hence we have
\[ (\DBCyc V)^{\otimes l} = (\BCyc V)^{\WGD \otimes l} =\bigl((\BCyc V)^{\otimes l}\bigr)^{\WGD} \]
for the weight-graded duals. Therefore, it suffices to show that $f_{klg}(\psi)$ vanishes on all but finitely many degrees and weights of $(\BCyc V)^{\otimes k}$. However, the relation $f_{klg}(\psi)(w) \neq 0$ for a generating word $w\in (\BCyc V)^{\otimes k}$ implies
\begin{equation}\label{Eq:UUUU}
\begin{aligned}
\Abs{w} &= \Abs{\psi} - e \Abs{\Prpg}\quad\text{and} \\
k(w) &= k(\psi) - 2 e,
\end{aligned}
\end{equation}
where $k$ denotes the weight, and hence $f_{klg}(\psi)\in (\DBCyc V)^{\otimes l}$ indeed holds.

\item The formulas~\eqref{Eq:DegreeForm} and~\eqref{Eq:FiltrDegreeForm} follow from~\eqref{Eq:UUUU} and from~\eqref{Eq:EulerFormula}.

As for the symmetry \eqref{Eq:SymmetryForm}, suppose that $L$ and $L'$ are compatible labelings of the same graph $\Gamma$ such that $L_1'$ differs from $L_1$ by a permutation $\mu\in \Perm_{k}$ of internal vertices and a permutation $\eta\in \Perm_l$ of boundary components. Viewing $\mu$ and $\eta$ as block permutations in the vertex and edge order, respectively, we get
\[ \sigma_{L'}(\Prpg^{\otimes e}\otimes w)=(-1)^{\Abs{\Prpg}(\eta+\mu)}\mu(\sigma_L(\Prpg^{\otimes e}\otimes \eta( w))). \]
The sign comes from the difference of $L_2$ and $L_2'$ which compensates the change of the orientation of~\eqref{Eq:OrientationComplex} caused by $\mu$ and $\eta$.\qedhere
\end{ProofList}
\end{proof}

Given $\mu\in \Perm_k$ and $\psi = \psi_1 \otimes \dotsb \otimes \psi_k \in (\DBCyc V)^{\otimes k}$, it is easy to see that
\[ \varepsilon(\mu, \Psi) = \varepsilon(\mu(\Susp),\mu(\psi)) \varepsilon(\mu,\Susp)\varepsilon(\Susp,\psi)\varepsilon(\mu,\psi), \]
where $\Psi = (\Susp \psi_1) \otimes \dotsb \otimes (\Susp \psi_k) \in (\DBCyc V[A])^{\otimes k}$ and $\varepsilon(\mu,\Susp) = (-1)^{\Abs{s}\mu}$. If $A= - \Abs{\Prpg}$, then we get from \eqref{Eq:SymmetryForm} that the degree shift $\HTP_{klg}: (\DBCyc V [A])^{\otimes k} \rightarrow (\DBCyc V[A])^{\otimes l}$ has the following symmetries:
\begin{equation}\label{Eq:SymMap}
\forall \mu \in \Perm_k, \eta\in \Perm_l: \quad \eta \circ \HTP_{klg} \circ \mu = \HTP_{klg}.
\end{equation}
Note that the degrees satisfy
\begin{equation}\label{Eq:DegDegShift}
\Abs{\HTP_{klg}} = \Abs{f_{klg}} + (k-l) A.
\end{equation}

\begin{Remark}[Symmetric maps versus maps on symmetric powers]\label{Rem:SymMaps}
In the situation above, we define~$\tilde{\HTP}_{klg}$ as the unique map such that the solid lines of the following diagram commute:
\[\begin{tikzcd}
(\DBCyc V[A])^{\otimes k} \arrow{r}{\HTP_{klg}} \arrow[two heads]{d}{\pi} & (\DBCyc V[A])^{\otimes l} \arrow[two heads,swap]{d}{\pi}\\
\arrow[bend left,dotted]{u}{\iota}\Sym_k \DBCyc V[A] \arrow{r}{\tilde{\HTP}_{klg}} & \Sym_l \DBCyc V[A].\arrow[bend right,swap,dotted]{u}{\iota}
\end{tikzcd}\]
The symmetry condition \eqref{Eq:SymMap} provides the existence of $\tilde{\HTP}_{klg}$ and implies commutativity of the dotted diagram as well. Moreover, for all $\psi_1$, $\dotsc$, $\psi_k \in \DBCyc V$ and $w_1$, $\dotsc$, $w_l\in \BCyc V$, we have
\[ \tilde{\HTP}_{klg}(\Susp^k \psi_1\cdots \psi_k)(\Susp^l w_1\cdots w_l) = \HTP_{klg}(\Susp^k \psi_1 \otimes \dotsb \otimes \psi_k)(\Susp^l w_1 \otimes \dotsb\otimes w_l), \]
where we use the pairing from Definition~\ref{Def:Pairings}. We denote $\tilde{\HTP}_{klg}$ again by $\HTP_{klg}$.
\end{Remark}

\begin{Definition}[Algebraic Schwartz kernel]\label{Def:LinSchw}
Let $V$ be a graded vector space and $\Pair: V\otimes V \rightarrow \R$ a non-degenerate pairing on $V$. We extend $\Pair$ to a non-degenerate pairing $\Pair: V^{\otimes k}\otimes V^{\otimes k}\rightarrow \R$ for $k\ge 1$ by setting
\begin{equation*}
%\label{Eq:TensorPairing}
\Pair(v_{11} \otimes \dotsb \otimes v_{1k}, v_{21}\otimes \dotsb \otimes v_{2k}) \coloneqq \varepsilon(v_1, v_2)\Pair(v_{11},v_{21}) \dots \Pair(v_{1k},v_{2k})
\end{equation*}
for all $v_{11}$, $\dotsc$, $v_{1k}$, $v_{21}$, $\dotsc$,  $v_{2k}\in V$, where $\varepsilon$ is the Koszul sign (see Definition~\ref{Def:Koszul}). For $k=0$, we let $\Pair: \R\otimes \R \rightarrow \R$ be the multiplication on $\R$.

For $k$, $l \ge 0$, we say that $\Kern_L\in V^{\otimes k + l}$ is the \emph{algebraic Schwartz kernel} of a linear operator $L: V^{\otimes k} \rightarrow V^{\otimes l}$ if the following is satisfied:
\begin{equation} \label{Eq:KernelEquation}
\forall w_1 \in V^{\otimes k},  w_2 \in V^{\otimes l}:\quad \Pair(L(w_1),w_2) = \Pair(\Kern_L, w_1\otimes w_2).
\end{equation}
We usually omit writing ``algebraic'' if it is clear from the context (i.e., if we do not consider any ``extensions'' of $V^{\otimes k}$).
\end{Definition}

In the situation of Definition~\ref{Def:LinSchw}, let $(e_i)\subset V$ be a basis and $(e^i)$ its dual basis such that $\Pair(e_i,e^j)=\delta_{ij}$. We define the coordinates $K_L^{ij} \in \R$ and $L^{ij}\in \R$ by
\[ \Kern_L = \sum_{i,j} K_L^{ij} e_i \otimes e_j\quad\text{and}\quad L^{ij} \coloneqq \Pair(L (e^i),e^j).\]
From \eqref{Eq:KernelEquation} we have
\begin{equation} \label{Eq:KernelCoordinates}
\Kern_L^{ij} = (-1)^{(\Abs{L}+1)(\Abs{\Pair}+\Abs{e_i})} L^{ij}\quad \text{for all }i,j.
\end{equation}

From now on, we will be in the situation of (A) and (B) in the Overview; in particular, we put $V[1]$ in place of $V$ in Definition~\ref{Def:LinSchw}. Let $\Kern_{\Id} \in V[1]^{\otimes 2}$ be the Schwartz kernel of the identity $\Id: V[1]\rightarrow V[1]$ and $\Kern_{\Htp} \in V[1]^{\otimes 2}$ the Schwartz kernel of the cochain homotopy $\Htp: V[1] \rightarrow V[1]$. From~\eqref{Eq:KernelCoordinates}, we get
\begin{equation*}
%\label{Eq:PropagatorsCoordinate}
\Kern_{\Htp}^{ij} = \Htp^{ij} \quad \text{and} \quad {\Kern_{\Id}}^{ij} = (-1)^{\Abs{e_i} + \Abs{\Pair}} \Pair(e^i,e^j)\quad\text{for all }i,j.
\end{equation*}
We see that the tensor $\TKer = \sum_{i,j} \TKer^{ij} e_i \otimes e_j$ from~\eqref{Eq:PropagatorT} can be expressed as
\begin{equation*}
%\label{Eq:GeomMeanT}
\TKer = (-1)^{n-2} \Kern_{\Id}.
\end{equation*}
This is the invariant meaning of $\TKer$. Note that the degrees satisfy
\[ \Abs{\TKer} = n-2\qquad\text{and}\qquad\Abs{\Kern_{\Htp}} = n - 3. \]

The assumption~\eqref{Eq:ConditionOnG} on $\Htp$ is equivalent to graded antisymmetry of the bilinear form $\Htp^+\coloneqq \Pair\circ (\Htp\otimes \Id): V[1]^{\otimes 2} \rightarrow \R$. This is further equivalent to
\[ \tau(\Kern_{\Htp}) = (-1)^{\Abs{\Kern_{\Htp}}}\Kern_{\Htp}. \]
Therefore, $\Kern_{\Htp}$ satisfies~\eqref{Eq:SymmetryCondition}, and hence it defines an admissible propagator for the construction of $f_{klg}$ for every $k$, $l\ge 1$, $g\ge 0$. We have from~\eqref{Eq:SymmetryForm} that the degree shift $\HTP_{klg}: (\DBCyc V[3-n])^{\otimes k} \rightarrow (\DBCyc V[3-n])^{\otimes l}$ is symmetric. Moreover, using~\eqref{Eq:DegreeForm},~\eqref{Eq:FiltrDegreeForm} and \eqref{Eq:DegDegShift}, we obtain
\[\begin{aligned}
\Abs{\HTP_{klg}} &= - 2d(k+g-1), \\
\Norm{\HTP_{klg}}&\ge \gamma (2-2g-k-l),
\end{aligned}\]
where $(d,\gamma) = (n-3,2)$. These are the degree and filtration conditions on an $\IBLInfty$-morphism from~\cite[Definition~2.8 and (8.3)]{Cieliebak2015}. In fact, our $\HTP=(\HTP_{klg})_{k,l\ge 1, g\ge 0}$ is precisely the $\IBLInfty$-homotopy from \cite[Theorem 11.3]{Cieliebak2015}.

Graded antisymmetry of $\Pair$ is equivalent to
\[ \tau(\TKer)= (-1)^{\Abs{\TKer}+1}\TKer. \]
Visibly, $\TKer$ does not satisfy~\eqref{Eq:SymmetryCondition}, and hence is not an admissible propagator. Nevertheless, we can still use it to define $f_{210}$ and~$f_{120}$ since the corresponding graphs $\Gamma$ (see Figure~\ref{Fig:GammasGraphs}) have only one internal edge $\mathrm{e}$, and, for a given $L_1$, there is a unique compatible~$L_2$ determined by the orientation of $\mathrm{e}$ (see Example~\ref{Ex:Canon} for the compatibility condition). As for the symmetry of the resulting maps, a transposition of internal vertices or boundary components in~\eqref{Eq:OrientationComplex} can be compensated only by $\mathrm{e}\mapsto-\mathrm{e}$, which produces $(-1)^{\Abs{\TKer}+1}$ (c.f.~the proof of Proposition~\ref{Prop:GraphPairing} (a)). Therefore, if we shift the degrees by $A= - \Abs{\TKer} +  1 = n- 3$, we obtain symmetric maps $\OPQ_{210}: (\DBCyc V[A])^{\otimes 2} \rightarrow \DBCyc V[A]$ and $\OPQ_{120}: \DBCyc V[A] \rightarrow (\DBCyc V[A])^{\otimes 2}$. We show in Example~\ref{Ex:Canon} below that these operations agree with those defined in Definition~\ref{Def:CanonicaldIBL}.

\begin{Example}[The canonical $\dIBL$-operations]\label{Ex:Canon}
We have
\begin{equation}\label{Eq:DefByGraphs}
\begin{aligned}
f_{210}(\psi_1 \otimes \psi_2)(w) & = \frac{1}{1!} \sum_{[\Gamma]\in \RRG_{210}} \frac{1}{\Abs{\Aut(\Gamma)}} \langle \psi_1 \otimes \psi_2 \mid w \rangle_\Gamma^{\Prpg}\quad\text{and} \\
f_{120}(\psi)(w_1 \otimes w_2) & = \frac{1}{2!} \sum_{[\Gamma]\in \RRG_{120}} \frac{1}{\Abs{\Aut(\Gamma)}} \langle \psi \mid w_1 \otimes w_2 \rangle_\Gamma^{\Prpg}.
\end{aligned}
\end{equation}
We parametrize $\RG_{210}$ by the ribbon graphs $\Gamma_{k_1, k_2}$ with $1\le k_1\le k_2$ and $\RG_{120}$ by the ribbon graphs $\Gamma^{s_1, s_2}$ with $0\le s_1 \le s_2$; these graphs are depicted in Figure~\ref{Fig:GammasGraphs}. We have
$\RRG_{210} = \RG_{210}\backslash \{[\Gamma_{1,1}]\}$ and $\RRG_{120} = \RG_{120}\backslash \{[\Gamma^{0,0}], [\Gamma^{0,1}]\}$. We also have
\[\Abs{\Aut(\Gamma_{k_1,k_2})} = \begin{cases} 1 & \text{if }k_1 \neq k_2, \\
              2 & \text{if }k_1 = k_2, \end{cases}\]
and likewise for $\Gamma^{s_1,s_2}$. We fix labelings~$L_{3}^v$ and parametrize $L_{3}^b$ by $c=1$, $\dotsc$, $k_1 + k_2 - 2$ for $\Gamma_{k_1,k_2}$ and by $c_1 = 1$, $\dotsc$, $s_1$ and $c_2 = 1$, $\dotsc$, $s_2$ for $\Gamma^{s_1,s_2}$ as it is indicated in Figure~\ref{Fig:GammasGraphs}.

There are two possible labelings $L_{1}^v$ for $\Gamma_{k_1, k_2}$ and two possible labelings $L_{1}^b$ for $\Gamma^{s_1, s_2}$; this is the only freedom in choosing a full labeling $L$ because $L_3$ is fixed and $L_2$ is just the orientation of the single internal edge, which is uniquely determined by $L_1$. For both $\Gamma_{k_1, k_2}$ and $\Gamma^{s_1, s_2}$, we will denote the two possible full labelings by $L^1$ and $L^2$. They can be depicted as follows:
\begin{figure}[t]
\centering
\input{\GraphicsFolder/gammad1d2.tex}
\quad
\input{\GraphicsFolder/gammas1s2.tex}
\caption[Ribbon graphs for $\OPQ_{210}$ and $\OPQ_{120}$.]{Graphs $\Gamma_{k_1,k_2}$ and $\Gamma^{s_1, s_2}$ with fixed labelings $L_3$.}
\label{Fig:GammasGraphs}
\end{figure}
\begin{equation}\label{Eq:OrientLabel}
\begin{tabular}{c|m{1.5cm} m{1.5cm}}
 & \makebox[1.5cm]{$\Gamma_{k_1,k_2}$} & \makebox[1.5cm]{$\Gamma^{s_1,s_2}$} \\\hline
\rule{0pt}{4ex}$L^1$ &\centering{\begin{tikzpicture}
\tikzset{->-/.style={decoration={markings,mark=at position #1 with {\arrow{>}}},postaction={decorate}}}
\node (V1) at (0,0) {1};
\node(V2) at (1,0) {2};
\draw[->-={0.5}] (V1) -- (V2);
\end{tikzpicture}}
& \begin{tikzpicture}
\tikzset{->-/.style={decoration={markings,mark=at position #1 with {\arrow{>}}},postaction={decorate}}}
\node (V1) at (1,0) {1};
\node(V2) at (0,0) {2};
\draw[->-={0.5}] (V1) -- (V2);
\end{tikzpicture}\\[1ex]
$L^2$ & \begin{tikzpicture}
\tikzset{->-/.style={decoration={markings,mark=at position #1 with {\arrow{>}}},postaction={decorate}}}
\def\Rad{0.4}
\node (B1) at (0,0) {1};
\node (B2) at (0.8,0) {2};
\draw[->-={0.5}] ($([shift=(0:\Rad)]B1)$) arc (0:360:\Rad);
\end{tikzpicture} & \begin{tikzpicture}
\tikzset{->-/.style={decoration={markings,mark=at position #1 with {\arrow{>}}},postaction={decorate}}}
\def\Rad{0.4}
\node (B1) at (0,0) {2};
\node (B2) at (0.8,0) {1};
\draw[->-={0.5}] ($([shift=(360:\Rad)]B1)$) arc (360:0:\Rad);
\end{tikzpicture}
\end{tabular}
\end{equation}
Let us check that the indicated $L_1$ and $L_2$ are compatible. For the complexes $C_2 \rightarrow C_1 \rightarrow C_0$ from \eqref{Eq:OrientationComplex}, we have the following:
\[\begin{aligned}
\Gamma_{k_1,k_2}:\qquad &\langle \mathrm{b} \rangle \xrightarrow{\Bdd_2 = 0}  \langle \mathrm{e} \rangle \xrightarrow{\Bdd_1} \langle \mathrm{v}_2 - \mathrm{v}_1 \rangle \oplus \langle \mathrm{v}_1 + \mathrm{v}_2 \rangle,  \\
\Gamma^{s_1, s_2}:\qquad &\langle \mathrm{b}_1 - \mathrm{b}_2 \rangle \oplus \langle \mathrm{b}_1 + \mathrm{b}_2 \rangle  \xrightarrow{\Bdd_2} \langle \mathrm{e} \rangle \xrightarrow{\Bdd_1 = 0} \langle \mathrm{v} \rangle.
\end{aligned}\]
As for $\Gamma_{k_1,k_2}$, the basis $\mathrm{v}_2 - \mathrm{v}_1$, $\mathrm{v}_1 + \mathrm{v}_2$ of $C_0$ is positively oriented with respect to the basis $\mathrm{v}_2$, $\mathrm{v}_1$. Therefore, $\mathrm{e}$ has to be oriented such that $\Bdd_1 \mathrm{e} = \mathrm{v}_2 - \mathrm{v}_1$; i.e., it is a path from $\mathrm{v}_1$ to $\mathrm{v}_2$. As for $\Gamma^{s_1,s_2}$, the basis $\mathrm{b}_1-\mathrm{b}_2$, $\mathrm{b}_1 + \mathrm{b}_2$ of $C_2$ is positively oriented with respect to $\mathrm{b}_1$, $\mathrm{b}_2$. Therefore, $\mathrm{e}$ has to be oriented such that $\mathrm{e} = \Bdd_2 (\mathrm{b}_1 - \mathrm{b}_2)$. Recall that we orient the boundary of a $2$-simplex by the ``outer normal first'' convention. We conclude that the labelings from~\eqref{Eq:OrientLabel} are indeed compatible.

As for $f_{210}$, the permutations $\sigma_1\coloneqq\sigma_{L^1}$ and $\sigma_2\coloneqq \sigma_{L^2}$ corresponding to the labelings $L^1$ and $L^2$, respectively, read
\[\begin{aligned}
\sigma_{1} &= \biggl(\begin{array}{cc|ccc}
 1 & 2     & \dots & c+2 & \dots \\
 1 & k_1+1 & \undermat{k_1+k_1-2}{\dots& c+2 & \dots}\dots & 2& \dots
\end{array}\biggr)\quad\text{and}\\[4ex]
\sigma_{2} &= \biggl(\begin{array}{cc|ccc}
 1 & 2     & \dots & c+2 & \dots  \\
 1 & k_2+1 & \undermat{k_1+k_2-2}{\dots & k_2 + 2 & \dots}\dots & k_2 + 2 & \dots
\end{array}\biggr).\\[3ex]
\end{aligned}\]
The underbracket marks the block which represents a cyclic permutation of the remaining indices. We see that
\[\begin{aligned}
\sigma_{1}&:  V^{\otimes 2} \otimes V^{\otimes s} \longrightarrow V^{\otimes k_1}\otimes V^{\otimes k_2},\quad e_i e_j w \longmapsto e_i w^1 e_j w^2,  \\
\sigma_{2}&: V^{\otimes 2}\otimes V^{\otimes s} \longrightarrow V^{\otimes k_2}\otimes V^{\otimes k_1},\quad e_i e_j w  \longmapsto e_i w^2 e_j w^1,
\end{aligned}\]
where $w^1 = w_c \dots w_{c+k_1-2}$, $w^2 = w_{c+k_1-1}\dots w_{c+k_1+k_2-3}$ and $s \coloneqq k_1 + k_2 - 2$. Defining $\tilde{w}^1 \coloneqq w^2$ and $\tilde{w}^2 \coloneqq w^1$, The Koszul sign of $\sigma_2$ can be written as
\[ \varepsilon(w\mapsto w^1 w^2) (-1)^{\Abs{w^2}\Abs{e_j} + \Abs{w^1}\Abs{w^2}} = \varepsilon(w\mapsto \tilde{w}^1 \tilde{w}^2 ) (-1)^{\Abs{\tilde{w}^1} \Abs{e_j}}. \]
We use these facts to rewrite \eqref{Eq:DefByGraphs} as follows:
\begin{align*}
& f_{210}(\psi_1 \otimes \psi_2)(w) \\
&\qquad=\begin{aligned}[t]
& \sum_{\substack{1\le k_1 < k_2}} \sum_{i,j} \TKer^{ij} \Bigl(\sum_{k(w^1) = k_1 - 1} \varepsilon(w\mapsto w^1 w^2)(-1)^{\Abs{w^1}\Abs{e_j}} \psi_1(e_i w^1) \psi_2(e_j w^2) \\
&{}+ \sum_{k(w^1) = k_2 - 1} \varepsilon(w\mapsto w^1 w^2) (-1)^{\Abs{w^2}\Abs{e_j} + \Abs{w^1}\Abs{w^2}} \psi_1(e_i w^2) \psi_2(e_j w^1)\Bigr) \\
&{}+ \sum_{1< k_1 = k_2} \frac{1}{2} \Bigl(\sum_{k(w^1) = k_1 - 1} \varepsilon(w\mapsto w^1 w^2)(-1)^{\Abs{w^1}\Abs{e_j}} \psi_1(e_i w^1) \psi_2(e_j w^2) \\
&{}+\sum_{k(w^1) = k_2 - 1} \varepsilon(w\mapsto w^1 w^2) (-1)^{\Abs{w^2}\Abs{e_j} + \Abs{w^1}\Abs{w^2}} \psi_1(e_i w^2) \psi_2(e_j w^1)\Bigr)\end{aligned}\\
&\qquad=\sum_{\substack{k_1, k_2 \ge 1 \\ k_1 + k_2 > 2}} \sum_{\substack{ k(w^1) = k_1 - 1 \\ k(w^2) = k_2 - 1}}  \TKer^{ij}\varepsilon(w \mapsto w^1 w^2) (-1)^{\Abs{w^1}\Abs{e_j}} \psi_1(e_i w^1) \psi_2(e_j w^2).
\end{align*}
This coincides with the formula from Definition~\ref{Def:CanonicaldIBL}.

As for $f_{120}$, the permutations $\sigma_1\coloneqq\sigma_{L^1}$ and $\sigma_2\coloneqq \sigma_{L^2}$ corresponding to the labelings $L^1$ and $L^2$, respectively, read
\[\begin{aligned}
\sigma_{1} &= \biggl(\begin{array}{cc|ccc|ccc}
1 & 2      & \dots & c_1+2 & \dots & \dots & c_2 + s_1 + 2 & \dots \\
 1 & s_1 + 2& \undermat{s_1}{\dots & c_1+2 & \dots}\dots & 2 & \dots &\undermat{s_2}{\dots & c_2 + s_1 + 2 & \dots}\dots & s_1 + 3 & \dots
\end{array}\biggr)\quad\text{and} \\[4ex]
\sigma_{2} &= \biggl(\begin{array}{cc|ccc|ccc}
 1       & 2 &  \dots  & c_2 + 2 & \dots & \dots & c_1 + s_2 + 2 & \dots  \\
 s_1 + 2 & 1 & \undermat{s_2}{\dots & c_2+2 & \dots} \dots  & s_1 + 3 & \dots & \undermat{s_1}{\dots & c_1 + s_2 + 2 & \dots} \dots &  2     & \dots
\end{array}\biggr), \\[3ex]
\end{aligned}\]
where the underbracketed blocks denote cyclic permutations of consecutive indices on the corresponding boundary component. We see that
\[\begin{aligned}
\sigma_1 &: V^{\otimes 2}\otimes V^{\otimes s_1}\otimes V^{\otimes s_2} \longrightarrow V^{\otimes k}, \quad e_i e_j w_1 w_2 \longmapsto e_i w_1^1 e_j w_2^1, \\
\sigma_2 &:  V^{\otimes 2}\otimes V^{\otimes s_2} \otimes V^{\otimes s_1} \longrightarrow V^{\otimes k}, \quad  e_i e_j w_1 w_2 \longmapsto e_j w_2^1 e_i w_1^1,
\end{aligned}\]
where $w_i^1$ denotes a cyclic permutation and $k\coloneqq s_1 + s_2 +2$. The Koszul sign of~$\sigma_2$ can be written as
\[\begin{aligned}&(-1)^{\Abs{e_i}\Abs{e_j} + \Abs{w_1}\Abs{w_2} + \Abs{e_i}\Abs{w_2}} \varepsilon(w_1 \mapsto w_1^1)\varepsilon(w_2\mapsto w_2^1)\\
&\qquad= (-1)^{(\Abs{e_i} + \Abs{w_1})(\Abs{e_j} + \Abs{w_2}) + \Abs{w_1} \Abs{e_j}}\varepsilon(w_1 \mapsto w_1^1)\varepsilon(w_2\mapsto w_2^1).\end{aligned}\]
We use this fact and the cyclic symmetry of $\psi$ to rewrite \eqref{Eq:DefByGraphs} as follows:
\[\begin{aligned}
&f_{120}(\psi)(w_1 \otimes w_2)
\\&\; = \begin{aligned}[t]
& \begin{multlined}[t]\sum_{0\le s_1 < s_2} \Bigl(\delta\bigl(\substack{k(w_1) = s_1 \\ k(w_2) = s_2}\bigr)\sum \TKer^{ij} \varepsilon(w_1\mapsto w_1^1) \varepsilon(w_2\mapsto w_2^1)(-1)^{\Abs{w_1} \Abs{e_j}} \\
\psi(e_i w_1^1 e_j w_2^1) +\delta\bigl(\substack{k(w_1) = s_2 \\ k(w_2) = s_1}\bigr) \sum \TKer^{ij} \varepsilon(w_1\mapsto w_1^1)\varepsilon(w_2\mapsto w_2^1)\\ (-1)^{\Abs{e_i} \Abs{e_j} + \Abs{w_2} \Abs{w_1} + \Abs{e_i}\Abs{w_2}} \psi(e_j w_2^1 e_i w_1^1) \Bigr)
\end{multlined}\\
&\begin{multlined}{}+\sum_{0< s_1 = s_2} \delta\bigl(\substack{k(w_1) = k(w_1) = s_1 \\ k(w_2) = k(w_2) = s_2}\bigr)
\frac{1}{2} \Bigl(\sum \TKer^{ij} \varepsilon(w_1\mapsto w_1^1) \varepsilon(w_2\mapsto w_2^1)\\ (-1)^{\Abs{w_1} \Abs{e_j}} \psi(e_i w_1^1 e_j w_2^1) + \sum \TKer^{ij} \varepsilon(w_1\mapsto w_1^1)\varepsilon(w_2\mapsto w_2^1) \\ (-1)^{\Abs{e_i} \Abs{e_j} + \Abs{w_2} \Abs{w_1} + \Abs{e_i}\Abs{w_2}} \psi(e_j w_2^1 e_i w_1^1) \Bigr) \end{multlined}
\end{aligned} \\
&\; = \begin{multlined}[t] \smash{\sum_{\substack{s_1, s_2 \ge 0 \\ s_1 + s_2 > 0}}} \delta\bigl(\substack{k(w_1) = s_1 \\ k(w_2) = s_2}\bigr) \sum \TKer^{ij} \varepsilon(w_1 \mapsto w_1^1)\varepsilon(w_2 \mapsto w_2^1)(-1)^{\Abs{w_1}\Abs{e_j}} \\ \psi(e_i w_1^1 e_j w_2^1).\end{multlined}
\end{aligned}\]
This coincides with the formula from Definition~\ref{Def:CanonicaldIBL}.
\end{Example}

\section{De Rham case}

We will now establish a formal analogy between the finite-dimensional and the de Rham case, which will explain the signs in Definition~\ref{Def:PushforwardMCdeRham}.

\textbf{The finite-dimensional case.} Consider the situation of (A) -- (D) in the Overview. To recall briefly, we have a finite-dimensional cyclic dga $(V,\Pair,m_1, m_2)$ and a subcomplex $\Harm \subset V$ such that there is a projection $\pi: V[1] \rightarrow \Harm[1]$ chain homotopic to $\Id$ via a chain homotopy $\Htp: V[1] \rightarrow V[1]$. Using~$m_2$, one constructs the canonical Maurer-Cartan element $\MC$ for $\dIBL(\CycC(V))$. The algebraic Schwartz kernel $\Kern_{\Htp}$ of $\Htp$ is an admissible propagator used to construct the $\IBLInfty$-quasi-isomorphism $\HTP = (\HTP_{klg}): \dIBL(\CycC(V)) \rightarrow \dIBL(\CycC(\Harm))$. The Maurer-Cartan element $\MC$ is then pushed forward along $\HTP$ to obtain the Maurer-Cartan element $\PMC \coloneqq \HTP_* \MC$  for $\dIBL(\CycC(\Harm))$ (see \cite[Lemma 9.5]{Cieliebak2015}). The formula for~$\PMC$ given in~\cite[Remark~12.10]{Cieliebak2015} reads
\begin{equation} \label{Eq:PushforwardMC}
\begin{aligned} &\PMC_{lg}(\Susp^l w_1 \otimes \dotsb \otimes  w_l) \\
&\qquad=\frac{1}{l!}\sum_{[\Gamma]\in \TRRG_{klg}} \frac{1}{\Abs{\Aut(\Gamma)}} (-1)^{k(n-2)}\langle (m_2^+)^{\otimes k}, w_1 \otimes \dotsb \otimes w_l \rangle_{\Gamma}^{\Kern_{\Htp}}.
\end{aligned}\end{equation}
Here the artificial sign $(-1)^{k(n-2)}$ is added because our sign conventions for $m_2^+$ differ (see Remark~\ref{Rem:mukplus}).
% where $w_i\in \BCyc V$ and $G\in V[1]^{\otimes 2}$ is the Schwartz kernel with respect to $m$ of the cochain homotopy $\Htp: V[1]\rightarrow V[1]$.
%Recall that  $\Prpg\in V[1]^{\otimes 2}$ is the Schwartz kernel of $\Htp: V[1] \rightarrow V[1]$.

\textbf{The de Rham case.}
We are in the setting of Definition~\ref{Def:PushforwardMCdeRham}. To recall briefly, we have the cyclic dga $(\DR(M), \Pair, m_1, m_2)$, the subspace of harmonic forms $\Harm\subset \DR$, the harmonic projection $\pi_\Harm: \DR\rightarrow \Harm$ and a Hodge propagator $\Prpg\in \DR(\Bl_\Diag(M\times M))$, which is the Schwartz kernel of a chain homotopy $\Htp: \DR \rightarrow \DR$ between $\pi_\Harm$ and~$\Id$. In analogy with the finite-dimensional case, the canonical Maurer-Cartan element~\eqref{Eq:CanonMC} for $\dIBL(\Harm)$ satisfies $\MC_{10} = (-1)^{n-2}m_2^+$ with $m_2^+ = \Pair(m_2 \otimes \Id)$. Because $\dim(\DR)$ is not of finite type, Definition~\ref{Def:CanonicaldIBL} does not give the canonical $\dIBL$-structure on $\CycC(\DR)$, and hence we have neither $\HTP$ nor~$\PMC$ in the standard sense.

In order to deduce the formal analogy, we embed $\DR(M)^{\otimes 2}$ into $\DR(\Bl_{\Diag}(M\times M))$ using the external wedge product $(\eta_1,\eta_2)\mapsto \tilde{\pi}_1^*\eta_1 \wedge \tilde{\pi}_2^*\eta_2$ and suppose that the Hodge propagator $\Prpg$ satisfies $\Prpg \in \DR^{\otimes 2}$. This never happens, so what follows is just a formal computation whose purpose is to deduce candidates for signs.%\footnote{If $\Prpg \in \DR(M)^{\otimes 2}$, then $\dim(\Im \Htp)< \infty$. From \eqref{Eq:gdef}, we get $\Dd \Htp \Dd = \Dd$, and hence $\dim(\Im \Dd)<\infty$. This is obviously not true for $\dim(M)>0$.}

\begin{Proposition}\label{Prop:FinDimAnalog}
In the de Rham case, suppose that $\Prpg\in \DR(M)^{\otimes 2}$. Then \eqref{Eq:PushforwardMC} reduces to \eqref{Eq:PushforwardMCdeRham}.
\end{Proposition}
\begin{proof}
Consider the intersection pairing $\tilde{\Pair}$ and its degree shift $\Pair$ (see Proposition~\ref{Prop:DGAs}). According to Definition~\ref{Def:LinSchw}, they extend to pairings on $\DR(M)^{\otimes k}$ and $\DR(M)[1]^{\otimes k}$ for all $k\ge 1$, respectively. For all $\eta_{11}$, $\eta_{12}$, $\eta_{21}$, $\eta_{22}\in \DR(M)$, we have:
\begin{equation}\label{Eq:PairComp}
\begin{aligned}
&\Pair(\SuspU^2 \eta_{11}\otimes \eta_{12},\SuspU^2 \eta_{21}\otimes \eta_{22}) \\
&\qquad = (-1)^{\eta_{11} + \eta_{21}} \Pair(\SuspU\eta_{11}\otimes \SuspU\eta_{12}, \SuspU \eta_{21}\otimes \SuspU\eta_{22}) \\
&\qquad = (-1)^{\eta_{11} + \eta_{21} + (1+\eta_{12})(1+\eta_{21})} \Pair(\SuspU \eta_{11}, \SuspU \eta_{21}) \Pair(\SuspU \eta_{12},\SuspU \eta_{22})\\
&\qquad =(-1)^{1+\eta_{12}\eta_{21}} \tilde{\Prpg}(\eta_{11},\eta_{21})\tilde{\Pair}(\eta_{12},\eta_{22}) \\
&\qquad = - \tilde{\Pair}(\eta_{11}\otimes \eta_{12}, \eta_{21}\otimes \eta_{22}).
\end{aligned}
\end{equation}
One can also check that
\[ \tilde{\Pair}(\eta_{11} \otimes \eta_{12}, \eta_{21}\otimes\eta_{22}) = \int_{x,y} \eta_{11}(x)\eta_{12}(y)\eta_{21}(x)\eta_{22}(y). \]

For the Hodge homotopy $\Htp: \DR(M) \rightarrow \DR(M)$ and its Hodge propagator $\Prpg \in \DR(M)^{\otimes 2}$, we have the following:
\[ \forall \eta_1, \eta_2\in \DR(M): \quad \tilde{\Pair}(\Htp(\eta_1),\eta_2) = \int_{x,y} \Prpg(x,y)\eta_1(x)\eta_2(y) = \tilde{\Pair}(\Prpg, \eta_1\otimes \eta_2). \]
%We will realize the degree shift $\DR(M)[1]$ as multiplication with the formal symbol $\SuspU$ of degree $-1$. Following
From this and \eqref{Eq:PairComp}, we obtain
\[\begin{aligned}
\Pair(\Htp(\SuspU \eta_1),\eta_2) &= \Pair(\SuspU \Htp(\eta_1),\SuspU \eta_2) = (-1)^{1+\eta_1} \tilde{\Pair}(\Htp(\eta_1),\eta_2) \\ &= (-1)^{1+\eta_1} \tilde{\Pair}(\Prpg,\eta_1\otimes \eta_2) = (-1)^{\eta_1}\Pair(\SuspU^2 \Prpg, \SuspU^2 \eta_1 \otimes \eta_2)  \\ &= \Pair(\SuspU^2 \Prpg, \SuspU\eta_1 \otimes \Susp\eta_2).
\end{aligned}\]
Therefore, the element $\SuspU^{2} \Prpg \in V[1]^{\otimes 2}$ corresponds to the Schwartz kernel $\Kern_{\Htp}$ of $\Htp: V[1] \rightarrow V[1]$. We write this correspondence as
\[ \Kern_{\Htp}\in V[1]^{\otimes 2}\ \sim\ \SuspU^2 \Prpg \in \Bl_\Diag(M\times M)[2]. \]

Let us check that $\SuspU^{2}\Prpg$ satisfies~\eqref{Eq:SymmetryCondition}. First of all, if we embed $\DR(M)^{\otimes k}$ into $\DR(M^{\times k})$ using the external wedge product $\eta_1 \otimes \dotsb \otimes \eta_k \mapsto \pi_1^*\eta_1 \wedge \dotsm \wedge \pi_k^* \eta_k \eqqcolon \eta_1(x_1)\wedge \dotsm \wedge \eta_k(x_k)$, then for all $\eta_1$, $\dotsc$, $\eta_k \in \DR(M)$ we have
\[\sigma(\eta_1 \otimes \dotsb \otimes \eta_k)(x_1,\dotsc,x_k) = \eta_1(x_{\sigma_1})\wedge \dotsc \wedge \eta_k(x_{\sigma_k}), \]
where the action on the left-hand side is given by~\eqref{Eq:Perm}. Now, the symmetry property \eqref{Eq:SymProp} implies
\[ \tau(\SuspU^{2}\Prpg) = - \SuspU^{2}\tau^*(\Prpg) = (-1)^{n+1} \SuspU^{2} \Prpg = (-1)^{\Abs{\SuspU^2 \Prpg}} \SuspU^2 \Prpg. \]
Therefore, the symmetry condition \eqref{Eq:SymmetryCondition} is indeed satisfied.

Let $\Gamma\in \TRRG_{klg}$, and let $L$ be a labeling of $\Gamma$. We abbreviate $\sigma\coloneqq \sigma_L \in \Perm_{3k} $. Given $\eta_{ij}\in \DR(M)$ for $j=1$, $\dotsc$, $s_i$ and $i=1$, $\dotsc$, $l$, where $s_i$ is the valency of the $i$-th boundary component, we set $\eta_i = \eta_{i1}\otimes \dotsb \otimes \eta_{is_i}$, $\eta =\eta_1 \otimes \dotsb \otimes \eta_l$, $\alpha_{ij}=\SuspU \eta_{ij}$, $\omega_i = \alpha_{i1}\otimes \dotsb \otimes \alpha_{i s_i}$ and $\omega = \omega_1\otimes \dotsb \otimes \omega_l$. We denote $s \coloneqq s_1 + \dotsb + s_l$, so that $3k = 2e + s$, where $e$ is the number of internal edges. We have
\begin{align*}(m_2^+)^{\otimes k} \bigl(\sigma((\SuspU^{ 2} \Prpg)^{\otimes e} \otimes \omega)\bigr) &= \varepsilon(\SuspU,\eta)(m_2^+)^{\otimes k} \bigl(\sigma((\SuspU^{ 2} \Prpg)^{\otimes e} \otimes \SuspU^{s} \eta)\bigr) \\
& = (-1)^{s e (n-1)}\varepsilon(\SuspU,\eta)(m_2^+)^{\otimes k} \bigl(\sigma(\SuspU^{2e+s}\Prpg^{\otimes e} \otimes \eta)\bigr) \\
&= \underbrace{(-1)^{\sigma + s e (n-1)} \varepsilon(\SuspU,\eta) }_{\eqqcolon \varepsilon_1}(m_2^+)^{\otimes k} \bigl(\SuspU^{2e+s} \sigma( \Prpg^{\otimes e} \otimes \eta)\bigr),
\end{align*}
where $\varepsilon(\SuspU,\eta)$ is the Koszul sign to order $\SuspU^s \eta_{11}\dots\eta_{l s_l} \mapsto \SuspU\eta_{11} \dots \SuspU \eta_{l s_l}$ and the operation $m_2^+: \DR(M)[1]^{\otimes 3} \rightarrow \R$ is given by $m_2^+ = \Pair(m_2 \otimes \Id)$. We denote $\kappa \coloneqq \Prpg^{\otimes e} \otimes \eta = \kappa_1 \otimes \dotsb \otimes \kappa_{3k}$, $\kappa_i \in \DR(M)[1]$ and compute
\begin{align*}
 & (m_2^+)^{\otimes k} (\SuspU^{3k} \sigma(\kappa)) \\
 &\quad \underset{\mathclap{\substack{\uparrow\rule{0pt}{1.7ex} \\ \ \ \ \Abs{m_2^+} = 3 - n}}}{=} \varepsilon(\sigma, \kappa) (m_2^+)^{\otimes k} (\SuspU^{3k} \kappa_{\sigma_1^{-1}} \otimes \dotsb \otimes \kappa_{\sigma_{3k}^{-1}})  \\
 & \quad = \begin{multlined}[t] (-1)^{\frac{1}{2}k(k-1)n}\varepsilon(\sigma,\kappa) (m_2^+)^{\otimes k}\bigl(\SuspU^3(\kappa_{\sigma_1^{-1}} \otimes \kappa_{\sigma_2^{-1}} \otimes \kappa_{\sigma_3^{-1}}) \otimes \dotsb \\ \otimes \SuspU^3(\kappa_{\sigma_{3k-2}^{-1}} \otimes \kappa_{\sigma_{3k-1}^{-1}} \otimes \kappa_{\sigma_{3k}^{-1}}) \bigr)\end{multlined}\\
 &\quad = \begin{multlined}[t]\overbrace{(-1)^{\frac{1}{2} k(k-1) n  + \kappa_{\sigma^{-1}_{2}} + \dotsb + \kappa_{\sigma^{-1}_{3k-1}}} \varepsilon(\sigma,\kappa)}^{\eqqcolon\varepsilon_2} (m_2^+)^{\otimes k} \bigl((\SuspU \kappa_{\sigma^{-1}_1} \otimes \SuspU \kappa_{\sigma^{-1}_2} \\ \otimes \SuspU \kappa_{\sigma^{-1}_3}) \otimes \dotsb \otimes (\SuspU \kappa_{\sigma^{-1}_{3k-2}} \otimes \SuspU \kappa_{\sigma^{-1}_{3k-1}} \otimes \SuspU \kappa_{\sigma^{-1}_{3k}})\bigr). \end{multlined}
\end{align*}
%where $\varepsilon(\sigma,\kappa)$ is the Koszul sign to permute $\kappa_1\dots \kappa_{3k} \mapsto \kappa_{\sigma^{-1}_1}\dots \kappa_{\sigma_{3k}^{-1}}$.
Next, using the formula \eqref{Eq:ChernSimons} for $m_2^+$, we get
\begin{align*}
&(m_2^+)^{\otimes k}\bigl((\SuspU \kappa_{\sigma^{-1}_{1}} \otimes \SuspU \kappa_{\sigma^{-1}_{2}} \otimes \SuspU \kappa_{\sigma^{-1}_{3}}) \otimes \dotsb \otimes (\SuspU \kappa_{\sigma^{-1}_{3k-2}} \otimes \SuspU \kappa_{\sigma^{-1}_{3k-1}} \otimes \SuspU \kappa_{\sigma^{-1}_{3k}})\bigr) \\
&\quad= \begin{multlined}[t](-1)^{\kappa_{\sigma^{-1}_{2}} + \dotsb + \kappa_{\sigma^{-1}_{3k-1}}} \Bigl(\int_{x_1} \kappa_{\sigma^{-1}_{1}}(x_1)\kappa_{\sigma^{-1}_{2}}(x_1)\kappa_{\sigma^{-1}_{3}}(x_1) \Bigr) \dotsm \\ \Bigl(\int_{x_k} \kappa_{\sigma^{-1}_{3k-2}}(x_k)\kappa_{\sigma^{-1}_{3k-1}}(x_k) \kappa_{\sigma^{-1}_{3k}}(x_k) \Bigr)\end{multlined} \\
&\quad= \begin{multlined}[t](-1)^{\kappa_{\sigma^{-1}_{2}} + \dotsb + \kappa_{\sigma^{-1}_{3k-1}}} \int_{x_1,\dotsc,x_k} \kappa_{\sigma_{1}^{-1}}(x_1)\kappa_{\sigma_{2}^{-1}}(x_1)\kappa_{\sigma_{3}^{-1}}(x_1) \dotsm \\ \kappa_{\sigma_{3k-2}^{-1}}(x_k) \kappa_{\sigma_{3k-1}^{-1}}(x_k) \kappa_{\sigma_{3k}^{-1}}(x_k) \end{multlined}\\
&\quad = \begin{multlined}[t]\overbrace{(-1)^{\kappa_{\sigma^{-1}_{2}} + \dotsb + \kappa_{\sigma^{-1}_{3k-1}}} \varepsilon(\sigma,\kappa)}^{\eqqcolon\varepsilon_3} \int_{x_1,\dotsc,x_k} \kappa_{1}(x_{\xi(\sigma_{1})})\kappa_{2}(x_{\xi(\sigma_{2})}) \kappa_{3}(x_{\xi(\sigma_{3})}) \dotsm \\ \kappa_{3k-2}(x_{\xi(\sigma_{3k-2})}) \kappa_{3k-1}(x_{\xi(\sigma_{3k-1})})\kappa_{3k}(x_{\xi(\sigma_{3k})}), \end{multlined}
\end{align*}
where $\xi(3j-2) = \xi(3j-1) = \xi(3j) = j$ for $j=1$, $\dotsc$, $k$ (see Definition~\ref{Def:PushforwardMCdeRham}). In total, we have
\begin{align*} &(m_2^+)^{\otimes k} \bigl(\sigma((\SuspU^{ 2} \Prpg)^{\otimes e} \otimes \omega)\bigr) \\
&\quad = \begin{multlined}[t]\varepsilon_1 \varepsilon_2 \varepsilon_3  \int_{x_1,\dotsc,x_k} \Prpg(x_{\xi(\sigma_1)},x_{\xi(\sigma_2)}) \dotsm  \Prpg(x_{\xi(\sigma_{2e-1})},x_{\xi(\sigma_{2e})}) \\ \alpha_{11}(x_{\xi(\sigma_{2e+1})}) \dotsm \alpha_{ls_{l}}(x_{\xi(\sigma_{2e+s})}),\end{multlined}
\end{align*}
where
\[ \varepsilon_1 \varepsilon_2 \varepsilon_3 = (-1)^{\sigma + s e (n-1)+ \frac{1}{2}k(k-1)n} \varepsilon(\SuspU,\eta). \]
Using~\eqref{Eq:EulerFormula} and~\eqref{Eq:TrivalentFormula} and $\varepsilon(\SuspU,\eta) = (-1)^{\Prpg(\omega)}$, we get the total sign
\[ (-1)^{k(n-2)}\varepsilon_1 \varepsilon_2 \varepsilon_3  = (-1)^{s(k,l) + \sigma + \Prpg(\omega)},  \]
where $(-1)^{k(n-2)}$ is the artificial sign from~\eqref{Eq:PushforwardMC}. This proves the proposition.
%Under $\DR(M)^{\otimes k}\subset \DR(M^{\times k})$, the action of $\sigma = \sigma_L$ from~\eqref{Eq:ActivePrmutation} corresponds to the pullback along
%\begin{equation} \label{Eq:PermutationsOfVariables}
%\begin{aligned}
%\sigma: M^{\times k} &\longrightarrow M^{\times k}\\
%(x_1,\dotsc,x_k) &\longmapsto (x_{\sigma_1}, \dotsc, x_{\sigma_k}).
%\end{aligned}
%\end{equation}
\end{proof}

\Add[noline,caption={Other de Rham formulas}]{What are the de Rham formulas for $\HTP_{klg}$, $\HTP^\MC_{klg}$, and so on? What kind of Feynman graphs? See Page 99 in Diary I.}

\begin{Remark}[Signs for the Fr\'echet $\dIBL$-structure on $\DR(M)$]\label{Rem:Frechet}
In \cite[Section~13]{Cieliebak2015}, they consider the weight-graded nuclear Fr\'echet space $\DBCyc \DR(M)_{\infty} \subset \DBCyc \DR(M)$ generated by $\varphi\in \DBCyc \DR(M)$ which have a smooth Schwartz kernel $k_\varphi\in \DR(M^{\times k})$; they showed that there is a canonical Fr\'echet $\dIBL$-structure on $\DBCyc \DR(M)_{\infty}[2-n]$. In order to deduce the signs, we can consider the subspace $\DBCyc \DR(M)_{\mathrm{alg}} \subset \DBCyc \DR(M)_\infty$ generated by $\varphi\in \DBCyc \DR(M)$ with an algebraic Schwartz kernel $\Kern_\varphi \in \DR(M)[1]^{\otimes k}$, rewrite \eqref{Eq:DefByGraphs} in terms of $\Kern_\varphi$ and extend the obtained formulas to $\DBCyc \DR(M)_\infty$.
\end{Remark}
\end{document}


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\newcommand{\teff}{$T_{\mathrm{eff}}$}
\newcommand{\logg}{$\log{g}$}
\newcommand{\teq}{$T_{\mathrm{eq}}$}
\newcommand{\Rearth}{$R_{\oplus}$}

\newcommand{\kepler}{{\it Kepler}}

\newcommand{\Plwr}{\ensuremath{P_{\mathrm{lwr}}}}
\newcommand{\Pupr}{\ensuremath{P_{\mathrm{upr}}}}

\shortauthors{Mullally et al.}

\begin{document}

\title{Optimal Choices for the Box Least Squares Algorithm}
\shorttitle{\kepler\ False Alarms}

\author{F.~Mullally\altaffilmark{1},
}

\altaffiltext{1}{SETI/NASA Ames Research Center, Moffett Field, CA 94035, USA}

\email{fergal.mullally@nasa.gov}

%``All of science is either physics or stamp collecting''\\
%- E. Rutherford

\begin{abstract}
\end{abstract}
\keywords{}

\setlength{\parskip}{1.0ex plus0.5ex minus0.2ex}

%\section*{Version}
%\begin{verbatim}
%$Id: ms.tex 194 2016-03-10 00:13:09Z fergalm $
%\end{verbatim}

\section{Introduction}

\section{Results}

Consider a transit with period $P$ and duration $\tau$ on a star observed continuously
for a timespan $T$. We will assume, for simplicity, that we make $N$ equally spaced observations across $T$.

\subsection{Trial Period Spacing}
To perform a BLS search across a range of periods \Plwr to \Pupr, the
$i^{\mathrm{th}}$ trial period to search is given by

\begin{equation}
P_i = \Plwr(1+ \tau/T)^i
\end{equation}

{\bf Justification}
Initially, we do not know the transit period, and we wish to select the smallest number of trial periods, $P_i$ such that at least one such $P_i$ is close enough to the true period to ensure detection of the transit signal. We define ``close enough'' to mean that the difference in phase for any two points when folded at the two periods is less than a transit duration. Let the phase of the 0th transit be $\phi_0$. If we assume we observe $n = T/P$ transits, the worst phase error will be
$t0 + n(P'-P)$. We therfore require

$$
n(P'-P) \leq \tau
$$

But $n = T/P$, so our requirement that all transits overlap implies
$$
P' < P + \leq \tau P/T
$$

Starting with \Plwr, our next trial period will be
\noindent
$P_1 = \Plwr\ + \tau \Plwr/T = \Plwr \times (1+ \tau/T)$\\
$P_2 = P_1\times (1+ \tau/T) = \Plwr \times (1+ \tau/T)^2$\\
...\\
$P_{i} = \Plwr \times (1+\tau/T)^i$\\

Choosing this period spacing will ensure that all transit mid-points lie within one transit duration for at least one trial period. In practice, this

Fold the lightcurve at a trial period $P'$.

\end{document}


%!TEX root = theory.tex
%
% =========================================================================
% -------------------------------------------------------------------------
% Transport (Advection - Dispersion):
% -------------------------------------------
%
%  This is a good place to outline key objectives of this section.
%
% -------------------------------------------------------------------------

\section{Transport Processes}
\label{sec:transport-processes}

\subsection{Overview}

Transport is one of the most important process that needs to be accurately
captured in the Environmental Management modeling tool set.
In what follows, we use "Transport" to refer to the set of physical processes
that lead to movement of dissolved and solid contaminants in the subsurface,
treating the chemical reactions that can affect the transport
rate through a retardation effect as a separate set of processes.
The principal transport processes to be considered are
\emph{advection}, \emph{mechanical dispersion}, and \emph{molecular diffusion}.
The equation for mass conservation of species $C$ can be written as
$$\label{eq:concervation of mass general}
  \frac{\partial ( \phi \sum_\alpha [s_{\alpha} C_{\alpha,i}] )}{\partial t } +
  \bnabla \cdot \bJ_{\text{adv}} \eq \bnabla \cdot \bJ_{\text{disp}} + \bnabla \cdot \bJ_{\text{diff}} + Q,
$$
where $C_{\alpha,i}$ is a concentration of $i$th species in phase $\alpha$,
$\bJ_{\text{adv}}$ is advective flux, $\bJ_{\text{disp}}$ is the dispersive flux,
$\bJ_{\text{diff}}$ is the diffusive flux (often grouped with the dispersive flux), and
$Q$ is the summation of various source terms which may include reactions.
%When using Richards equations we assume that the gas phase is not moving ...

The principal assumptions associated with the transport process models derive
from the continuum treatment of the porous medium.
Pore scale processes, including the resolution of variations in transport rates
within individual pores or pore networks \citep{li2008scale,kang2006lattice,lichtner-kang-2007},
are generally not resolved, although some capabilities for treating multi-scale
effects are discussed in Section~\ref{sec:transport-single-phase-dual-porosity}.
In general, it is assumed that within any one Representative Elementary Volume (REV)
corresponding to a grid cell all transport rates are the same.
It will be possible, however, to define overlapping continua with distinct transport
rates, as in the case where the fracture network and rock matrix are represented as separate continua.

Transport processes may be tightly coupled to both flow and reaction processes.
In the case of flow, one important coupling is associated with the transport of chemical
constituents that affect the density of the solution, which in turn affects flow rates through buoyancy.
In the case of chemical reactions, the coupling effect is normally very strong for reactive constituents.
Chemical reactions may consume components present in the gaseous phase (e.g., CO$_2$ or O$_2$), thus
modifying the saturation of the phase itself.
Reactions can strongly modify gradients, and thus transport rates, by either consuming
or producing various chemical species.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Transport in Porous Medium}
\label{sec:transport-single-phase}

The name of this section indicates that we consider the pransport processes in the soil with
pores, but no fractures.
At the same time we still consider the gas and solid phases in addition to the liquid pahse.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Process Model Equations}

The contaminants can be a part of one of three phases:
solid, liquid or gas.
When the contaminant $i$ is present in the solid phase (e.g. ice)
it is assumed not to be moving.
One of the key assumptions of Richards equations used for flow models
is that the gas phase is not moving.
%The transport in Amanzi neglects concentration of species in gaseous phase.
Therefore, the concervation of mass equation in its general form
\eqref{eq:concervation of mass general}
can be separated into equations
for the liquid and gas phases.
The equation for the species $i$ dissolved in the liquid phase is
\begin{equation}  \label{eq:MassConservation}
  \frac{\partial ( \phi s_l C_{l,i} )}{\partial t } + \bnabla \cdot \bJ_{l,i}^\text{adv}
  \eq
  \bnabla \cdot \bJ_{l,i}^\text{disp} + \bnabla \cdot \bJ_{l,i}^\text{diff} + Q_{l,i}
\end{equation}
and the equation for the pecies $i$ in the gas phase is
\begin{equation}  \label{eq:MassConservation gas}
  \frac{\partial ( \phi s_g C_{g,i} )}{\partial t }
  \eq
  \bnabla \cdot \bJ_{g,i}^\text{diff} + Q_{g,i}.
\end{equation}
The gas and liquid phases are related through an equation for saturations $s_l$ and $s_g$
$$
  s_g = 1-s_l.
$$
The transition of the polutants between phases can be modeled through the source terms $Q_*$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Assumptions and Applicability}

We consider pure advection process and exclude the attenuation mechanisms and microbial
behaviors that are discussed elsewhere.
Note that there are situations where the advection could be modified by attributes of
the transported mass and the pore structure.
One is the potential for nonreactive anions to be repulsed by negatively charged
solid surfaces into the center of pore throats where the velocity is faster.
Another is the advection of inorganic and organic colloids, and microorganisms,
whose movement can be affected by the geometry of pore throats.
In addition to being subject to the same physicochemical phenomena as abiotic colloids,
microorganisms have biological processes that can affect advection (e.g., temporal
changes in surface properties due to changes in metabolic state; chemotaxis; predation).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Advective Fluxes}
\label{sec:transport-advection}

Advection is the process where the bulk fluid motion transports mass and heat.
In the simplest conceptualization of advection, the mass of a component in a fluid
parcel simply moves with the velocity of the fluid parcel.
This assumes there are no other processes (e.g., diffusion, dispersion, reactions)
that can affect the component concentration in the fluid parcel.
Thus, advection can be a particulate or a dissolved species moving with the pore-water
whose velocity is governed by the flow processes (discussed elsewhere).
Continuum models have addressed these behaviors using bulk parameterizations
to characterize the pore-scale controls and controlling chemical gradients.

Numerical difficulties with the accuracy, robustness, and computation efficiency
of modeling the advection of moving steep concentration fronts, especially in
complex velocity fields, are well known.
In some cases, there are constraints on the Peclet and Courant numbers for the
useful application of a given technique.
The advective flux, $\bJ_{\text{adv}}$, of a dissolved species is described
mathematically as
\begin{equation} \label{eq:AdvectiveFlux}
  \bJ_{l,i}^\text{adv} = \phi s_l \bv_l C_{l,i},
\end{equation}
where $\phi$ is the porosity, $s_l$ is liquid saturation, $\bv_l$ is the
average linear velocity of the liquid, and $C_{l,i}$ is the concentration of the $i$th species in the liquid phase.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Dispersive Fluxes}
\label{sec:transport-dispersion}

Dispersion of a dissolved constituent refers to its spreading along tortuous pathways in a porous medium caused by mixing effects.
Dispersion takes place in the direction of the flow (longitudinal) and normal to the flow (transverse).
A conventional Eulerian Fickian representation of dispersion is assumed,
which may be taken as the asymptotic limiting form of the dispersion tensor \citep{neuman1990universal}.
It should be noted that issues related to the scale dependence of dispersion are not considered.
While this approach is known to have several limitations,
such as backward dispersion against the direction of flow and scale independence, nevertheless,
it is still widely used in practical applications.
Furthermore, it may be the only approach for representing local scale dispersion.

%\subsubsection{Process Model Equations}

The dispersive flux $\bJ_i^{\rm disp}$ for a variably saturated porous medium has the form
\EQ
  \bJ_i^{\rm disp} \eq -\phi s_\a \bD\bnabla C_i,
\EN
described in analogy to a Fickian process, where $\bD$ denotes the dispersion tensor.
The dispersion tensor takes different forms depending on whether the media is isotropic or anisotropic.

\paragraph{Isotropic Media.}
For an isotropic medium with no preferred axis of symmetry the dispersion tensor $\bD^{}$ has the well-known form \citep{bear-1972}
\BA\label{isotropic}
\renewcommand{\arraystretch}{2}
\bD^{} &\eq \a_T v \bI + \left(\a_L-\a_T \right)\frac{\bv\bv}{v},\\
&\eq \left(
\begin{array}{ccc}
\a_T v + (\a_L-\a_T)\dfrac{v_x^2}{v} & (\a_L-\a_T)\dfrac{v_x v_y}{v} & (\a_L-\a_T)\dfrac{v_x v_z}{v}\\
(\a_L-\a_T)\dfrac{v_y v_x}{v} & \a_T v + (\a_L-\a_T)\dfrac{v_y^2}{v} & (\a_L-\a_T)\dfrac{v_y v_z}{v}\\
(\a_L-\a_T)\dfrac{v_z v_x}{v} & (\a_L-\a_T)\dfrac{v_z v_y}{v} & \a_T v + (\a_L-\a_T) \dfrac{v_z^2}{v}
\end{array}
\right),
\EA
characterized by the two parameters $\a_L$ [m] and $\a_T$ [m] referred to the longitudinal and transverse dispersivity, respectively. The vector $\bv = (v_x,\,v_y,\,v_z)$ [m/s] denotes the average pore velocity with magnitude $v$, and $\bI$ is the identity matrix.

\paragraph{Anisotropic Media.}

The dispersion tensor for anisotropic media has not received much attention,  However, it has been shown that in an axi-symmetric medium with axis of symmetry $\blambda_s$, the dispersion tensor takes the general form \citep{lichtner_2002}
\BA     \label{gendisp}
\bD^{} \eq \a_T^H v \bI &+ \left[ \a_L^H-\a_T^V + \cos^2\!\theta \left( \a_L^V-\a_L^H+\a_T^V-\a_T^H \right) \right] \frac{\bv\bv}{v} \nonumber\\
& + \left(\a_T^V-\a_T^H \right) v \left[ \blambda_s\blambda_s - \dfrac{\cos\theta}{v} \left(\blambda_s\bv+\bv\blambda_s \right) \right],
\EA
where $\a_L^{H,\,V}$ and $\a_T^{H,\,V}$ refer to the longitudinal and transverse dispersivity in the horizontal and vertical directions, and $\theta$ denotes the angle between the axis of symmetry and the flow velocity.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Diffusive Fluxes}
\label{sec:transport-diffusion}

\noindent Molecular diffusion is often indistinguishable from mechanical dispersion as a process operating in porous media, and thus the two are often lumped together to form a hydrodynamic dispersion term.  Unlike dispersion, however, there is no effect of flow direction, so the potential difficulties associated with mismatches between flow and grid coordinate direction do not arise.  Molecular diffusion is an entropy-producing process in which the random motion of molecules causes spreading or homogenization of a concentration field.  Atomistic representations of molecular diffusion capture this random motion, but continuum models of the kind considered here typically represent only the average behavior of the molecules.  It is noteworthy, however, that atomistic and continuum models for molecular diffusion do agree if sufficiently long time scales with a sufficient number of molecules are considered \citep{bourg2008modeling}.

\noindent The principal distinctions in the treatment of molecular diffusion are between those based on Fick's Law, which states that the diffusive flux is linearly proportional to the concentration gradient, and multicomponent treatments that take into account the buildup of electrostatic forces as the individual charged species (ions) attempt to diffuse at their own rate.  In addition, a full treatment of molecular diffusion involves calculating fluxes in terms of gradients in chemical potential rather than concentration \citep{steefel2009fluid}.

\noindent In addition to the complexities associated with chemical interactions, it is also necessary to account for the effects of the porous medium through which diffusion occurs.  Corrections to the diffusive flux are often represented with a tortuosity (see Section~\ref{sec:tortuosity} below) based on an upscaled constitutive law intended to capture the heterogeneous pore geometries.  Since diffusion may be restricted or eliminated through narrow pore throats, the effective diffusivity for a specific ion may be quite different from its diffusivity in water alone.  Capturing the multiscale nature of the pore structure and its effect on molecular diffusion remains a challenge.

\paragraph{General Formulation for Molecular Diffusion.}

The most rigorous and general expression for molecular diffusion is given by
\begin{equation} \label{eq:OnsagerReciprocal}
  \boldsymbol{J}_{j}^{\rm diff} =-\sum_{i}{L_{ji}\boldsymbol{\nabla}\frac{\mu _{i}}{T}},
\end{equation}
where the $L_{ji}$ are the phenomenological coefficients introduced in
the theory of irreversible thermodynamics
\citep{onsager1931reciprocal,prigogine1968introduction,lasaga1998kinetic}
and $\mu_j$ is the chemical potential of the $i$th species.  Here, the
fluxes are linearly related to gradients in the chemical potentials of
the solutes rather than to their concentrations as in Fick's Law that
follows.  The phenomenological coefficients, $L_{ji}$, can be linked
back to measurable quantities by making use of the mobility as the
``velocity'' of a particle acted upon by a force, with the force in
this case provided by the chemical potential rather than the
concentration
\begin{equation} \label{eq:ChemicalPotentialGradient)}
  \boldsymbol{J}_{j}^{\rm diff} =-\boldsymbol{\nabla} ( M_{j} C_{j} \mu _{j}),
\end{equation}
where $u_{j}$ is the mobility of the $j$th ion defined by
\begin{equation} \label{eq:IonMobility}
  M_{j} \eq \frac{D_j}{RT}.
\end{equation}

\paragraph{Single Species Diffusion (Fick's Law).}

Molecular diffusion is usually described in terms of Fick's First Law, which states that the diffusive flux is proportional to the concentration gradient
\begin{equation} \label{eq:FicksFirstLaw}
\boldsymbol{J}_{i}^\text{diff} \eq -\boldsymbol{\nabla} \left( D_{i} C_{i} \right).
\end{equation}
$D_i$ is referred to as the diffusion coefficient and is specific to the chemical component considered as indicated by the subscript $i$.  Fick's First Law is a phenomenological theory for diffusion that relates diffusion to the ``driving force'' provided by the concentration gradient, although it can also be derived atomistically \citep{lasaga1998kinetic}.   In the case of diffusion in porous media, it is normally necessary to include a tortuosity correction as well (see discussion below).

\paragraph{Tortuosity.}\label{sec:tortuosity}
Since water-rock interaction commonly takes place in porous materials,
it is important to account for the effect of tortuosity (Figure ~\ref{fig:Tortuosity}),
which is defined as the ratio of the path length the solute would follow in water alone, $L$,
relative to the tortuous path length it would follow in porous media, $L_e$ \citep{bear-1972}
\begin{equation} \label{eq:Tortuosity}
  \tau_{L}
  =
  \left({\raise0.7ex\hbox{$ L $}\!\mathord{\left/ {\vphantom {L L_{e} }} \right.
  \kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$ L_{e}  $}} \right)^{2}.
\end{equation}
In this definition of tortuosity (sometimes the inverse of Equation \eqref{eq:Tortuosity} is used),
its value is always less than one and the effective diffusion coefficient in porous media
is obtained by multiplying the tortuosity by the diffusion coefficient for the solute in pure water.
\begin{figure}
\begin{center}
  \includegraphics[scale=0.5]{figs/Figure4-Steefel.pdf}
  \caption{Tortuous diffusion paths in porous media \citep{steefel2009fluid}}
  \label{fig:Tortuosity}
\end{center}
\end{figure}

With this formulation, the diffusion coefficient in porous media, $D_{i}^{*}$, is given by
\begin{equation} \label{eq:EffectiveDiffusion)}
  D_{i}^{*} =\tau_{L} D_{i}.
\end{equation}
The diffusive flux, then, is given by
\begin{equation} \label{eq:DiffusiveFluxPorousMedia)}
  \boldsymbol{J}_{j}^{\rm diff}
  \eq
  -\boldsymbol{\nabla} \left( \phi s_{\alpha} D_{j} \tau_{L} C{}_{j} \right)
  \eq
  -\boldsymbol{\nabla} \left( \phi s_{\alpha} D_{j}^{*} C{}_{j} \right) .
\end{equation}
Various approaches for calculating formation factors (and thus, the diffusion coefficient in porous medium) are in use,
with a formulation based on Archie's Law being the most common for fully saturated (single phase) systems
\begin{equation} \label{eq:CementationExponent}
  F_f  \eq \frac{1}{a\phi ^{m} } ,
\end{equation}
where $a$ is a unitless fitting constant and $m$ is unitless the cementation exponent.

For partially saturated systems, it is common to use the Millington-Quirk formulation
\citep{millington1961permeability, sumner-handbook, moldrup2000predicting}
\begin{equation}   \label{eq:MillingtonQuirk}
  D_{i}^{*} \eq \phi^{4/3} s_{\alpha}^{10/3}  D_{i},
\end{equation}
where the saturation, $s_{\alpha}$, can refer to either the gas or the aqueous phase.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Boundary Conditions}
\label{sec:transport-boundary-conditions}

\noindent
Boundary conditions have to be specified for each of the species concentrations $C_i$.
A number of boundary conditions are possible for the diffusive flux,
including a Dirichlet boundary conditions (prescribed concentration)
and Neumann boundary conditions (prescribed flux).
\todo[color=cyan]{GEH:  I suggest that we stick with third-type as the title for this boundary condition.
It is technically Robin (definitely not Cauchy), but third-type is commonly used in subsurface reactive transport.
I also suggest that we remove First-Type and Second-Type and simply refer to them as
Dirichlet (specified concentration) and Neumann (specified flux).}

\paragraph{Dirichlet Boundary Conditionis:}
These conditions specify a (typically fixed/constant) value of the concentration of the species, $C_0$,
at the boundary location:
\begin{equation}\label{eq:transport Dirichlet}
  \boldsymbol{C}(\boldsymbol{x})=\boldsymbol{C}_{0} .
\end{equation}

\paragraph{Neumann Boundary Conditions:}
These conditions specify a value of the flux
\begin{equation}\label{eq:transport Neuman flux}
  \boldsymbol{J}\cdot \bn = J_0^{\bn},
\end{equation}
or concentration gradient
\begin{equation}\label{eq:transport Neuman gradient}
  \frac{\partial \boldsymbol{C}}{\partial \bn} = C_0^{\bn},
\end{equation}
where either $J_0^{\bn}$ or $C_0^{\bn}$ is prescribed.
The flux term can be a total, advective, or diffusive flux.

\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Single-Phase Transport with Dual Porosity Model}
\label{sec:transport-single-phase-dual-porosity}

The multiscale nature of porous media and the transport processes associated is
arguably the most significant and largely unresolved challenge for simulation of
fate and transport in subsurface aquifers.
Transport actually operates at the pore scale where variations in flow velocity
and reaction rates can result in microscopic variability in transport rates.
Continuum treatments of transport in porous media cannot resolve such sub-grid
variations easily, although various upscaling techniques may be available for
capturing the smaller scale behavior.
In addition, multi-continuum or hybrid approaches may obviate the need for a formal
upscaling procedure, although there are significant computational difficulties and
expense associated with their implementation.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Process Model Equations}

The dual porosity formulation of the solute transport consists of two equations
for the fracture and matrix regions.
In the fracture region, we have \citep{simunek-vangenuchten_2008}
\begin{equation}  \label{eq:MassConservation}
  \frac{\partial (\phi_f s_{lf} C_f)}{\partial t}
  +
  \bnabla \cdot \bJ^\text{adv} \eq \bnabla \cdot \bJ^\text{disp}
  +
  \bnabla \cdot \bJ^\text{diff} - \Sigma_s + Q_f,
\end{equation}
where $s_{lf}$ is liquid saturation in fracture,
$\Sigma_s$ is the solute exchange term,
and $Q_f$ is source or sink term.
In the matrix region, we have
$$
  \frac{\partial (\phi_m s_{lm} C_m)}{\partial t}
  = \Sigma_s + Q_m,
$$
where $s_{lm}$ is liquid saturation in matrix, $Q_m$ is source or sink term.
The solute exchange term is defined as
$$
  \Sigma_s = \alpha_s (C_f - C_m) + \Sigma_w C^*,
$$
where $C^*$ is equal to $C_f$ if $\Sigma_w > 0$ and $C_m$ if $\Sigma_w < 0$.
The coefficient $\alpha_s$ is the first-order solute mass transfer coefficient [$\us^{-1}$].

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Boundary Conditions}
\label{sec:transport-boundary-conditions-dual-porosity}

\noindent
For the dual porosity model boundary conditions have to be specified
for the fracture and the matrix concentrations.
The choice of the boundary conditions for the fracture component $C_f$
are \eqref{eq:transport Dirichlet DP}-\eqref{eq:transport Neuman gradient DP}.
Yet, we repeat them here for ease of reference.
The boundary conditions for the matrix/pore component $C_m$
are limited only to  \eqref{eq:transport Dirichlet DP} and \eqref{eq:transport Neuman gradient DP},
and exclude the condition \eqref{eq:transport Neuman flux DP}
on the flux values.

\noindent
Boundary conditions have to be specified for each of the species concentrations $C_i$.
A number of boundary conditions are possible for the diffusive flux,
including a Dirichlet boundary conditions (prescribed concentration)
and Neumann boundary conditions (prescribed flux).
\todo[color=cyan]{GEH:  I suggest that we stick with third-type as the title for this boundary condition.
It is technically Robin (definitely not Cauchy), but third-type is commonly used in subsurface reactive transport.
I also suggest that we remove First-Type and Second-Type and simply refer to them as
Dirichlet (specified concentration) and Neumann (specified flux).}

\paragraph{Dirichlet Boundary Conditionis:}
These conditions specify a (typically fixed/constant) value of the concentration of the species, $C_0$,
at the boundary location:
\begin{equation}\label{eq:transport Dirichlet DP}
  \boldsymbol{C}(\boldsymbol{x})=\boldsymbol{C}_{0} .
\end{equation}

\paragraph{Neumann Boundary Conditions:}
These conditions specify a value of the flux
\begin{equation}\label{eq:transport Neuman flux DP}
  \boldsymbol{J}\cdot \bn = J_0^{\bn},
\end{equation}
or concentration gradient
\begin{equation}\label{eq:transport Neuman gradient DP}
  \frac{\partial \boldsymbol{C}}{\partial \bn} = C_0^{\bn},
\end{equation}
where either $J_0^{\bn}$ or $C_0^{\bn}$ is prescribed.
The flux term can be a total, advective, or diffusive flux.

\subsection{Data Needs}

\noindent For modeling of molecular diffusion in porous media, two kinds of data are needed:
%
\begin{enumerate}
\item
  Experimental data on the diffusivities of individual ions in aqueous solution;
\item
  Characterization of the tortuosity of the porous medium under consideration.
  The tortuosity of the medium may be determined by a number of methods,
  including transport experiments involving tracers \citep{navarre2009evolution},
  and microscopic imaging of the porous medium using synchrotron X-ray
  %~\todo{synchrotron X-ray? -Williamson}
  or related methods \citep{navarre2009evolution}, or estimation based on grain size and mineralogy of the materials.
  In addition, it may be possible to determine tortuosity from effective diffusion coefficients determined in field-scale experiments.
\end{enumerate}

\noindent
For the diffusivities of individual ions, there are some compilations in the literature \citep{lasaga1998kinetic,steefel2009fluid}.
While the diffusivity of individual ions could in theory be calibrated from field tests,
normally they should be determined independently so that a more accurate determination of the tortuosity can be carried out.


\section{DFS and BFS}

%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Turing Award}
  \begin{columns}
	\column{0.50\textwidth}
	  \fignocaption{width = 0.40\textwidth}{figs/hopcroft.jpg}{\centerline{John Hopcroft}}
	\column{0.50\textwidth}
	  \fignocaption{width = 0.35\textwidth}{figs/tarjan.png}{\centerline{Robert Tarjan}}
  \end{columns}

  \vspace{0.50cm}
  \begin{quote}
	\textcolor{blue}{``For fundamental achievements in the design and analysis of algorithms and data structures.''} \\
	\hfill --- Turing Award, 1986
  \end{quote}
\end{frame}
%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Depth-first search}
  \fignocaption{width = 0.55\textwidth}{figs/dfs-paper-tarjan}

  \uncover<2->{
	\begin{quote}
	  {\small ``We \textcolor{red}{have seen} how the depth-first search method may be used in the construction of
	  very efficient graph algorithms. $\dots$ \\
	  Depth-first search \textcolor{red}{is} a powerful technique with many applications.''}
	\end{quote}
  }

  \begin{alertblock}{Reference}
	\begin{itemize}
	  \item ``Depth-First Search And Linear Graph Algorithms'' by Robert Tarjan.
	\end{itemize}
  \end{alertblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%
\begin{frame}{The POWER of DFS}
  \begin{center}
	Graph decomposition vs. Graph traversal \\[10pt]
	Structures!
  \end{center}

  \pause
  \begin{enumerate}
	\item states of vertices
	\item types of edges
	\item lifetime of vertices (DFS)
	  \begin{itemize}
	    \item $v: \text{d}[v], \text{f}[v]$
	    \item $\text{f}[v]$: DAG, SCC
	    \item $\text{d}[v]$: biconnectivity
	  \end{itemize}
  \end{enumerate}
\end{frame}
%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Types of edges}
  \begin{definition}[Classifying edges]
    Given a DFS/BFS traversal $\Rightarrow$ DFS/BFS tree:
	\begin{description}[Forward edge:]
	  \item[Tree edge:] $\to$ child
	  \item[Back edge:] $\to$ ancestor
	  \item[Forward edge:] $\to$ \emph{nonchild} descendant
	  \item[Cross edge:] $\to$ neither ancestor nor descendant
    \end{description}
  \end{definition}

  \pause
  \begin{alertblock}{Remarks}
    \begin{itemize}
      \item applicable to both DFS and BFS
      \item w.r.t. DFS/BFS trees
    \end{itemize}
  \end{alertblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Types of edges (Problem 5.18)}
  \begin{figure}
	\begin{subfigure}{0.50\linewidth}
	  \centering
	  \includegraphics[width=0.50\textwidth]{figs/dfs-digraph.png}
	  \caption{DFS on directed graph.}
	\end{subfigure}%
	\begin{subfigure}{0.50\linewidth}
	  \centering
	  \includegraphics[width=0.50\textwidth]{figs/dfs-undirected.png}
	  \caption{DFS on undirected graph.}
	\end{subfigure}

	\begin{subfigure}{0.50\linewidth}
	  \centering
	  \includegraphics[width=0.50\textwidth]{figs/bfs-digraph.png}
	  \caption{BFS on directed graph.}
	\end{subfigure}%
	\begin{subfigure}{0.50\linewidth}
	  \centering
	  \includegraphics[width=0.60\textwidth]{figs/bfs-undirected.png}
	  \caption{BFS on undirected graph.}
	\end{subfigure}
  \end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Types of edges}
  \begin{exampleblock}{DFS tree and BFS tree coincide (Additional Problem)}
    \begin{itemize}
	  \item undirected connected graph $G = (V,E), v \in V$
	  \item DFS tree $T$ from $v$ $\equiv$ BFS tree $T'$ from $v$
      \item prove: $G = T$
    \end{itemize}
  \end{exampleblock}

  \pause
  \begin{center}
	$G_{\text{DFS}}$: tree + back \emph{vs.} $G_{\text{BFS}}$: tree + cross
  \end{center}

  \pause
  \begin{alertblock}{Question}
	\begin{itemize}
	  \item DFS\&BFS from different $v's$?
	  \item What if $G$ is a digraph?
	\end{itemize}
  \end{alertblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Lifttime of vertices in DFS}
  \fignocaption{width = 0.40\textwidth}{figs/active-interval.png}
\end{frame}
%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Lifttime of vertices in DFS}
  \begin{theorem}[Disjoint or contained]
	\begin{gather*}
	  \forall u,v: \\
	  [_{u} \; ]_{u} \cap [_{v} \; ]_{v} = \emptyset \\
	  \bigvee \\
	  ([_{u} \; ]_{u} \subsetneqq [_{v} \; ]_{v} \lor [_{v} \; ]_{v} \subsetneqq [_{u} \; ]_{u})
	\end{gather*}
  \end{theorem}

  \pause

  \begin{proof}
	\fignocaption{width = 0.30\textwidth}{figs/stack.png}
  \end{proof}
\end{frame}
%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Ancestor/descendant relation}
  \begin{exampleblock}{Preprocessing for ancestor/descendant relation (Problem 5.23)}
    \begin{itemize}
      \item binary tree $T = (V, E)$
      \item $r \in V$
    \end{itemize}
  \end{exampleblock}

  \[
	v: \text{d}[v], \text{f}[v]
  \]

  \pause
  \begin{alertblock}{Question}
    $\forall v$: how many descendants?
  \end{alertblock}

  \pause
  \begin{alertblock}{Remark}
	General (rooted) tree?
  \end{alertblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Edge types and lifetime of vertices in DFS}
  \begin{exampleblock}{Edge types and lifetime of vertices in DFS (Problem 5.2)}
    $\forall u \to v$:
    \begin{itemize}
      \item tree/forward edge: $[_{u}\; [_{v}\; ]_{v}\; ]_{u}$
      \item back edge: $[_{v}\; [_{u}\; ]_{u}\; ]_{v}$
      \item cross edge: $[_{v}\; ]_{v}\; [_{u}\; ]_{u}$
    \end{itemize}
  \end{exampleblock}

  \pause

  \begin{alertblock}{Remark}
    \begin{itemize}
      \item $\text{f}[v] < \text{d}[u]$: cross edge
      \item $\text{f}[u] < \text{f}[v]$: back edge
		\pause
		\[
		  u \to v \iff \text{f}[v] < \text{f}[u]
		\]
    \end{itemize}
  \end{alertblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Height and diameter of tree}
  \begin{exampleblock}{Height and diameter of tree (Problem 5.21)}
	Binary tree $T = (V, E)$ with $|V| = n$:
	\begin{itemize}
	  \item height ($O(n)$)
	  \item diameter ($O(n)$)
	\end{itemize}
  \end{exampleblock}

  \pause
  \vspace{0.30cm}
  \centerline{throught root or not?}

  \pause
  \vspace{0.50cm}
  \begin{alertblock}{Question}
	Diameter of a tree \emph{without} a designated root?
  \end{alertblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Perfect subtree}
  \begin{exampleblock}{Perfect subtree (Problem 5.22)}
	\begin{itemize}
	  \item binary tree $T = (V, E)$
	  \item root $r \in V$
	  \item goal: find all perfect subtrees
	\end{itemize}
  \end{exampleblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Counting shortest paths}
  \begin{exampleblock}{Counting shortest paths (Problem 5.26)}
	Counting \# of shortest paths in (un)directed graphs using BFS.
  \end{exampleblock}

  \pause
  \vspace{0.50cm}
  \centerline{Maybe in the next class$\dots$}
\end{frame}
%%%%%%%%%%%%%%%%%%%%


\subsection{Exchange rates}

\subsubsection{Interest rate parity}

What determines an exchange rate? Consider someone in the US choosing between investing at home or in the UK. Returns in the US are 4\% over the next year and otherwise identical investments in the UK offer 3\%.

Is the US investment better? Not necessarily - as the UK bond is valued in pounds, the investor will also consider changes to the exchange rate over the period. If the value of the pound is expected to increase sufficiently over the period then the 3\% bond would be a better investment.

If the investor can freely choose between the two –there are no capital controls - expected movements in the exchange rate equal current differences in returns. If the value of pounds was above this level then investors would sell pounds and buy US investments until this relationship was restored. This is known as interest rate parity.

\(Return_USD = Return_GBP /dfrac{Exchange rate_{next year}}{Exchange rate_today}\)

Next, let’s break down what the returns include. If inflation is 1\%, then a 3\% nominal return only gets you (roughly) 2\%.

\(Return \approx inflation + real\)

OK, so now we have a link between changes to the exchange rate and differences in inflation and real (inflation adjusted return).


\section{The Rust Model} \label{rust_model}

This section sets up the model whose input-output relationship I want to shed light on. I consider the single-agent dynamic discrete choice model of \citet{R87}.

\subsection{Bus-Engine Replacement Model}

Every period an agent needs to decide whether to replace the engine of a bus in the carpark or to conduct standard maintenance operations. The Rust model assumes optimal behaviour of the agent and asks what the implied deep economic parameters are, given this optimal behaviour.

The model output I consider is implied annual demand for engine replacements. Model inputs are engine replacement costs and a cost parameter that is part of the agent’s utility function. In the Rust model these variables are estimated but I consider them as inputs for the computation of implied annual demand. Since the inputs are estimated, they are subject to uncertainty. It is the influence of the inputs on the output I want to analyse. Applying sensitivity methods requires the simulation of the input variables. To that end, information about the distribution of the inputs is needed. The variance-covariance matrix of the inputs is estimated in \cref{model_setup} % especially for calculating the variance-covariance matrix and mean vector of the inputs.

In \citet{R87} an agent needs to manage a pool of buses for public transportation. Each period a bus exhibits an observable accumulated mileage state (odometer state) $x$ and other state variables $\varepsilon$, that are unobserved by the econometrician. The agent has to decide whether to perform standard maintenance or an expensive engine replacement. The agent’s decision in period $t$ is
\begin{equation*}
i_t\ =
\begin{cases}
0,\ \text{standard maintenance}, \\
1,\ \text{replace engine}.
\end{cases}
\end{equation*}

\noindent If replacement is chosen, the accumulated mileage state $x$ is set to zero. Otherwise, the bus is used for another period with standard maintenance and $x$ increases until the next period. Note that $x$ needs to be discretised for estimation purposes. I follow \citet{R87} and discretise the state space into 175 grids with a size of each bin of $2\,571$ miles.

The agent’s utility function given state $(x, \varepsilon)$ is described by
\begin{equation}
u(x, i, \varepsilon; \theta_1, RC)=v(x, i; \theta_1, RC)+\varepsilon(i), \label{utility}
\end{equation}

where
\begin{equation*}
v(x, i; \theta_1, RC)=
\begin{cases}
-c(x; \theta_1), & \text{if}\ i=0,\\
-RC-c(0;\theta_1), & \text{if}\ i=1,
\end{cases}
\end{equation*}

\noindent and $\varepsilon(i)$ denotes utility shocks. In the above expression, $RC$ represents the expected costs of an engine replacement, that is, net the possible payoff of selling the old engine. $c(x; \theta_1)$ is a cost function of operating a bus at state $x$. $\theta_1$ is a vector of structural parameters. I consider the linear cost function $c(x; \theta_{11}) = 0.001 \theta_{11} x$.
\cref{utility} reflects the trade-off between minimising maintenance costs and minimising unexpected, costly engine failures, e.g. due to loss of customer's goodwill.

How the state $(x, \varepsilon)$ changes over time, depends on the choice $i_t$ and is described by the Markovian transition density $p(x_{t+1}, \varepsilon_{t+1} \mid x_t, \varepsilon_t, i_t; \theta_2, \theta_3)$, where $\theta_2$ is assumed to be Euler’s constant and $\theta_3$ is a vector of transition probabilities. Markovian means that $p$ depends on $ (x_t, \varepsilon_t) $ only and not on other past states.

The discount factor is denoted $\beta \in (0,1) $. Given state $ (x_t, \varepsilon_t) $, the agent, choosing a sequence of decisions, maximises
\begin{equation*}
\max_{\{i_t, i_{t+1}, i_{t+2}, \dots\}}  \mathrm{E} \left[\sum_{\tau=1}^\infty \beta^{\tau-t} u(x_{\tau}, i_{\tau}, \varepsilon_{\tau}; \theta_1, RC) \right].
\end{equation*}

By $p$ being Markovian, the optimal decision is time-invariant. Thus, drop time indices. The decision problem is now defined by the solution to the Bellman equation
\begin{equation*}
	V(x, \varepsilon)=\max_i \{ v(x, i; \theta_1, RC)+\varepsilon(i) + \beta \int_{x'} \int_{\varepsilon'} V(x', \varepsilon')p(x', \varepsilon' \mid x, \varepsilon, i; \theta_2, \theta_3)dx' d \varepsilon' \}.
\end{equation*}
% \begin{multline*}
% V(x, \varepsilon)=\max_i \{ v(x, i; \theta_1, RC)+\varepsilon(i) + \\
% \beta \int_{x'} \int_{\varepsilon'} V(x', \varepsilon')p(x', \varepsilon' \mid x, \varepsilon, i; \theta_2, \theta_3)dx' d \varepsilon' \}.
% \end{multline*}

The above problem is simplified by imposing some further assumptions to ease the estimation process. The model is solved by using the so called Nested Fixed Point algorithm \citep{R87}. The parameters to be estimated are combined in a vector $\theta=(\theta_3, RC, \theta_{11}) $. The estimation procedure via Maximum Likelihood is conducted by first estimating the transition probabilities $\theta_3$. In a second step, the parameters $RC$ and $\theta_{11}$ are estimated.\footnote{I do not go into further details of how this model is solved. For details see \citet{R87}.}

\subsection{Distribution of the Input Variables} \label{model_setup}

The Maximum Likelihood estimator is asymptotically normally distributed \citep{R73}. Thus, I estimate the variance-covariance matrix of $\theta_{11}$ and $RC$ by a Monte Carlo simulation. Note that this general setup is used in \cref{comp_shap} and \cref{comp_morris} as well. For solving the model, I use the simulation and estimation capabilities of the Python \textit{ruspy} package \citep{OSE19}.\footnote{I chose \textit{scipy\_L-BFGS-B} as the solver for the Nested Fixed Point algorithm \citep{SP20}.} The estimation of the variance-covariance matrix is conducted in three steps.

Firstly, I set the true parameters in $\theta$. I chose to consider three transition probabilities, i.e. the mileage state $x$ can increase from the current period to the next by two grids at most. I set
\begin{align*}
\theta_3 &= (\theta_{30}, \theta_{31}, \theta_{32})=(0.39189182,\ 0.59529371,\ 0.01281447),\\
RC &= 10.07780762,\\
\theta_{11} &= 2.29417622.
\end{align*}

\noindent Note that $RC$ is scaled by $0.001$ for estimation purposes. I consider a total of 50 buses for 120 periods (months, i.e. in total ten years).

\begin{table}
	\centering
	\caption{Variance-Covariance Matrix of the Input Variables}
	\label{cov}
	\begin{threeparttable}
	\centering
	\input{../figures/covariance_matrix.tex}
	\begin{tablenotes}
	\small
	\item \textit{Notes:} Estimation based on $10\,000$ data sets.
	\end{tablenotes}
	\end{threeparttable}
\end{table}

Secondly, I simulate $10\,000$ data sets from the above true parameters and estimate $\hat{\theta}, \hat{\theta}$ being the vector of estimated structural parameters.

Thirdly, from the $10\,000$ parameter estimates of the replacement costs $\widehat{RC}$ and the slope parameter of the cost function ${\hat{\theta}}_{11}$, I calculate the sample variance-covariance matrix which is shown in \Cref{cov}. \Cref{cov} and \cref{correlation} show that $\widehat{RC}$ and ${\hat{\theta}}_{11}$ correlate, hinting that inputs in the Rust model are indeed dependent.

\subsection{Derivation of Implied Annual Demand}

\begin{figure}[t]
	\caption{Correlation Between the Input Variables}
    \label{correlation}
	\centering
	\vspace*{-4mm}
	\begin{centering}
	\includegraphics[scale=0.9]{../figures/correlation_rc_theta.png}
	\end{centering}

	\small
	\textit{Notes:} $10\,000$ estimated values of $RC$ and $\theta_{11}$.

\end{figure}

\citet{R87} shows how we can derive further quantities from the parameter estimates. He derives the implied annual demand (henceforth, demand) for bus-engine replacements, which is computed by
\begin{equation*}
\tilde{d}(RC)=\sum_{t=1}^{12} \sum_{i=1}^{M} {\tilde{i}}_t^m,
\end{equation*}

\noindent where $M$ is the number of buses and ${\tilde{i}}_t^m$ is a realisation of the process $(i_t^m, x_t^m)$, with $m$ being the bus index. Here, $M=50$. Since we consider the \textit{annual} demand, we sum over twelve months. Recall that a period is represented by one month, so we end up with the annual demand for 50 buses. $\tilde{d}(RC)$ is stochastic, since ${\tilde{i}}_t^m$ is. Let $\pi_m(x_0^m, i_0^m)$ denote the distribution of the initial state for each bus $m$. Given this, we can compute the probability distribution of $\tilde{d}(RC) $ by the transition probability $P(i_t \mid x_t, \theta)p(x_t \mid x_{t-1}, i_{t-1}, \theta_3) $. Note that $P(\cdot)$ depends on $\theta$, i.e. it depends on $RC$ as well. We can compute demand for a range of values for $RC$. In my analysis I compute demand for a single value of $RC$ only, which is arbitrarily set to $RC^{demand}=11.5$, i.e. $11\,500$ money units. \citet{R87} considers the expected demand, i.e. $d(RC)=E[\tilde{d}(RC)]$. Assuming that the initial state distribution $\pi$ is the long-run equilibrium distribution of $(i_t, x_t)$, $\pi$ is the solution to
\begin{equation*}
\pi(x, i)=\int_y \int_j P(i \mid x, \theta) p(x \mid y, j, \theta_3) \pi(dy, dj).
\end{equation*}

Note, that $\pi$ is a function of $\theta$. Assuming further that the processes $ (i_t^m, x_t^m)$ and $(i_t^k, x_t^k)$ are independent for $k \neq m$, i.e. the process of a bus is independent from the process of any other bus, we can write
\begin{equation*}
d(RC)=12 \cdot M \cdot \int_0^{\infty} \pi(dx, 1).
\end{equation*}

To see how the uncertainty in the inputs $(RC, \theta_{11})$ propagate to the output, demand, see \cref{uncertainty}. I again use \textit{ruspy} for the computation of the demand at $RC^{demand}$. I compute demand for $100\,000$ simulated data sets. The output is normally distributed as can be seen in \cref{uncertainty} with mean 58.67 and variance 41.83. Sensitivity analysis can identify the inputs that are responsible for this large variability in the model output.

Now, that the model, i.e. how the output depends on the inputs, has been clarified, we are ready to consider the sensitivity methods applied in this thesis.

\begin{figure}[t]
	\caption{Uncertainty Propagation}
    % \floatfoot{A note}
    \label{uncertainty}
	\begin{centering}
	\vspace*{-4mm}
	\begin{centering}
		\includegraphics[scale=0.9]{../figures/uncertainty_propagation_100000.png}
	\end{centering}
	\end{centering}

	\small
	\textit{Notes:} Estimated implied annual demand for $100\,000$ simulated data sets. The horizontal line shows the implied annual demand at the \textit{true} parameter values $(RC, \theta_{11})$. This \textit{true} demand amounts to $59.09$.
\end{figure}