text:
\section{Introduction}

In \cite{KW4} it was shown that the probabilities of topologically defined uniform spanning tree events can be computed as linear combinations of determinants of matrices whose entries involve the Green's function $G$ and the derivative $G'$ of the ``line-bundle Green's function''.  These probabilities were used to compute the intensity of loop-erased random walk \cite{KW4} and the probabilities of local events in the abelian sandpile model \cite{wilson:sand-local}.  We give another formula involving Pfaffians.  In addition to being (somewhat) computationally more efficient, the Pfaffian formula implies some structural properties of the polynomials in $G$ and $G'$.  First, it becomes apparent that the coefficients of the polynomials are integers --- previously the coefficients were only known to be half-integers.  Second, if each $G'_{u,v}$ variable is replaced with $G'_{u,v}+f(u)-f(v)$, then each polynomial is unchanged.  This invariance property was observed earlier for small sizes, but a general proof was missing until now, and it simplifies some of the sandpile calculations.

For background on the line-bundle Laplacian, response matrix, and Green's function for graphs embedded in surfaces, and their use in computing spanning tree probabilities, we refer the reader to \cite{KW4}.  Here we summarize the key facts that we use.

The line-bundle Green's function $\mathscr{G}_{u,v}(z)$ is a generalization of the usual Green's function $G_{u,v}$, where $u$ and $v$ are vertices of a graph $\mathcal G$, and $z\in\mathbb{C}$.  When $z=1$ it specializes to the usual Green's function:
\[
G_{u,v}=G_{v,u} =\mathscr{G}_{u,v}(1)\,.
\]
The line-bundle Green's function $\mathscr{G}$ has the symmetry $\mathscr{G}_{v,u}(z)=\mathscr{G}_{u,v}(1/z)$.  We define
\[
G'_{u,v}=\left[\frac{d}{dz} \mathscr{G}_{u,v}(z)\right]_{z=1}\,,
\]
which is antisymmetric, and is what we referred to as the derivative of the line-bundle Green's function.  From the symmetry of $\mathscr{G}$ it follows that $G'$ is antisymmetric: $G'_{v,u}=-G'_{u,v}$.
For the Green's function there is a designated sink vertex $s$ (which has been suppressed from the notation) for which
\[ G_{u,s} = 0 \]
for each vertex $u$.

There is another set of electrical variables that are useful to work with, the response matrix, or the Dirichlet-to-Neumann matrix $L_{u,v}$.  The response matrix is defined with respect to a designated set of vertices which we call \textbf{nodes}.  We think of the nodes as being ``boundary vertices'', and the other vertices as being internal, and the response matrix gives the linear map from voltages to current flows.  In the line bundle setting we denote the response matrix by $\L_{u,v}(z)$.  Here too $\L_{v,u}(z)=\L_{u,v}(1/z)$, and the line-bundle response matrix specializes to the usual response matrix when $z=1$,
\[
L_{u,v}=L_{v,u}=\left[\L_{u,v}(z)\right]_{z=1}\,,
\]
which is symmetric, and we define
\[
L'_{u,v}=\left[\frac{d}{dz} \L_{u,v}(z)\right]_{z=1}\,
\]
which is antisymmetric.  The response matrix variables satisfy the additional relation
\[ \sum_v L_{u,v} = 0
\]
for each vertex $u$.

Spanning tree probabilities for a graph $\mathcal G$ embedded on an annulus can be computed in terms of either set of variables, $\{G_{u,v},G'_{u,v}\}$ or $\{L_{u,v},L'_{u,v}\}$.

Suppose that graph $\mathcal G$ has $n$ nodes, which we label $\{1,\dots,n\}$.  A \textbf{grove} is a forest such that each tree contains at least one node.  Groves were first studied by Carroll and Speyer \cite{CS}, and then more systematically by Kenyon and Wilson \cite{KW1}, who gave this current definition.  Any grove induces a set partition $\sigma$ on the nodes where each set consists of the nodes from the same tree.  We let $Z[\sigma]$ denote the weighted sum of groves whose induced partition is $\sigma$.  The weighted sum of spanning trees $Z[\text{tree}]=Z[1,\dots,n]$ can be computed via the matrix-tree theorem, so we are interested in computing the ratios
\begin{equation} \label{eq:Zdot}
\dddot Z[\sigma] \colonequals \frac{Z[\sigma]}{Z[1|2|\cdots|n]}
\end{equation}
or
\begin{equation} \label{eq:Zbar}
\overline{Z}[\sigma] \colonequals \frac{Z[\sigma]}{Z[\text{tree}]}.
\end{equation}

Suppose that the graph $\mathcal G$ is embedded in an annulus so that the nodes $1,\dots,n-1$ are arranged in cyclic order on one boundary of the annulus, while node $n$ is on the other boundary of the annulus.  The grove partition function ratios \eqref{eq:Zdot} and \eqref{eq:Zbar} for these ``\textbf{annular-one graphs}'' $\mathcal G$ were used to compute probabilities for loop-erased random walk \cite{KW4} and for recurrent sandpile configurations \cite{wilson:sand-local}.  For annular-one graphs, $\dddot Z[\sigma]$ can be expressed in terms of a linear combination of determinants involving $L$ and $L'$, while $\overline{Z}_\sigma$ can be expressed in terms of a linear combination of determinants involving $G$ and $G'$ \cite{KW4}.  We shall re-express them as linear combinations of Pfaffians.

It turns out that $Z[\sigma]$ is itself a linear combination of $Z[\tau]$'s, where each $\tau$ is a ``partial pairing'' of the nodes $1,\dots,n$ \cite{KW4}.  A \textbf{partial pairing} is a set of pairs of nodes, singletons, and ``internalized'' nodes, which are not listed in the partition, but which may appear in any of the parts (like the other non-node vertices in a grove).  For example,
\begin{multline*}
   Z[2,6,9|3,4,5|7|1,8] = Z[2,9|3,5|7|1,8] - Z[2,9|3,6|7|1,8] \\- Z[2,9|3,5|6,7|1,8] - Z[2,9|3,5|1,6|7] + Z[2,9|3,5|1,6|7,8]\,.
\end{multline*}
It turns out that for the LERW and sandpile applications it suffices to assume that node $n$ is in a doubleton part.

Kenyon and Wilson \cite{KW4} showed that $Z[\tau]/Z[1|2|\cdots|n]$ can
be expressed as a linear combination of determinants involving the
$L_{i,j}$'s and $L'_{i,j}$'s, and that $Z[\tau]/Z[1,2,\cdots,n]$ can
be expressed as a linear combination of determinants involving the
$G_{i,j}$'s and $G'_{i,j}$'s.  We will give these determinant formulas
in the next section, since they are the starting point of the present work.

\subsection{Partial pairings in terms of Pfaffians}

For a partial pairing $\tau$ in which node $n$ is in a doubleton part,
we can encode $\tau$
by a string $\lambda$ of $n$ symbols, where the symbol at position $i$ encodes the role of node~$i$ in the partial pairing.
For bookkeeping purposes that will soon become apparent,
we label each symbol with the label of the node that it represents; when the labels
are $1,\dots,n$ we sometimes omit the labels.
For example, for the annular partial pairing
\[\tau=1,5|2|3,4|7,10|8,14|11|12,13 \quad=\quad
\begin{tikzpicture}[baseline=0.0cm-2.5pt]
\draw[fill=gray!40!white,draw=none] (0,0) circle(1.5);
\draw[fill=white,draw=none] (0,0) circle(.4);
    \foreach \x in {1,...,13} { \coordinate (\x) at ({1.5*cos((\x-0.5)*360/13)},{1.5*sin((\x-0.5)*360/13)});}
    \coordinate (14) at ({360*15/25}:.4);
    \draw[very thick] (14)--(8);
    \tkzDefPoint({-360/13}:1.5){1213};
    \tkzDefPoint({360*3/13}:1.5){34};
    \tkzDrawArc[very thick,color=black](1213,13)(12);
    \tkzDrawArc[very thick,color=black](34,4)(3);
    \draw[very thick] (1) .. controls +({360*13.5/26}:.5) and +({360*22.5/26}:.5) .. (5);
    \draw[very thick] (7) .. controls (65:1.2) and (15:1.2) .. (10);
    \foreach \x in {1,2,3,4,5,7,8,10,11,12,13,14} {\node [circle,fill=white,draw=black,inner sep=0.5pt] at (\x) {$\phantom{3}\mathclap{\x}\phantom{3}$};}
\end{tikzpicture}
\]
the associated (labeled) encoding string is
\begin{equation} \label{lambda}
\lambda =\lambda(\tau) = \U{1}\S{2}\U{3}\D{4}\D{5}\I{6}\D{7}\F{8}\I{9}\U{10}\S{1\!1}\U{12}\D{13}\OD{14}\,.
\end{equation}

Here node $n$, which is on the other boundary, is given the special symbol $\Osc{\odot}$.  The node paired with $n$ is also given a special symbol, \textsf{F}.  (So each $\lambda(\tau)$ has exactly one $\F{}$ and one $\Osc{\odot}$ symbol.)  \textsf{I} indicates that the node has been internalized, and \textsf{S} indicates a node in a singleton part.  The remaining nodes are assigned the symbols \textsf{U} and \textsf{D} so that when the \textsf{F} is cyclically rotated to the end, the substring $\lambda^\circ$ formed by the \textsf{U}'s and \textsf{D}'s defines a (labeled) Dyck path whose associated noncrossing matching is the pairing of the nodes in $\tau$.
In the above example,
\[ \lambda^\circ = \U{10}\U{12}\D{13}\U{1}\U{3}\D{4}\D{5}\D{7}
\quad=\quad
\begin{tikzpicture}[scale=0.4,baseline=0.0cm-1pt]
\draw[thick] (0,0) -- (1,1) -- (2,2) -- (3,1) -- (4,2) -- (5,3) -- (6,2) -- (7,1) -- (8,0);
\draw[fill,draw=none] (0,0) circle(.1);
\draw[fill,draw=none] (1,1) circle(.1);
\draw[fill,draw=none] (2,2) circle(.1);
\draw[fill,draw=none] (3,1) circle(.1);
\draw[fill,draw=none] (4,2) circle(.1);
\draw[fill,draw=none] (5,3) circle(.1);
\draw[fill,draw=none] (6,2) circle(.1);
\draw[fill,draw=none] (7,1) circle(.1);
\draw[fill,draw=none] (8,0) circle(.1);
\node at (0.5,0.5) {\contour{white}{$\scriptstyle 10$}};
\node at (1.5,1.5) {\contour{white}{$\scriptstyle 12$}};
\node at (2.5,1.5) {\contour{white}{$\scriptstyle 13$}};
\node at (3.5,1.5) {\contour{white}{$\scriptstyle 1$}};
\node at (4.5,2.5) {\contour{white}{$\scriptstyle 3$}};
\node at (5.5,2.5) {\contour{white}{$\scriptstyle 4$}};
\node at (6.5,1.5) {\contour{white}{$\scriptstyle 5$}};
\node at (7.5,0.5) {\contour{white}{$\scriptstyle 7$}};
\end{tikzpicture}
\quad=\quad
\begin{tikzpicture}[scale=0.4,baseline=0.0cm-1pt]
\draw[thick] (0,0) -- (1,1) -- (2,2) -- (3,1) -- (4,2) -- (5,3) -- (6,2) -- (7,1) -- (8,0);
\draw[fill,draw=none] (0,0) circle(.1);
\draw[fill,draw=none] (1,1) circle(.1);
\draw[fill,draw=none] (2,2) circle(.1);
\draw[fill,draw=none] (3,1) circle(.1);
\draw[fill,draw=none] (4,2) circle(.1);
\draw[fill,draw=none] (5,3) circle(.1);
\draw[fill,draw=none] (6,2) circle(.1);
\draw[fill,draw=none] (7,1) circle(.1);
\draw[fill,draw=none] (8,0) circle(.1);
\draw[thick,color=gray] (0.5,0.5)--(7.5,0.5);
\draw[thick,color=gray] (1.5,1.5)--(2.5,1.5);
\draw[thick,color=gray] (3.5,1.5)--(6.5,1.5);
\draw[thick,color=gray] (4.5,2.5)--(5.5,2.5);
\node at (0.5,0.5) {\contour{white}{$\scriptstyle 10$}};
\node at (1.5,1.5) {\contour{white}{$\scriptstyle 12$}};
\node at (2.5,1.5) {\contour{white}{$\scriptstyle 13$}};
\node at (3.5,1.5) {\contour{white}{$\scriptstyle 1$}};
\node at (4.5,2.5) {\contour{white}{$\scriptstyle 3$}};
\node at (5.5,2.5) {\contour{white}{$\scriptstyle 4$}};
\node at (6.5,1.5) {\contour{white}{$\scriptstyle 5$}};
\node at (7.5,0.5) {\contour{white}{$\scriptstyle 7$}};
\end{tikzpicture}
\,.
\]
We call the string $\lambda$ the \textbf{augmented cyclic Dyck path} associated with the partial pairing~$\tau$ --- ``augmented'' because it contains symbols not in the Dyck path $\lambda^\circ$, and ``cyclic'' because its start is determined by the location of the $\F{}$ symbol.

Given two labeled augmented cyclic Dyck paths $\lambda$ and $\mu$, we say that $\lambda\preceq\mu$ if they are the same length, have the same labels,
all the letters other than $\U{}$ and $\D{}$ are the same in both $\lambda$ and $\mu$, and as Dyck paths, $\lambda^\circ$ lies below $\mu^\circ$.

If $\lambda$ is a \textbf{labeled string}, we let $\mathbf\lambda_i$ denote its $i$th labeled symbol, and we let $\lambda(i)$ denote the label of $\lambda_i$.

For a labeled augmented cyclic Dyck path~$\mu$, we define $\mu^{\I{}}$ to be the labeled string obtained from~$\mu$ by deleting all the $\textsf{S}$ letters, and replacing each $\I{i}$ with the two letters $\II{i}$.  We also define $\mu^\textsf{S}$ to be the labeled string obtained from $\mu$ by deleting all the $\I{}$ letters, and replacing each $\S{i}$ with the two letters $\SS{i}$.  For example, if $\mu$ is the labeled augmented cyclic Dyck path in \eqref{lambda}, then
\[
\mu^{\I{}} = \U{1}\U{3}\D{4}\D{5}\II{6}\D{7}\F{8}\II{9}\U{10}\U{12}\D{13}\OD{14}
\]
and
\[
\mu^\textsf{S} = \U{1}\SS{2}\U{3}\D{4}\D{5}\D{7}\F{8}\U{10}\SS{1\!1}\U{12}\D{13}\OD{14}\,.
\]
Next we define $\dddot\mu$ and $\overline\mu$.  Recall that there is only one letter $\F{}$ in $\mu$; let $f$ be its label, so that $\F{f}\in\mu$.
For each letter $\U{i}$, $\D{i}$, $\F{f}$ in $\mu^\textsf{S}$, we make the substitutions
\begin{align*}
 \U{i} \mapsto \begin{cases} \OP{i} & i<f \\ \OC{i} & i>f \end{cases}
\quad\quad\quad\quad
 \D{i} \mapsto \begin{cases} \OC{i} & i<f \\ \OM{i} & i>f \end{cases}
\quad\quad\quad\quad
 \F{f} \mapsto \OC{f}
\end{align*}
to obtain $\overline\mu$.
We let $\dddot\mu$ be the result of these same substitutions applied to $\mu^{\I{}}$.
For our example,
\[
\dddot\mu = \OP{1}\OP{3}\OC{4}\OC{5}\II{6}\OC{7}\OC{8}\II{9}\OC{10}\OC{12}\OM{13}\OD{14}
\]
and
\[
\overline\mu = \OP{1}\SS{2}\OP{3}\OC{4}\OC{5}\OC{7}\OC{8}\OC{10}\SS{1\!1}\OC{12}\OM{13}\OD{14}\,.
\]
The original string $\mu$ can be recovered from either $\overline\mu$ or $\dddot\mu$.

Given a string $\sigma$ of $m$ labeled symbols $\Osc{\oplus}$, $\Osc{\ominus}$, $\Osc{\ocircle}$, $\Osc{\odot}$, such as as the ones above, we define an $m\times m$ matrix $M_\sigma(A,A')$ by
\[
M_\sigma(A,A') \colonequals \bordermatrix[{[]}]{
&\scriptstyle\sigma_j\neq\Osc{\odot} & \scriptstyle\sigma_j=\Osc{\odot} \cr
\scriptstyle\sigma_i\neq\Osc{\odot} &
-A'_{\sigma(i),\sigma(j)} + A_{\sigma(i),\sigma(j)}
\left(\!\begin{aligned}+1_{\sigma_i=\Osc{\oplus}}-1_{\sigma_j=\Osc{\oplus}}\\[-3pt]
-1_{\sigma_i=\Osc{\ominus}}+1_{\sigma_j=\Osc{\ominus}}\end{aligned}\right)
 &
A_{\sigma(i),\sigma(j)}
\cr
\scriptstyle\sigma_i=\Osc{\odot} &
-A_{\sigma(i),\sigma(j)}
& 0
}_{\ijNN{m}\,.}
\]
The symbols $\Osc{\oplus}$, $\Osc{\ominus}$, and $\Osc{\ocircle}$ are mnemonic for $+1$, $-1$, and $0$, which
go into the coefficient of $A_{\sigma(i),\sigma(j)}$ when $\sigma_i,\sigma_j\neq\Osc{\odot}$.
For example, when $\sigma$ is the above value for~$\overline\mu$, this matrix is
\[
\scalebox{0.7}{
\bordermatrix[{[]}]{
&\overset{-}{1}&2&\overset{\Osc{\odot}}{2}&\overset{-}{3}&4&5\ \ 7\ \ 8\ \ 10\ \ 11&\overset{\Osc{\odot}\,}{11}&12&\overset{+}{13}&\overset{\Osc{\odot}\,}{14}\cr
\mathllap{{\scriptstyle{}^+}\,1}&
  0 & A_{1,2}{-}A'_{1,2} & A_{1,2} & -A'_{1,3} & A_{1,4}{-}A'_{1,4}&\cdots\cdots\cdots&A_{1,11}&A_{1,12}{-}A'_{1,12}&2A_{1,13}{-}A'_{1,13}&A_{1.14}\cr
\mathllap{2}&
A'_{1,2}{-}A_{1,2}&0&A_{2,2}&-A_{2,3}{-}A'_{2,3}&-A'_{2,4}&\cdots\cdots\cdots&A_{2,11}&-A'_{2,12}&A_{2,13}{-}A'_{2,13}&A_{2,14}\cr
\mathllap{\raisebox{1pt}{$\scriptstyle\Osc{\odot}$}\,2}&
-A_{1,2}&-A_{2,2}&0&-A_{2,3}&-A_{2,4}&\cdots\cdots\cdots&0&-A_{2,12}&-A_{2,13}&0 \cr
\mathllap{{\scriptstyle{}^+}\,3}&
\vdots&\vdots&\ddots&0&A_{3,4}{-}A'_{3,4}&\cdots\cdots\cdots&A_{3,11}&A_{3,12}{-}A'_{3,12}&2A_{3,13}{-}A'_{3,13}&A_{3,14}\cr
\mathllap{4}&
&&&&0&\cdots\cdots\cdots&A_{4,11}&-A'_{4,12}&A_{4,13}{-}A'_{4,13}&A_{4,14}\cr
\mathllap{5}&
&&&&&\ddots&\vdots&\vdots&\vdots&\vdots\cr
\mathllap{7}\cr
\mathllap{\vdots}
}}
\]

We define $M_\mu^{(L)}=M_{\dddot\mu}(L,L')$, and $M_\mu^{(G)}=M_{\overline\mu}(G,G')$,
where $G_{i,n}$ is replaced with~$1$.
Both $M_\mu^{(L)}$ and $M_\mu^{(G)}$ are antisymmetric.  The new formulas involve Pfaffians of these matrices $M^{(L)}_\mu$ and $M^{(G)}_\mu$.

\begin{figure}[b!]
\centerline{\includegraphics[width=\textwidth]{skew-big-tile}}
\caption{Cover-inclusive Dyck tilings of a skew shape.  (This figure first appeared in \cite{KW3}.)}
\label{fig:dyck-tilings}
\end{figure}

The new formulas have coefficients that are defined in terms of ``\textbf{cover-inclusive Dyck tilings}'', which were first defined in \cite{KW3} and independently in \cite{MR2927185}, and were studied further in \cite{kim,KMPW,KW4,KW5,JVK},
and whose definition we now recall.
If $\lambda$ and $\mu$ are Dyck paths such that $\lambda$ is below $\mu$, then the region $\lambda/\mu$ is a skew Young diagram (rotated $45^\circ$).  A Dyck tile is a ribbon tile which is shaped like a Dyck path, i.e., a collection of $\sqrt{2}\times\sqrt{2}$ boxes rotated $45^\circ$ centered at the points of a Dyck path.  A Dyck tiling of $\lambda/\mu$ is a tiling of it by Dyck tiles.  We say that one Dyck tile covers another Dyck tile if it contains a box which is directly (not diagonally) above a box of the other tile.  A cover-inclusive Dyck tiling is one for which, whenever a Dyck tile $T_1$ covers another Dyck tile $T_2$, the range of $x$-coordinates of $T_1$ is a subset of the range of $x$-coordinates of $T_2$.  See Figure~\ref{fig:dyck-tilings} for a list of the Dyck tilings of a particular skew shape $\lambda/\mu$.

\begin{theorem} \label{thm:Pf-udfe}
Suppose $\tau$ is a partial pairing of the nodes of an annular-one graph with $n$ nodes, where node $n$ is paired in $\tau$.  Let $\lambda$ be the labeled augmented cyclic Dyck path which encodes $\tau$.  Then
\begin{equation} \label{dot-sum-PfL}
\frac{Z[\tau]}{Z[1|2|\cdots|n]} = \sum_{\mu\succeq\lambda} [\text{\# of c.i.\ Dyck tilings of $\lambda^\circ/\mu^\circ$}] \times \operatorname{Pf} M^{(L)}_\mu
\end{equation}
and
\begin{equation} \label{bar-sum-PfG}
\frac{Z[\tau]}{Z[1,2,\dots,n]} = \sum_{\mu\succeq\lambda} [\text{\# of c.i.\ Dyck tilings of $\lambda^\circ/\mu^\circ$}] \times \operatorname{Pf} M^{(G)}_\mu\,.
\end{equation}
\end{theorem}

\subsection{Examples}
\enlargethispage{12pt}
We give a couple of examples:

For the partial pairing $1,3|2,4$,
the encoding string $\lambda$ is $\D{1}\F{2}\U{3}\OD{4}$,
the only $\mu$ in the sum is $\mu=\D{1}\F{2}\U{3}\OD{4}$,
for which the skew Young diagram $\lambda^\circ/\mu^\circ$ has only the empty Dyck tiling, so the coefficient is $1$.  For this $\mu=\D{1}\F{2}\U{3}\OD{4}$, $\dddot\mu=\OC{1}\OC{2}\OC{3}\OD{4}$, so
\[
  \frac{Z[1,3|2,4]}{Z[1|2|3|4]} =
\operatorname{Pf}\underbrace{\begin{bmatrix}
0 & -L_ {1,2}' & -L_ {1,3}' & L_{1,4} \\
 \phantom{-} L_ {1,2}' & 0 & -L_ {2,3}' & L_{2,4} \\
 \phantom{-} L_ {1,3}' & \phantom{-} L_ {2,3}' & 0 & L_{3,4} \\
 -L_{1,4} & -L_{2,4} & -L_{3,4} & 0
\end{bmatrix}}_{M^{(L)}_{\textsf{DFU}\Osc{\odot}}}
= -L'_{1,2} L_{3,4} - L'_{2,3} L_{1,4} - L'_{3,1} L_{2,4}\,,
\]
which matches \cite[eqn.~5.5b]{KW4}, and $\overline\mu=\OC{1}\OC{2}\OC{3}\OD{4}$, so
\[
  \frac{Z[1,3|2,4]}{Z[1,2,3,4]} =
\operatorname{Pf}\underbrace{\begin{bmatrix}
 0 & -G_ {1,2}' & -G_ {1,3}' & 1 \\
 \phantom{-} G_ {1,2}' & 0 & -G_ {2,3}' & 1 \\
 \phantom{-} G_ {1,3}' & \phantom{-} G_ {2,3}' & 0 & 1 \\
 -1 & -1 & -1 & 0
\end{bmatrix}}_{M^{(G)}_{\textsf{DFU}\Osc{\odot}}}
= -G'_{1,2} - G'_{2,3} - G'_{3,1}\,,
\]
which matches \cite[eqn.~5.6b]{KW4}.

For the partial pairing $1,4|2|6,7$ the encoding string is $\lambda=
\U{1}\S{2}\I{3}\D{4}\I{5}\F{6}\OD{7}$,
the only $\mu\succeq\lambda$ is
$\mu=\U{1}\S{2}\I{3}\D{4}\I{5}\F{6}\OD{7}$,
and $\dddot\mu=\OP{1}\II{3}\OC{4}\II{5}\OC{6}\OD{7}$, so
\[
\frac{Z[1,4|2|6,7]}{Z[1|2|3|4|5|6|7]}
= \operatorname{Pf}\underbrace{\scalebox{.79}{$\begin{bmatrix}
 0 & L_{1,3} & L_{1,3}-L_ {1,3}' & L_{1,4}-L_ {1,4}' & L_{1,5} &L_{1,5}-L_ {1,5}' & L_{1,6}-L_ {1,6}' & L_{1,7} \\
 -L_{1,3} & 0 & -L_{3,3} & -L_{3,4} & 0 & -L_{3,5} & -L_{3,6} & 0 \\
 L_ {1,3}'-L_{1,3} & L_{3,3} & 0 & -L_ {3,4}' & L_{3,5} & -L_ {3,5}'& -L_ {3,6}' & L_{3,7} \\
 L_ {1,4}'-L_{1,4} & L_{3,4} & L_ {3,4}' & 0 & L_{4,5} & -L_ {4,5}' &-L_ {4,6}' & L_{4,7} \\
 -L_{1,5} & 0 & -L_{3,5} & -L_{4,5} & 0 & -L_{5,5} & -L_{5,6} & 0 \\
 L_ {1,5}'-L_{1,5} & L_{3,5} & L_ {3,5}' & L_ {4,5}' & L_{5,5} & 0 &-L_ {5,6}' & L_{5,7} \\
 L_ {1,6}'-L_{1,6} & L_{3,6} & L_ {3,6}' & L_ {4,6}' & L_{5,6} & L_{5,6}' & 0 & L_{6,7} \\
 -L_{1,7} & 0 & -L_{3,7} & -L_{4,7} & 0 & -L_{5,7} & -L_{6,7} & 0 \\
\end{bmatrix}$}}_{M^{(L)}_{\textsf{USIDIF}\Osc{\odot}}}
\]
while $\overline\mu=\OP{1}\SS{2}\OC{4}\OC{6}\OD{7}$, so
\[
\frac{Z[1,4|2|6,7]}{Z[1,2,3,4,5,6,7]}
= \operatorname{Pf}\underbrace{\scalebox{1}{$\displaystyle\begin{bmatrix}
 0 & G_{1,2}-G_ {1,2}' & G_{1,2} & G_{1,4}-G_ {1,4}' & G_{1,6}-G_{1,6}' & 1 \\
 G_ {1,2}'-G_{1,2} & 0 & G_{2,2} & -G_ {2,4}' & -G_ {2,6}' & 1 \\
 -G_{1,2} & -G_{2,2} & 0 & -G_{2,4} & -G_{2,6} & 0 \\
 G_ {1,4}'-G_{1,4} & G_ {2,4}' & G_{2,4} & 0 & -G_ {4,6}' & 1 \\
 G_ {1,6}'-G_{1,6} & G_ {2,6}' & G_{2,6} & G_ {4,6}' & 0 & 1 \\
 -1 & -1 & 0 & -1 & -1 & 0 \\
\end{bmatrix}$}}_{M^{(G)}_{\textsf{USIDIF}\Osc{\odot}}}
\]

For $1,2|3,7|4,6$ we have $\lambda=\textsf{UDFUID}\Osc{\odot}$, there are two $\mu$'s such that $\mu\succeq\lambda$:
\[
 \frac{Z[1,2|3,7|4,6]}{Z[1,2,3,4,5,6,7]}
= \operatorname{Pf} M^{(G)}_{\textsf{UDFUID}\Osc{\odot}} + \operatorname{Pf} M^{(G)}_{\textsf{DDFUIU}\Osc{\odot}}
\]

For the partial pairing $1,3|2|4,10|5,6|7,9$ we have $\lambda=\textsf{USDFUDUID}\Osc{\odot}$, there are five $\mu$'s such that $\mu\succeq\lambda$, and for one of these $\mu$'s the skew Young diagram $\lambda^\circ/\mu^\circ$ has two Dyck tilings:
\begin{multline*}
 \frac{Z[1,3|2|4,10|5,6|7,9]}{Z[1,2,3,4,5,6,7,8,9,10]}
= \operatorname{Pf} M^{(G)}_{\textsf{USDFUDUID}\Osc{\odot}} +\operatorname{Pf} M^{(G)}_{\textsf{USDFUUDID}\Osc{\odot}}\\ +\operatorname{Pf} M^{(G)}_{\textsf{DSDFUDUIU}\Osc{\odot}} +\operatorname{Pf} M^{(G)}_{\textsf{DSDFUUDIU}\Osc{\odot}} +2\times\operatorname{Pf} M^{(G)}_{\textsf{DSDFUUUID}\Osc{\odot}}
\end{multline*}

\subsection{Corollaries}

The formulas in Theorem~\ref{thm:Pf-udfe} immediately imply the following statement.
\begin{corollary}
For a partition $\tau$ on $\{1,\ldots,n\}$ in which $n$ is not in a singleton part, on an annular-one graph with $n$ nodes,
the ratio $\frac{Z[\tau]}{Z[1|2|\cdots|n]}$ is a polynomial in the variables $L$ and $L'$ with integer coefficients.  Similarly, $\frac{Z[\tau]}{Z[1,2,\cdots,n]}$ is a polynomial in $G$ and $G'$ with integer coefficients.
\end{corollary}

It was known that these ratios are polynomials in the $L$ and $L'$ variables, or the $G$ and $G'$ variables \cite{KW4}, but the integrality of the coefficients was previously a mystery.

Recalling
\[
  \frac{Z[1,3|2,4]}{Z[1,2,3,4]} = -G'_{1,2} - G'_{2,3} - G'_{3,1}\,,
\]
observe that this polynomial is invariant under the substitution $G'_{i,j}\to G'_{i,j}+f(i)-f(j)$.  The next corollary states that this is a general phenomenon for the $G$-$G'$-polynomials of any partition:
\begin{corollary} \label{coboundary}
For a partition $\tau$ on $\{1,\ldots,n\}$ in which $n$ is not in a singleton part, on an annular-one graph with $n$ nodes,
the $G$-$G'$-polynomial for $\frac{Z[\tau]}{Z[1,2,\cdots,n]}$ is invariant under replacing each $G'_{i,j}$ with $G'_{i,j}+f(i)-f(j)$.
\end{corollary}

\begin{proof}[Proof of Corollary~\ref{coboundary}]
Consider each Pfaffian in the formula from Theorem~\ref{thm:Pf-udfe}. Since the last column (row) is all $1$'s ($-1$'s), we can add an all-$f(i)$'s row to row $i$ and subtracting an all-$f(i)$'s-column from column $i$, without changing the value of the Pfaffian.  Since $G'_{i,j}$ occurs only in row $i$ and column $j$ (and row $j$ and column $i$), with coefficient $1$ (and $-1$), these operations replace each $G'_{i,j}$ with $G'_{i,j}+f(i)-f(j)$ and keep the Pfaffian invariant.
\end{proof}

We remark that it was known \cite{KW4} that substituting $G'_{i,j}\to G'_{i,j}+f(i)-f(j)$ and then evaluating the polynomial at the values of $G$ and $G'$ that arise from an annular-one graphs will give a result independent of $f$.  Corollary~\ref{coboundary} is a stronger statement, since it was  not known whether the values of $G$ and $G'$ that arise from annular-one graphs are full-dimensional or whether
they satisfy algebraic relations which cause the substituted $G$-$G'$-polynomials, when evaluated at these values, to be independent of $f$.

\section{Determinant formulas}
\newcommand{\mathbf{B}}{\mathbf{B}}

For an annular partial pairing $\tau$ on $n$ nodes, let $\lambda$ be its encoding string, let $T$ denote the set of internalized nodes, and $Q$ denote the set of singleton nodes.
Let $k$ denote the order of the Dyck path $\lambda^\circ$, i.e., half its length, so that $n=2k+2+|Q|+|T|$.

Let $S\subset\{1,\dots,n\}\setminus(Q\cup T)$ be a subset of the paired nodes which has size $k+1$ and includes $n$, and let $R = \{1,\dots,n\}\setminus(S\cup Q\cup T)$
be the complementary set of paired nodes.
Given $\lambda$ and $S$, Kenyon and Wilson \cite{KW4} defined
\begin{multline} \label{eq:A-inverse}
  \mathbf{B}_{\lambda,S}(\zeta) = \sum_{\mu\succeq\lambda} [\text{\# of c.i.\ Dyck tilings of $\lambda/\mu$}] \times \zeta^{\text{\# indices in $S$ at which $\mu$ has an up-step}
} \times \\
  \zeta^{-\text{\# indices in $S\setminus\{n\}$ after $\lambda$'s flat step}
   \,+\, \text{\# down steps of $\lambda$ after $\lambda$'s flat step}}\,,
\end{multline}
and showed how to use these polynomials $\mathbf{B}_{\lambda,S}$ to compute the ratios of grove partition functions.  Specifically
\begin{equation} \label{dot-sum-detL(z)}
  \frac{Z[\tau]}{Z[1|2|\cdots|n]} = (-1)^{|T|}\times\lim_{z\to 1} \sum_{R,S} \frac{\mathbf{B}_{\lambda,S}(z^2)}{(1-z^2)^k} \det \L_{R,T}^{S,T}\,,
\end{equation}
where $\L_{R,T}^{S,T}$ denotes the submatrix of $\L$ whose rows are indexed by $R$ and $T$ and whose columns are indexed by $S$ and $T$,
and we need to specify a pairing between the indices of $R$ and $S$ to determine the signs of the determinants.  We use the Dvoretzky-Motzkin cycle lemma bijection to make this pairing, as indicated below (figure taken from \cite{KW4}).  Essentially we make a path with period $2k+1$ which has an up step at each index in $R$ and a down step at each index in $S\setminus\{n\}$.  The up and down steps are the endpoints of chords underneath the path, and these chords define the pairing, where the extra up step is paired with $n$.
\[
 {}_{R=\{3,4,6,7,8,11\}}^{S=\{1,2,5,9,10,12\}}
 \!\Rightarrow\! \raisebox{-12pt}{\includegraphics[scale=0.43]{bijection-1}}
 \!\Rightarrow\! \raisebox{-12pt}{\includegraphics[scale=0.43]{bijection-2}}
 \!\Rightarrow\!
 {}_4^5|{}_6^2|{}_{\;7}^{10}|{}_8^9|{}_{11}^{\,1}|{}_{\;3}^{12}
\]

Recall that $\L_{i,j}=\L_{i,j}(z)$ is a function of $z$.
We change variables to $z=e^t$ (here we differ slightly from the notation
in \cite{KW4}, which used $\zeta=z^2=e^t$).
We expand $\L_{i,j}(e^t) = L_{i,j}+ L'_{i,j} t + \cdots$, and
let $\hat\L_{i,j}$ denote its linearized approximation $\hat\L_{i,j} = L_{i,j}+ L'_{i,j} t$.  In general the series expansion for $\L_{i,j}(e^t)$ will have more terms, but while
it is not \textit{a priori\/} obvious, the limit \eqref{dot-sum-detL(z)} can be evaluated using $\hat\L_{i,j}$ in place of $\L_{i,j}(e^t)$:
\begin{equation} \label{dot-sum-detL}
  \frac{Z[\tau]}{Z[1|2|\cdots|n]} = (-1)^{|T|}\times\lim_{t\to0} \frac{1}{(-2t)^k} \sum_{R,S} \mathbf{B}_{\lambda,S}(e^{2t}) \det \hat\L_{R,T}^{S,T}\,,
\end{equation}
and a similar formula
\begin{equation} \label{bar-sum-detG}
  \frac{Z[\tau]}{Z[1,2,\ldots,n]} = \lim_{t\to0} \frac{1}{(-2t)^k} \sum_{R,S} \mathbf{B}_{\lambda,S}(e^{2t}) \det \hat\mathscr{G}_{R,Q}^{S,Q}\,,
\end{equation}
holds, where $\hat\mathscr{G}_{i,j}=G_{i,j}+ G'_{i,j} t$ and each $\hat\mathscr{G}_{i,n}$ is replaced with $1$ \cite{KW4}.  From these formulas we derive the Pfaffian formulas.

\section{Pfaffian formulas}

We start in section~\ref{sec:pf-det} by showing that a Pfaffian can be expressed as a sum of determinants.  In section~\ref{sec:tripartite} we give an application of this identity to tripartite pairings.  Then we use the Pfaffian identity and equations \eqref{dot-sum-detL} and \eqref{bar-sum-detG} to prove Theorem~\ref{thm:Pf-udfe} in section~\ref{sec:main}.

\subsection{The Pfaffian as a sum of determinants} \label{sec:pf-det}

For any matching $M=(i_1,j_1),\ldots,(i_k,j_k)$, we define $\operatorname{sign}(M)=(-1)^{\operatorname{cr}(M)}$, where $\operatorname{cr}(M)$ is the number of crossings of arcs from $M$ when $M$ is drawn as $k$ arcs between the points $\{1,\ldots,2k\}$ on a line.  For the left endpoint of each arc we can associate an up-step, and for each right endpoint we can associate a down-step, which results in a Dyck path.  The down steps of the matching $M$ are the down steps of its Dyck path, i.e., $\{\max(i_1,j_1),\dots,\max(i_k,j_k)\}$.

Given a set of positive integers $R$ for which $R\subset\{1,\dots,2|R|\}$, we define $d_R$ as follows.  We let $n=2|R|$ and $S=\{1,\dots,n\}\setminus R$.  For an arbitrary matrix $A$ we define
\begin{equation} \label{dR(M)}
d_R(A) \colonequals \begin{cases}
 \det[A_{i,j}]_{i\in R}^{j\in S} & n\in S \\
 \det[-A_{i,j}]_{i\in R}^{j\in S} & n\in R\,,\end{cases}
\end{equation}
 where $R$ and $S$ are ordered according to the Dvoretzky-Motzkin bijection as described above.  For example,
\[
d_{\{3,4,6,7,8,11\}}(A) = \det[A_{i,j}]_{i=4,6,7,8,11,3}^{j=5,2,10,9,1,12}\,.
\]

\begin{lemma}\label{lemma:det_expansion_S}
Suppose $n\geq 0$ is even, $R\subset\{1,\ldots,n\}$, $|R|=n/2$, and $S=\{1,\ldots,n\}\setminus R$.
Let $A$ be an arbitrary $n\times n$ matrix.
Then
\[ d_R(A) = \sum_{\substack{\text{directed matchings $M$ s.t.}\\\text{$M$ matches $R$ to $S$}}} (-1)^{\operatorname{cr}(M)} \prod_{(r,s)\in M} (-1)^{1_{r>s}}A_{r,s} \,.\]
\end{lemma}

\begin{proof}
Let $k=n/2$.
The arrangement of elements of $R$ and $S$ in $d_R$
is given by the Dvoretzky-Motzkin cycle lemma bijection, and in particular corresponds to a matching
$M_0=\{(r_1,s_1),\dots,(r_k,s_k)\}$ (each $r_\ell\in R$ and $s_\ell\in S$) which has no crossings when drawn in the annulus.
By the determinant expansion,
\begin{equation}\label{det_expansion}
\det A_R^S=\sum_{\pi \in \mathfrak{S}_k} \operatorname{sign}(\pi)\prod_{\ell=1}^k A_{r_\ell,s_{\pi(\ell)}}\,.
\end{equation}
Suppose $n\notin R$.  When we draw matching $M_0$ on a line, there may be crossings of the arc $(j,n)$ from arcs $(a,b)$, such that $a>j>b$;
these are precisely the arcs whose starting point is larger than its endpoint.
When drawn on the line, the number of crossings is $\operatorname{cr}(M_0)=\sum_{(i,j)\in M_0} 1_{i>j}$.
If instead $n\in R$, then $\operatorname{cr}(M_0)=\sum_{(i,j)\in M_0} 1_{i<j}$.

For a permutation $\pi$ let the matching $M(\pi)$ be $M(\pi)=\{(r_1,s_{\pi(1)}),\ldots,r_k,s_{\pi(k)}\}$.
The matching $M_0$ corresponds to the identity permutation,  so at least when the permutation $\pi$ is the identity, we have
\[\operatorname{sign}(\pi) = (-1)^{\operatorname{cr}(M(\pi))} (-1)^{ \sum_{(r,s)\in M(\pi)} 1_{r>s}} (-1)^{(n/2)1_{n\in R}}\,,\]
a formula which we now verify for the other permutations.
Any permutation $\pi$ can be expressed as a sequence of transpositions, and it is a straightforward case analysis to verify
that any transposition changes the parity of the number of crossings in the matching plus the number of arcs directed backwards.
\end{proof}

\begin{theorem} \label{Pf}
Suppose $n\geq 0$ is even.
If $A$ is an arbitrary $n\times n$ matrix, and $d_R(A)$ is as defined in \eqref{dR(M)}, then
\begin{equation} \label{sum-dR=Pf(A)}
\sum_{\substack{R\subset\{1,\dots,n\}\\|R|=n/2}} d_R(A) =
\operatorname{Pf} \big[A-A^T\big]\,,
\end{equation}
where $A^T$ is the transpose of $A$.
\end{theorem}
\begin{proof}
From Lemma~\ref{lemma:det_expansion_S}, we see that the left-hand side of~\eqref{sum-dR=Pf(A)} equals
\[\sum_{\text{directed matchings $M$}} (-1)^{\operatorname{cr}(M)} \prod_{(r,s)\in M} (-1)^{1_{r>s}}A_{r,s}\,.\]

Let $n=2k$, and let $W_{i,j}=A_{i,j}-A_{j,i}$.
We can expand the Pfaffian as
\begin{equation*}\label{pfaffian_expansion}
\operatorname{Pf}[W] =  \sum_{\substack{\text{undirected matchings $M$}\\mathscr{M}=\{(i_1,j_1),\ldots,(i_k,j_k)\}\\i_1<j_1,\dots,i_k<j_k\\j_1<\cdots<j_k}} (-1)^{\operatorname{cr}(M)} \prod_{\ell=1}^k W_{i_\ell,j_\ell}.
\end{equation*}
When we make the substitution $W_{i,j}=A_{i,j}-A_{j,i}$, this has the effect of choosing directions for each pairing, converting the sum over undirected matchings into a sum over directed matchings:
\[ \operatorname{Pf}\big[A-A^T\big] = \sum_{\text{directed matchings $M$}} (-1)^{\operatorname{cr}(M)} \prod_{(r,s)\in M} (-1)^{1_{r>s}} A_{r,s}\,.\qedhere\]
\end{proof}

\subsection{Applications of the Pfaffian identity} \label{sec:tripartite}

Before continuing with our main result, we mention an interesting consequence of Theorem~\ref{Pf}.  Curtis, Ingerman, and Morrow \cite{CIM} gave an interpretation of the determinant $\det L_R^S$ when $R=\{r_1,\ldots,r_k\}$ and $S=\{s_1,\ldots,s_k\}$ are disjoint subsets of $\{1,\ldots,n\}$, which, when translated into the language of groves, asserts that
\begin{equation}\label{CIM}
  \det L_{r_1,\ldots,r_k}^{s_1,\ldots,s_k} = \sum_{\pi\in\mathfrak{S}_k} \operatorname{sign}(\pi)\frac{Z[r_1,s_{\pi(1)}|\cdots|r_k,s_{\pi(k)}|\text{(other nodes singletons)}]}{Z[1|\cdots|n]}\,.
\end{equation}
This formula holds for any graph.

If $B$ and $C$ are two disjoint sets of nodes, and we set
$A_{i,j}=0$ when $i\in C$ or $j\in B$ and otherwise set $A_{i,j}=L_{i,j}$,
then Theorem~\ref{Pf} with the above interpretation of the minors
implies $\operatorname{Pf}[A-A^T]$ is a sum over directed matchings for which the nodes in $B$ are sources and the nodes in $C$ are destinations, of the sign of the directed matching times the grove ratio associated with that matching.  In particular, nodes of $B$ are only paired with nodes not in $B$, and nodes in $C$ are only paired with nodes not in $C$.  If a matching $M$ contains a pair $(i,j)$ where $i,j\in B$ or $i,j\in C$, then $M$ is not included in the sum.  Notice that if $i,j\notin B\cup C$, then the matching $(M\setminus\{(i,j)\})\cup\{(j,i)\}$, in which the pair $(i,j)$ has been reversed, has the same weight as $M$ but opposite sign.
Thus
\begin{multline}\label{L[RGB]}
\operatorname{Pf}\bordermatrix[{[]}]{& \scriptstyle j\in B & \scriptstyle j\notin B\cup C & \scriptstyle j\in C\cr
\scriptstyle i\in B & 0 & \phantom{-} L_{i,j} & \phantom{-} L_{i,j} \cr
\scriptstyle i\notin B\cup C & -L_{i,j} & 0 & \phantom{-} L_{i,j} \cr
\scriptstyle i\in C & -L_{i,j} & -L_{i,j} & 0 \cr
} =\\[8pt]=
 \sum_{{\substack{\text{directed matchings $M$}\\\text{if $(i,j)\in M$ then}\\\text{$i\in B$ or $j\in C$ or both}}}}
\left( (-1)^{\operatorname{cr}(M)} \prod_{(i,j)\in M} (-1)^{1_{j<i}}\right)
 \dddot{Z}[i_1,j_1|\cdots|i_{n/2},j_{n/2}]\,.
\end{multline}
When the graph is circular planar (i.e., the nodes lie on the outer face of a planar graph), and $B=\{1,\ldots,|B|\}$ and $C=\{n+1-|C|,\ldots,n\}$, there is only one matching~$M$ for which $\dddot{Z}[M]\neq 0$, and the sign is positive, so $\dddot{Z}[M]$ is the Pfaffian.
For example,
\[
\dddot{Z}\left[
\begin{tikzpicture}[baseline=0.0cm-2.5pt]
\draw[fill=gray!40!white,draw=none] (0,0) circle(0.8);
    \foreach \x in {1,...,6} { \coordinate (\x) at ({0.8*cos((\x-0.5)*360/6)},{0.8*sin((\x-0.5)*360/6)});}
    \draw[very thick] (1) arc(120:240:{0.8/sqrt(3.0)});
    \draw[very thick] (3) arc(240:360:{0.8/sqrt(3.0)});
    \draw[very thick] (5) arc(0:120:{0.8/sqrt(3.0)});
    \node [circle,fill=white!80!red,draw=red,inner sep=0.5pt] at (1) {1};
    \node [circle,fill=white!80!red,draw=red,inner sep=0.5pt] at (2) {2};
    \node [circle,fill=white!80!green,draw=green,inner sep=0.5pt] at (3) {3};
    \node [circle,fill=white!80!green,draw=green,inner sep=0.5pt] at (4) {4};
    \node [circle,fill=white!80!blue,draw=blue,inner sep=0.5pt] at (5) {5};
    \node [circle,fill=white!80!blue,draw=blue,inner sep=0.5pt] at (6) {6};
\end{tikzpicture}
\right]=
\frac{Z[1,6|2,3|4,5]}{Z[1|2|3|4|5|6]} =
\operatorname{Pf}\begin{bmatrix}
    0     &    0     &\phantom{-} L_{1,3}&\phantom{-} L_{1,4}&\phantom{-} L_{1,5}&\phantom{-} L_{1,6}\\
    0     &    0     &\phantom{-} L_{2,3}&\phantom{-} L_{2,4}&\phantom{-} L_{2,5}&\phantom{-} L_{2,6}\\
   -L_{1,3}&   -L_{2,3}&    0     &    0     &\phantom{-} L_{3,5}&\phantom{-} L_{3,6}\\
   -L_{1,4}&   -L_{2,4}&    0     &    0     &\phantom{-} L_{4,5}&\phantom{-} L_{4,6}\\
   -L_{1,5}&   -L_{2,5}&   -L_{3,5}&   -L_{4,5}&    0     &    0     \\
   -L_{1,6}&   -L_{2,6}&   -L_{3,6}&   -L_{4,6}&    0     &    0
\end{bmatrix}\,.
\]
This is one of several tripartite matching formulas that
were derived earlier by Kenyon and Wilson \cite{KW2} using a different method \cite{KW1}.

The determinant formula \eqref{CIM} has been extended in several directions.  Kenyon and Wilson \cite{KW4} showed that if $Q=\{q_1,\ldots,q_\ell\}$ and $T=\{t_1,\ldots,t_m\}$,
and $Q,R,S,T$ partition $\{1,\ldots,n\}$, then
\begin{equation} \label{directed-grove}
  \det \L_{r_1,\ldots,r_k,t_1,\ldots,t_m}^{s_1,\ldots,s_k,t_1,\ldots,t_m}
   = (-1)^m \sum_{\pi\in\mathfrak{S}_k} \operatorname{sign}(\pi)\frac{\mathscr{Z}\big[{}_{r_1}^{s_{\pi(1)}}|\cdots|{}_{r_k}^{s_{\pi(k)}}|q_1|\cdots|q_\ell\big]}{\mathscr{Z}[1|\cdots|n]}\,,
\end{equation}
where the $\mathscr{Z}$'s give the weighted sum of ``cycle-rooted groves''.  (The cycle weights go to zero and $\mathscr{Z}$ converges to $Z$ when $z\to 1$, see \cite{KW4} for further explanation.)
When we combine Theorem~\ref{Pf} with the above formula, we obtain
the following:
\begin{theorem}
Suppose there are $n$ nodes, $P,Q,T$ partition $\{1,\ldots,n\}$, and $|P|=2k$ is even.  For each $i\in P$ let $\alpha_i$ and $\beta_i$ be parameters, and for $i\in T$ let $\alpha_i=\beta_i=1$.  List the nodes $p_1,\ldots,p_{2k},t'_1,t_1,\ldots,t'_m,t_m$, where $t'_i$ is a second copy of $t_i$, and let $T'=\{t'_1,\ldots,t'_m\}$.  Then
\newpage
\begin{multline}\label{Pf(LL)}
\operatorname{Pf}
\bordermatrix[{[]}]{& \scriptstyle j\in P\cup T & \scriptstyle j\in T'\cr
\scriptstyle i\in P\cup T &
\alpha_i\beta_j\L_{i,j}-\alpha_j\beta_i \L_{j,i} &
\alpha_i\L_{i,j} \cr
\scriptstyle i\in T' &
-\alpha_j \L_{j,i} &
0 \cr
}_{{\mathrlap{\raisebox{12pt}{$\scriptstyle j=p_1,\ldots,p_{2k},t'_1,t_1,\ldots,t'_m,t_m$}}i=p_1,\ldots,p_{2k},t'_1,t_1,\ldots,t'_m,t_m}}
=\\=
\sum_{\substack{\text{\rm directed matchings $M$ of $P$}\\mathscr{M}=\{(r_1,s_1),\ldots,(r_k,s_k)\}}} \left((-1)^{\operatorname{cr}(M)} \prod_{(r,s)\in M} (-1)^{1_{s<r}} \alpha_r\beta_s\right) \times\frac{\mathscr{Z}[{}_{r_1}^{s_{1}}|\cdots|{}_{r_k}^{s_{k}}|q_1|\cdots|q_\ell]}{\mathscr{Z}[1|\cdots|n]}\,.
\end{multline}
\end{theorem}
\begin{proof}
  If $i\in T'$ then take $\alpha_i=0$ and $\beta_i=1$.
  Then apply Theorem~\ref{Pf} with $A_{i,j} = \alpha_i \beta_j \L_{i,j}$,
  and use \eqref{directed-grove} to interpret the determinants.
  The factor of $(-1)^m$ in \eqref{directed-grove} is absorbed into the Pfaffian
  because we listed each $t'_i$ before $t_i$.
\end{proof}

Any of \eqref{CIM} or \eqref{L[RGB]} or \eqref{directed-grove} can be recovered from \eqref{Pf(LL)} by choosing the $\alpha$'s and $\beta$'s suitably and/or setting $z=1$.

\subsection{Proof of main theorem} \label{sec:main}

Our approach to proving the Pfaffian formulas in Theorem~\ref{thm:Pf-udfe} is to prove that the right hand sides of \eqref{dot-sum-detL} and \eqref{bar-sum-detG} are equal as polynomials in formal variables to the Pfaffian expressions.  We will not, for example, use the fact that $\sum_j L_{i,j}=0$ or other relations that the electrical network quantities might satisfy, since the $L_{i,j}$'s and the $G_{i,j}$'s satisfy different relations.  By working with formal variables that do not satisfy these extra relations, the same proof works for both the $L$-$L'$ polynomials and the $G$-$G'$ polynomials.

The roles of the $\textsf{S}$ and $\I{}$ symbols are reversed between the $L$-$L'$ polynomials for $\dddot Z[\tau]$ and the $G$-$G'$ polynomials for $\overline Z[\tau]$.  As a matter of convenience, we will give these symbols the roles they have for the $G$-$G'$ polynomials.  To obtain the $L$-$L'$ polynomials, we will at some point substitute $\I{}$ for $\textsf{S}$ and $\textsf{S}$ for $\I{}$.

Let $\lambda$ be a labeled augmented cyclic Dyck path with $n$ symbols.
Let $\lambda^*$ be the substring obtained from $\lambda$ by excising all $\textsf{S}$ and $\I{}$ symbols that it contains, let $n^*$ be the length of $\lambda^*$, and let $E_\lambda$ be the set of labels of $\textsf{S}$ symbols.
For our running example
\[
\lambda = \U{1}\S{2}\U{3}\D{4}\D{5}\I{6}\D{7}\F{8}\I{9}\U{10}\S{1\!1}\U{12}\D{13}\OD{14} \]
we have
\[
\lambda^* =
\U{1}\U{3}\D{4}\D{5}\D{7}\F{8}\U{10}\U{12}\D{13}\OD{14}\,,
\quad\quad\quad n^*=10\,,\quad\quad\quad E_\lambda=\{2,11\}\,.
\]
For an arbitrary $n\times n$ matrix $\mathscr{A}$ we define
\begin{equation} \label{Z_lambda}
\accentset\star\Z_\lambda(\mathscr{A})\colonequals
 \sum_{\substack{R^*\subset\{1,\dots,n^*\}\\|R^*|=n^*/2\\\{n^*\}\cap R=\varnothing\\S^*=\{1,\dots,n^*\}\setminus R^*}}
 \frac{\mathbf{B}_{\lambda^*,S^*}(e^{2t})}{(1-e^{2t})^k} \det \mathscr{A}_{\lambda^*(R^*),E_\lambda}^{\lambda^*(S^*),E_\lambda}
\end{equation}
Recall from \eqref{eq:A-inverse} that $\mathbf{B}_{\lambda,S}(\zeta)$ is a sum over $\mu\succeq\lambda$ of $(\text{\# of c.i.\ Dyck tilings of $\lambda^\circ/\mu^\circ$})$ times
$\zeta$ to the power
\begin{multline*}
\text{(\# up steps of $\mu$ in $S$ before flat step)}\\
-\text{(\# down steps of $\mu$ in $S$ after flat step) }\\
+\text{(\# down steps of $\lambda$ after flat step)}\,.
\end{multline*}
We define
\begin{multline} \label{Z^mu}
\accentset\star\Z^\mu(\mathscr{A})\colonequals \frac{1}{(1-e^{2t})^k}
 \sum_{\substack{R^*\subset\{1,\dots,n^*\}\\|R^*|=n^*/2\\\{n^*\}\cap R^*=\varnothing\\S^*=\{1,\dots,n^*\}\setminus R^*}}
\raisebox{-18pt}{$\displaystyle\begin{aligned}
&\det \mathscr{A}_{\mu^*(R^*),E_\mu}^{\mu^*(S^*),E_\mu}\times\\
&\exp\big[2t\,\big|\text{(up steps of $\mu^*$ before flat step)} \cap S^*\big|\big] \div\\
&\exp\big[2t\,\big|\text{(down steps of $\mu^*$ after flat step)} \cap S^*\big|\big]\,.
\end{aligned}$}
\end{multline}
Observe that if $\mu\succeq\lambda$ then $\mu^*(\cdot)=\lambda^*(\cdot)$ and $E_\mu=E_\lambda$, so
\begin{multline} \label{sum-Z^mu}
\accentset\star\Z_\lambda(\mathscr{A})=
\exp\big[2t\,(\text{\# down steps of $\lambda$ after flat step})\big]\times\\
\sum_{\mu\succeq\lambda} [\text{\# of c.i.\ Dyck tilings of $\lambda^\circ/\mu^\circ$}] \accentset\star\Z^\mu(\mathscr{A})\,.
\end{multline}

The following lemma will help us evaluate $\accentset\star\Z^\mu(\mathscr{A})$:

\newcommand{\widetilde{A}}{\widetilde{A}}
\begin{lemma} \label{lem:Pf(A[B,C,U,V])}
Suppose $n\geq0$ is even, $B,C,U,V\subset\{1,\dots,n\}$, $B\cap C=\varnothing$, and $U\cap V=\varnothing$.
Let $A$ be an arbitrary $n\times n$ matrix.
Then
\begin{multline} \label{Pf(A[B,C,U,V])}
 \sum_{\substack{R\subset\{1,\dots,n\}\\|R|=n/2\\mathbf{B}\subset R\\mathbb{C}\cap R=\varnothing\\S=\{1,\ldots,n\}\setminus R}}
  \exp\Big[2t\big(|S\cap U| -|S\cap V|\big)\Big]
  d_R(A)
=\\= \exp\big[t(|U|-|V|)\big]\times
\operatorname{Pf}\bordermatrix[{[]}]{
&\scriptstyle j\in B& \scriptstyle j\notin B\cup C&\scriptstyle j\in C\cr
\scriptstyle i\in B& 0&\widetilde{A}_{i,j}&\widetilde{A}_{i,j}\cr
\scriptstyle i\notin B\cup C   &-\widetilde{A}_{j,i}& \widetilde{A}_{i,j}-\widetilde{A}_{j,i} &\widetilde{A}_{i,j}\cr
\scriptstyle i\in C&-\widetilde{A}_{j,i}&-\widetilde{A}_{j,i}&0}_{\ijNN{n}\,,}
\end{multline}
where
\[
\widetilde{A}_{i,j}=A_{i,j} \exp\big[t(1_{j\in U}-1_{i\in U}-1_{j\in V}+1_{i\in V})\big]\,.
\]
\end{lemma}

\begin{proof}
  Observe that $2|S\cap U|=|S\cap U|-|R\cap U|+|U|$, and similarly for $2|S\cap V|$. Since $\widetilde{A}$ is obtained from $A$ by multiplying the $i$th row by $\exp[t(1_{i\in V}-1_{i \in U})]$ and $j$th column by $\exp[t(1_{j\in U}-1_{j\in V})]$, the determinants $d_R(A)$ and $d_R(\widetilde{A})$ differ by a factor depending on $R$ and $S$:
\[
\exp\Big[2t\big(|S\cap U| -|S\cap V|\big)\Big]\times
  d_R(A) = \exp\big[t(|U|-|V|)\big]\times d_R(\widetilde{A})\,.
\]
We can set $A_{i,j}=0$ whenever $j\in B$ or $i\in C$,
since these variables do not occur in equation~\eqref{Pf(A[B,C,U,V])}.
We then remove the restrictions $B\subset R$ and $C\cap R=\varnothing$ in the
summation on the left-hand side of equation~\eqref{Pf(A[B,C,U,V])},
since with the above variables zeroed out, $d_R(A)=0$ whenever $B \not \subset R$ or $C\cap R\neq \varnothing$.  Without these restrictions on the sum, we can apply Theorem~\ref{Pf} to sum up the $d_R(\widetilde{A})$'s to obtain~\eqref{Pf(A[B,C,U,V])}.
\end{proof}

\newcommand{\widetilde{\A}}{\widetilde{\mathscr{A}}}
\begin{lemma} \label{Pf(M_mu(A))}
Let $\mu$ be a labeled augmented cyclic Dyck path with $n$ symbols,
and suppose $\overline\mu$ has length $m$.
Let $U$ denote the set of labels in $\overline\mu$ above $\Osc{\oplus}$ symbols,
and let $V$ denote the set of labels in $\overline\mu$ above $\Osc{\ominus}$ symbols.
Let $\mathscr{A}$ be an arbitrary $n\times n$ matrix, and let
\begin{equation}\label{A-tilde}
\widetilde{\A}_{i,j}=\mathscr{A}_{i,j} \exp\big[t(1_{j\in U}-1_{i\in U}-1_{j\in V}+1_{i\in V})\big]\,.
\end{equation}
Then
\begin{equation} \label{Zsmu(A)}
\accentset\star\Z^\mu(\mathscr{A})=
\operatorname{Pf}\bordermatrix[{[]}]{
&  \scriptstyle\overline\mu_j\neq\Osc{\odot} & \scriptstyle\overline\mu_j=\Osc{\odot} \cr
\scriptstyle\overline\mu_i\neq\Osc{\odot} &
\frac{\displaystyle\widetilde{\A}_{\overline\mu(i),\overline\mu(j)}-\widetilde{\A}_{\overline\mu(j),\overline\mu(i)}}{\textstyle 1-e^{2t}}&\widetilde{\A}_{\overline\mu(i),\overline\mu(j)}\cr\cr
\scriptstyle\overline\mu_i=\Osc{\odot}&
-\widetilde{\A}_{\overline\mu(j),\overline\mu(i)}&0}_{\ijNN{m}\,.}
\end{equation}
\end{lemma}
\begin{proof}
Recall that $\mu^\textsf{S}$ is the string obtained from $\mu$ by replacing each $\S{i}$ symbol with $\SS{i}$ and omitting each $\I{}$ symbol.
Let $B_\mu$ denote the positions of these new $\Osc{\ocircle}$'s (replacing an~$\textsf{S}$) in $\mu^\textsf{S}$, let $C_\mu$ denote the positions of these new $\Osc{\odot}$'s in $\mu^\textsf{S}$.  The strings $\mu^\textsf{S}$ and $\overline\mu$ have the same length, which we are calling $m$.  If $\mu$ is our earlier example
\[\mu=\U{1}\S{2}\U{3}\D{4}\D{5}\I{6}\D{7}\F{8}\I{9}\U{10}\S{1\!1}\U{12}\D{13}\OD{14}\,,\]
then
\[
\mu^\textsf{S} = \U{1}\SS{2}\U{3}\D{4}\D{5}\D{7}\F{8}\U{10}\SS{1\!1}\U{12}\D{13}\OD{14}
\]
and
\[
B_\mu=\text{positions of }\{\OC{2},\OC{11}\}=\{2,10\}
\quad\quad\text{and}\quad\quad
C_\mu=\text{positions of }\{\OD{2},\OD{11}\}=\{3,11\}\,.
\]

Let $U_\mu$ denote the set of positions at which $\mu$ has an $\U{}$
before its $\F{}$, and let $V_\mu$ denote positions at which $\mu$ has
a $\D{}$ after its $\F{}$.

For a given $\mu$, the subsets $R^*$ of $\{1,\ldots,n^*\}$ for which $|R^*|=n^*/2$ are in
straightforward bijective correspondence with those subsets $R$ of
$\{1,\ldots,m\}$ for which $|R|=m/2$, $B_\mu\subset R$ and $C_\mu\cap R=\varnothing$,
i.e., $R=\overline{\mu}^{-1}(\mu^*(R^*))\cup B_\mu$.  Consider the pairing between $R^*$ and $S^*=\{1,\ldots,n^*\}\setminus R^*$ given by the cycle lemma bijection.  This pairing naturally extends to a pairing between $R$ and $S=\{1,\ldots,m\}\setminus R$, where a pair $(r^*,s^*)$ gets mapped to the pair $(\overline{\mu}^{-1}(\mu^*(r^*)),\overline{\mu}^{-1}(\mu^*(s^*)))$, with the pairing between $R$ and $S$ also containing the pairs $(b,b+1)$ for each $b\in B_\mu$.  Provided $n^*\notin R^*$, this extended pairing is precisely the pairing between $R$ and $S$ given by the cycle lemma bijection.
Thus
\begin{equation} \label{mu*-mu+}
 \sum_{\substack{R^*\subset\{1,\dots,n^*\}\\|R^*|=n^*/2\\\{n^*\}\cap R^*=\varnothing\\S^*=\{1,\dots,n^*\}\setminus R^*}}
\raisebox{-18pt}{$\displaystyle\left(\begin{aligned}
&\det \mathscr{A}_{\mu^*(R^*),E_\mu}^{\mu^*(S^*),E_\mu}\times\\
&\exp\big[2t\,\big| U_{\mu^*} \cap S^*\big|\big] \div\\
&\exp\big[2t\,\big| V_{\mu^*} \cap S^*\big|\big]\
\end{aligned}\right)$}
= \sum_{\substack{R\subset\{1,\dots,m\}\\|R|=m/2\\\{m\}\cap R=\varnothing\\mathbf{B}_\mu\subset R\\mathbb{C}_\mu\cap R=\varnothing\\S=\{1,\dots,m\}\setminus R}}
\raisebox{-18pt}{$\displaystyle\left(\begin{aligned}
&\det \mathscr{A}_{\mu^\textsf{S}(R)}^{\mu^\textsf{S}(S)}\times\\
&\exp\big[2t\,\big|U_{\mu^\textsf{S}}\cap S\big|\big] \div\\
&\exp\big[2t\,\big|V_{\mu^\textsf{S}}\cap S\big|\big]
\end{aligned}\right)_{\,.}$}
\end{equation}
We could apply Lemma~\ref{lem:Pf(A[B,C,U,V])} with $B=B_\mu$ and $C=C_\mu\cup\{m\}$ to evaluate the right-hand side of \eqref{mu*-mu+},
but it turns out to work better with $B=\varnothing$, $C=C_\mu\cup\{m\}$.
So long as $(C_\mu\cup\{m\})\cap R=\varnothing$, if $B_\mu\not\subseteq R$, then the determinant $\det \mathscr{A}_{\mu^\textsf{S}(R)}^{\mu^\textsf{S}(S)}$ has at least one repeated column and therefore does not contribute to the sum.
Applying Lemma~\ref{lem:Pf(A[B,C,U,V])} with $B=\varnothing$, $C=C_\mu\cup\{m\}$, $U=U_{\mu^\textsf{S}}$, $V=V_{\mu^\textsf{S}}$, and $n=m$, and then using the fact that $\mu^\textsf{S}(\cdot)=\overline\mu(\cdot)$, we see that the right-hand side of \eqref{mu*-mu+} equals
\[
\operatorname{Pf}\bordermatrix[{[]}]{
&  j\notin C& j\in C\cr
 i\notin C &
\widetilde{\A}_{\overline\mu(i),\overline\mu(j)}-\widetilde{\A}_{\overline\mu(j),\overline\mu(i)}&\widetilde{\A}_{\overline\mu(i),\overline\mu(j)}\cr
 i\in C&
-\widetilde{\A}_{\overline\mu(j),\overline\mu(i)}&0}_{\ijNN{m}\,,}
\]
with $\widetilde{\A}$ defined as in \eqref{A-tilde}.  Observe that $C_\mu\cup\{m\}$, $U_{\mu^\textsf{S}}$, and $V_{\mu^\textsf{S}}$ are the locations of $\Osc{\odot}$, $\Osc{\oplus}$, and $\Osc{\ominus}$ symbols in $\overline\mu$ respectively (which is of course the reason we defined $\overline\mu$ the way we did).

The definition of $\accentset\star\Z^\mu(\mathscr{A})$ also contains a factor of $1/(1-e^{2t})^{n^*/2-1}$. If for some $x$ we scale the rows and columns not in $C$ by a factor of $x^{1/2}$, and scale the rows and columns in $C$ by a factor of $x^{-1/2}$, the Pfaffian is scaled by a factor of $x^{[(m-|C|)-|C|]/2}$.
Now $m=n^*+2|E|$ and $|C|=|E|+1$, so $[(m-|C|)-|C|]/2=n^*/2-1$.
Upon taking $x=1/(1-e^{2t})$, we obtain \eqref{Zsmu(A)}.
\end{proof}

So far all these calculations are exact.  Next we take the limit $t\to 0$:
\begin{lemma} \label{Pf(M_mu(AA))}
Let $\mu$ be a labeled augmented cyclic Dyck path with $n$ symbols,
and suppose $\overline\mu$ has length $m$.
Let $\mathscr{A}$ be an $n\times n$ matrix of formal power series for which $\mathscr{A}_{i,j}(t)=\mathscr{A}_{j,i}(-t) = A_{i,j} + A'_{i,j} t + O(t^2)$.  Then
\[
\accentset\star\Z^\mu(\mathscr{A}) =
\operatorname{Pf}\phantom{\scriptstyle \overline\mu_i\neq\Osc{\odot}}\underbrace{\bordermatrix[{[]}]{
& \scriptstyle \overline\mu_j\neq\Osc{\odot}&\scriptstyle \overline\mu_j=\Osc{\odot}\cr
\mathllap{\scriptstyle \overline\mu_i\neq\Osc{\odot}} &
\left(\!\begin{aligned}
+1_{\overline\mu_i=\Osc{\oplus}}-1_{\overline\mu_j=\Osc{\oplus}}\\[-3pt]
-1_{\overline\mu_i=\Osc{\ominus}}+1_{\overline\mu_j=\Osc{\ominus}}
\end{aligned}\right)
A_{\overline\mu(i),\overline\mu(j)}-A'_{\overline\mu(i),\overline\mu(j)}&A_{\overline\mu(i),\overline\mu(j)}\cr
\mathllap{\scriptstyle \overline\mu_i=\Osc{\odot}}&
-A_{\overline\mu(i),\overline\mu(j)}&0}_{\mathrlap{\ijNN{m}}}}_{M_{\overline\mu}(A,A')}\ \ \ \ \ \ \raisebox{12pt}{+O(t)}
\]
\end{lemma}
\begin{proof}
Straightforward series expansion of the expression from Lemma~\ref{Pf(M_mu(A))}.
\end{proof}

It is also straightforward to extract the coefficients of higher powers of $t$ in the series expansion $\accentset\star\Z^\mu(\mathscr{A})$ using Lemma~\ref{Pf(M_mu(A))}.  As discussed earlier, the constant term is relevant for computing grove probabilities.  The term linear in $t$ is relevant for computing expected winding \cite{KW4}, and also depends on just the $A_{i,j}$'s and $A'_{i,j}$'s.

\begin{proof}[Proof of Theorem~\ref{thm:Pf-udfe}]
Immediate from \eqref{dot-sum-detL}, \eqref{bar-sum-detG}, \eqref{Z_lambda}, \eqref{Z^mu}, \eqref{sum-Z^mu} and Lemma~\ref{Pf(M_mu(A))}.
For the $G$-$G'$ polynomials we substitute $G$ for $A$ and $G'$ for $A'$.
For the $L$-$L'$ polynomials we first substitute $\I{}$ for $\S{}$ and $\S{}$ for $\I{}$, and then $L$ for $A$ and $L'$ for $A'$, and we absorb the factor of $(-1)^{|T|}$ from \eqref{dot-sum-detL} into the Pfaffian by writing $\II{i}$ rather than $\SS{i}$.
\end{proof}

\section{Open problems}

The coefficients in the
Pfaffian formulas in Theorem~\ref{thm:Pf-udfe} count Dyck tilings
whose lower path is $\lambda^\circ$ and whose upper path depends on
the summand.  It is known that the sum of these coefficients is the
number of increasing labelings of the planted plane tree associated
with the Dyck path $\lambda^\circ$ \cite{KMPW}.  Is there something more to
understand here?

Is there a polynomial-time algorithm for evaluating $Z[\tau]$?  For certain $\tau$'s there will be few or even just one Pfaffian, though for general $\tau$ the number of Pfaffians is exponentially large in the number of nodes.  But these Pfaffians are all closely related to one another, which suggests the possiblity that some clever linear algebra could be used to evaluate the sum without evaluating each individual Pfaffian.

\newcommand{\MRhref}[2]{\href{http://www.ams.org/mathscinet-getitem?mr=#1}{MR#2}}
\def\@rst #1 #2other{#1}
\newcommand\MR[1]{\relax\ifhmode\unskip\spacefactor3000 \space\fi
  \MRhref{\expandafter\@rst #1 other}{#1}}

\phantomsection
\pdfbookmark[1]{References}{bib}

\bibliographystyle{hmralphaabbrv}

redpajama_set_name: RedPajamaArXiv


Atlas's Details button does not work when the Titanic Rumbler Augment is in the build.
Any chance of adding EHP to the details page for frames so I can show people why using Redirection on armor heavy frames makes them worthy of slapping around?
Stradavar was added to the game.
Yep, got the mail from DE, there's also 3 new augment mods. I'll add this when I can, I'm really busy with my other project.
So, for whatever reason, slotting in the Titanic Rumbler augment for Atlas just results in the Detail part not working. Like, at all. As in the pop-in screen doesn't appear.
This mod is already in the database.
Seeing as the mechanics for 200%+ crit chance are known now, any chance you could update the formula the site uses?
Where for each +100% crit chance the critical damage multiplier increases by the "base crit multiplier" -1.
At 150% its 10.4x red crit and 5.7x normal crit, at 250% its 15.1x red crit and 10.4x normal crit, at 350% its 19.8x red crit and 15.1x normal crit, etc.
So 1 at 125%, 2 at 280%, 3 at 302% etc.
Also melee weapons with blood rush show the incorrect critical chance at 1.0x combo. Blood Rush is only active once you are at 1.5x combo or higher. SO before that they just have standard base crit chance for that weapon.
That's something I would want to fix, but I'm really busy at the moment so I don't know when it'll happen.
When you get the time could you also fix ember prime duration for accelerant, its staying at a constant 30sec even when you add or remove duration.
Are you sure it is affected by power duration ? It used to be no for a time if I recall correctly.
Duration base is 30s for all ranks but is affected by Power Duration.
yes im sure because i have been able to make it last 71secs at most but for a decent build i do it around 45-50secs.
Also another things that is confusing me is the snow globe for frost/prime specifically this build http://goo.gl/KoHcS0 when i use the exact same mods on live, the live stats shows 10,200 health while your detail stats says 13260 health but for the other abilities its correct so im trying to understand what is different.
Are you sure all mods and their ranks are the same? The only thing that would make a difference is that the build you linked has an Arcane Squall helmet and yours might not, but the difference would not be so big and that's why i ask.
I can confirm Snowglobes HP count on the site is wrong, too. The site for my build shows 25,063 HP on Snowglobe. (http://goo.gl/qtJ8JX) But currently looking at the same build in game it shows Snowglobe to only have 20,803 HP. Is it calculating something like how much damage can be absorbed in the invuln period to add to it's HP?
I'm using the formula on the wiki.
If I use the example from the wiki.
For example, with a maxed Steel Fiber and Intensify, rank-3 Snow Globe will have an initial health of (5000 + 5 × 300 × (1 + 1.1)) × (1 + 0.3) = 10595 before converting absorbed damage.
I'm getting the right value on WB.
1. The wiki is wrong and this formula is wrong, and this is a problem because this is the only formula I have.
2. The wiki is right and this formula is right, and there's something wrong with any of the 3 other mods from this build http://goo.gl/qtJ8JX that are modifying the HP. These mods are Transient Fortitude, Blinde Rage and armored Agility.
If we are in case 2, you guys can try to see which mod is causing the difference by adding them on by one in game and on WB, so we can see where's the error.
As for the updates, it's 'coming soon'.
So with zero mods installed in game right now the ability lists it as 5,000HP. BUT in WB zero mods = 6,500HP.
with max rank steel fiber + intensify in game HP is shown as 8,645. and if you take the equation listed on the wiki entry for Snowglobe, but change the value from the Base Armor Bonus from 1.1 to 0.1 you will get 8,645 instead of 10,595. but that doesn't make any sense because you're not adding 0.1 to it's base armor because then that would just be 330 armor, which is wrong because you have 630. so would this mean that the in-game HP value is shown incorrectly? i'm not so great with math so i honestly wouldn't know where to begin with helping out too much.
I think you found something, because if you look at the build you proposed before, the 25,063HP become 20,803 if you use 1.55 instead of 2.55 in the formula, for the armor bonus. So it seems that it works for the 2 builds. And this probably explains why you have 5000 in game and 6500 on Warfame Builder with no mods. It sounds like the game is subtractring 1 from the armor bonus, which leads, with 0 armor bonus, at a multiplier of 0 ((1 - 1) * 1500), instead of 1 on Warframe Builder ((1 + 0) * 1500), or it's simply not using the +1.
This does not tells us which formula is the right one, but at least, we know where the difference is coming from.
-Added the weapons Stardavar, Fragor Prime, Akstiletto Prime.
-Added the pictures for the following mods : Agility Drift, Coaction Drift, Cunning Drift, Endurance Drift, Power Drift, Speed Drift, Stealth Drift.
-Added the mods Fatal Teleport, Resonating Quake, Phoenix Renewal.
-Ember's Accelerant is now affected by power duration mods.
-Renamed Vauban's Bounce to Minelayer.
I'll update Minelayer later, when I'll have more data.
It seems you're missing the gimmic of Stradavar. Alt fire will let you switch back and fort between full-auto to semi-auto. When on semi-auto Stradavar's base critical chance is 20% rather than 10.


The Krone lightweights for increased profitability.
Saving vehicle weight means that you can gain more payload – Krone has followed this maxim in the development of the Ultra version, which is offered for the Coil Liner, Paper Liner, Profi Liner, and Mega Liner series. Through the use of new materials and a variety of design innovations, trailer tare weight has been significantly reduced.
Increased payload through lower tare weight.
Reduction in the number of journeys required.
Fuel savings and lower CO2 emissions through reduced transport requirement.
Lower transport costs through reduced fuel consumption and higher payload.
5,630 kg - this is the tare weight of the Profi Liner Ultra - made possible by a number of design innovations. From redesign of the chassis, to the use of light alloy, e.g. for the front bulkheads, rear walls and for weight optimisation of the trailer floor. The result: A significant increase in payload for your day-to-day transport, and up to 5,460 kg fork lift load bearing.
The new Mega Liner Ultra stands out as a real lightweight in the volume trailer sector with its tare weight of only 5,950 kg. The key factor in achieving its low weight is the completely redesigned chassis; thanks to the weight-optimised design of the main beams and cross members, it was also possible to reduce the tare weight without losing stability.
With the Paper Liner Ultra, KRONE offers a trailer that is ideally suited for the professional transport of paper rolls. The basis for this trailer is the modified KRONE chassis with short transverse beam spacing, equipped with running rails for pallet rollers where the rails are attached directly to the chassis. In addition, adjustable wedges can be placed at various intervals to further secure the rolls The tare weight of the Paper Liner Ultra is approx. 780 kg lighter than the standard version, at approx. 5,970 kg.
The Coil Liner Ultra has a dead weight of just 5,980 kg. By using fine-grained steel and a variety of design innovations, in the floor and the chassis for example, the dead weight of the Coil Liner has been considerably reduced again – while retaining the full strength and functionality of the semitrailer. The coil well is equipped with post pockets and two coil-securing supports as standard. In addition to that, the sliding curtain coil trailer has the Multi-Lock external frame as standard (with approx. 130 strapping points per side rave), as well as additional heavy duty strapping rings near the external frame, so that the load can always be secured in the best manner possible.


Risk management is about balancing risk and reward. Making prudent and responsible decisions is a critical part of financial planning and the overall wealth management process. Some risks are more difficult to mitigate than others, so insurance can be a very useful tool.
We help clients understand the complexities of their current policies and evaluate whether they are adequately protected so the risks are mitigated. Our goal is to make sure you have suitable levels and types of coverage in place.
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£50m of Regional Growth Fund money has been awarded to projects in Yorkshire and Humber, creating or safeguarding 5,500 jobs over ten years.
Successful bidders included Selby's Optare plc, Wakefield's Really Useful Products Limited and Leeds' Harrison Spinks Beds Ltd.
This latest round of RGF money aims to leverage £160m of further private sector investment in the next ten years.
To date the region has had nearly £310m in Growth Fund support which is estimated to have created or safeguarded around 56,000 jobs.


What is the air quality index in the Rogue Valley today?
Did you know that our air quality can be worse in the winter than at any other time of the year?
Learn more about temperature inversion and winter air pollution in the Rogue Valley.
Burn Wise Program of the EPA - Burn Wise is a partnership program of the U.S. Environmental Protection Agency that emphasizes the importance of burning the right wood, the right way, in the right wood-burning appliance to protect your home, health, and the air we breathe.
Copyright © 2016. Rogue Valley Clean Air.


Hack an online account of Facebook is an idea that interests many people. This is reflected in the huge amount of Google searches of "hack Facebook account". It is something that many people in the world would like to do, and everyone will have their reasons. But the truth is that few want it for ethical or honorable purposes. Also, entering a third party account is illegal, so you can be punished by law due to hacking into a Facebook account. As if that were not enough, it is also an action that requires a lot of experience, since it will be necessary to know about computer science and programming. Click here for more hacking Facebook.
Can I hack Facebook online?
3) Hacking a Facebook account requires a lot more than just the profile URL.
However, many people, in their ingenuity, think that these pages can give them access to a Facebook profile by just entering the URL and clicking a button. Although it seems ridiculous, it is very likely that the designers of that page are making some money with each naive person who falls into the trap. So do not be part of those who get carried away by curiosity. Do not believe in web pages that claim to be able to hack a Facebook account. Learn more about how to hack Facebook using Appspy.
HOW TO CHOOSE THE BEST FOREX DEMO ACCOUNT?


text:
Posts Tagged '3D'
Body Labs, An Osage University Partner, Receives US Patent for their 3D Body Modeling Technique and Algorithm
Posted in 3D Plotting Scaffolds, 3D Printing for Medical Application, BioPrinting in Regenerative Medicine, tagged 3D, 3D Modeling, 3D Scanning, computer algorithm on March 23, 2016| Leave a Comment »
Body Labs Secures Exclusive U.S. Patents And Licenses For 3D Body Modeling Technologies
Body Labs establishes leadership through proprietary technology developed from research led by world-class computer vision scientist Michael J. Black
Mar 21, 2016, 09:00 ET from Body Labs
Reported from PRNewswire by Stephen J. Williams, Ph.D.; Reporter
Source: http://www.prnewswire.com/news-releases/body-labs-secures-exclusive-us-patents-and-licenses-for-3d-body-modeling-technologies-300238518.html
NEW YORK, March 21, 2016 /PRNewswire/ — Body Labs (bodylabs.com), the provider of the world's most advanced technology for analyzing the human body's shape, pose and motion, announced today that it has secured the exclusive rights to two patents issued to Brown University and licenses to several new technologies developed at Max-Planck-Innovation GmbH.
U.S. Patents 9,189,886 B2 and 2013/0249908 A1 expand Body Labs' exclusive ability to create accurate 3D human models learned from natural shape and pose variations captured from data inputs such as images or range maps. These patents cover intellectual property included in Body Labs' statistical model of human shape, pose and motion. This statistical approach employs machine learning algorithms and the world's most comprehensive training set of human shape and pose to convert shape parameters (measurements or scans) into the most statistically-accurate body geometry currently available.
Body Labs also announced exclusive licenses to several new technologies developed at Max-Planck-Innovation GmbH. These technologies enable Body Labs to further expand its technical leadership into other areas of the human body such as hands, feet, faces and heads. Additionally, the new developments streamline the adoption of Body Labs' technology into other industry workflows such as animation pipelines, gaming, virtual reality (VR), augmented reality (AR), health, fitness and more. These technologies also provide never-before-seen detail into soft-tissue deformation trained on shape data and pose variations captured from 4D scanners.
"These exclusive patents and licenses enable us to unlock unprecedented personalization across the entire human body and ensure our technology is more accessible than ever to a growing list of industries," said William O'Farrell, co-founder and CEO of Body Labs. "It's been a privilege to be partnered with Brown University and the Max Planck Institute for the past three years. These patents and licenses enable us to further expand our relationship and drive rapid innovation in the 3D body modeling space."
According to Juniper Research, 60 million users across smartphones, tablets and smart glasses will use augmented reality apps in 2016. Gartner has also projected that 25 million units of virtual reality headsets will be in the hands of consumers by 2018. This year, tech giants such as Intel (Intel Capital led Body Labs' Series A round of financing) will be working with manufacturers to integrate depth-sensor technology into smartphones as well. This broad adoption will deliver entirely new developer platforms made for everyday consumers that can manage, process and host 3D data.
Body Labs is the only company to provide the body as a digital platform. Through Body Labs, brands and retailers can access API's for building consumer-facing experiences such as apparel sizing recommendations, bespoke clothing, fitness tracking, personalizing VR or AR experiences and more. For more information on how to personalize goods and services around human body shape visit:http://www.bodylabs.com/solutions.
About Body Labs
Founded in 2013 and headquartered in Manhattan, Body Labs collects, digitizes and organizes all of the data and information related to human body shape, pose and motion. Its mission is to transform the human body into a digital platform upon and around which goods and services can be designed, produced, bought and sold. For more information, please visit:www.bodylabs.com.
SOURCE Body Labs
http://www.bodylabs.com
Introduction to Tissue Engineering; Nanotechnology applications
Posted in Bio Instrumentation in Experimental Life Sciences Research, BioSimilars, Cardiac & Vascular Repair Tools Subsegment, Cell Biology, Signaling & Cell Circuits, Frontiers in Cardiology and Cardiovascular Disorders, Health Economics and Outcomes Research, Nanotechnology for Drug Delivery, Pharmaceutical R&D Investment, Technology Transfer: Biotech and Pharmaceutical, tagged 3D, ECM, Extracellular matrix, nanocomposites, nanostructure, nanotopographic, scaffold, Tissue engineering on January 1, 2013| 8 Comments »
Author, Editor and Curator: Tilda Barliya, PhD
Tissue Engineering is an emerging multidisciplinary field involving biology, medicine, and engineering that is likely to revolutionize the ways we improve the health and quality of life for millions of people by restoring, maintaining, or enhancing tissue and organ function. Tissue engineering emerged as organ transplantation is limited by the number of available donors and high cost process, leaving thousands of people each year on the transplant waiting lists in the United States alone. Many die before an organ donor becomes available. Dr. Tal Dvir from the Langer's lab at MIT have summarized this topic in his review (2. http://nextbigfuture.com/2011/01/nanotechnology-strategies-for-tissue.html)
Tissue engineering aims at developing functional substitutes for damaged tissues and organs, a process that involves the use of a combination of cells, engineering and material methods, including suitable biochemical and chemical factors to improve or replace biological functions. Rather than simply introducing cells into a diseased area to repopulate a defect and/or restore function, in tissue engineering the cells are often seeded in or onto biomaterials (scaffolds) before transplantation.
These biomaterial scaffold allows cells to attach and reorganize to form functional tissue by proliferating, synthesizing extracellular matrix, and migrating along the implant path (1,2,3) Figure 1.
Until recently, it was believed that the macroporous features of scaffolds used in tissue engineering mimicked the dimension scale of the extracellular matrix (ECM), and that the matrix itself (natural or artificial) only served as a support for the cells; morphogenesis was controlled passively by defining tissue boundaries. Emphasis was placed on critical engineering and material issues, such as improving mass transfer into the core of the cell constructs and designing biocompatible and biodegradable scaffolds with mechanical properties suitable for engineering various tissues. As the field evolved, attention focused on the biology of the scaffolds and how they affect various cell types.
Tissue engineers had recognized that some of the widely used scaffolds do not fairly recapitulate the cell microenvironment and that the ECM is a dynamic and hierarchically organized nanocomposite that regulates essential cellular functions such as:
morphogenesis,
Nanotechnological tools for tissue engineering may help design advanced nanocomposite scaffolds that can better mimic the ECM and eventually assemble more complex and larger functional tissues. In order to generate a functional tissue, effective organization of cells in the tissue is required with similar morphology and physiology of the parental tissue.
Morphogenesis in the three-dimensional (3D) scaffold should occur in a similar way to natural organ development. The cells reorganize owing to interaction with the ECM on the basis of:
topography,
mechanical properties (such as matrix stiffness, elasticity and viscosity)
concentration gradients of immobilized growth factors
ECM molecules.
Recently, Ott and co-workers (4) reported a study emphasizing the importance of the ECM structure in guiding the seeded cells and promoting morphogenesis. Rat hearts were decellularized by perfusion of detergents to preserve the underlying ECM and then reseeded with cardiac and endothelial cells (4). The cells migrated and self-organized in their natural location in the matrix and by day 8, under physiological load and electrical stimulation, the constructs were able to generate pump function (4). The importance of the ECM was shown for:
Arteries
So why is the Extracellular Martix (ECM) so important?
The ECM is composed of an intricate interweaving of protein fibres such as fibrillar collagens and elastins, ranging from 10 to several hundreds of nanometres. The mesh is covered with nanoscale adhesive proteins such as laminin and fibronectin that provide specific binding sites for cell adhesion (interacting with integrins, cadherins and so forth) and have been shown to regulate important cell behaviours such as growth, shape, migration and differentiation. Polysaccharides such as hyaluronic acid and heparan sulphate fill the interstitial space between the fibres and act as a compression buffer against the stress placed on the ECM or serve as a growth factor depot (Figure 2).
Scaffold design considerations
The ECMs of various tissues in the body differ in the composition and spatial organization of the collagens, elastins, proteoglycans and adhesion molecules, to maintain specific tissue morphologies and organ specific shape and function, and to supply specific instructive cues. Therefore, the design considerations for scaffolds should vary according to the desired engineered tissue. For example, the biochemical, electrical and mechanical functions of the heart are uniquely dependent on their biological nanostructures. The heart's 3D ECM network is composed of an intricate, micro- and nanoscale interweaving pattern of fibrillar collagen and elastin bundles that form a dense, elastic network with proteoglycans and with adhesive and non-adhesive molecules. In this defined mesh, the cardiomyocytes are forced to couple mechanically to each other, to form elongated and aligned cell bundles that interact with each other or with neighbouring capillaries and nerves.
Post-isolation cells lose their ultrastructural elongated morphology and their interaction with their surroundings, and they adopt a random distribution on the flat surface of the scaffold, which compromises many of their physiological properties. Therefore, the structure and support of the ECM is crucial. See Figure 2.
Limitations of the ECM:
Weak mechanical properties
Lack of electrical conductivity
Absence of adhesive and micoenvironment- defining moieties
Inability of cells to self-assemble to 3D tissue structure.
The rational behind incorporating nanostructures is to compensate for other scaffold limitations (Table 2) Ref.2
The Heart for example requires more than alignment and mechanical support (Boyang Zhang, Ref 5)
cell responses to micro- and nanopatterned topographical cues
cell responses to patterned biochemical cues
controlled 3D scaffolds
patterned tissue vascularization
electromechanical regulation (conductivity). of tissue assembly and function
Nanostructures can be used to record the electronic signals that are transmitted through cells such as neurons and cardiomyocytes. One way to record these signals is by lithographically defining nanostructures as field-effect transistors, which are sensitive to local electric field changes. In particular, silicon nanowire transistors are useful for measuring extracellular signals because they exhibit particularly exquisite field-effect sensitivity compared with conventional, planar devices; they are just tens of nanometres in diameter and can therefore interface with cells and tissue at a subcellular level; and they show nanotopographic features that encourage tight interfaces with biological systems.
This introduction reviewed some of the aspects required for tissue engineering with the affiliation to nanotechnology. In the next post, we will dive deeper into a specific tissue organ, the bioengineering aspect and how nanotechnology strategies may improve the design and outcome.
1. http://www.nanotech-now.com/news.cgi?story_id=35168
2. Dvir T., Timko BR., Kohane DS., and Langer R. Nanotechnological strategies for engineering complex tissues. Nature Nanotechnology 2010; 12():. http://nextbigfuture.com/2011/01/nanotechnology-strategies-for-tissue.html
3. http://www.nature.com/nnano/journal/v6/n1/abs/nnano.2010.246.html
4. Ott, H. C. et al. Perfusion-decellularized matrix: Using nature's platform to engineer a bioartificial heart. Nature Med. 14, 213–221 (2008).
5. Boyang Zhang, Yun Xiao, Anne Hsieh, Nimalan Thavandiran and Milica Radisic. Micro- and nanotechnology in cardiovascular tissue engineering. Nanotechnology 2011; 22(49): 494003
Targeted delivery of therapeutics to bone and connective tissues: current status and challenges- Part I
Posted in Biomarkers & Medical Diagnostics, Bone Disease and Musculoskeletal Disease, Disease Biology, Small Molecules in Development of Therapeutic Drugs, tagged 3D, advanced, bone, bone architecture, bone mineral density, bone physiology, bone repair, bone therapeutics, bone turnover, Cell therapy, delivery, nanoparticles, nanotechnology, nanotubes, novel, pharmaceutical drugs, theme, Therapeutic innovations, therapies, Therapy, treatment on September 23, 2012| 22 Comments »
Reporter Aviral Vatsa, PhD MBBS
Annual treatment costs for musculoskeletal diseases in the US are roughly 7.7% (~ $849 billion) of total gross domestic product. Such disorders are the main cause of physical disability in US. Almost half of all chronic conditions in people can be attributed to bone and joint disorders. In addition there is increasing ageing population and associated increases in osteoporosis and other diseases, rising incidences of degenerative intervertebral disk diseases and numbers of revision orthopedic arthroplasty surgeries, and increases in spinal fusions. All these factors contribute towards the increasing requirement of bone regeneration and reconstruction methods and products. Delivery of therapeutic grade products to bone has various challenges. Parenteral administration limits the efficient delivery of drugs to the required site of injury and local delivery methods are often expensive and invasive. The theme issue of Advance Drug Delivery reviews focuses on the current status of drug delivery to bone and the issues facing this field. Here is the first part of these reviews and research articles.
1. Demineralized bone matrix in bone repair: History and use
Demineralized bone matrix (DBM) is an osteoconductive and osteoinductive commercial biomaterial and approved medical device used in bone defects with a long track record of clinical use in diverse forms. True to its name and as an acid-extracted organic matrix from human bone sources, DBM retains much of the proteinaceous components native to bone, with small amounts of calcium-based solids, inorganic phosphates and some trace cell debris. Many of DBM's proteinaceous components (e.g., growth factors) are known to be potent osteogenic agents. Commercially sourced as putty, paste, sheets and flexible pieces, DBM provides a degradable matrix facilitating endogenous release of these compounds to the bone wound sites where it is surgically placed to fill bone defects, inducing new bone formation and accelerating healing. Given DBM's long clinical track record and commercial accessibility in standard forms and sources, opportunities to further develop and validate DBM as a versatile bone biomaterial in orthopedic repair and regenerative medicine contexts are attractive.
2. Biomimetic hydrogels for controlled biomolecule delivery to augment bone regeneration
The regeneration of large bone defects caused by trauma or disease remains a significant clinical problem. Although osteoinductive growth factors such as bone morphogenetic proteins have entered clinics, transplantation of autologous bone remains the gold standard to treat bone defects. The effective treatment of bone defects by protein therapeutics in humans requires quantities that exceed the physiological doses by several orders of magnitude. This not only results in very high treatment costs but also bears considerable risks for adverse side effects. These issues have motivated the development of biomaterials technologies allowing to better control biomolecule delivery from the solid phase. Here we review recent approaches to immobilize biomolecules by affinity binding or by covalent grafting to biomaterial matrices. We focus on biomaterials concepts that are inspired by extracellular matrix (ECM) biology and in particular the dynamic interaction of growth factors with the ECM. We highlight the value of synthetic, ECM-mimicking matrices for future technologies to study bone biology and develop the next generation of 'smart' implants.
3. Calcium phosphate cements as drug delivery materials
Calcium phosphate cements are used as synthetic bone grafts, with several advantages, such as their osteoconductivity and injectability. Moreover, their low-temperature setting reaction and intrinsic porosity allow for the incorporation of drugs and active principles in the material. It is the aim of the present work to: a) provide an overview of the different approaches taken in the application of calcium phosphate cements for drug delivery in the skeletal system, and b) identify the most significant achievements. The drugs or active principles associated to calcium phosphate cements are classified in three groups, i) low molecular weight drugs; ii) high molecular weight biomolecules; and iii) ions.
4. Silk constructs for delivery of musculoskeletal therapeutics
Silk fibroin (SF) is a biopolymer with distinguishing features from many other bio- as well as synthetic polymers. From a biomechanical and drug delivery perspective, SF combines remarkable versatility for scaffolding (solid implants, hydrogels, threads, solutions), with advanced mechanical properties and good stabilization and controlled delivery of entrapped protein and small molecule drugs, respectively. It is this combination of mechanical and pharmaceutical features which renders SF so exciting for biomedical applications. This pattern along with the versatility of this biopolymer has been translated into progress for musculoskeletal applications. We review the use and potential of silk fibroin for systemic and localized delivery of therapeutics in diseases affecting the musculoskeletal system. We also present future directions for this biopolymer as well as the necessary research and development steps for their achievement.
5. Demineralized bone matrix as a vehicle for delivering endogenous and exogenous therapeutics in bone repair
As a unique human bone extract approved for implant use, demineralized bone matrix (DBM) retains substantial amounts of endogenous osteoconductive and osteoinductive proteins. Commercial preparations of DBM represent a clinically accessible, familiar, widely used and degradable bone-filling device, available in composite solid, strip/piece, and semi-solid paste forms. Surgically placed and/or injected, DBM releases its constituent compounds to bone sites with some evidence for inducing new bone formation and accelerating healing. Significantly, DBM also has preclinical history as a drug carrier by direct loading and delivery of several important classes of therapeutics. Exogenous bioactive agents, including small molecule drugs, protein and peptide drugs, nucleic acid drugs and transgenes and therapeutic cells have been formulated within DBM and released to bone sites to enhance DBM's intrinsic biological activity. Local release of these agents from DBM directly to surgical sites in bone provides improved control of dosing and targeting of both endogenous and exogenous bioactivity in the context of bone healing using a clinically familiar product. Given DBM's long clinical track record and commercial accessibility in standard forms and sources, opportunities to formulate DBM as a versatile matrix to deliver therapeutic agents locally to bone sites in orthopedic repair and regenerative medicine contexts are attractive.
6. Nanofiber-based delivery of bioactive agents and stem cells to bone sites
Biodegradable nanofibers are important scaffolding materials for bone regeneration. In this paper, the basic concepts and recent advances of self-assembly, electrospinning and thermally induced phase separation techniques that are widely used to generate nanofibrous scaffolds are reviewed. In addition, surface functionalization and bioactive factor delivery within these nanofibrous scaffolds to enhance bone regeneration are also discussed. Moreover, recent progresses in applying these nanofiber-based scaffolds to deliver stem cells for bone regeneration are presented. Along with the significant advances, challenges and obstacles in the field as well as the future perspective are discussed.
7. Intra-operatively customized implant coating strategies for local and controlled drug delivery to bone
Bone is one of the few tissues in the human body with high endogenous healing capacity. However, failure of the healing process presents a significant clinical challenge; it is a tremendous burden for the individual and has related health and economic consequences. To overcome such healing deficits, various concepts for a local drug delivery to bone have been developed during the last decades. However, in many cases these concepts do not meet the specific requirements of either surgeons who must use these strategies or individual patients who might benefit from them. We describe currently available methods for local drug delivery and their limitations in therapy. Various solutions for drug delivery to bone focusing on clinical applications and intra-operative constraints are discussed and drug delivery by implant coating is highlighted. Finally, a new set of design and performance requirements for intra-operatively customized implant coatings for controlled drug delivery is proposed. In the future, these requirements may improve approaches for local and intra-operative treatment of patients.
8. Local delivery of small and large biomolecules in craniomaxillofacial bone
Current state of the art reconstruction of bony defects in the craniomaxillofacial (CMF) area involves transplantation of autogenous or allogenous bone grafts. However, the inherent drawbacks of this approach strongly urge clinicians and researchers to explore alternative treatment options. Currently, a wide interest exists in local delivery of biomolecules from synthetic biomaterials for CMF bone regeneration, in which small biomolecules are rapidly emerging in recent years as an interesting adjunct for upgrading the clinical treatment of CMF bone regeneration under compromised healing conditions. This review highlights recent advances in the local delivery small and large biomolecules for the clinical treatment of CMF bone defects. Further, it provides a perspective on the efficacy of biomolecule delivery in CMF bone regeneration by reviewing presently available reports of pre-clinical studies using various animal models.
9. Immobilized antibiotics to prevent orthopaedic implant infections
Many surgical procedures require the placement of an inert or tissue-derived implant deep within the body cavity. While the majority of these implants do not become colonized by bacteria, a small percentage develops a biofilm layer that harbors invasive microorganisms. In orthopaedic surgery, unresolved periprosthetic infections can lead to implant loosening, arthrodeses, amputations and sometimes death. The focus of this review is to describe development of an implant in which an antibiotic tethered to the metal surface is used to prevent bacterial colonization and biofilm formation. Building on well-established chemical syntheses, studies show that antibiotics can be linked to titanium through a self-assembled monolayer of siloxy amines. The stable metal–antibiotic construct resists bacterial colonization and biofilm formation while remaining amenable to osteoblastic cell adhesion and maturation. In an animal model, the antibiotic modified implant resists challenges by bacteria that are commonly present in periprosthetic infections. While the long-term efficacy and stability is still to be established, ongoing studies support the view that this novel type of bioactive surface has a real potential to mitigate or prevent the devastating consequences of orthopaedic infection.
10. Local delivery of nitric oxide: Targeted delivery of therapeutics to bone and connective tissues
Non-invasive treatment of injuries and disorders affecting bone and connective tissue remains a significant challenge facing the medical community. A treatment route that has recently been proposed is nitric oxide (NO) therapy. Nitric oxide plays several important roles in physiology with many conditions lacking adequate levels of NO. As NO is a radical, localized delivery via NO donors is essential to promoting biological activity. Herein, we review current literature related to therapeutic NO delivery in the treatment of bone, skin and tendon repair.
Demineralized bone matrix in bone repair: History and use
Biomimetic hydrogels for controlled biomolecule delivery to augment bone regeneration
Calcium phosphate cements as drug delivery materials
Silk constructs for delivery of musculoskeletal therapeutics
Demineralized bone matrix as a vehicle for delivering endogenous and exogenous therapeutics in bone repair
Nanofiber-based delivery of bioactive agents and stem cells to bone sites
Intra-operatively customized implant coating strategies for local and controlled drug delivery to bone
Immobilized antibiotics to prevent orthopaedic implant infections
Local delivery of nitric oxide: Targeted delivery of therapeutics to bone and connective tissues

redpajama_set_name: RedPajamaCommonCrawl


In Jesus we are now complete. Even as the architect's model is planned and prepared and completed in his office, so the perfect pattern of the life of holy service, for which Jesus has redeemed and called us, is now in Him in heaven. But now it must be formed in us and transferred to our earthly lives, and this is the Holy Spirit's work. He takes the gifts and graces of Christ and brings them into our lives as we need and receive them day by day, just as the various sections of a vessel may be reproduced in a distant land. Thus we receive of His fullness, even grace for grace-His grace for our grace, His supply for our needs, His strength for our strength, His body for our body, His Spirit for our spirit. He Himself of God is made unto us wisdom, and righteousness, and sanctification, and redemption (1 Corinthians 1:30). But it is much more than mere abstract help and grace, much more even than the Holy Spirit bringing us strength, and peace and purity. It is personal companionship with Jesus Himself! Lord, help us receive from Thee today that grace in all trials that shall result in our being made perfect.


If you're visiting Oklahoma City or just an OKC local, chances are you're going to want to visit Bricktown often. It houses the Bricktown Ballpark, the canal, the Chesapeake Arena (where the Oklahoma City Thunder play) retail shops, and of course - numerous fantastic eateries.
All About Cha's tea baristas server you up fantastically brewed unique tea blends, coffee, espresso and other coffee drinks, along with delicious breakfast, lunch and dinner items. Sip the local favorite - Green Con Panna, a combination of green tea and steamed milk topped with whipped cream. Feast on one of their freshly-made daily quiches, or a Chicken Avocado Croissant. Along with wraps and salads, Cha also has sushi rolls for lunch or dinner. Save room however, for one of their amazing desserts, including their Ultimate Chocolate Blackout Cake, a Pecan Tart or a Parisian Macaron.
Bricktown Brewery is a choice spot to head to for locally crafted beer and sinful comfort cuisine like burgers, BBQ and pizza. As one of the earliest eateries in Bricktown, their handcrafted beer is award-winning and pairs nicely with their delicious hickory smoked BBQ but there are a slew of other noteworthy menu items like the Hatch Green Chile Burger with white queso, hatch green chiles and more. Try an order of Sea Salt Caramel Fries or the Chopped & Chipped Nachos, corn tortilla chips topped with slow-smoked pork shoulder, BBQ, cheese, sour cream and capped off with a giant fried jalapeno. Round out your meal with a sizzling Apple Crisp or warm fresh-baked cookie and if you're around on Friday or Saturday, check out the live acoustic music!
Mickey Mantle's Steakhouse is a home-run if you're looking for elegant fine-dining in Bricktown. This spacious and lovely restaurant offers an extensive wine list of over 500 choices from all over the world along with grilled signature steaks, fresh seafood and more. Make sure and pick something from their Mac N' Cheese menu classic like jalapeno-bacon, or lobster! Head over to the adjoining #7 Lounge for a fine cigar or scotch or a fantastic cocktail after your meal. You can relax in a beautifully curtained private booth or take a complimentary limo ride to a Thunder game if you happen to be heading that way!
Crabtown's Cajun seafood-inspired menu is high on the list of visitors' dining choices in Oklahoma City. Their main feature is much loved - the Cajun Crab & Shrimp Boil that is turned out onto a butcher paper covered table. This snow crab and shrimp laden feast is fun to eat with a crowd! The heavily featured seafood menu offers other mouthwatering fare like Crab Pot Pie, Louisiana Maji Mahi, and Po Boys but Crabtown also features burgers, chicken fried steak, steak and red beans and rice. Wash down the heat with one of their specialty drinks and finish things off with one of their outstanding desserts like New Orleans Bread Pudding or Key Lime Pie.
Next time you're here for fun, a conference or a weekend away, we hope you dive in to the diverse food options Bricktown offers!


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Man Arrested after Series of Attacks on Churches in Rome
Date of incident: October 1, 2016
Police arrested a 39-year-old Ghanaian man suspected of attacking several churches in Rome over the previous two days. He is suspected of heavily damaging devotional statues, crucifixes, and candlesticks at the Basilica of Santa Prassede, the Church of San Martino ai Monti, and the Basilicas of San Giovanni de' Fiorentini and San Vitale.
According to reports, on the evening of September 30th, a man damaged a statue in the church of San Martino ai Monti and fled. Shortly afterwards, the same person entered the Basilica of Santa Prassede and "smash[ed] everything he found, especially statues and candlesticks, right before a group of parishioners who were quite scared. The two devotional statues of Saint Anthony of Padua and Saint Prassede were heavily damaged." The rector of the church said that there were about 15 worshippers present and, after a moment of shock, they chased after the perpetrator before he could enter the sanctuary to destroy the crucifix. Later, he is alleged to have entered the Church of San Martino ai Monti, where he smashed a devotional statue of the Madonna and Child. On the morning of October 1st, two further attacks occurred in the Basilicas of San Giovanni de' Fiorentini and San Vitale. Statues, candlesticks, crucifixes were smashed. According to a witness, "the perpetrator at this time had time to admit that it was not right that we worshipped in this way." After receiving descriptions of the perpetrator from priests and witnesses, police arrested a 39-year-old Ghanaian. His motives are under investigation. Sources: KathPress and The Roman Anglican (quoted witness & photo credit) A similar attack occurred in Vienna, Austria in 2014. Watch video here.


We are a family-owned company with a large experience in the production and trading of grains in Argentina. Andreoli Group is one of the few companies fully integrated since it has production in owned farms, a Service Division with an important storage capacity, supply warehouses and in house logistics division to give better support to the business. After years of hard and intensive work we have achieved a reputation of commitment, compromise and loyalty with our engagements, quality and with our social and environmental responsibility. We dedicate all our experience, resources and structure to satisfy the needs of our customers. Our goal is to be seen by our world clients as their local partner in South America.


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Philadelphia Marketplace > Press > CUPCAKES ON THE FLY AT PHILADELPHIA INTERNATIONAL AIRPORT – Local Food Truck Vendor, Cupcakeology, Arrives at PHL
CUPCAKES ON THE FLY AT PHILADELPHIA INTERNATIONAL AIRPORT – Local Food Truck Vendor, Cupcakeology, Arrives at PHL
CUPCAKES ON THE FLY AT PHILADELPHIA INTERNATIONAL AIRPORT
Local Food Truck Vendor, Cupcakeology, Arrives at PHL
PHILADELPHIA — September 9, 2013 — Traveling through Philadelphia International Airport (PHL) is now a piece of "cupcake" thanks to the addition of Cupcakeology. The airport's newest kiosk, located in Terminal B, is serving up an array of sweet treats for passengers.
Among the more than 40 flavors to choose from, some of Cupcakeology's leading sellers include: Lemon Drop, a light lemon cupcake filled with a lemon curd with lemon butter cream frosting; What's Up Doc, which features sweet carrots, walnuts and raisins, all packed into a cinnamon infused cupcake with a cream cheese frosting; and the Cookie Monster, which boasts Madagascar bourbon vanilla speckled with crushed Oreo cookies with an Oreo cookie and cream frosting.
"Opening a kiosk is a great opportunity for merchants to try their business in the airport environment, which sees more than 30 million passengers each year," said Mel Hannah, vice president and general manager of MarketPlace Philadelphia Management, the company responsible for the management and leasing operations of PHL's 160 retail locations. "We're pleased to welcome Cupcakeology to our diverse mix of merchants."
Cupcakeology is a local bakery based in Collingdale, Pa. The business was founded by husband and wife team, Robert and Jennifer Morris. Jennifer has been baking her own pound cakes since the age of nine, and Robert has commercial baking experience from past positions as a kitchen manager. Cupcakeology ships nationwide and also offers catering services.
For more information about specialty leasing opportunity at PHL and the MarketPlace Philadelphia Management retail program, please visit www.philamarketplace.com
EDITOR'S NOTE: A photo of the Cupcakeology kiosk at PHL is attached to this email.