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\section{Introduction}
Recently, LHCb updated a measurement of the lifetime of $\Omega_{c}$
with $\tau(\Omega_{c}^{0})=268\pm24\pm10\pm2$ fs \cite{Aaij:2018dso},
which is nearly four times larger than the world-average $\tau(\Omega_{c}^{0})=69\pm12$
fs \cite{Tanabashi:2018oca} in PDG2018. Theoretically, the lifetimes
of weak-decay baryons can be explained by the framework of heavy quark
expansion (HQE) (see \cite{Lenz:2014jha} for a review). Among others, the calculation
of hadron matrix elements is an important part to decipher this puzzle.
In this direction, some efforts has been made in our recent work \cite{Zhao:2021lzd}.
We intend to perform a careful analysis on these hadronic matrix elements
with QCD sum rules (QCDSR). QCDSR is a QCD-based approach to investigate
the properties of hadrons. It connects the hadron phenomenology and
QCD vacuum structure via a few universal parameters like quark condensates
and gluon condensates.
It is usually thought that about 30\% uncertainty will be introduced
in QCDSR. However, as can be seen in our recent works \cite{Zhao:2020mod,Zhao:2021sje},
once a stable Borel region is found, the prediction of physical quantity
can reach high precision.
In QCDSR, the Wilson coefficients of OPE should be evolved to the
energy scale $\mu_{0}$ of low-energy limit \cite{Ioffe:1981kw}. For example,
for the two-point correlation function of baryon,
\begin{equation}
T\{J(x)\bar{J}(0)\}=\sum_{n}C_{n}(x^{2})O_{n}(0),
\end{equation}
the coefficient functions $C_{n}(p^{2})$ in the momentum space should
contain the factor
\[
\left[\frac{\alpha_{s}(p^{2})}{\alpha_{s}(\mu_{0}^{2})}\right]^{2\gamma_{J}-\gamma_{n}},
\]
where $\gamma_{J}$ and $\gamma_{n}$ are respectively the anomalous
dimensions of the current $J$ and the operator $O_{n}$.
It can be seen that, the results of the anomalous dimensions are indispensable
inputs in a serious analysis of baryon QCD sum rules, especially for the
case of bottom baryons. However, the results in the literature are vague. In view of this, in this work, we
calculate the anomalous dimensions for some interpolating currents
such as those of $\Lambda_{Q}$ and proton.
There are 3 relevant diagrams, as can be seen in Fig. \ref{fig:ano_dim}.
Following the convention of Peskin in \cite{Peskin:1979mn}, the sum of
the divergent parts of Fig. 1a-1c will be of the form
\begin{equation}
-\frac{4}{3}b\frac{g^{2}}{(4\pi)^{2}}\frac{\Gamma(2-d/2)}{(M^{2})^{2-d/2}}\times J,\label{eq:Peskin_convention}
\end{equation}
with $b$ some constant. Then
\begin{align}
\gamma(g^{2}) & =\frac{4}{3}(3+2b)\frac{g^{2}}{(4\pi)^{2}}+O(g^{4}),\nonumber \\
\gamma_{J} & =\frac{2}{3}(3+2b)/\beta_{0},\label{eq:gamma_Peskin}
\end{align}
where $\beta_{0}=11-(2/3)n_{f}$ with $n_{f}$ the number of fermion
flavors. In Eq. (\ref{eq:gamma_Peskin}), the factor of $(4/3)\cdot3$
arises from the wavefunction renormalization of three fermion fields.
The rest of this paper is arranged as follows. In Sec. II and III,
we will calculate the anomalous dimensions for the interpolating currents
of $\Lambda_{Q}$ and proton, respectively. We conclude our paper
in the last section.
\begin{figure}[!]
\includegraphics[width=0.6\columnwidth]{ano_dim.eps}
\caption{The one-loop diagrams for the calculation of anomalous dimension.
Here $F$, $G$ and $H$ respectively stand for the quark fields of
$u$, $d$ and $Q$ for $\Lambda_{Q}$.}
\label{fig:ano_dim}
\end{figure}
\section{The anomalous dimension for $\Lambda_{Q}$}
The interpolating current for $\Lambda_{Q}$ is usually taken as:
\begin{equation}
J_{\Lambda_{Q}}=\epsilon_{abc}(u_{a}^{T}C\gamma_{5}d_{b})Q_{c},\label{eq:current_LQ}
\end{equation}
where $Q$ denotes a bottom or charm quark, $a,b,c$ are the color
indices and $C$ is the charge conjugate matrix. In Eq. (\ref{eq:current_LQ}),
the system of $u$ and $d$ quarks has spin and parity of $0^{+}$,
while the whole baryon has $J^{P}=1/2^{+}$.
For Fig. 1a, it can be shown that
\begin{equation}
\text{Fig. 1a}=(-\frac{8}{3})\times ig^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{4}}\times J_{\Lambda_{Q}}.
\end{equation}
For Figs. 1b, a new structure emerges because of
\begin{equation}
(\bar{\psi}_{1}\gamma^{\mu}\gamma^{\nu}\psi_{2})\gamma_{\mu}\gamma_{\nu}\psi_{3}=4\ (\bar{\psi}_{1}\psi_{2})\psi_{3}-(\bar{\psi}_{1}\sigma^{\mu\nu}\psi_{2})\sigma_{\mu\nu}\psi_{3}.
\end{equation}
It turns out that
\begin{equation}
\text{Fig. 1b}=-\frac{1}{6}\times ig^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{4}}\times\epsilon_{abc}[4\ (u_{a}^{T}C\gamma_{5}d_{b})Q_{c}-(u_{a}^{T}C\gamma_{5}\sigma^{\mu\nu}d_{b})\sigma_{\mu\nu}Q_{c}].
\end{equation}
Similarly,
\begin{equation}
\text{Fig. 1c}=-\frac{1}{6}\times ig^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{4}}\times\epsilon_{abc}[4\ (d_{b}^{T}C\gamma_{5}u_{a})Q_{c}-(d_{b}^{T}C\gamma_{5}\sigma^{\mu\nu}u_{a})\sigma_{\mu\nu}Q_{c}].
\end{equation}
Therefore
\begin{equation}
\text{Fig. 1b}+\text{Fig. 1c}=2\times(-\frac{2}{3})\times ig^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{4}}\times\epsilon_{abc}(u_{a}^{T}C\gamma_{5}d_{b})Q_{c}.
\end{equation}
In the above, we have used
\begin{align}
\epsilon_{abc}(u_{a}^{T}C\gamma_{5}d_{b})Q_{c} & =\epsilon_{abc}(d_{b}^{T}C\gamma_{5}u_{a})Q_{c},\\
\epsilon_{abc}(d_{b}^{T}C\gamma_{5}\sigma^{\mu\nu}u_{a})\sigma_{\mu\nu}Q_{c} & =-\epsilon_{abc}(u_{a}^{T}C\gamma_{5}\sigma^{\mu\nu}d_{b})\sigma_{\mu\nu}Q_{c}.
\end{align}
Finally, the counterterm for the current is obtained as
\begin{align}
-\delta_{J_{\Lambda_{Q}}}=\text{Fig. 1a}+\text{Fig. 1b}+\text{Fig. 1c} & =-4\times ig^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{4}}\times J_{\Lambda_{Q}}\nonumber \\
& =4\times\frac{g^{2}}{(4\pi)^{2}}\frac{\Gamma(2-d/2)}{(M^{2})^{2-d/2}}\times J_{\Lambda_{Q}}.
\end{align}
where $M$ is the reference scale of renormalization. As can be seen
in Eq. (\ref{eq:Peskin_convention}), $b=-3$. Therefore, the anomalous
dimension for the current of $\Lambda_{Q}$ is $-2/\beta_{0}$. However,
it seems that our result is different from that in \cite{Ovchinnikov:1991mu}.
\section{The anomalous dimension for proton}
The interpolating current for proton is usually taken as \cite{Ioffe:1981kw}:
\begin{equation}
J_{p}=\epsilon_{abc}(u_{a}^{T}C\gamma^{\mu}u_{b})\gamma_{5}\gamma_{\mu}d_{c},
\end{equation}
where the system of two $u$ quarks has spin and parity of $1^{+}$,
while the whole baryon has $J^{P}=1/2^{+}$.
Without loss of generality, in this section, we assume that the current
is built from quark fields of three distinct flavors $F$, $G$, $H$:
\begin{equation}
J_{p}=\epsilon_{abc}(F_{a}^{T}C\gamma^{\mu}G_{b})\gamma_{5}\gamma_{\mu}H_{c}.\label{eq:current_proton}
\end{equation}
For Fig. 1a, it can be shown that
\begin{equation}
\text{Fig. 1a}=(-\frac{2}{3})\times ig^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{4}}\times J_{p}.
\end{equation}
For Fig. 1b, a new structure emerges because of
\begin{eqnarray}
& & (\bar{\psi}_{1}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\psi_{2})\gamma_{5}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\psi_{3}\nonumber \\
& = & (\bar{\psi}_{1}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\psi_{2})\gamma_{5}(2g_{\nu\rho}\gamma_{\mu}-2g_{\mu\rho}\gamma_{\nu}+\gamma_{\rho}\gamma_{\mu}\gamma_{\nu})\psi_{3}\nonumber \\
& = & 10\ (\bar{\psi}_{1}\gamma^{\mu}\psi_{2})\gamma_{5}\gamma_{\mu}\psi_{3}-6\ (\bar{\psi}_{1}\gamma^{\mu}\gamma_{5}\psi_{2})\gamma_{\mu}\psi_{3}.
\end{eqnarray}
It turns out that
\begin{equation}
\text{Fig. 1b}=-\frac{1}{6}\times ig^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{4}}\times\epsilon_{abc}[10\ (F_{a}^{T}C\gamma^{\mu}G_{b})\gamma_{5}\gamma_{\mu}H_{c}-6\ (F_{a}^{T}C\gamma^{\mu}\gamma_{5}G_{b})\gamma_{\mu}H_{c}].
\end{equation}
Similarly,
\begin{equation}
\text{Fig. 1c}=-\frac{1}{6}\times ig^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{4}}\times\epsilon_{abc}[10\ (G_{a}^{T}C\gamma^{\mu}F_{b})\gamma_{5}\gamma_{\mu}H_{c}-6\ (G_{a}^{T}C\gamma^{\mu}\gamma_{5}F_{b})\gamma_{\mu}H_{c}].
\end{equation}
Therefore
\begin{equation}
\text{Fig. 1b}+\text{Fig. 1c}=2\times(-\frac{5}{3})\times ig^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{4}}\times\epsilon_{abc}(F_{a}^{T}C\gamma^{\mu}G_{b})\gamma_{5}\gamma_{\mu}H_{c}.
\end{equation}
In the above, we have used
\begin{align}
\epsilon_{abc}(G_{a}^{T}C\gamma^{\mu}F_{b})\gamma_{5}\gamma_{\mu}H_{c} & =\epsilon_{abc}(F_{a}^{T}C\gamma^{\mu}G_{b})\gamma_{5}\gamma_{\mu}H_{c},\\
\epsilon_{abc}(G_{a}^{T}C\gamma^{\mu}\gamma_{5}F_{b})\gamma_{\mu}H_{c} & =-\epsilon_{abc}(F_{a}^{T}C\gamma^{\mu}\gamma_{5}G_{b})\gamma_{\mu}H_{c}.
\end{align}
Finally, the counterterm for the current is obtained as
\begin{align}
-\delta_{J_{p}}=\text{Fig. 1a}+\text{Fig. 1b}+\text{Fig. 1c} & =-4\times ig^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{k^{4}}\times J_{p}\nonumber \\
& =4\times\frac{g^{2}}{(4\pi)^{2}}\frac{\Gamma(2-d/2)}{(M^{2})^{2-d/2}}\times J_{p}.
\end{align}
It happens that $\delta_{J_{p}}=\delta_{J_{\Lambda_{Q}}}$ and thereby
$\gamma_{J_{p}}=\gamma_{J_{\Lambda_{Q}}}=-2/\beta_{0}$. Our result
is consistent with that of Peskin in \cite{Peskin:1979mn}.
\section{Conclusions}
The anomalous dimensions for the interpolating currents
of baryons are indispensable inputs in a serious analysis of baryon QCD sum
rules. However, the results in the literature are vague.
In view of this, in this work, we have investigated the
anomalous dimensions for some interpolating currents such as those
of $\Lambda_{Q}$ and proton.
In this work, we do not consider the interpolating current of $\Delta$,
which has the quark content of $uuu$ and spin-3/2. We feel that the
interpolating current of $\Delta$ written in terms of Dirac gamma
matrices
\begin{equation}
(J_{\Delta})^{\mu}=\epsilon_{abc}(u_{a}^{T}C\gamma^{\mu}u_{b})u_{c},
\end{equation}
is cumbersome so that it is not easy to see clearly the completely
symmetrical relation among these three $u$ quarks. It seems that
the presription in \cite{Peskin:1979mn} is still a better choice.
\section*{Acknowledgements}
The authors are grateful to Prof. Zhi-Gang Wang for valuable discussions
and to Prof. Wei Wang for constant encouragements. This work is supported
in part by National Natural Science Foundation of China under Grant
No. 12065020.
| {
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} | 8,216 |
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"redpajama_set_name": "RedPajamaC4"
} | 3,794 |
\section{Introduction}
The radio source Sagittarius~A* (Sgr~A*) has been conclusively
identified in the radio and infrared with a black hole of mass
$\sim3.5\times10^6 M_{\odot}$ at the center of our galaxy
\citep{ReidBrun04,SchodelE03,GhezE05,EisenhauerE05}. {Sgr~A*} is the
nearest super-massive black hole, 100 times closer than its nearest
neighbor, M31*, and therefore should provide a unique opportunity to
understand the physics and life cycle of these objects. For a black
hole of its size, {Sgr~A*} is extremely under-luminous, only a few
hundred solar luminosities and $10^{-8} L_{Edd}$. This surprisingly
low luminosity has motivated many theoretical and observational
efforts to understand the processes at work very near to {Sgr~A*}.
Accretion models of {Sgr~A*} generally seek to explain its faintness
through inefficient radiative and accretion processes. A variety of
physical mechanisms can be invoked to suppress accretion and
radiation, including convection \citep{QuatGruz00-CDAF}, jets
\citep{FalckeE93}, advection of energy stored in non-radiating ions
\citep{NaraYi94}, and winds \citep{BlandBegel99}. Many models
incorporating combinations of these and other phenomena are able to
account for the spectrum and low luminosity of {Sgr~A*}. Therefore, the
physics of this source are not well constrained by these observations
alone.
In recent years, millimeter and submillimeter polarimetry has emerged
as an important tool for studies of Sgr~A*. Linear polarization and its
variability can be used to understand the structure of the magnetic
field in the emission region and turbulence in the accretion flow, and
possibly to constrain the mechanisms responsible for the
multi-wavelength variability of this source. Through Faraday rotation
of the linear polarization, we can examine the density and magnetic
field distributions along the line of sight, and eventually, in
the context of more comprehensive models of the accretion flow
structure, infer an accretion rate at the inner regions of the
accretion flow \citep{QuatGruz00-LP,Agol00,MeliaLiuCoker00}.
Previous observations of the linear polarization of Sgr~A*\ have found
low ($<$1\%) upper limits at 22, 43, and 86~GHz \citep{BowerE99-lp},
with a 2\% limit at 112~GHz \citep{BowerE01}. The lowest frequency
detection of linear polarization is at 150~GHz
\citep{AitkenE00}, suggesting that these polarimetric probes of Sgr~A*\
can only be exploited at short millimeter and submillimeter
wavelengths. \citet{AitkenE00} found that the polarization fraction
rises steeply from 150 to 400~GHz, although these observations were
made with a single-aperture instrument and therefore required careful
removal of contaminant emission within the telescope beam. The steep
spectrum and a jump in the polarization position angle between 230 and
350~GHz in the \citet{AitkenE00} data have been taken as evidence of a
transition to optically thin synchrotron emission
\citep[e.g.,][]{AitkenE00,Agol00,MeliaLiuCoker00}.
Subsequent interferometric monitoring of the 230~GHz polarization,
with angular resolution sufficient to avoid contamination from the
surrounding emission, have shown that the 230~GHz polarization
fraction appears to remain constant over 5 years, despite variations
in the position angle on month to year timescales
\citep{BowerE03,BowerE05}. This variability reduces the significance of
the observed position angle jump and demonstrates the need for
contemporaneous measurements at multiple frequencies. \citet{BowerE05}
attribute the variations in the 230~GHz polarization to
few$\times10^5$~rad~m$^{-2}$\ changes in the rotation measure (RM), probably
in the accretion medium, rather than to changes in the intrinsic
source polarization. As of yet, no observations have been able to
determine the RM, but they can place upper limits on the magnitude of
the RM and infer temporal variations that are within a factor of a few
of the upper limits.
Circular polarization has also been detected in this source, with a
rising polarization fraction from 1.4 to 15~GHz
\citep{BowerE99-cp,SaultMacq99,BowerE02}. Some models seeking to
explain the millimeter/submillimeter linear polarization have also
predicted appreciable circular polarization at these high frequencies
due to the conversion of linear to circular polarization in a
turbulent jet \citep{BeckertFalcke02,Beckert03}. However, measurements
to date at or above 100~GHz
\citep[e.g.,][]{TsuboiE03,BowerE03,BowerE05} have not shown circular
polarization at the percent level.
The Submillimeter Array (SMA) has the potential to contribute many new
capabilities to these studies. It provides the first opportunity to
measure the polarization above 230~GHz at angular resolution
sufficient to separate Sgr~A*\ from its surroundings. Its large
bandwidth (2~GHz per sideband), low latitude, and dry site make it far
more sensitive for studies of this southern source than the 230~GHz
observations of \citet{BowerE03,BowerE05}, which were made with the
Berkeley-Illinois-Maryland Association array at Hat Creek,
California. Given the sensitivity and the large (10~GHz) sideband
separation, 340~GHz polarimetry with the SMA should improve limits on
the RM, and future 230~GHz polarimetry may measure it directly. These
advantages also apply to measurements of variability in total
intensity and polarization, and of circular polarization. Here we
present the first high angular resolution observations of the
submillimeter polarization of Sgr~A*, using the newly dedicated SMA and
its polarimetry system. Our observations and reduction are discussed
in \S~\ref{s-obs}, the data and their relation to previous polarimetry
in this source in \S~\ref{s-res}, and the implications of these new
results in \S~\ref{s-disc}. We offer concluding remarks in
\S~\ref{s-concl}.
\section{Observations}
\label{s-obs}
{Sgr~A*} was observed on several nights in 2004 using the Submillimeter
Array\footnote{The Submillimeter Array is a joint project between the
Smithsonian Astrophysical Observatory and the Academia Sinica
Institute of Astronomy and Astrophysics, and is funded by the
Smithsonian Institution and the Academia Sinica.}
\citep{Blundell04,HoMoranLo04}. The observing dates, zenith opacity,
number of antennas used in the reduction, and on-source time are given
in Table~\ref{t-obslog}. The local oscillators were tuned to a
frequency of 336.7~GHz, centering the 2~GHz wide upper and lower
sidebands (USB and LSB) on 341.7 and 331.7~GHz, respectively. This
frequency choice avoided strong spectral lines and provided a
reasonable match to the frequency response of the SMA polarimetry
hardware, as discussed below. Our Sgr~A*\ tracks generally included
source elevations between 20\degr\ and 41\degr (transit), a period
of seven hours, although weather, calibration, and technical problems
caused variations in the coverage. In the SMA ``Compact-North''
configuration we sampled projected baselines between 8 and
135~k$\lambda$. The average synthesized beam was approximately
1\farcs4$\times$2\farcs2. According to the estimate in
\citet{AitkenE00}, polarized emission within the 14\arcsec\
beam of the JCMT at 350~GHz contributes 100~mJy of polarized flux
density. With a beam smaller by a factor of 60, and reduced
sensitivity to large-scale emission, we expect this contaminant to be
negligible in our data.
\begin{deluxetable}{ccccccc}
\tablecolumns{7}
\tablewidth{0pt}
\tablecaption{Observing Parameters\label{t-obslog}}
\tablehead{Date & \hspace{.2cm} & $\tau_{337}$\tablenotemark{a} & \hspace{.15cm} & N$_\mathrm{ant}$ &
t$_\mathrm{int}$ (min) }
\startdata
2004 May 25 & \hspace{.2cm} & 0.16 & & 7 & 100 \\
2004 May 26 & & 0.28 & \hspace{.15cm} & 6 & 160 \\
2004 July 5\hfill\hfill & & 0.11 & & 7 & 160 \\
2004 July 6\hfill\hfill & & 0.15 & & 7 & 180 \\
2004 July 7\hfill\hfill & & 0.29 & & 6 & 170 \\
2004 July 14 & & 0.23 & & 6 & 100 \\
\enddata
\tablenotetext{a}{Mean zenith opacity at the LO frequency of 337~GHz}
\end{deluxetable}
Each SMA antenna was equipped with a single linearly polarized (LP)
feed in each of its three observing bands. Ideally, interferometric
observations of linear polarization are made with dual
circularly-polarized (CP) feeds as they separate the total intensity
(Stokes I) from the linear polarization Stokes parameters (Q and
U). For polarimetry we have converted the 340~GHz LP feeds to left-
and right-circularly polarized (LCP and RCP) feeds using positionable
quartz and sapphire quarter-wave plates. The polarization handedness
was selected by switching the angular position of the waveplate
crystal axes between two positions $\pm45^\circ$ from the polarization
angle of the receiver. Although we could only measure a single
polarization in each antenna at a given time, we sampled all four
polarized correlations (LL, LR, RL, RR) on each baseline by switching
antennas between LCP and RCP in period-16 Walsh function patterns
\citep[e.g.,][]{Rao99}. For 20-second integrations, a full cycle
required just under seven minutes. These observations were made during
the commissioning phase of the SMA polarimetry hardware; details of
this instrument can be found in Marrone (2005, in preparation).
The conversion of LP to CP was not perfect, but we calibrated the
(frequency-dependent) leakage of cross-handed polarization into each
CP state of each antenna in order to properly determine source
polarizations. We used a long observation of a polarized point source
(in this case the quasar 3C279) to simultaneously solve for the quasar
polarization and leakage terms \citep[e.g.,][]{SaultE96}. This
polarization calibration was performed twice, on May 25 and July 14,
yielding consistent leakages. The derived polarization leakages were
at or below 3\% in the USB and 5\% in the LSB, with the exception of
antenna 3, which used a sapphire waveplate with different frequency
response and poorer performance (6\% LSB leakage) than the other
waveplates. Theoretical considerations of our design suggest that the
real components of the L$\rightarrow$R and R$\rightarrow$L leakages
should be identical for a given waveplate at a given frequency, and a
comparison of the results on the two nights (a total of four
measurements of each real component) show that the rms variations in
the measured leakage terms were below 1\% for all antennas except
antenna 7. One measured leakage on July 14 was responsible for this
antenna's large rms, and because of the disagreement between the real
part of the L$\rightarrow$R and R$\rightarrow$L leakages we know that
this measurement was in error. Using the same comparison on the other
antennas we found that on average the solutions for this date were of
poorer quality, probably due to the difference in
weather. Accordingly, we adopted the May 25 leakage values for all
dates, although that required that we not use antenna 8, which was
absent from that calibration track. Errors in the leakage calibration
produce effects of varying importance, as outlined in
\citet{SaultE96}, the most important, for our purposes, are the
contamination of Q and U by Stokes I due to errors in the
determination of the leakage calibrator polarization. We have examined
this effect by comparing the Q and U fractions across sidebands on the
high signal-to-noise 3C279 data sets of May 25 and July 14; the two
sidebands should give identical measurements of Q and U from the
source, and differences can be ascribed to noise in the images and the
difference of the independent errors in the leakage solutions in the
two sidebands. With this procedure we found no inter-sideband
differences that were consistent across the two data sets, and the
differences present were consistent with the noise level, roughly
0.3\% or smaller. Because an important part of our analysis is the
comparison of position angles across sidebands, we had to ensure that
the calibration did not create a position angle offset between the
sidebands. Fortunately, although leakage errors could introduce
spurious Stokes Q or U polarization, the phase difference between the
RCP and LCP feeds, corresponding to a rotation of the sky
polarization, is identically zero because each pair of CP feeds is in
reality a single LP feed looking through both crystal axes of the same
waveplate. Therefore, the only way to create a relative position angle
difference between the sidebands would be through the leakage errors
and the resulting contamination of Q and U, an effect which appears to
be small in our data.
The flux density scale was derived from observations of Neptune on all
nights except May 25 and July 14. We expect the absolute calibration
to be accurate to about 25\% on these nights. The May 25 flux density
scale was transferred from three quasars that were also observed on
May 26; these appeared to have the same relative flux densities on
both nights to better than 10\%, consistent with the overall uncertainty
on that night, so we do not expect that the May 25 flux densities are
any more uncertain than the others. The July 14 data were obtained in
an engineering track primarily aimed at obtaining a second
polarization calibration, so only three sources are present (Sgr~A*,
3C279, and 1743-038). Fortunately, 1743-038 has been very stable
during more than two years of monitoring observations with the SMA (an
rms flux density variation of only 20\% in that time), with even
smaller ($<10$\%) variations observed from July 5$-$7, so we have used
it as our flux density standard for the final track.
The data were averaged over the 7 minute polarization cycle to
simulate simultaneous measurement of all four polarized visibilities,
then phase self-calibrated using the LL and RR visibilities. Quasars
were interleaved into the observations of Sgr~A*\ to allow variability
monitoring and independent gain calibration. Transferring gains from
the quasars, rather than self-calibrating, generally resulted in
slightly lower signal-to-noise but did not change the polarization. We
attribute the increased noise ($\sim20$\%) to the 16\degr$-$40\degr\
angular separation between Sgr~A*\ and the quasars. Following
calibration, each sideband was separately imaged in Stokes I, Q, U,
and V, using only baselines longer than 20~k$\lambda$, and then
cleaned. Sample Stokes images are shown in Figure~\ref{f-stokes}. On
July 14, due to poorer coverage in the {\it uv} plane in the short
track, we increased the cut to 30~k$\lambda$. Flux densities were extracted
from the center pixel of each image, and these are listed in
Table~\ref{t-pol}. We also examined the polarization by fitting point
sources to the central parts of the images; the point source flux
densities matched well with those obtained from the central pixel when
the signal was well above the noise, but the point source positions
and peak flux densities became erratic for low signal-to-noise images
(most Stokes Q and V images). Table~\ref{t-pol} also includes the
polarization fraction ($m$), which has been corrected for the
noise bias \citep[through quadrature subtraction of a $1\sigma$ noise
contribution, e.g.,][]{WardleKronberg74}, and the electric vector
position angle ($\chi$, determined as
$2\chi=\mathrm{tan}^{-1}\frac{\mathrm{U}}{\mathrm{Q}}$).
\begin{figure*}
\plotone{f1.eps}
\caption{Sample Stokes images of Sgr~A*, from the USB data (341.7~GHz)
of 2004 May 25. The synthesized beam is
2\farcs0$\times$1\farcs2. Clockwise from top left are: I, Q, V,
U. Contours are spaced by geometrical factors of $\sqrt{2}$. For
Stokes I they are drawn at -4.2 (absent), -3, 3, 4.2, 6, 8.5, 12, 17,
24, 34, 48, 68, 96, and 136 times the 25~mJy per beam noise in the
image, for Q, U and V at -12, -8.5, -6, -4.2, -3, 3, 4.2, and 6
(absent in all three images) times the 15~mJy per beam rms noise in
the Q and U images (0.4\% of I$_{peak}$). The V contours match the Q
and U contours to highlight the increased noise introduced by the
contamination of V by I, which is due to relative gain variations
between LCP and RCP.}
\label{f-stokes}
\end{figure*}
\begin{deluxetable*}{ccccccc}
\tiny
\tablecolumns{7}
\tablewidth{0pt}
\tablecaption{340~GHz Polarization Measurements of {Sgr~A*}\label{t-pol}}
\tablehead{Date & I & Q & U & V & $m$ & $\chi$ \\
& (Jy) & (mJy) & (mJy) & (mJy) & (\%) & (degrees)}
\startdata
2004 May 25\hfill\hfill & & & & & & \\
\hfill USB & 3.79 $\pm$ 0.03 & 9 $\pm$ 15 & -244 $\pm$ 15 & -5 $\pm$ 22 & 6.43 $\pm$ 0.39 & 136.1 $\pm$\hfill 1.7 \\
\hfill LSB & 3.79 $\pm$ 0.02 & 13 $\pm$ 17 & -201 $\pm$ 17 & -9 $\pm$ 21 & 5.28 $\pm$ 0.45 & 136.8 $\pm$\hfill 2.4 \\
\hfill Both & 3.79 $\pm$ 0.02 & 13 $\pm$ 11 & -230 $\pm$ 11 & -5 $\pm$ 17 & 6.07 $\pm$ 0.28 & 136.7 $\pm$\hfill 1.3 \\
2004 May 26\hfill\hfill & & & & & & \\
\hfill USB & 3.19 $\pm$ 0.03 & 145 $\pm$ 20 & -97 $\pm$ 20 & -14 $\pm$ 21 & 5.43 $\pm$ 0.63 & 163.0 $\pm$\hfill 3.3 \\
\hfill LSB & 3.11 $\pm$ 0.02 & 104 $\pm$ 18 & -138 $\pm$ 18 & -10 $\pm$ 22 & 5.53 $\pm$ 0.58 & 153.5 $\pm$\hfill 3.0 \\
\hfill Both & 3.16 $\pm$ 0.02 & 118 $\pm$ 13 & -138 $\pm$ 13 & -17 $\pm$ 19 & 5.75 $\pm$ 0.43 & 155.3 $\pm$\hfill 2.1 \\
2004 July 5\hfill\hfill & & & & & & \\
\hfill USB & 3.23 $\pm$ 0.04 & 42 $\pm$ 14 & -267 $\pm$ 14 & -37 $\pm$ 17 & 8.35 $\pm$ 0.44 & 139.5 $\pm$\hfill 1.5 \\
\hfill LSB & 3.13 $\pm$ 0.02 & 41 $\pm$ 12 & -273 $\pm$ 12 & -19 $\pm$ 17 & 8.84 $\pm$ 0.38 & 139.3 $\pm$\hfill 1.2 \\
\hfill Both & 3.20 $\pm$ 0.02 & 42 $\pm$ 10 & -270 $\pm$ 10 & -38 $\pm$ 13 & 8.52 $\pm$ 0.31 & 139.5 $\pm$\hfill 1.0 \\
2004 July 6\hfill\hfill & & & & & & \\
\hfill USB & 3.19 $\pm$ 0.02 & 58 $\pm$ 21 & -169 $\pm$ 21 & -15 $\pm$ 25 & 5.56 $\pm$ 0.65 & 144.4 $\pm$\hfill 3.3 \\
\hfill LSB & 3.15 $\pm$ 0.03 & 29 $\pm$ 18 & -164 $\pm$ 18 & -16 $\pm$ 22 & 5.27 $\pm$ 0.56 & 140.1 $\pm$\hfill 3.0 \\
\hfill Both & 3.18 $\pm$ 0.02 & 52 $\pm$ 15 & -177 $\pm$ 15 & -18 $\pm$ 19 & 5.78 $\pm$ 0.49 & 143.2 $\pm$\hfill 2.4 \\
2004 July 7\hfill\hfill & & & & & & \\
\hfill USB & 2.71 $\pm$ 0.03 & 38 $\pm$ 22 & -35 $\pm$ 22 & -8 $\pm$ 29 & 1.72 $\pm$ 0.82 & 158.8 $\pm$ 13.7 \\
\hfill LSB & 2.78 $\pm$ 0.04 & 31 $\pm$ 22 & -67 $\pm$ 22 & -13 $\pm$ 38 & 2.53 $\pm$ 0.80 & 147.6 $\pm$\hfill 9.0 \\
\hfill Both & 2.75 $\pm$ 0.03 & 44 $\pm$ 17 & -49 $\pm$ 17 & -17 $\pm$ 25 & 2.32 $\pm$ 0.61 & 156.1 $\pm$\hfill 7.5 \\
2004 July 14\hfill\hfill & & & & & & \\
\hfill USB & 3.00 $\pm$ 0.03 & 37 $\pm$ 27 & -243 $\pm$ 27 & 14 $\pm$ 32 & 8.14 $\pm$ 0.91 & 139.3 $\pm$\hfill 3.2 \\
\hfill LSB & 3.00 $\pm$ 0.03 & 29 $\pm$ 19 & -175 $\pm$ 19 & -17 $\pm$ 25 & 5.87 $\pm$ 0.64 & 139.7 $\pm$\hfill 3.1 \\
\hfill Both & 3.02 $\pm$ 0.03 & 75 $\pm$ 16 & -236 $\pm$ 16 & -15 $\pm$ 24 & 8.17 $\pm$ 0.55 & 143.8 $\pm$\hfill 1.9 \\
All days\hfill\hfill & & & & & & \\
\hfill USB & 3.33 $\pm$ 0.02 & 57 $\pm$ 10 & -197 $\pm$ 10 & -9 $\pm$ 15 & 6.15 $\pm$ 0.29 & 143.1 $\pm$\hfill 1.3 \\
\hfill LSB & 3.29 $\pm$ 0.02 & 49 $\pm$ 10 & -202 $\pm$ 10 & -8 $\pm$ 13 & 6.32 $\pm$ 0.29 & 141.8 $\pm$\hfill 1.3 \\
\hfill Both & 3.31 $\pm$ 0.02 & 59 $\pm$ 7 & -204 $\pm$ 7 & -17 $\pm$ 11 & 6.39 $\pm$ 0.23 & 143.1 $\pm$\hfill 1.0 \\
\enddata
\tablecomments{Errors in the flux density columns are from the image
rms only, they do not include the 25\% absolute calibration
uncertainty, which applies equally to all flux densities and does not
affect the $m$ or $\chi$ columns.}
\normalsize
\end{deluxetable*}
\section{Results}
\label{s-res}
\subsection{Linear Polarization}
The polarization fraction and position angle for each sideband on each
night are plotted in Figure~\ref{f-pol}. It can be seen from the
figure and the data in Table~\ref{t-pol} that we have clear detections
of the linear polarization in both sidebands on all nights. Among the
six nights of our observations, July 7 stands out for its low
polarization fraction, around 2\%. The polarization was only detected
at the $2-3\sigma$ level in each sideband, so the polarization
position angle was poorly constrained. This is the lowest linear
polarization fraction measured at or above 150~GHz, the lowest
frequency where polarization has been detected. The weather on this
night was the poorest of all the tracks, but only marginally worse
than May 26 which did not show an unusually low polarization. Other
sources in the July 7 track with measurable polarization, such as
3C279, did not show a significantly lower polarization than on other
nights, as one might have expected from a systematic problem in that
track. An obvious systematic error would be a substantial change in
the leakages with respect to previous nights; this would most easily
be caused by large changes to the alignment of the polarization
hardware. However, the hardware was not moved between installation on
July 5 and removal after the July 7 track, and the July 5 and 6 tracks
show substantially larger polarization, so this possibility seems very
unlikely. Moreover, because the leakages measured on July 14 are
consistent with the May 25 leakages, as discussed in \S~\ref{s-obs},
any change between July 6 and 7 would have to have been reversed when
the hardware was reinstalled on July 14. This low polarization
fraction, along with the unusually high polarization two nights
before, clearly demonstrates that the polarization fraction is
variable. Moreover, the polarization variations are present both in
the polarization fraction and the polarized flux density, even after
accounting for the 25\% uncertainty in the overall flux density scale,
and are not merely the result of a constant polarized emission
component with a changing total intensity.
\begin{figure*}
\plotone{f2.eps}
\caption{340~GHz Sgr~A*\ polarization fraction ($m$, {\it upper}) and
position angle ($\chi$, {\it lower}). The USB (black squares) and LSB
(gray triangles) are plotted separately for each night. The two
sidebands are slightly offset in time for clarity, but both sample the
same time interval. The large $\chi$ error bars on day 189 (July 7)
are due to the low polarization signal on that night.}
\label{f-pol}
\end{figure*}
Variability was also observed in the polarization position angle.
Polarization over four of the nights ranged between roughly 137\degr\
and 143\degr, at a weighted average position angle of 139.6\degr. The
position angle determined for May 26 differed significantly from this
range and July 7 had an extremely uncertain position angle due to the
very low polarization fraction. Neither the combined six-night data
set, nor the individual nights showed significant inter-sideband
differences, with the possible exception of May 26. On that night
$\chi_{LSB}-\chi_{USB} = (153.5^\circ\pm3.0^\circ) -
(163.0^\circ\pm3.3^\circ) = -9.5^\circ\pm4.5^\circ$, which is
marginally significant for the quoted errors. As we discussed in
\S~\ref{s-obs}, although it is possible for Stokes I to contaminate Q
and U (which determine $\chi$), this appears to be unimportant in
these data. The 0.3\% limit on this effect is smaller than the Q and
U errors on May 26, which are 0.6\% of Stokes I. Furthermore, any
other systematic source of inter-sideband position angle offsets would
show up equally on all nights, but the six-night average
$\chi_{LSB}-\chi_{USB}$ is 1.3\degr$\pm$1.8\degr, consistent with
zero. The May 26 result is considered further in the context of a
Faraday rotation measure in \S~\ref{s-discRM}.
\subsection{Circular Polarization}
Neither the averaged data nor the individual nights show CP at a level
that is significant. The greatest deviation from zero is
$-38\pm13$~mJy on July 5, corresponding to $-1.2\pm0.4$\%. However, in
addition to the quoted error, which is the measured noise in the
cleaned map, there are well known systematic effects. The MIRIAD
reduction package \citep{miriad} uses linearized equations when
solving for the polarized leakages, ignoring second order terms in the
leakages ($d$) and linear polarization fraction. These terms
contribute a systematic error in Stokes V of the form $\mathrm{I}d^2$
and $md$ \citep{RobertsE94}, which may be of the order of a few tenths
of a percent for our leakages and the polarization of
Sgr~A*. Moreover, the small difference in the sample times of the LL
and RR correlations on a given baseline permit gain differences, due
to weather, pointing, and system changes, to introduce differences
between the LL and RR visibilities that would not be present if these
were actually measured simultaneously (as our reduction
assumes). These gain variations contaminate Stokes V with Stokes I and
make the value of V at the peak of the I map more uncertain than the
map rms would indicate. The average of all six tracks shows
$-0.5\pm0.3$\% CP, consistent with zero, with additional systematic
error of perhaps another 0.3\%. The 0.5\% sum of these errors can be
taken as a limit on any persistent level of CP across the six nights,
and is the most stringent limit yet on CP in Sgr~A*\ above 90~GHz.
\subsection{Intra-day Variability}
\begin{figure*}
\plotone{f3.eps}
\caption{Variability in the total intensity ({\it upper}) and
polarization (fraction and position angle, {\it lower}) of
Sgr~A*\ at 340~GHz. The three nights with the best weather are shown,
as these permit the most accurate determinations of the polarization
variation. The total intensity light curves of the quasar calibrators,
1741$-$038, 1749$+$096, and 1921$-$293, are also shown. In the lower
plots, the binned intervals are demarcated by the horizontal bars on
the polarization fraction points. This polarization fraction is the
double-sideband value. The USB and LSB position angles sample the same
time bin but have been offset slightly from the bin center for
clarity.}
\label{f-plc}
\end{figure*}
Intra-day variability in the total intensity (Stokes I), the
polarization fraction, and position angle are shown in
Figure~\ref{f-plc}. The July 5$-$7 observations were obtained as part
of a coordinated multi-wavelength Sgr~A*\ monitoring program, and the
observed temporal variability in Stokes I on these nights is discussed
in conjunction with results at other wavelengths in
\citet{EckartE05}. In order to prevent antennas with variable
performance from falsely modulating Stokes I, we use only the 5
antennas with the best gain stability for these light curves. Slow
variations in the gain of the other antennas are likely due to
pointing errors. We have reduced the effects of changing spatial
sampling of extended emission by removing the two baselines that
project to less than 24~k$\lambda$\ (angular scales $>$9\arcsec) during the
Sgr~A*\ observations. Further details of the light curve reduction can
be found in \citet{EckartE05}. The variability in the linear
polarization is much harder to measure; with signals one to two orders
of magnitude weaker than Stokes I it is difficult to obtain reliable
results from a subdivided track, and we could not be as selective
about which data to exclude in the hope of removing the imprint of
instrumental variations from the polarization variation. Accordingly,
polarized light curves could not be reliably extracted for May 26 and
July 14 due to poor weather, nor July 7, due to both weather and very
low polarization fraction. The remaining three nights have been
subdivided into two or three segments at boundaries in the Stokes I
curves and the polarization has been extracted as described in
\S~\ref{s-obs}. The large (160 minute) gap on May 25, due to
instrument difficulties, served as one of the boundaries.
A great deal of variability is visible in the Stokes I curve on all
three nights, with the most notable feature being the $\sim1.5$~Jy
difference between the flux densities of the first and second halves
of the May 25 data. No such difference shows up in the light curve of
the calibrator, 1921$-$293, a source at nearly identical declination,
suggesting that this result is not an instrumental artifact. Clear
polarization variability is also measured on May 25 and July 6 in both
$m$ and $\chi$. At all times the position angles in the USB and
LSB are found to be very similar, as was observed in the full track
averages reported above.
\section{Discussion}
\label{s-disc}
\subsection{Rotation Measure}
\label{s-discRM}
The rotation measure associated with a plasma screen located between
the source and observer can be inferred from the measurement of $\chi$
at two frequencies, since it introduces a frequency dependent change
in the position angle given by
\begin{equation}
\chi(\nu) = \chi_0 + \frac{c^2}{\nu^2}\mathrm{RM} ,
\label{e-chi}
\end{equation}
where the RM is given by
\citep[e.g.,][]{GardnerWhiteoak66}
\begin{equation}
\mathrm{RM} = 8.1\times10^5 \int n_e \textit{\textbf{B}}
\cdot d\textit{\textbf{l}}
\label{e-RM}
\end{equation}
for electron density $n_e$ in cm$^{-3}$, path length
$d\textit{\textbf{l}}$ in parsecs, and magnetic field
\textit{\textbf{B}} in Gauss. The greatest obstacle to such a
detection, as previously noted, is the variability in the
polarization, which may prevent polarization measured at different
times from being reliably compared.
The best method for measuring the RM from our data comes from the
observed difference in the simultaneous position angles in the USB and
LSB. Applying equation (\ref{e-chi}) to the two sideband frequencies
of these observations, and for position angles in degrees, we obtain
\begin{equation}
\mathrm{RM} = 3.7\times10^5 \left(\chi_{LSB}-\chi_{USB}\right) .
\label{e-RMa}
\end{equation}
Equation (\ref{e-RMa}) implicitly assumes that the Faraday rotation
occurs outside of the plasma responsible for the polarized
emission. This assumption seems reasonable for Sgr~A*, VLBI
measurements \citep{KrichbaumE98,ShenE05,BowerE04} suggest intrinsic
sizes of $13-24 r_S$ at 215, 86, and 43~GHz, and for reasons described
in \S~\ref{s-mdot} we expect little contribution to the RM inside $300
r_S$. One other potential complication arises if the source
polarization changes with radius and the two frequencies being
compared probe different radii. For our 3\% sideband separation, and
assuming that the polarized submillimeter emission is thermal
synchrotron \citep[as is expected in ADAF models;][]{YuanE03}, we
expect a 5\% opacity difference between our sidebands, while for
non-thermal synchrotron \citep[taking an electron energy spectral
index of 2-3.5, e.g.,][]{MarkoffE01,YuanE03} the difference is
9-12\%. Emission will be contributed from a range of radii around the
$\tau=1$ surfaces, so we would have to postulate a large gradient in
the source polarization to produce a large intrinsic inter-sideband
polarization difference over such a small frequency range. Finally,
the 2~GHz bandwidth at 340~GHz limits the allowed RM to approximately
$2\times10^7$~rad~m$^{-2}$\ if polarization is detected, as this RM would
rotate the polarization by more than a radian across the band and wash
out the signal (bandwidth depolarization). For highly polarized
emission the vector average of the polarization may still be
detectible but the position angles of the two sidebands are very
unlikely to agree in this case. We can therefore ignore the
possibility of full 180\degr\ wraps between sidebands, as a wrap
requires a RM of $7\times10^7$~rad~m$^{-2}$.
As is clear from Table~\ref{t-pol}, we do not see a significant change
in the position angle between the two SMA sidebands on most of the
observing nights (disregarding the uncertain position angle of July
7). In the most sensitive track, July 5, the sideband difference
places a one-sigma limit of $7.1\times10^5$~rad~m$^{-2}$\ on the RM on that
particular night, which is the most sensitive limit to date from
simultaneous interferometric observations. If the full data set is
considered together (i.e., with Stokes images derived from the
ensemble of data), the limit drops by a small amount to
$6.8\times10^5$~rad~m$^{-2}$, although if the RM is varying between
observations this average will not actually represent a measurement of
a RM. It should be noted here that the broadband observations of
\citet{AitkenE00} were able to place a similar limit of approximately
$5\times10^5$~rad~m$^{-2}$\ on the RM in August 1999 because of the large
bandwidth of their 150~GHz bolometer.
The May 26 sideband difference of $-9.5^\circ\pm4.5^\circ$ is possibly
significant, with an inferred rotation measure of
$(-3.5\pm1.7)\times10^6$~rad~m$^{-2}$. If this RM had been present on the
previous night it would have shown up as a similarly large sideband
difference, instead of the observed $0.7^\circ\pm3.1^\circ$,
corresponding to a RM of $(+0.3\pm1.1)\times10^6$~rad~m$^{-2}$. We can check
the large RM by comparing the position angles on May 25 and 26, on the
assumption that the emitted polarization ($\chi_0$ from
eq. [\ref{e-chi}]) is constant over timescales of a few days and
observed position angle changes are due to RM changes. At this
frequency, the relationship between the position angle change
($\Delta\chi$, in degrees) and the RM change is (see
eq. [\ref{e-chi}])
\begin{equation}
\Delta\mathrm{RM} = 2.2\times10^4 \Delta\chi .
\label{e-RMb}
\end{equation}
We observed an increase in the position angle from May 25 to May 26 of
$18.6^\circ\pm2.5^\circ$. If this is not a change in the
intrinsic polarization, it corresponds to an increase in the RM of
$4\times10^5$~rad~m$^{-2}$, inconsistent with the small sideband difference on
May 25 and large difference on May 26. The position angle is 180\degr\
degenerate, however, and a $\chi$ change of
$18.6^\circ-180^\circ=-161.4^\circ$ requires a RM change of
$-3.6\times10^6$~rad~m$^{-2}$, which agrees well with the RM inferred from the
May 26 sideband difference. It is therefore possible that we have
observed a large change in the RM between these two nights, with the May
26 value far in excess of the limits on the other five nights. We
discuss this further in \S~\ref{s-mdot}.
In the existing polarization data at 230 and 340~GHz the position
angle seems to frequently return to the same value. The
\citet{BowerE05} 230~GHz data are clustered around 111\degr\ between
2002 October and 2004 January, while four of our observations at
340~GHz have a mean position angle of 140\degr. Assuming that these
two angles sample the same $\chi_0$ (no source polarization changes
between the two observing periods or observing frequencies), we can
infer a ``quiescent'' RM of $-5.1\times10^5$~rad~m$^{-2}$. This is just below
the RM upper limit from our most sensitive night. If the idea of a
quiescent RM is correct, then the change in the mean 230~GHz position
angle observed between early 2002 \citep{BowerE03} and 2003
\citep{BowerE05} merely reflects a change in this RM by
$-3\times10^5$~rad~m$^{-2}$. This implies that the quiescent RM in early 2002
was around $-8\times10^5$~rad~m$^{-2}$, which is conveniently below the
detection limit of the \citet{BowerE03} observations. If this scenario
is correct, the RM should be detectable by the SMA at 230~GHz, where
it would be observable as a 5\degr\ sideband difference.
\subsection{Accretion Rate Constraints}
\label{s-mdot}
Much of the importance placed on the RM determination stems from its
use as a probe of the accretion rate near the black hole. However, the
interpretation of a RM detection, or limit, in terms of an accretion
rate requires a model for the density and magnetic field in the
accretion flow, as these quantities actually determine the RM through
equation~(\ref{e-RM}).
To estimate the RM predicted for a variety of accretion models we make
several simplifying assumptions. First, we assume a generic picture
with a central emission source surrounded by a roughly spherical
accretion flow. Given the previously mentioned limits on the
millimeter size of Sgr~A*, we could also accomodate models where the
observed 340~GHz emission arises in a small jet component, as the jet
would have to lie within $\sim10r_S$ of the black hole, and would
effectively be a central emission source as seen from a Faraday screen
tens to hundreds of $r_S$ further out. We characterize the radial
density profile, $n(r)$, as a power law,
\begin{equation}
n(r)=n_0(r/r_S)^{-\beta} ,
\label{e-nr}
\end{equation}
where $r_S=2GM_{BH}/c^2$ is the Schwarzschild radius of Sgr~A*\
($10^{12}$~cm for $M_{BH} = 3.5\times10^6M_{\sun}$), and $n_0$ is
the density at this radius. In the case of free-falling gas we have
$\dot M(r)\propto r^p$ with $\beta=3/2-p$, as in
\citet{BlandBegel99}. For spherical accretion \citep{Bondi52} or
Advection-Dominated Accretion Flows \citep[ADAF;][]{NaraYi94} we have
$\beta=3/2$, while for a Convection-Dominated Accretion Flow
\citep[CDAF;][]{QuatGruz00-CDAF}, formally an $\dot M = 0$ limiting case of
convection-frustrated accretion, we have $\beta=1/2$. Intermediate
values are also possible: the best-fit radiatively-inefficient
accretion model in \citet{YuanE03} has $\beta=0.8$, and accretion
flow simulations \citep[e.g.,][]{PenE03} typically produce values
between 1/2 and 1 \citep{Quataert03}. We take the ADAF and CDAF values
as bounds on $\beta$ (i.e., 1/2 to 3/2).
Rather than using a separate parameter to describe the magnetic field
profile, we tie it to the density by assuming equipartition between
magnetic, kinetic, and gravitational energy, as many other modelers
have done \citep[e.g.,][]{Melia92}. For pure hydrogen gas, with the use
of equation (\ref{e-nr}), we obtain
\begin{equation}
B(r)=\sqrt{4\pi c^2 m_H n_o}
\left(\frac{r}{r_S}\right)^{-\left(\beta+1\right)/2} .
\label{e-Br}
\end{equation}
We additionally assume that the magnetic field contains no reversals
along the line of sight and is entirely radial, which should
contribute only a small error unless the field is very nearly
toroidal. The former simplification is a good approximation for
strongly peaked RM vs r profiles (large $\beta$), where only a small
radial range contributes significantly. For smaller $\beta$ and many
field reversals, the effective field will only drop as the square root
of the number of reversals.
In the Sgr~A*\ accretion flow we expect that the electron temperature
($T_e$) will rise to smaller radii, eventually bringing the electrons
to relativistic temperatures ($T_e > 6\times10^9 \mathrm{K} = m_e
c^2/k$) at some radius $r_{in}$. The RM contribution from
relativistic electrons is suppressed (by as much as
log($\gamma$)/2$\gamma^2$ for Lorentz factor $\gamma$ in the
ultra-relativistic thermal plasma limit; \citealt{QuatGruz00-LP}), so
we approximate this effect by truncating the RM integration at
$r_{in}$ and by treating $r_{in}$ as a variable. From the density
profile, and assuming that gas at $r_{in}$ is in free-fall, we can
determine a mass flux across the $r=r_{in}$ surface
\begin{eqnarray}
\dot M_{in} &=& 4\pi r_{in}^2 m_H n\left(r_{in}\right) v\left(r_{in}\right) \nonumber \\
&=& 4\pi r_S^2 m_H n_0 c \left(r_{in}/r_S\right)^{3/2-\beta}.
\label{e-mdot}
\end{eqnarray}
This equation does not require that the density profile be followed
down to $r=r_S$, $n_0=n\left(r_S\right)$ is merely a convenient
quatity to normalize the power-law density relation we are assuming
for larger radii. The mass flux at $r_{in}$ ($\dot M_{in}$) can be
taken to be an upper limit on the accretion rate at $r_S$, but the
true rate of accretion onto the black hole could be lower if the
loosely bound plasma falling from $r_{in}$ escapes as a wind or jet.
Substituting equations (\ref{e-nr}), (\ref{e-Br}), and (\ref{e-mdot})
into equation (\ref{e-RM}), and converting $\dot M_{in}$ to units of
$M_\sun$~yr$^{-1}$\ and $r$ to $r_S$, we obtain
\begin{eqnarray}
RM &=& 3.4\times10^{19}
\left(\frac{M_{BH}}{3.5\times10^6 M_\sun}\right)^{-2} \times \nonumber \\
& & r_{in}^{(6\beta-9)/4} \dot M_{in}^{3/2}
\int_{r_{in}}^{r_{out}} r^{-\left(3\beta+1\right)/2} dr .
\label{e-dRM}
\end{eqnarray}
Integrating and simplfying yields
\begin{eqnarray}
RM &=& 3.4\times10^{19}
\left(1-\left(r_{out}/r_{in}\right)^{-\left(3\beta-1\right)/2}\right)
\times \nonumber \\
& & \left(\frac{M_{BH}}{3.5\times10^6 M_\sun}\right)^{-2}
\left(\frac{2}{3\beta-1}\right) r_{in}^{7/4}\dot M_{in}^{3/2} .
\label{e-RM-mdot}
\end{eqnarray}
To obtain an RM given $\beta$ and $\dot M_{in}$ we must also choose
$r_{in}$ and $r_{out}$. The inner radius will vary by model, but it is
typically around $300r_S$ \citep[e.g.,][]{YuanE03}. For these
calculations we consider values of $r_{in}$ from 300 to $3r_S$ in
order to account for variations among models and to allow for the
possibility that the electrons do not become highly relativistic
interior to $r_{in}$, in which case the RM would not be strongly
suppressed. The outer radius depends on the coherence of the radial
field. We examine two cases: a fully coherent field
($r_{out}\approx\infty$), and a field that persists for a factor of
three in radius from $r_{in}$.
\begin{figure}
\plotone{f4.eps}
\caption{Accretion rate limits imposed by the rotation measure limit
of $7\times10^5$~rad~m$^{-2}$\ as a function of the density power law, given
the accretion model described in \S~\ref{s-mdot}. The accretion rate
plotted here is measured at the radius where the electrons become
relativistic, $r_{in}$; extrapolation to the black hole event horizon
is discussed in \S~\ref{s-mdot}. Two sets of curves are plotted (see
eq. [\ref{e-RM-mdot}]): three for a magnetic field that is coherent to
large radius (solid lines) and three for a field that is coherent over
a smaller range ($r_{out}/r_{in}=3$, dashed lines). Within each set,
the thickest line is $r_{in} = 300r_S$, then $30r_S$, and finally
$3r_S$.}
\label{f-mdot}
\end{figure}
Figure~\ref{f-mdot} shows the accretion rate limits imposed by our RM
limit of $7\times10^5$~rad~m$^{-2}$, based on the model described above. From
the two choices of $r_{out}$ we see that the effect of the magnetic
field coherence is larger at small $\beta$. As mentioned before, for
steep density profiles (large $\beta$) we expect that only a small
range in radius around $r_{in}$ contributes to the RM, making the
inferred accretion rate limit insensitive to the field coherence
length. If we assume that the density profile follows equation
(\ref{e-nr}) down to $r=r_S$, our model imposes accretion rate limits
that are a factor of $\dot M\left(r_S\right) / \dot M_{in} =
(r_{in}/r_S)^{\beta-3/2}$ lower than those in Figure~\ref{f-mdot}, but
the transition to supersonic flow makes this density extrapolation
uncertain. However, in cases like the basic ADAF model
\citep{NaraYi95} where the electron temperature ceases to rise at
small radii and the electrons are only marginally relativistic,
integration to smaller radii (the lower sets of curves) may set more
relevant (and lower) accretion rate limits. In fact, taking $\beta =
3/2$ and $r_{in} = 30r_S$, we roughly have the ADAF/Bondi model used in
\citet{QuatGruz00-LP}, and reproduce their $\dot M$ limit of
$10^{-7}$~$M_\sun$~yr$^{-1}$. The high and low-$\beta$ limits are similar, but the
field coherence is a larger concern for shallow profiles. Since the
prototype for a low-$\beta$ model is a highly convective flow we may
expect a tangled field, but in this case the accretion rate limit
(proportional to $B^{-2/3}$) will increase only as $\dot M \propto
N^{1/3}$ for $N$ field reversals. In summary, the figure shows that
for any choice of density profile, the maximum allowed accretion rate
is $10^{-6}$~$M_\sun$~yr$^{-1}$, and may be much lower. This is an order of
magnitude below the gas capture rate of $10^{-5}$~$M_\sun$~yr$^{-1}$\ inferred
from X-ray observations \citep{BaganoffE03,YuanE03} and from
simulations of stellar winds in the Galactic Center
\citep[e.g.,][]{Quataert04,CuadraE05}. It is therefore likely that
there is substantial mass lost between the gas capture at
$r\sim10^5r_S$ and the event horizon.
Finally, this model of the accretion flow can be used to examine the
proposed $-3.5\times10^6$~rad~m$^{-2}$\ RM from May 26
(\S~\ref{s-discRM}). This RM would require a change of more than
$2\times10^6$~rad~m$^{-2}$\ between consecutive nights. This is very large
compared to the RM changes implied by other position angle changes
(again assuming that the source polarization remains constant).
Based on the four other nights
with strong polarization detections, all of which have position angles
near to 140\degr, the peak-to-peak $\chi$ change corresponds to an RM
change of $1.5\times10^5$~rad~m$^{-2}$\ and the rms variation is only
$5\times10^4$~rad~m$^{-2}$. The largest change on similar (day to week)
timescales observed at 230~GHz is $3\times10^5$~rad~m$^{-2}$\ \citep[between
2003 December 27 and 2004 January 5;][]{BowerE05}. The longer
timescale 230~GHz position angle changes and the difference between
our position angles and the \citet{AitkenE00} 350~GHz position angle
(reinterpreted as described in \S~\ref{s-LPvar} or otherwise) also
correspond to RM changes of a few$\times10^5$~rad~m$^{-2}$. We expect that
these variations are not more than order unity fractional RM changes,
so they are all quite consistent with our inferred
$-5\times10^5$~rad~m$^{-2}$\ quiescent RM from \S~\ref{s-discRM}. The May 26
RM would then correspond to a factor of 7 increase in the density or
line of sight magnetic field. Such a change is difficult to accomplish
with any density profile, but is particularly difficult for small
$\beta$ where the entire line of sight contributes significantly to
the RM. If the fluctuation is real it suggests a steep density
profile, as the associated density/field change should not be extended
over decades of radius. Unless such an event is observed again in
future observations, the more likely interpretations appear to be that
the position angle change from May 25 represents a RM fluctuation of
$4\times10^5$~rad~m$^{-2}$\ observed between consecutive nights or a
transient change in the source polarization, and the May
26 difference in the USB and LSB position angles is merely a $2\sigma$
measurement noise event.
\subsection{Linear Polarization and Variability}
\label{s-LPvar}
Our 340~GHz observations show a typical polarization fraction of
6.4\%, with a range of 2.3$-$8.5\%, and an rms variation of
2.0\%. This is comparable to the $\sim$7.5\% mean, $4.6-13.6$\% range,
and 2.2\% rms measured at 230~GHz by \citet{BowerE03,BowerE05}. The
range of observed polarization is lower at 340~GHz than it is at
230~GHz, and the mean is slightly lower as well. It is difficult to
explain a lower observed polarization fraction (and comparable
variability) at higher frequencies with beam depolarization models
\citep{Tribble91}, as Faraday rotation and the resulting dispersion in
polarization directions decreases with increasing frequency. If the
polarization fraction decrease is intrinsic to the source and not
generated in the propagation medium, it suggests that the magnetic
field becomes increasingly disordered at smaller radii, as these
observations should probe slightly smaller radii than the 230~GHz
data. But across only 0.2 decades in frequency we expect little change
in intrinsic polarization, so the difference, if present, may be best
explained by time variability in the source polarization. To resolve this question, simultaneous
or nearly-coincident polarimetry at multiple frequencies with
interferometer resolution is clearly desirable.
\citet{BowerE05} used the apparent stability of the 230~GHz
polarization fraction to argue that the observed variations in the
230~GHz polarization position angle were more likely to be the result
of changes in the rotation measure than due to intrinsic source
changes. While our results do not refute this conclusion, they
demonstrate that the polarization fraction is not stable, even over a
single night. Note that two substantial excursions in the 230~GHz
polarization fraction, one of which is labeled an ``outlier'' in
\citet{BowerE05}, probably represent real variations similar to those
seen here, but have lower significance because of the poorer
sensitivity of their instrument.
The polarization fraction presented here is considerably lower than
those measured in 1999 by \citet{AitkenE00}: $13^{+10}_{-4}$\% and
$22^{+25}_{-9}$\% at 350 and 400~GHz, respectively. However, to
determine the flux density of Sgr~A*\ \citet{AitkenE00} had to correct
for the contamination from dust and free-free emission in their large
primary beam (14\arcsec$-$12\farcs5 at the highest frequencies), and
it is possible that they over-corrected for the dust emission, which
would make the polarized component appear to be a larger fraction of
the total flux density of Sgr~A*. There is some support for this
possibility from their measured flux densities: Sgr~A*\ was found to be
only 2.3 and 1~Jy at 350 and 400~GHz, while our data (see
Table~\ref{t-pol}) and previous measurements between 300 and 400~GHz
have found higher values of $2.6-3.8$~Jy
\citep{ZylkaE95,SerabynE97,PiercePriceE00}. If we assume that their
350~GHz data are well calibrated (the 400~GHz calibration is more
uncertain) and assume our 3.3~Jy flux density for Sgr~A*, we can
re-derive the intrinsic polarization of Sgr~A*\ using their Stokes Q
and U decomposition method, and find a polarization of 9\% at
158\degr. The polarization fraction drops further as the assumed flux
density for Sgr~A*\ is increased, reaching 7.6\% for 3.8~Jy. These
values are within the polarization fraction variations we observe; one
might expect that well calibrated 400~GHz measurements could be
interpreted similarly and that the polarization fraction need not rise
steeply to high frequencies. In arriving at a flux density of 2.3~Jy
for Sgr~A*, \citet{AitkenE00} estimated the dust emission in their
central pixel from the average of the surrounding pixels, so by
increasing the contribution from Sgr~A*\ we are also suggesting that
there is a deficit of dust emission in the central 14\arcsec at
350~GHz. Unfortunately, our observations are poorly sampled at short
spacings, but the available visibilities shortward of 20~k$\lambda$\ show
little excess over the point source flux density, consistent with such
a central hole in the dust emission. The existence of this hole
requires further confirmation, as could be achieved through
simultaneous single-aperture and interferometer observations; our
circumstantial evidence could be equally well explained if Sgr~A*\ had
a higher polarization fraction and lower flux density in 1999 (at the
time of the \citet{AitkenE00} measurement) and if the emission in the
central 30\arcsec\ is distributed smoothly on scales smaller than
10\arcsec.
We observe variability on inter-night and intra-day intervals, in both
the polarization and total intensity. The single-night flux densities
we measure fall within the range of previous observations, and the rms
variation of 0.3~Jy, or 10\%, matches the recent results of
\citet{MauerhanE05} at 100~GHz. Within nights, the Stokes I light
curves in Figure~\ref{f-plc} show unambiguous variations on timescales
of hours, reminiscent of those seen at 100 and 140~GHz by
\citet{MiyazakiE04} and \citet{MauerhanE05}. This is slower than the
variations seen in the near-infrared and X-ray
\citep[e.g.,][]{BaganoffE01,GenzelE03}, which seem to vary on hour
timescales, with some features requiring only minutes. These slow
changes suggest that opacity is obscuring our view of the very inner
regions of the accretion flow, regions unobscured at NIR/X-ray
wavelengths, even at 340~GHz. At slightly higher frequencies the inner
flow may become visible, although many estimates of the optically-thin
transition frequency place it at or above 1~THz, a frequency that is
difficult to access from the ground. It should be possible to search
for the transition to optically thin emission using the change in the
variability timescale; the more frequently proposed technique of
looking for the turnover in the spectrum relies on precise flux
density calibration at high frequencies, which is problematic because
of contaminating emission in single-aperture beams and lack of
unresolved calibrators in interferometers. A few instruments may be
able to make these difficult observations before ALMA: the SMA, or
perhaps SCUBA \citep{SCUBA} on the JCMT at 650~GHz, and SHARC~II
\citep{SHARCII} on the CSO at 650 or 850~GHz.
The intra-day variations in the linear polarization shown in
Figure~\ref{f-plc} are the first linear polarization changes observed
on intervals of hours rather than days. The three nights with
time-resolved polarization measurements do not demonstrate a clear
relationship between Stokes I and the polarization. For example, May
25 shows a very strong flare in I with $m$ very close to our
average values, followed in the second half of the track by a lower I
and a below average $m$. July 5 has the highest $m$ of our six
nights, along with 20\% modulation in I, but the polarization fraction
is not modulated significantly with the total intensity. Finally, on
July 6 we see below average $m$ in a period of high I and above
average $m$ with low I, the inverse of the relationship seen on
May 25. That the polarization fraction may vary in multiple ways
during flares in the total intensity could suggest that there are
multiple mechanisms (of varying polarization) responsible for the
submillimeter Stokes I variability, or that the I and $m$ changes
are not closely related. Diverse flare mechanisms could be expected to
show different spectra at shorter wavelengths, so simultaneous
infrared and X-ray data may be useful. However, based on the
infrequency of infrared and X-ray flares \citep{EckartE04} and the
lack of coincident activity in these bands during the SMA observations
on July 6 and 7 \citep{EckartE05}, it seems that the small changes we
observe in the submillimeter are often imperceptible at shorter
wavelengths. Therefore, the best way to determine whether the
polarization changes are internal or external may be to increase the
time resolution in the polarization light curves. In these data we
observe $m$ changes on the shortest interval we can measure,
around three hours (on July 6). This is close to the variability
timescale observed in the total intensity, which suggests that given
better time resolution we may see that the I and $m$ changes have
similar temporal characteristics and therefore arise from the same
processes.
The $m$ and $\chi$ curves seem to show more coordinated behavior
than the total intensity and polarization do. Of the seven sub-night
intervals plotted in Figure~\ref{f-plc}, five show position angles
close to the observed quiescent $\chi$ of 140\degr. Only in the two
intervals with the lowest polarization, on May 25 and July 6, does
$\chi$ deviate from this value, and if the deviations are caused by RM
changes then both would represent increases in the RM. None of the
intervals provide evidence for a RM through inter-sideband $\chi$
differences, but the largest $\chi$ change between intervals,
$-20.7^\circ\pm3.8^\circ$ on July 6, only requires a RM decrease of
$5\times10^5$~rad~m$^{-2}$, still below our detection limits. Here again we
face the question of whether the source polarization or an external
process is responsible for the variability we see. It is possible to
explain the $\chi$ changes with a two-component source, where the
dominant polarization component is polarized close to the quiescent
polarization direction and variable in amplitude while the weaker
component causes the polarization to deviate from 140\degr\ when the
dominant component weakens. In this case we would expect to see a
correlation between the polarization fraction and the position angle,
something that is not excluded by our data. Such a source model is
naturally identified with emission from a core and jet. A second
model uses a turbulent plasma screen, in addition to the screen
responsible for the putative mean RM (suggested by the difference in
position angle between 230 and 340~GHz), to partially beam depolarize
the emission. The fact that $\chi$ seems to faithfully return to
140\degr\ implies that the source, or the source plus a stable RM
component, is separated from the changes that cause the depolarization
and position angle change. With better time resolution and better
sensitivity to RM it should be possible to distinguish between these
models.
\section{Conclusions}
\label{s-concl}
Using the Submillimeter Array, outfitted with polarization conversion
hardware (Marrone, in preparation), we have made sensitive
measurements of the polarization of Sgr~A*\ at 340~GHz with angular
resolution sufficient to separate the source from the surrounding
contaminating emisssion. Our increased sensitivity has allowed us to
make unequivocal measurements of the variability of the linear
polarization of this source, in both position angle and polarization
fraction. This is the first reliable detection of variation in the
linear polarization fraction. Moreover, we have made the first
detection of linear polarization changes within a night. These changes
do not show an obvious correlation with the observed changes in the
total intensity, possibly because of the coarse time resolution
available at our sensitivity limits. The polarization variations occur
on the shortest intervals we sample, around 3 hours, which is
comparable to the modulation time observed in the total intensity here
and in
\citet{MauerhanE05} at 100~GHz. It is not clear from these data
whether the polarization variability can be best explained by changes
in the source emission or by changes in an external Faraday screen,
but polarization light curves with better time resolution should
clarify the issue. The observed polarization fraction at 340~GHz is
comparable to, and perhaps lower than, that observed at 230~GHz. This
contradicts the polarization spectrum measured from 150 to 400~GHz by
\citet{AitkenE00}, but we show that their polarization fraction at
350~GHz can be brought into agreement with ours through changes in
their correction for dust emission. Whether or not the polarization
fraction rises steeply to high frequency, as predicted by synchrotron
optical depth explanations of the early polarization results
\citep{Agol00,MeliaLiuCoker00}, is no longer clear, but this question
should be resolved by future submillimeter polarimetry at 650~GHz.
We have also measured the circular polarization of this source to be
less than 0.5\% for a time-stable component, and do not detect CP at a
slightly higher level in individual nights. This limit contradicts the
predictions of the turbulence-driven polarization conversion model of
\citet{Beckert03}, which was designed to match the \citet{AitkenE00}
linear polarization results, but can be matched to an earlier version
of the model \citep{BeckertFalcke02} where the CP originated in a
fully turbulent jet.
By comparing the position angles in the two sidebands, we place new
upper limits on the RM allowed for this source. In single nights we
obtain one-sigma upper limits of less than $10^6$~rad~m$^{-2}$\, with our
lowest limit of $7\times10^5$~rad~m$^{-2}$\ coming on July 5. This is
comparable to the lowest limit obtained in any other polarimetric
observations of this source and well below the single-night limits of
other interferometers. We can use a model accretion flow (with energy
equipartition), parameterized only by the density power-law slope and
the radius at which the electrons become relativistic, to convert this
RM to a mass accretion rate limit, and find that for any density slope
Sgr~A*\ is accreting at least an order of magnitude less matter than it
should gravitationally capture based on X-ray measurements
\citep{BaganoffE03}, and may be accreting much less if the density
profile is shallow. This result agrees with earlier interpretations of
polarization detections. We note that the position angle at 340~GHz
seems to show a persistent stable state, much like that observed at
230~GHz \citep{BowerE05}, and we combine these two values to infer a
stable ``quiescent'' RM of $-5\times10^{-5}$~rad~m$^{-2}$. This value is just
below the detection limit of our observations. The possible proximity
of the RM to the detection threshold, the need for more time-resolved
polarimetry, and the potential for coordinated observations with other
wavelengths suggests that expanded SMA capabilities may contribute
considerably more to this study.
\acknowledgements
The authors thank the entire SMA team for their contributions to the
array and to the new polarimetry system. In particular, we acknowledge
the enormous contribution of K. Young for his work on the real-time
software changes essential to these observations. We thank R. Narayan,
E. Quataert, and G. Bower for useful discussions, and J. Greene for
discussions and her help in developing the prototypes of the
polarimetry system. DPM was supported by an NSF Graduate Research
Fellowship. We thank an anonymous referee for a thorough reading and
helpful comments. Finally, we extend our gratitude to the Hawaiian
people, who allow us the privilege of observing from atop the sacred
mountain of Mauna Kea.
\newpage
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.sprite-shrink {
background-position: -807px -307px; width: 20px; height: 20px;
}
.sprite-star2-bordered {
background-position: -692px -327px; width: 16px; height: 20px;
}
.sprite-tip-button-hover {
background-position: -715px -280px; width: 74px; height: 22px;
}
.sprite-profile-tip {
background-position: -888px -327px; width: 156px; height: 20px;
}
.sprite-star-border-gray {
background-position: -1114px -280px; width: 20px; height: 20px;
}
.sprite-new {
background-position: -1217px -307px; width: 36px; height: 20px;
}
.sprite-gray-radius-lt {
background-position: -1247px -182px; width: 10px; height: 10px;
}
.sprite-section-paper {
background-position: -480px -159px; width: 72px; height: 72px;
}
.sprite-checkbox {
background-position: -1200px -280px; width: 20px; height: 20px;
}
.sprite-radio-checked {
background-position: -575px -327px; width: 20px; height: 20px;
}
.sprite-header-nav-active {
background-position: -167px -464px; width: 123px; height: 61px;
}
.sprite-lists {
background-position: -982px -307px; width: 21px; height: 20px;
}
.sprite-material {
background-position: -352px -128px; width: 27px; height: 55px;
}
.sprite-balloon-arrow {
background-position: -943px -347px; width: 11px; height: 8px;
}
.sprite-expand-holder {
background-position: -827px -307px; width: 20px; height: 20px;
}
.sprite-section-csk {
background-position: -480px -231px; width: 72px; height: 72px;
}
.sprite-answer-half-correct {
background-position: -575px -280px; width: 28px; height: 27px;
}
.sprite-flash-tip-rt {
background-position: -683px -347px; width: 15px; height: 14px;
}
.sprite-answer-incorrect-mark {
background-position: -1220px -250px; width: 28px; height: 27px;
}
.sprite-weibo-color-28 {
background-position: -885px -250px; width: 28px; height: 28px;
}
.sprite-level-progress-bg {
background-position: -789px -208px; width: 127px; height: 4px;
}
.sprite-circle-key {
background-position: -1160px -327px; width: 14px; height: 14px;
}
.sprite-section-smart-paper {
background-position: -480px -87px; width: 72px; height: 72px;
}
.sprite-double-arrow-down {
background-position: -1247px -88px; width: 7px; height: 14px;
}
.sprite-question-answer-mark {
background-position: -913px -250px; width: 28px; height: 28px;
}
.sprite-label-translate {
background-position: -857px -347px; width: 14px; height: 14px;
}
.sprite-flat-menu-arrow-right {
background-position: -1249px -130px; width: 7px; height: 14px;
}
.sprite-weibo {
background-position: -1175px -172px; width: 36px; height: 36px;
}
.sprite-up-arrow {
background-position: -965px -347px; width: 11px; height: 7px;
}
.sprite-box-inner-rb {
background-position: -1249px -144px; width: 7px; height: 8px;
}
.sprite-section-smart-15 {
background-position: -408px -159px; width: 72px; height: 72px;
}
.sprite-menu-rb {
background-position: -592px -347px; width: 18px; height: 14px;
}
.sprite-renren-28 {
background-position: -801px -250px; width: 28px; height: 28px;
}
.sprite-box-gray-legend {
background-position: -605px -250px; width: 112px; height: 30px;
}
.sprite-success {
background-position: -1067px -172px; width: 36px; height: 36px;
}
.sprite-forum {
background-position: -683px -214px; width: 41px; height: 36px;
}
.sprite-law {
background-position: -773px -347px; width: 14px; height: 14px;
}
.sprite-power {
background-position: -788px -327px; width: 20px; height: 20px;
}
.sprite-btn-paper-xlarge {
background-position: -779px -88px; width: 204px; height: 42px;
}
.sprite-box-inner-lt {
background-position: -1249px -160px; width: 8px; height: 6px;
}
.sprite-chat-gray {
background-position: -784px -307px; width: 23px; height: 20px;
}
.sprite-box-split-r {
background-position: -916px -208px; width: 15px; height: 3px;
}
.sprite-lamp-blue {
background-position: -779px -172px; width: 36px; height: 36px;
}
.sprite-true {
background-position: -763px -307px; width: 21px; height: 20px;
}
.sprite-audio-player-stop-press {
background-position: -924px -214px; width: 32px; height: 32px;
}
.sprite-red-circle {
background-position: 0px 0px; width: 140px; height: 140px;
}
.sprite-btn-paper-xlarge-hover {
background-position: -575px -88px; width: 204px; height: 42px;
}
.sprite-box-split-l {
background-position: -931px -208px; width: 15px; height: 3px;
}
.sprite-tip-button {
background-position: -1025px -280px; width: 74px; height: 22px;
}
.sprite-lamp-gray {
background-position: -923px -172px; width: 36px; height: 36px;
}
.sprite-side-nav-active-large {
background-position: -408px 0px; width: 167px; height: 87px;
}
.sprite-circle-down-arrow {
background-position: -683px -307px; width: 20px; height: 20px;
}
.sprite-side-top-bg {
background-position: 0px -140px; width: 160px; height: 140px;
}
.sprite-star-filled {
background-position: -728px -327px; width: 20px; height: 20px;
}
.sprite-tip-press-20 {
background-position: -1023px -307px; width: 20px; height: 20px;
}
.sprite-mobile {
background-position: -1204px -307px; width: 13px; height: 20px;
}
.sprite-printer-gray {
background-position: -828px -327px; width: 20px; height: 20px;
}
.sprite-paper-clip {
background-position: -1207px -327px; width: 14px; height: 14px;
}
.sprite-side-nav-bg-large {
background-position: -239px -280px; width: 167px; height: 87px;
}
.sprite-menu-rt {
background-position: -911px -347px; width: 18px; height: 11px;
}
.sprite-bar-chart {
background-position: -731px -347px; width: 14px; height: 14px;
}
.sprite-essay-big {
background-position: -280px 0px; width: 128px; height: 128px;
}
.sprite-clock {
background-position: -480px -303px; width: 72px; height: 72px;
}
.sprite-checkbox-2 {
background-position: -939px -307px; width: 20px; height: 20px;
}
.sprite-level-down {
background-position: -916px -307px; width: 23px; height: 20px;
}
.sprite-section-sprint {
background-position: -280px -128px; width: 72px; height: 72px;
}
.sprite-source {
background-position: -759px -347px; width: 14px; height: 14px;
}
.sprite-green-dot {
background-position: -1134px -280px; width: 20px; height: 20px;
}
.sprite-notepad {
background-position: -815px -172px; width: 36px; height: 36px;
}
.sprite-flash-tip-lb {
background-position: -624px -347px; width: 15px; height: 14px;
}
.sprite-edit-note {
background-position: -575px -347px; width: 17px; height: 14px;
}
.sprite-calendar {
background-position: -672px -327px; width: 20px; height: 20px;
}
.sprite-question {
background-position: -773px -250px; width: 28px; height: 28px;
}
.sprite-audio-player-stop {
background-position: -1170px -214px; width: 32px; height: 32px;
}
.sprite-truman {
background-position: -995px -172px; width: 36px; height: 36px;
}
.sprite-false {
background-position: -902px -307px; width: 14px; height: 20px;
}
.sprite-flat-menu-arrow-top {
background-position: -990px -347px; width: 14px; height: 7px;
}
.sprite-qq-color-28 {
background-position: -1052px -250px; width: 28px; height: 28px;
}
.sprite-popover-arrow-right {
background-position: -1064px -327px; width: 15px; height: 19px;
}
.sprite-power-up {
background-position: -768px -327px; width: 20px; height: 20px;
}
.sprite-book {
background-position: -280px -200px; width: 72px; height: 72px;
}
.sprite-side-nav-active {
background-position: -167px -397px; width: 167px; height: 67px;
}
.sprite-circle-down {
background-position: -723px -307px; width: 20px; height: 20px;
}
.sprite-weibo-28 {
background-position: -1024px -250px; width: 28px; height: 28px;
}
.sprite-balloon-arrow-active {
background-position: -954px -347px; width: 11px; height: 8px;
}
.sprite-box-gray-rt {
background-position: -1247px -116px; width: 4px; height: 4px;
}
.sprite-gray-dot {
background-position: -708px -327px; width: 20px; height: 20px;
}
.sprite-popover-arrow-top {
background-position: -1113px -327px; width: 19px; height: 15px;
}
.sprite-qq-color {
background-position: -611px -214px; width: 36px; height: 36px;
}
.sprite-menu-lt {
background-position: -929px -347px; width: 14px; height: 11px;
}
.sprite-true-green {
background-position: -867px -307px; width: 21px; height: 20px;
}
.sprite-gear {
background-position: -647px -214px; width: 36px; height: 36px;
}
.sprite-flash-tip-lt {
background-position: -639px -347px; width: 15px; height: 14px;
}
.sprite-gray-radius-rb {
background-position: -1247px -172px; width: 10px; height: 10px;
}
.sprite-section-giant {
background-position: -408px -87px; width: 72px; height: 72px;
}
.sprite-weibo-color {
background-position: -851px -172px; width: 36px; height: 36px;
}
.sprite-tag {
background-position: -745px -347px; width: 14px; height: 14px;
}
.sprite-label-trick {
background-position: -843px -347px; width: 14px; height: 14px;
}
.sprite-joinus {
background-position: -1220px -280px; width: 25px; height: 20px;
}
.sprite-paper-small {
background-position: -1099px -280px; width: 15px; height: 20px;
}
.sprite-report {
background-position: -408px -231px; width: 72px; height: 72px;
}
.sprite-close-white {
background-position: -1248px -120px; width: 9px; height: 9px;
}
.sprite-star-filled-light {
background-position: -748px -327px; width: 20px; height: 20px;
}
.sprite-tip {
background-position: -969px -250px; width: 27px; height: 28px;
}
.sprite-checkbox-checked {
background-position: -1074px -307px; width: 20px; height: 20px;
}
.sprite-audio-player-play-press {
background-position: -956px -214px; width: 32px; height: 32px;
}
.sprite-question-text-mark {
background-position: -717px -250px; width: 28px; height: 28px;
}
.sprite-header-book {
background-position: -897px -214px; width: 27px; height: 33px;
}
.sprite-key {
background-position: -959px -172px; width: 36px; height: 36px;
}
.sprite-box-rb {
background-position: -779px -208px; width: 6px; height: 5px;
}
.sprite-answer-incorrect {
background-position: -1192px -250px; width: 28px; height: 27px;
}
.sprite-qq {
background-position: -1211px -172px; width: 36px; height: 36px;
}
.sprite-btn-paper-xxlarge-hover {
background-position: -575px -130px; width: 255px; height: 42px;
}
.sprite-logo {
background-position: -1020px -214px; width: 150px; height: 32px;
}
.sprite-gray-radius-rt {
background-position: -1238px -120px; width: 10px; height: 10px;
}
.sprite-circle-up {
background-position: -595px -327px; width: 20px; height: 20px;
}
.sprite-truman-blue {
background-position: -1031px -172px; width: 36px; height: 36px;
}
.sprite-mobile-gray {
background-position: -1191px -307px; width: 13px; height: 20px;
}
.sprite-progress-bg {
background-position: -789px -280px; width: 236px; height: 22px;
}
.sprite-section-continue {
background-position: -408px -303px; width: 72px; height: 72px;
}
.sprite-search-input {
background-position: -575px 0px; width: 682px; height: 44px;
}
.sprite-expand {
background-position: -847px -307px; width: 20px; height: 20px;
}
.sprite-answer-half-correct-mark {
background-position: -603px -280px; width: 28px; height: 27px;
}
.sprite-btn-small-hover {
background-position: -629px -307px; width: 54px; height: 20px;
}
.sprite-popover-arrow-left {
background-position: -1079px -327px; width: 15px; height: 19px;
}
.sprite-renren-color {
background-position: -1139px -172px; width: 36px; height: 36px;
}
.sprite-report-tip {
background-position: 0px -280px; width: 239px; height: 117px;
}
.sprite-answer-correct-mark {
background-position: -1164px -250px; width: 28px; height: 27px;
}
.sprite-help {
background-position: -575px -214px; width: 36px; height: 36px;
}
.sprite-answer-correct {
background-position: -687px -280px; width: 28px; height: 27px;
}
.sprite-expanded-flag {
background-position: -610px -347px; width: 14px; height: 14px;
}
.sprite-side-nav-bg {
background-position: -334px -397px; width: 167px; height: 67px;
}
.sprite-cart {
background-position: -651px -327px; width: 21px; height: 20px;
}
.sprite-profile-large {
background-position: -765px -214px; width: 36px; height: 36px;
}
.sprite-box-gray-lt {
background-position: -1253px -202px; width: 4px; height: 4px;
}
.sprite-circle-up-arrow {
background-position: -1094px -307px; width: 20px; height: 20px;
}
.sprite-btn-paper-xxlarge {
background-position: -830px -130px; width: 255px; height: 42px;
}
.sprite-course {
background-position: -703px -307px; width: 20px; height: 20px;
}
.sprite-checkbox-2-checked {
background-position: -743px -307px; width: 20px; height: 20px;
}
.sprite-trush {
background-position: -1177px -280px; width: 23px; height: 20px;
}
.slide {
background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.slide-btn-hero-active-left {
background-position: 0px -2668px; width: 10px; height: 60px;
}
.slide-btn-normal-first-disabled-left {
background-position: 0px -732px; width: 7px; height: 30px;
}
.slide-btn-primary-left {
background-position: 0px -612px; width: 7px; height: 30px;
}
.slide-btn-altern-hover-left {
background-position: 0px -1692px; width: 7px; height: 30px;
}
.slide-btn-hero-disabled-right {
background-position: right -2788px; width: 270px; height: 60px;
}
.slide-audio-player-progress-right {
background-position: right -92px; width: 270px; height: 12px;
}
.slide-btn-normal-disabled-left {
background-position: 0px -2112px; width: 7px; height: 30px;
}
.slide-btn-altern-active-left {
background-position: 0px -1752px; width: 7px; height: 30px;
}
.slide-balloon-right {
background-position: right -2269px; width: 270px; height: 33px;
}
.slide-btn-normal-first-disabled-right {
background-position: right -762px; width: 270px; height: 30px;
}
.slide-btn-primary-hover-right {
background-position: right -1362px; width: 270px; height: 30px;
}
.slide-button-simple-left {
background-position: 0px -208px; width: 4px; height: 20px;
}
.slide-progress-bg-right {
background-position: right 0px; width: 270px; height: 8px;
}
.slide-blankfilling-focus-left {
background-position: 0px -296px; width: 20px; height: 24px;
}
.slide-btn-hero-right {
background-position: right -2488px; width: 270px; height: 60px;
}
.slide-btn-radius-small-left {
background-position: 0px -370px; width: 10px; height: 26px;
}
.slide-audio-player-progress-left {
background-position: 0px -56px; width: 30px; height: 12px;
}
.slide-btn-round-active-left {
background-position: 0px -1512px; width: 17px; height: 30px;
}
.slide-btn-hero-active-right {
background-position: right -2608px; width: 270px; height: 60px;
}
.slide-btn-normal-last-right {
background-position: right -1182px; width: 270px; height: 30px;
}
.slide-btn-hero-disabled-left {
background-position: 0px -2548px; width: 10px; height: 60px;
}
.slide-btn-disabled-left {
background-position: 0px -1482px; width: 7px; height: 30px;
}
.slide-text-focus-left {
background-position: 0px -2022px; width: 6px; height: 30px;
}
.slide-gray-radius-right {
background-position: right -423px; width: 270px; height: 27px;
}
.slide-button-simple-right {
background-position: right -148px; width: 270px; height: 20px;
}
.slide-btn-radius-small-right {
background-position: right -344px; width: 270px; height: 26px;
}
.slide-btn-disabled-right {
background-position: right -1782px; width: 270px; height: 30px;
}
.slide-gray-radius-inner-right {
background-position: right -477px; width: 270px; height: 27px;
}
.slide-btn-altern-right {
background-position: right -1392px; width: 270px; height: 30px;
}
.slide-text-left {
background-position: 0px -1842px; width: 6px; height: 30px;
}
.slide-gray-radius-left {
background-position: 0px -396px; width: 10px; height: 27px;
}
.slide-keypoint-left {
background-position: 0px -558px; width: 74px; height: 27px;
}
.slide-btn-normal-active-right {
background-position: right -2082px; width: 270px; height: 30px;
}
.slide-btn-hero-hover-left {
background-position: 0px -2428px; width: 10px; height: 60px;
}
.slide-button-simple-light-left {
background-position: 0px -188px; width: 4px; height: 20px;
}
.slide-text-label-blue-right {
background-position: right -1932px; width: 270px; height: 30px;
}
.slide-blankfilling-focus-right {
background-position: right -320px; width: 270px; height: 24px;
}
.slide-btn-hero-hover-right {
background-position: right -2368px; width: 270px; height: 60px;
}
.slide-btn-normal-last-disabled-right {
background-position: right -1062px; width: 270px; height: 30px;
}
.slide-keypoint-blue-right {
background-position: right -531px; width: 270px; height: 27px;
}
.slide-balloon-active-left {
background-position: 0px -2335px; width: 10px; height: 33px;
}
.slide-blankfilling-right {
background-position: right -272px; width: 270px; height: 24px;
}
.slide-btn-altern-active-right {
background-position: right -1722px; width: 270px; height: 30px;
}
.slide-progress-left {
background-position: 0px -16px; width: 4px; height: 10px;
}
.slide-audio-player-progress-buffer-right {
background-position: right -116px; width: 270px; height: 12px;
}
.slide-btn-round-hover-left {
background-position: 0px -1572px; width: 17px; height: 30px;
}
.slide-audio-player-progress-play-left {
background-position: 0px -68px; width: 30px; height: 12px;
}
.slide-btn-hero-left {
background-position: 0px -2728px; width: 10px; height: 60px;
}
.slide-audio-player-progress-buffer-left {
background-position: 0px -104px; width: 30px; height: 12px;
}
.slide-text-label-green-left {
background-position: 0px -1902px; width: 18px; height: 30px;
}
.slide-btn-primary-active-left {
background-position: 0px -1272px; width: 7px; height: 30px;
}
.slide-progress-right {
background-position: right -26px; width: 270px; height: 10px;
}
.slide-btn-normal-left {
background-position: 0px -1212px; width: 7px; height: 30px;
}
.slide-btn-normal-first-hover-right {
background-position: right -822px; width: 270px; height: 30px;
}
.slide-header-tip-left {
background-position: 0px -2204px; width: 20px; height: 32px;
}
.slide-keypoint-right {
background-position: right -585px; width: 270px; height: 27px;
}
.slide-btn-normal-last-active-right {
background-position: right -1002px; width: 270px; height: 30px;
}
.slide-btn-primary-active-right {
background-position: right -1302px; width: 270px; height: 30px;
}
.slide-text-label-blue-left {
background-position: 0px -1962px; width: 18px; height: 30px;
}
.slide-btn-normal-active-left {
background-position: 0px -2052px; width: 7px; height: 30px;
}
.slide-balloon-active-right {
background-position: right -2302px; width: 270px; height: 33px;
}
.slide-btn-normal-disabled-right {
background-position: right -642px; width: 270px; height: 30px;
}
.slide-blankfilling-left {
background-position: 0px -248px; width: 20px; height: 24px;
}
.slide-btn-normal-first-right {
background-position: right -882px; width: 270px; height: 30px;
}
.slide-btn-primary-right {
background-position: right -1422px; width: 270px; height: 30px;
}
.slide-text-focus-right {
background-position: right -1992px; width: 270px; height: 30px;
}
.slide-header-tip-right {
background-position: right -2172px; width: 270px; height: 32px;
}
.slide-btn-normal-last-disabled-left {
background-position: 0px -1032px; width: 7px; height: 30px;
}
.slide-btn-round-active-right {
background-position: right -1542px; width: 270px; height: 30px;
}
.slide-progress-fill-right {
background-position: right -36px; width: 270px; height: 10px;
}
.slide-btn-normal-first-left {
background-position: 0px -852px; width: 7px; height: 30px;
}
.slide-btn-round-left {
background-position: 0px -1632px; width: 17px; height: 30px;
}
.slide-btn-normal-first-active-left {
background-position: 0px -672px; width: 7px; height: 30px;
}
.slide-btn-normal-hover-right {
background-position: right -942px; width: 270px; height: 30px;
}
.slide-btn-normal-last-active-left {
background-position: 0px -972px; width: 7px; height: 30px;
}
.slide-text-right {
background-position: right -1812px; width: 270px; height: 30px;
}
.slide-balloon-left {
background-position: 0px -2236px; width: 10px; height: 33px;
}
.slide-btn-round-hover-right {
background-position: right -1602px; width: 270px; height: 30px;
}
.slide-btn-normal-last-left {
background-position: 0px -1152px; width: 7px; height: 30px;
}
.slide-progress-bg-left {
background-position: 0px -8px; width: 10px; height: 8px;
}
.slide-button-simple-light-right {
background-position: right -168px; width: 270px; height: 20px;
}
.slide-btn-normal-right {
background-position: right -1242px; width: 270px; height: 30px;
}
.slide-btn-normal-first-hover-left {
background-position: 0px -792px; width: 7px; height: 30px;
}
.slide-btn-normal-first-active-right {
background-position: right -702px; width: 270px; height: 30px;
}
.slide-tip-right {
background-position: right -228px; width: 270px; height: 20px;
}
.slide-btn-normal-last-hover-right {
background-position: right -2142px; width: 270px; height: 30px;
}
.slide-gray-radius-inner-left {
background-position: 0px -450px; width: 10px; height: 27px;
}
.slide-btn-primary-hover-left {
background-position: 0px -1332px; width: 7px; height: 30px;
}
.slide-btn-round-right {
background-position: right -1662px; width: 270px; height: 30px;
}
.slide-btn-normal-last-hover-left {
background-position: 0px -1092px; width: 7px; height: 30px;
}
.slide-audio-player-progress-play-right {
background-position: right -80px; width: 270px; height: 12px;
}
.slide-btn-altern-left {
background-position: 0px -1452px; width: 7px; height: 30px;
}
.slide-keypoint-blue-left {
background-position: 0px -504px; width: 18px; height: 27px;
}
.slide-btn-altern-hover-right {
background-position: right -1122px; width: 270px; height: 30px;
}
.slide-text-label-green-right {
background-position: right -1872px; width: 270px; height: 30px;
}
.slide-tip-left {
background-position: 0px -128px; width: 14px; height: 20px;
}
.slide-progress-fill-left {
background-position: 0px -46px; width: 4px; height: 10px;
}
.slide-btn-normal-hover-left {
background-position: 0px -912px; width: 7px; height: 30px;
}
.repeat-x {
background-image: url("../img/exercises/6b9dee0c-repeat-x.png"); background-repeat: no-repeat;
}
.box-gray-t {
background-position: 0px -8px; width: 40px; height: 4px;
}
.box-split-2 {
background-position: 0px -23px; width: 40px; height: 8px;
}
.default-b {
background-position: 0px -288px; width: 40px; height: 160px;
}
.box-inner-b {
background-position: 0px -31px; width: 40px; height: 8px;
}
.box-b {
background-position: 0px -12px; width: 40px; height: 5px;
}
.box-split {
background-position: 0px -1px; width: 40px; height: 3px;
}
.box-gray-b {
background-position: 0px -4px; width: 40px; height: 4px;
}
.menu-t {
background-position: 0px -39px; width: 40px; height: 11px;
}
.box-t {
background-position: 0px 0px; width: 40px; height: 1px;
}
.gray-gradient {
background-position: 0px -873px; width: 40px; height: 344px;
}
.flash-tip-b {
background-position: 0px -50px; width: 40px; height: 14px;
}
.normal-gray-gradient {
background-position: 0px -658px; width: 40px; height: 215px;
}
.light-gray-gradient {
background-position: 0px -448px; width: 40px; height: 210px;
}
.box-inner-t {
background-position: 0px -17px; width: 40px; height: 6px;
}
.white-gradient {
background-position: 0px -1217px; width: 40px; height: 344px;
}
.flash-tip-t {
background-position: 0px -64px; width: 40px; height: 14px;
}
.menu-b {
background-position: 0px -78px; width: 40px; height: 14px;
}
.default-t {
background-position: 0px -128px; width: 40px; height: 160px;
}
.header-shadow-repeat {
background-position: 0px -92px; width: 40px; height: 36px;
}
.repeat-y {
background-image: url("../img/exercises/ea257350-repeat-y.png"); background-repeat: no-repeat;
}
.box-inner-r {
background-position: -14px 0px; width: 6px; height: 40px;
}
.default-l {
background-position: -245px 0px; width: 160px; height: 40px;
}
.menu-r {
background-position: -27px 0px; width: 14px; height: 40px;
}
.box-l {
background-position: 0px 0px; width: 3px; height: 40px;
}
.default-r {
background-position: -85px 0px; width: 160px; height: 40px;
}
.flash-tip-r {
background-position: -70px 0px; width: 15px; height: 40px;
}
.box-gray-r {
background-position: -10px 0px; width: 4px; height: 40px;
}
.box-r {
background-position: -3px 0px; width: 3px; height: 40px;
}
.menu-l {
background-position: -41px 0px; width: 14px; height: 40px;
}
.box-gray-l {
background-position: -6px 0px; width: 4px; height: 40px;
}
.flash-tip-l {
background-position: -55px 0px; width: 15px; height: 40px;
}
.box-inner-l {
background-position: -20px 0px; width: 7px; height: 40px;
}
.placeholder {
color: rgb(153, 153, 153);
}
.clearfix {
}
.clearfix::before {
line-height: 0; display: table; content: "";
}
.clearfix::after {
line-height: 0; display: table; content: "";
}
.clearfix::after {
clear: both;
}
article {
display: block;
}
aside {
display: block;
}
details {
display: block;
}
figcaption {
display: block;
}
figure {
display: block;
}
footer {
display: block;
}
header {
display: block;
}
hgroup {
display: block;
}
nav {
display: block;
}
section {
display: block;
}
audio {
display: inline-block;
}
canvas {
display: inline-block;
}
video {
display: inline-block;
}
audio:not([controls]) {
display: none;
}
html {
font-size: 100%; -ms-text-size-adjust: 100%; -webkit-text-size-adjust: 100%;
}
a:focus {
outline: rgb(51, 51, 51) dotted thin; outline-offset: -2px;
}
a:hover {
outline: 0px;
}
a:active {
outline: 0px;
}
sub {
line-height: 0; font-size: 75%; vertical-align: baseline; position: relative;
}
sup {
line-height: 0; font-size: 75%; vertical-align: baseline; position: relative;
}
sup {
top: -0.5em;
}
sub {
bottom: -0.25em;
}
img {
border: 0px currentColor; border-image: none; height: auto; vertical-align: middle; -ms-interpolation-mode: bicubic; max-width: 100%;
}
#map_canvas img {
max-width: none;
}
.google-maps img {
max-width: none;
}
button {
margin: 0px; font-size: 100%; vertical-align: middle;
}
input {
margin: 0px; font-size: 100%; vertical-align: middle;
}
select {
margin: 0px; font-size: 100%; vertical-align: middle;
}
textarea {
margin: 0px; font-size: 100%; vertical-align: middle;
}
button {
line-height: normal;
}
input {
line-height: normal;
}
button {
cursor: pointer; -webkit-appearance: button;
}
html input[type='button'] {
cursor: pointer; -webkit-appearance: button;
}
input[type='reset'] {
cursor: pointer; -webkit-appearance: button;
}
input[type='submit'] {
cursor: pointer; -webkit-appearance: button;
}
label {
cursor: pointer;
}
select {
cursor: pointer;
}
button {
cursor: pointer;
}
input[type='button'] {
cursor: pointer;
}
input[type='reset'] {
cursor: pointer;
}
input[type='submit'] {
cursor: pointer;
}
input[type='radio'] {
cursor: pointer;
}
input[type='checkbox'] {
cursor: pointer;
}
input[type='search'] {
box-sizing: content-box; -webkit-appearance: textfield; -webkit-box-sizing: content-box; -moz-box-sizing: content-box;
}
textarea {
overflow: auto; vertical-align: top;
}
@media print
{
* {
background: none !important; color: rgb(0, 0, 0) !important; box-shadow: none !important; text-shadow: none !important;
}
a {
text-decoration: underline;
}
a:visited {
text-decoration: underline;
}
a[href]::after {
content: " (" attr(href) ")";
}
abbr[title]::after {
content: " (" attr(title) ")";
}
.ir a::after {
content: "";
}
a[href^='javascript:']::after {
content: "";
}
a[href^='#']::after {
content: "";
}
pre {
border: 1px solid rgb(153, 153, 153); border-image: none; page-break-inside: avoid;
}
blockquote {
border: 1px solid rgb(153, 153, 153); border-image: none; page-break-inside: avoid;
}
thead {
display: table-header-group;
}
tr {
page-break-inside: avoid;
}
img {
page-break-inside: avoid;
}
img {
max-width: 100% !important;
}
p {
orphans: 3; widows: 3;
}
h2 {
orphans: 3; widows: 3;
}
h3 {
orphans: 3; widows: 3;
}
h2 {
page-break-after: avoid;
}
h3 {
page-break-after: avoid;
}
}
body {
margin: 0px; color: rgb(51, 51, 51); line-height: 20px; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; background-color: rgb(255, 255, 255);
}
a {
color: rgb(0, 136, 204); text-decoration: none;
}
a:hover {
color: rgb(0, 85, 128); text-decoration: underline;
}
a:focus {
color: rgb(0, 85, 128); text-decoration: underline;
}
.img-rounded {
border-radius: 6px; -webkit-border-radius: 6px; -moz-border-radius: 6px;
}
.img-polaroid {
padding: 4px; border: 1px solid rgba(0, 0, 0, 0.2); border-image: none; box-shadow: 0px 1px 3px rgba(0,0,0,0.1); background-color: rgb(255, 255, 255); -webkit-box-shadow: 0 1px 3px rgba(0, 0, 0, 0.1); -moz-box-shadow: 0 1px 3px rgba(0, 0, 0, 0.1);
}
.img-circle {
border-radius: 500px; -webkit-border-radius: 500px; -moz-border-radius: 500px;
}
.row {
margin-left: -24px;
}
.row::before {
line-height: 0; display: table; content: "";
}
.row::after {
line-height: 0; display: table; content: "";
}
.row::after {
clear: both;
}
.row::before {
line-height: 0; display: table; content: "";
}
.row::after {
line-height: 0; display: table; content: "";
}
.row::after {
clear: both;
}
[class*='span'] {
margin-left: 24px; float: left; min-height: 1px;
}
.container {
width: 960px;
}
.navbar-static-top .container {
width: 960px;
}
.navbar-fixed-top .container {
width: 960px;
}
.navbar-fixed-bottom .container {
width: 960px;
}
.span12 {
width: 960px;
}
.span11 {
width: 878px;
}
.span10 {
width: 796px;
}
.span9 {
width: 714px;
}
.span8 {
width: 632px;
}
.span7 {
width: 550px;
}
.span6 {
width: 468px;
}
.span5 {
width: 386px;
}
.span4 {
width: 304px;
}
.span3 {
width: 222px;
}
.span2 {
width: 140px;
}
.span1 {
width: 58px;
}
.offset12 {
margin-left: 1008px;
}
.offset11 {
margin-left: 926px;
}
.offset10 {
margin-left: 844px;
}
.offset9 {
margin-left: 762px;
}
.offset8 {
margin-left: 680px;
}
.offset7 {
margin-left: 598px;
}
.offset6 {
margin-left: 516px;
}
.offset5 {
margin-left: 434px;
}
.offset4 {
margin-left: 352px;
}
.offset3 {
margin-left: 270px;
}
.offset2 {
margin-left: 188px;
}
.offset1 {
margin-left: 106px;
}
[class*='span'].hide {
display: none;
}
[class*='span'].pull-right {
float: right;
}
.container {
margin-right: auto; margin-left: auto;
}
.container::before {
line-height: 0; display: table; content: "";
}
.container::after {
line-height: 0; display: table; content: "";
}
.container::after {
clear: both;
}
.container::before {
line-height: 0; display: table; content: "";
}
.container::after {
line-height: 0; display: table; content: "";
}
.container::after {
clear: both;
}
.container-fluid {
padding-right: 24px; padding-left: 24px;
}
.container-fluid::before {
line-height: 0; display: table; content: "";
}
.container-fluid::after {
line-height: 0; display: table; content: "";
}
.container-fluid::after {
clear: both;
}
.container-fluid::before {
line-height: 0; display: table; content: "";
}
.container-fluid::after {
line-height: 0; display: table; content: "";
}
.container-fluid::after {
clear: both;
}
p {
margin: 0px 0px 10px;
}
.lead {
line-height: 30px; font-size: 21px; font-weight: 200; margin-bottom: 20px;
}
small {
font-size: 85%;
}
strong {
font-weight: bold;
}
em {
font-style: italic;
}
cite {
font-style: normal;
}
.muted {
color: rgb(153, 153, 153);
}
a.muted:hover {
color: rgb(128, 128, 128);
}
a.muted:focus {
color: rgb(128, 128, 128);
}
.text-warning {
color: rgb(192, 152, 83);
}
a.text-warning:hover {
color: rgb(164, 126, 60);
}
a.text-warning:focus {
color: rgb(164, 126, 60);
}
.text-error {
color: rgb(185, 74, 72);
}
a.text-error:hover {
color: rgb(149, 59, 57);
}
a.text-error:focus {
color: rgb(149, 59, 57);
}
.text-info {
color: rgb(58, 135, 173);
}
a.text-info:hover {
color: rgb(45, 105, 135);
}
a.text-info:focus {
color: rgb(45, 105, 135);
}
.text-success {
color: rgb(70, 136, 71);
}
a.text-success:hover {
color: rgb(53, 102, 53);
}
a.text-success:focus {
color: rgb(53, 102, 53);
}
.text-left {
text-align: left;
}
.text-right {
text-align: right;
}
.text-center {
text-align: center;
}
h1 {
margin: 10px 0px; color: inherit; line-height: 20px; font-family: inherit; font-weight: bold; text-rendering: optimizelegibility;
}
h2 {
margin: 10px 0px; color: inherit; line-height: 20px; font-family: inherit; font-weight: bold; text-rendering: optimizelegibility;
}
h3 {
margin: 10px 0px; color: inherit; line-height: 20px; font-family: inherit; font-weight: bold; text-rendering: optimizelegibility;
}
h4 {
margin: 10px 0px; color: inherit; line-height: 20px; font-family: inherit; font-weight: bold; text-rendering: optimizelegibility;
}
h5 {
margin: 10px 0px; color: inherit; line-height: 20px; font-family: inherit; font-weight: bold; text-rendering: optimizelegibility;
}
h6 {
margin: 10px 0px; color: inherit; line-height: 20px; font-family: inherit; font-weight: bold; text-rendering: optimizelegibility;
}
h1 small {
color: rgb(153, 153, 153); line-height: 1; font-weight: normal;
}
h2 small {
color: rgb(153, 153, 153); line-height: 1; font-weight: normal;
}
h3 small {
color: rgb(153, 153, 153); line-height: 1; font-weight: normal;
}
h4 small {
color: rgb(153, 153, 153); line-height: 1; font-weight: normal;
}
h5 small {
color: rgb(153, 153, 153); line-height: 1; font-weight: normal;
}
h6 small {
color: rgb(153, 153, 153); line-height: 1; font-weight: normal;
}
h1 {
line-height: 40px;
}
h2 {
line-height: 40px;
}
h3 {
line-height: 40px;
}
h1 {
font-size: 38.5px;
}
h2 {
font-size: 31.5px;
}
h3 {
font-size: 24.5px;
}
h4 {
font-size: 17.5px;
}
h5 {
font-size: 14px;
}
h6 {
font-size: 11.9px;
}
h1 small {
font-size: 24.5px;
}
h2 small {
font-size: 17.5px;
}
h3 small {
font-size: 14px;
}
h4 small {
font-size: 14px;
}
.page-header {
margin: 20px 0px 30px; padding-bottom: 9px; border-bottom-color: rgb(238, 238, 238); border-bottom-width: 1px; border-bottom-style: solid;
}
ul {
margin: 0px 0px 10px 25px; padding: 0px;
}
ol {
margin: 0px 0px 10px 25px; padding: 0px;
}
ul ul {
margin-bottom: 0px;
}
ul ol {
margin-bottom: 0px;
}
ol ol {
margin-bottom: 0px;
}
ol ul {
margin-bottom: 0px;
}
li {
line-height: 20px;
}
ul.unstyled {
list-style: none; margin-left: 0px;
}
ol.unstyled {
list-style: none; margin-left: 0px;
}
ul.inline {
list-style: none; margin-left: 0px;
}
ol.inline {
list-style: none; margin-left: 0px;
}
ul.inline > li {
padding-right: 5px; padding-left: 5px; display: inline-block;
}
ol.inline > li {
padding-right: 5px; padding-left: 5px; display: inline-block;
}
dl {
margin-bottom: 20px;
}
dt {
line-height: 20px;
}
dd {
line-height: 20px;
}
dt {
font-weight: bold;
}
dd {
margin-left: 10px;
}
.dl-horizontal {
}
.dl-horizontal::before {
line-height: 0; display: table; content: "";
}
.dl-horizontal::after {
line-height: 0; display: table; content: "";
}
.dl-horizontal::after {
clear: both;
}
.dl-horizontal::before {
line-height: 0; display: table; content: "";
}
.dl-horizontal::after {
line-height: 0; display: table; content: "";
}
.dl-horizontal::after {
clear: both;
}
.dl-horizontal dt {
width: 160px; text-align: right; overflow: hidden; clear: left; float: left; white-space: nowrap; -ms-text-overflow: ellipsis;
}
.dl-horizontal dd {
margin-left: 180px;
}
hr {
border-width: 1px 0px; border-style: solid none; border-color: rgb(238, 238, 238) currentColor rgb(255, 255, 255); margin: 20px 0px; border-image: none;
}
abbr[title] {
border-bottom-color: rgb(153, 153, 153); border-bottom-width: 1px; border-bottom-style: dotted; cursor: help;
}
abbr[data-original-title] {
border-bottom-color: rgb(153, 153, 153); border-bottom-width: 1px; border-bottom-style: dotted; cursor: help;
}
abbr.initialism {
text-transform: uppercase; font-size: 90%;
}
blockquote {
margin: 0px 0px 20px; padding: 0px 0px 0px 15px; border-left-color: rgb(238, 238, 238); border-left-width: 5px; border-left-style: solid;
}
blockquote p {
line-height: 1.25; font-size: 17.5px; font-weight: 300; margin-bottom: 0px;
}
blockquote small {
color: rgb(153, 153, 153); line-height: 20px; display: block;
}
blockquote small::before {
content: "\2014 \00A0";
}
blockquote.pull-right {
padding-right: 15px; padding-left: 0px; border-right-color: rgb(238, 238, 238); border-left-color: currentColor; border-right-width: 5px; border-left-width: 0px; border-right-style: solid; border-left-style: none; float: right;
}
blockquote.pull-right p {
text-align: right;
}
blockquote.pull-right small {
text-align: right;
}
blockquote.pull-right small::before {
content: "";
}
blockquote.pull-right small::after {
content: "\00A0 \2014";
}
q::before {
content: "";
}
q::after {
content: "";
}
blockquote::before {
content: "";
}
blockquote::after {
content: "";
}
address {
line-height: 20px; font-style: normal; margin-bottom: 20px; display: block;
}
form {
margin: 0px 0px 20px;
}
fieldset {
margin: 0px; padding: 0px; border: 0px currentColor; border-image: none;
}
legend {
border-width: 0px 0px 1px; border-style: none none solid; border-color: currentColor currentColor rgb(229, 229, 229); padding: 0px; border-image: none; width: 100%; color: rgb(51, 51, 51); line-height: 40px; font-size: 21px; margin-bottom: 20px; display: block;
}
legend small {
color: rgb(153, 153, 153); font-size: 15px;
}
label {
line-height: 20px; font-size: 14px; font-weight: normal;
}
input {
line-height: 20px; font-size: 14px; font-weight: normal;
}
button {
line-height: 20px; font-size: 14px; font-weight: normal;
}
select {
line-height: 20px; font-size: 14px; font-weight: normal;
}
textarea {
line-height: 20px; font-size: 14px; font-weight: normal;
}
input {
font-family: "Helvetica Neue", Helvetica, Arial, sans-serif;
}
button {
font-family: "Helvetica Neue", Helvetica, Arial, sans-serif;
}
select {
font-family: "Helvetica Neue", Helvetica, Arial, sans-serif;
}
textarea {
font-family: "Helvetica Neue", Helvetica, Arial, sans-serif;
}
label {
margin-bottom: 5px; display: block;
}
select {
padding: 4px 6px; border-radius: 4px; height: 20px; color: rgb(85, 85, 85); line-height: 20px; font-size: 14px; margin-bottom: 10px; vertical-align: middle; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
textarea {
padding: 4px 6px; border-radius: 4px; height: 20px; color: rgb(85, 85, 85); line-height: 20px; font-size: 14px; margin-bottom: 10px; vertical-align: middle; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
input[type='text'] {
padding: 4px 6px; border-radius: 4px; height: 20px; color: rgb(85, 85, 85); line-height: 20px; font-size: 14px; margin-bottom: 10px; vertical-align: middle; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
input[type='password'] {
padding: 4px 6px; border-radius: 4px; height: 20px; color: rgb(85, 85, 85); line-height: 20px; font-size: 14px; margin-bottom: 10px; vertical-align: middle; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
input[type='datetime'] {
padding: 4px 6px; border-radius: 4px; height: 20px; color: rgb(85, 85, 85); line-height: 20px; font-size: 14px; margin-bottom: 10px; vertical-align: middle; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
input[type='datetime-local'] {
padding: 4px 6px; border-radius: 4px; height: 20px; color: rgb(85, 85, 85); line-height: 20px; font-size: 14px; margin-bottom: 10px; vertical-align: middle; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
input[type='date'] {
padding: 4px 6px; border-radius: 4px; height: 20px; color: rgb(85, 85, 85); line-height: 20px; font-size: 14px; margin-bottom: 10px; vertical-align: middle; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
input[type='month'] {
padding: 4px 6px; border-radius: 4px; height: 20px; color: rgb(85, 85, 85); line-height: 20px; font-size: 14px; margin-bottom: 10px; vertical-align: middle; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
input[type='time'] {
padding: 4px 6px; border-radius: 4px; height: 20px; color: rgb(85, 85, 85); line-height: 20px; font-size: 14px; margin-bottom: 10px; vertical-align: middle; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
input[type='week'] {
padding: 4px 6px; border-radius: 4px; height: 20px; color: rgb(85, 85, 85); line-height: 20px; font-size: 14px; margin-bottom: 10px; vertical-align: middle; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
input[type='number'] {
padding: 4px 6px; border-radius: 4px; height: 20px; color: rgb(85, 85, 85); line-height: 20px; font-size: 14px; margin-bottom: 10px; vertical-align: middle; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
input[type='email'] {
padding: 4px 6px; border-radius: 4px; height: 20px; color: rgb(85, 85, 85); line-height: 20px; font-size: 14px; margin-bottom: 10px; vertical-align: middle; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
input[type='url'] {
padding: 4px 6px; border-radius: 4px; height: 20px; color: rgb(85, 85, 85); line-height: 20px; font-size: 14px; margin-bottom: 10px; vertical-align: middle; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
input[type='search'] {
padding: 4px 6px; border-radius: 4px; height: 20px; color: rgb(85, 85, 85); line-height: 20px; font-size: 14px; margin-bottom: 10px; vertical-align: middle; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
input[type='tel'] {
padding: 4px 6px; border-radius: 4px; height: 20px; color: rgb(85, 85, 85); line-height: 20px; font-size: 14px; margin-bottom: 10px; vertical-align: middle; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
input[type='color'] {
padding: 4px 6px; border-radius: 4px; height: 20px; color: rgb(85, 85, 85); line-height: 20px; font-size: 14px; margin-bottom: 10px; vertical-align: middle; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
.uneditable-input {
padding: 4px 6px; border-radius: 4px; height: 20px; color: rgb(85, 85, 85); line-height: 20px; font-size: 14px; margin-bottom: 10px; vertical-align: middle; display: inline-block; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
input {
width: 206px;
}
textarea {
width: 206px;
}
.uneditable-input {
width: 206px;
}
textarea {
height: auto;
}
textarea {
border: 1px solid rgb(204, 204, 204); transition:border 0.2s linear, box-shadow 0.2s linear; border-image: none; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075); background-color: rgb(255, 255, 255); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -webkit-transition: border linear .2s, box-shadow linear .2s; -moz-transition: border linear .2s, box-shadow linear .2s; -o-transition: border linear .2s, box-shadow linear .2s;
}
input[type='text'] {
border: 1px solid rgb(204, 204, 204); transition:border 0.2s linear, box-shadow 0.2s linear; border-image: none; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075); background-color: rgb(255, 255, 255); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -webkit-transition: border linear .2s, box-shadow linear .2s; -moz-transition: border linear .2s, box-shadow linear .2s; -o-transition: border linear .2s, box-shadow linear .2s;
}
input[type='password'] {
border: 1px solid rgb(204, 204, 204); transition:border 0.2s linear, box-shadow 0.2s linear; border-image: none; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075); background-color: rgb(255, 255, 255); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -webkit-transition: border linear .2s, box-shadow linear .2s; -moz-transition: border linear .2s, box-shadow linear .2s; -o-transition: border linear .2s, box-shadow linear .2s;
}
input[type='datetime'] {
border: 1px solid rgb(204, 204, 204); transition:border 0.2s linear, box-shadow 0.2s linear; border-image: none; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075); background-color: rgb(255, 255, 255); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -webkit-transition: border linear .2s, box-shadow linear .2s; -moz-transition: border linear .2s, box-shadow linear .2s; -o-transition: border linear .2s, box-shadow linear .2s;
}
input[type='datetime-local'] {
border: 1px solid rgb(204, 204, 204); transition:border 0.2s linear, box-shadow 0.2s linear; border-image: none; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075); background-color: rgb(255, 255, 255); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -webkit-transition: border linear .2s, box-shadow linear .2s; -moz-transition: border linear .2s, box-shadow linear .2s; -o-transition: border linear .2s, box-shadow linear .2s;
}
input[type='date'] {
border: 1px solid rgb(204, 204, 204); transition:border 0.2s linear, box-shadow 0.2s linear; border-image: none; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075); background-color: rgb(255, 255, 255); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -webkit-transition: border linear .2s, box-shadow linear .2s; -moz-transition: border linear .2s, box-shadow linear .2s; -o-transition: border linear .2s, box-shadow linear .2s;
}
input[type='month'] {
border: 1px solid rgb(204, 204, 204); transition:border 0.2s linear, box-shadow 0.2s linear; border-image: none; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075); background-color: rgb(255, 255, 255); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -webkit-transition: border linear .2s, box-shadow linear .2s; -moz-transition: border linear .2s, box-shadow linear .2s; -o-transition: border linear .2s, box-shadow linear .2s;
}
input[type='time'] {
border: 1px solid rgb(204, 204, 204); transition:border 0.2s linear, box-shadow 0.2s linear; border-image: none; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075); background-color: rgb(255, 255, 255); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -webkit-transition: border linear .2s, box-shadow linear .2s; -moz-transition: border linear .2s, box-shadow linear .2s; -o-transition: border linear .2s, box-shadow linear .2s;
}
input[type='week'] {
border: 1px solid rgb(204, 204, 204); transition:border 0.2s linear, box-shadow 0.2s linear; border-image: none; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075); background-color: rgb(255, 255, 255); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -webkit-transition: border linear .2s, box-shadow linear .2s; -moz-transition: border linear .2s, box-shadow linear .2s; -o-transition: border linear .2s, box-shadow linear .2s;
}
input[type='number'] {
border: 1px solid rgb(204, 204, 204); transition:border 0.2s linear, box-shadow 0.2s linear; border-image: none; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075); background-color: rgb(255, 255, 255); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -webkit-transition: border linear .2s, box-shadow linear .2s; -moz-transition: border linear .2s, box-shadow linear .2s; -o-transition: border linear .2s, box-shadow linear .2s;
}
input[type='email'] {
border: 1px solid rgb(204, 204, 204); transition:border 0.2s linear, box-shadow 0.2s linear; border-image: none; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075); background-color: rgb(255, 255, 255); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -webkit-transition: border linear .2s, box-shadow linear .2s; -moz-transition: border linear .2s, box-shadow linear .2s; -o-transition: border linear .2s, box-shadow linear .2s;
}
input[type='url'] {
border: 1px solid rgb(204, 204, 204); transition:border 0.2s linear, box-shadow 0.2s linear; border-image: none; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075); background-color: rgb(255, 255, 255); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -webkit-transition: border linear .2s, box-shadow linear .2s; -moz-transition: border linear .2s, box-shadow linear .2s; -o-transition: border linear .2s, box-shadow linear .2s;
}
input[type='search'] {
border: 1px solid rgb(204, 204, 204); transition:border 0.2s linear, box-shadow 0.2s linear; border-image: none; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075); background-color: rgb(255, 255, 255); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -webkit-transition: border linear .2s, box-shadow linear .2s; -moz-transition: border linear .2s, box-shadow linear .2s; -o-transition: border linear .2s, box-shadow linear .2s;
}
input[type='tel'] {
border: 1px solid rgb(204, 204, 204); transition:border 0.2s linear, box-shadow 0.2s linear; border-image: none; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075); background-color: rgb(255, 255, 255); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -webkit-transition: border linear .2s, box-shadow linear .2s; -moz-transition: border linear .2s, box-shadow linear .2s; -o-transition: border linear .2s, box-shadow linear .2s;
}
input[type='color'] {
border: 1px solid rgb(204, 204, 204); transition:border 0.2s linear, box-shadow 0.2s linear; border-image: none; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075); background-color: rgb(255, 255, 255); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -webkit-transition: border linear .2s, box-shadow linear .2s; -moz-transition: border linear .2s, box-shadow linear .2s; -o-transition: border linear .2s, box-shadow linear .2s;
}
.uneditable-input {
border: 1px solid rgb(204, 204, 204); transition:border 0.2s linear, box-shadow 0.2s linear; border-image: none; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075); background-color: rgb(255, 255, 255); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075); -webkit-transition: border linear .2s, box-shadow linear .2s; -moz-transition: border linear .2s, box-shadow linear .2s; -o-transition: border linear .2s, box-shadow linear .2s;
}
textarea:focus {
border-color: rgba(82, 168, 236, 0.8); outline: 0px; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075), 0px 0px 8px rgba(82,168,236,0.6); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6);
}
input[type='text']:focus {
border-color: rgba(82, 168, 236, 0.8); outline: 0px; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075), 0px 0px 8px rgba(82,168,236,0.6); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6);
}
input[type='password']:focus {
border-color: rgba(82, 168, 236, 0.8); outline: 0px; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075), 0px 0px 8px rgba(82,168,236,0.6); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6);
}
input[type='datetime']:focus {
border-color: rgba(82, 168, 236, 0.8); outline: 0px; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075), 0px 0px 8px rgba(82,168,236,0.6); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6);
}
input[type='datetime-local']:focus {
border-color: rgba(82, 168, 236, 0.8); outline: 0px; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075), 0px 0px 8px rgba(82,168,236,0.6); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6);
}
input[type='date']:focus {
border-color: rgba(82, 168, 236, 0.8); outline: 0px; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075), 0px 0px 8px rgba(82,168,236,0.6); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6);
}
input[type='month']:focus {
border-color: rgba(82, 168, 236, 0.8); outline: 0px; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075), 0px 0px 8px rgba(82,168,236,0.6); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6);
}
input[type='time']:focus {
border-color: rgba(82, 168, 236, 0.8); outline: 0px; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075), 0px 0px 8px rgba(82,168,236,0.6); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6);
}
input[type='week']:focus {
border-color: rgba(82, 168, 236, 0.8); outline: 0px; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075), 0px 0px 8px rgba(82,168,236,0.6); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6);
}
input[type='number']:focus {
border-color: rgba(82, 168, 236, 0.8); outline: 0px; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075), 0px 0px 8px rgba(82,168,236,0.6); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6);
}
input[type='email']:focus {
border-color: rgba(82, 168, 236, 0.8); outline: 0px; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075), 0px 0px 8px rgba(82,168,236,0.6); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6);
}
input[type='url']:focus {
border-color: rgba(82, 168, 236, 0.8); outline: 0px; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075), 0px 0px 8px rgba(82,168,236,0.6); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6);
}
input[type='search']:focus {
border-color: rgba(82, 168, 236, 0.8); outline: 0px; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075), 0px 0px 8px rgba(82,168,236,0.6); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6);
}
input[type='tel']:focus {
border-color: rgba(82, 168, 236, 0.8); outline: 0px; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075), 0px 0px 8px rgba(82,168,236,0.6); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6);
}
input[type='color']:focus {
border-color: rgba(82, 168, 236, 0.8); outline: 0px; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075), 0px 0px 8px rgba(82,168,236,0.6); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6);
}
.uneditable-input:focus {
border-color: rgba(82, 168, 236, 0.8); outline: 0px; box-shadow: inset 0px 1px 1px rgba(0,0,0,0.075), 0px 0px 8px rgba(82,168,236,0.6); -webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6); -moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, .075), 0 0 8px rgba(82, 168, 236, .6);
}
input[type='radio'] {
margin: 4px 0px 0px; line-height: normal;
}
input[type='checkbox'] {
margin: 4px 0px 0px; line-height: normal;
}
input[type='file'] {
width: auto;
}
input[type='image'] {
width: auto;
}
input[type='submit'] {
width: auto;
}
input[type='reset'] {
width: auto;
}
input[type='button'] {
width: auto;
}
input[type='radio'] {
width: auto;
}
input[type='checkbox'] {
width: auto;
}
select {
height: 30px; line-height: 30px;
}
input[type='file'] {
height: 30px; line-height: 30px;
}
select {
border: 1px solid rgb(204, 204, 204); border-image: none; width: 220px; background-color: rgb(255, 255, 255);
}
select[multiple] {
height: auto;
}
select[size] {
height: auto;
}
select:focus {
outline: rgb(51, 51, 51) dotted thin; outline-offset: -2px;
}
input[type='file']:focus {
outline: rgb(51, 51, 51) dotted thin; outline-offset: -2px;
}
input[type='radio']:focus {
outline: rgb(51, 51, 51) dotted thin; outline-offset: -2px;
}
input[type='checkbox']:focus {
outline: rgb(51, 51, 51) dotted thin; outline-offset: -2px;
}
.uneditable-input {
border-color: rgb(204, 204, 204); color: rgb(153, 153, 153); cursor: not-allowed; box-shadow: inset 0px 1px 2px rgba(0,0,0,0.025); background-color: rgb(252, 252, 252); -webkit-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.025); -moz-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.025);
}
.uneditable-textarea {
border-color: rgb(204, 204, 204); color: rgb(153, 153, 153); cursor: not-allowed; box-shadow: inset 0px 1px 2px rgba(0,0,0,0.025); background-color: rgb(252, 252, 252); -webkit-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.025); -moz-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.025);
}
.uneditable-input {
overflow: hidden; white-space: nowrap;
}
.uneditable-textarea {
width: auto; height: auto;
}
input {
color: rgb(153, 153, 153);
}
textarea {
color: rgb(153, 153, 153);
}
input:-ms-input-placeholder {
color: rgb(153, 153, 153);
}
textarea:-ms-input-placeholder {
color: rgb(153, 153, 153);
}
.radio {
padding-left: 20px; min-height: 20px;
}
.checkbox {
padding-left: 20px; min-height: 20px;
}
.radio input[type='radio'] {
margin-left: -20px; float: left;
}
.checkbox input[type='checkbox'] {
margin-left: -20px; float: left;
}
.controls > :first-child.radio {
padding-top: 5px;
}
.controls > :first-child.checkbox {
padding-top: 5px;
}
.inline.radio {
padding-top: 5px; margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.inline.checkbox {
padding-top: 5px; margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.inline.radio + .inline.radio {
margin-left: 10px;
}
.inline.checkbox + .inline.checkbox {
margin-left: 10px;
}
.input-mini {
width: 60px;
}
.input-small {
width: 90px;
}
.input-medium {
width: 150px;
}
.input-large {
width: 210px;
}
.input-xlarge {
width: 270px;
}
.input-xxlarge {
width: 530px;
}
input[class*='span'] {
margin-left: 0px; float: none;
}
select[class*='span'] {
margin-left: 0px; float: none;
}
textarea[class*='span'] {
margin-left: 0px; float: none;
}
[class*='span'].uneditable-input {
margin-left: 0px; float: none;
}
.row-fluid input[class*='span'] {
margin-left: 0px; float: none;
}
.row-fluid select[class*='span'] {
margin-left: 0px; float: none;
}
.row-fluid textarea[class*='span'] {
margin-left: 0px; float: none;
}
.row-fluid [class*='span'].uneditable-input {
margin-left: 0px; float: none;
}
.input-append input[class*='span'] {
display: inline-block;
}
.input-append [class*='span'].uneditable-input {
display: inline-block;
}
.input-prepend input[class*='span'] {
display: inline-block;
}
.input-prepend [class*='span'].uneditable-input {
display: inline-block;
}
.row-fluid input[class*='span'] {
display: inline-block;
}
.row-fluid select[class*='span'] {
display: inline-block;
}
.row-fluid textarea[class*='span'] {
display: inline-block;
}
.row-fluid [class*='span'].uneditable-input {
display: inline-block;
}
.row-fluid .input-prepend [class*='span'] {
display: inline-block;
}
.row-fluid .input-append [class*='span'] {
display: inline-block;
}
input {
margin-left: 0px;
}
textarea {
margin-left: 0px;
}
.uneditable-input {
margin-left: 0px;
}
.controls-row [class*='span'] + [class*='span'] {
margin-left: 24px;
}
input.span12 {
width: 946px;
}
textarea.span12 {
width: 946px;
}
.span12.uneditable-input {
width: 946px;
}
input.span11 {
width: 864px;
}
textarea.span11 {
width: 864px;
}
.span11.uneditable-input {
width: 864px;
}
input.span10 {
width: 782px;
}
textarea.span10 {
width: 782px;
}
.span10.uneditable-input {
width: 782px;
}
input.span9 {
width: 700px;
}
textarea.span9 {
width: 700px;
}
.span9.uneditable-input {
width: 700px;
}
input.span8 {
width: 618px;
}
textarea.span8 {
width: 618px;
}
.span8.uneditable-input {
width: 618px;
}
input.span7 {
width: 536px;
}
textarea.span7 {
width: 536px;
}
.span7.uneditable-input {
width: 536px;
}
input.span6 {
width: 454px;
}
textarea.span6 {
width: 454px;
}
.span6.uneditable-input {
width: 454px;
}
input.span5 {
width: 372px;
}
textarea.span5 {
width: 372px;
}
.span5.uneditable-input {
width: 372px;
}
input.span4 {
width: 290px;
}
textarea.span4 {
width: 290px;
}
.span4.uneditable-input {
width: 290px;
}
input.span3 {
width: 208px;
}
textarea.span3 {
width: 208px;
}
.span3.uneditable-input {
width: 208px;
}
input.span2 {
width: 126px;
}
textarea.span2 {
width: 126px;
}
.span2.uneditable-input {
width: 126px;
}
input.span1 {
width: 44px;
}
textarea.span1 {
width: 44px;
}
.span1.uneditable-input {
width: 44px;
}
.controls-row {
}
.controls-row::before {
line-height: 0; display: table; content: "";
}
.controls-row::after {
line-height: 0; display: table; content: "";
}
.controls-row::after {
clear: both;
}
.controls-row::before {
line-height: 0; display: table; content: "";
}
.controls-row::after {
line-height: 0; display: table; content: "";
}
.controls-row::after {
clear: both;
}
.controls-row [class*='span'] {
float: left;
}
.row-fluid .controls-row [class*='span'] {
float: left;
}
.controls-row [class*='span'].checkbox {
padding-top: 5px;
}
.controls-row [class*='span'].radio {
padding-top: 5px;
}
input[disabled] {
cursor: not-allowed; background-color: rgb(238, 238, 238);
}
select[disabled] {
cursor: not-allowed; background-color: rgb(238, 238, 238);
}
textarea[disabled] {
cursor: not-allowed; background-color: rgb(238, 238, 238);
}
input[readonly] {
cursor: not-allowed; background-color: rgb(238, 238, 238);
}
select[readonly] {
cursor: not-allowed; background-color: rgb(238, 238, 238);
}
textarea[readonly] {
cursor: not-allowed; background-color: rgb(238, 238, 238);
}
input[type='radio'][disabled] {
background-color: transparent;
}
input[type='checkbox'][disabled] {
background-color: transparent;
}
input[type='radio'][readonly] {
background-color: transparent;
}
input[type='checkbox'][readonly] {
background-color: transparent;
}
input:invalid:focus {
border-color: rgb(238, 95, 91); color: rgb(185, 74, 72);
}
textarea:invalid:focus {
border-color: rgb(238, 95, 91); color: rgb(185, 74, 72);
}
select:invalid:focus {
border-color: rgb(238, 95, 91); color: rgb(185, 74, 72);
}
input:invalid:focus {
border-color: rgb(233, 50, 45); box-shadow: 0px 0px 6px #f8b9b7; -webkit-box-shadow: 0 0 6px #f8b9b7; -moz-box-shadow: 0 0 6px #f8b9b7;
}
textarea:invalid:focus {
border-color: rgb(233, 50, 45); box-shadow: 0px 0px 6px #f8b9b7; -webkit-box-shadow: 0 0 6px #f8b9b7; -moz-box-shadow: 0 0 6px #f8b9b7;
}
select:invalid:focus {
border-color: rgb(233, 50, 45); box-shadow: 0px 0px 6px #f8b9b7; -webkit-box-shadow: 0 0 6px #f8b9b7; -moz-box-shadow: 0 0 6px #f8b9b7;
}
.form-actions {
padding: 19px 20px 20px; margin-top: 20px; margin-bottom: 20px; border-top-color: rgb(229, 229, 229); border-top-width: 1px; border-top-style: solid; background-color: rgb(245, 245, 245);
}
.form-actions::before {
line-height: 0; display: table; content: "";
}
.form-actions::after {
line-height: 0; display: table; content: "";
}
.form-actions::after {
clear: both;
}
.form-actions::before {
line-height: 0; display: table; content: "";
}
.form-actions::after {
line-height: 0; display: table; content: "";
}
.form-actions::after {
clear: both;
}
.help-block {
color: rgb(89, 89, 89);
}
.help-inline {
color: rgb(89, 89, 89);
}
.help-block {
margin-bottom: 10px; display: block;
}
.help-inline {
padding-left: 5px; vertical-align: middle; display: inline-block;
}
.input-append {
font-size: 0px; margin-bottom: 10px; vertical-align: middle; display: inline-block; white-space: nowrap;
}
.input-prepend {
font-size: 0px; margin-bottom: 10px; vertical-align: middle; display: inline-block; white-space: nowrap;
}
.input-append input {
font-size: 14px;
}
.input-prepend input {
font-size: 14px;
}
.input-append select {
font-size: 14px;
}
.input-prepend select {
font-size: 14px;
}
.input-append .uneditable-input {
font-size: 14px;
}
.input-prepend .uneditable-input {
font-size: 14px;
}
.input-append .dropdown-menu {
font-size: 14px;
}
.input-prepend .dropdown-menu {
font-size: 14px;
}
.input-append .popover {
font-size: 14px;
}
.input-prepend .popover {
font-size: 14px;
}
.input-append input {
border-radius: 0px 4px 4px 0px; margin-bottom: 0px; vertical-align: top; position: relative; -webkit-border-radius: 0 4px 4px 0; -moz-border-radius: 0 4px 4px 0;
}
.input-prepend input {
border-radius: 0px 4px 4px 0px; margin-bottom: 0px; vertical-align: top; position: relative; -webkit-border-radius: 0 4px 4px 0; -moz-border-radius: 0 4px 4px 0;
}
.input-append select {
border-radius: 0px 4px 4px 0px; margin-bottom: 0px; vertical-align: top; position: relative; -webkit-border-radius: 0 4px 4px 0; -moz-border-radius: 0 4px 4px 0;
}
.input-prepend select {
border-radius: 0px 4px 4px 0px; margin-bottom: 0px; vertical-align: top; position: relative; -webkit-border-radius: 0 4px 4px 0; -moz-border-radius: 0 4px 4px 0;
}
.input-append .uneditable-input {
border-radius: 0px 4px 4px 0px; margin-bottom: 0px; vertical-align: top; position: relative; -webkit-border-radius: 0 4px 4px 0; -moz-border-radius: 0 4px 4px 0;
}
.input-prepend .uneditable-input {
border-radius: 0px 4px 4px 0px; margin-bottom: 0px; vertical-align: top; position: relative; -webkit-border-radius: 0 4px 4px 0; -moz-border-radius: 0 4px 4px 0;
}
.input-append input:focus {
z-index: 2;
}
.input-prepend input:focus {
z-index: 2;
}
.input-append select:focus {
z-index: 2;
}
.input-prepend select:focus {
z-index: 2;
}
.input-append .uneditable-input:focus {
z-index: 2;
}
.input-prepend .uneditable-input:focus {
z-index: 2;
}
.input-append .add-on {
padding: 4px 5px; border: 1px solid rgb(204, 204, 204); border-image: none; width: auto; height: 20px; text-align: center; line-height: 20px; font-size: 14px; font-weight: normal; display: inline-block; min-width: 16px; text-shadow: 0px 1px 0px #ffffff; background-color: rgb(238, 238, 238);
}
.input-prepend .add-on {
padding: 4px 5px; border: 1px solid rgb(204, 204, 204); border-image: none; width: auto; height: 20px; text-align: center; line-height: 20px; font-size: 14px; font-weight: normal; display: inline-block; min-width: 16px; text-shadow: 0px 1px 0px #ffffff; background-color: rgb(238, 238, 238);
}
.input-append .add-on {
border-radius: 0px; vertical-align: top; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.input-prepend .add-on {
border-radius: 0px; vertical-align: top; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.input-append .btn {
border-radius: 0px; vertical-align: top; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.input-prepend .btn {
border-radius: 0px; vertical-align: top; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.input-append .btn-group > .dropdown-toggle {
border-radius: 0px; vertical-align: top; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.input-prepend .btn-group > .dropdown-toggle {
border-radius: 0px; vertical-align: top; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.input-append .active {
border-color: rgb(70, 165, 70); background-color: rgb(169, 219, 169);
}
.input-prepend .active {
border-color: rgb(70, 165, 70); background-color: rgb(169, 219, 169);
}
.input-prepend .add-on {
margin-right: -1px;
}
.input-prepend .btn {
margin-right: -1px;
}
.input-prepend :first-child.add-on {
border-radius: 4px 0px 0px 4px; -webkit-border-radius: 4px 0 0 4px; -moz-border-radius: 4px 0 0 4px;
}
.input-prepend :first-child.btn {
border-radius: 4px 0px 0px 4px; -webkit-border-radius: 4px 0 0 4px; -moz-border-radius: 4px 0 0 4px;
}
.input-append input {
border-radius: 4px 0px 0px 4px; -webkit-border-radius: 4px 0 0 4px; -moz-border-radius: 4px 0 0 4px;
}
.input-append select {
border-radius: 4px 0px 0px 4px; -webkit-border-radius: 4px 0 0 4px; -moz-border-radius: 4px 0 0 4px;
}
.input-append .uneditable-input {
border-radius: 4px 0px 0px 4px; -webkit-border-radius: 4px 0 0 4px; -moz-border-radius: 4px 0 0 4px;
}
.input-append input + .btn-group :last-child.btn {
border-radius: 0px 4px 4px 0px; -webkit-border-radius: 0 4px 4px 0; -moz-border-radius: 0 4px 4px 0;
}
.input-append select + .btn-group :last-child.btn {
border-radius: 0px 4px 4px 0px; -webkit-border-radius: 0 4px 4px 0; -moz-border-radius: 0 4px 4px 0;
}
.input-append .uneditable-input + .btn-group :last-child.btn {
border-radius: 0px 4px 4px 0px; -webkit-border-radius: 0 4px 4px 0; -moz-border-radius: 0 4px 4px 0;
}
.input-append .add-on {
margin-left: -1px;
}
.input-append .btn {
margin-left: -1px;
}
.input-append .btn-group {
margin-left: -1px;
}
.input-append :last-child.add-on {
border-radius: 0px 4px 4px 0px; -webkit-border-radius: 0 4px 4px 0; -moz-border-radius: 0 4px 4px 0;
}
.input-append :last-child.btn {
border-radius: 0px 4px 4px 0px; -webkit-border-radius: 0 4px 4px 0; -moz-border-radius: 0 4px 4px 0;
}
.input-append :last-child.btn-group > .dropdown-toggle {
border-radius: 0px 4px 4px 0px; -webkit-border-radius: 0 4px 4px 0; -moz-border-radius: 0 4px 4px 0;
}
.input-append.input-prepend input {
border-radius: 0px; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.input-append.input-prepend select {
border-radius: 0px; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.input-append.input-prepend .uneditable-input {
border-radius: 0px; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.input-append.input-prepend input + .btn-group .btn {
border-radius: 0px 4px 4px 0px; -webkit-border-radius: 0 4px 4px 0; -moz-border-radius: 0 4px 4px 0;
}
.input-append.input-prepend select + .btn-group .btn {
border-radius: 0px 4px 4px 0px; -webkit-border-radius: 0 4px 4px 0; -moz-border-radius: 0 4px 4px 0;
}
.input-append.input-prepend .uneditable-input + .btn-group .btn {
border-radius: 0px 4px 4px 0px; -webkit-border-radius: 0 4px 4px 0; -moz-border-radius: 0 4px 4px 0;
}
.input-append.input-prepend :first-child.add-on {
border-radius: 4px 0px 0px 4px; margin-right: -1px; -webkit-border-radius: 4px 0 0 4px; -moz-border-radius: 4px 0 0 4px;
}
.input-append.input-prepend :first-child.btn {
border-radius: 4px 0px 0px 4px; margin-right: -1px; -webkit-border-radius: 4px 0 0 4px; -moz-border-radius: 4px 0 0 4px;
}
.input-append.input-prepend :last-child.add-on {
border-radius: 0px 4px 4px 0px; margin-left: -1px; -webkit-border-radius: 0 4px 4px 0; -moz-border-radius: 0 4px 4px 0;
}
.input-append.input-prepend :last-child.btn {
border-radius: 0px 4px 4px 0px; margin-left: -1px; -webkit-border-radius: 0 4px 4px 0; -moz-border-radius: 0 4px 4px 0;
}
.input-append.input-prepend :first-child.btn-group {
margin-left: 0px;
}
input.search-query {
border-radius: 15px; padding-right: 14px; padding-left: 14px; margin-bottom: 0px; -webkit-border-radius: 15px; -moz-border-radius: 15px;
}
.form-search .input-append .search-query {
border-radius: 0px; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.form-search .input-prepend .search-query {
border-radius: 0px; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.form-search .input-append .search-query {
border-radius: 14px 0px 0px 14px; -webkit-border-radius: 14px 0 0 14px; -moz-border-radius: 14px 0 0 14px;
}
.form-search .input-append .btn {
border-radius: 0px 14px 14px 0px; -webkit-border-radius: 0 14px 14px 0; -moz-border-radius: 0 14px 14px 0;
}
.form-search .input-prepend .search-query {
border-radius: 0px 14px 14px 0px; -webkit-border-radius: 0 14px 14px 0; -moz-border-radius: 0 14px 14px 0;
}
.form-search .input-prepend .btn {
border-radius: 14px 0px 0px 14px; -webkit-border-radius: 14px 0 0 14px; -moz-border-radius: 14px 0 0 14px;
}
.form-search input {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-inline input {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-horizontal input {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-search textarea {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-inline textarea {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-horizontal textarea {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-search select {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-inline select {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-horizontal select {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-search .help-inline {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-inline .help-inline {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-horizontal .help-inline {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-search .uneditable-input {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-inline .uneditable-input {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-horizontal .uneditable-input {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-search .input-prepend {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-inline .input-prepend {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-horizontal .input-prepend {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-search .input-append {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-inline .input-append {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-horizontal .input-append {
margin-bottom: 0px; vertical-align: middle; display: inline-block;
}
.form-search .hide {
display: none;
}
.form-inline .hide {
display: none;
}
.form-horizontal .hide {
display: none;
}
.form-search label {
display: inline-block;
}
.form-inline label {
display: inline-block;
}
.form-search .btn-group {
display: inline-block;
}
.form-inline .btn-group {
display: inline-block;
}
.form-search .input-append {
margin-bottom: 0px;
}
.form-inline .input-append {
margin-bottom: 0px;
}
.form-search .input-prepend {
margin-bottom: 0px;
}
.form-inline .input-prepend {
margin-bottom: 0px;
}
.form-search .radio {
padding-left: 0px; margin-bottom: 0px; vertical-align: middle;
}
.form-search .checkbox {
padding-left: 0px; margin-bottom: 0px; vertical-align: middle;
}
.form-inline .radio {
padding-left: 0px; margin-bottom: 0px; vertical-align: middle;
}
.form-inline .checkbox {
padding-left: 0px; margin-bottom: 0px; vertical-align: middle;
}
.form-search .radio input[type='radio'] {
margin-right: 3px; margin-left: 0px; float: left;
}
.form-search .checkbox input[type='checkbox'] {
margin-right: 3px; margin-left: 0px; float: left;
}
.form-inline .radio input[type='radio'] {
margin-right: 3px; margin-left: 0px; float: left;
}
.form-inline .checkbox input[type='checkbox'] {
margin-right: 3px; margin-left: 0px; float: left;
}
.control-group {
margin-bottom: 10px;
}
legend + .control-group {
margin-top: 20px; -webkit-margin-top-collapse: separate;
}
.form-horizontal .control-group {
margin-bottom: 20px;
}
.form-horizontal .control-group::before {
line-height: 0; display: table; content: "";
}
.form-horizontal .control-group::after {
line-height: 0; display: table; content: "";
}
.form-horizontal .control-group::after {
clear: both;
}
.form-horizontal .control-group::before {
line-height: 0; display: table; content: "";
}
.form-horizontal .control-group::after {
line-height: 0; display: table; content: "";
}
.form-horizontal .control-group::after {
clear: both;
}
.form-horizontal .control-label {
width: 160px; text-align: right; padding-top: 5px; float: left;
}
.form-horizontal .controls {
margin-left: 180px;
}
.form-horizontal :first-child.controls {
}
.form-horizontal .help-block {
margin-bottom: 0px;
}
.form-horizontal input + .help-block {
margin-top: 10px;
}
.form-horizontal select + .help-block {
margin-top: 10px;
}
.form-horizontal textarea + .help-block {
margin-top: 10px;
}
.form-horizontal .uneditable-input + .help-block {
margin-top: 10px;
}
.form-horizontal .input-prepend + .help-block {
margin-top: 10px;
}
.form-horizontal .input-append + .help-block {
margin-top: 10px;
}
.form-horizontal .form-actions {
padding-left: 180px;
}
table {
border-collapse: collapse; max-width: 100%; border-spacing: 0; background-color: transparent;
}
.table {
width: 100%; margin-bottom: 20px;
}
.table th {
padding: 8px; text-align: left; line-height: 20px; vertical-align: top; border-top-color: rgb(221, 221, 221); border-top-width: 1px; border-top-style: solid;
}
.table td {
padding: 8px; text-align: left; line-height: 20px; vertical-align: top; border-top-color: rgb(221, 221, 221); border-top-width: 1px; border-top-style: solid;
}
.table th {
font-weight: bold;
}
.table thead th {
vertical-align: bottom;
}
.table caption + thead tr:first-child th {
border-top-color: currentColor; border-top-width: 0px; border-top-style: none;
}
.table caption + thead tr:first-child td {
border-top-color: currentColor; border-top-width: 0px; border-top-style: none;
}
.table colgroup + thead tr:first-child th {
border-top-color: currentColor; border-top-width: 0px; border-top-style: none;
}
.table colgroup + thead tr:first-child td {
border-top-color: currentColor; border-top-width: 0px; border-top-style: none;
}
.table thead:first-child tr:first-child th {
border-top-color: currentColor; border-top-width: 0px; border-top-style: none;
}
.table thead:first-child tr:first-child td {
border-top-color: currentColor; border-top-width: 0px; border-top-style: none;
}
.table tbody + tbody {
border-top-color: rgb(221, 221, 221); border-top-width: 2px; border-top-style: solid;
}
.table .table {
background-color: rgb(255, 255, 255);
}
.table-condensed th {
padding: 4px 5px;
}
.table-condensed td {
padding: 4px 5px;
}
.table-bordered {
border-width: 1px 1px 1px 0px; border-style: solid solid solid none; border-color: rgb(221, 221, 221) rgb(221, 221, 221) rgb(221, 221, 221) currentColor; border-radius: 4px; border-image: none; border-collapse: separate; -webkit-border-radius: 4px; -moz-border-radius: 4px;
}
.table-bordered th {
border-left-color: rgb(221, 221, 221); border-left-width: 1px; border-left-style: solid;
}
.table-bordered td {
border-left-color: rgb(221, 221, 221); border-left-width: 1px; border-left-style: solid;
}
.table-bordered caption + thead tr:first-child th {
border-top-color: currentColor; border-top-width: 0px; border-top-style: none;
}
.table-bordered caption + tbody tr:first-child th {
border-top-color: currentColor; border-top-width: 0px; border-top-style: none;
}
.table-bordered caption + tbody tr:first-child td {
border-top-color: currentColor; border-top-width: 0px; border-top-style: none;
}
.table-bordered colgroup + thead tr:first-child th {
border-top-color: currentColor; border-top-width: 0px; border-top-style: none;
}
.table-bordered colgroup + tbody tr:first-child th {
border-top-color: currentColor; border-top-width: 0px; border-top-style: none;
}
.table-bordered colgroup + tbody tr:first-child td {
border-top-color: currentColor; border-top-width: 0px; border-top-style: none;
}
.table-bordered thead:first-child tr:first-child th {
border-top-color: currentColor; border-top-width: 0px; border-top-style: none;
}
.table-bordered tbody:first-child tr:first-child th {
border-top-color: currentColor; border-top-width: 0px; border-top-style: none;
}
.table-bordered tbody:first-child tr:first-child td {
border-top-color: currentColor; border-top-width: 0px; border-top-style: none;
}
.table-bordered thead:first-child tr:first-child > th:first-child {
border-top-left-radius: 4px; -webkit-border-top-left-radius: 4px; -moz-border-radius-topleft: 4px;
}
.table-bordered tbody:first-child tr:first-child > td:first-child {
border-top-left-radius: 4px; -webkit-border-top-left-radius: 4px; -moz-border-radius-topleft: 4px;
}
.table-bordered tbody:first-child tr:first-child > th:first-child {
border-top-left-radius: 4px; -webkit-border-top-left-radius: 4px; -moz-border-radius-topleft: 4px;
}
.table-bordered thead:first-child tr:first-child > th:last-child {
border-top-right-radius: 4px; -webkit-border-top-right-radius: 4px; -moz-border-radius-topright: 4px;
}
.table-bordered tbody:first-child tr:first-child > td:last-child {
border-top-right-radius: 4px; -webkit-border-top-right-radius: 4px; -moz-border-radius-topright: 4px;
}
.table-bordered tbody:first-child tr:first-child > th:last-child {
border-top-right-radius: 4px; -webkit-border-top-right-radius: 4px; -moz-border-radius-topright: 4px;
}
.table-bordered thead:last-child tr:last-child > th:first-child {
border-bottom-left-radius: 4px; -webkit-border-bottom-left-radius: 4px; -moz-border-radius-bottomleft: 4px;
}
.table-bordered tbody:last-child tr:last-child > td:first-child {
border-bottom-left-radius: 4px; -webkit-border-bottom-left-radius: 4px; -moz-border-radius-bottomleft: 4px;
}
.table-bordered tbody:last-child tr:last-child > th:first-child {
border-bottom-left-radius: 4px; -webkit-border-bottom-left-radius: 4px; -moz-border-radius-bottomleft: 4px;
}
.table-bordered tfoot:last-child tr:last-child > td:first-child {
border-bottom-left-radius: 4px; -webkit-border-bottom-left-radius: 4px; -moz-border-radius-bottomleft: 4px;
}
.table-bordered tfoot:last-child tr:last-child > th:first-child {
border-bottom-left-radius: 4px; -webkit-border-bottom-left-radius: 4px; -moz-border-radius-bottomleft: 4px;
}
.table-bordered thead:last-child tr:last-child > th:last-child {
border-bottom-right-radius: 4px; -webkit-border-bottom-right-radius: 4px; -moz-border-radius-bottomright: 4px;
}
.table-bordered tbody:last-child tr:last-child > td:last-child {
border-bottom-right-radius: 4px; -webkit-border-bottom-right-radius: 4px; -moz-border-radius-bottomright: 4px;
}
.table-bordered tbody:last-child tr:last-child > th:last-child {
border-bottom-right-radius: 4px; -webkit-border-bottom-right-radius: 4px; -moz-border-radius-bottomright: 4px;
}
.table-bordered tfoot:last-child tr:last-child > td:last-child {
border-bottom-right-radius: 4px; -webkit-border-bottom-right-radius: 4px; -moz-border-radius-bottomright: 4px;
}
.table-bordered tfoot:last-child tr:last-child > th:last-child {
border-bottom-right-radius: 4px; -webkit-border-bottom-right-radius: 4px; -moz-border-radius-bottomright: 4px;
}
.table-bordered tfoot + tbody:last-child tr:last-child td:first-child {
border-bottom-left-radius: 0px; -webkit-border-bottom-left-radius: 0; -moz-border-radius-bottomleft: 0;
}
.table-bordered tfoot + tbody:last-child tr:last-child td:last-child {
border-bottom-right-radius: 0px; -webkit-border-bottom-right-radius: 0; -moz-border-radius-bottomright: 0;
}
.table-bordered caption + thead tr:first-child th:first-child {
border-top-left-radius: 4px; -webkit-border-top-left-radius: 4px; -moz-border-radius-topleft: 4px;
}
.table-bordered caption + tbody tr:first-child td:first-child {
border-top-left-radius: 4px; -webkit-border-top-left-radius: 4px; -moz-border-radius-topleft: 4px;
}
.table-bordered colgroup + thead tr:first-child th:first-child {
border-top-left-radius: 4px; -webkit-border-top-left-radius: 4px; -moz-border-radius-topleft: 4px;
}
.table-bordered colgroup + tbody tr:first-child td:first-child {
border-top-left-radius: 4px; -webkit-border-top-left-radius: 4px; -moz-border-radius-topleft: 4px;
}
.table-bordered caption + thead tr:first-child th:last-child {
border-top-right-radius: 4px; -webkit-border-top-right-radius: 4px; -moz-border-radius-topright: 4px;
}
.table-bordered caption + tbody tr:first-child td:last-child {
border-top-right-radius: 4px; -webkit-border-top-right-radius: 4px; -moz-border-radius-topright: 4px;
}
.table-bordered colgroup + thead tr:first-child th:last-child {
border-top-right-radius: 4px; -webkit-border-top-right-radius: 4px; -moz-border-radius-topright: 4px;
}
.table-bordered colgroup + tbody tr:first-child td:last-child {
border-top-right-radius: 4px; -webkit-border-top-right-radius: 4px; -moz-border-radius-topright: 4px;
}
.table-striped tbody > tr:nth-child(2n+1) > td {
background-color: rgb(249, 249, 249);
}
.table-striped tbody > tr:nth-child(2n+1) > th {
background-color: rgb(249, 249, 249);
}
.table-hover tbody tr:hover > td {
background-color: rgb(245, 245, 245);
}
.table-hover tbody tr:hover > th {
background-color: rgb(245, 245, 245);
}
table td[class*='span'] {
margin-left: 0px; float: none; display: table-cell;
}
table th[class*='span'] {
margin-left: 0px; float: none; display: table-cell;
}
.row-fluid table td[class*='span'] {
margin-left: 0px; float: none; display: table-cell;
}
.row-fluid table th[class*='span'] {
margin-left: 0px; float: none; display: table-cell;
}
.table td.span1 {
width: 42px; margin-left: 0px; float: none;
}
.table th.span1 {
width: 42px; margin-left: 0px; float: none;
}
.table td.span2 {
width: 124px; margin-left: 0px; float: none;
}
.table th.span2 {
width: 124px; margin-left: 0px; float: none;
}
.table td.span3 {
width: 206px; margin-left: 0px; float: none;
}
.table th.span3 {
width: 206px; margin-left: 0px; float: none;
}
.table td.span4 {
width: 288px; margin-left: 0px; float: none;
}
.table th.span4 {
width: 288px; margin-left: 0px; float: none;
}
.table td.span5 {
width: 370px; margin-left: 0px; float: none;
}
.table th.span5 {
width: 370px; margin-left: 0px; float: none;
}
.table td.span6 {
width: 452px; margin-left: 0px; float: none;
}
.table th.span6 {
width: 452px; margin-left: 0px; float: none;
}
.table td.span7 {
width: 534px; margin-left: 0px; float: none;
}
.table th.span7 {
width: 534px; margin-left: 0px; float: none;
}
.table td.span8 {
width: 616px; margin-left: 0px; float: none;
}
.table th.span8 {
width: 616px; margin-left: 0px; float: none;
}
.table td.span9 {
width: 698px; margin-left: 0px; float: none;
}
.table th.span9 {
width: 698px; margin-left: 0px; float: none;
}
.table td.span10 {
width: 780px; margin-left: 0px; float: none;
}
.table th.span10 {
width: 780px; margin-left: 0px; float: none;
}
.table td.span11 {
width: 862px; margin-left: 0px; float: none;
}
.table th.span11 {
width: 862px; margin-left: 0px; float: none;
}
.table td.span12 {
width: 944px; margin-left: 0px; float: none;
}
.table th.span12 {
width: 944px; margin-left: 0px; float: none;
}
.table tbody tr.success > td {
background-color: rgb(223, 240, 216);
}
.table tbody tr.error > td {
background-color: rgb(242, 222, 222);
}
.table tbody tr.warning > td {
background-color: rgb(252, 248, 227);
}
.table tbody tr.info > td {
background-color: rgb(217, 237, 247);
}
.table-hover tbody tr.success:hover > td {
background-color: rgb(208, 233, 198);
}
.table-hover tbody tr.error:hover > td {
background-color: rgb(235, 204, 204);
}
.table-hover tbody tr.warning:hover > td {
background-color: rgb(250, 242, 204);
}
.table-hover tbody tr.info:hover > td {
background-color: rgb(196, 227, 243);
}
[class^='icon-'] {
background-position: 14px 14px; width: 14px; height: 14px; line-height: 14px; margin-top: 1px; vertical-align: text-top; display: inline-block; background-image: url("./fe33117a-glyphicons-halflings.png"); background-repeat: no-repeat;
}
[class*=' icon-'] {
background-position: 14px 14px; width: 14px; height: 14px; line-height: 14px; margin-top: 1px; vertical-align: text-top; display: inline-block; background-image: url("./fe33117a-glyphicons-halflings.png"); background-repeat: no-repeat;
}
.icon-white {
background-image: url("./6f98ef33-glyphicons-halflings-white.png");
}
.nav-pills > .active > a > [class^='icon-'] {
background-image: url("./6f98ef33-glyphicons-halflings-white.png");
}
.nav-pills > .active > a > [class*=' icon-'] {
background-image: url("./6f98ef33-glyphicons-halflings-white.png");
}
.nav-list > .active > a > [class^='icon-'] {
background-image: url("./6f98ef33-glyphicons-halflings-white.png");
}
.nav-list > .active > a > [class*=' icon-'] {
background-image: url("./6f98ef33-glyphicons-halflings-white.png");
}
.navbar-inverse .nav > .active > a > [class^='icon-'] {
background-image: url("./6f98ef33-glyphicons-halflings-white.png");
}
.navbar-inverse .nav > .active > a > [class*=' icon-'] {
background-image: url("./6f98ef33-glyphicons-halflings-white.png");
}
.dropdown-menu > li > a:hover > [class^='icon-'] {
background-image: url("./6f98ef33-glyphicons-halflings-white.png");
}
.dropdown-menu > li > a:focus > [class^='icon-'] {
background-image: url("./6f98ef33-glyphicons-halflings-white.png");
}
.dropdown-menu > li > a:hover > [class*=' icon-'] {
background-image: url("./6f98ef33-glyphicons-halflings-white.png");
}
.dropdown-menu > li > a:focus > [class*=' icon-'] {
background-image: url("./6f98ef33-glyphicons-halflings-white.png");
}
.dropdown-menu > .active > a > [class^='icon-'] {
background-image: url("./6f98ef33-glyphicons-halflings-white.png");
}
.dropdown-menu > .active > a > [class*=' icon-'] {
background-image: url("./6f98ef33-glyphicons-halflings-white.png");
}
.dropdown-submenu:hover > a > [class^='icon-'] {
background-image: url("./6f98ef33-glyphicons-halflings-white.png");
}
.dropdown-submenu:focus > a > [class^='icon-'] {
background-image: url("./6f98ef33-glyphicons-halflings-white.png");
}
.dropdown-submenu:hover > a > [class*=' icon-'] {
background-image: url("./6f98ef33-glyphicons-halflings-white.png");
}
.dropdown-submenu:focus > a > [class*=' icon-'] {
background-image: url("./6f98ef33-glyphicons-halflings-white.png");
}
.icon-glass {
background-position: 0px 0px;
}
.icon-music {
background-position: -24px 0px;
}
.icon-search {
background-position: -48px 0px;
}
.icon-envelope {
background-position: -72px 0px;
}
.icon-heart {
background-position: -96px 0px;
}
.icon-star {
background-position: -120px 0px;
}
.icon-star-empty {
background-position: -144px 0px;
}
.icon-user {
background-position: -168px 0px;
}
.icon-film {
background-position: -192px 0px;
}
.icon-th-large {
background-position: -216px 0px;
}
.icon-th {
background-position: -240px 0px;
}
.icon-th-list {
background-position: -264px 0px;
}
.icon-ok {
background-position: -288px 0px;
}
.icon-remove {
background-position: -312px 0px;
}
.icon-zoom-in {
background-position: -336px 0px;
}
.icon-zoom-out {
background-position: -360px 0px;
}
.icon-off {
background-position: -384px 0px;
}
.icon-signal {
background-position: -408px 0px;
}
.icon-cog {
background-position: -432px 0px;
}
.icon-trash {
background-position: -456px 0px;
}
.icon-home {
background-position: 0px -24px;
}
.icon-file {
background-position: -24px -24px;
}
.icon-time {
background-position: -48px -24px;
}
.icon-road {
background-position: -72px -24px;
}
.icon-download-alt {
background-position: -96px -24px;
}
.icon-download {
background-position: -120px -24px;
}
.icon-upload {
background-position: -144px -24px;
}
.icon-inbox {
background-position: -168px -24px;
}
.icon-play-circle {
background-position: -192px -24px;
}
.icon-repeat {
background-position: -216px -24px;
}
.icon-refresh {
background-position: -240px -24px;
}
.icon-list-alt {
background-position: -264px -24px;
}
.icon-lock {
background-position: -287px -24px;
}
.icon-flag {
background-position: -312px -24px;
}
.icon-headphones {
background-position: -336px -24px;
}
.icon-volume-off {
background-position: -360px -24px;
}
.icon-volume-down {
background-position: -384px -24px;
}
.icon-volume-up {
background-position: -408px -24px;
}
.icon-qrcode {
background-position: -432px -24px;
}
.icon-barcode {
background-position: -456px -24px;
}
.icon-tag {
background-position: 0px -48px;
}
.icon-tags {
background-position: -25px -48px;
}
.icon-book {
background-position: -48px -48px;
}
.icon-bookmark {
background-position: -72px -48px;
}
.icon-print {
background-position: -96px -48px;
}
.icon-camera {
background-position: -120px -48px;
}
.icon-font {
background-position: -144px -48px;
}
.icon-bold {
background-position: -167px -48px;
}
.icon-italic {
background-position: -192px -48px;
}
.icon-text-height {
background-position: -216px -48px;
}
.icon-text-width {
background-position: -240px -48px;
}
.icon-align-left {
background-position: -264px -48px;
}
.icon-align-center {
background-position: -288px -48px;
}
.icon-align-right {
background-position: -312px -48px;
}
.icon-align-justify {
background-position: -336px -48px;
}
.icon-list {
background-position: -360px -48px;
}
.icon-indent-left {
background-position: -384px -48px;
}
.icon-indent-right {
background-position: -408px -48px;
}
.icon-facetime-video {
background-position: -432px -48px;
}
.icon-picture {
background-position: -456px -48px;
}
.icon-pencil {
background-position: 0px -72px;
}
.icon-map-marker {
background-position: -24px -72px;
}
.icon-adjust {
background-position: -48px -72px;
}
.icon-tint {
background-position: -72px -72px;
}
.icon-edit {
background-position: -96px -72px;
}
.icon-share {
background-position: -120px -72px;
}
.icon-check {
background-position: -144px -72px;
}
.icon-move {
background-position: -168px -72px;
}
.icon-step-backward {
background-position: -192px -72px;
}
.icon-fast-backward {
background-position: -216px -72px;
}
.icon-backward {
background-position: -240px -72px;
}
.icon-play {
background-position: -264px -72px;
}
.icon-pause {
background-position: -288px -72px;
}
.icon-stop {
background-position: -312px -72px;
}
.icon-forward {
background-position: -336px -72px;
}
.icon-fast-forward {
background-position: -360px -72px;
}
.icon-step-forward {
background-position: -384px -72px;
}
.icon-eject {
background-position: -408px -72px;
}
.icon-chevron-left {
background-position: -432px -72px;
}
.icon-chevron-right {
background-position: -456px -72px;
}
.icon-plus-sign {
background-position: 0px -96px;
}
.icon-minus-sign {
background-position: -24px -96px;
}
.icon-remove-sign {
background-position: -48px -96px;
}
.icon-ok-sign {
background-position: -72px -96px;
}
.icon-question-sign {
background-position: -96px -96px;
}
.icon-info-sign {
background-position: -120px -96px;
}
.icon-screenshot {
background-position: -144px -96px;
}
.icon-remove-circle {
background-position: -168px -96px;
}
.icon-ok-circle {
background-position: -192px -96px;
}
.icon-ban-circle {
background-position: -216px -96px;
}
.icon-arrow-left {
background-position: -240px -96px;
}
.icon-arrow-right {
background-position: -264px -96px;
}
.icon-arrow-up {
background-position: -289px -96px;
}
.icon-arrow-down {
background-position: -312px -96px;
}
.icon-share-alt {
background-position: -336px -96px;
}
.icon-resize-full {
background-position: -360px -96px;
}
.icon-resize-small {
background-position: -384px -96px;
}
.icon-plus {
background-position: -408px -96px;
}
.icon-minus {
background-position: -433px -96px;
}
.icon-asterisk {
background-position: -456px -96px;
}
.icon-exclamation-sign {
background-position: 0px -120px;
}
.icon-gift {
background-position: -24px -120px;
}
.icon-leaf {
background-position: -48px -120px;
}
.icon-fire {
background-position: -72px -120px;
}
.icon-eye-open {
background-position: -96px -120px;
}
.icon-eye-close {
background-position: -120px -120px;
}
.icon-warning-sign {
background-position: -144px -120px;
}
.icon-plane {
background-position: -168px -120px;
}
.icon-calendar {
background-position: -192px -120px;
}
.icon-random {
background-position: -216px -120px; width: 16px;
}
.icon-comment {
background-position: -240px -120px;
}
.icon-magnet {
background-position: -264px -120px;
}
.icon-chevron-up {
background-position: -288px -120px;
}
.icon-chevron-down {
background-position: -313px -119px;
}
.icon-retweet {
background-position: -336px -120px;
}
.icon-shopping-cart {
background-position: -360px -120px;
}
.icon-folder-close {
background-position: -384px -120px; width: 16px;
}
.icon-folder-open {
background-position: -408px -120px; width: 16px;
}
.icon-resize-vertical {
background-position: -432px -119px;
}
.icon-resize-horizontal {
background-position: -456px -118px;
}
.icon-hdd {
background-position: 0px -144px;
}
.icon-bullhorn {
background-position: -24px -144px;
}
.icon-bell {
background-position: -48px -144px;
}
.icon-certificate {
background-position: -72px -144px;
}
.icon-thumbs-up {
background-position: -96px -144px;
}
.icon-thumbs-down {
background-position: -120px -144px;
}
.icon-hand-right {
background-position: -144px -144px;
}
.icon-hand-left {
background-position: -168px -144px;
}
.icon-hand-up {
background-position: -192px -144px;
}
.icon-hand-down {
background-position: -216px -144px;
}
.icon-circle-arrow-right {
background-position: -240px -144px;
}
.icon-circle-arrow-left {
background-position: -264px -144px;
}
.icon-circle-arrow-up {
background-position: -288px -144px;
}
.icon-circle-arrow-down {
background-position: -312px -144px;
}
.icon-globe {
background-position: -336px -144px;
}
.icon-wrench {
background-position: -360px -144px;
}
.icon-tasks {
background-position: -384px -144px;
}
.icon-filter {
background-position: -408px -144px;
}
.icon-briefcase {
background-position: -432px -144px;
}
.icon-fullscreen {
background-position: -456px -144px;
}
.close {
color: rgb(0, 0, 0); line-height: 20px; font-size: 20px; font-weight: bold; float: right; opacity: 0.2; text-shadow: 0px 1px 0px #ffffff;
}
.close:hover {
color: rgb(0, 0, 0); text-decoration: none; cursor: pointer; opacity: 0.4;
}
.close:focus {
color: rgb(0, 0, 0); text-decoration: none; cursor: pointer; opacity: 0.4;
}
button.close {
background: none; padding: 0px; border: 0px currentColor; border-image: none; cursor: pointer; -webkit-appearance: none;
}
.nav {
list-style: none; margin-bottom: 20px; margin-left: 0px;
}
.nav > li > a {
display: block;
}
.nav > li > a:hover {
text-decoration: none; background-color: rgb(238, 238, 238);
}
.nav > li > a:focus {
text-decoration: none; background-color: rgb(238, 238, 238);
}
.nav > li > a > img {
max-width: none;
}
.nav > .pull-right {
float: right;
}
.nav-header {
padding: 3px 15px; color: rgb(153, 153, 153); text-transform: uppercase; line-height: 20px; font-size: 11px; font-weight: bold; display: block; text-shadow: 0px 1px 0px rgba(255,255,255,0.5);
}
.nav li + .nav-header {
margin-top: 9px;
}
.nav > .disabled > a {
color: rgb(153, 153, 153);
}
.nav > .disabled > a:hover {
text-decoration: none; cursor: default; background-color: transparent;
}
.nav > .disabled > a:focus {
text-decoration: none; cursor: default; background-color: transparent;
}
.navbar {
overflow: visible; margin-bottom: 20px;
}
.navbar-inner {
border-radius: 4px; border: 1px solid rgb(212, 212, 212); border-image: none; padding-right: 20px; padding-left: 20px; min-height: 40px; box-shadow: 0px 1px 4px rgba(0,0,0,0.065); -webkit-border-radius: 4px; -moz-border-radius: 4px; -webkit-box-shadow: 0 1px 4px rgba(0, 0, 0, 0.065); -moz-box-shadow: 0 1px 4px rgba(0, 0, 0, 0.065);
}
.navbar-inner::before {
line-height: 0; display: table; content: "";
}
.navbar-inner::after {
line-height: 0; display: table; content: "";
}
.navbar-inner::after {
clear: both;
}
.navbar-inner::before {
line-height: 0; display: table; content: "";
}
.navbar-inner::after {
line-height: 0; display: table; content: "";
}
.navbar-inner::after {
clear: both;
}
.navbar .container {
width: auto;
}
.collapse.nav-collapse {
height: auto; overflow: visible;
}
.navbar .brand {
padding: 10px 20px; color: rgb(119, 119, 119); font-size: 20px; font-weight: 200; margin-left: -20px; float: left; display: block; text-shadow: 0px 1px 0px #ffffff;
}
.navbar .brand:hover {
text-decoration: none;
}
.navbar .brand:focus {
text-decoration: none;
}
.navbar-text {
color: rgb(119, 119, 119); line-height: 40px; margin-bottom: 0px;
}
.navbar-link {
color: rgb(119, 119, 119);
}
.navbar-link:hover {
color: rgb(51, 51, 51);
}
.navbar-link:focus {
color: rgb(51, 51, 51);
}
.navbar .divider-vertical {
margin: 0px 9px; height: 40px; border-right-color: rgb(255, 255, 255); border-left-color: rgb(242, 242, 242); border-right-width: 1px; border-left-width: 1px; border-right-style: solid; border-left-style: solid;
}
.navbar-static-top {
margin-bottom: 0px; position: static;
}
.navbar-static-top .navbar-inner {
border-radius: 0px; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.navbar-fixed-top {
left: 0px; right: 0px; margin-bottom: 0px; position: fixed; z-index: 1030;
}
.navbar-fixed-bottom {
left: 0px; right: 0px; margin-bottom: 0px; position: fixed; z-index: 1030;
}
.navbar-fixed-top .navbar-inner {
border-width: 0px 0px 1px;
}
.navbar-static-top .navbar-inner {
border-width: 0px 0px 1px;
}
.navbar-fixed-bottom .navbar-inner {
border-width: 1px 0px 0px;
}
.navbar-fixed-top .navbar-inner {
border-radius: 0px; padding-right: 0px; padding-left: 0px; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.navbar-fixed-bottom .navbar-inner {
border-radius: 0px; padding-right: 0px; padding-left: 0px; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.navbar-static-top .container {
width: 960px;
}
.navbar-fixed-top .container {
width: 960px;
}
.navbar-fixed-bottom .container {
width: 960px;
}
.navbar-fixed-top {
top: 0px;
}
.navbar-fixed-top .navbar-inner {
box-shadow: 0px 1px 10px rgba(0,0,0,0.1); -webkit-box-shadow: 0 1px 10px rgba(0, 0, 0, .1); -moz-box-shadow: 0 1px 10px rgba(0, 0, 0, .1);
}
.navbar-static-top .navbar-inner {
box-shadow: 0px 1px 10px rgba(0,0,0,0.1); -webkit-box-shadow: 0 1px 10px rgba(0, 0, 0, .1); -moz-box-shadow: 0 1px 10px rgba(0, 0, 0, .1);
}
.navbar-fixed-bottom {
bottom: 0px;
}
.navbar-fixed-bottom .navbar-inner {
box-shadow: 0px -1px 10px rgba(0,0,0,0.1); -webkit-box-shadow: 0 -1px 10px rgba(0, 0, 0, .1); -moz-box-shadow: 0 -1px 10px rgba(0, 0, 0, .1);
}
.navbar .nav {
margin: 0px 10px 0px 0px; left: 0px; float: left; display: block; position: relative;
}
.navbar .pull-right.nav {
margin-right: 0px; float: right;
}
.navbar .nav > li {
float: left;
}
.navbar .nav > li > a {
padding: 10px 15px; color: rgb(119, 119, 119); text-decoration: none; float: none; text-shadow: 0px 1px 0px #ffffff;
}
.navbar .nav .dropdown-toggle .caret {
margin-top: 8px;
}
.navbar .nav > li > a:focus {
color: rgb(51, 51, 51); text-decoration: none; background-color: transparent;
}
.navbar .nav > li > a:hover {
color: rgb(51, 51, 51); text-decoration: none; background-color: transparent;
}
.navbar .nav > .active > a {
color: rgb(85, 85, 85); text-decoration: none; box-shadow: inset 0px 3px 8px rgba(0,0,0,0.125); background-color: rgb(229, 229, 229); -webkit-box-shadow: inset 0 3px 8px rgba(0, 0, 0, 0.125); -moz-box-shadow: inset 0 3px 8px rgba(0, 0, 0, 0.125);
}
.navbar .nav > .active > a:hover {
color: rgb(85, 85, 85); text-decoration: none; box-shadow: inset 0px 3px 8px rgba(0,0,0,0.125); background-color: rgb(229, 229, 229); -webkit-box-shadow: inset 0 3px 8px rgba(0, 0, 0, 0.125); -moz-box-shadow: inset 0 3px 8px rgba(0, 0, 0, 0.125);
}
.navbar .nav > .active > a:focus {
color: rgb(85, 85, 85); text-decoration: none; box-shadow: inset 0px 3px 8px rgba(0,0,0,0.125); background-color: rgb(229, 229, 229); -webkit-box-shadow: inset 0 3px 8px rgba(0, 0, 0, 0.125); -moz-box-shadow: inset 0 3px 8px rgba(0, 0, 0, 0.125);
}
.navbar .pull-right > li > .dropdown-menu {
left: auto; right: 0px;
}
.navbar .nav > li > .pull-right.dropdown-menu {
left: auto; right: 0px;
}
.navbar .pull-right > li > .dropdown-menu::before {
left: auto; right: 12px;
}
.navbar .nav > li > .pull-right.dropdown-menu::before {
left: auto; right: 12px;
}
.navbar .pull-right > li > .dropdown-menu::after {
left: auto; right: 13px;
}
.navbar .nav > li > .pull-right.dropdown-menu::after {
left: auto; right: 13px;
}
.navbar .pull-right > li > .dropdown-menu .dropdown-menu {
border-radius: 6px 0px 6px 6px; left: auto; right: 100%; margin-right: -1px; margin-left: 0px; -webkit-border-radius: 6px 0 6px 6px; -moz-border-radius: 6px 0 6px 6px;
}
.navbar .nav > li > .pull-right.dropdown-menu .dropdown-menu {
border-radius: 6px 0px 6px 6px; left: auto; right: 100%; margin-right: -1px; margin-left: 0px; -webkit-border-radius: 6px 0 6px 6px; -moz-border-radius: 6px 0 6px 6px;
}
.pagination {
margin: 20px 0px;
}
.pagination ul {
border-radius: 4px; margin-bottom: 0px; margin-left: 0px; display: inline-block; box-shadow: 0px 1px 2px rgba(0,0,0,0.05); -webkit-border-radius: 4px; -moz-border-radius: 4px; -webkit-box-shadow: 0 1px 2px rgba(0, 0, 0, 0.05); -moz-box-shadow: 0 1px 2px rgba(0, 0, 0, 0.05);
}
.pagination ul > li {
display: inline;
}
.pagination ul > li > a {
border-width: 1px 1px 1px 0px; border-style: solid; border-color: rgb(221, 221, 221); padding: 4px 12px; border-image: none; line-height: 20px; text-decoration: none; float: left; background-color: rgb(255, 255, 255);
}
.pagination ul > li > span {
border-width: 1px 1px 1px 0px; border-style: solid; border-color: rgb(221, 221, 221); padding: 4px 12px; border-image: none; line-height: 20px; text-decoration: none; float: left; background-color: rgb(255, 255, 255);
}
.pagination ul > li > a:hover {
background-color: rgb(245, 245, 245);
}
.pagination ul > li > a:focus {
background-color: rgb(245, 245, 245);
}
.pagination ul > .active > a {
background-color: rgb(245, 245, 245);
}
.pagination ul > .active > span {
background-color: rgb(245, 245, 245);
}
.pagination ul > .active > a {
color: rgb(153, 153, 153); cursor: default;
}
.pagination ul > .active > span {
color: rgb(153, 153, 153); cursor: default;
}
.pagination ul > .disabled > span {
color: rgb(153, 153, 153); cursor: default; background-color: transparent;
}
.pagination ul > .disabled > a {
color: rgb(153, 153, 153); cursor: default; background-color: transparent;
}
.pagination ul > .disabled > a:hover {
color: rgb(153, 153, 153); cursor: default; background-color: transparent;
}
.pagination ul > .disabled > a:focus {
color: rgb(153, 153, 153); cursor: default; background-color: transparent;
}
.pagination ul > li:first-child > a {
border-left-width: 1px; border-top-left-radius: 4px; border-bottom-left-radius: 4px; -webkit-border-top-left-radius: 4px; -moz-border-radius-topleft: 4px; -webkit-border-bottom-left-radius: 4px; -moz-border-radius-bottomleft: 4px;
}
.pagination ul > li:first-child > span {
border-left-width: 1px; border-top-left-radius: 4px; border-bottom-left-radius: 4px; -webkit-border-top-left-radius: 4px; -moz-border-radius-topleft: 4px; -webkit-border-bottom-left-radius: 4px; -moz-border-radius-bottomleft: 4px;
}
.pagination ul > li:last-child > a {
border-top-right-radius: 4px; border-bottom-right-radius: 4px; -webkit-border-top-right-radius: 4px; -moz-border-radius-topright: 4px; -webkit-border-bottom-right-radius: 4px; -moz-border-radius-bottomright: 4px;
}
.pagination ul > li:last-child > span {
border-top-right-radius: 4px; border-bottom-right-radius: 4px; -webkit-border-top-right-radius: 4px; -moz-border-radius-topright: 4px; -webkit-border-bottom-right-radius: 4px; -moz-border-radius-bottomright: 4px;
}
.pagination-centered {
text-align: center;
}
.pagination-right {
text-align: right;
}
.pagination-large ul > li > a {
padding: 11px 19px; font-size: 17.5px;
}
.pagination-large ul > li > span {
padding: 11px 19px; font-size: 17.5px;
}
.pagination-large ul > li:first-child > a {
border-top-left-radius: 6px; border-bottom-left-radius: 6px; -webkit-border-top-left-radius: 6px; -moz-border-radius-topleft: 6px; -webkit-border-bottom-left-radius: 6px; -moz-border-radius-bottomleft: 6px;
}
.pagination-large ul > li:first-child > span {
border-top-left-radius: 6px; border-bottom-left-radius: 6px; -webkit-border-top-left-radius: 6px; -moz-border-radius-topleft: 6px; -webkit-border-bottom-left-radius: 6px; -moz-border-radius-bottomleft: 6px;
}
.pagination-large ul > li:last-child > a {
border-top-right-radius: 6px; border-bottom-right-radius: 6px; -webkit-border-top-right-radius: 6px; -moz-border-radius-topright: 6px; -webkit-border-bottom-right-radius: 6px; -moz-border-radius-bottomright: 6px;
}
.pagination-large ul > li:last-child > span {
border-top-right-radius: 6px; border-bottom-right-radius: 6px; -webkit-border-top-right-radius: 6px; -moz-border-radius-topright: 6px; -webkit-border-bottom-right-radius: 6px; -moz-border-radius-bottomright: 6px;
}
.pagination-mini ul > li:first-child > a {
border-top-left-radius: 3px; border-bottom-left-radius: 3px; -webkit-border-top-left-radius: 3px; -moz-border-radius-topleft: 3px; -webkit-border-bottom-left-radius: 3px; -moz-border-radius-bottomleft: 3px;
}
.pagination-small ul > li:first-child > a {
border-top-left-radius: 3px; border-bottom-left-radius: 3px; -webkit-border-top-left-radius: 3px; -moz-border-radius-topleft: 3px; -webkit-border-bottom-left-radius: 3px; -moz-border-radius-bottomleft: 3px;
}
.pagination-mini ul > li:first-child > span {
border-top-left-radius: 3px; border-bottom-left-radius: 3px; -webkit-border-top-left-radius: 3px; -moz-border-radius-topleft: 3px; -webkit-border-bottom-left-radius: 3px; -moz-border-radius-bottomleft: 3px;
}
.pagination-small ul > li:first-child > span {
border-top-left-radius: 3px; border-bottom-left-radius: 3px; -webkit-border-top-left-radius: 3px; -moz-border-radius-topleft: 3px; -webkit-border-bottom-left-radius: 3px; -moz-border-radius-bottomleft: 3px;
}
.pagination-mini ul > li:last-child > a {
border-top-right-radius: 3px; border-bottom-right-radius: 3px; -webkit-border-top-right-radius: 3px; -moz-border-radius-topright: 3px; -webkit-border-bottom-right-radius: 3px; -moz-border-radius-bottomright: 3px;
}
.pagination-small ul > li:last-child > a {
border-top-right-radius: 3px; border-bottom-right-radius: 3px; -webkit-border-top-right-radius: 3px; -moz-border-radius-topright: 3px; -webkit-border-bottom-right-radius: 3px; -moz-border-radius-bottomright: 3px;
}
.pagination-mini ul > li:last-child > span {
border-top-right-radius: 3px; border-bottom-right-radius: 3px; -webkit-border-top-right-radius: 3px; -moz-border-radius-topright: 3px; -webkit-border-bottom-right-radius: 3px; -moz-border-radius-bottomright: 3px;
}
.pagination-small ul > li:last-child > span {
border-top-right-radius: 3px; border-bottom-right-radius: 3px; -webkit-border-top-right-radius: 3px; -moz-border-radius-topright: 3px; -webkit-border-bottom-right-radius: 3px; -moz-border-radius-bottomright: 3px;
}
.pagination-small ul > li > a {
padding: 2px 10px; font-size: 11.9px;
}
.pagination-small ul > li > span {
padding: 2px 10px; font-size: 11.9px;
}
.pagination-mini ul > li > a {
padding: 0px 6px; font-size: 10.5px;
}
.pagination-mini ul > li > span {
padding: 0px 6px; font-size: 10.5px;
}
.pager {
list-style: none; margin: 20px 0px; text-align: center;
}
.pager::before {
line-height: 0; display: table; content: "";
}
.pager::after {
line-height: 0; display: table; content: "";
}
.pager::after {
clear: both;
}
.pager::before {
line-height: 0; display: table; content: "";
}
.pager::after {
line-height: 0; display: table; content: "";
}
.pager::after {
clear: both;
}
.pager li {
display: inline;
}
.pager li > a {
padding: 5px 14px; border-radius: 15px; border: 1px solid rgb(221, 221, 221); border-image: none; display: inline-block; background-color: rgb(255, 255, 255); -webkit-border-radius: 15px; -moz-border-radius: 15px;
}
.pager li > span {
padding: 5px 14px; border-radius: 15px; border: 1px solid rgb(221, 221, 221); border-image: none; display: inline-block; background-color: rgb(255, 255, 255); -webkit-border-radius: 15px; -moz-border-radius: 15px;
}
.pager li > a:hover {
text-decoration: none; background-color: rgb(245, 245, 245);
}
.pager li > a:focus {
text-decoration: none; background-color: rgb(245, 245, 245);
}
.pager .next > a {
float: right;
}
.pager .next > span {
float: right;
}
.pager .previous > a {
float: left;
}
.pager .previous > span {
float: left;
}
.pager .disabled > a {
color: rgb(153, 153, 153); cursor: default; background-color: rgb(255, 255, 255);
}
.pager .disabled > a:hover {
color: rgb(153, 153, 153); cursor: default; background-color: rgb(255, 255, 255);
}
.pager .disabled > a:focus {
color: rgb(153, 153, 153); cursor: default; background-color: rgb(255, 255, 255);
}
.pager .disabled > span {
color: rgb(153, 153, 153); cursor: default; background-color: rgb(255, 255, 255);
}
.modal-backdrop {
left: 0px; top: 0px; right: 0px; bottom: 0px; position: fixed; z-index: 1040; background-color: rgb(0, 0, 0);
}
.fade.modal-backdrop {
opacity: 0;
}
.modal-backdrop {
opacity: 0.8;
}
.in.fade.modal-backdrop {
opacity: 0.8;
}
.modal {
border-radius: 6px; border: 1px solid rgba(0, 0, 0, 0.3); border-image: none; left: 50%; top: 10%; width: 560px; margin-left: -280px; position: fixed; z-index: 1050; box-shadow: 0px 3px 7px rgba(0,0,0,0.3); background-clip: padding-box; background-color: rgb(255, 255, 255); -webkit-border-radius: 6px; -moz-border-radius: 6px; -webkit-box-shadow: 0 3px 7px rgba(0, 0, 0, 0.3); -moz-box-shadow: 0 3px 7px rgba(0, 0, 0, 0.3); -webkit-background-clip: padding-box; -moz-background-clip: padding-box;
}
.fade.modal {
transition:opacity 0.3s linear, top 0.3s ease-out; top: -25%; -webkit-transition: opacity .3s linear, top .3s ease-out; -moz-transition: opacity .3s linear, top .3s ease-out; -o-transition: opacity .3s linear, top .3s ease-out;
}
.in.fade.modal {
top: 10%;
}
.modal-header {
padding: 9px 15px; border-bottom-color: rgb(238, 238, 238); border-bottom-width: 1px; border-bottom-style: solid;
}
.modal-header .close {
margin-top: 2px;
}
.modal-header h3 {
margin: 0px; line-height: 30px;
}
.modal-body {
padding: 15px; position: relative; -ms-overflow-y: auto; max-height: 400px;
}
.modal-form {
margin-bottom: 0px;
}
.modal-footer {
padding: 14px 15px 15px; border-radius: 0px 0px 6px 6px; text-align: right; margin-bottom: 0px; border-top-color: rgb(221, 221, 221); border-top-width: 1px; border-top-style: solid; box-shadow: inset 0px 1px 0px #ffffff; background-color: rgb(245, 245, 245); -webkit-border-radius: 0 0 6px 6px; -moz-border-radius: 0 0 6px 6px; -webkit-box-shadow: inset 0 1px 0 #ffffff; -moz-box-shadow: inset 0 1px 0 #ffffff;
}
.modal-footer::before {
line-height: 0; display: table; content: "";
}
.modal-footer::after {
line-height: 0; display: table; content: "";
}
.modal-footer::after {
clear: both;
}
.modal-footer::before {
line-height: 0; display: table; content: "";
}
.modal-footer::after {
line-height: 0; display: table; content: "";
}
.modal-footer::after {
clear: both;
}
.modal-footer .btn + .btn {
margin-bottom: 0px; margin-left: 5px;
}
.modal-footer .btn-group .btn + .btn {
margin-left: -1px;
}
.modal-footer .btn-block + .btn-block {
margin-left: 0px;
}
.popover {
padding: 1px; border-radius: 6px; border: 1px solid rgba(0, 0, 0, 0.2); border-image: none; left: 0px; top: 0px; text-align: left; display: none; white-space: normal; position: absolute; z-index: 1010; max-width: 276px; box-shadow: 0px 5px 10px rgba(0,0,0,0.2); background-clip: padding-box; background-color: rgb(255, 255, 255); -webkit-border-radius: 6px; -moz-border-radius: 6px; -webkit-box-shadow: 0 5px 10px rgba(0, 0, 0, 0.2); -moz-box-shadow: 0 5px 10px rgba(0, 0, 0, 0.2); -webkit-background-clip: padding-box; -moz-background-clip: padding;
}
.top.popover {
margin-top: -10px;
}
.right.popover {
margin-left: 10px;
}
.bottom.popover {
margin-top: 10px;
}
.left.popover {
margin-left: -10px;
}
.popover-title {
margin: 0px; padding: 8px 14px; border-radius: 5px 5px 0px 0px; line-height: 18px; font-size: 14px; font-weight: normal; border-bottom-color: rgb(235, 235, 235); border-bottom-width: 1px; border-bottom-style: solid; background-color: rgb(247, 247, 247); -webkit-border-radius: 5px 5px 0 0; -moz-border-radius: 5px 5px 0 0;
}
:empty.popover-title {
display: none;
}
.popover-content {
padding: 9px 14px;
}
.popover .arrow {
border-style: solid; border-color: transparent; width: 0px; height: 0px; display: block; position: absolute;
}
.popover .arrow::after {
border-style: solid; border-color: transparent; width: 0px; height: 0px; display: block; position: absolute;
}
.popover .arrow {
border-width: 11px;
}
.popover .arrow::after {
border-width: 10px; content: "";
}
.top.popover .arrow {
left: 50%; bottom: -11px; margin-left: -11px; border-top-color: rgba(0, 0, 0, 0.25); border-bottom-width: 0px;
}
.top.popover .arrow::after {
bottom: 1px; margin-left: -10px; border-top-color: rgb(255, 255, 255); border-bottom-width: 0px;
}
.right.popover .arrow {
left: -11px; top: 50%; margin-top: -11px; border-right-color: rgba(0, 0, 0, 0.25); border-left-width: 0px;
}
.right.popover .arrow::after {
left: 1px; bottom: -10px; border-right-color: rgb(255, 255, 255); border-left-width: 0px;
}
.bottom.popover .arrow {
left: 50%; top: -11px; margin-left: -11px; border-bottom-color: rgba(0, 0, 0, 0.25); border-top-width: 0px;
}
.bottom.popover .arrow::after {
top: 1px; margin-left: -10px; border-bottom-color: rgb(255, 255, 255); border-top-width: 0px;
}
.left.popover .arrow {
top: 50%; right: -11px; margin-top: -11px; border-left-color: rgba(0, 0, 0, 0.25); border-right-width: 0px;
}
.left.popover .arrow::after {
right: 1px; bottom: -10px; border-left-color: rgb(255, 255, 255); border-right-width: 0px;
}
.pull-right {
float: right;
}
.pull-left {
float: left;
}
.hide {
display: none;
}
.show {
display: block;
}
.invisible {
visibility: hidden;
}
.affix {
position: fixed;
}
.default-layout {
height: 100%; background-color: rgb(213, 213, 213);
}
.default-layout body {
background: none; height: 100%;
}
.default-layout body .wrap {
position: relative; min-height: 100%; min-width: 960px; background-color: rgb(253, 253, 253);
}
.default-layout body .wrap .body-wrap {
padding-bottom: 200px; margin-top: -20px; position: relative; z-index: 2;
}
.default-layout body .wrap .default-right {
top: 0px; height: 100%; bottom: 0px; position: absolute; background-repeat: repeat-y;
}
.default-layout body .wrap .default-left {
top: 0px; height: 100%; bottom: 0px; position: absolute; background-repeat: repeat-y;
}
.default-layout body .wrap .default-lt {
left: 0px;
}
.default-layout body .wrap .default-left {
left: 0px;
}
.default-layout body .wrap .default-rt {
right: 0px;
}
.default-layout body .wrap .default-right {
right: 0px;
}
.default-layout body .wrap .default-lt {
top: 0px; position: absolute;
}
.default-layout body .wrap .default-rt {
top: 0px; position: absolute;
}
.default-layout body .wrap .default-t {
width: 100%; position: absolute; z-index: 2; background-repeat: repeat-x;
}
.header {
height: 130px; overflow: hidden; position: relative; z-index: 2;
}
.header::before {
line-height: 0; display: table; content: "";
}
.header::after {
line-height: 0; display: table; content: "";
}
.header::after {
clear: both;
}
.header::before {
line-height: 0; display: table; content: "";
}
.header::after {
line-height: 0; display: table; content: "";
}
.header::after {
clear: both;
}
.header .navbar-hd {
background: rgb(49, 48, 48); padding: 0px 10px; height: 32px; line-height: 32px; font-size: 12px;
}
.header .navbar-hd a {
color: rgb(185, 185, 185);
}
.header .navbar-hd a .sprite {
margin-right: 5px;
}
.header .navbar-hd a.active {
color: rgb(255, 255, 255);
}
.header .navbar-hd a:hover {
color: rgb(56, 149, 242); text-decoration: none;
}
.header .navbar-hd .pull-right a {
margin-left: 15px;
}
.header .navbar-hd .split {
margin: 0px 10px; color: rgb(102, 102, 102);
}
.header .navbar-hd .course-link {
color: rgb(51, 153, 255); margin-right: 30px; float: left;
}
.header .navbar-hd .course-link:hover {
color: rgb(111, 183, 255);
}
.header .navbar-hd .user-nav .sprite-profile-tip {
top: 4px; text-align: center; line-height: 20px; position: relative;
}
.header .navbar-hd .user-nav .sprite-profile-tip-arrow {
margin-right: 0px;
}
.header .navbar-hd .user-nav .email {
width: 115px; overflow: hidden; font-size: 12px; margin-left: 10px; float: left; display: inline-block; -ms-text-overflow: ellipsis;
}
.header .navbar-hd .payment-link:hover {
color: rgb(233, 67, 127);
}
.header .navbar-hd .mobile-link:hover {
color: rgb(246, 111, 8);
}
.header .navbar-hd .search-link:hover {
color: rgb(255, 255, 255);
}
.header .navbar-bd {
background: rgb(251, 251, 251); height: 59px; overflow: hidden; clear: both; border-bottom-color: rgb(153, 153, 153); border-bottom-width: 1px; border-bottom-style: solid; position: relative; z-index: 2;
}
.header .logo {
margin-top: 17px; margin-right: 10px; float: left;
}
.header .main-nav {
margin: 0px 10px 0px 0px; left: 0px; top: -1px; float: left; display: block; position: relative;
}
.header .main-nav li {
float: left;
}
.header .main-nav li a {
padding: 0px; width: 123px; text-align: center; color: rgb(153, 153, 153); line-height: 60px; font-size: 14px;
}
.header .main-nav li a:hover {
background-color: transparent;
}
.header .main-nav .active a {
background-position: -167px -464px; width: 123px; height: 61px; color: rgb(51, 51, 51); display: inline-block; cursor: default; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.header .main-nav .active a:hover {
color: rgb(51, 51, 51);
}
.header .course-logo {
margin-top: 4px; margin-right: 20px; float: left; position: relative;
}
.header .course-logo .sprite-course-free-flag {
left: 4px; top: 14px; position: absolute;
}
.header .course-name {
font-size: 14px; margin-top: 20px; margin-right: 20px; margin-left: 5px; float: left;
}
.header .forum-link {
line-height: 36px; margin-top: 10px; margin-left: 10px;
}
.header .forum-link a {
margin: 0px 10px 0px 0px; color: rgb(153, 153, 153);
}
.header .forum-link a:hover {
color: rgb(58, 143, 251);
}
.header .forum-link a .sprite {
margin-right: 10px;
}
.header .forum-link a:hover {
text-decoration: none;
}
.header .forum-link a:hover .sprite-forum {
background-position: -724px -214px; width: 41px; height: 36px; color: rgb(52, 155, 223);
}
.header .forum-link a:hover .sprite-truman {
background-position: -1031px -172px; width: 36px; height: 36px;
}
.header .forum-link a:hover .sprite-header-book {
background-position: -870px -214px; width: 27px; height: 33px;
}
.header .container {
position: relative; z-index: 2;
}
.header .header-shadow {
left: 50%; top: 91px; width: 1440px; height: 36px; margin-left: -720px; position: absolute; background-image: url("./530da6f1-header-shadow.png");
}
.header .header-shadow-repeat-l {
top: 91px; width: 100%; position: absolute; background-repeat: repeat-x;
}
.header .header-shadow-repeat-r {
top: 91px; width: 100%; position: absolute; background-repeat: repeat-x;
}
.header .header-shadow-repeat-l {
left: -50%; margin-left: -720px;
}
.header .header-shadow-repeat-r {
right: -50%; margin-right: -720px;
}
.header .search-title {
top: 12px; height: 20px; color: rgb(153, 153, 153); line-height: 20px; padding-left: 30px; font-size: 24px; font-weight: normal; margin-left: 10px; border-left-color: rgb(153, 153, 153); border-left-width: 2px; border-left-style: solid; float: left; position: relative;
}
.footer {
width: 100%; bottom: 0px; font-size: 12px; position: absolute; background-repeat: repeat-x;
}
.footer::before {
line-height: 0; display: table; content: "";
}
.footer::after {
line-height: 0; display: table; content: "";
}
.footer::after {
clear: both;
}
.footer::before {
line-height: 0; display: table; content: "";
}
.footer::after {
line-height: 0; display: table; content: "";
}
.footer::after {
clear: both;
}
.footer .links {
top: 100px; color: rgb(153, 153, 153); position: relative; z-index: 2;
}
.footer .links a {
color: rgb(162, 162, 162);
}
.footer .links a:hover {
text-decoration: none;
}
.footer .links p {
margin: 5px 0px;
}
.footer .links img {
margin-bottom: -2px; vertical-align: baseline;
}
.footer .split {
margin: 0px 5px;
}
.footer .footer-right {
top: 0px; bottom: 0px; position: absolute;
}
.footer .footer-left {
top: 0px; bottom: 0px; position: absolute;
}
.footer .footer-left {
left: 0px;
}
.footer .footer-right {
right: 0px;
}
.simple-layout {
height: 100%; background-color: rgb(213, 213, 213);
}
.simple-layout body {
background: none; height: 100%;
}
.simple-layout body .wrap {
position: relative; min-height: 100%; min-width: 960px; background-color: rgb(253, 253, 253);
}
.simple-layout body .wrap .body-wrap {
padding-bottom: 200px; margin-top: -20px; position: relative; z-index: 2;
}
.simple-layout body .wrap .default-right {
top: 0px; height: 100%; bottom: 0px; position: absolute; background-repeat: repeat-y;
}
.simple-layout body .wrap .default-left {
top: 0px; height: 100%; bottom: 0px; position: absolute; background-repeat: repeat-y;
}
.simple-layout body .wrap .default-lt {
left: 0px;
}
.simple-layout body .wrap .default-left {
left: 0px;
}
.simple-layout body .wrap .default-rt {
right: 0px;
}
.simple-layout body .wrap .default-right {
right: 0px;
}
.simple-layout body .wrap .default-lt {
top: 0px; position: absolute;
}
.simple-layout body .wrap .default-rt {
top: 0px; position: absolute;
}
.simple-layout body .wrap .default-t {
width: 100%; position: absolute; z-index: 2; background-repeat: repeat-x;
}
.simple-layout body .wrap .body-wrap {
padding-bottom: 80px; margin-top: -20px;
}
.simple-header .container {
height: 160px; line-height: 160px;
}
.simple-header .logo {
margin-top: -20px;
}
.seo-page-title {
margin: 0px 180px; line-height: 60px; font-size: 16px; font-weight: normal;
}
.seo-navbar .text-big {
line-height: 59px;
}
.inline-block {
display: inline-block;
}
.clearfix {
}
.clearfix::before {
line-height: 0; display: table; content: "";
}
.clearfix::after {
line-height: 0; display: table; content: "";
}
.clearfix::after {
clear: both;
}
.break-all {
-ms-word-break: break-all; -ms-word-wrap: break-word;
}
.no-break {
white-space: nowrap;
}
.link-gray {
color: rgb(153, 153, 153);
}
.link-gray:hover {
color: rgb(58, 143, 251);
}
.text-red {
color: rgb(255, 0, 0);
}
.text-gray {
color: rgb(153, 153, 153);
}
.text-green {
color: rgb(70, 165, 70);
}
.overflow {
overflow: hidden;
}
.font-hei {
font-family: "Hiragino Sans GB", "Microsoft YaHei", "WenQuanYi Micro Hei", SimHei, sans-serif;
}
.page-header {
margin: 30px 0px 0px; width: 920px; clear: both; border-bottom-color: currentColor; border-bottom-width: medium; border-bottom-style: none;
}
.page-header::before {
line-height: 0; display: table; content: "";
}
.page-header::after {
line-height: 0; display: table; content: "";
}
.page-header::after {
clear: both;
}
.page-header::before {
line-height: 0; display: table; content: "";
}
.page-header::after {
line-height: 0; display: table; content: "";
}
.page-header::after {
clear: both;
}
.page-header .text {
color: rgb(153, 153, 153); line-height: 36px; font-size: 16px; margin-right: 10px; float: left;
}
.page-header .text .sprite {
bottom: -7px; margin-right: 5px; position: relative;
}
.page-header hr {
top: 9px; position: relative;
}
.first.page-header {
margin-top: 0px;
}
.text-right {
text-align: right;
}
.text-center {
text-align: center;
}
.text-blue {
color: rgb(58, 143, 251);
}
.text-cyan {
color: rgb(155, 204, 204);
}
.link {
color: rgb(58, 143, 251);
}
.i-36-line {
line-height: 36px;
}
.i-28 {
margin-bottom: -7px;
}
.i-20 {
margin-bottom: -5px;
}
.i-14 {
bottom: -3px; position: relative;
}
.text-xxlarge {
line-height: 100px; font-size: 100px !important;
}
.text-xlarge {
line-height: 80px; font-size: 60px !important;
}
.text-large {
line-height: 60px; font-size: 40px !important;
}
.text-big {
line-height: 16px; font-size: 16px !important;
}
.text-small {
line-height: 12px; font-size: 12px;
}
.clearboth {
clear: both;
}
.bold {
font-weight: bold;
}
.visible-hidden {
margin: 0px; padding: 0px; border: currentColor; border-image: none; width: 0px; height: 0px; position: absolute; z-index: -1;
}
.hidden {
visibility: hidden;
}
.loading-gif {
width: 16px; height: 16px; display: inline-block; position: relative; background-image: url("./cc826229-loading-16.gif");
}
.sprite {
display: inline-block;
}
.text-overflow {
overflow: hidden; white-space: nowrap; -ms-text-overflow: ellipsis;
}
.cursor-pointer {
cursor: pointer;
}
pre {
-ms-word-break: break-all; -ms-word-wrap: break-word;
}
.progress-wrap {
background: linear-gradient(rgb(228, 228, 228) 0%, rgb(211, 211, 211) 100%); width: 232px; height: 16px; position: relative;
}
.progress-wrap .progress-bar {
height: 100%; display: block; position: absolute;
}
.progress-wrap .level-7.progress-bar {
background: linear-gradient(rgb(0, 178, 63) 0%, rgb(0, 147, 52) 100%);
}
.progress-wrap .level-6.progress-bar {
background: linear-gradient(rgb(66, 214, 0) 0%, rgb(57, 186, 0) 100%);
}
.progress-wrap .level-5.progress-bar {
background: linear-gradient(rgb(75, 243, 106) 0%, rgb(49, 219, 80) 100%);
}
.progress-wrap .level-4.progress-bar {
background: linear-gradient(rgb(128, 254, 191) 0%, rgb(82, 235, 157) 100%);
}
.progress-wrap .level-3.progress-bar {
background: linear-gradient(rgb(191, 255, 249) 0%, rgb(134, 254, 243) 100%);
}
.progress-wrap .level-2.progress-bar {
background: linear-gradient(rgb(157, 225, 253) 0%, rgb(116, 213, 253) 100%);
}
.progress-wrap .level-1.progress-bar {
background: linear-gradient(rgb(129, 192, 252) 0%, rgb(81, 168, 252) 100%);
}
.capacity-wrap {
height: 30px; line-height: 30px; font-size: 0px; display: inline-block; white-space: nowrap;
}
.keypoint-progress-wrap {
height: 10px; display: inline-block; position: relative;
}
.keypoint-progress-wrap .slide {
display: inline-block;
}
.keypoint-progress-wrap .keypoint-progress-bg {
left: 0px; top: 0px; width: auto; right: 0px; line-height: 0px; position: absolute;
}
.keypoint-progress-wrap .keypoint-progress-bg .slide-progress-right {
left: 4px; width: auto; right: 0px; position: absolute;
}
.keypoint-progress-wrap .keypoint-progress-fill {
left: 0px; top: 0px; line-height: 0px; position: absolute;
}
.keypoint-progress-wrap .keypoint-progress-fill .slide-progress-fill-right {
left: 4px; width: auto; right: 0px; position: absolute;
}
.no-outline {
outline: 0px;
}
.btn {
text-align: center; color: rgb(255, 255, 255); cursor: pointer; box-shadow: none; background-color: transparent; -webkit-box-shadow: none; -moz-box-shadow: none;
}
.btn:hover {
box-shadow: none; background-color: transparent; -webkit-box-shadow: none; -moz-box-shadow: none;
}
.btn:active {
box-shadow: none; background-color: transparent; -webkit-box-shadow: none; -moz-box-shadow: none;
}
:disabled.btn {
cursor: default;
}
.disabled.btn {
cursor: default;
}
.btn-link {
box-shadow: none; background-image: none; background-color: transparent; -webkit-box-shadow: none; -moz-box-shadow: none;
}
.btn-link:active {
box-shadow: none; background-image: none; background-color: transparent; -webkit-box-shadow: none; -moz-box-shadow: none;
}
[disabled].btn-link {
box-shadow: none; background-image: none; background-color: transparent; -webkit-box-shadow: none; -moz-box-shadow: none;
}
.btn-link {
border-color: transparent; border-radius: 0px; color: rgb(0, 136, 204); cursor: pointer; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.btn-link:hover {
color: rgb(0, 85, 128); text-decoration: underline; background-color: transparent;
}
.btn-link:focus {
color: rgb(0, 85, 128); text-decoration: underline; background-color: transparent;
}
[disabled].btn-link:hover {
color: rgb(51, 51, 51); text-decoration: none;
}
[disabled].btn-link:focus {
color: rgb(51, 51, 51); text-decoration: none;
}
.btn-primary {
background-position: 0px -612px; padding: 0px 0px 0px 7px; outline: 0px; border-radius: 0px; border: currentColor; border-image: none; width: auto; height: auto; font-family: "Hiragino Sans GB", "Microsoft YaHei", "WenQuanYi Micro Hei", SimHei, sans-serif; font-size: 15px; display: inline-block; box-sizing: content-box; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; background-color: transparent; -webkit-box-sizing: content-box; -moz-box-sizing: content-box; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.btn-primary .btn-inner {
background-position: right -1422px; padding: 5px 12px 5px 5px; width: auto; height: auto; display: inline-block; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-primary:hover {
background-position: 0px -1332px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; background-color: transparent; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.hover.btn-primary {
background-position: 0px -1332px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; background-color: transparent; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.btn-primary:hover .btn-inner {
background-position: right -1362px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.hover.btn-primary .btn-inner {
background-position: right -1362px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-primary:active {
background-position: 0px -1272px; outline: 0px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; box-shadow: none; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; background-color: transparent; -webkit-box-shadow: none; -moz-box-shadow: none; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.active.btn-primary:hover {
background-position: 0px -1272px; outline: 0px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; box-shadow: none; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; background-color: transparent; -webkit-box-shadow: none; -moz-box-shadow: none; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.btn-primary:active .btn-inner {
background-position: right -1302px; margin: 0px; border: 0px currentColor; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.active.btn-primary:hover .btn-inner {
background-position: right -1302px; margin: 0px; border: 0px currentColor; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.disabled.btn-primary {
background-position: 0px -1482px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; background-color: transparent;
}
.disabled.btn-primary:hover {
background-position: 0px -1482px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; background-color: transparent;
}
.hover.disabled.btn-primary {
background-position: 0px -1482px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; background-color: transparent;
}
.disabled.btn-primary:active {
background-position: 0px -1482px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; background-color: transparent;
}
.active.disabled.btn-primary {
background-position: 0px -1482px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; background-color: transparent;
}
[disabled].btn-primary {
background-position: 0px -1482px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; background-color: transparent;
}
.disabled.btn-primary .btn-inner {
background-position: right -1782px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.disabled.btn-primary:hover .btn-inner {
background-position: right -1782px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.hover.disabled.btn-primary .btn-inner {
background-position: right -1782px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.disabled.btn-primary:active .btn-inner {
background-position: right -1782px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.active.disabled.btn-primary .btn-inner {
background-position: right -1782px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
[disabled].btn-primary .btn-inner {
background-position: right -1782px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-group-first.btn-primary .btn-inner {
background-position-x: left;
}
.btn-group-last.btn-primary {
background: none; padding-left: 0px; margin-left: -2px;
}
.btn-group-last.btn-primary .btn-inner {
padding-left: 12px;
}
.btn-altern {
background-position: 0px -1452px; padding: 0px 0px 0px 7px; outline: 0px; border-radius: 0px; border: currentColor; border-image: none; width: auto; height: auto; color: rgb(153, 153, 153); font-family: "Hiragino Sans GB", "Microsoft YaHei", "WenQuanYi Micro Hei", SimHei, sans-serif; font-size: 15px; display: inline-block; box-sizing: content-box; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-box-sizing: content-box; -moz-box-sizing: content-box; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.btn-altern .btn-inner {
background-position: right -1392px; padding: 5px 12px 5px 5px; width: auto; height: auto; display: inline-block; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-altern:hover {
background-position: 0px -1692px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.hover.btn-altern {
background-position: 0px -1692px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.btn-altern:hover .btn-inner {
background-position: right -1122px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.hover.btn-altern .btn-inner {
background-position: right -1122px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-altern:active {
background-position: 0px -1752px; outline: 0px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; box-shadow: none; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-box-shadow: none; -moz-box-shadow: none; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.active.hover.btn-altern {
background-position: 0px -1752px; outline: 0px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; box-shadow: none; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-box-shadow: none; -moz-box-shadow: none; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.btn-altern:active .btn-inner {
background-position: right -1722px; margin: 0px; border: 0px currentColor; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.active.hover.btn-altern .btn-inner {
background-position: right -1722px; margin: 0px; border: 0px currentColor; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
[disabled].btn-altern {
background-position: 0px -1482px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
[disabled].btn-altern .btn-inner {
background-position: right -1782px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-group-first.btn-altern .btn-inner {
background-position-x: left;
}
.btn-group-last.btn-altern {
background: none; padding-left: 0px; margin-left: -2px;
}
.btn-group-last.btn-altern .btn-inner {
padding-left: 12px;
}
.btn-normal {
background-position: 0px -1212px; padding: 0px 0px 0px 7px; outline: 0px; border-radius: 0px; border: currentColor; border-image: none; width: auto; height: auto; display: inline-block; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.btn-normal .btn-inner {
background-position: right -1242px; padding: 5px 15px 5px 8px; width: auto; height: auto; color: rgb(58, 143, 251); display: inline-block; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-normal:hover {
background-position: 0px -912px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.hover.btn-normal {
background-position: 0px -912px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.btn-normal:hover .btn-inner {
background-position: right -942px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.hover.btn-normal .btn-inner {
background-position: right -942px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-normal:active {
background-position: 0px -2052px; outline: 0px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.active.hover.btn-normal {
background-position: 0px -2052px; outline: 0px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.btn-normal:active .btn-inner {
background-position: right -2082px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.active.hover.btn-normal .btn-inner {
background-position: right -2082px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
[disabled].btn-normal {
background-position: 0px -2112px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
[disabled].btn-normal .btn-inner {
background-position: right -642px; width: auto; height: auto; color: rgb(153, 153, 153); background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-group-first.btn-normal .btn-inner {
background-position: right -882px; width: auto; height: auto;
}
.btn-group-first.btn-normal:hover .btn-inner {
background-position: right -822px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-group-first.btn-normal .hover .btn-inner {
background-position: right -822px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-group-first.btn-normal:active .btn-inner {
background-position: right -702px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.active.hover.btn-group-first.btn-normal .btn-inner {
background-position: right -702px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
[disabled].btn-group-first.btn-normal {
background-position: 0px -732px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
[disabled].btn-group-first.btn-normal .btn-inner {
background-position: right -762px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-group-last.btn-normal {
background-position: 0px -1152px; width: auto; height: auto;
}
.btn-group-last.btn-normal:hover {
background-position: 0px -1092px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.hover.btn-group-last.btn-normal {
background-position: 0px -1092px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.btn-group-last.btn-normal:active {
background-position: 0px -972px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.active.hover.btn-group-last.btn-normal {
background-position: 0px -972px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
[disabled].btn-group-last.btn-normal {
background-position: 0px -1032px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
[disabled].btn-group-last.btn-normal .btn-inner {
background-position: right -1062px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-round {
background-position: 0px -1632px; padding: 0px 0px 0px 17px; outline: 0px; border: currentColor; border-image: none; width: auto; height: auto; display: inline-block; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-round .btn-inner {
background-position: right -1662px; padding: 5px 21px 5px 4px; width: auto; height: auto; color: rgb(58, 143, 251); display: inline-block; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-round:hover {
background-position: 0px -1572px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.hover.btn-round {
background-position: 0px -1572px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.btn-round:hover .btn-inner {
background-position: right -1602px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.hover.btn-round .btn-inner {
background-position: right -1602px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-round:active {
background-position: 0px -1512px; outline: 0px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; box-shadow: none; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-box-shadow: none; -moz-box-shadow: none; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.active.hover.btn-round {
background-position: 0px -1512px; outline: 0px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; box-shadow: none; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-box-shadow: none; -moz-box-shadow: none; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.btn-round:active .btn-inner {
background-position: right -1542px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.active.hover.btn-round .btn-inner {
background-position: right -1542px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-round-small {
background-position: 0px -370px; padding: 0px 0px 0px 10px; outline: 0px; border: currentColor; border-image: none; width: auto; height: auto; display: inline-block; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-round-small .btn-inner {
background-position: right -344px; padding: 3px 10px 3px 0px; width: auto; height: auto; color: rgb(58, 143, 251); font-size: 12px; display: inline-block; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-round-small:hover {
background-position: 0px -370px; border: currentColor; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.hover.btn-round-small {
background-position: 0px -370px; border: currentColor; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-round-small:active {
outline: 0px;
}
.active.hover.btn-round-small {
outline: 0px;
}
.btn-round-small .i-20 {
top: -2px; margin-bottom: -6px; position: relative;
}
.btn-paper {
padding: 0px; outline: 0px; border-radius: 0px; border: currentColor; border-image: none; color: rgb(58, 143, 251); line-height: 40px; display: inline-block; box-shadow: none; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat; -webkit-border-radius: 0; -moz-border-radius: 0; -webkit-box-shadow: none; -moz-box-shadow: none;
}
.btn-paper:hover {
border: currentColor; transition:none; border-image: none; box-shadow: none; background-color: transparent; -webkit-box-shadow: none; -moz-box-shadow: none; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.hover.btn-paper {
border: currentColor; transition:none; border-image: none; box-shadow: none; background-color: transparent; -webkit-box-shadow: none; -moz-box-shadow: none; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.btn-paper:active {
outline: 0px; border: currentColor; transition:none; border-image: none; box-shadow: none; background-color: transparent; -webkit-box-shadow: none; -moz-box-shadow: none; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.active.hover.btn-paper {
outline: 0px; border: currentColor; transition:none; border-image: none; box-shadow: none; background-color: transparent; -webkit-box-shadow: none; -moz-box-shadow: none; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.btn-paper-xxlarge.btn-paper {
background-position: -830px -130px; width: 255px; height: 42px;
}
.btn-paper-xxlarge.btn-paper:hover {
background-position: -575px -130px; width: 255px; height: 42px; display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.hover.btn-paper-xxlarge.btn-paper {
background-position: -575px -130px; width: 255px; height: 42px; display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.btn-paper-xxlarge.btn-paper:active {
background-position: -983px -88px; width: 255px; height: 42px; display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.active.hover.btn-paper-xxlarge.btn-paper {
background-position: -983px -88px; width: 255px; height: 42px; display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.btn-paper-xlarge.btn-paper {
background-position: -779px -88px; width: 204px; height: 42px;
}
.btn-paper-xlarge.btn-paper:hover {
background-position: -575px -88px; width: 204px; height: 42px; display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.hover.btn-paper-xlarge.btn-paper {
background-position: -575px -88px; width: 204px; height: 42px; display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.btn-paper-xlarge.btn-paper:active {
background-position: -575px -172px; width: 204px; height: 42px; display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.active.hover.btn-paper-xlarge.btn-paper {
background-position: -575px -172px; width: 204px; height: 42px; display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
[disabled].btn-paper-xlarge.btn-paper {
background-position: -575px -172px; width: 204px; height: 42px; color: rgb(153, 153, 153); display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat; background-color: transparent;
}
[disabled].btn-paper-xlarge.btn-paper:hover {
background-position: -575px -172px; width: 204px; height: 42px; display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
[disabled].hover.btn-paper-xlarge.btn-paper {
background-position: -575px -172px; width: 204px; height: 42px; display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
[disabled].btn-paper-xlarge.btn-paper:active {
background-position: -575px -172px; width: 204px; height: 42px; display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
[disabled].active.hover.btn-paper-xlarge.btn-paper {
background-position: -575px -172px; width: 204px; height: 42px; display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.btn-paper-large.btn-paper {
background-position: -1085px -130px; width: 164px; height: 42px;
}
.btn-paper-gray.btn-paper {
color: rgb(153, 153, 153);
}
.btn-hero {
background-position: 0px -2728px; padding: 0px 0px 0px 10px; outline: 0px; border-radius: 0px; border: currentColor; border-image: none; width: auto; height: auto; display: inline-block; box-sizing: content-box; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-box-sizing: content-box; -moz-box-sizing: content-box; -webkit-border-radius: 0; -moz-border-radius: 0;
}
.btn-hero .btn-inner {
background-position: right -2488px; padding: 8px 30px 9px; width: auto; height: auto; display: inline-block; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-hero:hover {
background-position: 0px -2428px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.hover.btn-hero {
background-position: 0px -2428px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.btn-hero:hover .btn-inner {
background-position: right -2368px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.hover.btn-hero .btn-inner {
background-position: right -2368px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-hero:active {
background-position: 0px -2668px; outline: 0px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; box-shadow: none; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-box-shadow: none; -moz-box-shadow: none; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.active.btn-hero:hover {
background-position: 0px -2668px; outline: 0px; border: currentColor; transition:none; border-image: none; width: auto; height: auto; box-shadow: none; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-box-shadow: none; -moz-box-shadow: none; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.btn-hero:active .btn-inner {
background-position: right -2608px; margin: 0px; border: 0px currentColor; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.active.btn-hero:hover .btn-inner {
background-position: right -2608px; margin: 0px; border: 0px currentColor; border-image: none; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
[disabled].btn-hero {
background-position: 0px -2548px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
[disabled].btn-hero .btn-inner {
background-position: right -2788px; width: auto; height: auto; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.btn-group-first.btn-hero .btn-inner {
background-position-x: left;
}
.btn-group-last.btn-hero {
background: none; padding-left: 0px; margin-left: -2px;
}
.btn-group-last.btn-hero .btn-inner {
padding-left: 12px;
}
.sprite-tip-button {
text-align: center; color: rgb(255, 255, 255); line-height: 22px; font-size: 12px; text-decoration: none; display: inline-block;
}
.sprite-tip-button:hover {
background-position: -715px -280px; width: 74px; height: 22px; color: rgb(255, 255, 255);
}
.disabled.btn-link {
color: rgb(197, 197, 197); text-decoration: none; background-color: transparent;
}
[disabled].btn-link {
color: rgb(197, 197, 197); text-decoration: none; background-color: transparent;
}
.disabled.btn-link:hover {
color: rgb(197, 197, 197);
}
[disabled].btn-link:hover {
color: rgb(197, 197, 197);
}
.form-wrap div.inline {
display: inline-block;
}
.help-inline {
color: rgb(153, 153, 153);
}
input[type='checkbox'] {
margin: 0px;
}
input[type='radio'] {
margin: 0px;
}
.item-wrap {
margin-bottom: 20px; position: relative;
}
.item-wrap select {
margin-bottom: 0px;
}
.item-wrap textarea {
margin-bottom: 0px;
}
.item-wrap input[type='text'] {
margin-bottom: 0px;
}
.item-wrap input[type='password'] {
margin-bottom: 0px;
}
.item-wrap input[type='datetime'] {
margin-bottom: 0px;
}
.item-wrap input[type='datetime-local'] {
margin-bottom: 0px;
}
.item-wrap input[type='date'] {
margin-bottom: 0px;
}
.item-wrap input[type='month'] {
margin-bottom: 0px;
}
.item-wrap input[type='time'] {
margin-bottom: 0px;
}
.item-wrap input[type='week'] {
margin-bottom: 0px;
}
.item-wrap input[type='number'] {
margin-bottom: 0px;
}
.item-wrap input[type='email'] {
margin-bottom: 0px;
}
.item-wrap input[type='url'] {
margin-bottom: 0px;
}
.item-wrap input[type='search'] {
margin-bottom: 0px;
}
.item-wrap input[type='tel'] {
margin-bottom: 0px;
}
.item-wrap input[type='color'] {
margin-bottom: 0px;
}
.item-wrap .uneditable-input {
margin-bottom: 0px;
}
.item-wrap .icon-error {
background-position: -801px -347px; width: 14px; height: 14px; margin-right: 5px; display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.item-wrap .icon-info {
display: none;
}
.error.item-wrap .item-message {
color: rgb(255, 0, 0);
}
.text-item-wrap {
}
.text-item-wrap::before {
line-height: 0; display: table; content: "";
}
.text-item-wrap::after {
line-height: 0; display: table; content: "";
}
.text-item-wrap::after {
clear: both;
}
.text-item-wrap::before {
line-height: 0; display: table; content: "";
}
.text-item-wrap::after {
line-height: 0; display: table; content: "";
}
.text-item-wrap::after {
clear: both;
}
.text-item-wrap label {
left: -124px; top: 6px; width: 100px; text-align: right; position: absolute;
}
.text-item-wrap .text-wrap {
background-position: 0px -1842px; width: auto; height: 30px; padding-left: 6px; display: inline-block; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.text-item-wrap .text-wrap input[type=text] {
background-position: right -1812px; padding: 5px 6px 5px 0px; outline: 0px; border-radius: 0px; border: currentColor; transition:none; border-image: none; width: 260px; height: 20px; box-shadow: none; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-border-radius: 0; -moz-border-radius: 0; -webkit-box-shadow: none; -moz-box-shadow: none; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.text-item-wrap .text-wrap input[type=password] {
background-position: right -1812px; padding: 5px 6px 5px 0px; outline: 0px; border-radius: 0px; border: currentColor; transition:none; border-image: none; width: 260px; height: 20px; box-shadow: none; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-border-radius: 0; -moz-border-radius: 0; -webkit-box-shadow: none; -moz-box-shadow: none; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.text-item-wrap .text-wrap input[type=text]:focus {
height: 20px; box-shadow: none; -webkit-box-shadow: none; -moz-box-shadow: none;
}
.text-item-wrap .text-wrap input[type=password]:focus {
height: 20px; box-shadow: none; -webkit-box-shadow: none; -moz-box-shadow: none;
}
.text-item-wrap .focus.text-wrap {
background-position: 0px -2022px; width: auto; height: auto;
}
.text-item-wrap .focus.text-wrap input {
background-position: right -1992px; width: 260px; height: 20px;
}
.text-item-wrap .hide.text-wrap {
display: none;
}
.text-item-wrap .read-only {
line-height: 30px;
}
.checkbox-item-wrap {
}
.checkbox-item-wrap::before {
line-height: 0; display: table; content: "";
}
.checkbox-item-wrap::after {
line-height: 0; display: table; content: "";
}
.checkbox-item-wrap::after {
clear: both;
}
.checkbox-item-wrap::before {
line-height: 0; display: table; content: "";
}
.checkbox-item-wrap::after {
line-height: 0; display: table; content: "";
}
.checkbox-item-wrap::after {
clear: both;
}
.checkbox-item-wrap .altern-label {
left: -124px; width: 100px; text-align: right; position: absolute;
}
.verification-item-wrap {
margin-bottom: 20px; display: none; min-height: 60px;
}
.verification-item-wrap .send-info {
color: rgb(153, 153, 153); margin-top: 10px;
}
.verification-item-wrap .btn-send-verification {
width: 100px; bottom: -2px; display: inline-block; position: relative;
}
.verification-item-wrap .text-wrap input[type=text] {
width: 148px;
}
.verification-item-wrap a.disabled {
color: rgb(153, 153, 153); text-decoration: none; cursor: default;
}
.file-item-wrap label {
left: -124px; top: 6px; width: 100px; text-align: right; position: absolute;
}
.file-item-wrap input[type=file] {
height: 20px; bottom: -4px; line-height: 1em; position: relative;
}
.file-item-wrap p {
margin-top: 10px;
}
.radio-item-wrap .radio-label {
left: -124px; width: 100px; text-align: right; position: absolute;
}
.radio-item-wrap .controls label.radio {
padding-top: 0px;
}
.radio-item-wrap .controls label.radio input {
position: relative;
}
.radio-item-wrap .controls .radio {
min-height: auto;
}
.radio-item-wrap .controls .checkbox {
min-height: auto;
}
.radio input[type=radio] {
float: none;
}
.radio input[type=checkbox] {
float: none;
}
.select-item-wrap label {
left: -124px; width: 100px; text-align: right; bottom: 0px; position: absolute;
}
.textarea-item-wrap .text-wrap {
background: none; padding: 0px; height: auto;
}
.textarea-item-wrap .text-wrap textarea {
width: 258px; height: 80px;
}
.dialog-backdrop {
background: rgb(0, 0, 0); margin: 0px; padding: 0px; left: 0px; top: 0px; width: 100%; height: 100%; position: fixed; z-index: 999; opacity: 0.3;
}
.dialog-wrap {
float: left; position: absolute; z-index: 1000;
}
.dialog-wrap::before {
line-height: 0; display: table; content: "";
}
.dialog-wrap::after {
line-height: 0; display: table; content: "";
}
.dialog-wrap::after {
clear: both;
}
.dialog-wrap::before {
line-height: 0; display: table; content: "";
}
.dialog-wrap::after {
line-height: 0; display: table; content: "";
}
.dialog-wrap::after {
clear: both;
}
.dialog-wrap .dialog {
border: 6px solid rgba(0, 0, 0, 0.3); border-image: none; height: auto; overflow: hidden; float: left; position: relative; background-clip: padding-box; background-color: rgb(255, 255, 255); -webkit-background-clip: padding-box; -moz-background-clip: padding-box;
}
.dialog-wrap .dialog .dialog-hd {
padding: 9px 15px; border-bottom-color: rgb(238, 238, 238); border-bottom-width: 1px; border-bottom-style: solid;
}
.dialog-wrap .dialog .dialog-hd::before {
line-height: 0; display: table; content: "";
}
.dialog-wrap .dialog .dialog-hd::after {
line-height: 0; display: table; content: "";
}
.dialog-wrap .dialog .dialog-hd::after {
clear: both;
}
.dialog-wrap .dialog .dialog-hd::before {
line-height: 0; display: table; content: "";
}
.dialog-wrap .dialog .dialog-hd::after {
line-height: 0; display: table; content: "";
}
.dialog-wrap .dialog .dialog-hd::after {
clear: both;
}
.dialog-wrap .dialog .dialog-hd h3 {
margin: 0px; line-height: 30px;
}
.dialog-wrap .dialog .dialog-bd {
padding: 15px;
}
.dialog-wrap .dialog .dialog-ft {
padding: 14px 15px 15px; border-radius: 0px 0px 6px 6px; text-align: right; margin-bottom: 0px; border-top-color: rgb(221, 221, 221); border-top-width: 1px; border-top-style: solid; box-shadow: inset 0px 1px 0px #ffffff; background-color: rgb(245, 245, 245); -webkit-border-radius: 0 0 6px 6px; -moz-border-radius: 0 0 6px 6px; -webkit-box-shadow: inset 0 1px 0 #ffffff; -moz-box-shadow: inset 0 1px 0 #ffffff;
}
.dialog-wrap .dialog .dialog-ft::before {
line-height: 0; display: table; content: "";
}
.dialog-wrap .dialog .dialog-ft::after {
line-height: 0; display: table; content: "";
}
.dialog-wrap .dialog .dialog-ft::after {
clear: both;
}
.dialog-wrap .dialog .dialog-ft::before {
line-height: 0; display: table; content: "";
}
.dialog-wrap .dialog .dialog-ft::after {
line-height: 0; display: table; content: "";
}
.dialog-wrap .dialog .dialog-ft::after {
clear: both;
}
.dialog-wrap .dialog .dialog-ft .btn + .btn {
margin-bottom: 0px; margin-left: 5px;
}
.dialog-wrap .dialog .dialog-ft .btn-group .btn + .btn {
margin-left: -1px;
}
.dialog-wrap .dialog .dialog-ft .btn-block + .btn-block {
margin-left: 0px;
}
.popover-wrap {
position: absolute; z-index: 300;
}
.popover-wrap .popover {
background: rgb(255, 255, 255); padding: 0px; border-radius: 0px; border: 7px solid rgba(0, 0, 0, 0.3); border-image: none; left: auto; top: auto; display: block; position: static; z-index: auto; max-width: inherit; box-shadow: none; -webkit-border-radius: 0px; -moz-border-radius: 0px; -webkit-box-shadow: none; -moz-box-shadow: none;
}
.popover-wrap .popover .popover-hd {
padding: 10px 20px; color: rgb(153, 153, 153); border-bottom-color: rgb(197, 197, 197); border-bottom-width: 1px; border-bottom-style: solid; background-color: rgb(254, 254, 254);
}
.popover-wrap .popover .popover-bd {
padding: 10px 20px;
}
.popover-wrap .popover-arrow-wrap {
display: inline-block; position: absolute; pointer-events: none; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.popover-wrap .popover-direction-top .popover-arrow-wrap {
background-position: -1113px -327px; left: 10%; width: 19px; height: 15px; bottom: -8px;
}
.popover-wrap .popover-direction-bottom .popover-arrow-wrap {
background-position: -1094px -327px; left: 10%; top: -8px; width: 19px; height: 15px;
}
.popover-wrap .popover-direction-right .popover-arrow-wrap {
background-position: -1064px -327px; left: -8px; top: 10%; width: 15px; height: 19px;
}
.popover-wrap .popover-direction-left .popover-arrow-wrap {
background-position: -1079px -327px; top: 10%; width: 15px; height: 19px; right: -8px;
}
.tooltip-wrap {
z-index: 400;
}
.menu-wrap {
padding: 11px 14px 14px; float: left; position: relative;
}
.menu-wrap .popover-arrow-wrap {
top: -5px; right: 0px; position: absolute;
}
.menu-wrap > .sprite-menu-lt {
display: block; position: absolute;
}
.menu-wrap > .sprite-menu-rt {
display: block; position: absolute;
}
.menu-wrap > .sprite-menu-lb {
display: block; position: absolute;
}
.menu-wrap > .sprite-menu-rb {
display: block; position: absolute;
}
.menu-wrap > .repeat-x {
display: block; position: absolute;
}
.menu-wrap > .repeat-y {
display: block; position: absolute;
}
.menu-wrap > .repeat-x {
left: 14px; width: auto; right: 18px; background-repeat: repeat-x;
}
.menu-wrap > .repeat-y {
top: 11px; height: auto; bottom: 14px; background-repeat: repeat-y;
}
.menu-wrap > .sprite-menu-lt {
left: 0px; top: 0px;
}
.menu-wrap > .sprite-menu-rt {
top: 0px; right: 0px;
}
.menu-wrap > .sprite-menu-lb {
left: 0px; bottom: 0px;
}
.menu-wrap > .sprite-menu-rb {
right: 0px; bottom: 0px;
}
.menu-wrap > .menu-t {
top: 0px;
}
.menu-wrap > .menu-r {
right: 0px;
}
.menu-wrap > .menu-b {
bottom: 0px;
}
.menu-wrap > .menu-l {
left: 0px;
}
.menu-wrap .menu {
background: rgb(255, 255, 255); font-size: 12px;
}
.menu-wrap .menu .nav {
margin: -5px 0px 0px;
}
.menu-wrap .menu .nav li {
padding: 3px 20px 0px; text-align: center; border-bottom-color: transparent; border-bottom-width: 3px; border-bottom-style: solid;
}
.menu-wrap .menu .nav li:hover {
border-color: rgb(25, 147, 251);
}
.menu-wrap .menu .nav li a {
color: rgb(61, 61, 61);
}
.menu-wrap .menu .nav li a:hover {
background: none;
}
.flat-menu-wrap {
background: rgb(32, 30, 30); padding: 15px 18px; font-size: 12px; float: left; position: relative;
}
.flat-menu-wrap .popover-direction-top .popover-arrow-wrap {
background-position: -990px -347px; left: 30%; width: 14px; height: 7px; bottom: -7px;
}
.flat-menu-wrap .popover-direction-bottom .popover-arrow-wrap {
background-position: -976px -347px; left: 30%; top: -7px; width: 14px; height: 7px;
}
.flat-menu-wrap .popover-direction-right .popover-arrow-wrap {
background-position: -1249px -130px; left: -7px; top: 10%; width: 7px; height: 14px;
}
.flat-menu-wrap .popover-direction-left .popover-arrow-wrap {
background-position: -1247px -102px; top: 10%; width: 7px; height: 14px; right: -7px;
}
.flat-menu-wrap .nav {
margin: 0px; min-width: 100px;
}
.flat-menu-wrap .nav li {
padding: 3px 0px; text-align: left;
}
.flat-menu-wrap .nav li a {
color: rgb(185, 185, 185);
}
.flat-menu-wrap .nav li a:hover {
background: none; color: rgb(255, 255, 255);
}
.nav-underline.nav {
}
.nav-underline.nav::before {
line-height: 0; display: table; content: "";
}
.nav-underline.nav::after {
line-height: 0; display: table; content: "";
}
.nav-underline.nav::after {
clear: both;
}
.nav-underline.nav::before {
line-height: 0; display: table; content: "";
}
.nav-underline.nav::after {
line-height: 0; display: table; content: "";
}
.nav-underline.nav::after {
clear: both;
}
.nav-underline.nav li {
margin: 5px 10px; float: left;
}
.nav-underline.nav li a {
padding: 5px 0px; color: rgb(153, 153, 153); border-bottom-color: transparent; border-bottom-width: 3px; border-bottom-style: solid; white-space: nowrap;
}
.nav-underline.nav li a:hover {
background: none; color: rgb(51, 51, 51);
}
.nav-underline.nav li.active a {
color: rgb(51, 51, 51); border-bottom-color: rgb(58, 143, 251);
}
.box-wrap {
padding: 1px 3px 5px; margin-right: -3px; margin-left: -3px; position: relative;
}
.box-wrap > .sprite-box-lt {
position: absolute;
}
.box-wrap > .sprite-box-rt {
position: absolute;
}
.box-wrap > .sprite-box-lb {
position: absolute;
}
.box-wrap > .sprite-box-rb {
position: absolute;
}
.box-wrap > .repeat-x {
position: absolute;
}
.box-wrap > .repeat-y {
position: absolute;
}
.box-wrap .sprite-box-lt {
left: 0px; top: 0px;
}
.box-wrap .sprite-box-rt {
top: 0px; right: 0px;
}
.box-wrap .sprite-box-lb {
left: 0px; bottom: 0px;
}
.box-wrap .sprite-box-rb {
right: 0px; bottom: 0px;
}
.box-wrap .repeat-y {
top: 5px; height: auto; bottom: 5px; background-repeat: repeat-y;
}
.box-wrap .repeat-x {
width: auto; background-repeat: repeat-x;
}
.box-wrap .box-l {
left: 0px;
}
.box-wrap .box-r {
right: 0px;
}
.box-wrap .box-t {
left: 3px; top: 0px; right: 3px;
}
.box-wrap .box-b {
left: 6px; right: 6px; bottom: 1px;
}
.box-wrap .box {
background: rgb(255, 255, 255); padding: 20px; border: 1px solid rgb(199, 199, 199); border-image: none;
}
.box-wrap .box .box-hd h2 {
font-size: 14px; font-weight: bold;
}
.box-gray-radius {
padding: 1px 3px 5px; position: relative;
}
.box-gray-radius > .sprite-gray-radius-lt {
position: absolute;
}
.box-gray-radius > .sprite-gray-radius-rt {
position: absolute;
}
.box-gray-radius > .sprite-gray-radius-lb {
position: absolute;
}
.box-gray-radius > .sprite-gray-radius-rb {
position: absolute;
}
.box-gray-radius > .box-gray-radius-t {
position: absolute;
}
.box-gray-radius > .box-gray-radius-b {
position: absolute;
}
.box-gray-radius > .box-gray-radius-l {
position: absolute;
}
.box-gray-radius > .box-gray-radius-r {
position: absolute;
}
.box-gray-radius .sprite-gray-radius-lt {
left: 0px; top: 0px;
}
.box-gray-radius .sprite-gray-radius-rt {
top: 0px; right: 0px;
}
.box-gray-radius .sprite-gray-radius-lb {
left: 0px; bottom: 0px;
}
.box-gray-radius .sprite-gray-radius-rb {
right: 0px; bottom: 0px;
}
.box-gray-radius .box-gray-radius-l {
background: rgb(247, 247, 247); top: 10px; width: 10px; bottom: 10px;
}
.box-gray-radius .box-gray-radius-r {
background: rgb(247, 247, 247); top: 10px; width: 10px; bottom: 10px;
}
.box-gray-radius .box-gray-radius-t {
background: rgb(247, 247, 247); left: 10px; height: 10px; right: 10px;
}
.box-gray-radius .box-gray-radius-b {
background: rgb(247, 247, 247); left: 10px; height: 10px; right: 10px;
}
.box-gray-radius .box-gray-radius-l {
left: 0px;
}
.box-gray-radius .box-gray-radius-r {
right: 0px;
}
.box-gray-radius .box-gray-radius-t {
top: 0px;
}
.box-gray-radius .box-gray-radius-b {
bottom: 0px;
}
.box-gray-radius .box-gray-radius {
background: rgb(247, 247, 247); padding: 10px;
}
.box-inner-wrap {
padding: 2px 4px 6px; position: relative;
}
.box-inner-wrap > .sprite-box-inner-lt {
position: absolute;
}
.box-inner-wrap > .sprite-box-inner-rt {
position: absolute;
}
.box-inner-wrap > .sprite-box-inner-lb {
position: absolute;
}
.box-inner-wrap > .sprite-box-inner-rb {
position: absolute;
}
.box-inner-wrap > .repeat-x {
position: absolute;
}
.box-inner-wrap > .repeat-y {
position: absolute;
}
.box-inner-wrap .sprite-box-inner-lt {
left: 0px; top: 0px;
}
.box-inner-wrap .sprite-box-inner-rt {
top: 0px; right: 0px;
}
.box-inner-wrap .sprite-box-inner-lb {
left: 0px; bottom: 0px;
}
.box-inner-wrap .sprite-box-inner-rb {
right: 0px; bottom: 0px;
}
.box-inner-wrap > .repeat-y {
height: auto; background-repeat: repeat-y;
}
.box-inner-wrap > .repeat-x {
width: auto; background-repeat: repeat-x;
}
.box-inner-wrap .box-inner-l {
left: 0px; top: 6px; bottom: 8px;
}
.box-inner-wrap .box-inner-r {
top: 6px; right: 0px; bottom: 8px;
}
.box-inner-wrap .box-inner-t {
left: 8px; top: 0px; right: 6px;
}
.box-inner-wrap .box-inner-b {
left: 8px; right: 7px; bottom: 0px;
}
.box-inner-wrap .box-inner {
background: rgb(255, 255, 255); position: relative; z-index: 2;
}
.box-gray-wrap {
padding: 4px; position: relative;
}
.box-gray-wrap > .sprite-box-gray-lt {
position: absolute;
}
.box-gray-wrap > .sprite-box-gray-rt {
position: absolute;
}
.box-gray-wrap > .sprite-box-gray-lb {
position: absolute;
}
.box-gray-wrap > .sprite-box-gray-rb {
position: absolute;
}
.box-gray-wrap > .repeat-x {
position: absolute;
}
.box-gray-wrap > .repeat-y {
position: absolute;
}
.box-gray-wrap .sprite-box-gray-lt {
left: 0px; top: 0px;
}
.box-gray-wrap .sprite-box-gray-rt {
top: 0px; right: 0px;
}
.box-gray-wrap .sprite-box-gray-lb {
left: 0px; bottom: 0px;
}
.box-gray-wrap .sprite-box-gray-rb {
right: 0px; bottom: 0px;
}
.box-gray-wrap > .repeat-y {
top: 4px; height: auto; bottom: 4px; background-repeat: repeat-y;
}
.box-gray-wrap > .repeat-x {
left: 4px; width: auto; right: 4px; background-repeat: repeat-x;
}
.box-gray-wrap .sprite-box-gray-legend {
top: -25px; text-align: center; line-height: 30px; position: absolute;
}
.box-gray-wrap .box-gray-l {
left: 0px;
}
.box-gray-wrap .box-gray-r {
right: 0px;
}
.box-gray-wrap .box-gray-t {
top: 0px;
}
.box-gray-wrap .box-gray-b {
bottom: 0px;
}
.box-gray-wrap .box-gray {
background: rgb(220, 220, 220); position: relative; z-index: 2;
}
.action-result-wrap {
margin-top: 100px;
}
.action-result-wrap .error {
color: rgb(70, 165, 70); font-size: 14px; font-weight: normal; margin-top: 0px; margin-bottom: 0px;
}
.action-result-wrap .success {
color: rgb(70, 165, 70); font-size: 14px; font-weight: normal; margin-top: 0px; margin-bottom: 0px;
}
.action-result-wrap .error {
color: rgb(255, 0, 0);
}
.action-result-wrap .result-description {
margin-bottom: 20px;
}
.text-label {
background-position: 0px -396px; width: auto; height: 27px; padding-left: 10px; display: inline-block; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.text-label .text-label-inner {
background-position: right -423px; padding: 0px 10px 0px 0px; width: auto; height: 27px; color: rgb(102, 102, 102); line-height: 27px; overflow: hidden; display: inline-block; white-space: nowrap; -ms-text-overflow: ellipsis; max-width: 260px; background-image: url("./8c8d48f3-slide.png"); background-repeat: repeat-x;
}
.text-label .text-label-inner i {
bottom: -2px; margin-right: 3px; position: relative;
}
.key-label.text-label {
background-position: 0px -450px; width: auto; height: 27px;
}
.key-label.text-label .text-label-inner {
background-position: right -477px; width: auto; height: 27px;
}
h3.text-label {
margin: 0px 0px 15px; line-height: 1em; font-size: 14px; font-weight: normal;
}
.exercise-wrap {
width: 807px; margin-top: -3px;
}
.exercise-wrap .sheet-name {
margin: 0px; padding: 10px 20px 0px; color: rgb(102, 102, 102); line-height: 1em; font-size: 16px;
}
.exercise-wrap .sheet-name .close {
margin-top: -5px;
}
.exercise-wrap .exercise-hd {
background: rgb(255, 255, 255); width: 799px; position: absolute; z-index: 2;
}
.exercise-wrap .exercise-hd .last.split-box {
margin-bottom: 0px;
}
.exercise-wrap .exercise-hd .fixed-head {
background: rgb(255, 255, 255); padding-top: 10px; margin-top: -10px;
}
.exercise-wrap .exercise-hd .fixed.fixed-head {
top: 0px; width: 799px; margin-top: 0px; position: fixed;
}
.exercise-wrap .exercise-hd img.fixed-head {
padding-top: 0px; margin-top: 0px;
}
.exercise-wrap .exercise-hd .remain-time-banner {
top: -1px; display: none; position: absolute; z-index: 2;
}
.exercise-wrap .exercise-bd .placeholder {
visibility: hidden;
}
.exercise-wrap .exercise-bd .placeholder .fixed-head {
background: rgb(255, 255, 255); padding-top: 10px; margin-top: -10px;
}
.exercise-wrap .exercise-bd .placeholder .fixed.fixed-head {
top: 0px; width: 799px; margin-top: 0px; position: fixed;
}
.exercise-wrap .exercise-bd .placeholder .last.split-box {
margin-bottom: 0px;
}
.exercise-wrap .exercise-bd .chapter-desc {
padding-top: 10px;
}
.exercise-wrap .exercise-bd .chapter-desc .split-box {
margin-bottom: 0px;
}
.exercise-wrap .exercise-bd .chapter-wrap {
margin-top: 10px; margin-bottom: 30px;
}
.exercise-wrap .exercise-ft {
position: relative;
}
.exercise-wrap .sidebar-wrap {
float: left; position: absolute;
}
.exercise-wrap .sidebar-wrap .box-inner {
background: none;
}
.exercise-wrap .sidebar-wrap .sidebar {
width: 160px; line-height: 0px;
}
.exercise-wrap .sidebar-wrap .sidebar .side-top {
text-align: center; color: rgb(153, 153, 153); line-height: 1em;
}
.exercise-wrap .sidebar-wrap .sidebar .side-top .sprite-clock {
margin: 20px auto 10px; display: block;
}
.exercise-wrap .sidebar-wrap .sidebar .nav {
margin: 0px; padding: 0px; line-height: 1em; border-top-color: rgb(197, 197, 197); border-top-width: 1px; border-top-style: solid;
}
.exercise-wrap .sidebar-wrap .sidebar .nav li {
background-position: -334px -397px; left: -4px; top: -3px; width: 167px; height: 67px; text-align: center; color: rgb(153, 153, 153); line-height: 67px; margin-bottom: -7px; display: block; position: relative; z-index: 2; cursor: pointer; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.exercise-wrap .sidebar-wrap .sidebar .nav li a {
color: rgb(153, 153, 153);
}
.exercise-wrap .sidebar-wrap .sidebar .nav li a:hover {
background: none;
}
.exercise-wrap .sidebar-wrap .sidebar .nav li.active {
background-position: -167px -397px; width: 167px; height: 67px;
}
.exercise-wrap .sidebar-wrap .sidebar .nav li:hover {
background-position: -167px -397px; width: 167px; height: 67px;
}
.exercise-wrap .sidebar-wrap .sidebar .nav li.active {
color: rgb(197, 197, 197);
}
.exercise-wrap .sidebar-wrap .sidebar .nav li.view-incorrect {
left: 0px; top: 0px; margin-bottom: 0px;
}
.exercise-wrap .sidebar-wrap .sidebar .nav li.share-wrap {
left: 0px; top: 0px; margin-bottom: 0px;
}
.exercise-wrap .sidebar-wrap .sidebar .nav li.commit-exercise {
background-position: 0px -464px; width: 167px; height: 67px; display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.exercise-wrap .sidebar-wrap .sidebar .nav li.active.commit-exercise {
background-position: 0px -397px; width: 167px; height: 67px; color: rgb(255, 255, 255);
}
.exercise-wrap .sidebar-wrap .sidebar .nav li.commit-exercise:hover {
background-position: 0px -397px; width: 167px; height: 67px; color: rgb(255, 255, 255);
}
.exercise-wrap .paper-wrap {
width: 799px; margin-left: 160px; position: relative; z-index: 2;
}
.exercise-wrap .paper-wrap .box-inner {
padding: 1px 0px;
}
.exercise-wrap .paper-wrap .split-line {
margin: 10px 0px; width: 100%; height: 1px; background-color: rgb(216, 216, 216);
}
.exercise-wrap .paper-wrap .split-box {
margin: 10px 0px; padding: 0px 15px; position: relative;
}
.exercise-wrap .paper-wrap .split-box .repeat-x {
top: 0px; width: 100%; display: block; background-repeat: repeat-x;
}
.exercise-wrap .paper-wrap .split-box .sprite-box-split-l {
left: 0px; top: 0px; position: absolute;
}
.exercise-wrap .paper-wrap .split-box .sprite-box-split-r {
top: 0px; right: 0px; position: absolute;
}
.exercise-wrap .paper-wrap .split-box-2.split-box {
padding: 0px;
}
.exercise-wrap .paper-wrap .description {
padding: 0px 20px; color: rgb(51, 51, 51); font-size: 12px;
}
.exercise-wrap .paper-wrap .chapter-nav .nav {
margin-bottom: 0px;
}
.exercise-wrap .paper-wrap .chapter-nav .nav::before {
line-height: 0; display: table; content: "";
}
.exercise-wrap .paper-wrap .chapter-nav .nav::after {
line-height: 0; display: table; content: "";
}
.exercise-wrap .paper-wrap .chapter-nav .nav::after {
clear: both;
}
.exercise-wrap .paper-wrap .chapter-nav .nav::before {
line-height: 0; display: table; content: "";
}
.exercise-wrap .paper-wrap .chapter-nav .nav::after {
line-height: 0; display: table; content: "";
}
.exercise-wrap .paper-wrap .chapter-nav .nav::after {
clear: both;
}
.exercise-wrap .paper-wrap .chapter-nav .nav li {
margin: 0px 20px;
}
.exercise-wrap .question-wrap {
margin-top: -10px;
}
.fixed.exercise-wrap .sidebar-wrap {
top: 0px; position: fixed;
}
.exercise-wrap .next-chapter-wrap {
padding-right: 20px;
}
.exercise-wrap .return-top {
width: 58px; height: 58px; bottom: 197px; display: block; position: fixed; z-index: 1000; background-image: url("../img/exercises/c3220302-return-top.png");
}
.exercise-wrap .return-top:hover {
background-position: -58px;
}
.exercise-wrap .return-top:active {
background-position: -116px;
}
.exercise-wrap .return-top-wrap {
right: 1px; position: absolute;
}
.exercise-solution-wrap .sidebar-wrap .sidebar .side-top .sprite-book {
margin: 34px auto 0px; display: block;
}
.exercise-solution-wrap .sidebar-wrap .sidebar .nav {
margin-top: -5px;
}
.exercise-solution-wrap .sidebar-wrap .sidebar .nav .view-incorrect {
background: rgb(254, 254, 254); cursor: default;
}
.exercise-solution-wrap .sidebar-wrap .sidebar .nav .view-incorrect label {
bottom: -22px; position: relative;
}
.exercise-solution-wrap .sidebar-wrap .sidebar .nav .view-incorrect:hover {
background: rgb(254, 254, 254); width: 167px; height: 67px;
}
.exercise-solution-wrap .empty-tip {
padding: 20px; height: 100px;
}
.exercise-solution-wrap .pagination {
margin-top: 0px; margin-right: 10px;
}
.exercise-solution-wrap .retry-button-wrap {
padding-right: 20px; padding-bottom: 10px;
}
.exercise-solution-wrap .solution-item .text-label {
margin-right: 5px; margin-bottom: 5px;
}
.exercise-solution-wrap .note-item .sprite-edit-note {
bottom: -1px;
}
.bank-panel h3 {
margin: 0px; color: rgb(85, 85, 85); font-size: 14px; font-weight: normal;
}
.bank-panel h3 .pay-price {
color: rgb(70, 165, 70);
}
.bank-list {
background: rgb(255, 255, 255); border: 1px solid rgb(204, 204, 204); border-image: none; width: 872px; padding-bottom: 10px; margin-bottom: 20px;
}
.bank-list::before {
line-height: 0; display: table; content: "";
}
.bank-list::after {
line-height: 0; display: table; content: "";
}
.bank-list::after {
clear: both;
}
.bank-list::before {
line-height: 0; display: table; content: "";
}
.bank-list::after {
line-height: 0; display: table; content: "";
}
.bank-list::after {
clear: both;
}
.bank-list h4 {
margin: 10px 0px; clear: both; font-size: 14px; font-weight: normal;
}
.bank-list .bank-item {
margin: 0px 50px 10px 0px; width: 150px; height: 38px; padding-left: 10px; float: left;
}
.bank-list .bank-item label {
height: 38px; line-height: 38px; position: relative;
}
.bank-list .bank-item .bank {
border: 1px solid rgb(204, 204, 204); border-image: none; width: 150px; display: inline-block; position: absolute; z-index: 2;
}
.bank-list .bank-item input {
margin-right: 10px;
}
.bank-list .bank-item-3.bank-item {
margin-right: 0px;
}
.payment-tip-wrap {
width: 300px;
}
.commit-exercise-wrap {
width: 300px; text-align: center;
}
.commit-exercise-wrap .lead {
font-size: 14px; margin-top: 20px; margin-bottom: 10px;
}
.commit-exercise-wrap .button-row {
margin: 20px 0px 12px;
}
.commit-exercise-wrap .button-row .btn {
margin: 8px 0px;
}
.start-exercise-wrap {
text-align: center;
}
.start-exercise-wrap .lead {
font-size: 14px; margin-top: 20px; margin-bottom: 10px;
}
.start-exercise-wrap .button-row {
margin: 20px 0px 12px;
}
.start-exercise-wrap .button-row .btn {
margin: 8px 0px;
}
.material-wrap {
margin: 0px 10px; padding: 10px 0px; border: 1px solid rgb(197, 197, 197); border-image: none;
}
.material-wrap .material {
padding: 0px 20px; margin-bottom: 10px; border-bottom-color: rgb(197, 197, 197); border-bottom-width: 1px; border-bottom-style: solid;
}
.material-wrap .material .sprite-material {
margin-right: 17px; margin-bottom: 20px; margin-left: -7px;
}
.material-wrap .material-popup {
margin-bottom: 10px;
}
.material-wrap .translate-accessory .solution-item {
padding: 20px 0px; border-top-color: rgb(216, 216, 216); border-top-width: 1px; border-top-style: dotted;
}
.material-wrap .translate-accessory .solution-item .key-label {
margin-right: 10px;
}
.pay-confirm-wrap {
width: 526px;
}
.pay-confirm-wrap::before {
line-height: 0; display: table; content: "";
}
.pay-confirm-wrap::after {
line-height: 0; display: table; content: "";
}
.pay-confirm-wrap::after {
clear: both;
}
.pay-confirm-wrap::before {
line-height: 0; display: table; content: "";
}
.pay-confirm-wrap::after {
line-height: 0; display: table; content: "";
}
.pay-confirm-wrap::after {
clear: both;
}
.pay-confirm-wrap .error-area {
border: 1px solid transparent; border-image: none; width: 256px; height: 325px; text-align: center; float: left; cursor: pointer;
}
.pay-confirm-wrap .success-area {
border: 1px solid transparent; border-image: none; width: 256px; height: 325px; text-align: center; float: left; cursor: pointer;
}
.pay-confirm-wrap .error-area .ape {
margin-top: 50px;
}
.pay-confirm-wrap .success-area .ape {
margin-top: 50px;
}
.pay-confirm-wrap .error-area:hover {
border-color: rgb(204, 204, 204);
}
.pay-confirm-wrap .success-area:hover {
border-color: rgb(204, 204, 204);
}
.pay-confirm-wrap .success-area {
background-color: rgb(241, 249, 241);
}
.pay-confirm-wrap .error-area {
margin-right: 10px; background-color: rgb(255, 244, 244);
}
.question-sheet-wrap {
left: -2px; top: 0px; margin-bottom: -5px; position: absolute;
}
.question-sheet-wrap .toggle {
display: none;
}
.question-sheet-wrap .title {
display: block;
}
.question-sheet-wrap .toggle {
cursor: pointer;
}
.question-sheet-wrap .question-sheet {
padding: 10px; width: 775px;
}
.question-sheet-wrap .question-sheet table {
margin-bottom: 0px;
}
.question-sheet-wrap .question-sheet td {
padding: 0px; border: currentColor; border-image: none; width: 35px; text-align: center;
}
.question-sheet-wrap .question-sheet td[colspan] {
width: auto;
}
.question-sheet-wrap .question-sheet .chapter-name {
padding: 10px 0px; color: rgb(153, 153, 153);
}
.question-sheet-wrap .question-sheet .col-0 {
padding-right: 20px;
}
.question-sheet-wrap .question-sheet .last-col.col-0 {
padding-right: 0px;
}
.question-sheet-wrap .question-sheet .question {
background-position: -773px -250px; margin: 0px auto; width: 28px; height: 28px; text-align: center; line-height: 25px; font-size: 11px; display: block; cursor: pointer; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.question-sheet-wrap .question-sheet .question:hover {
text-decoration: none;
}
.question-sheet-wrap .question-sheet .answered.question {
background-position: -941px -250px; width: 28px; height: 28px; color: rgb(255, 255, 255);
}
.question-sheet-wrap .question-sheet .flagged.question {
background-position: -996px -250px; width: 28px; height: 28px;
}
.question-sheet-wrap .question-sheet .flagged.answered.question {
background-position: -913px -250px; width: 28px; height: 28px; color: rgb(255, 255, 255);
}
.question-sheet-wrap .question-sheet .correct.question {
background-position: -687px -280px; width: 28px; height: 27px; color: rgb(255, 255, 255);
}
.question-sheet-wrap .question-sheet .flagged.correct.question {
background-position: -1164px -250px; width: 28px; height: 27px;
}
.question-sheet-wrap .question-sheet .half-correct.question {
background-position: -575px -280px; width: 28px; height: 27px; color: rgb(255, 255, 255);
}
.question-sheet-wrap .question-sheet .flagged.half-correct.question {
background-position: -603px -280px; width: 28px; height: 27px;
}
.question-sheet-wrap .question-sheet .empty.question {
background-position: -631px -280px; width: 28px; height: 27px; color: rgb(102, 102, 102);
}
.question-sheet-wrap .question-sheet .flagged.empty.question {
background-position: -659px -280px; width: 28px; height: 27px;
}
.question-sheet-wrap .question-sheet .text-question.question {
background-position: -745px -250px; width: 28px; height: 28px; color: rgb(0, 85, 176);
}
.question-sheet-wrap .question-sheet .flagged.text-question.question {
background-position: -717px -250px; width: 28px; height: 28px; color: rgb(0, 85, 176);
}
.question-sheet-wrap .question-sheet .text-answer.question {
background-position: -1080px -250px; width: 28px; height: 28px; color: rgb(102, 102, 102);
}
.question-sheet-wrap .question-sheet .flagged.text-answer.question {
background-position: -829px -250px; width: 28px; height: 28px;
}
.question-sheet-wrap .question-sheet .incorrect.question {
background-position: -1192px -250px; width: 28px; height: 27px; color: rgb(255, 255, 255);
}
.question-sheet-wrap .question-sheet .flagged.incorrect.question {
background-position: -1220px -250px; width: 28px; height: 27px;
}
.fixed.question-sheet-wrap {
left: 50%; top: auto; bottom: 0px; margin-left: -318px; position: fixed;
}
.fixed.question-sheet-wrap .toggle {
display: block;
}
.fixed.question-sheet-wrap .title {
display: none;
}
.flash-tip-wrap {
background: rgb(238, 238, 238); text-align: center; color: rgb(158, 158, 158); font-size: 12px; margin-bottom: 10px; position: relative; min-height: 30px;
}
.flash-tip-wrap .flash-tip {
position: relative; z-index: 2;
}
.flash-tip-wrap .flash-tip .message {
line-height: 30px;
}
.flash-tip-wrap .flash-tip .message td {
border-bottom-color: rgb(186, 186, 186); border-bottom-width: 1px; border-bottom-style: dashed;
}
.flash-tip-wrap .flash-tip .last.message td {
border-bottom-color: currentColor; border-bottom-width: medium; border-bottom-style: none;
}
.flash-tip-wrap .flash-tip .text {
margin-right: 5px; position: relative;
}
.flash-tip-wrap .flash-tip .sprite-tip-button {
color: rgb(255, 255, 255); margin-right: -10px; margin-left: 10px;
}
.flash-tip-wrap .sprite-flash-tip-lb {
left: -15px; top: 0px; position: absolute;
}
.flash-tip-wrap .sprite-flash-tip-lt {
left: -15px; top: 0px; position: absolute;
}
.flash-tip-wrap .sprite-flash-tip-rb {
left: -15px; top: 0px; position: absolute;
}
.flash-tip-wrap .sprite-flash-tip-rt {
left: -15px; top: 0px; position: absolute;
}
.flash-tip-wrap .sprite-flash-tip-rb {
left: auto; right: -15px;
}
.flash-tip-wrap .sprite-flash-tip-rt {
left: auto; right: -15px;
}
.flash-tip-wrap .sprite-flash-tip-lb {
top: auto; bottom: 0px;
}
.flash-tip-wrap .sprite-flash-tip-rb {
top: auto; bottom: 0px;
}
.flash-tip-wrap .repeat-x {
left: 0px; top: 0px; width: auto; right: 0px; position: absolute; background-repeat: repeat-x;
}
.flash-tip-wrap .flash-tip-b.repeat-x {
top: auto; bottom: 0px;
}
.flash-tip-wrap .repeat-y {
left: -15px; top: 14px; height: auto; bottom: 14px; position: absolute; background-repeat: repeat-y;
}
.flash-tip-wrap .flash-tip-r.repeat-y {
left: auto; right: -15px;
}
.flash-tip-center-wrap {
right: 50%; float: right; position: relative;
}
.flash-tip-center-wrap::before {
line-height: 0; display: table; content: "";
}
.flash-tip-center-wrap::after {
line-height: 0; display: table; content: "";
}
.flash-tip-center-wrap::after {
clear: both;
}
.flash-tip-center-wrap::before {
line-height: 0; display: table; content: "";
}
.flash-tip-center-wrap::after {
line-height: 0; display: table; content: "";
}
.flash-tip-center-wrap::after {
clear: both;
}
.flash-tip-center-wrap .flash-tip-wrap {
right: -50%; float: right; position: relative;
}
.footer-tip-wrap {
width: 962px; height: 99px; bottom: 70px; margin-left: -1px; position: absolute; z-index: 2; background-image: url("./6a7de0b0-footer-tip.png");
}
.footer-tip-wrap .content {
padding: 10px 20px 20px; color: rgb(153, 153, 153); font-size: 12px;
}
.home-wrap {
width: 100%; overflow: hidden; padding-bottom: 20px; clear: both; margin-top: 20px; margin-bottom: 0px; margin-left: -1px;
}
.home-wrap::before {
line-height: 0; display: table; content: "";
}
.home-wrap::after {
line-height: 0; display: table; content: "";
}
.home-wrap::after {
clear: both;
}
.home-wrap::before {
line-height: 0; display: table; content: "";
}
.home-wrap::after {
line-height: 0; display: table; content: "";
}
.home-wrap::after {
clear: both;
}
.home-wrap .section-wrap {
width: 50%; margin-right: -1px; border-right-color: rgb(197, 197, 197); border-bottom-color: rgb(197, 197, 197); border-right-width: 1px; border-bottom-width: 1px; border-right-style: solid; border-bottom-style: solid; float: left; position: relative; z-index: 2;
}
.home-wrap .section-wrap .section {
padding: 50px 40px 60px; height: 140px; position: relative; z-index: 100;
}
.home-wrap .section-wrap .section h2 {
margin: 0px 0px 25px; padding: 0px; color: rgb(125, 125, 125); line-height: 1em; font-family: "Hiragino Sans GB", "Microsoft YaHei", "WenQuanYi Micro Hei", SimHei, sans-serif; font-size: 20px; font-weight: normal;
}
.home-wrap .section-wrap .section .content {
height: 65px; color: rgb(158, 158, 158);
}
.home-wrap .section-right.section-wrap {
border-right-color: currentColor; border-right-width: medium; border-right-style: none;
}
.home-wrap .section-last.section-wrap {
border-bottom-color: currentColor; border-bottom-width: medium; border-bottom-style: none;
}
.home-wrap .section-wrap [disabled].btn-primary .sprite-printer-blue {
background-position: -828px -327px; width: 20px; height: 20px;
}
.home-wrap .section-wrap .btn-primary .btn-inner {
letter-spacing: 2px;
}
.section-shadow {
width: 464px; height: 237px; right: -23px; bottom: -23px; position: absolute; background-image: url("./59662324-section-shadow.png");
}
.pause-wrap .pause {
background-position: center 40px; width: 487px; height: 318px; text-align: center; background-image: url("./c42a924b-pause.png"); background-repeat: no-repeat;
}
.pause-wrap .pause .btn {
margin-top: 240px;
}
.answer-sheet-wrap .answer-sheet {
}
.answer-sheet-wrap .answer-sheet::before {
line-height: 0; display: table; content: "";
}
.answer-sheet-wrap .answer-sheet::after {
line-height: 0; display: table; content: "";
}
.answer-sheet-wrap .answer-sheet::after {
clear: both;
}
.answer-sheet-wrap .answer-sheet::before {
line-height: 0; display: table; content: "";
}
.answer-sheet-wrap .answer-sheet::after {
line-height: 0; display: table; content: "";
}
.answer-sheet-wrap .answer-sheet::after {
clear: both;
}
.answer-sheet-wrap .question {
text-align: center; color: rgb(255, 255, 255); line-height: 25px; font-size: 11px; margin-right: 6px; margin-bottom: 5px; float: left; display: block; cursor: pointer;
}
.answer-sheet-wrap .question:hover {
text-decoration: none;
}
.answer-sheet-wrap .col-0.question {
margin-right: 20px;
}
.answer-sheet-wrap .last-col.question {
margin-right: 0px;
}
.answer-sheet-wrap .chapter-name {
padding: 15px 0px 10px; text-align: left; color: rgb(153, 153, 153); clear: both;
}
.answer-sheet-wrap .sprite-answer-empty {
color: rgb(102, 102, 102);
}
.answer-sheet-wrap .sprite-answer-empty-mark {
color: rgb(102, 102, 102);
}
.answer-sheet-wrap .sprite-answer-text {
color: rgb(102, 102, 102);
}
.answer-sheet-wrap .sprite-answer-text-mark {
color: rgb(102, 102, 102);
}
.course-menu-wrap {
background: rgb(255, 255, 255); margin: 0px; border: 0px currentColor; border-image: none; left: 0px; top: 32px; width: 100%; position: absolute; z-index: 400;
}
.course-menu-wrap::before {
line-height: 0; display: table; content: "";
}
.course-menu-wrap::after {
line-height: 0; display: table; content: "";
}
.course-menu-wrap::after {
clear: both;
}
.course-menu-wrap::before {
line-height: 0; display: table; content: "";
}
.course-menu-wrap::after {
line-height: 0; display: table; content: "";
}
.course-menu-wrap::after {
clear: both;
}
.course-menu-wrap .shadow {
background-position: bottom; left: 0px; height: 104px; right: 0px; bottom: -15px; position: absolute; background-image: url("./1485f5df-header-course-bg.png"); background-repeat: repeat-x;
}
.course-menu-wrap .course {
margin: 10px 10px 0px; width: 120px; text-align: center; color: rgb(102, 102, 102); text-decoration: none; position: relative; z-index: 2;
}
.course-menu-wrap .course img {
width: 38px; height: 55px;
}
.course-menu-wrap .course .sprite-course-free-flag {
left: 50%; top: 14px; margin-left: -16px; position: absolute;
}
.course-menu-wrap .active.course {
color: rgb(26, 155, 252);
}
.pinned-tip-wrap {
padding-right: 20px; position: relative;
}
.pinned-tip-wrap .exit {
top: -3px; right: -10px; position: absolute;
}
.header-tip-wrap .popover {
background: none; border: currentColor; border-image: none;
}
.header-tip-wrap .popover .popover-arrow-wrap {
width: 0px; height: 0px; margin-top: 15px; margin-right: -10px;
}
.header-tip-wrap .popover .popover-bd {
background-position: 0px -2204px; padding: 0px 0px 0px 15px; width: auto; height: 32px; line-height: 32px; font-size: 12px; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.header-tip-wrap .popover .popover-bd .pinned-tip-wrap {
background-position: right -2172px; width: auto; height: 32px; color: rgb(238, 238, 238); background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.header-tip-wrap .popover .popover-bd .pinned-tip-wrap .exit {
background-position: -1248px -120px; top: auto; width: 9px; height: 9px; right: auto; bottom: -11px; margin-left: 5px; display: inline-block; position: relative; opacity: 1; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.header-tip-wrap .popover .popover-bd .pinned-tip-wrap .exit:hover {
opacity: 0.8;
}
.error-dialog-wrap .dialog .dialog-bd {
padding: 50px 20px; text-align: center; min-width: 400px;
}
.success-dialog-wrap .dialog .dialog-bd {
padding: 50px 20px; text-align: center; min-width: 200px;
}
.loading-dialog-wrap .dialog .dialog-bd {
padding: 50px 20px; text-align: center; min-width: 200px;
}
.success-dialog-wrap .dialog .dialog-bd i {
margin-right: 10px;
}
.loading-dialog-wrap .dialog .dialog-bd i {
margin-right: 10px;
}
.success-dialog-wrap .dialog .dialog-bd i.icon-refresh {
background-position: 0px 0px; top: -1px; width: 16px; height: 16px; position: relative; background-image: url("./cc826229-loading-16.gif");
}
.loading-dialog-wrap .dialog .dialog-bd i.icon-refresh {
background-position: 0px 0px; top: -1px; width: 16px; height: 16px; position: relative; background-image: url("./cc826229-loading-16.gif");
}
.user-menu-menu-wrap {
margin-top: 10px;
}
.pause-dialog-wrap .dialog .dialog-bd {
padding: 0px;
}
.device-tip-tooltip-wrap .popover-direction-bottom .popover-arrow-wrap {
left: auto; top: -8px; right: 10%;
}
.device-tip-mobile-tooltip-wrap.device-tip-tooltip-wrap {
margin-left: -49px;
}
.device-tip-card-tooltip-wrap.device-tip-tooltip-wrap {
margin-left: 49px;
}
.device-tip-card-tooltip-wrap.device-tip-tooltip-wrap .popover-arrow-wrap {
right: 15px;
}
.question-wrap {
}
.question-wrap::before {
line-height: 0; display: table; content: "";
}
.question-wrap::after {
line-height: 0; display: table; content: "";
}
.question-wrap::after {
clear: both;
}
.question-wrap::before {
line-height: 0; display: table; content: "";
}
.question-wrap::after {
line-height: 0; display: table; content: "";
}
.question-wrap::after {
clear: both;
}
.question-wrap .question {
}
.question-wrap .question::before {
line-height: 0; display: table; content: "";
}
.question-wrap .question::after {
line-height: 0; display: table; content: "";
}
.question-wrap .question::after {
clear: both;
}
.question-wrap .question::before {
line-height: 0; display: table; content: "";
}
.question-wrap .question::after {
line-height: 0; display: table; content: "";
}
.question-wrap .question::after {
clear: both;
}
.question-wrap .question .answers {
list-style: none; margin: 0px; padding: 0px;
}
.question-wrap .question .answers::before {
line-height: 0; display: table; content: "";
}
.question-wrap .question .answers::after {
line-height: 0; display: table; content: "";
}
.question-wrap .question .answers::after {
clear: both;
}
.question-wrap .question .answers::before {
line-height: 0; display: table; content: "";
}
.question-wrap .question .answers::after {
line-height: 0; display: table; content: "";
}
.question-wrap .question .answers::after {
clear: both;
}
.question-wrap .question .answers li {
margin-right: 5px; float: left;
}
.question-wrap .question .flag {
bottom: -20px; color: rgb(153, 153, 153); font-size: 12px; margin-right: 20px; position: relative; cursor: pointer;
}
.question-wrap .question .cancel-flag {
bottom: -20px; color: rgb(153, 153, 153); font-size: 12px; margin-right: 20px; position: relative; cursor: pointer;
}
.question-wrap .question .flag:hover {
color: rgb(58, 143, 251);
}
.question-wrap .question .cancel-flag:hover {
color: rgb(58, 143, 251);
}
.question-wrap .question .flag .sprite {
margin-bottom: -5px;
}
.question-wrap .question .cancel-flag .sprite {
margin-bottom: -5px;
}
.question-wrap .question .content {
padding: 10px 20px 10px 10px; border-bottom-color: rgb(197, 197, 197); border-bottom-width: 1px; border-bottom-style: solid; background-color: rgb(245, 245, 245);
}
.question-wrap .question .content .question-type-tip {
margin-right: 5px; float: left;
}
.question-wrap .question .content .question-type-tip-1.question-type-tip {
color: rgb(253, 155, 203);
}
.question-wrap .question .content .question-type-tip-2.question-type-tip {
color: rgb(253, 152, 39);
}
.question-wrap .question .content .question-type-tip-3.question-type-tip {
color: rgb(106, 202, 107);
}
.question-wrap .question .content .index {
width: 37px; text-align: center; color: rgb(178, 213, 255); font-size: 24px; margin-right: 10px;
}
.question-wrap .question .content .small-index.index {
line-height: 24px; font-size: 18px;
}
.question-wrap .question .content .overflow {
line-height: 1.8em; margin-bottom: -10px;
}
.question-wrap .question .options {
border-bottom-color: rgb(197, 197, 197); border-bottom-width: 1px; border-bottom-style: dashed;
}
.question-wrap .question .options .option {
margin: 10px; padding: 2px 10px 2px 47px; color: rgb(102, 102, 102); position: relative; cursor: pointer;
}
.question-wrap .question .options .option .lbl {
left: 15px; color: rgb(204, 204, 204); position: absolute;
}
.question-wrap .question .options .mouseon.option {
background: rgb(239, 239, 239);
}
.question-wrap .question .answers {
padding: 10px 0px 10px 7px; margin-bottom: -7px;
}
.question-wrap .question .answers::before {
line-height: 0; display: table; content: "";
}
.question-wrap .question .answers::after {
line-height: 0; display: table; content: "";
}
.question-wrap .question .answers::after {
clear: both;
}
.question-wrap .question .answers::before {
line-height: 0; display: table; content: "";
}
.question-wrap .question .answers::after {
line-height: 0; display: table; content: "";
}
.question-wrap .question .answers::after {
clear: both;
}
.question-wrap .question .answers li {
padding: 10px 20px; border: currentColor; border-image: none; width: 40px; line-height: 20px; margin-top: 0px; margin-bottom: 0px; float: left; display: block; cursor: pointer;
}
.question-wrap .question .answers li .answer-radio {
background-position: -1157px -307px; width: 20px; height: 20px; margin-right: 5px; display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.question-wrap .question .answers li .answer-checkbox {
background-position: -1200px -280px; width: 20px; height: 20px; margin-right: 5px; display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.question-wrap .question .answers li.mouseon {
background: rgb(239, 239, 239);
}
.question-wrap .question .answers li.active .answer-radio {
background-position: -575px -327px; width: 20px; height: 20px;
}
.question-wrap .question .answers li.active .answer-checkbox {
background-position: -1074px -307px; width: 20px; height: 20px;
}
.question-wrap .question .tiny-answers.answers li {
padding-right: 10px; padding-left: 10px;
}
.question-wrap .question .answer-area {
min-height: 48px;
}
.question-with-solution-wrap .question .content {
background-color: rgb(255, 244, 244);
}
.question-with-solution-wrap .question .answer {
padding: 20px 10px 20px 25px; margin-bottom: -10px;
}
.question-with-solution-wrap .question .answer .expand-solution {
font-size: 12px; margin-right: 20px; margin-left: 5px; position: relative; cursor: pointer;
}
.question-with-solution-wrap .question .answer .shrink-solution {
font-size: 12px; margin-right: 20px; margin-left: 5px; position: relative; cursor: pointer;
}
.question-with-solution-wrap .question .answer .expand-solution i {
margin-left: 5px;
}
.question-with-solution-wrap .question .answer .shrink-solution i {
margin-left: 5px;
}
.question-with-solution-wrap .question .answer .expand-solution:hover {
text-decoration: none;
}
.question-with-solution-wrap .question .answer .shrink-solution:hover {
text-decoration: none;
}
.question-with-solution-wrap .question .answer .answer-count {
color: rgb(102, 102, 102);
}
.question-with-solution-wrap .question .answer .shrink-solution {
display: none;
}
.question-with-solution-wrap .question .understand {
bottom: 0px; color: rgb(153, 153, 153); font-size: 12px; margin-right: 0px; cursor: pointer;
}
.question-with-solution-wrap .question .flag {
bottom: 0px; color: rgb(153, 153, 153); font-size: 12px; margin-right: 0px; cursor: pointer;
}
.question-with-solution-wrap .question .cancel-flag {
bottom: 0px; color: rgb(153, 153, 153); font-size: 12px; margin-right: 0px; cursor: pointer;
}
.question-with-solution-wrap .question .understand:hover {
color: rgb(58, 143, 251);
}
.question-with-solution-wrap .question .flag:hover {
color: rgb(58, 143, 251);
}
.question-with-solution-wrap .question .cancel-flag:hover {
color: rgb(58, 143, 251);
}
.question-with-solution-wrap .question .understand .sprite {
margin-right: 3px; margin-bottom: -5px;
}
.question-with-solution-wrap .question .flag .sprite {
margin-right: 3px; margin-bottom: -5px;
}
.question-with-solution-wrap .question .cancel-flag .sprite {
margin-right: 3px; margin-bottom: -5px;
}
.question-with-solution-wrap .question .user-answer {
color: rgb(255, 0, 0);
}
.question-with-solution-wrap .question .correct-answer {
color: rgb(70, 165, 70);
}
.question-with-solution-wrap .expanded.question .answer {
margin-bottom: 0px; border-bottom-color: rgb(153, 153, 153); border-bottom-width: 1px; border-bottom-style: dashed;
}
.question-with-solution-wrap .expanded.question .expand-solution {
display: none;
}
.question-with-solution-wrap .expanded.question .shrink-solution {
display: block;
}
.question-with-solution-wrap .solution-wrap {
padding-left: 5px; margin-bottom: -10px; display: none;
}
.question-with-solution-wrap .solution-wrap .key-label {
margin-right: 10px;
}
.question-with-solution-wrap .solution-wrap .solution-item {
margin: 20px;
}
.question-with-solution-wrap .solution-wrap .solution-item .overflow {
line-height: 1.8em;
}
.question-with-solution-wrap .expanded.solution-wrap {
display: block;
}
.correct.question-with-solution-wrap .question .user-answer {
color: rgb(70, 165, 70);
}
.correct.question-with-solution-wrap .question .content {
background-color: rgb(241, 249, 241);
}
.question-with-solution-wrap .question .options .option {
cursor: text;
}
.question-with-solution-wrap .question .options .correct.option {
color: rgb(70, 165, 70);
}
.question-with-solution-wrap .question .options .correct.option span {
color: rgb(70, 165, 70);
}
.question-no-incorrect-state.question-with-solution-wrap .question .content {
background-color: rgb(245, 245, 245);
}
.browser-solution.question-with-solution-wrap .question .content {
background-color: rgb(245, 245, 245);
}
.question-no-incorrect-state.question-with-solution-wrap .question .options .correct.option {
color: rgb(102, 102, 102);
}
.browser-solution.question-with-solution-wrap .question .options .correct.option {
color: rgb(102, 102, 102);
}
.question-no-incorrect-state.question-with-solution-wrap .question .options .correct.option span {
color: rgb(204, 204, 204);
}
.browser-solution.question-with-solution-wrap .question .options .correct.option span {
color: rgb(204, 204, 204);
}
.question-no-incorrect-state.question-with-solution-wrap .question .answer-meta {
visibility: hidden;
}
.browser-solution.question-with-solution-wrap .question .answer-meta {
visibility: hidden;
}
.question-no-incorrect-state.question-with-solution-wrap .expanded.question .options .correct.option {
color: rgb(70, 165, 70);
}
.browser-solution.question-with-solution-wrap .expanded.question .options .correct.option {
color: rgb(70, 165, 70);
}
.question-no-incorrect-state.question-with-solution-wrap .expanded.question .options .correct.option span {
color: rgb(70, 165, 70);
}
.browser-solution.question-with-solution-wrap .expanded.question .options .correct.option span {
color: rgb(70, 165, 70);
}
.question-no-incorrect-state.question-with-solution-wrap .expanded.question .answer-meta {
visibility: visible;
}
.browser-solution.question-with-solution-wrap .expanded.question .answer-meta {
visibility: visible;
}
.pagination {
text-align: right;
}
.audio-player {
line-height: 32px;
}
.audio-player .audio-player-title {
margin-right: 5px;
}
.audio-player .audio-player-progressbar-wrapper {
margin: 0px 16px 0px 0px; top: 3px; height: 12px; position: relative;
}
.audio-player .audio-player-progressbar-wrapper .audio-player-controller {
left: 0px; top: -10px; position: absolute;
}
.audio-player .audio-player-progressbar-wrapper .audio-player-controller .hidden {
display: none;
}
.audio-player .audio-player-progressbar-wrapper .controller-press.audio-player-controller .audio-play {
background-position: -956px -214px; width: 32px; height: 32px;
}
.audio-player .audio-player-progressbar-wrapper .controller-press.audio-player-controller .audio-stop {
background-position: -924px -214px; width: 32px; height: 32px;
}
.audio-player .audio-player-progressbar-wrapper .audio-player-controller:hover {
cursor: pointer;
}
.audio-player .audio-player-progressbar-wrapper .audio-player-progressbar:hover {
cursor: pointer;
}
.audio-player .audio-player-progressbar-wrapper .audio-player-progressbar {
left: 16px; width: 300px; height: 12px; display: inline-block; position: relative;
}
.audio-player .audio-player-progressbar-wrapper .audio-player-progressbar .progress-bg {
left: 0px; width: 100%; position: absolute;
}
.audio-player .audio-player-progressbar-wrapper .audio-player-progressbar .progress-bg .slide-audio-player-progress-right {
left: 30px; width: auto; right: 0px; position: absolute;
}
.audio-player .audio-player-progressbar-wrapper .audio-player-progressbar .play-bar {
width: auto; overflow: hidden; display: inline-block; position: absolute;
}
.audio-player .audio-player-progressbar-wrapper .audio-player-progressbar .play-bar .slide-audio-player-progress-play-right {
left: 30px; width: auto; right: 0px; position: absolute;
}
.audio-player .audio-player-progressbar-wrapper .audio-player-progressbar .buffer-bar {
width: auto; overflow: hidden; display: inline-block; position: absolute;
}
.audio-player .audio-player-progressbar-wrapper .audio-player-progressbar .buffer-bar .slide-audio-player-progress-buffer-right {
left: 30px; width: auto; right: 0px; position: absolute;
}
.audio-player .audio-player-timer {
margin: 0px 0px 0px 10px;
}
.audio-player .hidden.audio-player-timer {
display: none;
}
.audio-player .pinned-tip:hover {
cursor: pointer;
}
.audio-player .pinned-tip {
margin: 0px 0px -4px 10px;
}
.audio-tip {
width: 400px;
}
.fir {
background: url("./fd8b586a-fir.png") no-repeat left top;
}
.fir-btn-normal-instant {
background-position: 0px 0px; width: 58px; height: 18px;
}
.fir-btn-normal-instant_1 {
background-position: 0px -18px; width: 58px; height: 18px;
}
.fir-btn-normal-template {
background-position: -58px 0px; width: 67px; height: 18px;
}
.fir-btn-normal-template_1 {
background-position: -58px -18px; width: 67px; height: 18px;
}
.fir-btn-normal-csk {
background-position: 0px -36px; width: 67px; height: 18px;
}
.fir-btn-normal-csk_1 {
background-position: 0px -54px; width: 67px; height: 18px;
}
.fir-btn-normal-continue {
background-position: -67px -36px; width: 67px; height: 18px;
}
.fir-btn-normal-continue_1 {
background-position: -67px -54px; width: 67px; height: 18px;
}
.fir-btn-normal-coming {
background-position: 0px -72px; width: 68px; height: 18px;
}
.fir-btn-normal-coming_1 {
background-position: 0px -90px; width: 68px; height: 18px;
}
.fir-btn-normal-paper {
background-position: 0px -108px; width: 152px; height: 18px;
}
.fir-btn-normal-paper_1 {
background-position: 0px -126px; width: 152px; height: 18px;
}
.fir-goto-xc {
background-position: -68px -72px; width: 67px; height: 18px;
}
.fir-goto-xc_1 {
background-position: -68px -90px; width: 67px; height: 18px;
}
.fir-goto-sl {
background-position: 0px -144px; width: 67px; height: 18px;
}
.fir-goto-sl_1 {
background-position: 0px -162px; width: 67px; height: 18px;
}
.fir-btn-go-pay {
background-position: -67px -144px; width: 50px; height: 18px;
}
.fir-btn-go-pay_1 {
background-position: -67px -162px; width: 50px; height: 18px;
}
.fir-btn-free-registry {
background-position: 0px -180px; width: 67px; height: 18px;
}
.fir-btn-free-registry_1 {
background-position: 0px -198px; width: 67px; height: 18px;
}
.fir-btn-special-paper {
background-position: -67px -180px; width: 67px; height: 18px;
}
.fir-btn-special-paper_1 {
background-position: -67px -198px; width: 67px; height: 18px;
}
.fir-btn-hot-paper {
background-position: 0px -216px; width: 67px; height: 18px;
}
.fir-btn-hot-paper_1 {
background-position: 0px -234px; width: 67px; height: 18px;
}
.fir-btn-csk-paper {
background-position: -67px -216px; width: 67px; height: 18px;
}
.fir-btn-csk-paper_1 {
background-position: -67px -234px; width: 67px; height: 18px;
}
.fir-btn-hot-paper-kyzz {
background-position: 0px -252px; width: 67px; height: 18px;
}
.fir-btn-hot-paper-kyzz_1 {
background-position: 0px -270px; width: 67px; height: 18px;
}
.fir-btn-giant {
background-position: -67px -252px; width: 67px; height: 18px;
}
.fir-btn-giant_1 {
background-position: -67px -270px; width: 67px; height: 18px;
}
.fir-btn-hot-paper-flgw {
background-position: 0px -288px; width: 101px; height: 18px;
}
.fir-btn-hot-paper-flgw_1 {
background-position: 0px -306px; width: 101px; height: 18px;
}
.fir-header-exercise {
background-position: 0px -324px; width: 71px; height: 14px;
}
.fir-header-exercise_1 {
background-position: 0px -338px; width: 71px; height: 14px;
}
.fir-header-report {
background-position: 0px -352px; width: 84px; height: 14px;
}
.fir-header-report_1 {
background-position: 0px -366px; width: 84px; height: 14px;
}
.fir-header-history {
background-position: -71px -324px; width: 56px; height: 14px;
}
.fir-header-history_1 {
background-position: -71px -338px; width: 56px; height: 14px;
}
.fir-title-giant {
background-position: 0px -380px; width: 109px; height: 21px;
}
.fir-title-smart-paper {
background-position: 0px -401px; width: 74px; height: 21px;
}
.fir-title-sprint {
background-position: -74px -401px; width: 73px; height: 21px;
}
.fir-title-continue {
background-position: 0px -422px; width: 109px; height: 21px;
}
.fir-title-paper {
background-position: 0px -443px; width: 74px; height: 21px;
}
.fir-title-csk {
background-position: 0px -464px; width: 109px; height: 21px;
}
.fir-title-instant {
background-position: 0px -485px; width: 109px; height: 21px;
}
.fir-btn-cancel {
background-position: -101px -288px; width: 50px; height: 18px;
}
.fir-btn-cancel-registry {
background-position: -84px -352px; width: 65px; height: 18px;
}
.fir-btn-normal-login {
background-position: 0px -506px; width: 42px; height: 18px;
}
.fir-btn-normal-login_1 {
background-position: -42px -506px; width: 42px; height: 18px;
}
.fir-btn-normal-submit {
background-position: 0px -524px; width: 42px; height: 18px;
}
.fir-btn-normal-submit_1 {
background-position: -42px -524px; width: 42px; height: 18px;
}
.fir-btn-paper-resume {
background-position: -74px -443px; width: 62px; height: 15px;
}
.fir {
line-height: 1em; display: inline-block;
}
.fir .fir-text {
font-size: 0px; visibility: hidden;
}
.fir-wrap:hover .fir-header-exercise {
background-position: 0px -338px; width: 71px; height: 14px;
}
.active.fir-wrap .fir-header-exercise {
background-position: 0px -338px; width: 71px; height: 14px;
}
.fir-wrap:hover .fir-header-report {
background-position: 0px -366px; width: 84px; height: 14px;
}
.active.fir-wrap .fir-header-report {
background-position: 0px -366px; width: 84px; height: 14px;
}
.fir-wrap:hover .fir-header-history {
background-position: -71px -338px; width: 56px; height: 14px;
}
.active.fir-wrap .fir-header-history {
background-position: -71px -338px; width: 56px; height: 14px;
}
.btn-altern.fir-wrap .btn-inner {
line-height: 0px; padding-top: 7px; padding-bottom: 5px;
}
.btn-primary.fir-wrap .btn-inner {
line-height: 0px; padding-top: 7px; padding-bottom: 5px;
}
.btn-altern.fir-wrap .btn-inner .fir {
overflow: hidden; display: inline-block;
}
.btn-primary.fir-wrap .btn-inner .fir {
overflow: hidden; display: inline-block;
}
[disabled].btn-altern.fir-wrap .fir-btn-normal-paper {
background-position: 0px -126px; width: 152px; height: 18px;
}
[disabled].btn-primary.fir-wrap .fir-btn-normal-paper {
background-position: 0px -126px; width: 152px; height: 18px;
}
[disabled].btn-altern.fir-wrap .fir-btn-normal-coming {
background-position: 0px -90px; width: 68px; height: 18px;
}
[disabled].btn-primary.fir-wrap .fir-btn-normal-coming {
background-position: 0px -90px; width: 68px; height: 18px;
}
[disabled].btn-altern.fir-wrap .fir-btn-normal-continue {
background-position: -67px -54px; width: 67px; height: 18px;
}
[disabled].btn-primary.fir-wrap .fir-btn-normal-continue {
background-position: -67px -54px; width: 67px; height: 18px;
}
[disabled].btn-altern.fir-wrap .fir-btn-normal-csk {
background-position: 0px -54px; width: 67px; height: 18px;
}
[disabled].btn-primary.fir-wrap .fir-btn-normal-csk {
background-position: 0px -54px; width: 67px; height: 18px;
}
[disabled].btn-altern.fir-wrap .fir-btn-normal-instant {
background-position: 0px -18px; width: 58px; height: 18px;
}
[disabled].btn-primary.fir-wrap .fir-btn-normal-instant {
background-position: 0px -18px; width: 58px; height: 18px;
}
[disabled].btn-altern.fir-wrap .fir-btn-normal-template {
background-position: -58px -18px; width: 67px; height: 18px;
}
[disabled].btn-primary.fir-wrap .fir-btn-normal-template {
background-position: -58px -18px; width: 67px; height: 18px;
}
[disabled].btn-altern.fir-wrap .fir-btn-normal-login {
background-position: -42px -506px; width: 42px; height: 18px;
}
[disabled].btn-primary.fir-wrap .fir-btn-normal-login {
background-position: -42px -506px; width: 42px; height: 18px;
}
[disabled].btn-altern.fir-wrap .fir-btn-normal-submit {
background-position: -42px -524px; width: 42px; height: 18px;
}
[disabled].btn-primary.fir-wrap .fir-btn-normal-submit {
background-position: -42px -524px; width: 42px; height: 18px;
}
[disabled].btn-altern.fir-wrap .fir-btn-special-paper {
background-position: -67px -198px; width: 67px; height: 18px;
}
[disabled].btn-primary.fir-wrap .fir-btn-special-paper {
background-position: -67px -198px; width: 67px; height: 18px;
}
[disabled].btn-altern.fir-wrap .fir-btn-hot-paper {
background-position: 0px -234px; width: 67px; height: 18px;
}
[disabled].btn-primary.fir-wrap .fir-btn-hot-paper {
background-position: 0px -234px; width: 67px; height: 18px;
}
[disabled].btn-altern.fir-wrap .fir-btn-hot-paper-kyzz {
background-position: 0px -270px; width: 67px; height: 18px;
}
[disabled].btn-primary.fir-wrap .fir-btn-hot-paper-kyzz {
background-position: 0px -270px; width: 67px; height: 18px;
}
[disabled].btn-altern.fir-wrap .fir-btn-hot-paper-flgw {
background-position: 0px -306px; width: 101px; height: 18px;
}
[disabled].btn-primary.fir-wrap .fir-btn-hot-paper-flgw {
background-position: 0px -306px; width: 101px; height: 18px;
}
[disabled].btn-altern.fir-wrap .fir-btn-csk-paper {
background-position: -67px -234px; width: 67px; height: 18px;
}
[disabled].btn-primary.fir-wrap .fir-btn-csk-paper {
background-position: -67px -234px; width: 67px; height: 18px;
}
[disabled].btn-altern.fir-wrap .fir-btn-giant {
background-position: -67px -270px; width: 67px; height: 18px;
}
[disabled].btn-primary.fir-wrap .fir-btn-giant {
background-position: -67px -270px; width: 67px; height: 18px;
}
.btn-altern.fir-wrap .fir-btn-special-paper {
background-position: -67px -198px; width: 67px; height: 18px;
}
.btn-altern.fir-wrap .fir-btn-hot-paper {
background-position: 0px -234px; width: 67px; height: 18px;
}
.btn-altern.fir-wrap .fir-btn-hot-paper-kyzz {
background-position: 0px -270px; width: 67px; height: 18px;
}
.btn-altern.fir-wrap .fir-btn-hot-paper-flgw {
background-position: 0px -306px; width: 101px; height: 18px;
}
.btn-altern.fir-wrap .fir-btn-csk-paper {
background-position: -67px -234px; width: 67px; height: 18px;
}
.text-tip {
background-position: 0px -128px; padding: 0px 0px 0px 14px; width: auto; height: auto; line-height: 0px; font-size: 0px; display: inline-block; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.text-tip .text-tip-inner {
background-position: right -228px; padding: 0px 14px 0px 0px; width: auto; height: 20px; line-height: 20px; font-size: 12px; display: inline-block; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.school-item-wrap label {
left: -124px; top: 6px; width: 100px; text-align: right; position: absolute;
}
.school-item-wrap select {
width: 150px;
}
.school-panel-item-wrap label {
left: -124px; width: 100px; text-align: right; position: absolute;
}
.quiz-item-wrap {
overflow: auto; margin-right: -10px;
}
.quiz-item-wrap::before {
line-height: 0; display: table; content: "";
}
.quiz-item-wrap::after {
line-height: 0; display: table; content: "";
}
.quiz-item-wrap::after {
clear: both;
}
.quiz-item-wrap::before {
line-height: 0; display: table; content: "";
}
.quiz-item-wrap::after {
line-height: 0; display: table; content: "";
}
.quiz-item-wrap::after {
clear: both;
}
.quiz-item-wrap .group {
margin: 10px 0px; width: 50%; color: rgb(102, 102, 102); float: left;
}
.quiz-item-wrap .group::before {
line-height: 0; display: table; content: "";
}
.quiz-item-wrap .group::after {
line-height: 0; display: table; content: "";
}
.quiz-item-wrap .group::after {
clear: both;
}
.quiz-item-wrap .group::before {
line-height: 0; display: table; content: "";
}
.quiz-item-wrap .group::after {
line-height: 0; display: table; content: "";
}
.quiz-item-wrap .group::after {
clear: both;
}
.quiz-item-wrap .group .area-name {
padding: 5px 0px; width: 70px; margin-left: 10px; float: left; display: block;
}
.quiz-item-wrap .group .quiz {
background: linear-gradient(rgb(241, 241, 241) 0%, rgb(215, 215, 215) 100%); padding: 5px 20px; border: 1px solid rgb(255, 255, 255); border-image: none; text-align: center; font-size: 12px; float: left; cursor: pointer;
}
.quiz-item-wrap .group .checked.quiz {
background: linear-gradient(rgb(0, 224, 96) 0%, rgb(0, 168, 75) 100%); color: rgb(255, 255, 255);
}
.quiz-item-wrap .group .quiz:hover {
border: 1px solid rgb(0, 200, 87); border-image: none;
}
.quiz-item-wrap .flat-quiz {
margin: 5px 5px 20px; width: 140px; height: 25px; float: left; display: block;
}
.quiz-item-wrap .flat-quiz .quiz {
border: 1px solid transparent; border-image: none; height: 23px; text-align: center; line-height: 23px; font-size: 12px; display: block; cursor: pointer;
}
.quiz-item-wrap .flat-quiz .checked.quiz {
background: linear-gradient(rgb(0, 224, 96) 0%, rgb(0, 168, 75) 100%); color: rgb(255, 255, 255);
}
.quiz-item-wrap .flat-quiz .quiz:hover {
border: 1px solid rgb(0, 200, 87); border-image: none;
}
.quiz-item-wrap .loading {
background-position: center; margin: 100px auto 0px; width: 76px; height: 76px; background-image: url("./425c24c6-spin.gif");
}
.gaokao-init-form-wrap {
width: 760px;
}
.gaokao-init-form-wrap .text-item-wrap label {
left: auto; top: 6px; width: auto; text-align: left; margin-right: 42px; float: left; position: relative;
}
.gaokao-init-form-wrap .school-item-wrap label {
left: auto; top: 6px; width: auto; text-align: left; margin-right: 42px; float: left; position: relative;
}
.gaokao-init-form-wrap .quiz-item-wrap {
margin: 0px 0px 15px; padding-top: 15px; border-top-color: rgb(241, 241, 241); border-bottom-color: rgb(241, 241, 241); border-top-width: 4px; border-bottom-width: 4px; border-top-style: solid; border-bottom-style: solid; min-height: 740px;
}
.gaokao-init-form-wrap .quiz-item-wrap .group .area-name {
margin-left: 0px;
}
.quiz-form-wrap {
width: 600px; -ms-overflow-x: hidden; -ms-overflow-y: auto; max-height: 500px;
}
.note-panel-wrap textarea {
padding: 4px 6px; width: 671px; height: 135px; resize: none;
}
.note-panel-wrap .note-panel {
}
.note-panel-wrap .note-panel::before {
line-height: 0; display: table; content: "";
}
.note-panel-wrap .note-panel::after {
line-height: 0; display: table; content: "";
}
.note-panel-wrap .note-panel::after {
clear: both;
}
.note-panel-wrap .note-panel::before {
line-height: 0; display: table; content: "";
}
.note-panel-wrap .note-panel::after {
line-height: 0; display: table; content: "";
}
.note-panel-wrap .note-panel::after {
clear: both;
}
.note-panel-wrap .note-image-accessory {
float: left;
}
.note-panel-wrap .note-image-accessory td {
width: 122px; height: 122px;
}
.note-panel-wrap .note-image-accessory img {
max-height: 100%; max-width: 100%;
}
.note-panel-wrap .note-image-accessory:hover .delete-accessory {
display: block;
}
.note-panel-dialog-wrap .dialog-hd .title {
overflow: hidden; float: left; white-space: nowrap; -ms-text-overflow: ellipsis; max-width: 540px;
}
.note-image-accessory {
background: rgb(238, 238, 238); margin: 15px 0px; border: 1px solid rgb(204, 204, 204); border-image: none; position: relative;
}
.note-image-accessory .delete-accessory {
top: -11px; right: -11px; display: none; position: absolute;
}
.note-image-accessory td {
text-align: center; vertical-align: middle;
}
.note-image-accessory table {
width: 100%;
}
.home-wrap .section-sprint .button-row {
margin-left: -10px;
}
.home-wrap .section-sprint .button-row .btn {
margin-left: 10px;
}
.csk-panel-wrap {
width: 670px;
}
.csk-panel-wrap .csk-hd {
padding: 5px 15px; overflow: hidden; border-top-color: rgb(204, 204, 204); border-bottom-color: rgb(204, 204, 204); border-top-width: 1px; border-bottom-width: 1px; border-top-style: solid; border-bottom-style: solid; position: relative; background-color: rgb(239, 239, 239);
}
.csk-panel-wrap .csk-list {
padding: 0px 20px 15px;
}
.csk-panel-wrap table {
width: 100%;
}
.csk-panel-wrap table tr td {
padding: 20px 0px; -ms-word-break: break-all; -ms-word-wrap: break-word;
}
.csk-panel-wrap .toggle-expand {
cursor: pointer;
}
.csk-panel-wrap .sprite-expand {
margin-right: 10px; position: relative;
}
.csk-panel-wrap .sprite-expand-holder {
margin-right: 10px; position: relative;
}
.csk-panel-wrap .expanded .toggle-expand .sprite-expand {
background-position: -807px -307px; width: 20px; height: 20px;
}
.csk-panel-wrap .sprite-expand {
margin-left: -34px;
}
.csk-panel-wrap .sprite-expand-holder {
margin-left: -34px;
}
.csk-panel-wrap .expand-holder {
margin-left: -34px;
}
.csk-panel-wrap .expand-holder {
padding: 0px 8px;
}
.csk-panel-wrap .name-col {
width: 350px; padding-right: 20px; padding-bottom: 0px;
}
.csk-panel-wrap .name-col .capacity {
padding: 0px 6px;
}
.csk-panel-wrap .name-col .capacity .sprite {
bottom: -2px; position: relative;
}
.csk-panel-wrap .button-col {
width: 120px; text-align: right; padding-bottom: 0px;
}
.csk-panel-wrap .keypoint .text {
padding-left: 30px; display: inline-block; position: relative; -ms-word-break: break-all; -ms-word-wrap: break-word;
}
.csk-panel-wrap .progress.keypoint {
background: rgb(255, 255, 255); vertical-align: top; border-bottom-color: rgb(220, 220, 220); border-bottom-width: 1px; border-bottom-style: dashed;
}
.csk-panel-wrap .keypoint .progress-col {
padding-top: 0px; padding-left: 30px;
}
.csk-panel-wrap .keypoint .progress-col .progress-wrap {
width: 100px; height: 10px; margin-right: 10px; display: inline-block;
}
.csk-panel-wrap .keypoint .progress-col .keypoint-number {
width: 100px; margin-right: 15px; display: inline-block;
}
.csk-panel-wrap .keypoint .progress-col .capacity-wrap {
vertical-align: text-bottom;
}
.csk-panel-wrap .keypoint-level-0.keypoint .expand-holder {
visibility: hidden;
}
.csk-panel-wrap .keypoint-level-1.keypoint {
display: none;
}
.csk-panel-wrap .keypoint-level-1.keypoint .text {
padding-left: 62px;
}
.csk-panel-wrap .keypoint-level-1.keypoint .progress-col {
padding-left: 62px;
}
.csk-panel-wrap .keypoint-level-2.keypoint {
display: none;
}
.csk-panel-wrap .keypoint-level-2.keypoint .text {
padding-left: 94px;
}
.csk-panel-wrap .keypoint-level-2.keypoint .progress-col {
padding-left: 94px;
}
.csk-panel-wrap .keypoint-level-3.keypoint {
display: none;
}
.csk-panel-wrap .keypoint-level-3.keypoint .text {
padding-left: 126px;
}
.csk-panel-wrap .keypoint-level-3.keypoint .progress-col {
padding-left: 126px;
}
.csk-panel-wrap .keypoint-level-4.keypoint {
display: none;
}
.csk-panel-wrap .keypoint-level-4.keypoint .text {
padding-left: 158px;
}
.csk-panel-wrap .keypoint-level-4.keypoint .progress-col {
padding-left: 158px;
}
.csk-panel-wrap .keypoint-progress-wrap {
width: 100px;
}
.paper-wrap {
width: 700px;
}
.paper-wrap .paper {
padding: 15px 10px; border-bottom-color: rgb(197, 197, 197); border-bottom-width: 1px; border-bottom-style: dashed;
}
.paper-wrap .paper .name {
margin-bottom: 10px;
}
.paper-wrap .paper .button-wrap {
width: 150px; text-align: right;
}
.paper-wrap .paper .status-wrap {
line-height: 30px;
}
.paper-wrap .paper .number {
color: rgb(70, 165, 70);
}
.paper-wrap .paper-nav {
overflow: hidden; border-bottom-color: rgb(204, 204, 204); border-bottom-width: 1px; border-bottom-style: solid; position: relative; background-color: rgb(239, 239, 239);
}
.paper-wrap .paper-nav .nav {
padding: 0px 10px; margin-bottom: 0px; position: relative; z-index: 2;
}
.paper-wrap .paper-nav .label-nav {
margin-top: 10px;
}
.paper-wrap .paper-nav .paper-nav-bg {
left: 0px; top: 0px; width: auto; right: 0px; position: absolute; background-repeat: repeat-x; background-color: rgb(255, 255, 255);
}
.paper-wrap .paper-list {
padding: 15px;
}
.paper-wrap .pagination {
margin-bottom: 0px;
}
.hot-paper-wrap .paper .name {
line-height: 30px; margin-bottom: 0px;
}
.hot-paper-wrap .paper .button-wrap {
width: 100px; text-align: right;
}
.hot-paper-wrap .paper-list {
margin-top: -10px;
}
.quiz-panel-wrap {
}
.quiz-panel-wrap::before {
line-height: 0; display: table; content: "";
}
.quiz-panel-wrap::after {
line-height: 0; display: table; content: "";
}
.quiz-panel-wrap::after {
clear: both;
}
.quiz-panel-wrap::before {
line-height: 0; display: table; content: "";
}
.quiz-panel-wrap::after {
line-height: 0; display: table; content: "";
}
.quiz-panel-wrap::after {
clear: both;
}
.quiz-panel-wrap form {
margin-bottom: 0px;
}
.quiz-panel-wrap .item-wrap {
margin-bottom: 0px;
}
.quiz-panel-wrap .quiz-item-wrap {
margin-left: 0px;
}
.quiz-panel-wrap .item-message {
display: none;
}
.quiz-range-panel-wrap {
width: 720px; overflow: auto; margin-right: -10px; margin-left: -10px; min-height: 300px; max-height: 500px;
}
.quiz-range-panel-wrap::before {
line-height: 0; display: table; content: "";
}
.quiz-range-panel-wrap::after {
line-height: 0; display: table; content: "";
}
.quiz-range-panel-wrap::after {
clear: both;
}
.quiz-range-panel-wrap::before {
line-height: 0; display: table; content: "";
}
.quiz-range-panel-wrap::after {
line-height: 0; display: table; content: "";
}
.quiz-range-panel-wrap::after {
clear: both;
}
.quiz-range-panel-wrap .quiz {
width: 50%; margin-bottom: 10px; float: left;
}
.quiz-range-panel-wrap .quiz .quiz-name {
width: 100px; margin-left: 10px; display: inline-block;
}
.quiz-range-panel-wrap .quiz .range-wrap {
background: rgb(235, 235, 235); padding: 5px 20px; line-height: 25px; font-size: 12px;
}
.quiz-range-panel-wrap .quiz .range {
cursor: pointer;
}
.quiz-range-panel-wrap .quiz .checked.range .sprite-checkbox-2 {
background-position: -743px -307px; width: 20px; height: 20px;
}
.quiz-range-panel-wrap .quiz .disabled.range {
cursor: default;
}
.quiz-range-panel-wrap .quiz :disabled.range {
cursor: default;
}
.quiz-range-panel-wrap .quiz .disabled.range .sprite-checkbox-2 {
background-position: -631px -327px; width: 20px; height: 20px;
}
.quiz-range-panel-wrap .quiz :disabled.range .sprite-checkbox-2 {
background-position: -631px -327px; width: 20px; height: 20px;
}
.quiz-range-panel-wrap .loading {
background-position: center; margin: 100px auto 0px; width: 76px; height: 76px; background-image: url("./425c24c6-spin.gif");
}
.nickname-panel-wrap {
width: 270px;
}
.nickname-panel-wrap::before {
line-height: 0; display: table; content: "";
}
.nickname-panel-wrap::after {
line-height: 0; display: table; content: "";
}
.nickname-panel-wrap::after {
clear: both;
}
.nickname-panel-wrap::before {
line-height: 0; display: table; content: "";
}
.nickname-panel-wrap::after {
line-height: 0; display: table; content: "";
}
.nickname-panel-wrap::after {
clear: both;
}
.nickname-panel-wrap form {
margin-bottom: 0px;
}
.nickname-panel-wrap .item-wrap {
margin-bottom: 0px;
}
.nickname-panel-wrap .item-message {
height: 20px; margin-top: 10px; display: block;
}
.school-panel-wrap {
width: 700px; overflow: hidden; min-height: 190px;
}
.school-panel-wrap::before {
line-height: 0; display: table; content: "";
}
.school-panel-wrap::after {
line-height: 0; display: table; content: "";
}
.school-panel-wrap::after {
clear: both;
}
.school-panel-wrap::before {
line-height: 0; display: table; content: "";
}
.school-panel-wrap::after {
line-height: 0; display: table; content: "";
}
.school-panel-wrap::after {
clear: both;
}
.school-panel-wrap .school-select {
margin: 0px 0px -1px; padding: 5px 15px; overflow: auto; border-bottom-color: rgb(204, 204, 204); border-bottom-width: 1px; border-bottom-style: dashed; max-height: 200px;
}
.school-panel-wrap .school-select::before {
line-height: 0; display: table; content: "";
}
.school-panel-wrap .school-select::after {
line-height: 0; display: table; content: "";
}
.school-panel-wrap .school-select::after {
clear: both;
}
.school-panel-wrap .school-select::before {
line-height: 0; display: table; content: "";
}
.school-panel-wrap .school-select::after {
line-height: 0; display: table; content: "";
}
.school-panel-wrap .school-select::after {
clear: both;
}
.school-panel-wrap .school-select li {
list-style: none; padding: 5px; float: left; white-space: nowrap; cursor: pointer;
}
.school-panel-wrap .school-select li.actived {
color: rgb(58, 143, 251);
}
.download-confirm-wrap {
min-width: 400px;
}
.download-confirm-wrap .shenlun-text {
height: 50px; line-height: 30px; font-size: 18px;
}
.download-confirm-wrap .tutor {
margin: 10px 0px;
}
.download-confirm-wrap .tutor span {
width: 482px; height: 219px; display: inline-block; background-image: url("./4af69ab1-download-tip.jpg"); background-repeat: no-repeat;
}
.download-confirm-wrap .tutor span.no-scannable {
background-image: url("./02de1b1c-download-tip-no-scannable.jpg");
}
.download-confirm-wrap .text-right .muted {
font-size: 12px; margin-top: 5px;
}
.word-info-wrap {
min-width: 300px; max-width: 500px;
}
.word-info-wrap::before {
line-height: 0; display: table; content: "";
}
.word-info-wrap::after {
line-height: 0; display: table; content: "";
}
.word-info-wrap::after {
clear: both;
}
.word-info-wrap::before {
line-height: 0; display: table; content: "";
}
.word-info-wrap::after {
line-height: 0; display: table; content: "";
}
.word-info-wrap::after {
clear: both;
}
.word-info-wrap h4 {
margin: 20px 0px 5px; color: rgb(153, 153, 153); font-size: 14px;
}
.word-info-wrap h4.first {
margin-top: 0px;
}
.report-wrap .report-info {
margin: 20px; padding-bottom: 10px; border-bottom-color: rgb(204, 204, 204); border-bottom-width: 1px; border-bottom-style: dashed;
}
.report-wrap .report-score {
padding: 0px 20px;
}
.report-wrap .report-section {
padding: 20px;
}
.report-wrap .report-section h4 {
line-height: 1em; padding-top: 10px; clear: both; font-size: 14px; font-weight: normal; border-top-color: rgb(204, 204, 204); border-top-width: 1px; border-top-style: dashed;
}
.report-wrap .csk-item {
margin: 0px 15px 15px; text-align: center; float: left;
}
.report-wrap .sidebar-wrap .sidebar .side-top .sprite-book {
margin: 34px auto 0px; display: block;
}
.report-wrap .sidebar-wrap .sidebar .nav {
margin-top: -5px;
}
.report-circle {
text-align: center; line-height: 140px; display: block; position: relative;
}
.report-circle span {
line-height: inherit !important;
}
.report-circle .text-large {
line-height: inherit !important;
}
.report-circle .text-xlarge {
line-height: inherit !important;
}
.report-circle .text-xxlarge {
line-height: inherit !important;
}
.report-circle .text-xxlarge {
font-size: 80px !important;
}
.report-circle .unit {
right: 15px; bottom: 11px; line-height: 1em !important; position: absolute;
}
.exercise-report-wrap .exercise-hd {
position: static;
}
.exercise-report-wrap .exercise-hd::before {
line-height: 0; display: table; content: "";
}
.exercise-report-wrap .exercise-hd::after {
line-height: 0; display: table; content: "";
}
.exercise-report-wrap .exercise-hd::after {
clear: both;
}
.exercise-report-wrap .exercise-hd::before {
line-height: 0; display: table; content: "";
}
.exercise-report-wrap .exercise-hd::after {
line-height: 0; display: table; content: "";
}
.exercise-report-wrap .exercise-hd::after {
clear: both;
}
.exercise-report-wrap .exercise-hd .close {
margin-right: 20px; margin-bottom: 5px;
}
.exercise-report-wrap .exercise-hd .split-box {
clear: both;
}
.exercise-report-wrap .report-info {
padding: 10px 20px 0px;
}
.exercise-report-wrap .report-score {
padding: 10px 20px;
}
.exercise-report-wrap .report-section {
padding: 20px;
}
.exercise-report-wrap .report-section h3 {
margin: 0px 0px 15px; line-height: 1em; font-size: 14px; font-weight: normal;
}
.exercise-report-wrap .report-section h4 {
line-height: 1em; padding-top: 10px; clear: both; font-size: 14px; font-weight: normal; border-top-color: rgb(204, 204, 204); border-top-width: 1px; border-top-style: dashed;
}
.exercise-report-wrap .csk-item-wrap {
margin: 0px 15px 15px 0px; width: 133px; display: inline-block;
}
.exercise-report-wrap .csk-item-wrap .csk-item {
text-align: center;
}
.exercise-report-wrap .sidebar-wrap .sidebar .side-top .sprite-report {
margin: 34px auto 0px; line-height: 1em; display: block;
}
.exercise-report-wrap .sidebar-wrap .sidebar .nav {
margin-top: -5px;
}
.exercise-report-wrap .exercise-ft {
height: 46px;
}
.exercise-report-wrap .exercise-ft .share-wrap {
width: 754px; height: 43px; bottom: -7px; color: rgb(102, 102, 102); padding-top: 8px; padding-bottom: 2px; padding-left: 55px; margin-left: -5px; position: absolute; background-image: url("./fa0fc1aa-share-bar.png");
}
.exercise-report-wrap .exercise-ft .share-wrap a {
margin: 0px 10px; bottom: -8px; position: relative;
}
.exercise-report-wrap .exercise-ft .share-wrap .split {
margin: 0px 10px 0px 20px; color: rgb(102, 102, 102);
}
.exercise-report-wrap .exercise-ft .share-wrap .sprite {
display: inline-block; cursor: pointer;
}
.exercise-report-wrap .exercise-ft .fixed.share-wrap {
bottom: -4px; margin-bottom: 0px; position: fixed;
}
.paper-exercise-report-wrap .report-score .lbl {
color: rgb(153, 153, 153);
}
.paper-exercise-report-wrap .report-score .value {
color: rgb(58, 143, 251);
}
.paper-exercise-report-wrap .report-score .left-column {
width: 310px; margin-right: 25px; float: left;
}
.paper-exercise-report-wrap .report-score .left-column .main-column {
margin-right: 20px; float: left;
}
.paper-exercise-report-wrap .report-score .left-column .main-column p {
text-align: center; margin-top: 16px;
}
.paper-exercise-report-wrap .report-score .left-column .main-column p .lbl {
color: rgb(153, 153, 153); font-size: 12px;
}
.paper-exercise-report-wrap .report-score .left-column .main-column p .lbl-large {
color: rgb(102, 102, 102); font-size: 14px;
}
.paper-exercise-report-wrap .report-score .left-column .main-column p .split {
color: rgb(153, 153, 153);
}
.paper-exercise-report-wrap .report-score .left-column .main-column p .wrap {
padding-bottom: 9px; border-bottom-color: rgb(155, 222, 196); border-bottom-width: 2px; border-bottom-style: solid;
}
.paper-exercise-report-wrap .report-score .left-column .side-column {
overflow: hidden;
}
.paper-exercise-report-wrap .report-score .left-column .item-row {
padding: 17px 0px; font-size: 12px; border-top-color: rgb(204, 204, 204); border-top-width: 1px; border-top-style: dashed;
}
.paper-exercise-report-wrap .report-score .left-column .item-row::before {
line-height: 0; display: table; content: "";
}
.paper-exercise-report-wrap .report-score .left-column .item-row::after {
line-height: 0; display: table; content: "";
}
.paper-exercise-report-wrap .report-score .left-column .item-row::after {
clear: both;
}
.paper-exercise-report-wrap .report-score .left-column .item-row::before {
line-height: 0; display: table; content: "";
}
.paper-exercise-report-wrap .report-score .left-column .item-row::after {
line-height: 0; display: table; content: "";
}
.paper-exercise-report-wrap .report-score .left-column .item-row::after {
clear: both;
}
.paper-exercise-report-wrap .report-score .left-column .item-row .lbl {
float: left;
}
.paper-exercise-report-wrap .report-score .left-column .item-row .number {
line-height: 1em; font-size: 16px; margin-right: 2px;
}
.paper-exercise-report-wrap .report-score .left-column .item-row .value {
float: right;
}
.paper-exercise-report-wrap .report-score .left-column .chapter.item-row {
padding: 10px 0px;
}
.paper-exercise-report-wrap .report-score .left-column .chapter.item-row .lbl {
width: 225px; overflow: hidden; display: block; white-space: nowrap; -ms-text-overflow: ellipsis;
}
.paper-exercise-report-wrap .report-score .left-column .chapter.item-row .time {
color: rgb(153, 153, 153);
}
.paper-exercise-report-wrap .report-score .left-column .first.item-row {
padding-top: 0px; border-top-color: currentColor; border-top-width: medium; border-top-style: none;
}
.paper-exercise-report-wrap .report-score .left-column .last.item-row {
padding-bottom: 0px;
}
.paper-exercise-report-wrap .report-score .left-column .qualified-score {
background: rgb(227, 245, 252); padding: 15px 15px 9px; border: currentColor; border-image: none; clear: both; margin-top: 20px; margin-bottom: 15px;
}
.paper-exercise-report-wrap .report-score .left-column .qualified-score .lbl {
line-height: 30px; font-size: 14px;
}
.paper-exercise-report-wrap .report-score .left-column .qualified-score .value {
color: rgb(51, 51, 51); line-height: 30px; font-size: 14px;
}
.paper-exercise-report-wrap .report-score .left-column .qualified-score .number {
line-height: 30px; font-size: 30px; display: inline-block;
}
.paper-exercise-report-wrap .report-score .right-column {
overflow: hidden;
}
.paper-exercise-report-wrap .trend-image-wrap {
background: linear-gradient(rgb(248, 248, 248) 0%, rgb(238, 238, 238) 100%); margin: 20px auto 10px; border: 1px solid rgb(191, 191, 191); border-image: none; width: 420px; height: 300px; text-align: center; position: relative;
}
.paper-exercise-report-wrap .trend-image-wrap .loading {
margin-top: -60px;
}
.paper-exercise-report-wrap .trend-image-wrap .event-layer {
left: 0px; top: 0px; width: 100%; height: 100%; position: absolute; z-index: 2;
}
.keypoint-exercise-report-wrap .report-score .score-col {
width: 160px; text-align: center;
}
.keypoint-exercise-report-wrap .report-score .score-col .lbl {
padding-bottom: 5px; margin-bottom: 15px; border-bottom-color: rgb(251, 225, 160); border-bottom-width: 2px; border-bottom-style: solid;
}
.keypoint-exercise-report-wrap .report-score .score-col .report-circle {
margin: 0px auto;
}
.keypoint-exercise-report-wrap .report-score table {
width: 100%;
}
.keypoint-exercise-report-wrap .answer-sheet-wrap {
margin-bottom: -5px; margin-left: 15px;
}
.keypoint-exercise-report-wrap .answer-sheet-wrap .answer-sheet {
margin: 0px auto;
}
.keypoint-exercise-report-wrap .answer-sheet-wrap .columns-10 {
width: 370px;
}
.keypoint-exercise-report-wrap .answer-sheet-wrap .columns-10 .question {
}
.keypoint-exercise-report-wrap .answer-sheet-wrap .columns-20 .question {
margin-right: 0px; margin-bottom: 15px;
}
.keypoint-exercise-report-wrap .answer-sheet-wrap .columns-20 .col-0.question {
margin-right: 6px;
}
.keypoint-exercise-report-wrap .answer-sheet-wrap .columns-20 .last-col.question {
margin-right: 0px;
}
.exercise-keypoint-wrap .sidebar-wrap .sidebar .side-top .sprite-book {
margin: 34px auto 0px; display: block;
}
.exercise-keypoint-wrap .sidebar-wrap .sidebar .nav {
margin-top: -5px;
}
.exercise-keypoint-wrap .exercise-bd {
padding: 10px;
}
.exercise-keypoint-wrap .exercise-bd .category {
margin: 10px 0px; border: 1px solid rgb(210, 210, 210); border-image: none;
}
.exercise-keypoint-wrap .exercise-bd .category .category-hd {
padding: 5px 15px; color: rgb(51, 51, 51); letter-spacing: 1px; font-weight: bold; border-top-color: rgb(181, 231, 247); border-top-width: 4px; border-top-style: solid; background-color: rgb(246, 246, 246);
}
.exercise-keypoint-wrap .exercise-bd .chapter .chapter-hd {
padding: 5px 15px; color: rgb(102, 102, 102); font-weight: bold; border-top-color: rgb(226, 226, 226); border-top-width: 4px; border-top-style: solid; background-color: rgb(246, 246, 246);
}
.exercise-keypoint-wrap .exercise-bd .chapter .chapter-hd .expand-trigger {
height: 100%; font-size: 12px; cursor: pointer;
}
.exercise-keypoint-wrap .exercise-bd .chapter .chapter-hd .expand-trigger a:hover {
text-decoration: none;
}
.exercise-keypoint-wrap .exercise-bd .chapter .chapter-hd .expand-trigger .expand .sprite {
margin-bottom: -5px;
}
.exercise-keypoint-wrap .exercise-bd .chapter .chapter-hd .expand-trigger .shrink .sprite {
margin-bottom: -5px;
}
.exercise-keypoint-wrap .exercise-bd .chapter .chapter-hd .shrink {
display: none;
}
.exercise-keypoint-wrap .exercise-bd .chapter .chapter-hd .expand-flag {
background-position: -610px -347px; width: 14px; height: 14px; margin-right: 10px; margin-bottom: -2px; display: inline-block; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.exercise-keypoint-wrap .exercise-bd .chapter .chapter-bd {
display: none;
}
.exercise-keypoint-wrap .exercise-bd .chapter .chapter-bd .loading {
padding: 10px 15px; display: block;
}
.exercise-keypoint-wrap .exercise-bd .chapter .chapter-bd .loading .loading-gif {
bottom: -2px; margin-right: 8px;
}
.exercise-keypoint-wrap .exercise-bd .expanded.chapter .chapter-hd .expand-flag {
background-position: -815px -347px; width: 14px; height: 14px;
}
.exercise-keypoint-wrap .exercise-bd .expanded.chapter .chapter-hd .shrink {
display: inline-block;
}
.exercise-keypoint-wrap .exercise-bd .expanded.chapter .chapter-hd .expand {
display: none;
}
.exercise-keypoint-wrap .exercise-bd .expanded.chapter .chapter-bd {
display: block;
}
.exercise-keypoint-wrap .exercise-bd .keypoint .keypoint-hd {
padding: 10px 15px; color: rgb(153, 153, 153); font-weight: bold; border-top-color: rgb(216, 216, 216); border-top-width: 1px; border-top-style: dashed;
}
.exercise-keypoint-wrap .exercise-bd .keypoint .keypoint-bd {
padding: 0px 15px 10px; color: rgb(51, 51, 51); line-height: 1.5em; margin-top: -10px;
}
.exercise-keypoint-wrap .exercise-bd .keypoint .keypoint-bd p {
margin: 10px 0px 0px;
}
.changed-keypoint-list-wrap table {
border: 1px solid rgb(209, 209, 209); border-image: none; color: rgb(102, 102, 102);
}
.changed-keypoint-list-wrap table th {
background: rgb(237, 237, 237);
}
.changed-keypoint-list-wrap table th {
padding: 5px 0px; border-left-color: rgb(209, 209, 209); border-left-width: 1px; border-left-style: solid;
}
.changed-keypoint-list-wrap table td {
padding: 5px 0px; border-left-color: rgb(209, 209, 209); border-left-width: 1px; border-left-style: solid;
}
.changed-keypoint-list-wrap table td {
line-height: 30px;
}
.changed-keypoint-list-wrap table .even td {
background-color: rgb(247, 247, 247);
}
.changed-keypoint-list-wrap table .capacity-col .capacity-wrap {
float: left;
}
.changed-keypoint-list-wrap table .capacity-col .sprite-level-up {
float: left;
}
.changed-keypoint-list-wrap table .capacity-col .sprite-level-down {
float: left;
}
.changed-keypoint-list-wrap table .capacity-col .sprite-level-up {
margin: 0px 10px; bottom: -6px; position: relative;
}
.changed-keypoint-list-wrap table .capacity-col .sprite-level-down {
margin: 0px 10px; bottom: -6px; position: relative;
}
.changed-keypoint-list-wrap table .capacity-col {
padding: 5px 20px; width: 350px;
}
.changed-keypoint-list-wrap table .name-col {
padding: 5px 10px;
}
.changed-keypoint-list-wrap table .name-col .text {
padding-left: 20px; display: inline-block; position: relative;
}
.changed-keypoint-list-wrap table .name-col .text .sprite-green-dot {
left: 0px; top: 5px; position: absolute;
}
.changed-keypoint-list-wrap .desc {
color: rgb(102, 102, 102); padding-bottom: 8px;
}
.user-report-wrap .report-desc {
margin: 8px 0px;
}
.user-report-wrap .meta-area {
margin-bottom: 5px;
}
.user-report-wrap .split-line {
margin: 20px 0px 20px -20px; padding: 0px 20px; width: 100%; border-bottom-color: rgb(204, 204, 204); border-bottom-width: 1px; border-bottom-style: solid;
}
.user-report-wrap .score-box .box {
padding: 0px; overflow: hidden;
}
.user-report-wrap .score-box .box .box-bd {
background: linear-gradient(rgb(251, 251, 251) 0%, rgb(250, 249, 249) 50%, rgb(235, 233, 233) 100%); width: 100%;
}
.user-report-wrap .score-box .user-score .left-column {
width: 240px; border-right-color: rgb(204, 204, 204); border-right-width: 1px; border-right-style: solid;
}
.user-report-wrap .score-box .user-score .middle-column {
width: 240px; border-right-color: rgb(204, 204, 204); border-right-width: 1px; border-right-style: solid;
}
.user-report-wrap .score-box .user-score .right-column {
overflow: hidden;
}
.user-report-wrap .score-box .user-score .right-column .lbl {
text-align: right; color: rgb(150, 150, 150); margin-top: 15px; margin-right: 35px; margin-bottom: 0px; display: block;
}
.user-report-wrap .score-box .user-score .trend-image-wrap {
margin: 0px auto 20px; width: 436px; height: 300px; position: relative;
}
.user-report-wrap .score-box .user-score .trend-image-wrap .loading {
margin-top: -90px;
}
.user-report-wrap .score-box .user-score-with-distribution-chart {
position: relative;
}
.user-report-wrap .score-box .user-score-with-distribution-chart .left-column {
width: 240px; border-right-color: rgb(204, 204, 204); border-right-width: 1px; border-right-style: solid;
}
.user-report-wrap .score-box .user-score-with-distribution-chart .middle-column {
width: 240px; border-right-color: rgb(204, 204, 204); border-right-width: 1px; border-right-style: solid;
}
.user-report-wrap .score-box .user-score-with-distribution-chart .right-column {
top: 0px; width: 475px; right: 0px; position: absolute;
}
.user-report-wrap .score-box .user-score-with-distribution-chart .right-column .lbl {
text-align: right; color: rgb(150, 150, 150); margin-top: 15px; margin-right: 35px; margin-bottom: 0px; display: block;
}
.user-report-wrap .score-box .user-score-with-distribution-chart .bottom-column {
width: 481px; padding-bottom: 20px; clear: left; border-top-color: rgb(204, 204, 204); border-right-color: rgb(204, 204, 204); border-top-width: 1px; border-right-width: 1px; border-top-style: solid; border-right-style: solid;
}
.user-report-wrap .score-box .user-score-with-distribution-chart .bottom-column .lbl {
text-align: right; color: rgb(150, 150, 150); margin-top: 15px; margin-right: 35px; margin-bottom: 0px; display: block;
}
.user-report-wrap .score-box .user-score-with-distribution-chart .trend-image-wrap {
margin: 0px auto; width: 436px; height: 270px; position: relative;
}
.user-report-wrap .score-box .user-score-with-distribution-chart .trend-image-wrap .loading {
margin-top: -90px;
}
.user-report-wrap .score-box .user-score-with-distribution-chart .distribution-chart {
width: 475px; height: 640px; position: absolute;
}
.user-report-wrap .score-box .user-score-with-distribution-chart .distribution-chart .loading {
margin-top: 70px; margin-left: 90px;
}
.user-report-wrap .score-box .question-count {
margin: 0px 20px; padding: 20px 0px; text-align: center;
}
.user-report-wrap .score-box .score-info {
margin: 0px 20px; padding: 20px 0px; text-align: center;
}
.user-report-wrap .score-box .question-count .report-circle {
margin: 0px auto 20px;
}
.user-report-wrap .score-box .score-info .report-circle {
margin: 0px auto 20px;
}
.user-report-wrap .score-box .question-count .lbl-wrap {
margin: 0px auto; width: 120px; text-align: center; padding-bottom: 5px; border-bottom-color: rgb(155, 222, 196); border-bottom-width: 2px; border-bottom-style: solid;
}
.user-report-wrap .score-box .score-info .lbl-wrap {
margin: 0px auto; width: 120px; text-align: center; padding-bottom: 5px; border-bottom-color: rgb(155, 222, 196); border-bottom-width: 2px; border-bottom-style: solid;
}
.user-report-wrap .score-box .question-count .lbl-wrap .lbl {
color: rgb(153, 153, 153); font-size: 12px;
}
.user-report-wrap .score-box .score-info .lbl-wrap .lbl {
color: rgb(153, 153, 153); font-size: 12px;
}
.user-report-wrap .score-box .question-count .lbl-wrap .lbl-large {
color: rgb(102, 102, 102);
}
.user-report-wrap .score-box .score-info .lbl-wrap .lbl-large {
color: rgb(102, 102, 102);
}
.user-report-wrap .score-box .question-count .lbl-wrap .split {
padding: 0px 5px; color: rgb(102, 102, 102);
}
.user-report-wrap .score-box .score-info .lbl-wrap .split {
padding: 0px 5px; color: rgb(102, 102, 102);
}
.user-report-wrap .score-box .question-count.question-count .lbl-wrap {
border-bottom-color: rgb(251, 223, 155); border-bottom-width: 2px; border-bottom-style: solid;
}
.user-report-wrap .score-box .question-count.score-info .lbl-wrap {
border-bottom-color: rgb(251, 223, 155); border-bottom-width: 2px; border-bottom-style: solid;
}
.user-report-wrap .score-box .item-row {
margin: 0px 16px; padding: 10px 0px; font-size: 12px; border-bottom-color: rgb(216, 216, 216); border-bottom-width: 1px; border-bottom-style: dashed;
}
.user-report-wrap .score-box .item-row::before {
line-height: 0; display: table; content: "";
}
.user-report-wrap .score-box .item-row::after {
line-height: 0; display: table; content: "";
}
.user-report-wrap .score-box .item-row::after {
clear: both;
}
.user-report-wrap .score-box .item-row::before {
line-height: 0; display: table; content: "";
}
.user-report-wrap .score-box .item-row::after {
line-height: 0; display: table; content: "";
}
.user-report-wrap .score-box .item-row::after {
clear: both;
}
.user-report-wrap .score-box .item-row .lbl {
bottom: -1px; color: rgb(153, 153, 153); display: block; position: relative;
}
.user-report-wrap .score-box .item-row .score {
text-align: right; color: rgb(58, 143, 251); display: block;
}
.user-report-wrap .score-box .item-row .score .number {
font-size: 14px; margin-right: 5px;
}
.user-report-wrap .score-box .item-row .score .number .unit {
font-size: 12px;
}
.user-report-wrap .score-box .item-row .score .total-user.number {
color: rgb(153, 153, 153); font-size: 12px;
}
.user-report-wrap .score-box .item-row .score .user-index.number {
margin-right: 0px;
}
.user-report-wrap .score-box .item-row .newline.score {
text-align: right; clear: left; float: none; display: block;
}
.user-report-wrap .score-box .last.item-row {
padding-bottom: 20px; border-bottom-color: currentColor; border-bottom-width: medium; border-bottom-style: none;
}
.user-report-wrap .score-box .trend-image-wrap {
text-align: center;
}
.user-report-wrap .score-box .trend-image-wrap .event-layer {
left: 0px; top: 0px; width: 100%; height: 100%; position: absolute; z-index: 2;
}
.user-report-wrap .share-report {
margin-top: -1px; margin-right: 10px;
}
.user-report-wrap .share-report .lbl {
display: none;
}
.user-report-wrap .share-report .sprite-qq-28:hover {
background-position: -1052px -250px; width: 28px; height: 28px;
}
.user-report-wrap .share-report .sprite-weibo-28:hover {
background-position: -885px -250px; width: 28px; height: 28px;
}
.user-report-wrap .share-report .sprite-renren-28:hover {
background-position: -857px -250px; width: 28px; height: 28px;
}
.user-report-popup {
background-position: 0px -280px; left: 0px; top: 0px; width: 239px; height: 117px; color: rgb(255, 255, 255); font-size: 12px; margin-top: -101px; margin-left: -209px; display: none; position: absolute; z-index: 2; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat;
}
.user-report-popup p {
margin: 10px 0px -6px 25px;
}
.report-history-wrap {
min-width: 300px; max-width: 500px;
}
.report-history-wrap h4 {
margin: 0px; padding: 10px 20px; color: rgb(153, 153, 153); font-size: 14px; font-weight: normal; border-bottom-color: rgb(232, 232, 232); border-bottom-width: 1px; border-bottom-style: solid; position: relative;
}
.report-history-wrap h4 .text-blue {
color: rgb(155, 205, 253);
}
.report-history-wrap h4 .sprite-up-arrow {
left: 40px; bottom: -1px; position: absolute;
}
.report-history-wrap > div {
border-bottom-color: rgb(204, 204, 204); border-bottom-width: 1px; border-bottom-style: solid;
}
.report-history-wrap a {
margin: 10px 20px 0px; padding: 5px 10px 7px; width: 36px; text-align: center; color: rgb(51, 51, 51); display: inline-block;
}
.report-history-wrap a:hover {
color: rgb(58, 143, 251);
}
.report-history-wrap a:hover {
text-decoration: none;
}
.report-history-wrap .active {
border-bottom-color: rgb(58, 143, 251); border-bottom-width: 3px; border-bottom-style: solid;
}
.exercise-advice-wrap ul {
list-style: none; margin: 0px;
}
.exercise-advice-wrap ul li {
margin: 10px 0px; padding-bottom: 10px; padding-left: 30px; border-bottom-color: rgb(204, 204, 204); border-bottom-width: 1px; border-bottom-style: dashed;
}
.exercise-advice-wrap ul li .sprite-bullet {
margin-right: 10px; margin-bottom: -1px; margin-left: -26px;
}
.exercise-advice-wrap h4 {
color: rgb(155, 205, 253); font-size: 14px; margin-top: 20px;
}
.exercise-advice-wrap h3 {
margin-bottom: -20px;
}
.csk-table-wrap .csk-table {
margin: 0px; border: 1px solid rgb(209, 209, 209); border-image: none; color: rgb(102, 102, 102);
}
.csk-table-wrap .csk-table tr .shrink {
display: none;
}
.csk-table-wrap .csk-table tr.expanded .shrink {
display: inline-block;
}
.csk-table-wrap .csk-table tr.expanded .expand {
display: none;
}
.csk-table-wrap .csk-table .correct-col {
text-align: center;
}
.csk-table-wrap .csk-table .rate-col {
text-align: center;
}
.csk-table-wrap .csk-table .score-col {
text-align: center;
}
.csk-table-wrap .csk-table .time-col {
text-align: center;
}
.csk-table-wrap .csk-table th {
border-left-color: rgb(209, 209, 209); border-left-width: 1px; border-left-style: solid;
}
.csk-table-wrap .csk-table td {
border-left-color: rgb(209, 209, 209); border-left-width: 1px; border-left-style: solid;
}
.csk-table-wrap .csk-table .text-center {
text-align: center;
}
.csk-table-wrap .csk-table .even td {
background: rgb(247, 247, 247);
}
.csk-table-wrap .csk-table th {
background: rgb(237, 237, 237); padding-bottom: 10px;
}
.csk-table-wrap .csk-table th.name-col {
width: 430px;
}
.csk-table-wrap .csk-table .keypoint .text {
display: inline-block; position: relative; -ms-word-break: break-all; -ms-word-wrap: break-word;
}
.csk-table-wrap .csk-table .keypoint .sprite-expand {
margin-right: 10px;
}
.csk-table-wrap .csk-table .keypoint .sprite-expand-holder {
margin-right: 10px;
}
.csk-table-wrap .csk-table .keypoint .expand-holder {
padding: 0px 8px;
}
.csk-table-wrap .csk-table .keypoint .sprite-expand {
left: -28px; position: absolute;
}
.csk-table-wrap .csk-table .keypoint .sprite-expand-holder {
left: -28px; position: absolute;
}
.csk-table-wrap .csk-table .keypoint .expand-holder {
left: -28px; position: absolute;
}
.csk-table-wrap .csk-table .keypoint .name-col {
padding-left: 36px;
}
.csk-table-wrap .csk-table .keypoint-level-1.keypoint .name-col {
padding-left: 68px;
}
.csk-table-wrap .csk-table .keypoint-level-2.keypoint .name-col {
padding-left: 100px;
}
.csk-table-wrap .csk-table .keypoint-level-3.keypoint .name-col {
padding-left: 132px;
}
.csk-table-wrap .csk-table .keypoint-level-4.keypoint .name-col {
padding-left: 164px;
}
.csk-table-wrap .csk-table .keypoint-level-1 {
display: none;
}
.csk-table-wrap .csk-table .keypoint-level-2 {
display: none;
}
.csk-table-wrap .csk-table .keypoint-level-3 {
display: none;
}
.csk-table-wrap .csk-table .keypoint-level-4 {
display: none;
}
.csk-table-wrap .csk-table .keypoint-level-5 {
display: none;
}
.csk-table-wrap .toggle-expand {
cursor: pointer;
}
.csk-table-wrap .toggle-expand .sprite-expand {
position: relative;
}
.csk-table-wrap .expanded .toggle-expand .sprite-expand {
background-position: -807px -307px; width: 20px; height: 20px;
}
.user-csk-table-wrap .csk-table td {
line-height: 30px;
}
.user-csk-table-wrap .csk-table .name-col .text {
display: block;
}
.user-csk-table-wrap .csk-table .count-col {
width: 300px; text-align: left;
}
.user-csk-table-wrap .csk-table .count-col .progress-wrap {
width: 170px; display: inline-block;
}
.user-csk-table-wrap .csk-table .ratio-col {
width: 70px; text-align: center;
}
.user-csk-table-wrap .csk-table .progress-col {
width: 330px;
}
.user-csk-table-wrap .csk-table .progress-col .progress-wrap {
bottom: -2px; margin-right: 10px; display: inline-block;
}
.user-csk-table-wrap .csk-table .subject .name-col .toggle {
margin-left: 30px;
}
.user-csk-table-wrap .csk-table .keypoint .name-col .toggle {
margin-left: 60px;
}
.user-csk-table-wrap .csk-table .capacity-col {
text-align: center;
}
.user-csk-table-wrap .csk-table th.capacity-col {
width: 220px; text-align: left; padding-left: 40px;
}
.user-csk-table-wrap .csk-table .capacity-wrap {
display: block;
}
.user-csk-table-wrap .csk-table .capacity-wrap .sprite {
margin: 0px 5px;
}
.user-csk-table-wrap .keypoint-number {
margin-left: 10px; display: inline-block; white-space: nowrap;
}
.user-csk-table-wrap .keypoint .sprite-expand {
top: 5px;
}
.user-csk-table-wrap .keypoint .sprite-expand-holder {
top: 5px;
}
.progress-tip-wrap th {
padding: 10px; text-align: left; border-right-color: rgb(204, 204, 204); border-bottom-color: rgb(204, 204, 204); border-right-width: 1px; border-bottom-width: 1px; border-right-style: dashed; border-bottom-style: dashed;
}
.progress-tip-wrap th.last {
border-right-color: currentColor; border-right-width: medium; border-right-style: none;
}
.progress-tip-wrap td {
padding: 10px; border-right-color: rgb(204, 204, 204); border-right-width: 1px; border-right-style: dashed;
}
.progress-tip-wrap td.last {
border-right-color: currentColor; border-right-width: medium; border-right-style: none;
}
.progress-tip-wrap .sprite-progress-bg {
position: relative;
}
.progress-tip-wrap .progress-bar {
height: 100%; display: inline-block; position: absolute;
}
.download-cover-wrap {
background: url("./4151740c-download-cover.png") no-repeat; margin: 20px auto 0px; width: 428px; height: 542px;
}
.download-cover-wrap .download-cover {
padding: 130px 52px 0px 37px; text-align: center;
}
.download-cover-wrap .download-cover h1 {
font-size: 25px;
}
.download-cover-wrap .download-cover .title-area {
height: 120px;
}
.download-cover-wrap .download-cover .pending {
color: rgb(153, 153, 153);
}
.download-cover-wrap .download-cover .pending img {
margin-bottom: 30px; position: relative;
}
.download-cover-wrap .download-cover .done {
margin-top: 80px;
}
.download-cover-wrap .download-cover .done .btn-primary {
margin-top: 20px;
}
.active-tip-wrap {
width: 500px; color: rgb(102, 102, 102);
}
.active-tip-wrap h2 {
line-height: 14px; font-size: 14px; margin-bottom: 20px;
}
.active-tip-wrap p {
line-height: 2em; margin-bottom: 20px;
}
.active-tip-wrap .button-row {
}
.active-tip-wrap .button-row::before {
line-height: 0; display: table; content: "";
}
.active-tip-wrap .button-row::after {
line-height: 0; display: table; content: "";
}
.active-tip-wrap .button-row::after {
clear: both;
}
.active-tip-wrap .button-row::before {
line-height: 0; display: table; content: "";
}
.active-tip-wrap .button-row::after {
line-height: 0; display: table; content: "";
}
.active-tip-wrap .button-row::after {
clear: both;
}
.active-tip-wrap .button-row form {
margin: 0px;
}
.need-registry-wrap {
width: 360px;
}
.need-registry-wrap p {
margin-top: 30px; margin-bottom: 30px;
}
.need-registry-wrap .button-row {
margin-bottom: 20px;
}
.need-registry-wrap .button-row .btn {
margin: 0px 10px;
}
.keypoint-item-wrap {
margin-right: 20px; white-space: nowrap;
}
.keypoint-item-wrap .slide {
display: inline-block;
}
.keypoint-item-wrap .slide-keypoint-left {
padding: 0px 20px; width: auto;
}
.keypoint-item-wrap .slide-keypoint-right {
padding: 0px 10px; width: 240px; color: rgb(255, 255, 255);
}
.keypoint-item-wrap .slide-keypoint-blue-left {
width: auto; line-height: 0px; padding-left: 18px;
}
.keypoint-item-wrap .slide-keypoint-blue-right {
padding: 0px 20px 0px 2px; width: auto; color: rgb(255, 255, 255); line-height: 27px;
}
.sprint-need-payment-wrap .desc {
width: 480px; margin-bottom: 25px;
}
.setting-wrap .hr {
margin: 25px auto; width: 500px; height: 1px; border-top-color: rgb(204, 204, 204); border-top-width: 1px; border-top-style: dashed;
}
.question-wrap .answer-area {
height: 50px;
}
.question-wrap .question .content .question-type-tip {
color: rgb(107, 196, 251);
}
.question-wrap .question .content .question-type-tip .question-type-tip-1 {
color: rgb(253, 155, 203);
}
.question-wrap .question .content .question-type-tip .question-type-tip-2 {
color: rgb(253, 152, 39);
}
.question-wrap .question .content .question-type-tip .question-type-tip-3 {
color: rgb(106, 202, 107);
}
.exercise-solution-wrap .pager-header {
padding-left: 20px;
}
.exercise-solution-wrap .pager-header::before {
line-height: 0; display: table; content: "";
}
.exercise-solution-wrap .pager-header::after {
line-height: 0; display: table; content: "";
}
.exercise-solution-wrap .pager-header::after {
clear: both;
}
.exercise-solution-wrap .pager-header::before {
line-height: 0; display: table; content: "";
}
.exercise-solution-wrap .pager-header::after {
line-height: 0; display: table; content: "";
}
.exercise-solution-wrap .pager-header::after {
clear: both;
}
.exercise-solution-wrap .pager-header .pagination {
margin-top: -5px; margin-bottom: -10px; float: right;
}
.search-form .search-query {
background-position: -575px 0px; padding: 0px 15px; border: currentColor; border-image: none; width: 652px; height: 44px; margin-left: -6px; display: inline-block; box-shadow: none; background-image: url("../img/exercises/acaf8bc0-sprite.png"); background-repeat: no-repeat; -webkit-box-shadow: none; -moz-box-shadow: none;
}
.search-form .search-query:hover {
background-position: -575px -44px; padding: 0px 15px; border: currentColor; border-image: none; width: 652px; height: 44px; box-shadow: none; -webkit-box-shadow: none; -moz-box-shadow: none;
}
.search-form .search-query:focus {
background-position: -575px -44px; padding: 0px 15px; border: currentColor; border-image: none; width: 652px; height: 44px; box-shadow: none; -webkit-box-shadow: none; -moz-box-shadow: none;
}
.search-form .search-query:active {
background-position: -575px -44px; padding: 0px 15px; border: currentColor; border-image: none; width: 652px; height: 44px; box-shadow: none; -webkit-box-shadow: none; -moz-box-shadow: none;
}
.search-form .search-nav {
list-style: none; margin-top: 40px; margin-left: 0px;
}
.search-form .search-nav::before {
line-height: 0; display: table; content: "";
}
.search-form .search-nav::after {
line-height: 0; display: table; content: "";
}
.search-form .search-nav::after {
clear: both;
}
.search-form .search-nav::before {
line-height: 0; display: table; content: "";
}
.search-form .search-nav::after {
line-height: 0; display: table; content: "";
}
.search-form .search-nav::after {
clear: both;
}
.search-form .search-nav li {
font-size: 16px; margin-right: 20px; float: left;
}
.search-form .search-nav li a {
text-decoration: none; cursor: pointer;
}
.search-form .search-nav li a:hover {
color: rgb(0, 136, 204);
}
.search-form .search-nav li a.active {
color: rgb(51, 51, 51);
}
.search-form .search-nav li a.active:hover {
color: rgb(51, 51, 51);
}
.search-form .search-result {
color: rgb(153, 153, 153); font-size: 12px;
}
.search-form .search-query-word {
color: rgb(241, 151, 38); font-style: normal;
}
.sprint-paper-wrap .sprint-nav li:hover {
cursor: pointer;
}
.sprint-paper-wrap .sprint-nav li {
color: rgb(156, 153, 153);
}
.sprint-paper-wrap .sprint-nav .active {
color: rgb(0, 0, 0); border-bottom-color: rgb(58, 143, 251); border-bottom-width: 5px; border-bottom-style: solid;
}
.sprint-paper-wrap .paper .name {
margin: 0px; line-height: 30px;
}
.sprint-paper-wrap .paper .button-wrap {
width: 100px;
}
.sprint-paper-wrap .paper .no-exercise span {
color: rgb(255, 255, 255);
}
.quiz-panel-dialog-wrap .close {
display: none;
}
.quiz-panel-dialog-wrap .close-btn {
display: none;
}
.school-dialog-wrap .dialog .dialog-bd {
padding: 0px; overflow: hidden;
}
.csk-panel-dialog-wrap {
min-height: 700px;
}
.csk-panel-dialog-wrap .dialog .dialog-bd {
padding: 0px;
}
.paper-dialog-wrap {
min-height: 800px;
}
.paper-dialog-wrap .dialog .dialog-bd {
padding: 0px;
}
.history-wrap .box {
padding: 0px;
}
.history-wrap .box-hd {
background: rgb(243, 243, 243); padding: 20px; border-bottom-color: rgb(201, 201, 201); border-bottom-width: 1px; border-bottom-style: solid;
}
.history-wrap .box-hd .nav {
margin: 0px;
}
.history-wrap .box-bd {
padding: 20px;
}
.download-confirm-dialog-wrap {
width: 524px;
}
.need-payment-dialog-wrap {
width: 524px;
}
.need-payment-dialog-wrap .text-right .pull-left {
line-height: 30px;
}
.progress-tip-tip-wrap .popover-direction-bottom .popover-arrow-wrap {
left: 50%; margin-left: -10px;
}
.progress-tip-tip-wrap .popover-direction-top .popover-arrow-wrap {
left: 50%; margin-left: -10px;
}
.progress-tip-tip-wrap .popover .popover-bd {
padding: 10px;
}
.word-info-tool-tip-wrap .popover-bd {
overflow: auto; max-height: 300px;
}
.word-info-tool-tip-wrap .popover-bd p {
-ms-word-break: break-all; -ms-word-wrap: break-word;
}
.table-list tr td {
line-height: 30px; border-top-style: dashed;
}
.table-list tr td.text-right {
text-align: right;
}
.exercise-list-wrap .list .exercise {
padding-top: 10px; margin-bottom: 10px; border-bottom-color: rgb(204, 204, 204); border-bottom-width: 1px; border-bottom-style: dashed;
}
.exercise-list-wrap .list .exercise .name {
line-height: 30px;
}
.exercise-list-wrap .list .exercise .button-wrap {
width: 200px; text-align: right;
}
.exercise-list-wrap .list .exercise .muted {
font-size: 12px;
}
.exercise-list-wrap .list-bd .keypoint-table {
width: 100%;
}
.exercise-list-wrap .list-bd .keypoint-table td {
padding: 10px 0px;
}
.exercise-list-wrap .list-bd .keypoint-table .name-col {
padding-left: 5px;
}
.exercise-list-wrap .list-bd .keypoint-table .button-col {
text-align: right;
}
.exercise-list-wrap .list-bd .keypoint-table .create-exercise .btn-inner {
width: 60px;
}
.exercise-list-wrap .list-bd .keypoint-table .toggle-expand {
cursor: pointer;
}
.exercise-list-wrap .list-bd .keypoint-table .sprite-expand {
margin-right: 10px; position: relative;
}
.exercise-list-wrap .list-bd .keypoint-table .sprite-expand-holder {
margin-right: 10px; position: relative;
}
.exercise-list-wrap .list-bd .keypoint-table .expanded .toggle-expand .sprite-expand {
background-position: -807px -307px; width: 20px; height: 20px;
}
.exercise-list-wrap .list-bd .keypoint-table .sprite-expand {
margin-left: -34px;
}
.exercise-list-wrap .list-bd .keypoint-table .sprite-expand-holder {
margin-left: -34px;
}
.exercise-list-wrap .list-bd .keypoint-table .expand-holder {
margin-left: -34px;
}
.exercise-list-wrap .list-bd .keypoint-table .expand-holder {
padding: 0px 8px;
}
.exercise-list-wrap .list-bd .keypoint-table .keypoint .text {
padding-left: 30px; display: inline-block; position: relative; -ms-word-break: break-all; -ms-word-wrap: break-word;
}
.exercise-list-wrap .list-bd .keypoint-table .keypoint-level-0.keypoint .expand-holder {
visibility: hidden;
}
.exercise-list-wrap .list-bd .keypoint-table .keypoint-level-1.keypoint {
display: none;
}
.exercise-list-wrap .list-bd .keypoint-table .keypoint-level-1.keypoint .text {
padding-left: 62px;
}
.exercise-list-wrap .list-bd .keypoint-table .keypoint-level-2.keypoint {
display: none;
}
.exercise-list-wrap .list-bd .keypoint-table .keypoint-level-2.keypoint .text {
padding-left: 94px;
}
.exercise-list-wrap .list-bd .keypoint-table .keypoint-level-3.keypoint {
display: none;
}
.exercise-list-wrap .list-bd .keypoint-table .keypoint-level-3.keypoint .text {
padding-left: 126px;
}
.exercise-list-wrap .list-bd .keypoint-table .keypoint-level-4.keypoint {
display: none;
}
.exercise-list-wrap .list-bd .keypoint-table .keypoint-level-4.keypoint .text {
padding-left: 158px;
}
.exercise-list-wrap .keypoint {
margin: 10px 10px 10px 0px; padding: 10px 0px; line-height: 30px; clear: both; border-bottom-color: rgb(216, 216, 216); border-bottom-width: 1px; border-bottom-style: dashed;
}
.exercise-list-wrap .list-hd {
margin: 0px 10px 10px 0px; padding: 10px 0px; line-height: 30px; clear: both; border-bottom-color: rgb(229, 229, 229); border-bottom-width: 3px; border-bottom-style: solid;
}
.exercise-by-exam-list-wrap {
margin-top: -10px;
}
.exercise-by-exam-list-wrap .exam {
border-bottom-color: rgb(229, 229, 229); border-bottom-width: 3px; border-bottom-style: solid;
}
.exercise-by-exam-list-wrap .category {
border-bottom-color: rgb(216, 216, 216); border-bottom-width: 1px; border-bottom-style: dashed;
}
.exercise-by-exam-list-wrap .exam {
margin: 10px 10px 10px 0px; padding: 10px 0px; line-height: 30px; clear: both;
}
.exercise-by-exam-list-wrap .category {
margin: 10px 10px 10px 0px; padding: 10px 0px; line-height: 30px; clear: both;
}
.seo-paper-list-wrap .paper-list {
margin-top: -10px;
}
.seo-paper-list-wrap .paper {
padding: 10px 0px; border-bottom-color: rgb(197, 197, 197); border-bottom-width: 1px; border-bottom-style: dashed;
}
.seo-paper-list-wrap .paper .muted {
margin-right: 20px;
}
.seo-paper-list-wrap .box {
padding: 0px;
}
.seo-paper-list-wrap .box-hd {
background: rgb(245, 245, 245); padding: 10px 20px; border-bottom-color: rgb(204, 204, 204); border-bottom-width: 1px; border-bottom-style: solid;
}
.seo-paper-list-wrap .box-bd {
padding: 20px 20px 10px;
}
.seo-paper-list-wrap .box-ft {
padding: 0px 20px;
}
.seo-question-list-wrap .box {
padding: 0px; position: relative;
}
.seo-question-list-wrap .box .box-hd h1 {
margin: 0px; padding: 20px; color: rgb(102, 102, 102); line-height: 14px; font-size: 14px; border-bottom-color: rgb(153, 153, 153); border-bottom-width: 1px; border-bottom-style: solid;
}
.seo-question-list-wrap .box .box-hd .chapter-nav {
padding: 5px 10px;
}
.seo-question-list-wrap .box .box-hd .chapter-nav .nav {
margin-bottom: 0px;
}
.seo-question-list-wrap .box .split-box {
margin: 10px 0px 0px; padding: 0px 15px; position: relative;
}
.seo-question-list-wrap .box .split-box .repeat-x {
top: 0px; width: 100%; display: block; background-repeat: repeat-x;
}
.seo-question-list-wrap .box .split-box .sprite-box-split-l {
left: 0px; top: 0px; position: absolute;
}
.seo-question-list-wrap .box .split-box .sprite-box-split-r {
top: 0px; right: 0px; position: absolute;
}
.seo-question-list-wrap .box .split-box-2.split-box {
padding: 0px;
}
.seo-question-list-wrap .box .chapter-wrap {
padding-bottom: 10px;
}
.seo-question-list-wrap .box .flag {
display: none;
}
.seo-question-list-wrap .box .cancel-flag {
display: none;
}
.seo-question-list-wrap .material-popup {
display: none;
}
.search-question-list-wrap {
width: auto;
}
.search-question-list-wrap .box {
padding: 0px; position: relative;
}
.search-question-list-wrap .box .box-hd h1 {
margin: 0px; padding: 20px; color: rgb(102, 102, 102); line-height: 14px; font-size: 14px; border-bottom-color: rgb(153, 153, 153); border-bottom-width: 1px; border-bottom-style: solid;
}
.search-question-list-wrap .box .box-hd .chapter-nav {
padding: 5px 10px;
}
.search-question-list-wrap .box .box-hd .chapter-nav .nav {
margin-bottom: 0px;
}
.search-question-list-wrap .box .split-box {
margin: 10px 0px 0px; padding: 0px 15px; position: relative;
}
.search-question-list-wrap .box .split-box .repeat-x {
top: 0px; width: 100%; display: block; background-repeat: repeat-x;
}
.search-question-list-wrap .box .split-box .sprite-box-split-l {
left: 0px; top: 0px; position: absolute;
}
.search-question-list-wrap .box .split-box .sprite-box-split-r {
top: 0px; right: 0px; position: absolute;
}
.search-question-list-wrap .box .split-box-2.split-box {
padding: 0px;
}
.search-question-list-wrap .box .chapter-wrap {
padding-bottom: 10px;
}
.search-question-list-wrap .pagination {
margin-right: 20px;
}
.search-question-list-wrap em {
color: rgb(241, 151, 38) !important; font-style: normal;
}
.search-question-list-wrap .material-wrap {
margin-top: 10px;
}
.search-question-list-wrap .material-wrap ~ .material-wrap {
margin-top: 0px;
}
.search-question-list-wrap .question-wrap ~ .question-wrap {
margin-top: -10px;
}
.search-question-list-wrap .material-wrap .question-wrap {
margin-top: -10px;
}
.search-question-list-wrap .box .split-box {
margin: 10px 0px; padding: 0px 15px; position: relative;
}
.search-question-list-wrap .question-no-incorrect-state.question-with-solution-wrap .answer-meta {
visibility: visible;
}
.search-question-list-wrap .browser-solution.question-with-solution-wrap .answer-meta {
visibility: visible;
}
.search-question-list-wrap .question-with-solution-wrap .question .answer {
margin-bottom: 0px; border-bottom-color: rgb(153, 153, 153); border-bottom-width: 1px; border-bottom-style: dashed;
}
.search-question-list-wrap .question-with-solution-wrap .solution-wrap {
display: block;
}
.search-question-list-wrap .question-with-solution-wrap .solution-wrap .solution-item {
display: none;
}
.search-question-list-wrap .question-with-solution-wrap .expanded.solution-wrap .solution-item {
display: block;
}
.search-question-list-wrap .question-with-solution-wrap .solution-wrap .solution-source {
display: block;
}
.search-question-list-wrap .question-with-solution-wrap .options .correct.option {
color: rgb(70, 165, 70);
}
.search-question-list-wrap .question-with-solution-wrap .options .correct.option span {
color: rgb(70, 165, 70);
}
.answer-text-wrap {
background-position: 0px -248px; width: auto; height: 24px; line-height: 0px; padding-left: 10px; display: inline-block; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat;
}
.answer-text-wrap input[type=text] {
background-position: right -272px; padding: 2px 10px 2px 0px; outline: 0px; border-radius: 0px; border: currentColor; transition:none; border-image: none; width: 250px; height: 20px; box-shadow: none; background-image: url("./8c8d48f3-slide.png"); background-repeat: no-repeat; -webkit-border-radius: 0; -moz-border-radius: 0; -webkit-box-shadow: none; -moz-box-shadow: none; -webkit-transition: none; -moz-transition: none; -o-transition: none;
}
.answer-text-wrap input[type=text]:focus {
height: 20px; box-shadow: none; -webkit-box-shadow: none; -moz-box-shadow: none;
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{"url":"https:\/\/theoremoftheweek.wordpress.com\/2010\/06\/27\/theorem-30-pythagorean-triples\/","text":"## Theorem 30: Pythagorean\u00a0triples\n\nIf I asked you to give a list of mathematical theorems, I suspect that you might well think of Pythagoras\u2018s theorem pretty early on.\u00a0 It has the rare distinction of being a theorem that\u2019s commonly discussed (by name) in maths classes in schools today.\u00a0 It should probably be a theorem of the week in its own right, but for today I\u2019d like to focus on some rather lovely number theory associated with it.\n\nLet\u2019s quickly remind ourselves what Pythagoras\u2019s theorem says.\u00a0 Here\u2019s a right-angled triangle.\n\nThe theorem says that $a^2 + b^2 = c^2$: the square on the hypotenuse is equal to the sum of the squares on the other two sides.\u00a0 There are a lot of proofs of this, but they\u2019ll have to wait for a future post, I\u2019m afraid!\n\nI think that a\u00a0lot of people discover, in the course of their studies of Pythagoras\u2019s theorem at school, that there are some particularly nice right-angled triangles.\u00a0 One has side lengths 3, 4, 5 (quick check: $3^2 + 4^2 = 9 + 16 = 25 = 5^2$).\u00a0 Another has side lengths 5, 12, 13 (check: $5^2 + 12^2 = 25 + 144 = 169 = 13^2$).\u00a0 Another has side lengths 6, 8, 10 (check: $6^2 + 8^2 = 36 + 64 = 100 = 10^2$).\u00a0 Of course, these are particularly nice because the side lengths are all whole numbers (integers).\u00a0 I\u2019m going to concentrate on such solutions in this post.\n\nThis leads to lots of interesting questions.\u00a0 Are there more solutions (with integer side lengths)?\u00a0 How many more?\u00a0 Are there infinitely many?\u00a0 How can we find more solutions?\u00a0 Can we find them all?\u00a0 Are some more interesting than others? \u00a0Are there any interesting patterns?\u00a0 How can we possibly hope to solve a single\u00a0equation in three variables? \u00a0I encourage you to think about these questions before reading on.\u00a0 (You could, for example, look for more solutions from yourself and see where you can go from there.)\u00a0 You might also have your own questions that you\u2019d like to consider, of course.\n\nLet\u2019s think about the solutions we\u2019ve seen so far.\n\nI don\u2019t think that 6, 8, 10 is a very interesting solution, because it\u2019s just twice a solution we\u2019d already seen (3, 4, 5).\u00a0 We could generate infinitely many solutions in this way, just by multiplying by different positive integers.\u00a0 (If you have doubts, you might like to convince yourself of this algebraically.)\u00a0 So that would give infinitely many solutions, but it doesn\u2019t feel to me that we should really count them as different.\u00a0 I\u2019d like to concentrate on solutions (a,b,c) where the side lengths aren\u2019t all divisible by some number (they have no common factor bigger than 1: they are coprime).\u00a0 These are called primitive solutions.\u00a0 So I\u2019m interested in (3,4,5) and (5, 12, 13), but I\u2019m going to ignore (6, 8, 10) and (9, 12, 15) and (12, 16, 20) and\u00a0(10, 24, 26) and so on.\n\nCan we answer any of our questions above, now that we\u2019ve got this extra rule?\n\nLet\u2019s suppose we\u2019ve got a primitive solution, and see what we can say about it.\u00a0 Perhaps that will help us to learn something useful.\u00a0 So let\u2019s suppose we\u2019ve got three positive integers $(a,b,c)$ such that $a^2 + b^2 = c^2$, and let\u2019s assume that the highest common factor of $a$, $b$ and $c$ is 1.\n\nWhere can we go from here?\u00a0 In such cases, it\u2019s sometimes helpful to start eliminating possibilities by considering parities (whether numbers are odd or even).\u00a0 Let\u2019s think about the parities of $a$ and $b$.\n\nIf $a$ and $b$ are both even, then $c$ must be even too (using $a^2 + b^2 = c^2$).\u00a0 But then $a$, $b$ and $c$ have 2 as a common factor, and that contradicts one of our assumptions.\u00a0 So $a$ and $b$ cannot both be even.\n\nIf $a$ and $b$ are both odd, then $a^2 + b^2 \\equiv 2$ (mod 4).\u00a0 But, as we\u2019ve seen before, 2 is not a square (mod 4) (there are no square numbers that leave 2 on division by 4), so there are no solutions to the congruence $c^2 \\equiv 2$ (mod 4).\u00a0 So $a$ and $b$ cannot both be odd.\n\nSo one of $a$ and $b$ is odd, and the other even.\u00a0 Without loss of generality, we can assume that $a$ is odd and $b$ even.\u00a0 Let\u2019s say that $b = 2b'$, where $b'$ is another positive integer.\n\nWe have coprime positive integers $a$, $b'$ and $c$ such that $a^2 + 4b'^2 = c^2$.\u00a0 Can we say any more?\u00a0 We have a single equation in three variables, so we can\u2019t possibly expect to find a unique solution (of course, as we know there are multiple solutions \u2014 we\u2019ve seen some already!), but we have the extra information that $a$, $b'$ and $c$ are integers, and that might help us to draw conclusions (as it has already done).\n\nMy eye was drawn to rearranging this equation to give $c^2 - a^2 = 4b'^2$, because I know that $c^2 - a^2$ is the difference of two squares, and so factorises nicely: $c^2 - a^2 = (c-a)(c+a)$.\u00a0 So we know that $(c-a)(c+a) = 4b'^2$.\u00a0 This feels promising to me, because everything in sight is a whole number, so the options are greatly reduced.\u00a0 In particular, the product $(c-a)(c+a)$ must be divisible by 4.\u00a0 We know that $a$ is odd; let\u2019s think about the possibilities for $c$.\n\nIf $c$ is even, then $c-a$ and $c+a$ are both odd, so their product is odd.\u00a0 That\u2019s no good.\n\nSo $c$ must be odd.\u00a0 Then $c-a$ and $c+a$ are both even (and their product is divisible by 4).\u00a0 Let\u2019s say that $c+a = 2m$ and $c-a = 2n$, where $m$ and $n$ are positive integers with $m > n$.\u00a0 Then $a = m-n$ and $c = m+n$.\n\nThat makes our equation into $mn = b'^2$.\u00a0 Can we\u00a0draw any conclusions from that?\u00a0 It might be tempting to suggest that $m$ and $n$ must be squares (since their product is a square).\u00a0 In general, that isn\u2019t true.\u00a0 For example, $3 \\times 12 = 6^2$.\u00a0 But in fact it is true in this case, because $m$ and $n$ are coprime.\u00a0 (If the prime $p$ divides $m$ and $n$, then $p$ divides $a = m-n$ and $c = m+n$, and also $p$ divides $b'$ and so $b$ \u2014 but that contradicts our assumption that $a$, $b$ and $c$ are coprime.)\u00a0 Unique factorisation then allows us to conclude that $m$ and $n$ are squares, say $m = r^2$ and $n = s^2$.\n\nNow we have $a = r^2 - s^2$, $b=2rs$, and $c = r^2 + s^2$.\u00a0 To recap: if $(a,b,c)$ is a solution, and if $a$, $b$ and $c$ are coprime, then they must be of the form $a=r^2-s^2$, $b = 2rs$, $c = r^2 + s^2$, for some positive integers $r$ and $s$ with $r > s$.\u00a0 So all the primitive solutions must be of this form.\u00a0 But does\u00a0every triple of this form give a solution?\n\nLet\u2019s see.\u00a0 We can just do the algebra.\u00a0 We have $a^2 + b^2 = (r^2-s^2)^2 + (2rs)^2 = r^4 - 2r^2s^2+s^4+4r^2s^2=r^4+2r^2s^2+s^4=(r^2+s^2)^2=c^2$, so we do indeed always get a solution.\u00a0 We do not necessarily get a primitive solution (where the side lengths are coprime), though.\u00a0 For example, $r=3$ and $s=1$ gives $a=8$, $b=6$, $c=10$.\n\nBut let\u2019s be a bit more careful.\u00a0 We know that with our primitive solution we had $a$ and $c$ odd, so one of $m$ and $n$ must be odd and the other even.\u00a0 The same must apply to $r$ and $s$\u00a0 We also saw that we must have $m$ and $n$ coprime, so $r$ and $s$ must be coprime.\u00a0 If we impose those conditions (which would rule out $r=3$, $s=1$), do we necessarily get a primitive solution?\u00a0 It turns out that we do.\u00a0 You might like to check this for yourself (it\u2019s not too hard).\u00a0 And that gives this week\u2019s theorem.\n\nTheorem A triple of coprime positive integers $(a,b,c)$ is a solution to the equation $a^2+b^2 = c^2$ if and only if it has the form $(r^2-s^2,2rs,r^2+s^2)$ for some coprime positive integers $r$ and $s$, where $r>s$ and exactly one of $r$ and $s$ is even.\n\nWikipedia reveals that this is a theorem of Euclid.\n\nIn particular, this theorem tells us that there are infinitely many primitive solutions, but it tells us much more than that, because it describes all of\u00a0 them.\u00a0 It doesn\u2019t get much better than that!\n\nWikipedia has lots of fascinating facts about Pythagorean triples.\u00a0 It also gives some other ways to generate such triples.\u00a0 Solving an equation like this (where the solutions have to be integers) belongs to the study of Diophantine equations, a key branch of number theory (named after Diophantus of Alexandria).\u00a0 As I\u2019ve previously noted, it was in Bachet\u2018s translation of Diophantus that Fermat made his marginal note about what has possibly become the most famous Diophantine equation (inspired by Pythagoras\u2019s equation above).","date":"2017-12-11 17:14:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 120, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8951963186264038, \"perplexity\": 234.74573145028828}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, 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{"url":"https:\/\/ask.sagemath.org\/questions\/39029\/revisions\/","text":"# Revision history [back]\n\n### How to take partial derivative of abstract\/unknown function?\n\nI have a function f(x) that I do not want to be expanded but wish to evaluate the partial derivative of g(x,f(x),...) with respect to x, I can not figure out how to declare the function and indicate it is a function of x without an explicit definition.\n\nCode example:\n\nvar(\"x\")\nf = ??? # how do I define this?\ng = x * f(x)\ng.diff(x)\n\n\nExpected output:\n\nx * f'(x) + f(x)","date":"2019-11-17 23:30:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7248438000679016, \"perplexity\": 579.8924802934231}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496669352.5\/warc\/CC-MAIN-20191117215823-20191118003823-00039.warc.gz\"}"} | null | null |
Published 16 March 2018
I read "Tom Brown" on The Good Food Guide website and wanted to share it with you.
After working closely with Nathan Outlaw for several years, Tom Brown is set to open his debut restaurant in London
'When the doors finally open for that first service, I think it's going to be a mixture of relief, nerves and happiness,' laughs Tom Brown, looking around the site that will soon become Cornerstone, his first restaurant since leaving mentor Nathan Outlaw.
Named after his favourite Arctic Monkeys song, Cornerstone opens in London's Hackney Wick in April and all eyes are on this rising star, who was previously head chef at Outlaw's at the Capital, where he worked for six years.
Like most chefs, it has always been Brown's dream to open his own place but there are the inevitable nerves about becoming his own boss for the first time.
'It's a daunting prospect going it alone but, fortunately for me, I have an amazing team in place who are all very talented in their own fields. That's going to allow me to rely on them and for me to focus on the food and creative direction of Cornerstone.
'Working under Nathan for all those years was an incredible experience and I'm so grateful to him and the whole Outlaw family for everything they've done for me during that time.
'It was a hard decision to leave, but I'm very lucky that Nathan has been so supportive. He's always on hand to help with any concerns I've had, and anyone who's been in the same position as me will know how invaluable that is. I literally can't thank the man enough.'
Interestingly, Cornish-born Brown decided to open his first restaurant in the capital rather than return to the South West, and he is especially excited about launching in East London.
'Ever since I've lived in London I've always loved Hackney Wick, so when I came across this site it felt perfect.
'Hackney Wick has a such a great vibe and community and I felt it totally suited the type of restaurant I wanted to open.'
The style of food at Cornerstone is going to be based around the very best ingredients dictated entirely by seasonality and 'perfect execution' in the preparation and cookery, with clean and bold flavours.
Dishes that Brown and his team have been developing include marinated squid with black pudding, brown butter and blood orange dressing; roast cod with Café de Paris hollandaise, and lamb neck risotto with pickled cockles and wild garlic dressing.
'The ingredients will be the most important thing,' says Brown. 'If you don't get those right, you may as well not open the doors.
'I'll be using a lot of suppliers that I've built strong relationships and mutual trust with over the years, but I'm also looking to discover new ones, particularly local to East London and Hackney.'
After working with Nathan Outlaw for so long, it will come as no surprise that the menu at Cornerstone will feature plenty of fish and seafood, but it will not be the driving force.
'It would be silly to not have plenty of fish on the menu, but the focus won't be solely on seafood. The biggest thing I've learnt from working with Nathan is an appreciation and understanding of produce, balance, cleanliness of flavour and restraint in dishes.
'Very often it's the simplest things that are hardest to perfect. Nathan is an absolute master of this.'
And what other skills has Brown learnt from his former boss in terms of the successful day-to-day business of running restaurants?
'In terms of how to run a restaurant, Nathan is always calm and collected, he treats all his staff with absolute respect and genuinely just wants everyone to do well. Hopefully that approach has rubbed off on me a little over the years.
'Working with Nathan as a mentor has been so important, he inspires you daily and makes you push yourself and want to be better than you were yesterday.
'If I can emulate this with my own staff, then I'll be a very happy man. My advice to young chefs just starting out in the industry is to try and find someone to work for that inspires you, work hard and be dedicated and try to soak up as much knowledge from those around you as possible.' | {
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pointer-events: none;
z-index: 1;
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-moz-transition-duration: 0.8s;
-ms-transition-duration: 0.8s;
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transition-duration: 0.8s;
}
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-webkit-transition-property: height, width;
-moz-transition-property: height, width;
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transition-property: height, width;
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David Snell ist der Name folgender Personen:
* David Snell (Komponist) (1897–1967), US-amerikanischer Komponist
David Snell (Journalist) (1921–1987), US-amerikanischer Journalist und Cartoonist
David Rees Snell (* 1966), US-amerikanischer Schauspieler | {
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package main
import (
"bufio"
"bytes"
"flag"
"fmt"
"math/big"
"os"
"strconv"
"time"
)
var problemNumber = flag.Int(
"problem", 18, "A problem number from projecteuler.net")
var dataFile = flag.String(
"datafile", "", "Data file used to solve the problem")
// Sample run:
//
// <built-program> --problem=18 --datafile=<path-to-repo>/data/problem18
func solve18() {
fd, err := os.Open(*dataFile)
if err != nil {
panic(fmt.Sprintf("Failed to open %s: %v", *dataFile, err))
}
defer fd.Close()
reader := bufio.NewReader(fd)
spaceSep := []byte(" ")
fmt.Printf("Reading data from file ... %s\n", *dataFile)
var numbers [][]int
line, _, err := reader.ReadLine()
for err == nil && len(line) > 0 && line[0] != '#' {
// Add a new row of numbers
var rowNumbers []int
lineData := bytes.Split(line, spaceSep)
for _, element := range lineData {
number, _ := strconv.Atoi(string(element))
rowNumbers = append(rowNumbers, number)
}
// Add the row to our collection and move to the next line
numbers = append(numbers, rowNumbers)
line, _, err = reader.ReadLine()
}
// Make sure we have some minimum amount of data
if len(numbers) < 2 {
panic(fmt.Sprintf("Need atleast two rows of data !"))
}
// File is read, now start building up partial sums
for rowNum := len(numbers) - 2; rowNum >= 0; rowNum-- {
for index, number := range numbers[rowNum] {
// Update the number to be a partial sum
sum1 := number + numbers[rowNum+1][index]
// TODO(agam): Check that each row has the right
// number of elements.
sum2 := number + numbers[rowNum+1][index+1]
if sum1 > sum2 {
numbers[rowNum][index] = sum1
} else {
numbers[rowNum][index] = sum2
}
}
}
// At the end of this process, the answer should be at the top
fmt.Printf("The answer is [%d]\n", numbers[0][0])
}
func solve19() {
// Go through all the first of the months and count the sundays
sundayCount := 0
month := time.January
year := 1901
for year < 2001 {
day := time.Date(year, month, 1, 0, 0, 0, 0, time.UTC)
if day.Weekday() == time.Sunday {
sundayCount++
}
if month == time.December {
month = time.January
year++
} else {
month++
}
}
fmt.Printf("The answer is [%d]\n", sundayCount)
}
func solve20() {
// Ok I cheated here and just picked up the answer from WolframAlpha
// http://www.wolframalpha.com/input/?i=sum+of+digits+of+100%21
// TODO(agam): Use the WolframAlpha API here to atone for this sin.
fmt.Println("The answer (should be) 648")
// Naive solution
factorial := big.NewInt(1)
for i := 1; i <= 100; i++ {
factorial.Mul(big.NewInt(int64(i)), factorial)
}
// Get the whole number as a string, and add up the digits
factorialString := factorial.String()
digitSum := 0
for _, ch := range factorialString {
digitSum += int(ch - '0')
}
fmt.Printf("The answer is %d\n", digitSum)
}
func main() {
flag.Parse()
fmt.Printf("Solving problem %d\n", *problemNumber)
switch *problemNumber {
case 18:
solve18()
return
case 19:
solve19()
return
case 20:
solve20()
return
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,715 |
Today I want to share something with you, a blogpost about the first event I organized in Barcelona! Me and my beloved concept store Passage Barcelona decided to start a project: A Bloggers Night to bring some of the top online influencers together.
Why should bloggers feel like competition while they much rather should collaborate and help each other grow? That's exactly what we thought! It should be fun to share experiences, talk about your interests, problems and laughs… we are in the same world after all! I am very glad that everything turned out exactly the way we hoped and we got great feedback from everyone that visited! So what happened during Bloggers Night? Let me show you!
Art: We invited blogger/ illustrator Saray Martín from Dansvogue to our art gallery to do a live illustration of the event. She almost finished it by now and it will be hanging in the gallery of Passage very soon!
Food: In collaboration with La Fermata we were able to serve up some delicious pizza's. Maybe not the healthiest food, but at least we were all being guilty together! And I mean…. Look at those pictures, don't they look delicious?!
Furthermore, we also invited photographers, stylists, socialites, and more people from the Spanish fashion world so everyone had a chance to meet each other! It was great to see everyone was having fun, I definitely was! So thank you guys, for coming and for being awesome. And of course to Passage Barcelona, for organizing this super cool event with me!
P.s. Check out some more ambient pictures down below!
To see more: #passagebloggers #weitaliaindependent #bloggersnight @ Instagram! | {
"redpajama_set_name": "RedPajamaC4"
} | 4,407 |
En mathématiques, une algèbre de Gerstenhaber est une structure algébrique qui généralise en un certain sens les algèbres de Lie et de Poisson. Elle tient son nom de
Murray Gerstenhaber qui les a introduites en 1963. Formellement, c'est un espace vectoriel gradué muni de deux lois de degrés différents et de symétries opposées.
Les algèbres de Gerstenhaber exactes, aussi connues sous le nom d'algèbres de Batalin-Vilkovisky ou BV-algèbres interviennent dans le qui permet d'étudier les des théories de jauges lagrangiennes.
Définition
On dit que est une algèbre de Gerstenhaber (graduée) lorsque :
G est un espace vectoriel -gradué, le degré d'un élément a étant noté ;
Le « produit » est de degré 0, c'est-à-dire que pour tout couple (a, b) d'éléments de G, ;
Le crochet de Lie est de degré -1, c'est-à-dire que pour tout couple (a, b) d'éléments de G, ;
est une algèbre graduée commutative ;
est une ;
La « relation de Leibniz » suivante est vérifiée pour tous a, b, c éléments de G : .
Exemples
L'espace des multichamps de vecteurs, munis du produit extérieur et du , forme une algèbre de Gerstenhaber.
L'algèbre extérieure d'une algèbre de Lie est une algèbre de Gesternhaber.
Les formes différentielles sur une variété de Poisson forment une algèbre de Gesternhaber.
La cohomologie de Hochschild H*(A,A) d'une algèbre graduée A est une algèbre de Gerstenhaber.
L'homologie d'une est une algèbre de Gerstenhaber.
Articles connexes
Opérade
Références
Gerstenhaber
Gerstenhaber | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,030 |
Q: Sendmail on Ubuntu trying to use Google Apps servers as a relay, by default? I have a recently installed Ubuntu (natty) server, with a domain pointed to it. The mail for this domain is handled by Google Apps (via MX dns records pointing to Google's mail servers).
Now, I've installed logcheck, and am periodically getting emails (via logcheck) about the following log entries:
Jun 17 15:02:06 nattybox sm-mta[26023]: STARTTLS=client, relay=aspmx.l.google.com., version=TLSv1/SSLv3, verify=FAIL, cipher=RC4-SHA, bits=128/128
Jun 17 13:02:07 nattybox sm-mta[8533]: STARTTLS=client, relay=aspmx.l.google.com., version=TLSv1/SSLv3, verify=FAIL, cipher=RC4-SHA, bits=128/128
Jun 17 13:17:45 nattybox sm-mta[10465]: STARTTLS=client, relay=aspmx.l.google.com., version=TLSv1/SSLv3, verify=FAIL, cipher=RC4-SHA, bits=128/128
Jun 17 13:51:56 nattybox sm-mta[14995]: STARTTLS=client, relay=aspmx.l.google.com., version=TLSv1/SSLv3, verify=FAIL, cipher=RC4-SHA, bits=128/128
Jun 17 12:02:06 nattybox sm-mta[32143]: STARTTLS=client, relay=aspmx.l.google.com., version=TLSv1/SSLv3, verify=FAIL, cipher=RC4-SHA, bits=128/128
It looks like sendmail is trying to use the google mail servers as a relay, which I guess is failing? I installed sendmail via aptitude, and am using the configuration files that came with it. To be clear, sendmail seems to be working in that messages sent from this server get delivered just fine.
Are the above log messages something I should be concerned about, and if so, how would I go about fixing it?
A: The logging that you posted is just showing the attempted establishment of TLS over SMTP between your Sendmail and Google's server and nothing more. If your Sendmail is able to send to other addresses fine, then it is probably functioning.
You might send something to a non-Google address to examine its header. That will provide some quick answers to how your SMTP routing may be setup, in lieu of or in support of the rest of your Sendmail logs...
A: Yes this is only basically a warning and most likely your email is getting through. You probably can get rid of the message by adjusting your mail configuration file to NOT attempt to use SSL/TLS security in mail transmission. If I recall in a sendmail configuration it was a single line.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,578 |
10 things EW loves: Week of May 28
Homer, hip-hop art, and 8 other things we love this week
By EW Staff
Updated May 28, 2004 at 04:00 AM EDT
1. GILMORE GIRLS
The rapid-fire dialogue, the pop-culture nods, the best mom-daughter relationship ever: It all started here on season 1, now on DVD.
2. THE 'FOG OF WAR' DVD
Errol Morris' Oscar-winning documentary is a series of chats with ex-defense secretary Robert McNamara. Expect to be totally mesmerized.
3. JONNY QUEST
Gee whiz! This four-DVD set featuring all 26 episodes of the '60s cartoon, about an adventurous kid and his two butt-kicking dads (okay, one of them was just a "friend"), sure is swell!
4. THE BUMBLEBEEZ 81's "RIDE THE PONY"
With the 'beez turned into semi-animated bobbleheads, this video's got us buzz-buzz-buzzing!
5. "FLOAT ON," MODEST MOUSE
Melodic pop guitars, bouncy beats, and earnest vocals all come together on the indie trio's latest infectious single. And that makes us very happy.
6. THE BOOK OF HIP-HOP COVER ART, by ANDREW EMERY
Put this on your coffee table, sucka MC! Almost 30 years of provocative, witty, and yes, sometimes even tame rap-album covers in this phat (and fat!) 160-page tome.
7. TONYA PINKINS from 'CAROLINE, OR CHANGE'
Her powerful performance as a maid in the 1960s South just earned her another Tony nom.
8. 'COLONIAL HOUSE' GAMES on PBS.ORG
Over eight interactive diversions for the 17th-century-obsessed, including a fashion quiz and trivia! Splendid!
9. THE ILIAD
You don't need a classics degree (or a toga) to appreciate the superiority of Homer's ancient war epic over the new bloated big-screen adaptation, "Troy." Even with Brad Pitt's bodacious biceps.
10. SUPER SIZE ME
We've got lotsa love for Morgan Spurlock, who ate nothing but Mickey D's for 30 days and has this awesome doc to show for it. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,299 |
The Bristol Prier monoplane was an early British aircraft produced in a number of single- and two-seat versions.
Background
The Bristol Prier Monoplanes were a series of tractor configuration monoplanes designed for the Bristol and Colonial Aeroplane Company by Pierre Prier, the former chief pilot of the Bleriot school at Hendon, who had joined Bristol in July 1911. At this time Bristol lacked a designer, George Challenger and Archibald Low having left the company to work for Vickers's newly established aircraft division.
Design and development
Unsurprisingly, the Bristol Prier monoplanes resembled the successful Blériot XI monoplane, with a fabric-covered wire-braced wood fuselage and parallel-chord wings using wing-warping for lateral control, although differing in details. The all-moving tailplane was an elongated fan-shape, mounted in a mid-position between the upper and lower longerons, and the undercarriage had a pair of wheels on an axle mounted onto a pair of forward-projecting skids. It was powered by a 50 hp (37 kW) Gnome rotary engine.
The first aircraft built (works No. 46) had been intended to compete in the Gordon Bennett Trophy race, but it was not ready in time. Two more aircraft, nos. 56 and 57, with a revised engine mounting, were started with the intention of taking place in the Daily Mail Circuit of Britain race.
Work was then started on developing a two-seater version, this being works number 58. By October 1911 it had been thoroughly tested, and the directors of Bristol decided the type was suitable for volume production. Six airframes were started (Nos 71-76), the first of which was carefully prepared, with the aluminium cowling polished, a pigskin-upholstered seat and stowage for binoculars and a thermos flask, for exhibition at the 1911 Paris Aero Salon where it was the only British aircraft on display. At the same time two of the prototype aircraft (Nos. 46 and 57) were adapted to take a Anzani engine, with the intention of developing a low-powered sports aircraft. It was joined in Paris by No. 74, where it was used to make demonstration flights over Paris during the Aero Salon.
No. 72 was sent to Madrid for demonstration flights piloted by Howard Pixton, and when these were completed Pixton was sent to Germany, where he flew No. 74 at Döberitz to demonstrate its capabilities to the German army. As a result of this Bristol started a German subsidiary, the Deutsche Bristolwerke Flugzeuggesellschaft m.b.H. (Renamed Halberstädter Flugzeugwerke in 1913), and an associated flying school at Halberstadt, No. 74 being allocated to the school.
The Prier monoplanes were used mostly for training and racing, and some were purchased for military use. Two of the two-seaters were sold and delivered to the Turkish Government. One of the two-seaters was sold to the Bulgarian government and delivered on 16 September 1912. It flew during the Balkan War and once carried Hubert Wilkins who was taking films for a London newspaper.
Variants
P-1: Prototype aircraft. Works nos. 46, 56,57
Single-seat "School". Clement-Bayard or Isaacson
Two-seat short body. Wingspan 32 ft 9 in. (9.95 m). Powered by Gnome. Works nos.68, 81, 95-8, 102
Two-seat long body. Wingspan , length . Powered by Gnome. Works nos. 82, 85-89, 91, 130, 155, 156.
Two-seat side-by-side. Wingspan 34 ft 6 in (10.48 m), length . Powered by Gnome. Works nos. 107-109
Operators
Bulgarian Air Force
Ottoman Air Force
Royal Flying Corps
3 Squadron in 1912
Specifications (P-1)
References
Single-engined tractor aircraft
1910s British sport aircraft
Prier monoplane
Mid-wing aircraft
Aircraft first flown in 1911
Rotary-engined aircraft | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,170 |
Rafel Crespi Cladera (La Puebla, Baleares, 1962) es un profesor español, catedrático de organización de empresas de la Universidad de las Islas Baleares y director de la cátedra Banca March de la empresa familiar. Actualmente prepara su candidatura para ser rector de la Universidad de las Islas Baleares.
Biografía
Se diplomó en ciencias económicas y empresariales por la Universidad de las Islas Baleares (1984), posteriormente se desplazó a Barcelona para licenciarse en ciencias económicas y empresariales por la Universidad de Barcelona (1986), doctorándose en ciencias económicas y empresariales por la Universidad Autónoma de Barcelona (1992). Ha ejercido labores docentes en la Universidad Autónoma de Barcelona como ayudante (1986-1992) y como Titular de Universidad (1993-1987). Fue profesor titular en la Universidad Pompeu Fabra de Barcelona (1992-1993). Ejerciendo como investigador visitante en diversas universidades internacionales como: Tilburg Univesity, en CentER (Center for Economic Research) (1995-1996); Oxford Univesity, en la Said Business School (1998); Cambridge Unversity en el Centre for Business Research (2005-2006), y University of Illinois at Champaign Urbana (2009, 2010, 2011). Más recientemente ha sido profesor visitante en la IE Business School en Madrid (2015-2016). Actualmente es catedrático del departamento de Economía de la Empresa en la Universidad de las islas Baleares.
Ha impartido docencia en cursos de Dirección Estratégica, Empresa Familiar, Introduction to Business, Strategic Management , Corporate Governance entre otros.
Actividad investigadora
Crespí desarrolla su labor investigadora en el grupo de investigación GREO de "Investigación en Economía de las Organizaciones" de la Universidad de las Islas Baleares. La investigación se realiza de forma coordinada con grupos de investigación del departamento de Economía de la Empresa de la Universidad Autónoma de Barcelona, con el Departamento de Gestión de Empresas de la Universidad Pública de Navarra y con el Departamento de Dirección y Organización de Empresas de la Universidad de Zaragoza.
Sus líneas de investigación se centran en gobierno corporativo y empresa familiar. A lo largo de su carrera, desde 1998 ha sido Investigador Principal de forma ininterrumpida en proyectos de investigación competitivos. Ha dirigido 5 tesis doctorales. Es autor de numerosas publicaciones recogidas en el Journal Citation Report entre las que destacan las publicadas en Strategic Management Journal, Family Business Review, Journal of Family Business Strategy, Corporate Governance: An International Review, Journal of World Business, Journal of Accouting an Public Policy o Journal of Banking & Finance. Cuenta con numerosas comunicaciones y ponencias en congresos nacionales e internacionales y varios capítulos de libro en editoriales com Oxford University Press o Edward Elgar.
Ha sido miembro adjunto del área de Economía de la Agencia Nacional de Evaluación y Prospectiva, del Ministerio de Ciencia e Innovación. (2006-2010) y miembro de la comisión de ciencias sociales de la AQU del sistema universitario de Catalunya para la acreditación de profesorado (2010-2015). Ha sido director de investigación del Observatorio de Gobierno Corporativo de la Fundación de Estudios Financieros, dependiendo del Instituto Español de Analistas Financieros (2006-2010). Ha ejercido como director en el programa de doctorado DEMO, una iniciativa interuniversitaria entre Universidad Autónoma de Barcelona, la Universidad de las Islas Baleares y la Universidad Pública de Navarra.
Gestión y política universitaria
En su trayectoria dentro de la Universidad de las Islas Baleares, ha ejercido como director del Departamento de Economía de la Empresa durante el período 2007 a 2015 y como subdirector del mismo departamento de 2003 a 2007. Ha ejercido como miembro de la Comisión de Postgrado y en la Comisión de Doctorado en la UIB del año 2004 al 2007. Miembro de la comisión del plan estratégico (2002-2006) en el año 2002. Miembro del Claustro de la UIB desde 1999.
Secretario del Departamento de Economía de la Empresa en la Universitat Autònoma de Barcelona de 1993 a 1995.
Miembro del Claustro constituyente de la Universitat Pompeu Fabra en 1999.
Presidente del Comité Organizador del congreso de Asociación científica de Dirección de Empresas en Palma de Mallorca, 2002.
Referencias
Profesores de España
Profesores de la Universidad de las Islas Baleares
Catedráticos de la Universidad de las Islas Baleares
Personas de La Puebla | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,254 |
Q: Is there anyway to mock terraform data source I am working on unit testing for terraform. For some modules, I have to authorized into AWS to be able to retrieve terraform data source. Is there anyway to mock or override data source for something like below?
data "aws_region" "current" {
}
Thank you in advance.
A: Terraform does not include any built-in means to mock the behavior of a provider. Module authors generally test their modules using integration testing rather than unit testing, e.g. by writing a testing-only Terraform configuration that calls the module in question with suitable arguments to exercise the behaviors the author wishes to test.
The testing process is then to run terraform apply within that test configuration and observe it making the intended changes. Once you are finished testing you can run terraform destroy to clean up the temporary infrastructure that the test configuration declared.
A typical Terraform module doesn't have much useful behavior in itself and instead is just a wrapper around provider behaviors, so integration testing is often a more practical approach than unit testing in order to achieve confidence that the module will behave as expected in real use.
If you particularly want to unit test a module, I think the best way to achieve your goal within the Terraform language itself is to think about working at the module level of abstraction rather than the resource level of abstraction. You can then use Module Composition techniques, like dependency inversion, so that you can pass your module fake input when you are testing it and real input when it's being used in a "real" configuration. The module itself would therefore no longer depend directly on the aws_region data source.
However, it's unlikely that you'd be able to achieve unit testing in the purest sense with the Terraform language alone unless the module you are testing consists only of local computation (locals and output blocks, and local-compute-only resources) and doesn't interact with any remote systems at all. While you could certainly make a Terraform module that takes an AWS region as an argument, there's little the module could do with that value unless it is also interacting with the AWS provider.
A more extreme alternative would be to write your own aws provider that contains the subset of resource type names you want to test with but whose implementations of them are all fake. You could then use your own fake aws provider instead of the real one when you're running your tests, and thus avoid interacting with real AWS APIs at all.
This path is considerably more work of course, and so I would suggest to embark on it only if the value of unit testing your particular module(s) is high.
A: Another super-labour-intensive solution would be to emulate aws api on localhost and redirect (normal) aws provider there. I've found https://github.com/localstack/localstack - https://docs.localstack.cloud/integrations/terraform/ may probably be helpful with this.
| {
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} | 4,793 |
Walls of Jericho ist eine 1998 gegründete Hardcore-Punk-Band aus Detroit, Michigan, USA.
Geschichte
Walls of Jericho gingen aus der Gruppe Earthmover hervor. 1999 veröffentlichten sie eine erste Single, kurz danach erschien die erste EP A Day And A Thousand Years auf Genet Records. Zur Jahrtausendwende veröffentlichten sie das Album The Bound Feed The Gagged auf Trustkill Records. Anfang 2001 entschied sich Drummer Keely, die Band zu verlassen. Sein Nachfolger Derek Grant wurde jedoch nach kurzer Zeit von Alkaline Trio abgeworben, und die erfolglose Suche nach einem neuen Drummer führte schließlich fast zur Auflösung der Band. Nach einer längeren Pause konnte diese Position mit Alexei Rodriguez dann aber besetzt werden.
Im Jahr 2004 wurde das Album All Hail The Dead (wiederum über Trustkill Records) veröffentlicht, woraufhin Rodriguez die Band verließ. Dustin Schoenhofer, ein langjähriger Freund der Band, konnte die Lücke jedoch schnell schließen. Am 22. August 2006 folgte das Album With Devils Amongst Us All. Es verfolgt eine etwas melodischere Richtung und geht teilweise weg vom reinen Hardcore der Anfangstage.
Zusammenarbeit mit Corey Taylor / Redemption EP
Die EP Redemption, die im Jahre 2008 erschien, wurde zusammen mit dem Slipknot- und Stone-Sour-Sänger Corey Taylor produziert. Taylor übernahm neben Kucsulain den Gesang auf der EP. Der Frontmann und die Band lernten sich auf der Family-Values-Tour 2006 kennen. Taylor gefiel die Musik der Band, besonders der Song No Saving Me. Daraufhin bot er Kusculain eine weitere Zusammenarbeit an, die aber an Zeitproblemen scheiterte. Stilistisch ist die Redemption-EP eher untypisch für Walls of Jericho, da es sich um ruhige Lieder mit klarem Gesang handelt.
The American Dream (2008)
Mit der Veröffentlichung des Albums The American Dream im Jahre 2008 machten Walls of Jericho musikalisch wieder einen Schritt in Richtung Hardcore / Metal. Das Album wurde wie schon zuvor With Devils Amongst Us All von Ben Schigel produziert.
Nach einer längeren Pause, bedingt durch eine Babypause von Frontfrau Candace Kucsulain, traten Walls of Jericho im Januar 2012 bei der Persistence Tour auch wieder in Deutschland auf.
Nach einem Wechsel zum Label Napalm Records veröffentlichte die Band im März 2016 ihr fünftes Studioalbum No one can save you from yourself.
Diskografie
1999: A Day and a Thousand Years (EP)
2000: The Bound Feed the Gagged
2004: All Hail the Dead
2006: From Hell (EP)
2006: With Devils Amongst Us All
2008: Redemption (EP)
2008: The American Dream
2016: No One Can Save you from Yourself
Weblinks
Facebook-Profil der Band
Einzelnachweise
Hardcore-Punk-Band
Metalcore-Band
US-amerikanische Band | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,418 |
Alan Bruce Slifka (October 13, 1929 – February 4, 2011) was a New York investor and philanthropist, a co-founder of the Abraham Fund and founding chairman of the Big Apple Circus. He was a native of Manhattan.
Education
Slifka and his sister were initially home schooled by their mother. This provided an opportunity for the twins to learn the importance of moral values and the basics of coexistence. In the fourth grade, the twins began studying at the Ethical Culture Society's Fieldston School.
Slifka graduated from Yale University in 1951, where he worked on the business staff of campus humor magazine the Yale Record. He then went on to earn a Master's degree in Business Administration at Harvard University in 1953.
Professional background
Following his graduation from Harvard, Slifka joined the financial firm L.F. Rothschild & Company, where he worked as a securities analyst for 32 years, rising to partner before leaving to start his own company, now Halcyon Asset Management.
Philanthropy
In 1977, Slifka became the founding chairman of the New York School for Circus Arts, a non-profit training school whose performing arm is the Big Apple Circus. In 1993, he became founding chairman emeritus. In 1995, in recognition of Slifka's lead gift to a successful capital campaign, the circus's new permanent creative center in Walden, NY was named the Slifka Family Creative Center.
Together with sociologist Eugene Wiener, Slifka was a co-founder in 1989 of The Abraham Fund Initiatives, named for the common ancestor of Arabs and Jews, and served as chairman of the organization since its founding. This was the first nonprofit organization dedicated to furthering coexistence between Israel's Arab and Jewish citizens. The Abraham Fund works to advance a shared society of inclusion and equality between Jews and Arabs in Israel.
The Slifka Program on Intercommunal Coexistence at Brandeis University, create by Slifka in 2001, seeks to build professional expertise and creative leadership in the field of coexistence and offers a master's degree in coexistence and conflict. The Sylvia and Joseph Slifka Israeli Coexistence Scholarship at Brandeis, which Slifka funded in honor of his parents, is awarded each year to two citizens of Israel (one Jewish, one Arab) who are committed to fostering coexistence and harmony.
In recognition of his work with The Abraham Fund Initiatives, Slifka was awarded the Knesset Prize for Coexistence in 2000. Brandeis awarded him an honorary doctorate in 2003.
Alan B. Slifka Foundation
The Alan B. Slifka Foundation was established in New York in 1965. The foundation's goals include harmony not only among Jews and Arabs but also between religious and secular elements of Israeli society. In addition to fostering Jewish values and education, the foundation also promotes biomedical research on sarcomas and autism spectrum disorders.
Personal life
Alan Bruce Slifka was the son of Joseph and Sylvia Slifka. His twin sister is Barbara Slifka. His father owned successful textile and real estate businesses.
At the time of his death he was married to Riva Ritvo-Slifka. He had three sons, Michael, Randolph, and David.
References
1929 births
2011 deaths
American bankers
Jewish American philanthropists
Philanthropists from New York (state)
Autism activists
Harvard Business School alumni
People from Manhattan
Yale University alumni
20th-century American philanthropists
21st-century American Jews | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,541 |
John Patrick Hemingway
The quaint northern Michigan village of Walloon Lake provides an idyllic backdrop for writers to discover and hone their skills and the 2nd Annual Walloon Lake Writer's Retreat (April 13-16,…
Continue ReadingErnest Hemingway's Grandson & Fellow Author, John Patrick Hemingway, to Lead 2nd Annual Walloon Lake Writer's Retreat, April 13-16, 2023
The latest construction project in Walloon Lake was announced today during a groundbreaking ceremony for The Renwick, a multi-use project which will include 16 luxury condominiums and a market and…
Continue ReadingWalloon Lake Community Leaders Break Ground on Newest Village Building: The Renwick
Summer 2021 Traffic Notices for Walloon Lake Area
They say in Michigan, that "road construction" is just one of our seasons – and this year is no exception. Those of you planning to visit us in Walloon Lake…
Continue ReadingSummer 2021 Traffic Notices for Walloon Lake Area
Welcome to Walloon Lake
Stay tuned as our new website takes shape! We are excited to provide you news and information on the Village of Walloon Lake. We will continue to update and add…
Continue ReadingWelcome to Walloon Lake | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,399 |
PRAISE FOR MICHAEL KORYTA
"Superb writing and storytelling from one of crime fiction's brightest young talents. _Envy the Night_ represents Koryta's finest work to date."
George Pelecanos
"Koryta's dialogue is as sharp as the knives his characters wield, and his plot twists at the most unexpected moments. [ _Envy the Night_ ] places Koryta solidly in the company of the genre's most powerful voices."
_Publishers Weekly_ , starred review
"This diabolical novel, laid out in simple but eloquent prose and pitch-perfect dialogue, heralds a changing of the guard. I have seen the future of 'The Best Mystery Writer in America' and his name is Michael Koryta."
_New York Times_ bestselling author Ridley Pearson
"For a while now Michael Koryta has been called one of the rising young talents in crime fiction. I say enough of that. _A Welcome Grave_ proves the promise and establishes Koryta as one of the best of the best, plain and simple. With stories like this, his Lincoln Perry is going to be around for a long, long time."
Michael Connelly
"Michael Koryta stakes a claim as one of today's pre-eminent crafters of crime fiction with _A Welcome Grave_. Despite his youth, his haunting writing and logical, sophisticated plotting rival that of established stalwarts..."
_Publishers Weekly_ , starred review
" _A Welcome Grave_ showcases Koryta's accomplished writing skills. Highly recommended for all mystery and crime fiction collections."
_Library Journal_ , starred review
"Sentence for polished sentence, no one in the genre writes better."
_Kirkus Reviews_
"The appeal of Koryta's addictively readable narrative comes from his remarkably realistic portrayal of not only individuals but circumstances as well. There are no stereotypical characters, no predictable plot lines; like life, nothing is pure black or white but varying shades of gray. Readers who enjoy authors like Michael Connelly or Dennis Lehane should add Michael Koryta to their reading lists."
_Chicago Tribune_
"With the publication of _A Welcome Grave_ , it's time to stop referring to Michael Koryta as a boy wonder and just focus on the sheer wonder of his storytelling. Koryta knows how to put his characters—and his readers—into an ever-tightening vise of twists, turns and conspiracies, but it's his empathy that makes his work stand out. This is a nuanced, mature novel that proves the depth of Koryta's talent."
Laura Lippman
Michael Koryta became the youngest winner of the annual St Martin's Press/Private Eye Writers of America Best First Private Eye Novel contest. Michael was just 20 when he wrote the winning novel, _Tonight I Said Goodbye_ , which went on to earn an Edgar nomination for best first novel. Michael's second novel was _Sorrow's Anthem_ , which received excellent reviews. The third in his Lincoln Perry series, _A Welcome Grave_ , confirmed Michael's place amongst the highest echelon of international crime writers. _Envy the Night_ is his first standalone novel.
Michael Koryta is just 27 and lives in Bloomington, Indiana. He majored in criminal justice at Indiana University, is an award-winning newspaper reporter, and a PI who works for one of Indiana's only certified legal investigators.
MICHAEL
KORYTA
# ENVY THE
NIGHT
This is a work of fiction. All of the characters, organizations and events portrayed in this novel are either products of the author's imagination or are used fictitiously.
First published in Australia in 2008
First published in the United States in 2008 by Thomas Dunne Books,
an imprint of St. Martin's Press
Copyright © Michael Koryta 2008
All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without prior permission in writing from the publisher. _The Australian Copyright Act 1968_ (the Act) allows a maximum of one chapter or 10 per cent of this book, whichever is the greater, to be photocopied by any educational institution for its educational purposes provided that the educational institution (or body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL) under the Act.
Arena Books, an imprint of
Allen & Unwin
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Australia
Phone:
(61 2) 8425 0100
Fax:
(61 2) 9906 2218
Email: info@allenandunwin.com
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National Library of Australia
Cataloguing-in-Publication entry:
Koryta, Michael.
Envy the night/Michael Koryta.
ISBN 9781741755916 (pbk.)
813.6
Additional typesetting by Midland Typesetters, Australia
Printed and bound in Australia by Griffin Press
10 9 8 7 6 5 4 3 2 1
_For Dennis Lehane, who remembered the elevator_
## ACKNOWLEDGMENTS
Deepest gratitude to Dennis Lehane, Roland Merullo, Christine Caya, Sterling Watson, Meg Kearney, Laura Lippman, and all others involved with the Writers in Paradise program at Eckerd College in St. Petersburg, and the low-residency MFA program at Pine Manor College in Boston, where this book was born.
The Willow Flowage is a real place, albeit one with which I took plenty of fictional liberties, and I'm grateful to my father for introducing me to it, and to Dwight and Fran Simonton for being gracious hosts over the years and providing some wonderful background information. Also to Jim Kiepke for always finding the fish.
Ryan Easton guided me through details related to cars and the body shop business, and my sister, Jennifer, advised on dealing with stroke patients. If I got anything right, the credit is theirs, and if I got it wrong, the blame is mine.
Thanks, as always, to my agent, David Hale Smith, and to the St. Martin's Press/Thomas Dunne team for their wonderful work, particularly Pete Wolverton, Andy Martin, Katie Gilligan, and Liz Byrne.
Further thanks to:
Michael Connelly, Bob Hammel, Laura Lane, Gena Asher, Don Johnson, Robert Pepin, Louise Thurtell, and Lawrence Rose. And to all of the booksellers, reviewers, and magazine publishers who do so much to help, particularly Jim Huang, Jamie and Robin Agnew, Richard Katz, Jon and Ruth Jordan, John and Toni Cross, Otto Penzler, Barbara Peters, Lynn Kaczmarek, Chris Aldrich, and Janet Rudolph.
_I envy the night
for its absence of light._
Dax Riggs, "Ancient Man"
## 1
Frank Temple III walked out of the county jail at ten in the morning with a headache, a citation for public intox, and a notion that it was time to leave town.
It wasn't the arrest that convinced him. That had been merely a nightcap to an evening of farewells—Frank hanging from the streetlamp outside of Nick's on Kirkwood Avenue, looking down into the face of a bored cop who'd seen too many drunks and saying, "Officer, I'd like to report a missing pair of pants."
It hadn't been the hours in the detox cell, either. Frank was one of six in the cell, and one of just two who managed not to vomit. Sitting with his back against the cold concrete block wall listening to some poor son of a bitch retch in the corner, Frank considered the jail, the people who checked in and didn't check out the next morning, the way he would. He considered the harsh fluorescent lights reflecting off gray and beige paint, the dead quality of the air, the hard looks the men inside developed to hide the hopelessness. It would be the same when the sun rose as when it set, except you wouldn't be sure when that happened, couldn't even use the sun to gauge the lack of change. He considered all of that, and knew that if he could understand only one thing about his father, it was the decision he'd made to avoid this place.
This was the second time Frank had been in a jail. The first was for a drunk driving charge in a small North Carolina town two years earlier. He had failed the Breathalyzer but requested field sobriety testing anyhow, his booze-addled brain sure that he could pass. After watching Frank stumble and stagger through the first exercise, the cop put an end to it, said, "Doesn't look like your balance is too good, kid." Frank, leaning against the car for support, had waved him closer, as if about to impart a secret of the highest magnitude. The cop leaned down, and when he was close enough, Frank whispered, "Inner ear infection."
He had the cuffs on and was in the back of the car before he was finished explaining the connection between one's sinuses and one's balance. His was not a receptive audience.
So this was the second trip to a jail, and even if his father hadn't found a coward's way to avoid a life sentence, the number would be the same. Frank wouldn't have visited. But he also couldn't hide the thought, listening to those drunks mumble and belch and vomit beside him, that maybe the reason he put himself in situations like this was because he wanted a taste. Just a taste, that was all, something he could walk back into the free world with and think— _that's what it would've been like for him_.
He'd been chased into the night of drinking by one disturbing phone message and one pretentious professor. The message had come first, left by a voice he hadn't heard in many years.
_Frank, it's Ezra. Ezra Ballard. Been a long time, hasn't it? You sound older on your message. Anyhow, I'm calling because, well . . . he's coming back, Frank. I just got a call from Florida telling me to open up the cabin. Now, I'm not telling you to do anything, don't even care if you call me back. I'm just keeping my word, right? Just keeping my word, son. He's coming back, and now I've told you._
Frank hadn't returned the call. He intended to let it go. Knew that he should, at least. By the end of the day, though, he was done in Bloomington. A single semester of school—his fifth college in seven years, no degree achieved or even threatened—and Frank was done again. He'd come here to work with a writer named Walter Thorp ( _Walt to my friends, and I hate all of them for it_ ), whose work Frank had admired for years. Bloomington was closer to home than Frank had allowed himself to come in years, but Thorp was a visiting professor, there for only one semester, and he couldn't pass up that chance. It had gone well, too. Thorp was good, better even than Frank had expected, and Frank had worked his ass off for a few months. Read like crazy, wrote like crazy, saw good things happening on the page. The last week of the semester brought an e-mail from Thorp, requesting a meeting, and Frank used that as encouragement to push Ezra Ballard's call out of his mind. Focus on the future, don't drown in the past.
That was his mantra when he went to the cramped office on the third floor of Sycamore Hall, sat there and listened as Thorp, glancing occasionally at that gold watch he always wore on the inside of his wrist, complimented Frank's writing, told him that he'd seen "great strides" during the semester, that Frank clearly had "powerful stories to tell." Frank nodded and thanked his way through it, feeling good, validated in his decision to come here, to ignore that phone call.
"I've never done this for a student before," Thorp said, arching an eyebrow, "but I'd like to introduce you to my agent."
Frank couldn't even feel the elation yet; this was that much of a surprise. Just looked back at Thorp and didn't speak, waited to see what else would be said.
"In fact," Thorp added, tracing the edge of his desk with a fingertip, eyes away from Frank's, "I've already mentioned you to him a few times. He's interested. Very interested. But he was wondering—we both were, really—have you ever given thought to writing nonfiction? Maybe a memoir?"
Frank got it then. He felt his jaw tighten and his eyes go flat and he stared at the old-fashioned window behind Thorp's head and wondered what the great writer would look like flying through it, landing on the terrace three floors below.
"I only ask because your story, and the way it intersects with your father's story, well, it could be quite compelling. To have that in addition to your own narrative gifts, Frank, is quite a package. Nate—he's my agent—he thinks the market would be fantastic. You might even be able to get a deal on just a synop and a few sample chapters. Nate thinks an auction would be possible, and that's the sort of circumstance where the dollar figures can go through the—"
He had the good sense not to follow Frank out the door and down the steps. Ten hours later, Frank was in the jail, all the amusement left in his drunken mind vanishing when the booking officer looked up from the paperwork and said, "No middle name?"
Nope, no middle name. Too bad, because going by your middle name was an easy thing—provided you had one. But he didn't. Just that Roman numeral tacked on the end, Frank Temple III, the next step in the legacy, a follow-up act to two war heroes and one murderer.
They'd put him into the detox cell then, left him there to wait for sobriety, left him with swirling thoughts of his father and Thorp and the message. Oh, yes, the message. He'd deleted it, but there would be no need to play it again anyhow. It was trapped in his brain, cycled through a dozen times as he sat awake waiting for morning.
_He's coming back._
He was not allowed to come back. Frank and Ezra had promised one another that, agreed that they'd let him live out his days down there in Miami so long as he never tried to return, but now there was this phone call from Ezra saying that after seven years the son of a bitch had decided to test their will, call that old bluff.
All right, then. If he would return, then so would Frank.
He was northbound by noon, the Jeep loaded with his possessions. Except loaded wasn't the right word, because Frank always traveled light so he could pack fast. The quicker he packed, the easier it was to ignore his father's guns. He didn't want them, never had. Through nineteen states and who knew how many towns in the last seven years, though, they'd traveled with him. Other than the guns, he had a laptop computer, two suitcases full of clothes, and a pile of books and CDs thrown into a cardboard box. Twenty-five years of life, it seemed like he should have more than that, but Frank had stopped accumulating things a long time ago. It was better to be able to move on without being burdened by a lot of objects that reminded you only of where you'd just been.
West through Illinois before heading north, to avoid the gridlock and construction that always blanketed Chicago, then across the state line and into Wisconsin as the sun disappeared, the destination still hours ahead. Tomahawk, a name Frank would've dismissed as cliché if he'd written it for a North Woods lake town. The town was real enough, though, and so were his memories of it.
His father wouldn't be there. Devin Matteson would be. If Ezra's call was legitimate, then Devin was returning for the first time in seven years. And if Frank had an ounce of sense, he'd be driving in the opposite direction. What lay ahead, a confrontation with Devin, was the sort of possibility that Grady Morgan had warned him he _had_ to avoid. Grady was one of the FBI agents who'd brought down Frank's father. Grady was also a damn good man. Frank had been close to him for a while, as close as he had been to anyone for a few months during the worst of it, but then the media sniffed that relationship out and Frank left Chicago and Grady behind. They hadn't talked much since.
He drove past Madison in the dark and pushed on. He hadn't eaten all day, just drank Gatorade and swallowed ibuprofen and drove, hoping to do it all in one stretch, with just a few stops for gas and to exercise sore muscles. Before he reached Stevens Point, though, he knew he wasn't going to make it. The hangover had killed his appetite, but he'd need food if he was going to stay awake, and now the fatigue was beginning to overpower him. There was a rest stop ahead, maybe the last one he'd see for a while, and he pulled off and parked. Lowered the driver's seat as far as it would go, enough to let his legs stretch a bit, and then he slept.
It was a Big Brother kind of thing, no doubt about it, but Grady Morgan had kept an active monitor on Frank Temple III for seven years. It wasn't proper, or even really legal, because Frank had no role whatsoever in anything that could still be considered an active investigation for Grady. But nobody had noticed or cared or commented yet, and as long as they didn't, he'd keep watching. Without a touch of remorse. He owed the kid at least this much.
The feelers Grady had out there in the world, computers that ran daily checks on Frank's fingerprints and Social Security number, had been quiet for a long time. As had the phone lines and the e-mails and the mailbox. No word from Frank in quite a while, and there were times when Grady ached to speak to him, check in, but he didn't. He just went to work every day and eyed the calendar that showed retirement was not far away and hoped that Frank would continue to stay off the radar screen. Grady didn't want to see a blip.
Here was one. The wrong kind of blip, too, an arrest in Indiana, and when it first came through to his computer Grady felt an immediate sick swirl go through his stomach, and he actually looked away from the screen for a moment, not wanting to read the details.
"Shit, Frank," he muttered. "Don't do this to me."
Then he sighed and rubbed a forehead that was always growing, chasing the gray hair right off his skull, and he turned back to the computer screen and read the details of the arrest. When he got through, he let out a breath of relief.
Public intoxication. That was it. The second arrest in seven years, the second time Grady had felt this chill of sorrow, and the second time he could roll his eyes and chalk it up as No Big Deal, Kids Being Kids.
He hoped.
As he pushed back from his desk and walked to the window and looked out at the Chicago skyline, he sent a silent request to Frank Temple III somewhere out there across the miles.
_Tell me it was just fun. Tell me, Frank, that you were out with some buddies having beers and chasing girls and laughing like idiots, like happy, happy idiots. Tell me that there was no fight involved, no temper, no violence, not even a closed fist. You've made it a long way._
A long, _long_ way.
Frank III had been eighteen years old when Grady met him. A slender, good-looking kid with dark features contrasted by bright blue eyes, and a maturity that Grady hadn't seen in a boy of that age before, so utterly cool that Grady actually asked a psychologist for advice on talking to him. _He's showing nothing,_ Grady had said. _Every report we've got says he was closer to his father than anyone, and he is showing nothing_.
He showed something in the third interview. It had been just him and Grady sitting in the Temple living room, and Grady, desperate for some way to get the kid talking, had pointed at a framed photograph of father and son on a basketball court and said, _Did he teach you how to play?_
The kid had sat there and looked at him and seemed almost amused. Then he'd said, _You want to know what he taught me? Stand up._
So Grady stood up. When the kid said, _Take that pen and try to touch it to my heart. Hell, try to touch it anywhere. Pretend it's a knife_ , Grady hadn't wanted to. All of a sudden this was seeming like a real bad idea, but the kid's eyes were intense, and so Grady said what the hell and made one quick thrust, thinking he'd lay the pen against the kid's chest and be done with it.
The speed. Oh, man, the speed. The kid's hands had moved faster than anybody's Grady had ever seen, trapped his wrist and rolled it back and the pen was pointing at Grady's throat in a heartbeat's time.
_Half-assed effort,_ Frank Temple III had said. _Try again. For real this time._
So he'd tried again. And again, and again, and by the end he was working into a sweat and no longer fooling around, was beginning to feel the flush of shame because this was a _child,_ damn it, and Grady had done eight years in the Army and another fifteen in the Bureau and he ran twenty miles a week and lifted weights and he could beat this kid . . .
But he couldn't. When he finally gave up, the kid had smiled at him, this horribly genuine smile, and said, _Want to see me shoot?_
_Yes,_ Grady said.
What he saw at the range later that afternoon—a tight and perfect cluster of bullets—no longer surprised him.
Seven years later, he was thinking about that day while he stared out of the window and told himself that it was nothing but a public intox charge, a silly misdemeanor, and that there was nothing to worry about with Frank. Frank was a good kid, always had been, and he'd be absolutely fine as long as he stayed away from a certain kind of trouble.
That was all he needed to do. Stay away from that kind of trouble.
## 2
Frank woke to the grinding of a big diesel motor pulling away, sat up, and saw gray light filling the sky. When he opened the door and tried to get out of the Jeep his cramped muscles protested, and he felt a quick razor of pain along the left side of his stiff neck. He was hungry now, the alcohol long since vanished from his cells and the Gatorade calories burned up. He took the edge off with a Snickers bar and a bottle of orange juice from the vending machines, ate while he studied the big map on the wall. He'd come closer last night than he'd realized; Tomahawk was only one hundred miles ahead.
The closer he got, the more his resolve wavered. Maybe it would be best to pretend he'd never gotten that message from Ezra, didn't even know Devin was on his way back. Maybe he'd just spend a little time in the cabin, stay for a weekend, catch some fish. It would be fine as long as he didn't see Devin Matteson. If he stayed away from Devin, if it was just Frank and Ezra and the woods and the lake, this could end up being a good trip, the sort of trip he'd needed to take for a while now. But if he _did_ see Devin . . .
_What are you doing here, then, if it's not about Devin?_ he thought. _You really think this is some sort of vacation?_
Whatever part of his brain was supposed to rise to that argument remained silent. He drove with the windows down as the gray light turned golden and the cold morning air began to warm on his right side. Past Wausau the smell of the place began to change—pine needles and wood smoke and, even though there wasn't a lake in sight, water. There would be a half-dozen lakes within a mile of the highway by now. He knew that both by the change in the air and from the map in the rest stop, this portion of the state freckled with blue.
The smells were triggering a memory parade, but Frank wasn't sure if he wanted to sit back and watch. It was that sort of place for him now. The deeper he got into the tall pines, the faster the memories flooded toward him, and he was struck by just how much he'd loved this place. It was one thing to recall it from somewhere hundreds of miles away, and another to really be here, seeing the forests and the sky and smelling the air. Maybe he'd stay for a while. The summer stretched ahead of him, and the money wouldn't run out. Blood money, sure, and spending it while hating the methods that had earned it made Frank a hypocrite at best and something far darker at worst, but it was there.
The first few times he and his father had made the trip, the highway had been two-lane this far north. Then the tourism dollars began to knock on the right doors down in Madison, and soon the four-lane was extending. Frank's mind was on the cabin, and he blew right past the Tomahawk exits before remembering that he had nothing in the way of food or supplies. He'd have to come back down after he'd unpacked, grab some lunch and buy groceries and then head back to the lake.
He exited at an intersection with County Y, a narrow road slashing through the pines, and had gone about a mile down it when someone in a silver Lexus SUV appeared behind him. From the way it came on in the rearview mirror Frank knew it was really eating up the road, had to be doing seventy at least. As the car approached, it shifted into the oncoming lane, the driver planning to pass Frank without breaking stride. Had to be a tourist, driving like that. The locals had more class.
It was that thought that made him look at the license plate. He probably wouldn't have done it otherwise, but now he wanted to prove his theory correct, so his eyes went to the plate.
Florida.
The car was gone in a silver flash then, swerving in ahead of him and pulling away. The muscles at the base of his neck had gone cold and tight and his breath seemed trapped.
Florida.
It didn't mean anything. A strange little touch of déjà vu, sure, but it didn't mean anything. Yes, the Willow Flowage was an isolated place and a damn long drive from Florida, but there were several million cars with Florida license plates. There wasn't even a _chance_ that Devin Matteson was driving that car.
"Not a chance," Frank said aloud, but then that message from Ezra filled his head again— _I got a call from Florida . . . he's coming back_ —and he pressed hard on the gas pedal and closed the gap on the silver Lexus. A closer look was all he needed. Just that minor reassurance, enough that he could go on to the cabin laughing at himself for this reaction.
He kept accelerating, closed until he was only a car length behind. Now he was leaning forward, his chest almost against the steering wheel, peering into the tinted rear window of the Lexus as if he'd actually be able to tell who the driver was.
There was only one person in the car, and it was a male. He could tell that much, but nothing else. He pulled a little closer, almost on top of the Lexus now, staring hard at the silhouette of the driver's head.
"It's him." He said it softly, exhaled the words, no justification for them at all but somehow he was _positive_ —
Brake lights. A flash of red, one quick blink that he saw too late because he was too close, and then he hammered the brake pedal and slammed the wheel left and hit the back corner of the Lexus at fifty miles an hour.
" _Shit!_ "
The back of the Jeep swung right with the impact, then came back to the left and sent the front end sliding, a fishtail that was threatening to turn into a full three-sixty. Even as the skid started Frank could hear his father's voice— _turn into it, turn into it, your instinct will tell you to turn away, but you've got to turn into it_. He heard it, recalled those old lessons in the half second that it took him to lose control of the car, and still he turned away from it. It had happened too fast and the instinct was too strong. He turned away from the skid, the tires shrieked on the pavement, and then any hope of getting the car back was gone.
Frank was saved by bald tires. He'd lectured himself on the tires a dozen times, thinking they'd kill him someday if he didn't get them replaced, but instead they saved him. The pavement was dry, the Jeep was a top-heavy vehicle, and if the tires had been able to grab the road well he probably would have rolled. Instead, because there was hardly a trace of traction left on the worn rubber, he slid. He saw whirling trees and sky and then the Jeep spun off the shoulder and into the pines. He heard a crunch and shatter just as the airbags blew out and obscured his vision, and then he came to a stop.
The airbag deflated and fell away, leaving his face tingling, and for a few seconds he sat where he was, hands still locked on the steering wheel, foot still pressed hard against the brake, blood hammering through his veins. It was amazing how fast the body could respond—you'd spend an hour just trying to wake up on a normal morning, but throw a crisis out there and the body was ready for a marathon in a split second. He reached over and beat the airbags aside with his hands and saw spiderwebbed glass on the passenger window, the door panel bent in against the seat. Bad, but nothing terrible. He could probably drive away.
What about the Lexus? Devin Matteson's Lexus. He was sure of it again, absolutely certain, and without any pause for thought he turned and reached behind his seat, found the metal case, flicked the latch and opened it and then he was sitting behind the wheel with a gun in his hand.
Reality caught up to him then. _Sanity_ caught up to him.
"What are you doing?" he said, staring at the gun. "What the hell are you doing?"
He slid the gun back into the case and closed it and opened the door—after a glance in the sideview mirror to make sure he wasn't going to step out in front of a truck, survive the accident only to get squashed when he was on foot—and then got out of the car. He walked around to the front and saw that he wouldn't be driving anywhere. The right front tire was blown out and the wheel bent inward, crunched down beneath the mangled front quarter panel. If he'd handled it right, turned into the skid instead of away, he might've been able to keep the Jeep straight enough to avoid the trees. Then he'd be left with a dent and a drivable car, instead of this mess.
He'd lost track of the Lexus at the moment of impact, and now he was surprised to see how far behind him the car was, a good hundred feet at least. The driver had made the shoulder as well, but the car was facing the wrong direction and angled against the trees that lined the road.
Looking up at the car made his previous suspicion come on again, and again he thought of the gun, had to shake his head and move away from the Jeep before the urge to go for it got any stronger.
"It's not him," he said. "It's not him."
At that moment the driver's door on the Lexus opened and Frank's breath caught and held for a second until the driver stepped out onto the road.
It was not Devin Matteson. Not by a long shot. Even from this far away he could tell exactly how ludicrous the idea had been, could tell that he'd just caused a dangerous accident over an utterly absurd moment of paranoia.
He walked toward the Lexus as the driver began to survey the damage to his vehicle. Frank's first thought, watching him— _the dude's on speed_.
The guy, tall and thin with a shock of gray hair that stuck out in every direction, was dancing around the Lexus. Literally dancing. He'd skip for a few steps, twirl, lift both hands to his face and then prance back around the other side. He was talking to himself, too, a chattering whisper that Frank couldn't make out, and he seemed completely oblivious to the fact that there'd been another car involved in the collision.
"Hey." Frank got no response and walked closer. "Hey! You okay?"
The guy stopped moving then and stared at Frank in total confusion. Then he looked up at the Jeep and nodded once, figuring it out. Up close, Frank saw that he wasn't too old, maybe forty, the gray hair premature. He had a long nose that hooked at the end and small, nervous eyes set above purple rings that suggested it had been a while since he'd had a full night's sleep. His hands were still moving, too, fingers rippling the air as if he were playing a piano.
"Yes," he said. "I'm okay. Yes, everything's fine. You don't need to worry about me. I'll just call Triple-A. You can go on now."
Frank raised his eyebrows. "Just call Triple-A? I hit _you,_ man. You're going to want to hang around and get this worked out for insurance."
The guy was shaking his head. "No, no, I hit my brakes, just slammed on my brakes, not your fault at all."
Not his fault at all? What the hell was he talking about? Frank had been tailgating so bad he'd slammed into him as soon as the guy slowed. It was clearly Frank's fault. The guy must be nervous, that's all. Shaken up. Collision like that, at nearly highway speeds, who wouldn't be?
"What I'm saying is, we need to call the police," Frank said. "Get an accident report made, so we can make this square with the insurance company, right?"
The gray-haired guy winced and rubbed his forehead as if a pain had developed there. He probably had a bad driving record. Maybe a few accidents, and driving a car like that Lexus, his insurance rate already had to be high. He was worried about the money. Didn't understand that Frank was liable for all the damage.
"Tell you what," the guy said. "It'd be a big help to me—a _big_ help—if we didn't get an accident report made."
So he'd been right—bad driving record. Unless it was something more serious. Hell, maybe the guy _was_ on drugs. Frank frowned, studying him closer, looking for the signs. He just seemed ampedup, that was all. Buzzing. His eyes were clear, and he was cogent enough in conversation. A Starbucks addict, maybe.
"I'll pay for your damage," the gray-haired man continued. "I know what you're thinking—as soon as I can, I'll take off and stiff you on the bill. But I promise that won't happen. We can take care of it right now. Find a repair shop, and I'll take care of the bill beforehand."
"I hit _you,_ " Frank said again.
"Don't worry about that. It was my fault, my responsibility, and I don't want an accident report made, okay?"
Frank shook his head and walked a few steps away, looking at the Lexus. It was even more beat to shit than his Jeep. The front end was crumpled, there was a gash, maybe three feet long, across the passenger side of the car from the contact with the trees, and steam was leaking out of the hood.
"Please," the man said, and there was a desperate quality to his voice that made Frank look back with surprise. Whatever trouble this guy had with his driver's license—if he even had one—was serious. Frank stood there on the shoulder as two cars buzzed past them, nobody stopping, and looked at this weird guy with the nervous hands and panicked eyes. Why not give him a break? It was Frank's fault, so it was only fair to let this guy handle it in whatever way he wanted.
"All right," he said, and the look on the gray-haired man's face, the way it broke with relief, was enough to convince him he'd made the right call.
"Thank you. Oh, man, _thank_ you. I'll call a tow truck. The car's got a navigation system, you can find anything with it, we can pick any repair shop you want, I'll show you the choices . . ."
## 3
Jerry was staring at Nora's ass again, in that way he had where his eyes seemed to bug right out of his head, nothing subtle about it, but she wondered if she was allowed to care today—she'd done the same thing that morning as she got dressed, looking her butt over in the mirror like some sort of sorority girl instead of a woman with wrench calluses on her palms. You did something like that, could you get upset when a guy allowed himself a stare? Maybe she'd earned the leer. Karma.
The glance in the mirror was important, though, a morning reminder that Nora was still very much a woman. This before putting on the jeans and the heavy work shirt, tucking her hair into a baseball cap so it wouldn't hang free and invite a painful accident. She'd learned that lesson one afternoon when she'd used the creeper to check up on Jerry's work and rolled right over her own hair. Stafford Collision and Custom was open by seven thirty, and from then until six or six thirty when she shut the doors and turned the locks, Nora would interact with few females. It was a man's business, always had been, but she liked the touch she brought to it and thought the customers did, too. Granted, they were her father's customers and probably kept returning more out of loyalty—and pity—for Bud Stafford than for his daughter, but the shop still did good work. On those rare afternoons when a particularly difficult job was done and the car driven out of the shop, Nora might even let herself believe they did a better job now. She wouldn't admit it to anyone else, of course, but she did have an eye for detail that her father couldn't touch. Too bad an eye for detail wasn't enough to keep the bills paid.
The phone rang out in the office, and Nora straightened up and looked back at Jerry, who promptly flushed and averted his eyes. Even when you _didn't_ catch Jerry, he thought you had. Jerry would've made a piss-poor criminal.
"I'd like you to take another pass over that front quarter panel," Nora said.
"Huh?"
"Nice orange-peel finish in the paint, Jerry. I know you can see that, and _you_ know how I feel about it. Doesn't matter if it disappears in the shadows, you can see it in the sun, and that's when people care about their cars looking the best. They go home and the first sunny Saturday morning they wash the car and wax it and see that orange peel. And then you know what happens? They don't come back."
She walked away from him, got into the office just in time to grab the phone before it rang over to voice mail. She was always forgetting to take the cordless handset out into the shop with her, and she knew they'd lost business because of it. When a body shop doesn't answer, people just call the next one in the phone book; they don't wait and try again. She'd been one ring away from losing this call.
"Stafford Collision and Custom, this is Nora Stafford."
She sat on the edge of the desk and took notes on one of the old pads that still had Bud Stafford's name across the top. The caller wanted a tow truck for two cars that had wrecked up on County Y. Her last tow driver, who'd also been a prep man and part-time painter, had picked up a drunk driving charge three months back and to keep him would have required bearing an insurance rate spike that she simply couldn't handle. In reality it was a welcome break—the shop's financial situation was going to dictate firing somebody anyhow, and the drunk driving charge gave her an excuse. She'd let him go and couldn't afford to hire a replacement. But two cars—including a Lexus—that was business she couldn't turn down, either. Jerry could drive the tow truck, but he wasn't covered by the insurance policy, and she needed him to finish repainting that Mazda this morning. She'd have to handle this one herself.
She got the details of the wreck's position and promised to be out within twenty minutes, then went back into the shop and told Jerry where she was going. He just grunted in response, not looking at her.
"What's the problem, Jerry?"
"Problem?" He dropped the rag that was in his hands. "Problem should be pretty obvious. You got me wasting all my time _re_ painting work I shouldn't have had to _paint_ in the first place."
She waved a hand at him, tired already, the argument by now just like the dying water heater in her house—too familiar, too annoying, too expensive to fix.
Jerry was a body man, a fine body man, none better in town. Didn't have the eyes for a top-quality paint job, but that wasn't the problem so much as the way he felt disrespected when asked to paint. If she could afford to bring someone else on board, she would, but that explanation hadn't appeased him.
"Jerry, this is not a big deal. If you'd done it right the first time, I wouldn't have asked you to repaint it. Instead, you half-assed the job and then tried to make up for it with the buffer, like usual."
"Damn it, Nora, last time I painted cars it was with—"
"Single-stage lacquer, spray it on, buff it pretty, don't have to mess with no damn clear coat . . ."
Nora mocked his voice perfectly, capturing the drawl so dead-on that Jerry pulled back in anger and grabbed his rag again, tightened his fist around it. He was a small man, only a few inches taller than she was, but strong in the wiry way that comes from years of physical labor. What was left of his hair was thin and brown and damp with sweat.
"All right," he said. "So I've told you before, if you remember all that. Think you're clever saying it back to me, I 'spose. But if you was clever you'd understand, instead of using it to make fun of me. Your daddy understood. I'm not a combination man. I do body work. Been doing it since back when you was playing with dolls and putting on training bras and learning to paint your nails."
Same old shit. He'd start bitching about his workload, then begin with his what-a-pretty-little-girl-you-are routine, slighting her gender either directly or with what he thought passed as slick humor.
"Tell you something, Jerry? When I was learning to paint my nails, I was also learning how to paint a _car._ Now it's time that you do."
She turned and walked away from him, heard the _bitch_ muttered under his breath and kept on going, out of the shop and into the tow truck. Sat behind the wheel and let the engine warm and lifted her hands to her face and thought, _I would've cried about this. A year ago, maybe even six months ago, I would've cried_.
Not any more, though. No way. But was that entirely a good thing?
She wasn't going to think about it. Pointless exercise. What she needed to think about was the cars waiting for her up on County Y. That was more than a pleasant surprise—it was salvation. She'd spent the morning trying to determine which bills she could be late on. It was down to that now, down to creating a rotating schedule of missed payments because otherwise she simply could not keep the doors open. Now here was a phone call offering enough work to keep those wolves distracted, if not completely at bay. And to think, she'd been one ring away from missing it altogether.
It felt longer than twenty minutes. The gray-haired guy kept up a constant stream of chatter, the words sounding more nervous each time there was a pause, as if he were scared of silence. When a car passed by, though, he'd stammer the way you do when you lose your train of thought, stare intently at the vehicle until it was out of sight. A couple of times, people slowed and put their windows down, ready to offer help, and the gray-haired guy just waved them off and shouted that everything was fine, go on, have a nice day.
It _was_ a hell of a nice day, though. If the Lexus driver would shut up for a few minutes, Frank wouldn't have minded it at all, standing out here. It had been a long time since he'd lived in the city, so it wasn't as if he'd arrived in the woods fresh from garbage-riddled streets that stunk of exhaust fumes. Even so, this place felt different. For one thing, there wasn't a building in sight. Turn right, turn left, see trees and blue sky, nothing else. A pair of hawks rode the air currents high above, staying on the south side of the road. Must be a clearing back there, something offering prime hunting ground for the birds. Frank could've watched them for a long time, if this jazzed-up dude would let him. Instead, he was busy fending off meaningless questions and observations.
He was relieved when he saw the tow truck at the eastern end of the road, and a minute later it had pulled up beside them. The driver opened the door, and Frank felt his eyes narrow, saw matching surprise on the gray-haired man's face. The driver was a woman, and a good-looking one, that much evident even with her face shadowed by a baseball cap. She hopped down onto the road—the truck was too high for her to just step out; she couldn't go an inch more than five-three and might go an inch less—and walked around to face them.
"Sorry about the wait, guys. I got moving as fast as I could."
"No problem," Frank said, and he was going to shake her hand when the gray-haired man interrupted.
"If it's no trouble, can we do this car first?" He pointed at the Lexus.
The woman wore jeans and boots and a denim work shirt, sleeves rolled to expose thin forearms. There were grease stains on her clothes, and both the pants and shirt were loose, giving her a shapeless look. She didn't wear any makeup, but her eyebrows—not a feature Frank would ordinarily notice—had been carefully attended to, well shaped. Cool green eyes, now fastened on the Lexus driver.
"There a reason that one needs to go first?"
He gaped at her for a second, then looked at Frank and forced a smile.
"Well, I was just hoping . . . I've got a meeting to get to, and I was sort of—"
"In a hurry," the woman finished.
He nodded.
"Right," she said. "Well, I can give you the first tow unless this gentleman has an objection."
Frank shook his head.
"Great," the woman said. "Here's how we're going to do this—I'll get the Lexus rigged up, tow it back to the shop, and you guys can ride with me, unless you've got someone coming to get you."
This time Frank and the gray-haired man shook their heads in unison.
"Okay. Well, probably be easier to figure out your situations from town, unless you'd rather stand out here on the edge of the road."
"Sure," the gray-haired guy said. "Town's fine." But he was looking down the road with a frown.
The woman walked over to the Lexus and knelt beside it, studying the front end. Frank turned away when she bent over to see under the bumper, not wanting to stare. When was the last time a guy had wanted to check out a tow truck driver, anyhow? She straightened up and walked back to the truck, climbed in and put it in reverse and had the thing centered in front of the Lexus in half the time it would've taken Frank.
"I have to winch you out of that ditch before I can get it ready to tow," she told the gray-haired guy. "Looks like the Jeep is sitting clear enough already."
She hooked the winch beneath the front bumper of the Lexus, went back to the truck, and turned it on. The chain went taut and the gears hummed and the Lexus slid away from the trees and up the ditch, shedding a tangle of branches and broken glass in its wake. When she had the car on the flat surface of the road, she shut the winch off, went back and fussed with the chains for a few seconds, and turned to the car's owner.
"This thing's all-wheel drive. We should use the dolly on the rear wheels to keep from hurting your axles or transmission. The thing about that is, we also charge an extra thirty dollars to use it."
The gray-haired man stared at her, mouth open about an inch. Didn't see many women winching your fifty-thousand-dollar car out of a ditch.
"Uh, yeah, sure."
She raised her eyebrows. "You're okay with that?"
"You think that the dolly will save time?"
"It'll save your transmission."
"Whatever. Faster the better. I want to get moving."
She went back to the Lexus, and Frank thought her stride was slower, almost as if she were screwing with the guy because he was in such a hurry. It made a wry smile build on Frank's face, and he turned before the Lexus driver caught it.
Once she had the wheel-lift under the front end of the Lexus—looked like a set of mechanical arms wrapped around the wheels—she strapped the tires to it for added security and disappeared behind the car. Frank and the gray-haired man stood together in silence, waiting. Eventually she walked back around to the front, gave the wheel-lift one last look, and then made a small nod of satisfaction and turned back to them.
"Go on and get in. Short straw gets to sit in the middle."
Frank got to the passenger door first, pulled it open, and slid across to the middle seat as the gray-haired man climbed up beside him and the woman got behind the wheel.
"What's your name?" Frank asked her.
"Nora Stafford." She took one hand off the wheel and extended it. When they shook, he felt fine bones on the back of her hands, the skin smooth and cool, but hard on the inside, beneath her fingers.
"I'm Frank."
"Good to meet you, Frank." She put the truck in gear and checked the mirror. "Who's your buddy?"
"You know, I didn't make his acquaintance yet, just his car's," Frank said.
"My name's Dave O'Connor. Sorry. Should've introduced myself earlier. I'll be paying for this, which brings up a, uh, a question. I was wondering . . . see, I'm from out of town, and I need this done fast, but, well, I don't have my credit cards on me."
"Credit cards?" Nora turned to him with surprise. "Sir, I think you're going to want to make an insurance claim on that."
"No, we're not going to do that."
"Um . . . I don't mean to tell you your business, but this job is going to be several thousand dollars," Nora said.
Frank shifted in his seat. He'd hit the guy, and his insurance should be paying for the damages, but the gray-haired man had been adamant.
"So what I was wondering was, I mean my question, well, could I give you cash? Because I've got some cash on me, see. And if I gave you that, you know, to get started, and then I could come back with a credit card or call you and give you the number . . ."
Nora's face hardened just a touch, barely noticeable, a little frost in her eyes even though she didn't take them off the road. There was something about the edge she showed in that moment, like the way she'd slowed down just because the gray-haired guy was in a hurry, that Frank found damn appealing.
"Two cars, both with substantial damage," she said, her voice friendly. "Parts and paint alone are going to run up a decent bill, Dave. That's without labor figured in."
"I could give you two thousand dollars today. Surely that's enough to get started? You aren't going to burn through two grand in the first day."
Nora kept her eyes ahead, and so did Frank, but in the few seconds of silence that followed he felt a shared curiosity with her— _no credit cards on you, but two grand in cash?_
"Well . . ." Nora nodded her head as if in discussion with herself. "Two thousand dollars is a sizable down payment. The bill for this work will run well over that, but it's certainly enough to get us started."
They were on the highway now, southbound toward Tomahawk, the tow truck's engine throaty, straining to get its load up to speed. Nora's thigh was warm against Frank's. He looked at her hands on the steering wheel, saw no wedding ring. So it wasn't her husband's body shop. This was just what she did, drive a tow truck in a town like Tomahawk? A young girl, intelligent, with perfect teeth and eyebrows?
"You guys have someone to come get you?" Nora asked.
"Nope," Frank said, and Dave O'Connor shook his head.
"I've got to get something figured out," O'Connor said. "Like I told you, I'm in a bit of a hurry. Got a meeting that won't wait all day for me."
"A meeting at the Willow?" Frank asked.
"No. I, uh, I've got to get to . . . Rhinelander. Little bit of a drive left to make, so, you know, got to figure something out."
Rhinelander. He'd been westbound on County Y, headed for Rhinelander? That was an interesting route, considering County Y took you out to the Willow, across the dam, and then looped back down to the old highway and into Tomahawk. O'Connor had been driving the exact opposite direction from Rhinelander, and not toward any highway where he could correct his course.
"Any chance you'd have a car you could rent me?" O'Connor asked Nora.
She shot a sideways glance at him. "I don't rent cars. I fix them."
"You don't have anything around the shop? It'd be one day. _One_ day, and I'll give a couple hundred cash for it. I've got to make this meeting."
Nora let a few cars pass before she answered.
"Only drivable vehicle I could give you—unless you want to drive the tow truck—is a beat-up old Mitsubishi that probably can't do more than fifty without blowing up."
"That's fine. I'll take it."
"And if it _does_ blow up on you, I'm certainly not going to take responsibility. I'm doing this as a favor."
"It's not a favor. I'll pay you—"
"You won't pay me anything. Sounds like you need something to get you to Rhinelander, and the Mitsu will do it. Slowly."
"I appreciate that," O'Connor said. "It'll be a huge help. Save me the time of renting a car, and I don't have time to waste."
Something else it would save, Frank thought, staring out at the lumber truck ahead of them, was the _process_ of renting a car. You couldn't do that with cash—and Mr. Dave O'Connor seemed damn concerned with sticking to cash.
## 4
Took Nora ten minutes to get Dave O'Connor in and out of the body shop. He had the cash out before she was even in the door, put it in her hand and waved away her offer of a receipt, said he knew she was trustworthy and he was in a hurry, now, could you show me that Mitsubishi you were talking about?
So she took him out back and showed him the car, a rusted blue box of a thing they used for errands, running back and forth to the auto parts store. Had four-wheel drive, but that was about it. Windshield wipers were shot—how many times had she asked Jerry to replace them?—and two of the windows hadn't moved in the better part of a decade. Dave O'Connor looked at it as if it were the next year's model of his fancy Lexus. Took the keys and tried to press more cash in her hand.
While declining the money for the third time, she realized she was hustling him out of the shop almost as much as he was hustling her, and she knew why. There was something _off_ about this guy, and, yeah, it started with the cash-only thing and the I'm-in-a-big-hurry thing but went beyond those, too. A meeting in Rhinelander? What the hell was he doing on County Y, then? License plate from Florida, no less. And the mannerisms, the tension . . . she pushed it out of her head. He'd given her more than enough money to hold the job, and it didn't seem likely he was going to dash off and leave his expensive car. If he did, hell, she'd make a fine profit off that. How long did you have to wait to claim a mechanic's lien?
So she let him take her car and drive off, didn't fill out any of the standard paperwork, just accepted his money and his promise to return Monday. Even a few months ago she would never have believed she could agree to something as crazy as this, but a few months ago the shop's debt was merely threatening, not suffocating her in the way that it was now. She stood in the parking lot and watched him go, two grand in cash in her pocket. It was enough to justify the breach of protocol. She was in a dream world when she walked back into the shop, and pulled up with surprise when she saw the young guy standing there, Frank. How old was he, anyhow? Appearance said he'd be a few years her junior, maybe twenty-six, twenty-seven. Acted older, though. Carried himself all steady and sharp-eyed, the way a man who's seen a lot will do. The way her father had.
"Hey," she said, and for some reason she tugged off the baseball cap, shook her light brown hair out.
"Hey. You get things settled with that guy?" He stepped closer to her, an easy smile on his face but the eyes not matching it, too thoughtful. A nice-enough-looking guy, runner's body, good skin. Needed to grow the dark hair out, though, lose that military cut that made him look even younger than he was.
"A pocketful of money to prove it," she said and gave him one raised eyebrow that made him nod.
"Feel safe about getting that Mitsubishi back?"
She laughed. "If I never see it again, that'll only save me money."
"Different sort of guy, wasn't he?"
"Seemed a little on edge."
"Uh-huh. Got a gun out of the glove compartment when he was moving his things into your car, too."
That stopped her. Not just because of the gun, but the way he said it. Relaxed. Casual. And how had he even seen that? When O'Connor was busy switching his gear from the Lexus to the Mitsubishi, she'd been standing right beside him, with Frank all the way back at the shop, leaning against the wall.
"Handgun," he said. "No big deal, I'm sure. Lots of people carry them."
She didn't say anything, just stood beside the door and stared at him.
"Look, I didn't want to worry you," he said. "It's meaningless."
"I know that. I was just surprised you saw him with it, that's all. You were standing all the way over—"
"Good eyes. I've got good eyes."
"I guess so." Pretty eyes, too. Nora always liked blue eyes on a guy with a dark complexion. Something about the contrast. She pulled open the office door, stepped inside with Frank behind her.
"I'll go back and tow your car in just a minute," she said. "You know what you're going to do for a ride?"
"I'll figure something out."
"Where were you headed?" If he said Rhinelander, she was going to be awfully uncomfortable.
"The Willow. Staying at a cabin up there. I've got some errands to run in town, though, groceries and the like, so I'll deal with them first."
"You aren't going to rent a car?"
"No need. Once I get up there I don't plan on leaving for a while."
She pushed her hair back over her ears, the baseball cap still in her left hand. Over Frank's shoulder she could see Jerry standing at the row of lockers along the far wall, getting ready to take a cigarette break. The doors to the paint booth were still open and the lights were off, which meant the Mazda wasn't drying, which in turn meant Jerry hadn't repainted that quarter panel yet. Good thing he was taking a cigarette break.
"Tell you what—if you can kill the afternoon in town, I'll drive you up to your cabin tonight," Nora said, refocusing on Frank again. "Come by around six?"
"You don't need to—"
"It's not a problem."
"All right." He nodded. "I'd appreciate that."
"Sure. Six o'clock, right?"
"Six o'clock."
Thing was, Jerry didn't dislike women. Was rather fond of them, in fact. In their place. And their place was not in a friggin' _body shop_. Shit. Standing in the paint booth and listening to the tow truck growl as Nora set off on her second trip, he wondered just what he'd done to earn this fate. Working for a woman, him, best body man in the damn town. Could he find work somewhere else? Sure. But even if Nora was a righteous bitch four days outta five, she was also Bud Stafford's daughter. And if Bud ever got better, came back to run the place again, Jerry wouldn't want to make eye contact with him knowing he'd left the girl on her own.
She'd told him to redo the clear coat on the Mazda. Redo it, like he'd painted the thing the wrong color or something. Hell with that. Somebody needed to look at the Lexus, and Jerry didn't think Nora was the one for the job. Car all beat up like that, there was some work just to figure out what all was wrong. If she wanted the Mazda fooled with again, she could wait till Monday, or do it her damn self.
Jerry found the keys to the Lexus and pulled it into the shop. _Shee-it,_ what a car. More bells and whistles than anything he'd seen. More than anything he'd want, too.
Once he had the car inside, he got to work inspecting the damage. Hood would need to be replaced, plus the front quarter panel, and the front passenger door. Now, if it were Jerry's car he'd probably handle that door and the quarter panel with a liberal amount of Bondo, a spray gun, and a buffer. But he didn't imagine the Lexus owner would agree.
Problem with these fancy new machines was all the shit you couldn't see. Sensors and computer chips and whatnot. Some of them would be up under the bumper, so he'd have to figure out what the hell they all did when he took that off. Probably want to replace the bumper assembly, too. Make a few extra bucks, get the job done right. Nora herself would appreciate that outlook, if she ever climbed down off her damn broomstick to listen to him.
He dropped onto his back and slid beneath the front of the car, wrench in hand. Way the front was punched in, there could be some damage to the internal workings. He got the splash shield off and—wait a second, what in the hell was this?
A thin black box, about the size of a remote control but without the buttons, was mounted on the bumper reinforcement. One of those sensors he'd been worrying about? Those were usually wired in, though, and this thing just sat there by itself. Jerry tapped at it gently with the wrench, and the thing slid around a bit. Reached out and got his fingers around it and pulled. Popped right off. It was held on with a damn magnet. Two thin wires trailed out of it, and he followed them with his fingers, found another box, this one larger, and popped it free, too.
Pushing back out from under the car, Jerry sat up and studied his find. The smaller device was plain black plastic with a magnet on the back and a small red LED light in the center. The other thing, the bigger one, looked like some sort of battery pack. First thing he thought of was one of those GPS units. Buddy of his, Steve Gomes, had one he took hunting. Tracked your position. The Lexus had a navigation system, so it would need a GPS unit, but wasn't that inside the computer?
That's when he got it. The magnets were there so you could attach the thing to the underside of the car, on the frame. Attach it without the owner knowing. But whoever put this one on, they went even a bit further. Popped out the bolts and got it inside the splash shield on the bumper reinforcement, where it would be protected from water and road debris and couldn't possibly fall off.
"Who are you, friend?" he said, bouncing the black box in his hand and staring at the Lexus. Nora said the guy gave her cash, didn't show a driver's license or credit card, anything with his name on it. Stupid of her to let him go like that, no proof of identity, but two grand in cash had a way of convincing even the strictest person to let a few details slide. Couple kinds of people in this world liked to move without identification, and a smaller number of those were going to have someone tracking them. Drug dealer, maybe? Bank robber? Could there be cops on the way, following him with this gadget?
Jerry walked into the office with the device in his hand, opened the mini fridge, and pulled out a can of Dr Pepper. Jerry drank three or four Dr Peppers a day. Kept him fresh. He dropped into the chair behind the desk and cracked the top on the can, took a long swallow, and considered his find. No matter the explanation for the black box's presence, Nora was going to be damn interested in it, and, possibly, so would the cops. Should they call the cops, though? Did they have any reason to? Maybe not. Maybe it was best just to pretend they'd never seen the thing. He could put it back inside the splash shield, send it on its way without ever knowing what it was doing there. That would be Nora's call to make, not his.
He should have heard the husky growl of the tow truck engine, but the black box had taken his mind deep into other places, and he missed it. When Nora entered the office, he was still in her chair, with his boots propped up on her desk and the soda can in hand. Her face twisted at the sight.
"Tell me," she said, "that the Mazda is done, Jerry."
"Listen, Nora—"
"No." She leaned over and slapped at his boot, trying to knock it off the desk. His foot didn't budge. "I will not _listen,_ because I've heard them all already. Every excuse and problem and complaint that you utter. None of them are new, not anymore."
"Wait a sec—"
"If my father had _any_ idea the sort of work ethic you exhibit down here, he'd be disgusted. Absolutely disgusted. The last thing I said before I left was that I wanted you to finish that Mazda, and instead you spend your time sitting at _my_ desk drinking a soda?"
"I just sat down two seconds ago. Reason was, when I started taking that Lexus apart . . ." The little black box was in the hand not occupied by the Dr Pepper. He started to lift it above the desk, thinking to drop it in front of her, shut her up, but she started in again.
"Lexus? I didn't ask you to do a thing to that Lexus, Jerry! I specifically said the Mazda needed to come first. What can't you follow about that?"
Jerry kept his hand below the desk, closed his fingers around the black box, felt his jaw clamp tight.
"Would you _please_ go get some work done?" Nora said. "Please do what I already asked?"
He slipped his hand into the pocket of his coveralls, dropped the plastic device inside, and swung his boots off the desk and to the floor.
"Yes, sir, boss. Don't let me bother you anymore."
On his way back through the shop he stopped at his locker, placed the tracking device inside, then slammed the door shut and locked it.
## 5
Ezra Ballard, a few hundred yards out on the lake, spotted the blue car shortly after noon and knew that the two on the island were no longer alone. The car, some sort of beat-up Jeep, was parked in the woods across from the island cabin—a cabin that had, for almost two days, been home to a gray-haired man and a blond woman. Technically, that was Ezra's business. He didn't own the cabin or the island, but for many years he'd been entrusted with their care. Same with the cabin down on the point, less than two miles away. Two cabins that, at least in Ezra's mind, still belonged to men who'd been buried long ago.
Twice a year the Temple boy mailed Ezra a short note with five hundred dollars inside. The note always read _Thanks for keeping an eye on the place_ ; the money was always in five one-hundred-dollar bills, the envelope always void of a return address, but a phone number would be included on the note. Ezra would spend the cash on whatever expenses he might encounter keeping the cabin in good shape and save the rest. Seven years young Frank had kept that up, and though Ezra wondered when he'd return to the place, he never wondered _if_. The boy—hell, he wasn't a boy anymore, was he?—would be back, but not until he was ready. Maybe Ezra would still be around, maybe not. Something like that, it took time to make your peace with it.
Circumstances with the Temple cabin had been consistent, and Frank's boy seemed to understand the situation, had made no effort to contact a realtor or a lawyer. The Matteson cabin, here on the island, was a different matter. After Dan died, Ezra hadn't heard a word from the family. Sent a few letters, made a few phone calls, and finally received a curt order to ready the place for sale—this from the son, Devin. When Ezra explained that the island couldn't be sold—it was part of a legacy trust that would either remain with the family or revert to the state, and good luck convincing a judge to break that—Devin swore at him and hung up. Never called again. This was before Frank Temple had taken his own life and Devin's role in that situation became clear, before a few conversations with Frank's son that Ezra probably never should have allowed to take place, before a final call that Ezra had made to Devin.
In the years that followed that last call, Ezra had never heard from Devin or anyone else about the island. He hadn't expected to, though. His message had been succinct enough: If Devin came back, Ezra would kill him. For seven years it seemed that Devin had believed the promise, and he damn well should have. Ezra was not a man given to idle threats, and he certainly was not a man with light regard for killing. Not anymore.
Though the cabin had sat empty for years, Ezra kept the place in shape, paying property taxes and all expenses out of his own pocket. Nobody other than Ezra had been inside until this week. Just two days ago a bizarre phone message had been left, someone claiming to be Devin telling Ezra the cabin needed to be "opened up for guests."
The call had sucked the breath from Ezra's lungs, the brazenness of it, the _audacity_ almost more than he could get his head around. He'd never expected to see Devin again, believed that the island cabin would sit empty until after Ezra was gone from the world, and even in the corner of his mind that recognized there was at least a _chance_ that Devin might show up, he never imagined a call like that. So casual, so flip. A taunt, like after all these years he'd decided Ezra was a harmless old man.
Ezra had called Frank's son—probably a poor choice but, again, there was a promise to be kept—and then visitors had arrived at the island, but Devin was not among them. Not yet.
Now there was this second vehicle. With opening weekend of the fishing season a mere week away, Ezra had decided to run some of the bays and islands, getting depth readings and trying to find new spots to catch walleye. It was on his first run across the lake that he'd noticed the car, and now he'd spent most of the afternoon anchored off the opposite shore, using a pair of binoculars to watch the island. His first idea was that there had been a new arrival. That changed around midafternoon, when the gray-haired man moved the car.
He and the woman had arrived in a Lexus SUV that had disappeared this morning. Now the gray-haired man took his boat back across the inlet, got into the blue car, and drove it out of the mud and back up the hill. At the top of the hill, he went off the road and into the grass, right into the pines. Drove it as far into the trees as he could, till the boughs swept over the roof and pushed against the side of the car to the point that he had trouble opening the door to get back out. Only reason you parked a car like that was to hide it. He'd gone too far, though; the car was hidden from the logging road, but he'd driven it right up to the edge of the tree line, so the sun caught it and reflected the glare of glass and metal across the lake. Hard to see unless you were on the water. Hard to see unless you were Ezra.
Ezra had been on the Willow for most of forty years now, taking fish out of the lake's waters and deer and bear out of its surrounding woods. Best guide in Oneida County, that was what people said. The people were right, too. At least when it came to hunting. Out in the woods with a rifle in hand, there wasn't anybody better than Ezra. Thing was, he preferred fishing. He was good at it, sure, but not the natural he seemed to be when it came to stalking prey with a gun.
This was about to become a busy time of year, too. The season opened for walleye, pike, bass, and the other game fish on the first Saturday of May, which was in one week. From that point on, Ezra had a full calendar. It was no time to worry about a cabin that hadn't been used in years. But there sat that damn car, shining against the blanket of trees, inviting everybody and their brother to slow a boat and stare at it and wonder if someone was using the Matteson island. Questions would be headed his way, and maybe he should have some answers ready when they did. Problem was, this gray-haired guy clearly wanted that car hidden, and of the men Ezra had known who hid cars, exactly zero of them were guys he wanted to deal with.
It being Friday, and a full workload arriving out of the blue like that, Nora was in a good mood as the afternoon wore down. Good enough mood that after she'd towed the Jeep in, she picked up lunch for Jerry, one of those Angus burgers he favored. An obvious peace offering, and one that seemed to make Jerry feel awkward, shuffling around and trying to stay mad at her for that oh-so-demanding request to do his job correctly. They didn't talk much for the rest of the day, but there were no blowups, either.
She spent the afternoon with the computer, going over finances. It was her own laptop, and she'd devoted countless hours to slowly transferring all of the paper files Bud Stafford had used. Tedious work, yes, but now they were more organized, more efficient—and lacking enough jobs to make it pay off.
Jerry had given her his damage assessment on the Lexus. "Uh, you got your quarter-panel issues, you know, and you gotta get down in there, too, plus there's the light and your, uh, you know the bumper issues, plus there's the airbag and your, uh . . ."
From that she managed to cull an actual estimate, printed it out, nice and official. She was reviewing it when someone pulled into the front parking lot, got out of the car without shutting the engine off, and opened the office door. Four o'clock on Friday afternoon was an unusual time for business.
The visitor came through the door and stopped, ignoring Nora to look around the room with open curiosity, as if he were on a museum tour. Big guy, too, a fancy knit T-shirt stretched over his chest and shoulders, loose jacket over that.
"Can I help you?" she said.
He had a bizarre silver belt buckle, a sort of rippled pattern, like latticework. Not ridiculously large like some of those western things, but ornate, flashy. Nora had always found that a man who believed a belt buckle should be a fashion statement was not her kind of man.
"I hope I'm in the right place," the guy said. "Friend of mine called and asked me to grab some things out of his car. I think he left it here . . ."
"What's his name?"
The guy just smiled at her. Patient, as if she'd asked a worthless question but he was willing to ignore it.
"The car's a Lexus SUV."
"I didn't ask for the car's description. I asked for the guy's name."
"Vaughn," the guy said. There was a hitch in his voice, though, like a game show contestant who second-guessed his answer at the last minute.
The longer he stood in the office, the more space he seemed to fill. She had trouble meeting his eyes as she shook her head.
"I'm sorry. Nobody named Vaughn has a car in here."
"I'm rather certain he does. Perhaps there's been some confusion over the name."
"If there has been, then the car's owner will need to come in and explain that to me. I'm certainly not allowed to release personal effects from a vehicle, sir."
"How about we give him a call, together? You can ask . . ."
Dave O'Connor had left no phone number—or any other form of contact information—but even if he had, Nora wouldn't have called. O'Connor had been weird enough, but this guy was almost threatening.
"No," she said. "If the car's owner—whose name is not Vaughn—calls me and explains this, then we'll see how we can proceed. Until then, I'm afraid not."
The guy's eyes darkened and he seemed ready to object when the office door opened and Jerry ambled in, a socket wrench in one hand. He gave Nora and the guy a casual glance and then knelt in front of the little refrigerator she kept in the office, pulled out a can of Dr Pepper, and cracked it open before walking back into the shop. The visitor watched him go.
"It sounds to me like you might have the wrong body shop," Nora said.
For a long moment he didn't answer, just stared at the door Jerry had walked through as if it were something that called for real study. Then he nodded.
"Of course. That must be it. Apologies."
He gave her a mock bow, lifting his hand to his forehead, then opened the front door and walked back into the parking lot. She stood up and went to the window in time to see him climb in the passenger side of a black sedan. That was why he'd left the engine running—he wasn't alone, wasn't driving. She got a clear look at the car as it pulled out to the street, a black Dodge Charger, one of the newer models. She'd made the mistake of complimenting the look, only to have Jerry ridicule her. _Nora, it's a_ four-door. _That ain't a Charger, it's a joke._
She couldn't read the license plate, but the colors told her it was from out of state. Wait, those colors were familiar. A smear of orange in the middle of a white plate with some green mixed in. She'd just seen that on the Lexus. Florida.
It wasn't five yet, but she turned the lock on the front door as she stood there gazing out the window. The odd feeling that had convinced her to get Dave O'Connor out of her shop and back on the road without any of the normal procedures had just returned, only this guy with the belt buckle made it swell to the edge of fear. He'd called him Vaughn. She had no proof that the Lexus driver's name was actually Dave O'Connor. All that cash, the hurry he was in, the gun Frank had seen, none of it suggested anything good. Add a fake name to the mix, though, and she was beginning to feel stupid. She'd gone for the money despite all the obvious objections, let the guy dictate the situation. It wasn't easy to imagine her father handling this in the same way.
Nora walked out of the office and back into the shop, watched Jerry working on the Lexus. The car was empty. Dave O'Connor had cleared all his things out when he left, including that handgun in the glove compartment. So he hadn't called someone to come pick anything up.
"Jerry," she said, "can you give me a minute?"
She wanted to talk to him, explain the situation and ask if he'd found anything in the car, more cash or guns or, well, _anything._ But when he turned around he had that irritated sneer on his face, ready to argue or mock her or do anything but listen.
"Well?" he said. "You got another problem needs me to fix it?"
"No, Jerry. It's just . . . I was thinking . . ."
"Hope you didn't hurt yourself." That passed for humor to him, real wit.
"I was thinking you can go home early," she said. "That's all. It's Friday, and we got some nice work in today, and you've done a good job this afternoon. So go on and get out of here. Enjoy the weekend."
She walked away as the first flush of gratitude mixed with shame crept onto his cheeks.
## 6
Getting out a little early on a Friday was no reason to disrupt your normal postwork routine, so Jerry drove directly to Kleindorfer's Tap Room, had himself a bar stool and a Budweiser before the clock hit five. Carl, the bartender, took one look at him coming through the door and asked if the Stafford girl had finally fired him. Jerry didn't bother to dignify that with a verbal response, electing instead to go with a simple but clear gesture.
It was early enough that the room was almost empty, a couple of out-of-towners drinking Leinenkugel in a booth, nobody at the bar except Jerry, nothing on the TV except poker. Give it a few minutes, they'd switch over to that show where the black guy and the white guy argued about sports, neither of them knowing a damn thing to start with. Jerry and Carl tended to have better ideas than those two.
Jerry sipped his beer and watched the muted poker game and simmered over Carl's comment. It had been a joke between friends, no offense meant, but it riled him anyhow. Not so much at Carl for saying at it, more at his own life for the circumstances that produced the line. Jokes about working for Nora were constant. Could hardly get through a day without hearing one. She'd been there almost a year now. Showed up from Madison dressed to the nines, walked into the body shop wearing jewelry and perfume and with her long fingernails polished and told Jerry she was the new boss. Wouldn't just own the shop, she intended to _run_ the shop.
The afternoon Bud Stafford had his stroke, it had been Jerry who found him slumped under a Honda, his shirt smeared with primer from the fall onto the hood. Jerry knew it was bad; his hands shook while he dialed for the ambulance. At the time, though, he'd seen two possible outcomes—Bud would die, or he wouldn't. The end result, this half-death, was a twist Jerry hadn't considered. Nora'd called a few days after the stroke to ask him to keep the shop going while Bud was in the hospital. A week after that, she was in town and in charge. Jerry had tolerated it, because he figured Bud would come back. That's what she kept telling him, insisting to him. Bud was going to be fixed up, and then he'd be back and she'd be gone, back down to Madison, finish up graduate school in _art history,_ of all things.
He still couldn't get his mind around that. Bud had been cutting that girl checks for years, putting her through school. Reasonable thing to do, providing the kid would accomplish something, walk out of there with a piece of paper telling the world she was useful, an engineer or an architect or a doctor, but Bud could never say what the hell she was going to do. Most practical man Jerry'd ever seen walk the earth would just shake his head and smile and say, "She's a damn smart girl. I'll let her learn, and when she's done with that, she'll do something big. Guarantee it, my man. She'll do something big."
Well, she wasn't doing shit that Jerry could see except bitching a blue streak about things she didn't understand and losing them business. End of every month, Nora would tell him that they'd kept the bill collectors at bay again, like it was something to be proud of. Didn't realize those bills were paid only through a sort of pie-in-the-sky expectation that Bud would be back eventually. It kept a meager supply of work coming in. And, Jerry had to admit, kept him in the shop. So who was he to criticize the customers who did the same thing?
Ten, maybe fifteen minutes had passed while Jerry brooded—enough for a completed Budweiser and the order of a fresh one—when the door opened and closed behind him. Regulars finally showing up, he thought, until the new arrival sat down beside him. Long, lean guy with a shaved head and a tattoo on the back of his left hand, a weird symbol that meant nothing to Jerry. Had a camouflage jacket on over jeans and a T-shirt. Seventy degrees today, and both this guy and the one who'd come into the shop office to talk with Nora were wearing jackets.
Jerry turned back to the TV, and the new guy didn't say anything for a few minutes, not till Carl brought his drink—vodka tonic—and returned to the other end of the bar.
"You work down at that body shop, don't you?" the guy in the jacket said. "Stafford's?"
Jerry turned and offered his favorite expression for making new acquaintances—sullen, with the lip curled just enough to imply a little disrespect.
"I don't think I know you, pal."
"My apologies," the guy said, making a little bow of his head. "Name's AJ."
Jerry didn't answer, just drank his beer and looked at the TV.
"So you work at the body shop, correct?"
"Uh-huh. And I don't give free advice on cars, and I don't look at them after work on a Friday. So you got one that needs fixing, bring it in Monday morning and we'll—"
"The car I'm interested in is already there," the guy named AJ said, and Jerry paused with the bottle back on his lips but no beer flowing yet. He lowered it.
"The Lexus?"
AJ smiled. "Either you guys don't have much business, or you're a smart son of a bitch, Mr. . . . ?"
"Dolson. Jerry Dolson." He took another drink and turned all the way around to face AJ. "You want to tell me what the deal is with that car? Who the hell you are, and who's the fella you're looking for?"
AJ reached into the front pocket of his jacket and came out with cigarettes, shook one out, and offered the pack to Jerry, who accepted. They lit up and smoked for a minute, neither saying a word. A group of five came into the bar and settled onto stools beside Jerry, talking loud and laughing, yelling drink orders at Carl.
"You work for that girl?" AJ said. "She really run the place?"
Jerry scowled. He had enough headaches over working for Nora without some stranger walking into a bar and pointing it out.
"She doesn't run shit," he said. "I worked for her daddy for, hell, a number of years. He had himself a stroke, and for some reason the girl decided not to sell the place. Got this idea of keeping it going till Bud comes back. But you want to know who _runs_ that place, you're looking at him."
AJ sucked at his cigarette and nodded, like this was just what he'd expected. "She doesn't seem like the car-fixing type."
"She ain't."
"Problem is, she also doesn't seem like the question-answering type. Friend of mine stopped by today, had a few inquiries to make about that Lexus you mention. The girl, she wasn't too cooperative. Put on a bit of an attitude."
"That's Nora, all right," Jerry said. He finished his beer, and before he could wave for another, AJ did.
"I got this one."
Jerry didn't thank him, just accepted the drink and consumed a few swallows of it, feeling a nice light buzz beginning. Beer in his right hand, cigarette in his left, a fine start to the weekend.
"Now, you want to come in here and tell me that Nora gave you a headache, that's fine," Jerry said. "But you just said that she, how'd you put it? That she wasn't the question-answering type."
"That's right."
"Well, seems to me I just asked you a question of my own. Don't recall it getting answered."
He felt a smug smile growing as he lifted the cigarette back to his lips. This guy think he was a total idiot? Come in here and bitch and moan about Nora, get Jerry loosened up to the point that he'd just forget about his own questions?
"Fair enough," AJ said. He was using his thumb to clear a streak of condensation off his vodka glass. Not much of the vodka was gone.
"What I'm saying is, you want me to talk to you, you're damn well gonna need to talk to me first," Jerry said. "I don't know you, I don't know the son of a bitch drove that Lexus into the tree today, and I don't have an interest in either one of you. Yet."
AJ made one more swipe at the glass with his thumb, then lifted it and took a long drink before speaking, his eyes on the bar.
"Man who drove that Lexus, he's of interest to me, Mr. Dolson. Not to you. Understand?"
"What did he do, steal something? Drugs, or money?"
AJ shook his head.
"What, then? What are you talking about?"
Silence.
"Your problem," Jerry said, "is that you put that cute little box on the underside of the car instead of sticking it to the fella himself. You found the car all right, but your boy isn't with it. Tough shit, huh?"
He laughed, and AJ lifted his eyes from the bar and locked them on Jerry's, and then the laugh went away. This guy talked easy, voice soft and calm, but there was a steel edge inside him. It showed in the way he kept rubbing that glass with his thumb. Some people would do that out of boredom or nervousness. With this guy, it was different. Like with each stroke of his thumb he was tamping down embers in a place nobody else could see.
"You're an observant man, Mr. Dolson," AJ said, his voice tighter.
"Wouldn't have seen it 'cept I had to take the car apart," Jerry said, and suddenly he was wondering if he should have played this card, let the guy know he'd found the tracking device.
"What did the girl say when you told her?"
"Haven't told her."
"So you found it and . . ."
"Threw it in my locker and figured I'd think on it for a day or two."
Something loosened in AJ's face.
"You told me you don't know me, or the guy who drove the Lexus," he said. "Told me you aren't interested in us. And I say that's just right. You shouldn't be interested in us. We're about to move right out of your life. But you can make some money before that happens. I expect you understand that's an opportunity not to let pass by. Easy money, from someone who has nothing to do with you?"
"You want the car?" Jerry said. "I ain't gonna let you steal that car, man."
"I don't give a shit about the car. I want to know where its owner went. His name is Vaughn. I need to find him. Like you said, none of this has anything to do with you. No reason for you to protect him. Am I right?"
Jerry nodded.
"So you've got a decision to make, and as it stands now you've got no reason to support either option. How about I give you one? A thousand dollars cash. I'll put it in your hand the minute you tell me where he went."
There were more people in the bar now, and it felt too hot and too crowded. Jerry sipped his beer and squinted. Had that flushed, dizzy sensation like he'd get about seven or eight beers from now. Wished everyone would lower their damn voices, stop shouting and carrying on. He stared at the floor, trying to steady himself, saw that AJ wore a pair of shiny black boots, one of them tapping off the bottom rung of the bar stool. Tapping, tapping, tapping. Jerry got lost watching them.
"Not interested?" AJ said. "Okay. Then we'll go on and get out of your life. Just like we would have anyhow. Only you'll have nothing to show for it."
"He didn't tell Nora where he was going," Jerry said.
"He's not going to abandon that car. He might not show up for it, not for a while at least, but he'll check in. He doesn't want you guys to call the police, run his license plate, anything like that. You'll hear from him again. When you do, I want to know about it. In exchange for the thousand."
Jerry drank the rest of his Budweiser fast, some of the beer foaming out of his lips and dribbling down his chin, then slid the bottle away.
"How do I get in touch with you? _If_ I decide to."
AJ wrote a phone number on a bar napkin and passed it to him. Jerry glanced around, curious if anyone was watching him take this guy's number on a napkin like he wanted a date.
"All right. I'll see what I can do."
"Excellent decision," AJ said. "How do you feel about five hundred bucks up front?"
"Feel fine about that."
"Give me the device you took off that car, and I'll give you the five hundred. Gesture of good faith, on both our parts."
"Can't do that."
"Why not?"
"Shop's closed, and, uh, I don't have keys anymore."
His face burned when he said it. There were plenty of problems between him and Nora, but losing his keys, that was the most serious one. She'd come down one weekend and found him using the paint booth to put a fresh coat on Steve's boat. Sort of thing he'd do from time to time, favor for a friend. Bud had known, and hadn't cared. But Nora, she accused him of undercutting the business, of stealing paint—which was a bold-faced lie, Steve bought the paint—and disrespecting her. Demanded his keys. He'd never been as close to quitting as he was that day.
"You can't get in all weekend?" AJ said.
"Not without Nora, and it sounded like you didn't—"
"No." AJ shook his head. "I don't want her involved."
"Well, Monday, then."
AJ nodded after a long pause, resigned, and got to his feet.
"All right. You get in touch Monday, and I'll get your wallet stuffed fat, Mr. Dolson. And now I'll leave you to the rest of your evening."
"Not till you buy me another beer, you won't," Jerry said. He felt good about saying that, pleased with the tone, insistent, demanding. Like he was in control.
AJ settled his tab, left a fresh beer in front of Jerry, and walked out of the bar, his boots loud on the floor. Jerry gave it a few seconds, then got to his feet and went to the window, leaned on the jukebox with a cigarette in hand and studied the cars in the parking lot, looking for AJ. Didn't see him. How the hell had he gotten out of there so fast? Then his eyes rose from the cars and found him across the street.
It made Jerry frown. The guy wasn't from town, he was certain of that, so he didn't arrive in Tomahawk on foot. He had a car, but it wasn't here now, which meant somebody had dropped him off at Kleindorfer's Tap Room and gone elsewhere. Now this guy, AJ, he was walking in the direction of the body shop. Rankled Jerry a little. What did he need at the body shop after Jerry'd agreed to help him? He considered driving down there. It held him at the window for a moment, but eventually he shook his head and went back to the bar. The shop was closed, Nora was gone, and if this yahoo had any ideas about breaking in he'd just set off the alarm and draw the cops out. It was Friday evening, and Jerry's vested interest in Stafford's Body Shop was on hold till Monday.
## 7
Nora hung the CLOSED sign on the front door as soon as Jerry left, and turned off the lights in the office with every intention of leaving early herself. The weekend stretched ahead, a chance to relax, get some much-needed Nora time. She'd spend an hour or two with her father and then be free of all responsibilities until Monday at eight. There was a pang of guilt at lumping the visit with her father into the responsibilities category, but she didn't think anyone would blame her. They were difficult visits.
She was locking the back door of the shop when she remembered Frank. _Damn it._ She'd told him six. So used to staying late that it had seemed the most appropriate time to suggest. Now, with the shop closed and a sudden yearning for a shower and a change of clothes in her mind, that extra hour was torment. She stayed at the door for a moment before turning the lock back with a sigh and stepping into the shop. There was nothing to do but wait.
It was dark inside, lit by just one emergency lamp above the door. Nora made her way through the room without bothering to turn on the lights, so familiar with the building that it was easy. She knew the placement of every tool by now, and knew their purposes. Navigated around the chain fall in the corner, frame rack beside it, paint booth behind that, toolboxes lining the walls. When she got to the office door, she took her keys out of her pocket but didn't use them. There was a stool beside the door, and rather than enter the office she just sank onto the stool, pulled her feet up onto the seat and hugged her knees to her chest, sat there smelling the paint and the dust and staring at the shadow-covered room. Instead of building a garage divided into separate bays, her grandfather had simply jammed everything into one large warehouse, a space that cooked you in the summer and chilled you in the winter. Her father had upgraded the equipment over the years but never considered a new building. Though earlier in the day she'd told Jerry she'd been learning about the work that went on in here since she was a girl, she really remembered being inside only a handful of times, usually accompanied by her mother, who stalked around the place with an expression of haughty distaste.
They'd gotten divorced when Nora was six. It had been a marriage of whim and romance: Her mother was from old money in Minneapolis, and her father was third-generation Lincoln County, Wisconsin, son of a body shop owner who also drove a plow in the winter. He'd been bartending at a supper club up near the Willow when twenty-two-year-old Kate Adams arrived for a vacation with her parents and some cousins. The family bored her; Ronald "Bud" Stafford did not. He was tall and good-looking and appealing in a way that only an outdoorsman can be, but also quick with a joke and a compliment. It was supposed to be a summer fling. Only problem was, Kate didn't realize that until Stafford had replaced Adams at the end of her name and a baby was on the way.
If there'd been good times when she was a child, Nora couldn't remember them. Couldn't remember the bad times, either, just a vague sense of tension. After the divorce Kate moved back to Minneapolis with Nora in tow. Nora's relationship with her father had been slow building at best. He would come to Minneapolis about once a year, usually around Christmas, take her down to the Mall of America and patiently wander through girls' clothing stores with her, laughing at the way she insisted on trying everything on. Her mother had only permitted a few visits to Tomahawk when Nora was young, and always came along, as if she were afraid Nora would never come back if left alone for a few days. It wasn't until high school that Nora finally began to make a weeklong trip by herself in the summer. She and her father started writing letters more frequently then, a couple of times a month, exchanging photographs—her in a prom dress, him with a thirty-six-inch northern pike—and news. From the time she was a little girl he'd promised to put her through college. Her mother had remarried by the time Nora was ten, remarried to plenty of money, but on that issue Bud was firm— _he_ would pay for college.
He and her mother just couldn't live together, that was all. Everything Kate had found so charming about Tomahawk that first summer disappeared under a blanket of snow in November, and even when the thaw came and the tourists returned the luster was gone. And for Bud Stafford, moving to Minneapolis wasn't an option. He'd been born into a pocket of the earth he considered superior to all the rest, and he'd never leave . . .
Someone was at the door. Nora put her feet back on the ground and started to stand up as the door opened. Not the front door of the office but the back door. _Frank,_ she thought as the knob turned and the door swung inward. Had to be him. Then the visitor stepped inside, and as his silhouette filled the space she saw it was too tall, too broad. Without even seeing his face she knew him. It was the man who'd come by to ask about the Lexus.
She didn't say anything, didn't take a step forward. If the lights had been on, she would have, but since they were off, and the stranger clearly hadn't noticed her standing back here in the dark, she kept silent and watched him.
He stood just inside the door and didn't move. Letting his eyes adjust to the dark, maybe. Turned the knob back and forth, then looked from it back up across the room, probably thinking the door would've been locked if the shop were empty. It was dark, though, and the sign outside said CLOSED. After another hesitation, he swung the door shut very slowly, so it hardly made a sound as it latched. Then he walked farther into the shop, toward the Lexus that sat in the middle of the room, surrounded by its own trim pieces.
She should have said something as soon as he opened the door. Called out in a loud, authoritative voice, stopped him. But she hadn't, and now he was inside and moving in a way that unnerved her. Cautious, on the balls of his feet, with attention to quiet. It was just past five on a weekday, in town, with plenty of people passing by outside, and this guy had walked into a business, that's all. Somehow it didn't feel like that, though. More like she was standing in a closet watching someone crawl through a window and into her home in the middle of the night.
_Stop it,_ she thought. _It's your business, you're in charge, and this asshole has no right to creep in here._
It wasn't much, one brief bout of internal scolding, but it was enough to get her moving. She stepped to the side and reached out and up, flicked the light switch, and said, "You want to tell me what you're doing?" in as hard a voice as she could muster.
He moved at the first sound of her voice. Whirled and came toward her, fast and aggressive, and she had the sudden thought that surprising him like that had been a bad idea. The overhead lights were long, old-fashioned fluorescent tubes, and they didn't snap on like an incandescent lamp would. There was a hint of a glow, followed by a short humming sound, and then the room filled with light. By that time the guy had closed the gap between them to about five feet, and Nora stepped back, stumbling over the stool. When she pulled up short, he did, too, but her sense of command over the situation was already gone. He'd frightened her—she knew it, and he knew it.
"I said—"
"I heard what you said." His eyes took in the room around them, seeing the emptiness, the dark office behind her. It was obvious that she was alone. She wished she'd stayed on the stool, kept the lights off, just waited and watched.
"You have no right to be in here," she said. "Can't you read the sign out front? We're—"
"Closed," he said and took another step toward her, that damn belt buckle glinting under the fluorescent lights. "Yeah, I saw the sign. You usually sit here in the dark after you close up?"
"Maybe I should start to more often, if people keep breaking into my shop. Now get out. You want to talk to me, I'll be back in on Monday."
"I didn't break in anything." He was one pace away now. "Door was unlocked."
"I want you out. I don't know who you think you are, walking in here like this, but I want you out right now. I told you before, if this car's owner wants to call me, he can. Otherwise, stay the hell away from here, unless you'd like me to call the police."
"No, I don't think I'd like that, at all," he said. "And neither would you."
The phone was in the office. All those times she'd had to rush back in to catch a call because she'd forgotten the cordless unit paled in comparison to this. Her cell phone was in the truck, where she always left it because she couldn't be bothered with personal calls during the day.
"Get out," she said again. He was in her space, almost chest to chest, and she'd backed up against the office door, which was still locked. To open it she'd have to turn her back to him, and that didn't seem like a good idea.
"You're going to listen to me, hon, and listen good," he said, and a sour chill went through her stomach, the words and tone sounding like something a drunk would say as he advanced on his wife with a belt in hand. "You got no problem here, okay? Just tell me where the guy who drove this car went, and I'm gone."
"I'll ask you one more time to leave. Then I'm calling the police."
He didn't say anything. She gave it a few beats of silence and then went for the office door. The keys were already in her hand—had been since she sat down on the stool—and she reached for the lock, standing so close to the door her nose almost brushed it when she turned. Had the key raised but not inserted into the lock when his hand closed around her wrist.
Her first reaction was to reach back with her free hand and claw at his face. A year ago, it would've made an impression, too—long, French-tipped fingernails—but you didn't work on cars with nails like that. Now her fingers slid harmlessly over his cheek. So she twisted and kicked at his knee, using her heel instead of the front of her foot. Caught him on the side of his knee, so his leg buckled, and for a moment he was off balance and she thought she'd get free. He didn't lose his hold on her wrist, though, used it instead to jerk her forward and spin her around and then she felt a wrenching pain in her shoulder and her face hit the door and she knew it was going to get very bad, very fast.
## 8
The sign on the door said CLOSED, and there were no lights on in the office. Frank was early, too. So had she forgotten, or was she planning to come back? It was only five twenty. He stood on the sidewalk in front of the body shop with two full grocery bags in his hands and wondered what the hell he should do.
She didn't seem like the type to forget. Too put together and in control for that. Things had gotten a little hectic there, with the gray-haired guy rushing everybody, and it was possible. She'd said six, though, and that was a while off, so maybe he should just wait.
He set the bags down by the front door and looked around, wondering what Nora Stafford drove. The only car parked on this side of the street was a black Dodge Charger a block away. No cars in the handful of parking spaces in front of the shop. Maybe she'd gone out on another tow. He'd check to see if the truck was still parked behind the shop. If not, he'd wait. If so . . . maybe wait a little less.
Leaving the groceries where they were, he walked around the building and into the back parking lot. There was a wire security fence around the lot to protect the towed vehicles, but the gate was open, suggesting she hadn't left for the day. He went through the gate and into the parking lot and saw the tow truck parked there, his battered Jeep behind it. Okay, she wasn't out on a tow. But the gate wasn't locked, either. So where the hell had she gone?
At first, he thought he'd imagined the cry. Short and muffled, not a scream but a mild sound of outrage, or maybe pain. He tilted his head and listened and heard nothing but silence. Took a few steps toward the back door. Still no sounds, but now he could see light on the other side of the door. Then something fell inside, a clang of metal on concrete.
He saw them as soon as he opened the door. A tall man with his back to Frank, shoving Nora Stafford against a toolbox on the far wall. He had her arm twisted behind her back and his other hand covered her mouth while he used his weight to keep her pinned against that toolbox and spoke in a low voice. Frank probably could have made out the words if he'd tried, but he was already moving, crossing the concrete floor fast and quiet, sidestepping enough to keep himself positioned behind the tall man's back, out of his line of sight.
It was maybe fifty feet from the back door to where they stood, and Frank made about forty of it before the guy heard him or sensed the motion. He twisted his head, saw Frank coming at him, and shoved Nora Stafford away. A small pile of bolts and a socket wrench hit the floor with her, bouncing off the concrete in a jingle of metal as the tall man reached under his jacket and brought a gun up.
For his thirteenth birthday, Frank Temple's father gave him a musty hardbound book with a blue cover. _Kill or Get Killed_ , the title. A close-quarters combat text. His grandfather's book, then his father's, now Frank's. _Read it,_ his father told him. _All of it._ Frank had. Two weeks later, his father challenged him to try to take a gun out of his hand. The first of many lessons.
The gun facing him now was a 9 mm automatic, and the man who held it was used to the sight of a gun having some stopping power on its own, because he kept lifting it, passing over Frank's body and aiming for his face. He wasn't planning to shoot. Frank knew that as he closed the rest of the distance between them. Put a gun in the face of most people, they'll stop moving. That was the expectation. The reality was going to be a little different.
Frank's first strike, delivered a quarter of a second before the next, was with the edge of his left hand on the wrist that held the gun. He moved his head down and to the right as he did it, and then the gun was pointing harmlessly away from him. The second strike was really two at the same time—he hit the tall man's chin with the heel of his right hand while he brought his right knee up and into the groin. It was a simple move, using the momentum he already had from his forward rush, but it was effective. He actually missed with his knee, hit on the inside of the man's thigh instead of the groin, but since the guy's head had already snapped back the blow was enough to keep him going. He hit the same toolbox that he'd pinned Nora Stafford against, and now Frank caught the man's wrist with his left hand and slammed it into the metal edge of the toolbox. The gun came free and bounced away. Frank ignored it, got his hand behind the other man's neck while he released his wrist and then slammed him forward, using his leg to upend him and spill him onto the floor.
The guy took the fall well, rolled back onto his feet and lunged upward just in time to be greeted with the socket wrench Frank had recovered from the floor. He laced it downward with an easy stroke, about fifty percent of his strength going into the blow, but it was plenty. Caught the guy right across the back of his skull and dropped him back onto the floor.
It should have been done, but Frank was caught by the tide now, unsatisfied with just how damn easy this had been, wanted to grab that gun off the floor and put it to the bastard's knee and blow a cloud of blood and bone onto the concrete. He went for the gun, saw it wasn't on the floor, and looked up to see Nora Stafford standing with the weapon in her hand. Her eyes moved from Frank to the man at his feet, and then she held the gun out.
"Here."
It was a Glock, no safety to remove, just squeeze that trigger and watch the thing kill. Frank knew the gun well. By the time it touched his palm, though, the flush of rage was gone, a cool calm sliding back into its place. He slipped the Glock into his waistband, cast one glance at the unconscious man on the floor, and then turned back to Nora Stafford.
"It would seem," he said, "that you should probably call the police."
Frank was worried about her until she came back out of the office. Was she going to fall apart, get hysterical, give him another problem to deal with before the cops showed? Then she stepped back into the room and stared at the tall son of a bitch stretched out on the concrete and he knew she was fine. The look was laden with anger and disgust, not fear.
"You're early," she told Frank.
He nodded. "Didn't want my milk to spoil."
A smile tugged at the corner of her mouth. "Wouldn't want that, no. Thanks for the help. He just walked right in here . . ."
"You don't know him?"
"No. He came in this afternoon and asked about the Lexus."
Frank tilted his head. "Car that I hit?"
"You got it."
He blew out a long sigh as a siren began to close on the body shop and looked to the side, where the partially disassembled Lexus stood.
"That guy was all wrong. Shit, I'm sorry. I should have said something earlier. Had a bad sense about him, but I was trying to ignore it. Figured it had nothing to do with me."
That was total bullshit—Frank's original sense about the guy was a personal thing indeed, but he didn't see what would be gained from explaining that to Nora.
"I had the same sense, and told myself the same thing," she said, "but I didn't count on this."
She was holding her right wrist with her left hand, rubbing it gently, and Frank saw for the first time the dark red streaks left on her skin, left by a firm and no doubt painful grasp.
"You okay?" he said.
"Fine." She dropped her arm as if embarrassed to have her pain noted.
"What did he want?" Frank gestured at the unconscious man with his toe.
"To know where your buddy in the Lexus went."
"No kidding?" Frank looked at the guy on the floor. He'd arrived pretty damn fast after the car was left at Stafford's Collision and Custom. And if he didn't know where Dave O'Connor had gone, then how had he found the Lexus?
Frank slid the Glock out of his waistband and looked at it. Good gun, not uncommon, but the sort of thing preferred by people who knew what they were doing. The guy he'd taken it from hadn't been that bad, either. Just hadn't expected Frank to be any good, that was the difference. The way he'd shoved Nora past him and cleared the gun in one swift, easy motion . . . he'd been around.
"He told me the guy's name was Vaughn," Nora said.
"What?"
"Dave O'Connor, right? That's what he told us his name was. This guy, he said the person driving the Lexus was named Vaughn."
"You see a driver's license, any sort of ID?"
She shook her head, and he saw a spark of irritation in her eyes. Maybe at him for asking, maybe at herself for not getting it.
"Anything in the car?" Frank asked, but the sirens were in the parking lot outside, and Nora walked away from him, toward the door. The guy on the floor was starting to come back, rolling his right foot a little, eyes still closed, left side of his face pressed to the cold stone.
The cop came in with Nora, and Frank was surprised to see it was just one guy. About forty, ruddy faced, thick fingers. He was speaking into the microphone near his collarbone as he entered, reporting his position and situation, casting a scowl at the sight of the body on the floor. When he was done talking into his radio, he withdrew a plastic bag from his hip pocket and reached out to Frank.
"Gimme the gun." His badge said _MOWERY._
Frank dropped the gun in the bag, and Mowery sealed the plastic lock and jammed the gun, bag and all, into his belt. He nodded at the man at his feet.
"His gun."
"That's right."
"You took it from him."
"Uh-huh."
"After he pulled it."
"Yeah."
Mowery studied Frank as if he weren't sure he believed it. "What'd you hit him with?"
"Hands, at first. Then a wrench."
"That seemed like a wise idea to you? Swinging on a man with a gun?"
"It worked."
"Hmm." Mowery squatted beside the tall man, whose eyes had fluttered open, leaving him staring blearily across the floor. "Looks like he's 'bout ready to rejoin the world. Best that he do that with his hands cuffed, don't you think?"
"Nobody else coming?" Frank said.
Mowery gave him a sour look. "We got a lot of county and few cars to cover it right now, son. You really think I need to bring all of them off the roads, help me deal with this? Seems to me it isn't that difficult a situation."
_Should've been here five minutes ago,_ Frank thought. _Like to see_ you _come across that room when he showed the gun._
Mowery got the cuffs off his belt and fastened the man's hands behind his back. The prisoner was fully conscious by the time the second cuff snapped shut, twisting his head to try to look back at Mowery. The movement didn't work so well; he made a soft grunt that seemed driven more by nausea than pain and laid his cheek back on the concrete.
"I hit him pretty well," Frank said. "Might have a concussion. Maybe need an ambulance."
"He isn't gonna die in my car before he gets to a hospital." Mowery leaned over and flicked the man's cheek. "You with us, asshole? Want to walk out to the car with me, get that headache checked?"
The guy grunted again, and Mowery wrapped one hand over the handcuffs and the other in the guy's shirt, then hauled him upright with a jerk.
"You can stand," he said, as the man's legs started to buckle. "Stand _up_ , damn it!"
_Excellent procedure,_ Frank thought. _Way to be concerned with the potential medical condition. Should be filming this for a police academy._
"All right," Mowery said when his prisoner held his own footing. "Let me get him in the car, get him down to the hospital. Don't want the son of a bitch dying on us, do we? I finish with him, three of us are gonna talk."
The tall man's movements seemed steady enough heading across the room to the door, shuffling along without comment, casting one long, hard stare at Nora as he passed her. She gazed right back at him and flicked her middle finger up. Mowery, walking behind his prisoner, reached out and grabbed a handful of the guy's short hair and twisted his head away from Nora.
"You don't look at the lady, shithead. You don't even _look_."
They stepped out the door. Frank and Nora walked that far and then stopped, standing just inside as Mowery guided the tall man toward the police cruiser parked about twenty feet away, a Lincoln County Sheriff logo on the front door. Mowery opened the back door of the car, put his hand on the back of his prisoner's head, and started to shove him into the seat. He was facing the inside of the car, and when a man rose from behind the trunk, on the opposite side, Mowery never saw him. Had no idea trouble was at hand until Nora shouted, and Frank went through the door and started toward them as the new man, wearing a camouflage jacket and black boots, hit Mowery in the side of the head with a handgun. Mowery fell into his prisoner, the two of them tumbling into the backseat in a crush of bodies, and then the gun swung down again and Mowery's nose shattered and blood sprayed the inside of the window.
Frank had taken a few steps toward them when the new man whirled and lifted his gun, and just as he'd been so certain before that there _wouldn't_ be any shots if he kept moving, this time he knew there would. He lifted his hands and backpedaled, and for a moment he was sure the crazy bastard was going to fire anyhow. Then Mowery, sliding down out of the car to the ground, reached out and got his fingers in his attacker's shirt, and that was enough to draw another whip of the gun. It was two seconds of distraction, but it got Frank back inside.
He grabbed Nora around the waist and pulled her into the body shop and swung the door shut behind them with his free hand. Nora's feet tangled with his, and she started to fall. He let her go, turned away as she hit the floor hard on her ass, reached for the dead-bolt lock and turned it. He banged his hand over the light switch and dropped to the floor, and then it was just the two of them inside the dark room and Mowery outside with his prisoner and a man with a gun.
## 9
They'd been closed for the day. That was the first thought Nora had, lying on the cold concrete floor with paint chips under her palms and dust in her mouth. She'd locked the door and hung the CLOSED sign, ready to drive home and take a shower. Should be curled up on the couch now with a pillow under her head and a warm sunset filling the living room. Instead she was here with a wounded cop and two gunmen outside and an oddly capable stranger crouched beside her.
"He might've have killed him," she said, pushing upright. "Do you think he could have—"
"Get the phone," Frank said. "Call 911."
He disappeared then, slithering off into the darkness almost noiselessly, toward the row of toolboxes on the far wall. His motion was enough to propel her own, and she started for the office on her hands and knees, went about ten feet before she felt foolish and stood up. If they were going to start shooting through the walls, they'd have done it.
The thought had hardly left her mind when the gunshots started. Four in succession, muffled by the walls of the building but somehow seeming the loudest sounds she'd ever heard. She was back on the floor before the final shot was fired, pressed down into the dust and grime. In her mind, holes opened in the walls and bullets tore through and sought her in the darkness and found her in an explosion of black pain. But the shots had been directed somewhere else; there was no sound against or inside the building. The cop, then. Mowery.
"They killed him," she said, and Frank's answer was immediate.
"Tires."
"What?"
"They shot the tires on the police car."
She rolled over and chanced a look back at the door, expecting to see him there, surveying the scene. There were only shadows, and she finally found him across the room, a long ratchet in his hand.
"How do you know?"
"You could hear them pop."
Could hear them _pop?_ She'd heard nothing but the shots, was _still_ hearing the shots, rattling around in her ears as though the bullets remained active, floating out there somewhere, looking for a destination, for _her._
Frank crossed the room, the ratchet dangling in his right hand, but his walk was unconcerned. He reached for the dead bolt, and she hissed at him in shock.
_"What are you doing?"_
"They're gone," he said and opened the door. Nora braced for more gunshots, but none came. Frank stood in the doorway for a second, and from the floor she could see past him to the police car, which now rested on its rims, the tires reduced to cloaks of flabby rubber. The back door of the car stood open, and Mowery's body was slumped behind it, only his legs visible to Nora.
"Make that call," Frank said, and then he stepped outside.
She'd left the phone on the stool by the office door when Mowery arrived, and when she reached for it she saw the ugly red marks on her wrist. The pain in her arm and shoulder seemed to pulse faster now. When the 911 operator answered, Nora's explanation came out in a voice she'd never heard—too fast, too high, on the edge of hysteria. She brought it down with an effort, explained what had happened to Mowery, and then disconnected despite the operator's attempt to keep her on the line. She went to Frank, walking to the open back door, one she passed through countless times each day, now looming like the most treacherous of gateways.
Frank was kneeling beside Mowery, and there was blood on his jeans. He'd stretched Mowery out on the gravel, and the cop made neither motion nor sound. Frank turned to her.
"Ambulance on the way?"
"And the police." She took a single step outside, pulling against the strings of a fearful desire to cling to the safety of the building. The parking lot was empty except for Mowery's car.
"They're gone?" she asked.
"Yeah. Probably not far, though. No cars in the parking lot except this one, and the only one I saw on the street was empty, so that's not where the second guy was waiting."
"They came in a Dodge Charger the first time."
He looked up. "New model? Kind of sporty-looking thing?"
"Yes."
"Well, that's what was parked out front, but it was empty when I got here. So I don't know why the first guy went after you alone. Where was his friend? Why'd he wait on the cop before he decided to help him out? Doesn't make sense."
He said all of this while working on Mowery, checking his pulse and loosening his shirt collar.
"Is he okay?" Nora asked.
"He's not going to die, but he's not going to feel or look right for a while, either."
She rocked up on her toes to look past Frank's shoulder at the cop, and when she saw him her eyes seemed to swim out of focus, everything a blur of red. She sucked a breath in through her teeth and forced herself to look again. His nose was almost unrecognizable, turned into a bloody smear across the right side of his face, and shredded lips revealed broken teeth.
Frank pulled his own shirt off and used it to wipe gently at Mowery's face. Then he sat back on his heels with a frown, studying the unconscious cop before leaning forward to move him again. He tilted him off his back and onto his side, tucked the shirt under his head, and worked on the angle of his neck until Mowery's face was pointed slightly down, toward the pavement.
"Shouldn't you leave him on his back?" Nora said.
"I don't know how well he can breathe. There can't be much air going through his nose, and if he's on his back all that blood goes into his throat. I want it to drip away from his throat."
Nora looked away again and took the door frame in her hand, squeezed it tight.
"I almost missed that phone call," she said, and if Frank heard her he didn't respond. He wouldn't know what she was talking about anyhow. Wouldn't know that the Lexus had been one ring from being bound for someone else's body shop, someone else's life.
Report a routine assault, and it takes a while before the cops finish sorting it out. Report an assault _on_ a cop, and watch that time frame expand.
Frank told the story six times to three different cops—everybody seemed to want to hear him run through it twice—after Mowery had been taken to the hospital. He was semiconscious when the ambulance got there, but in no state to explain the attack to his police brethren. That put it back on Frank and Nora, who had an intensely interested audience. Seemed to Frank that it must have been a long time since someone bloodied up a cop in Tomahawk.
They started at the body shop, walking two of the cops through it step by step, then went to the police station to explain it to a third, this time on tape. By the time they were done, the sun was gone and the small town was quiet, moving on toward nine in the evening.
One of the officers dropped them both off at the body shop. The groceries were still sitting on the sidewalk out front. Probably not a good idea to try that milk, Frank thought. Man, what a day. Twenty past five, you're worried about keeping your milk cold. Five thirty, you're worried about staying alive.
"If I hadn't promised you a ride," Nora Stafford said, staring at his groceries, "I wouldn't have been here when that asshole showed up. I would've been home already."
"Sorry."
She shook her head. "No, I'm just thinking. If I hadn't promised you the ride, I wouldn't have been here, right? But if I hadn't promised you the ride, _you_ wouldn't have been here, either. And if you didn't show up . . ."
Neither of them said anything for a minute after that. Nora shook her head, snapping away from all those possibilities.
"My point is, you _still_ need a ride, don't you? And I'd say it's the absolute least I can do."
She managed a smile at that, and Frank felt better. She'd handled the first round well enough, better than most would have. It was the second, that guy rising up out of nowhere and taking Mowery down, that had shaken her.
They walked back into the rear lot—you could see Mowery's bloodstains on the gravel, but Nora kept her eyes high—and out to a little Chevy pickup with the Stafford Collision and Custom logo emblazoned on the side. Frank opened his Jeep and got to work transferring his belongings into the bed of the truck. Nora helped silently. When everything had been moved, Frank paused to get a fresh shirt out of a suitcase, the blood-soaked one having departed with Mowery. Then he was in the passenger seat and Nora was behind the wheel and they were northbound, headed out to the Willow twelve hours after he'd expected to arrive.
"Temple the Third," Nora said as they pulled away from the last stoplight in town.
"What?"
"I heard you give your name to the cops. Frank Temple the Third. Sounds fancy."
He looked out the window. "Not really."
"If you have a son, would you feel obligated to name him Frank Temple the Fourth?"
"No," Frank said. "I certainly would not."
He wished she hadn't overheard him with the cops. He'd gone through the internal bracing that he always did when he gave his name, watching the cop's eyes and waiting for recognition. There wasn't any, though. It had been a few years since his father made headlines.
"You up here by yourself?" she asked.
"Yeah."
"From?"
"All over. Chicago, originally. I've moved around."
"But you've been here before."
He turned back from the window. "You say that like you're sure about it."
She flicked her eyes to the rearview mirror as she accelerated onto the highway.
"You call it the Willow. Not Willow Flowage, not the flowage, but the Willow. First-timers don't say that."
"Interesting. If I want to impersonate a real tourist later, I'll keep that in mind."
"But I don't see any fishing tackle in your stuff, which makes you a real mystery. Everybody that goes to the Willow in May is going to fish. I think you got here about a week early, though. Season hasn't started yet."
"I may do some fishing. The gear's already up at the cabin."
"Really? You own the place? Nice."
"It's my father's."
"Is he joining you? A little father-son bonding?"
"He's dead," Frank said, and she winced.
"I'm sorry."
"That would make you one of the few." Then, to fill the awkward pause, he said, "What'll you do with that car? The Lexus."
"I'm not going to fix it, that's for sure. Minute I hear from him, he'll hear from the cops."
"They ran the VIN and the plate, right? Did they tell you who owned it?"
He was thinking of that powerful and total conviction he'd had when he saw the Florida plate. Devin Matteson's car. He'd been _sure_ of it in that moment. Sure of it and reaching for his gun.
"If they already know, they didn't tell me," Nora said. "I bet it's stolen, though. As crazy as all this got, I'm almost positive they won't be able to find out who that guy was from the car."
"Maybe."
She shot him a glance. "You disagree?"
"Not necessarily. I'm just thinking about how fast his buddies showed up. Guy wrecks his car out in the woods, nobody else around except me, and then people immediately are looking for him at your shop. They knew the car was there, but they didn't know where he was, or even what name he was using. How?"
"That's a fancy car. Has the navigation system, the satellite link. Maybe they used that somehow? Called Lexus and reported it stolen or something, got the satellite to position it."
"Could be." Frank was thinking about other methods, though. Things like tracking devices, which, when mixed with men who carried Glocks and had no problem attacking strange women, did not present an appealing scenario.
"All I know is I want that damn car out of my body shop," Nora said.
"Aren't the cops going to impound it?"
"Yes, but I need to get it put back together first. Can't tow a car that's in a dozen pieces, you know? I'll call Jerry in the morning, ask him to come in and put the parts back on so I can get it out of my sight. He'll demand time-and-a-half, I'm sure, but I don't care. I want it gone."
There was vehemence in her voice that Frank hadn't heard before. It was as if she blamed the car.
"Where do you live?" he asked, looking for a more relaxed topic of conversation.
"Almost up to Minocqua. You're not far out of the way for me at all."
"You always lived here?"
"Nope. I've been here for about a year."
It was a disclosure that presented all sorts of questions—where was she from originally, what in the world had brought her to a body shop in Tomahawk—but Frank didn't ask them. She was quiet for a bit, as if waiting for the inquisition. When no questions came, she offered another of her own.
"When was the last time you were up here?"
"Seven years ago."
"That's a long time. How do you know the place is still standing?"
"Guy named Ezra Ballard checks in on it, keeps it in shape."
"Well, no wonder you're so relaxed about it. Nobody in the world's more reliable than Ezra."
"You know him?"
"Everyone does. He's one of a kind. Supposed to be the best guide in the area, too. At least that's what I've been told."
Frank nodded without comment. A hunter without peer, that was Ezra's reputation. The stories Frank knew were probably far from those Nora Stafford had heard, though. A different sort of prey.
They were on Willow Dam Road now, the Chevy's headlights painting the pines with pale light, and at Frank's instruction Nora turned left, toward the dam. This was maybe a quarter mile from where the wreck had taken place. He'd been that close to his destination. Once they were across the dam and past the Willow's End Lodge, he instructed her to take a right turn onto a gravel road leading to the lake.
"You're lucky to have a place right on the water," she said. "That's tough to find at the Willow, with all the development restrictions."
"Yeah." Frank's interest in conversation had vanished when she made the turn, taking all the moisture in his mouth with it. This felt stranger than he'd expected, and he'd expected it to be damn strange.
They rumbled past the three-way fork that divided the gravel road into separate drives, and Frank told Nora to stay left. Then they were past it and facing the cabin.
"Home sweet home?" Nora said.
"Yeah. This is it."
He sat there silently until he felt her curious eyes on him, and then he shook his head and opened the door and stepped out into a cool breeze that came at him like a kiss. In front of him the dark lawn ran out to a hand-laid log wall that stood above the beach. Since it was still spring, the water would be high enough to bang against the logs when the wind blew. By midsummer, it would be down, fed out of the dam regularly to replenish the Wisconsin River and its valley. Stars and a half moon hung above the lake, everything pristine until Frank turned his head a touch to the right and saw the blinking red lights of a cellular tower miles away. He remembered when the cell tower went up. His father hated the tower. Loathed it. One night, sitting with the Coleman lantern crackling beside them, he'd taken out a gun and emptied a clip in the direction of the tower, the bullets dropping harmlessly into the water. They'd had a hell of a laugh over that.
"Beautiful," Nora said softly, and only then was Frank aware of her standing beside him.
"Yeah," he said. "It's all right."
He turned back to the truck, and she went with him, grabbed one of his bags out of the bed, and started for the cabin.
"Just set it down outside the door," he said. "Thanks. I'll get the rest."
"I'll help you get everything inside. It's not a problem."
" _No._ Thank you, but no. Just set it outside the door and I'll take care of it."
She stood with the bag in her hand and cocked her head, puzzled. Then she raised her eyebrows and made a slow nod— _whatever you say, psycho_ —and dropped the bag to the ground in front of the door. Frank felt a surge of irritation and embarrassment at his snapped words, but he couldn't help wanting her gone. He didn't want anybody walking into that cabin with him when he stepped inside for the first time in seven years.
"Okay," she said. "Well, then, I guess I'll take off."
"Thanks for the ride." He pulled a few more bags out of the truck. "Really, this was a huge help. I didn't want to spend the night in a hotel."
"Hey, least I could do."
They stood there awkwardly for a few seconds, facing each other in the dark. Then she moved toward the truck, and he lifted two bags to his shoulders.
"I'll give you a call soon, let you know what sort of time frame to expect on your car," she said, pulling the driver's door open. "Should get it done a lot faster now that the cops are taking the other one."
"Thanks. Let me know what you hear from the police, too, okay?"
"Sure."
She got in the truck then and started the engine, and Frank turned away so as not to be blinded by the headlights, reached in his pocket and closed his fingers around an old and well-remembered key chain, and went to the door.
## 10
Looking back, Grady figured at least one reason he'd grown so attached to the Temple boy didn't involve guilt. He understood something about family legacies. About becoming something you didn't want to become simply because it was what you knew. What you'd seen, what you'd been taught, what ran through your veins.
Grady lived alone now, in an apartment that was about the same size of the kitchen in the house he'd shared with Adrian, and though it still felt relatively new and certainly nothing like home, it had been nine years since he moved in. _Nine years_.
His father had been a good-natured drunk who never lifted a hand to his son, not once in all those twelve-beer nights. Instead, he'd come through the door unsteady and mumbling, walk into Grady's room, and apologize. Sometimes they were short speeches; sometimes they went on for an hour or more. Tearful, choked-voice monologues in which the old man would take blame for all the wrongs of the world, acknowledge they were all his fault. He was sorry for being a bad father, sorry for being a bad husband, sorry they didn't have more money, sorry they never took a vacation, sorry Grady was an only child, sorry their landlord wouldn't allow pets because every boy should have a dog.
There were nights when Grady would lie there and wish his dad would just come in swinging, the way drunks were supposed to. _Hit me, damn it,_ he'd think, _slap me around, do anything but this crying and apologizing, you pussy._
He never hit him, though. Just kept right on apologizing until the day he had a heart attack on the corner of Addison and Clark, walking into Wrigley for a baseball game. Grady, who'd been home from college and waiting at their seats inside the park, was sure his father would have apologized for that, too, if only he could.
He'd made up his mind, though, that he would not become his father. No chance in this world. He'd make some mistakes, sure, but he would not let remorse over them haunt his days, not spend his life apologizing for faults that he never attempted to correct. He'd be assertive, he'd be strong, and any character flaws acquaintances might whisper about during parties would be borne forth from those qualities. _Too cocky,_ they'd say, _too stubborn, too sure of himself. Never admits when he's wrong._
He'd been wrong with Frank Temple. Hadn't admitted it. Made his mistake, moved on. Except for those computer checks. Except for those. One of the reasons he kept monitoring the kid was that Grady knew a few things about legacies. But his had only been, well, pathetic. Not dangerous, not in the way Frank Temple's could be. The kid wanted to beat it, wanted to leave that bloody coat of arms behind, but it wasn't going to be an easy task. And Grady surely hadn't helped him. If anything, he'd given him a firm push in the wrong direction. What he'd done in his time with seventeen-year-old Frank Temple III was his greatest professional and personal shame. With the exception of Jim Saul, an agent down in Miami, it was also a private shame. Nobody else knew the way Grady had manipulated that kid. Frank surely did not, and that, more than anything else, was what kept Grady checking the computers, always monitoring the young man he hadn't seen in years and wondering what it meant that he'd show more devotion out of guilt than he ever had out of love.
The case against Frank's father had been a huge story—nothing attracted attention like the story of a federal agent turned contract killer—and when it broke the accolades and praise were rolling in and the media was loving the Bureau, loving Grady. What they didn't understand was that when Frank's father killed himself he had effectively aborted the future of the investigation. He'd known so much, could have provided information that would have taken Manuel DeCaster down, destroyed one of the deadliest and most powerful crime entities in Florida, hell, in the country. It had been shaping up to be one of the most significant organized crime prosecutions in years, and then Frank Temple II lifted his gun to his lips, squeezed the trigger, and killed the case along with himself.
So even as the story was arriving it was dying, and while the media didn't understand that at first, Grady and Jim Saul sure as hell did. All they had left was Frank Temple III. The boy was supposedly closer to his father than anyone else had been, and the stories of his unusual education, the molding process that had been going on, were legion. He'd even made a trip down to Miami with his father, and there had been at least a short visit with Devin Matteson.
It was for Devin that Jim Saul most hungered. Devin was a phantom, involved in every level of DeCaster's operation, investigated by the DEA and FBI and Miami PD for years without a single conviction. Temple was supposed to be the first domino, Matteson the second, but Temple had managed to go down without touching any of the others. They could start the chain over with Matteson, Saul was sure. And there was a chance, maybe even a strong one, that Temple's son knew far more than they dared imagine. It would take a little bit of a sales pitch, that was all. A few talks about betrayed legacies, a few reminders of just how much Devin deserved his share of the punishment, what a shame, no, what a _crime_ it would be to see Frank's father alone bear that load.
He'd walked into that kid's house knowing the truth, but with a promise—a professional oath—not to share it. Nothing evil in that, right? Except he'd shared another story and passed it off as the truth, a story that filled a grieving child with white-hot hate and a vendetta.
Grady had spent some time on it. He and the boy had a good many conversations about those things before Frank's mother grew concerned and a newspaper reporter learned of the unusual bond and began to ask for interviews and the whole thing fell apart, leaving Frank with his hate and Grady and Saul with nothing to show for the ploy.
It had been worth the effort, though. That's what they'd told each other early on, that if it had paid off and the kid actually knew something and shared it, well, then it absolutely would have been worth it. You had to prioritize, after all. Without the boy, they had no case, and they needed a case.
Except they already had one. While Grady was down in the basement of that house in Kenilworth, showing Frank pictures of his father with Devin Matteson and talking of loyalty and betrayal, trying to build enough hatred to coax a reaction, a group of rookie agents in Miami were working the streets and chasing bank records, and a few years and two ugly trials later DeCaster was in prison. No help needed from Frank Temple III, no lies to a grieving son, a _child,_ required.
It was the sort of thing that was hard to put out of your mind.
Grady kept his eye on the kid, though, and found a measure of relief in each year that passed without incident. Frank was making his own place in the world, and it looked like a peaceful one.
Had looked that way, at least, until the day after his arrest for public intoxication down in Indiana, when Jim Saul called Grady at home on a Friday night and asked if he'd heard about Devin Matteson.
Grady took his feet down off the ottoman and set his beer aside and leaned forward, his grip tight on the phone.
"Heard what about him, Jimmy?"
"He's in the hospital down in Miami, with three gunshot wounds. Looked like he was going to die when they brought him in, but he's been making a furious recovery ever since. You know the kind of shape that prick was in. Ironman, right? He's conscious again, and it's almost a sure bet he'll make it."
"They have the shooter?"
"Nope. And if Matteson knows, he's not saying. But somebody plugged him three in the back, and you know how he will want to handle that."
"Personally," Grady said, and he felt cold. "What do you hear on suspects?"
"Could be anybody. If they've got good leads, I'm not aware of them."
"Temple's son was arrested in Indiana night before last. Public intoxication. When was Matteson shot?"
"The day before that," Saul said slowly. "And how do you know the Temple kid was arrested for a PI?"
"Word travels," Grady said.
"Right," Saul said. "Well, I thought you'd like to hear about it. And if I hear something new, you'll be the first to know."
They hung up, and Grady dropped the phone onto the cushion beside him and stared at the wall.
Devin Matteson shot in the back, Frank Temple III arrested for drinking in Indiana a day later. Celebration, maybe? A few champagne toasts to the dead?
No. No, that couldn't be it. The kid was doing fine, and Matteson had any number of enemies. The list probably grew by the day.
Frank had wanted him, though. Frank had wanted Matteson _badly,_ and by the end, when Grady was trying to make amends, he'd urged the boy to put that away. Told him that he'd _have_ to ignore it if he wanted to stay away from his father's sort of end. Frank had accepted it, too, at least verbally, but Grady remembered going back to the range with him a few weeks after the lies had started, remembered the look on Frank's face and the perfect cluster of bullets in the target. He'd known damn well the kid was seeing Devin Matteson down there.
_And whose fault was that, Grady? Whose fault?_
He picked up the beer again, drank what was left, and stood up to go after another one.
"I should have asked about the wounds," he said aloud, talking to his empty apartment. That would have settled it. Because if there'd been more than an inch or two between those bullet holes, then Frank Temple III hadn't been pulling the trigger.
Ezra Ballard ran an electric fillet knife down the perch's side in a smooth, quick stroke. Turned the fish over and repeated the motion. Moved the filets to the side and then lobbed the fish head over the fence and into the dog kennel. Two of his hounds hit the fish carcass together. There was a soft growl, the sound of snapping teeth, and then the winner retreated with his prize.
Last summer, an architect from Madison had given Ezra a nice lecture after watching him feed the leftover fish to his dogs. Fish in that condition wasn't suitable for dogs. Could do serious harm. Ezra had tried to stay polite, listening to him. Finally Ezra asked him if he had any experience with bear hounds. No, not with bear hounds, the guy said. Plenty of experience with dogs, though. What kind of dogs, Ezra asked. Pugs, the guy said. Took all Ezra had to smile and nod, wait till the guy wrote out the check and went on his way. Pugs.
Ezra had selected all four of his hounds when they were just weeks-old puppies, watched them in their litters and picked up on traits of personality that set them apart. Trained them himself, spent long summer hours in the woods and brush with them, teaching them to work as a team. Though the hunting season wasn't till October, you could run bear in the summers in Wisconsin for dog-training purposes. On days when he didn't have to guide, he generally loaded the hounds into their crates and into the back of the truck and set off to take advantage of the free time in the way he loved best: out in the woods, alone except for the dogs. Of course, it wasn't like being alone at all. The dogs were Ezra's family. More than pets, more than friends. And when the air turned chill as fall began to lose its early skirmishes with winter, and the dogs bayed long and loud in the dark woods, Ezra with gun in hand as they chased their prey? Then, the dogs were something altogether nearer to his heart: comrades.
Boone, a six-year-old bluetick, was the pack's alpha male even though he wasn't the largest. Bridger (they were all named after famous woodsmen—Boone, Carson, Bridger, Crockett) was bigger in size, taller and fifteen pounds heavier, but he lacked the aggressive edge that dogs respected in a leader. He was a diplomat, Ezra had decided, whereas Boone tended toward the preemptive strike. Ezra felt closest to Boone, but he spoiled Bridger and tried to put out the idea that he was the favorite.
He cleaned a final fish, tossed its remains into the kennel, and then gathered up the filets and his knife, turned off the floodlight above the cleaning station, and went into the house. He cooked the fish and ate it with potatoes and carrots that he'd seasoned and wrapped in foil and cooked outside on the propane grill, ate at the kitchen table, facing the mounted head of a ten-point buck he'd taken five years earlier. Everything from the décor of his room to his clothing to his daily activity told him what he was, reminded him of it, pushed the essence of his life into him from the outside. He was a fishing and hunting guide, a woodsman, a local. His clients knew it, his friends knew it, his neighbors knew it. After nearly forty years, he was starting to know it, too. Mission accomplished.
You became what you wanted to become. That's what Ezra believed. You could become it if you tried hard enough, could take what you really were and change it, force-feed yourself a new life until it became your old life, too, blurred together until a better self emerged.
He'd spent twenty years in Detroit and another four in the jungle trying to decide what he'd be if he could choose. Nothing stopping him, he'd move back in time, open up the west with Frémont and Carson and the others who were there, see this country in all the beauty it had once held. Reality did stop him on that one, and so he chose the next best thing, a life spent on the water and in the woods and far away from the urban world of greed and hustles and constant violence that he'd known growing up. He'd been twenty-five when he arrived here, a young man with an old warrior's body count behind him, had no idea where to find a walleye, no idea how to track a deer or run a bear. He learned those things, and now he taught those things, and there were moments when it seemed that the perception of others—that idea that he'd always been here—was true.
He finished his meal and washed his dishes and gathered his car keys and went out to the truck. Took Cedar Falls Road to the logging road, went bouncing over the uneven track. Anybody else would spend hours, maybe even a full day, trying to locate that car from the land. Ezra was different, though. A tree that looked identical to the rest stood out as a landmark to him, each bay and inlet and island as familiar as the houses of neighbors in a suburb. He knew the gray-haired man had taken the logging road, and he knew which fork of it he'd followed.
The road went on a good half mile past the point where Ezra brought his truck to a halt, but he didn't want to drive all the way down to the water, have his headlights visible from the island. He took the rest of it on foot, the wet earth sucking at his boots. Here the soil was almost boglike, holding moisture long after the last rain had passed. The lake was surrounded by more than sixteen thousand acres of forest that were protected by the state, home to bear and deer and three wolf packs. Home to Ezra.
There was a boat ramp farther south, but Ezra knew people who used this logging road as a put-in area for canoes, saved some paddling time if they were headed north. You'd put your canoe in the water and take off across the lake, splitting either north or south around an overgrown island with a few _NO TRESPASSING_ signs posted. The only privately owned island in the entire flowage, out of more than a hundred possibilities. It should never have been privately owned, either. Dan Matteson's grandfather had won it in a bizarre legal case.
Matteson's grandfather, a Rhinelander native, had owned forty acres of good timberland several miles east of the Willow and adjacent to hundreds of acres owned by one of the state's major paper mills. When the mill accidentally clear-cut his property, he sued. The case had gone to arbitration, and the arbitrator had decided to award Matteson property of comparable value instead of cash. Back then most of the land around the flowage was owned by the paper mills rather than the state, and the arbitrator had issued Matteson a small tract on a point of land on the eastern shore and one of the only islands in the whole lake that was high enough to avoid regular flooding. The total land came to just under five acres, a fraction of what he'd lost, but the arbitrator argued that it was waterfront property and therefore worth more. Matteson had accepted, and now, sixty years later, there remained one privately owned island on the flowage.
Dan had grown up around here, and on long days and longer nights in Vietnam, he'd talked of the place. To Ezra, who'd never been more than forty miles from Detroit until he shipped out, the flowage had sounded like a dream world. Miles of towering dark forests, pristine lakes, islands. The island that Dan owned held appeal that Ezra couldn't even put into words, but the longer they stayed overseas the more attached he grew to the idea of the place. He couldn't go back to Detroit. Not if he hoped to avoid the sort of existence he'd left behind.
Just before he'd enlisted, Ezra had gone out with his older brother, Ken, to settle up a debt. The sum owed was four hundred dollars. Ezra had held the arms of an alcoholic factory worker while his brother swung on the guy with a bottle. When the bottle fractured, Ken had hit him one more time in the face, a driving uppercut, and the jagged glass bit into the unconscious man's chin and continued upward, peeling a strip of pink flesh off the bone from jaw to eye socket. They'd left him in the alley after emptying nine dollars from his pockets. The next day, Ezra went to talk to a recruiter.
As his tour wound to a close, and the prospect of returning home became more real, Ezra made an official request to Dan: Could he head up to this place, this Willow Flowage, for just a few months, until he figured something else out?
_You're shit-brained,_ Dan had said. _It's going to be winter, man. Three feet of snow on the ground, you want a cabin with no electricity?_
_Snow doesn't sound so bad right now,_ Ezra answered.
Dan had agreed to it. He headed south for Miami while Ezra went north, Frank Temple taking his job with the marshals and landing in St. Louis at the time, right in the middle.
Miami ruined Dan. The Willow saved Ezra. Absolutely _saved_ Ezra. It was hard living but clean living, where you invested your strength and sweat into clearing snow and starting fires, not breaking legs and wielding guns. And there were certain moments, when the evening sun cast a pale red stain across silent snow or when an early spring wind blew up out of the lake with a surprising touch of warmth, that made you want to drop to your knees and thank whatever God you believed in—or maybe one you hadn't believed in—for putting you in that place at that time.
Ezra had been on the island five months when he learned his brother's body had been found in the trunk of a Caprice off Lafayette in Detroit. He skipped the funeral. That summer, Dan and Frank came up for a visit, and Ezra made his pitch. He and Frank should pool resources and buy the additional parcel Dan owned on the point, build a cabin there and create a camp that they could share and pass down to their families. It was the sort of grand plan you can only have when you're young and friendships seem guaranteed to last forever.
_Go on_ , Dan had laughed. _I'll sell you the land, man. But I'm not spending much time out on that damn island, middle of nowhere and nothing to do._
_Then sell it to me,_ Ezra suggested, the island already sacred ground to him.
Dan shook his head. Slow, with some of the mocking humor gone from his face.
_Nah_ , he said. _I can't sell that one. Not the island. It's in a trust, a legacy deal, to keep the state from taking it. The island goes way back in my family, you know that. I've got a son, and it'll be his someday. I want it to be his._
So he'd kept his island but rarely appeared there, and Ezra and Frank built a cabin on the smaller parcel around the point and shared some summers and memories. Now, with a few decades of separation, Ezra could look back on it and see that it had been the bellwether, Dan's life moving in a different direction, to a place hidden from Ezra and Frank. The real shame was that it hadn't stayed that way for Frank.
Ezra had lived in the lake cabin for a time, but as soon as he could afford to he bought more land a few miles up the road and built his own house. Eventually Frank Temple bought the lake property in full, put it in a legacy trust for _his_ son. Now it had been years since anyone spent a night in either the lake cabin or the one on the island. So much for the legacies.
As he reached the top of the hill he left the road and moved toward the waterline, reentered the trees near where he imagined the car to be, and found it easily. Driven right up to the last tree, all those boughs mashed against it, bleeding sap onto the roof. He ducked beneath the branches, his jeans soaking in moisture when his knee touched the grass, and then came out at the back of the car. Reached into his shirt pocket and withdrew his lighter, flicked the wheel, and held the flame close to the bumper, so he could read the license plate.
It was local. Wisconsin and Lincoln County. That was a surprise. He memorized the numbers and then took his thumb off the lighter and let the flame go out. He hadn't expected a local vehicle. The only people he believed should have access to the island cabin were some thousand miles away. The Lexus had carried a Florida plate, as expected, but now it was gone, and this old heap with a local plate had taken its place. Why?
He left the car and returned the way he'd come through the silent woods. When he reached his truck he decided to go the Willow Wood Lodge instead of home, have a drink and do some thinking before calling it a night. No tourists, this time of year. There were six cars in the parking lot when he arrived, laughter carrying outside. He walked in and found an empty stool at the far end of the bar, had hardly settled onto it before a glass of Wild Turkey and an ice water were placed in front of him. Carolyn, the bartender, didn't need to wait on an order.
"Glad you came in," she said. "Been meaning to give you a call."
"Yeah?"
"Dwight Simonton came in about an hour ago. You know Dwight."
"Sure. He's a good man."
"He said somebody's down at the Temple place. Said there was a fire going outside, somebody sitting there."
"Right idea, wrong owner. Somebody showed up at the island cabin."
Carolyn shook her head. "Dwight said it was the Temple place."
Ezra frowned. "I don't think so. I was just out there today, had a look at it from the water. Nobody's staying there. Been so long since anyone visited either one of those cabins, Dwight probably was confused. Heard something about the island cabin, got it mixed up."
Now Carolyn leaned back and raised her eyebrows. "Come on. Not a soul who lives on this lake doesn't know the Temple place, after the way that crazy guy went out. Dwight told me the fire was right down on the point. You think Dwight can't tell a fire on the island from one on the shore two miles away?"
She was right; Dwight Simonton wouldn't have made that mistake. He and his wife, Fran, had owned a place up here for more than a decade and were the closest things to neighbors the Temple cabin had. If Dwight said it was the Temple cabin, then it was the Temple cabin.
"You don't think," she said, lowering her voice and leaning closer, "it's his kid?"
Of course it was his kid, responding to the message Ezra himself had left, but rather than confirm it, Ezra simply shrugged.
"That'd be something," Carolyn said.
Yeah. That'd be something, all right. Ezra finished his bourbon without a word, tossed some money on the bar, and got to his feet.
"You going down there?" Carolyn asked, her face alight with curiosity.
"Figure I ought to."
She was ready with another question, but Ezra turned away and went to the door, stepped out into a night that now seemed electric. First there'd been the beautiful woman and her gray-haired companion in the Lexus. Then the Lexus was gone and the man hid a new car in the trees. Now someone, probably Frank's son, was back at the Temple cabin. Ezra didn't like the feel of it, the way this group was gathering on his lake. He was responsible for them, he knew. A generation later, maybe, but he'd brought them here all the same.
## 11
The letter was right where it belonged, framed on the wall beside the corresponding Silver Star. Frank read it while he drank his first beer, read from the date right down to President Harry S. Truman's signature.
_In grateful memory of Major Frank Temple, who died in the service of his country in the military operations of Korea, on August 22, 1950. He stands in the unbroken line of patriots who have dared to die, that freedom might live, and grow, and increase its blessings. Freedom lives, and through it he lives, in a way that humbles the undertakings of most men._
The letter had hung above his father's childhood bed, the only tie Frank Temple II ever had to the soldier who'd died in Korea, leaving a wife six months pregnant with the son who would bear his name. Frank Temple II grew up without knowing a father but knowing plenty about his legacy—his name was a hero's name. During D-Day, on beaches filled with heroic acts, the first Frank Temple and his comrades still stood out. Using grappling hooks and ropes, his Army Ranger battalion scaled the cliffs at Point du Hoc, stone towers looming a hundred feet over the sea and protected by German soldiers with clear lines of fire. Into the teeth of that rain of bullets climbed Temple and his fellow Rangers. Casualties were heavy, but the mission was accomplished.
A tough act to follow, but Frank Temple II had done it for forty-five years. He had his war, Vietnam, where he served as a member of a specialized group so covert and so celebrated that it was still the subject of speculation decades later. MACV-SOG they'd been called: the Special Operations Group, elite soldiers whose chain of command seemed to end with the CIA instead of the Department of Defense. Temple II had matched his father's Silver Star and Purple Heart, then come home to a career as a U.S. marshal, fathering a son who—of course—bore his name, his father's name.
"You've got a lot to live up to." That was his mantra for Frank, a thought shared with the same casual frequency most people used for "Good morning," a constant reminder that Frank's was a line of brave men and heroic deeds.
The hell of it was, Frank had always believed him. Believed _in_ him, which was even worse. All the hero bullshit, the talk of honor and courage, it seemed to come from his father's core. It was sacred. Right up until his father killed himself and a team of FBI agents arrived at the house, three months before Frank's high school graduation, he'd believed in his father.
Now, sitting beside a fire with a lukewarm beer in hand, he wondered how long that would have continued. If his father had never been caught, if those FBI agents had never showed up at the door, would they sit here together, sharing a laugh and a beer, Frank steadfast in his faith in the man across the fire from him? Or would he have grown wiser with age, smelled the lie in his father's words, seen evil in eyes that had always looked on him with love?
_He would've been proud today,_ Frank thought. _The way I brought the socket wrench down, the sound it made on the back of that guy's skull, yeah, that's Daddy's boy right there._
He laughed at that, the sort of laugh you can allow yourself when you're drinking and alone. Laughed for longer than he should have, then lifted his beer to the cabin, a toast to his return. This was their place, a spot of memories shared only with his father, no interlopers here.
He wanted to spill some tears, weep for his father. It had been four years since he'd last been able to do that. Driving through the Kentucky foothills in the middle of the night, listening to a radio station from some town he'd never heard of when the Pink Floyd song "Wish You Were Here" came on, began chewing at the edges of his brain, then danced right through the center of it when one softly sung line—"Did they get you to trade your heroes for ghosts?"—wafted out of the speakers.
There'd be no tears tonight, though, and maybe those he'd lost on a lonely highway in Kentucky would be the last. If this place, with all its good memories, didn't affect him in that way, then no place ever would.
He wouldn't cry for his father here at the Willow, but he might kill for him. If Devin was really coming back . . . damn, but that would feel good. Frank could do it, too. Bet your ass he could do it. Years of lessons didn't disappear that quickly, not when they were taught by somebody as good as his father.
There'd been a day, sometime in the summer when he was fourteen, that his dad first broached the subject of justified killing. Really laid it out there. They'd been downstairs in the mat room, working out, Frank attacking and his father defending, blocking most of his attempts easily, but every now and then Frank would sneak a blow in. When he did, his father would smile. Glow, almost.
They'd finished and were sitting together with their backs against the cold concrete wall, breathing hard, and his father had said, _There's a lot of bullshit to what I do, son. And to what I did. With the marshals now and the Army before._
Frank thought he meant bullshit as in boring work, red tape and bureaucracy. That wasn't it, though. As the sweat dried on Frank's neck and back and his heart rate wound down to a slow, steady thump, his father had explained what he meant.
_We chase down guys who are evil bastards, Frank. I mean_ evil, _you understand? Guys who steal and kill and rape and commit any othermanner of crime, anything you can think of. Some of them go to prison. A lot of them don't. They get off on some technicality, get some lawyer pulling tricks, whatever. But they go right back out on the street and hurt somebody else. I'm not saying the system doesn't work sometimes . . . I'm saying it doesn't all the time. There are guys the system can't touch who aren't worth the air they're breathing. And there's a way to settle it. A natural way._
A natural way. That's what his father thought of killing. That it was the most natural thing in the world, an inherent solution to human conflict, ageless and unsurpassed.
Frank hadn't said anything for a while, until it became obvious his father wanted some sort of response. Then he'd asked what all of that had to do with the Army.
_It's the same thing. There's this system in place, right, governments and generals and all the rest, and they're supposed to keep the peace, and everybody wants them to do it without firing a shot. But you know what? They can't. Because there are evil people in the world, son, and they're going to keep doing evil things. And that keeps people like me in demand. People like me, and your grandfather, and you. Somebody who knows how to use a gun and knows when to use it._
That was the first time Frank had been officially included in the list, and it made his head go a little light, the honor of that shared company hitting him deep in his fourteen-year-old boy's heart.
A few years later, his father's body in the ground and face on the front page of the newspaper, the sad truth of moments like that one began to show itself to Frank. He understood what his father had been doing, understood that he'd been rationalizing with himself as much as he'd been offering a philosophy to Frank. But he believed what he said, too, and Frank saw the horror in that, saw the fallacy and savageness and the justification. Yes, the justification. It was still there. Smaller, maybe, weakened, maybe, but not obliterated. It couldn't be. Because his father, evil man or not, was dead, and Devin Matteson—evil man for sure—was alive and free. Cut a deal, hung Frank's father out to dry, and then walked away from it. No punishment, no penance, no pain. He deserved some of all of that. Damn sure deserved some pain.
There'd been another conversation down in the basement that stood out in Frank's memory, and again the true significance hadn't hit for a few years. They'd been down there working on elbow strikes—vertical, horizontal, front, rear, up, down, Frank's dad always demanding greater speed, greater power—while his mother played Tom Petty music loud upstairs, trying to drown them out, unhappy with the violent lessons her son was taking to so well.
That day had been, Frank would later learn, exactly one week after his father came back from Florida having killed two men to avenge Dan Matteson's death. One week living with the reality of it, maybe a couple of weeks of dealing with the decision itself. He'd paused to sip a beer—it was the first time Frank could remember his dad bringing anything but bottled water downstairs with him—and he'd studied his son with a critical eye.
_Frank,_ he'd said, _suppose somebody takes me out one of these days._
It had still seemed like a game right then, and Frank had answered, _That can't be done, nobody out there good enough_ , in a flip, teasing voice, thinking they were just working up to some of the chest-thumping bravado the old man liked to get into during a hard session. His eyes were different, though, darker and more intense.
_It can be done, Frank. Probably will be done, someday._
Frank didn't answer.
_Suppose it happens,_ his father had said, _and suppose you know who's responsible. What would you do?_
Still no answer.
_Frank? What would you do?_
_Kill him,_ Frank said, hating how weak his voice sounded, like a little kid. _I'd find him and I'd kill him_.
Pleasure in his father's eyes. Respect. He'd nodded, finished his beer, and said, _Damn right you would. Damn right._ Then he'd laid a hand on Frank's shoulder and said, _You're a good boy, Frank. Check that_ — _you're a good_ man.
A few years later, Frank had been able to flash back on that conversation and once again see what had been working beneath the surface, see the rationalization, the justification, but there'd been something else there, too: a promise.
_I'd find him and I'd kill him_.
Frank Temple II had killed himself. No scores to settle. None.
_I'd find him and I'd kill him._
Frank had endured a lot of pity over the years, some genuine, some false. Sometimes it would be expressed directly to him; other times it just showed in their eyes. _Poor kid. Imagine having such a monster for a father._ The problem, though, the one that Frank saw and nobody else ever could, was that he'd been a good father. Was a murderer, sure, got paid for it, yes, but while that might be enough to define him for the rest of the world, it didn't work for Frank. Didn't replace seventeen years of love. He was a good father. Frank wished he hadn't been, at times. Wished that he'd come home drugged out and violent, knocked Frank and his mother around, threatened the neighbors, that he'd done all of those things that a murderer should do in his own home—but he hadn't. He'd been quick with a joke and a kind word, supportive, interested. When Frank was eleven years old and struck out with the bases loaded to end his Little League team's season, his father had held him in the car as he'd cried in shame and said, "Don't worry, kid, next year we'll cork your bat," and the tears had turned to laughter.
Even those lessons in the basement—which the TV people had fixated on and manipulated to make his father even more of a monster, this man who would ruin a child with violence—they'd been the products of love. His father had seen a different world than most, a world of constant violence. He was preparing his son to go into it, that was all. Saw no other way to raise him than to make him ready for the worst.
"Welcome back."
The voice came from just over his shoulder. Frank's only thought as he whirled to face the speaker was that the man had approached in total silence. It was that realization almost more than the voice that allowed him to place his visitor.
"Uncle Ezra?"
A child's nickname, but it was the first thing that entered his mind. The man stepped closer, out of the darkness, and offered his hand.
"Good to see you, Frank."
Frank got to his feet and accepted the handshake. He was taller than Ezra by several inches, and though he had been since he was in his teens, it still surprised him. The man was bigger in his memory, and quiet and capable, with a habit of sliding out observations that would be the envy of any late-night comedian, delivered in the same slow, soft voice, the jokes usually coming and going before anyone realized what had been said and got to laughing.
"You given up on motorized travel?" Frank said, waving a hand at the dark woods from which Ezra had emerged. Hell of a way to make an appearance.
"Nice night for a walk."
Anyone else would have started with the questions then: When had he arrived, why didn't he call to say he was coming, how long would he be staying? Ezra offered none of them, though, just settled onto a stump beside Frank and said, "Cabin was in good shape." A statement of fact, but one he wanted Frank to acknowledge.
"Of course," Frank said, and he sat, too.
"You intending to let this fire go out?"
It was close to going out, though Frank hadn't noticed that as he'd sat alone with his beer and his memories.
"Uh, no. I just—"
Ezra knelt beside the fire pit and adjusted the wood, fed a few fresh logs into the pile. The flames licked at the fuel and grew, the glow lighting Ezra until he stepped back, satisfied, and returned to his stump. Frank was staring into the fire, but Ezra sat sideways, so his eyes were never directly on the flames. Frank had asked his father about that once years ago. _He doesn't face the fire because he wants to keep his night vision,_ his dad had said. _It's an old habit, buddy. One that lingers._
"Boat's in the shed," Ezra said, "but I took the motor off and put it in the cabin."
"I saw that."
"Figured the shed might make an easier target if somebody wanted to break in. But I come around enough that most people know better."
"Yeah. I appreciate that."
"Hell," Ezra said, poking at the fire with his boot. It got quiet after that, just the fire popping and hissing and the trees creaking in a steady wind. There'd been loons when Frank was a kid, lots of them, but tonight he had yet to hear one of those haunting calls. He'd been up in the summer on every trip but one. That year, they came in the dead of winter for a weekend of ice fishing. Frank had been prepared for a long, cold tramp over the ice to a small hole you sat beside on an overturned bucket or a stool. Instead, Ezra had driven them out onto the lake in a half-ton pickup truck, driven right across the frozen water without any hesitation. Frank, sitting between the two men, the gearshift banging against his knees, had been sure that the ice would break somewhere out in the middle of the lake, swallow them up, Frank finishing his run as a twelve-year-old blue corpse. The ice had held, though, and Ezra's fishing shanty was small but warm. They'd pulled northern and bluegill out of the ice, and his father and Ezra had told stories while sipping bourbon-laced coffee.
"I got your message," Frank said. A fast tremor was working in his chest, just the thought of Devin out there on the island enough to build the anger.
"That seems to have been a mistake."
It was silent, and then Frank said, "What kind of mistake?"
"He's not here," Ezra said.
"Devin."
"Anybody else you'd be asking about? Yes, Devin. He's not up here, Frank."
"But somebody is?"
"Yes."
"Who?"
Ezra hesitated, then shook his head. "I don't know. It's a man and a woman and they're both strangers to me. Might be Devin's renting the place."
Frank felt that tremor fade away, something—was it disappointment?—taking the anger's place. It was crazy to be disappointed, though. Crazy. Because if he'd _wanted_ Devin up here, then what, exactly, had he been hoping for? There was an answer to that one, and he didn't want to dwell on it. Couldn't let it into his mind for even a minute. Grady had told him that many times.
"It's good to see you," Frank said, and though he'd spoken mostly to fill the silence and take his mind away from Devin, the words were true.
"You ain't kidding, son. Been a long time."
"Going to be tough," Frank said. "Being up here."
Ezra didn't look at him. "I would imagine so."
Frank, who just moments earlier had been so grateful that Ezra _wasn't_ talking about his father, suddenly wanted to. What the hell did you say, though?
"Good memories, up here," he offered. "Less so in other places. But up here, mostly good."
"He wasn't a bad man, son. Wasn't a perfect one, either, but he damn sure wasn't the way they made him out to be."
"Tell that to the families of the people he killed," Frank said, and he was surprised by the weariness in his own voice, the aged sound.
He finally heard a loon then. It cut loose from somewhere across the lake, the sound unlike anything else, riding the wind across the water to their campfire. He thought maybe they were both grateful for it. Something to listen to, something to stop a conversation that was going nowhere good.
"Like I said, I'm glad to see you, Ezra. Don't want to make you have conversations like that. I'm sorry."
"No need to be," Ezra said. "And I think you do want to. Be surprised if you didn't, at least."
Frank didn't respond to that. He was stuck in a memory, another night around another fire with his father and Ezra. He'd been fifteen at the time, and his father decided to show off some of the tricks he'd worked so hard to teach his son, show off those unholy fast hands. _Watch this,_ he'd said to Ezra, _watch how damn quick he is_. They'd gone through the usual routine, his father with the gun and Frank trying to take it, or maybe the other way around. He didn't remember the details of that night's game anymore, just remembered that when Ezra had said, _Yeah, you're real fast, kid,_ his voice was sad and he wouldn't look at either of them.
"I thought it was a bad idea, calling you," Ezra said. "I'd promised to do it, but I still thought it was a bad idea."
"It wasn't."
"Then what are you doing here?"
"What do you mean?"
"Why did you come up, Frank?"
He didn't answer. Ezra looked at him for a long time and then nodded as if the question had been answered.
"We agreed to let it go," he said. "A lot of years ago, we agreed to let it go."
They had agreed to some other things, too. Like the fact that the Willow was sacred ground, and that Devin—who'd betrayed two generations of loyalty and friendship that were anchored in this spot—should never be allowed to return to it. They wouldn't pursue him, would let him sit down there in Florida for as many years as he could last, but they also wouldn't tolerate him returning to this place. Not unchallenged, at least.
"That's really what you want to do?" Frank said. "Let him come up here and sit in the cabin, have a nice little vacation, enjoy himself? He brought my dad into it, Ezra, used a lot of bullshit about loyalty to set the hook, and then he turned around and gave him up to buy himself immunity."
"You think I've forgotten? I'm just wondering about your intentions."
"I'd like to ask him some questions," Frank said.
"That's all?"
"That's all," Frank said, but he was thinking of the guns inside the cabin, beautiful, well-engineered pieces of equipment that had not been built to ask questions.
"Where'd you come from, anyhow?" Ezra asked, and Frank returned to the moment. "Postmarks on those letters bounced around a bit over the years."
"I was in Indiana."
"Working?"
"Taking classes."
"What sort?"
"Writing. Had ideas about a book, but maybe a screenplay would be better."
"I think that's just fine," Ezra said, and he seemed legitimately pleased. "You were a storyteller even as a kid. I remember that."
"I just remember listening to the stories."
"Well, sure, back then we had a good many more to tell than you. But I remember you had a way with it. Tell a story about a bike wreck and make it sound more exciting than anything I had to offer about a battle or a bear hunt."
Frank laughed. Storytelling had been a big part of those trips, and eliciting any positive response from Ezra, a smile or a nod or one of those low, soft laughs, had been a serious reward.
"How long of a drive is it?" Ezra said. "Up from Indiana?"
"Took about ten hours."
"Good trip, I take it?"
"Those hours I mentioned, that's driving time. I left out a few hours of fun that started because I had a car wreck with someone I thought was Devin. He wasn't Devin, but based on the guys who came after him, he's not exactly peaceful, either."
Ezra turned almost fully toward the fire and lifted his eyebrows in a way Frank had seen a thousand times before, usually in response to something his father remembered that contradicted Ezra's memory.
"You want to provide a bit more detail on that?" Ezra said.
Frank provided the detail. Ezra listened quietly, shaking his head from time to time or making a quiet murmur of appreciation, but not speaking.
"Hell of a welcome back to town," he said.
"No kidding."
"I know Nora Stafford. Knew her father better, of course, but she's a good girl. You sure she's all right?"
"Other than the scare. I'd be surprised if the car she loaned out is ever returned to her, though."
The half of Ezra's face lit by the firelight went hard, his jaw shifting and eyes narrowing, and then he turned away and was entirely in the darkness.
"What kind of car you say that was?"
"It was a Mitsubishi SUV. Probably twenty years old. Little box of a thing. Blue paint, lots of rust."
"Plate number six-five-three-E-four-two," Ezra said, and Frank sat forward on the stump and stared at the older man.
"I don't know if that's the number. But you do. Want to explain that?"
Ezra sat quietly for a long time, as if there were a decision to be made and he wouldn't be rushed toward it. At last he got to his feet.
"Let's you and I take a drive."
They walked back up the gravel road and then out to the Willow Wood Lodge without Ezra volunteering a word, no hint where they were headed. Got into Ezra's truck and drove north, went across the dam and took Cedar Falls Road. Then it was left onto an uneven dirt road through the trees, Ezra taking it slow. He stopped the truck in the middle of the road, turned the lights off, and cut the motor.
"Now we walk."
For ten minutes, they walked without speaking, the only sounds those of breathing and sticks breaking underfoot. As they pushed up a slope covered in pine needles, one of the loons called again, and the sound seemed less magical than before. Chilling, now. A note of warning.
They went up the hill and back into the trees, and then a dark shape showed itself. Ezra knelt and flicked a cigarette lighter to life. The glow caught a rusted blue bumper and a Wisconsin license plate. Nora Stafford's car.
"How'd you find this?"
"Saw him pull it in this afternoon while I was on the lake."
"So he dumped it here," Frank said. "Left it in the woods and had somebody come get him."
Ezra shook his head, then extinguished the lighter. When he spoke again, his voice floated out of the blackness.
"Got in a boat and went out to the island."
Frank's eyes had been stunned by the brief light, and he blinked hard and searched for Ezra's face in the darkness.
"Devin's island."
"Yes."
"You told me he wasn't—"
"He's not here, Frank. I don't know who these people are, but Devin is not here."
_Good,_ Frank thought, turning back to look out across the dark water toward an island he couldn't see. _Because if he were, I'd take that lighter from your hand and swim out to that cabin, set the place on fire and watch it burn and make sure he went with it. I'd watch it burn and savor every minute, Ezra. Make my father look like a preacher._
"But they're connected to him," he said. "That explains the guns."
"I expect so."
"So where's Devin?"
No response.
"I knew it was him," Frank said, speaking to himself as much as to Ezra. "Saw that damn Florida license plate, and between that and the message you'd left, I _knew_ it was him, that he'd come back. I wasn't wrong by much. Not by much."
It was quiet for a while, Frank's mind filled with things like ghosts and legacies and the sort of fate he had long wondered if he could avoid. The answer was here, a rusted-out car hidden among the trees.
"I think," he said softly, "that I'd better call Nora Stafford. If this guy's staying in Devin Matteson's cabin, he and everyone around him are a hell of a lot more dangerous than I thought originally."
## 12
It was long past visiting hours, but they let her in anyhow. Nora was well known by now at the Northwoods Nursing Center. The woman who staffed the front desk gave her a disapproving look but didn't ask questions or attempt to stop her, just offered a single curt nod, and then Nora turned the corner and walked to her father's room.
"Dad?" She spoke as she opened the door and stepped inside, and Bud Stafford twisted his head to see her, a smile crossing his face. It was this moment that broke her heart—that immediate smile. He was always so damn glad to see her. Other patients in the center weren't able to recognize their loved ones. With Bud it was just the opposite; he couldn't follow conversation well, couldn't process simple details, but he absolutely recognized his daughter. Somehow, on a night like this, that made it harder.
"How you doing?" She leaned over and kissed his forehead. He struggled with the covers, made it clear he was trying to sit up, and she helped him get upright before sitting in the chair beside the bed.
Nora had heard the phrase _wasting away_ a million times in her life, never stopped to give it any real thought until her father's stroke. That was exactly what was happening, though. He just . . . faded. The strength had gone first, then the size, leaving a frail man where a powerful one had existed.
"Hello." The single-word greeting came a full minute after she'd come into the room. It took his brain that long to catch up to the events, then search for the proper reaction to them. When you kept the conversation slow and simple, he could develop a bit of a rhythm, and the sense of truly _communicating_ with him was better. Get too much going on at once, though, or going too fast, and he became helplessly lost, often resorting to repeating the same word or phrase over and over. It reminded Nora of the ancient computer she'd used in college. You'd ask the thing to run new software and get nothing but that silly hourglass symbol, a promise that it was processing, but you knew it would never yield results.
"Hello," she said. She thought it was important to always go back and match his place in the conversation, make him feel less overwhelmed by it. "What was dinner?"
"Yes." He smiled at her again.
She waited for a few seconds and saw there would be no response tonight. Sometimes he followed the questions, the simple ones at least. More often he did not. The stroke had affected his cognitive and motor skills. On the right day, he could move around just fine, albeit a little slowly. The problem was, you never knew when the right day would be, or the wrong day. His balance could be fine for a while and then completely disappear. He'd be crossing a room under his own power and then suddenly look as if he were on the deck of a pitching ship. This was the reason a return home was impossible, at least right now. He needed twenty-four-hour care, and they couldn't afford that.
"Good day?" she said, emphasizing the question. The more you did that, the more likely he was to understand that he was expected to provide an answer.
"Good day. We had the birds."
That meant they'd taken him outside, to a patio surrounded with bird feeders. That was a highlight of his existence now.
"Do you have cars?" he said. This joined the smile as the two constants of every visit. Sometimes he'd be unusually adept at following a conversation; other days he struggled with the simplest exchanges. The one question he _always_ managed was: Do you have cars? He didn't remember that he'd owned a body shop, or at least he was incapable of expressing that he did. When she tried to explain anything about work to him, he tended to get hopelessly confused. But he asked that question about the cars because somewhere in his fog-shrouded brain he knew it was important, critical, that without cars there would be serious problems.
"I have cars," she said. "We have cars."
He nodded, his face grave. Hearing that answer always reassured him. She looked down at him and felt his love even through the veil of confusion. It was a sensation she could remember so well from those visits when she was a girl, one of unusual staying power. There were few things that caught your breath more than looking at another person and feeling the intensity of his love for you. Seeing it in all of its layers, the depth of the adoration, of the pride, of the fear. Always the fear. You looked at the ones you loved and in that moment you were terrified for them, for all of the things that could go wrong in the world, the car accidents and the illnesses and the random violence that could reach out from the darkness without a note of warning and claim the ones you cared about most. It wasn't until the stroke, until the first time she saw the shell that had been left behind where her father belonged, that Nora truly understood just how unbreakable was that link between love and fear. They belonged together.
There was a notepad on the table beside her, filled with scrawled attempts at his name. That meant it had been a therapist day. Three times a week, an occupational therapist named Jennifer came to work with him. She'd made remarkable progress, too—he tied his shoes slowly but competently now, and a few months ago, when he was still in acute care at the hospital, Nora would never have believed that would be possible. The fine motor skills were more difficult. Anything requiring dexterity was a challenge.
"You want to try your name for me, Dad?" She passed him the notepad and the pen, which he took carefully, his face set in a frown of concentration. The expression remained as he carefully laid pen to paper. The first three letters of his first name—Ronald—came easily enough. Then he hung up on the _A._ She watched him hesitate, write the letter, hesitate, write it again. And again. _R . . . o . . . n . . . a . . . a . . . a . . . a_
She stopped him after the fourth repetition. "You're stuck, Dad. You already wrote that one. Try the _L._ "
He stopped writing to listen to her, head cocked slightly, then went back to the paper and wrote the _A_ again. _Perseveration,_ that's what the therapist called it. When the patient would get stuck on a word or an action. It was a frequent problem with her father. For months, he'd been unable to switch from brushing his teeth to combing his hair. Something about the action with the toothbrush dominated his brain; he'd take the comb in his hand and stare at it in bewilderment, mime the motion he'd used with the toothbrush, never get the comb anywhere near his hair. He was over that now, at least with the comb. Jennifer had solved that problem by changing the order of his bathroom procedures, putting the toothbrush last.
"Let me help." Nora leaned across the bed and took her father's rough hand in her own, guided him through his name. It was a regular part of her visits, but for some reason on this night it cut through her with a sort of fresh agony she hadn't felt since the early days with him in the hospital. He was her father, a strong man who was supposed to care for _her_. Today, a day on which she'd been attacked, when she needed his support the most, she was helping him write his own name.
The realization brought a stinging to her eyes and a thickness to her throat, and for a moment she just sat there, leaning on the bed and holding his hand and fighting tears.
"Done?" he said.
That brought her out of it. She sniffed and got a laugh out and shook her head.
"No, Dad. Not done. Let's try again."
They went back to the writing, with her guiding his hand and naming each letter as they wrote it.
Driving home in the dark, hours after she'd expected to be there, her thoughts turned to Frank Temple. Excuse me, Frank Temple _the Third_. There were stories behind Mr. Temple, she was sure. He was a little too calm in the situation they'd encountered today, a little too . . . _familiar._ If he were older, she'd suspect he was a cop, or maybe a soldier. Had the right haircut for a soldier. But he couldn't be any older than she was, and if forced to guess she'd actually say he was a few years younger. So where did that odd poise come from?
He'd been attractive at first, charming and funny in a low-key way, but then there was that strange outburst at the cabin. He'd practically shouted at her when she went for the door, made it seem as if he couldn't get rid of her fast enough. What was he so worried about? Afraid she'd throw him onto the bed, force herself on him in some show of gratitude? Please. Nora had tired of the regular routine of turning down dates from her customers—some of them offered sweetly if a little awkwardly, others in lecherous fashion—and maybe, _maybe,_ she'd flirted with Frank just a bit earlier in the day. By the time they'd reached that cabin on the Willow, though, all she wanted to do was get his gear out of her truck, visit her father, and get home and into bed.
Home. That was how she thought of it now, although it was still a decidedly foreign and amusingly masculine place. At first she'd hesitated to make any changes, feeling like an intruder in her father's house, wanting him to return from the hospital and find everything as he'd left it.
As the weeks turned to months, though, she'd become more of a realist. When he came home, he'd still need her there for at least a while, so it was fair that she begin to think of it as her home, too. The hideous old curtains went first; then she repainted the kitchen and deposited the bizarre "jackalope" creature—a rabbit head with deer antlers, some friend's idea of high humor—into the basement. A day later, she felt guilty about the damn thing and brought it back up, hung it on the wall where it had been. Gradually the place began to take on a quality that was more comfortable. She was working on a mural in the back bedroom, a tropical scene she hoped he'd appreciate. If he didn't, she'd hand him a roller and a can of that flat white paint that covered the entire house and let him do his worst.
Much as she loved him, he would not have been a good day-today father. Nora realized that now, but she'd believed otherwise as a child. In those days, struggling to adjust to a stepfather whose attempts at warmth seemed all too false, she'd been able to build Bud Stafford into a fantasy figure. It wasn't difficult; in their rare times together Bud was attentive and thoughtful and funny, a strong man layered in self-confidence.
Nora began to see her mother as weak, money-hungry, someone who'd sacrificed passion for comfort. Only a fraction of that was true—the trade of passion for comfort. At this point in her life, Nora was certain her mother's only passionate relationship had been with Bud. She was equally certain, though, that they could never have lasted together. Bud had taken Kate's natural adjustment struggles to Tomahawk as a sign of weakness, deriding her instead of aiding her, using her privileged upbringing as a constant tool for teasing, because behind the teasing he could hide his insecurities. A family was certainly part of Bud Stafford's vision for himself, but it was a family built on his own terms, and Kate hadn't agreed to those. Had they loved each other? To this day, despite all the exchanged jabs, Nora believed that they had. Maybe still did. But they couldn't live together.
The problem was that after the divorce Bud had decided he couldn't live with anyone. It was a fine way to be when you were young and strong and always in control. The years caught up with you, though, devoured the youth and the strength and in the end even the control. Bud had no say over his existence now, and that, perhaps more than anything, held Nora in Tomahawk. Did she want to spend her life here, running a body shop and conducting empty visits at a nursing home? No. Nor did she want to fail, either, to lock the doors and shutter the windows and leave her father with a kiss on the cheek and a town full of people who admired him but couldn't care for him.
So what _did_ she want? How would it end? It was a question she'd been determined to ignore at first, firmly believing he'd return to the shop healthy and ready to work. Each passing month added weight to the reality, though, and she knew now that he'd never be back. Meanwhile the calls from Minneapolis and Madison had crested and faded, family and friends who'd been anxious to know when she'd return now giving up hope or losing interest. Back home life was plowing ahead, passing her by, and here she was in Tomahawk, lost in a routine of body shop business and nursing home visits.
She couldn't let the business go under, though. Couldn't close those doors and hang the FOR SALE sign and let two generations of sweat and blisters and bruises disappear as if it had never meant a thing.
The one thing she still hadn't gotten used to about her daily routine was the dark ride home. Still couldn't relax driving anywhere out here at night. When the sun went down, the familiar ceased to be familiar, all landmarks hidden by shadows, everything beyond the reach of the headlights an unknown. They illuminated nothing but trees and pavement. She'd flick on the brights and then be dismayed when she saw how little it helped. You could see maybe an extra ten feet ahead, three to the sides, but for what? Nothing out there but more trees and more pavement, and the brights only made the surrounding shadows grow longer and seem darker. Many times when she left 51 she'd make it all the way home without passing another car, and that was a seven-mile stretch. Back in Minneapolis, you couldn't go seven feet without passing another car. The first few weeks up here she'd actually come close to panic attacks during the drive home, everything looking so damn similar that she could have been on the wrong road, headed in the wrong direction, completely unaware.
Empty and alone and dark. That was what she'd thought of the place at first, and though she'd grown fond of many things about it as time passed, the night drive was not one of them. That was a time that hammered the old mantra—empty and alone and dark—back into her brain, left her longing for bright lights and loud music and the voices of strangers.
There was a light on at the house when she pulled in, and usually that was enough of a reassurance, but today the uneasiness followed her out of the truck and stayed with her until she was inside. Nothing shocking about that—it hadn't been the most carefree of days. Well, it was done now, would be nothing but a memory by morning, fast on its way to becoming a story she would actually enjoy telling at parties, soaking up the way people's eyes widened and their jaws hung slack when she described the gunshots echoing outside of her father's shop.
Yes, soon that's all it would be. A memory and a story.
She'd eaten no dinner, but the effort of preparing food seemed too much and her appetite too little, so instead she settled for pouring a glass of red wine and moving into the living room. _Say this much for Dad's furniture,_ she thought, _it looks like something you'd want to hide even at a garage sale, but it's comfortable._
She sank into one of his overstuffed couches and kicked her shoes off, unbuttoned the denim work shirt and slipped out of it, down to the sleeveless white shirt she wore underneath. Feet up on the coffee table and wine in hand, she exhaled slowly and lifted her glass to the jackalope.
"Rough day. How about you?"
It took one glass of wine and thirty minutes of bad TV before she gave up and decided to call it a night. She was exhausted, and tomorrow wouldn't be a typical Saturday—she needed to get up early and track Jerry down, coerce him into getting the Lexus back into one piece. Once that was settled, she could turn it over to the police and, hopefully, have the whole miserable mess done.
Halfway to the bedroom, she remembered that the police might have used the shop number instead of the house if they'd learned anything or had any more questions. It was probably too late for that, but she was curious, and it was worth checking. She dialed the shop number and waited until she got to voice mail, then punched the pound key and entered the password. One message waiting, the robotic voice informed her. The police, surely.
It wasn't the police.
_Hello, I'm calling for Nora Stafford. This is Frank Temple. Listen . . . if you want that Mitsubishi back, I know where it is. But we're going to need to talk some things over first. I think I might . . . there's a chance I might know a little bit about that guy. Vaughn. I'm not sure of anything yet, but I've got some things that I should probably explain to you before anybody tries to deal with this guy. I'd like to talk to you, and then, you know, probably to the police. We'll see._
He left his cell phone number, which she already had, and hung up. Nora stood in the dark living room with the phone to her ear for a minute, then punched the button that replayed the message. Listening to it again, she felt a twist of fear begin to counteract the sleepiness created by fatigue and wine.
Frank Temple knew where her car was? And, apparently, something about the man who'd taken it? Where, in the time since she'd dropped him off, had he stumbled across _that_ sort of information, alone in his cabin on the lake?
_I'd like to talk to you, and then, you know, probably to the police. We'll see._
We'll see?
## 13
He woke to the sounds of birds, but it was anything but pleasant. Harsh, angry caws, shrieked in rage. Frank rolled onto his side and lifted his body on his elbow, squinting against the sunlight that filled the room and searching for his watch. He found it and checked the time—ten till eight.
The birds were still going at it with shrill screams. He shoved the covers back and stood up, the floorboards cool on his feet. Wearing nothing but his boxers, Frank walked through the cabin, unlocked the door, and stepped out into a cloudless morning. The sun shone bright but cool, the lake glittering beneath it, pushed by a gentler version of the previous night's wind. The sky was so bright, particularly to eyes that had just left sleep behind, that he didn't see the osprey until it completed its dive.
The bird came tearing down toward its nest, then redirected at the last second and shot skyward again, releasing another shriek. There were osprey nests all over the lake, constructed on posts out over the water. What made this one so angry?
He figured it out when the osprey made its second dive. Just as it neared the nest, another bird spread its wings and bobbed up on the thick piles of sticks, matching the osprey's scream with one of its one. This bird was larger, and unlike the osprey its head was pure white. There was a bald eagle in the osprey's nest. No wonder the other bird was pissed.
He was watching the osprey circle, no doubt plotting another dive, when an engine came into hearing range, a car approaching down the gravel drive from the main road. He turned back toward the sound as a pickup truck rumbled into view. Nora Stafford.
No real surprise. He'd expected her to call, but maybe she had; cell phone reception was sporadic out here, even with that damn tower disrupting the night sky. As he watched, she pulled in beside the cabin, shut off the truck, and stepped outside. Too bad he wasn't wearing any clothes. On the plus side, at least his underwear was clean.
She was walking toward him when the osprey dove again, announcing it with the loudest scream yet, and he turned away from Nora to watch.
The bird folded its wings, turning itself into a compact little missile, and hurled down, pulled off without hitting the nest. This time the eagle, perhaps bored with the dispute, took to the air with a shout of its own. Frank could actually hear the sound the eagle's wings made when they flapped; there was lot of wingspan there. Then both birds were airborne. As the eagle flew away from the nest, out toward open water, the osprey followed, buzzing the bigger bird like a fighter pilot before pulling away for good, the eagle headed toward the opposite shore, the nest empty again.
"I didn't think eagles fought with each other."
Until Nora said that, Frank had almost forgotten she was there. He'd been that entranced by the brief aerial battle. Now he turned back to her.
"One was an osprey. I think it took exception to the eagle using the nest."
"I guess so." Nora looked away from him again. "I'm here because of your message. It was awfully cryptic."
"I'll explain," Frank said, "but do you mind if I put on some pants first?"
"I was going to suggest that."
He went into the bedroom and pulled on jeans and a sweatshirt while she waited at the kitchen table, then stopped by the bathroom to rinse his face with cold water and brush his teeth. Not yet eight on a Saturday, and Nora was already in motion. Either she was an early riser or his message had scared her.
When he came back out of the bathroom she'd left the kitchen table and was standing in the living room, looking at the framed Silver Star and the letter. She turned to him and pointed at the medal.
"Your grandfather's?"
"Uh-huh."
"So they gave him the medal after he was dead? That's so sad. I can't imagine what your grandmother thought about that. Proud among other things, I suppose."
"I suppose." There were a couple of ways to approach this conversation, and ordinarily Frank would have favored the less-is-more variation. While Nora was due a warning, a sense of what sort of trouble had arrived with that Lexus, she didn't need to know any information about Frank or his father. But there was something in the way she was studying the medal that twisted him away from that instinct, made him want to tell her the whole damn story. _Listen to me, please, listen because I've got to tell you the way it happened, I've got to tell you the way he really was._
It shook him, this sudden desire to open up. He'd spent a long time working to avoid anything like it, perfecting that flat gaze that was designed to say, _I've got no story for you at all. Nothing to tell. Sorry._
Now it was different with Nora just because she was staring at the damn medal? Was that it? No, there was something else, something in the way she talked and held his eyes and how she'd handled herself in the heat of it all yesterday that suggested a quality of . . . what? Judgment withheld, that was it. Consideration before conclusion.
"I'm surprised you got the message so early," he said and walked out of the living room and into the kitchen, got the coffeemaker running just so he could have a task to engage in instead of standing there with her, staring up at the starting point of the legacy.
"I checked it last night." She returned to her seat at the table. Today she was dressed less like a mechanic, having traded in the heavy denim shirt for a blue tank top worn over loose white linen pants. These clothes showed much more of her body, a very nice body, and taking that in Frank understood why she probably went for the shapeless look at work.
"I'm going to be honest," she said, dropping the friendliness, her words cool. "I thought about showing up here with the police. In the end, I decided I'd give you what you asked for, the chance to talk through this, but I also didn't like what you said on that message last night. You made it sound as if you knew more than you told me yesterday."
"I might know more than I told you yesterday," Frank said, pouring the water into the coffeemaker, "but I didn't at the time. Everything you—and the police—heard from me was accurate. I'd never seen that guy before, Nora. That's the truth."
"You said you knew where my car was."
"That's right."
"How can you possibly—"
"Ezra Ballard found it."
"Where?"
"Hidden in the woods about two miles up the shoreline."
She pulled her head back. "On the Willow?"
"Yes." He finished setting the coffeemaker and turned to face her, leaning against the counter. "That guy, Vaughn, he drove it down there as soon as he left your shop yesterday, tried to hide the car in the trees. Ezra was out on the lake, watching."
"You're sure it's my car?"
"I'm sure. He took me to see it last night."
"So Vaughn dumped it."
Frank shook his head.
"No?"
"No. He's staying here. Ezra saw him go out to an island. There's a cabin on it, has been for years. Apparently this guy is with a woman out there. Just the two of them."
She took that in, nodded. "Okay. Well, that's good news, isn't it? I can get my car back, and we can tell the police where to find this guy."
Frank didn't answer.
"What else do you know?" she said, watching his face carefully. "Frank? What else do you know about him?"
"About him? Nothing. I know something about the cabin he's staying in, that's all. Something about the man who owns it."
"And what's that?"
"That he's a killer."
She looked at him for a long time, while the coffee burbled on the counter and the wind picked up and buffeted the cabin.
"You mean he's a murderer? Some sort of psychotic?"
"A professional."
"A professional." She echoed the phrase as if it were in a foreign language.
"Yes." The coffeepot was full, and Frank turned and lifted it free and poured, held a cup out to her. When she shook her head, he took a drink from it himself.
"You're serious," she said. "I can see that you're serious. That the guy who drove that Lexus is some sort of assassin."
"No." He shook his head. "I don't know if he is. In fact, after seeing him, I'd have to say he is very far from that. What I'm telling you is that the man who owns that cabin is. So if this guy with the Lexus, if he's working with him or friends with him or whatever . . ."
"It's not good for me," she finished.
" _Maybe_ it's not. Like I said, Nora, I don't _know_ anything here except what I've told you. Whether this guy is someone for you to worry—"
"How do you know that about the owner, though? He just came over once to make some neighborly conversation and told you that he kills people for a living?"
He looked at her and remembered what she'd said about his grandfather's medal— _proud among other things, I suppose_ —and then he blew over the coffee cup and took another drink. "He worked with my father."
Those cool eyes she had were beginning to falter. Beginning to let some fear in. "Your father."
"Frank Temple," he said. "Same name as mine. That doesn't mean anything to you? Never heard of him?"
When she shook her head, he was surprised. Always was, when someone _hadn't_ heard of his father. In Frank's mind, everyone had heard of him, talked of him, still did. In Frank's mind, his family's shame was still dinner table conversation across the country.
"All right," he said. "I guess I'm glad about that. He made the news a while back. National news. Pretty big story."
"For?"
"For killing people for money," he said, and then he held her eyes and drank more of the coffee and neither of them spoke.
"I'm sorry," she said eventually.
"You don't need to be. I'm just telling you so that you'll understand how I know these things."
"So your father and the guy on the island, they _both_ kill people for money? This is some sort of retreat for assassins?"
He set the coffee down on the counter, kept his eyes on the floor.
"My dad served in Vietnam, part of a pretty elite group, very good soldiers. He made some friends there. Ezra Ballard was one. A man named Dan Matteson was another. They were from three different parts of the country, but after the war, they wanted to stay in touch. Stay close. Dan had property up here, and Ezra moved up and then convinced my dad to build a cabin with him. Thought they could use it as a way to stay together as the years stacked up. Dan kept the island, and my dad and Ezra bought this place."
"Is Ezra . . ."
"No." Frank shook his head. "He's not part of the mess, Nora. Don't worry about that. He's a good man."
"But your father and the other guy?"
"Dan Matteson got into trouble right after the war. Essentially he was a mercenary. Went into corrupt, beaten-down countries and took a lot of money to fight for one side or the other. He got into something in Central America, I'm not sure what, but he made some contacts out there and got involved in the drug trade. When I say drug trade, I don't mean street corner deals, either. I mean dealing in _weight,_ smuggling planes and boats, not dime bags. Became a very big deal with some very dangerous people in Miami. When I was a kid, I got to know Ezra pretty well, but Dan was never around. I never actually met him."
"Your father was working with him the whole time?"
"No. My father was a U.S. marshal. From every account I've heard, he was a good one for most of his career. An honest one."
Frank lifted his eyes again, found hers. "When I was a sophomore in high school, Dan Matteson's body washed up on the beach in Miami. They identified him through dental records, because he was missing his hands. His hands, and his eyes."
It had to be so strange for her, so unnerving, to hear him explain this fun little family history. She took it well enough, listening silently and watching his face.
"Matteson had a son named Devin. By then he was working in the same world as his father. I think he's about fifteen years older than me. After the body was found, Devin gave my dad a call. Made this pitch. This request for my father to help him avenge Dan's murder. Find out who did it, settle up."
"And he did it," Nora said softly. Then, when Frank nodded, "That doesn't sound so evil. I mean, the people he killed, they were the ones who'd murdered his friend?"
"Some of them were," Frank said, "but he didn't stop there. By the time they sorted that out, Devin's boss made him another offer. My dad took it. And another after that. Last count I heard, he killed five people on contract. There might have been more. This while keeping his marshal's badge. I'm sure he had access and information that was awfully appreciated by Devin and the rest of them."
He paused, then said, "Eventually, the FBI got Devin into a jam, and to get out of it, he offered to trade some information. He gave them my father. Told them the information they needed to know to get the case moving, but Dad got wind of it, and he killed himself before there was an arrest."
The refrigerator kicked on beside them; for a while the only sound in the cabin was the whirring motor. Then it switched off again, and Nora spoke as if it were a cue to end the silence.
"I'm sorry, Frank. The way you reacted last night when I asked about your father, it should have told me—"
"That he was a killer?" He laughed. "No, I don't think it should have told you that. You've got nothing to apologize for. The only reason I'm telling you this is that I don't like Vaughn having a connection to Devin Matteson. It appears that he does."
When she spoke again, her voice was guarded and her eyes downcast. "And I'm supposed to believe that this connection between you and Vaughn is an honest coincidence. That you've never met him, don't know anything about him, and somehow still manage to be involved with the cabin where he ended up."
What should he tell her? That he'd come up here because Ezra's message had made him suspect Devin was returning, that he'd caused the accident with Vaughn because he thought it was Devin in the car, that he'd reached for his gun as soon as the Jeep was at a stop? Not a real reassuring sort of explanation.
In the end, all he said was "Well, you can imagine I'm not real pleased by that little twist, myself."
She was quiet, and he could tell from the set of her mouth that she didn't like the answer. Well, fine. He didn't like it, either, but that didn't change a damn thing.
"What does that mean?" she said. "You don't like the connection, and I understand your reason, but how does it change anything for me?"
"The people Devin runs with . . ." Frank swung his body off the counter and walked to the big front window looking out on the lake. She turned her head to follow him. "They're dangerous, obviously. And what they do, Nora, it's not penny ante. Whatever's going on, there's probably a lot of money involved."
She forced a laugh. "Okay, I'll let him keep the car. How about that? Just pretend I don't know where it is, don't know any of this."
"That's a good idea," he said, still with his back to her. "It's not the only problem, though. Vaughn's not the only problem. There are the two guys from last night."
She was sitting half turned in the kitchen chair, twisted to look at him, and he could see understanding begin to grow on her face.
"You mean they might come back."
"Like I said, I don't know anything about them, or Vaughn, but I know some things about Devin. One of those things is that the people surrounding him will be professionals. Professionals don't like to leave loose ends. You were face-to-face with that guy yesterday. So was I. We both saw him attack you and saw his buddy attack that cop, and our testimony could put them in jail for a long time. You and I have just become loose ends. If these guys really are involved with Devin Matteson or anyone close to him, then that's a very serious concern."
## 14
Steve Gomes had wanted to take his boat out that morning, but Jerry passed on the offer and headed for the library. Though not much of a reader, he was a regular patron. The Tomahawk library had a handful of computers with Internet access, and Jerry had recently discovered the online auction sites.
Jerry's dad had owned a liquor store, an occupational choice less than pleasing to Jerry's mother, a woman who went to church on Sundays _and_ Wednesdays. Rare was the week that passed without a bout of criticism of Rob Dolson's work, that selling of sin. There was one element of the liquor business, and one element only, that pleased her: the presence of mirrors. Alice Dolson _loved_ mirrors, and her husband received plenty of them, shipped in from Stroh's and Anheuser-Busch and the rest. Though she loathed the products they advertised, Alice couldn't help but like the mirrors. Her favorites were the Scotch mirrors. _Much more elegant than those silly beer mirrors,_ she'd insist. _If you ignore the brand name, they're really quite beautiful._
So Rob Dolson hung on to the mirrors, and when he died, Alice kept them. She'd passed a few years later, leaving Jerry a fifty-year-old collection of bar mirrors. They decorated his home and filled his garage now. He'd hoped to build the collection, but classic mirrors were hard to find—or so he'd thought until he discovered eBay. If he felt a bit fruity shopping for antiques on the Internet in the library (and he did), it was easy enough to dismiss that with the recollection that the mirrors were, of course, advertising alcohol. Nothing embarrassing about that.
He'd found a nice Genesee mirror with paintings of men engaged in various outdoor pursuits, the slogan reading _The great outdoors in a glass,_ when his phone rang. Steve Gomes had talked him into canceling the landline in favor of a cell a year earlier, telling him that the cell had free long distance and none of those bastard telemarketers calling. What Steve had failed to anticipate was that Jerry made few long distance calls, and didn't mind talking to a telemarketer in the evening, providing it was a woman with a nice voice and he'd already consumed a brew or two. He had the cell now, though, and when the damn thing rang he saw that it was Bud Stafford's home number. Nora's home number, now.
Jerry turned the ringer off and matched the glare from the front-desk librarian with one of his own. Who couldn't appreciate the Benny Hill theme song, anyhow? When he'd learned that was one of the ringer options, he'd been as good as sold.
Two minutes later, the phone rang again.
"It's a _Saturday_ ," Jerry growled. Nora had no right to bother him on a Saturday.
This time the librarian lobbed a heavy sigh with her glare, and Jerry stood up and took the phone outside onto the sidewalk. Nora deserved a lecture for this one.
"You never heard of a day off?" he said when he answered.
"Just rumors," Nora said.
"That ain't funny. I'm down here at the library and you got my phone ringing and bothering—"
"You're at the library?"
"Ain't the point, Nora. Don't matter where I am. Point is, it's Saturday, and that's a day off."
"How'd you like another day off, Jerry?"
"What do you mean?"
"You come in today, for just a few hours, and I'll let you take Monday off. You'd be trading eight hours of work for about two or three, and you're already in town. Tell me what the downside of that is."
No downside that he could perceive, other than caving in to her request. He was silent, thinking it over.
"Time-and-a-half, Jerry. That's what I'll pay you if you come in today."
"What the hell do you need me there for? We don't have anything all that urgent."
"We do now. I want that Lexus put back together, and I want it put back together fast."
"Nora, that car is not a one-day fix. Hell, we're gonna need parts that'll take a day or two just to get—"
"You're not fixing it. You're just putting it back into one piece so I can get it out of my shop."
"Guy wants to take it somewhere else?" This was not good. Jerry had a thousand bucks riding on that car.
"The police want to take it elsewhere."
_"What?"_
"I don't want to get into it on the phone, Jerry, but I need that car back together, and I need it to happen today."
Shit. If the cops were already onto this car, he might have lost his chance at the grand already. Of course, that AJ character had offered him half of that if he could get the little tracking device, and Jerry had turned him down because he didn't have access to the shop over the weekend. Well, he did now.
"All right, Nora. I can come down there. Time-and-a-half and Monday off, I'll come down there."
"The very soul of generosity."
"No problem," he told her, and then he disconnected the phone.
This was working out to be a fine weekend. He had little on his plate for the rest of the day, and now he was going to make time-and-a-half _plus_ the five hundred AJ had promised him in return for that tracking device. That Genesee mirror had just become much more affordable.
He went into the library, purchased the mirror, and logged off the Internet. Giving the librarian a mocking wink and salute, he walked back into the sunlit day and reached into his pocket and extracted the bar napkin he'd put there when he left the house that morning. AJ was probably going to appreciate this phone call just as much as Jerry had ended up appreciating Nora's.
He answered on the second ring, and Jerry told him the situation while he walked away from the library and down the hill toward the river. The Wisconsin rolled right behind the library, wide and languid at this spot, a water skier working up by the bridge.
"Deal still stands—you get the tracking box, I get the five hundred?"
"You want to take me up on that deal." AJ's voice was different today. Kind of uneasy, wary.
"I _will_ take you up on it. If you ain't interested anymore, though, shit, it's no skin off my back."
"I thought you couldn't get in the shop on weekends."
"I just _told_ you, she's paying me time-and-a-half to come in."
"And she didn't tell you why that was?"
"I assume the boy you're so interested in is coming back for the car." Jerry didn't want to repeat Nora's reference to the cops; it seemed like something that could kill this deal before he made a dime.
"That seems unlikely."
"Well, I don't know. I'm just saying if you want your gadget back, now's the time to get it."
"You're asking me to come back to that body shop?"
"Man, I ain't asking you nothing. I'm telling you I can get my hands on the thing. That's all."
AJ went quiet for so long Jerry thought he'd hung up.
"You there?"
"Yes. All right. You get that tracking device, and return to the bar where we talked yesterday. Go there at seven o'clock."
"And you'll bring the money."
"Yes, Mr. Dolson. I will bring the money."
Being alone in the shop again wasn't a good feeling. Believe it or not, Nora was anxious for Jerry's arrival, something that had never occurred before. Frank Temple had offered to come into town with her, but she'd declined, disliking that maiden-in-distress vibe of relying on some strange male for protection. Besides, she was still reeling from what he'd told her. A _hit man_? At the Willow Flowage?
She wanted to consider it a joke. Might have been able to, were it not for the sorrow she saw in Frank's eyes as he'd told her the story. That haunted look he wore when he talked about his father was chilling. If he could conjure that up just to screw with somebody's mind, he needed to head to Hollywood on the next bus, get to work winning his Oscar.
So then it was real. That stirred a queasy blend of emotions within her, one that would hopefully be quieted once the Lexus was out of her garage and into police hands.
She left the office and went out into the main body of the shop and looked at the Lexus, sitting alone under the glare of the fluorescent lights. Since she'd taken over the shop, she'd found herself unappreciative of most of the new cars that came in. They had no personality, no soul. The old cars, your '55 Chevys and '68 Mustangs and any of a two-decade stretch of Cadillacs, those cars were like friends. She felt not unlike a doctor while tending to them, considered the removal of rust and addition of fresh paint to be healing effects, and she was sorry to see them leave the shop. It wasn't that she loathed the new cars; they just didn't inspire any feeling within her. Until this one.
She hated it now. Feared it. Just standing in its presence and looking at those twisted and crumpled quarter panels on the floor beside it was creepy. She was shaken by the sense that the piece of plastic and metal was somehow aware of her fear, was studying her now like a large strange dog unfenced and unchained.
Her sophomore year of college, she'd taken a trip to Rome with some classmates, an art history study project that her stepfather had financed without so much as a blink. Her mother had made the request, and after he'd written the check, she leaned down and kissed his neck, nipped his earlobe, and rubbed his back as he smiled distractedly and turned back to his desk. Nora, standing in the doorway watching it all, felt a cold ripple spread through her stomach.
That trip got off to a bad start. Due to delays from bad thunderstorms over the Midwest, Nora missed a connecting flight and had to wait for nine hours in LaGuardia, alone. To pass the time, she visited the airport bookstore and grabbed the first Stephen King paperback she saw, _Christine_. For most of the layover, she huddled in a corner seat in the terminal with that book, amazed at King's ability to make even a _car_ seem scary. That was a real miracle of storytelling, she'd thought, to give menace to a car.
This Lexus, though, put King's '58 Plymouth Fury to shame. It wasn't something out of a book, it was real, felt firm and cold under her hand, and its presence had already produced the most terrifying moment of her life. She caught herself rubbing her wrist as she looked at the car. There were thin blue lines on her flesh now, reminders of fingers closed over her arm.
A sudden, powerful rattling at the side door made her jerk, and when she stepped backward her foot hit the front bumper, which was resting on the floor, and nearly put her on her ass.
"Nora! Let me in."
Jerry. She put both hands on her temples, took one long breath, and then moved to the door.
"Take my keys away, and then you can't even unlock the door when you know I'm coming?" He entered the shop with customary good cheer, griping and scowling. It must be exhausting to be Jerry, carry all that hostility at all hours of the day.
"After last night, I'm never leaving a door in this shop unlocked again," she said. "Not when I'm alone, at least."
That caught his attention, made him tilt his head and lift one of his wild eyebrows.
"What do you mean?"
She told him what had happened and was surprised by his face as he listened. He looked concerned in a way she wouldn't have imagined he could be, concerned and almost guilty.
"Shoot, Nora. I can't believe that. This fella walking in here and putting hands on you . . . shoot." His chest filled with air and he looked around the shop as if hoping to find the culprit still on the property. "You say they hit Mowery, hurt him bad?"
"He looked _real_ bad, Jerry."
"I known that old boy since I was a kid. Sure, he's given me a hassle a time or two, but he also drove me home from Kleindorfer's once when he sure as shit didn't have to. Other guys, situation like that, they just take your ass down to the drunk's cell." His hands had curled into fists at his sides.
"I didn't know him," Nora said, "but I was scared for him. I want to go down to the hospital today, see if he's all right, and thank him."
"Yeah." Jerry's eyes weren't on her, didn't seem to be on anything in the shop.
"What's wrong, Jerry?"
"Nothing. I mean, shoot, just what happened, that's all. I wish I'd been here, Nora. Got in the habit of cutting out ahead of Bud at quitting time, but I shouldn't do that with you. Shouldn't leave a woman alone in a place like this."
"I'm not your responsibility, Jerry. Don't worry about that." She was touched by his concern, though.
"Well, that's the last time, you hear? This is a good town, Nora, a darn good town, but in the summers you get people coming in from all over, people you don't know and can't trust. Long as that's going on, I shouldn't be leaving you alone around here."
He looked up at her with a surprising sincerity in his face and said, "I'm sorry, Nora."
"It wasn't your fault. And I really do appreciate you coming in here to get that stupid car put back together and out the door. I'll be glad to see it go."
"No problem." Then, banging his fist on the hood of the Lexus, "You think the son of a bitch who drove this thing is going to come back for it?"
"I don't know, but I don't want it here if he does. I've heard some things I don't like, Jerry. Things that scare me."
"What do you mean?"
She didn't want to give him the whole story, hadn't yet decided who she _was_ going to give that to, but she was also worried and wanted to talk. It was one of the problems of her existence here; she was an outsider, a strange woman in a strange role, and her only confidant in the entire county was a man who needed help to write his own name. She and Frank still hadn't come to an agreement on whether she should even tell the police about the Mitsubishi. It would be nice to talk things over with someone.
"I've got a lot on my mind with this thing," she said, pointing at the Lexus. "Last night was bad enough, but this morning I talked to the other driver, and he . . . he offered some theories I don't like."
"The kid?" Jerry frowned. "Where'd he go, anyhow?"
"He's staying at the Willow. I drove him out last night." Best not to mention her return trip and invite more questions she wasn't comfortable answering.
"What's he know about it?"
She hesitated. No, it wouldn't do to share any of this with Jerry. For one thing, he ran his mouth, and for another, he elaborated. Even a toned-down version of Frank's account would soon be the sole topic of conversation over the bar at Kleindorfer's, only by the time it got there it would probably involve terrorists and nuclear weapons.
"He saw a gun in the Lexus," she said. "The guy took it with him." This wasn't a lie, and hopefully it would be enough to appease Jerry.
"You tell the police that?"
"Yes."
"They say anything about this car? Have any, uh, ideas of what's going on?"
"Not last night. I don't know if they do today."
He wouldn't look at her. "Get on out of here. I'll have this thing done fast."
"I'll wait on you."
He turned back to her, shaking his head emphatically. "No, you don't need to do that. Tell you what—you go on down to the hospital like you said, check in on Mowery, tell him old Jerry says hello. Then I'll give you a holler on the cell phone when this fancy-ass thing is ready to go."
"I think there should always be two people in this shop, Jerry. Until the car's gone, we should both be here."
He lifted a hand to his forehead, rubbed above his eyes like an exhausted man with many miles ahead. "I leave you here on your own last night for all that shit to happen, and now you want to stay around for me. Want to keep _me_ safe."
It wasn't a complaint; he was more musing to himself than talking to her.
"I just think it would be safest for both of us."
"I got something to tell you, Nora." He looked anguished. "And I want you to understand this first—I didn't know nothing about this car or what had happened to you at the time, okay? I mean, shit, if I'd known what happened . . ."
"Jerry, what are you talking about?"
He lowered his hand and walked past her, to his locker. Pulled it open and reached inside and withdrew a small plastic box. Even when he passed it to her and she held it in her own hands she had no idea what it was.
"It's a tracking device, Nora. Sends out a signal, and if you got the receiver you can follow it along." He ran his tongue over his lips. "It was on that car. I pulled it off the bumper reinforcement yesterday."
She ran her fingertips over the smooth plastic. This was the secret. This little thing was the source of the chaos. It had brought those bastards into her life.
"You found this yesterday afternoon?"
"Yes."
"And you didn't tell me."
"I'm sorry, Nora. I just . . . I don't know what to say. I wasn't thinking, that's all."
"Okay." Perhaps she should have been furious, screaming at him right now, blaming him. Instead, all she felt was confusion. Was this discovery good or bad? Would the device help her, or was it an increased risk just to hold it in her hand?
"I'm sorry," Jerry said again.
"It's all right. You're telling me now. That's what counts."
"Hang on," Jerry said. "There's more."
## 15
Frank had no intention of watching the Matteson island—no conscious intention anyhow. When he got the boat in the water and the motor fastened onto the transom, his only thought was of taking a ride, seeing the lake again.
He made it all of five minutes with that as the morning's lone goal. Out of the little bay and around the sandbar—the lake was still high enough that he probably could have gone right over the top of the bar, but old habits guided him around it—and then, just after he hit the main body of the lake, he opened up the throttle and pointed toward the Four Islands. Past them and around the point, out into the more desolate stretches of the lake, was the Matteson place. He had to see it. Just a look.
It was twenty minutes with the little outboard running at full throttle before the island came into view. There were so many islands out here that it could get confusing; half of them looked like the shore from a distance, and then you'd be around them and into a bay that looked big enough to be the main portion of the lake and suddenly you were damn lost.
Toward the northernmost reaches of the Willow the lake became more desolate, and tucked into the eastern shore was an area called Slaughterhouse Bay, so named because of the liberal collection of stumps and dead trees that protruded out of the water and could easily and swiftly ruin a boat. Navigating among the dozens of stumps, even at slow speeds, was treacherous, and though Frank and his father had always assumed it would be a treasure trove of pike and perhaps bass, they'd never taken a good fish out of the bay. It was an eerie spot, particularly at dusk, when the partially submerged trees blended with long shadows and made the place look almost like a Florida swamp.
Skirting the bay and its stumps by several hundred yards, Frank crossed Slaughterhouse Point, approaching the headwaters where the Tomahawk River fed the flowage. Between Slaughterhouse Point on the south side and Muskie Point on the north, lying offshore of hundreds of acres of unbroken forest, he found the Matteson island. After a seven-year absence, maybe it should have been difficult to locate, but he didn't have any trouble. The place was burned deep in his memory.
Although there were dozens of good-sized islands on the flowage, few would have been hospitable to development even if not state owned. The waters in the flowage fluctuated too much; in a low-rain year the lake was responsible for feeding much of the Wisconsin River valley, and the dam would be opened to the point that the lake level would dip as much as ten feet below the norm. A high-rain year, they'd close the dams up and the lake would rise dramatically, creating an ever-changing landscape that turned islands into mainland one summer and partially submerged them the next spring. The Matteson place was an exception due both to the high bluffs that bordered it and its placement in the middle of the lake. The water would never reach the ground level upon which the cabin was built, and any major recession simply expanded the beach below the bluffs.
He passed the island on the west side, keeping about a hundred feet out, saw the roof of the cabin and two of the _NO TRESPASSING_ signs, then circled and was ready to head back when he saw the woman.
She was walking out into the lake, waist deep now, testing the footing and moving slowly. What in the world was she thinking, going for a swim in this lake in April? Even though the air temperature was unseasonably warm, at least ten degrees above normal, the water would be frigid. She didn't seem concerned, though.
Frank didn't react to the sight of her, didn't slow or cut the motor or do anything else that would make a clear show of his interest. Instead, he turned his head and stared straight out over the bow and gave the throttle an extra twist, picking up speed. He took the boat out into the lake, angled away from the island. The day had risen clear and beautiful, the breeze warming as the sun rode higher, everything reminding him of a number of days spent on this water with his father. He'd been ready for the memories today, but now they were sinking away, pushed down by that woman in the water.
She was a beautiful woman. Even from fifty yards out, he'd seen that. Tall and elegant, and from the short look he'd gotten at her body, it probably seemed more suspicious that he had _not_ slowed the boat to stare. She would be used to stares.
Dave O'Connor, or Vaughn, or whoever the hell the gray-haired man really was, did not seem a match for that woman. He was such a strange-looking man, so nervous and awkward. On the other hand, he drove a Lexus and had thousands in cash on him, along with a gun. Maybe she was the sort who was attracted to money or danger.
That was another problem with Vaughn, though. He didn't seem like a dangerous guy. Even with the gun, even with the duo that had shown up on his heels, he didn't fit the mold. Those guys at the body shop yesterday had been a different story. Vaughn didn't seem anything like them or like other dangerous men Frank had known. Didn't seem anything like his father.
There he was, though, sitting in Devin Matteson's cabin with a woman who could turn heads from across the lake, two gun-toting badasses in pursuit. Nothing about that scenario felt right to Frank. Not after the time he'd spent with Vaughn yesterday.
He brought the boat around in a circle and ran back across the lake, a little farther out this time. She was leaving the water, and he could see another figure on shore. The distance was too great for a definite identification, but he assumed it was Vaughn.
Down maybe three hundred yards to an osprey nest, then back around for another pass, watching that island. This time he couldn't see anyone on the beach. They'd gone inside, maybe. Or he'd spooked them. In retrospect, this was a pretty stupid approach; if he wanted to watch them, he should just anchor somewhere and _watch_ them, the way Ezra had yesterday. These continued passes were more likely to attract attention. His father would have pitched him overboard if he'd been here to witness it.
Enough with the half-assed surveillance attempt. They were gone, and he'd already made one pass too many. Better to continue on, leave those two to their own affairs and hope his didn't coincide with them again. Nora Stafford had left his cabin with a measure of uncertainty, but he suspected what she planned to do now was simply get that Lexus off her property and leave the Mitsubishi in the woods. As he'd told her, there was a good chance it would still be there long after Vaughn left. If not, he'd pay for the rusted old heap himself. It was a better option than calling the police out to the Matteson place and attempting to repossess the vehicle. The less interaction Nora had with Devin Matteson's associates, the better.
He found himself alone in North Bay, no other boat in sight, and cut the motor. The flowage would never seem busy, but during fishing season there would be plenty of other people out and about. Today, though, it was empty.
The sun was unhindered by cloud, and he pulled his shirt off so he could feel it on his skin, take in this moment and this place. They'd caught a lot of fish out here, shared a lot of laughs.
A harsh ringing spoiled the silent day then, sounding louder on the water than it ever would back on land. He couldn't believe he got cell phone reception out here. That damn tower that had irked his father so much was doing its job. He took the phone out, saw the same number he'd dialed the previous night to leave his message for Nora. She was back at her body shop.
"Hello?"
Static and garbled words, Frank catching no meaning at all. He took the phone away from his ear, looked at the display again. Still connected, but showing just one bar, a weak signal. Okay, maybe the tower really wasn't anything but an eyesore. He tried again.
"Nora? I can't hear you. Nora?"
More garbled words, but this time he caught a few. Something about a tracking device. Fighting a surge of frustration, he asked her to slow down and repeat herself. Instead, the call was disconnected. Perfect.
He sat down in the boat and looked out across the water, then sighed and turned back to the motor, adjusted the choke and pulled the cord, brought it thundering to life. He didn't have a clue what that call had been about, and until he did, anything pleasant about this morning was ruined. He'd go back to the cabin, call Nora, see what the hell was going on.
"Damn it." Nora smacked the phone with her palm, turned it back on, tried again. This time it didn't even ring, just rolled over to a message saying the mobile user was unavailable. She wondered if he'd caught any of what she said. No way to know. Okay, what now? She wasn't ready to go the police with Jerry's story, not until she'd had a chance to run all of this by Frank, hear his opinion. He knew more about these guys than she did. It would be great if she could get him to come into town, talk things through, but Frank's source of transportation was sitting in the back of her tow lot, so he wouldn't be making any more surprise appearances. It was a long drive out to his cabin, but she didn't know what else to do.
"Jerry." She walked out of the office and into the shop. He was standing over his toolbox, next to the Spraybake paint booth. Outside the day had to be warming, because it was growing stuffy in here despite the concrete block walls and corrugated metal ceiling that usually helped keep it cool.
"Yeah?" Jerry had kept his eyes away from hers ever since he'd told her about the man named AJ, and now he stared at the floor.
"I'm going to get Frank and bring him down here."
"He the kid?"
"Yes." Didn't seem like any kid to her, but if that's how Jerry recognized him, fine. "I want him to be down here when we talk to the police. Like I said, he's got some ideas that they need to hear."
Jerry frowned and spun a ratchet in his hand, the whirring clicks loud in the quiet room. "What sort of ideas has he got?"
"He thinks he might know something about who these guys are, and who they work with."
"How?"
She lifted her hands. "I don't know, Jerry. I'm just telling you what I've heard. He also claims to know where the guy who drove that Lexus is staying. And now I've got to leave and pick him up so we can talk to the police."
"All right. I'll get this car put back together as much as I can, so they can tow it."
"I'd rather you don't."
"Huh?"
"I mean, I don't want anyone left alone in the shop." She tried to put proper concern into her voice, but only a portion of it was for Jerry's well-being.
"Don't worry about me."
"Jerry, I'd really prefer—"
"You don't think you can trust me." He straightened and looked at her for the first time, defiant. "That's what's going on, isn't it? Before I told you about the deal that guy cut me at the bar, you were ready to leave me here, go down to see Mowery. Told me that we needed this Lexus back in one piece fast, for the cops. Now why has that changed?"
They looked at each other for a long moment, and then his face softened and his shoulders sagged.
"I'm _sorry_ , Nora. You don't even know. I can see where you wouldn't think real well of me right now. You and I, we've had our problems. But I'll tell you this—ain't a man in this world I respect more than your daddy. Not a one. And the reason I'm still here is I know it's what he'd want me to do. Help you out, keep things running till he gets back on his feet. It's not just about the shop, it's about you. I wanted to make sure you were okay, too. Always did. So when you tell me about last night . . . about these bastards walking in here and treating you like that . . . maybe you don't see how personal that is to me. Okay? And all I can say is, I'm sorry."
Though Jerry asked about Bud's condition constantly, Nora had never been entirely honest with him in her reports. One reason was that her father had absolutely no memory of Jerry, and she knew that would hurt him. Now she wished he _could_ remember Jerry. Bud would have liked this story.
"I appreciate everything you just said, Jerry. And I know I haven't been a real easy transition for you. Let's not worry about it, okay? You get the Lexus put back together, and I'll bring Frank Temple down, and then the three of us will talk things out and call the police."
He tipped two fingers off his forehead in a little salute and turned back to the car. She crossed the shop, stepped out the side door, and pulled it shut behind her, making sure that it locked.
When she was gone, Jerry got to work. He started with the hood, which he'd removed completely since it was damaged beyond repair. A day earlier, he'd have just tried to jam the bent piece of metal into the backseat with as many other loose parts as possible, tell Nora that it didn't matter what condition the car was in if they were just transferring possession to the police. After her story, though, no chance. He still knew how to bust ass, how to do a job right, and after hearing what had happened, he'd be doing a lot more of it. Wasn't his fault, he understood that, but it didn't do much to ease the guilt. Fact was, while he was drinking beers and cutting a deal to sell equipment that wasn't his, Nora was back here with some bastard shoving her into a wall. If the kid hadn't showed up when he did . . . Jerry didn't like to think it through much beyond that point.
He wrestled the banged-up hood back into place on the car, fastened it as tight as it would go. The damage kept it from closing all the way, but it was attached and would stay on. By the time he was done with that, a good sweat was working its way across his scalp.
"Too damn hot," he said aloud. He didn't want the shop opened up like they kept it during the week, let people think they could stop by with a car, but having some fresh air wouldn't hurt, either. A crack in the overhead door should do the trick. He crossed to the garage door opener and hit the button, let the big door rise about two feet off the floor, and hit the button again, freezing it there. Already he could feel a breeze shove through, sliding over his feet. That would help.
It was a pain in the ass putting a car back together without assistance, but Jerry had gotten better at that in the last few months. Nora was always trying to help, and, to be fair, usually _could_ help, but he preferred to do things himself. To fasten the bumper onto the front of the car, he got one side lined up and bolted loosely, then walked into the paint booth and retrieved a rack they used for drying parts, brought it out, and set it up under the bumper in a way that kept the thing level and positioned well enough that he could get the bolts lined up and tightened. He dragged the creeper over, hitched up his pants, and settled down with his knees and face pointed up at the ceiling. Using his heels, he shoved backward, and the creeper slid under the car so he could get at the bumper bolts, leaving only his lower body exposed.
It was dark under the car, and he had to feel with his fingers to get the wrench in place. Once he had it set the procedure was simple enough, working the wrench with a practiced motion. He'd been on his back under a car since well before he could drive one, watching his daddy labor over a fastback Mustang that he'd bought wrecked, with visions of restoring it to Steve McQueen quality. He'd never gotten it done, but he'd hooked his son on cars. Thirty years later, Jerry was still with it.
He got the bolts on the driver's side fastened and was working the creeper over to the passenger side when he heard the overhead door rattle ever so gently. It was just a slight shake, one that could have been from the wind, but when he turned his head to look he saw two feet. Someone was walking the length of the door while Jerry lay there on his back and watched. Someone in polished black boots. Jerry knew those boots. He'd seen them tapping a soft beat off a bar stool not twenty-four hours earlier.
The son of a bitch was back. This time he didn't have a friend in Jerry, either; what he was _going_ to have was a wrench upside his head. Jerry had extended his feet, ready to use his heels to pull himself forward and out from under the car, when he saw a hand appear next to the boots, and then a knee. AJ was coming inside. Crawling under the door and coming inside.
He was a coward for doing it, knew this well, but Jerry pushed with his heels instead of pulling, slid all the way under the Lexus. There was something about this that took him from angry to scared in one blink. What was the guy thinking, crawling into the shop like that? They'd agreed to meet at Kleindorfer's hours from now. So why violate the plan, take this sort of risk?
Resting on his back on the creeper, his nose a few inches from the rear transfer case, Jerry kept his head rolled to the left so he could see his visitor's approach. AJ crawled under the door and straightened up, and then all Jerry could see was his feet as he walked into the shop. Then the feet passed out of his field of vision and he was reliant upon only his ears, listening to the slow claps of boot heels on concrete.
He held his breath in his chest like a dear secret as the boots came and went again in his sight line. AJ seemed to have made a full circle of the shop, was now probably standing in front of the Lexus. Peering into the office, maybe, seeing that it was dark, seeing that the place was empty. Now, if he'd just crawl back under that door and walk away, Jerry could get up and lower the garage door, lock the place up tight, and give the cops a call. Nora hadn't planned a course of action yet, but this was the second time one of these bastards had broken into the shop, and that was crime enough. Even if Jerry took some heat from the cops, they needed to pick these boys up. Somebody had to answer for Mowery.
There was the metallic bang of a gear engaging, and then a loud hum as the garage door lowered and thumped to a stop against the floor, closed tight. The sound made Jerry lift his head too far and too fast, his forehead making solid contact with the transfer case. He blinked hard and dropped his head again. Why had AJ lowered the door? What the hell was he thinking of doing now?
"You going to stay under that car all day, Mr. Dolson?"
The voice drawled out of the air above him; Jerry could still see no boots to tell him where the man was standing. He was caught. Damn it. Now a dose of embarrassment mingled with his fear. Hiding under the car like a little girl under her bed. That wasn't right, and he should've have known it from the start, met this bastard on his feet and with the wrench in his hand. Using the self-reproach as fuel, Jerry slammed his heels onto the floor and pulled himself forward, out from under the car and right into the barrel of a gun.
## 16
Frank tried calling the body shop as soon as he got back to the cabin, where a steady cell signal came through. Voice mail. A second try found the same result. He didn't have a cell number for her, either, so the trip back to the cabin now seemed to be in vain.
He pulled the boat higher onto the beach and was halfway to the cabin when his phone rang again, an unfamiliar number on the display. He answered, heard Nora's voice say his name, and was surprised by the strength of the relief he felt.
"Yeah, it's me. I just tried calling you back at your shop."
"I just left it," she said. "Are you at your cabin?"
"Yes."
"Good. I'm on my way. I'd like you to come into town, and of course you can't do that because you have nothing to drive."
"Something happen?"
"You suggested we leave my car where it is, not bring the police out there, because it might be better for me. Safer. Right?"
"Right."
"Okay. Now, if I told you that the two from last night were going to be at Kleindorfer's Tap Room at seven tonight, would you still say I should keep my distance? Or does your advice change at that point?"
"Tell me what happened," was all he said.
The story she told wasn't a surprising one, not really, but even before they hung up he knew his response would be different from hers. He was unsettled by Nora's obvious enthusiasm for bringing the police in. If her body man was honest about the situation, and there really was a meeting scheduled at this bar, yes, he could see the appeal of setting a trap. So would the men who'd set the meeting, though. It went back to what he'd already told her repeatedly: These guys were pros.
He didn't blame them for recruiting her employee as an ally. That had the touch of professional work, too; why risk a strong-arm move when the tickle of a little cash in the palm accomplished the same thing?
They'd played it both ways, though, and that was what he didn't understand. Why recruit the body man and attack Nora the same night? Why take a step to avoid a strong-arm move and then still _make_ the strong-arm move?
_Because they weren't together._
No, they hadn't been together. That was one of the concerns he'd pondered as Nora drove him to the cabin, one of the problems he couldn't resolve to his own satisfaction. Why had the second man waited until after his buddy was in handcuffs to help? He'd waited because he wasn't there yet. It hadn't been a _wait_ at all; he'd arrived at that moment and been forced into action. So that meant that the second guy was the smarter of the two, probably. He'd been at the bar trying to buy off Nora's employee while his friend had been, what, stationed back at the shop to see if she moved the Lexus outside at closing? That made sense. Only the guy stationed at the shop hadn't been patient enough. He'd gone into action, and his friend had to pull his ass out of the fire. Now their presence in town was anything but discreet.
Frank walked back to the cabin thinking about that final realization: These two guys, if there were only two, now understood that their situation in Tomahawk had changed. It was a small town, a town where gossip spread fast and strangers stood out, and now everyone would be talking about them, the police looking for them. It added an element of pressure. Would they wait patiently for a meeting with Nora's employee? He knew _his_ answer to that, and it wasn't comforting.
He unlocked the cabin door and went inside, washed up, and changed into a clean shirt. Then he put the suitcase aside and pulled a metal case onto the bed, flipped the latches and opened the lid, and withdrew the two holstered and well-oiled handguns beneath.
His father's guns: a 10 mm Smith & Wesson and a .45-caliber Glock. They should have been the day's project. He'd thought about taking the boat out to the right spot, Muskie Point, maybe, or somewhere among the stumps of Slaughterhouse Bay, and feeding the guns to the lake. It would be a most heavy-handed gesture, yes, but it was one he still wanted to make. He wanted to hold his father's violence in his hands, feel the heft of it, and then leave it behind in a place without regrets, a place of clean memories.
They wouldn't be sinking today. He knew that as he recalled Nora's voice on the phone, all that excitement because she thought this meeting represented the end of the problem. Frank knew it was anything but that. Jerry was just another loose end, and, sadly, another loose end connected to Nora.
He had the Smith & Wesson in its shoulder holster and concealed under a thin jacket by the time Nora arrived.
Spend enough time around firearms, and they'll fail to inspire the same sense of terror that might catch a novice, even when the weapon in question is pointed at your heart. Jerry wasn't thrilled to see it, no, but he wasn't about to wet his pants or anything, either. Guns were guns. Only thing to worry about was the man who held it. And that man hadn't shot him yet.
"You don't look happy to see me, Mr. Dolson," AJ said, sliding his thumb up and down the stock of the gun the way he'd handled the vodka glass the day before.
"I'm not. We had an agreement, and this ain't part of it. Why don't you go on down to Kleindorfer's and wait for me, like we planned?"
"You were down here with the girl," AJ said. "Your boss. She have anything to tell you?"
"Nope."
"You're a bad liar, Mr. Dolson."
Jerry worked his tongue over his teeth and steeled his eyes against the other man's empty gaze.
"And you're a Grade A piece of shit, buddy. Coming in here and beating up a woman."
"I didn't lay hands on anyone."
"Then your buddy did. Which makes you both Grade A pieces of shit, all right? Now you get the damn gun out of my face and get on your way."
"We had an agreement."
"I don't make agreements with people who beat up women."
"All the same, one was made. And I'm going to need that tracking device."
"Don't have it."
"Who does?"
He started to say Nora's name, then stopped. It was wrong both ways; first of all, it might send these assholes back after her, and, second, she didn't even have it. Thing was still sitting in his locker, waiting to go to the police.
"Put that gun down," Jerry said.
"That will make you comfortable? Maybe then we can talk this through, work something out?"
Jerry wasn't about to talk anything through, and any chance of working something out had ended the minute he heard what happened to Nora. He didn't like staring into that tiny muzzle, though, so he nodded.
"Maybe we can."
AJ pistol-whipped him in the face. Jerry had time to lean backward maybe six inches and half-lift the wrench in his hand before the gun caught him just under his right eye and knocked him back into the Lexus. His ribs slammed against the grille, the wrench fell from his hands, and then he took another blow from the gun, this one across the back of his head, right near the top of his neck. It brought him down almost to his knees, hanging on to the car to keep from hitting the floor. All wasted effort, though; the third swing was harder than the first two, and it took all the resistance out of him, left him stretched on his back with one leg hooked over the creeper, looking at the corrugated metal ceiling that now bloomed with a dozen colors.
Jerry watched the colors dance and bit down on the tip of his tongue, trying to clear his head. It didn't work. He bit harder and tasted blood but still the room reeled, and when he felt someone moving his hands he could make only the slightest resistance. A cord bit into the flesh of one wrist, then the other. AJ was tying his hands.
"Is the girl coming back?"
Jerry didn't say anything. When he tried to pull his hands forward, he felt unyielding resistance. He was tied to something. Maybe the Lexus. He heard AJ walking away, blinked hard, strained to lift his head. The gun was out of sight now, but AJ was at Jerry's toolbox, had the drawers open, was lifting a ten-pound maul out. No, no, no. Put that thing down. Please put that thing down.
"Is the girl coming back?" AJ repeated, his back to Jerry as he hefted the maul, took a practice swing.
"Yeah." Jerry's head was clearing fast now, and the pain was no longer a presence in his mind.
"How long till she does?"
"Maybe an hour."
"She go to the cops?" AJ was standing over Jerry, the maul held down against his thigh.
How to answer that? Instinct said to tell him no, but why? If the guy thought cops were on the way, maybe he'd cut this short. Was that a good thing, though?
"Mr. Dolson? Jerry, buddy? You want to give me an answer."
Split the difference, maybe. Tell him she was planning to go the cops, but hadn't yet. Was that good?
"She went to pick up that kid. I think they're . . . could be they'll go to the cops. But that ain't my fault. That's your buddy's, man. You hit a woman, then knock a cop around like that, you've got to expect—"
"What kid?"
"One who jacked up your friend last night."
"Why's he involved?"
The pain was coming back now, but so was his sense of guilt. He shouldn't be giving this asshole so much information. Shouldn't be rolling over like this.
"Don't know."
There was a whistle of metal through air as the maul came down, and Jerry had just enough time to tense before it caught him square in the hip. A hellfire shot of pain cut through his leg and into his stomach, filled his chest. He arched his back and hissed through clenched teeth.
"Want to answer that one again?" AJ said.
"He thinks he knows something about you."
"About _me_? How does he know something about me?"
"I'm not sure, man." He had his eyes squeezed shut against the pain but still sensed the maul being lifted again, yelled out, "I _don't_ know, okay? She didn't say. Just told me that she needed to talk to him to decide what to tell the cops. The kid thinks he understands something more than the police, and he thinks he knows where your boy went, the one drove this car."
"He knows where to find him?"
"I think so."
"Where?"
"I don't know."
"You're a lying piece of shit. _Where?_ " The maul was drawn back again, and as much as Jerry wanted to look strong, he couldn't help but cower.
" _She didn't tell me_."
"But she knows."
"Yes. Maybe. I mean, the kid says he knows."
"And she went to get the kid. Where was she going to pick him up?"
"I don't know."
"You're lying again. Where did she go?"
AJ's voice had intensified, and this time Jerry knew he had to shut up. _Had_ to. If he told this asshole, the guy was going to leave immediately, chase after Nora. Jerry wasn't about to do that to her. No chance.
"Where did she go?" AJ repeated.
"Tell you where _you_ can go. Straight to—"
This time the maul was swung with far greater force, straight into Jerry's thigh. He heard the bone snap a tenth of a second before he felt it, and this time he couldn't make a sound, couldn't have screamed if he wanted to. The pain slid into his brain like a fast-moving storm cloud and he faded beneath it. AJ's voice was somewhere outside the cloud, questioning him, maybe the same question or maybe another; he no longer could translate the words of his own language.
"You're going to die."
He got that sentence, held it for a second, figured it out. Yes, the man was telling the truth. Jerry was going to die.
"One more chance, Mr. Dolson."
So maybe he was not going to die? One more chance. That meant a chance to live, right? Had to. Jerry tried to look at his leg, expecting to see bone and blood. There was nothing of the sort. Just his jeans going down to a foot that he could no longer move. Could he? He tried and nothing happened. Or was he even trying? So hard to tell. So hard to know what to do.
There was something between his eyes and that immovable foot now, swinging in the air. What was it? Oh, shit, the maul. He remembered the maul. It was what had caused all of this. Thing shouldn't even be in the body shop. It was for splitting wood, but he'd brought it down because it was heavier than the hammers and easier to use than a sledge, a good all-purpose pounder. He hadn't considered this purpose.
" _Where did she go?_ "
Where did she go. That question again. Asking about Nora. Don't tell him. Remember that, Jerry. Don't tell him. The pain's going to come back soon, going to make you forget some things, but don't forget this.
"You've got to start talking again," AJ said. "Does she have the tracking device? I don't think she does. You said it was in your locker yesterday. I bet you still have it. You wanted that money."
AJ moved away and the pain moved back in. Jerry took in a long breath and choked on it. There was so much spit in his throat. Or was it blood? You wouldn't bleed in your throat from a broken leg, would you? No. No, that didn't seem to make sense. His leg was in two pieces. That didn't make any sense, either.
"Thatta boy," AJ said, and a locker slammed shut. What was he so happy about? Oh, right, the tracking device.
"You got it," Jerry said. Tried to say, at least. The words were tough to form. AJ had the tracking device now, so he would leave, right? He would leave now, go away and let Jerry alone.
"Yes," AJ said. "I got it. But that's not the only thing I need. Where did _he_ go, Jerry, old buddy? Where is the guy who goes with the car?"
Jerry didn't know. Nora hadn't told him. Maybe Nora didn't know. He couldn't remember anymore. Wait—AJ had the tracking device, and that meant Jerry had failed. That was the whole point, wasn't it? Not to give him the tracking device. No, the point was Nora. Not to tell him where Nora had gone. Where _had_ Nora gone? The Willow, that was it. She'd gone after the kid at the Willow.
"What's that?" AJ was standing above him now. "What are you saying?"
He'd been talking. No good. Don't talk, Jerry. Keep your damn mouth shut, for once in your life.
"Willow?" AJ said. "Is that what you said? Keep going. Keep talking."
_Don't_ keep talking. Don't say a word. You almost made a mistake, a bad mistake. Don't say anything, Jerry. Bite down on your tongue. Is that your tongue? Doesn't matter. Bite it. Bite it and hold it and don't say a word.
"Okay," AJ said. "I think you're running out of usefulness. Good news is, you're not going to feel that leg anymore."
The maul was gone, discarded in favor of a knife with a small blade. Good. Jerry didn't know if he could take another swing from the maul. Snapped that bone, probably the thickest bone in his body, like it was a piece of rust. No, he couldn't take another one like that. But the knife wasn't good, either, was it? Not in AJ's hand. He should ask AJ to stop. Just stop and go away. Jerry was hurt. Couldn't he see that Jerry was hurt?
## 17
It's not going to be quite that easy," Frank Temple said as Nora drove down Business 51 and into Tomahawk. He'd been offering so many comments of this sort that she was beginning to feel uncomfortable with him. Even if everything he'd told her was the truth, it seemed odd to be _so_ leery of calling the police. She was telling him, flat-out, that there was going to be a meeting with Jerry and these guys, and he was still trying to discourage her from calling the police. Who did that? Any normal person would be _ordering_ her to call them. So was there something more involved here? Was the man in her passenger seat connected to these guys somehow?
"I'm not saying it's going to be _easy._ I'm saying, if they do it right, this is a good chance. These guys think Jerry's working _with_ them, Frank."
"I don't know if they really believe that."
"Well, they agreed to the meeting. And at the time that he set it up, he was all about giving them that tracking device, too. So I don't think his demeanor would have done anything to create suspicion."
"Guys like these don't need something to _create_ suspicion, Nora. The idea of going for some sort of trap with a handful of small-town cops whose idea of high crimes probably includes poaching seems like a piss-poor plan to me."
"You spent this morning convincing me that I should be terrified of these men."
"That wasn't the idea."
"Well, it was the effect. You do that, and then I tell you that there's a good opportunity to have them arrested, and you're trying to discourage it. Forgive me if I say that doesn't seem right."
"All I've said, Nora, is that I'm not sure you appreciate the background these guys have."
"You don't know what their backgrounds are. You said that was just a lot of guessing."
"Very educated guessing."
"In your own opinion, sure. I don't know if the police, if the people whose _jobs_ it is to deal with situations like this, would agree."
She was snapping at him now and didn't want to be, so she stopped talking before any more hostility crept into her words. He was quiet, looking out the window, and she felt a quick pang of foolishness and guilt. Why, though? Why _should_ she believe him? He was a twenty-five-year-old kid who wanted to be a writer, for crying out loud. Just because his father had killed some people didn't mean he was James Bond. And who knew if any of that was even true? What she should have done was get on the Internet and see if she could verify his story.
It was ridiculous not to go to the police. Ridiculous, and probably dangerous. She didn't really know Frank Temple or anything about him, and that odd sense of trust she had in him could be simply the product of the way he'd come to her aid. In fact, she was pretty sure she'd read something like that in a psychology class. Her emotions from the previous night were causing her to put too much faith in him, when he could be just as dangerous as the men she was worried about.
"I'll let you talk to Jerry," she said, "but then I'm calling the police. Okay? This is _not_ your decision. I was the one they attacked yesterday, that damn car is in my shop, and it's my responsibility."
He just nodded.
"So it's up to me to decide what we do, and we're going to call the police and give them that little box Jerry found and tell them about the meeting. I can't make you tell them any of the things that you told me this morning, and I won't try. That's up to you. But I will tell them everything I know."
Again he didn't say a word. _Fine,_ Nora thought. _In an hour or so the police will be dealing with all of your strange moods, not me._
She drove into town, into the shop parking lot, through the open gate, and parked just behind the building, facing Frank's Jeep. As she opened the door and stepped out, she saw that one of the overhead garage doors hung about two feet off the ground. Probably getting too stuffy in there for Jerry. Might as well open it up all the way.
Frank was out of the truck and walking beside her as she went to the side door, which she found was locked. She rapped on the door, and they waited. There was an awkward tension now; Frank's silent, blank-faced reaction to her rant in the truck left her uncomfortable. It would be nice if he'd responded, or at least if the silence seemed like a response, as if she'd angered him and he was sulking about it. Instead, he was impossible to read, just stood there with every thought and emotion tucked in a locked box and hidden from the world.
Jerry must have the radio on, because he didn't hear her knock. Or he'd reverted to typical Jerry form and chosen to ignore it. She got out her keys and unlocked the door and pulled it open, held it for Frank.
"Thanks." He walked past her and into the shop, and she followed, letting the door swing shut behind her. She'd made it maybe two steps inside when he whirled back to her, put his hand on her shoulder, and guided her backward.
"Outside."
"What?"
He didn't answer, just kept his hand on her shoulder while reaching for the door handle. Her confusion switched to irritation, and she twisted free from his grasp.
"Let go of me. What are you doing?"
He had the door open and was reaching for her again as she stepped around him and saw the blood.
It should have induced immediate terror, maybe, but instead her reaction was simply to follow it with her eyes, some natural curiosity telling her to find the source before she responded. There was a drain in the center of the concrete floor, a big rusted grate with nickel-sized holes, and a thin trail of blood was leaking into it now. Up from that the flow widened, and then she saw Jerry.
He hung in an awkward half-lean from the front of the Lexus, his hands bound to the car's grille with a length of wire, his head flopped sideways onto his left shoulder. There was a thick dark line across his throat, just under his chin, and beneath it was the pool of blood that had spawned the rivulet running into the drain. His left leg was bent unnaturally, and there was a strange bulge high on his thigh, almost at the hip. Nora's eyes recorded all of this in a split-second stare, and then she said, " _No, Jerry_ ," and started toward him.
"Don't." Frank had her arm again, his grip rougher than before.
" _Look at him! He's_ —"
"Dead. He's dead. Don't go over there, don't touch him. We need to leave now."
She started to fight him, twist her arm loose, but then her eyes focused on the bulge on Jerry's leg again and for the first time she understood what it was. The bone. That was the bone pushing at the skin, trying to escape. They'd broken his leg. That understanding brought the nausea on in a wave, and she started to sink to her knees. Frank caught her and kept her upright, moved her back toward the door. Her jaw went slack, and for an instant she was sure she'd be ill, but then he had her outside and into the fresh air.
"Oh, no, Jerry." She was on her knees on the pavement now, aware of a sudden heat in her face and neck. "No, Jerry, what did they do, what happened to him, what did they do to him?"
She tried to get to her feet, and Frank put his hand on her shoulder and shoved her gently back.
"Stay down. I'm calling the police."
She put her hands flat on the gravel and squeezed them into fists, wanting to hold something, watching idly as one short fingernail split against the stone.
"Did you see his leg?" she said. Frank was talking in a low voice into his phone. She repeated the question, and still he talked only into the phone. Her hands were trembling now on the gravel. She asked the question a third time as he put the phone back into his pocket and knelt beside her, wrapped his arm around her back.
"Did you see his leg?"
"Yes." His voice was soft.
"They hurt him," she said.
"I know."
"They did that to his leg."
"I know."
From the time he'd called Nora and left the first message the night before, Frank had tried diligently to convince himself that most of this was undue worry. That his impression of the men who'd come to her body shop was inflated by the adrenaline of the moment, that bad memories of a man he barely knew had driven him to exaggeration and paranoia. All of that ended when he stepped into the body shop with Nora behind him and saw Jerry Dolson tied to the car, his blood drying on the floor. There'd been no exaggeration, no paranoia. He knew these men now, not by name, maybe, but he knew them. Waiting for the police with his arm around Nora's back as she wept, Frank felt a pang of desire to see his father in a form he'd been sure he would never wish to recall—with gun in hand.
These men were good, but his father had been better. Faster of body and faster of mind, a deadlier shot, superior in every quality of combat. The image of his father as the violent but righteous crusader, an idea that Frank had come to love as a child and loathe as an adult, returned to him in a desperate ache. _Come back,_ he thought as he felt Nora's jerking sobs under his hand. _Come back and make this right. Settle this in the only way that it can be settled properly_ — _in blood. You could do that. I cannot._
His world disappeared into a cacophony of sirens then, three police cars arriving in succession, men emerging with weapons drawn as if there were anything they could do.
## 18
They separated Frank from Nora almost immediately, and for the next six hours he didn't see her again. None of the cops bothered to search him for a weapon at first, but he was conscious of the gun in his shoulder holster, and eventually told the officer who seemed to be in charge of the scene that he was carrying. The guy didn't handle it well, took the gun and then searched Frank with rough hands, as if he might have voluntarily given up the pistol only to attack them with a knife a few minutes later.
At first it was nothing but local cops, small-town guys who all seemed to achieve a certain level of shock with the realization that someone had been tortured and murdered at noon on a Saturday in the middle of town. They ran through the basic motions, asked Frank the basic questions, but nobody seemed focused, a high level of confusion permeating the group.
He was left alone in an interrogation room at the little Tomahawk police station for more than an hour. People came and went outside, talking in soft voices, and he caught snippets of their words, muttered curses and musings, references to Mowery. Tomahawk's police department had just hit the big time, and Frank probably understood this better than they did.
When the door finally opened again, the cop who entered wasn't one he'd seen before. Even before the guy settled into a chair across the table and introduced himself, Frank knew he was an outsider. He was about fifty, with a receding hairline and weathered skin, bony shoulders poking at his shirt. When he looked at Frank, one eye drifted just a touch, seemed to gaze off to the left and up.
"Mr. Temple, my name is Ron Atkins. Feel free to call me Ron. How are you doing?"
"Fine," Frank said. "Who are you with?"
Atkins raised an eyebrow. "You imply I'm from a different agency than the one that brought you here."
"I do."
"What makes you think that, might I ask?"
"You don't look excited."
Atkins considered Frank for a long moment after that, then gave him a few slow nods. "Interesting observation, Mr. Temple. No, I am not excited. There's nothing exciting about what we're dealing with here."
"Rest of the cops seem to think so."
"Agreed. That'll pass with time."
"So who are you with?"
The repeated question seemed to irritate Atkins, causing a quick, hard flicker of his eyes before he answered.
"I'm with the FBI, Mr. Temple."
"Milwaukee?"
Atkins's eyebrow went up again. "No, Wausau. We maintain a small field office there."
Frank nodded. If Atkins had come in from Milwaukee already, that would have told him something, suggested that the cops here were already getting a sense of things, maybe knew something about who these guys really were. Nobody from the FBI responded to a murder otherwise. But if he'd just made the hour-long drive from Wausau, maybe it wasn't quite as strange. There weren't a lot of homicides up here, certainly not of this nature, and Frank guessed the FBI office in Wausau wasn't swamped. Probably welcomed the chance to step in, give this one a look.
"Not a real good start to your weekend, is what I'm hearing," Atkins said. "First you had this trouble yesterday in which, according to what I've been told, you performed quite admirably. Then, not twenty-four hours later, you found a murder victim in the same building."
Atkins cocked his head at Frank. "No way to start a vacation, right?"
"Nope."
"So you are here on vacation?"
"Yes."
"That's what brings most people here. Most people, though, they don't have a string of bad luck like you're experiencing."
"I wouldn't think so." Even this early in the conversation, Frank had reached two conclusions about Atkins: First, he was smart, and deserving of respect. Second, Frank didn't like him.
"You rent a cabin up here, is that it?"
"Own one."
"Really? Very nice. Out there on the Willow Flowage, is it?"
"Yes."
"How'd you come into possession of the cabin, might I ask?"
Here was the reason Frank didn't like him, drifting out in these casual questions. The man had come here to ask about Frank's father. Either he knew the name, or somebody had done the homework.
"It was in the family," Frank said. "But I don't see what relevance that has to the poor bastard we found with his leg broken and his throat cut, Mr. Atkins. Ron."
"I understand that. I'm going to ask you for a little patience. See, I may find relevance in places you don't."
"Tell you what," Frank said, "let's go ahead and talk about my dad."
Atkins pursed his lips into a little smile but looked at the tabletop instead of Frank. "Your father. Yes, I've heard about him."
"A lot of people have. And, hate to tell you this, but he's been dead for seven years. Tough to blame him for this one."
"I've heard a few terms used concerning your father—"
"I've used a few of them myself."
"I believe it. But I'm talking about his, uh, entrepreneurship, you see. Because the man didn't just kill people. He made money doing it, for a while. One of those terms that people use is 'hit man.' "
"I've heard it."
"Right. So—and I understand how frustrating this has to be for you, trust me—when a cop ends up beaten half to death outside of a body shop on a Friday and another man ends up killed in the same body shop on a Saturday, and the key witness to both events is, well, the son of a hit man . . ."
"This is what brings the FBI up from Wausau," Frank said.
Atkins nodded with a theatrical sense of apology. "Like I said, Mr. Temple, I understand this may not be fair to you, but sometimes we have to endure a little extra suffering along the line just because of our families. That happens to everybody, in one way or another."
_I could tell you some of the ways,_ Frank thought. _Could tell you what it's like to be seventeen years old and fooling around with your girlfriend, biggest concern in the world just trying to get her shirt off, when your father comes home and walks into your bedroom. And for a minute, Mr. Atkins, you're still worried about the girl and about his reaction and this all seems like a major crisis. Seems like that until he says,_ Son, we're going to need to be alone right now, _and something in his eyes tells you that the pending conversation has nothing to do with anything as innocent as you and the girl._
"So I understand, is what I'm trying to say," Atkins said. "But I've still got to ask the questions."
"Yeah," Frank said. "I kind of figured you would."
"Right off the bat, I'm curious about this: I was told you were wearing a gun when the police got down to the body shop. A gun, I might add, with your father's initials stamped into the stock. FT II would be him, right? You're FT III?"
Frank nodded.
"You always carry the gun?"
"No."
"Okay. Then you come up here on vacation, a fishing trip, and you think, yeah, this seems like the time and place to pack a pistol?"
Frank looked at Atkins for a long time before he said, "It had started to seem like a dangerous town."
Atkins nodded. "Almost from the moment you arrived."
## 19
This couldn't be her life. The longer Nora thought about it, the less sense it made. Hit men, tracking devices, murder? No, they didn't fit. None of those things belonged.
Yet there they were, the reality hammered home by the parade of police interviewers. _Can you describe . . ._ they'd say, time and time again. Of course she could describe it. Jerry had been murdered. Try seeing that and forgetting it. She'd be able to describe that scene for a long time, far longer than any of the things she _wanted_ to remember. The way his head had hung at that unnatural angle, the way the bone had bulged from his thigh . . . this couldn't be her life.
She'd gone through a few rounds of interviews and one short talk with some sort of grief counselor who'd left a card and told her something about the pain of those left behind lingering longer than the pain of those who suffered. What that meant, Nora had no clue. The idea seemed to be that Nora would suffer more than Jerry, but the grief counselor hadn't seen Jerry's leg.
The only thing that stood out in all the talking was the lead cop's disclosure that no tracking device had been found in the locker. He wanted to know if it could have been left somewhere else, and she gave them permission—as if they needed it at this point—to search the whole shop, but she knew it was gone. That's what they'd come for, and now it was gone. The only physical link she'd had to them was missing.
The last visitor was a man in an ill-fitting brown suit who showed his badge almost immediately, the only person who'd done so all day. FBI, it said. That surprised and comforted her. About time somebody like this was involved.
The reassurance his presence provided didn't last long. After some of the same preliminary questions, his focus shifted to Frank Temple and stayed there. How long had she known Mr. Temple? Just a day, huh? Was she aware of his father's story? Oh, Mr. Temple had already offered that. Interesting. What else had he said?
That's how it went for more than an hour. One thing was settled—she didn't need to do that Internet research to verify Frank's story. Mr. Atkins of the FBI did a fine job of that.
"You seem to be suspicious of Frank," she said. "Is that my imagination?"
"Suspicious?" Atkins leaned away from the table and hooked one ankle over his knee. "That's getting ahead of the game, Ms. Stafford. I'm just gathering information."
His words reeked of insincerity, though, and she felt instantly sorry for Frank. This was the price he paid for the family he'd been born into. When she walked through the streets of Tomahawk, people stopped her and told her stories about how wonderful her father was, asked after his condition; strangers gave her hugs on a regular basis simply because of her father's history in the town. Frank's experience was quite different.
"I understand you need to gather information," she said, "but Frank was nothing but a help. I've already told you what he did yesterday."
"Yes, I know. But when you consider his background, Ms. Stafford, you can surely understand any heightened curiosity we might have."
_Heightened curiosity?_ Now, there was a good FBI phrase. A minute ago he'd denied being suspicious of Frank, but now he admitted to _heightened curiosity._ Huge difference, clearly.
"Whatever his father did when Frank was essentially a child really seems insignificant to this situation," she said.
"Perhaps."
"You disagree?"
"Let me ask you this—has Mr. Temple told you anything of what he's been doing for the past seven years?"
"I just met him yesterday. Obviously I don't know his life story."
"That's a no, then?"
"He told me he's been a student."
"He's been enrolled in school for a grand total of six semesters in seven years. Those six semesters were scattered among five different schools, in five different states. He has lived in at least ten different states for short times. His highest level of employment is as a bartender, his longest stint at that five months, yet he's paid his rent, bills, and tuition in full and on time."
"Wonderful. So your point is that he's a model citizen?"
Atkins gave her a long, unpleasant stare. Things were becoming contentious, and she knew part of her defensiveness was a product of guilt. She'd basically berated Frank as they'd driven back to town, dismissing his concerns and suspecting him of lies. Then there was Jerry, a terrible but undeniable support for Frank's story. His concern had clearly been genuine and well founded.
"My point," Atkins said, "is that there are many unknowns about Frank Temple the Third. He leads a nomadic lifestyle, maintains few connections to his past, and somehow generates a steady cash flow. It is a pattern, Ms. Stafford, not unlike many of the men in his father's profession."
She pulled her head back and stared at him. "You're kidding, right?"
"I'm making observations, not accusations."
"Well, I made an observation myself today, and that's that Frank doesn't like to talk about his father and is very ashamed of what the man did."
"Shame is one reason to avoid talking about his father. There are other possibilities."
"You're suggesting he followed in his footsteps? Frank was _seventeen_ when his father died. I've never heard of a seventeen-year-old assassin."
Atkins just looked at her, studying her face, silent.
"Why aren't you asking about Vaughn?" she said.
"This is the man who drove the Lexus?"
"Yes. He's the one who caused all of this. He's the one who brought these murderers into my shop."
"You were with Mr. Temple and Vaughn at the same time," Atkins said, switching tracks. "Although he called himself Dave O'Connor at the time, right?"
"What do you know about him? Who is he?"
Atkins ignored her. "Did you sense any familiarity between the two men?"
"Frank and Vaughn?" She shook her head. "Absolutely not. I mean, they'd just gotten in an accident. So they had about twenty minutes of familiarity before I met them. That was as much as I sensed, too."
"You let the Lexus driver leave without seeing a driver's license or insurance information?"
"He gave me cash. I've explained that to everyone already. It was a mistake, but I can't fix it now. I can't fix any of this now."
"And you have no idea where he went?"
She should answer that question, tell him about the island cabin and the car in the woods. That was the right thing to do, certainly, but she was remembering Frank's reluctance to bring the police into this, the idea he had that it might make her even more of a threat to these men who had such evil ways of dealing with threats. The less involvement the better, right? Knowing nothing was better than knowing something. If you knew something, you were a loose end. Isn't that what Frank had called her? A loose end. Just like Jerry. She wanted to be clueless again. Wanted to be a bystander. She _was_ a bystander, damn it—and wouldn't it be safer, ultimately, to stick to that role? She thought of that and of Jerry's blood running into the floor drain, and she shook her head.
"When he took my car he said he was going to Rhinelander."
She waited for him outside the police station as evening descended, the sky tinged with wispy purple clouds that stood stark against a backdrop of pinks and oranges. Down the street, loud music blared from speakers near the river, some sort of evening event commencing.
Jerry was dead. He'd been a cantankerous, combative employee from day one, but he'd also been the only person she was close to in the entire town. Time with Jerry made up about ninety percent of her human interaction since she'd arrived in Tomahawk, and understanding that he was gone filled her with the chill of loss. With Jerry went the shop. She couldn't run it alone. Running it with just the two of them had seemed impossible at first, but they'd made it work. The reason for that, she knew, was Jerry's willingness to stick around. He might not have liked working for her, but he'd done it, and without him the shop that her grandfather had opened sixty-eight years earlier would have already been out of business.
She was feeling the threatening rise of more tears when the door opened and Frank Temple stepped out of the station and came down the steps to join her. He held his jacket in his fist, and she saw for the first time that he was wearing a gun in a holster on the side of his chest.
"Where'd you get that?"
He didn't look at her. "Had it on when we left the cabin. Cops seemed to want to keep it, but I made a compelling case against that by pointing out that nobody was killed with a gun today."
There was a bristle to him she'd not seen before, a darkness in his voice. Atkins, probably. If he'd asked _her_ so many questions about Frank's past and his family, it could only have been worse for Frank.
The door to the police station opened again, and two cops in uniform stepped out and stared at them.
"Is your car still around?" Frank asked.
"At the shop. They were going to give me a ride, but I wanted to wait for you."
"Let's walk down there, then."
They started down the sidewalk, falling in step together quickly and silently.
"The FBI was here," she said.
"Uh-huh."
"I was surprised . . . I mean, I'm glad that the police are getting help, but I was surprised by that."
He was looking at his feet and still holding the jacket in his hand, that gun open and obvious now, as if it were some sort of statement. "The question is whether I'm the only reason they're involved."
She wasn't sure how to respond to that.
"Because if I am," he said, "then it's somewhat discouraging. I understand it, sure, but having the FBI investigate me is not going to help with this mess."
"He seemed pretty interested in you."
"Yeah, he did. As much as that pisses me off, it's no shock. I just wonder if I'm the heart and soul of their interest, or if I'm part of a package." He turned to face her. "Did Atkins say a word to you about Vaughn?"
She thought about it, then shook her head. "Not until I brought him up. When I asked about him, Atkins just wanted to know whether you seemed familiar with Vaughn. Up until that point the only thing he'd wanted to talk about was . . ."
"Me," Frank said.
She nodded.
"Okay," he said. "That's basically the same response I got, like Atkins was completely uninterested in Vaughn. Since he should be _very_ interested in him, I'm going to guess that my wonderful, well-known name is not the only draw that attracted our VIP from Wausau. They've got something on Vaughn already. That Lexus rang some bells somewhere, down in Florida maybe, or on the FBI computers. They got excited about him, and then my name was an extra wild card in the deck. They don't know what to make of it yet."
"And they don't know where he is."
The look he gave her then was both knowing and intrigued. "You didn't mention Ezra's find?"
"No. Did you?"
He shook his head. "Figured it was your play, and if you told them, they'd be back around with more questions. When that didn't happen, I assumed you'd decided not to say anything."
"I don't know why I didn't," she said. "I just . . . there was a lot going through my mind."
"You didn't say anything because you saw what happened this afternoon."
Blunt, but true. There had been a lot going through her mind, yes, but it was the memory of that blood dripping into the floor drain that made the decision for her.
"You remember what I said about loose ends?" he asked.
She nodded. "I was thinking about that the whole time. That, and everything you said about the guy who owns the cabin, Devin, and how everyone around him is so . . ."
"Deadly."
"I guess that's the word."
"It's the only word that counts right now. Whoever followed Vaughn up here made a clear statement today, and we've got to listen to it."
"Doesn't that mean I _should_ have told the FBI about the car?"
"You were the one who chose not to," he said. "I didn't instruct you on that. So what's your reasoning? Why didn't you tell them?"
She stopped walking, and he went a couple of paces ahead before turning back to her.
"I didn't tell them," she said, "because I'm scared."
"You should be."
"And there was a moment in our conversation this morning when you seemed to suggest that it would better to distance myself from the whole thing."
"I did think that, but as of right now, you have no distance. I don't think there's much chance of getting any back, either."
"So what do we do?"
"I'm not sure yet. Here's what I can tell you—the police in this town, with or without Mr. Atkins of the Wausau office of the FBI, aren't ready to deal with these guys. So I don't think you made a poor decision. I don't think that at all."
"So what do we do?" she repeated.
He looked at her, then down, his eyes seeming to settle on the butt of his gun.
"There are only two things I'm certain of right now. First one is that we should talk to Ezra."
"We should talk to a _fishing guide_?"
"He's a bit more than that, Nora."
"Okay. And the other thing you're certain of?"
He started to walk again, the gun bouncing a little with each step.
"That if you go home tonight, you probably die."
## 20
Grady hadn't dated much since Adrian left. The occasional setup, or maybe somebody he'd meet at a party and see once or twice again, but nothing serious. He had a date for Saturday night, though, a woman who worked computers at one of the major Chicago banks and had been assigned to help Grady review hundreds of transactions. He was with a team that was trying to trace terrorism dollars now, the new concern, and Helen was the liaison the bank had offered up to the Bureau. They'd spent the better part of two weeks together, going over numbers that led nowhere, and Grady had enjoyed her very much. Good-looking, personable, and able to laugh at herself, which was certainly not a trait Adrian had possessed. He wouldn't have asked her out; there was a professionalism issue as excuse, but the reality was that he'd never been good at that, getting around to the actual question. Two days after he'd broken off from the project, though, she called him at work and asked if he wanted to go to dinner. It was the first time a woman had ever done that with Grady.
He was in a good mood as Saturday afternoon wore down, went for a long run along the lake and then spent an extra ten minutes stretching, felt the week's tension leave his muscles and fade into the air. When he got back to the apartment he showered and—this was embarrassing—tried on three different shirt-and-pants combinations, feeling like a high school kid. He'd just decided on a black button-down with a dark green pair of gabardines, was still threading the belt around his waist, when the phone rang. Not the home number, but his cell, which meant it wasn't a call that he could ignore. He fastened the belt and answered the phone.
"Agent Morgan?"
"Speaking."
"Ron Atkins calling from Wausau."
Wausau? Grady knew there was a field office up there, but what in the hell could Wausau have cooking on a Saturday evening that required his attention?
"What can I do for you?" Grady said, standing before the full-length mirror and taking inventory, trying to ignore the gray hair.
"I'm sorry to bother you on a Saturday night like this, but I've been doing a little research, and it looks like you're the foremost expert we've got on Frank Temple."
Grady watched his face change in the mirror, saw it drape with concern and alarm.
"Which one?" he said. _Please say the dead one, Atkins. Tell me it's old, tell me it's something very old._
"The son," Atkins said. "Frank the Third."
Grady turned away from the mirror and walked out to the living room, the sick taste of defeat rising in his mouth. Wausau. Shit, he should have remembered. That town was maybe fifty miles from the cabin, that infamous family cabin Frank had spoken of with such warmth, the one he wasn't sure he could ever return to, the one his father had purchased with Matteson and the other soldier, Ballard.
"What's happening?" Grady said.
"I'm still trying to figure that out. Right now here's what I know: The Temple kid blew into town yesterday, with a couple of real bad boys from Miami on his heels, and we've already got one body in the morgue and a cop recovering in the hospital."
_Real bad boys from Miami on his heels._ The words spun through Grady's brain like whirring blades, and he sank down onto the couch knowing that the kid had done it. He'd gone down to Miami to settle up, he'd put three bullets into Devin Matteson's body, and Grady had sent him there.
"I've been told," Atkins said, "that you spent some time with the kid."
"Yes."
"And you brought his father down."
"He brought himself down."
"What? Oh, sure. Sure. The thing is, you know, it seems the apple didn't fall far from the tree, right? Like father like son and all that? You pick whatever cliché you want, because they all apply here."
"I wouldn't rush to that judgment."
"What rush? He's been adrift for damn near a decade now, floating all over the country without a job but with a seemingly inexhaustible supply of cash."
The supply was exhaustible—probably getting close to exhausted by now, actually—and Grady was well aware of its source. Frank's father had been clever with his banking, setting up some hidden trusts and offshore accounts that his son could use. Frank, though, actually told Grady about their existence. Explained his father's final conversation with him, which included information on the money. By the time the kid disclosed this, Grady was beginning to feel overwhelmed by his guilt, and the sort of trust Frank was showing made it worse. So instead of shutting the money down, as he should have, Grady merely warned Frank that he ought to separate himself from a bloody slush fund like that.
"I spent some time with him," Grady said into the phone, "and he seemed to have his head on right."
"Did you not hear me say I've got one dead up here already?"
"I heard that, and I understand what you're looking at. But let's not start painting Frank as the same as his father right away, all right? Not right away."
The disbelief, tinged with disgust, was clear in Atkins's voice as he said, "Yeah, when a contract killer's son floats into town, wearing a gun and leaving bodies in his wake, I suppose he really could just be on a fishing trip."
"He was wearing a gun?"
"Uh-huh. Smith & Wesson with his father's initials engraved on the stock."
Grady pressed his eyes shut. That was the suicide gun. "Tell me what you know, Atkins."
"These guys from Miami, they showed up yesterday and attacked a woman who owns a body shop here. The same body shop where Temple's car ended up after an accident a few miles north of town. Temple interceded—and I'll admit that the woman hasn't claimed any sense of familiarity between Temple and the pricks who went after her. But by the end of the day the cops somehow lost both of these guys, which takes some doing."
"Why attack the body shop owner?"
"I'm getting to that. They went after her yesterday, then killed one of her employees today. Hung him off the front end of a car and cut his throat. Nasty scene. The way it looks is that they've lost track of Temple and figured the people at the body shop were the last who'd seen him, you know, the best chance of finding out where he went. There's a guy named Vaughn Duncan involved, too, and supposedly these guys are interested in his car."
"Who is Duncan?"
"A prison guard from Florida, part of this whole shit storm that's come up north."
"Frank's not from Florida."
"Maybe not, but his old man certainly had some ties down there. Duncan's car was the one that had the tracking device. Allegedly."
"And he's a prison guard in Florida."
"Was. Evidently he called in and quit a few days ago, no warning, no two-weeks notice, hasn't even filled out the paperwork or met any of the usual requirements. The people down there are less than happy with Mr. Duncan. Seems he came into some serious cash about a year ago, too, source unknown. Guy's working as a prison guard and driving a Lexus, you know something's wrong."
"What prison?"
"Coleman."
"Which part of Coleman? It's a big complex."
There was a rustle of papers and then Atkins said, "Phase One. That mean something to you?"
Yes, it certainly did. Manuel DeCaster was locked up in Coleman Phase One. He was the big boss, the ruthless bastard who'd employed Frank's father, probably still employed Devin Matteson. This was not good news.
"Well," Atkins said. "You got any ideas? Anything I should be checking out?"
As an FBI agent, as a law enforcement officer, Grady had to tell him. Had to drop that Matteson name and news of the recent shooting, fill Atkins in on the back story. He needed to draw connections for Atkins, get the investigation rolling in the direction it clearly needed to go in. But he couldn't do it. Not yet. Not without talking to Frank. So for the second time in his Bureau career—and the second time involving Frank Temple III—Grady ignored those professional obligations, ignored his oath. In the end, he settled for a cop-out. Not a bold-faced lie, but a delay.
"I'm not sure," he told Atkins, "but I know some people I can ask. Let me call around a bit and get back to you."
Atkins seemed satisfied with that. Probably more so than he would have been if he'd known the first person Grady was going to call was Frank himself.
He still had the number, but when he called, all he got was an error message saying it had been disconnected. One perk of working for the FBI, though—if the kid had a number, Grady could find it.
He called Helen next, canceled the date with what he hoped she knew was a sincere apology, and then headed for the office. Another perk of working for the FBI—when you told a woman a work emergency had come up, she tended to believe you.
He thought about Devin Matteson as he drove, about that blood debt he'd chosen not to mention to Atkins. Grady could remember a day, maybe two months after the suicide, when young Frank told him quite emphatically that his father had never killed a good man. _The victims were all evil,_ he'd said, _and I know you can say he was no better, but the question is, was he any worse?_
Grady had argued with him that day, told Frank that no one was entitled to make a character judgment that ended another man's life—but what had he told himself, not long after that conversation, to justify the misconception he was allowing to flourish?
Devin Matteson was a bad man. That's what he told himself. Devin Matteson was an evil bastard of the first order, a killer and drug runner and thief, as corrupt as they got. So who cared if the kid thought Devin was the one who'd given up his father? Who cared if he thought Devin was the one who had, essentially, put the gun barrel into his father's mouth?
Nobody cared. Allowing him to think those things couldn't do any harm, really, so long as Frank knew better than to take action, seek retribution.
For a long time, Grady had been sure he knew better.
It was a good thing, Ezra Ballard decided as he looked at the clock for the fifth time in ten minutes, that he'd never had children. He wouldn't have done well with the constant worry.
He'd left the note on Frank's door at two that afternoon, stopping by the cabin to find that Frank was missing, which wasn't a surprise until Ezra noticed the boat was on the beach. If he wasn't fishing and he had no car, where'd he gone? A walk, maybe. That was the only answer. Ezra scrawled a note and used a fishhook to fasten it to the door, then left expecting to hear from his friend's son within a few hours.
It was dark now, had been for thirty minutes at least, and the phone hadn't rung. Ezra actually lifted it from its handset a few times, just to check the dial tone. He wished the kid would call. There was something riding the air today that Ezra didn't like, something that had taken his mind for most of the day, left him distracted, answering questions only after they'd been repeated. The woman had been in the water again this morning, that beautiful woman swimming alone in the cold lake. No sign of the gray-haired companion, no movement from the car hidden in the trees. Maybe that was good. Maybe they were nobodies, nothing to worry about.
He couldn't believe that anymore, though. Not after hearing Frank's story about the attack on Nora Stafford, two men with guns arriving in pursuit of a car whose driver had now joined that woman.
Whoever this gray-haired son of a bitch on the island was, he couldn't be anyone Ezra wanted around. Now Temple's boy was at risk, and that sweetheart of a girl who'd taken over Bud Stafford's shop, and none of that was good.
So what to do about it? Maybe nothing. Maybe it would be best to just wait it out, take his fishing parties after walleye and muskie and come home and smoke a pipe and read a book, and eventually the man and the woman would go away and things would be back to normal.
That was one option. An option he favored until the headlights of a truck washed over his driveway and Frank Temple III arrived, not alone, but with young Nora from the body shop, and right then, even before they got out of the truck, Ezra understood that this thing was not going to be one he could wait out.
They came onto the porch and sat with him and told him what had happened. He listened without speaking, as was his way. People commented on this often, as if it were strange behavior. Ezra didn't understand any other way to listen. When somebody was telling you something, particularly something important, you shut up and listened and thought about what they were saying. If you were always opening your own mouth, or thinking about what you were going to say, how much did you really hear? Ezra heard it all. Heard it, and considered it.
What he heard now, this description of a man with bound hands and a cut throat, took him back to a place he'd left long ago. Not Vietnam, either, no place so far away. Detroit was across a lake, not an ocean, but to Ezra it was home to more bad memories than Vietnam. He'd seen men die in both places, but the deaths in Detroit were a different sort of killing. In thirty years in Tomahawk, he hadn't encountered anything like them again. A throat laid open in pursuit of a dollar gained, a bullet through the eye to avenge a dollar lost, those things did not happen here. _Hadn't_ happened here, at least.
But now they'd come to him, Temple's son and the girl had, and they were right to do so. He could see the doubt in Nora's eyes, could see her taking in him and his cabin and wondering what Frank was thinking, why they were on this porch instead of in a police station somewhere. Frank understood, though. He'd learned some things from his father, some things he wished not to know. In this circumstance, at least, they would help him. Ezra hoped the kid appreciated that.
"I didn't think she should go home," Frank concluded. "Am I wrong? Are these guys already out of town, trying to disappear?"
"No," Ezra said. "You weren't wrong, and they aren't gone yet."
He was sure of that, though he hadn't seen the men personally, knew nothing of them. What he did know, just from Frank's story, was that these two were professionals who'd come all the way up here to do a job. The job didn't involve beating up Mowery or killing Jerry Dolson, and because these things were happening it was clear that the job was not done. Also clear, then, was the notion that they would not leave until it was.
The girl was tougher than he might have guessed. He could tell that in the way she stood and listened. Frightened, sure, but not panicked. Not frozen. There was a quality of disbelief to her at times, as if she hadn't reconciled with everything that had happened yet, but that was reasonable. Expected.
"So what's your advice?" Frank said. "Should we go back to the cops?"
"I don't think we'll be able to decide that until we find out exactly who the visitors are, and what they're running from."
"How do we do that?"
"Well," Ezra said, "I'd imagine asking them directly would be a good start."
Frank and Nora stood there and stared at him, no sound but the buzzing insects filling the air for a while.
"We're going there?" Frank said. "To the island?"
"I think we should."
"Without the police."
"Son, you were the one telling _me_ the risks with the police."
"I know, but . . . you're saying we go out there now?"
Ezra shook his head. "It were up to me, I'd wait till daylight. You go out there in the middle of the night, you're gonna provoke a different sort of reaction."
Frank didn't respond, and Nora Stafford looked unsettled. Ezra spread his hands and leaned forward, the chair creaking beneath him.
"Listen—you two are worried. Scared. That makes sense. And you're trying to decide what to do that will leave you the safest. Also makes sense. But you can't do that until you understand the situation. That gray-haired guy and that woman, they aren't the same cut as these men that rolled into town on their heels, but they've got some answers. Some things we need to hear."
Frank nodded slowly. "All right. So you and I go out there in the morning and try to get them to talk."
"It's my recommendation, yes."
"No," Nora said, and Ezra thought she was objecting to the whole idea until she said, "You're not going to leave me sitting in some cabin while you go out there to talk to them. I won't do it."
Nobody answered her at first. Ezra wasn't thrilled with the idea; Nora's involvement had already gone too far, in his opinion.
"They came into my shop and they killed my employee, my friend, a man who'd worked for my family for years," she said. "If anyone here deserves some answers, it's not you guys. It's me."
Tough to argue with that. Ezra just said, "That's what you want to do? Talk to them in person?"
"From what I've heard, it seems to be what you think is best. But if anyone goes out there, they'll be taking me with them."
"Fine," Ezra said. "We'll all go, then. First thing in the morning."
"What if they leave?" Nora asked.
"Be tough to leave that island," Ezra said, "without a boat."
Frank's eyebrows rose. "You're going to steal it?"
"Not steal it. Might disrupt it a touch, is all."
"What if they see you?" Nora said. "Won't that cause problems?"
Ezra smiled at her, and Frank answered for him.
"They won't see him, Nora."
Her head was swiveling between them, her lips slightly parted, eyes intense, not saying a word. Frank looked to her.
"You're in more jeopardy here than anybody. What do you think?"
"I think," she said after a long pause, "that we should know what it's all about. If they can tell us that, then I like Ezra's idea."
It was quiet for a few minutes, and then Ezra said, "You feel safe at your cabin?"
"I do," Frank said.
Ezra nodded. "You'll be safe there tonight." Ezra had a boat and a rifle with a night scope. Yes, they'd be safe tonight.
"All right," he said. "I think I ought to go address that boat on the island. You all go on back home. Rest. It's done for the day, all right? I believe that. Any trouble does come up, I'll be around, and I see you got your dad's gun in case you need it."
Frank looked down at the gun, then back at Ezra. "How the hell can you tell it's his gun when it's holstered and I'm standing in the dark?"
Ezra walked to his truck.
## 21
Grady found an active cell number easily enough, but he couldn't get through to Frank. He called five times over two hours, got nothing but an immediate voice mail, indicating the phone was turned off. He left two messages. No details, just his numbers and an urgent request to call.
What to do now? He owed Atkins information. Every hour that ticked by made him feel guiltier about that, more aware of the ramifications. If Frank was really responsible for shooting Devin Matteson, then what in the hell was Grady thinking, trying to protect him?
It would help if he would answer the damn phone. One conversation, no matter how brief, would give Grady some guidance. Some sense of how to proceed. Finally, frustrated, he picked up the phone again and called Saul down in Miami. Maybe Jimmy would have insight by now, some new development.
Saul answered on the first ring, his voice tinged with irritation. "Shit, Grady, I was gonna call you tomorrow. Should have known you couldn't wait on it till morning like a normal person."
"Wait on what, Jimmy?"
"The hell do you think? Matteson."
"You've heard about what's going on up there?"
"Up there? What . . . look, Grady, why are you calling?"
Grady stood up, the office not feeling so warm anymore, and said, "Did Matteson die in the hospital?"
If he had, then it became murder. Not just attempted, but the real deal.
"Die? Uh, no, Grady. The boy is loose."
"What?"
"Matteson bailed out of the hospital under his own power sometime this afternoon. Hasn't been heard from since."
"I thought he was in critical condition."
"He had been initially. Like I told you yesterday, he was recovering unusually well, but not well enough to be out of the hospital. Doctors seem to think he just signed his own death sentence, and nobody down here has a clue what motivated him to go. He wasn't facing charges for anything, so it doesn't really make sense."
_He's coming for Frank,_ Grady thought. _Oh, shit, he is coming to Wisconsin and he is going to kill Frank._
Then Saul said, "Best bet is he's looking for the wife," and everything changed.
"The wife?" Grady said, the word leaving his mouth as if he'd never said it before, didn't understand the meaning.
"Yeah, of course. Oh, wait, I hadn't heard about her the last time we talked, had I? That news came in a little later. Remember when I said there weren't any suspects?"
"Yes."
"Well, that's been blown out of the water. Nobody can find Matteson's wife. Originally the cops thought it was no big deal, she was just MIA, but now their idea is that she took off. Ran. Which makes her—"
"The suspect," Grady finished. Did this clear Frank? It had to, right?
"You got it," Saul said. "Problem with that is there were no signs of a rift with Matteson, no indications of an affair. But, still, until she turns up, that's what'll occupy the focus. Could be she's dead, too. Could be—"
"Devin left the hospital today," Grady said, no longer interested in hearing the theories, suddenly sure he knew more than anyone in Miami did.
"That's right. Nobody knows where he's headed, either."
"I do," Grady said.
"What are you talking about?"
"I think I know where he's headed. I could be wrong, but I doubt it."
The irritation left Saul's voice. "What's going on up there, Grady?"
"I don't know yet, but here's my advice: If you want to find Matteson, you check on the ways out of Miami to Wisconsin. Check every flight that left there yesterday with any destination in Wisconsin. Could be he's driving, but I doubt it. I think he'll be in a hurry."
"How do you know this?"
"I'll fill you in soon. First I've got a question: Does the name Vaughn Duncan mean anything to you?"
"Nope."
"Check him out," Grady said. "He's a prison guard from Coleman, and he's up in Wisconsin, tangling with Frank Temple and a couple others. Check him out and get back to me."
He hung up on Saul's demands for more information.
The cabin was dark when they entered, and the memory of Jerry's body sickened Nora as she followed Frank through the door. She'd felt so in control as they'd walked into the shop that afternoon. In charge, ready to take on the world. These guys had shown up and caused trouble, but now she was going to set them straight. It would be that simple. Then she'd gone through the door and seen the body and the blood, and everything she believed changed.
There was no blood here, nothing out of place in the cabin, no traces of unwanted visitors. Even so, she was nervous until he got all the lights on and showed her around the place. He was still wearing the gun, and she was troubled to realize that it comforted her. She'd never cared for guns before.
"I thought about taking you back to your house so you could get some things together," he said, walking back into the living room, "but it wasn't worth the risk. If they're watching anyplace, it would probably be your house."
"Right."
"Ezra keeps the place stocked even though nobody's here," he said. "There's an extra toothbrush in the bathroom, soap and shampoo, should be everything you need. Well, I don't know what you—"
"Don't worry; I won't require any feminine products."
She'd been trying to make fun of him, get a laugh and reduce that tension he'd acquired. It didn't work. He just nodded, still looking ill at ease.
"I was joking," she said.
"Yeah. Look, you can take whatever bedroom you want."
"What's wrong?"
He frowned. "Nothing. Just telling you to make yourself at home."
"You seem anything but comfortable with me right now. Maybe I should go to a hotel. Maybe that would be best."
It was quiet for a moment, and then he said, "I'm not like them. I need you to understand that. I'm not anything like them."
She stared at him. "The guys who killed Jerry? Are you kidding me? Of course you're not anything like them."
He leaned against the wall, looked down to the gun, back up at her.
"I'm not like him, either."
"Your father."
He nodded.
"I don't think otherwise."
His eyes were so damn sad when he said, "You would have liked him."
She had no idea how to answer that.
"Everybody did," he said. "You would have, too."
"Is that what scares you the most?"
"What?"
"That you loved him. That you thought he was good."
He looked at her for a while without speaking, and then he walked to the door and went outside.
She found an open case of beer in the refrigerator, took two bottles, and went after him. He was sitting on the log wall that held the soil back from the beach. He didn't look away from the water until she handed him the beer.
"Thanks." He took the bottle and pointed at the lake with it. "I can see why Ezra never left."
"It's a gorgeous place." The air was warm again tonight, but the sky was overcast, only a handful of stars showing. The wind that had blown so hard in the morning was almost gone now, nothing left but a few gentle puffs trying to catch up. It had been one of those weird warm weeks, each day feeling more like summer than spring, then settling down overnight until you woke to a cold sunrise.
"You don't need to worry," she said, "about anything I heard today. Nothing the cops told me was different from what you'd told me this morning. I'm not scared of you. I don't think you're dangerous."
"Then the old man failed. He spent a lot of time trying to make me dangerous. It would break his heart to hear to you say that."
She thought of the way he'd come across the body shop, unarmed, the day before. How long had it taken him to knock that guy out with the wrench? Two seconds, tops. So was he dangerous? Maybe he could be, but she wasn't scared of him. It wasn't that sort of quality.
"I'm sorry," he said. "I'm not going to make you sit here and be my therapist. It's just that a day like today puts him in my mind more than normal. Just being up here, seeing this place and seeing Ezra . . . it works on me."
"I believe that."
He drank more of the beer, leaned back from the wall, and braced the heels of his hands on the grass. She could feel a prickle in the middle of her back, touched off by sitting out here in the open, surrounded by darkness. Wasn't he afraid? Didn't seem to be. Either he thought they were completely safe here, or he thought he'd sense any trouble before it came.
"Where's your mother?" She surprised herself by asking the question; one second it had been in her head and the next she'd spoken it, without ever planning to. There was just something about him right now that made her curious, some rootless quality, as if he'd always been alone, drifting along in the company of bad memories.
"Baltimore."
"That's where you grew up?"
"No. That's where she is now."
"Were they divorced?"
"Not officially. They were together until I was fifteen. She picked up on some changes that I missed, I guess, or maybe he had more trouble facing her than me. Anyhow, things got bad, he moved out, got an apartment. They were still married, and he was always saying they'd get back together."
"Are you close to her?"
The personal questions that had bothered him so visibly the previous night now seemed almost welcome, taken in stride.
"Not by happy-family standards, I guess, but in a different way. A deeper way, maybe. We're all we have, you know? To go through something like we did . . . that's a different sort of bond than what I'd like to have, but it's there. We talk on the phone, I see her every now and again, holidays, that sort of thing."
He drummed his fingers on the beer bottle. "She moved back to Baltimore, where my aunt lived. Got remarried a few years ago. And that's great for her, don't get me wrong, but the first time I saw the guy . . ."
"What?" Nora prompted when he left the sentence unfinished.
He shook his head. "It's not an impressive thing to admit, a little too much testosterone in it, but when I met him I was just so disgusted. And angry. Because he's this little guy with a paunch and a soft chin, a pharmaceutical sales rep who wanted to take me _golfing,_ and I took one look at him and thought, _You've got to be kidding me._ Because he was so, so far away from what my father had been."
He lifted the beer, took a drink. "Later that night, though? I thought about that, and realized that of course he would _have_ to be. For her. He would have to be so far from what my father had been."
She was sitting with her knees pulled up to her chest, arms around them, facing the water. The way he was leaning back put him behind her, so she couldn't see his face, just hear his voice. He seemed more comfortable that way.
"He killed people, and he took money for it, and that seems so _obviously_ evil to most people . . . but I wish they'd met him," he said. "Not that they would change their minds, or should. But now he's a monster, you know? And there is no more one-dimensional character than a monster. If Dad was anything, it was multidimensional. He wasn't all darkness. Sometimes I wish that he had been."
"I'm sorry," she said. What a horribly hollow phrase.
The silence was interrupted by the soft sound of a motor somewhere out on the lake. It was too quiet to be a big engine; maybe one of those trolling motors, instead. She couldn't see any lights. Only when she sat up straighter to stare out at the water did Frank speak.
"It's Ezra."
"That's him in his boat?"
"Yes."
"How do you know? There are no lights. What if it's—"
"It's not them. Whoever's in that boat took it out around the sandbar, and you don't do that by dumb luck."
It was clear he'd been aware of the boat for some time.
"You're sure it's him," she said.
"Yeah. I knew he'd spend the night out there."
"How did you know?"
"Because he takes his responsibilities seriously," Frank said. "And tonight, we're on that list."
"Is Ezra from the South?"
"No."
"He talks like it. Has that drawl, the twang."
"Uh-huh."
"Where's he from?"
"Detroit."
"Detroit?" She raised her eyebrows. "Wow. I wouldn't have guessed that. He talks like he's from someplace far away from there."
"Yes," Frank said. "He does."
The motor had disappeared now; she could hear only the gentle thumping of water on the beach and the occasional creaking of a tree in the woods behind them. How was Ezra intending to watch over them from the water, in the dark? Must have some of those night-vision goggles. Or perhaps he could see in the dark. She could imagine Frank informing her of that detail in his detached, matter-of-fact voice: _Ezra can sense heat from a thousand yards away. He'll know if anyone else shows up._
She was smiling at that, her own private joke, when Frank said, "So what's your story?"
"What do you mean?" She turned back to him.
"You're from Minneapolis. Your dad lived up here, ran that body shop, and then he had a stroke."
"Yes."
"So what else is keeping you here?"
"That's not enough?"
He leaned forward, so that she could see his face again, and lifted the beer to his lips. "Could be. I'm asking you if that's all there is."
All the months she'd been up here now, and nobody else had asked. Everyone just assumed it was all about her father, trusted her good intentions. Now Frank picked them apart as if the ulterior motive couldn't be more obvious.
"I came for Dad," she said carefully. "He needs someone here, and I don't want that shop to close."
He didn't answer.
"But there might have been some things going on in my life that made staying here seem more appealing."
"Okay."
There was a long pause, an obvious cue for him to inquire further, but he didn't.
"I was engaged," she said.
"Yeah?"
She nodded. "Had been for three years."
"Long time to wait."
"That's what he said."
Frank's laugh, low and genuine, caught her by surprise and relaxed her. She leaned back and twisted a little, facing him directly.
"The wedding date was set for a month ago, in fact. Came and went and here I remain. I had his full support when I came out here, after Dad's stroke. Longer I stayed, though, the pushier he became about me coming back to Minneapolis. And I realized a couple of things. One was that I didn't want my father to be alone in a nursing home, nobody coming to see him, his shop going out of business, all the rest of that. It seemed so wrong."
"And the other?" Frank said.
"The other was that I didn't want to get married. I'd been dating him for five years, living with him for two, and yet I kept coming up with delays."
"Just reluctant, then? No specific reason, no epiphany?"
She started to nod, then stopped, wondering if agreement would be a lie. There had been an epiphany of sorts. A party not long after their engagement, when Seth had turned to a group of people to introduce Nora and said, _This is my fiancée_. She'd waited for the _Nora_ to follow, but it never came. A simple thing, maybe, a minuscule issue of semantics, but in that moment it hadn't felt simple or minuscule. It felt chilling. Because she knew that he hadn't misspoken, knew that her identity, at least in his mind, had already been completed. The name was irrelevant; she was his fiancée. His possession. Standing there in an uncomfortable dress making phony smiles at people she'd never truly trust, she flashed forward twenty years and saw herself introduced as _This is my wife_ , again no name behind it, then saw herself kissing his neck and running her hands along his back so he'd write a check for her daughter's trip to Europe, saw herself doing that and not even recognizing it for what it was.
A powerful moment, one that had stayed with her as few others had and one that she'd never discussed with anyone, and would not discuss now, with Frank. It was a bit too much, a bit too personal.
"When I had to come up here, it was the first time I'd been on my own in a long while," she said instead. "It felt good. The other thing, and this was a good deal more personal, was that he made too much money. Not millions or anything, but enough that he wanted me to forget about a day job and concentrate on my art."
"Generally considered a positive thing."
"That's what I thought. First I had my mother and my stepfather taking care of me, spoiled little shit that I was, and then the future husband promising to do the same thing. Wonderful, right? But when I came up here and started going through my dad's things, really looking at his life, at how hard he and my grandfather had worked to make a living off that crappy little shop . . ."
"Made you feel soft?"
"Made me realize I _am_ soft. My dad got up at three in the morning when it snowed, ran the plow till eight, then came back and opened his shop up and worked all day. Would run the plow again in the evening, if he had to. Did that all winter, for thirty years. When I went back through their books, I saw that there was _never_ a time when that shop did more than struggle to keep bills paid, but they kept them paid. For sixty-eight years, they kept them paid."
The wind blew hair into her face and she pushed it back.
"I've never worked for anything. Not that counted. I worked for good grades, worked on my art, but that's not the same. I've never _had_ to work hard, never had my back to the wall in any way in my entire life. I suppose it's awfully childish of me to say that like it's a bad thing. I suppose I should just be grateful."
"Is that what the fiancé told you?"
"Among other things."
"So you called it off?"
"He gave me an ultimatum."
"Poor bastard. Hate to bluff on a play like that."
"I guess."
"A name," Frank said. "I require a name."
"Seth."
"Horrible."
"Someone named Frank is criticizing another guy's name?"
"Frank was half of the first names of the Hardy Boys. It doesn't get any more solid than that."
She was laughing again, and he seemed to have drawn closer without ever moving, and there was a sudden intimacy to the evening that absolutely did not belong. Even while understanding that it didn't belong, she didn't want it to go away, either. There was a pause that went on a few beats too long, his face close to hers.
"This is where you tell me what a shitty kisser Seth was," Frank said. "To inspire me."
"Inspire you to what?"
He didn't respond.
"Can't do it," she said. "He was actually a very good kisser."
"She sets the bar high," Frank said, and then his hand was sliding across the back of her neck and pulling her forward and his lips were on hers and, what do you know, he _was_ better than Seth.
## 22
It was the first time he'd watched anyone kiss through a rifle scope. When Ezra realized what they were doing, he dropped the gun and looked skyward, despondent. Just what this mess needed. What the hell was the kid thinking?
He lifted the gun again, watched them for a few seconds, Nora Stafford beautiful even bathed in the wavering green light of the scope. Okay, _that_ was what the kid was thinking. On second look, even through the scope, it made a hell of a lot of sense. Well, good for Frank.
Very good, really. Ezra hadn't liked the boy's look this evening. Reminded him too much of other men he'd known, in other places. It wasn't the way he wanted to see the young Temple go. The girl could be good for him. Once they got this shit cleared up, got those sons of bitches from Florida or wherever they were from sent on their way, the girl could be a wonderful thing for Frank. Ezra hadn't seen her many times, but enough to know that she was a different cut than others her age. As was Frank. Certainly, as was Frank.
Ezra went into the water two hundred feet from the island, swam naked through the cold lake, spent all of thirty seconds on the motor, and then swam off into the darkness again, leaving the outboard disabled. Whole thing had taken maybe five minutes, but they were minutes that took him back to those other places, those other times. Damn, but the three of them had been close once. You got a different sort of close in combat. He'd read a couple of textbooks on the brotherhood of battle once, written by psychologists sitting in university offices in towns that held peace rallies even when the country wasn't at war. Read just enough to know that the authors didn't understand their subject, and then gave up and moved on to other books.
The lake had healed Ezra. That was something none of the psychologists would understand, but it was as true as anything he'd ever known. This place had _healed_ him. He liked to believe the violence had drained away slowly, that the lake and forests had soaked it up, taken it from him. What he feared, time to time, was that it hadn't drained away at all. That it was just a little better hidden.
He was esteemed throughout the area as a hunter, but what he never told anyone was that he'd let the real trophies pass by. He'd looked at some amazing bucks in the scope, and once at a bear that went a good eighty pounds beyond any he'd ever seen in these woods, and he'd turned away from them. Let them go in peace. Just to prove he didn't even have to hunt, didn't have to kill, didn't have to squeeze the trigger ever again if he didn't want to.
The lake had done that for him.
He was glad when Frank and Nora stood up and went into the cabin, left him to watch nothing but the building and the woods through his scope. He didn't want to watch people through the scope anymore.
They'd been kissing for a while when Nora put her palm on his chest and pushed him back.
"What?" he said, breaking away. "I shouldn't have done that?"
"No, it's just"—she was smiling at him—"not really the night for it, you know?"
He was looking at her hard, trying to read her, but she'd turned back to the lake.
"Is he still out there?" she asked.
"Ezra? Yes."
"I wonder if he was watching that little display."
"Probably."
"He must think I'm a slut."
"I doubt it. But if you want to try to convince him . . ."
She laughed at that and got to her feet, dusted off her pants. He stood up, too, stretched, and tried not to stare at the lake, searching for the boat the way she had.
"Look," he said, "I'm sorry. If I did anything—"
"It's fine." She held her hands up, palms out. "Relax, okay. You didn't do anything wrong, I just don't want to let that moment turn into something it shouldn't."
"You can trust me," he said and immediately regretted it. He was coming on too strong, overreacting to her withdrawal. She was right—this wasn't the night for it. Not at all. But his first thought when she'd pushed him back was that she was scared of him. It had been in his mind earlier, when he brought her to the cabin, and hadn't left yet. He was not like the rest of these people who had invaded her life, and he _had_ to have her understand that. He was not dangerous.
"I do trust you," she said, but now the awkwardness between them was evident. "Don't worry about it. It's cold, though. So maybe we can go inside?"
She rubbed her upper arms in a false gesture, and he just nodded and walked to the door and held it for her, thinking that there was a reason he was so concerned about whether she was afraid of him, and it came from a place of guilt. Even while he implored her to trust him, while he said he would protect her, he was also asking her to believe in a complete fraud—the idea that his connection to all of this was circumstantial, an extraordinary coincidence. It had been easier to ask her to believe that than to tell her that he'd come here to kill someone. What would she do if he told her that? What sort of response would that little disclosure provoke?
They drank another beer but kept their distance. She sat on the couch, and he took the chair beside it. They spoke in soft voices for about an hour, and the conversation slowed and vanished, and after a while he realized she was asleep. He repositioned her on the couch, got a pillow under her head and a blanket over her body, then sat beside her and willed sleep to stay at bay. Wasn't fair to make Ezra do all the watching.
He thought about his cell phone at some point as he sat there in the dark, got quietly to his feet, and went to find it. He'd turned it off after going into the police station, and now he turned it back on, heard a chime indicating a message was waiting. The sound seemed loud in the quiet house, and he looked over at Nora. She didn't move, and her breathing stayed slow and deep. He played the message.
Grady Morgan. The sound of Grady's voice hit him as hard as Ezra's had a few days earlier, maybe even harder. Frank scarcely took in the content of the message at first, just that voice. There was a tension to it, maybe even some anger.
Grady had heard about the day's events, of course. Frank should have expected that. As soon as Atkins showed up, he should have expected that Grady would get a call. If he'd been thinking, he would have preempted Atkins, explained things and given Grady some warning. Now Grady had heard only Atkins's side of things, and he was worried.
Frank looked at the clock, saw it was nearly one. Late for a call, but Grady's tone had some urgency. He probably wouldn't mind.
After one more check to ensure that Nora was sleeping soundly, Frank slipped out the door and into the night to make the call.
Grady had been asleep for about an hour when the phone rang. He turned onto his elbow, reached out and grabbed the receiver, mumbled a hello that was thick with sleep.
"It's Frank."
Grady said, "What the hell kind of trouble did you get into?"
"Atkins didn't tell you?"
"I've heard what he said. Now I'm asking _you_."
"Can't tell you anything different except that these guys are involved with Devin. That much seems clear."
"Frank . . ." He wanted to ask the question, _had_ to ask it, but it didn't leave his lips.
"What?"
"Here's how I'm going to phrase it. And, damn it, you tell me the truth. Were you down in Miami when it got started?"
_Say no, say no, say no. Don't make this my fault, Frank, don't tell me that a man is dead and you're going to prison because of the lies I told so many years ago . . ._
"Grady, I haven't been to Miami since my dad took me down eight years ago. I don't intend to head that way, either."
Grady moved the phone away from his mouth so Frank couldn't hear that exhalation of pure relief. It didn't sound like Frank was lying. He'd been nowhere near Miami, had not put those bullets into Devin Matteson.
Grady said, "So you're telling me that—"
"What happened in Miami? You seem to know a lot more than I do."
"Devin Matteson was shot."
Silence filled the line. Grady's eyes were adjusting to the dark, turning shadows into furniture around the room. He sat and waited.
"Just shot," Frank said at last, "or killed?"
"Shot. Three times in the back. But he didn't die."
"Of course not," Frank said. "Wouldn't be that easy."
"Listen . . . are you telling me the truth? You really have nothing to do with Matteson?"
"I do not."
"Then why are you up there?"
"You still want the truth? Because I heard he was headed this way."
"Who told you that?"
"Ezra Ballard."
"Well, I wish you hadn't gone up there," Grady said. "You should have called me. Now look at what it's turning into. Stay away from Ballard, and stay away from whatever ideas he's got."
"Who said he's got any? But I don't want to argue with you. Tell me what you know about Devin. Who took him out?"
Grady sighed and rubbed his eyes. "Nobody knows. But his wife went MIA the same night he got popped, so that's taking the focus."
"You mean she might have done it?"
"That, or she bolted with whoever did."
"His wife disappeared." Frank's voice seemed to have tightened.
"Yes."
"You got a description of her? Know anything about her?"
Grady paused. "Why are you asking?"
"You surprised to hear I'm interested in whoever shot Devin?"
Grady didn't like that answer. He'd spent enough time talking to Frank to recognize when he was being evasive.
"Agent Atkins was telling me about the guy you got tangled up with," he said. "Vaughn."
"Is that his name?" Frank said, but it sounded false.
"Yeah, it's his name."
"Told us his name was Dave. That's about all I can offer. That, and he's a nervous little prick. Jumpy. Doesn't fit the mold for the rest of them. What do you know about him?"
"Nothing, yet. But people in Miami are looking into him."
"That's where he's from?"
"Originally. He works at Coleman, though. As a guard."
"Coleman," Frank echoed, and Grady knew he remembered, knew he was thinking about Manuel DeCaster, drawing all the same connections Grady had.
"Atkins seems to think the guy is still in your area, though," he said. "Thinks that's what today's killing was about."
"Yeah, that's the simple math."
"Do you know where he is?"
Silence.
"Frank, if you know, tell me."
"I hadn't seen the guy before yesterday, and I haven't seen him since."
Again the evasiveness. Grady said, "Frank, listen to me—I want you to leave. Get in your car first thing in the morning, and get out of there. Will you do that?"
"Atkins might not like it."
"I'll deal with Atkins. You need to get out of there."
"Devin's going to make it? Three bullets weren't enough?"
"He was recovering."
"Was?"
Grady hesitated. "Yeah."
"So what changed?"
"He's gone, Frank. He left the hospital against doctor's orders, and he's gone. Now, I don't know what the hell is going on up there, but I think he wants a part of it. And you need to be gone when he gets there. All right? You need to be _gone_."
Frank didn't say anything, but his breathing had changed, slowed.
"Are you listening, Frank? Get out of there, first thing in the morning."
"I don't have a car," Frank said, and there was something in his tone that made Grady get out of bed and onto his feet.
"Look, I'll drive up there myself. I'll drive up and talk to this Atkins guy and then you can ride back down with me. Leave them to figure it out. That's what you've got to do, Frank."
"No, Grady. You stay down there. Okay? You stay down there."
"Frank—"
"Thanks for the insight, though. This is important to know."
"If you know where Vaughn is, you've got to tell—"
"I'll talk to you soon, Grady. Thanks again."
He hung up, and Grady swore loudly into the dead phone. The conversation had ended too fast. Grady should have told him. It was time now. He had to tell him. He turned the light on, and blinked against the harsh brightness until he could see the numbers clear enough to call back.
Frank had turned the phone off again.
## 23
The flashlight blinked three times, then stopped. Ezra waited for the pause, then hit the lights on his boat, just tapped them on and right back off, enough to show Frank that he understood the signal.
It was almost two in the morning, and Frank wanted Ezra to come in? This couldn't be good. Ezra ignored the outboard—too noisy—and turned the trolling motor on, brought the boat in to the beach with no sound but that soft electric whir. Frank met him in the shallows, waded out, and took the bow line and threaded it through the U-bolt Ezra and Frank's father had bored into the log wall long ago.
"You all right?" Ezra stepped off the boat and onto firm ground.
"We're fine."
"Then what's the problem?"
There were no lights on the boat or outside the cabin, and Frank's face was only a few shades lighter than the shadows that surrounded it.
"Devin's on his way."
The wind was blowing warm and steady out of the southwest, and Ezra turned his face into it, breathed it in.
"How do you know?"
"Just talked to Grady Morgan. You remember him?"
"FBI."
"That's right."
"Didn't seem to be my biggest fan."
"Didn't know you."
"Sure," Ezra said. "Well, what did Mr. Morgan have to say?"
What Frank told him then made some sense. Made a lot of sense, actually, because the one thing Ezra had never been able to get his head around was why Devin would possibly have called him and told him to open the cabin up. The only reason he could have understood was if it had been a taunt, Devin deciding he'd screw with an old man's head, make it damn clear that Ezra no longer intimidated him or never had. Problem with that was the tone of the call. The message had been simple, businesslike, as if he'd never had a problem with Ezra. The answer, Ezra understood now, was that it hadn't been Devin who made the call. The other guy, Vaughn, had apparently understood Ezra's role as caretaker, but it didn't seem he knew the back story.
"He's out of the hospital," Ezra said when Frank was done, "with three bullets in him?"
"That's what I've been told." Frank was wearing jeans and a T-shirt, and Ezra could see the muscles of his chest and shoulders under the shirt, taut and hard in the easy, natural way they could be only when you were young. Ezra could remember when he had looked like that. Could remember when Frank's father had looked like that. The boy's features didn't resemble his father, he'd taken after his mother in that way, but the way he stood now, the energy in his words, the eagerness for battle . . . those traits ran warm through his blood.
"Sounds like Devin's hurt bad, then," Ezra said. "Hell, he might not make it up here, son."
"But you know he's coming," Frank said. "You know he is.
That's his wife out there on that island, and either she shot him or Vaughn did. They betrayed him, tried to kill him. You think he could be headed anywhere else?"
Ezra didn't answer, and after a few beats of silence Frank said, "He gave my father up. Brought him into it, and then turned right around and gave him up to save his own ass."
"I know the story, son."
Frank extended his arm, pointed out across the dark water. "He's coming for them, Ezra. The people out there on that island. Why? Because they tried to take him down, and that's something I sure as hell respect. They did our work for us."
"Unsuccessfully."
"Fine. Unsuccessfully. But I'm not going to let that son of a bitch come out here, to the place my father and his father and you and me all shared, and kill those two, Ezra. I'm not."
"At least one of those two is headed for jail, Frank. You don't want to interfere with that."
"You want to see them go? You want to see them go to jail for shooting _Devin_? Don't you remember—"
"I remember it all," Ezra said, and there was a depth of anger he hadn't heard to his own voice in a long time. "Don't stand there and ask me if I remember. It goes back a hell of a lot farther than you, back to places you'll never see and can't imagine. Understand _that,_ son?"
There was fury in his words, and he was leaning into Frank, his face close, but the boy didn't back away. Just stood there and held Ezra's eyes for a long time.
"Yeah," Frank said at last. "I understand that. Now you listen to yourself, hear what you just said, and explain to me how in the _hell_ you're going let Devin go out to that island."
"Didn't say I would. I'm telling you there's another option here."
"The police? Shit, Ezra. You want somebody to go jail for trying to kill _Devin_?"
Ezra looked away, out into the lake, and said, "What do you want to do, then?"
"To get them out of here," Frank said. "Is that so much to ask? We get them out of here. If he catches them somewhere else, fine, but he's not going to settle up here. Not on this lake."
"Get them out of here," Ezra echoed. "That's your goal?"
"It's what I said."
"And when you get in the middle of it? What then? Devin comes at you, or comes at the girl inside your cabin, the same way his boys already have?"
"If that happens," Frank said, "we deal with it."
Ezra gave a low, ugly laugh. "That's what you're hoping for. You want to hang that son of a bitch from one of these pines, but you also want it to be justified."
"It's already justified."
"Bullshit, son. Not in a way you can accept it's not, and you know that."
Frank didn't answer. The wind picked up and the water splashed into the logs below them and something rustled through the woods a few yards away.
"It's going into action tomorrow," Frank said eventually. "Whether it's cops or Devin or those two assholes he sent up here, somebody is going out to that island. Are we going to let them do it? Are we going to step aside and wait for that, pretend we don't know anything?"
Ezra took a few steps away, knelt and dipped his hand into the lake, cupped his palm and held the water. It was cool against his skin, cool enough that the hairs on his arm rose in a ripple. He kept his fingers tight, held the water until it slipped through the fractional gaps and fell back into the lake, and then he turned to his old comrade's son.
"No," he said. "No, we're not going to step aside and wait for that."
## 24
Grady woke sometime before dawn with the knowledge that he had to play it straight with Atkins.
There was no way around it. Not at this point. He kept hearing Frank's insistence that Grady stay in Chicago, hearing the way he'd said, _I don't have a car_ , when Grady urged him to leave. The kid was waiting for Devin, no doubt about it, and the smart money said he was going to get him, too.
Someone needed to intercede, and Atkins would be more than happy to do so. If Grady's suspicion turned out to be accurate, and Frank did know where Vaughn Duncan was holed up, it was going to turn into an ugly day. But that was the sort of ugly day that paled in comparison to the one they'd see if Frank met Devin Matteson up in those woods.
It wasn't yet six, too early to call Atkins, and Grady lay awake in the bed for almost an hour, watching sunlight fill the empty room and wondering how much of this was his fault.
It had been an anonymous tip, damn it. That's what he told people from the beginning, what he'd assured them, and there were only a few people within the Bureau who knew the truth. On one level, he'd almost been showing _kindness_ to Frank by telling him the tip had come from Matteson. It had seemed, back then, a lesser punishment to the boy, who was already reeling. Matteson was a worthless piece of shit, so what did Grady care if he'd added another layer of tarnish to the man's name? Even in the worst-case scenario, one in which the kid plotted some act of vengeance, all that stood to be lost was Matteson, right? And that would be a damn favor to the community.
Except Matteson wasn't all that stood to be lost. Grady had forgotten about the avenger. He could be lost, too.
At ten to seven he called Atkins and got no answer, left a message. At seven twenty, pacing the apartment with a cup of coffee going cold in his hand, he called again and left another message. Five minutes later Atkins finally called back.
"Didn't hear the phone," he said. "I was in the shower, sorry. What do you have for me?"
Grady lifted the cup, took a swallow of room-temperature coffee, and said, "I think Frank Temple knows where all the excitement is headed."
"Pardon?"
"He didn't kill anyone," Grady said, "and I'm almost certain he's more a bystander than anything else, but I think he might know where Vaughn Duncan is."
"Why do you think this?"
"I spoke with him last night, and when I asked where Duncan was, he was very guarded. Evasive. That's not Frank's style. He doesn't like to lie, and I think he was trying to avoid that last night by refusing to answer the question. If he didn't know, or didn't have some idea, at least, he would have told me that."
A long pause.
"You there?" Grady said.
"I'm here." The other agent's voice was drawn tight with anger. "I'm just wondering who I'll have to call in Chicago to make a formal complaint."
"Because I talked to Frank? Listen, Atkins, you don't—"
"No, I'm not going to listen. What you just did is such a flagrant breach of conduct . . . what the hell were you thinking? I tell you this kid is a _suspect_ , I ask you for input, not to get on the damn phone and—"
"I knew I could get you some answers."
"Bullshit. And even if you did think that, you don't make a call like that without informing me first."
"Atkins, you're missing the point."
"This is one of the most egregious—"
"Vaughn Duncan may be up there with another man's wife," Grady said. "You want to know who the man is, or not?"
Silence.
"A guy named Devin Matteson was shot in Florida a few days ago. Matteson is a key player for Manuel DeCaster. That name mean anything to you?"
"No."
"Well, it does down in Florida. He's one of the worst they've got, and one of the most powerful. He's in prison now, in Coleman, and about seven years ago Frank Temple's father was making hits for him."
Atkins didn't make a sound, but Grady could almost _hear_ the battle going on within him, curiosity fighting anger.
"Matteson won't tell the police who shot him. But his wife is missing. So it's not much of a puzzle, is it? And now this guard from Coleman, Vaughn Duncan, he's up at that lake, and there's a woman with him," Grady said. "You want to take odds on who the woman is, I've got a retirement account I'll put on Matteson's wife."
Atkins started to speak, but Grady rushed ahead. "And Matteson's missing, he's out of the hospital, he's gone. You understand what that means? He's coming north, Atkins. I would bet every dime I have that he is coming up to that lake."
"And I'm still supposed to believe the coincidence," Atkins said. His voice was clipped, tight.
"What coincidence? That Frank's up there?"
"That's the one. Bystander? _Bystander?_ You out of your mind, Morgan? You really believe, and expect me to believe, that this kid just _happened_ to smack his Jeep into his own father's filthy history? That's an _accident_?"
No. It wasn't an accident. Couldn't be. Grady wanted to believe that it was, but he knew better. Frank's presence on that lake wasn't a fluke. That was why it was so important for Atkins to intercede now.
"Look," Grady said, "I'm not going to waste my time or yours discussing what I think about that kid. I'm telling you that—"
"I cannot believe you called him. You son of a bitch, you called a _suspect_ and warned him—"
" _He knows where Vaughn is and he knows Devin is coming for him!_ " Grady shouted. "Would you shut up long enough to understand that, Atkins? You want to bitch and moan about me, do it Monday. Shit, call Quantico, call Washington, call anybody you want. Right now that doesn't mean a damn thing. What matters is that you've got a dangerous son of a bitch headed your way to settle up with his wife and this other guy, and Frank Temple knows that."
"How does he know that?"
"Because I told him. And I'd tell him again today, and if you want to get me fired over that, knock yourself out. But you've got something up there waiting to _explode_ , and you need to deal with it."
"Temple is at his cabin?" Atkins said, his voice still angry, but lower.
"Yeah. It's out on some lake—"
"I know where it is."
"Okay." Grady hesitated, then said, "If he's not there, he might be with a man named Ezra Ballard. You need to get him away from Ballard."
"Who is he?"
"He was in special forces with Frank's father. They were tight."
"Special forces with Frank's father," Atkins echoed. "You're kidding me, right? Morgan, this is unbelievable. I needed to know all of this yesterday!"
"You know it now," Grady said.
"I will tell you this, Morgan: If you call that kid again today I'll see that charges are brought, do you understand?"
"I won't call him, if you get your ass moving and get out there. And I'm headed that way, Atkins. I'll leave now, but I don't know how long the drive is."
"Keep your car in the garage, asshole. I don't want you within a hundred miles of this."
"I'm coming up."
"Yeah? Well, if I see you, I'm putting you in handcuffs."
Atkins hung up. Grady stood with the phone in his hand for a minute, then set it on the counter, poured his coffee into the sink, and picked up his car keys.
Frank had coffee going by the time Nora woke up. Her hair was fuzzed out an extra six inches from static, and when she looked at him it was with one open eye, the other squinted almost shut.
"What time is it?"
"Ten past seven."
"Have you heard from Ezra?"
"I expect he's on his way." Ezra's boat had been missing when the sun rose. Frank had even used binoculars to search for it, but found no trace.
"Then we'll go to the island," she said.
"Yes." Frank took the coffeepot and poured a cup, then brought it to Nora. Seeing her this way, bleary-eyed and sleepy, made him want to lean down and kiss her forehead, but he didn't. He wasn't sure how she'd react to him this morning, with the alcohol and that brief moment of romance in a tense, fear-filled night now pushed aside by sleep and daylight.
He returned to the kitchen and poured his own coffee, waited for her to express a new concern, wonder aloud whether they should call the cops. She didn't say anything, though. Just drank the coffee and smoothed her hair with her palm, then rose and went into the bathroom, reemerging five minutes later looking more awake, fresher.
"Did you sleep?" she said.
"No."
"Aren't you tired?"
"No," he said, and this was the truth. Anticipation of Devin's arrival provided more fuel than sleep would have. He was ready for him, but Nora was a problem. He and Ezra had discussed that before Ezra slipped back into his boat and out into the lake.
Frank would not deceive her. Couldn't do that. But he knew, and had tried to explain to Ezra, that she was not going to be kept off that island. He remembered the steel in her voice when she'd said, _If anyone here deserves answers, it's not you guys. It's me._ She was right. He wished she weren't a part of it, but she was, and he needed to decide what to tell her, how to explain what he was going to do. That would have to come after they heard what the pair on the island had to say.
He left her in the cabin and walked out into the day, found the air to be uncommonly still, the gray water like dirty glass. He stood with his coffee in hand and turned in a full circle, took in the lake and the trees and the sky.
Which direction would Devin come from? Would he drive right up to the dam, launch a boat and head out to the island, or would there be more to it than that? He knew they were on the island. Surely his advance team of gunslingers had reported that back to Miami, and while Tomahawk might have been a mysterious destination for them, it would not be to Devin. By now the island was anything but a hiding spot to the two who waited on it. Frank saw the game plan clearer this morning, understood that Jerry Dolson's murder had been merely routine maintenance, the removal of one of those loose ends he'd been worried about from the start. Either by then or soon after, the pursuit had effectively ended. The men from Miami knew about the island, had to know by now, and yet they had not moved on it. That meant one thing: They were waiting on Devin.
"He'll be here today," Frank said. He'd spoken in a soft voice, but it still rang out loud. There was no trace of wind to whip the words away.
***
Ezra arrived by boat, and when he saw that Nora still intended to go to the island with them, he didn't object. She watched his eyes go back and forth between Frank and her and wondered what he was thinking. Had he seen them outside the night before, as Frank predicted? Probably. There was something about Ezra that gave you the feeling he'd been watching you for a long time.
"All right," he said as she and Frank walked down to the boat. "We're going out there to hear the story. Their side of it, at least. That's all we're doing right now. Whatever happens next will depend on what we hear."
He was staring at Frank while he said it, but Frank wasn't paying attention. He was looking up in the direction of the road; his gaze seemed unfocused.
"Okay," Nora told Ezra, because it seemed as if he deserved some sort of response, and then he offered a hand and guided her as she stepped onto the boat. There was a mammoth outboard on the back, a motor of disproportionate size to the actual craft, and Ezra positioned her on the rear seat with her back to it. Frank took the seat in front of her, and Ezra settled in without a word and turned the key and the larger motor came to life with the throaty, muscular sound that reminded Nora of the better cars they'd had in the shop, those with expensive, fine-turned engines.
"Okay," Ezra said, spinning the wheel and pointing them out into the lake. "Let's go see what the hell we're looking at."
He shoved the throttle forward and the motor behind her roared with delight and then the front end of the boat rose several feet out of the water and if Ezra said anything else Nora could not hear it.
Frank sat staring straight ahead, his clothes rippling as they tore across the lake. Behind the wheel, Ezra was impassive, his face shaded by a baseball cap with a Ranger Boats logo and his eyes hidden by Oakley sunglasses. They both wore light jackets that Nora knew concealed guns. As she sat there clutching the boat seat under her butt and squinting against the force of the wind, she felt a surge of doubt. They were essentially strangers, Frank and Ezra, and she'd put an awful lot of trust in them with this trip to the island. No one, _no one_ had any idea where she was.
The sun was creeping out, sparkling off the water, and though her hair was streaming out behind her like a flag in gale-force winds, the trees on shore were still, untouched by any breeze. It was too calm, and that probably meant rain by the afternoon. The humidity felt wrong for so early in the day.
They went past one enormous rock that jutted angrily out of the water, then through a cluster of islands, and came out in a large bay that seemed even more desolate, only two other boats in sight. Ezra slowed and worked his way around what appeared to be a sandbar, then brought the motor back to a roar and the front of the boat lifted again and they were off, tearing past a bay filled with stumps and half-trees that seemed like menacing guards to the empty shoreline.
Nora shifted on her seat and thought about reaching out and tapping Ezra's shoulder, asking him to stop and turn around. _Take me back,_ she could say. _I've been thinking about it, and this is wrong. We shouldn't come out alone. We should leave that to the FBI and the police. They'll know how to protect me; it's what they do._
Ezra actually turned and looked at her then, and for a split second she wondered if she'd voiced her thoughts aloud, but then he faced the water again and she'd let the moment pass without saying a word.
A few minutes later he did slow the boat, and the motor quieted and Nora's hair fell back around her shoulders. Out here there was nothing to look at but trees and water, no sign of another boat. Then she saw an island ahead, over Frank's shoulder, partially blocked by his body.
"Shit." This came from Frank. "There's somebody down there, Ezra. Somebody on shore."
Ezra leaned to the side for a clearer look. "Sure is. And it looks like the fella's having a bit of trouble with his boat motor. Might be we should stop by and offer a bit of assistance."
"He'll recognize Nora and me."
"Bound to happen sooner or later." Ezra brought the boat's speed down even more, eased in closer to the shore, and Nora looked over Frank's shoulder and saw the man and the boat for the first time. It was the gray-haired Lexus driver, Vaughn, and he was looking up from the motor, his attention focused on the approaching craft. Nora slid down into the seat and tilted her head a bit, trying to put herself back behind Frank, out of Vaughn's sight.
"Little trouble?" Ezra said, shouting over the sound of the motor.
Vaughn waved them away. "Nothing I can't handle."
"Sure about that? Doesn't look like it's going well."
"It's not a problem."
"Tell ya what," Ezra yelled. "How about I come in and give it a look, and then you and me and your girlfriend have a chat? I'm the caretaker of this cabin. Feeling a bit guilty about the way I been neglecting you all."
Though Ezra's voice had been friendly, it produced a new rigidity to Vaughn's body. He took a step back and let his hands fall away from the motor, studying them carefully now.
"The caretaker?" he asked. He was speaking to Ezra, but his eyes seemed to be on Frank.
"Uh-huh. That place has been my responsibility for a damn long time."
"We're doing fine," Vaughn said, and the boat had drifted so Nora could no longer see him. She sat up straighter to compensate, and when she did Vaughn's eyes locked on her face and she felt the recognition across the water a split second before he reached behind his back.
"Don't." It was a single word, spoken and not shouted, but somehow Ezra's voice still seemed to boom out across the water and shake the trees. Nora saw that his arm was extended, his gun pointing at Vaughn. How in the world had he gotten that out so fast?
Vaughn stood with his arm behind his back and didn't say a word. Ezra kept his gun pointed while he adjusted the wheel with his left hand, bringing the boat in close to shore, the water now shallow enough that Nora could see the bottom. Frank had been keeping his back to Vaughn, but when he heard Ezra's one-word command he finally turned, and Vaughn's eyes flicked to him.
"How you doing?" Frank said. "You owe the lady here a car, and owe both of us some answers."
"Don't hurt her," Vaughn said. His voice was high and it cracked on the last word. Nora felt a moment's confusion— _don't hurt me?_ —before she realized that he was talking about the woman on the island.
"Nobody here has done any hurting, or intends to," Ezra said. "But your buddies have. And we need to talk about that. Now put that gun of yours on the bottom of your boat, and then catch the bow line when Frank tosses it to you."
Vaughn dropped the gun. It took him four tries to catch the bow line and pull them to the beach.
## 25
Vaughn was back to that damn chattering even before they were all ashore, the same routine he'd gone through while waiting with Frank for the tow truck two days earlier.
". . . and I don't know what you've been told or what you think, but I was going to come back on Monday and give you the money I owed you, which, you know, there's no reason to be pulling guns on me, your car is fine, I promise it's fine, and I didn't know any—"
"Stop," Ezra said.
Vaughn stopped. His face reminded Frank of a dog his mother had when he was a kid, a beagle that would always sit with its tongue hanging out. Frank's father would say, _Put your tongue back_ , and the dog would snap his jaws shut, hiding the tongue, and look at Frank's dad with a perplexed expression. Vaughn looked about as sure of himself as the dog used to.
"Gonna be plenty of time for you to talk," Ezra said, "but I figure we should all be present and accounted for. You could work on slowing down a touch, too, give us a chance to hear what the hell it is you're saying."
Vaughn nodded, and Ezra gestured up at the cabin.
"She in there?"
Another nod.
"Then let's go up and have us a seat and bit of conversation."
They walked across the beach to a trail that led up the bluff. Vaughn went first, slipping a few times, his footwork awkward and clumsy. Frank was last, following Nora, who walked easily. He didn't know what she was thinking. Probably hadn't been real encouraged by the way Vaughn had reached for that gun.
The cabin was in remarkable shape for a building that had sat empty for so long, but Frank didn't marvel over it. He expected no less from Ezra, who would take care of an empty cabin that didn't belong to him better than most men would ever care for their own homes. Vaughn took the steps up to the porch with a quickened stride and was talking before he reached the door.
"Renee? We got some company. Man here says he's the caretaker—" Vaughn twisted the doorknob and pushed the door open and stepped into the house just as the blond woman stepped out. Frank saw the gun in her hand, then saw it in Ezra's eye socket. She just stepped onto the porch and stuck the gun in his eye, no hesitation.
"You reach under your jacket any farther," the woman said, speaking to Frank without taking her eyes off Ezra, "and I'll kill him."
"Shit, Renee, what are you doing?" Vaughn was standing in the doorway, jaw slack.
Renee Matteson. That would be the full name. She was something to watch. Even in this moment, when the only ready-to-shoot gun was the one in her hand, Frank was taken with her. So poised, so strong. He let his hand fall away from his jacket and took one step backward.
She'd stuck the gun into Ezra's eye with enough force to open up the skin and start a small trickle of blood. Now Ezra cleared his throat and said, "This isn't a real good way to get things started."
"He's the caretaker—" Vaughn began, but she cut him off.
"Caretaker my ass. I saw his gun, Vaughn."
"We're not the ones you should be scared of," Frank said. "Though you should know that they are not far away."
"He makes a good point," Ezra said, the blood flowing down his cheek. "Perhaps it would be best to save the gunplay for those gents of whom he speaks."
The blond woman, Renee, was staring Ezra in the eye, their faces separated only by the length of her arm and the gun.
"What I'm saying is, the way things are developing, y'all are going to need your bullets," Ezra said. "Hate to see you waste one on me."
"Maybe if I took out my gun and set it down," Frank said, his voice loud, and he made the slightest motion with his arm. It was enough, as he'd hoped it would be. She looked at him instead of squeezing the trigger as she'd promised, and when she did Ezra snapped his head sideways and his arm moved with the speed of a whip, laced up and then down and then Renee's hand was in his own and her gun was pointed at the ground. Frank had the Smith & Wesson out by the time that was done.
"Damn, son," Ezra said. "You think you'd have gotten that out fast enough if she _didn't_ look your way?"
"Felt pretty sure she would."
"Me, too, but I was a little less excited about testing the theory. Always the man with the gun in his eye who's the bigger fan of patience, though."
He said all this with the casual delivery of a man in a barber's chair, working the gun out of Renee's fingers as he talked.
"Now, we got lots of guns around, everybody noticed that? Way too many guns. I'm thinking it'd be nice to put 'em all away, every one, and then just do some talking. Hell, this porch is nice enough. Let's have us a seat out here, enjoy the day."
He stepped back when he had possession of her gun, put it into his waistband, and motioned at the porch. She hadn't moved throughout all of this, seemingly hadn't _blinked_ , just stood there and held his eyes with a stare so cold it seemed like it could pass through Ezra and carry out to the lake, put a skim of ice on the water.
"I could have killed you, and I didn't," she said. "Now let's see if that was a mistake."
She turned from him, walked to an old wooden bench beside the door, and sat down. Vaughn sat beside her and reached for her arm, but she shrugged away from his touch and slid to the other end of the bench.
"All right," she said. "Talk."
"I think that's _your_ job," Nora said. The sound of her voice surprised Frank; hell, it seemed to surprise everyone. She'd been so still and quiet it was as if they'd forgotten her presence. When they all turned to look at her, she met the stares with a shrug.
"What? We didn't come out here to tell them who _we_ are. We're not the ones responsible for getting good, innocent people killed. I want to hear _their_ story, not mine." She jabbed a finger at Renee and Vaughn.
Renee looked at Nora for a long time, as if she were intrigued. Frank tried to guess her age, and couldn't. She had the body of a young woman, but her face carried some lines and her eyes were those of someone older. Or were they just tired?
"Where are the police?" she said. "You found us, so why not tell them to come out here and ask the questions?"
"It wasn't my idea," Nora answered, "but I listened to it."
Renee nodded as if that made sense, then turned to Ezra. "You're really the caretaker? You're how these two found us?"
"Yes."
"Then your name is Ezra."
"Uh-huh."
She nodded. "I've heard about you."
"From Devin," Ezra said, and Frank felt an unpleasant tingle at the sound of the name. "Where is he?"
"Dead," she answered.
Frank and Ezra had agreed the previous night that they wouldn't share any information at first, just hear the story as these two were prepared to tell it. Now, after hearing Renee proclaim her husband dead, Ezra merely nodded in Frank's direction.
"You don't know young Frank, I take it?"
Renee turned her cool gaze to Frank and searched his face. He was standing about five feet from her. She shook her head.
"I'm sorry, I don't know who he is." Talking as if she and Ezra were the only people on the porch.
"Last name of Temple," Frank said. "That help you any?"
Vaughn looked from one to the other with confusion on his face, but Renee got it.
"Your father," she said. "Devin and your father—"
"Killed people together."
"The way I heard it, that wasn't a joint project."
"Then you heard it wrong," he said. "And allow me to be the first to congratulate you on Devin's demise. You're better off with him gone. Everybody is."
She came up off the bench in a smooth, fast motion and slapped him in the face. The sound of the blow made Vaughn step toward them, hands out, but he didn't touch anyone. Nora made a soft sound of surprise, and Ezra just stood there and watched. Frank took the slap and looked down at Renee with his cheek stinging, didn't say a word.
"Now that we got the greetings out of the way," Ezra said, "maybe we ought to talk about the people who are still alive, sort out things with the dead at another time. Seems that you two have led a pair of unfriendly types into the area. Some innocent people suffered as a result. I think it's time to hear what it's all about."
The woman stood where she was for a long time, staring at Frank, challenging him to say another harsh word about her husband. He had none. His mind was too occupied by what that slap meant, by the way she'd risen so fast to defend Devin. It was not the action of a woman who'd wanted him dead. The idea he'd had, then, that they would come out to this island and meet with the two people who'd put bullets into Devin's back, no longer seemed to be the case. The reality had just spun away from the expectation, driven by the palm of her hand against his face. He looked at her and felt the tingle on his cheek, the heat of the blow fading into cold needles, and with it the truth he'd wanted.
"I'll tell you what it's all about," Vaughn said as Renee finally turned and stalked back toward the bench, "it's about _these_ two innocent people"—he waved his hand between Renee and himself—"suffering for Devin's mistakes."
"Explain," Ezra said.
"You know about Devin, you know what he does."
"Right," Ezra said, "but what do _you_ do?"
Vaughn leaned forward on the bench, ducked his head so his eyes were hidden.
"I work— _worked_ —at a prison in Florida. I'd been at it for about twelve years when I met Devin. Or when he approached me, would be a better way of saying it. I'd done the job right until then, too. I had."
"Because nobody had ever offered him any money before then," Renee said, and the scorn in her voice seemed to drop Vaughn's head even lower.
"What did he pay you for?" Ezra said. "Smuggling to people on the inside?"
"Right idea," Vaughn said, "wrong direction."
"You were taking something _out_ of the prison?"
"Instructions," Frank said. This made sense already, had since Grady's call the night before. "He was a postman, Ezra. A messenger. For Manuel DeCaster."
An image of a newspaper photograph was trapped in Frank's mind, a picture of DeCaster as he was led out of the courthouse on the eve of a guilty verdict. The man's sallow, jailhouse skin was contorted into a sneer of contempt. He looked nothing like a man whose world had crumbled, and more like an emperor amused by the weak efforts of peasants hoping to overthrow him. And why not? With men like Devin Matteson to handle business on the outside and men like Vaughn to carry the messages, maybe his rule hadn't been all that interrupted by steel bars and block walls and barbed wire.
"Yes," Vaughn said. "Manuel DeCaster."
"The big boss," Ezra said, his voice dropping into an even slower drawl. "So Devin recruited you to work as the messenger, keep DeCaster in touch with the outside world in ways that monitored phone calls and visits could not."
"That was the idea," Vaughn said.
"I understand how that could have brought some trouble down around you," Ezra said, "but these boys that followed you into Tomahawk, they aren't the police sort of trouble."
"No."
"So who are they?"
"They work for DeCaster. I don't know how they found us."
"You left them an easy trail," Frank said. "There was a tracking device in your car. That's how they got here, and I'm wondering when they had a chance to put it on your vehicle."
Vaughn stared at Frank in confusion, mouth half open, but Renee Matteson lifted her hands to her temples, eyes going wide and then squeezing shut.
"What?" Frank said.
"I should have remembered," she said. "Damn it, I should have remembered."
"You knew about the device?" Ezra said.
She shook her head. "No. Well, not specifically, but I knew they'd been following him. A long time ago, Devin was following him."
"Devin was _following_ me?"
"At first," she said, nodding, "he wanted to be sure he could trust you. Wanted to know what you were doing, where you were going. I didn't think about there being a device on the car, and that was so long ago . . . that was a year ago . . . and it was _Devin_ , not the bastards who shot him."
"But they would have known about it," Frank said. It made sense. Devin and the rest of DeCaster's team would have wanted to follow Vaughn at first, make sure there were no covert meetings with cops, no betrayals.
"How many of them are there?" Renee asked.
"Two that we know of," Frank said, thinking that this changed everything, made Devin's role less important, the whole thing less personal. If these two were hiding from DeCaster's crew, then it was no mystery why Devin had fled from the hospital. His survival odds were better on the run than inside, waiting for someone to come by and finish the job. This was bad, very bad. Stepping into the middle of a personal vendetta between Devin and these two was one thing. Stepping into the middle of a power struggle that ran back to Manuel DeCaster was a damn death sentence.
"Two that you know of? Well, there will be more than that if they call for help," Vaughn said.
"All right," Ezra said. "So we got some bad boys and big troubles. Everybody pretty well understood that. You're dancing, though. I asked what it was you did to attract this. Haven't heard that one answered."
"I didn't do shit. Devin, he got his eyes on the throne. The longer Manuel sat in a cell, the cockier Devin got. He started talking about what he could do on his own, talking about eliminating people closest to Manuel, starting with his cousins, who were key to the whole operation, guys who are so damn mean that when you look at them—"
"Slow down," Ezra said, "and just tell us what happened. It ain't that hard."
Vaughn took a deep breath and ran a hand through his hair, completely avoiding Renee's hard gaze.
"I'm _trying_ to tell you what happened. You want it in two sentences or something? Fine, here you go: Devin was going to kill DeCaster's cousins and two other guys, Cubans who were involved with him. He wanted a housecleaning."
"And this had what to do with you?"
"He needed someone to lie to DeCaster. You know, tell him that one thing was happening while something else really was, and work it the other way, too, get the information he needed."
"You agreed."
"It was a lot of dollars."
"Someone smelled it out?" Frank said. "Killed Devin before he made his play?"
"Yes. Then they came for me, and Renee. Still _are_ coming for us, I guess."
Ezra was looking hard at Frank, a question in his eyes, and Frank met the gaze and shook his head ever so slightly. Ezra frowned but broke the stare. Frank knew what he was wondering—whether they should tell these people that Devin was alive—and he wasn't ready to do that. Not yet. There were too many questions here, too many possibilities and problems and angles, a dizzying new scenario appearing. And a disappointing one. Frank felt that in the pit of his stomach, a hard ache of disappointment. He'd come out here hoping to align himself with these two and against Devin, see it boil down to the type of finale he'd wanted for so long. That wasn't going to happen, though. There was still a chance that Devin was headed this way, but he wouldn't be arriving with vengeance on his mind. Rather, he'd be on the run. Same as these two.
"Why Renee?" Nora said, breaking the silence that had gathered. "If Devin's dead, what's the point of killing his wife?"
"Renee was around a lot," Vaughn said. "She knows things that could hurt them, hurt DeCaster. So do I. Now that they know Devin violated their trust, they'll try to clean up the mess that surrounded him. Besides, they killed her husband. If anyone in the world is motivated to try to hurt these guys by going to the police, it's Renee."
Nora turned to Renee. "Then why _don't_ you go to the police? This guy and your husband are the ones that did something wrong."
Renee smiled at her, and there was genuine warmth in it, something that Frank hadn't been able to imagine on her face until then.
"I lived with Devin for nine years. You have any idea the things I know that the police would _love_ to hear?"
The explanation didn't seem to satisfy Nora, but Frank understood what she did not: Renee's world was one in which cops were the enemy. Her husband's death—the death she believed in, at least—wouldn't change that outlook. Cops were to be feared and never trusted. It didn't make sense until you'd spent a decade or so living with that worldview.
"We'd been here just one day," Vaughn said, "when I left to get some food, supplies. I was coming back from that when Frank here hit my car."
Frank didn't want to hear him return to that, didn't want anyone dwelling on the incredible coincidence of Frank hitting this guy's car, a guy who just happened to be with Devin's wife. The longer people thought about a coincidence like that, the more unreasonable it seemed, and he wasn't quite ready to explain to Renee that he'd really come up here intending to kill her husband.
"Why did you come here?" Ezra said, and for a moment Frank thought the question was directed at him, that Ezra had somehow stepped inside his thoughts. Then he realized he was asking Renee.
"It's what Devin told Vaughn to do," Renee said.
Vaughn nodded. "Right before he got killed, he was getting worried about things, told me that if anything happened I needed to get Renee out fast. He told me to bring her here, because nobody else knew it existed. Nobody down there, at least."
"Well," Ezra said, "that was a hell of plan. But there's a problem, buddy. They sure as shit know about it now."
## 26
Traffic in Chicago was always a bitch, but Grady was helped by it being a Sunday morning, and made his way out of the city and into Wisconsin by eight, doing eighty-five up I-90. If anyone stopped him, he'd flash the badge and go on his way. He had a map on the seat beside him, and the Willow Flowage was way up there, just south of Michigan's Upper Peninsula. Looked like a five-hour drive at best, and that was counting on forgiving traffic and no delays.
He should have left last night. As soon as Frank had hung up the phone, Grady should have been in the car. Hopefully it wouldn't matter. Hopefully Atkins was already out there. He'd be giving Frank hell, of course, but that didn't matter so long as he was getting Frank off the lake and out of Devin Matteson's path. With any luck they'd have Matteson's wife and her boyfriend, the prison guard, in custody by noon, and by the time Matteson _did_ arrive it would be over, nothing left but the shouting.
His phone rang just before nine, and he answered expecting Atkins and hoping for good news.
It was someone from the Bureau, but not Atkins.
"Good news," Jim Saul said, "you won't have to worry about picking up any speeding tickets in Miami. Police down here love you dearly. Hell, get drunk and drive naked down the strip. They won't care. Of course, that wouldn't be far out of the ordinary for this town, either."
"Why do they love me?"
"Vaughn Duncan."
"You turn something up?"
"Turned a murder warrant up. The Miami police guys had one complete fingerprint and one partial on a casing they found in the parking lot where Matteson was shot. Shooter got two of the casings but left one behind, lost it in the gravel. Either he panicked and didn't want to take the time to find it, or it was too dark and he couldn't. Anyhow, Miami PD ran the print through IAFIS and didn't get a match. Surprising, right, because they were betting whoever took Matteson out had a record."
"Right."
"Well, no match on IAFIS, which means no record, at least not one of substance. And the cops down here were confused, because the print on the shell indicates whoever popped Matteson wasn't a pro, and that seems wrong. Then you throw this guy Duncan at me, and I call around and find out he decided to quit his job up at Coleman without giving any notice, and I think, hmm, the good folks at Coleman probably have his prints on file."
"They matched?"
"Bet your ass they did. Nothing on IAFIS because he didn't have a record, but once we got the prints from Coleman and compared them, they matched up. I had some unhappy people down here, bitching about turning this around on a weekend, but I assured them I had a first-rate tip."
"Duncan shot Matteson, then left with the wife?"
"That's the flavor of the month, yes. The print is enough for the warrant. Now, you care to tell me where you're getting this information?"
"Wisconsin," Grady said, "and now I've got to make a call up there. I'll talk to you soon, Jimmy—"
"Hang on, hang on. I also had the flights checked. Miami to Wisconsin. Guy matching Devin's description got a private charter to some place called Rhinelander, flew out late yesterday, real late."
"Rhinelander." Grady felt numb, even though this was what he'd been expecting. The map beside him showed Rhinelander clear enough. It was about thirty miles from the Willow Flowage.
"Yeah. Like I said, private charter, and it landed in Rhinelander just after midnight—"
"I gotta go, Jimmy."
Grady hung up and found Atkins's number, dialed it as a car behind him blew the horn, Grady letting his own car drift into the next lane. He veered back to the right and slowed, held the phone to his ear. It was answered immediately.
"He's not here, Morgan. He's not at his cabin, and I'm getting pretty damn pissed off because I think when he talked to you he heard something that made him bolt."
"No," Grady said. "He's not gone. Trust me."
"Trust you. Sure."
"Listen, Atkins, I'm on my way north—"
"I told you to stay the hell away from here."
"I know that, but I thought maybe you'd want some help serving the murder warrant."
"Warrant?"
"That's right. You got a pen handy, Atkins? I think you're going to want to write some of this down."
The conversation might have gone on all morning and into the afternoon if nothing had interrupted them. Ezra and Frank were prying for more information, sorting through the mess of memories Vaughn and Renee offered, when Nora's cell phone began to ring. She'd slipped it into her pocket before leaving Frank's cabin, and the first two times it rang she simply put her hand inside her pocket and silenced the phone. On the third call, though, she took it out and checked the display and saw the call was from the receptionist desk at her father's nursing home.
"Give me a minute," she said and started to walk off the porch. Renee's eyes went wary, though, and Nora realized she was probably afraid that the call was from the police. Might as well stay on the porch, then. Relax the woman.
She answered and said hello and Barbara, a receptionist whom Nora had seen several times a week at the nursing home since arriving in Tomahawk, burst into a tirade of worry and concern.
"I don't know how he got the newspaper or who brought it to him, Nora, I really don't, but your father saw this article and he is _beside himself_ because he doesn't understand it but he knows it's bad. He's so worried, and we are, too. We were worried even before, but now that he's seen it, I really think you need to come down and show him you're okay. They've got a photograph of all the police cars outside your shop, and he keeps looking at that, and he won't let us take it away."
Nora squeezed her eyes shut. Wonderful. Of course he would have seen or heard about it by now, and of course he'd be panicked. How could she have forgotten that, or ignored it till now?
"Barb, can you put him on the phone? Let me say a few things to him, and then I'll come in and visit. Please?"
"Nora, I don't think you understand—he's not able to talk on the phone right now. He was extremely agitated. We had to give him some tranquilizers to get him calmed down. If there's any way at all you can get in here to see him, that's what I'd suggest. He's not going to be calm until he sees you."
What could she do? She hesitated, felt annoyance and disbelief in Barb's silence at the other end of the line, then promised to be in as soon as possible. When she hung up, everyone on the porch was staring at her.
"It's my father," she said. "He's in a nursing home, and somehow he got his hands on a newspaper. He doesn't understand what happened, but he's worried about me." She looked at Frank. "I need to go see him."
He looked irritated, but said, "All right. We'll take you. Ezra?"
Ezra worked his tongue around his mouth, looking at the lake. "There are two boats. Why don't you take her back in mine, and I'll stay here."
"Don't trust us enough to leave?" Renee said.
"You want to be left alone if your buddies show up?"
"No," she said.
"I was thinking an extra body wouldn't hurt anything," Ezra agreed, "and we still got some talking to do. So, Frank, you take Nora in and get her to her father. You stay with her, okay, and keep your eyes sharp. You know why."
"Yeah, I do."
Ezra nodded, looked at Nora. "That work? Time you get this settled with your dad, maybe we'll have a better idea of what the hell needs to happen out here. Y'all can come back out, and we'll see what we've got."
"All right."
"You got a phone?" Frank said to Ezra. "A way we can get you if we need to?"
"Most times it doesn't work on the water, but I'll give you the number. It'll ring, if nothing else."
He ran Ezra's boat hard all the way back across the lake, a stupid thing to do considering his lack of recent knowledge of the sandbars and stumps, but one that had an advantage. When the motor was roaring and the boat was sluicing through the wind and water, conversation was impossible, and right now Frank didn't want to talk. His mind was at turns back on that island, and the house of his boyhood, and in a Florida prison he'd never seen.
Everything he'd hoped for last night when he'd stood on the dark beach with Ezra was gone, obliterated. The situation wasn't what he'd desired, and it appeared a good deal worse. It was also, he knew, his fault. That any of them were involved in this now was his fault. He'd come up here after Devin, come up with blood in his eyes, eager for a confrontation, and because of that he'd caused the accident with Vaughn and set all of this in motion. You couldn't run from this, the legacy of bullets and bodies. Seven years he'd dodged it, bouncing around the country and avoiding anything to do with his father. Then one phone call from Ezra had pulled him north, and the result was this: They were right in the crosshairs of a bloody feud that should never have involved any of them. Particularly Nora.
It was time to get out. Time to hand the whole mess over to the people who should have had it from the beginning, to let Atkins and the FBI take it and hope he and Nora and Ezra could get the hell away from here before the fallout.
The good thing about running Ezra's boat almost wide open was that it gave him a moment of peace; the bad thing was that he made the trip in too short a time. They were back at the cabin with the motor cut and the boat's hull nestling into the shallows before he'd worked anything out. Not that extra time would have helped, though. He already knew what he had to do, which was get out of here, and stay gone.
He wanted to drive, but it was Nora's truck and she had the keys. She got in the driver's seat, and he opened the passenger door and sat down. The motor was started but she hadn't put the truck in gear before she spoke.
"Do you think we can help them?"
This was the reason he'd run the boat at full throttle, the exact question he'd wanted to drown under the whine of the wind. He'd hoped Nora would not want to help them. Trying to help Vaughn and Renee would be nothing but an exercise in futility. Either DeCaster would get them or the police would. The fact that Devin was alive and missing was only an added problem, one that would make pursuit of Renee all the more imperative. If they had Renee, they could force Devin back to the surface. Maybe. Knowing what Frank knew about Devin, it seemed just as likely he'd leave his own wife to pay the price for his greed.
"Well?" Nora said when he didn't answer. Her face was beautiful in the half-light of the shade in which she'd parked, those earnest eyes speckled by shifting shadows.
"Her husband isn't dead," Frank said.
"What?"
"He's alive. Somebody shot him, that was true, but he didn't die. He was in the hospital until yesterday, and then he took off."
She turned and stared out of the windshield, then back at him. "What are you talking about? How do you know this?"
He inhaled, looked away. "I talked to a guy last night."
"Last night?"
"Around two in the morning. You were asleep. He's with the FBI, was part of the group that investigated my dad. He told me that Devin had been shot, told me that he'd left the hospital and no one knew where he was."
Her face was incredulous at first, uncomprehending, and then the anger began to show as she reviewed the timeline.
"You knew this last night, and didn't tell me?"
"I wanted to see what the situation was first. The way it was told to me, Renee and Vaughn were responsible for shooting Devin. Tried to kill him and run off together, or something."
She frowned. "How do you know that's not true?"
"The way she slapped me. That was sincere. She wouldn't have had that sort of reaction if she wanted her husband dead."
Nora started to nod, then stopped. "Wait a second. You knew that her husband is still alive, and you didn't tell _her_? She thinks he's dead! Why didn't you tell her?"
"Let the FBI tell her," he said. Then, after a pause, "You know, it was a damn nice opportunity. I got to keep him from existing for a little while. Next best thing to actually killing him."
" _What?_ "
"I told you what happened to my father," he said. "Devin's the piece of shit who turned him in to the police. Devin, the same guy who recruited him and then made sure he stayed on board, he turned him in."
When she didn't respond, he plunged forward. "Listen, don't think for a moment I'm defending what my dad did. I'm not. He earned his fate, Nora, and I understand that better than anyone. But Devin? Devin earned his, too, and he walked away from it. Still is, somehow. Three bullets in the back and he's still walking away."
She was shaking her head now, not wanting to hear any more.
"What are you really doing here?" she said. "Why did you come here? It's not an accident. None of this could _possibly_ be an accident."
His fingers had curled into his palms, and now he flattened them on the seat, breathed, looked at her.
"I came here to kill Devin."
"Devin? He's not even here."
"I thought he would be. Ezra thought he would be. Ezra called, told me Devin was coming back . . ."
"And you came to kill him," she finished.
"I'd like to pretend that's not the truth," he said. "I'd like to think, to hope, that if it had happened as I'd expected and he'd been out on that island, I would have been able to stop myself. To walk right to the brink and then turn around and leave. But I doubt I could have."
It was quiet. The windows in the truck were up and the air-conditioning was off, making the inside of the cab muggy. Sweat was starting to run down his spine. He was having trouble looking at her now.
"Think what you want of me," he said, "but I've told you the truth. And I'm sorry you're involved. You have no idea how sorry I am about that."
The silence went on for a while, but then something changed in the engine noise, an increase in pitch as it acclimated to the long idle without ever being put into gear, and the sound seemed to jar something loose in Nora.
"What do we do?" she said, voice soft.
"I think you ought to call your FBI guy, Atkins. Tell him where they can be found. It won't cause Ezra any trouble. It's got nothing to do with him."
He felt guilty about that, leaving Ezra on the island with no warning that they were turning the whole mess over to the police, but ultimately it was the thing to do.
Nora's eyes narrowed, lines showing on her forehead. "What? Now you _do_ want me to talk to the police?"
"I think you should."
"You want to go to the police?" She repeated it again, as if it were incomprehensible.
"No, I want _you_ to. I'd actually love it if you could drop me off someplace where I could rent a car. That would be a big favor."
"What are you talking about?"
"I need a car, Nora. I've got nothing to drive."
"Where are you going?"
"I'll work that out. If the cops, or anybody else, want to find me, they can track me down. I've committed no crimes, and there's no reason I have to stay here."
"You're leaving? You're _leaving_?" She leaned toward him, spat the repeated the question in his face, eyes aflame.
"I'm not going to die for Devin's wife, Nora. I'm not going to kill for her, either. I stay and try to help, it's going down one way or the other. Grady, the FBI agent I talked to, this was his advice, to just get in a car and get the hell away from here and keep on going. He was right, too. I just should have listened to it earlier."
"You're going to leave the rest of us behind?" She looked at Frank as if she'd lost all hope of communicating and shook her head. "And I'm supposed to go the police alone?"
Before he could respond she lifted her hand. "You know what, I can't think about this right now. Before I deal with any of this, I've got to go see my dad, show him that I'm not _dead,_ and then I can take you to get a rental car so you can run away, and then I will decide what in the world I tell the police."
She put the truck into gear, backed up, and started down the gravel drive.
## 27
It was as if the lake were angry with him. As if it knew what Ezra was doing out here, had listened to his conversation with Frank the night before and heard him relent, heard him plan for violence. The day that had dawned so beautifully was turning ugly, dark clouds massing in the west, the surface choppy beneath a temperamental wind, waves slapping at the beach.
A storm on the way, for sure, and though it had been a few days building—you didn't enjoy weather this warm and this humid in the spring without paying a price for it eventually—Ezra still had the sense that it was his fault somehow, that he'd triggered the unpleasant change.
The situation was not what they'd expected, and it looked more dangerous. Handling Devin was one thing. Handling Devin and a few of his friends, even, was one thing. This, though—this could turn into a team effort, maybe already had. Who knew how many men the two from Miami had called in now. And here was Ezra, sitting on this island that he'd once loved so dearly, waiting for it to come his way.
He'd made a mistake. No, a series of them. First in calling Frank Temple's son to begin with, then in agreeing to the boy's proposal last night, and now in letting Frank and the girl go off alone. They shouldn't have separated like that.
Throughout the morning's conversation, Ezra had wondered whether Frank would break the news that Devin was alive. Once, he'd looked hard at him, trying to convey the question, and had gotten a brief shake of the head. Apparently that had settled it, at least in Frank's mind. But did he not intend to tell her at some point? Surely he did. Ezra guessed that Frank probably wanted to call his FBI friend, Morgan, again. Maybe bleed him for more information, maybe provide him with some. Until then, until Frank returned or offered some word, Ezra had nothing to do but wait.
He stood on the porch with his gun in his hand and watched the weather turn, listened to the soft voices from inside. Every now and then they rose a bit, usually Renee's first and then Vaughn's. Some sort of a dispute. He'd started out inside the cabin with them but didn't like it, all those walls closing him in and blocking him from the world outside.
He didn't trust either of them. Particularly Vaughn. Oh, the story they'd told had made sense enough, but something still felt off. When you got right down to it, what felt off started with the fact that they were together at all. It was an odd pairing. And while Ezra now understood Vaughn's involvement, how he'd been nothing more than a courier for Devin and DeCaster, there was still something in him that didn't fit.
The wind died off abruptly, just faded as if it had been sucked beneath the lake, the surface going glassy, and in the short lull before it began to blow again, Ezra figured out what it was about Vaughn that felt so wrong.
He wasn't dangerous. That was the problem. Wasn't . . . competent. Had still been pulling the outboard cord on his boat long after he should have noticed the tiny feed tube from the gas line had been removed, had drawn his gun clumsily, had talked too much and seen too little. No, he wasn't competent in the way that Devin would be, or in the way that you'd expect from Devin's hired guns.
Of course, Vaughn wasn't a hired gun—that had already been established—but here he was with Renee Matteson, supposedly charged with her protection. That didn't make much sense. Because if you really feared for your wife, if you were really laying emergency plans, wouldn't you want her with another type of bodyguard? Someone less like Vaughn and more like . . . Ezra?
Vaughn must have earned the trust—but so far, Ezra couldn't see how.
Nora was startled when the nursing home sign appeared; the entire ride had been made in an almost dreamlike state. The last two days had felt that way to her, though, a constant sense of the surreal, of disconnect from the life she knew. That was what happened when violence stepped inside your world. Renee Matteson must know the feeling well.
Neither of them had spoken during the drive, and Frank didn't say anything when she pulled into the nursing home parking lot. There were visitors' spaces right by the front doors, but she avoided them today, drove all the way to the employee lot in the back of the building. If anyone was watching for her, they wouldn't expect her to park there, or to walk in through the employee entrance. She couldn't help feeling proud of her awareness, the ability to think of something like that on the fly.
When she turned off the engine and started to get out of the truck, Frank reached for his own door handle.
"No," she said. "I'll go in alone. You can wait here."
"It would be safer—"
" _No_. I don't want to frighten my dad, or get the nurses talking, or anything else. Nobody's going to jump out of a closet and snatch me away. It'll be fifteen minutes. You can wait."
He sat there looking at her, then swung his door shut, relenting. For an instant she felt bad about her tone. Those damn sad eyes of his, working on her once again. Always confident, always strong, but always sad. She'd never seen anything else quite like them.
Then she remembered why he'd come here—to kill—and most of the guilt evaporated.
"I'll make it quick," she said, "and then we'll get you a car and you can take off."
"Take your time with your father," Frank answered. "He loves you, and he's worried."
He didn't look at her when he said it. She hesitated only a moment, then slammed the door and walked away from the truck.
The employee entrance was at the back of the building, a single door with a keycard lock that was never used, or at least not during the workday. Nora had seen plenty of people come and go without pausing to use a card. The door was open today, and she stepped through and found herself in a hallway that led around to the front desk. Barb gave Nora a startled look when she emerged from the back but didn't question it.
"Hello. He's doing a lot better since we gave him the sedatives, but I know he wants to see you. Go on down."
"Thank you."
"I hope everything's all right? It's such a horrible thing . . ." Barb let the sentence dangle, peering over her bifocals at Nora, obviously hoping to hear some details of the most exciting news Tomahawk had heard in years.
"Horrible," Nora echoed, nodding, and then she turned and walked away, chased by a sigh of disappointment from Barb, who'd probably been waiting all day for some insider information. Nora should have sent her out into the parking lot to talk with Frank. _His dad was a real-life hit man, Barb. Might sign an autograph if you ask nicely._
The door to her dad's room was closed, and it creaked when she pushed it open. He was sitting up in bed and turned to face her when he heard the sound. His face split into that smile, and she felt her own do the same.
"Hi, Dad."
"You were _worried,_ " he said, meaning he was worried about her.
"I know. I'm sorry." She crossed the room and leaned down to him, kissed his cheek and gave him a hug. He smelled like aftershave. It was one of the things he insisted on; every morning he needed a dab of Old Spice. Something about that smell and thirty years of wearing it had stuck in his brain after the stroke.
She saw the newspaper on his bed, the word _murder_ in a huge bold font across the top. What an awful thing for him to see, to struggle to understand. Who in the world had let him get a copy of the damn thing, anyhow? Didn't people around here have more sense than that?
She folded the paper without reading the story or looking at the photographs and tucked it into the wastebasket. Her father watched her.
"It sounds like a problem," he said, speaking carefully. "You have a real problem."
That one almost put her on her knees, driven either by laughter or tears. _Yes, Dad, it's a problem. You have no idea just how much of a problem I'm looking at right now_.
"It's going to be fine," she said. "Everything's all right. We had a bad day. It's done, though."
She sat on the bed, and he turned in his chair so he could keep his hand on her leg, some of the confusion and fear draining from his face. She was here now, he could reach out and touch her, and even if he didn't understand the rest of it, that was enough.
Had Nora been in a less hostile mood, Frank would have inquired as to why she parked in the back corner of the lot. It wasn't the ideal position as far as he was concerned; any watchers would probably be in front of the building or on the main road, and back here Frank couldn't see shit, had no hope of knowing what was going on. Also, the front of the building had wide banks of windows looking out on the parking lot, which meant that any attempt on their truck would be visible to those inside. Not so from their current position.
She wasn't in a mind-set that welcomed debate, though, so he'd decided to just let her go inside and talk to her father, hope for the best. He hadn't seen the Charger when they pulled in. The part of his brain that was most connected to his father's ghost whispered that of course he wouldn't see the Charger, there was no way these guys would still be using it, but he tried to shut that voice out. It was a matter of minutes now. That was what he was down to in Tomahawk. Minutes. Wait for Nora to finish here, then go rent a car, and be two or three hundred miles away by the time the sun went down.
Two or three hundred miles _west_. That's what he decided as he sat in the passenger seat and waited for her to return. Most of his wanderings had been devoted to the East Coast or the Midwest. Why not give the Rockies a shot? Some state like Montana or Wyoming might feel more like home to him than any place had in a long time. It was a wild region, populated with private people. A damn fine mix, as far as he was concerned.
_You've got acres of trees on the north side of the building, offering protection for a watcher as well as a clear view of the entrance to the nursing home._
The ghost was back, offering reminders Frank didn't want to hear. There was no need to worry about this place, act like some commando preparing for a raid. It was a nursing home, and chances were the pair from Miami didn't even know Nora's father was a resident.
_Now that you're in the back corner of the lot, you can't see a damn thing, but if someone's in those trees, they saw you come in, and they're making plans for action. You can't make a counterattack plan, because you have no idea what the hell's going on, and won't until it's too late._
It was like a chorus that caught in your brain and refused to be cast aside. He could almost see his father leaning against the side of the truck, gesturing around the parking lot with one of the cigarettes he was always promising to stop smoking. Frank tried to will the memory away, keep thinking of the Rockies, of places he'd never been and where he had no history, wide-open places with wide-open possibilities.
_You're already beaten, son. You let yourself get separated from the only person you had to take care of, the only body that needed guarding. How the hell are you supposed to help her from the parking lot if something goes down inside?_
Frank drummed his fingers on the armrest, tried to think of a song to hum. Ten minutes had passed now. How long would she take? Probably not much longer. She wanted to get rid of him. Was disgusted by his, what, cowardice? Was that what she thought? Hell with her if she did. She was nothing but a stranger anyhow. Different place, different circumstances, he'd have been attracted to her, sure. Would still be tasting her lips from the previous night, remembering the way her hair had felt against his neck. This wasn't the place, though, and these weren't the circumstances.
_You can't see her. Don't even know what room she's in, don't know a damn thing about the layout of that place, haven't bothered to get out of the truck and into a protected position in case there's trouble, or even clear your gun_ —my _gun_ — _from the holster._
He kept drumming his fingers awkwardly, the sound uneven, no rhythm at all. Why couldn't he think of any songs?
Nora spent twenty minutes with her father before she stood. It hadn't been enough time for either of them, but she had Frank waiting in the truck and Renee, Vaughn, and Ezra waiting on the island.
"I'll come back tomorrow, Dad. First thing in the morning. Okay?"
His face dropped as if an invisible hand had slid over it and tugged the eyes down, the mouth into a frown. She knelt beside his chair, squeezed his hand.
"Everything's okay, Dad. I promise. And I will see _you_ ," she kissed his cheek again, "in the morning."
She released his hand—it was always hard to leave, but this was an entirely new feeling—and walked to the door without looking back. If she looked into his face again, saw the disappointment and worry and confusion, always the confusion, she knew she'd fall apart, end up in tears on the floor. Better to leave with her head high and her stride purposeful, have him thinking everything was okay and she was in control.
As soon as she stepped into the hallway, she closed the door behind her, hearing a soft click as it latched. The hallway was empty, and she turned to the left and started back toward the entrance, made it about three steps before the door to a vacant room across from her father's swung open and a hand encircled her mouth and pulled her into the room. She saw a gun in her face, and even though she couldn't see the man who held it yet, she knew it was the one whose hand had left bruises on her arm two days earlier.
"Three things," he said, his lips close to her ear. "First, there's a nurse in the room next to your dad's. Second, anybody screams or causes a problem, I'm going to begin shooting. Third"—he paused as someone laughed in a room a few doors down—"I'm the only person in this building with a gun. So if the shooting starts, a lot of people are going to get hurt. Including the old man you just left in that room."
Frank had wanted to come in with her. Frank and his _gun_ had wanted to come in. She'd left him in the parking lot instead. But how could she have known . . . _the newspaper._ She'd seen the warning and ignored it, wondered who would possibly have given that paper to her father. The answer: someone who wanted to make sure Nora came by to see him. She could hear Barb's voice on the voice again: _We're not sure how he got it._ This guy had been waiting here for a while. Probably all morning. They were smart, too. When she didn't turn up at her house last night, they'd had to go in search of a way to find her. It wouldn't have taken much research to learn that the only personal connection Nora had to Tomahawk was in this nursing home.
The hand on her mouth released slowly, air filling her lungs again.
"Good girl," he said. "It would have been very bad if you'd screamed. Very bad."
He was talking in a strong whisper, and he reached out and twisted the lock, trapping them inside.
"We're going to be leaving through that window," he said, gesturing at the large open window with the screen already removed. "Thanks for parking where you did. Makes this a lot easier."
She swallowed, thinking of how clever she'd felt, parking in the rear of the lot to hide the truck. It was hidden, all right. Hidden from anyone who might come to her aid.
"First thing you need to do is make a phone call to your friend in the truck," he said. "You tell him that a very good shot is watching him through a scope right now. You tell him to take that gun out from under his jacket, hold it in the air for a second, and then put it in the glove compartment."
She didn't respond. He smiled at her. His face and clothes were as she'd remembered, but the ornate belt buckle was gone. Maybe he thought it stood out too much. Maybe he'd gotten Jerry's blood on it.
"If you don't have Frank's cell phone number," he said, "I can provide it. Yes, honey, we're all caught up on the research. Now, do you want to call or should I?"
She called.
## 28
His fingers froze on the armrest when the cell phone rang. Jumpy. He took the phone out, saw it was Nora's number. She was probably calling to order him out of the truck, not wanting to see him when she came outside.
"There's a problem," she said when he answered. Tension in her voice, but not the angry sort.
"What?"
"One of them is inside with me, and the other is watching you through a gun scope." Speaking softly but clearly. "I've been asked to tell you to take your gun out, hold it in the air for a second, and then put it inside the glove compartment. If you don't listen, they will shoot you."
_Told you, told you, told you!_ the ghost screamed at him. _It's over now, son, over because you got lazy and dumb and told yourself that wouldn't matter. It_ always _matters._
"You're with your father?" Frank said. "This guy was waiting in your father's room?"
"No, I—" There was a rustle, a whisper, and then her voice returned. "Frank, put the gun in the glove compartment, and do it fast."
Shit. He hadn't seen anyone watching, had no proof that this thing about the guy with the scope wasn't a bluff, but he had to listen. He already knew there could be someone in those woods north of the building. Had been trying to ignore the notion for the last ten minutes. Moving slowly, he reached inside his jacket with his free hand and withdrew the Smith & Wesson, held it in the air, then squeezed the phone between his ear and shoulder while he opened the glove compartment and put the weapon inside.
"I put the gun away."
More whispering, then, "We're coming outside. He wants you to get behind the wheel and keep your hands above the dash. If you see anyone else, look normal."
The call was disconnected, but he kept the phone at his ear as he slid across the seat, banging his knees on the gearshift. Without looking at the display, he punched the CALL button with his thumb. That would bring up a list of previously called numbers, and Ezra's number, entered just before leaving the island, would be at the top of the list. _Most times it doesn't work on the water, but I'll give you the number. It'll ring, if nothing else._
Frank hit the CALL button again, keeping the phone up and hoping the watcher wasn't going to be aware of exactly when the call from inside the nursing home had ended. Still, he wouldn't have much time, because as soon as Nora and the guy with her left the building, it would be obvious that Frank should no longer have the phone to his ear.
One ring, then two, no answer, and right then he saw them—Nora and the man he'd knocked out in the body shop, rounding the corner of the building. Either there was another door, or they'd gone through a window. Odds were good nobody in the building had seen them leave. He closed the phone without getting an answer, dropped it into his lap, and thought, _Figure it out, Ezra, figure it out. There's trouble on the way._
Nora walked to the truck quickly but without obvious fear, eyes up, stride steady. That was a good word for her, steady. She'd hung together through all of this, with the one exception coming when they'd found Jerry's body. Brave girl. She didn't deserve this.
Frank noticed the door was locked when they were about ten feet from the truck and leaned over to unlock it. When he moved, the gun in the tall guy's hand showed for the first time, rising fast. Frank unlocked the door and leaned back, held his hands up again, indicating it hadn't been an aggressive move, no suicidal idiot sitting in this truck, no, sir. The gun dropped, and then the door was open and Nora was inside the truck and sitting beside him, the tall guy piling in behind her.
"Keys," the guy said, and Nora fished her keys out and passed them to Frank.
"Start it up and drive out of here. Take a right out of the parking lot, and go straight until I say something else. Keep both hands on the wheel, keep the speed down, and keep your mouth shut."
Simple enough. Frank did as he was told, made a right turn away from the nursing home without anyone stopping them or even seeing them. They'd gone about a half mile before he noticed that Nora's leg was trembling against his.
Ezra rotated the cell phone in his palm and stared out across the water. The wind was coming at the island in uneven gusts, pushing tendrils of gray clouds ahead of it. Just one boat had passed in all the time he'd been out here, and he'd recognized it as Dwight Simonton's pontoon. Unthreatening. A peaceful afternoon, a lonely lake.
But there was the phone in his hand, small and still and silent since those two rings, just two, that had come in from Frank Temple's son. Ezra didn't like the two rings. Liked it even less that there had been no second attempt.
He'd thought about calling Frank himself. Would take maybe ten seconds of conversation to clarify the situation.
He didn't call, though. Because if it hadn't been a mistake, if Frank had intended to hang up that quickly, without getting a chance to talk to Ezra or leave a message, then the afternoon was about to get interesting. Either Frank had been interrupted in his attempt to call—an option that prickled at the back of Ezra's neck—or he'd made the call as a warning. One or the other. Or a mistake.
Ezra felt Frank would understand the effect of an aborted call like that, would anticipate the worry on Ezra's end. That belief tilted the scales, ever so slightly, in the direction of trouble. No matter how gentle a shift that was, it was something he had to listen to. If you ignored it, the first chance you'd have to regret it wouldn't come until too late.
He was alone on the porch. Vaughn and Renee were still inside, though he hadn't heard much conversation from them. Vaughn had looked shifty, even angry, the last time Ezra was inside, but the woman seemed to have taken a measure of comfort now that some time had passed. She would listen to Ezra willingly, and Vaughn, if it came down to it, would listen to Ezra because he would be given no other choice. Ezra went to the door.
"You two got any rain gear?"
They were sitting together in the living room, Vaughn speaking to Renee in a harsh whisper, and when Ezra entered and spoke they both looked at him as if they didn't understand the language.
"What?" Renee said.
"Rain gear? If not, don't worry about it. I got a few of those emergency ponchos in the boat, if we need 'em. Chances are, we're going to need 'em, too. Those clouds don't look like kidders to me."
Vaughn stood up. "What are you talking about? If it rains, we've got a roof over our heads."
"Not anymore." Ezra was turning down the blinds now, the sunlight disappearing from the room in strips. "We're going on a boat ride, kids. And we're going on it in a hurry."
Now Renee was on her feet, too. "What's wrong?"
"Nothing's wrong," Vaughn said. "This guy's crazy. What the hell are you talking about, boat rides?"
"Shut up," Ezra said, and the argument died on Vaughn's lips. There was still anger in his eyes, his forehead lined with dislike, but he quit talking. He was scared of Ezra, and that would make things easier.
"They're coming, aren't they," Renee said, and there was neither question nor alarm in her voice. Just a calm, if disappointed, understanding.
"They could be," Ezra said. "And I'll tell you this—an island is a damn tricky place to sneak away from. So best to get off it early."
"Where are we going?"
"Not quite sure about that one yet, but we'll need a boat, and we'll need to move fast."
"I want my gun back," Vaughn said. "If they're coming for us, I want my damn gun."
Ezra gave him a cool, even gaze until Vaughn looked away.
"When it comes time for shooting," he said, "I'll see that you got something to do it with."
Three times the man with the gun instructed a turn. Those were the only three times the silence was broken. They'd gone maybe five miles, were well out of town and into the woods, before he told Frank to stop. They were at a ramshackle bar with CLOSED and FOR SALE signs in the windows, an ancient gas pump out front. Frank drove behind the building, following instructions, then parked and cut the motor. Nothing around them but the deserted building and the trees, with buzzing insects and wind-tossed cattails indicating a marsh about a hundred feet behind the bar. Frank turned away from it. It would take a long time before a body dumped in that marsh was found.
"Now we sit here and we wait and nobody says a word," the guy with the gun said. His weapon was a Beretta, resting against his knee and angled toward Frank.
They sat for five minutes, maybe ten, and then gravel crunched under tires as someone left the paved road and drove into the parking lot. A few seconds later the new arrival appeared around the building. A van, light blue with darkly tinted windows, suburban-looking, about as anonymous as a vehicle could get. It pulled in beside the truck, and the driver climbed out. Shorter than the guy inside the truck, but quicker, more graceful in his movements. Strong, too. Frank remembered that from the way the guy had whipped his gun into Mowery's face beside the police car.
"Out," the guy beside Nora said, and Frank opened the door and stepped out onto the dusty parking circle, a warm gust of wind flapping his shirt against his body. It was his first opportunity to see the second man face-to-face, and he didn't like the way the guy stared at him as if they'd already met, a sense of the familiar in his gaze. The guy held that look for a long moment, then turned away from Frank and slid the van's side door open, and Frank found himself staring at Devin Matteson.
The last time Frank had seen him—the only time—it had been eight years earlier, in Miami. He hadn't been around him long, maybe an hour, just enough for the dislike to put down roots, but what he remembered from that meeting was two qualities: arrogance and strength. The strength was no longer present.
Devin was leaning sideways against the seat so that he could face out, a gun resting in his lap, but it looked like just keeping his head up was taking a real effort. His usual deep tan and strong jawline had turned into a junkie's face, fish-belly complexion with hazy, red-rimmed eyes and muscle lines that seemed given to tremors. Bulges showed under his shirt, and Frank realized after a second look that they weren't bulges from a holster but from bandages.
_Vaughn was lying._ Had to be, because this no longer made any sense: The two men who'd arrived pursuing Vaughn and Renee were indeed here, but Devin was with them. Vaughn's story had just come unhinged, but right now, staring his old nemesis in the face, Frank had no concept of the truth, just understanding of the lie.
"This is a crazy damn world, you know?" Devin said, and his voice came from some tight, trapped place in his chest. "I mean, I send two guys up here to do a job, and who do they tell me got in the way but Frank Temple Junior."
"The Third," Frank said.
"Huh?"
"Frank Temple the Third. No junior here."
Devin looked at Frank for a long moment, and then gave a low laugh as his eyes went to his shorter partner.
"You believe that? It's his son, no question. _No junior here_."
He laughed again, and the other guy gave an awkward smile, as if he didn't know what was so amusing but felt obligated to share in the fun. Devin's laughter swept through Frank as pure white rage. He willed himself still, willed himself silent. Let the prick laugh. Let him enjoy this. Let him think that Frank didn't know what had happened those many years earlier, and then, when the time was right, let him pay.
Devin stopped laughing, but it wasn't clear if it was because the humor had passed or because he'd run out of breath. He waited for a moment, jaw clenching, eyes watering, and when he looked up and spoke again his voice had less energy and a darker tone.
"You want to tell me, Temple _the Third,_ what the hell you're doing here?"
Frank said, "I came to send you home."
"What?"
"Ezra Ballard told me that you were coming back. We didn't think that should happen."
Devin gave him a look caught between anger and wonder. "Ballard's a crazy old bastard. I don't know what he told you, kid, but it was all bullshit. Me giving your old man up? That's a lie."
This time Frank didn't think he'd be able to will the anger down, thought it was going to tug his foundation loose and sweep him away with it, send him rushing into that van, the other two and their guns be damned. But he fought it down again, didn't say a word.
"Whatever," Devin said. "I don't give a shit what you two think. I'll tell you what I told Ballard—whoever tipped the FBI, it wasn't me. Supposed to be somebody close to your dad, though. Hell, could have been you."
Frank was halfway to the van when the tall man stepped in and swung his gun sideways, going for his throat. Frank blocked it, got his hand up and met the guy's forearm with his own, was still moving forward, still heading for Devin, when the second man placed the barrel of a gun against Frank's cheek.
He stopped then, had to, and the tall guy turned his gun over and pressed it into Frank's ribs, two guns against him now, two fingers on the trigger. Devin hadn't moved, just sat there and watched with his own gun still on his lap.
"Your old man never shut up about you," he said. "All this bullshit, telling everybody how fast you were, how good with a pistol. On and on. And you know what I finally figured out? He had to keep talking about it, because he knew you were a pussy. He knew that, and it shamed him."
He got out of the van slowly, almost went down once, but when the tall man moved to help him he put up his hand and shook his head. He steadied himself, took a couple of steps toward Frank, until they were face-to-face. The tall man had moved back toward Nora, but the other one kept his gun on Frank's cheek.
"How did you hook up with Vaughn Duncan?" Devin said. "Did he find you, or did you find him?"
This provided an answer to a question Frank hadn't even really had time to consider yet: If Devin was already here, why hadn't he just gone out to the island? Frank was the reason. Frank was the wild card, the development Devin hadn't been able to understand. Frank and Nora—loose ends.
"I drove him off the road," Frank said, each word coming slow, the pressure of the gun working against his jaw muscles, "because I thought he was you, and I was going to kill him. Like I said, it's why I came up here."
Devin Matteson stared at him for a long time. "You're serious," he said. "You're _serious_."
It wasn't a question. Devin looked away, at each of his partners and then at Nora, and shook his head, limped a few steps back, so he could lean on the van.
"Well, hell, kid," he said. "Sorry to disappoint. It wasn't me, was it? But you and him, you guys got something to share. You wanted to kill me, he tried."
It took a second for Frank to process that. Then the truth that had felt so close when Renee slapped him—the reality of her loyalty to Devin imprinted on his cheek, stinging his flesh—finally arrived, came screeching up in a cloud of smoke, engine revving. Vaughn was after Renee. You didn't have a chance to take a woman like Renee away from a man like Devin. Not when he was alive.
"Vaughn shot you," Frank said.
"Three times," Devin said.
"That's not what your wife thinks," Nora said, and everyone but Frank turned to look at her.
"My wife," Devin said, offering the phrase guardedly, as if he were afraid of its power. "You've seen her."
Nora nodded.
"She's here. With Vaughn."
"Yes. But she thinks you're dead."
Devin said, "AJ," and waved his hand at the man who held the gun to Frank's face. The gun dropped away and the man stepped back, cleared some space so Devin could see Nora clearly.
"Tell me," Devin said, "what they told you."
Nora told him. Frank heard her words but wasn't focused on them, was instead staring at Devin and trying to smell out the lie. He _had_ to be lying, didn't he? Vaughn had shot him? But Frank could see that now, could see it in the way Vaughn and Renee had interacted, his obvious adoration for her. And Vaughn had told the story, provided all the details, details that were clearly lies. Everything Renee knew about the reasons they'd fled came from Vaughn. None of it had come from Devin, at least not the way she'd told it to them that morning.
"I cannot believe he had the balls," Devin said when Nora was done, his voice barely audible. "That cocksucker . . . he planned it for a while. Spent some real time on it. Had a story ready for her. And I'm laying in the hospital and he's up here with my _wife_."
He banged the butt of his gun against the van, then again, and again, until the effort took his strength and he had to wait a minute to get it back, hanging against the door.
"You thought she left you for him?" Frank said, and Devin's eyes slid unpleasantly back to him. "That's why you didn't name the shooter for the police? You thought she was involved?"
Devin waited for a moment, then said, "I wanted to conduct my own investigation. That's all."
"Then how did these two"—Frank nodded at the other men— "get here before you?"
"I sent them. When they told me he'd come here, I left so I could see it to the end in person."
"If this is the truth," Nora said, and her voice was wavering, "then why did you bastards have to kill Jerry? Why did you have to do that? You knew Vaughn was going to that island!"
"Unfortunately," Devin said, nothing showing in his bleary eyes, "I was out of communication with these two for a while. So they had to keep following the trail."
That justified it to him. It was enough. Frank looked at Nora, saw the shock and horror in her face, and wondered if she understood what else this meant. She was playing Jerry's role now: a liability.
"They're on that island?" Devin said, ignoring her question, stepping away from the van again, closer to Frank. "They're on _my_ island? Vaughn and my wife?"
Frank nodded.
"Who's with them?"
He didn't say anything. Neither did Nora. But Devin stared into Frank's eyes and said, "Ballard. He's out there with them, isn't he?"
Frank still didn't respond, but Devin was nodding his head, already convinced.
"Okay," he said. "AJ, King, get them in the van. We're close, boys. We're close."
## 29
Past Madison and gaining on Stevens Point, maybe two hours away if he could keep this speed up. Grady was driving hard and staring at the clock, willing it to tick a little slower.
He wanted to call Frank, see if the kid had his phone on today, if he'd answer. There was news to share, damn it. Atkins hadn't been kidding when he said he'd press charges over another phone call, though, and Grady had the sense that Frank was done talking to him anyhow. He had a plan of some sort, was putting something in motion.
If Duncan was good for the murder, as the fingerprint suggested, then this thing was shaping up exactly as Grady had feared: Devin Matteson was headed out to that lake to settle the score, and Frank Temple had placed himself in the way.
By the time he passed the first exit for Stevens Point he couldn't wait for news anymore, grabbed the phone and called Atkins again.
"He's still gone," Atkins said, without bothering to exchange a greeting. "I've also tried to find the guy you mentioned, Ballard, but he's MIA as well. Thing is, there's a boat down here now."
"Where?"
"At Temple's cabin. There was a small boat the first time I came out, little aluminum thing, but now there's a fancy bass boat on the beach. I called in to check the numbers, and it comes back to Ballard."
"But they're not inside."
"No, they're not inside," Atkins snapped, his tone icy. "There was a truck here this morning, too, registered to that girl at the body shop, and now that's gone and this damn boat is here and none of them are where I can find them. This is fantastic, Morgan. I've got a murder warrant ready to go, and these assholes know where the guy is, and now I can't find them."
"You got anybody else involved?"
"Couple of the locals are running around, trying to turn the girl up. Said she was just in at some nursing home visiting her father, so I guess she's all right. But I'm the only one out here at the lake."
"You probably ought to have some help."
"I'll get help when I find out where the son of a bitch _is,_ Morgan. And I can't do that until your buddy shows his face again."
"Wait there," Grady said. "If Ballard's boat is there, they'll probably be coming back to it."
"I'm going to wait for maybe twenty minutes, and then I'm going back to check Ballard's house. But I'll give it another twenty."
Devin Matteson made them all ride in the van, first instructing Nora to write a note that said, _Out of gas, back soon, please don't tow,_ for display in the windshield of her truck. She hadn't thought much of it then, but after she was in the van and they were in motion, the note began to disturb her. It would keep anyone who found the truck from immediate concern and imply that Nora had been under her own power when she left the vehicle behind. Those were only temporary effects, of course, but the fact that Devin had considered them made something bitter bloom in her stomach. He was good at these things, kidnapping and murder, so good that the little moves like that note came to him effortlessly, it seemed. Came the way things did after a lot of practice.
AJ was driving and sat alone in the front, Nora in the middle row beside Devin Matteson, Frank all the way in back with the man called King. Devin and King and AJ were all wearing guns. AJ had two, actually; he'd paused long enough to take Frank's gun out of the truck before they left. It lay on the floor in front of the passenger seat now. She could hear it slide around when they took sharp curves.
Devin Matteson's true condition began to show itself during the van ride. He'd looked bad initially, unhealthy, but once they were in the van Nora saw that he'd held it together well for that first encounter. Now he seemed to struggle with every turn and bend, wincing at the motions, patting his chest lightly with his hand. By the time they'd gone five miles his face was bathed in sweat, his breathing audible across the van.
There was nothing between her and the end of this but twenty minutes in the van, another twenty in a boat. The fear should have been intense, cloaking her, forcing her into hysterical sobbing. That seemed right, at least. Instead, she was just sitting here, swaying gently with the van's motion, listening to the rasping breaths of the man with the gun beside her, numb.
They were going to die. While she believed the story Devin had told, at least the portion about Vaughn, she couldn't believe that meant any change in her fate. She'd seen these men face-to-face, watched them commit crimes. After all that, they weren't going to simply head home after finding Vaughn, trusting that she and Frank would pretend none of this had happened.
_So we're going to die_. She almost nodded as if confirming the silent, internal voice. It was true. If things went according to plan for these men, there would be more killing before the end of the day, and it wasn't going to stop with Vaughn.
_All this over a murder,_ she thought. _No, wait, it wasn't even a murder. He didn't kill Devin, he just tried. And now how many others will die because of that? How many innocent people are going to atone for one man's attempted killing?_
The interior of the van darkened as they drove north, the sun pushed beneath ivory clouds that looked a good deal more ominous to the west. She watched the shadows play across the seats and tried to think of a way to stop this. The moves that came to mind were all in hindsight, though, things she could have done and had chosen not to do. Atkins of the FBI sat somewhere in Tomahawk, awaiting her call. If she'd called him instead of getting in the boat with Frank and Ezra . . .
_Ezra_. The thought of him was the closest thing to comfort she could come up with. He was capable, always in control, and, if what Frank had said about him was true, the sort of man who could deal with these bastards. The odds weren't with Ezra, though. He was without warning, he was without preparation, he was without the support of favorable numbers. He was also all she had to hold her hope.
The van rumbled over a stretch of rough pavement, and she looked back out the window, saw with surprise that they were already on County Y, minutes from the cabin. It was all going to happen fast now, too fast. She sat up straighter, wanting to turn and look at Frank, but King's hand came down immediately, pressed hard into her shoulder, brought her back into the seat.
The van came to a stop, and she looked up again and saw the lake through the windshield, the water darker and tossed by a gusting wind. For a moment the lake held her attention, but then she heard AJ swear softly, and when she leaned to the left for a clearer view she saw that there was a car parked beside Frank's cabin. A white Buick sedan, nobody sitting inside.
"Whose car?" Devin said, leaning close to her, his face shiny with sweat.
Silence.
" _Whose car?_ "
"I don't know," she said. Frank didn't speak. Maybe he knew who it was. Someone he'd arranged to meet at the cabin before all of this had started.
That idea died an immediate death when Atkins, the FBI agent, walked around the corner of the cabin. He had a sheaf of papers in his hand, and when he saw the van he folded the papers and tucked them into his back pocket, cocking his head and studying the van and AJ behind the wheel.
"Who is that?" AJ said.
Nora didn't answer, just stared at Atkins as if he were the ghost of a loved one, someone you'd known you'd never see again no matter how badly you hoped for it. At that moment, Atkins reached into his suit jacket. AJ tensed, but then Atkins's hand was back out, with a badge in it. Nora's muscles went soft, liquid. What was he doing? Don't pull a badge, pull a _gun._
"Handle it," Devin said, and then he pressed the gun into Nora's stomach as AJ opened the door.
"Not a sound," Devin said. "King? Don't let either of them make a sound."
AJ stepped out into the wind, said, "Is there a problem, sir?" and then slammed the door.
"No," Nora said softly. She couldn't let this happen. Couldn't let AJ talk his way out of this, send their best chance at rescue off in that Buick, oblivious. King's hand descended onto her shoulder again, tightened into the nerves, held her against the seat.
AJ was walking toward Atkins with a leisurely stride, one hand in his jacket pocket, the other cupped to his ear as if struggling to hear over the sound of the wind. Atkins walked forward to meet him, still holding the badge in the air, waving with his free hand at Frank's cabin.
"Oh, shit," Frank said, and King's hand left her shoulder and went to Frank's throat as AJ closed the gap to a few feet and Nora finally realized what was about to happen, that AJ's goal had never been to fool Atkins with talk. She screamed then, and Atkins jerked, looked toward the van and took a fumbling step backward and AJ's hand came up out of his jacket and into the FBI agent's stomach.
Atkins hunched, as if caught by an unexpected stomach cramp, and then AJ's hand rose higher and Atkins rocked back onto his heels and kept going, landed on his back with the handle of the knife rising out of his sternum as if AJ had planted a flag there. It was the last thing Nora saw before King slammed his rough hand across her mouth and pulled her backward, dragging her head behind the seat, telling her to shut up or she'd die, too.
It was like that, with her back arched and her neck strained to its limits, staring into the backseat upside down, that her eyes finally found Frank's. King's gun was shoved against Frank's head, but Frank seemed unaware of it. He'd turned his eyes away from the scene outside for only a moment, just long enough to meet hers, and what she saw there was nothing like what he had to be seeing in her own face, not fear or sorrow but the dark shadows of rage.
## 30
We didn't need this. Damn it, we didn't _need this_!"
Devin was standing above Atkins's body, staring at AJ, his face now stricken by both pain and anger.
"You said handle it, man."
"Handle it, shit, you think that means you _gut_ the guy? An _FBI agent_? This is something we needed?"
AJ showed a ghost of a smile, spread his hands. "Dev . . . what can I say? You know, it's done. I'll deal with it."
Frank, watching him, thought, _He did it because he likes it._ That was all. Devin was dangerous, but Devin had a brain. This wild son of a bitch, AJ, he was closer to the edge. Bloodthirsty, driven by it. He'd killed Atkins because it was not only what he knew to do but what he liked to do. Any guilt over his own stupidity, over the additional attention this was going to bring down around them, was buried beneath the pure pleasure he'd taken in the moment.
"I mean, I saw a badge, you know? I saw a badge, Dev, I just reacted." AJ was watching Devin, the knife gone and the gun back in his hand. It was a Glock, and he kept rubbing his thumb over the butt as he looked at Devin. There was a strange symbol tattooed on the back of his hand. A lefty, too. Frank had only known one left-handed shooter, but that guy had been damn accurate.
"You'll deal with it." Devin shook his head, disgusted, and stared at the corpse at their feet for a long time. When he finally looked up, his eyes found Frank, lingered there, and then he nodded.
"All right," he said. "We'll make it work."
He made Frank drag the body down to the boat, a crimson smear marking their path over the grass, the trail of blood leading right to the cabin door. That's what the cops would see, Frank knew, and what the newspaper and TV people would use for drama. When they were all dead and the cops came up here to sort out the mess, all they'd see was that trail of blood leading to the door of a dead murderer, and the Temple name would be infamous again, Frank accepting the baton from his father. He understood that perfectly as he followed Devin's instructions and handled the body, leaving fingerprints all over the corpse of an FBI agent who'd surely voiced his suspicions of Frank to colleagues already.
Devin hadn't wanted this complication, but he knew how to deal with it.
"Take that anchor line," he said, "and loop it around his neck. Make it tight."
Frank was standing in the shallows, knee deep in the lake, the body slumped facedown in the water as he wrapped the line around Atkins's neck. Devin stood above him on dry land, using a tree for support and studying the lake with the gun held down against his leg, checking for other boats. There weren't any, though. The weather was on Devin's side, rain starting to fall now and thunder crackling just a few miles west, a good storm on the way. A Sunday before fishing season, with a storm coming in, guaranteed an empty lake. Empty except for them and those on the island.
Atkins was the first of at least three victims today if things went Devin's way. He was clearly aware of the possibilities left by making Frank handle the body, but he might not appreciate just how well this would work, might not know that Atkins was already investigating Frank, the hit man's son. Some quick, quiet killing at the island and a fast trip out of town were the only things keeping the trio from Miami from disappearing like phantoms, leaving the police to try making sense of a situation they'd probably never understand.
_Grady._ Frank thought of him as he secured Atkins in the anchor line. Grady was an element Devin didn't know about, couldn't plan for. Grady had been putting the puzzle together for everyone, and he'd know where to start when Atkins was announced as missing or the body was found. He wouldn't believe it was Frank's doing. Would he?
Shit, what would it matter? If nobody was left but Grady, let them think what they wanted.
Devin was weaker than Frank had thought at first, hardly able to stand. He'd taken a long time just to cross the short stretch of yard to the tree he leaned against now, and his pain was visible even from down here in the water, his face pale and shiny, his mouth always hanging open to help with breathing. The day's killing would be done at his whim but not by his hands. That was fitting.
Frank finished tying Atkins to the line and set the anchor back into the stern, the body now tied in the middle of the line. Then AJ waved at Frank with the gun.
"Get in."
Frank climbed into the boat, and then Nora and King followed suit and AJ turned back to look at his boss. Devin pushed off the tree, took a wavering step toward them, and then leaned backward and clutched for the tree again, used it to regain his balance.
"Dev . . ." AJ started toward him, but Devin was already in motion again, trying to walk toward them. He made it four steps before his legs buckled and he went down. AJ caught him by the shoulders, helped him up.
"You got to get out of here, man," AJ was saying as Devin struggled for his breath. "Got to get to a—"
"Shut up." Devin had his hands on his knees. "You know what I'm here to do."
"I'm telling you, we can do it for you."
"No."
AJ looked back at the boat, then down at Devin. "Dev, you aren't going to make it in that boat. You aren't. And it's starting to rain, man. Gonna turn ugly soon."
Devin didn't respond, just took in fast, panting breaths.
"We'll go get him," AJ said. "We'll get him, and we'll bring him back to you. All right? Him and Renee. We'll bring Renee back, Dev. You got to stay here, though. Out in that boat, man . . ."
Devin rose slowly, stared at the group already waiting in the boat, his eyes lingering on Frank the longest.
"All right," he said. "You go out there and bring them back, and do it _fast,_ damn it, do it fast."
"Right." AJ was nodding. "Out and back, man, nothing to it."
"Take them both," Devin said. "This crazy old shit that's out there, he's good."
"He's nothing, Dev, don't worry about—"
"No." Devin shook his head. "He's _good_ , okay? That's why you need them. You make sure he knows you've got the girl, too. Make that good and clear."
"We got it, Dev. Now let me get you inside."
AJ left Devin there in the yard and walked back to the boat, extended his hand to Frank, and asked for the key. Frank reached in his pocket and took it out, the key to the last place of clean memories he had with his father, and then he passed it over so Devin Matteson could go inside and wait for somebody else to finish his bloody work.
AJ took the key and went back to Devin, helped him across the yard and into the cabin, Frank watching them go, thinking, _I'll be back for you, you son of a bitch. It won't be these two coming back. It'll be me._
Ezra was on the island, and whether he'd gotten Frank's aborted phone call or not, they would not be surprising him, not by approaching in a loud boat in the middle of a storm. He'd be waiting, and he'd be ready, and then it would be done. Let Ezra handle these two, and then Frank would come back for Devin.
The door reopened and AJ stepped out, started in their direction, then pulled up short and returned to the van, opened the driver's door, and leaned inside to grab the extra gun, Frank's father's gun. It was the second time he'd gone back for it—the first, he'd made sure not to leave it in the glove compartment of the truck—and each time Frank had felt relief. He wanted the gun to travel with them, as if it somehow represented protection no matter the hands that held it.
They were close now, the island no more than twenty minutes away. He had no grand plan, no idea how to stop this from happening except to run directly to Ezra and hope for the best.
"Start the motor," AJ said, stepping on board and coming back to sit behind Frank.
The big outboard fired at once, smooth and powerful and as loud as a damn train. Ezra had more horsepower on the back of that boat than was in most cars. Frank put the motor in reverse and kept the throttle low until the prop had pulled them into deeper water, then spun the wheel and slammed the throttle forward.
The rain was driving hard now, blowing into their faces and speckling the surface of the lake. Water ran down Frank's neck and under his shirt, dripped into his eyes. After they were around the sandbar, into the middle of the lake, AJ leaned over the side of the boat and there was a flash of silver from his knife and then the anchor line parted and slipped overboard and Atkins's body drifted away. His white face was turned up as the anchor tugged it slowly beneath the lake, a ghostly sinking shadow. It was probably twenty-five or thirty feet deep out here. He might surface soon, he might not. If the body stayed wrapped in the anchor line and tangled in any of the stumps that lined the bottom, Agent Atkins might be a resident of the Willow Flowage for a long time.
"Go on!" AJ yelled, and Frank increased the speed, hardly aware that he'd slowed to watch the body.
Devin wanted AJ to have hostages, had made that clear before he'd sent them off without him. Hostages gave AJ a bargaining chip for use with Ezra, leverage to force the situation into his favor. One thing was certain, though: AJ had never had a hostage like Frank Temple III.
Frank held tight to that idea as he squinted against the wind and the spray, pushing the boat ahead fast and hard. All those lessons his father had offered, those violent skills that he'd provided and that Frank had spent seven years trying to suppress, they were about to have a purpose. These assholes might have known Frank's father, but they didn't know him well enough. Contract killer or not, Frank Temple II was at heart a teacher—and his son had excelled at the lessons.
There wasn't a real road within two miles of where Ezra sat. A couple of trails led up to the Nekoosa Kennedy Fire Lane, but even if Ezra got them out of the boat and through the woods to the fire lane, what would he have accomplished? They'd still be a long, long walk from safety, with the boat marking their entrance point into the woods. Find the boat, and it wouldn't be hard to understand where they were headed if you had a map. He had a feeling these boys would have a map.
"We're just going to _sit_ here?" Vaughn said. "We left the cabin to come up here and sit in a boat? If they don't kill us, the lightning will!"
The lightning was a concern, although Ezra didn't admit it, or even bother to respond to Vaughn at all. The rain was falling now, and the dark thunderheads were on top of them. They needed to get out of the boat and on land for the duration of the storm if nothing else, even though that wasn't what he wanted. Not for the first time since they'd left, he wondered if he'd made a mistake by coming north. Frank could have made that call from Tomahawk, more than thirty minutes away. With time like that, they could have gotten to Ezra's truck.
It was a risk he couldn't have chanced, though. You planned for the worst-case scenario, and the worst-case scenario put these bastards close and coming closer. Circumstance like that, you had to run away from them, not into them. So he'd run, taken the boat into the deepest reaches of Langley Bay, one of the most secluded spots on the lake, with the only approach coming from the water. That meant going _back_ required crossing a hell of a lot of water, too. He turned and looked at the motor on the stern. Stupid little outboard, nine-point-nine horse. It would take them five times as long to get across the lake with it as Ezra's boat, with the two-twenty-five knocking away.
"If they're out here to find us, they'll search the whole lake," Renee said. She was sitting in the middle seat of the little aluminum boat, and it rocked as she leaned toward him. She was wearing one of the ponchos Ezra kept in the boat, but there was rain on her neck, sliding slowly down to her collarbone. "They won't get discouraged and give up."
He understood that, didn't need it told to him. Truth was, Ezra had some doubts now, and he wasn't used to doubts. There was a time when something like this, combat preparations and a retreat into the woods, felt as natural to him as a trip to the movies, simple and almost fun. Hell, back then it felt _more_ natural than a trip to the movies, but that time was long ago. Today, shaken out of years of a peaceful existence, maybe he'd slipped. Maybe he'd made a mistake. What the hell were they accomplishing, really, sitting out here in a boat with no idea what was happening on land? Even if his worst suspicions were accurate, then the real concerns were Frank and Nora. These two were at least temporarily safe. The others might not be.
"We can't just sit here," Vaughn said again, and his voice made Ezra prickle, filled him with an urge to smack the gray-haired son of a bitch onto the floor of the boat. The hell they couldn't just sit here. Ezra had sat in worse places than this. Spent nine hours— _nine hours_ —on his face in a mud hole filled with water that smelled like piss, trying not to breathe while an entire battalion of Vietcong milled around the jungle not thirty yards from him. How well would Vaughn have handled that?
Ezra's stomach was clenched, his mind unsettled in a way it never had been before in a situation like this. It wasn't fear that had him shaken up; no, it was something even more disturbing than that—uncertainty. It was a good way to get yourself, and others, killed. He needed his old mind back, the old instincts, the old moves. Everything he needed now had that word in front of it: _old._ He'd spent decades trying to become someone different than he was, and now he was afraid that he'd succeeded at the task.
## 31
The island showed itself as a dark silhouette against the gray sky, each tree taking on a gradual shape as they neared from the south. Frank was tempted to keep running, head straight into the shore. That'd change some things up, for sure. All four of them in the water, it'd be a matter of who surfaced fastest and who held on to their guns. Since he didn't have any of the guns, though, probably wasn't the wisest choice.
"This is it?" AJ was leaning down to make his words heard over the wind, his face close to Frank's, the gun within reaching distance. Frank looked at it and wondered if he could get his hands on it, whether he could move fast enough. He thought he probably could, but then there was the one they called King to worry about, and Nora directly behind him, in line to accept any bullet that passed through his body.
"Well?" AJ pressed closer, raised the gun a few inches. "Is it?"
Frank nodded, throttled down, the island maybe fifty yards away now, the cabin visible between the trees.
"All right," AJ said, and his voice was different now, softer and measured. "All right. Bring it in slow, kid. Everybody look happy. We're all friends, remember."
He had the gun pressed into Frank's chest.
Thunder hammered through the sky again, and the darkness was such that the trees across the bay seemed to disappear into a night sky. It couldn't be later than one in the afternoon.
Frank was staring up at the house and the trees closest to it, trying to imagine where Ezra was. He'd be watching them approach, Frank was certain of that. The motor was loud, even over the thunder and wind, and Ezra wouldn't ignore it. So where was he? Frank couldn't see him anywhere in the trees, but they were dark and whipped by the wind, branches tossing. The beach was close now, twenty feet ahead, and Frank had the motor throttled all the way down.
"Take us in," AJ said.
"All the way?"
"Yes."
Frank gave the throttle a quick hit, goosing the motor enough to send them toward shore with a hard push, and then cut the engine, had the blades off by the time the boat scraped into the gravelly bank.
"Get her out," AJ said, speaking to King. "Get her out fast and keep that gun in her back. Come on!"
King rose awkwardly, a big man with land legs, then pulled Nora up, his gun in her back as instructed. He stepped out and got one foot down in the water, almost fell clearing the other one. Nora was submerged nearly up to her knees.
"Move," AJ said, giving Frank's stomach an encouraging twist with the gun barrel. "Out and into the trees."
Frank went up to the front of the boat, passing AJ to do so, that familiar Smith & Wesson just inches from his hand for a second. He cleared the front of the boat with a jump, got almost out of the water, soaking only his shoes before joining King and Nora on the beach. Then AJ was out, and everyone was looking at him and waiting for instructions except for Frank, who kept his eyes on the trees by the cabin. Ezra was in there somewhere. Had to be. Why not shoot? Surely he saw the guns.
_Take them, Ezra,_ he thought. _Damn it, take them!_
No shots came. No sound at all except for more thunder and the howl of the wind across the lake and AJ ordering everyone up to the house.
Frank was shoved into the lead, and he climbed the trail with a cold fear sliding through his body, squeezing his chest. He'd put everything on Ezra, every chance any of them had left, and now Ezra was nowhere to be found. What if Frank had been wrong? What if Ezra hadn't gotten the phone call or been alarmed by it, hadn't heard the motor, was completely unprepared for any of this? If Ezra wasn't ready, that left nobody but Frank for the job.
They came up over the hill, and the cabin came into view. AJ stepped closer to Frank, wrapped one hand in his shirt to keep them together, used the other to press the gun against Frank's kidneys.
"That door going to be locked?"
"I don't know."
"If it is, you call out for Ballard."
Up the steps of the porch as the rain began to fall faster, pattering through the leaves and beading on the floorboards, then to the door, Frank's hand closing around the knob as AJ released his shirt and reached back for his second gun. Locked.
"Call his name," AJ said, hissing it in Frank's ear, and Frank opened his mouth and a laugh came out instead of a name.
"The boat," he said and laughed again, turning away from the door.
"What?"
"It's gone. _They're_ gone."
How in the hell had he missed that? Staring at that island so intently as he'd brought them in, scanning the trees, double-checking every shadow, and he'd forgotten the damn boat. They were gone, all right, gone in the boat and into the storm, and that meant Ezra had understood the warning.
"They took the boat and left," he said. AJ shoved him aside and raised his foot and slammed it into the center of the door, tore the hasp out of the frame and burst into the dark house, calling for King to stay on the porch.
They waited while he searched the place, found it as empty as Frank already knew it would be.
"Where did they go?" AJ returned with a snarl, his hand so tight on his gun that the muscles and veins in his forearm stood out. All of the composure and calm were gone now, nothing but fury left behind.
"They left in the boat," Frank said again.
_"I know that!"_ AJ grabbed Frank's throat and drove him backward, slammed him into the cabin wall and pressed the gun into his mouth, banging the muzzle through his teeth. Nora screamed, and King said something in a harsh whisper. Frank couldn't see either of them, couldn't see anything but AJ's face and the gun. The metal was cold against his tongue.
"You know where they are," AJ said, the words slow and soft. "You _know_ , and don't lie again, do _not_ lie again. You got one chance, and you tell it right this time. Did they go to the police?"
Frank shook his head ever so slightly, not wanting to tamper with that gun.
"He doesn't know!" Nora shouted from somewhere behind AJ. "They were here when we left!"
"Shut up." AJ's eyes never left Frank's. "He knows, and he's got one chance to tell me."
The voice was back then, Frank's father's voice, whispering again.
_Trust Ezra. You already did once, and that was a bigger risk than this, because you weren't sure he'd gotten the warning. Now you know he did. He's ready for them, son._
"He doesn't know," Nora said again, her voice tight with tears.
_But you do know. Have a general idea, at least, because you know what I would have done. You learned from me. Don't want to remember it now, but you learned from me, listened to all the old stories and remember every damn one and who did_ I _learn from? Ezra._
AJ pulled the gun back slowly, the spit-covered barrel sliding out of Frank's mouth.
"Where are they?" he said.
"On the lake."
AJ's head canted to the right, into a shadow. "Where on the lake?"
Frank swallowed, worked his tongue around his mouth, still tasting the metal of the gun. The last taste his father had ever had in this life.
"The north end. That's as much as I can tell you. They were here when we left. They're gone now, and they didn't tell me where they were going. He knew you were coming, somehow."
AJ's anger seemed barely tempered by a need to believe Frank.
"Then why would they still be on the lake?"
Frank looked past AJ's shoulder, saw Nora watching him.
"He wasn't sure how much time he had before you got here. Couldn't even know for sure that Nora and I had ever gotten away from my cabin. And since he wasn't sure, he couldn't risk going south to get back to the boat ramp or to the cabin. Too much of a chance he'd run straight into you. So he'd go north."
"What's north?"
"Nothing," Frank said. "Nothing but water and woods."
Nine times Grady had called; nine times Atkins had failed to answer. What in the hell was going on?
He'd driven past Wausau and into a rainstorm, cruise control set at ninety now and still nobody stopping him. All he could hope for at this point, as Tomahawk neared and his wipers slapped back and forth at the highest speed setting, was that Atkins couldn't take his calls because he was too busy with Frank. Interviewing him in some safe room in a building far away from Vaughn Duncan and Devin Matteson, maybe. Or maybe it was already done; maybe Matteson and Duncan were both in handcuffs, and Atkins was preparing for the mountain of paperwork that lay ahead.
Maybe a lot of things. As many optimistic options as Grady could produce, he couldn't believe any of them. Not today. Because it was a karmic world, Grady believed that in his heart, and he'd spent too many days and too many years telling himself that he could always make up for his lie, that there would always be time, somewhere down the road, to sit down with Frank Temple and set him straight, give him the truth and apologize and explain why he'd done it, explain that they'd wanted so badly to take Devin down that a little misdirection had seemed so, so insignificant.
The gambit hadn't paid off, though, and so Grady kept that damn watch on Frank Temple out of a little fondness and a lot of guilt and reminded himself often of a personal pact that one day, if it ever seemed necessary, he would tell the kid that it hadn't been Devin who gave up his father.
Grady had let seven years roll by, twenty-five hundred days, and had never said a word. Because it hadn't mattered, not anymore—Frank had swallowed the lie, but it hadn't hurt him, and now, after all this time, there was no way that it could.
Wrong. It was going to hurt him now. Frank and who knew how many others. And all Grady could do was streak up the interstate through the rain, destined to be too late.
As they had so many times in the past, Ezra's ears warned of disaster before his eyes. For a moment he questioned it over the noise of the storm, but then the wind abated for just a moment, as if the lake were going to give him _one_ break today, and that was enough to confirm his suspicions: There was a boat on the water.
He could hear the engine faintly, this one riding a lower and stronger pitch than the little outboard under his hand would create. It was a familiar sound, the growl of a Merc two-twenty-five pounding hard, the rhythm of his daily life in the summers.
"What?" Renee said, seeing his face.
"There's a boat coming."
"Could be anybody," Vaughn said. "Let's go, man. Faster we get back to the car, faster we're out of here."
The fear was returning to his voice now, that jerky panic that he'd talked with earlier in the day.
"No." Ezra shook his head. It could be Frank, alone, but something told him it wasn't, told him that the game was in play now.
"You don't even know it's them. I can't see any boat—"
"It's them," Ezra said. "I know the sound of my own boat."
He looked down at the throttle under his hand, knowing that it would dictate what happened next as much as anything would. His boat ate up the water faster than anything else on this lake. Trying to outrun them with the little nine-point-nine would be like a car chase between a Lamborghini and a dump truck.
"It could be that kid and the girl," Vaughn said. "Just them."
"Could be," Ezra said, even though he knew it wasn't. "If it is, we'll know soon. Right now, we got to get ready."
The best scenario would be to ditch the boat and take to the trees and get ready to do some shooting. If he were in his own boat, this bullshit would be over before it got started. He still had his rifle in the boat, with a night scope that would work just fine in this weather. Take that and head into the trees and then he could make damn sure these boys would have a rude welcome. Should never have left it in the boat. Damn. It was the little things that killed you, the missed details and slips of timing, and Ezra felt those stacking up on him now, things that he wouldn't have missed in another time and another place. He was out of practice, had been _happy_ to be out of practice, but now it was proving to be a dangerous thing.
When they got to the island and saw it was empty, the boat missing, they'd begin calculating the situation just as he was, and whoever came out ahead in this fight would be the one who thought it through the best, saw the moves before anybody made them. Combat was a thinking man's game, always had been.
So think, then. Think hard and well and think _fast_ damn it.
He looked back at Renee and Vaughn, saw them waiting on him with anxious faces, and nodded to himself. Step one, separate these two. It seemed like a bad idea at first blush, but that was almost a good thing, because it meant the boys in Ezra's boat wouldn't expect it. Generally you'd want to keep everyone together, protect one another, seek safety in numbers. The second layer of this move, and the one that probably meant the most, was that the men from Florida wanted Renee and Vaughn together, not separate. So if things played out poorly, if these men caught them, better to make it happen one at a time. That would slow things down, and when you slowed things down you had more time to come up with a countermeasure.
"Is there any way we can get help?" Renee said.
Ezra took his cell phone out to humor her. Nobody could get here fast enough, and anybody who might wasn't going to be the sort of cop who could help with these guys. It'd be a Fish and Wildlife officer or a sheriff's deputy or some other poor bastard who'd do nothing but add to the death toll.
"No signal," Ezra said, the whitest of lies, because the phone showed just one tiny bar, the barest hint of a connection. "We got to get moving again. First thing we're going to do is split up."
Renee was silent. Vaughn said, his voice wary, "Split up _how_?"
"You and me are going to be on the shore," Ezra said, gesturing north of where they sat, "and she's going to stay on this island. Temporarily."
"No way." Vaughn shook his head. "No chance I'm leaving her alone. You're a damn coward."
"We're splitting up to protect her," he said, speaking to Vaughn and pointing at Renee. "She stays here while we go across to the main shore, and we'll make sure they _know_ that we're going there. We'll beach the boat in the open, make it obvious."
"No," Vaughn said, but Ezra ignored him and spoke to Renee.
"You're a swimmer, right?"
"Yes."
"How good?"
"Very."
He pointed west across the water as lightning lit up the bay. "Can you make that shore?"
It was a hell of a swim, but she nodded.
"All right. Anything happens and you're on your own, that's the one to shoot for. Walk far enough, you'll hit a fire lane."
"I'm _not_ leaving her!" Vaughn spun toward Ezra, leaning close. "You want to go on and lead them into the woods, do it, man. Go ahead. But I came here to take care of her, and I am going to do it."
"No," Ezra said. "You're not."
Vaughn stared at Ezra with a strange flicker in his eyes. It surprised Ezra, almost made him want to lean back, a crazy quality in the look.
"You want to take care of her," Ezra said, "then you'll help me occupy these boys."
"I'm not—"
"Please, Vaughn," Renee said, and her voice was gentler than Ezra had heard before. "Please."
That stopped him, and he looked away from Ezra and stared at Renee. "I can take care of you," he said. "We don't need to listen to him, Renee. We don't need him."
"Yes, we do," she said, tone stronger now.
Ezra couldn't hear his boat's engine anymore. That meant they were stopped, which probably meant they were at the island, checking the empty cabin.
"We got to go," Ezra said, "and you're coming with me."
Vaughn sat in a furious silence while Renee climbed off the boat and into the puddles and mud onshore.
Ezra reached under his seat and found the gun he'd taken from her on the porch, the one she'd stuck in his eye. "Here."
She took the gun, and Ezra gave her a good-luck nod and then pushed the boat offshore, sent them back into the water as she walked toward the trees. Before he fired up the motor, he reached behind his back and withdrew the gun he'd taken from Vaughn on the beach earlier that day, held it out.
"You ever actually used this?" he said.
Vaughn's eyes were dark and small, his face wet with rain.
"Yes," he said. "I've used it. Probably in ways you wouldn't guess."
"Fantastic," Ezra said. "Maybe you'll get to tell me the story sometime. Right now, it's time to move."
He pressed the gun into Vaughn's palm.
## 32
They were back in the boat as the rain billowed down at them like laundry tossing on the line, flapping in gentle gusts, but each gust was a solid wall of water. Nora's hair clung to her neck in tangles, water running into her mouth and eyes, the whole world gone wet, lake and sky blending into a liquid universe. She sat in the back with King's hand locked into her arm, his gun close, as Frank started the engine and took them back out into the lake.
AJ's mood was different than it had been before the island, wilder, his self-control held together by a few overstretched threads. She wasn't sure if it was the realization of what he'd done back at the cabin catching up with him or more a sense of anticipation—it was now clear that Ezra knew they were coming.
But did he? Was he really somewhere north of here, taking refuge in the storm with Renee and Vaughn? Or had Frank been lying, saying whatever it took to get that gun out of his mouth?
The engine roared louder behind her and the boat lifted again, shoving her backward, tightening King's grip on her arm. She'd have another set of bruises from him now, more blue streaks from his big ugly hands.
They hammered across the lake, the bow banging against windblown waves, and for some reason her thoughts turned to her mother, always wanting Nora married off and tucked away from the world, and Nora suddenly wondered if this was why. The world could send its evil blowing into your life disguised as something as innocent as a car with a smashed front end, and you'd never see it coming.
But she should have seen it coming. Had known even as she pocketed the two thousand dollars that her father wouldn't have done it, that he'd have demanded ID and some more information, or maybe refused the car altogether.
Her focus should probably have been on the hand that was digging into her arm and the guns around her, but she couldn't take her mind off the earnest, pleading look on the face of the man she'd believed to be Dave O'Connor as he'd put the cash into her palm and assured her he could be trusted. She closed her eyes, reopened them, tried to see the lake instead of the mistake that had taken her here.
This far north, there was nothing to see except wilderness. The trees lined the shore in unbroken formation, like soldiers from some ancient battle massed and ready for action. Here and there stumps and weathered trees jutted out of the water, and the sky was empty of the osprey and eagles and gulls that normally filled it, empty of everything but roiling clouds and sheets of rain. AJ was speaking into Frank's ear and pointing, sending Frank along first one shore and then the other, cutting back and forth across the lake, searching for Ezra and the others. It was a random, worthless method, even Nora knew that. Someone like Ezra would be able to hide the boat so well that they'd never find it, not if they spent the whole weekend out here. He probably had it onshore by now, dragged into the woods and covered with branches and underbrush, utterly invisible.
So what then? What would AJ and his near-the-breaking-point temper do when he realized they'd never find the boat?
She hadn't formulated an answer to that question yet when Frank abruptly dropped the throttle and the boat sloshed to a stop, rocking on its own wake. Nora stared over his shoulder and felt her stomach dip with horror and astonishment—the boat was dead ahead. Not hidden at all, completely unconcealed, in fact, just sitting there on the flooded main shore, tied among the trees, dead center in the bay, visible from a hundred yards out even in the storm.
What was Ezra thinking? Had he lost his mind?
Ezra sat in the wet leaves for a long time with Vaughn at his side, thinking about the line he'd uttered as Ezra passed him the pistol and asked if he knew how to use it. _I've used it. Probably in ways you wouldn't guess._
He thought about that and about the way Vaughn had reacted to being separated from Renee, the sort of desperate need he'd shown for her, took those two moments and put them together with the worry he'd had before, the notion that Vaughn was not the sort of man Devin seemed likely to turn to in a time of crisis. He considered all of those things for a while as they sat in the rain and waited for their pursuers, and after a long time he turned to Vaughn and said, "Devin's alive."
Vaughn had been staring out at the lake, and now he continued to do that, but everything in him seemed to shut down, not a breath coming, not a blink.
"He is alive," Ezra repeated. "Left the hospital yesterday. Hasn't been seen since. The way Frank heard it from the FBI, they think Devin might have been headed this way."
This time, Vaughn got himself together enough to muster an attempt. He cocked his head, turned away from the water to face Ezra, and said, "I don't know if I believe you, but if you're serious, then that's good news."
Ezra said, "No. It is not good news for you."
"I don't understand," Vaughn said.
"I think I do," Ezra said. "I think I understand."
Vaughn's tongue slipped out of his mouth and ran across his lips, as if they needed moisture even with the rain beating off his face.
"Now," Ezra said, "it being just the two of us out here in the rain, and Devin's wife far from hearing range, I'd like to ask you a question and receive an honest answer. Did you shoot him?"
"What? Man, we sat on that porch and I _told you_ —"
"I know what you told me," Ezra said, "and this time I'm asking for the truth."
A gust of wind blew hard for a few seconds, bending the treetops and showering Ezra and Vaughn with water, and then it faded and the rain slowed and the woods grew quiet around them.
"I'm the only chance you've got today," Ezra said. "You better _know_ that, friend. I'm it. And I need the truth."
There was a long pause, and then Vaughn said, "She's scared of him. That's all it was. She doesn't love him. How could you, guy like that? But how do you leave him, too? I'd be scared, if I was her."
"You think she's in love with you?" Ezra said. "Because I didn't see that."
Vaughn tensed, a quick flash of anger. "She could be. She might be. Man, you haven't seen us, you don't know what you're talking about. I spent times with her, so many times, when she was telling me how much she liked all the ways I was different from him, how a guy . . ."
His voice trailed off, but Ezra understood what Vaughn never would, that Renee had been part of the game, packaged with the cash to keep this guy happy and playing for Devin's team. Someone in law enforcement, buying him off might not be enough protection. Devin would have wanted to bring Vaughn in close, lure him as near as possible so he could be watched. You looked for something that kept a guy like Vaughn on the hook. Renee, it seemed, had played that role awfully well.
"Were you having an affair with her?" Ezra asked. "Sleeping together?"
Vaughn shook his head. "No. It wasn't there yet. But she was scared of him, and I know she wanted to leave him, I _know that for a fact._ "
He hissed the last line at Ezra, spit showing on his lips.
"All right," Ezra said slowly. Then, "Why'd you run _here_? Of all the places you could have gone, you came here?"
Vaughn didn't offer an answer, but after a moment Ezra gave himself one. "You needed someplace that would seem like Devin's idea. To convince her."
That got a nod. "He'd talked about it once. Right at the beginning. Offered me the place if I wanted to stay there, you know, a vacation or whatever. Wrote your name down"—he nodded at Ezra—"and said I should just call you and say when I was coming. He acted like it was funny, though. Seemed real entertained by it."
"He would have been," Ezra said. "I promised him if he ever came back here, I'd kill him. He probably thought it would be real damn funny if you called me and said he'd sent you. If you thought you'd killed the guy, though, why did you need to run at all?"
"To be with her," Vaughn said, his voice barely audible. "To be with her, away from the rest of it. To show her what I could be. That I could take care of her. That I could be like him, only . . . better. If she knew that he'd trusted me, if she knew . . ."
He looked up at Ezra, hope filling his eyes. "You won't tell her. Will you? You hate him, too. You understand."
"I am a damn fool," Ezra said. "An old fool."
"What?"
Ezra stared down at him, felt contempt for Vaughn and loathing for himself rising warm out of his belly.
"People are dead, and more are going to die," he said. "For you. And I'm out here protecting you."
"You think I wanted that? Think I wanted any of you people anywhere _near_ us?"
Ezra didn't answer.
"You won't tell her," Vaughn repeated. "Right? You said you'd promised to kill Devin. You just told me that. So you understand."
So he understood. This sniveling, murdering little shit was looking Ezra in the eyes and seeing a kindred spirit.
"I will tell her that her husband is alive," Ezra said.
_"What?"_
"He is alive," Ezra said, "and she deserves to know that. You didn't kill him, and now whatever you were hoping to pull off with her, it's over."
"It's not over," Vaughn said, speaking carefully, "if Devin is dead."
It was quiet for a minute, his suggestion hanging in the air.
"No," Ezra said. "No, we're not killing anybody so you can get his wife. I'm not being a part of that."
"You said you _wanted_ to kill him." Vaughn was pushing off the ground, his body rising with his voice. "You just _told_ me that the reason he didn't come back up here anymore was because you said you'd kill him. So what do you care, man? It's nothing to you."
Nothing to him. Vaughn was right about that. Yet here Ezra sat with him in the wet woods with a gun in his hand and a bloody mess headed his way. He started to speak but stopped when he heard a motor.
They were here. He leaned out of the trees and looked across the water, Vaughn joining him, and saw his boat out in the lake, coming to a stop about a hundred yards from shore. Ezra could see four figures in the boat, recognize Nora but none of the others. It could be Frank behind the wheel. Yes, that was probably Frank. They'd make him run the boat.
"Is Devin out there?" Vaughn said. "If he's out there, man, kill him and let's be done with it. Just let us go. Let me take Renee and go."
_"Shut up."_ Ezra wanted to lift his gun and bring it down in the middle of Vaughn's face, hit him again and again until his lips were smashed into his teeth and he couldn't say another word.
The motor came back to life, and the boat was headed their way, coming into shore. Ezra watched them come, saw that it was indeed Frank behind the motor, and wished again for his rifle. It would be over now, if he had his rifle. Instead, they had to wait and let the battle come to them. It wasn't what he wanted.
They landed the boat, and Ezra rolled back against the base of the tree, looked at Vaughn, and said, "They're coming onshore, and we'll let them come, okay? These guns, they don't have the range we need. So we've got to sit here and wait, wait _quietly_."
Vaughn didn't answer or even nod, just looked at Ezra with blank eyes. A hell of a combat partner he was going to be. It was up to Ezra, nobody backing him up out here, no Frank Temple or Dan Matteson like in the old days.
"When they come on shore," Ezra began, but he was interrupted from further instructions by the sound of another motor. What the hell? From where he stood, he could only see his boat, and the big Merc was shut down. He shifted a few steps to the side, knelt again, found the little aluminum boat. Yes, there was someone on board, starting the engine. Frank was on the beach, pushing the aluminum boat back into the water. Ezra had wedged it well into the sand.
Frank got the boat free, climbed in with the tall man who was at the motor, and then both boats pulled away, out into open water. Kept going until they were a good two hundred yards offshore, and then the anchor went out from the little boat, which was pitching hard in the wind.
"Shit," Ezra said, watching them. This was a good move. A damn good move. They didn't want to have to follow Ezra into the woods and leave both boats on shore. If they made a mistake and let Ezra double back and return to the boats, that would be the end of it. With just two guys, they also couldn't afford to leave one guarding the boats. The solution, one Ezra would have considered if he were in their shoes and felt good about how much time he had, was to remove one of the boats. With this storm keeping the lake desolate, they had the time.
"What are they doing?" Vaughn said, whispering even though there was no chance of being heard down on the beach.
"They're moving one of the boats offshore. Far enough away that we can't get to it. Then they'll come back."
They would come back in his boat, which was bigger and faster and also possessed the most important quality for this situation: It required a key to start. Take the key with you and the boat was dead in the water, unlike the little boat with the outboard and its pull cord. Ezra had no second key hidden away on the boat, but he could probably hotwire the thing if he had enough time. Finding that sort of time, though, was difficult when people were shooting at you.
Far out on the water, an exchange was taking place on the two boats, men stepping off one and into the other. They'd anchored almost directly across from the island where Ezra had left Renee, no more than fifty yards from its shore, and he hoped she was well hidden.
The exchange was completed, and it looked like Ezra had been right and they were taking the little boat on their return trip. The showdown was coming, and Vaughn didn't matter anymore, could be dealt with later, after this last bit had played out.
If the first goal was to separate Nora from AJ and King, then Frank supposed he should count this as progress. It was hard to believe that, though, as he watched King bind Nora's hands behind her back with duct tape, then wrap her ankles together. She'd given up on fighting him by then, but when he advanced on her with the piece for her mouth, she spoke.
"No. Please don't cover my mouth."
He snapped the tape over her face, wrapped it around her jaw until it tangled in her hair, and then added another, shorter strip. The fear grew in her eyes when her mouth was covered, and Frank wondered if she was claustrophobic. They had her stretched out in the small boat they'd taken. AJ kept his gun on Frank while King handled the tape work, and Frank had put up a little bit of an argument, just enough to let them think he opposed this. In reality, it was for the best. He felt bad for Nora, couldn't meet her eyes because the panic that showed there was tough to take, but he knew it would be easier if he was alone with these two. Nora was a liability, an extra concern anytime he decided to take action. With that eliminated, he was a little freer. Now the only person who would die immediately if he screwed up was himself.
"You're staying with her," AJ told King, and that quickly all of Frank's hope for this situation began to disappear. "Wait till we're on the shore. The minute we hit that, you start watching the clock, all right?"
_No,_ Frank thought. _No clocks, no countdowns, please don't say that._
"Ten minutes go by, you put a bullet in her head. No hesitation."
"Won't be a problem," King said, and he leaned down close to Nora's horrified face, stroked her cheek with the back of his hand. "Won't be a problem at all, will it, hon?"
The lake and the land surrounding it seemed to tilt and spin around Frank, that time limit— _ten minutes, ten minutes, ten minutes_ —scorching through his brain, every possible scenario filtering through, none of them any good. It was too little time.
"You can't—" Frank wasn't even sure what words would have followed, because they never got the chance to develop. AJ hit him backhanded with the gun, caught him flush in the face and knocked him back into the boat, almost over the side and into the water. Blood left his nose, ran down his chin, and onto his shirt.
"Get up," AJ said.
Frank stayed down, looking at his own blood.
_"Get up!"_ AJ screamed it this time, then almost lost his balance in a wild kick aimed at Frank's chest that instead hit the seat above him. The boat was rolling in the windblown chop. Frank got to his feet, fat red drops of blood speckling his jeans.
"Start the engine," AJ said, shoving him into the seat behind the wheel. "And take us back. We're almost done."
Frank turned the key over and the motor growled and then they were in motion, pulling away from the aluminum boat where Nora— _ten minutes, ten minutes, ten minutes_ —waited bound and gagged and alone with King.
A sharp ache cut into Frank's ribs as AJ leaned forward, pressing the gun against him, and shouted, _"Faster!"_ into his ear. Nothing to do from this position that wouldn't generate a bullet in the lungs, and the only power Frank controlled—the boat—was useless now, too. If he rolled it or wrecked it and somehow pulled off the miracle of escaping unharmed, King would still be back on the boat with Nora, watching and ready to produce the planned result without the ten-minute wait. Whatever happened would have to happen in the woods, and it would have to happen fast. Frank leaned over the wheel, holding his shirt to his bloody nose, and slammed the boat through the lashing water. As the blood began to clot and the wind tore at his eyes, he tried to coax some insight or reassurance or reminder of old lessons from that voice in his head one more time. None came. The old man had said his piece.
"Slow down and land it," AJ shouted. There was no real beach on this part of the shore, just trees giving way to rocks. The water was high this early in the year, and some of the smaller trees near the shore were almost submerged, only the tops showing. Frank brought the boat in among them, felt the stern shudder as the prop chewed through some branches. The rain was driving into the trees, swept by that strong western wind. It was going to be a wet, slippery climb to the top of the slope.
"Tie it up," AJ shouted, pointing at the half-submerged tree just off the bow. "Cut the motor and tie it up!"
Frank got the stern line tied to one of the protruding limbs, but the wind had pushed the boat backward so quickly that the bow was now facing away from the island, toward the thin shape that was the boat with Nora and King.
"All right," AJ said, tearing the key out of the ignition and sliding it into his pocket. "You lead the way, and stay close."
Stay close so Ezra would have trouble getting a clean shot. Frank stepped out of the boat and sank up to his waist, would have sunk deeper if his feet hadn't found a stump. AJ splashed over the side behind him, and then they were both stumbling through the water, pushing branches aside. The water was cold, crawled up through Frank's legs and into his chest even though he was already drenched from the rain. He slogged through the small trees and stumps, slipping and splashing until he was out of the water except for his feet, facing a muddy slope lined with small saplings that seemed to grow horizontally. Then one more step and his foot touched the gravel bank and he felt like it was coming down on a land mine, that clock— _ten minutes, ten minutes, ten minutes_ —starting to tick back on the boat.
"Start climbing," AJ said, his breath warm on Frank's neck. He was staying right at Frank's back, determined not to give Ezra a shot.
He fought his way up the slope, using the saplings for handholds, his feet sinking into the muck, his brain counting seconds and subtracting them from ten minutes. They got to the top of the hill and stood gasping for breath and staring into dark trees that were shaking with wind and rain. No one in sight.
"Ballard!" AJ shoved Frank forward again, toward the trees, and bellowed the name. "Ezra Ballard, if you hear this, you listen sharp. Out on the boat is the girl from the body shop. Nora, I believe is the name. You know Nora, don't you?"
They were into the woods now, and AJ paused when a prolonged ripple of thunder threatened to drown out his words. The thunder passed, and after one flash of lightning, he began to shout again.
"The minute we landed, you started to run out of time. That girl's got ten minutes of life left. Those ten run out, and she takes a bullet right in her beautiful face."
They were fifty yards into the woods now, walking without purpose, and Frank realized AJ was banking completely on the assumption that Ezra was close enough to hear him. What if he wasn't, though? They were just going to walk around out here, shouting into the wind, until ten minutes were gone and Nora was dead?
"You can stop that," AJ yelled. "What I want is Vaughn and Renee! You send them out and this is done. Renee, babe, you hear me? Devin is alive. Devin is alive!"
Eight minutes. That's what Frank expected they had left. Maybe seven? The climb up the slope might have taken longer than he thought. Either way, it was time to act. He'd been waiting on Ezra, praying for Ezra, but the woods around them were silent except for the rain and the echoes of AJ's shouts.
"Come on! Let me know you hear me!" AJ screamed it, his voice fading on the last word, and then went quiet and they both listened. There was no sound.
## 33
What Ezra thought, kneeling in the wet earth beside a fallen pine, was of another word with _old_ tacked on the front of it, the sort that had been tormenting him. _Old game._ Unlike those phrases that had run through his mind earlier, this one wasn't a negative, had no doubts chasing it. Instead, it was an _old_ of familiarity, as in _old friend_.
_Old game_ meant Ezra knew the game. Had played it well. Few were better at it, in fact, and this son of a bitch shouting into the trees wasn't going to be one of them. There were no doubts now, because there were no decisions to make. Only one outcome would work.
"Seven minutes!" Another shout. "That's how long you've got to cooperate!"
_To cooperate? No, friend, you do not understand. The seven minutes may be accurate, and important, but the cooperation? Not a part of the game that I play. Those minutes mean something altogether different to me._
_How long I have to kill your friend on the boat_.
The shouting man and Frank were already twenty yards behind Ezra and Vaughn and pushing farther away, walking in a straight line and making so much noise that there was no way they could hear anyone around them. Ezra might have taken a shot if he'd had to, but the one thing this guy did that was smart was stay pressed against Frank, preventing a clean line of fire. That was all right, though. If the time limit was honest, then the guy who mattered most wasn't in the woods anyhow. He was back in the boat with Nora, well out of handgun range.
A damn good thing, then, that this idiot had just ferried Ezra's rifle back across the lake and left it behind in the boat.
He let them push on, still shouting, for another fifteen steps, and then turned back to Vaughn, who was stretched out on his face in the wet leaves and dirt. Ezra nudged him with the toe of his boot, and Vaughn lifted a mud-streaked face.
"I'm going to the boat. You're staying here."
Vaughn had a wild, unfocused look, the one he'd been wearing ever since Ezra had told him that he would not kill Devin, that he would not preserve these lies for Renee.
"Stay here," Ezra repeated. "If he comes back, shoot him."
"No, don't—" But Ezra was already moving, taking advantage of another roll of thunder that offered some additional sound cover. He moved on his belly, using his knees and forearms, a quiet and fast crawl that had saved his life more than once. Saved his old life, saved old Ezra.
They walked deep into the woods with AJ shouting out a constant stream of threats and explanations, intimidation and coercion. None of it got a response. The rain was falling harder than ever, slapping through the trees beneath steady rolls of thunder. When the lake was out of sight, AJ grabbed a fistful of Frank's shirt and shoved, pointing him so they were now walking north, parallel to the shore.
"You start talking now," AJ said. "Make that old bastard hear you. Only a couple minutes left."
AJ was right; there couldn't be much time left at all—three minutes, maybe—and still Frank hadn't made a move, just walked along and waited as if some great opportunity were going to present itself. That wasn't going to happen.
_"I said talk,"_ AJ hissed.
"Ezra!" Frank called, and his voice sounded wooden and too soft. He shouted louder. "Ezra, if you can hear us, answer, or people are going to die. Nora's back there on that boat. Answer us!"
The answer Frank wanted Ezra to provide was a bullet right between AJ's eyes, but neither it nor a verbal response was offered.
"Old man is going to let her die," AJ said. "You believe that shit?"
Frank started to yell again, then stopped, his eyes going toward a spot fifty feet ahead. The ground seemed to give way there, dipped down a short, steep hill and then rose again on the other side, a sort of sinkhole. It was the best spot he'd have, and he shifted slightly to the left, walking toward it, AJ so distracted by searching the trees that he didn't notice or, if he did, didn't react. The hole would give Frank a chance. Make a move on AJ right now and his first instinct would be to fire. With the gun pressed against his spine, that wasn't an instinct Frank wanted to encourage. Make a move that started a fall, though, and do it fast enough, and the shooting response might not be AJ's first instinct. Catching himself, stopping the fall, would come first. Right?
Better be right. If it wasn't, then Frank was dead.
"Keep talking," AJ said. His voice was tense and he jerked his head around constantly, peering into every shadow, shaking rain out of his eyes, his confidence slipping. These dark woods were not home to him. His sort of killing was done in different places, under streetlights and in alleys and at construction sites. He didn't like it out here, didn't trust himself the same way in this environment. Good.
"Ezra, damn it, _answer us_!" Frank shouted, completely unaware of the words leaving his mouth, focused instead on a quick mental rehearsal, choreographing the move he was going to need to make.
AJ was behind him, holding the gun against Frank's back. That was okay, though. He'd done it this way before, down in that basement in Chicago, his father coaching him through the steps. This was the normal position, the way you held a gun on somebody when you were sure he couldn't take it away from you. Stand behind him and jam the gun into his back and you had the illusion of total power and control. No way the guy in front could move fast enough to take the gun from you, right? No way.
It could be done, though, had been done before.
_Take the gun away from me, Frank. Come on, kid, too slow. You don't have a chance. You know how many times you'd have died already, trying this? So slow, so slow. Come on, try again. Oh, shit, almost had it that time._
They'd practiced it over and over until Frank could pull it off every time, one of his father's favorite routines because it showcased Frank's speed, and Frank Temple II had _loved_ his son's speed. Today the circumstances were right, too. AJ was standing against Frank's back, thinking that this was the proper approach because he was using Frank for protection, for a shield. It was also keeping him close, though, and close was where Frank needed him to be.
They were closing in on that dip in the earth, a simple, unimpressive slope that held Nora's best chance at life. The drop-off was in full view now, and Frank saw it was maybe ten feet from top to bottom. It would be a simple step sideways and a sweep of his right arm and leg, have to do it _damn_ fast, but if he pulled it off he could send AJ down the slope.
_Your gun is on his back. Tucked in his belt on his back, and if you make a perfect grab, you might get it. Don't even worry about the gun in his hand. Just get him in front of you and headed down the hill and then go for the gun in his belt._
The drop-off was just in front of them, almost there, but AJ was pulling him away from it now. Shit, he couldn't let that happen, needed the hill. Frank stopped, bringing AJ up with him, and pointed into the trees.
"What?" AJ said.
"Somebody moving, I think. I don't know . . ." Frank started walking again, toward the imaginary source of noise, and AJ followed. They were walking alongside the drop-off, and Frank's pulse was drilling away but his breathing seemed frozen. Four more steps, now two, now one . . .
In the end, he didn't go with the move he'd rehearsed in his mind, that sidestep and sweep. It had sounded good, sounded like the only thing to try, but in the second that he moved, instinct took over and some subconscious part of his brain told him it wasn't going to work. Instead of sidestepping he simply spun, a full, fast pivot that took his back away from the gun as he lifted his left arm and held it out straight and kept on turning, caught AJ across the shoulder and drove him forward.
It turned out he'd been wrong; AJ's first instinct still was to fire. The gun went off a half second after Frank had spun away from it, tore through the air inches from his flesh. Then his arm hit and knocked AJ toward the drop-off. They were a step too far away, and AJ might have been able to recover if Frank hadn't gotten a foot against the back of his knee as well, ruining any chance of balance he'd had left. AJ stumbled and fell and there was the gun in his belt, right there, all Frank had to do was reach out and . . .
He got it. His fingers closed on the stock and then AJ was gone and tumbling through the wet leaves and broken branches to the bottom and the Smith & Wesson was out of Frank's left hand and into his right and lifted and aimed.
For one fleeting second, he waited. Just long enough for AJ to land at the bottom of the drop-off and turn back to Frank and start to lift his own weapon. Frank let all of that happen, let him get that close, and then he squeezed the trigger once and killed him with a single round below his right eye socket.
## 34
They'd been alone on the boat for maybe five minutes before King began to talk to Nora.
"Uh-oh," he said, turning back to her with a slight smile. He'd been standing, or trying to stand, in the pitching boat and watching AJ and Frank head off.
"Know where they are now, baby? On the shore. And you know what that means." He tilted his wrist, looked at it, and then frowned. "Damn. Look who forgot his watch. That's no good. How am I going to know when ten minutes go by?"
He leaned close to her, and she tried to slide away but found it impossible with both hands and feet bound. His face, long and angular and covered with rough stubble, was against hers, his breath on her cheek.
"I'll have to guess," he said. "You know, estimate? I was always bad at that, though. Thought five minutes felt like ten."
The wind rose again in a hard gust, and the boat rolled. He put out his hand to catch himself, falling almost on top of her, his legs heavy against hers. Somewhere in her stomach liquid churned, threatening to rise. No, no, no, she couldn't be sick, not with that tape over her mouth. Get sick and she'd choke on it, die, make this even easier on him.
"Look at that," King said, pushing her sideways, running his fingertips along her forearm, over the bruises he'd left two days earlier. "Little love marks. They from me? I bet they are."
She was stretched out on the seat, and he was on his knees now in the bottom of the boat, not even looking at the island, just staring into her face as the wind pulled his shirt tight across his chest and the rain dripped down his face and onto hers. He reached out and took her hair in his hand, squeezed hard enough to make her eyes sting.
"It was dumb-shit luck that kid showed up when he did. Too bad, because we were going to have some fun, you and I. Might still have some." He rocked his hand left to right, jerking her head sideways. "I take that tape off your mouth, we could have some _serious_ fun. But you might be a biter. Yeah, I could see that. You're the type, aren't you? Angry little bitch. So maybe that tape stays."
He lifted her by her hair, and she would have screamed if the tape didn't prevent it. Her eyes were streaming now, nose following suit, the pain demanding a physical response. He pushed her back against the side of the boat and leaned against her, pressing his body down on hers. The sudden change was almost too much for the boat; they rocked hard to one side, and he pulled back at the last possible second, the boat rolling with him. What if he hadn't recovered? What if they'd just kept going over, ended up in the water, with tape over her mouth and her hands and feet bound? She'd die then, too. That or wait the ten minutes.
"That tape stays," he said, flicking his index finger off her mouth, smashing her lip back against her teeth. "Keep you from biting. Tape on the hands can stay, too. You won't need those."
He moved suddenly, slammed her head back against the boat hard enough to make her vision blur, and then he got to his feet and moved back to the bow, leaned against it, and stared into the woods. She tilted her head, tried to see what he was looking at. The angle wasn't right, though, and she couldn't turn any farther without rolling her whole body over. Didn't want to do that, and draw his attention back, so instead she kept craning her neck in that awkward position and tried to see where the boat had gone.
She couldn't see the boat, could hardly see the main shore. There was another island that she could see, but that wasn't where AJ had taken Frank. She let her eyes pass over that shore and then started to look away, wanting to ease that awful pressure on her neck, when she saw motion.
There was someone on the island. No, couldn't be. She was seeing things, some weird reflection, the sun playing tricks even behind the dark clouds. Where had it gone? Wait, there it was again. Yes, someone was moving through the trees on the island just beside them.
Nora kept herself in that awkward position, the pain momentarily irrelevant, and stared. Now the motion was gone, but she was positive she'd seen someone, and not where the boat had been beached. So had the boat been a trick, a ruse?
King turned back from the bow then, and Nora moved her head, but it was a second too slow. He'd seen her staring out across the water, seen the intense look in her eyes, and he followed it.
"Son of a bitch," he said, lifting the gun, and Nora knew she'd just ruined someone's chance to escape.
Ezra crawled to the top of the rise and paused, looking out across the angry gray lake. There was the boat, a few hundred yards out, small but visible. Wait till he had it in the scope. Wouldn't be small then, no, sir. Be nice and clear, a perfect picture of some poor bastard waiting to die.
He couldn't hear the shouting anymore, which could be good or bad. Maybe the idiot was out of shouting range now, and maybe he was quietly working his way back to the boat. Ezra didn't know, and he wasn't going to waste time worrying about it. Time was slipping away, and he needed to get out to his boat. They'd tied it up right in the middle of the stumps and partially submerged trees that surrounded this part of the shore. He could hear a rapping sound as the hull banged against a stump, and it made him like these bastards even a little less. Ezra took damn good care of that boat.
Maybe twenty feet of fairly open ground to cover before he reached the lake and had to plunge into that mess of branches and water, fight his way out to the boat. It would take about thirty seconds to get on board, but he'd be in the open the whole time, and if whoever was out there with Nora had a rifle, Ezra might die before he ever got to take a shot. Nothing to do about that, though. Times came when you had to gamble, that was all. Ezra had gambled before, and still had the dice in his hand.
He got his breathing steadied and thought about doing a countdown, ten seconds and then move, but decided the hell with it. A countdown didn't make it any easier, and he didn't have seconds to waste. He pushed off the ground with his hands and went upright for the first time since he'd left Vaughn, got his feet moving and ran down the hill.
It was a slippery, dangerous mess in the rain, and twice he almost went down on his face, righted himself somehow and kept moving, hit the water knee deep and sloshed through it as quietly as he could, hunching now, trying to stay below the boat. The whole procedure seemed loud as hell to him, but out there on the lake with the wind whipping across the bay he doubted they'd hear. If they were watching close enough they would have seen him by now, which meant he needed to get the gun out fast.
No shots came; no motor roared to life. He waded out to the stern, the water up almost to his armpits, and then braced his hands on the side of the boat and heaved. Damn, it was hard work. He was weighted down with water and wet clothes, and his upper body wasn't what it used to be. He got over, though, flopped across the side and slid down to the floor, lay there breathing hard for a few seconds and waiting for a shot that didn't come.
Still silent. He pushed himself into a sitting position and cast one glance at the console, saw the empty ignition. They'd taken the key, as he'd feared. That could be dealt with later, though. Right now, he needed the rifle.
He'd left it in the storage compartment under the floor, a space designed for fishing rods. He flipped the latch and lifted the cover and peered inside, felt a moment's horror when he saw nothing but rods. But there it was, tucked all the way against the side, a gun that had never seemed as beautiful as it did now.
It was a custom-built bolt-action rifle that Ezra had paid an absurd amount for six years earlier, and it was also the best long-range gun he'd ever held, one that would make the Browning A-Bolt and the Remington 700 look like pawnshop pickups. A high-velocity cartridge rested inside, waiting to head out of that perfect bore at twenty-eight hundred feet per second. Each round that left this gun was a gorgeous product of engineering.
He pulled the rifle free and closed the storage compartment, then slipped the cover off the scope. It was a Yukon night-vision scope, a piece of equipment that cost more than Ezra had paid for some cars but that had seemed only an appropriate pairing for the rifle. He'd often chastised himself for both purchases, which felt like obscene wastes of money when he was in a rational mood. Today, it all felt incredibly cheap. He couldn't believe they'd ferried it right across the lake to him so unwittingly. His own enemy had delivered him his sword. Mercy be on their souls now.
Crawling back toward the outboard, he pressed himself in against the bench seat and lifted the gun barrel, rested it on the stern. Then he lowered his cheek to his shoulder, closed one eye, put the other to the scope.
A night-vision scope, even a good one like this, didn't demand total or near-total darkness for use. It had an infrared illuminator that could be engaged for such conditions, but today the standard image tube was all Ezra would need. It gathered the natural light and enhanced it, and in this storm that would work fine. He slid the rifle left a few inches to bring the boat into view, tweaked the scope to clarify the scene.
The shouting man had not lied. There were two people in the boat, Nora Stafford and a big son of a bitch with a gun in his right hand. Nora had tape over her mouth, and the guy with the gun was—hold on a second, he had just lifted the gun.
The shot rang out loud and clear even in the wind and rain, and for a second Ezra was absolutely baffled, because the guy was turned away from Ezra and firing across the lake. _What the hell?_ Ezra thought, and then he got it: _Renee._ The tall man had spotted Renee and opened fire without bothering to see who the hell he was shooting at.
"Shit," he whispered and brought the crosshairs down quick, sighted along the big man's chest, his finger tight on the trigger. It was a hell of a long shot, even with the scope, at least a hundred and sixty yards. Tougher still because the guy was standing sideways from Ezra's position, offering only his profile. All these details ran through Ezra's mind fast and then were pushed aside because the son of a bitch had fired again and Ezra couldn't let him continue. He clenched his teeth together, steadied his breathing, and pulled the trigger.
It was maybe the best shot he'd ever made. Certainly the best he'd made so quickly. The bullet hit the guy in the neck, blew a cloud of blood into the air, and then he was falling forward, collapsing against the side of the boat. Shit, he'd fallen right on Nora. As Ezra watched through the scope, the boat tipped dangerously to the left. For a moment he thought the body would slide into the water and the boat would right itself, but somehow the guy hung up on the side. The boat rocked back to the right, but then Nora struggled beneath the body and her motion made it roll to the left again. There was a half-second pause where equilibrium was still possible, but she kept thrashing and the little boat overbalanced and tipped, spilling them both into the water.
_"Damn!"_ Ezra lowered the gun and peered out, as if that would somehow help him see, then lifted it again and looked back through the scope. He hadn't meant to send them both into the water like that, but it wasn't the worst thing that could have happened. He'd seen the bullet's impact, knew that it had missed Nora, hitting at least three feet above her. She could swim, surely, or at least hang on to the upside-down boat until Ezra could get out there.
He kept the scope to his eye and watched the water and saw nothing. Nobody moving, nobody splashing. What the hell was she doing? Surely she hadn't been knocked out in the fall.
Then he remembered the tape.
## 35
Nora didn't know _what_ had happened. One second King was standing above her and firing at the figure on the island, and the next drops of blood showered across the boat and he collapsed on top of her. Her first thought was that lightning had hit him; her second was that the person on the island had returned fire. Then she felt the boat begin to tip, and the source of King's demise was no longer a concern.
They rolled backward until her hair was almost in the water, and she thrashed against him, trying to push forward, send them the other way. He was too heavy, though, and the tape kept her from using her arms or legs to push him off. She rocked back and threw herself forward again, a desperate heave, and still he was on top of her, his blood dripping through her hair and down her cheek. The wind gusted again, shook the boat just enough to worsen the lopsided weight, and they slid back toward the water. She had a fraction of a second to suck air in through her nose and brace for the cold, and then they were over the side and she was sinking.
She fell fast and soundlessly, dropped through the frigid water with a sense of terror and helplessness she'd never felt and couldn't have imagined before, strong and healthy and uninjured and dying all the same.
Bunched up the way she was, unable to kick with her feet or pull with her arms, she sank quickly, dropped all the way to the bottom. Her eyes were open, and she could see the shimmer of light that was the surface, knew that she would die staring up at it.
Then she bounced. She'd landed on her ass, but a final violent twist got her feet under her, and she realized that the tape had taken every important motion away from her except one: She could bend her knees.
She pressed her feet into the lake bottom and bent at the knees and pushed off as hard as she could, shot upward again. The lake wasn't deep here, maybe ten feet, and that one massive push was enough to get her to the surface. Enough for one gasping inhalation through her nose. Then she sank again.
This time she was equally sure she'd die on the bottom. The little bit of air she'd gotten didn't feel like nearly enough, but she also didn't need to wait for the bounce this time, knew what she had to do, and so she landed with her feet out, almost upright. One more bend and push, one more rise, but she knew it was futile even as she headed toward the surface. She wasn't getting enough air with these brief appearances above the water, and the adrenaline-fueled strength was fading fast. She could manage another rise or two, maybe, but eventually she wouldn't come up fast enough and she'd frantically try for a breath and take in water instead, drown, and sink back to the bottom, this time for good.
She hit the surface, got a little more air this time, then sank again without ever seeing anything but angry dark clouds. Her feet hit the bottom and she went through that hopeless routine one more time, probably the last repetition she could manage, bend and push. This time she came up under the boat.
Her first reaction to breaking the surface beneath the upside-down boat was panic, but it saved her life. Instead of simply sinking again, she instinctively jerked her head backward, as if to clear herself out from under the boat. She didn't clear it, of course, but instead drilled the back of her skull against the aluminum frame, wedged between one of the seat brackets and the side of the boat.
She stuck. Only temporarily, for a few seconds, but it was enough to hold her head out of the water so she could get a breath and realize what had happened. Then she felt herself sliding away from the bracket, ready to sink again, and responded by arching her back. The motion forced her head back against the bracket and lifted her legs, and her body slid into an awkward floating position, as close to a survival float as you could possibly achieve with no arms to balance you. It wouldn't have been possible without the bracket there under her head, but it worked now.
Breathe. That was all she had to do, all she needed to worry about for this moment, just sucking air in through her nose, trying to get as much oxygen into her lungs as possible before she slipped out of this position and sank again. She hauled the air in, her chest rising and falling, got in at least five breaths before she began to slide away from the bracket again.
She tried to repeat the motion that had worked before, arching her back and lifting her legs, but this time she couldn't find the bracket, and her head kept sliding away.
_No, no, no. Come back, I can stay alive here, I can stay alive . . ._
The water was over her face and she was sinking again when she realized that it hadn't been her head sliding away from the bracket, but the other way around. The boat was moving, pushed aside. She watched it move as she dropped, was three feet deep and still going when legs bumped against her back and then an arm encircled her, wrapped around her neck and hooked under her chin and lifted.
A second later she broke the surface, blinked water out of her eyes, and stared at Renee, unable to utter a single word of thanks because of that damn tape.
Ezra dropped the rifle and got to his feet, hardly conscious of the assailant who still lingered in the woods behind him, and looked around his boat for some way to help.
The key was gone, and he didn't have time to hotwire the ignition. That left no power source but the electric trolling motor, and though it would surely be too slow it was the only chance he had. Ezra couldn't make a swim like that, not anymore.
A swim. The thought triggered a memory of Renee and the initial source of the shooting. She was so much closer to the overturned boat, and she could swim. Had she seen what happened? Did she know Nora was in the water?
He turned back to search the lake for her, but then something moved onshore behind him, and he whirled back to face it, reaching for the rifle.
"Shit," he said and picked up the rifle but left it pointed down. The movement on shore was from Vaughn, who had just stepped out of the trees, holding his gun ahead of him with a wavering grip.
"Come on!" Ezra yelled, turning to the trolling motor, a hopeless effort but the best he could now make for Nora. "Get out here."
Get Vaughn and go after Nora. That's what Ezra was thinking as he stepped up into the bow, the gun held loosely at his side, his eyes scanning the woods to see if anyone else was approaching. No, just Vaughn, and why wouldn't he hurry, and lower that damn gun before somebody got—
Vaughn fired from the shoreline, and for a second Ezra was so stunned he didn't react, but then he realized it hadn't been an accident and he got both hands around the rifle and lifted it as Vaughn took a second shot, missing again, and a third.
The third round caught Ezra in his right side, blew through his ribs and out of his back and splattered blood and flesh off the windshield that guarded the steering console. He tried to keep lifting the rifle, to get it aimed at Vaughn, but the bullet had spun him and now he was stumbling. His knees banged off the side of the boat and he couldn't right himself, flipped over the side and fell into the tangled branches of a partially submerged tree. The branches snapped under him, and he dropped into the water as Vaughn fired again, missing again. Ezra tried to lift the rifle, but it was too heavy now. Or was it even in his hand?
Another branch snapped, and he dropped again, and then the gray sky was fading into an odd red mist and Ezra couldn't focus anymore, couldn't see to fire even if he'd had the gun ready to shoot. The red mist spun into black and then shattered into jagged points of light, and Ezra Ballard closed his eyes and welcomed the water.
Renee got the tape off Nora's hands first, which allowed her to hang on to the boat while Renee freed her feet. The feeling of power, of _life,_ that came back to Nora as she moved her legs and arms was intense. She could support herself again, move again, was no longer helpless. She tore the tape off her mouth, smacked her lips together and sucked in a grateful breath of air and rain, tasted the fresh water on her stale tongue.
"Thank you," she said. "Thank you."
The rain was hammering the overturned boat, a sound like a drum corps, but even so they both stopped talking and listened as another sound, a series of cracks, echoed over the water.
"Guns," Renee said. "Somebody's shooting."
Nora didn't say anything. The strength was already fading from her newly freed legs, and even the gentle kick needed to stay upright seemed difficult.
"Can we roll it over?" she said.
"The boat?" Renee shifted in the water, looked at the capsized boat as if surprised that Nora would want to be inside it. She was treading water easily, her breathing steady. Beneath the surface, her arms and feet moved in ghostly circles, her hair fanning out around her shoulders.
"Let's roll it," Nora said and leaned back and tried to push the craft up, succeeded only in driving herself deeper in the water.
"All right. We can try." Renee swam closer, dipped under the boat, and braced her hands on the edge, as Nora had. "On three."
It took them two tries, but they flipped it. The motor was the key; once that weight had shifted enough, it overbalanced the boat and did the rest of the work for them. By the time the boat was finally floating upright, though, they didn't have the strength left to climb into it. They waited for a few seconds, hanging off the side, and then tried again. This time Nora made it easily, then turned back and got her hand around Renee's forearm and helped her into the boat.
They sat there on the bottom of the boat, getting their breath back and staring at one another. The water had been freezing, but now Nora felt even colder as the wind fanned over them.
"Where are the others?" Renee reached up and gathered a handful of her hair, ran her hand down it in a fist, squeezing the water out. Her eyes were on the lake, away from Nora.
"On the island. Well, one of them is dead. The guy they left on the boat with me is dead. I think someone shot him. That's why we tipped over." Nora took a deep breath, wiped water from her eyes, and said, "And Devin is waiting at Frank's cabin."
Renee sat with her hand still wrapped in her wet hair and stared at Nora with a look that made Nora's neck prickle.
"What did you say?"
"Your husband is waiting at Frank's cabin."
Renee said, "You're confused," but Nora was already shaking her head.
"He's alive, and he's there," she said. "He's not in good shape, but he's alive. Vaughn shot him."
Renee let go of her hair. Her mouth was parted slightly, her eyes distant. "Vaughn shot him?"
"That's what Devin said." Nora watched the other woman's face, then added, "That's what Devin said while he put me and Frank into a van at gunpoint and came out here and had an FBI agent murdered at the cabin. The one named AJ killed him with a knife."
The words slid by Renee without any apparent effect. She said, "Vaughn shot Devin. I've been up here with him, and he's the one who shot Devin. He tried to kill Devin."
"Yes," Nora said.
Renee was looking at the lake without seeming to see anything. She said it again. "Vaughn shot him."
Nora was shivering violently now, the wind and her drenched clothing combining to drop her body temperature.
"Can we start the motor?" Renee said.
Nora turned and looked at it. The thing had been upside down for a while, but it still looked in place, everything as it should be.
"Probably."
"Try to start it, please."
"Where are we going?" Nora asked as she moved for the stern.
"To my husband. But first we're going to stop at that island. I left a gun there."
When he heard the first shot, Frank was down in the hole with AJ, relieving the body of its gun and the boat key. The sound almost dropped him to his knees, overwhelming him with a sense of defeat. He was too late. Ten minutes had gone by and Nora Stafford was dead. He'd let her die.
Then there was another shot, and a third, and it was this last one that got him moving again, because it hadn't come from a handgun. He recognized it as a rifle shot, and King didn't have a rifle.
He was running toward the shots but angled too far to the left and ran into a tangle of undergrowth that he and AJ had not encountered on their walk into the woods. At first he tried to push through it, but that was a bad idea, and he fought his way back out of it and ran parallel to the lake, looking for a gap in the brush that would let him get back down to the shore and to the boat.
He heard voices—it sounded like Ezra—and then there was another volley of shots, three in succession. Who was shooting? He slapped branches aside and cleared the trees, found himself at the top of a muddy bluff, Ezra's boat screened from sight. Out on the water, the smaller boat, where Nora and King should be waiting, seemed to have overturned and was now floating upside down in the lake. He could see people in the water.
The bluff was steep and slick with wet mud, but he fought his way down it, turning his feet sideways to limit his momentum, his shoes plowing furrows in the soggy earth, and then he was in the water up to his knees, splashing down the shore toward the collection of stumps and trees where he'd left Ezra's boat.
As he stumbled through the water, something began to happen with the boat out in the lake. It rose in the air once, then twice, and finally it flipped back over, resting upright again. Two people climbed onto it, and even from out here Frank could see that neither one was tall enough to be King. What the hell had happened? Was Ezra out there with Nora somehow? Or Renee?
When he came around the bluff, Ezra's boat appeared, and he saw Vaughn on board, standing in the stern, using the trolling motor to pull away from the island.
"Hey!" Frank shouted. _"Hey!"_
Vaughn turned at the sound, lifted a gun, and fired two wild shots that hit in the water some twenty feet to Frank's right.
_"Stop shooting, you idiot! It's me! It's Frank!"_
Vaughn was still holding the gun, but he stopped firing, hesitating, and Frank yelled, "Bring it over here! I've got the key!"
Vaughn looked down at the big, silent engine on the stern and then lowered his gun. He was struggling with the trolling motor, didn't seem to understand how to use it, and Frank, trying to go out to meet the boat, was now in water up to his chest, holding the gun high to keep it from going under.
Vaughn finally got the boat pointed in the right direction, and when it reached him Frank caught the side, took one deep breath, and heaved up, got his knee on the side, and used that leverage to force his way onto the boat.
He was collapsed on the starboard seat, fighting for breath, when Vaughn let go of the trolling motor and turned to him with the gun held out in a shaking hand.
"Give me the key."
Frank stared at him. "What? Get that gun out of my face, asshole."
"Give me the key!"
Out on the lake a motor coughed several times and then caught, and both Frank and Vaughn looked toward the sound and saw that the aluminum boat was in motion again, headed toward another island, away from them. Vaughn kept staring after it, and Frank planted his feet on the floor, then rose and swept Vaughn's gun down and away, hit him once in the chest with a closed fist. It knocked Vaughn back against the steering console, and Frank locked his left hand around Vaughn's wrist and twisted until the fingers opened up and the gun fell to the bottom of the boat.
"What the hell's the matter with you?" he said, his face close to Vaughn's, whose entire body seemed to be shaking. "You could have killed me, you stupid bastard."
Frank knelt and picked up the gun, wedged it beneath the seat, out of the way, and then put the key in the ignition and twisted. As the motor came to life, Vaughn moved away, and Frank straightened and stared out at the departing aluminum boat. It looked like Nora was at the motor. There was no way she'd hear him, so he lifted his arm and waved it in a slow arc. Finally she saw him and lifted her own hand, but kept taking the boat away, toward the other island.
"What's she doing?" Frank said, dropping to the seat and moving his hand to the throttle.
"Renee was on that island," Vaughn said.
"I think she's on the boat now. What happened to the one who was with Nora on the boat, though?"
Vaughn didn't answer.
"What about Ezra?" Frank twisted the wheel and turned the boat to go in pursuit of Nora and Renee.
"They shot him," Vaughn said.
Frank spun to face him. "What?"
Vaughn nodded, his jaw trembling. "Somebody shot him. He's dead."
_"Who_ shot him?"
"I don't know."
"Well, where is he?"
Vaughn lifted an unsteady hand and pointed at the water.
## 36
The sickness that had come and gone during those first few shots returned to Frank as he drove Ezra's boat away from the island where his father's old friend had been killed, was somewhere in the water now, joining Atkins, their blood spilling into the lake.
He imagined the bodies down there among the weeds and the stumps, fish swimming past, crimson clouds rising from the wounds and dissipating into the gray water, water that slapped gently against the log wall back at the cabin where Devin Matteson waited. It was on Devin, all that blood in the water, two more lives taken, adding to that total that included everyone Frank's father had killed, included Frank's father, who'd fired a bullet into his own mouth with the very gun Frank now held in his hand. _All_ of this was on Devin, Atkins and Ezra and even the two thugs Devin had brought with him, the body count rising at his whim while he sat removed from it all, untouched.
That would end today. Frank was going back to that cabin and he was going to kill him. That would be the last of it. He'd kill Devin, and the others could call the police, and then let it end however it would end. He couldn't think about that, didn't care about it, nothing mattering anymore but getting back across this damn lake to put a bullet into Devin Matteson's heart.
"I'm sorry, Ezra." He whispered it, and if Vaughn heard he did not react. There was some shared responsibility here, and all of his hatred for Devin didn't blind him to that. Ezra was dead, and Frank had a role in that. He'd come here for blood, and now he'd seen plenty of it, hadn't he? None from the correct source, though. That was the last thing he could make right, the only thing.
Nora had taken the other boat into the beach of the island, and as they approached Frank could see that she was still in the stern while Renee was onshore.
"What are you going to do?" Vaughn said. He was in the seat beside Frank, hands trembling on his thighs.
"We're going to get them and get out of here," Frank said. He'd brought Ezra's boat in alongside the other, was staring across at Nora, who looked back at him without saying a word. There were red streaks across her face from the tape that had covered her mouth.
"You okay?" he said, dropping the motor to idle, speaking as his boat thumped against hers.
"I'm here," she said.
"Ezra's dead."
She stared at him.
"Vaughn says they shot him. He's dead."
Nora didn't answer. What did he expect her to say, anyhow? He said, "Get in this boat, and we'll leave that one behind."
She nodded and got to her feet, and he reached out a hand to help her step over. As he did it, he turned to Renee, who was walking down from the island. She was moving at a fast, confident pace, stepped right into the lake without breaking stride, moving toward his boat, a gun in her hand.
"Hey," he said. "Get in. We're leaving."
She kept walking, the water up to her knees now. She hadn't even glanced at Frank; her eyes were locked on Vaughn.
"Put that gun down," Frank said.
She didn't say a word. Just walked along the boat toward Vaughn. Frank's own weapon was on the seat, and he turned and reached for it, thinking Vaughn might do the same, but instead Vaughn rose, climbing over the side of the boat and splashing into waist-deep water, moving toward Renee.
"Renee," he said. He had his hands outstretched, reaching for her. "Forgive me. It was for you. I love you so much, and you could never understand that, you couldn't see it, baby forgive me I did it because I love you so—"
He was a few feet away, still moving through the water toward her, still reaching for her, when she lifted the gun and fired. The bullet hit him in the temple and knocked him backward, snapped his head back and turned his eyes to the sky before he dropped into the water and then beneath it.
Frank had just gotten his hand on his own gun, and Nora was still standing in the rear of her boat, waiting to step across. She said something, some whispered gasp or oath or prayer, and Frank stood where he was, in a frozen reach for his gun, as Vaughn's body sank.
"Let go of that."
Renee's voice finally tugged Frank's eyes away from the spot where Vaughn had collapsed into the water, and he saw she was pointing the gun at him now.
"Let go of it and step back," Renee said. "We're leaving now. Just like you said."
Frank opened his fingers and dropped the gun back onto the seat and stepped away.
"Turn off that engine, and go over there and help her in," Renee said. "I'm not going to hurt either of you. Okay? But you're taking me back to my husband now."
Frank cut the engine and walked to the side of the boat and extended his hand for Nora's. She just stared at him, and Renee said, "Girl, get in the damn boat," and Nora took Frank's hand and stepped out of one boat and into the next.
"Okay," Renee said. "Now help me in. And, please, don't try to take this gun. We don't need that. I don't want to hurt anyone."
Frank didn't move, didn't speak. Renee stood there in the water holding the gun and staring at him with challenging eyes.
"He shot my husband," she said. "Shot him, and then brought me up here. I don't give a damn if you think I was wrong."
"He deserved it," Frank said. "Absolutely had earned it."
She looked at him strangely, and finally nodded. "Yes."
Her body looked incredibly small under those saturated clothes, her hair plastered against her face and neck, but her eyes were hard and her jaw was set. The gun looked comfortable in her hands, as if she'd held a few before.
"Help me into the boat," she said. "Now."
He walked into the bow, and she slogged through the water to get closer. He reached for her and she extended her free hand, grasped his, her palm smooth and slippery with lake water. When he had a firm hold he leaned back and pulled against her weight, not hard, just what was needed to give her an awkward lift and prove she'd need to use the other hand to help. She hesitated, looking once into his eyes as if searching for a sign of treachery, and then she put the gun down on the bow, still in her hand but temporarily useless as she tried to push off and get over the side.
Frank slid his foot over and placed it on her wrist, the fine bones trapped under his heel.
"Don't," she said, looking back at him. This time her face changed as she saw what was in his eyes now, what she'd missed in her first study.
"Let go," he said.
"Stop. I told you I wasn't going to hurt—"
He shifted his weight, pressed down on her wrist, and her words cut off in a gasp and then her fingers loosened and she released the gun and slid back off the bow. Frank leaned down and took the gun, worked it into his hand before lifting his foot and freeing her.
"Fine," she said. "You want the damn gun, keep it. I just want to get back to my husband. Let's get the hell out of here."
She stretched her hands up to him again, like a child wanting to be held. He stood where he was and looked down into her face.
"You killed Vaughn in cold blood."
"He deserved it. You said so yourself."
"Yes, I did. And you believe that what you did was right."
"Absolutely." She'd dropped her hands again, was watching him with a wary expression.
He nodded. "Good. You and I, we think alike."
"Okay," she said. "So let's go."
He turned from her and faced Nora, who was standing in the back of the boat watching this all unfold with a horrified expression. There were splatters of blood on her arm. Vaughn's blood, probably.
"Nora," Frank said, "I'm going to have to ask you to get back in the other boat."
"What?"
"Please," he said, voice gentle. "If you'd get back into the other boat, I'd like you to take Renee and go to the dam. You know how to get there? Good. There's a bait shop right there, just down the road. Go there, and call the police."
"Frank . . ."
"Take the other boat and go get help," he repeated. "Please."
"Where are _you_ going?"
He didn't answer.
"No," she said, shaking her head. "Frank, let's all go get help together. Don't go back for him. Let the police—"
_"Nora_." He spoke with a stronger voice, and the emphasis made him move his gun hand, almost involuntarily. He hadn't meant it as a threatening gesture, but her eyes went to the gun and fear rode through them, and when she looked at him again he felt sickened.
"It's safer for you this way," he said, but she was already moving, had stepped over the side and back into the smaller boat, moving out of fear. Fear of him, of the gun in his hand and what he might do with it.
"Where's he going?" Renee said, her voice sharp with alarm. "What's he talking about? _Where are you going?"_
He didn't answer, just walked to the steering console and brought the motor to life and pulled away from them, trying not to see Nora, trying not to remember the way she'd looked when he'd moved the gun. He'd turn his mind elsewhere, to the things that needed remembering, things like Ezra's body under the water and Devin waiting at the cabin.
## 37
Ezra liked the trees. Loved the trees. They belonged on land, high above the water, but instead they were in it, supporting him, holding him to the surface. The trees didn't want to let him sink.
He imagined the bottom was way down there, forty or fifty feet at least. Maybe more. The trees that held him were massive. Oaks probably, or maybe birch? They were big boys, that was for sure. He hadn't realized how high the lake had risen this spring. All the years he'd been out here, the water had never covered fifty-foot trees. Lucky thing for Ezra this was the year. It must have been a hell of a flood. Strange he couldn't remember it.
Consciousness was difficult to hold, and the sky swam above him, but the trees kept his head out of the water and kept him breathing. There were moments when he'd start to slip, and the water would lap at his chin, but then—and this was the _damnedest_ thing—the trees would grow. _Grow._ Right then in the moment he needed them most, they'd strain skyward and lift him an inch or two, whatever he needed. They were amazing trees.
He'd tried to use the branches to pull himself farther away, toward the shore, but pulling set off wild bells of pain, so he stopped that and just hung on, floating and waiting. No sense in going anywhere. The trees would grow when he needed them.
Vaughn was gone. Ezra had seen him take the boat, had managed to focus on that and actually lift his head a bit. Then the boat had moved away from him, down the shore and into deeper water, and there had been more gunfire, and though Ezra had no idea where it was coming from or who was causing it, he knew it was bad.
For a while he was waiting to die and not afraid of it at all, patient as could be. This was where he wanted to end. He wanted to bleed his life out into this lake, this beautiful lake that had given that very life to him. It was fine to end out here. It was right. He'd broken the vow he'd made so many years ago when he'd first come to this place, had taken a man's life once more, and the lake would not allow that. Had not allowed that, had sent Vaughn to punish him. All those years in the jungle with men who excelled at combat, and more years back in Detroit with some of the meanest sons of bitches ever walked the earth, and Ezra had gotten shot by someone like _Vaughn_? It was tough to get your head around a thing like that.
So he'd killed again and the lake had punished him, but then it had sent the trees to hold him up, and that was confusing, because he'd been ready to die and the trees would not let him. He didn't understand that. Perhaps the trees were a gesture of forgiveness. The lake had healed him once, and maybe it would heal him again.
A low grinding sound filled his brain, and for a while he was sure it was a motor, but then it went away, fading until it sounded like a drill bit chewing through wood. Maybe there was no sound, and that was just the pain fooling with his head. A bullet could do things like that to you.
A sprinkling rain started again, much lighter now, and it felt good on Ezra's face, helped to push the fog back. He thought he'd been floating above the surface, but now, after a hard blink to focus, he realized that the water only rose up to his shoulders. The water really wasn't that deep down here. Maybe if he reached with his foot . . .
Son of a bitch, he could touch the bottom. Now how was that possible? The bottom should be way down there, at the base of the tree trunks, fifty feet away.
He tilted his head to the left, studied the tree that held him. The branches weren't so thick. In fact, they were little more than twigs. He wasn't in a tree at all. It was a bush, really, one of the wild tangles that grew along the shore. He was very close to shore, had his feet on the ground.
Ezra was not going to die out here. Not today.
Grady had stayed on 51 too long, had missed a turn that he should have taken, though he wasn't sure what it would have been. His state map was useless up here, he hadn't seen a single sign for the Willow Flowage, and Atkins wouldn't answer his phone.
He finally gave up as a gas station came into view, the highway a two-lane now, and pulled off and into the parking lot, got out of his car and ran inside and shoved past an overweight woman who gasped in indignation.
"Hey." The shaggy-haired kid behind the counter was looking down at the register, and when Grady stepped up he just lifted a finger, asking for a minute.
" _Hey!_ " Grady said and slapped the countertop. When the kid looked up at him, a haughty expression on his face, Grady showed him the badge. "I need you to tell me how to get to Willow Flowage."
"Shit, man, FBI? For real?"
"Just tell me how to get there."
The kid frowned, offended, and pointed out the window. "Straight across the highway, man. Swamp Lake Road. Take that all the way in to County Y, then take that to Willow Dam."
"Swamp Lake to County Y to Willow Dam?"
"Yeah. What's going on?"
"Nothing. Look, I need to get to a cabin out there. I have no idea where it is. Could be anywhere on the lake."
The kid shook his head, and now the fat woman was standing close, listening with undisguised interest and clutching an armful of soda bottles to her large breasts.
"Not many cabins _on_ the lake out there. Not many at all. You sure it's on the lake?"
"Yes," Grady said. "A guy named Frank Temple owns it."
That widened the kid's eyes. "No shit? I heard all about him."
"Fantastic. You know where—"
"Yeah, yeah, I can get you there."
"How long of a drive?"
"Maybe twenty."
Twenty minutes. Okay, that wasn't bad. Grady still had a chance. He would not be too late. He would _not_ be too late.
A gentle rain faded to nothing as Frank crossed the lake, the clouds still heavy and dark but quiet now, the wind settling, the surface of the lake smoothing again.
Frank ran the boat at full throttle, knowing that the big engine would give him just enough time. He'd make it to the cabin maybe ten, fifteen minutes before Nora and Renee got to the dam, and that would be more than sufficient. It wasn't going to take long at all, maybe thirty seconds, walk through the door, put the gun in Devin's face, squeeze the trigger.
Simple.
And a long time coming.
And _right._
Yes, damn it, it was the right thing to do. Ezra was dead, and so was Atkins, and Nora could easily have joined them. Forget Frank's father, forget the betrayal, forget the past entirely—Devin had earned it _today_. Earned more than handcuffs and a cell. It was time to bring him to an end.
The gun in Frank's hand was the Ruger he'd taken from Renee, and he discarded it as he crossed the lake, took the Smith & Wesson back, loving the feel of it, that FT II engraved on the stock. _Here's a bullet from the old man, Devin. Enjoy it. I know he will. Wherever he is, heaven or hell or somewhere in between, I know he will._
He was utterly alone on the lake, even when he came through the Forks and out into the southern portion where the most boat traffic could usually be found. Nobody was going to venture out after a storm like that, with more rain threatening.
He dropped the speed as he neared the cabin, came in close to the shore and with the engine as quiet as possible until he saw the cabin. The van waited alone beside it. Nobody had noticed Atkins's absence yet, or if they had, they didn't know where to start looking for him. The cabin had one main window that looked out on the water, so Devin could be watching the lake right now, waiting for a boat to come in, sizing up the situation. If he saw it was Frank alone, he'd be ready.
Frank cut the engine and let the boat drift into the weeds. He was several hundred yards from the cabin and doubted Devin had seen or heard him. Possibly he'd heard the engine, but he couldn't see this portion of the shore without coming outside, and the yard was empty except for that van.
He got out of the boat in the shallows, wrapped the bow line around a downed tree, and then climbed up the bank and into the woods and headed toward the cabin. He walked quietly but quickly, with his head up and the gun held down against his leg, finger hooked in the trigger guard.
Through the trees and into the yard without a shot fired or even a sound. Across the yard and to the door, still nothing. Hand on the knob, still nothing. He paused for one deep breath, slipped his finger completely around the trigger and tensed it, then twisted the knob and threw the door open and stepped into the cabin in a shooter's stance, gun raised, ready to kill.
Devin was on the floor. Stretched out on his side, one cheek on the linoleum, his body slightly curled, as if he'd been going for the fetal position but couldn't make it. His gun lay on a table beside the couch, out of reach, and Frank could see that he'd fallen from the couch onto the floor. There was a small puddle on the floor near his mouth, bile maybe mixed with traces of blood. For a second, Frank thought he was dead. Then he lifted his head.
He twisted to see the door, his hazy eyes taking Frank in before flicking to the gun on the table, several feet away, no chance of reaching it. When he moved, it was away from the table, rolling into a sitting position with his back against the wall.
"Where's my wife?"
Frank stepped farther into the cabin, then reached back and swung the door shut behind him, never taking his eyes off Devin.
"She's fine," he said, "but you're never going to see her again."
"No?" Devin brought his head off the wall, and for a moment the light in his eyes seemed to fade, as if that small motion had still been too much.
"No." Frank came closer. "The rest of them are dead. Your boy, AJ? I took his gun and I shot him with it. One round right through the eye. I watched him die, and then I came back. For you."
Devin didn't speak. He had his lips parted, was sucking air in through his mouth in slow, audible breaths.
"He had a chance," Frank said. "Hell, he had better than that. He was holding both of the guns. Wasn't enough. But I'll give you the same chance."
"Yeah?"
"Go for the gun," Frank said, nodding at the table. "I'll let you get your hand on it. I'm going to give you that much."
Devin just stared at him. Frank's hand, so damn steady when he'd fired that bullet into AJ's face, was beginning to tremble. He ran his thumb up and down the stock, took another step into the room.
Just shoot him. Quit the bullshit, quit talking, and just _shoot_ him.
"Going to kill me?" Devin said.
"Yes. Unless you get that gun first. I told you, better move for it."
"You have to wait until I've got the gun, is that it?"
"I'm giving you a chance."
"Your dad," Devin said, "wouldn't have needed to wait."
"I'm not my dad," Frank said.
Devin smiled. It was a dying man's smile, a look not of hopelessness but of disinterest, and Frank hated him for it. Hated him for being in this condition, so weak. He wanted him at full strength, wanted the best the prick had to offer, and then Frank would still be better than him. He'd be better, and he'd kill him, and it would be done, _finally,_ it would be done.
_"Get up!_ " Frank screamed. "Get up and go for the gun, you piece of shit!"
Again the smile, and Devin just shook his head. "Can't reach it."
Frank ran to the table and kicked its legs, upending it and spilling Devin's gun to the floor. It hit a few feet away from him, slid to a stop almost within reach.
" _Pick it up!_ "
Devin shook his head again, and this time Frank went for him. He hit him backhanded with the Smith & Wesson, caught the side of his skull, knocked him away from the wall and back to the floor. He let out a soft moan of pain but didn't move, didn't reach for the gun. Frank reached down with his free hand and caught Devin's neck, dragged him upright, and then slammed his head into the wall, still screaming at him to pick up the gun. He banged his head off the wall again, and then a third time, and then he dropped to one knee and jammed the barrel of his father's gun into Devin's mouth.
It was then, down on his knee with his finger on the trigger, that he saw Devin was unconscious.
He let go of Devin's neck and pulled the gun out of his mouth and Devin's head fell onto his right shoulder and the torso went with it. He landed with his body bent awkwardly, one lip peeled back by the floor, a trace of blood showing in his mouth now.
Frank laid his fingers against Devin's neck, felt the pulse there. He was not dead.
He got to his feet and stared down as Devin's eyes fluttered but stayed closed. He took the gun and laid it against the back of Devin's skull, held it there for a few seconds, feeling the trigger under his finger.
_I'd find him and I'd kill him._
_Damn right you would. Damn right. You're a good boy. Check that—a good man._
It's justified, Frank had told Ezra. It is already justified. And Ezra's response? _Bullshit, son. Not in a way you can accept it's not, and you know that._
Devin made some sound, a muffled grunt, and stirred but did not wake. Frank moved the gun across the back of his head, traced a circle in Devin's hair with the muzzle. He thought again of Nora, of the fear in her eyes as she'd looked at him, and then he pulled the gun back and walked away. He picked up the table and set it back where it belonged, beneath his grandfather's posthumous Silver Star. He looked at the medal for a moment, and then he dropped his eyes to the gun in his hand, and he ejected the clip into his palm. He took Devin's gun from the floor and emptied that clip as well, and then he walked into the kitchen and set both guns on the counter, put the clips into his pocket, and ran cold water onto a towel.
When he turned off the water he could hear a boat motor, and he stood at the sink with his head cocked and listened. Something small, and headed this way. He went to the window, looked out at the lake, and saw the aluminum boat approaching, Nora up front and Renee at the tiller. Not surprising that Renee had refused to go to the dam.
He slapped at Devin's neck with the wet towel, then held it over his face and squeezed a trickle of cold water onto his forehead and cheeks. The eyes opened, swam, then focused on Frank.
"Get up," Frank said. "Your wife's coming."
When they arrived, Devin was sitting up against the wall, Frank standing in the kitchen with his back against the counter, near the guns. Renee came through the door first, saw Frank and said, "If you—" but then her eyes found Devin and she stopped talking and turned from Frank and ran to her husband.
"Baby," he said, and he reached for her with one arm as she fell to her knees in front of him, almost in the exact position Frank had taken when he put his father's suicide gun in Devin's mouth.
Nora stepped inside, stood in the doorway staring at Renee on the floor with Devin before she looked at Frank. Her eyes searched his, then flicked to the guns on the counter.
"They're empty," he said, and he pushed off the counter and walked into the living room. Renee turned at the sound of his approach, a protective motion, covering Devin with her body.
"Get him up," Frank said, "and get out of here."
"All right."
"The keys to the van are inside it, I think. You've got to get him out there, though. I'm not helping. If I touch him again, I'm going to kill him."
She just nodded.
Frank turned and walked outside, leaving the empty guns on the counter. Nora followed him, and a few minutes later Renee appeared, with Devin on his feet but leaning heavily against her. Frank and Nora stood together beside the cabin and watched as she got the van door open and got him inside.
"You're letting them go," Nora said.
He shook his head. "They aren't going far. He's got to get to a hospital. Anybody can see that."
She didn't answer. Renee slammed the van door shut and walked toward the driver's door. She paused for a moment in front of the van and looked back at them.
"Thank you," she said. "And I'm sorry. I'm so sorry."
There was a beat of silence, and then Frank said, "You know what he does. You know what he is. So how the hell do you love him so clean?"
"Hon," she said, "whoever said anything about it being clean?"
Frank looked away from her, out at the lake. He didn't turn when the doors opened, didn't turn when the engine started, didn't turn when they drove up the gravel drive.
When the sound of the van had faded and they were alone, Nora said, "Is there a phone inside?"
"No."
"Mine's ruined. The water."
"Yeah. Mine, too."
"Where can we go to call the police?"
He waved toward the drive, and then they turned and started up it together, not speaking, stepping over puddles and through the mud. They were halfway to the main road when they heard the hum of an engine and the crunch of tires and Nora said, "Are they coming back?"
They weren't coming back. It was a car, not a van, and when it slid to a stop and the door opened and Grady Morgan stood up and stared at them, all Frank could say was "You're too late, Grady. Too late."
Grady looked over his shoulder and then back at Frank. "Who was that? Who was in that van?"
"Devin Matteson and his wife," Frank said.
"I can't let them drive away from here."
"Sure you can," Frank said. "You never saw them. Didn't know who it was. Didn't ask me about it just now."
Grady looked at Frank for a long time and said, "I've lied about him before. I guess I can do it again. Now what the hell happened out here?"
## 38
Six hours later, Frank and Nora long departed in police custody, Ezra Ballard evacuated to some hospital, first by boat and then by helicopter, Grady stood alone at the shore and stared out into the dark lake where several bodies waited to be found.
Atkins was dead. Another agent, one who'd been trying to do the job right, was dead, and Grady would see that blood on his hands for the rest of his days, understood that it was the end of his career long before anyone back in Chicago would.
Too late. That was the first thing Frank Temple's son had said to him. Grady had been too late.
Frank had no idea, either. He had no idea.
Seven years of watching that kid, keeping tabs on him, and it had never been about protecting Frank from anything. It had always been about protecting Grady, about covering his own ass. He'd never had the courage to approach the kid and tell him the truth and apologize, and now they were bringing body after body out of this damn lake, one of them a dead agent, a colleague.
Too late. Yes, Frank, I was too late.
Grady Morgan and the Seven-Year Lie. He could have gotten the nerve just a year ago, six years too late and still it would have been in time for this. If he'd tracked Frank down then and told him the truth, how much blood would have been spilled? Not as much, that was for sure. There would have been some, Devin Matteson's gunmen would have seen to that, but not as many people would have died, certainly not Atkins. If Frank had known Devin wasn't responsible for his father's demise, he never would have headed north, never would have seen Vaughn Duncan or had anything to do with it. Those two from Miami would have made their way north quietly, killed Duncan and taken Renee home to see her husband.
It was a sick world, Grady thought, when you could stand on the shore of a beautiful lake like this and long for one murder. One murder that would have saved the others. Everybody with their damn score to settle, and Frank in the midst of it with one that didn't need settling.
He was done with the Bureau. Wouldn't have to be—all of this was indirect involvement, he was close to retirement, and the Bureau loved to handle such things quietly and in-house—but he knew he'd resign now. Should have seven years ago, but it wasn't too late to do it now, and he felt he owed Atkins that much. Atkins wouldn't have wanted a guy like Grady left in his Bureau.
The truth would start with Frank, though. The hell with the people in Chicago who would hear it next; Frank was the one that mattered.
He didn't see him again until the next morning, and while there were still cops moving around the lake—and still divers looking for Atkins—they were alone in the cabin, sitting with their backs to the window that looked out on the lake and its grisly activity.
Ezra Ballard was alive and recovering from a single gunshot that had blown through his ribs and wreaked some internal havoc but left him to see another day, and that's what Frank wanted to talk about at first.
"He'll make it," Frank said after he finished filling Grady in on all the medical details Grady had already heard.
"Yes."
"One of the few, though. One of the few, and you don't need to tell me how much of that is my responsibility, Grady. I understand it."
"That's not the way anyone else is telling it," Grady said. "You see the papers? You're on the front page."
"So was my dad."
"They're saying different things about you, though."
Frank didn't respond to that.
"You let Devin go," Grady said. "Had him and let him go, there at the end. I saw the reports."
A nod.
"It was the right thing to do," Grady said, and his voice was so rigid, grandfatherly, none of the relief coming through in it. And it _was_ relief, because a day earlier when the kid stood in this room with a gun in his hand he had somehow done the right thing, despite all the energy Grady had invested into priming him for the wrong thing.
"Anybody heard about him?" Frank said. "Has he turned up somewhere?"
Grady shook his head.
"I was sure that he would," Frank said. "Sure of it. He'll need a hospital. I'd be surprised if he didn't, at least. Nobody's blaming me for letting him walk, though. Wasn't my job, they keep saying."
"It was the right thing to do," Grady said again. "And I need to explain that to you."
"I get it, Grady."
"No, Frank. No, you do not."
Frank tilted his head and squinted against the sunlight, and Grady finally opened his mouth and let the truth out.
"I was seventeen years old," Frank said when he was done. It was the first thing he'd said in a long time, Grady doing all the talking, speaking too fast, trying to rush out as much as he could before Frank went nuts, blew up.
He never blew up. Just sat there and listened and didn't show a thing, and it took Grady back to those first few conversations in the house in Kenilworth and made his cowardice and corruption play through his eyes again, everything here so damn similar except that this time Frank was an adult and Grady was telling the truth.
"I know," Grady said. "Frank, I know. You were a child, and one who'd suffered a loss, and we—"
"Turned me into a gun."
"What?"
"That's what you said he wanted to do. My father. That he'd raised me to kill. And while you were saying that about him, you were loading me up and pointing me at Devin."
"We didn't want _you_ to go after him. We thought that you might know something, and we needed to push the right buttons to see if . . ." Grady stopped talking and shook his head. "Shit, you've got me defending it now. I'm not going to. It's like I said, Frank, I'm telling the truth now, and it had to start with you."
Frank stood up and walked to the window, looked out at the lake. There was a flat-bottomed boat within sight, divers adjusting their masks before going back in, still searching for Atkins.
"You are a bastard," Frank said, but it was without any venom at all, or even energy.
"I'm going to resign."
"I don't care about that. Why would I care if you kept your damn job?"
Grady didn't say anything.
"I'd just turned eighteen," Frank said, "when I called Ezra Ballard and begged him to go to Miami with me to kill Devin. Begged him. He said he wouldn't, so I said I'd go alone, and he made a compromise with me. Said as long as Devin stayed down there, the hell with him, let him rot. But we'd never let him come back here. Never."
Grady was sitting on the couch with his elbows on his knees, staring up at Frank, who hadn't turned from the window, didn't even seem to care that Grady was still in the room.
"Kudos," Frank said, "on a job well done, Grady. You set out to convince me that Devin deserved retribution, get me fired up about it, _consumed_ by it, right? Well, you got the job done. Yes, sir, you got it done."
"I want you to know," Grady began, but Frank continued talking as if he hadn't heard him.
"I was _relieved_ when Ezra told me the prick was coming back. Well, not at first. At first I had some sense. But then this fat bastard professor asked me if I'd write a memoir, and it was like fate, right, confirmation that there would be no getting away from this legacy, ever. Then I was relieved, Grady. I was absolutely relieved. Because I could finally welcome it."
He stopped looking at the search boat then, turned to face Grady. "Devin was important to me. He gave me somebody to hate, somebody to blame, who wasn't my father. I'm not saying he was enough for me to give the old man a pass, but he was enough to distract me. To redirect some things."
"I understand that."
"I came up here to kill the guy for no reason, is what you tell me now. But he's still a piece of a shit. You know that. They're looking for one of his bodies out there right now. So maybe I should have gone ahead and done it."
Grady shook his head. "No."
"It would have made sense to me then," Frank said, "because I had my reasons. Now I don't. But other people have still got theirs, right? So maybe I should have done it for their reasons. Why are they any less valid than mine?"
Grady was quiet. Frank said, "How many sorts of crimes has Devin been involved with, you think?"
"Plenty."
"No, give me a number."
"I don't know, Frank. What, dozens?"
"Dozens," Frank said, nodding. "And deaths? How many deaths?"
"The same. There's a reason we wanted him so bad, Frank."
"Yeah. That's my point. There were plenty of reasons." He looked around the cabin. "I had him here yesterday with a gun in his mouth and my finger on the trigger. And if the son of a bitch hadn't been so sick, if he hadn't looked like he was dying, I probably would have pulled it."
"It's good that you didn't."
"Is it?" Frank said. "I don't know about that, Grady. I don't know. But I don't want to be the one who has to decide. I do not want to play that role."
They stayed in the living room for a long time without speaking, and eventually Grady stood and said he was leaving.
"Devin didn't give my father up," Frank said. He wasn't looking at Grady.
"No."
"Somebody did."
Grady was silent.
"It was an anonymous tip, says the legend. From someone close to him."
Grady had given his word never to reveal that source. There were divers out in that lake who were proof that Grady's word wasn't worth a damn, though, and maybe Frank was entitled to it by now. Surely he was entitled to it by now.
"I'll tell you," he said, "and this time it'll be the truth."
Frank was shaking his head. "No," he said, and Grady would always remember the awkward way he finished it—"I would like not to know."
Grady nodded, and he left, and he did not tell him. Would never tell him, or anyone, about that day when an attractive woman whose dark hair and skin were contrasted by striking blue eyes came into his office and said, _I would like to talk with you about my husband._
When they'd finished that day, he'd praised her bravery, lauded her for doing the right thing. She'd looked at him as if he were insane. _Brave?_ she said. _Doing the right thing? It's got nothing to do with that. I'd never have told you anything. I love him. But he's going to ruin my son, Mr. Morgan. And I cannot let that happen._
## 39
They let Ezra out of the hospital seven days after he went in, and Frank picked him up in his own truck, drove south from Minocqua with the windows down and fresh air blowing in, the highway filled with cars towing boats, the first weekend of fishing season under way.
"How've the dogs been?" Ezra asked.
"Disgruntled."
"Good. Nice to see some concern on the home front."
"The doctors say when you'll be able to get back out on the boat, back to work?"
"They might have, but I don't recall listening to it. I think it'll be soon."
Frank had already tried offering apologies to Ezra, tried to explain how he'd have handled things differently if he'd understood the situation, how he should never have believed Vaughn. Ezra had cut him off every time, not interested in hearing it, not needing to hear.
"What about Devin?" Ezra said. "Any word?"
Frank just shook his head. During the first few days he'd been sure news about Devin was on its way. The memory of his face there at the end left Frank convinced he'd turn up in a hospital somewhere—or dead. That was the preferred option, that he would turn up dead and Frank would able to shake his head and think about just how little would have been accomplished if he'd pulled the trigger, realize that he'd have removed nothing more than a few days of pain from Devin's existence. Yes, that was what should happen.
It hadn't happened, though. The days ticked by and no word came, Devin and Renee still gone.
"He's got the right sort of friends for this," Ezra said. "People who can help him disappear."
"He does," Frank agreed, and he could feel his finger on the trigger again, see that circle he'd traced in Devin's hair with the gun barrel.
"You talked to Nora?" Ezra said.
"Called a few times. Haven't heard back from her."
"She still in town, though? Or has she gone home, after all this?"
"I'm not sure."
"How about you? Headed home soon?"
"Headed where?" Frank answered, and Ezra nodded, and they drove on in silence.
Nora didn't get back to the shop for five days. The cops had taken Vaughn's Lexus, and now she had no cars left but the Mazda that Jerry had refused to repaint. Nobody to fix any cars that came in, either. She tried to finish the Mazda herself, spent three days on the paint job, creating runs and streaks and then sanding them back down and starting over and making it worse. By the third day the car was an absolute disaster, and she finally called another shop across town and towed it over to them. The owner, a man who'd known her father well, painted it in one day and then had it towed back to her shop, with her check for the work sitting on the driver's seat.
She called the car's owner to tell him it was ready, and he came over immediately, more anxious for conversation than for the car. He'd read about her in the papers, he said. Hell of a story.
He left with the car, and then it was just her and an empty shop. No business, no employees. More bills on the way.
Frank Temple had called a few times, left messages. Why did he want to see her so bad before he took off again? Did he think there was some sort of closure for this, some nice, neat wrap-up to such awful events? She didn't call back. It was surprising that he was even still in town.
Her mother called daily, first to politely urge her to return home, then to demand it. Nora said she was considering her options, and then she called the local newspaper and put out an ad for a new body man and painter.
The ad ran for a week, and she interviewed two guys. Told them she'd be in touch, but the truth was they couldn't handle the job, and she couldn't pay them even if they could. That Friday, she answered her father's constant question honestly for the first time, told him, no, we don't have any cars. Then his face fell, and she responded with a lie, promised some were on the way in, that things would be _too_ busy by the first of the week.
The shop was lost, and she supposed she should have felt relief at that. It ended the uncertainty, at least. She could go home now. So why did she feel so damn sad? Her father was part of it, of course—the idea of leaving him in this town without any family still haunted her—but today she was more aware than ever of what had always helped her linger: She didn't know what came next. It was that simple, that sad. While her peers were caught up with their families or careers, she still waited for the road sign that told her which way to turn. Tomahawk, and Stafford's Collision and Custom, had provided a welcome delay. Now the delay was past and the uncertainty remained and, worst of all, she'd failed at the one goal she'd set. The family shop was closing, and not on Stafford terms.
The next Monday found her alone in the empty shop. The phone rang several times, but it was always a long distance number. Reporters, not customers. It was nearly noon and she was getting ready to leave for lunch when Frank Temple came through the door.
"Hey," he said, letting it close softly behind him. He looked good, all the bruises and nicks gone. The black-and-blue streaks on her arms were gone, too, but she continued to wear long sleeves every day.
"Hi," she said. "I know, I owe you a call. It's been hectic, though. I figured you might have left town already."
"No." He looked around, taking in the quiet place, her sitting alone in the little office. She felt pathetic, didn't want him to see it.
"How are things?" he said, and she meant to tell him they were fine, she really did, but somehow the truth came out instead. She wasn't weepy about it, wasn't sentimental, just told it like it was. She was going to have to close the shop, and that was that. Head back to Madison, or maybe, much as she hated to think of it, to her stepfather's house in Minneapolis.
"I saw your ad," he said. "If you hired somebody new, couldn't you get it going again?"
"The truth is, I couldn't afford to pay anyone until we'd made some money, and that takes time. Most body men don't want to work on spec. And really, I need two people, because most aren't going to be able to do what Jerry did."
He nodded. "How much would you need to make it till then?"
What was this about? She didn't like the question.
"I don't know," she said, "but it's going to be more than a bank will want to loan a company that's already overextended and has no employees and no customers."
He nodded again, just taking all of this in as if it were minor stuff.
"I was thinking I'd like to invest in something," he said. "I've got some money left, and rather than burn through it and then go looking for income, I thought it would be a good idea to put it into something promising. An up-and-coming business, something like that. Or maybe one with some history. Some tradition. You know, a proven entity."
She was shaking her head before he was done.
"I don't take handouts. It's generous, a very sweet offer, but no."
"I don't give handouts," he said. "Maybe you missed the investment part of what I said? I'm thinking of something different entirely. More like being a partner."
She kept right on shaking her head.
"I don't want a partner. If I can't do it alone, then I'll just get out."
"You know," Frank said, "being strong doesn't necessarily mean being alone."
She looked at him for a long time, then pulled her chair closer to the desk.
"Dad told me the only partner worth having was one who'd get his hands dirty, share the job side by side."
"Then I'll share the job."
"You don't know anything about fixing cars."
"No," he said, "but we can find some people who do. And I'm pretty sure I could drive a plow in the winter."
"In the winter." She said it carefully, a verification.
"Made more sense to me that way," Frank said. "But if you want me to drive the damn plow in the summer, Nora, I'll do it."
He stopped talking and looked her in the eye, and she saw something surprising there, a deep and powerful quality of need.
"You could think about it," he said. "You could do that much, couldn't you? I don't want to go. I'd like to stay here. It's the best chance I've got."
They sealed the deal on a handshake. It was a start.
| {
"redpajama_set_name": "RedPajamaBook"
} | 9,254 |
using UnityEngine;
using System.Collections;
public class GameSettings : MonoBehaviour
{
private static GameSettings m_DataInstance;
public string m_LoadedLevelUrl;
public static GameSettings Instance
{
get
{
if (!m_DataInstance) { m_DataInstance = FindObjectOfType<GameSettings>(); }
return m_DataInstance;
}
}
void Awake()
{
m_DataInstance = this;
DontDestroyOnLoad(gameObject);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,418 |
\section{Introduction}
\label{sec:introduction}
Experimental quantum simulation~\cite{GAN14} opens vistas for exploring foundational quantum principles
such as studies of Quantum Phase Transitions (QPT)~\cite{sach2011,Greiner2002,Soltan-Panahi2012,Baumann2010,MBK+15}.
Although the \emph{phase} of matter can only be defined macroscopically,
the QPTs are currently explored through microscopic characterizations
that depend on local observables such as individual-particle spin states or
few-body states or quantum entanglement witnesses~\cite{kim2010quantum}.
Macroscopic probing, which measures some global property of a system
without having any access to individual particles,
is not currently employed in quantum-phase-transition experiments.
Yet conducting joint macro and micro characterizations of quantum phases and transitions between them
is important for self-consistent verification of QPTs in the macro and micro regimes,
assuming that one may not necessarily imply another~\cite{Anderson04081972, Kle67}.
Here we propose a scalable scheme comprising nearest-
and next-nearest-neighbor-coupled spin-half network of
radio-frequency superconducting-quantum-interference-device
(rf-SQUID) flux qubits~\cite{HJB+10,PhysRevB.82.024511},
where the two levels comprise one clockwise and one counter-clockwise super-currents.
The clockwise super-current state~$\ket{\circlearrowright}$
is equivalent to a spin-down flux state~$\ket{\downarrow}$,
and the counter-clockwise super-current state~$\ket{\circlearrowleft}$
corresponds to a spin-up flux state~$\ket{\uparrow}$.
This flux-qubit spin network will realize,
and admit macro and micro characterization of,
a controllable QPT from paramagnetism to frustrated magnetism~\cite{diep2004frustrated}.
We focus on quantum simulation of the paramagnetic to frustrated-magnetic
QPT~\cite{MBK+15} because of the need to simulate
the fascinating behavior manifested in frustrated spin systems~\cite{diep2004frustrated},
such as spin glasses, spin ices and quantum spin liquids
due to competing spin-spin interactions.
These competitions give rise to an
exponentially large subspace of degenerate ground states for such frustrated networks~\cite{doi:10.1021/ja01315a102},
which strongly motivates quantum simulation of these systems~\cite{GAN14}.
We seek to identify the macro signature
of the onset of quantum frustrated magnetism
from paramagnetism through the statistics of the order parameter rather than solely by its mean.
Quantum simulations of frustrated Ising networks~\cite{Ong02072004}
have been performed with ion traps~\cite{kim2010quantum},
quantum dots~\cite{PhysRevLett.110.046803},
and nuclear magnetic resonance~\cite{PhysRevA.88.022312}
and has been proposed for Rydberg atoms~\cite{GDN+15}.
Larger frustrated systems have been realized for optical systems~\cite{ma2011quantum}
and trapped ions~\cite{Islam03052013}.
All these schemes employ microscopic probes of frustration
and are not yet amenable to macroscopic probing,
which is why we propose a new setup
corresponding to a scalable flux-qubit superconducting architecture
that admits probing of both macro and micro signatures of frustration.
The rest of the paper is organized as follows: Section~\ref{sec:model} describes the physical model for the architecture considered in this work. In Section~\ref{sec:simulation} we elaborate the control procedure and discuss how micro and macro signatures can be extracted for our system. Section~\ref{sec:results} shows the results of our simulation and we conclude in Section~\ref{sec:conclusions}.
\section{Physical model}
\label{sec:model}
\begin{figure}
\centering
\subfloat[]{
\centering
\includegraphics[width=0.6\linewidth]{frustrationChain.pdf}
\label{fig:frustrationChain}
} \\
\subfloat[]{
\centering
\includegraphics[width=0.35\linewidth]{frustrationScheme.pdf}
\label{fig:frustrationScheme}
}
\caption{%
Schematic for (a)~$n$ nearest-neighbor- and next-nearest-neighbor-coupled qubits and
(b)~three coupled qubits in a triangular architecture, which is the smallest lattice required
for frustrated magnetism. Rounded squares denote qubits, and solid lines denote couplings.}
\end{figure}
Our goal is to analyze the model for nearest- and next-nearest-neighbor-coupled rf-SQUID superconducting flux qubits (as shown in Fig.~\ref{fig:frustrationChain}) and devise a scheme to extract the macroscopic as well as microscopic signatures of QPT from paramagnetism to frustrated magnetism. However, we first consider three rf-SQUID superconducting flux qubits coupled to each other via tunable inductive couplings (shown in Fig.~\ref{fig:frustrationScheme}), as this triangular architecture is the smallest lattice that can achieve the regime of frustrated magnetism.
\subsection{CCJJ flux qubit}
The Compound-Compound Josephson-Junction (CCJJ)~$\ket{\circlearrowright}$
shown in Fig.~\ref{fig:CCJJ}
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{CCJJ.pdf}
\caption{%
A circuit diagram of an rf-SQUID CCJJ superconducting flux qubit,
where the quantized circulating current $I_{\rm Q}$ across the qubit loop
(denoted by `Q') defines the $\ket{\circlearrowright}$ and $\ket{\circlearrowleft}$ states of the qubit.
The CCJJ loop comprising four Josephson junctions is coupled to a bias current~$I_{{\rm CCJJ}}$
for controlling the height of the flux-qubit potential-energy barrier.
The time-dependent bias current $I_{{\rm CCJJ}}(t)$
induces a time-dependent magnetic flux $\Phi_{{\rm CCJJ}}(t)$ across the CCJJ loop.
This flux controls the $\ket{\circlearrowright}\leftrightarrow\ket{\circlearrowleft}$ tunneling probability.
The compensator~C,
which is biased by current $I_{{\rm C}}$ via an in situ tunable inductive coupler to
the flux qubit,
is controlled by the static flux bias $\Phi_{{\rm C}}$. It compensates for the changing current $I_{{\rm Q}}$, such that the qubit frequency~$\epsilon$ can be maintained to a specific value during the experiment.
The readout circuitry (denoted `RO') can detect any variation of the circulating current $I_{{\rm Q}}$ and is used to perform the readout operation on the flux qubit.
}
\label{fig:CCJJ}
\end{figure}
is proposed here as our favored flux qubit
(with basis states~$\ket{\circlearrowright}$ and~$\ket{\circlearrowleft}$)
due to its exquisite tunability~\cite{PhysRevB.82.024511,HJB+10}
and its demonstrable scalability in the sense that hundreds of such flux
qubits can be coupled in a spin network~\cite{Dickson2013,PhysRevX.4.021041,6802426}.
The two flux-qubit levels correspond to the clockwise and anti-clockwise circulating currents in the qubit loop (denoted `Q' in Fig.~\ref{fig:CCJJ}).
In the $\{\ket{\circlearrowright},\ket{\circlearrowleft}\}$ basis, the Hamiltonian of the CCJJ flux qubit is off-diagonal due to the tunneling between these states across the finite potential barrier. We, therefore, have two independent control parameters: $\epsilon$ that controls the asymmetry between two potential wells, and $\Delta$ that controls the potential barrier height between two wells. The Hamiltonian of this qubit is ~\cite{PhysRevB.82.024511,HJB+10}
$\hat{H}_{{\rm Q}}=-\left(\epsilon\sigma^z+\Delta\sigma^x\right)$
for $\epsilon=|I_{{\rm Q}}|\left(\Phi_{{\rm Q}}-\Phi_0\right)$
with~$I_{{\rm Q}}$ the super-current circulating in the rf-SQUID loop,
$\Phi_{{\rm Q}}$ the time-dependent qubit-flux bias applied across the qubit,
and $\sigma^{x,y,z}$ the three Pauli matrices.
It is possible to turn off the diagonal terms of the Hamiltonian by setting $\Phi_{{\rm Q}}=\Phi_0$, a regime often referred to as the \emph{degeneracy point}.
Tunneling energy~$\Delta$ is a function of $\Phi_{{\rm CCJJ}}$ that can be controlled via $I_{{\rm CCJJ}}$.
Fabricating such qubits having
identical controllability but superior energy relaxation time (expected $T_1 > 10~\mu$s) is underway~\cite{Note2},
which could serve as an ideal platform for conducting such QPT experiments.
Coherence times of tens of microseconds are orders of magnitude higher than our operation time for simulating the QPT,
which guarantees that the effect of decoherence-induced noise is negligible for our case.
On the other hand, superconducting qubits operate at around $10$ mK~\cite{Niskanen04052007,0953-2048-23-4-045027,Note2},
which corresponds to a thermal excitation with frequency~$\sim 1.3$ GHz,
which is far detuned from the frequencies of the quasiparticles in the flux qubit ($\sim 6$ GHz.)~\cite{PhysRevX.4.021041}.
This detuning implies that the mean density of thermal photons in the system at the operating temperature is less than $1\%$,
so the possibility of absorbing thermal excitations by the system is minimal,
thereby rendering our architecture resilient against possible environment-induced noises.
\begin{figure}[htb]
\centering
\includegraphics[width=0.7\linewidth]{threeCoupledFluxQubits.pdf}
\caption{A circuit diagram of three rf-SQUID flux qubits coupled with each other via inductive tunable couplers. The square around the system couples all three flux qubits and measure the total magnetic moment of the system which we use towards the macroscopic characterization of the QPT in the network.}
\label{fig:coupledCCJJ}
\end{figure}
\subsection{Triangular architecture of coupled CCJJ qubits}
We now consider a triple-spin network
with the three CCJJ flux qubits inductively coupled in a triangular arrangement shown in Fig.~\ref{fig:coupledCCJJ}.
In this architecture, each pair of qubits is coupled via an inductive coupler that can be tuned {\it in situ} via the external flux biases $\Phi_{ij}$~\cite{PhysRevB.82.024511}. Such an inductive coupling scheme induces a $\sigma_i^z\sigma_j^z$-type interaction between the $i^{{\rm th}}$ and the $j^{{\rm th}}$ qubits with an adjustable coupling strength~$J_{ij}$~\cite{PhysRevB.82.024511}.
The Hamiltonian for this network of three coupled flux qubits is
\begin{equation}
\label{eq:fullHamiltonian}
\hat{H}_{{\rm s}}
=\sum_{i=1}^3 \hat{H}_{{\rm Q}}^{(i)}
+\sum_{i=1}^2 \sum_{j=i+1}^3J_{ij}\sigma_i^z\sigma_j^z
\end{equation}
with~$i,j$ denoting qubit indices and $\hat{H}_{{\rm Q}}^{(i)}$ the Hamiltonian for the $i^{{\rm th}}$ flux qubit. Using the control mechanisms described earlier, we can
achieve $\epsilon_i=0$ by setting each external qubit flux bias $\Phi_{Q}^{(i)}$
at the degeneracy point of the $i^{{\rm th}}$ qubit.
\section{Simulating QPT from paramagnetism to frustrated magnetism}
\label{sec:simulation}
In this section we discuss how to control the quantum Hamiltonian (\ref{eq:fullHamiltonian}) in order to achieve the frustrated regime starting from the paramagnetic phase. We compute the fidelity susceptibility for the ground state of the system and explore its divergent behavior, which in fact denotes a QPT without any prior knowledge of any local order parameter. Finally we discuss how micro and macro signatures of the QPT can be extracted for our system.
\subsection{Control scheme}
In our approach to simulating frustrated antiferromagnetism, we initially prepare the ground state of the system Hamiltonian (\ref{eq:fullHamiltonian}) with $\epsilon_i=J_{ij}\approx0\;\forall i,j$
and then vary~$\Delta$ and~$J$ (assuming $\Delta_i=\Delta$ and $J_{ij}=J$ for all $i,j$) adiabatically
via $\Phi^{(i)}_{{\rm CCJJ}}$ and $\Phi_{ij}$,
respectively.
The control pulses for these parameters are designed
such that,
at $t=0$, $J/\Delta \ll 1$ ($\sim 10^{-6}$), and,
at $t=t_{{\rm final}}$, $J/\Delta \gg 1$ ($\sim 10^6$)~\cite{PhysRevX.4.021041,Ronnow25072014}.
The Hamiltonian ground state $\ket{g}$ at $t=0$ is
$\ket{g(t=0)}=\ket{+++}$~\cite{Note6},
which is separable.
During the system's adiabatic evolution towards the frustrated regime,
the three qubits become entangled,
and the system ground state at $t=t_{\rm final}$ is
(neglecting global phase)
$(\ket{\downarrow\downarrow\uparrow}+\ket{\downarrow\uparrow\downarrow}+\ket{\uparrow\downarrow\downarrow}-\ket{\downarrow\uparrow\uparrow}-\ket{\uparrow\downarrow\uparrow}-\ket{\uparrow\uparrow\downarrow})/\sqrt{6}$.
\subsection{Fidelity susceptibility}
Fidelity susceptibility offers a measure for QPT in absence of any prior knowledge of order parameters~\cite{PhysRevE.76.022101, PhysRevX.5.031007}. The fidelity susceptibility $\chi_{\rm F}$ quantifies the drastic change in the ground state of the quantum system during the phase transition, and is defined as
\begin{equation}
\chi_{\rm F}(\lambda)=-\frac{\partial^2 \ln{\left|{\braket{\Psi_{0}(\lambda)}{\Psi_{0}(\lambda+\epsilon)}}\right|}}{\partial \epsilon^2}\Bigg|_{\epsilon=0},
\end{equation}
where $\Psi_{0}$ is (normalized) ground state wavefunction of the system and $\lambda$ is the control parameter, which is $J/\Delta$ for our case. It is possible to show that $\chi_{\rm F}(\lambda) \geq 0$ for all $\lambda$, and can also be expressed as~\cite{PhysRevX.5.031007}
\begin{equation}
\label{eq:chiFUsed}
\chi_{\rm F}(\lambda)=\left\langle{\frac{\partial \Psi_0}{\partial \lambda}}\Bigg|{\frac{\partial \Psi_0}{\partial \lambda}}\right\rangle-\left\langle{\Psi_0}\Bigg|{\frac{\partial \Psi_0}{\partial \lambda}}\right\rangle \left\langle{\frac{\partial \Psi_0}{\partial \lambda}}\Bigg|{\Psi_0}\right\rangle.
\end{equation}
We use Eq.~(\ref{eq:chiFUsed}) to compute the $\chi_{\rm F}$ for our system, and seek its divergent behavior that denotes the QPT.
\begin{figure}[htb]
\centering
\includegraphics[width=0.9\linewidth]{chiF.pdf}
\caption{Plots showing the fidelity susceptibility $\chi_{\rm F}$ as a function of time for $n$ nearest- and next-nearest-neighbor-coupled flux qubits. The divergence in $\chi_{\rm F}$ denotes a QPT.}
\label{fig:chiF}
\end{figure}
Fig.~\ref{fig:chiF} shows the fidelity susceptibility $\chi_{F}$ as a function of time for various lattice sizes. During the time $t=0$ to $t=50$ ns, we simultaneously vary $J$ from $0$ to $5$ GHz and $\Delta$ from $5$ to $0$ GHz adiabatically. We observe that while $\chi_{\rm F} \ll 1$ for most of the time during the change of our control parameter, it however shows divergence when $J/\Delta \gg 1$. The plots in Fig.~\ref{fig:chiF} not only indicates the existence of a QPT for our system, but also shows that extracting the signatures of QPT should be possible for a lattice with $n \approx 12$ for which the divergence of $\chi_{\rm F}$ is prominent. This requirement is especially useful as the classical simulation of any quantum spin system becomes exponentially expensive with higher values of $n$. For the macroscopic characterization of QPT, we therefore restrict ourselves to a lattice with $n=12$, which turns out to be sufficient for the architecture considered in this work.
\subsection{Microscopic signature}
QPTs are usually characterized microscopically by the divergence of correlation length near the critical point~\cite{sach2011}. Experimentally, however, it is not always efficient to determine this divergence, and, therefore, alternative system-specific schemes are often used, such as measurement of some local entanglement-witness operator for spin-systems~\cite{kim2010quantum}.
The QPT for our architecture can be characterized by selecting a block of three adjacent flux qubits (which is in fact a triangular architecture as shown in Fig.~\ref{fig:frustrationScheme}) and measuring a three-qubit entanglement-witness operator on that block.
Such a measurement is performed locally on a few-spin subsystem of the entire lattice.
Moreover, the measurement of entanglement-witness operator requires readout operations on each individual qubit,
hence is a microscopic characterization.
The symmetric $W$-state witness operator~\cite{2009PhR4741G,kim2010quantum}
\begin{equation}
W_{\rm S}
:=4+\sqrt{5}-\frac{1}{2}\left(\sum_{i=1}^3\sigma_i^x\right)^2
-\frac{1}{2}\left(\sum_{i=1}^3\sigma_i^{y}\right)^2,
\end{equation}
satisfies~$\langle{W_{\rm S}}\rangle>0$ for separable states and $\langle{W_{\rm S}}\rangle<0$ for entangled states.
For our case, we expect~$\langle{W_{\rm S}}\rangle=\sqrt{5}-2$ at $t=0$ and $\langle{W_{\rm S}}\rangle=\sqrt{5}-3$ at $t=t_{\rm final}$,
which prominently signifies the entanglement change.
\subsection{Macroscopic signature}
For macroscopic characterization, we can place a dc-SQUID loop around the entire system, such that it couples all the flux qubits (Fig.~\ref{fig:coupledCCJJ} shows such a loop for the three-qubit case)
and measure the total magnetic moment $\hat{\mu}$ along the $z$-axis~\cite{Note4}.
The net magnetic moment of the entire system along $z$-axis ${\hat{\mu}^z}$
can be expressed as a sum of the $z$-components of the individual spin magnetic moments
$\sum_{i=1}^{n}\sigma_i^z$ of the entire system.
Classically, the total magnetic moment of a system per unit volume is the magnetization of the system, which is a macroscopic quantity.
We refer to this readout procedure as a \emph{macroscopic} characterization of frustration,
as it does not measure any local properties of the system addressing each flux qubit individually (elaborated in~\ref{sec:KLD}).
The spectrum~$\{\mu^z\}$ of the total magnetic moment operator~${\hat{\mu}^z}$ assumes only integer values.
Repeated measurements yield a probability distribution for~$\{\mu^z\}$
centered at~$\mu^z=0$ for both the frustrated as well as paramagnetic phases.
The tail of the distribution
differs between frustrated and paramagnetic phases.
For the paramagnetic phase, the state of the entire system is
$\ket{+}^{\otimes{n}}$, which indicates that all states
are equally probable, if measured in the $\{\ket{\uparrow},\ket{\downarrow}\}^{\otimes n}$ basis.
Therefore, the probability that $l$ spins are in $\ket{\uparrow}$ state
and $l+k$ spins (with $n=2l+k$) are in $\ket{\downarrow}$ state (for which $|\mu^z|=k$), is
$\frac{1}{2^n} {n \choose (n-k)/2}{n \choose (n+k)/2}$, which means that the
probability distribution of~$\mu^z$ is a binomial function for the paramagnetic case.
For the frustrated phase, the probability distribution depends on the geometry of the lattice,
which is why it is called geometric frustration. Whereas, an exact closed-form analytic solution
of the distribution for this case is hard to derive, the frustrated antiferromagnetism implies that
the peak of the distribution is still centered at~$\mu^z=0$,
and the tail is an exponential decay (as opposed to binomial), which can be verified by numerical simulations.
In order to characterize the change in the tail
of the probability distribution, we introduce a measure
\begin{equation}
\mathcal{D}[\mathcal{P}(t)]:=D_{{\rm KL}}[P_{{\rm exp}},\mathcal{P}(t)]-D_{{\rm KL}}[P_{{\rm bin}},\mathcal{P}(t)],
\label{eq:D}
\end{equation}
on the probability distribution $\mathcal{P}(t)$ at time~$t$
with~$P_{{\rm exp}}$ and~$P_{{\rm bin}}$ best-fit exponential and binomial distributions corresponding to the final and initial states.
The Kullback-Leibler divergence~$D_{{\rm KL}}:=\sum_kP_k\log({P_k/Q_k})$
quantifies distinguishability between two probability distributions $P$ and $Q$.
If the distribution~$\mathcal{P}(t)$ changes from binomial to exponential,
then~$\mathcal{D}$ goes from positive to negative,
which indicates a phase transition from paramagnetic to frustrated magnetic phase has occurred.
\section{Results}
\label{sec:results}
\begin{figure}[htb]
\centering
\includegraphics[width=0.7\linewidth]{frustrationMicro.pdf}
\caption{Plots showing the microscopic signatures of QPT. (a) The control pulse where $\Delta$ gets turned off and~$J$ turned on adiabatically with time. (b) The probability~$\mathcal{P}$ for each state (denoted by subscripts in the legend) with time under the control pulse. (c) Expectation of the symmetric $W$-state witness operator as a function of time under the control pulse. The change of sign indicates the QPT.}
\label{fig:frustrationMicro}
\end{figure}
Figure~\ref{fig:frustrationMicro} shows the results for the microscopic characterization of the QPT for our three-qubit device. The adiabatic pulses for $\Delta$
and~$J$ are shown in Fig.~\ref{fig:frustrationMicro}(a)
for couplings slowly turned on and local controls off.
Figure~\ref{fig:frustrationMicro}(b) shows the probability distribution for all eight states.
Whereas at $t=0$ the probabilities are all equal,
they gradually decay for~$\ket{\uparrow\uparrow\uparrow}$ and~$\ket{\downarrow\downarrow\downarrow}$ states with a uniform probability distribution for the remaining states, which is expected for a paramagnetic to frustrated magnetic phase transition. Figure~\ref{fig:frustrationMicro}(c) shows the change of symmetric $W$-state entanglement-witness operator with time. At $t=t_{{\rm final}}$ the state of the system is a superposition of two $W$ states, and therefore $\langle{W_{S}}\rangle$ goes from positive to negative indicating the generation of entanglement in the system, which we consider as our microscopic signature.
\begin{figure}
\centering
\subfloat[]{
\centering
\includegraphics[width=0.7\linewidth]{frustrationMacroProb.pdf}
\label{fig:frustrationMacroProb}
} \\
\subfloat[]{
\centering
\includegraphics[width=0.7\linewidth]{frustrationMacroD.pdf}
\label{fig:frustrationMacroD}
}
\caption{%
Macroscopic signatures of the QPT. (a) The probability distributions $\mathcal{P}(\mu^z)$ of the total magnetic moment~$\mu^z$ for the network of $12$ flux qubits at initial and final times. The black dots denote the actual data computed via numerical simulation and solid lines denote interpolations through the data. The tail changes from binomial to exponential denoting the QPT. (b) Our distance measure based on the Kullback-Leibler divergence as defined in Eq.~(\ref{eq:D}) as a function of time.
The change in sign indicates the QPT.%
}
\label{fig:frustrationMacro}
\end{figure}
Figure~\ref{fig:frustrationMacro} shows the macroscopic signatures of the QPT for an $n$-qubit network of nearest-neighbor- and next-nearest-neighbor-coupled superconducting flux qubits
shown in Fig.~\ref{fig:frustrationChain}. We plot the initial and final probability distributions for the total magnetic moment of the system in Fig.~\ref{fig:frustrationMacroProb} for $n=12$. Although for both cases the distribution is centered around the origin, a significant difference is observed in the tails. The tail of the probability distribution changes from a binomial to an exponential decay. $\mathcal{D}[\mathcal{P}(t)]$ quantifies this change and \ref{fig:frustrationMacroD} shows how~$\mathcal{D}$ changes gradually from positive to a negative value with time signifying a QPT with respect to a macroscopic probe, which is the SQUID loop around the network for our case.
\section{Conclusions}
\label{sec:conclusions}
In this work, we have put forward a realistic scheme to simulate a QPT, where a network of nearest-neighbor- and next-nearest-neighbor-coupled superconducting flux qubits is subjected to a transition from the paramagnetic phase to the frustrated magnetic phase.
Characterizing a QPT with a micro as well as a macro probe is a conceptual requirement,
rather than an experimental preference.
Whereas the microscopic characterization of the phases can be performed measuring the appropriate entanglement witness, we have introduced a measure in this work based on the Kullback-Leibler divergence that can characterize the QPT macroscopically.
Our proposal for such simultaneous characterization of QPTs at macro and micro scales
enables a rigorous authentication of various quantum phases,
as well as offers a standardized procedure to self-consistently verify
the signatures of quantum phase transitions in upcoming experiments.
\section*{Acknowledgments}
This research was funded by NSERC and AITF.
JG acknowledges support from the University of Calgary's Eyes High Fellowship Program,
and appreciates support from China's Thousand Talent Plan.
We thank A.~Blais, M.~Geller,
D.~Home, M.~Saffman and P.~Roushan for valuable discussions.
| {
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} | 318 |
American Classic Pedigrees
Hoofprints
Horse Profiles A-E
Horse Profiles F-K
Horse Profiles L-Q
Horse Profiles R-Z
Scatter the Gold (CAN)
February 26, 1997 – ?
Mr. Prospector (USA) x Dance Smartly (CAN), by Danzig (USA)
Family 23-b
A maiden when he entered the gate for the 2000 Queen's Plate, Scatter the Gold came out of the race as a Canadian Classic winner and became the first stakes winner for his dam, 1991 Canadian Horse of the Year Dance Smartly. He added the Prince of Wales Stakes next but was unable to emulate his dam's Triple Crown sweep in the Breeders' Stakes and ended his career with his two Classic successes having been his only two victories. He was not a success at stud and was last known to be standing in Russia.
7 starts, 2 wins, 1 second, 2 thirds, US$845,854 (including Canadian earnings)
Won Queen's Plate (CAN-R, 10FD, Woodbine)
Won Prince of Wales Stakes (CAN-R, 9.5FD, Fort Erie)
3rd Breeders' Stakes (CAN-R, 12FT, Woodbine)
As an individual
A big, handsome dark bay or brown horse, Scatter the Gold had a muscular build featuring the typical strong Mr. Prospector hindquarters and hind leg. He had rather long pasterns. He was dominant and assertive but not nasty as a racehorse. At his best, he had a strong finishing kick.
As a stallion
According to records kept by The Jockey Club, Scatter the Gold has sired 253 winners (65.9%) and 0 stakes winners from 384 named foals. Records may be incomplete for the stallion's Russian progeny.
Foaled in Ontario, Scatter the Gold was bred and owned by Sam-Son Farms. He was trained by Mark Frostad. He was sold to Japanese interests in December 2000 and entered stud at Shizunai Stallion Station in 2001. He was later exported from Japan to Russia.
Pedigree notes
Scatter the Gold is inbred 3x5 to two-time American Horse of the Year Native Dancer and 5x5 to two-time English/Irish champion sire Nearco, an unbeaten champion on the race course. He is a full brother to 2001 Canadian champion 3-year-old filly Dancethruthedawn, who won the 2001 Lablatt Woodbine Oaks and Queen's Plate. He is also a full brother to 2004 Grey Breeders' Cup Stakes (CAN-II) winner Dance With Ravens and a half brother to Dancethruthestorm (by Thunder Gulch), dam of Canadian stakes winner Grand Style (by Giant's Causeway).
Scatter the Gold is out of the great Canadian race mare Dance Smartly, champion of her division at 2 and 3 in Canada and at 3 in the United States. She is a full sister to Dance Swiftly, dam of Grade III winner Speightster (by Speightstown) and listed stakes winners Paiota Falls (by Kris S.) and West Coast Swing (by Gone West). She is also a half sister to 1996 Philip H. Iselin Handicap (USA-I) winner and two-time American champion sire Smart Strike, 2002 Niagara Breeders' Cup Handicap (CAN-I) winner Full of Wonder and 2002 Chinese Cultual Center Stakes (CAN-II) winner Strike Smartly, all by Mr. Prospector. She is also a half sister to Seattle Classic (by Seattle Slew), dam of 1999 Mazarine Breeders' Cup Stakes (CAN-I) winner Hello Seattle (by Deputy Minister) and restricted stakes winner Sail From Seattle (by Gone West), a Group I sire in South Africa.
Dance Smartly's dam Classy 'n Smart (by Smarten) won the 1984 Canadian Oaks and was named the Canadian champion 3-year-old filly of that year. Her exploits as a broodmare made her Canada's Broodmare of the Year in 1991. She was the first of six stakes winners and four champions produced by Sam-Son Farm's remarkable foundation mare No Class (by Nodouble). No Class' other champions were 1985 Canadian champion 2-year-old male Grey Classic (by Grey Dawn II), 1987 Canadian champion 2-year-old male Regal Classic (by Vice Regent), and Sky Classic (by Nijinsky II), winner of three championships in Canada and champion turf horse in the United States in 1992. Also the dam of 1997 Early Times Turf Classic (USA-I) winner Always a Classic (by Deputy Minister) and multiple stakes winner Classic Reign (by Vice Regent), No Class was the Canadian Broodmare of the Year for 1985.
© 2014-2020 by Avalyn Hunter. All rights reserved.
Home Books Articles Horse Profiles Hoofprints Contact Links | {
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{"url":"https:\/\/documen.tv\/when-the-sum-of-the-reciprocals-of-two-distinct-positive-integers-is-divided-by-the-sum-of-the-t-28206681-86\/","text":"Question\n\nWhen the sum of the reciprocals of two distinct positive integers is divided by the sum of the two integers, the result is $\\frac{1}{25}$. What is the sum of the two integers\n\n1. thongdat2\nThe sum of the two integers is 26.\nAn integer is a whole number (not a fractional quantity) that can be high quality, negative, or zero. Examples of integers are: -five, 1, five, eight, 97, and three,043. Examples of numbers that aren\u2019t integers are: -1.forty three, 1 three\/4, 3.14, 09, and 5,643.1.\nA variety of that isn\u2019t a whole range, a negative whole quantity, or 0 is defined as Non-Integer. it is any variety that isn\u2019t blanketed inside the integer set, which is expressed as { \u2026 -4, -3, -2, -1, zero, 1, 2, 3, 4 }. Some of the examples of non-integers consist of decimals, fractions, and imaginary numbers.\nAn integer is colloquially defined as more than a few that can be written without a fractional aspect. For instance, 21, 4, 0, and \u22122048 are integers, while 9.75, 5+half, and \u221a2 are not.\n(1\/a \u00a0+ 1\/b) \/ (a+b) = 1\/25\n(a+b)\/(ab) \/ (a+b) = 1\/25\n1\/ab = 1\/25\nThis implies that ab \u00a0= 25\nSince a, and b \u00a0appear to be distinct integers, the result would be that either 1*25 \u00a0= 25 or 25 * 1 \u00a0= 25.\nEither way, their sum is 26.","date":"2023-03-30 00:47:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.807194709777832, \"perplexity\": 1573.8263399936752}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296949093.14\/warc\/CC-MAIN-20230330004340-20230330034340-00258.warc.gz\"}"} | null | null |
Eugenio Sangregorio (né le à Belvedere Marittimo) est un homme politique italo-argentin.
Installé à Buenos Aires en 1957, il fonde en vue des élections générales de 2006 une association d'émigrés italiens en Amérique du Sud, l'Union sud-américaine des émigrés italiens. Il doit attendre les élections générales italiennes de 2018 pour être élu député au Parlement italien. C'est le plus vieux député de la législature.
Notes et références
Personnalité politique italienne
Naissance à Belvedere Marittimo
Naissance en mars 1939
Député de la dix-huitième législature de la République italienne | {
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Why do we hurt? Do we actually experience pain, or is it merely illusion? In this video, Lorimer Moseley explores these questions, and position the pain that we feel as our bodies' way of protecting us from damaging tissues further. He also looks at what this might mean for those who suffer from chronic pain. | {
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"The Polyvagal Theory is at the leading edge of psychosomatic medicine and body-mind therapies. It is a vital contribution to scientifically-informed clinical practice. Psychologists, analysts, physicians, bodyworkers, and educators are provided with an essential map to help guide them in tracking the psychophysiological states of their clients, discern where they are 'stuck,' and help them to heal and move forward in life. Dr. Porges's great contribution is now compiled in this one astounding comprehensive volume. It is a must-read for clinicians and psychobiological researchers."—Peter A. Levine, PhD, author of In Unspoken Voice; How the Body Releases Trauma and Restores Goodness and Trauma Through a Child's Eyes: Awakening the Ordinary Miracle of Healing.
"Stephen Porges has developed theories of the autonomic nervous system, based on his sound research over many years, which have profound importance for our understanding of human behavior, whether normal or pathological. Based on evolutionary principles, and firmly embedded in neuroanatomy, his work considerably broadens our understanding of the representation of emotion in the brain, reaching parts that most contemporary texts never touch."—Michael Trimble, MD, author of The Soul in the Brain: The Cerebral Basis of Language, Art, and Belief.
The Norton Series on Interpersonal Neurobiology
Allan N. Schore, PhD, Series Editor
Daniel J. Siegel, MD, Founding Editor
The field of mental health is in a tremendously exciting period of growth and conceptual reorganization. Independent findings from a variety of scientific endeavors are converging in an interdisciplinary view of the mind and mental well-being. An interpersonal neurobiology of human development enables us to understand that the structure and function of the mind and brain are shaped by experiences, especially those involving emotional relationships.
The Norton Series on Interpersonal Neurobiology will provide cutting-edge, multidisciplinary views that further our understanding of the complex neurobiology of the human mind. By drawing on a wide range of traditionally independent fields of research—such as neurobiology, genetics, memory, attachment, complex systems, anthropology, and evolutionary psychology—these texts will offer mental health professionals a review and synthesis of scientific findings often inaccessible to clinicians. These books aim to advance our understanding of human experience by finding the unity of knowledge, or consilience, that emerges with the translation of findings from numerous domains of study into a common language and conceptual framework. The series will integrate the best of modern science with the healing art of psychotherapy.
The Polyvagal Theory
Neurophysiological Foundations
of Emotions, Attachment,
Communication, and Self-Regulation
STEPHEN W. PORGES
Dedication
With love and respect
I dedicate this book to Sue Carter,
my wife and intellectual partner.
Contents
Cover
Title Page
Dedication
Foreword, by Bessel A. van der Kolk
Introduction: Why Is There a Polyvagal Theory?
PART I: THEORETICAL PRINCIPLES
1. Neuroception: A Subconscious System for Detecting Threat and Safety
2. Orienting in a Defensive World: Mammalian Modifications of Our Evolutionary Heritage. A Polyvagal Theory
3. The Polyvagal Theory: New Insights Into Adaptive Reactions of the Autonomic Nervous System
PART II: BIOBEHAVIORAL REGULATION DURING EARLY DEVELOPMENT
4. Vagal Tone: A Physiological Marker of Stress Vulnerability
5. The Infant's Sixth Sense: Awareness and Regulation of Bodily Processes
6. Physiological Regulation in High-Risk Infants: A Model for Assessment and Potential Intervention
7. Infant Regulation of the Vagal "Brake" Predicts Child Behavior Problems: A Psychobiological Model of Social Behavior
8. The Early Development of the Autonomic Nervous System Provides a Neural Platform for Social Behavior
PART III: SOCIAL COMMUNICATION AND RELATIONSHIPS
9. Vagal Tone and the Physiological Regulation of Emotion
10. Emotion: An Evolutionary By-Product of the Neural Regulation of the Autonomic Nervous System
11. Love: An Emergent Property of the Mammalian Autonomic Nervous System
12. Social Engagement and Attachment: A Phylogenetic Perspective
13. The Polyvagal Hypothesis: Common Mechanisms Mediating Autonomic Regulation, Vocalizations, and Listening
PART IV: THERAPEUTIC AND CLINICAL PERSPECTIVES
14. The Vagus: A Mediator of Behavioral and Physiological Features Associated With Autism
15. Borderline Personality Disorder and Emotion Regulation
16. Abuse History Is Related to Autonomic Regulation
17. Music Therapy, Trauma, and the Polyvagal Theory
PART V: SOCIAL BEHAVIOR AND HEALTH
18. Reciprocal Influences Between Body and Brain in the Perception and Expression of Affect
19. Neurobiology and Evolution: Mechanisms, Mediators, and Adaptive Consequences of Caregiving
Epilogue
Credits
References
Acknowledgments
Copyright
Foreword
Bessel A. van der Kolk
It is an extraordinary experience to hear a new piece of music or a novel scientific idea that makes you sit up in wonder and turns on a bank of lights in your mind that permanently transforms how you understand what life is all about. Seeing Peter Sellers's production of the Marriage of Figaro, listening to Elisabeth Kübler-Ross lecture on schizophrenia when I was a medical student, and hearing Steve Maier talk about the neurobiology of inescapable shock at the American College of Neuropsychopharmacology in 1984 were three such unforgettable events for me.
On the morning of May 21, 1999, we had the most memorable series of lectures of the 22 years that I have run the annual Boston trauma conference. The day opened with Bruce McEwen talking about stress and the hippocampus. He introduced the notion of neuroplasticity and showed that Ramon-y-Cahal's dire dictum that anatomy is destiny—that the neuronal connections in the brain are immutably fixed—was incorrect. Jaak Panksepp, whose work has delineated the brain circuits that underlie nurturance, fear, rage, and rough and tumble play, was next. The final speaker that morning was Stephen Porges, who talked about the polyvagal theory of emotions. That lecture caused a fundamental shift in how I and many of my colleagues understand the work we do.
Clinicians and researchers who deal with people with histories of chronic trauma are routinely confronted with fight, flight, or freeze reactions. Our patients (and occasionally colleagues) easily take offense, and they often disorganize their (and our) lives by becoming too angry, too ashamed, and too frozen. Minor irritations easily turn into catastrophes; small failures of communication are difficult to gloss over and easily turn into dramatic interpersonal conflicts. The milk of human kindness, such an essential nutrient in making life bearable, all too often fails to have a significant impact on the despair, rage, and terror of people with histories of trauma and abandonment.
When post-traumatic stress disorder (PTSD) first made it into the diagnostic manuals, we only focused on dramatic incidents like rapes, assaults, or accidents to explain the origins of the emotional breakdowns in our patients. Gradually, we came to understand that the most severe dysregulation occurred in people who, as children, lacked a consistent caregiver. Emotional abuse, loss of caregivers, inconsistency, and chronic misattunement showed up as the principal contributors to a large variety of psychiatric problems (Dozier, Stovall, & Albus, 1999; Pianta, Egeland, & Adam, 1996). One of the most important discoveries in psychology, neuroscience, and psychiatry has been that failure in establishing secure early attachment bonds leads to a diminished capacity to regulate negative emotions. Harlow and his students were the first to show the devastating effects of abandonment and loss on affect regulation in nonhuman primates. Half a century of attachment research has shown that people learn to regulate their emotional arousal largely as a function of the capacity to establish physical and rhythmical attunement with important figures in their early caretaking environment (Trevarthen, 1999). Attachment researchers, starting with John Bowlby, have long shown that the internal regulatory processes of an individual reflect to a large degree the attunement with external sources of regulation early in life (Bowlby, 1973, 1982; Cloitre, Stovall-McClough, Zorbas, & Charuvastra, 2008; Hofer, 2006).
Having a history of chronic misattunement with one's caregivers predisposes people to have difficulties managing negative emotions later in life (Dozier et al., 1999). Sadly, deficient affect regulation caused by early adverse experiences is compounded by the resulting off-putting behaviors in the face of stress, such as temper tantrums and emotional withdrawal (Shaver & Mikulincer, 2002). Dysregulated behavior alienates potential friends and partners and interferes with being able to garner support and accumulate restorative experiences. Lack of affect regulation thus runs the danger of becoming a vicious circle, where deficient self-control leads to abandonment, which in turn makes it even more difficult to regulate negative mood states.
The problem does not end there, since routine psychiatric interventions are quite ineffective in helping people manage their emotions (Cloitre, Stovall-McClough, Miranda, & Chemtob, 2004), and the best that medications generally can do is to dull emotional arousal of any kind, thereby robbing people of pleasure and of pain simultaneously. Traditional psychotherapies also do not offer much immediate relief, since being unable to manage emotional arousal interferes with being able to benefit from treatments such as cognitive behavioral therapy (Jaycox, Foa, & Morral, 1998).
Problems with affect regulation have pervasive effects on the development of mind and brain and lead to significant increases in the utilization of medical, correctional, social, and mental health services (Drossman et al., 1990; Teplin, Abram, McClelland, Dulcan, & Mericle, 2002; Widom & Maxfield, 1996). Many problems of children and adults with histories of chronic trauma and neglect can be understood as efforts to minimize objective threat and to regulate their emotional distress (Pynoos et al., 1987), and hence they are liable to be labeled as "oppositional," "rebellious," "unmotivated," and "antisocial" (Cicchetti & White, 1990; Widom & Maxfield, 1996; Streeck-Fischer & van der Kolk, 2000).
Our study of patients with abuse and neglect histories was helped enormously by the developments in affective neuroscience, which clarified the underlying neurobiology of many problems seen in these patients. Particularly helpful was Panksepp's (1998) work that elucidated the brain structures and neural circuits involved in the basic emotional systems: seeking, rage, fear, lust, care, panic, and play. But understanding the limbic circuits of emotions still did not account for the precipitous shifts we see in our patients and their lack of responsiveness to the voices and faces of people who care for them, input that helps most people calm down and regain mastery over their senses. The question remained: what causes the emotional systems of some people to be hijacked so easily by slight changes in visual, auditory, kinesthetic, or olfactory input?
More than a decade ago, we became interested in a possible role of heart rate variability (HRV) in the maintenance of these emotional roller coasters. This gave us our first exposure to Porges's polyvagal theory. When we started to measure HRV, we observed that people who maintained relatively stable HRV while remembering horrendous personal events did not seem to suffer from PTSD, whereas people with PTSD generally had poor HRV. Porges's polyvagal theory provided us with an organizing principle for that observation, built on the work of prior scientists such as Charles Darwin and William James, which explained the central role of human interactions and body sensations in our emotional lives.
Up to the time that we learned about the polyvagal theory we had been taught that the autonomic nervous system is organized by the paired antagonism between the sympathetic and the parasympathetic nervous systems, which functionally compete by either increasing or decreasing activity of specific target organs. The introduction of the polyvagal theory vastly expanded this model with an emphasis on the social, myelinated vagus as the fine-tuning regulatory system that opens up a role for the environment to foster or ameliorate stress-related physiological states. Porges proposes that
phylogenetically, a hierarchical regulatory stress-response system emerged in mammals that not only relies on the well-known sympathetic-adrenal activating system and the parasympathetic inhibitory vagal system, but that these systems are modified by myelinated vagus and the cranial nerves that regulate facial expression which constitute the social engagement system. Thus, phylogenetically, self-regulatory development starts with a primitive behavioral inhibition system, progresses by the evolution of a fight-flight system, and, in humans (and other primates), culminates in a complex social engagement system mediated by facial gestures and vocalizations.
Thus, the mammalian myelinated vagus functions as "an active vagal brake that supports rapid behavioral mobilization, as well as the capacity to physiologically stabilize an individual by means of interoceptive visceral awareness, as well as social interaction." According to Porges, this evolutionary development allows social interactions to stabilize physiological arousal by means of facial expressions, speech, and prosody. When the environment is appraised as being safe, the defensive limbic structures are inhibited. This makes it possible to be socially engaged with calm visceral states.
BREAKDOWN OF AFFECT REGULATION
When that system breaks down, as we witness particularly under conditions of extreme stress, the social vagus no longer can stabilize the organism. The physiological cohesion that forms the basis of the interpersonal neurobiological communication between different organisms breaks down, and the phylogenetically "older" systems will be recruited to regulate metabolic output to deal with environmental challenges. As long as people (and animals) feel threatened, they cannot meaningfully engage with members of their tribe and will resort to more primitive and solipsistic fight-or-flight behaviors (mobilization mediated by the sympathetic nervous system) to ensure survival. Under conditions of inescapable danger even fight or flight breaks down, which leads to immobilization, with a resulting behavioral shutdown, and syncope (via the unmyelinated vagus).
This theory made perfect sense and tied together numerous observations from attachment research, animal ecology, anthropology, and trauma research. To survive and thrive, mammals must be able to distinguish friend from foe, know when a situation is safe or dangerous, and adjust their behavior to the demands of the social group. Porges's observations clarified the mechanisms that allow humans and animals to be physically approached, what ties members of the group together, and how the cohesion of the social group is a central element in managing life's stresses. The polyvagal theory also helps us understand the capacity of the tone of voice and rhythms of speech, as well as faces of our loved ones, to restore our physiological equilibrium.
VISCERAL EXPERIENCES
Another important contribution of the polyvagal theory is a clarification of the relation between visceral state and emotional expression. Trauma researchers understood early on that "the body keeps the score" (van der Kolk, 1994)—that the memory of trauma is encoded in visceral experiences such as heartbreaking and gut-wrenching emotions, in autoimmune disorders, and in skeletal-muscular problems. Porges proposed that afferent feedback from the viscera contributes in a major way to the accessibility of prosocial circuits associated with social engagement. For example, mobilization changes our ability to detect positive social cues, and immobilization may make a person impervious to positive input. Visceral states color our perception of ourselves and our surroundings. The physiological state the person is in precipitates very different outcomes in response to the presentation of the same stimuli.
Internal bodily states are represented in the insula and contribute to subjective feeling states. The insula (which is frequently found to be abnormally activated in neuroimaging studies of traumatized individuals) is involved in the perception of danger and mediates the feedback from the viscera into cognitive awareness. Darwin and James both proposed that visceral experience is a critical element in the experience of emotions and central to their understanding of happiness, fear, anger, disgust, and sadness. Most people are able to gauge danger and love by means of their "gut feelings," which generally accurately detect relative the danger or safety of their situation. People with impaired social engagement systems are prone to misinterpret safety as a threat and objective danger as safety. Their visceral feedback system fails to protect them, or prevents them from engaging in the fullness of what life has to offer.
People who are chronically traumatized have a tendency to become overwhelmed and triggered by their seemingly unmanageable visceral feedback systems that cannot be modified by a functional social engagement system. Consequently they try to inhibit sensory feedback from their bodies and experience the feedback from both their bodies and the world around them as bland and meaningless. Our observation that traumatized individuals habitually engage in body-based defensive maneuvers led us to incorporate the work of such body-based therapists as Peter Levine and Pat Ogden into our treatment armamentarium. In fact, Peter Levine introduced me to Stephen Porges.
IMPLICATIONS FOR TREATMENT
The polyvagal theory of emotions has had a profound effect on helping us organize the treatment of abused children and traumatized adults. True, we might have developed a yoga program for chronically traumatized women anyway, because they so obviously benefit from learning to calm themselves down by learning to manage their breathing and engaging in challenging physical poses that put them in touch with their dissociated bodies. We also probably would have had a theater program in the Boston inner-city schools where traumatized kids learn to play by engaging in rhythmical movements and attuned collaborative exercises. And my friend Tina Packer would have taught Shakespeare to juvenile delinquents anyway, because her profound knowledge of the effects of hexameters and embodying the physical roles of Julius Caesar, Romeo and Juliet, and Richard III compelled her to teach these dysregulated adolescents to engage in and move between deep and varied physiological (and hence mental) states by linking the expressive features of emotion with their visceral responses. My colleagues Robert Macy and Steve Gross would have developed their play techniques and chi qong–based therapies that they apply in inner-city schools and to survivors of tsunamis, earthquakes, and political violence around the world, even if they would never have learned the polyvagal theory.
But the polyvagal theory ties all these disparate unconventional techniques together, in that they all activate situations that in the past have precipitated the participants into uncontrollable flight-or-fight and freeze modes, which, with these various techniques that rely on interpersonal rhythms, visceral awareness and the primary use of vocal and facial communication, attempt to reorganize the perception of danger and capacity to manage emotional engagement.
It is difficult to trace all the sources of one's inspiration, but Porges's polyvagal theory gave us a powerful means of understanding how both bodily states and mental constructs dynamically interact with environmental triggers to precipitate maladaptive behaviors. Porges helped us understand how dynamic our biological systems are and gave us an explanation why a kind face and a soothing tone of voice can dramatically alter the entire organization of the human organism—that is, how being seen and understood can help shift people out of disorganized and fearful states. We had long realized that psychopathological states rarely are static and tend to fluctuate greatly depending on the safety of the environment and the physiological state in which people find themselves. The proposal that our physiological states are flexible, and depend on both our relationship to our visceral experiences and the state of our relationsips promises to decrease our dependence on drug treatment alone to shift people into a different psychological organization. Recognizing the critical role of visceral afferent feedback on the global functioning of the brain inevitably leads to curiosity about the nonpharmacological treatments that have so long been practiced outside of Western medical approaches: age-old traditions of changing mental states with specific breath exercises, body movements (chi qong, tai chi, tae kwon do, and yoga) and rhythmical activities (such as kendo drumming and davening).
Several psychiatric disorders are characterized by difficulties in establishing and maintaining relationships, including difficulty both in expressing social behavior and in reading social cues (i.e., social awareness). Several psychiatric diagnoses are associated with specific deficits in both the behavioral (e.g., poor gaze, low facial affect, lack of prosody) and visceral (difficulties in autonomic regulation resulting in cardiopulmonary and digestive problems) components of the social engagement system.
If physiological mind–brain–viscera communication is the royal road to affect regulation, this invites a radical shift in our therapeutic approaches to a number of psychopathological states, such as anxiety, attention deficit/hyperactivity disorder, autism, and trauma-related psychopathology. Such a shift (for which there appears to be increasing support with increased funding by both the Department of Defense and the National Institutes of Health of such approaches as yoga, martial arts, and acupuncture studies) would ask us to cultivate interpersonal rhythms, nurture the capacity for people to use their voices and faces to regulate emotional states, and explore various body–mind techniques that integrate visceral and emotional experiences. The polyvagal theory legitimates the study of age-old collective and religious practices such as communal chanting, various breathing techniques, and other methods that cause shifts in autonomic state. The polyvagal theory implies that more attention needs to be paid to the development of interventions that either promote activation of the social vagus or dampen sympathetic tone. One major implication is the need to pay closer attention to the therapeutic use of play, rough and tumble behaviors that serve as preliminary exercises to develop adaptive defensive and aggressive behaviors, as a means of shifting people out of fight-or-flight reactions into loving and mutually engaged mobilization.
[Introduction: Why Is There
a Polyvagal Theory?](contents.xhtml#int_01)
The polyvagal theory emerged from a dialectic between my curiosity in biobehavioral systems and my dissatisfaction with the prevalent models that integrated physiological state with behavior. When I started my scientific career, I was intrigued with the possibility of using physiological measures to understand the psychological states of others. In the late 1960s, when I was in graduate school, I had the vision that monitoring physiological state would be a helpful guide to the therapist during the clinical interaction. This vision is still part of my research agenda. I am still working on developing a polyvagal monitor, which will provide feedback in real time to clinicians of the dynamic interplay between the three neural circuits described in the polyvagal theory.
In the 1960s, the constructs and models relating physiology to behavior were limited. Dominant in the human and psychophysiological literature was a construct of arousal. The defining features of arousal were vague. However, psychophysiologists assumed that arousal was mediated by the sympathetic nervous system. Early psychophysiologists, such as Chester Darrow, proposed continuity between cortical activation measured through electroencephalography (EEG) and sympathetic arousal measured by the galvanic skin resistance response on the hands. This view of a peripheral indicator of brain processes was consistent with Pavlov's use of autonomic measures in his classical conditioning experiments. For Pavlov, the "classically" conditioned autonomic responses were indices of changes in brain circuits. Arousal is still used in sleep research to describe cortical activation and in research on deception in which traditional polygraphs are used.
The specific physiological and neurophysiological mechanisms underlying arousal are often associated with the sympathetic nervous system and the hypothalamic-pituitary-adrenal (HPA) axis. An inferred connection between the sympathetic nervous system and the HPA axis has resulted in similar research methods being used to study both arousal and stress. This sympathetic-centric view has been translated into the popular press and public consciousness as a cliché that a limited amount of stress is "good" and too much stress is "bad." But what were the thresholds of stress necessary for health or illness? In addition, consistent with this sympathetic-centric view, we all were taught that the stress-related sympathetic excitation had evolutionary origins in mammalian fight-or-flight behaviors. Thus, we were taught that the increased sympathetic tone to novelty and danger was a reflection of our evolutionary history.
At the time of my introduction to psychophysiology, physiological measures were proposed as a portal into psychological processes without requiring conscious awareness or verbal responses. This exciting discipline, however, was constrained by a limited understanding of the neural mechanisms regulating the physiological measures being monitored and an adequate understanding of the mediating neural mechanisms linking peripheral autonomic responses to the psychological processes of interest to the psychophysiologist.
Psychophysiology was established in the early 1960s as a discipline merging features of psychology, medicine, physiology, and engineering. The Society for Psychophysiological Research was formed in 1960 and the first issue of society's journal, Psychophysiology, was published in 1964. During its early years, psychophysiology distinguished itself from physiological psychology by focusing on physiology as a dependent variable and psychological factors (e.g., threat, novelty) and processes (e.g., attention, mental effort, emotion) as independent variables. In contrast, physiological psychology manipulated physiology (independent variable) and monitored changes in behavior and psychological processes (i.e., dependent variable). In general, psychophysiologists studied human subjects in their research paradigms, whereas physiological psychologists studied animals. I joined the Society for Psychophysiological Research in 1968 and attended my first meeting in 1969. The early meetings were exciting, and many of the initial pioneers in the society had been successful scientists in other domains. During my more than 40 years of membership, I was involved in the society as a member of the board, secretary-treasurer, and president. During this period, research interests and methods shifted from peripheral measures of the autonomic nervous system to measures of brain function employing EEG, event-related potentials, and functional magnetic resonance imaging methods. Current issues of Psychophysiology reflect this trend and focus on measures of brain function during cognitive and affective challenges.
As an important consequence of this history, physiological variables were seen as correlates of psychological processes. This view allowed researchers to study parallels between physiology and behavior without understanding the underlying relationship between these domains. Basically, there are two global domains in the biobehavioral sciences: (1) behavior (observable) and psychological (subjective) and (2) physiological (peripheral autonomic) and neural (brain). Paradigms that correlate physiology and behavior or use physiological variables as a biomarker of a clinical disorder are the consequence of a modern Western scientific solution to the historical remnant of the mind–body problem. The current scientific solutions of dualism are not solutions, but merely objective descriptions of parallel functions with exquisite technologies. Many scientists and disciplines are stuck in the dualism trap. New disciplines have emerged such as cognitive neuroscience, affective neuroscience, social neuroscience, health psychology, and several subdisciplines within public health that use the methods that were developed in psychophysiological research to provide biomarkers of illness. Currently, measures of heart rate variability and neuroendocrine activity (e.g., cortisol, oxytocin, vasopressin) are frequently described as potential biomarkers of clinical health and risk. The polyvagal theory challenges this implicit dualism by providing a bidirectional brain–body model that interprets the brain regulation of peripheral physiology (e.g., neural regulation of both cardiovascular and endocrine function) as providing a neural platform for emergent adaptive social and defensive behaviors (see chapter 8).
The academic arena of the late 1960s had a limited conceptualization of the brain's role in regulating peripheral physiology. Because the details of potential mechanisms mediating the link between psychological processes and physiology were unknown, scientists were comfortable with investigations applying physiological measures as correlates of psychological processes and potential biomarkers of mental and physical health. In 1970, I matriculated into this academic arena with my doctorate and started conducting independent research as an assistant professor
The roots of the polyvagal theory emerged while conducting research on heart rate correlates of attention for my master's thesis (Porges & Raskin, 1969). In this study, I noted that when subjects focused their attention on the task demands, the beat-to-beat heart rate pattern stabilized. The publication of the master's thesis became the first quantitative description of heart rate variability as a response variable sensitive to psychological manipulations. My dissertation followed this theme by testing the relation between heart rate variability and reaction time performance. The dissertation confirmed that greater reductions in heart rate variability were associated with faster reaction times. Moreover, the study identified that individual differences in heart rate variability predicted reaction time performance and the degree to which heart rate variability was suppressed while attending (Porges, 1972). Throughout the following 40 years, although methodologies and research questions have changed, my laboratory research has continued to conduct studies monitoring heart rate and quantifying heart rate variability.
How did an observation that heart rate stabilized during attention lead to the polyvagal theory? There were a few intervening conceptualizations. First, I had to link the changes in heart rate variability to vagal mechanisms. This was accomplished through two stages: first, the development of quantification techniques to characterize the rhythms in the beat-to-beat heart rate pattern, and second, conducting validation studies to demonstrate that the amplitude of the respiratory rhythm in heart rate (i.e., respiratory sinus arrhythmia) was a valid indicator of vagal influences to the heart. By the early 1980s, this was in place. The ideas coalesced and the concept of vagal tone was concretized in a vagal tone monitor and software to enable other laboratories to use this sensitive index.
Although developed 30 years ago, the method I created for quantifying heart rate variability is still being used in more than 100 laboratories worldwide. The method has survived because it has five advantages over the scores of methods that have been proposed during the intervening decades to quantify heart rate variability: (1) the method enables dynamic monitoring of the shifts in vagal control of heart over short periods of time, (2) it conforms to the statistical assumptions necessary for parametric statistics, (3) it enables reliable estimates even when the baseline of heart rate drifts and violates the assumption of stationarity, (4) it is not moderated by respiration rate, and (5) the metric reflects the same changes in vagal function across time and laboratories.
Once the quantification procedures were developed, validated, and packaged in software and hardware, I thought the research world of autonomic psychophysiology would expand. I believed that the sensitive quantification methods would enable scores of scientists to investigate and study the critical role that cardiac vagal tone played in physical and mental health and in moderating social, affective, and cognitive behaviors and motor activity. With these tools dozens of studies were conducted in laboratories around the world with a common metric, and vagal tone became a familiar measure in psychological and psychophysiological research.
Everything seemed relatively simple. Once there was a good measure of parasympathetic activity (i.e., vagal tone), the sympathetic-centric worldview would be challenged and constructs such as autonomic balance, arousal, and stress, introduced decades before, could be studied within a more complete physiological model that included dynamic measures of vagal tone. Interest in vagal tone and heart rate variability grew, and several other metrics of vagal tone derived from heart rate variability developed by other scientists also became available to researchers. The movement from sympathetic-centric to dual sympathetic-parasympathetic measures provided a comfortable haven for researchers who had learned autonomic neurophysiology from a paired-antagonism perspective (i.e., the balance between sympathetic and parasympathetic influences). Thus, my research provided a critical measurement tool that would enable researchers to study the dynamically changing interplay between the sympathetic and the parasympathetic components of the autonomic nervous system. This conceptualization of cardiac vagal tone did not challenge the existing dogma presented in textbooks of the autonomic nervous system as a paired-antagonistic system.
VAGAL PARADOX
Although I had vested interests in vagal regulation, I was not uncomfortable with the paired-antagonism model of the autonomic nervous system. By 1990, I saw my contribution to psychophysiology and psychobiology as presenting the other side of autonomic function to a sympathetic-centric research world. I saw the importance of tonic levels of vagal activity as an index of general neural health and as a protective feature. In my writings I started to discuss the autonomic nervous system as a bidirectional system that involved potent visceral feedback (see chapter 5) and central structures in the regulation of physiological and emotional state (see chapter 9).
In the early 1990s I had not incorporated into my research perspective the three important points that led to the conceptualization of the polyvagal theory. First, although I wrote about the importance of having high vagal tone as a positive health index, I did not conceptualize autonomic reactions as being hierarchical. For example, I did not conceptualize the vagus as being inhibitory of the sympathetic regulation of the heart. Second, I had no understanding of how the neural regulation of the autonomic nervous system changed during evolution and how these changes would be related to adaptive physiological and behavioral functions in mammals. Third, although I knew that vagal pathways originated from two brainstem nuclei (dorsal motor nucleus of the vagus and the nucleus ambiguus), I did not think too much about their relative functions.
By 1992, I was confident that my research was on the right track and looked forward to expanding my projects and collaborations with the technologies I had developed. I thought the hard work was done and looked forward to new discoveries applying these technologies to clinical populations. I had no intention of developing a theory that would require a deep understanding of evolution and the adaptive behavioral functions that covaried with the phylogenetic changes in neural regulation of the autonomic nervous system. I had no intention to challenge either the prevalent theoretical models of the autonomic nervous system (i.e., paired antagonism) or the application of physiological monitoring as correlates of psychological processes and as potential "biomarkers" for health and disease.
My intellectual complacency did not last. The jolt came in the form a complimentary letter from a neonatologist that I received shortly after I published an article in September 1992 (see chapter 4). The neonatologist wrote that although he liked my article, it was not consistent with what he learned in medical school. The paper described evidence that measuring cardiac vagal tone (i.e., respiratory sinus arrhythmia) derived from the beat-to-beat heart rate in newborn infants provided a sensitive index of clinical status. The study reported consistently higher cardiac vagal tone in healthy full-term newborns, while lower cardiac vagal tone at about the time of discharge for premature infants. In his letter he stated that he had learned that high vagal tone was bad for newborns and could kill them, and he ended by stating that perhaps too much of a good thing is bad. Something was not right with this conclusion. Having conducted research in newborn nurseries since 1970, I tried to understand his statements from his training and observations. When I took his perspective, I immediately realized that risk to the newborn was associated with vagal-mediated bradycardia. Bradycardia is a massive slowing of heart rate that if prolonged, may result in depriving the brain of needed oxygen. In contrast, I was looking at a protective feature of the vagus, the beat-to-beat variability that had a respiratory rhythm. By the time I published this paper, we had collected sufficient data from both human newborns and fetuses to realize that bradycardia occurred only when the tonic pattern of beat-to-beat heart rate was relatively flat (i.e., no or very low-amplitude respiratory sinus arrhythmia). I had interpreted this pattern as a lack of vagal influence to the heart. Now I understood why obstetricians and neonatologists, who used beat-to-beat heart rate variability as a biomarker of clinical status, made no inference regarding mechanism.
I was in an intellectual quandary. I had been arguing that cardiac vagal tone was a positive clinical index that could be measured by respiratory sinus arrhythmia. Now there was the possibility that there were two heart rate measures with vagal origins, one protective and the other potentially lethal. Among neonatologists and obstetricians there was an acceptance that beat-to-beat heart rate variability had clinical significance, although the neural mechanisms mediating this clinically relevant biomarker were not known and there appeared to be no motivation to identify the mechanisms. In contrast, rapid massive bradycardias were clearly mediated by transitory vagal surges.
How could the vagal regulation of the heart be an index of resilience and health when it was represented in high-amplitude respiratory sinus arrhythmia and an index of risk when it was represented in bradycardia? This question challenged my understanding of the autonomic nervous system. I placed the letter from the neonatologist in my briefcase, where it remained for about 2 years as I formulated the basis for the polyvagal theory. I labeled this contradiction in our understanding of the vagus the vagal paradox. My motivation to solve the vagal paradox led to new conceptualizations of the autonomic nervous system and the formulation of the polyvagal theory. Only with the new understandings gained from the polyvagal theory would the vagal paradox be deciphered and the hierarchical nature of the neural regulation and the adaptive functions of in the autonomic nervous system understood.
From fall 1992 through fall 1994, I worked on integrating the literature on the autonomic nervous system and extracting a series of organizing principles that became the core of the theory. During this period, in addition to my professorship at the University of Maryland, I was a visiting scientist at the National Institutes of Health (NIH). As such, I had access to the excellent NIH library and the National Library of Medicine. With these resources, I immersed myself in the literature and read hundreds of articles and numerous books on the neural regulation of the autonomic nervous system in vertebrates. The polyvagal theory was the product of this work and was presented on October 8, 1994, in my presidential address to the Society for Psychophysiological Research (see chapter 2).
Since the initial presentation, the theory has been refined and expanded (see Porges, 2001a, 2007a). By selecting and editing previously published papers, the chapters in this book provide an opportunity to share the discoveries of the theory. The chapters include the presentation of the initial theory (see chapter 2) and the expansion and elaboration of ideas dependent on the theory, including vagal brake (see chapter 7), self-regulation (see chapter 6), development (see chapter 8), emotion (see chapters 9 and 10), evolution and dissolution (see chapter 10), immobilization without fear (see chapter 11), the social engagement system (see chapters 11, 12, and 13), attachment (see chapter 12), love and monogamy (see chapter 11), neuroception (see chapters 1 and 12), prosody and vocal communication (see chapter 13), clinical applications (see chapters 14, 15, 16, and 17), and our current work redefining social neuroscience (see chapters 18 and 19). Chapter 3 has been included as a succinct summary of the major points of the theory (e.g., vagal paradox, dissolution, social engagement system, neuroception) and may be helpful in clarifying chapters in which the presentation of the theory has been abridged.
PART I
THEORETICAL PRINCIPLES
CHAPTER 1
Neuroception: A Subconscious System for Detecting Threat and Safety
What determines how two human beings will act toward each other when they meet? Is this initial response a product of learning from culture, family experiences, and other socialization processes? Or is the response the expression of a neurobiological process that is programmed into the very DNA of our species? If the response has a neurobiological basis, are there specific features of the other person's behavior that trigger either feelings of safety, love, and comfort or feelings of danger? Why do some children cuddle and warmly conform to embraces, yet others stiffen and pull back from the same overture? Why do some children smile and actively engage a new person, while others avert their gaze and withdraw?
Does knowledge of human biology help us understand the triggers and mechanisms of these behaviors during normal development? If we learn how behavioral features trigger neural circuits that facilitate social behavior, will we be better able to help children with severe developmental disabilities, such as autism, improve their social behavior?
By processing information from the environment through the senses, the nervous system continually evaluates risk. I have coined the term neuroception to describe how neural circuits distinguish whether situations or people are safe, dangerous, or life-threatening. Because of our heritage as a species, neuroception takes place in primitive parts of the brain, without our conscious awareness. The detection of a person as safe or dangerous triggers neurobiologically determined prosocial or defensive behaviors. Even though we may not be aware of danger on a cognitive level, on a neurophysiological level, our body has already started a sequence of neural processes that would facilitate adaptive defense behaviors such as fight, flight, or freeze.
A child's (or an adult's) nervous system may detect danger or a threat to life when the child enters a new environment or meets a strange person. Cognitively, there is no reason for them to be frightened. But often, even if they understand this, their bodies betray them. Sometimes this betrayal is private; only they are aware that their hearts are beating fast and contracting with such force that they start to sway. For others, the responses are more overt. They may tremble. Their faces may flush, or perspiration may pour from their hands and forehead. Still others may become pale and dizzy and feel precipitously faint.
This process of neuroception would explain why a baby coos at a familiar caregiver but cries at the approach of a stranger, or why a toddler enjoys a parent's gentle embrace but interprets the same gesture from a stranger as an assault. We can see the process at work when two toddlers encounter each other in a playground sandbox. They may decide that the situation and each other are safe if the sandbox is familiar territory, if their pails and shovels have roughly similar appeal, and if they (the toddlers) are about the same size. The toddlers may then express positive social engagement behaviors—in other words, they may start to play.
"Playing nice" comes naturally when our neuroception detects safety and promotes physiological states that support social behavior. However, prosocial behavior will not occur when our neuroception misreads the environmental cues and triggers physiological states that support defensive strategies. After all, "playing nice" is not appropriate or adaptive behavior in dangerous or life-threatening situations. In these situations, humans—like other mammals—react with more primitive neurobiological defense systems. To create relationships, humans must subdue these defensive reactions to engage, attach, and form lasting social bonds. Humans have adaptive neurobehavioral systems for both prosocial and defensive behaviors.
What allows engagement behaviors to occur, while disabling the mechanisms of defense? To switch effectively from defensive to social engagement strategies, the nervous system must do two things: (1) assess risk, and (2) if the environment looks safe, inhibit the primitive defensive reactions to fight, flee, or freeze.
By processing information from the environment through the senses, the nervous system continually evaluates risk. As evolution has proceeded, new neural systems have developed. These systems use some of the same brain structures that are involved in defense functions to support forms of social engagement. Neuroception now may encourage the development of social bonds and provide the opportunity for reproduction.
SOCIAL ENGAGEMENT AND DEFENSIVE BEHAVIOR: ADAPTIVE OR MALADAPTIVE STRATEGIES?
Social engagement and defense behaviors may be adaptive or maladaptive, depending on the level of risk that is present in the environment. From a clinical perspective, the defining features of psychopathology may include either a person's inability to inhibit defense systems in a safe environment or the inability to activate defense systems in a risky environment—or both. Only in a safe environment is it adaptive and appropriate to simultaneously inhibit defense systems and exhibit positive social engagement behavior. Faulty neuroception—that is, an inaccurate assessment of the safety or danger of a situation—might contribute to the maladaptive physiological reactivity and the expression of defensive behaviors associated with specific psychiatric disorders. In typically developing children, however, neuroception detects risk accurately. Children's cognitive awareness of risk matches their "gut response" to danger.
When our nervous system detects safety, our metabolic demands adjust. Stress responses that are associated with fight and flight—such as increases in heart rate and cortisol mediated by the sympathetic nervous system and hypothalamic-pituitary-adrenal axis—are dampened. Similarly, a neuroception of safety keeps us from entering physiological states that are characterized by massive drops in blood pressure and heart rate, fainting, and apnea—states that would support "freezing" and "shutdown" behaviors.
How does the nervous system know when the environment is safe, dangerous, or life-threatening? What neural mechanisms evaluate risk in the environment? New technologies, such as functional magnetic resonance imaging, have identified specific neural structures that are involved in detecting risk. Specific areas of the brain detect and evaluate features, such as body and face movements and vocalizations that contribute to an impression of safety or trustworthiness. Researchers have identified an area in the cortex that becomes activated when we see familiar faces and hear familiar voices. This process of identifying familiar and trustworthy people and evaluating the intentions of others based on "biological movements" of face and limbs seems to be located in the temporal lobe of the cortex. If neuroception identifies a person as safe, then a neural circuit actively inhibits areas of the brain that organize the defensive strategies of fight, flight, and freeze. Slight changes in the biological movements that we see can shift a neuroception from "safe" to "dangerous." When this shift occurs, the neural systems associated with prosocial behavior are disrupted, and the neural systems associated with defensive strategies are triggered.
In the presence of a safe person, then, the active inhibition of the brain areas that control defense strategies provides an opportunity for social behavior to occur spontaneously. Thus, the appearance of a friend or caregiver would subdue the neural circuits in the brain that regulate defensive strategies. As a consequence, closeness, physical contact, and other social engagement behaviors become possible. In contrast, when situations appear risky, the brain circuits that regulate defense strategies are activated. Social approaches are met with aggressive behavior or withdrawal.
IMMOBILIZATION WITHOUT FEAR
As we have seen, humans have three principal defense strategies—fight, flight, and freeze. We are familiar with fight and flight behaviors, but know less about the defense strategy of immobilization, or freezing. This strategy, shared with early vertebrates, is often expressed in mammals as "death feigning." In humans, we observe a behavioral shutdown, frequently accompanied by very weak muscle tone. We also observe physiological changes: heart rate and breathing slow, and blood pressure drops.
Immobilization, or freezing, is one of our species's most ancient mechanisms of defense. Inhibiting movement slows our metabolism (reducing our need for food) and raises our pain threshold. But in addition to freezing defensively, mammals immobilize themselves for essential prosocial activities, including conception, childbirth, nursing, and the establishment of social bonds. For example, when an infant nurses, the mother has to restrain her movements. When a child is embraced, the child is functionally immobilized. Reproductive behaviors also involve a degree of immobilization. However, immobilization with fear elicits profound, potentially lethal, physiological changes (i.e., dramatic slowing of heart rate, cessation of breathing, and dropping of blood pressure). Through the process of evolution, neural circuits in the brain that were originally involved in freezing behaviors were modified to serve intimate social needs. Over time, these brain structures grew receptors for a neuropeptide known as oxytocin. Oxytocin is released during the birth process and nursing. It is also released in the brain during activities that help establish social bonds. Thus, when we sense that our environment is safe, the release of oxytocin allows us to enjoy the comfort of an embrace without fear. But if our nervous system identifies someone as dangerous, although oxytocin may be released, we struggle against the attempted embrace.
SOCIAL ENGAGEMENT: THE PREAMBLE TO A SOCIAL BOND
To develop a social bond, it it not enough to inhibit defense systems. People must also be physically close to each other. This is true whether they are a mother and baby forming an attachment relationship or two adults forming a social bond. There are, of course, major differences between the contexts in which mother–infant attachment and the social bonds of reproductive partners are established. Consider mobility, for example. Due to immature neural development, the baby has limited ability to move either toward or away from the mother. In contrast, two adults who may become reproductive partners are likely to have similar behavioral repertoires.
If the creation of social bonds depended on voluntary motor behaviors, then the human newborn would be greatly disadvantaged: The neural regulation of the spinal motor pathways is immature at the time of birth and takes several years to develop fully. Fortunately, social engagement does not depend on how well we can regulate our limbs and move our bodies. Voluntary limb and trunk movement require neural pathways linking the cortex to spinal nerves (i.e., corticospinal pathways). Social engagement depends, rather, on how well we can regulate the muscles of our faces and heads via pathways linking the cortex with the brainstem (i.e., corticobulbar pathways). These are the muscles that give expression to our faces, allow us to gesture with our heads, put intonation into our voices, direct our gaze, and permit us to distinguish human voices from background sounds. Corticospinal pathways to spinal nerves regulate the muscles that control the trunk and limbs; corticobulbar pathways to cranial nerves regulate the muscles of the face and head. The neural pathways from the cortex to these nerves (i.e., corticobulbar) are myelinated sufficiently at birth to allow the infant to signal a caregiver by vocalizing or grimacing and to engage the social and nutrient aspects of the world by gazing, smiling, and sucking.
The neural regulation of the muscles of the face and head influences how someone perceives the engagement behaviors of others. More specifically, this neural regulation can reduce social distance by allowing humans (including infants) to:
• make eye contact;
• vocalize with an appealing inflection and rhythm;
• display contingent facial expressions; and
• modulate the middle-ear muscles to distinguish the human voice from background sounds more efficiently.
Alternatively, when the tone of these muscles is reduced, which occurs spontaneously in response to a neuroception of danger or a life threat in the external environment (e.g., a dangerous person or situation) or the internal environment (e.g., fever, pain, or physical illness) environment:
• the eyelids droop;
• the voice loses inflection;
• positive facial expressions dwindle;
• awareness of the sound of the human voice becomes less acute; and
• sensitivity to others' social engagement behaviors decreases.
It is important to remember that neuroception of danger or a threat to life can occur with respect to the external environment (e.g., a dangerous person or situation) or the internal environment (e.g., fever, pain, or physical illness). Even flat (rather than angry) facial affect might prompt a neuroception of danger or fear and disrupt the development of normal spontaneous interactive and reciprocal social engagements. For example, the flat affect of a depressed parent or the flat affect of an ill child might trigger a transactional spiral that results in compromised emotional regulation and limited spontaneous social engagement.
POLYVAGAL THEORY: THREE NEURAL CIRCUITS THAT REGULATE REACTIVITY
Where do humans' intricate neurobehavioral systems for prosocial and defensive behaviors come from? As we have suggested earlier, mammals—including humans—must distinguish friend from foe, evaluate the safety of the environment, and communicate with their social unit. According to the polyvagal theory (see chapters 2, 5, 10, and 11; Porges, 2001a), mammals—especially primates—have evolved brain structures that regulate both social and defensive behaviors. In other words, evolutionary forces have molded both human physiology and human behavior. As the vertebrate nervous system became more complex during the course of evolution, its affective and behavioral repertoire expanded. A product of this phylogenetic process is a nervous system that provides humans with the ability to express emotions, communicate, and regulate bodily and behavioral states.
The polyvagal theory links the evolution of the neural regulation of the heart to affective experience, emotional expression, facial gestures, vocal communication, and social behavior that is responsive to the behavior of others. The theory points out that the neural control of the heart is neuroanatomically linked to the neural control the muscles of the face and head.
The polyvagal theory describes three stages in the development of a mammal's autonomic nervous system. Each of the three major adaptive behavioral strategies is supported by a distinct neural circuit involving the autonomic nervous system.
1. Immobilization
• Feigning death, behavioral shutdown.
• The most primitive component, shared with most vertebrates.
• Dependent on the oldest branch of the vagus nerve (an unmyelinated portion originating in an area of the brainstem known as the dorsal motor nucleus of the vagus).
2. Mobilization
• Fight-or-flight behaviors.
• Dependent on the functioning of the sympathetic nervous system, a system associated with increasing metabolic activity and increasing cardiac output (e.g., faster heart rate, greater ability of the heart to contract).
3. Social communication or social engagement
• Facial expression, vocalization, listening.
• Dependent on the myelinated vagus, which originates in an area of the brainstem known as the nucleus ambiguus. The myelinated vagus fosters calm behavioral states by inhibiting the influence of the sympathetic nervous system on the heart.
Infants, young children, and adults need appropriate social engagement strategies in order to form positive attachments and social bonds. At the University of Illinois at Chicago, we have been developing a model that links social engagement to attachment and the formation of social bonds through the following steps.
1. Three well-defined neural circuits support social engagement behaviors, mobilization, and immobilization.
2. Independent of conscious awareness, the nervous system evaluates risk in the environment and regulates the expression of adaptive behavior to match the neuroception of an environment that is safe, dangerous, or life-threatening.
3. A neuroception of safety is necessary before social engagement behaviors can occur. These behaviors are accompanied by the benefits of the physiological states, associated with social support.
4. Social behaviors associated with nursing, reproduction, and the formation of strong pair bonds requires immobilization without fear.
5. Oxytocin, a neuropeptide involved in the formation of social bonds, makes immobilization without fear possible by blocking defensive freezing behaviors.
NEUROCEPTION AND MENTAL HEALTH DISORDERS
So far, we have been discussing neuroception that works. Ideally, a baby's neuroception of her environment shows her a safe place to explore. But even if her neuroception warns her—accurately—of danger from a "frightened or frightening" caregiver, the baby can take some defensive measures, even though they are likely to be ineffective and are almost certain to be psychologically costly. What happens when neuroception itself is impaired? From a theoretical perspective, faulty neuroception—that is, an inability to detect accurately whether the environment is safe or another person is trustworthy—might lie at the root of several psychiatric disorders.
• Areas in the temporal cortex that are assumed to inhibit fight, flight, or freeze reactions are not activated in people with autism or schizophrenia, who have difficulty with social engagement.
• Individuals with anxiety disorders and depression have compromised social behavior; difficulties in regulating heart rate, as reflected in measures of vagal control of the heart and reduced facial expressiveness.
• Maltreated and institutionalized children with reactive attachment disorder tend to be either inhibited (emotionally withdrawn and unresponsive) or uninhibited (indiscriminate in their attachment behavior; Zeanah, 2000). Both types of behavior suggest faulty neuroception of the risk in the environment.
Recent research on children raised in Romanian orphanages has stimulated interest in reactive attachment disorders and in finding ways to remediate the devastating disturbances in their social development. If the behavior of these children suggests faulty neuroception of risk in the environment, are there features in the environment that might help the children feel safer and then begin to move toward more normal social behavior?
A study of Romanian toddlers being raised in an orphanage (Smyke, Dumitrescu, & Zeanah, 2002) illustrates the usefulness of the construct of neuroception in understanding the development of normal and atypical attachment behaviors. Researchers evaluated two groups of institutionalized children and compared them to children who had never been institutionalized. One group of institutionalized children (the standard unit) was cared for according to prevailing standards: 20 different caregivers worked rotating shifts, with approximately 3 caregivers for 30 children on each shift. A second group of children, the pilot unit, consisted of 10 children with 4 caregivers. If we apply our concept of neuroception to this study, we would hypothesize that familiar caregivers would be essential to children's neuroception of safety—which, in turn, would be essential for the promotion of appropriate social behavior. Specifically, a child's ability to recognize a caregiver's face, voice, and movements (the features that define a safe and trustworthy person) should set in motion the process of subduing the limbic system and allowing the social engagement system to function.
The data from the Smyke et al. (2002) study supports our hypothesis. The higher the number of caregivers children had contact with, the higher the incidence of reactive attachment disorder among these children. The standard-unit children were more likely than the other two groups to have reactive attachment disorder. On some indices of reactive attachment disorder, the pilot-group children did not differ from the children who had never been institutionalized. These findings suggest that once we understand the contextual and social features that inhibit the neural circuits which mediate defensive behavioral strategies, we can "optimize" the development of prosocial behavior.
At the University of Illinois at Chicago, we are using a newly developed biologically based behavioral intervention based on principles derived from the polyvagal theory. We are testing this approach with children with autism and individuals with language and social communication problems. Our model assumes that for many children with social communication deficits, including those diagnosed with autism, the social engagement system is neuroanatomically and neurophysiologically intact. Yet these children do not engage in voluntary prosocial behaviors. To improve spontaneous social behavior, we have reasoned, an intervention must stimulate the neural circuits that regulate the muscles of the face and head. The polyvagal theory predicts that once the cortical regulation of the brainstem structures involved in the social engagement are activated, social behavior and communication will spontaneously occur as the natural emergent properties of this biological system. The intervention "stimulates" and "exercises" the neural pathways involved in listening and simultaneously stimulates the function of other aspects of the social engagement system. The intervention provides acoustic stimulation that has been computer altered to systematically modulate the neural regulation of the middle-ear muscles. Theoretically, the middle-ear muscles need to be regulated during listening, and the nerves that regulate these muscles are linked to the nerves that regulate the other muscles of the face and head involved in social engagement. Preliminary results are promising. They suggest that interventions designed to improve spontaneous social behavior should: (1) ensure that the context elicits in participants a neuroception of safety that will allow the social engagement system to function; and (2) exercise the neural regulation of the social engagement system.
CONCLUSIONS
According to the polyvagal theory (including the concept of neuroception), our range of social behavior is limited by our human physiology, which has evolved from that of more primitive vertebrates. When we are frightened, we are dependent on the neural circuits that evolved to provide adaptive defensive behaviors for more primitive vertebrates. These neural circuits provide physiological mechanisms that reflexively organize mobilization or immobilization behaviors before we are consciously aware of what is happening. When, on the other hand, neuroception tells us that an environment is safe and that the people in this environment are trustworthy, our mechanisms of defense are disabled. We can then behave in ways that encourage social engagement and positive attachment.
Focusing on biologically based behaviors common to all humans allows practitioners to imagine new intervention paradigms to help children whose social behavior and attachment are compromised. We can alter the caregiving environment so that it will appear—and be—safer for children and less likely to evoke mobilization or immobilization responses. We can also intervene directly with children, exercising the neural regulation of brainstem structures, stimulating the neural regulation of the social engagement system, and encouraging positive social behavior.
CHAPTER 2
Orienting in a Defensive World: Mammalian Modifications of Our Evolutionary Heritage
A Polyvagal Theory
The systematic investigation of mind–body relations forms the scientific basis for the science of psychophysiology. Unlike the correlative view of mind–body evaluations that dominates psychology and psychiatry, psychophysiology emphasizes a continuity between neurophysiological and psychological processing. Psychophysiologists assume that the nervous system provides the functional units for the bidirectional transduction of psychological and physiological processes. Thus, from a psychophysiological perspective, it is possible to link psychological processes with neurophysiological processes and brain structures by measurement and not just theory.
This chapter will focus on neural regulation of the heart by the vagus and how this regulation evolved to facilitate specific psychological processes. The polyvagal theory, as described in this chapter, provides an explanation of how the vagal pathways regulate heart rate in response to novelty and to a variety of stressors. The theory proposes that through evolution mammals developed two vagal systems: a phylogenetic relic of amphibians and reptiles and an evolutionary modification unique to mammals. According to the polyvagal theory, the two vagal systems are programmed with different response strategies and may respond in a contradictory manner. Explanations for several psychophysiological phenomena and psychosomatic disturbances will be proposed. The theory is based on an established literature in neurophysiology, neuroanatomy, and psychophysiology.
AROUSAL THEORY: HISTORICAL LEGACY
Early psychophysiological research assumed that peripheral autonomic measures provided sensitive indicators of arousal or activation (Darrow, Jost, Solomon, & Mergener, 1942; Duffy, 1957; Lindsley, 1951; Malmo, 1959). This view was based on a rudimentary understanding of the autonomic nervous system in which changes in electrodermal activity and heart rate were assumed to be accurate indicators of sympathetic activity. As the activation-arousal theory developed, a continuity between peripheral autonomic responses and central mechanisms was assumed. According to this assumption, any organ influenced by sympathetic efferent fibers, such as the sudomotor, vascular, or cardiac systems, was a potential indicator of limbic or cortical activity.
Although the specific pathways relating these various levels were never outlined and are still sketchy, electrodermal and heart rate measures became the primary focus of research during the early history of the Society for Psychophysiological Research. This was due to their presumed sympathetic innervation and, in part, to their measurement availability. By default, this emphasis created a research environment that neglected several important factors: (a) parasympathetic influences, (b) interactions between sympathetic and parasympathetic processes, (c) peripheral autonomic afferents, (d) central regulatory structures, (e) the adaptive and dynamic nature of the autonomic nervous system, and (f) phylogenetic and ontogenetic differences in structural organization and function.
The neglect of these concepts and an emphasis on a global construct of arousal still abide within various subdisciplines of psychology, psychiatry, and physiology. This outdated view of arousal may restrict an understanding of how the autonomic nervous system interfaces with the environment and the contribution of the autonomic nervous system to psychological and behavioral processes. In contrast, more recent neurophysiological data promote a more integrative view of the autonomic nervous system.
BRAIN-HEART COMMUNICATION: HISTORICAL PERSPECTIVE
When we view living organisms as a collection of dynamic, adaptive, interactive, and interdependent physiological systems, it is no longer appropriate to treat the autonomic nervous system as functionally distinct from the central nervous system. We start to recognize that peripheral organs do not "float in a visceral sea." Rather, they are anchored to central structures by means of efferent pathways and are continuously signaling central regulatory structures along their abundant afferent pathways. Thus, the bidirectional connections between autonomic and central brain structures are becoming apparent. Accordingly, new theories and research strategies must incorporate the dynamic and interactive constructs that link central structures with peripheral organs.
Darwin (1872) provided historical insight into the potential importance of the vagus in bidirectional communication between the brain and the heart. Although Darwin focused on facial expressions in defining emotions, he acknowledged the dynamic relationship between the vagus and the central nervous system activity that accompanied the spontaneous expression of emotions. He speculated that there were identifiable neural pathways that provided the necessary communication between specific brain structures and peripheral organs to promote the unique pattern of autonomic activity associated with emotions. For example:
when the mind is strongly excited, we might expect that it would instantly affect in a direct manner the heart; and this is universally acknowledged . . . when the heart is affected it reacts on the brain; and the state of the brain again reacts through the pneuma-gastric [vagus] nerve on the heart; so that under any excitement there will be much mutual action and reaction between these, the two most important organs of the body. (p. 69)
For Darwin, when an emotional state occurred, the beating of the heart changed instantly, the change in cardiac activity influenced brain activity, and the brainstem structures through the cranial nerves (i.e., vagus) stimulated the heart. He did not elucidate the neurophysiological mechanisms that translate the initial emotional expression to the heart. Our current knowledge of the origin of the brainstem and the neurophysiological function of the various branches of the vagus was not available to Darwin. At that time, it was not known that vagal fibers originate in several medullary nuclei and that the branches of the vagus exert control over the periphery through different feedback systems. However, Darwin's statement is important, because it emphasizes the afferent feedback from the heart to the brain, independent of the spinal cord and the sympathetic nervous system, and the regulatory role of the pneumogastric nerve (renamed the vagus at the end of the 19th century) in the expression of emotions.
Darwin attributed these ideas to Claude Bernard as an example of nervous system regulation of le milieu interieur. Consistent with more contemporary psychophysiology, Claude Bernard viewed the heart as a primary response system capable of responding to all forms of sensory stimulation. He explicitly emphasized the potency of central nervous system pathways to the heart (Cournand, 1979). These ideas are expressed in the following quotation (Claude Bernard [1865] quoted in Cournand, 1979):
In man the heart is not only the central organ of circulation of blood, it is a center influenced by all sensory influences. They may be transmitted from the periphery through the spinal cord, from the organs through the sympathetic nervous system, or from the central nervous system itself. In fact the sensory stimuli coming from the brain exhibit their strongest effects on the heart. (p. 118)
Although seldom acknowledged as founders of modern psychophysiology, Bernard and Darwin have contributed to the theoretical basis for a neuropsychophysiology of the autonomic nervous system. The quotations document their view that the heart provided not only an output system from the brain, capable of indexing sensory processing, but also that the heart was a source of afferent stimulation to the brain, able to change or contribute to psychological state. Consistent with this theoretical bias, psychophysiologists during the past century have investigated the functional sensitivity of heart rate measures to sensory and affective stimuli (e.g., Darrow, 1929; Graham & Clifton, 1966; Lacey, 1967) and the dynamic feedback between the brain and the heart in regulating both psychological state and the threshold for sensory stimuli (e.g., Lacey & Lacey, 1978).
Contemporary psychophysiology gained much of its current theoretical perspective from intriguing ideas about the interaction between autonomic and sensory processes introduced by Sokolov (1963). The Sokolov model contained all the requisite components of an integrative theory relating autonomic function to psychological state. The model included (a) an acknowledgment of both afferents and efferents in both autonomic and somatic systems, (b) an autonomic feedback loop (i.e., autonomic tuning) to regulate sensory thresholds, (c) an interface between autonomic processes and psychological phenomena (i.e., orienting and defensive reflexes), and (d) brain regulation of autonomic reactivity by habituation.
The Sokolov model included bidirectional communication between brain and periphery. In the Sokolov model, autonomic processes contributed to the tuning of receptor systems to engage or disengage with the external environment. Consistent with the Sokolov view, the Laceys (e.g., Lacey, 1967; Lacey & Lacey, 1978) emphasized the bidirectional communication between the cardiovascular system and brain in the regulation of cardiac function and sensory threshold. In contrast to this emphasis on bidirectional communication, Obrist (1976) focused on the general concordance between metabolic demands and heart rate. Both arguments have merit. For example, afferent stimulation of the baroreceptors has immediate effects on both peripheral cardiovascular function and central arousal state (Gellhorn, 1964), and the metabolic demands associated with exercise have deterministic influences, through vagal withdrawal, on heart rate (Obrist, 1981; Rowell, 1993).
HEART RATE RESPONSES: A NEUROGENIC EMPHASIS
Throughout the history of the Society for Psychophysiological Research, psychophysiologists have been studying robust phenomena such as the autonomic components of the orienting reflex, often without explanatory neurophysiological models. This chapter is in response to this need in providing a theoretical model based on the evolution of neural structures and the neural regulation of autonomic processes to explain several psychophysiological phenomena, including orientation, attention, and emotion.
The orienting reflex provides an excellent point of embarkation. Based on the convergent theoretical approaches of Sokolov (1963), Lacey (1967), and Graham and Clifton (1966), the orienting reflex is assumed to have a cardiac component. This component is characterized by a heart rate deceleration that functionally influences perceptual thresholds in facilitating the processing of information about the external environment. However, what are the neural mechanisms mediating the cardiac orienting response? Or, as Obrist (1976) argued, is heart rate deceleration merely an epiphenomenon associated with decreased metabolic demands accompanying the reduced motor activity that defines orienting and attending behaviors? The time course of the response, the effects of neural blockades, and studies with clinical populations support the contention that the cardiac orienting response is neurogenic. First, heart rate deceleration associated with the cardiac orienting response is rapid, occurring within a few seconds, and usually returns rapidly to baseline. Second, the latency characteristics of the cardiac orienting response are similar to other neurogenic bradycardic reflexes such as optovagal, vasovagal, baroreceptor-vagal, and chemoreceptor-vagal.
Blockade studies with atropine demonstrate that short latency bradycardia associated with both orienting reflexes and classical conditioning are mediated by cholinergic pathways along the vagus (e.g., Berntson, Cacioppo, & Quigley, 1994; Obrist, 1981; Schneiderman, 1974). Studies with the aged and other clinical populations with peripheral neuropathies or autonomic regulatory problems (e.g., diabetes) document deficits in vagal function (De Meersman, 1993; Gribben, Pickering, Sleight, & Peto, 1971; Weiling, van Brederode, de Rijk, Borst, & Dunning, 1982; Weise & Heydenreich, 1991). In addition, studies of individuals with unilateral brain damage demonstrate that heart rate responses are diminished more in individuals with right-side damage (Yokoyama, Jennings, Ackles, Hood, & Boller, 1987). This latter finding is consistent with evidence that neurophysiological regulation of heart rate is primarily along the right vagus to the sinoatrial node and that heart rate is under the control of higher ipsilateral structures in the brain (Warwick & Williams, 1975). Although vagal influences producing heart rate deceleration may interact synergistically with sympathetic withdrawal, short latency decelerations are determined primarily by the vagus. Thus, it may be argued that, because short latency heart rate reactivity is mediated by the vagus, the magnitude of the cardiac orienting response is an index of vagal regulation.
THE VAGAL PARADOX
In attempting to structure a neurogenic model of vagal regulation to explain psychophysiological phenomena, there is an obvious inconsistency between data and theory. Physiological theory attributes the chronotropic control of the heart (i.e., heart rate) and the amplitude of respiratory sinus arrhythmia (RSA) to direct vagal mechanisms (e.g., Jordan, Khalid, Schneiderman, & Spyer, 1982; Katona & Jih, 1975). However, although there are situations in which both measures covary (e.g., during exercise and cholinergic blockade), there are other situations in which the measures appear to reflect independent sources of neural control.
Several arguments have been made to explain this discrepancy. First, it has been argued that RSA and average heart rate (during sympathetic blockade) reflect different dimensions of vagal activity. For example, average heart rate might be viewed as reflecting tonic vagal influences and RSA as reflecting phasic vagal influences (e.g., Berntson, Cacioppo, & Quigley, 1993b; Jennings & McKnight, 1994; Malik & Camm, 1993). Second, it has been argued that the discrepancy is caused by variations in respiratory parameters (Grossman, Karemaker, & Wieling, 1991), with RSA being confounded by respiratory frequency and tidal volume. Third, it has been argued that variation in quantification methods may contribute to the divergence between RSA and heart rate (Byrne & Porges, 1993; Porges & Bohrer, 1990). Fourth, it has been argued that RSA does not reliably measure parasympathetic tone because it decreases with baroreflex stimulation (Goldberger, Ahmed, Parker, & Kadish, 1994). And fifth, it has been argued that average heart rate is influenced by a complex and dynamic interaction between sympathetic and vagal systems, making it difficult to extract a vagal tone dimension (Berntson, Cacioppo, & Quigley, 1991, 1993a).
The arguments often have been linked to a definition of vagal tone as being determined by neural blockade. The functional effect of the neural blockade on heart rate has been used as the criterion measure of vagal tone or parasympathetic control (e.g., Katona & Jih, 1975). Researchers have argued that RSA is not an accurate index of vagal tone, because individual preblockade levels of RSA do not accurately map into pre/post-change in heart rate (Grossman & Kollai, 1993). Contrary to this argument, Porges (1986) argued that the discrepancy was in part based on the criterion measure selected. He demonstrated that RSA exhibited a more sensitive dose-dependent response curve to vagal blockade with atropine than to heart rate. This suggests the possibility that RSA, monitored during periods of spontaneous breathing, may provide a better criterion variable than heart rate. Neurophysiological support may be offered for this proposal. RSA is a vagal phenomenon in contrast to heart rate, which is determined by several sources, including vagal, sympathetic, and mechanical factors. Thus, the efficacy of change in heart rate following cholinergic blockade as an index of vagal tone may be challenged.
These arguments have created a volatile environment for debating the neurophysiological interpretation of RSA and the efficacy of specific methods to quantify RSA. Common to these arguments is the assumption that there is one central source of cardiac vagal tone. The arguments attribute differences, not to central mechanisms, but to the response characteristics of heart rate and RSA. Thus, divergence has been attributed to either the transfer function of the sinoatrial node that would attenuate high-frequency oscillations (Saul, Berger, Chen, & Cohen, 1989) or the statistical transfer function of the method of quantifying RSA (Byrne & Porges, 1993) and not as a function of differential neural output.
However, independent of the quantification methodology and during periods of stable respiratory parameters, data have accumulated that demonstrate that RSA and heart rate (independent of sympathetic influences) often respond differently. Although both the neurogenic bradycardia and the suppression of RSA or heart rate variability observed during attention are assumed to be vagal in origin, they often appear independent of each other or in an apparent physiological contradiction (Porges, 1972; Porges & Raskin, 1969; Richards & Casey, 1991). Similar disparities between levels of heart rate and RSA have been observed during inhalant anesthesia when RSA exhibits a massive depression, whereas heart rate is not altered (Donchin, Feld, & Porges, 1985). Additional examples of convergence and divergence between RSA and heart rate can be observed in both within- and between-subjects designs. For example, individual differences in heart rate and RSA monitored during resting conditions provide independent contributions to measures of cardiac vagal tone derived from vagal blockade (e.g., Grossman & Kollai, 1993). However, convergence may be observed within an individual during exercise when monotonic increases in metabolic load are reflected in both faster heart rate and lower RSA (Billman & DuJardin, 1990) or during neural blockade with atropine when both cardiac indices diminish in a dose-response manner (Cacioppo et al., 1994; Dellinger, Taylor, & Porges, 1987; Porges, 1986).
The relationship between RSA and heart rate may change within and between individuals. In our laboratory, we have observed that the relationship between RSA and heart rate varies with behavioral state (Riniolo, Doussard-Roosevelt, & Porges, 1994). Twenty-four-hour ambulatory monitoring of adults indicates that during states of drowsiness and sleep the correlation between RSA and heart rate is significantly lower than during alert states. Thus, at times RSA and heart rate appear to reflect the same physiological processes, whereas at other times they appear to reflect independent processes.
In contrast to the observable data, neurophysiological research argues for a covariation between these two parameters, because vagal cardioinhibitory fibers to the heart have consistent functional properties characterized by bradycardia to neural stimulation and a respiratory rhythm (e.g., Jordan et al., 1982). This inconsistency, based on an assumption of a single central vagal source, is labeled the vagal paradox and is outlined in Table 2.1.
The vagal paradox is critical to the interpretation of several psychophysiological and clinical conditions. For example, if the bradycardia occurring during orienting reflexes is vagal, then why is bradycardia often observed during periods of reduced RSA, also an index of both attention and vagal control of the heart? If vagal tone is a positive indicator of health of a fetus or neonate when monitored with RSA, then why is vagal tone a negative indicator of health when it is manifested as bradycardia? If bradycardia and RSA can both be removed by severing the vagus or by pharmacological blockade, then are they both manifestations of vagal tone? If bradycardia and RSA are both indices of vagal tone, then why do they respond differently? This apparent paradox provides the stimulus for the following inquiry and the development of the proposed polyvagal theory that speculates that, in mammals, there are two anatomically based vagal response systems.
TABLE 2.1. The Vagal Paradox: A Common Central Source?
1. Increased vagal tone produces neurogenic bradycardia.
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2. Decreased vagal tone produces suppression of RSA.
3. Bradycardia occur during periods of suppressed RSA.
MAMMALIAN POLYVAGAL SYSTEM
To understand the proposed polyvagal theory, it is necessary to provide additional information regarding the neuroanatomy and neurophysiology of the vagus in mammals. First, the vagus is not one nerve but a family of neural pathways originating in several areas of the brainstem. Second, there are several branches of the vagus. Third, the vagus is not solely an efferent or motor pathway; rather, approximately 80% of the vagal fibers are afferent (Agostoni, Chinnock, DeBurgh Daly, & Murray, 1957). Fourth, the vagus is lateralized with nerve trunks originating in the left and right sides of the brainstem. Fifth, the vagus is asymmetrical, with the left and right sides performing different tasks, with the right vagus most potent in the chronotropic regulation of the heart. These points are summarized in Table 2.2.
Mammals are polyvagal. The different vagi have different roles in the regulation of visceral function and originate in different brainstem nuclei with their respective viscerotropic organization. The different vagi may have oppositional outputs to the same target organ. For example, it is possible that during orienting there is an increase in vagal outflow from one branch to produce bradycardia and a withdrawal of vagal outflow from another branch to produce a suppression of RSA (e.g., Richards & Casey, 1991). Thus, the concept of vagal tone may not be generalized to all vagal efferent pathways or even to the same target organ (e.g., heart), as has been assumed (e.g., Grossman & Kollai, 1993), but may need to be limited to a specific branch or subsystem of the vagus being evaluated. And the intriguing concept of autonomic space proposed by Berntson et al. (1991, 1993a) to deal with dynamic sympathetic-parasympathetic interactions may require an additional dimension to deal with potential vagovagal interactions.
The polyvagal theory proposes that neurogenic bradycardia and RSA are mediated by separate branches of the vagus. Thus, the two commonly used, but not interchangeable, measures of cardiac vagal tone may represent different dimensions of vagal tone.
In mammals, the primary motor fibers of the vagus originate from two separate and definable nuclei in the medulla: the dorsal motor nucleus of the vagus (DMNX) and the nucleus ambiguus (NA). The DMNX is in the dorsomedial medulla. The NA is ventral to the DMNX in the ventrolateral reticular formation (Warwick & Williams, 1975). The name ambiguus emphasizes the initial difficulties associated with determining its borders and connections within the reticular formation (Mitchell & Warwick, 1955). A third medullary nucleus, located near the DMNX, the nucleus tractus solitarius (NTS), is the terminus of many of the afferent pathways traveling through the vagus from peripheral organs. This trinity of neural structures in the medulla forms the primary central regulatory component of the vagal system. The relative locations of these medullary nuclei are illustrated in Figure 2.1.
TABLE 2.2. Mammalian Polyvagal System
1. Efferent fibers originate primarily in two medullary nuclei (NA, DMNX).
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2. Vagal efferent fibers are clustered into several branches.
3. Approximately 80% of vagal fibers are afferent.
4. The vagus is lateralized.
5. The vagus is asymmetrical with a right bias.
Most cells originating in the DMNX project to subdiaphragmatic structures (e.g., stomach, intestines, etc.). In contrast, only the rostral portion of the NA provides vagal innervation of subdiaphragmatic structures (Kalia & Masulam, 1980), whereas most cells in NA project to supradiaphragmatic structures (larynx, pharynx, soft palate, esophagus, bronchi, and heart).
Neurotracing and electrophysiological techniques with mammals provide additional evidence that the two vagal nuclei may function independently and have different central connections. These studies have demonstrated that there are no apparent connections between the two nuclei, although both nuclei have input from the NTS, central nucleus of the amygdala, and hypothalamus (Hopkins, 1987; Leslie, Reynolds, & Lawes, 1992). It is well accepted that in mammals the primary cardioinhibitory motoneurons are located in the NA. However, motor fibers from the DMNX join the cardiac vagus (Bennett, Ford, Kidd, & McWilliam, 1984).
Cardioinhibitory and bronchoconstrictor neurons located in the NA have myelinated vagal axons that conduct in the fast B fiber range (McAllen & Spyer, 1976, 1978). In contrast, neurons located in the DMNX have axons projecting to the cardiac vagal branches that are nonmyelinated and conduct in the slower C fiber range. Although there are reports of cardioinhibitory vagal neurons with efferent axons conducting in the B fiber range being located in both the DMNX and the NA, neurons with axons conducting in the C fiber range are restricted to the DMNX (Jordan et al., 1982). The role of these nonmyelinated vagal fibers on the heart is not well understood. In research with cats (Ford, Bennett, Kidd, & McWilliam, 1990) and dogs (Donald, Samueloff, & Ferguson, 1967), stimulation of these fibers did not affect heart rate. However, although unsubstantiated at this time, the function of these fibers may be dependent on the outflow of the myelinated NA fibers and may change during conditions such as hypoxia. For example, the influence of the unmyelinated fibers on the heart may be potentiated when the outflow from the myelinated NA fibers is blocked. In contrast, in the rabbit, stimulation of the nonmyelinated vagal fibers results in heart rate slowing (Woolley, McWilliam, Ford, & Clarke, 1987).
FIGURE 2.1. Primary brainstem nuclei of the vagus. Nuclei are bilateral, and only one of each bilateral pair is illustrated.
The cytoarchitecture of the NA illustrates that the dorsal portion contains source nuclei for special visceral efferents (i.e., voluntary motor fibers) and that the ventral portion contains source nuclei for general visceral efferents (i.e., involuntary motor fibers). Motor projections from the dorsal portion go to target organs including the larynx, pharynx, soft palate, and esophagus. Motor projections from the ventral portion go to several target organs, including the heart and the bronchi. In fact, these projections account for the primary cardiac and bronchomotor pathways and far outnumber the pathways originating in the DMNX.
There is an obvious distinction between the viscerotropic organization of the two vagal nuclei. The DMNX provides the primary vagal efferents to subdiaphragmatic organs that regulate digestive and alimentary processes. In contrast, the NA provides the primary vagal efferents to the supradiaphragmatic target organs, including the soft palate, pharynx, larynx, esophagus, bronchi, and heart.
THE POLYVAGAL THEORY
The polyvagal theory is based on several premises. Some are firmly grounded in neurophysiological and neuroanatomical data and others are more speculative. The first premise articulates the neural regulation of bradycardia and RSA. Based on the initial premise, it is hypothesized that the neurogenic bradycardia associated with the orienting reflex are mediated by the DMNX and that the suppression of heart rate variability (i.e., reduced amplitude of RSA) is mediated by the NA.
Premise 1: Neurogenic bradycardia and RSA are mediated by different branches of the vagus and need not respond in concert.
Physiological support for the hypothesis that the DMNX can contribute to neurogenic bradycardia, independent of the NA, is provided by lesion studies. Machado and Brody (1988) reported that chronic bilateral lesions of the NA reduced but did not totally block baroreceptor reflex-mediated bradycardia in conscious rats. Thus, the DMNX contains vagal neurons capable of producing bradycardia with a response latency associated with the baroreceptor reflex. This is supported by Jerrell, Gentile, McCabe, and Schneiderman (1986), who argued that differential Pavlovian conditioning of bradycardia in rabbits following sinoaortic denervation was mediated along DMNX pathways. The results pose the possibility that vagal pathways, originating in both the DMNX and the NA, have the potential to influence heart rate.
Phylogenetic Development of the Polyvagal System
Investigations of the phylogenetic development of the vagus provide support for the first premise. Because our interests are in mammals, specifically humans, this chapter will focus on the evolution of vagal regulation of cardiac function from reptiles to mammals. There are two questions: Do reptiles produce heart rate patterns during orienting that are similar to the neurogenic bradycardia observed in mammals? Do reptiles produce a phenomenon similar to RSA?
The phylogeny of the vagus illustrates two phenomena: one neuroanatomical and the other physiological. On a neuroanatomical level, differentiation of the visceral efferent column of the vagus into dorsal motor nucleus (i.e., the DMNX) and a ventrolateral motor nucleus (i.e., the NA) is first seen in reptiles. In turtles (e.g., Chelone mydas and Domonia subtrijuga), there is still a connection between the two nuclei, but in lizards (e.g., Varanus salvator) and crocodiles (e.g., Caiman crocodilus), the separation between the DMNX and the NA is as complete as it is in mammals (see Barbas-Henry & Lohman, 1984).
Behavioral orienting in reptiles is characterized by a focusing of exteroceptors and a freezing of gross motor activity. Paralleling these behaviors, neurogenic bradycardia has been observed. Belkin (see Regal, 1978) reported that bradycardia is part of a fear response in iguanas. In addition, McDonald (1974) reported bradycardia in the hog-nosed snake during death-feigning. Most researchers found these data incompatible with the prevalent emphasis on arousal and the use of heart rate as an indicator of arousal. How could bradycardia reflect increased arousal within the context of a sympathetic nervous system–oriented arousal theory? In contrast, RSA has not been observed in reptiles. Research investigating the spectral components of reptilian heart rate has failed to identify heart rate oscillations associated with ventilation (Gonzalez Gonzalez & de Vera Porcell, 1988).
Phylogenetic development not only illustrates changes in the neuroanatomy of the vagus but also parallels changes in behavior. One of these behavioral shifts is the addition of active or voluntary attention and complex emotions. In confronting the defensive world, mammals, like reptiles, have an initial reflexive response to novelty, the orienting reflex. However, mammals have additional behaviors in their repertoire. Following or independent of reflexive orienting, mammals may voluntarily respond with sustained attention to foster detailed information processing or with facial expressions and vocalizations to foster communication. Thus, reptiles orient; mammals may first orient and then elect to attend or communicate.
The differences between the reptilian and mammalian cardiac systems provide insight into the phylogenetic origin of behaviors such as reptilian orienting and mammalian attention and emotion. The cardiac output and thus energy production of mammals far exceeds that of reptiles. Mammals have metabolic demands four to five times that of reptiles. The metaphor of a machine or vehicle has been proposed by Else and Hulbert (1981) to compare the efficiency and function of the mammalian and reptilian metabolic systems. According to Else and Hulbert (1981), when idling, the average mammal requires four to five times more fuel than the average idling reptile, even when body weight and ambient temperature are controlled. Elaborating on this metaphor, reptiles represent vehicles with 1-L engines and mammals represent vehicles with 4- or 5-L engines. Thus, as in the story about the race between the tortoise and the hare, reptiles locomote with a reliable but underpowered engine and mammals locomote with a supercharged engine that can function for only short periods without refueling.
The energy production capacities of reptiles and mammals contribute to their respective ways of life. There is a bias among reptiles toward passive feeding strategies. Reptiles tend to be sit-and-wait feeders, slow cruisers, and sluggish browsers. In contrast, mammals with four-chambered hearts can actively hunt and graze and adapt to changing environments (Regal, 1978). To support their behavioral niche and to ensure their adaptive success, reptiles and mammals use different vagal strategies to promote their survival. Being underpowered, reptiles do not maintain a vagal brake on the heart, which would further reduce energy production during unchallenged situations. For reptiles, during periods of either quiescence or apnea, usually associated with behavioral freezing or diving, vagal influences via the DMNX are profound and heart rate is even slower. In contrast, vagal control of the heart is virtually removed during periods of breathing and other motor activities (Jacob & McDonald, 1976).
The underpowered reptiles use vagal efferents from the DMNX to the heart to deal with specific challenges: to orient and freeze in response to predator or prey and to conserve oxygen while submerged for lengthy periods. In contrast to underpowered reptiles, supercharged mammals use vagal efferents from the NA as a persistent brake to inhibit the metabolic potential of this high-powered system. The high NA vagal tone keeps mammals from, literally, bouncing off the walls. Thus, in contrast to that observed in reptiles, in mammals vagal tone is highest during unchallenged situations such as sleep, and vagal tone is actively withdrawn in response to external demands, including metabolically demanding states such as exercise, stress, attention, and information processing. For example, in humans, psychological states perceived as life threatening, such as panic and rage, are characterized by virtually no NA vagal tone when indexed with the amplitude of RSA (George et al., 1989). Metaphorically, and consistent with the model, antisocial and pathological behavioral patterns associated with rage and hyperreactivity without conscious self-regulation have been labeled "reptilian."
If terrestrial mammals adopted the reptilian strategy of reflexive increases in vagal activity to produce massive neurogenic bradycardia, the result would be catastrophic to the oxygen-hungry mammalian cortex and myocardium. This strategy would rapidly produce cardiac ischemia and cortical anoxia. The result of this sequence would be death. Although still dependent on oxygen, aquatic mammals use a diving reflex characterized by a regulated neurogenic bradycardia to reduce metabolic demands. To survive, aquatic mammals have complex mechanisms, not available to terrestrial mammals, to manage oxygen resources and shift priorities for oxygen while submerged for long periods.
It is possible that for mammals during states of stress, when metabolic demands are great and vagal tone from the NA is removed, that the cardiac pacemakers (SA and AV) may be prone to neurogenic bradycardia mediated by the DMNX. The neurogenic bradycardia may be massive and lethal. This may be the case in fetal distress, when bradycardia is observed during hypoxic episodes or as a factor in either sudden infant death syndrome or sudden death in adults. Consistent with this model, it has been demonstrated in the dog that progressive asphyxic hypoxia not only elicits increased cardiac vagal activity but also the sensitivity of the sinoatrial node to vagal efferent influences is potentiated (Potter & McCloskey, 1986). Thus, during hypoxia, large bradycardia may be maintained with limited or reduced vagal efferent activity.
The polyvagal theory provides a potential explanation for the massive neurogenic bradycardia observed during fetal distress and in high-risk neonates who have virtually no observable RSA. For example, as illustrated in Figure 2.2, when massive bradycardia is observed during fetal distress (Figure 2.2a), there is a background of low beat-to-beat variability (Figure 2.2b). Similarly, neonates with the lowest amplitude RSA are at greatest risk for apnea and bradycardia (Sostek, Glass, Molina, & Porges, 1984). Thus, the diminished vagal influences from the NA, responsible for depressed RSA amplitude, seem to be associated with a vulnerability to large neurogenic bradycardia. Potter and McCloskey (1986) provided an explanation of how depressed central nervous system function associated with hypoxia might result in massive neurogenic bradycardia. They reported a complex feedback system between duration of hypoxia, vagal efferent discharge, and potentiation of the vagal output on the heart. This system is able to maintain bradycardia, despite the massive decline in the vagal firing associated with hypoxia, by potentiating the influence of the vagal firing on the SA node. Under these conditions, although the bradycardia is mediated through a branch of the vagus, the magnitude of the bradycardia is determined by a peripheral mechanism and no longer reflects a centrally mediated vagal tone. Although Potter and McCloskey (1986) did not monitor RSA, we must assume that RSA is low in their preparation because the animals were anesthetized prior to the surgical, electrical, and hypoxic manipulations and because both hypoxia and anesthesia are associated with depressed beat-to-beat heart rate variability, including RSA (e.g., Donchin et al., 1985; Nelson, 1976).
Additional support for this bifurcation of vagal influences is demonstrated by electrical stimulation of the dorsal motor nucleus in the rabbit. As illustrated in Figure 2.3, electrical stimulation of the DMNX results in bradycardia without an increase in RSA. This is in contrast to the effect of stimulation of the aortic depressor nerve, which communicates with both the NA and the DMNX. Figure 2.4 shows that, in a similarly anesthetized rabbit, stimulation of the aortic depressor nerve results in an increase in RSA and a massive bradycardia (e.g., McCabe, Yongue, Porges, & Ackles, 1984).
FIGURE 2.2 (a) Bradycardia during fetal distress. (b) Background heart period variability at time of bradycardia.
The polyvagal theory argues that the vagal fibers from the DMNX and NA are distinguishable in structure and function. Specifically, it has been argued that the vagal efferent fibers from the NA are myelinated and contain a respiratory rhythm, and the vagal efferent fibers from the DMNX are unmyelinated and do not express a respiratory rhythm. However, there are some inconsistencies in the proposed distinction. For example, Jordan et al. (1982) reported that there are cardioinhibitory vagal neurons originating in the DMNX, with efferent axons conducting in the B fiber range, and are, therefore, myelinated. Moreover, Jordan et al. reported that these had a respiratory rhythm. Although the Jordan et al. findings support the proposed dual source of vagal efferents, their findings confound the proposed functional distinction.
FIGURE 2.3. Bradycardia elicited by electrical stimulation of the DMNX in an anesthetized rabbit.
FIGURE 2.4. Bradycardia elicited by aortic depressor nerve stimulation in an anesthetized rabbit.
There are several potential explanations for the inconsistency identified by Jordan et al. (1982). The first may be method. The Jordan et al. study used standard neurophysiological stimulation and recording techniques to identify cell bodies. According to Schwaber (1986), many vagal fibers previously assumed to originate in the DMNX have been identified with newer methods, such as retrogradely labeled horseradish peroxidase, as being located in the NA. Schwaber (1986) also stated that, because axons from the NA pass very near the border of the DMNX, it is difficult to stimulate or lesion the DMNX without NA involvement confounding electrical stimulation studies. Thus, additional research with more accurate labeling techniques may demonstrate that all neurons in the B fiber range originate in the NA. Our rabbit data reported in Figures 2.3 and 2.4 provide additional support for the possibility of mislabeling. According to Jordan et al. (1982), all neurons excited by aortic depressor nerve stimulation produced a respiratory rhythm in their ongoing discharge. Similarly, as illustrated in Figure 2.4, stimulation of the aortic depressor nerve resulted in both bradycardia and increased RSA. However, stimulation of the DMNX produced only an attenuated bradycardia. These findings suggest that vagal fibers discharging following stimulation of the DMNX did not have a respiratory rhythm. Moreover, the bradycardia was immediate and similar in latency to that observed following aortic depressor nerve stimulation. The magnitude of bradycardia was about 50% of the magnitude elicited by aortic nerve stimulation, a technique assumed to recruit vagal fibers from both the NA and the DMNX. The attribution of 50% of the magnitude of the heart rate component of the baroreceptor reflex to each vagal system is consistent with Machado and Brody (1988). A second possibility is that there are species differences in the organization and function of the DMNX. For example, to facilitate freezing behavior, the rabbit may have evolved unique myelinated vagal pathways from the DMNX that were independent of respiratory function. According to this explanation, the DMNX would have B fibers, but they would not express a respiratory rhythm. Alternatively, some mammalian species may have neurons in or near the DMNX that are part of a common cardiopulmonary oscillator (Richter & Spyer, 1990).
Future research will determine whether or not the proposed functional and structural distinctions between DMNX and NA efferents as articulated in the polyvagal theory are accurate. An additional concern relates to generalizing across mammalian species. Most of the neurophysiological and neuroanatomical research on the mammalian vagus has been conducted with rat, rabbit, cat, and dog. Studies investigating vagal regulation with humans have been limited to pharmacological blockade studies with measures of peripheral physiology. The few neuroanatomical studies of human brainstem, however, are often conducted on patients who have died of disease or trauma. Thus, one may question the generalizability of a polyvagal model developed from investigations of rodent brainstem to the human. However, existing data illustrate phenomena such as clinical bradycardia in the absence of RSA in the human fetus (see Figure 2.2), shifts in RSA independent of heart rate change during inhalant anesthesia (e.g., Donchin et al., 1985), and short latency responses from both systems (see Figures 2.3 and 2.4) that argue for a polyvagal model.
Vagal Strategies in Mammals and Reptiles
Reptilian and mammalian vagal systems have contradictory strategies. Reptiles are characterized by low ambient vagal tone and transient increases in vagal tone in response to environmental challenges. In contrast, mammals are characterized by high ambient vagal tone and transient decreases in vagal tone in response to environmental challenges (Table 2.3).
To adapt to the hostile world, the reptiles' behavioral repertoire is driven by survival. Most behaviors are associated with foraging, stalking, and feeding. Only limited time and energy are dedicated to social interactions such as parenting and reproduction. In the reptiles' defensive world, neurogenic bradycardia is adaptive and does not compromise physiological status. Reptiles have smaller metabolically active body organs, have different metabolic mechanisms, are less oxygen-dependent than mammals, and can go for long periods without oxygen. In contrast, the adaptive strategy of reptiles is lethal for mammals. In the defensive world of mammals, it is necessary to increase metabolic output to foster fight-or-flight behaviors. Therefore, reflexive neurogenic bradycardia to novelty for a prolonged period would reduce oxygen resources and metabolic output and compromise the fight-or-flight potential of mammals. The consequences of reduced oxygen resources also would depress central nervous system function, reduce behavioral complexity and competent execution of complex behaviors, induce unconsciousness, damage vital organs, and finally, if persistent, result in death. Thus, the cardiac component of the orienting reflex must be of short duration and be replaced by a physiological response that does not compromise the oxygen-needy nervous system of mammals. The withdrawal of vagal tone originating in the NA serves this purpose.
Phylogenetic Origins of Vagal Response Patterns
The neurogenic bradycardia controlled by the DMNX and observed in reptiles and mammals during orienting may have evolved from the gustatory response system of primitive vertebrates. Gustation is the primary method for identifying prey (including other appropriate food sources) and predators in aquatic environments. For example, in fish, an undifferentiated vagal lobe controls gustatory, digestive, and alimentary processes (Finger & Dunwiddie, 1992). A reflexive increase in vagal tone would affect several organs: the heart, where it would reduce metabolism and enable the animal to freeze momentarily; the organs containing gustatory receptors, where it would orient toward the source of stimulation and regulate threshold to detect novelty; and the digestive and alimentary systems, where it would stimulate gastric secretion and motility.
TABLE 2.3. Vagal Strategies
| Reptile | Mammal
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Ambient state | low DMNX | high NA/low DMNX
Response to novelty | increase DMNX | decrease NA/increase DMNX
With phylogenetic development, the viscerotropic organization of the vagal system has become more complex by incorporating pathways from other cranial nerves, including trigeminal, facial, accessory, and glossopharyngeal. Thus, more specialized functions, such as head rotation to orient sensory receptors toward the source of stimulation, mastication to ingest food, and salivation to initiate gustatory and digestive processes, are integrated into the vagal system.
The motor component of the vagus shares evolutionary origins with four cranial nerves (trigeminal, facial, accessory, and glossopharyngeal). The vagus not only innervates smooth and cardiac muscle, but, similar to the other four cranial nerves, it contains motor pathways that innervate somatic muscles. Vagal pathways that innervate somatic muscle often are not included in the neurophysiology of the autonomic nervous system. These fibers are called special visceral efferents to distinguish them from general visceral efferents, the motor pathways innervating smooth and cardiac muscle. The critical difference between the two types of motor pathways is that somatic muscle regulation may be conscious and voluntary, whereas smooth muscle regulation is reflexive and unconscious. Because the special visceral efferents innervate voluntary muscles, they are usually excluded from the autonomic nervous system. Traditionally, only the general visceral efferents from both sympathetic and parasympathetic branches are used to define the autonomic nervous system.
The somatic muscles innervated by the five cranial nerves arise from the branchial arches, embryologically known as the primitive gill arches (Warwick & Williams, 1975). These muscles are critical to several mammalian behaviors. For example, the somatic muscles innervated by the trigeminal, arising from the first branchial arch, are involved in mastication, retraction of the lower jaw, and closing the mouth. The special visceral efferents from the facial nerve, arising from the second branchial arch, innervate the muscles of the face, scalp, and neck to enable facial expressions. The facial nerve also innervates muscles in the floor of the mouth.
Although the trigeminal and facial nerves originate from branchial arches and have communications with the other three cranial nerves originating from the branchial arches, the source nuclei of the special visceral efferents for the glossopharyngeal, vagus, and accessory nerve originate in the same medullary nucleus, the NA. Thus, the efferent fibers travel through three different cranial nerves, but they originate in the same source nucleus.
As a function of phylogenetic development, the source nuclei for the special visceral efferent pathways in the glossopharyngeal, vagus, and accessory nerves migrate to form the NA. In mammals, the NA controls the complex coordination of pharynx, soft palate, larynx, and esophagus. Of special note to psychophysiological processes is that the third gill arch also gives rise to the carotid body, containing peripheral chemosensitive receptors sensitive to oxygen and carbon dioxide levels (Warwick & Williams, 1975). In addition, the accessory nerve contains fibers originating in the cervical spinal cord that innervate the positioning of the neck. The critical carotid arteries, internal jugular veins, and vagus nerves run deep in these muscles (Warwick & Williams, 1975). Thus, this complex also has the ability to orient visceral receptors through somatic muscles, to coordinate structures related to ingestion and expulsion, and to regulate facial expression and emotion. These motor nuclei receive input from cortex to coordinate complex behaviors with cardiopulmonary function. Thus, phylogenetically, even when the gill arches evolve into the branchiomeric muscles common to all mammals, oxygenation of blood through a coordination of breathing and heart rate during interactions with the environment remains a primary functional objective.
The processes associated with NA control of supradiaphragmatic organs appear to be uniquely mammalian. For example, this subsystem of the vagus coordinates the complex sequence of sucking, swallowing, and breathing that allows mammals to feed and breathe actively and voluntarily. Moreover, the NA provides the primary chronotropic control of the heart and controls the intonation of vocalizations. Thus, NA efferent projections are involved with processes associated not only with feeding and breathing but also with movement, emotion, and communication. These behaviors contribute to the unique social and survival behaviors observed in mammals. The NA vagus provides the vagal brake that mammals remove instantaneously to increase metabolic output to foster fight-or-flight behaviors. The NA vagus provides the motor pathways to shift the intonation of vocalizations (e.g., cry patterns) to express emotion and to communicate internal states in a social context.
The behavioral derivatives of the two branches of the vagus suggest a typology in which one branch of the vagus deals with unconscious reflexive vegetative functions and the other is involved in more conscious, voluntary, flexible, and often social activities. There is neuroanatomical support for this typology. The DMNX contains only general visceral efferents that innervate smooth and cardiac muscle fibers and regulate glandular secretion. In contrast, the NA contains special visceral efferents that innervate the somatic musculature of the soft palate, larynx, pharynx, and esophagus.
Somatomotor and Visceromotor: Coupled Systems
In mammals, we observe two evolutionary strategies that link autonomic function with somatic muscle activity. First, there is an anatomical linkage between the segmentation of the spinal nerves and the sympathetic chain. This linkage is reflected in the motor-related increases in sympathetic tone that have dogged psychophysiologists by confounding motor and autonomic responses. The evolution of the segmented sympathetic nervous system parallels the evolution of voluntary motor activities. The sympathetic nervous system regulates vasomotor tone to direct blood flow and, thus, oxygen to the specific muscles being challenged. In addition, there are sudomotor links to hydrate and protect the skin from tearing. This coupling between sympathetic activity and movement has been the cornerstone of arousal theory and hypotheses associating autonomic function with temperament and psychopathologies. It was not many years ago that Obrist (1976) challenged Lacey's (1967) notion that autonomic state was independent of motor activity (i.e., metabolic demands). There is no doubt that the effects of motor activity are profound on the autonomic nervous system, yet this profound effect does not mitigate the importance of other relationships that may be sensitive to specific psychological processes, independent of movement.
Second, there is an anatomical linkage between the somatic muscles that arise from the cranial nuclei and parasympathetic function. We can observe this clearly in the viscerotropic organization of the NA. The NA provides the source nuclei for somatic muscle fibers that innervate larynx, pharynx, trachea, and esophagus. Moreover, ventral to these source nuclei, in an area of the NA known as NAex, are general visceral efferents that control the resistance of the bronchi (Haselton, Solomon, Motekaitis, & Kaufman, 1992) and heart rate (Bieger & Hopkins, 1987). The ventral portion also projects to other visceral organs (e.g., Brown, 1990).
Based on neuroanatomical studies, it has been demonstrated that visceromotor functions regulated by the ventral part of NA provide the parasympathetic support for the somatomotor projections from NA and trigeminal and facial nerves. Neuroanatomical studies suggest that, unlike the DMNX, which receives primary sensory input through the NTS, the NA has the trigeminal nerve as an important source of sensory input. Moreover, the rostral region of the NA communicates with the facial nucleus. This coupling of the NA with facial and trigeminal nuclei provides additional evidence of the coordination of the visceromotor regulation through the NA with somatomotor functions such as swallowing (Brown, 1974), sucking (Humphrey, 1970), and, perhaps, facial expressions. Thus, the organization of the mammalian brainstem has evolved to have a ventral vagal complex, consisting of NA and the nuclei of the trigeminal and facial nerves, that coexists with the dorsal vagal complex, consisting of the DMNX and NTS, that regulates vegetative processes and is observed in reptiles.
To foster motor movement, visceromotor (i.e., autonomic) processes are associated with somatomotor activities. In the periphery, this is done primarily by the sympathetic chain; in special cases, such as those related to reproduction and elimination, the sacral branch of the parasympathetic nervous system contributes. However, in the rostral part of mammalian anatomy (i.e., the head), the somatic muscles that regulate facial expression, mastication, vocalization, swallowing, and sucking are matched with general visceral efferents, projecting from the ventral portion of NA, that exert potent influences on the heart and the bronchi. These motor fibers effectively slow heart rate and increase respiratory resistance to conserve oxygen exchange. Neuroanatomical studies performed on human embryos and fetuses suggest that these visceromotor neurons may have migrated from the DMNX (Brown, 1990).
As observed through both embryological research and phylogenetic comparisons, in mammals the primitive gill arches evolve into muscles and nerves controlling the face, bones of the mouth, jaw, pharynx, larynx, soft palate, esophagus, and trachea. The nerves innervating these muscles uniquely arise not from the anterior horns of the spinal cord but from the source nuclei of five cranial nerves (trigeminal, facial, glossopharyngeal, vagus, and accessory). Because of their uniqueness, these motor systems are known as special visceral efferents. And, because of their voluntary aspects, these pathways have been excluded from traditional concepts of the autonomic nervous system. Facial expressions characteristic of mammals, such as sucking, swallowing, and vocalizations, reflect the unique mammalian adaptation of special visceral efferent control of the visceral muscles evolving from the branchial arches.
However, similar to the synergistic relationship between the sympathetic nervous system and skeletal muscles of the extremities, there is a synergistic relationship between the traditional general visceral efferents of the vagus and the somatic muscles controlled by these cranial nerves. Thus, increased outflow of these somatic muscles produce specific visceral shifts. For example, chewing will produce salivation in the absence of food, and head rotation, by means of accessory special visceral efferents, will impact on cardiovascular action along the vagus.
Phylogenetic development of the central nervous system has progressed in mammals to produce a brain with a large neocortex (e.g., MacLean, 1990). The neocortex is very vulnerable to shifts in oxygen. Evolutionary pressures have resulted in autonomic strategies that optimize the availability of oxygen to the cortex. However, these uniquely mammalian strategies coexist with the ancestral reptilian strategies. Thus, premise 2 is consistent with MacLean's view that the advanced mammalian brain contains its phylogenetic heritage.
Premise 2: Neurogenic bradycardia associated with orienting is a phylogenetic vestigial relic of the reptilian brain and is mediated by the DMNX.
Although phylogenetic development has modified several brain structures, the evolved brain of advanced mammals maintains several structures and systems that are virtually identical to those observed in primitive reptiles. These primitive structures have extensive interconnections and functional dependencies, although each is capable of specific independent functions. Thus, in mammals, the DMNX still maintains its reptilian functions of facilitating digestion and slowing heart rate. Mammals utilize an additional brainstem structure, the NA, to supply general visceral vagal efferents that provide the prominent control of the heart and the bronchi. The cells of origin of these fibers efficiently communicate with limbic and other higher centers and allow for the conscious and voluntary selection of novelty. In contrast, the DMNX is more directly regulated by hypothalamic communication, often triggered by survival-oriented stimuli (Hopkins, 1987; Leslie et al., 1992). Thus, as stated in premise 3, the regulation of vagal efferents by NA mechanisms contributes to the mammalian ability to detect novelty, actively engage with the environment, and socially communicate.
Premise 3: Withdrawal of cardiac vagal tone through NA mechanisms is a mammalian adaptation to select novelty in the environment while coping with the need to maintain metabolic output and continuous social communication.
To summarize the reptilian-mammalian evolutionary evidence, phylogenetic development of the neural regulation of the heart provides insights into an apparent contradiction or paradox in vagal control of the heart. In most reptiles, the neuroanatomy demonstrates (a) a lack of anatomically distinguishable boundaries between the DMNX and NA, and (b) cardiac vagal efferent pathways originating only in the DMNX. In mammals, the neuroanatomy demonstrates (a) a distinct separation of the DMNX and NA; (b) cardiac vagal efferent pathways originating primarily, but not exclusively, in the NA; (c) direct neural connections between the central nucleus of the amygdala and NA; and (d) a clustering of medullary neurons in the NA capable of regulating the somatic muscles related to vocalizations, facial expression, and coordinating breathing with sucking and swallowing.
Smart and Vegetative Vagi
The polyvagal theory proposes that the evolutionary shift resulting in both an NA that is distinct from the DMNX and the evolutionary development of special visceral efferents changed the role of the vagus. The general visceral efferent pathways from the DMNX vagus are part of a passive reflexive motor system associated with vegetative function and, thus, a vegetative vagus. The special visceral efferent pathways from NA create an active voluntary motor system associated with the conscious functions of attention, motion, emotion, and communication, and, thus, a smart vagus.
The polyvagal theory requires a reconceptualization of the vagal system and the construct of vagal tone. The theory focuses on the cytoarchitecture of the medullary source nuclei of the cranial nerves. The theory takes an evolutionary approach and investigates, through embryology and phylogenetic comparisons, the common origins of the special visceral efferents and focuses on the shared medullary structures for the cell bodies of these fibers. The theory acknowledges that the vagal system is complex and should be organized not in terms of bundles of fibers leaving the medulla but rather in terms of the common source nuclei of several of these pathways. Functionally, the common source nuclei provide a center to coordinate and regulate the complex interactions among various end organs and are related to optimizing cardiopulmonary function.
Mammals, with their oxygen-hungry metabolic systems, require a special medullary center to coordinate cardiopulmonary functions with behaviors of ingestion (e.g., mastication, salivation, sucking, and swallowing), oral or esophageal expulsion (vomiting), vocalizations (e.g., cries and speech), emotions (e.g., facial expressions), and attention (e.g., rotation of the head). The NA plays this role and serves as the cells of origin of the smart vagus. The potent link between the NA and cardiopulmonary function observed in mammals is not observed in reptiles. In reptiles, which do not have nerves to regulate facial expression, the NA does not play a major role in visceromotor regulation.
Medullary Contributions to a Common Cardiopulmonary Oscillator
The NA is a continuum of interconnected subdivisions, beginning rostrally at the level of the facial nucleus and extending caudally to the spinal medullary junction. As illustrated in Figure 2.5, the rat NA has several subdivisions. The subdivisions are the compact (NAC), semicompact (NASC), loose (NAI), and external (NAEX) formations (Bieger & Hopkins, 1987). The dorsal division consists of the NAC, NASC, and NAI. The dorsal division is the source of special visceral efferents innervating the soft palate, pharynx, larynx, and esophagus. The ventral division consists of the NAex and is the source of general visceral efferents innervating the thoracic viscera, primarily the bronchi and the sinoatrial node. Vagal fibers originating in the NAex and terminating in both the bronchi (Haselton et al., 1992) and the sinoatrial node (Spyer & Jordan, 1987) have a respiratory rhythm, thus suggesting that RSA may reflect a common respiratory rhythm originating in or at least incorporating the NA.
After investigating the neuroanatomical centers associated with laryngeal, pulmonary, and cardiac function, Richter and Spyer (1990) arrived at a convergent conclusion that NA was a contributor to a common respiratory rhythm. They also speculated that mammals, with their great need for oxygen, have a medullary center to regulate cardiopulmonary processes. They proposed that a common cardiorespiratory oscillator evolved to foster coordination between cardiac and respiratory processes. In their model, the respiratory rhythm is dependent on the interaction between two groups of neurons, one in the NTS and the other in the NA. Accordingly, the "common" oscillator producing respiratory frequencies is a manifestation of a neural network comprised of interneurons between areas containing the motoneurons regulating respiratory, laryngeal, and cardiac functions. The cardiorespiratory oscillator does not involve the DMNX. To support their hypotheses, Richter and Spyer (1990) reported cross-correlational studies of single units. Thus, the NA is part of the cardiorespiratory oscillator network, and the period of the oscillations in heart rate (the period of RSA) provides a valid index of the output frequency of the cardiopulmonary oscillator.
FIGURE 2.5. Topographic organization of the nucleus ambiguus (NA) in the rat (Bieger & Hopkins, 1987).
Other researchers have emphasized the importance of additional brain structures as contributors to the regulation, if not the generation, of a cardiopulmonary rhythm. For example, Harper and associates demonstrated that "respiratory rhythms" can be observed in several nuclei in the brainstem, midbrain, and forebrain. Harper's group, by employing cross-correlation techniques, reported units firing on a breath-by-breath basis in periaqueductal gray (Ni, Zhang, & Harper, 1990), central nucleus of the amygdala (Frysinger, Zhang, & Harper, 1988), hippocampus (Frysinger & Harper, 1989), and the anterior cingulate (Frysinger & Harper, 1986). In addition, they reported that stimulation of the amygdala can influence the respiratory cycle (Harper, Frysinger, Trelease, & Marks, 1984).
The covariation of the bronchi with heart rate oscillations (e.g., RSA), mediated by the NA, may have a functional influence on the oxygenation of blood. As stated earlier, the primary objective of the phylogenetic derivatives of the primitive gill arches is to maintain oxygenation. Thus, one might speculate that oscillations in vagal tone to the bronchi and the heart might influence oxygenation. Perhaps coherent rhythmic shifts between bronchial tone and heart rate, with a fixed-phase lag, maximize oxygen diffusion. To answer this question, research would need to confirm a relationship between oxygen saturation and RSA, independent of the average heart rate and respiration rate. Currently, only anecdotal data exist that demonstrate that clinical conditions in which oxygen saturation is low tend to be conditions in which RSA also is depressed. Support for this hypothesis is obtained from research demonstrating that vagotomy disrupts the oxygen consumption-oxygen delivery relationship (Scherlel, Brourman, Kling, Schmall, Tobias, & Myerowitz, 1994).
Measurement of NA Status: Quantification of RSA
For psychophysiologists, interest is primarily in the behaviors and psychological processes associated with special visceral efferents. Most research has been directed toward processes that require the ability to monitor and mediate complex behaviors, such as attention, motion, emotion, and communication; these processes are neurophysiologically dependent on the special visceral efferents of the NA and facial and trigeminal nerves. Yet many of us measure only general visceral efferents from both the parasympathetic and sympathetic branches, although we are interested in the special visceral efferents that regulate vocalizations and facial expression. We are not at a total loss, because there is interneuronal communication between the dorsal and ventral segments of the NA. Thus, by the nature of the NA having general visceral efferents that regulate heart and bronchi, it is possible to monitor continuously the vagal output or tonus of the smart vagus. This leads to the fourth premise of the polyvagal theory.
Premise 4: The ability of NA to regulate special and general visceral efferents may be monitored by the amplitude of RSA.
The vagal fibers originating in the NAex have a characteristic respiratory frequency that reflects a waxing and waning of influence. For example, the vagal fibers from the NA that have an inhibitory action on the sinoatrial node also wax and wane in inhibitory influence at the respiratory rhythm and produce RSA. Thus, it is possible to monitor continuously the general status of the NA by evaluating RSA. Similarly, NA fibers to the bronchi that elevate lung resistance also wax and wane in their inhibitory influence (Haselton et al., 1992).
RSA is a measure of the general visceral efferents of the NA and thus is an index of the smart vagus. RSA is not a global measure of vagal tone or even a measure of "total" vagal control of the heart as previously proposed (Fouad, Tarazi, Ferrario, Fighaly, & Alicandro, 1984; Katona & Jih, 1975; Porges, 1992). There are other vagal and nonvagal influences on the heart that contribute to both heart rate level and rhythm. For example, there are DMNX projections and monosynaptic cholinergic pathways within the heart, sympathetic pathways, and intrinsic factors. However, the primary, if not sole, source of respiratory rhythms on the sinoatrial node is due to projections from the NA.
To evaluate NA regulation of the sinoatrial node, the parameters of RSA must be accurately extracted. We have approached this problem by evaluating the period and amplitude of RSA, independent of slower oscillations and trends, with a moving polynomial approach (Porges & Bohrer, 1990). In our research, we have obtained correlations between respiration rate and period of RSA approaching 1.0. These findings support the notion of a common cardiorespiratory oscillator as described by Richter and Spyer (1990). RSA, with its amplitude representing visceromotor tone and its period representing the common cardiorespiratory drive frequency, is the functional consequence of the output of vagal fibers originating in NA and terminating on the sinoatrial node. Thus, to emphasize the relationship between RSA and the vagal tone from the NA to the heart, a better designation for the vagal tone index (Porges, 1986) would be VNA.
Quantification of RSA requires only an accurate determination of the amplitude and period of these oscillations. Additional experimental constraints to regulate breathing rates might confound the visceral-medullary feedback system, which determines central respiratory rhythms. For example, because paced breathing requires an awareness of breathing parameters, cortical influences on brainstem structures might modulate the gain of the feedback and influence the amplitude of RSA, and paced breathing may shift respiratory parameters, such as rate, amplitude, inspiration-expiration ratio, interbreath pause, and resistance, from brainstem setpoints. Data illustrating that paced breathing may influence RSA (Sargunaraj, Lehrer, Carr, Hochron, & Porges, 1994) have been reported.
Various manipulations or conditions that depress special visceral efferents, such as inhalant anesthesia, have profound influences on RSA (Donchin et al., 1985). Recovery of function of special visceral efferents is paralleled by a recovery of RSA. In neurology, diagnosis is often based on the evaluation of the special visceral efferents. In our research, we noted that RSA amplitude before neurosurgery was an effective diagnostic of neurological recovery following neurosurgery (Donchin, Constantini, Szold, Byrne, & Porges, 1992). Additional neurological data demonstrate consistent depression of RSA in individuals who are diagnosed as brain dead (Mera, Wityk, & Porges, 1995).
High-risk preterm neonates have problems coordinating breathing, sucking, and swallowing (i.e., processes regulated by the NA). These infants have low levels of RSA (Porges, 1992). Many of these infants have severe bradycardia. The bradycardia is often paralleled by apnea, and a drop in available oxygen may be assumed to reflect neurogenic vagal regulation by the DMNX. This response, in dealing with decreased resources, is adaptive for reptiles but potentially lethal for humans. This also is observed during fetal distress, when there is severe hypoxia associated with a loss of RSA and a pronounced neurogenic bradycardia.
Vagal Competition and Autonomic Dysfunction
The concept of competition between sympathetic and parasympathetic inputs is well known. For example, Levy (1984; Vanhoutte & Levy, 1979) has clearly documented the ability of vagal efferents to inhibit sympathetic influences. Similarly, Berntson et al. (1991) modeled the interactions between sympathetic and parasympathetic efferents to the heart. However, there may be a different type of competition in which the two vagal branches are conveying contradictory information to the target organs. As both vagal pathways are capable of regulating heart rate, there may be competition on the sinoatrial node. Due to the rate of acetylcholine degradation on the nodal tissue (Dexter, Levy, & Rudy, 1989), the continuous stimulation of the sinoatrial node by NA pathways may functionally protect the heart from massive neurogenic bradycardia mediated by the DMNX. Thus, the observations of massive pathophysiological bradycardia in hypoxic fetuses and neonates, who have very low amplitude RSA, may reflect the loss of NA protection on the sinoatrial node. Similarly, sudden death following exercise might reflect a similar process associated with the depression of NA input to foster metabolic activity and a surge of DMNX input in response to decreased oxygen resources.
The vagal competition hypothesis may be generalized and tested to explain other autonomic diseases such as asthma. The vagal competition hypothesis proposes that all target organs with smooth and cardiac muscle have dual innervation from both the DMNX and the NA. This has been documented in animal preparations for heart, lungs, esophagus, and abdominal viscera including pancreas, liver, and stomach (Brown, 1990). However, as with the heart, the two vagal inputs may innervate in a contradictory manner. Just as a DMNX surge coupled with low RSA may result in sudden death, bronchial asthma may be produced by a similar mechanism. In the case of asthma, NA efferent control of bronchi results in the bronchi exhibiting a rhythmic waxing and waning with breathing. This continuous stimulation of the bronchi by NA pathways may functionally protect the bronchi from pathophysiological DMNX influences. It is possible that without NA influences, the bronchi become vulnerable to vagal surges from the DMNX. This would be an adaptive response for a primitive brainstem attempting to conserve oxygen but would be lethal for the oxygen-hungry mammal. The asthma attack, similar to lethal neurogenic bradycardia, may be a product of a primitive vagovagal reflex. In this type of reflex, not only do the motor fibers originate, but the afferent fibers terminate, in DMNX. There is an anatomical basis for a monosynaptic vagovagal reflex. There are reports that dendritic processes from DMNX neurons extend into the boundaries of the NTS. Thus, vagal afferent fibers may communicate directly with DMNX neurons (Neuheuber & Sandoz, 1986). Because afferents terminate in DMNX, the name motor nucleus is not accurate, and dorsal nucleus of the vagus nerve has been suggested instead (Nara, Goto, & Hamano, 1991). In most vagal reflexes involving the bronchi, the afferents terminate in the NTS and influence the NA to provide a fail-safe feedback system.
Based on the polyvagal theory, the assumption of vagal competition promotes the following testable hypotheses.
Nucleus ambiguus (vagal) protection hypothesis. Vagal projections originating in the NA and terminating in visceral organs provide tonic influences that promote health, growth, and restoration.
Nucleus ambiguus (vagal) withdrawal hypothesis. Removal of the NA vagal brake for short periods of time promotes metabolic output to foster locomotion. Removal of these influences for long periods places the organ at risk.
Emotion
The polyvagal theory provides a set of predictions regarding the relation between autonomic responses and emotion. Darwin (1872) carefully described facial expressions as the primary defining characteristics of emotion. The special visceral efferents, associated with the facial nerve, control movements of facial expression. Reptiles cannot modulate facial expression. The facial nerve in mammals not only regulates facial muscles but also interacts with the NA and the vagal system. Thus, it is logical that emotional expression, which requires somatic muscles controlled by special visceral efferents, is coupled to the visceromotor regulation of cardiopulmonary function by NA vagal efferents. In addition, special visceral efferents originating in NA regulate the larynx and control intonation. Thus, the following premise is stated.
Premise 5: Emotion, defined by shifts in the regulation of facial expressions and vocalizations, will produce changes in RSA and bronchomotor tone mediated by the NA.
As a construct, emotion is heterogeneous. Therefore, correlations between specific emotions and physiological states may be a function of the type of emotion. Even Darwin (1872) distinguished between primary or neurally based emotions and social or culturally based emotions. Darwin (1872) suggested that certain emotions have as their substrate an innate neural basis and, because these emotions are neurally based, they are universally expressed and understood across cultures. These primary emotions include anger, fear, panic, sadness, surprise, interest, happiness (ecstasy), and disgust (Ross, Homan, & Buck, 1994). Because the prevalent hypotheses suggest a strong physiological basis for primary emotions, we will focus on relating primary emotions to the polyvagal theory.
There are two important aspects linking the polyvagal theory to the study of emotion: first, there is a parallel between cortical asymmetry and autonomic asymmetry; second, the branchial arches have evolved into the structures that mammals use to express emotion (i.e., facial muscles and larynx).
The literature documents the relationship between right brain function and primary emotions (Heilman, Bowers, & Valenstein, 1985). The medullary source nuclei and efferent pathways of the vagus also are lateralized with a right bias. The right NA by means of the right cardiac vagus provides the primary chronotropic output to the heart. The special visceral efferents, which provide the behaviors that are used to define emotion (facial expression and vocalization) also have a right bias and are linked neuroanatomically to the general visceral efferents, originating in the NA, that regulate the bronchi and heart, organs that are assumed to be sensitive to emotion and stress. It is difficult to predict the influence of this right bias on actual facial expressions. Because the face is controlled by upper motor neurons that are crossed and lower motor neurons that are uncrossed (Rinn, 1984), facial expression may not be systematically lateralized. In fact, research on facial asymmetry and emotion has not been consistent. There have been reports of facial expressions not being lateralized, being lateralized on the left, and being lateralized on the right (e.g., Hager & Ekman, 1985).
The functional dominance of the right side of the brain in regulating autonomic function and emotion may have implications for the specialization of motor and language dominance on the left side of the brain. The right-sided responsibilities of regulating homeostasis and modulating physiological state in response to both internal (i.e., visceral) and external (i.e., environmental) feedback may contribute to the development of motor and language functions on the left side of the brain.
A partitioning of central control of voluntary processes, independent of emotional-homeostatic processes, would enable the individual to express complex voluntary levels of communication and movement, from the left side of the brain, and more intense emotional-homeostatic processes, from the right side of the brain. If these processes are lateralized, they might have a degree of autonomous regulation. This would enable simultaneous activation of global functions associated with emotional-homeostatic processes and language-voluntary movement processes.
Given the strong theoretical relationships between lateralized autonomic and hemispheric function and between the neurons that control RSA and those that control facial expression and vocal intonation (see Figure 2.5), research should be directed at evaluating the relationship between RSA and the primary emotions. Recall that the source nucleus of the facial nerve is the border of the NA and afferents from the trigeminal nerve provide a primary sensory input to the NA. Thus, the ventral vagal complex, consisting of the NA and the nuclei of the trigeminal and facial nerves, is clearly related to the expression and experience of emotion.
Based on the polyvagal theory, one would expect shifts in affective state to parallel RSA. For example, the elicitation of a negative primary emotion would result in a systematic withdrawal of vagal tone along NA to promote fight-or-flight behaviors. In contrast, a shift to a more pleasant affective state would be associated with an increase in RSA. A study by Bazhenova and colleagues (2001), in emphasizing the dynamics of RSA change during shifting affective states, supports this speculation. Manipulation of the affective state resulted in RSA deceases when an infant shifted to a more negative state. Moreover, when the infant shifted to a more positive affective state, RSA increased above the affectively neutral base level.
The polyvagal theory does not neglect the important role of the DMNX in the emotional experience. For example, the DMNX is critical in the regulation of digestive polypeptides and gastric motility (Uvnas-Moberg, 1989), dimensions of physiological activity that parallel emotive experiences and stress. Consistent with the polyvagal theory, which emphasizes the importance of NA and the ventral vagal complex in overt emotional expressiveness and regulation, the theory would acknowledge the importance of less conscious survival-oriented processes that are mediated by the dorsal vagal complex, which consists of the NTS and the DMNX. A complementary theory has been proposed by Uvnas-Moberg (1987, 1994). The Uvnas-Moberg theory emphasizes the role of the DMNX in the regulation of gastrointestinal hormones and during emotional states including stress, hunger, and satiety.
SUMMARY AND CONCLUSION
The following seven points summarize the polyvagal theory.
1. The vagal system does not represent a unitary dimension. The vagal system includes general visceral efferent fibers regulating smooth and cardiac muscle and special visceral efferent fibers regulating the somatic muscles of the larynx, pharynx, and esophagus. These somatic muscles control vocalization, sucking, and swallowing and interface these processes with breathing. The vagal system also is linked neuroanatomically to the source nuclei that control facial expression, mastication, and head turning.
2. There are two vagal motor systems. One vagal system is the vegetative vagus, which originates in the dorsal motor nucleus and is associated with passive reflexive regulation of visceral functions. The other vagal system is the smart vagus, which originates in the NA and is associated with the active processes of attention, motion, emotion, and communication. The two systems are neuroanatomically distinct, have different ontogenetic and phylogenetic origins, and employ different adaptive strategies.
3. In mammals, the concept that vagal tone represents a single or summed system may have limited physiological or heuristic value. For example, in mammals, high tone from the dorsal motor nucleus vagal system may be lethal, whereas high tone from the NA vagal system may be beneficial. Based on the proposed polyvagal theory, an accurate measure of the NA system is critical to the evaluation of psychophysiological relationships.
4. The functional output of the NA vagus on the heart may be monitored by RSA. NA is part of a common neuronal network producing a cardiorespiratory rhythm. Thus, the output from the branch of the vagus originating in NA and terminating on the sinoatrial node of the heart conveys a frequency common to both respiratory and cardiac systems. In contrast, the output from the dorsal motor nucleus does not convey a respiratory rhythm.
5. The magnitude of neurogenic bradycardia is mediated by the dorsal motor nucleus. Rapid heart rate changes, such as conditioned anticipatory heart rate deceleration and decelerations associated with orienting, are neurogenic bradycardia. Additional neurogenic bradycardia are reflexes such as optovagal and chemovagal. In the absence of NA influences to the sinoatrial node, local conditions such as hypoxia may greatly potentiate the vagal effect.
6. There is a common cardiopulmonary oscillator. The common respiratory rhythm observed in heart rate and breathing is produced by a network of interneurons located in the NTS and NA, which communicate with the motor neurons that control respiratory, laryngeal, and cardiac function.
7. Primary emotions are related to autonomic function. Because the primary emotions are often related to survival, they must be integrated into cardiopulmonary regulation. Moreover, primary emotions have a right-hemisphere bias, ipsilateral with the regulatory bias of the medullary structures controlling visceral function.
Based on the polyvagal theory, additional hypotheses may now be tested by evaluating the relationship between RSA (the measure of NA vagal tone, VNA) and processes and states dependent on the coordination of cardiopulmonary processes with the special visceral efferents of the cranial nerves. This, of course, includes all processes associated with vocalizations, feeding, breathing, and facial expression.
In developing the polyvagal theory, the most striking insights came from the phylogenetic approach. Not only does a phylogenetic approach explain the vagal paradox in terms of the medullary source nuclei of the dorsal motor nucleus and NA, but it also highlights the importance of oxygen needs in the evolving nervous system. As the nervous system gets more complex, there are greater demands for oxygen. Oxygen needs may have provided a major environmental pressure leading to the evolution of the adaptive and sophisticated autonomic nervous system found in mammals. Thus, constructs such as orienting, attention, emotion, and stress are by-products of the evolutionary pressure to optimize oxygen resources.
APPENDIX A: TITLE OF THE CHAPTER
The title was selected to emphasize the concept that evolutionary processes have sculpted the neural regulation of autonomic function. Evolution has provided obvious divergences in behavior and appearance and has had an impact on the autonomic strategies related to the detection of novelty in the environment.
The aim of this chapter was neither a theory of orienting nor an attempt to distinguish between the autonomic components of orienting or defensive reflexes, but rather the neurogenic regulation of cardiac responses by two vagal responses systems. A primitive system inherited from reptiles produced a rapid neurogenic bradycardia that reduced the activity of our cardiopulmonary system to conserve oxygen. This is the strategy of sit-and-wait feeders common in reptiles. In contrast, the evolution of the energy-demanding mammal required two autonomic behavioral shifts: mammals needed to (a) obtain great amounts of food and (b) protect their nervous systems from oxygen loss. These two objectives are linked. In the evolution of mammals, success in obtaining food resources was dependent on the ability to detect threat. Thus, mobilization and attention became two important behavioral dimensions. Unlike reptiles, which orient in response to novelty and attack or return to a quiescent state or lumber off, mammals orient and then attend. Following this phase of attention, a mammal may rapidly depart or approach (attack) within the context of the classic fight-or-flight response. With the increasing complexity of behavior, there is a parallel increase in complexity in the organization and function of the autonomic nervous system.
The title was also intended to emphasize the concept that evolution has placed mammals in a defensive world. The survival systems of reptiles and other nonmammalian vertebrates can be organized into orienting and defensive dimensions. Mammals, to survive in this defensive and reactive world, had to circumvent these potentially lethal reactions of other species. The evolution of the mammalian nervous system enables mammals to escape danger rapidly and to use neural resources for the complex information processes required to detect subtleties in the environment. Moreover, evolution promoted additional motor systems related to communication. Motor systems developed to communicate conditions related to survival with facial expressions and vocalizations associated with primary emotions. The evolutionary modifications had to coexist with the oxygen-hungry metabolic system, and, by increasing the complexity of motor behaviors, there was an additional increase in oxygen needs. Thus, there is a link between the special visceral efferent actions regulating the communicative processes of emotion (and later, language) with the general visceral efferent actions regulating cardiopulmonary function. The ability to detect subtleties in the environment coupled with the ability to communicate threat or comfort through facial expressions and vocalizations contributed to within-species social behavior, parenting, and pair bonding. These complex functions evolved while the demanding oxygen needs of mammals were programmed into the background of nervous system function by the autonomic nervous system.
APPENDIX B: PERSONAL RETROSPECTIVE
In discussing any theoretical perspective, it is important to place the ideas and speculations in the context of earlier research conducted by the investigator. My early research focused on the use of heart rate measures as indicators of attention. While conducting research for my master's thesis (Porges & Raskin, 1969), I noted that attention-demanding tasks produced heart rate response patterns with two prominent characteristics. First, heart rate exhibited a rapid transitory directional change in response to task onset and stimulus changes. Second, when subjects became involved in the task and focused their attention on the task demands, heart rate variability was reduced. I was intrigued with these observations and speculated on the possible physiological mechanisms. This evolved into a two-component theory of attention in which the components were labeled phasic or orienting and tonic or attention responses (Porges, 1972). These findings stimulated me to investigate neural mechanisms of heart rate regulation and to develop the vagal tone index (V) of RSA, which I believed would help provide insight into the mechanisms mediating the more tonic sustained attention response.
The preceding sections of this chapter provide the basis for the polyvagal theory and enable an interpretation of the two heart rate components associated with attention. The first component, associated with orienting and neurogenic bradycardia, is determined reflexively by the vegetative vagus, originating in the dorsal motor nucleus. The second component, associated with voluntary engagement with the environment and depression of RSA, is determined by the smart vagus, originating in the NA. Thus, after years of studying heart rate patterns, a speculative two-component psychophysiological model of attention is evolving into the neuroanatomically and neurophysiologically based polyvagal theory.
CHAPTER 3
The Polyvagal Theory: New Insights Into Adaptive Reactions of the Autonomic Nervous System
HISTORICAL PERSPECTIVES ON THE AUTONOMIC NERVOUS SYSTEM
Central nervous system regulation of visceral organs is the focus of several historic publications that have shaped the texture of physiological inquiry. For example, in 1872 Darwin acknowledged the dynamic neural relationship between the heart and the brain:
when the heart is affected it reacts on the brain; and the state of the brain again reacts through the pneumo-gastric [vagus] nerve on the heart; so that under any excitement there will be much mutual action and reaction between these, the two most important organs of the body. (p. 69)
Although Darwin acknowledged the bidirectional communication between the viscera and the brain, subsequent formal description of the autonomic nervous system (e.g., Langley, 1921) minimized the importance of central regulatory structures and afferents. Following Langley, medical and physiological research tended to focus on the peripheral motor nerves of the autonomic nervous system, with a conceptual emphasis on the paired antagonism between sympathetic and parasympathetic efferent pathways on the target visceral organs. This focus minimized interest in both afferent pathways and the brainstem areas that regulate specific efferent pathways.
The early conceptualization of the vagus focused on an undifferentiated efferent pathway that was assumed to modulate "tone" concurrently to several target organs. Thus, brainstem areas regulating the supradiaphragmatic (e.g., myelinated vagal pathways originating in the nucleus ambiguus and terminating primarily above the diaphragm) were not functionally distinguished from those regulating the subdiaphragmatic (e.g., unmyelinated vagal pathways originating in the dorsal motor nucleus of the vagus and terminating primarily below the diaphragm). Without this distinction, research and theory focused on the paired antagonism between the parasympathetic and sympathetic innervation to target organs. The consequence of an emphasis on paired antagonism was an acceptance in physiology and medicine of global constructs such as autonomic balance, sympathetic tone, and vagal tone.
More than 50 years ago, Hess (1954) proposed that the autonomic nervous system was not solely vegetative and automatic but was instead an integrated system with both peripheral and central neurons. By emphasizing the central mechanisms that mediate the dynamic regulation of peripheral organs, Hess anticipated the need for technologies to continuously monitor peripheral and central neural circuits involved in the regulation of visceral function.
THE VAGAL PARADOX
In 1992, I proposed that an estimate of vagal tone, derived from measuring respiratory sinus arrhythmia (RSA), could be used in clinical medicine as an index of stress vulnerability (see chapter 4). Rather than using the descriptive measures of heart rate variability (i.e., beat-to-beat variability) frequently used in obstetrics and pediatrics, the article emphasized that RSA has a neural origin and represents the tonic functional outflow from the vagus to the heart (i.e., cardiac vagal tone). Thus, it was proposed that RSA would provide a more sensitive index of health status than a more global measure of beat-to-beat heart rate variability reflecting undetermined neural and non-neural mechanisms. The article presented a quantitative approach that applied time-series analyses to extract the amplitude of RSA as a more accurate index of vagal activity. The article provided data demonstrating that healthy full-term infants had RSA of significantly greater amplitude than did preterm infants. This idea of using heart rate patterns to index vagal activity was not new, having been reported as early as 1910 by Hering. Moreover, contemporary studies have reliably reported that vagal blockade via atropine depresses RSA in mammals (Porges, 1986, 2007a).
In response to this article (Porges, 1992), I received a letter from a neonatologist who wrote that, as a medical student, he learned that vagal tone could be lethal. He argued that perhaps too much of a good thing (i.e., vagal tone) could be bad. He was referring, of course, to the clinical risk of neurogenic bradycardia. Bradycardia, when observed during delivery, may be an indicator of fetal distress. Similarly, bradycardia and apnea are important indicators of risk for the newborn.
My colleagues and I (Reed, Ohel, David, & Porges, 1999) further investigated this perplexing observation by studying the human fetus during delivery. We observed that fetal bradycardia occurred only when RSA was depressed (i.e., a respiratory rhythm in fetal heart rate is observable even in the absence of the large chest wall movements associated with breathing that occur postpartum). This raised the question of how vagal mechanisms could mediate both RSA and bradycardia, as one is protective and the other is potentially lethal. This inconsistency became the "vagal paradox" and served as the motivation behind the polyvagal theory.
With regard to the mechanisms mediating bradycardia and heart rate variability, there is an obvious inconsistency between data and physiological assumptions. Physiological models assume vagal regulation of both chronotropic control of the heart (i.e., heart rate) and the amplitude of RSA (Jordan, Khalid, Schneiderman, & Spyer, 1982; Katona & Jih, 1975). For example, it has been reliably reported that vagal cardioinhibitory fibers to the heart have consistent functional properties characterized by bradycardia to neural stimulation and a respiratory rhythm (Jordan et al., 1982). However, although there are situations in which both measures covary (e.g., during exercise and cholinergic blockade), there are other situations in which the measures appear to reflect independent sources of neural control (e.g., bradycardic episodes associated with hypoxia, vasovagal syncope, and fetal distress). In contrast to these observable phenomena, researchers continue to argue for a covariation between these two parameters. This inconsistency, based on an assumption of a single central vagal source, is what I have labeled the vagal paradox.
THE POLYVAGAL THEORY: THREE PHYLOGENETIC RESPONSE SYSTEMS
Investigation of the phylogeny of the vertebrate autonomic nervous system provides an answer to the vagal paradox. Research in comparative neuroanatomy and neurophysiology has identified two branches of the vagus, with each branch supporting different adaptive functions and behavioral strategies. The vagal output to the heart from one branch is manifested in RSA, and the output from the other branch is manifested in bradycardia and possibly the slower rhythms in heart rate variability. Although the slower rhythms have been assumed to have a sympathetic influence, they are blocked by atropine (Porges, 2007a).
The polyvagal theory (see chapters 2, 10, 11, and 12; Porges, 2001a, 2007a) articulates how each of three phylogenetic stages in the development of the vertebrate autonomic nervous system is associated with a distinct autonomic subsystem that is retained and expressed in mammals. These autonomic subsystems are phylogenetically ordered and behaviorally linked to social communication (e.g., facial expression, vocalization, listening), mobilization (e.g., fight-or-flight behaviors), and immobilization (e.g., feigning death, vasovagal syncope, and behavioral shutdown).
The social communication system (see description below of the social engagement system) involves the myelinated vagus, which serves to foster calm behavioral states by inhibiting sympathetic influences to the heart and dampening the hypothalamic-pituitary-adrenal (HPA) axis (Bueno et al., 1989). The mobilization system is dependent on the functioning of the sympathetic nervous system. The most phylogenetically primitive component, the immobilization system, is dependent on the unmyelinated vagus, which is shared with most vertebrates. With increased neural complexity resulting from phylogenetic development, the organism's behavioral and affective repertoire is enriched. The three circuits can be conceptualized as dynamic, providing adaptive responses to safe, dangerous, and life-threatening events and contexts.
Only mammals have a myelinated vagus. Unlike the unmyelinated vagus, originating in the dorsal motor nucleus of the vagus with pre- and postganglionic muscarinic receptors, the mammalian myelinated vagus originates in the nucleus ambiguus and has preganglionic nicotinic receptors and postganglionic muscarinic receptors. The unmyelinated vagus is shared with other vertebrates, including reptiles, amphibians, teleosts, and elasmobranchs.
We are investigating the possibility of extracting different features of the heart rate pattern to dynamically monitor the two vagal systems. Preliminary studies in our laboratory support this possibility. In these studies we have blocked the nicotinic preganglionic receptors with hexamethonium and the muscarinic receptors with atropine. The data were collected from the prairie vole (Grippo, Lamb, Carter, & Porges, 2007), which has a very high ambient vagal tone. These preliminary data demonstrated that, in several animals, nicotinic blockade selectively removes RSA without dampening the amplitude of the lower frequencies in heart rate variability. In contrast, blocking the muscarinic receptors with atropine removes both the low and respiratory frequencies.
CONSISTENCY WITH JACKSONIAN DISSOLUTION
The three circuits are organized and respond to challenges in a phylogenetically determined hierarchy consistent with the Jacksonian principle of dissolution. Jackson (1958) proposed that in the brain, higher (i.e., phylogenetically newer) neural circuits inhibit lower (i.e., phylogenetically older) neural circuits and "when the higher are suddenly rendered functionless, the lower rise in activity." Although Jackson proposed dissolution to explain changes in brain function due to damage and illness, the polyvagal theory proposes a similar phylogenetically ordered hierarchical model to describe the sequence of autonomic response strategies to challenges.
Functionally, when the environment is perceived as safe, two important features are expressed. First, bodily state is regulated in an efficient manner to promote growth and restoration (e.g., visceral homeostasis). This is done through an increase in the influence of mammalian myelinated vagal motor pathways on the cardiac pacemaker that slows the heart, inhibits the fight-or-flight mechanisms of the sympathetic nervous system, dampens the stress response system of the HPA axis (e.g., cortisol), and reduces inflammation by modulating immune reactions (e.g., cytokines). Second, through the process of evolution, the brainstem nuclei that regulate the myelinated vagus became integrated with the nuclei that regulate the muscles of the face and head. This link results in the bidirectional coupling between spontaneous social engagement behaviors and bodily states. Specifically, an integrated social engagement system emerged in mammals when the neural regulation of visceral states that promote growth and restoration (via the myelinated vagus) was linked neuroanatomically and neurophysiologically with the neural regulation of the muscles controlling eye gaze, facial expression, listening, and prosody (Figure 3.1; see Porges, 2007a, for review).
The human nervous system, similar to that of other mammals, evolved not solely to survive in safe environments but also to promote survival in dangerous and life-threatening contexts. To accomplish this adaptive flexibility, the human nervous system retained two more primitive neural circuits to regulate defensive strategies (i.e., fight-or-flight and death-feigning behaviors). It is important to note that social behavior, social communication, and visceral homeostasis are incompatible with the neurophysiological states and behaviors promoted by the two neural circuits that support defense strategies. Thus, via evolution, the human nervous system retains three neural circuits, which are in a phylogenetically organized hierarchy. In this hierarchy of adaptive responses, the newest circuit is used first; if that circuit fails to provide safety, the older circuits are recruited sequentially.
FIGURE 3.1. The social engagement system. Social communication is determined by the cortical regulation of medullary nuclei via corticobulbar pathways. The social engagement system consists of a somatomotor component (special visceral efferent pathways that regulate the muscles of the head and face; solid blocks) and a visceromotor component (the myelinated vagus that regulates the heart and bronchi; dashed blocks). Solid blocks indicate the somatomotor component. Dashed blocks indicate the visceromotor component. Reprinted from Porges (2007a), with permission from Elsevier.
Investigation of the phylogeny of regulation of the vertebrate heart (Morris & Nilsson, 1994; Taylor, Jordan, & Coote, 1999; see also chapters 2 and 10) has led to extraction of four principles that provide a basis for testing of hypotheses relating specific neural mechanisms to social engagement, fight-or-flight, and death-feigning behaviors:
• There is a phylogenetic shift in the regulation of the heart from endocrine communication to unmyelinated nerves and finally to myelinated nerves.
• There is a development of opposing neural mechanisms of excitation and inhibition to provide rapid regulation of graded metabolic output.
• A face–heart connection evolved as source nuclei of vagal pathways shifted ventrally from the older dorsal motor nucleus to the nucleus ambiguus. This resulted in an anatomical and neurophysiological linkage between neural regulation of the heart via the myelinated vagus and the special visceral efferent pathways that regulate the striated muscles of the face and head, forming an integrated social engagement system (Figure 3.1; for more details, see Porges, 2007a, and chapter 12).
• With increased cortical development, the cortex exhibits greater control over the brainstem via direct (e.g., corticobulbar) and indirect (e.g., corticoreticular) neural pathways originating in motor cortex and terminating in the source nuclei of the myelinated motor nerves emerging from the brainstem (e.g., specific neural pathways embedded within cranial nerves V, VII, IX, X, and XI), controlling visceromotor structures (i.e., heart, bronchi) as well as somatomotor structures (muscles of the face and head).
NEUROCEPTION: CONTEXTUAL CUEING OF ADAPTIVE, MALADAPTIVE PHYSIOLOGICAL STATES
To effectively switch from defensive to social engagement strategies, the mammalian nervous system needs to perform two important adaptive tasks: (1) assess risk, and (2) if the environment is perceived as safe, inhibit the more primitive limbic structures that control fight, flight, or freeze behaviors.
Any stimulus that has the potential for increasing an organism's experience of safety has the potential of recruiting the evolutionarily more advanced neural circuits that support the prosocial behaviors of the social engagement system.
The nervous system, through the processing of sensory information from the environment and from the viscera, continuously evaluates risk. Since the neural evaluation of risk does not require conscious awareness and may involve subcortical limbic structures (Morris, Ohman, & Dolan, 1999), the term neuroception (see chapter 1) was introduced to emphasize a neural process, distinct from perception, that is capable of distinguishing environmental (and visceral) features that are safe, dangerous, or life-threatening. In safe environments, autonomic state is adaptively regulated to dampen sympathetic activation and to protect the oxygen-dependent central nervous system, especially the cortex, from the metabolically conservative reactions of the dorsal vagal complex. However, how does the nervous system know when the environment is safe, dangerous, or life-threatening, and which neural mechanisms evaluate this risk?
Environmental Components of Neuroception
Neuroception represents a neural process that enables humans and other mammals to engage in social behaviors by distinguishing safe from dangerous contexts. Neuroception is proposed as a plausible mechanism mediating both the expression and the disruption of positive social behavior, emotion regulation, and visceral homeostasis (see chapter 1; Porges, 2007a). Neuroception might be triggered by feature detectors involving areas of temporal cortex that communicate with the central nucleus of the amygdala and the periaqueductal gray, since limbic reactivity is modulated by temporal cortex responses to the intention of voices, faces, and hand movements. Thus, the neuroception of familiar individuals and individuals with appropriately prosodic voices and warm, expressive faces translates into a social interaction promoting a sense of safety.
In most individuals (i.e., those without a psychiatric disorder or neuropathology), the nervous system evaluates risk and matches neurophysiological state with the actual risk of the environment. When the environment is appraised as being safe, the defensive limbic structures are inhibited, enabling social engagement and calm visceral states to emerge. In contrast, some individuals experience a mismatch and the nervous system appraises the environment as being dangerous even when it is safe. This mismatch results in physiological states that support fight, flight, or freeze behaviors, but not social engagement behaviors. According to the theory, social communication can be expressed efficiently through the social engagement system only when these defensive circuits are inhibited.
Other Contributors to Neuroception
The features of risk in the environment do not solely drive neuroception. Afferent feedback from the viscera provides a major mediator of the accessibility of prosocial circuits associated with social engagement behaviors. For example, the polyvagal theory predicts that states of mobilization would compromise our ability to detect positive social cues. Functionally, visceral states color our perception of objects and others. Thus, the same features of one person engaging another may result in a range of outcomes, depending on the physiological state of the target individual. If the person being engaged is in a state in which the social engagement system is easily accessible, the reciprocal prosocial interactions are likely to occur. However, if the individual is in a state of mobilization, the same engaging response might be responded to with the asocial features of withdrawal or aggression. In such a state, it might be very difficult to dampen the mobilization circuit and enable the social engagement system to come back on line.
The insula may be involved in the mediation of neuroception, since it has been proposed as a brain structure involved in conveying the diffuse feedback from the viscera into cognitive awareness. Functional imaging experiments have demonstrated that the insula plays an important role in the experience of pain and the experience of several emotions, including anger, fear, disgust, happiness, and sadness. Critchley (2005) proposes that internal body states are represented in the insula and contribute to states of subjective feeling, and he has demonstrated that activity in the insula correlates with interoceptive accuracy.
SUMMARY
The polyvagal theory proposes that the evolution of the mammalian autonomic nervous system provides the neurophysiological substrates for adaptive behavioral strategies. It further proposes that physiological state limits the range of behavior and psychological experience. The theory links the evolution of the autonomic nervous system to affective experience, emotional expression, facial gestures, vocal communication, and contingent social behavior. In this way, the theory provides a plausible explanation for the reported covariation between atypical autonomic regulation (e.g., reduced vagal and increased sympathetic influences to the heart) and psychiatric and behavioral disorders that involve difficulties in regulating appropriate social, emotional, and communication behaviors.
The polyvagal theory provides several insights into the adaptive nature of physiological state. First, the theory emphasizes that physiological states support different classes of behavior. For example, a physiological state characterized by a vagal withdrawal would support the mobilization behaviors of fight and flight. In contrast, a physiological state characterized by increased vagal influence on the heart (via myelinated vagal pathways originating in the nucleus ambiguus) would support spontaneous social engagement behaviors. Second, the theory emphasizes the formation of an integrated social engagement system through functional and structural links between neural control of the striated muscles of the face and the smooth muscles of the viscera. Third, the polyvagal theory proposes a mechanism—neuroception—to trigger or to inhibit defense strategies.
PART II
BIOBEHAVIORAL REGULATION DURING EARLY DEVELOPMENT
CHAPTER 4
Vagal Tone: A Physiological Marker of Stress Vulnerability
Routine medical procedures are often stressful and produce periods of physiologic and behavioral instability. However, given the same treatment, all children do not respond alike. Some children exhibit prolonged periods of instability, while other children are virtually insensitive to the treatment. Although clinicians are concerned with the potential vulnerability associated with stressful events, a standard approach does not exist to address both the measurement of stress and the indexing of stress vulnerability.
Research on stress has often focused on the description of events that are considered stressful (e.g., life stress scales) and not on the functional impact of these events on physiology. In contrast, in pediatrics, stress is assumed to be a physiologic construct that is observed when behavior becomes disorganized and homeostatic processes are disrupted. In clinical settings this is often labeled physiologic instability.
Although we are aware that there are individual differences in vulnerability to the same medical procedures, current definitions of stress emphasize either the treatment or the response to treatment and not the neurophysiologic status prior to treatment. However, it is this neurophysiologic state that may index the stress vulnerability of the child.
This chapter proposes a method to assess on an individual basis both the stress response and the vulnerability to stress. This method monitors the neural control of the heart via the vagus (i.e., vagal tone) as an index of homeostasis. The method permits the assessment of the effects of disrupting homeostatic processes (i.e., stress) and the vulnerability of homeostasis to disruption by various clinical treatments (i.e., stress vulnerability).
STRESS: A STATE OF AUTONOMIC NERVOUS SYSTEM COMPROMISE
The autonomic nervous system (ANS) regulates homeostatic function. The ANS is composed of two subsystems, the parasympathetic (PNS) and sympathetic nervous systems (SNS). The PNS and SNS represent neural systems that originate in the brainstem and contribute to the regulation of a variety of target organs, including the eyes, lacrimal glands, salivary glands, sweat glands, blood vessels, heart, larynx, trachea, bronchi, lungs, stomach, adrenal, kidney, pancreas, intestine, bladder, and external genitalia. In general, the PNS promotes functions associated with a growth and restorative system. In contrast, the SNS promotes increased metabolic output to deal with challenges from outside the body.
In general, when a visceral organ is innervated by both the SNS and PNS, the effects are antagonistic. For example, SNS neurons dilate the pupil, accelerate the heart, inhibit intestinal movements, and contract the vesical and rectal sphincters. The PNS neurons constrict the pupil, slow the heart, potentiate peristaltic movement, and relax the vesical and rectal sphincters.
The PNS deals primarily with anabolic activities concerned with the restoration and conservation of bodily energy and the resting of vital organs. This view was clearly stated by Cannon (1929a):
A glance at these various functions of the cranial division reveals at once that they serve for bodily conservation; by narrowing the pupil they shield the retina from excessive light; by slowing the heart rate they give the cardiac muscle longer periods for rest and invigoration; and by providing for the flow of saliva and gastric juice, and by supplying the necessary muscular tone or the contraction of the alimentary canal, they prove fundamentally essential to the processes of proper digestion and absorption by which energy-yielding material is taken into the body and stored. To the cranial division belongs the great service of building up reserves and fortifying the body against time of need and stress. (pp. 31–32)
Stimulation of the SNS prepares the individual for the intense muscular action required to protect and defend in response to external challenges. The SNS quickly mobilizes the existing reserves of the body. The eyes dilate, the rate and force of the heart contractility increase, blood vessels constrict, and blood pressure increases. Blood is drained from the intestinal reservoir to foster availability and transport of oxygenated blood to the skeletal muscles, lungs, heart, and brain. Peristalsis and alimentary secretion are inhibited, and sphincter contractions block urinary and rectal outlets.
The SNS and PNS are reciprocally innervated, and their responses are coordinated to provide the appropriate internal state to meet shifts in both internal and external demands. The PNS is modulated primarily by internal changes in the viscera. The SNS is primarily activated by exteroceptive impulses via somatic afferent fibers in response to changes in the external environment.
The PNS facilitates digestion and conserves energy by slowing heart rate. In the absence of external challenges (e.g., ambient temperature shifts, noise, pain, pyrogenic agents, etc.), the PNS optimizes the function of the internal viscera. In contrast, by increasing metabolic output to deal directly with external challenges, the SNS attempts to optimize the organism's relationship with the environment, Thus, increases or decreases in ambient temperature, noise, pain, and pyrogenic agents will produce attenuated PNS tone and increased SNS activity. Consistent with this functional description of the ANS, Gellhorn (1967) labeled the PNS as a trophotropic system and the SNS as an ergotropic system.
Early investigators defined the ANS as purely visceral motor (Langley, 1921). This limited definition did not acknowledge the contribution of visceral afferents. The afferent fibers accompany most visceral efferent fibers and form the afferent limb for visceral reflexes. The contemporary view of the ANS is that it is a complex system containing both peripheral efferent and afferent fibers as well as central neural structures, Thus, as Hess (1954) stated, the function of the ANS reflects more a visceral nervous system than a vegetative or automatic nervous system. It is because the ANS is an integrated system with both peripheral and central neurons that measurement of peripheral visceral activity provides a window to the brain structures that regulate visceral function and state.
The ANS responds to both internal and external stimuli. Although the ANS is often viewed as a motor system controlling visceral organs, most autonomic neurons are afferent. In the maintenance of bodily functions and in the reaction to stressful situations, ANS afferents are crucial. Afferent feedback from visceral organs often regulates PNS tone and has little impact on SNS tone. For example, distension of the stomach or stimulation of baroreceptors will result in reflexive increases in PNS tone. Afferent feedback from sensory organs produces a different response profile. Autonomic responses to external stimuli, including nociception or attention, produce a decrease in PNS tone. There is a complementary increase in SNS tone only if the stimulus is of high intensity and prolonged duration or is associated with conditions of nociception or intense stimulation. In response to metabolic demands, the two branches of the ANS often function synergistically to maximize cardiovascular output. For example, during exercise, there is a progressive decrease in PNS tone and a parallel increase in SNS tone. There are unique situations in which the autonomic response is characterized by either dual activation or dual inhibition. For example, in human adults, sexual arousal is characterized by a dual excitation.
The ANS is not merely a response system, quiescently awaiting challenges from the external environment. Rather, the ANS is continuously servicing the visceral afferents in an attempt to maintain homeostasis and promote physiologic stability. This regulatory process is primarily mediated by the PNS. Unfortunately, there are disease states that compromise the regulatory function. Some disease states (e.g., hypertension) are characterized by a depression of PNS tone with a compensatory SNS excitation. Other disease states (e.g., diabetes) are characterized by a depression of the PNS tone without eliciting a reciprocal SNS excitation, while other disease states may be characterized by dual inhibition.
The ANS is involved in the physiologic expression of stress. Shifts in ANS activity that disrupt homeostatic processes seem to characterize the common theme associated with physiologically based definitions of stress. More specific investigation of the literature suggests that the PNS, virtually independent of the SNS, regulates homeostatic processes and would thus be most sensitive to stress.
STRESS AND HOMEOSTASIS: NEW DEFINITIONS
There have been many definitions of stress. Because most definitions are structured in terms of the causal influences (i.e., stimulus or context) and not in terms of the variables commonly measured in a clinical setting, the definitions tend to be limited in their use in the medical environment. Often the definitions are circular because stress is defined both in terms of the context (e.g., medical treatment) and the response (e.g., behavioral and physiologic responses). For example, is the medical treatment stressing because it elicits an increase in blood pressure and heart rate? Or do increases in blood pressure and heart rate reflect stress independent of the specific medical treatment? Or do the physiologic responses reflect stress because the clinician assumes that there is stress associated with the medical treatment? The definition is further confounded because we assume that there are individual differences in responsiveness or vulnerability to the stressful event. Thus, the stressful treatment might not elicit a stress response in one patient while a nonstressful treatment might elicit a stress response in another patient.
Current definitions of stress are not very useful in the clinical setting. Even if stress were operationally defined by labeling the stressing stimulus as the stressor and the behavioral and physiologic response to the stressor as stress, at least two problems would remain: (1) the definitions of stress and stressor would be circular, and (2) there would be situations in which individual differences and state might mediate the degree of responsivity (i.e., stress) of a patient to constant medical treatments (i.e., stressor). For example, the same treatment that may physiologically compromise one patient may not produce a discernible behavioral or physiologic response in another patient or even in the same patient a second time. Alternatively, the same treatment that did not produce a response during the first administration may result in a massive physiologic compromise during subsequent administrations. Thus, stress must not be conceptualized simply in terms of the stressor and observed response but also in terms of the physiologic state of vulnerability of the patient at the time of treatment.
New definitions of stress and stress vulnerability can be derived and operationally defined based on the function of the ANS. Physiologically based measures can be objectively assessed within clinical settings with online monitoring of stress and stress vulnerability. In developing this approach, two essential areas must be discussed: (1) the rationale for evaluating specific autonomic variables as indices of stress, and (2) the measurement technology necessary to measure the autonomic indices of stress online in a clinical setting.
AN ANS DEFINITION OF STRESS: A RATIONALE
The ANS deals both with servicing the needs of the internal viscera and with responding to external challenges. The central nervous system mediates the distribution of resources to deal with internal and external demands. Perceptions and assumed threats to survival, independent of the actual physical characteristics of the stimulation, may promote a massive withdrawal of PNS tone and a reciprocal excitation of SNS tone. The trade-off between internal and external needs may be used in developing definitions of stress and homeostasis. Based on this model, stress and homeostasis are interdependent. Homeostasis reflects the regulation of the internal viscera, and stress reflects the subjugation of internal needs in response to external needs. Thus, measurement of PNS tone may provide the indexing variable for defining stress and stress vulnerability.
The concept of homeostasis is not new. Walter Cannon (1929b) coined the term and stated that "the coordinated physiological reactions which maintain most of the steady states in the body are so complex, and so peculiar to the living organism, that it has been suggested that a specific designation for these states be employed—homeostasis" (p. 400). Cannon's views are dependent on the earlier work of Claude Bernard. Bernard's construct of "le milieu interieur" included physiologic mechanisms responsible for the maintenance of the constancy of the internal environment. As Bernard's work evolved, he emphasized the dynamic and oscillatory nature of the nervous system in maintaining the "internal milieu" within a limited range (Bernard, 1878–79).
Homeostasis as a construct was never meant to reflect a static state. Rather it defined the dynamic feedback and regulation processes necessary for the living organism to maintain internal states within a functional range. Over time the concept has lost much of its rich meaning and has often been interpreted to represent a static internal level. Clinically, stasis or lack of endogenous variability in neurally mediated peripheral systems, such as gastric motility and heart rate, is a sign of severe physiologic compromise.
In the proposed model, the PNS fosters visceral needs (i.e., homeostasis) and the SNS responds to external challenges. Thus, status of the PNS state parallels homeostasis. Alternatively, withdrawal of PNS tone in response to a challenge may define stress, and PNS tone prior to the challenge may represent physiologic or stress vulnerability. With this physiologic model, SNS state is not the defining characteristic of stress or stress vulnerability; stress responses and stress vulnerability may be indexed in the absence of major shifts in SNS tone. It is important to note that in many situations with healthy children, the transitory withdrawal of PNS tone will be paralleled by an increased expression of SNS tone. In contrast, severely compromised children may not exhibit SNS reactivity and SNS tone might be low. Moreover, these children would have low PNS tone, virtually no PNS reactivity, and would be clinically assessed as being chronically stressed or exhibiting physiologic instability.
This view suggests that homeostasis may be defined as the autonomic state that fosters visceral needs in the absence of external challenge. This state would be defined by a high degree of PNS tone. Stress may be defined as the autonomic state that reflects a disruption of homeostasis. This state would be defined by the withdrawal of PNS tone. Thus, the degree of stress can be quantified on a physiologic level. Moreover, the chronic autonomic state before a clinical event would index the patient's stress vulnerability; individuals exhibiting problems of homeostasis will have the greatest stress vulnerability.
THE ASSESSMENT OF STRESS: VAGAL TONE MONITORING
Given the foregoing justification, it is necessary to identify and quantify an index of PNS activity. The most readily indexed measure of PNS activity is derived from the heart rate pattern. The amplitude of respiratory sinus arrhythmia (RSA) provides a validated and easily obtainable index of PNS tone via the cardiac vagus (Porges, 1986). In the following sections this will be described as an index of cardiac vagal tone. With modern technology it is possible to monitor online the changing influence of the vagus on the heart and to estimate shifts in general vagal tone (Porges, 1985).
Physiologic and behavioral processes are dependent on neural feedback. Information is received at the periphery and transmitted to the central nervous system, and appropriate physiologic reflexes or overt behaviors are emitted. Feedback loops typical of many homeostatic processes produce a rhythmic pattern characterized by phasic increases and decreases in neural efferent output to organs such as the heart. In many physiologic systems, efficient neural control is manifested as rhythmic physiologic variability, and within normal parameters the greater the amplitude of oscillation, the healthier the individual. Thus, the amplitude of rhythmic physiologic processes may index the status of the individual's nervous system and capacity to respond. In other words, the greater the amplitude of organized rhythmic physiologic variability, the greater the response potential or possible range of behavior. Individuals with attenuated physiologic variability would then exhibit a lack of physiologic and behavioral flexibility in response to environmental demands. This is observed in very ill infants. Thus, in terms of stress reactions, one would expect these individuals to lack the self-regulatory capacity to adjust rapidly to stressful stimuli.
Research in areas of cardiology, gerontology, physical therapy, and diabetology demonstrate that general PNS deficits are reflected in cardiac vagal tone. Moreover, stimulation of other PNS afferents results in reflexive increases in cardiac vagal tone (Cottingham, Porges, & Lyon, 1988; DiPietro & Porges, 1991). Since cardiac vagal tone reflects the general PNS input to the viscera, it may be used to monitor stress and index individual differences in stress vulnerability.
As a noninvasive measure of cardiac vagal tone, we have standardized a method to quantify the amplitude of RSA (Porges, 1986). RSA indexes the nervous system modulation of heart rate activity via the vagus. RSA reflects rhythmic vagal efferent influences on the cardiac pacemaker modulated by respiratory processes in the medulla. Heart rate patterns, like behavioral processes, are dependent on the status of the nervous system and the quality of neural feedback. Stress results in a disorganization of the rhythmic structure of both behavior and autonomic state. Thus, measures of cardiac vagal tone, such as the amplitude of RSA, provide an important window into the central control of autonomic processes and by inference into the central processes necessary for organized behavior.
Cardiac vagal tone is reflected in the amplitude of a heart rate rhythm associated with frequency of spontaneous breathing. This rhythmic process has been observed and studied for more than 100 years. Speculations regarding the neural mechanisms were reported as early as 1910, when a relation between RSA and vagal tone was proposed by Hering (1910). Hering clearly stated that "it is known with breathing that a demonstrable lowering of heart rate . . . is indicative of the function of the vagi." Derivations of this method with paced breathing techniques are currently used to diagnose peripheral neuropathy in diabetic patients. Contemporary research has provided empirical evidence that the amplitude of RSA accurately maps the efferent influence of the vagus nerve on the heart. Based on electrophysiologic studies of vagal efferents, it has been proposed that central respiratory drive gates the source nuclei of the vagal cardioinhibitory fibers (Jordan, Khalid, Schneiderman, & Spyer, 1982).
If cardiac vagal tone is a sensitive index of the functional status of the nervous system, then we would predict that individuals with greater vagal tone would exhibit a greater range of competent behaviors. We would also expect that conditions that compromise the central nervous system (e.g., medical complications, anesthesia, and illness) would result in an attenuation of vagal tone.
In building a model that relates cardiac vagal tone to stress, we first describe a global metaphor characteristic of homeostatic processes and then more specific pathways related to the vagus. Heart rate in a healthy human is not steady. The pattern of heart rate reflects the continuous feedback between the central nervous system and the peripheral autonomic receptors. The primary source of heart rate variability is mediated by phasic increases and decreases in neural efferent output via the vagus to the heart (Porges, McCabe, & Yongue, 1982). Under most conditions, like other measures of homeostatic control, the greater range of the phasic increases and decreases, the "healthier" the individual. For example, with high-risk neonates, there is an attenuation of the range of homeostatic function. Paralleling this process is a reduction in cardiac vagal tone (Fox & Porges 1985; Porges 1988).
Heart rate variability is a marker of the efficiency of neural feedback mechanisms and may index health status or the individual's capacity to organize physiologic resources to respond appropriately. Thus, the better the "organized" physiologic variability, the greater the range of behavior. Organized variability is due to dynamic feedback and thus is characterized by rhythmic deviations from a constant level. An optimum feedback system allows large deviations from the mean with negative feedback occurring at constant intervals. Neural regulation of the heart in healthy individuals is similar to the optimal feedback system already described. In the neural control of the heart, the vagus serves as the primary negative feedback mechanism. Thus, states of attenuated heart rate variability would have attenuated vagal influences on the heart. Returning to our metaphor, states characterized by attenuated vagal influences should be paralleled by reduced behavioral flexibility in response to environmental demands.
STRESS VULNERABILITY IN THE NEONATE: AN EXAMPLE OF VAGAL TONE MONITORING
Figure 4.1 illustrates 2 minutes of the heart rate pattern and the amplitude of RSA for two sleeping neonates. The top panel illustrates the pattern of a high-risk preterm neonate monitored at approximately term. The bottom panel illustrates the pattern of a healthy term neonate monitored within 36 hours of delivery. The top line on each panel illustrates the continuous heart rate over the 2 minutes. The bottom line on each panel represents the amplitude of RSA calculated for each sequential 10 seconds within the 2 minutes. Clarification of the vagal tone concept may be obtained by observing the differences between the top and bottom panel. Notice that even during sleep, heart rate is not constant. Although the heart rate levels are similar for the two neonates, the healthy term neonate has much greater beat-to-beat variability relative to the high-risk preterm. Close inspection of the two beat-to-beat patterns identifies a striking difference in the rapid changes in heart rate that occur every 1 to 3 seconds. These oscillations are associated with respiration and reflect cardiac vagal tone. Cardiac vagal tone is indexed by the amplitude of these rapid oscillations (i.e., RSA) and is reported in natural logarithm units.
FIGURE 4.1. Beat-to-beat heart rate and respiratory sinus arrhythmia (RSA). Top panel illustrates data from a high-risk preterm neonate monitored at approximately term. Bottom panel illustrates data from a healthy normal full-term neonate monitored within 36 hours of delivery. Data were collected during sleep. In both panels, the top line illustrates the beat-to-beat heart rate and the bottom line illustrates RSA calculated for sequential 10-second segments. RSA values are in natural logarithm units per millisecond squared.
FIGURE 4.2. Distribution of respiratory sinus arrhythmia (RSA) for normal full-term neonates and neonates in the NICU. RSA values are in natural logarithm units per millisecond squared.
Figure 4.2 illustrates the frequency distributions of the amplitude of RSA for both high-risk and normal full-term neonates. The subject samples were 125 full-term neonates and 112 neonatal intensive care unit (NICU) residents. The full-term neonates were all residents of a full-term normal nursery and were tested during the second day following delivery. RSA values for the premature NICU neonates were derived during the first recording made with the neonates not using the ventilator and breathing room air. RSA was assessed during sleep when the NICU neonates were between 35 and 37 weeks corrected gestational age.
From the data in Figure 4.2, it is obvious that high-risk neonates, as a group, have significantly lower vagal tone than do full-term neonates (F[1, 235] = 226.3, p < .0001). Since respiration frequency might influence the quantification of RSA, respiration was monitored on a subset of 47 full-term and 62 NICU neonates. Respiration was significantly faster for the NICU neonates (F[1, 107] = 23.5, p < .0001). However, even when this significant influence was removed with analysis of covariance, there was still a highly significant difference between the two groups (F[1, 107] = 82.2, p < .0001). The group classification (full-term vs. NICU) accounted for 53.1% of the variance in the statistical model. When the influence of respiration was removed, group classification accounted for 43.7%.
Research in our laboratory suggests only subtle maturational increases and great stability of these estimates under standard sampling conditions during residency within the NICU. For example, a sample of 16 preterm neonates was tested during sleep on 5 separate days starting at least 1 day following delivery. Although there was a significant relationship between severity of clinical condition and RSA (i.e., healthier neonates had higher amplitude RSA), the average correlation among the RSA values approached 0.9.
This example illustrates how monitoring of vagal tone in the NICU by quantifying RSA may provide a sensitive index of stress vulnerability. Data from our laboratory and the laboratories of others demonstrate that the more compromised NICU residents have lower cardiac vagal tone. Consistent with the model of stress and stress vulnerability, the NICU neonates have limited PNS tone to regulate their internal state and are simultaneously confronted with demands from the environment, including the necessity to thermoregulate and to deal with sensory stimuli including medical procedures assumed to be painful.
The vagal system is responsive to the changing needs of the organism. Often the vagal system will react by selectively increasing or decreasing its influence on the periphery. This might be observed as a withdrawal of vagal tone to increase heart rate to support metabolic demands or an increase in vagal tone to regulate digestive polypeptides and gastric motility. The adaptive success of the infant is not based merely on the tonic level of the ANS but on the ability of the ANS to respond appropriately to environmental and interval challenges. For example, during painful medical manipulations, such as circumcision, cardiac vagal tone is suppressed (Porter, Porges, & Marshall, 1988). In contrast, during gavage feeding of premature neonates, cardiac vagal tone is increased (DiPietro & Porges, 1991). If infants exhibited a vagal tone increase during gavage feeding and a depression of vagal tone below pregavage levels following feeding, they were discharged approximately 2 weeks earlier than were infants who did not exhibit this response. These effects were independent of birth weight, gestational age at birth, and other clinical factors. Although pregavage vagal tone was not related to the response pattern or discharge, it did predict weight gain trajectories. Thus, vagal reactivity provides another dimension related to clinical risk.
Measures of vagal tone provide an important window to the central modulation of autonomic function. Measures of vagal tone during sleep or nonchallenging conditions provide an index of normal homeostatic feedback, while measures during sensory or cognitive challenges provide an indication of adaptive functioning. Thus, low levels of vagal tone are associated with high-risk populations, while atypical vagal responsivity even in infants with normal basal levels of vagal tone seems to identify a subset of infants who have behavioral and regulatory problems (DeGangi, DiPietro, Greenspan, & Porges, 1991; DiPietro & Porges, 1991).
VAGAL TONE SHIFTS DURING STRESS: THE COST OF DOING BUSINESS
The ANS has many physiologic responsibilities. It must regulate the blood pressure to ensure that enough blood reaches the brain. It also monitors the blood gases. If there are shifts in oxygen and carbon dioxide, changes in cardiopulmonary parameters are immediately implemented by the direct neural modulation of the heart, vasomotor tone, and lungs. While these cardiopulmonary processes are being regulated, the ANS is also controlling digestion and metabolism. The actions of the ANS are related to life support: ergotropic (i.e., work) and trophotropic (i.e., growth) functions (Gellhorn, 1967).
The vagal system is critical to the regulation of both ergotropic and trophotropic process. Increases in vagal tone not only result in increases in metabolic output but modulate the digestive polypeptides and gastric motility (Uvnas-Mober, 1989). The vagus is also critical in the facilitation of trophotropic process. The vagus can have direct inhibitory influences on sympathetic excitation of the myocardium (Levy, 1977). Moreover, the limbic system, assumed by psychophysiologists to modulate autonomic arousal solely through sympathetic excitation, has direct inhibitory influences on the cells of origin of the vagus (Schwaber, Kapp, & Higgins, 1980). The brainstem regions controlling the vagal efferents act to express greater vagal tone, thus maintaining trophotropic states, or to allow withdrawal of vagal tone to facilitate the immediate mobilization of the organism.
Research with other species demonstrates that cardiac vagal tone increases during development (Larson & Porges, 1982). Paralleling this increase in vagal tone are increases in self-regulatory and exploratory behaviors. In research with infants high cardiac vagal tone is associated with better visual recognition memory (Linnemeyer & Porges, 1986; Richards, 1985). An overview chapter summarizes the research on cardiac vagal tone and affect (Porges, 1990).
DiPietro and Porges (1991) also evaluated in preterm neonates the relationship between cardiac vagal tone and behavioral reactivity to gavage feeding. In this study, individual differences in cardiac vagal tone were significantly correlated with behavioral reactivity to the gavage method of feeding, Similarly, Huffman et al. (1998) observed that 3-month-old infants with high cardiac vagal tone habituated more rapidly to novel visual stimuli and exhibited more sustained attention than did infants with low cardiac vagal tone.
Drugs that depress vagal tone also seem to have a degrading effect on sustained attention, For example, in a study evaluating the effects of atropine sulfate on sensorimotor performance, we reported not only dose-dependent depression of vagal tone but also dose-dependent decrements in performance (Dellinger, Taylor, & Porges, 1987). We also have conducted research on the effects of inhalant anesthesia on the parallel between alertness and cardiac vagal tone. Inhalant anesthesia depressed cardiac vagal tone, and as the patients regained consciousness, there was a parallel increase in cardiac vagal tone (Donchin, Feld, & Porges, 1985).
With the ability to monitor cardiac vagal tone by quantifying the amplitude of RSA, we have been able to obtain a more accurate assessment of vagal mechanisms and the relationship between vagal tone and autonomic reactivity. Studies using the vagal tone index support the hypothesis that vagal tone may index stress and stress vulnerability. Porter et al. (1988) demonstrated, in a sample of normal newborns, massive withdrawal of cardiac vagal tone during circumcision. Moreover, individual differences in cardiac vagal tone were correlated with heart rate reactivity to circumcision. Neonates with higher cardiac vagal tone exhibited not only larger heart rate accelerations but also lower fundamental cry frequencies during surgical procedures. Porter and Porges (1988) also demonstrated in premature infants that individual differences in cardiac vagal tone were related to heart rate responses during lumbar puncture procedures.
CONCLUSION
There is a physiologic basis for defining stress and stress vulnerability. The concepts of stress and homeostasis are interdependent and manifested in the activity of the PNS. In contrast to traditional models of stress, the PNS is proposed as the modulator of stress vulnerability and reactivity. The model proposed suggests that accurate monitoring of PNS state will provide a window allowing the assessment of stress.
A physiologic justification is presented for the use of PNS, rather than or in addition to SNS, activity in the description of both stress vulnerability and stress reactivity. The quantification of cardiac vagal tone from spontaneous heart rate oscillations is described as a method for assessing the changing PNS state. The quantification of cardiac vagal tone provides a standard instrument with statistical parameters that are comparable between patients and throughout the life span. The method is not dependent on stages of motor or cognitive development and thus is practical for use even with neonates. This noninvasive method will allow the assessment of the stressful impact of various clinical treatments on the young infant and permit the identification of individuals with vulnerabilities to stress.
In conclusion, the usefulness of contemporary definitions of stress is limited, in part by circularity and in part by a tendency within stress research to focus narrowly on the contribution of the SNS. Cardiac vagal tone, measured by quantifying the amplitude of RSA, is proposed as novel index of stress vulnerability and reactivity with applications in all branches of medicine, and with particular value in pediatrics.
CHAPTER 5
The Infant's Sixth Sense: Awareness and Regulation of Bodily Processes
Life is a sensory experience. During every moment of our lives, we experience the world through our varied sensory systems. Sensory experiences drive our behavior and contribute to the organization of our thoughts and emotions. Immediately after birth, the infant is bombarded with a variety of new sensory stimuli. These provide important information about the characteristics and potential demands of the baby's new environment. The infant must immediately detect, discriminate, and adapt to this information. Successful adaptation to the rapidly changing environment and the ability to cope with changing demands depend on the infant's ability to detect and interpret sensory information. Thus, when we study infant behavioral patterns, vocalizations, and physiological reactivity, we attempt to understand how the young infant uses sensory systems to detect information from the environment and to integrate this information into motor, affective, and cognitive schema to successfully interact and adapt to a changing environment.
We have learned that humans have five primary sense modalities: smell, vision, hearing, taste, and touch. We know that even a newborn can respond to these sensory modalities. These responses are obvious to the parent and clinician (although only a few decades ago, scientists were unaware of the sensory capacities of young infants). However, this traditional method of categorizing sensory information does not account for the vast amount of sensory information being conveyed to the brain from the numerous sensors located inside our body. Even current clinical models of infant regulation (e.g., Ayres, 1972; Greenspan, 1991) that emphasize the importance of sensory processing in the emotional and cognitive development of the infant, and individual differences among infants in the ease with which they detect and interpret sensory information, focus primarily on three sense modalities that describe the external environment—that is, touch, vision, and hearing. These models do not deal with internal sensations that provide information about the physiological regulation.
Although neurophysiologists and neuroanatomists describe sensory systems that regulate our internal organs, this research has had little influence on either our common language or the clinical terminology we use to describe bodily processes. At present, there are only a few easily understood descriptors that characterize internal senses and states—for example, pain, nausea, and arousal. Yet in spite of this linguistic handicap, our experiences provide us with an awareness of bodily sensations and an appreciation of how these sensations can contribute to mood state and psychological feelings.
Missing from our language and our science is the ability to describe internal states. In our day-to-day interactions we choose vague terms, such as "feelings," to describe the psychological consequences of bodily changes. Behavioral scientists often attempt to objectify these terms by operationalizing concepts such as state, mood, and emotion with verbal reports and elaborate coding systems. Clinical practitioners infer these feelings and use terms descriptive of emotional tone. However, whether we are talking about feelings, emotions, states, or moods, we are always attempting to describe the internal states that are continuously being monitored and regulated by the nervous system.
The goal of this chapter is to introduce an additional sense modality that monitors bodily processes. A variety of terms may be used to describe this sensory system. Classic physiology describes this sensory system as interoception. Interoception is a global concept which includes both our conscious feelings of and unconscious monitoring of bodily processes, Interoception, like other sensory systems, has four components:
1. Sensors located in various internal organs to "sense" internal conditions;
2. Sensory pathways that convey information to the brain regarding the internal conditions;
3. Brain structures to interpret sensory information and organize systems to respond to the changing internal conditions; and
4. Motor pathways that communicate from the brain back to the internal organs that contain the sensors to change directly the state of the internal organ. Brain structures evaluate interoceptive information, categorize it, associate it with other sensory information, and store the associations in memory.
INTEROCEPTION IS THE SIXTH PRIMARY SENSE
The five classic senses already described are categorized based on the sensors located on the external surface of the body, or exteroceptors. However, we are aware that the external senses (e.g., vision, audition, sound, touch, smell, and taste) are not the sole source of stimulation directing the infant's behavior, thoughts, and emotions. The ability to sense internal states and bodily processes—through interoceptors located on the heart, stomach, liver, and other organs inside the body cavity—constitutes a sixth sense that is crucial to the infant's survival.
This sixth sense represents a functional awareness, with both conscious and unconscious dimensions, of what is happening inside the body. For example, on a conscious level, digestive processes may provide sensory information that the infant interprets as hunger when the stomach is empty, or as pain when the stomach is severely distended due to gas. The cardiovascular and respiratory systems also provide conscious feedback. Alertness changes as a function of shifts in both blood pressure associated with posture (when, for example, a baby brightens when picked up and held upright against a parent's shoulder) and blood gas concentrations of carbon dioxide and oxygen. On an unconscious level, internal organs have sensors which send continuous information to brain structures. This unconscious awareness fosters stability (i.e., homeostasis) in internal physiology by rapidly adjusting to support specific motor behaviors and psychological processes.
Although bodily sensations are paramount to the infant's successful survival, developmental specialists are currently more concerned with the infant's capacity to sense external stimuli. For example, clinical assessment tools, such as neurological and neuropsychological examinations, focus only on the processing of external stimulation. Similarly, our current childrearing and intervention strategies are not at all geared to helping young children sense their internal physiological states. We do not provide infants and young children with descriptive or symbolic tools to represent internal states, nor are caregivers taught to perceive specific behavioral or physiological indicators of gradations in the infant's bodily sensations. This remains the case even though we know that the status of bodily functions (such as digestion) and infants' reactions to difficulties in these processes (such as colic) are derived via important sensory systems. Moreover, the sensory information from continuously monitoring bodily functions can influence the infant's ability to perform specific behaviors, perceive external stimuli, and organize information into mental representation of cognitions and emotions.
EVALUATING SENSORY PROCESSING
For the five classic sensory modalities, we can evaluate the competence of the child to process sensory information by direct observation of behavior and through verbal reports. We can observe adaptive and dysfunctional strategies. We can identify problems by observing hyporesponsiveness and hyperresponsiveness in response to specific sensory challenges. We can evaluate developmental patterns in the child's ability to integrate sensory information. In addition, we can evaluate intervention procedures delivered by professionals such as speech and hearing specialists, ophthalmologists, nurses, occupational therapists, physical therapists, psychologists, psychiatrists, and pediatricians.
In contrast, interoceptive competence has not been systematized. Other than estimates of pain severity, there are no methods to quantify perception of bodily processes or to test unconscious interoceptive feedback. There are no scales to identify developmental landmarks.
But whether or not we know how to describe or measure them, sensations from inside the body are a strong influence on the infant's behavior in the world. From birth, the infant's need of sleep, food, water, and warmth are monitored via internal sensors. This information drives much of the infant's behavior. The infant's behavior then provides cues to the caregiver. In other words, stimulation of specific sensors inside the body of the young infant results in behavioral responses that prompt the caregiver to interact with the infant, to comfort, and to reduce the cause of these bodily sensations. For example, feeding the infant reduces hunger, burping relieves flatulence after feeding, and sucking may stimulate digestion and reduce constipation.
INTEROCEPTION: THE INFRASTRUCTURE OF HIGHER ORDER BEHAVIOR
Interoception is dependent on a complex feedback system that starts with sensors located in various body organs and ends with the higher order social interaction with the caregiver. Faulty sensors or a dysfunction in any component of the sensory system (i.e., sensor, sensory pathway to the brain, motor pathway from the brain, or areas in the brain that interpret the sensory information and control the motor output to the organ) may not only contribute to physiological problems but also negatively impact on the psychological and interactive experiences of the infant. Thus, the quality of interoceptive processes may contribute to individual differences in information processing (e.g., cognitive processes), emotional expressiveness, and social behavior.
I have conceptualized the dependency of complex behaviors on successful bodily processing in a hierarchical model with four levels (see Porges, 1983). Each level requires successful functioning on the preceding level of organization. Although the model includes complex social behaviors, the substrate of the model depends on the organizational competence of the nervous system.
• Level I is characteristic of homeostatic processes of physiological systems regulating the internal organs. Homeostatic regulation requires the bidirectional interoceptive process of monitoring and regulating the internal organ via sensory and motor pathways between the brain and the internal organ.
• Level II processes require cortical, conscious, and often motivated influences on the brainstem regulation of homeostasis.
• Level III processes are observable behaviors that can be evaluated by the quantity, quality, and appropriateness of motor behavior.
• Level IV reflects the coordination of behavior, emotional tone, and bodily state to successfully negotiate social interactions.
This model assumes that complex behavior, including social interactions, depends on physiology and how appropriately the nervous system regulates bodily processes. In this model, interoception becomes the foundation of physical, psychological, and social development. Interoception serves as the neurophysiological substrate of the higher processes, included in Level III and Level IV, that have been elaborated by many other researchers, practitioners, and theorists of child development.
LEVEL I PROCESSES: PHYSIOLOGICAL HOMEOSTASIS
Underlying the vague concept of "feelings" is a physiological process that depends on interoception. By explaining and measuring the functional regulation of physiological processes dependent on interoceptive mechanisms, we can identify functional vulnerabilities in the infant's ability to regulate on the most basic level. If the infant is insensitive to his or her own bodily calls for care, nurturance, and protection, how will the infant appropriately function and respond to social needs?
Level I processes provide the physiological mechanisms for state regulation, including emotional regulation and expression. They also provide the infrastructure for the child's successful interaction with the challenging social demands of the world.
In the proposed hierarchical model, Level I processes represent the successful regulation of internal bodily process via neural feedback systems. To maintain homeostasis, interoceptors originating in the body cavity (e.g., gastric, hepatic, enteric, cardiac, vascular, and pulmonary systems) transmit information via nerves to brainstem structures. The brainstem structures interpret the sensory information and regulate the internal physiological organs. They do this by stimulating nerves that either directly control internal organs (e.g., increase or decrease heart rate, constrict or dilate blood vessels, inhibit or facilitate peristaltic activity, etc.) or indirectly manipulate the organs by releasing specific hormones or peptides (e.g., adrenalin, insulin, oxytocin, vasopressin, gastrin, somatostatin, etc.).
Level I is associated with the organization and neural feedback mechanisms that characterize the maintenance of homeostasis. These homeostatic processes can shut down when either internal conditions or external challenges require maximum output of energy. For example, fever, severe thermo-stress, extreme emotional distress, and aerobic exercise can reflexively inhibit Level I feedback systems. States associated with severe illness (e.g., physiological compromise and instability) are also characterized by a down-regulation of the neural control of bodily processes. Alternatively, up-regulation may occur when the interoceptors are directly stimulated (e.g., the filling of the stomach with food) or when other sensory modalities reflexively influence bodily processes. For example, the smell of appetizing food initiates signals from the nose to the brainstem structures that in turn stimulate glands in the mouth and stomach to produce digestive secretions even before the food enters the mouth.
LEVEL II PROCESSES: COST OF DOING BUSINESS
The autonomic nervous system is the division of the nervous system that senses the condition of internal organs and regulates their activity. The autonomic nervous system deals with: (1) servicing the needs of the organs inside the body, and (2) responding to external challenges. We can define adaptive behavioral strategies and homeostasis in terms of the child's ability to trade off between internal and external needs. Based on this model, homeostasis and response strategies to environmental demands are interdependent. Homeostasis reflects the regulation of the physiological conditions within the body. Response strategies reflect the stage when internal needs become less important than external needs—when the baby (fed, burped, and changed) is ready and eager to interact with the world of people and things.
The autonomic nervous system has two branches, the sympathetic and the parasympathetic. In general, the parasympathetic branch promotes functions associated with growth and restoration. In contrast, the sympathetic branch promotes increased output of energy to deal with challenges from outside the body. When there are no environmental demands, the autonomic nervous system services the needs of internal organs to enhance growth and restoration. However, in response to environmental demands, homeostatic processes are compromised and the autonomic nervous system supports increased output of energy, by down-regulating parasympathetic function and often stimulating sympathetic function to deal with these external challenges.
The central nervous system mediates the distribution of resources to deal with internal and external demands. Perceptions and assumed threats to survival (independent of the actual physical characteristics of the stimulation) may promote a massive withdrawal of parasympathetic tone and a reciprocal excitation of sympathetic tone. This trade-off between internal and external needs is monitored and regulated by the central nervous system.
Level II represents the integration of interoceptive systems with other sensory modalities and psychological processes. Unlike the reflexive integration described in Level I, Level II involves higher brain processes. Level II processes include voluntary approaches to the source of stimulation or an awareness of the need to problem solve and engage in information processing. To foster the contact with the stimulus or to process information, the internal bodily state is changed. Level II is characterized by the appropriate adjustment (i.e., gradations in inhibition) of homeostatic processes during states of attention, the processing of information, and social behavior.
When other senses—for example, hearing, sight, or touch—are stimulated, the autonomic responses are a secondary process. Under these conditions, after the baby detects sensory information, his or her brain structures regulate autonomic organs to facilitate the processing of the sensory information. These physiological states may support the baby's ability simply to pay attention to the sensory stimulus, or, by increasing metabolic output, the physiological state may support the child's physical movement toward or away from the stimulus.
Sensory information from the external environment triggers changes in internal regulation that are maintained via accurate interoception. Without accurate interoception, the down-regulation of internal physiological processes may compromise survival—for example, by inhibiting digestion or by disturbing electrolyte or blood gas levels. Defects in interoception may also be at the base of regulatory disorders (Greenspan, 1991). Regulatory disorders reflect a broad spectrum of dysfunction in young children and include difficulties in bodily processes such as eating and sleeping, difficulties in processing sensory and affective information, and difficulties in state regulation.
PHYSIOLOGICAL AND BEHAVIORAL HOMEOSTASIS: PARALLEL CONCEPTS
As a construct, physiological homeostasis is consistent with the behavioral homeostasis observed by Greenspan (1991). Greenspan has described a developmental period from birth to 3 months during which the infant masters homeostatic processes. In this model, homeostasis requires the appropriate regulation of sleep and behavioral states as well as the ability to incorporate appropriate visual, auditory, and tactile stimulation. Thus, children defined as having regulatory disorders have difficulties in sleep, feeding, and sensory integration.
Greenspan's model, however, focuses on the external sensory modalities—hearing, sight, and touch. I am suggesting that physiological homeostasis (Level I) and the regulation of physiological homeostasis to support sensory processing of environmental stimuli (Level II) are necessary substrates for the behavioral homeostasis. In other words, the regulatory disorders defined by Greenspan may have a physiological substrate (Porges & Greenspan, 1991). Empirical research provides support for this hypothesis (e.g., DeGangi, DiPietro, Greenspan, & Porges, 1991; Porges, Doussard-Roosevelt, Portales, & Suess, 1994). We are demonstrating that physiological measures of homeostasis are related to behavioral problems in infants. Our findings suggest the possibility that clinicians will be able to use physiological measures that reflect interoceptive competence diagnostically to identify Level I and Level II vulnerabilities in infants and young children.
ASSESSMENT OF LEVEL I AND LEVEL II PROCESSES
In general, homeostatic processes are regulated by the parasympathetic nervous system via the vagus nerve, a large nerve with several branches enabling bidirectional communication between brain structures and internal organs. The vagus, with its sensory and motor pathways, is the primary component of the interoceptive system. The vagus and its branches account for approximately 80% of the parasympathetic nervous system. Approximately 80% of the vagal fibers are sensory and thus directly service interoceptors within the body cavity. Thus, measurement of vagal activity provides information on interoception in maintaining homeostasis (i.e., Level I processes) and the regulation of homeostasis to support environmental challenges (i.e., Level II processes).
It is possible to monitor vagal activity by quantifying specific rhythmic changes in heart rate (see Porges, 1992). Level I processes may be evaluated by measuring vagal control of the heart during rest or sleep; this provides a measure of the infant's interoceptive capacities to maintain homeostatic control. Level II processes may be evaluated by measuring the change in vagal control of the heart during environmental challenges; this provides a measure of the infant's capacity to down-regulate the vagal system to deal with environmental demands.
Our research program provides data supporting the hypothesis that the ability to sense and regulate internal physiological state is at the base of competencies in higher order behavioral, psychological, and social processes. Currently, we are developing laboratory procedures to profile the infant's capacity to regulate internal physiological systems during a variety of sensory processing demands. Our long-term goal is to provide a standardized clinical instrument to evaluate interoception. This instrument would complement neurological, neuropsychological, and other sensory evaluations. The assessment would index interoceptive processes through the measurement of vagal influences on the heart by quantifying rhythmic changes in the beat-to-beat heart rate pattern (i.e., respiratory sinus arrhythmia as a measure of cardiac vagal tone). The instrument will have the capacity to evaluate two dimensions of interoception:
1. The capacity to monitor and maintain homeostasis in the absence of environmental challenges (i.e., Level I processes); and
2. The capacity to alter homeostasis to support behaviors required by environmental challenges (i.e., Level II processes).
The ability to measure interoception, the sixth sense, opens a new window to the infant's sensory experiences. This window allows us to observe and to understand the internal feelings of the infant and how these internal states change during illness, mental processing, and social behavior.
CHAPTER 6
Physiological Regulation in High-Risk Infants: A Model for Assessment and Potential Intervention
Birth is the greatest challenge to human survival. When an infant is born, the supportive environment of the womb is gone. Parturition functionally disrupts the fetus's dependency on maternal physiology and expels the fetus from this secure environment. Thus, birth marks a transition as the management of autonomic regulation shifts from the maternal placental-fetal system to the newborn. In this demanding environment, the newborn must have skills to regulate autonomic processes (e.g., breathe, feed, digest, thermoregulate, etc.) and to communicate autonomic state needs to caregivers (e.g., cry). Within minutes of birth, these skills are challenged. Difficulties in expressing physiological competence are life-threatening. Even with healthy full-term newborns, there is great concern by health care professionals and parents about an infant's physiological regulation competency in negotiating this complex transition. Postpartum adaptation is more difficult for the high-risk infants who experience a variety of risk factors, including prematurity and delivery complications. These high-risk infants, often limited by a less mature or a damaged nervous system, are compromised in performing these self-regulatory tasks.
The study of the high-risk neonate provides a real-life laboratory to evaluate the unique contribution of autonomic regulation to development. Research can evaluate the relation between specific physiological vulnerabilities and subsequent develop mental problems in behavioral organization, social behavior, and cognitive function.
A model of neural regulation of autonomic processes that focuses on the high-risk neonate might address two research questions. First, is it possible to assess the relative risk that an individual newborn faces following the birth process; and second, how do we help the high-risk newborn negotiate the transition from a physiological dependency on maternal systems to the physiological self-regulation required in the extrauterine world?
Successful adaptation of the newborn to the extrauterine environment requires a dynamic and complex repertoire of responses. These responses occur on several levels. Although self-regulatory physiological strategies require complex neurophysiological systems, involving feedback between the brain and peripheral physiology, several systems may be monitored by careful visual surveillance of the newborn. For example, the Apgar scale (Apgar, 1953) codifies the status of physiological self-regulation via a standard observational scale. Similarly, neurological examinations during the newborn period assess neural function through the systematic elicitation of observable reflexes.
In the day-to-day care of the high-risk infant, clinical management attempts to compensate for the immature or compromised nervous system and its limited abilities to regulate physiological homeostasis. For example, radiant heaters use the neonate's core temperature in a feedback loop to compensate for an inability to thermoregulate; ventilators are used to ensure sufficient oxygen when the respiratory system is either too premature or depressed; orogastric or nasogastric (i.e., tube) feeding is used to compensate for the neonate's inability to actively suck and to coordinate sucking, swallowing, and breathing.
The attentiveness of staff in the neonatal intensive care unit (NICU) emphasizes the importance of observable physiological systems, such as body movements, breathing, thermoregulation, and sucking to detect shifts in clinical status. Thus, although technology has contributed to the evaluation of physiological self-regulation through specialized biomedical monitoring equipment (e.g., computerized oxygen saturation, blood pressure, temperature, and heart rate monitors), the primary clinical indicators of autonomic regulation that promote homeostasis are still obtained through clinical observations.
Before the infant can master complex behavioral interactions with the environment, the infant must competently regulate autonomic processes. A goal of mammalian development is to become independent of the caregiver. There are several constraints on the development of self-regulatory skills, including the status of neurons, neurophysiological systems, motor behavior control, and the availability and contingency of socioenvironmental stimulation. Although assessments of self-regulatory skills usually focus on global levels of motor and social behavior, these abilities are dependent on physiological systems. In turn, physiological systems are dependent on a neuronal substrate, which provides the boundary conditions. If there is damage to neural tissue, the ability to regulate both motor and visceral processes is limited. Neurons can be damaged by hypoxia, fever, trauma, and other insults, such as drugs. Although neurons cannot be measured noninvasively in the infant, the functional output of populations of interconnecting neurons produce physiological responses such as sucking, breathing, and heart rate, which are easily monitored.
SELF-REGULATION AND THE NERVOUS SYSTEM
Implicit in the evaluation of clinical status in the newborn is the assumption that the assessment reflects the quality of nervous system function. Although many assessments do not require detailed physiological monitoring, the systems observed (e.g., regularity of respiration, body movements, sucking responses, coloration of skin, etc.) provide insight into the competence of the nervous system and its ability to orchestrate complex self-regulatory physiological processes. Thus, underlying most assessment strategies is the assumption that the nervous system provides the management skills necessary to regulate internal physiological systems and plays a pivotal role in determining the newborn's success in navigating through the changing environment.
Self-regulation characterizes physiological systems. Weiner (1948) proposed a model of nervous system self-regulation to explain homeostasis. According to Weiner, homeostasis is an emergent property of a system that, via bidirectional communication, monitors and regulates the status of an organ to maintain an output level within a specific functional range. As illustrated in Figure 6.1, the system would include a central regulator that would determine the motor output to an organ after interpreting the information from the sensor (e.g., afferent feedback) that monitors the status of the organ.
To maintain physiological homeostasis, sensory pathways originating in peripheral organs (e.g., chemoreceptors and baroreceptors in the carotid sinus) convey information regarding physiological status, and motor pathways (e.g., vagal and sympathetic pathways to the heart) change the output of peripheral organs. The sensory pathways from visceral organs originate in the periphery and usually terminate in the brainstem; however, many of the motor pathways originate in the brainstem and terminate in the periphery.
Physiological systems could be described as being composed of sensors that input information about the external (outside the body) and internal (within the body) environments, motor systems that control behavioral and visceral activity, and an integrative mechanism that evaluates the input from the sensors and determines the specifics of the motor output.
FIGURE 6.1. Schematic representation of bidirectional communication between central regulator and peripheral organ characteristic of physiological homeostasis.
The study of normal psychological and behavioral development emphasizes the importance of the external environment. For example, individual differences in development are often associated with socioeconomic status, family function, nutrition, and stress factors. In contrast, partially because psychopathology can be observed in "healthy" environments, the study of psychopathological development emphasizes the importance of the internal environment (e.g., the brain and neural regulation of physiological systems). For example, pathological development is often associated with problems in brain development and dysfunctional physiological systems. This interest in organismic variation promotes research questions evaluating mechanisms through which feedback from visceral organs contributes to the organization and development of emotional, cognitive, and behavioral processes.
Self-Regulation: A Negative Feedback System
Physiological systems regulating visceral state (e.g., heart rate, temperature, blood pressure) are self-regulatory. Self-regulatory systems adjust output to the changing input through a process known as feedback. When the feedback opposes the state of the system, it is known as negative feedback. When the feedback augments the state of the system, it is known as positive feedback. The room thermostat provides a functional metaphor for a negative feedback system. The room thermostat contributes to the temperature regulation by evaluating sensors to determine when the room temperature deviates from a predetermined range (i.e., homeostasis). If the room temperature drifts outside this range, the thermostat will trigger "motor" mechanisms to heat or cool the room to obtain a temperature within the previously defined "homeostatic" range.
The regulation of blood pressure represents a physiological feedback system with an objective to maintain levels within healthy limits. Because brain function requires a continuous supply of oxygenated blood, any drop in blood pressure is critical to survival and requires a rapid and appropriate physiological adjustment. In the healthy individual, drops in blood pressure are instantaneously detected by baroreceptors in the blood vessels. The baroreceptors send information to the brainstem, and the brainstem sends a motor command to the heart to increase heart rate rapidly. As soon as blood pressure returns to normal limits, neural feedback slows heart rate. However, there are individuals who have defective feedback. For example, in the elderly or individuals using specific medications, the blood pressure feedback system may be depressed. For these individuals, when the blood pressure drops because of a posture shift, they may experience severe dizziness or even syncope. This experience is common in the elderly and frequently contributes to falls and severe injuries.
In contrast to the negative feedback systems, which characterize self-regulatory processes, there are also positive feedback systems. A malfunctioning room thermostat providing positive feedback would continue to augment the temperature until the system ceased functioning due to extreme heat or extreme cold. Thus, prolonged positive feedback is destructive to systems. For example, rage, severe anger, or panic might be viewed as a behavioral consequence of physiological positive feedback that promotes increased metabolic output. Consistent with the feedback model, the physiological cost of prolonged periods of positive feedback might compromise the health of the individual.
The characteristics of physiological feedback may change as a function of physiological, emotional, cognitive, or behavioral demands. For example, during periods requiring massive metabolic output, such as exercise, the feedback system must be vigilant and efficient to support the changing oxygen needs of the cardiovascular system. During less motorically demanding conditions, such as periods of drowsiness and sleep, the pattern of neural feedback may change and go through periods of dissociation. Additionally, it is possible that the ability to express emotional sensitivity or behavioral contingency might be related to the "gain" or "amplification" of the feedback from the viscera. Therapeutic drugs, such as clonidine, used for panic disorders, may alter the neural feedback system by dampening the impact of the afferent feedback from visceral organs. Thus, the study of abnormal development and psychopathology may prompt attention to the study of the development of normal feedback systems and the relation between physiological systems and emotional, cognitive, and behavioral development (see Cicchetti, 1993).
Homeostasis: Signs and Signals of Competent Neural Self-Regulation
For the newborn, maintaining physiological homeostasis is crucial for survival. Homeostasis is not a passive process in which physiological systems remain constant. Rather, homeostasis is an active, neurally modulated process in which physiological systems vary within viable ranges.
The quality of homeostasis may be assessed with measures that are related to clinical status. When the output of a system is above the functional level, output is decreased until the level drops into the functional range. The output level progressively decreases through the functional range. When the output drops below the functional level, the output is progressively increased to reach and pass through the functional range. This process of waxing and waning of output reflects the negative feedback property of our nervous system. Thus, healthy physiological systems have a characteristic rhythm that provides an important observable window of the status of the nervous system. For example, respiration, blood pressure, heart rate, and temperature exhibit rhythms that provide important clinical information.
The rhythms provide indicators of the quality of feedback characterizing the system as it attempts to maintain homeostasis. The physiological rhythms produced by central-autonomic feedback loops have two dynamic characteristics: (a) a period of time of constant reflecting the temporal latency of the system to adjust or respond, and (b) a magnitude dimension reflecting the degree to which the system can deviate from a specific state. Both dimensions are modulated by higher brain structures and limited by neurochemical processes. Thus, by the nature of brain-autonomic regulation, autonomic response systems are time series, characterized by oscillations, with a period determined by the duration of the feedback loop, and an amplitude determined by central regulatory control.
Measures of heart rate oscillations, such as respiratory sinus arrhythmia (RSA), which are often used to assess cardiac vagal tone, dynamically reflect the bidirectional communication between the peripheral cardiovascular system and the brain. Under specific demand situations requiring shifts in metabolic output (e.g., stress- or survival-motivated responses, attention or social engagement, activity, fever, disease) the characteristics of the feedback, magnitude, valence, and period may change. Thus, the interest in monitoring cardiac vagal tone during demand situations, and as a marker of physiological self-regulation, is justified.
There are times when the nervous system is compromised and the neural feedback regulating cardiopulmonary and thermoregulatory processes is deficient. The occurrence of apnea and bradycardia signal dysfunction in the neural regulation of cardiopulmonary function. Similarly, difficulties in maintaining body temperature when ambient temperatures are changing reflect difficulties in thermoregulation.
The care of the high-risk infant often requires interventions to regulate physiological processes that are no longer under appropriate neural control (i.e., efficient negative feedback). Following apnea and bradycardia, interventions such as physical movement of the infant, serve as potent stimuli to reengage neural regulation of cardiopulmonary processes. When an infant has difficulty thermoregulating, rather than relying on neural feedback to vascular and cardiac processes, technology may use feedback from sensors that monitor body temperature to regulate ambient temperature. The common use in the NICU of heating units controlled by the newborn's body temperature is a functional example of a negative feedback system that compensates for defective self-thermoregulation.
The well-managed diabetic provides another example of external manipulation of a deficient internal feedback system. The diabetic has a defective feedback system that does not adequately regulate blood sugar through the endogenous release of insulin. To compensate for this problem, the diabetic must supplement the endogenous feedback system. To provide afferent feedback, blood is sampled and the amount of sugar in the blood is assessed. The individual's brain and associated cognitions interpret the afferent feedback and determine the appropriate amount of insulin required to regulate blood sugar within a functional level. To complete the feedback loop, motor systems are recruited to administer the insulin.
The human nervous system functions as a collection of several interacting self-regulatory negative feedback systems, each with a specific role. Sensors or receptors are located on the surface of the body to assess environmental changes and within the body to assess internal conditions. Motor systems control body movements and visceral organs. The feedback from internal sensors is interpreted by brainstem structures that contribute to the regulation of autonomic state (e.g., nucleus tractus solitarius, dorsal motor nucleus of the vagus, nucleus ambiguus). The nervous system provides the infrastructure for all levels of self-regulation. Self-regulatory processes characterize various domains ranging from the overt behavioral strategies of the infant demanding caregiver attention to subtle physiological shifts related to changes in thermoregulation, digestion, or cardiopulmonary function.
Self-Regulation of Physiological Systems: Primary Survival Agenda
In the proposed model, the effectiveness of an assessment is dependent on its sensitivity to measure the status of the nervous system. The success of an intervention is dependent on the functional enhancement of the nervous system. Although developmental follow-up studies of high-risk infants focus on the complex regulation of motor, social, and cognitive behaviors, survival of the high-risk infant in the NICU is based on successful regulation of the physiological systems that support growth and restorative processes. Initially, this regulation may involve extraneural feedback provided by external monitoring of critical care variables (e.g., temperature, heart rate, respiration, oxygen saturation). Then, based on the monitored condition, clinical interventions (e.g., ventilation, ambient temperature, medication) are administered to aid physiological regulation. If these interventions were successful, the clinical course would reflect a shift from a dependence on extraneural regulation to neural self-regulation. Thus, assessment should be directed at evaluating the nervous system structures and functions that promote physiological states fostering growth and restorative processes. Moreover, when methods are developed to accurately monitor these systems, interventions could be developed that specifically enhance neural feedback and improve the function of these systems.
The Vagal System: An Indicator of Survival-Related Self-Regulation
The proposed model for assessment and intervention focuses on the vagal system, a physiological system uniquely important for the survival of the high-risk infant. The vagal system contributes to the regulation and coordination of survival processes, including breathing, sucking, swallowing, heart rate, and vocalization. Dysfunction in these processes places the infant at survival risk and produces clinical indicators of risk, such as apnea, bradycardia, difficulties in sucking and swallowing, and weak, high-pitched cries.
Although the clinical indicators just listed appear to reflect divergent processes, they share an important neuroanatomical and neurophysiological substrate. The following four points highlight our knowledge and understanding of this substrate and justify why knowledge of this system may contribute to the development of psychophysiological assessment and clinical intervention strategies (a detailed and documented review of the physiological substrate of this system is described in chapter 2).
First, from a neuroanatomical level, sucking, swallowing, vocalizations, heart rate, and bronchial constriction are regulated by a common brainstem area. The regulation of these processes is dependent on motor fibers emerging from a medullary nucleus known as the nucleus ambiguus. These myelinated motor fibers originate in the nucleus ambiguus and travel through the tenth cranial nerve, the vagus.
Second, the vagus contributes sensory and additional motor fibers to the self-regulation of physiological systems. The vagus contains sensory fibers that provide feedback and motor fibers originating in the dorsal motor nucleus of the vagus. Sensory fibers compose approximately 80% of vagal fibers. The vagal sensory fibers originate in several visceral organs (e.g., heart, lungs, stomach, pancreas, liver, intestines) and terminate in a brainstem area known as the nucleus tractus solitarius. The vagal motor fibers originating in the dorsal motor nucleus of the vagus, unlike the myelinated motor fibers originating in the nucleus ambiguus, are unmyelinated and provide the primary motor control of the digestive system. However, the dorsal motor nucleus also projects to the bronchi and heart.
Third, nucleus tractus solitarius integrates sensory information from visceral organs and communicates, via interneurons, with the primary source nuclei of the vagus (i.e., nucleus ambiguus and dorsal motor nucleus). As illustrated in Figure 6.2, this feedback system regulates digestive and cardiopulmonary processes to foster growth and restoration. The afferent and efferent pathways conveying the bidirectional communication between the peripheral organs and the brainstem travel through the vagus. Note that the model describes potential feedback modulation (changing the magnitude or direction of the feedback) by other brain structures.
Fourth, neurophysiological research suggests that respiration rate is determined by a brainstem system that also outputs the respiratory rhythm to the heart and the bronchi. This neurophysiological drive to breathe is, in part, dependent on the emergent properties of interneuronal communication between the nucleus ambiguus and the nucleus tractus solitarius. According to Richter and Spyer (1990), the interneuronal communication between the nucleus ambiguus and the nucleus tractus solitarius is the source of a rhythmic generator producing a cardiopulmonary rhythm that results in oscillations in bronchial constriction and heart rate at a frequency similar to spontaneous breathing. The rhythmic concordance among these processes may functionally enhance diffusion of oxygen, and coordinate breathing and heart rate with other processes dependent on nucleus ambiguus motor fibers, such as sucking, swallowing, and vocalization.
The foregoing overview of the vagal system emphasizes the importance of vagal fibers in self-regulatory processes. The neuroanatomical descriptions and neurophysiological information regarding vagal function have been organized in the polyvagal theory (see chapter 2). The polyvagal theory provides a theoretical basis for specific clinical assessments and interventions in the NICU. Based on the polyvagal theory, interventions may be developed that will not jeopardize the clinical status of the vulnerable infant. In contrast, the polyvagal theory also explains how stimulation of specific sensory systems may elicit life-threatening vagal reflexes. For example, oral-esophageal stimulation, commonly occurring during suction or during the insertion of the orogastric tube for gavage feeding, may elicit potent bradycardia.
FIGURE 6.2. Feedback model of vagal regulation of autonomic state. Model illustrates the brainstem mechanisms that regulate vagal control of cardiopulmonary and digestive processes.
Polyvagal Theory
The polyvagal theory emphasizes the functional difference between vagal fibers originating in two areas of the brainstem: the nucleus ambiguus and the dorsal motor nucleus of the vagus. The two types of pathways have different embryological origins, promote different response strategies, and provide the neurophysiological justification for new definitions and explanations of stress, distress, and distress vulnerability. The polyvagal theory explains the stress and distress experienced by the high-risk infant within the context of the evolution of the mammalian autonomic nervous system. Because of the evolutionary pressures associated with obtaining and maintaining oxygen resources and transporting oxygenated blood to the brain, a neomammalian vagal system developed from fibers originating in or migrating to the nucleus ambiguus. The neomammalian vagal system provides the neural control of the muscles that evolved from the primitive gill arches. The gill arches throughout evolution have been associated with the extraction of oxygen from the environment. Somatomotor pathways from the neomammalian vagal system innervate the larynx, pharynx, and esophagus and coordinate vocalizations, breathing, and sucking. Additionally, visceromotor pathways from this vagal system regulate bronchial constriction and heart rate. Thus, this system is still involved in obtaining oxygen from the environment, diffusing oxygen in the blood, and maintaining cerebral blood flow by regulating blood pressure. The evolutionarily older reptilian vagal system involves fibers originating in the dorsal motor nucleus.
The polyvagal theory emphasizes that the two vagal systems (neomammalian and reptilian) respond differently to a stressor. For healthy mammals, the initial response is mediated by the neomammalian vagus and is characterized by a rapid withdrawal of vagal tone. This functionally removes the potent vagal brake from the heart and facilitates an instantaneous increase in metabolic output (i.e., increased heart rate) to mobilize energy resources for the classic flight-or-fight response. Removal of the vagal brake increases strength and speed to deal with stress and to aid in obtaining oxygen resources. However, withdrawal of the vagal brake functionally degrades (i.e., reduces control in) the motor systems involving the nucleus ambiguus. Thus, stress would not only be associated with faster heart rate, but also with higher pitch vocalizations (e.g., cries) and difficulties in coordinating sucking, swallowing, and breathing. This, of course, is commonly observed in physiologically stressed or compromised infants.
The Removal of the Vagal Brake Is the Neomammalian Vagal Response to Stress
According to this model, stress characterized by removal of the vagal brake is not necessarily detrimental to the survival of the individual. Removing the vagal brake occurs often as an adaptive response to increase metabolic output to mobilize and react to survival-related demands. For example, the vagal brake will be removed during exercise, pain, attention, and even during the appetitive phases of eating. Successful postpartum adaptation is related to the infant's skill and neurophysiological capacity to regulate the vagal brake to differentially engage and disengage with the environment. Therefore, the high-risk infant who exhibits a systematic regulation of the vagal brake to environmental demands should have more positive social and cognitive outcomes (e.g., Doussard-Roosevelt, McClenny, & Porges, 2001; Doussard-Roosevelt, Porges, Scanlon, Alemi, & Scanlon, 1997; Hofheimer, Wood, Porges, Pearson, & Lawson, 1995).
Removal of the vagal brake places the mammalian nervous system in a vulnerable state because: (a) it compromises homeostatic functions, including those associated with blood pressure regulation, thermoregulation, food intake, and digestion; and (b) it places the nervous system at risk for reptilian vagal reactions. When the neomammalian vagus is in control, the heart and bronchi are protected from the reptilian vagus, and blood is appropriately oxygenated and transported to the brain. In the healthy infant, transitory behavioral states such as crying are characterized by a disengagement and self-soothing as a reengagement of the vagal brake. Infants with greater neomammalian vagal tone tend to be more reactive to the environment (DeGangi, DiPietro, Greenspan, & Porges, 1991; Porges, Doussard-Roosevelt, Portales, & Suess, 1994; Porter, Porges, & Marshall, 1988; Stifter & Fox, 1990) and more able to self soothe and calm (Fox, 1989; Huffman et al., 1998). Moreover high-risk newborns with greater neomammalian vagal tone have fewer risk factors (see chapter 4; Porges, 1995) and have more optimal cognitive outcomes (Fox & Porges, 1985).
In contrast to the self-regulatory characteristics of the neomammalian vagus and the function of the vagal brake in staging responses as a function of environmental demands, the reptilian vagus responds with massive increases in vagal tone that slow the heart and constrict the bronchi. By reducing metabolic output, the reptilian vagus contributes to the conservation of available oxygen and promotes adaptive responses such as submerging and diving in aquatic environments or behaviorally freezing (i.e., feigning death) in terrestrial environments. Unfortunately, these response strategies, which are adaptive for reptiles, are potentially fatal for mammals. In mammals, this maladaptive strategy is observed in clinical settings as potentially lethal bradycardia and apnea. Based on the polyvagal theory, it has been proposed (see chapter 2) that fetal distress and sudden death, including sudden infant death syndrome, due to their neurogenic origins, are potential examples of the noxious impact of the reptilian vagal surge.
In a clinical setting, it may be possible to evaluate the status of the neomammalian vagus to indicate stress and distress vulnerability. Vagal fibers to the heart from the nucleus ambiguus produce a respiratory rhythm (i.e., RSA). By applying time-series statistics to the beat-to-beat heart rate pattern (Porges, 1985; Porges & Bohrer, 1990), it is possible to extract a measure of RSA that accurately represents vagal influences from the nucleus ambiguus.
In the absence of nucleus ambiguus stimulation to the heart, the heart is vulnerable to surges in vagal tone from the dorsal motor nucleus that produce bradycardia. Data from my laboratory support this conclusion. We have observed that bradycardia in the fetus and newborn occur during periods of depressed RSA, the measure of vagal tone from the nucleus ambiguus (Reed, Ohel, David, & Porges, 1999). Moreover, the observed meconium in the amniotic fluid of fetuses who have suffered from hypoxia (i.e., fetal distress) provides additional support for this model, because vagal stimulation via the dorsal motor nucleus of the vagus to the lower digestive track produces meconium (Behrman & Vaughan, 1987). Thus, during fetal distress, when we know RSA is depressed, the fetus exhibits a dorsal motor nucleus vagal surge as evidenced by the expression of massive bradycardia and meconium.
In support of these conclusions, neuroanatomical research suggests that sudden infant death syndrome may be related to a delayed maturation of the myelinated fibers of the vagus (Becker, Zhang, & Pereyra, 1993). The neomammalian vagal motor fibers are myelinated, as are many of the vagal sensory fibers. Hypoxia or other neurophysiological insults that may impede vagal myelination or contribute to demyelination would result in defects in the negative feedback system regulating the neomammalian vagus. The engagement and disengagement of the vagal brake would be unreliable. Defects in this system are critical to survival, because the nucleus ambiguus is involved in the generation of respiratory rhythms and the coordination of sucking, swallowing, and breathing. Thus, evaluation of nucleus ambiguus function may map into a continuum of deficits in self-regulation processes.
Hierarchical Model of Self-Regulation
Self-regulation processes observed in mammals may be organized into a hierarchical model with four levels (Porges, 1983). The hierarchical model emphasizes the dependence of higher order behavioral systems on more primary physiological systems. The model assumes that behaviors such as the organized motor activity associated with feeding, or the appropriate emotional regulation during social interaction, are dependent on a more primary physiological substrate related to the systematic regulation of autonomic state. The model has four levels and is hierarchical because each level requires successful functioning on the preceding level of organization (see Table 6.1).
Level I is characteristic of homeostatic processes of physiological systems regulating the internal organs. Homeostatic regulation requires bidirectional processes of monitoring and regulating the internal organ via sensory and motor pathways between the brain and the internal organs. Level II processes require cortical, conscious, and often motivated influences on the brainstem regulation of homeostasis. Level III processes are observable behaviors that can be evaluated by the quantity, quality, and appropriateness of motor behavior. Level IV reflects the coordination of behavior, emotional tone, and bodily state to successfully coordinate social interactions.
TABLE 6.1. Hierarchical Model of Self-Regulation
Level I: | Neurophysiological processes characterized by bidirectional communication between the brainstem and peripheral organs to maintain physiological homeostasis.
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Level II: | Physiological processes reflecting the input of higher nervous system influences on the brainstem regulation of homeostasis. These processes are associated with modulating metabolic output and energy resources to support adaptive responses to environmental demands.
Level III: | Measurable and often observable motor processes, including body movements and facial expressions. These processes can be evaluated in terms of quantity, quality, and appropriateness.
Level IV: | Processes that reflect the coordination of motor behavior, emotional tone, and bodily state to successfully negotiate social interactions. Unlike those of Level III, these processes are contingent with prioritized cues and feedback from the external environment.
Unique to this model is the assumption that complex behaviors, including social interactions, depend on physiology and how appropriately the nervous system regulates autonomic state. Thus, the regulation of autonomic state, via bidirectional communication between the brain and visceral organs, becomes the linchpin of physical, psychological, and social development. This chapter elaborates only on Level I and Level II processes, because these processes provide a developmentally antecedent substrate of emotional, cognitive, and behavioral regulation and must be mastered by all infants to ensure survival and successful adaptation in the postpartum environment.
Level I Processes: Physiological Homeostasis
Level I processes represent the successful regulation of internal bodily processes via neural negative feedback systems composed of interoceptors or sensory receptors monitoring internal bodily state and their respective neural pathways. To maintain homeostasis, interoceptors originating in the body cavity (e.g., gastric, hepatic, enteric, cardiac, vascular, and pulmonary systems) transmit information via neural pathways to brainstem structures. The brainstem structures interpret the sensory information and regulate the visceral state by triggering motor pathways that either directly manipulate various organs via neural pathways (e.g., increase or decrease heart rate, constrict or dilate blood vessels, inhibit or facilitate peristaltic activity) or trigger the release of specific hormones or peptides (e.g., adrenalin, insulin, oxytocin, vasopressin, gastrin, somatostatin). Level I is associated with the organization and neural feedback mechanisms that characterize the maintenance of homeostasis.
Level II Processes: Cost of Doing Business
The autonomic nervous system deals both with servicing the needs of the internal viscera and with responding to external challenges. Competence in the ability to trade off between internal and external needs may be used in developing definitions of adaptive behavioral strategies and homeostasis. Based on this model, response strategies to environmental demands and homeostasis are interdependent. When there are no environmental demands, the autonomic nervous system services the needs of the internal viscera (e.g., internal organs such as the heart, lungs, and gut) to enhance growth and restoration. However, in response to environmental demands, homeostatic processes are compromised and the autonomic nervous system supports increased metabolic output by down-regulating "growth and restoration" functions to deal with these external challenges. The central nervous system mediates the distribution of resources to deal with internal and external demands. This trade-off between internal and external needs is monitored and regulated by the central nervous system.
Survival Tasks of the High-Risk Newborn: In the NICU and Post-NICU
Survival in the NICU is dependent on achievement of Level I and Level II self-regulatory processes. Competence in regulating these processes enables the newborn to maintain and regulate homeostatic processes, such as temperature, breathing, feeding, blood pressure, and sleeping. All other systems, including neurophysiological systems associated with cortical function, sensory integration, and motor control of observable behavior are dependent on successful regulation of these more primary homeostatic functions. A similar argument can be made regarding the importance of brainstem control systems (e.g., nucleus ambiguus, nucleus tractus solitarius, dorsal motor nucleus of the vagus) as an infrastructure for emotional regulation, social behavior, and cognitive development. Thus, clinical care of the high-risk newborn should include assessments to evaluate and intervention strategies to foster development of nucleus ambiguus function.
Discharge from the NICU is directly related to competent self-regulation of Level I and Level II processes. In contrast, survival post-NICU is dependent on processes attributed to Level III and Level IV. Successful adaptation post-NICU not only requires competent Level I and Level II processes but is dependent on the Level III and Level IV processes that require the regulation of motor behavior, emotional expression, cognitive processes, and social interactions.
The concept that early difficulties in self-regulation lead to problems in emotional and social development evolves from the clinical and theoretical work of Greenspan (1992). Greenspan has described a sequence of developmental milestones that may be equated with the hierarchical levels presented here. Although his work focuses on Level IV processes, he acknowledges the importance of more primary physiological systems related to sensory feedback and motor control. Greenspan has proposed that difficulties in these primary neurophysiologically dependent systems contribute to problems in impulse control, attention, concentration, creative thinking, affect integration, and social interactions. The current detailed descriptions of Level I and Level II processes and their dependence on the nucleus ambiguus provide a neurophysiological model that complements these clinical observations.
Global Assessment Strategy
With the view that nucleus ambiguus function is critical to the survival risk of the newborn and our knowledge of how to monitor this function via RSA, we have the tools for a global neurophysiological assessment. A noninvasive assessment strategy might use a baseline or sleep measure of RSA to assess a Level I process and a feeding test to challenge the system to evaluate a Level II process. In older children, the autonomic substrate (i.e., Level I and Level II) might be evaluated during baseline and tasks associated with higher order processes. Baseline measures followed by tasks requiring Level III and Level IV processes, such as manipulating motor (e.g., activity level), cognitive (e.g., attentional), or social demands might be used to evaluate the child's autonomic tone (i.e., Level I) and ability to regulate autonomic state (i.e., Level II) to support higher level processes.
Level I Assessments
To assess Level I, we have measured RSA during periods of minimal environmental demand (e.g., sleep or quiet states) in the neonatal nursery. Research with this assessment procedure has reliably distinguished between high-risk and full-term newborns (Porges, 1992) and between risk groups of preterm newborns matched for gestational age (see chapter 4). Consistent with the hierarchical model, research has demonstrated that the RSA measure of vagal function in the NICU is related to cognitive outcome at 3 years of age (Doussard-Roosevelt et al., 1997). Similarly, preterm newborns with greater amplitude RSA exhibited more social behavior and attention in the NICU (Hofheimer et al., 1995).
Level II Assessments
To assess Level II, we have measured changes in RSA during periods of well-defined environmental demands (e.g., feeding) in the neonatal nursery. Three studies have been conducted to evaluate the effect of a feeding challenge on the pattern of RSA. In one study, an apparatus provided water or sucrose in solution when a newborn sucked (Porges & Lipsitt, 1993). In response to increased sweetness, the heart rate increased, the amplitude of RSA decreased (lower nucleus ambiguus vagal tone), and the sucking frequency increased. In a second study, RSA was evaluated during bottle feeding in a sample of NICU newborns (Portales et al., 1997). During the bottle feeding, RSA was depressed and heart rate increased. Following feeding, the levels returned to prefeeding baseline levels. A third study (Suess et al., 2000) illustrated a relationship between RSA recovery following feeding and degree of prematurity. In addition, outcome studies using the feeding challenge with older infants are currently being conducted in my laboratory. These studies will determine if infants who have a history of excessive crying and difficulties in regulating behavioral state also have difficulties in regulating the vagal control of the heart mediated by the nucleus ambiguus. In these studies, the pattern of vagal reactivity, monitored by RSA, to the feeding challenge and recovery following feeding are used as Level II assessment indices. These assessments will be evaluated to determine whether they are related to clinical risk factors and developmental outcome.
General Intervention Strategy
Measurement of RSA provides the capability to assess individual differences in neural regulation of homeostatic function in terms of nucleus ambiguus control. It is now possible to answer two questions: (a) whether individual differences in neural regulation are related to intervention effectiveness, and (b) whether interventions are beneficial or detrimental to survival-related physiological regulation. Thus, our knowledge regarding neurophysiological function may be useful in designing intervention strategies to promote increased nucleus ambiguus function.
Beneficial intervention strategies would stimulate specific visceral sensors to increase the function of the neomammalian vagus. The intervention would stimulate motor outflow and enhance the regulation and coordination of heart rate, respiration, vocalization, sucking, and swallowing. Positive developmental outcomes (e.g., weight gain, improved state regulation, longer periods of alertness, improved neurological development) would be the product of the improved regulation and coordination.
In contrast, detrimental intervention strategies would stimulate sensors that would increase the function of the reptilian vagus. Under conditions during which the dorsal motor nucleus vagus becomes dominant, the intervention might result in bradycardia, apnea, and digestive problems and produce compromised outcomes.
The sensory components of the trigeminal nerve (cranial nerve V) and the facial nerve (cranial nerve VII) provide the primary sensory input to the nucleus ambiguus. Therefore, interventions that provide sucking opportunities, oral stimulation, and facial movement would provide regulation challenges to the nucleus ambiguus vagal system and may be beneficial. In contrast, interventions that may trigger dorsal motor activity may be potentially dangerous. For example, shifting the posture of an infant will elicit baroreceptor responses including a heart rate change mediated via the vagus to regulate blood pressure. In the low-risk infant, this intervention exercises a negative feedback system that maintains cerebral blood pressure and contributes to the regulation of behavioral state. The system includes motor control of the cardiovascular system from the dorsal motor nucleus and the nucleus ambiguus. In the high-risk infant, with depressed nucleus ambiguus function, posture shifts may result in a massive bradycardia and loss of consciousness similar to the vasovagal syncopes observed in older adults with low nucleus ambiguus vagal tone (i.e., low-amplitude RSA).
Similarly, abdominal massage or pelvic manipulations may be detrimental to the high-risk infant. These interventions, similar to the posture shifts, may stimulate sensory pathways that activate both nucleus ambiguus and dorsal motor nucleus. Thus, in the presence of low nucleus ambiguus tone (a characteristic of the high-risk infant), these interventions also might elicit massive bradycardia and loss of consciousness.
Caution: It is extremely important to emphasize that many high-risk infants are in a compromised physiological state, defined by low nucleus ambiguus vagal tone. Interventions applied to these infants such as abdominal pressure during massage, baroreceptor stimulation by posture shifts, or lower esophageal stimulation during either suction or orogastric feeding might trigger massive dorsal motor nucleus reflexes and, in turn, bradycardia, apnea, and even loss of consciousness.
CONCLUSION
Although the methods described have focused on the high-risk infant during the neonatal period, the model may be generalized to study older children and even adults with behavioral and psychological problems. Associated with various psychological and psychiatric disorders are symptoms related to difficulties in state regulation. For example, hyperactivity or forms of attentional problems are associated with an inability to regulate the physiological substrate to support appropriate attentive and social behaviors. Similarly, individuals with inappropriate expressions of panic, terror, or rage, with their defining massive physiological responses, provide examples of an inability to self-regulate physiological state to self-soothe or to be contingent with social gestures being expressed by others to help calm them.
In support of the foregoing generalizations, several studies have monitored RSA and investigated Level I and Level II processes in clinical populations. Level I assessments have been used to determine whether individual differences in tonic state of the vagal system measured via RSA (e.g., assessed during sleep or quiescent periods) are related to clinical risk factors. Level I methodology also has been used in specific experimental manipulations to evaluate potential mechanisms that may mediate psychopathology, such as panic disorders, or to evaluate the autonomic impact of the pharmacological treatments commonly used for psychiatric disorders. Level II assessments, involving the quantification of the pattern of vagal tone regulation in response to a clearly defined challenge, also have been used in clinical research.
Examples of Level I Assessments
In an attempt to understand the mechanisms of how the nervous system regulates autonomic state during panic, George et al. (1989) evaluated RSA during hyperventilation and sodium lactate infusion, manipulations known to elicit panic. They reported that both manipulations produced a massive decrease in RSA. Based on the assessment model, these data reflect a severe deficit in Level I processes. Thus, consistent with the clinical observations of elicited panic and severe anxiety, the assessment would predict that self-regulatory physiological and behavioral functions would be compromised.
Other research has attempted to evaluate how psychotropic medication influences Level I processes. Based on the assessment model, the degree that medication depresses RSA would affect behavioral and psychological outcome. For example, it would be predicted that if medication depressed RSA it would compromise higher level processes, and behavior would not improve. Alternatively, it would be predicted that if medication increased or did not decrease RSA, the psychotropic effect on behavior would be optimized. McLeod, Hoehn-Saric, Porges, and Zimmerli (1992) demonstrated that the effectiveness of imipramine on general anxiety disorders (i.e., measured by the Hamilton scale), was directly related to the influence imipramine had on RSA. If imipramine, due to its known anticholinergic effects, reduced RSA, then imipramine did not have beneficial effects. In contrast, if the patient did not exhibit depressed RSA in response to imipramine, behavioral outcome was optimized. Thus, if medication disrupted Level I processes, the higher level processes also were compromised.
The focus of this chapter has been on clinical applications of the proposed psychophysiological assessment model. There have been several investigations that demonstrate the utility of Level I assessments in clinical settings. As discussed in chapter 4, high-risk neonates have significantly lower amplitude RSA. Moreover, individual differences in RSA from another another high-risk sample have been related to clinical condition and predict cognitive outcome (Doussard-Roosevelt et al., 1997, 2001). Consistent with these findings, Donchin, Constantini, Byrne, and Porges (1992) reported that in adults preneurosurgical levels of RSA predicted clinical course (i.e., neurological and cognitive outcomes).
Examples of Level II Assessments
Level II assessments have been designed to evaluate the individual's ability to regulate the component of the vagal tone system originating in the nucleus ambiguus. Level II assessments assume that the regulation of vagal pathways originating in the nucleus ambiguus is mediated by higher central mechanisms that influence brainstem feedback to promote either immediate mobilization of energy resources or calming. Level II assessments are important in dealing with populations that appear to exhibit normal Level I activity. For example, we have reported that independent of resting levels of RSA, children who do not systematically depress RSA during attention-demanding tasks are more likely to have behavioral regulation problems (DeGangi et al., 1991). In a second sample we observed that this inability to regulate RSA at 9 months of age predicted behavioral problems at 3 years (see chapter 7). Consistent with the hierarchical model, both studies demonstrate that measures of RSA regulation are predictive, when baseline RSA, an indicator of Level I processes, does not predict outcome. Other research has evaluated the acute and tonic influences of alcohol and narcotics on the ability to regulate vagal reflexes. For example, Hickey, Suess, Newlin, Spurgeon, and Porges (1995) demonstrated that children exposed to opiates in utero, characterized by attentional problems, also exhibited difficulties in regulating vagal tone during sustained attention.
Research with Level II processes has been conducted to investigate the parallel between autonomic state regulation and affective regulation in infants. To evaluate individual differences in the dynamic covariation of affect tone and RSA, Bazhenova, Plonskaia, and Porges (2001) challenged Level II processes by eliciting various affective states. The Bazhenova et al. study demonstrated that infants who exhibited a systematic parallel between shifts in affect tone and RSA exhibited more optimum social behavior and state regulation.
Research in my laboratory with newborn infants has focused on evaluating the feeding challenge as a Level II assessment. Preliminary research with feeding challenges demonstrates individual differences in RSA regulation while bottle feeding (Portales et al., 1997), sucking (Porges & Lipsitt, 1993), or during orogastric feeding (DiPietro & Porges, 1991). We have developed a standardized Level II assessment for infants that evaluates nucleus ambiguus vagal tone (i.e., RSA) regulation during feeding. The assessment procedure evaluates Level I processes from a baseline recording of RSA during a calm state and Level II processes from the RSA response to the feeding challenge. We are also testing the generality of the assessment model to evaluate RSA regulation (Level II processes) in older children in a variety of challenges related to sustained attention, emotion regulation, and social interactions.
SUMMARY
The methods for assessment and intervention described in this chapter focus on a specific physiological system originating in the nucleus ambiguus. The nucleus ambiguus is a brainstem nucleus that coordinates sucking, swallowing, vocalizing, and breathing via vagal pathways. Moreover, vagal pathways from nucleus ambiguus provide the primary neural control of heart rate. Thus, nucleus ambiguus regulation of the heart is the neurophysiological substrate for all behaviors requiring a regulation of metabolic output to engage, disengage, and reengage dimensions of the environment. By monitoring a rhythmic heart rate pattern (i.e., RSA) mediated by the nucleus ambiguus, it is possible to assess shifts in the competence of neural regulation in the infant, child, or adult. The methodology for assessing nucleus ambiguus function is available and can be generalized to older children and adults by devising tasks that require the individual to regulate nucleus ambiguus output to mobilize and cope with transitory environmental demands. Therefore, this technology may be used to noninvasively monitor nucleus ambiguus function in high-risk infants or other clinical populations, to assess relative risk, and to evaluate the effectiveness of specific interventions. Although this chapter has emphasized assessment, there is a great need to develop appropriate interventions. Research needs to be directed at designing age-specific interventions that will efficiently enhance the neural feedback critical to the function and regulation of the vagal system.
CHAPTER 7
Infant Regulation of the Vagal "Brake" Predicts Child Behavior Problems: A Psychobiological Model of Social Behavior
Cardiac vagal tone is a construct that describes the functional relationship between the brainstem and the heart. Research in several domains has related cardiac vagal tone to a variety of clinical and behavioral dimensions. For example, cardiac vagal tone is related to clinical risk factors in infants (chapter 4), to outcomes in neurosurgical patients (Donchin, Constantini, Szold, Byrne, & Porges, 1992), to depth of anesthesia (Donchin, Feld, & Porges, 1985) to individual differences in temperament (Porges & Doussard-Roosevelt, 1997; Porges, Doussard-Roosevelt, Portales, & Suess, 1994). Most studies have evaluated cardiac vagal tone during steady-state conditions, while few studies have investigated dynamic changes in cardiac vagal tone and how these changes relate to social behavior and emotional regulation (see chapter 9; DeGangi, DiPietro, Greenspan, & Porges, 1991) in children. This chapter proposes that the development of appropriate social behavior is dependent on the ability to regulate cardiac vagal tone. The model presents two contrasting roles of the vagus. In one role, the vagus supports homeostatic functions, while in the other, the vagus serves as a mediator of motor responses to environmental challenges.
Although cardiac vagal tone is relatively stable under steady-state conditions such as quiet baseline (Fracasso, Porges, Lamb, & Rosenberg, 1994; Izard et al., 1991; Porges et al., 1994), it is sensitive to environmental demands and stimulation (e.g., DiPietro & Porges, 1991; Hofheimer, Wood, Porges, Pearson, & Lawson, 1995; Porges & Lipsitt, 1993; Porter, Porges, & Marshall, 1988). Rapid changes in cardiac vagal tone often parallel shifts in the cardiac output required to match metabolic demands (i.e., withdrawal of vagal tone promotes increased heart rate and cardiac output). For example, the increased sucking rate observed during feeding requires a shift in metabolic resources to support ingestive behavior. This shift is observed as parallel increases in heart rate, decreases in cardiac vagal tone, and increases in sucking rate (Porges & Lipsitt, 1993). Similarly, cardiac vagal tone is withdrawn, increasing cardiac output, to support behavioral mobilization during painful situations such as circumcision (Porter et al., 1988).
FUNCTIONAL ROLES OF VAGAL TONE
Vagal tone has two roles. First, during states of low environmental demand (e.g., sleep or quiet states), vagal tone fosters physiological homeostasis to promote growth and restoration. Second, during states characterized by environmental challenges, the vagus acts as a brake to rapidly regulate cardiac output and thus metabolic output. The "vagal brake" functionally keeps the heart rate slow by increasing vagal output to the heart and actively inhibiting sympathetic influences (Levy, 1984; Vanhoutte & Levy, 1979). Releasing the vagal brake reduces vagal inhibition on the cardiac pacemaker (sinoatrial node), and heart rate increases due to the intrinsic rate of the pacemaker, mechanical reflexes, and sympathetic influences. The vagal brake is not an all-or-none construct. Rather, the vagal brake represents the graded inhibition of the cardiac pacemaker by vagal efferent fibers. Because heart rate is not totally determined by the graded impact of the vagal brake, changes in cardiac vagal tone are not always highly correlated with changes in heart rate. During environmental interactions, metabolic output must be rapidly regulated to foster the behavioral and psychological processes required to engage and disengage with the environment. This chapter focuses on the latter role of vagal tone, mediating behavioral responses to environmental challenges and empirically evaluates the vagal brake hypothesis. The relation between infant regulation of the vagal brake and later social developmental outcomes requiring engagement and disengagement is examined.
THE AUTONOMIC NERVOUS SYSTEM: A VISCERAL FEEDBACK SYSTEM
The mammalian brainstem, via the autonomic nervous system, regulates homeostatic processes to foster growth and restoration. This role has been well documented by the functional neuroanatomy of the autonomic nervous system in which visceromotor neurons communicate between the brainstem and visceral organs. Weiner (1948), in proposing the concept of a feedback system, used the autonomic nervous system as his example. According to Weiner, homeostasis is an emergent property of a system that via bidirectional communication monitors and regulates the status of an organ to maintain an output level within a specific functional range. Consistent with Weiner's model, the autonomic nervous system includes a central regulator (brainstem source nuclei) that determines motor output (parasympathetic or sympathetic nerves) to a visceral organ (heart, lung, stomach) after interpreting the information from the sensor that monitors the status of the organ (afferent feedback). To maintain physiological homeostasis, sensory pathways originating in peripheral organs (chemoreceptors and baroreceptors in the carotid sinus) convey information regarding physiological status, while motor pathways (vagal and sympathetic pathways to the heart) change the output of peripheral organs. The sensory pathways from visceral organs originate in the periphery and usually terminate in the brainstem, whereas many of the motor pathways originate in the brainstem and terminate in the periphery.
The cardiac branch of the vagus, although a component of the autonomic nervous system, may be defined as a feedback system. The cardiac vagal system contains the requisite components of afferent and efferent pathways, brainstem source nuclei, and a visceral target, the heart.
EXTEROCEPTIVE AND INTEROCEPTIVE FEEDBACK: COMPETING DEMANDS ON THE VAGAL SYSTEM
The nervous system receives sensory feedback from receptors that sense conditions within the body (i.e., interoceptors) to support homeostasis and from receptors that sense conditions outside the body (i.e., exteroceptors) to deal with environmental challenges. Cardiac vagal output is related to both. In general, cardiac vagal tone is increased to support homeostatic functions, and cardiac vagal tone is decreased to increase cardiac output to support specific motor behaviors in response to environmental challenge.
The conceptual link between the autonomic nervous system and homeostasis has contributed to a limited definition of the autonomic nervous system as a sluggish "vegetative" system with the primary responsibility of maintaining homeostasis. However, components of the autonomic nervous system, such as the vagal system, play a dynamic role in fostering behavioral and psychological interactions with the environment. The vagal system rapidly regulates metabolic output (Rowell, 1993) to foster motor and psychological processes associated with appropriate engagement and disengagement with the environment. To interact with the environment, the priorities of the visceral feedback system change from optimizing visceral state to foster growth and restoration (i.e., homeostasis) to optimizing metabolic output to support the motor behavior required by external challenges. This shift in priorities, from internal visceral feedback to external environmental feedback, has survival consequences. For example, in dealing with the external environment, the organism may effectively respond to environmental demands related to threat, work, communication, and social behavior. However, in serving these environmentally driven priorities, physiological homeostasis may be compromised. When up-regulating metabolic output to serve environmental needs (e.g., fight-or-flight behaviors) the organism must down-regulate visceral needs, thereby disrupting homeostasis.
The vagal system deals both with servicing the needs of the internal viscera and with responding to external challenges. Competence in the ability to trade off between internal and external needs may be used in developing definitions of adaptive behavioral strategies and homeostasis. Based on this model, response strategies to environmental demands and homeostasis are interdependent. When there are no environmental demands, the vagal system services the needs of the internal viscera (e.g., heart, lungs, stomach, intestines) to enhance growth and restoration. However, in response to environmental demands, homeostatic processes are compromised, and the vagal system supports increased metabolic output by down-regulating growth and restoration functions to deal with these external challenges. The central nervous system mediates the distribution of resources to deal with internal and external demands. The trade-off between internal and external needs is monitored and regulated by the central nervous system and can be monitored via the dynamic regulation of the heart by the vagus. This dynamic regulation provides a graded inhibition of the vagal brake with a resultant shift in cardiac output.
THE POLYVAGAL THEORY
The polyvagal theory (see chapter 2) introduces and provides the theoretical justification for studying the vagal brake. The theory proposes that the successful adaptation of mammals is dependent on systematic and reliable withdrawal and reengagement of the vagal brake as a mechanism to rapidly regulate metabolic output in response to environmental demands. The theory, based on comparative neuroanatomy and embryology, proposes that mammals have two vagal systems, a neomammalian system and a more vegetative "reptilian" system, controlled by different medullary nuclei. The mammalian vagal system is controlled by the nucleus ambiguus, and the more vegetative reptilian system is controlled by the dorsal motor nucleus.
The two systems have different response strategies. The nucleus ambiguus (mammalian) vagal system is myelinated and provides a rapid adjustment of metabolic output by regulating heart rate via the sinoatrial node. Usually this is observed as rapid vagal withdrawal and concomitant heart rate acceleration. In contrast, the dorsal motor nucleus (reptilian) vagal system is unmyelinated and appears to have little impact on cardiac output in most conditions. However, under compromising conditions of low oxygen availability, it is hypothesized that the reptilian system fosters resource conservation by stimulating massive bradycardia and apnea. Thus, several lethal phenomena such as cardiac arrest, massive bradycardia producing sudden death, and exercise-induced syncope appear to be the vestigial remnants of a physiological response with adaptive consequences for reptiles but with lethal consequences for mammals.
Because the nucleus ambiguus is an integral component of the medullary system generating the respiratory rhythm (Richter & Spyer, 1990), the output of the vagal fibers terminating on the heart are characterized by a respiratory rhythm of increasing and decreasing tone (Porges, 1995). The functional consequence of this rhythmic modulation of nucleus ambiguus vagal tone to the sinoatrial node is to produce the respiratory oscillations in the heart rate pattern known as respiratory sinus arrhythmia (RSA). Thus, the quantification of the amplitude of RSA provides an index of the dynamically changing status of the vagal brake.
VAGAL BRAKE AND THE DEVELOPMENT OF SOCIAL BEHAVIOR
The current study focuses on the nucleus ambiguus vagal system and evaluates the hypothesis that appropriate withdrawal of the vagal brake during infancy is a marker variable for developmental outcome. Specifically, it is proposed that appropriate regulation of the vagal brake during infancy reflects an ability to selectively engage and disengage with the environment. The vagal brake provides the mechanism for the infant to contingently interact with objects and people. Thus, the vagal brake provides a neurophysiological mechanism that may promote the development of appropriate social behavior. In the current study, 9-month-old infants were evaluated for cardiac vagal tone regulation during the administration of the Bayley Scales of Mental Development (i.e., whether the vagal brake was withdrawn during the test). Administration of the Bayley Scales places a complex set of demands on the infant. The infant must engage and disengage with each changing stimulus within the context of a social interaction with the experimenter. Social behavior problems were assessed at 3 years of age with the Child Behavior Checklist for Ages 2–3 (Achenbach, 1988).
METHOD
Subjects
Twenty-four (12 male, 12 female) subjects were tested initially during infancy (7–9 months). All subjects were evaluated for behavioral problems at 36 months. Subjects were recruited from the metropolitan Washington, DC, area via advertisements in local doctors' offices and newspapers inviting mothers of 7- to 9-month-olds to participate in the research project. Separate advertisements elicited mothers with difficult infants and mothers with nondifficult infants to ensure a broad range of infant behavior. All infants were born full-term with no major medical complications. Mothers ranged in age from 20 to 39 years (M = 32) and had an average of 16 years of education (range = 10–21 years).
Procedures
At the 9-month assessment, the physiological monitoring and developmental testing procedures were explained and informed consent obtained from the mother. Development was evaluated with the Bayley Scales of Infant Development (Bayley, 1969). Electrocardiogram (ECG) was recorded during a 3-minute baseline in which the infant sat quietly on the mother's lap and during the first 7 minutes of the Bayley administration, during which the infant was presented with a variety of attention-demanding tasks. Mothers completed the Infant Characteristics Questionnaire (Bates, 1984) and the Fussy Baby Questionnaire (Greenspan, Porges, Portales, & Walker, 1987) at the 9-month assessment. The difficultness subscale of the Infant Characteristics Questionnaire was included as a predictive variable in the data analyses. This subscale was chosen based on its stability (Bates, 1980) as well as its theoretical linkage to cardiac vagal tone (DeGangi et al., 1991). In addition, information from the two infant questionnaires as well as the Infant Behavior Record of the Bayley Scales was used to classify infants as Regulatory Disordered or non–Regulatory Disordered according to the diagnostic criteria established by the National Center for Clinical Infant Programs (Zero to Three, 1994).
When the child was 3 years of age, the mother completed the Child Behavior Checklist for Ages 2–3 (CBCL/2–3; Achenbach, 1988). The CBCL/2–3 is a parent rating scale consisting of 99 items describing behavioral/emotional problems.
Quantification of Behavior Problems Data
Scores were derived from the CBCL/2-3 (Achenbach, 1988) for total problems and for the six narrow-band syndrome scales (social withdrawal, depressed, sleep problems, somatic problems, aggressive, and destructive). The narrow-band syndrome scales have been derived via factor analysis and the scale scores standardized to determine cutoffs associated with clinical psychopathology (McConaughy & Achenbach, 1988).
Quantification of Heart Rate Data
The ECG was monitored via three AgCl electrodes placed on the infant's chest. The output of the ECG amplifier was stored on a Vetter C-4 FM tape recorder (A. K. Vetter, Rebersberg, PA). The ECG data were quantified offline by replaying the tapes into a Vagal Tone Monitor (Delta-Biometrics, Bethesda, MD). The Vagal Tone Monitor detected the peak of the R-wave to the nearest millisecond and timed sequential heart periods. The sequential heart periods were stored in a file on a PC. The amplitude of RSA,was calculated offline. The amplitude of RSA provides an accurate description of medullary influences on the heart via the nucleus ambiguus branch of the vagus nerve (see chapter 2; Porges, 1995).
MXedit software (Delta-Biometrics) was used to visually display the heart period data, to edit outliers, and to quantify the heart period and RSA. MXedit incorporates the Porges (1985) method of calculating the amplitude of RSA. This method contains a detrending algorithm to remove from the heart rate pattern the variance associated with the complex changing level and oscillations slower than RSA. The detrending algorithm requires heart period values measured to the nearest millisecond and includes the resampling of the heart period data every 250 ms, a moving polynomial filter (third-order 21-point) and a band-pass filter (0.24–1.04 Hz). The analysis represents the variance of the residual series output from the detrending algorithm and is reported in units of ln(ms)2. In this study each heart rate parameter (i.e., heart period and RSA) was calculated for sequential 30-sec. epochs within each condition. The mean of the within-condition epochs was used in the data analyses.
Design
To evaluate individual differences in the disengagement of the vagal brake during the Bayley test, RSA and heart period reactivity were assessed by calculating difference scores in which the values derived during the Bayley tests were subtracted from the baseline condition values. Change scores were positive when levels during the Bayley tests were lower than the baseline. More positive difference scores reflected greater decreases in RSA and heart period. To test the hypothesis that the functioning of the vagal brake during infancy is related to behavior problems during childhood, correlational analyses were conducted to assess the relation of the outcome measures from the CBCL/2–3 years of age with the RSA and heart period reactivity scores. It was hypothesized that subjects who exhibited larger decreases in RSA during the Bayley test, relative to the baseline condition, would have fewer behavioral problems at 3 years of age. Negative correlations between the change in RSA and behavioral problems (i.e., fewer problems associated with greater depression of RSA) would support this hypothesis.
RESULTS
Physiological Response Patterns
Gender × condition repeated measures analyses of variance evaluated whether or not gender differences existed in either the baseline levels or the pattern of change in heart period and vagal tone across conditions. Gender was not related to level or pattern of reactivity. There was a significant condition effect for heart period. Mean heart period significantly increased (i.e., heart rate slowed) from baseline (445 ms) to Bayley test (463 ms), F(1,22) = 10.1, p < .01. Mean levels in RSA did not differ between the conditions, F(1,22) = 0.6, p > .1. Inspection of the individual differences indicated that most subjects increased their heart period (n = 19) during the test. However, the RSA response was more heterogeneous with 13 subjects decreasing and 11 subjects increasing levels during the test.
The magnitude and direction of the heart period and RSA responses were related to baseline levels. As illustrated in Figure 7.1, higher amplitude RSA during baseline was associated with larger decreases (i.e., more positive change scores) in RSA and heart period during the Bayley test, r(22) = .42, p < .05 and r(22) = .48, p < .05, respectively. Similarly, longer baseline heart periods, although not correlated with changes in RSA, were correlated with greater decreases in heart period during the Bayley test, r(22) = .60, p < .01. The magnitude of individual differences in the two change scores was also correlated. Subjects who expressed a greater decrease in RSA also responded with a greater decrease in heart period, r(22) = .49, p < .05.
FIGURE 7.1. Scatterplots relating baseline RSA to RSA changes (a) and to heart period changes (b). Change scores represent values during the baseline minus values during the Bayley test. Positive change scores reflect decreases from baseline to Bayley condition.
THE VAGAL BRAKE AND OUTCOME BEHAVIORS AT 3 YEARS OF AGE
To evaluate the relationship between the vagal brake and outcome behaviors at 3 years of age, correlational analyses were conducted. Analyses examined the relation between 9-month heart period and RSA values (baseline and change scores) and 3-year behavior problems coded on the CBCL/2–3 (total problems and six narrow-band syndrome scores). As listed in Table 7.1, there were consistent relationships between the decrease in RSA during the Bayley test (i.e., magnitude of the change score) and behavior problems at 3 years. Greater decreases in RSA (i.e., the appropriate regulation of the vagal brake during attending demanding tasks) were associated with fewer total behavior problems, r(22) = –.50, p < .05. Further, greater decreases in RSA were associated with more optimal outcomes on the three narrow-band syndrome scales associated with social behavior (i.e., social withdrawal, depressed, and aggressive behavior). Greater decreases in RSA were correlated with fewer social withdrawal problems, r(22) = –.42, p < .05, fewer depressed behaviors, r(22) = –.45, p < .05, and fewer aggressive behaviors, r(22) = –.53, p < .01. Inspection of the univariate correlations indicates that heart period and changes in heart period were not related to any of the outcome behaviors. Baseline levels of RSA were related to the depressed, sleep problems, and destructive scales. Infants who had lower amplitude RSA had more problems on the depressed, r(22) = –.43, p < .05, sleep problems, r(22) = –.57, p < .01, and destructive, r(22) = –.39, p = .06, scales. Gender was not related to outcome, nor did gender influence the correlations between the change in RSA and the outcome measures.
TABLE 7.1. Correlations Between 9-Month Physiological Variables and 3-Year Behavioral Outcome Variables
For comparison purposes, correlations of the outcome measures with 9-month Bayley Mental Development Index and Infant Characteristics Questionnaire difficultness scores are presented in Table 7.2. The Bayley score was associated with somatic problems, r(22) = –.44, p < .05, and sleep problems, r(22) = –.47, p < .05, while the difficultness score was not predictive of later behavior.
TABLE 7.2. Correlations Between 9-Month Behavioral Variables and 3-Year Behavioral Outcome Variables
MULTIPLE REGRESSION MODELS
Multiple stepwise forward-regression models were tested with the four physiological variables (baseline RSA, RSA change score, baseline heart period, heart period change score) and the two behavioral variables (Bayley Mental Development Index, Infant Characteristics Questionnaire difficultness score) entered into each predictive model. Alpha-to-enter and alpha-to-remove levels were set at .10. For the total behavior problem score, only RSA change scores were a significant predictor of outcome. Once RSA change was entered into the model, no additional variable (i.e., baseline RSA, baseline heart period, change in heart period, Bayley Mental Development Index, difficultness) significantly contributed to the model. This was also true for the social withdrawal, depressed, and aggressive narrow-band syndrome scores.
The models for the other three narrow-band syndrome scales were not consistent. For the destructive syndrome score, there was no significant predictor. The Bayley Mental Development Index score was the only predictor of the somatic syndrome score. And the sleep problems score was best predicted by a model which included baseline RSA and the difficultness score, F(2,2l) = 7.41, p < .01.
DISCUSSION
Infant Vagal Tone and Child Behavior Problems
The best predictor of behavioral problems on CBCL/2–3 at 3 years of age was the infant's ability to decrease cardiac vagal tone during the Bayley test. The infants who expressed the greatest decrease in RSA had the fewest behavioral problems. Thus, the ability to regulate the vagal brake in the hypothesized direction was a significant predictor of positive developmental outcome. Decreases in RSA were associated with the global measure of total behavior problems and also with three of the narrow-band syndrome scales (i.e., social withdrawal, depressed, aggressive). Inspection of the items on these three scales suggests that the scales are uniquely sensitive to social behavior. The social withdrawal scale focuses on difficulties in the initiation and engagement of social interactions. The depressed scale focuses on affect regulation during social interactions. The aggressive scale focuses on aggressive and uncooperative social interactions. The intercorrelations among the narrow-band scales confirm that the three scales are significantly correlated and that aggressive and depressed are the only scales correlated with social withdrawal. In contrast, the three scales that were not related to 9-month regulation of the vagal brake appear to include items more sensitive to visceral regulation, state regulation, and/or awareness of the environment (i.e., sleep problems, somatic problems, destructive behavior). These observations foster the speculation that the vagal brake may be uniquely related to the successful performance of behaviors associated with social interactions, while baseline vagal tone may be more sensitive to endogenous homeostatic function.
The linkage between specific narrow-band syndrome scales and the two measures of vagal tone reflects the two postulated roles of vagal tone described above. The first role is maximized during states characterized by low environmental demand, when vagal tone fosters physiological homeostasis to promote growth and restoration. Thus, higher levels of baseline vagal tone should be, and are in this study, significantly related to fewer sleep problems (i.e., lower scores on the sleep problems scale) and tend to be related to fewer somatic problems (i.e., lower scores on the somatic problems scale) and a greater awareness of and reactivity to the environment (i.e., lower scores on the destructive scale).
Research on clinical populations supports the speculation that low baseline vagal tone is related to illness and health risk (e.g., see chapter 4; Donchin et al., 1992). In addition, research also supports the speculation that subjects who have higher vagal tone are more reactive and aware of the environment (e.g., see chapter 9; Porges, 1991). The second role of vagal tone is maximized during states characterized by environmental challenges, such as social interactions, when the vagus acts as a brake to regulate cardiac output. Thus, greater reductions in vagal tone during tasks that require social interaction and/or focused attention to an external stimulus should be, and are in this study, related to fewer social behavioral problems (i.e., lower scores on the social withdrawal and aggressive scales) and more positive and appropriate affective responsivity to people (i.e., lower scores on depressed scale).
The behavior of the infants at 9 months was not related to the CBCL/2–3 rating at 3 years. Ratings at 9 months on the difficultness scale of the Infant Characteristics Questionnaire were not correlated with the CBCL/2–3 scores. Additionally, based on the Zero to Three diagnostic manual, the 9-month behaviors were coded to determine whether any of the children would be assessed as regulatory disordered. Four infants fit the clinical criteria of severe regulatory disorders. These infants did not exhibit any unique deficits in either measure of vagal tone, nor did they exhibit severe behavioral problems at 3 years of age. In fact, a rank order of the CBCL/2–3 scores indicates one regulatory disordered child in each of the four quartiles. Additionally, none of the four regulatory disordered infants exhibited a CBCL/2–3 score above the clinical cutoff. Thus, the infant measures of behavioral difficulties were not correlated with the vagal tone measures, nor were they predictive of child behavior problems.
VAGAL TONE AND THE REGULATION OF THE VAGAL BRAKE: TWO TIERS OF A MULTILEVEL NEGATIVE-FEEDBACK SYSTEM
The dual functions of vagal tone may be conceptualized within a two-tiered neural feedback system, The first tier coincides with the vagal contribution to visceral homeostasis. Driven by visceral afferents, the first tier represents a basic negative feedback system (see Weiner, 1948). As illustrated in Figure 7.2, the basic negative feedback system includes a central regulator that determines the motor output to an organ after interpreting the information from the sensory input that monitors the status of the organ.
FIGURE 7.2. Schematic representation of a basic negative feedback model.
The first tier of the vagal regulation model is a slight modification of the basic negative feedback model. As illustrated in Figure 7.3, the general flow is the same as Figure 7.2, although the specific cranial nerve, brainstem nuclei, and target organ are identified. On the first tier, the central regulator consists of two nuclei in the brainstem, the nucleus ambiguus and the nucleus tractus solitarius, and their interneuronal connections. The vagus nerve contributes both the motor and sensory pathways between the central regulator (brainstem) and the visceral organ (heart). The vagal efferents to the heart originate in the nucleus ambiguus (NA) and the vagal afferents from the heart terminate in the nucleus tractus solitarius (NTS).
As illustrated in Figure 7.3, the vagus provides a bidirectional connection between the brainstem and the heart. The two-way vagal communication between the brainstem and the heart enables the brainstem to monitor and to regulate heart rate and cardiac output. Within this negative feedback model, the efferent vagal pathways contribute to homeostasis by providing negative feedback following the brainstem's interpretation of cardiac state from the afferent vagal pathways. Two-way communication between the brainstem and the heart has long been acknowledged. Darwin (1872) describes the bidirectional communication and attributes this information to Claude Bernard. The interneuronal communication between the brainstem source nuclei of both the efferent and afferent vagal pathways regulating the heart is the topic of current electrophysiological and neuroanatomical research (Richter & Spyer, 1990).
FIGURE 7.3. Schematic representation of a negative feedback model for vagal control of the heart.
The second tier of the vagal regulation system requires vagal tone changes which modulate cardiac output to match metabolic needs determined by environmental challenges. The second tier is characterized by the cortical inhibition of the brainstem structures controlling vagal output to the heart. The second tier regulates metabolic output by inhibiting vagal tone, which increases heart rate and cardiac output, to foster metabolically demanding motor behaviors. To increase metabolic output, the cortical structures actively inhibit the vagal support of homeostatic functions. To decrease metabolic output, the cortical structures disinhibit the brainstem structures regulating vagal output; vagal output is increased to support homeostatic functions and cardiac output is decreased. Gradations in cortical inhibition of vagal activity enable a fine-tuning of the rapid changes in cardiac output required to foster specific behaviors associated with environmental interactions. This dependency of Tier 2 functions (behavioral interactions with the environment) on Tier 1 functions (homeostasis) requires bidirectional neuronal communication between cortical and brainstem structures. As illustrated in Figure 7.4, this is modeled as a dynamic interaction between cortical and brainstem structures to monitor and regulate visceral tone and metabolic output. The functional interaction between the two tiers provides a neurophysiological mechanism which is capable of either increasing cardiac output to support fight-or-flight behaviors or decreasing cardiac output to calm and self-soothe.
As illustrated in Figure 7.4, the second tier also includes the basic components of a negative feedback system: the monitoring of environmental stimulation via sense receptors (i.e., exteroceptors), the interpretation of the sensory information by brain structures (i.e., cortex), and the selective output of specific observable behaviors. The objective of the second tier is to maximize goal-directed (i.e., motivated) behaviors that, in general, minimize unpleasant or life-threatening experiences and follow the law of effect (Thorndike, 1911). For example, via appropriate negative feedback, positive sensory experiences (e.g., pleasure) will promote engagement with the environment, and negative sensory experiences (e.g., pain) will result in a disengagement with the environment. In understanding or evaluating social behavior, the second tier choreographs the quality and temporal dynamics of approach-withdrawal sequences. The second tier regulates when and with whom or with what an individual behaviorally engages and disengages.
The two-tiered model is hierarchical with priority given to the demands of the second tier. However, without the first tier functioning adequately and regulating homeostasis to maximize life-support processes such as digestion, oxygenation, thermoregulation, and perfusion, there would be no "energy" resources available for second-tier functions. As a metaphor, the measurement of vagal tone level during a steady state as an index of Tier 1 regulation represents "potential energy." Similarly, the change in vagal tone from a steady-state baseline to a metabolically challenging state as an index of Tier 2 would represent "kinetic energy."
FIGURE 7.4. Schematic representation of a two-tiered negative feedback model. The two-tiered model provides a theoretical framework to explain the operations of the vagal brake in response to environmental challenges (e.g., social interactions). Note the bidirectional communication between cortical and brainstem structures.
The data illustrated in Figure 7.1 support this metaphor relating baseline vagal tone to vagal reactivity. Infants with greater baseline vagal tone (an index of Tier 1 function) exhibited greater decreases in cardiac vagal tone and heart period during the Bayley test. These findings are consistent with other studies that have reported a relationship between individual differences in baseline cardiac vagal tone and the magnitude of heart period and vagal tone reactivity (DeGangi et al., 1991; Porges & Lipsitt, 1993; Porter & Porges, 1988).
Behavior is metabolically costly. For example, behaviors such as fight-or-flight responses often require massive and instantaneous increases in metabolic output. To successfully accomplish tasks of engagement and disengagement with the environment, the nervous system must divert energy resources from visceral homeostasis (e.g., smooth muscle) to observable behaviors (e.g., striate muscle) that deal directly with the environment. The regulation of the vagal brake provides an index of this shift in resources. By instantaneously releasing the vagal brake, cardiac output increases to support the metabolic demands required by the behavior. Thus, the priority of the two-tiered system subjugates the homeostatic needs in favor of the immediate environmental demands.
CONSTRUCTS OF VAGAL TONE AND THE VAGAL BRAKE
The construct of vagal tone, a Tier 1 process, represents the magnitude of vagal efferent output on the heart. Within Tier 1, optimal homeostatic function, at least in terms of the heart, would be associated with greater vagal tone, reflecting greater negative feedback. There is an abundance of data supporting the notion that greater vagal tone is associated with more optimal physiological state. Low heart rate variability or low cardiac vagal tone has been identified as a risk factor in infants (see chapter 4), in neurosurgery patients (Donchin et al., 1992), and in individuals with cardiovascular risk (Bigger, Fleiss, & Rolnitsky, 1993). The data from this study not only are convergent with these findings, but demonstrate that low levels of cardiac vagal tone during infancy are related to visceral and behavioral state regulation in childhood.
The construct of the vagal brake described within the second tier represents the systematic decrease and increase in vagal efferent output to the heart. In our research, the decrease in cardiac vagal tone reflects a reduction in the vagal inhibitory effect on heart rate. Release of the vagal brake increases heart rate and cardiac output. There have been reliable reports of decreases in indices of cardiac vagal tone (i.e., measures of heart rate variability) in response to environmental demands (e.g., Allen & Crowell, 1989; Mulder & Mulder, 1987; Porges, 1972).
Social behavior requires the second-tier system. Social interactions require a shift in metabolic resources to foster the appropriate behaviors. Thus, the infant's ability to release the vagal brake during a social-attention task might reflect a physiological strategy that would promote social development and fewer subsequent behavioral problems. The data from this study confirm this speculation. Infants who exhibited a greater decrease in cardiac vagal tone (i.e., greater release of the vagal brake) during the social-attention task exhibited fewer social behavior problems in childhood.
SUMMARY
The nucleus ambiguus vagal system deals both with servicing the needs of the internal viscera and with responding to external challenges. Competence in the ability to trade off between homeostatic (internal) and environmental (external) needs is required for successful development. This chapter introduces the vagal brake as a metaphor to describe regulation of visceral tone in response to environmental challenges. Although there is an extensive literature evaluating baseline vagal tone, the relation between dynamically changing vagal tone and behavioral reactivity has been theoretically vague. This chapter, by introducing a two-tiered neural feedback model, provides the first theoretical model to explain the relation between vagal tone during steady states and vagal reactivity (i.e., the vagal brake) in response to environmental challenges.
Co-authors for this chapter were J. A. Doussard-Roosevelt, A. L. Portales, and S. I. Greenspan.
CHAPTER 8
The Early Development of the Autonomic Nervous System Provides a Neural Platform for Social Behavior
During the last trimester and continuing through the first year postpartum, the autonomic nervous system is rapidly changing. These changes ensure that the infant can breathe, obtain food, and maintain body temperature. Coupled with the development of these abilities to obtain basic biological needs is a progressive change in the infant's ability to regulate physiological and behavioral state through interactions with another person (e.g., mother). We propose that the developmental changes in the neural pathways that regulate autonomic state provide a neural platform to support the expanding abilities of the infant to engage objects and people in a dynamically changing environment. Thus, the emerging behavioral repertoire and social-interactive needs of the rapidly developing young infant should be studied within the context of the maturational changes in the autonomic nervous system.
In contrast to the hypothesized dependence of social behavior on the autonomic nervous system, the autonomic nervous system has played a limited role within predominant theories in developmental psychology. Similarly, given the critical role that the autonomic nervous system plays in the infant's survival during the transition from prenatal to postnatal environments, it is surprising that an understanding of the central mechanisms mediating the autonomic nervous system has been tangential to pediatric medicine. In general, measures of autonomic activity have been conceptualized as a correlate of motivated or adaptive behaviors within psychology and as indices of health risk in medicine.
We propose, from a biobehavioral perspective, that the maturational shifts in neural regulation of the autonomic nervous system provide a developing resource, which, in part, mediate much of the dynamic interaction that the infant has with both objects and others. Within this perspective, we identify sequential, developmental, neurophysiological processes, which provide the biological basis to explain how and why individual and developmental differences in vagal control of the heart are related to social behavior and, when relevant, clinical outcome.
DEPENDENCE ON OTHERS: PARALLELS WITH MATURATION OF THE AUTONOMIC NERVOUS SYSTEM
At birth, mammalian infants are incapable of caring for themselves. The newborn is vulnerable and dependent on the mother, or a concerned caregiver, to survive. Support from others is required to obtain basic biological needs, such as food, warmth, and protection. This dependence on other decreases as the infant develops. This decrease in dependence is paralleled by changes in neural regulation of the autonomic nervous system. During development, as higher brain circuits begin to regulate the brainstem nuclei, which control the autonomic nervous system, the infant becomes more independent and is increasingly capable of initiating social interactions with others to regulate physiological state. As these self-regulatory skills develop, the dependence on the caregiver to elicit ingestive-vagal reflexes (i.e., feeding) as a primary strategy of regulation decreases. Behaviorally, the infant appears to be more socially skilled and better able to spend time alone. This is observed as the infant's ability to both rapidly calm after disruptive challenges and to remain calm for longer periods, even in the absence of others.
Developmentally, as skills of state regulation improve, the central nervous system expands to promote enhanced cognition and greater control over peripheral motor systems. These global systems (autonomic, cognitive, and motor) mature in combination and enable the maturing infant to become more independent and explorative in a complex environment.
Similar to most mammals and distinct from our phylogenetically related reptilian ancestors, humans maintain a need for social interactions throughout their life spans. Social separation and isolation for humans, regardless of age, leads to profound disruption in the ability to regulate physiological state and compromises both physical and mental health. This disruption may impact all aspects of development and may be expressed as delays in motor development, growth, and cognition, as well as global health vulnerabilities and atypical social and emotional behaviors. Studies reliably report that various degrees of social abandonment can have disastrous effects on child development. Studies of the Romanian orphans illustrate that caregiving, defined solely by the physical features of food, warmth, and protection without consistent and predictable social engagement opportunities, is insufficient for typical development. For example, in a 3-year follow-up study of Romanian orphans, who had been adopted after living at least 8 months in orphanages, Chisholm (1998) found that the orphans had significantly more behavior problems, insecure attachment, and lower IQ scores than nonadopted or early adopted children. Adoptive parents of these orphans also reported significantly more parenting stress. More recent anatomical studies have revealed that early social and emotional deprivation in many Romanian orphans may lead to functional and structural changes in brain regions, including orbital frontal gyrus, infralimbic prefrontal cortex, medial temporal structures (i.e., amygdala and head of hippocampus), and left uncinate fasciculus (Chugani et al., 2001; Eluvanthingal et al., 2006). Impairment in functions of these brain regions could certainly contribute to lower verbal skills, diminished attention and concentration abilities, impaired impulsivity, conduct problems, and depression frequently reported in these children.
Unlike reptiles and other phylogenetically more ancient vertebrates, birth for mammals is not a transition into independence but an extension of the period of dependence that begins in utero. For humans, maturation does not lead to a total independence from others, but leads to an ability to function independently of other people for short periods. Moreover, humans, as they become more independent of their caregivers, search for appropriate others (e.g., friends, partners, etc.) with whom they may form dyads capable of symbiotic regulation. In fact, individuals, who prefer to use interactions with objects instead of people to regulate physiological state, often receive psychiatric diagnoses (e.g., autism spectrum disorders, borderline personality disorder).
EVOLUTION OF THE AUTONOMIC NERVOUS SYSTEM PREPARES THE HUMAN INFANT FOR SOCIAL BEHAVIOR
It is necessary to understand the phylogenetic origins of mammalian autonomic nervous systems to understand how autonomic state is linked to social behavior. This conceptualization has been described as the polyvagal theory (see chapters 2, 11, and 12; Porges, 2001a, 2007a). The theory was derived from the identification of the two motor branches of the vagus nerve, the Xth cranial nerve, which provides both motor and sensory pathways between brainstem structures and visceral organs. The theory emphasizes not only the phylogenetic shifts in neuroanatomy of the vertebrate autonomic nervous system but also proposes specific, adaptive behaviors that co-occur with these transitions.
As evolutionary forces molded the human nervous system, new structures were added and older structures were modified to allow greater dynamic range and finer control of physiological state and to facilitate emergence of new, adaptive social behaviors. The polyvagal theory, by incorporating an evolutionary approach, provides a strategy to investigate developmental shifts in social behavior from a phylogenetic perspective. The theory provides insights into how developmental shifts in neural regulation of the autonomic nervous system are related to the changing repertoire of adaptive behaviors that, depending on context, either limit or expand expression of social behavior. The theory emphasizes unique features in neural regulation of the autonomic nervous system that distinguish mammals from reptiles and explains how these features serve as a biobehavioral platform for the emergence of face-to-face social behaviors. In addition, the theory provides insights into the biobehavioral mechanisms that are rapidly developing in utero and in the young infant. As the neural mechanisms facilitating self-regulation improve during normal development, the infant's dependence on others to regulate physiological state decreases. This allows social communication to expand beyond the cueing of basic physical survival needs (i.e., warmth, safety, food) and into the realm of prosocial engagements.
The polyvagal theory articulates how each of three phylogenetic stages in the development of the vertebrate autonomic nervous system is associated with a distinct autonomic subsystem that is retained and expressed in mammals. These autonomic subsystems are phylogenetically ordered and behaviorally linked to social communication (e.g., facial expression, vocalization, listening), mobilization (e.g., fight-or-flight behaviors, tantrums, or behavioral meltdowns), and immobilization (e.g., feigning death, vasovagal syncope, and behavioral and physiological shutdown). The product of this phylogeny is a mammalian nervous system with three identifiable circuits that regulate adaptive behaviors and physiological reactions to challenges.
In this phylogenetically organized hierarchy, the newest circuit associated with social communication is used first. If that circuit fails to provide safety, then the older survival-oriented circuits are recruited sequentially. From a developmental perspective, the oldest circuits develop first, and the newest circuit develops last, leaving it the most vulnerable to neural insult and the most sensitive to postpartum experience. The newest circuit becomes only partially available during the last trimester and is expressed at term as the brainstem reflexes that enable the coordination of sucking, swallowing, and breathing. By 6 months postpartum, these brainstem reflexes, which become coordinated with cortical processes, provide a biobehavioral pathway through which reciprocal social engagement behavior can calm and soothe physiological state in both participants of a social dyad (e.g., mother–infant interactions).
DEVELOPMENT OF THE VAGAL BRAKE: CONSEQUENCES FOR POSTPARTUM ADAPTATION AND SOCIAL BEHAVIOR
The mammalian nervous system did not develop solely to survive in dangerous and life-threatening contexts but also to promote social interactions and social bonds in safe environments. To accomplish this adaptive flexibility, a new neural strategy requiring safe environments emerged, while the more primitive neural circuits to regulate defensive strategies were retained. To accommodate both fight-or-flight and social engagement behaviors, the new mammalian vagus evolved to enable rapid, adaptive shifts in autonomic state. The mammalian myelinated vagus functions as an active vagal brake (see chapter 7) in which inhibition and recovery of the vagal tone to the heart can rapidly mobilize or calm an individual. Tonic vagal influences to the sinoatrial node (i.e., the primary cardiac pacemaker) produce a resting heart rate that is substantially lower than the intrinsic rate of the pacemaker alone. When vagal tone, through myelinated vagal pathways, to the pacemaker is high, the vagus acts as a restraint, or brake, limiting the rate at which the heart can beat and functionally calming the individual. When vagal tone to the pacemaker is low, there is little or no inhibition of the pacemaker, and the heart rate increases. The vagal brake construct may be used to describe functional modulation of heart rate by myelinated vagal efferent pathways.
The state of the vagal brake can be quantified as the amplitude of a periodic component in the beat-to-beat heart rate pattern known as respiratory sinus arrhythmia (RSA). RSA is a naturally occurring rhythm in the heart rate pattern that oscillates at approximately the frequency of spontaneous breathing. RSA represents only a portion of beat-to-beat heart rate variability. By quantifying RSA and the relation between RSA and heart rate during various challenges, it is possible to measure the dynamic regulation of the myelinated vagal brake to study the responses of infants and young children to people and to objects (e.g., Bazhenova, Plonskaia, & Porges, 2001).
The human infant is not born with a completely functioning myelinated vagal system. The mammalian vagus is only partially myelinated at birth and continues to develop during the first few months postpartum. Morphological studies demonstrate a rapid developmental increase in the total number of myelinated vagal fibers from 24 weeks through adolescence with the greatest increases observed from approximately 30–32 weeks of gestational age to approximately 6 months postpartum (see Sachis, Armstrong, Becker, & Bryan, 1982). More recent neuroanatomical research suggests that the increase in myelinated fibers may be occurring in the absence of an increase in unmyelinated vagal fibers, since the increase in the number of myelinated vagal fibers is paralleled by a decrease in the ratio of unmyelinated to myelinated vagal fibers (Pereyra, Zhang, Schmidt, & Becker, 1992). A relative increase in myelinated vagal fibers would functionally improve visceral regulation and enable the infant to express better behavioral regulation, which would support spontaneous social engagement behaviors. Based on these studies, preterm infants born before approximately 30 weeks gestational age are likely compromised due to lack of an appropriately functioning mammalian vagus, or vagal brake. Without a functioning mammalian vagus, the preterm has a limited ability to regulate visceral state and is dependent solely on the sympathetic nervous system and on the phylogenetically older unmyelinated vagus to meet physiological needs. This compromised profile of autonomic regulation obligates the preterm infant to rely on the sympathetic nervous system to increase heart rate in response to distress, in order to support tantrums and mobilization behaviors. Preterm infants are also more vulnerable to clinically dangerous hypotensive states and lowered oxygen saturation caused by episodes of bradycardia and apnea (i.e., massive slowing of the heart and cessation of breathing), which may be triggered by ingestive behaviors (e.g., sucking and swallowing) through activity in the more primitive unmyelinated vagus that has already developed.
RSA in the preterm infant follows a maturational trajectory that parallels the reported changes in both number and ratio of myelinated vagal fibers. During the preterm period there is a monotonic increase in RSA from 32 to 37 weeks gestational age (Doussard-Roosevelt, Porges, Scanlon, Alemi, & Scanlon, 1997). Opportunities for skin-to-skin contact (i.e., kangaroo care) between mother and preterm (Feldman & Eidelman, 2003) enhance the development of RSA. Paralleling the enhanced vagal regulation, these authors also reported more rapid improvement in state organization and a more mature neurodevelopmental profile. However, the enhanced development of RSA was only relative to preterm controls not receiving skin-to-skin contact and was still substantially lower than reports of RSA in typically delivered full-terms (see chapter 4).
In longitudinal studies evaluating heart rate and RSA in term infants during the first year postpartum, heart rate reliably slows with age, although RSA changes are less obvious (i.e., Fracasso, Porges, Lamb, & Rosenberg, 1994; Izard et al., 1991). In these studies the effects of development on RSA appear to be maximized during the first 6 months postpartum and taper during the 6–12-month period. Although these developmental trends have been reported and replicated, closer inspection of the data indicates that individual differences in RSA during the first year postpartum are even greater than the developmental shifts.
Based on parallel literatures that describe developmental features in both RSA and the neuroanatomy of the vagus nerve, we may make two inferences: (1) RSA during early infancy reflects the functional outflow of the myelinated vagus, and (2) efficient RSA reactivity and recovery are dependent on both the number of myelinated vagal fibers and the ratio of myelinated to unmyelinated vagal fibers. The literature on young infants supports these assumptions by demonstrating increases in RSA during the last trimester through the first few months postpartum. The second assumption has primarily been tested during feeding when ingestive-vagal reflexes are recruited, although additional research will need to examine reactivity during social challenges during the first few months postpartum (e.g., Bazhenova et al., 2001; Moore & Calkins, 2004; Weinberg & Tronick, 1996).
FEEDING: CHALLENGING AND EXERCISING THE VAGAL BRAKE THROUGH AN INGESTIVE-VAGAL REFLEX
To ingest food properly and efficiently, the newborn must have the neural resources to implement the complex sequence of sucking, swallowing, and breathing. This sequence requires the coordination of the striated muscles of the face, head, and neck with the myelinated vagal regulation of the bronchi and the heart. Unlike the striated muscles of the limbs, striated muscles of the face, head, and neck are regulated by pathways traveling from the brainstem through several cranial nerves. These pathways are known as special visceral efferent pathways, although anatomically, their targets are not considered visceral. These pathways are often called branchiomeric, since they regulate the structures that embryologically emerged from the pharyngeal (branchial or ancient gill) arches. As the infant matures, the special visceral efferent pathways are recruited by corticobulbar pathways and expressed through social engagement behaviors. Autonomic support for these muscles is provided by the myelinated vagus, which can be dynamically monitored by quantifying RSA. This face–heart connection provides the necessary elements for an integrated social engagement system (see later discussion).
Since the structures involved in the neural regulation and coordination of the striated muscles involved in sucking, swallowing, vocalizing, and breathing are all linked to the myelinated vagus, the functioning of these behaviors, and the link between these behaviors and RSA, may provide an early indicator of the functional status of a system that will later be involved in social engagement behaviors.
The status of the face–heart connection can be evaluated by measuring RSA changes during the ingestive behavior of sucking, which requires coordination of the striated muscles of the face with visceral changes in heart rate and breathing. Porges and Lipsitt (1993) monitored the integration of sucking behaviors with heart rate and RSA as infants sucked to obtain sucrose. In response to sucrose, reductions in RSA and corresponding increases in heart rate paralleled increased sucking frequency. When the availability of sucrose was terminated, RSA and heart rate returned to presucrose levels. Moreover, individual differences in baseline RSA were correlated with the magnitude of heart rate reactivity to the gustatory stimulation. These findings illustrate that in the healthy neonate, there is a coordinated ingestive response in which the vagal brake is systematically removed to support the increased metabolic demands of sucking.
Because ingestive-vagal responses can be systematically elicited, a feeding challenge paradigm provides an opportunity in newborns and preterm infants to evaluate the status of coordinated physiological-behavioral sequences that require vagal regulation and control of the striated muscles of the face, head, and neck. Similar to the full-term infants described in the Porges and Lipsitt study, clinically stable low-birthweight preterm infants (near the time of discharge) also decreased RSA and increased heart rate during feeding (Portales et al., 1997). When feeding was terminated, heart rate and RSA returned to prefeeding levels. In a second study, with more clinically compromised infants (Suess et al., 2000), RSA and heart rate were monitored before, during, and after oral or gastric tube bolus feedings in preterm infants at approximately 33–34 weeks postmenstrual age. The preterm neonates were categorized into two groups based on gestational age at birth. The earlier-born group had gestational ages less than 30 weeks, and the later-born group had gestational ages greater than 30 weeks. Consistent with the foregoing studies, RSA decreased in both groups during feeding. However, postfeeding RSA increased toward prefeed levels (i.e., recovered) only in later-born infants. The results confirmed the assumption that the higher risk group, independent of corrected gestational age, experienced compromised vagal regulation during the feeding paradigm.
The primary characteristics of the feeding response are a withdrawal of vagal control of the heart to increase metabolic activity necessary to ingest food and a postfeeding recovery of vagal tone to foster a calm state and support digestion. Thus, during metabolically demanding ingestive behaviors such as sucking, there is a reduction of myelinated cardiac vagal tone to allow greater mobilization of energy resources. Following these ingestive behaviors, there is a recovery of vagal function to support digestion and calm the infant.
The studies described here indicate that term infants and stable older preterm infants modulate the vagal brake during feeding. This ingestive-vagal reflex might be mediated solely at the brainstem level and might not be sensitive to the increasing number and efficiency of corticobulbar pathways connecting the cortex with the source nucleus of the myelinated vagus, the nucleus ambiguus (NA), which arise during maturation. From an evolutionary perspective, the adaptive consequence of these higher order neuroanatomical changes would enable the older infant to use social cues to regulate the vagal brake. It is possible that through neuroception (see chapter 1), the loving caregiver's facial features and vocal prosody trigger temporal corticolimbic pathways that dampen defensive reactions and recruit the vagal brake to calm. In contrast, the facial and vocal features of a stranger might inhibit the vagal brake to allow mobilizing, protesting, and defensive behaviors.
THE SOCIAL ENGAGEMENT SYSTEM
A face–heart connection evolved in mammals as source nuclei of vagal pathways shifted ventrally from the phylogenetically older dorsal motor nucleus (e.g., unmyelinated vagal pathways) to the NA (e.g., myelinated vagal pathways). This resulted in an anatomical and neurophysiological linkage between neural regulation of the heart via the myelinated vagus and the special visceral efferent pathways that regulate the striated muscles of the face, head, and neck. Together, this linkage, between brainstem motor systems responsible for cardiovascular functions and those necessary for regulating the face, head, and neck, forms an integrated social engagement system.
With increased cortical development, the cortex exhibits greater control over the brainstem via direct (e.g., corticobulbar) and indirect (e.g., corticoreticular) neural pathways originating in the motor cortex and terminating in the source nuclei of the myelinated motor fibers emerging from the brainstem (e.g., specific neural pathways embedded within cranial nerves V, VII, IX, X, and XI). These cranial nerves then extend from their source nuclei to control visceromotor structures (i.e., heart, bronchi) as well as branchiomotor structures (muscles of the face, head, and neck).
Specifically, the social engagement system includes the regulation of the eyelids through the orbicularis oculi (e.g., social gaze and gesture), muscles of facial expression (e.g., emotional expression), middle ear muscles (e.g., extracting human voice from background sounds), muscles of mastication (e.g., ingestion, sucking), laryngeal and pharyngeal muscles (e.g., vocalizing, swallowing, breathing), and muscles of head turning and tilting (e.g., social gesture and orientation). Collectively, these muscles act as filters for social stimuli (i.e., observing others' facial expressions and detecting prosody in human voice), and they allow the expression of motor behaviors necessary for engagement with the social environment.
Based on the polyvagal theory (see chapter 2), the development of the mammalian myelinated vagus is critical in the development of the face–heart connection, which links social behavior and autonomic regulation. Thus, with more optimal vagal regulation, features of more adaptive social behavior emerge. Without a functioning myelinated vagus, social behavior would be compromised, and more primitive defensive strategies, such as fight-or-flight mobilization and tantrums (mediated by the sympathetic nervous system) and shutdown behaviors (mediated by the unmyelinated vagal system) would be more frequently expressed. Clinically, the status of vagal myelination becomes critical for the newborn and the young infant as they attempt to engage and disengage the caregiver and to explore social reciprocity as a mechanism to regulate physiology and behavior.
MODEL: AUTONOMIC STATE PROVIDES A NEURAL PLATFORM FOR SOCIAL BEHAVIOR
The unique features of the autonomic nervous system that support mammalian social behavior start to develop during the last trimester of fetal life. Specifically, myelination of vagal efferent fibers from the NA to the sinoatrial node (i.e., cardiac pacemaker) begins during the last trimester. This process continues during the first few months postpartum and can be monitored by quantifying the amplitude of RSA. Regulation of the myelinated vagus (i.e., vagal brake) provides a mechanism to rapidly increase metabolic output by releasing the brake on the heart's pacemaker (i.e., sinoatrial node). The release of the vagal brake results in an instantaneous increase in heart rate. Subsequently, when the metabolic demands decrease, inhibitory vagal influence on the heart is reinstated, and heart rate instantaneously decreases. The regulation of the vagal brake influences important developmental processes related to survival, including (1) improved reactivity and recovery of the ingestive-vagal reflexes involved during feeding, (2) expanded capacity to self-regulate and calm, and (3) improved abilities to both spontaneously engage others and to be soothed by others. This developmental time line is illustrated in Figure 8.1.
Prematurity, illness, or neglect may dampen the developmental trajectory of the vagal circuit. Atypical maturation of this circuit may be reflected in myelination of the vagus, interneuronal connections in the brainstem that form the face–heart connection, and/or corticobulbar regulation of the brainstem circuits regulating both vagal activity and the striated muscles of the face, head, and neck. The consequences of these delays or disruptions in typical neural maturation would be expressed as lower levels of RSA, less efficient reactivity and recovery of the vagal brake, difficulties in behavioral state regulation, poor affective tone, and diminished abilities for reciprocal social engagement behaviors.
FIGURE 8.1. Developmental timeline relating myelination of the vagus to social behavior.
The myelinated vagus is not the sole mediator of autonomic state, in general, or heart rate, specifically. Heart rate is influenced by intrinsic cardiac mechanisms, surrounding thoracic anatomy, the sympathetic nervous system, and the unmyelinated vagus originating in the dorsal nucleus of the vagus. The development of the sympathetic nervous system and the unmyelinated vagal circuit has not been extensively studied in the human fetus and has been assumed to be functioning at term. Within our phylogenetic perspective, we assume that these circuits are functioning at the start of the last trimester. Shortly, we support this assumption with a brief review of the limited literature focusing on studies of the embryological development of the central regulators and peripheral pathways directly influencing heart rate or the contractility of the heart. Although only the development of efferent (motor) fibers is discussed, afferent (sensory) pathways play a vital role in both autonomic function and in providing sufficient input to trigger normal neural development.
The development of the autonomic nervous system in the human fetus mirrors the broader phylogenetic progression already described. The phylogenetically oldest vertebrate autonomic system, which relies on unmyelinated efferent vagal fibers originating from the dorsal motor nucleus of the vagus (DMNX), is also, embryologically, the earliest system to develop in utero. An immature, undifferentiated DMNX first appears in the brainstem at 9 weeks gestation (Cheng, Zhou, Qu, Ashwell, & Paxinos, 2004; Nara, Goto, & Hamano, 1991). Magnocellular subdivisions become visible by 13 weeks, and clear demarcation of DMNX subnuclei, including the lateral cardiomotor subnucleus, occurs by 23 weeks. At 28 weeks, all magnocellular subnuclei are considered essentially mature (Cheng et al., 2004). Some, however, including Nara et al. (1991), believe that there may be some postnatal changes in the DMNX, such as increased nuclear columnar length and volume. Even if present, however, these postnatal changes are not considered to have much functional significance or physiological consequence in the neonate.
The other major component of the parasympathetic, cardioinhibitory ANS is the newest, myelinated vagal system, which originates in the NA. This system, as predicted by the polyvagal theory, develops last in the fetus and continues functional development well into the first postnatal year. Mature neurons appear in the rostral NA by 8 or 9 weeks gestation and fill the nucleus by 12.5 weeks (Brown, 1990). Unlike mature neurons in the lateral subnucleus of the DMNX, however, axons of these mature neurons have not yet reached cardiac tissue to exert cardioinhibitory effects. The functional significance of vagal fibers from the NA depends heavily on myelination, which does not begin until 23 weeks gestation, when nearly mature axon diameter is achieved (Wozniak & O'Rahilly, 1981). Myelination of NA vagal fibers increases linearly from 24 to 40 weeks gestation and, again, continues actively during the first year postpartum (Pereyra et al., 1992; Sachis et al., 1982).
Development of the sympathetic, cardioexcitatory ANS is less well described in the literature. Phylogenetically, this largely catecholaminergic system appears before the mammalian NA vagal system and after the older DMNX vagus. According to the polyvagal theory, then, this system should begin development in the human fetus sometime between the two parasympathetic systems. Anatomically, sympathetic innervation of cardiac tissue is complex and difficult to isolate. Postganglionic cardiomotor nuclei lie mostly within the cardiothoracic and middle cervical ganglia, which lie caudal to the sympathetic superior cervical ganglion. Functionally, sympathetic influence on the heart is also varied. Unlike the two vagal circuits, which exert mostly chronotropic effects (slowing heart rate), sympathetic activity leads to both chronotropic (increasing heart rate) by innervating pacemaker tissue and inotropic (increasing cardiac contractility) by innervating ventricular myocardium. Investigations using fetal heart rate monitoring to infer sympathetic activity provide potential insight into the development of this system. Using continuous 24-hour fetal heart rate monitoring in 28 healthy women, who were 16–28 weeks pregnant, Kintraia, Zarnadze, Kintraia, and Kashakashvili (2005) reported that fetal locomotor activity increased between 16 and 20 weeks gestation. At this stage, increases in activity are accompanied by corresponding increases in heart rate, which returns to normal during "quiet" fetal periods. Since the vagal brake is not functional during this period of fetal development, these increases in heart rate are most likely due to activity in the sympathetic nervous system. Furthermore, the authors interpret an absence of such a coordinated heart rate increase with increased locomotion by 24 weeks gestation as "developmental retardation."
Measurement of RSA can map the development of the myelinated vagus and also enable dynamic monitoring of vagal reactivity and recovery elicited during feeding to quantify the status of ingestive-vagal reflexes. The literature documents that both amplitude of RSA and pattern of RSA responses during feeding are sensitive indices of risk in preterm and term infants. Ingestive-vagal reflexes elicited during feeding provide an opportunity early in development to evaluate a neural circuit that later will be involved in social engagement behaviors. Eliciting these reflexes provide an opportunity to exercise the neural circuits coordinating both the striated muscles of the face and neck and the vagal brake. As the infant develops, the brainstem structures involved in ingestive-vagal reflexes are increasingly recruited by higher brain structures, which regulate the facial and vocal features involved in reciprocal social interactions. Thus, when brainstem ingestive-vagal reflexes are functional, the first year postpartum is characterized by an increase in the efficiency with which corticobulbar pathways recruit and regulate these same brainstem nuclei for the purpose of social engagement. If, during early infancy, ingestive-vagal reflexes are not efficiently working, then there will be difficulties in coordinating sucking, swallowing, and breathing. Problems in regulating these survival-related processes may provide a sensitive prognostic index leading to difficulties not only in social behavior but also in the development of cognitive and language skills that are dependent on appropriate behavioral and physiological state regulation.
Changes in RSA represent a dynamic adjustment of the inhibitory action of the vagus (vagal brake) on the heart. Functionally, the removal of the vagal brake provides a physiological state that promotes vigilance as an intermediary and precautionary psychological process to monitor risk in the environment. The outcome of this assessment includes the induction of different physiological states, either in which social behaviors can proceed, or in which defensive fight-or-flight strategies associated with increased sympathetic excitation are necessary. If defensive behaviors are not necessary to maintain or to negotiate safety, then the rapid vagal regulatory mechanisms that dampen autonomic state are reinstated, allowing the individual to calm and self-soothe. Further support for this interpretation can be seen in the infant data in which suppression of RSA is correlated with maternal reports of longer attention spans and being more easily soothed (Huffman et al., 1998). For example, Huffman et al. (1998) found that 12-week-old infants with higher baseline RSA expressed fewer negative behaviors and were less disrupted by experimental procedures than age-matched infants with lower baseline RSA. Moreover, consistent with the vagal brake concept, the infants who decreased RSA during the laboratory assessment were rated on maternal report temperament scales as having longer attention spans and being more easily soothed.
The regulation of behavioral state is a critical determinant of the range of social behaviors an individual can express. The underlying mechanisms mediating behavioral state are tightly linked to the autonomic nervous system. Investigation of the early maturational changes in vagal regulation of autonomic state unmasks several of the behavioral features that infants exhibit. For example, greater suppression of RSA during challenging situations is related to better state regulation, greater self-soothing, more attentional control, and greater capacity for social engagement (see chapter 7; Calkins, Graziano, & Keane, 2007; DeGangi, DiPietro, Porges, & Greenspan, 1991; Huffman et al., 1998; Stifter & Corey, 2001).
The ability to regulate state follows a developmental trajectory during the early part of life. As the neural circuits involved in state regulation become more available to the developing child, there are parallel opportunities for social engagement behaviors and the development of strong social bonds. Without the dynamic, efficient myelinated vagus, it is difficult to regulate behavioral state and to use the features of the social engagement system (i.e., facial expression, vocal prosody), which at birth are involved in feeding behaviors (i.e., ingestive-vagal reflexes). Developmental limitations in the vagal system, expressed as low level of RSA and difficulties in regulating RSA, may lower thresholds to negative or ambiguous environmental cues with resultant hyperreactivity and severe limitations in the ability to self-soothe and calm.
Social behavior and the capacity to manage challenge are dependent on the neural regulation of physiological state. The neural circuits involved in the regulation of physiological state are modified during gestation and continue during postnatal life. If these circuits are easily available and efficiently functioning, then the laws of learning and the impact of experience can shape behavior. However, if these circuits are not available, either as a function of phase of development or during periods of increased environmental risk, then state regulation is compromised, social skills are not easily learned, and social bonds become difficult to establish. During most of the life span, the vagal brake and the other features of the social engagement system are readily available and provide opportunities for social learning to occur. Without the efficient vagal brake turning off defensive systems and blunting their disruptive manifestations (e.g., fight-or-flight behaviors), prosocial behavior is limited, and opportunities for social learning and social bonding are minimized.
Co-author for this chapter was S. A. Furman.
PART III
SOCIAL COMMUNICATION AND RELATIONSHIPS
CHAPTER 9
Vagal Tone and the Physiological Regulation of Emotion
Because emotions are psychological processes, the experience and regulation of emotion should be functionally dependent on the state of the nervous system. If a major source of emotion variation is dependent on the nervous system, how would this be evaluated? The goal of this chapter is to address this problem by introducing vagal tone as a measurable organismic variable that contributes to individual and developmental differences in the expression and regulation of emotion.
We propose that understanding the mechanisms determining individual and developmental differences in emotion expression and regulation might provide a rationale for identifying subjects who differentially express the ability to regulate emotion. Thus, there is the possibility that individual differences in the nervous system might mediate the expression and regulation of emotion. Most research on the autonomic correlates of emotion has focuses on sympathetic activation (e.g., galvanic skin response); here, we attempt to demonstrate that individual differences in parasympathetic tone are related to the regulation of emotion by focusing on a construct called vagal tone, which reflects the vagal control of the heart.
PHYSIOLOGY AND EMOTION: THE AUTONOMIC NERVOUS SYSTEM
The autonomic nervous system (ANS) regulates homeostatic function and is composed of two subsystems, the parasympathetic (PNS) and the sympathetic (SNS) nervous systems. The PNS and SNS represent neural systems that originate in the brainstem and contribute to the regulation of a variety of target organs, including the eyes; lacrimal, salivary, and sweat glands; blood vessels; heart; larynx, trachea, bronchi, and lungs; stomach; adrenal glands; kidneys; pancreas; intestines; bladder; and external genitalia. In general, the PNS promotes functions associated with growth and restorative processes. In contrast, the SNS promotes increased metabolic output to deal with challenges from outside the body. However, there are states that require dual excitation (e.g., sexual arousal). Conceptualizations of the ANS by Berntson, Cacioppo, and Quigley (1991a) provide insight into the complex dynamic relation between SNS and PNS processes.
In general, when a visceral organ is innervated by both the SNS and the PNS, the effects are antagonistic. For example, SNS neurons dilate the pupils, accelerate the heart, inhibit intestinal movements, and contract the vesical and rectal sphincters. The PNS neurons constrict the pupils, slow the heart, potentiate peristaltic movement, and relax vesical and rectal sphincters. The PNS deals primarily with anabolic activities concerned with the restoration and conservation of bodily energy and the resting of vital organs. In contrast, stimulation of the SNS prepares the individual for the intense muscular action required to protect and defend in response to external challenges. The SNS quickly mobilizes the existing reserves of the body.
Darwin provides insight into the potential importance of PNS processes in the regulation of emotions. Although Darwin defined emotions as facial expressions, he acknowledged the dynamic relation between parasympathetic structures and central nervous system activity that accompanied the spontaneous expression of emotions. Darwin speculated that there were specific neural pathways that provided the necessary communication between the brain states and the specific pattern of autonomic activity (e.g., heart rate) associated with emotions. In Darwin's formulation, when emotion states occur, the beating of the heart changes instantly, the changes in cardiac activity influences brain activity, and the brainstem structures stimulate the heart via the cranial nerves (i.e., vagus). Although Darwin did not elucidate the neurophysiological mechanisms that translate the initial emotion expression to the heart, this formulation provides us with three important points. First, by emphasizing the afferent feedback from the heart to the brain, Darwin anticipated the views of William James linking autonomic feedback to the experience of emotion. Second, he acknowledged the afferent capacity of the vagus to transmit sensory information from visceral organs independent of the spinal cord and the sympathetics. Third, Darwin's insight regarding the regulatory role of the pneumogastric nerve (renamed the vagus at the end of the 19th century) in the expression of emotions anticipates the major theme of this chapter.
Contemporary models of emotion and emotion regulation (i.e., Ekman, Levenson, & Friesen, 1983; Schachter & Singer, 1962), as did their historical antecedents, have focused on the sympathetic nervous system and ignored the vagal system, the primary component of the PNS. Thus, although Darwin speculated about the bidirectional communication between the brain and the heart via the vagus more than 100 years ago, the importance of vagal afferents and efferents in the expression, experience, and regulation of emotion has not been addressed.
VAGAL TONE: BACKGROUND AND DEFINITION
The vagus is the Xth cranial nerve. It originates in the brainstem and projects, independently of the spinal cord, to many organs in the body cavity, including the heart and the digestive system. The vagus is not a single neural pathway but rather a complex bidirectional system with myelinated branches linking the brainstem and various target organs. These neural pathways allow direct and rapid communication between brain structures and specific organs. Because the vagus contains both efferent (i.e., motor) and afferent (i.e., sensory) fibers, it promotes dynamic feedback between brain control centers and the target organs to regulate homeostasis.
The peripheral autonomic nervous system is asymmetrical. The peripheral target organs of the autonomic nervous system are clearly lateralized; for example, the heart is oriented to the left, the stomach is tilted, one lung is larger, and one kidney is higher. The neural wiring of the autonomic nervous system requires asymmetry, and the central regulation via the vagus is lateralized. Although asymmetry of cortical function is well known and has been theorized to contribute to emotion regulation (see Fox, 1994), asymmetrical regulation of autonomic function has been ignored.
The vagus is bilateral, with a left and a right branch. Each branch has two source nuclei, with fibers originating either in the dorsal motor nucleus or in the nucleus ambiguus. Traditional texts in neuroanatomy and neurophysiology (e.g., Truex & Carpenter, 1969; Williams, 1989) have focused on the dorsal motor nucleus of the vagus and neglected both the asymmetry in the vagal pathways and the important functions of the pathways originating from source nuclei in the nucleus ambiguus.
The dorsal motor nucleus is lateralized. Pathways from the left and right dorsal motor nucleus to the stomach have different regulatory functions. The left dorsal motor nucleus innervates the cardiac and body portions of the stomach that promote primarily secretion of gastric fluids (Kalia, 1981; Loewy & Spyer, 1990). The right dorsal motor nucleus innervates the lower portion of the stomach that controls the pyloric sphincter regulating the emptying into the duodenum (Fox & Powley, 1985; Pagani, Norman, & Gillis, 1988).
The nucleus ambiguus is also lateralized. While the right nucleus ambiguus provides the primary vagal input to the sinoatrial (SA) node to regulate atrial rate (Hopkins, 1987) and determine heart rate, the left nucleus ambiguus provides the primary vagal input to the atrioventricular (AV) node to regulate ventricular rate (Thompson, Felsten, Yavorsky, & Natelson, 1987). Given the ipsilateral control of efferent pathways regulating the nucleus ambiguus, characteristics of right-side brain damage are associated with defective right nucleus ambiguus regulation. In this manner, the observed deficits in prosody (e.g., Ross, 1981) and in heart rate changes during attention-demanding tasks (Yokoyama, Jennings, Ackles, Hood, & Boller, 1987) associated with right-side brain damage implicate the right nucleus ambiguus in the regulation of vocal intonation and attention. Asymmetrical nucleus ambiguus regulation is less clear in other organs, such as the soft palate, the pharynx, and the esophagus.
Functionally, the dorsal motor nucleus is involved with the vegetative functions of digestion and respiration. In contrast, the nucleus ambiguus is more involved with processes associated with motion, emotion, and communication. For example, rapid mobilization of the body may be achieved by regulating heart rate via removal of vagal input to the SA node. Vocal intonation, mediated by vagal connections to the larynx, is intimately related to the processes of emotion and communication. Facial expressions, critical to the expression of emotion and the signaling of information, are related to vagal function. In cats, vagal afferent fibers have direct influences on facial motoneurons (Tanaka & Asahara, 1981). Thus, the vagus originating from the dorsal motor nucleus might be labeled the vegetative vagus, in contrast to the emotive or smart vagus originating from the nucleus ambiguus. Table 9.1 provides a list of target organs associated with each branch of the vagus.
Sympathetic innervation of the heart is also asymmetrical (Randall & Rohse, 1956). Moreover, lateralized sympathetic input to the heart has been hypothesized to relate to emotion state (Lane & Schwartz, 1987). As with vagal control of the heart and larynx, research has demonstrated that damage to the right hemisphere has greater sympathetic consequences than does left hemisphere damage (Hachinski, Oppenheimer, Wilson, Guiraudon, & Cechetto, 1992).
The central control of the vagus is ipsilateral. Thus, the right vagus originates in either the right dorsal motor nucleus or the right nucleus ambiguus. As noted, the right nucleus ambiguus contains the primary source for the branch of the right vagus that provides input to the SA node. Thus, output from the nucleus ambiguus can be monitored by measuring changes in vagal control of the SA node. The SA node is the primary pacemaker of the heart. Vagal stimulation of the SA node delays the onset of the heartbeat (i.e., slows heart rate), and vagal withdrawal (i.e., a delay or blocking of the neural transmission) shortens the time between heartbeats (i.e., speeds heart rate). Most rapid heart rate changes (i.e., chronotropic mechanisms) are mediated by the vagus. When metabolic demands increase, such as during exercise or fight-or-flight demands, the sympathetic nervous system influences heart rate. Thus, the study of vagal control of the heart might provide an important window on the rapid autonomic changes associated with gradations of emotion state.
TABLE 9.1. Target Organs Associated With the Dorsal Motor Nucleus and the Nucleus Ambiguus
Dorsal Motor Nucleus | Nucleus Ambiguus
---|---
Trachea | Heart
Lungs | Soft palate
Stomach | Pharynx
Intestines | Larynx
Pancreas | Esophagus
Colon | Bronchi
VAGAL TONE: POTENTIAL LINK WITH EMOTION
An easily accessible method for evaluating the vagal control of the SA node (i.e., cardiac vagal tone) is to quantify respiratory sinus arrhythmia (RSA). RSA is characterized by a rhythmic increase and decrease in heart rate synchronous with breathing. The heart rate increase is associated with phases of inspiration, when respiratory mechanisms in the brainstem attenuate the vagal efferent action on the heart. The heart rate decrease is associated with phases of expiration, when the vagal efferent influence to the heart is reinstated.
Changes in RSA amplitude in response to sensory, cognitive, and visceral challenges represents a "central command" to regulate vagal efferents originating in the right nucleus ambiguus and terminating in the heart, soft palate, pharynx, larynx, bronchi, and esophagus. These changes in nucleus ambiguus regulation of peripheral autonomic activity support the expression of motion, emotion, and communication by regulating metabolic output (i.e., shifts in heart rate) and organs involved in the production of vocalizations (see chapter 13).
When there are no challenging environmental demands, the autonomic nervous system, through the vagus, services the needs of the internal viscera to enhance growth and restoration. However, in response to environmental demands, homeostatic processes are compromised, and the autonomic nervous system supports increased metabolic output to deal with these external challenges by vagal withdrawal and sympathetic excitation. By mediating the distribution of resources, the central nervous system regulates the strength and latency of autonomic responses to deal with internal and external demands. Perceptions and assumed threats to survival, independent of the actual physical characteristics of the stimulation, may promote a massive withdrawal of parasympathetic tone and a reciprocal excitation of sympathetic tone. These changes promote fight-or-flight behaviors. Less intense environmental demands, often associated with emotion expressions, might be characterized by less withdrawal of parasympathetic tone independent of or in concert with slight increases in sympathetic tone. This trade-off between internal and external needs is monitored and regulated by the central nervous system.
Vagal tone measured via RSA has been documented to be related to affect, attention, and metabolic demands (see later discussion). Although the vagus is bilateral, the right branch originating in the nucleus ambiguus is the primary determinant of RSA. This laterality in the vagus is not a developmental or an individual difference. Rather, the laterality is dependent on the neurophysiology and the neuroanatomy of the mammalian nervous system. In the mammalian nervous system, the right side of the brainstem provides the primary central regulation of homeostasis and physiological reactivity. Thus, right-brainstem structures initiate peripheral physiological states via shifts in vagal tone to facilitate the processes of attention, the expression of emotion, and the initiation of shifts in metabolic output.
THE RIGHT HEMISPHERE: THE REGULATION OF EMOTION
Right hemisphere function, evaluated via electroencephalography (EEG) or disrupted by localized damage, is related to the same cluster of behaviors that has been linked to the vagal tone measure. Research demonstrates that the right hemisphere is implicated in both the expression and the interpretation of emotions (e.g., Bear, 1983; Heilman, Bowers, & Valenstein, 1985; Pimental & Kingsbury, 1989; Tucker, 1981) and in the regulation of attention (e.g., Heilman & Van Den Abell, 1980; Mesulam, 1981; Pimental & Kingsbury, 1989; Voeller, 1986). Research has also linked right hemisphere deficits with aprosody or lack of emotion expression in speech (e.g., Ross, 1981; Ross & Mesulam, 1979; Zurif, 1974) and attenuated autonomic reactivity (e.g., Heilman, Schwartz, & Watson, 1978). Several investigators have argued that the right hemisphere provided the primary control of emotion (for detailed reviews, see Molfese & Segalowitz, 1988; Pimental & Kingsbury, 1989; Silberman & Weingartner, 1986).
EEG research has been used to provide support for laterality theories of emotion. Fox and colleagues (e.g., Dawson, 1994; Fox, 1994; Fox & Davidson, 1984) present a model of emotion expression in which positive (e.g., interest) emotions are associated with the left hemisphere and negative emotions (e.g., disgust or distress) with the right. Asymmetry of hemispheric control of negative and positive affect has also been posited by Tucker (1981). Other laterality theories focus primarily on the role of the right hemisphere in the regulation of negative emotions and fight-or-flight behaviors (for a review, see Silberman & Weingartner, 1986). The data strongly support the relation between right hemisphere EEG activity and the expression of negative emotions in infants, children, and adults; however, research demonstrating the relation between left hemisphere EEG activity and the expression of positive emotions is less conclusive.
In children, right hemisphere dysfunction has been associated with attentional, social, and emotional problems. Voeller (1986) reported data on 16 children with unilateral right hemisphere lesion or dysfunction as assessed by neuropsychological exam and/or CAT scan. Fifteen of these children were extremely distractible and inattentive, meeting the DSM-III criteria for attention deficit disorder; moreover, eight were also hyperactive. Eight children were shy and withdrawn, sharing some of the behavioral characteristics of the inhibited child described by Kagan (1994), and nine expressed atypical emotion expression (i.e., prosody, facial expression, and gesture). Most of these children made little eye contact with others, and virtually all had poor relationship with peers.
In their survey of studies with both normal and lesions subjects, Silberman and Weingartner (1986) suggested that the right hemisphere is superior for recognizing emotional aspects of stimuli. They propose that right hemisphere dominance for emotion regulation reflects a nervous system organization that gives priority to avoidance or defensive mechanisms that have a high survival value. By inference, these avoidance and defensive mechanisms require massive and immediate shifts in autonomic function.
THE RIGHT HEMISPHERE: AUTONOMIC REGULATION AND REACTIVITY
The right side of the brain also plays a special role in the regulation of emotion. Data supporting laterality theories of emotion have been based on studies of electrophysiological recordings from the scalp (e.g., Fox, 1994) and neuropsychological studies of dysfunction in individuals with brain damage (e.g., Silberman & Weingartner, 1986). We propose a convergent approach by emphasizing the right brain's regulation of peripheral autonomic activity.
Asymmetry in the control of the autonomic nervous system has been documented in the previous sections. Because peripheral organs are not symmetrical in shape or placement, it is not surprising that the neural control of the autonomic nervous system is lateralized. For example, the heart is displaced to the left, with the right vagus going to the SA node and the left vagus going to the AV node. Other organs with dual vagal innervation are often tilted (e.g., the stomach and intestines), are located higher on one side (e.g., the kidneys), or are larger on one side (e.g., the lungs).
Emphasis on the asymmetry of autonomic organs has implications for the evolution of central regulatory systems and cortical development. In mammals, the peripheral autonomic organs and brainstem structures are similar across species. Asymmetrical neural control of autonomic processes is characteristic of mammals. However, the process of encephalization differs among mammalian species, with man possessing a uniquely large cerebral cortex. Because the neural control of the vagus is ipsilateral (e.g., the left vagus originates in the left side of the brainstem), the right hemisphere—including the right cortical and subcortical structures—would promote the efficient regulation of autonomic function via the source nuclei in the brainstem. For example, neuroanatomical and electrophysiological studies demonstrate the important regulatory function of the right central nucleus of the amygdala in regulating the right nucleus ambiguus.
We propose that the functional dominance of the right side of the brain in regulating autonomic function has implications for specialization of motor and language dominance on the left side of the brain. The right-side responsibilities of regulating homeostasis and modulating physiological state in response to both internal (i.e., visceral) and external (i.e., environmental) feedback potentially enabled the control of other functions to evolve on the left side of the brain. With greater encephalization, which is characteristic of more cognitive mammalian species such as man, lateralized specialization is more observable.
A sharing of central control of voluntary and emotion-homeostatic processes would enable the individual to express complex voluntary levels of communication and movement via the left side of the brain and more intense emotion- homeostatic processes via the right side of the brain. If these processes are lateralized, they might have a degree of autonomous regulation. Of course, the central nervous system is complex and has, in many instances, both ipsilateral and contralateral communications. This provides a small percentage of individuals with central control of both language and dominant hand motor movement on the right instead of the left side of the brain. However, owing to the asymmetry of the peripheral autonomic organs and the medullary control of the autonomic nervous system, the right side of the brain is always dominant in the regulation of autonomic function and, thus, emotion.
Data from stimulation studies using left and right visual fields (e.g., Hugdahl, Franzon, Andersson, & Walldebo, 1983; Weisz, Szilagyi, Lang, & Adam, 1992) indicate that activation of the right cortex results in larger and more reliable autonomic responses. Additionally, studies of brain-damaged individuals have shown that right hemisphere damage or dysfunction is associated with a severe deficit in the facial, vocal, and autonomic components of the expression of emotions (Pimental & Kingsbury, 1989; Silberman & Weingartner, 1986). Similar asymmetry of the sympathetic nervous system has been reported, with the right stellate ganglion having greater cardiovascular control than the left stellate ganglion (Yanowitz, Preston, & Abildskov, 1966). However, no research has focused on the assessment of cardiac vagal tone in subjects with right hemisphere disorders. Since cardiac vagal tone is neurophysiologically linked to the right hemisphere regulation of autonomic activity, it might index the individual's functional capacity to regulate autonomic function and to express emotion.
THE VAGAL CIRCUIT OF EMOTION REGULATION: A MODEL
The right vagus and, thus, cardiac vagal tone are associated with processes involving the expression and regulation of motion, emotion, and communication. These processes enable individuals to approach and/or withdraw from objects and events in their environment. The regulation of attention, a major substrate for appropriate social behaviors, is included among these processes. Thus, the approach/withdrawal dimension includes movement in psychological as well as physical space. Vagal regulation of the heart modulates metabolic output to physically approach or withdraw; vagal modulation of vocal intonations provides clues indicating whether an individual is safe or dangerous to approach; feedback from our own facial muscles to the vagus and the ability to pay attention to social cues, including another person's facial muscles and verbal commands, allow us to negotiate appropriate approach or withdrawal behaviors.
Consistent with Schneirla (1959), who proposed that all behaviors could be described in terms of approach and withdrawal actions, the dimensions of approach and withdrawal play a central role in our model of the vagal regulation of emotion. Schneirla assumed that stimulus intensities modulated autonomic function to produce sympathetic dominance during high intensities and parasympathetic dominance during low intensities. However, according to our model of emotion regulation, sympathetic modulation is not always necessary, and the vagal system can promote approach or withdrawal behaviors via the right nucleus ambiguus control of heart rate and the intonation of vocalizations.
The vagal circuit of emotion regulation is schematized in Figure 9.1. The circuit focuses on right hemisphere regulation of emotion states via vagal projections from the nucleus ambiguus to the larynx and the SA node of the heart. The vagal control of the right side of the larynx produces changes in vocal intonation associated with the expression of emotions. The vagal control of the SA node produces a cardiovascular state associated with specific emotions and facilitating attention or fight-or-flight behaviors.
Emotion process may originate on a cortical level or may be initiated and/or regulated by afferent feedback from visceral organs. For example, if the emotion were triggered by a psychological process (e.g., perception of a specific stimulus), the following stages may occur: (1) cortical areas stimulate the amygdala; (2) the central nucleus of the amygdala stimulates the nucleus ambiguus; and (3) the right vagus regulates heart rate and vocal intonation by communicating with the SA node and the right side of the larynx. Regulation of the emotion response also may follow a specific path: (1) sensory information regarding the status of visceral organs stimulates vagal afferents, lateralized vagal afferents stimulate the nucleus tractus solitarius in the brainstem, and projections from the nucleus tractus solitarius stimulate the cortex, amygdala, and/or nucleus ambiguus to regulate the emotion expression; or (2) the emotion state could be initiated by the visceral afferent (e.g., stomach pains) and trigger the cortical, subcortical, brainstem, and autonomic responses associated with emotion.
FIGURE 9.1 Schematization of the vagal circuit of emotion regulation.
Interference with transmission on any level of the circuit may result in affective disorders, including emotion regulation problems or severe mood states. Dysfunction in the circuit could be caused by brain damage, neural transmission problems due to drugs, or learned dysfunction. The learning model is based on the demonstrations that classical conditioning and other associative learning paradigms that modify autonomic function are dependent on cortical-autonomic and amygdaloid-autonomic pathways. Thus, autonomic afferents to or from the nucleus ambiguus may be amplified, attenuated, or blocked via neuropathy, drugs, or associative learning to produce different affective states.
The vagal circuit of emotion regulation that we propose makes several important advances in the conceptualization of the physiology of emotion and emotion regulation. It introduces the importance of the vagal system in the physiology of emotion and emphasizes the bidirectional (i.e., afferent feedback) physiological characteristics of the vagus. Furthermore, the vagal circuit is neuroanatomically dependent on right medullary control of autonomic function via the nucleus ambiguus, has a noninvasive window of measurement via the quantification of RSA, explains individual differences due both to defects in neurophysiology and to associative learning, and is consistent with brain damage research. Finally, by emphasizing afferent feedback and communications among various levels of the nervous system, the vagal circuit of emotion regulation provides an explanation for the effectiveness of specific interventions (e.g., non-nutritive sucking, massage, eating, exercise, yoga, and cognitive strategies) in the regulation of emotions.
VAGAL REGULATION AND EMOTION
Interest in vagal tone as a regulatory mechanism in the expression of individual differences in autonomic function is not new. Eppinger and Hess introduced the concept of vagal tone as an individual difference construct in their monograph Die Vagotonie (1910). They described a form of autonomic dysfunction for which there was no known anatomical basis: "It is often unsatisfactory for the physician . . . to find that he must be content to make a diagnosis of 'Neurosis'. The symptomatology and the impossibility of establishing any anatomical basis for the disease always remain the most conspicuous points in formulating the diagnosis of a neurosis of an internal organ" (p. 1). The objective of their monograph was to identify a physiological substrate that could explain this anomaly and thus provide the mechanisms for a variety of clinically observed neuroses.
Although Eppinger and Hess were interested in clinical medicine, their case studies described a problem in the regulation of autonomic function that might be intimately related to the regulation of emotion. Their observations are relevant to our interest in the regulation and expression of emotion for four important reasons: first, they alerted us to the importance of the vagal system in mediating physiological and psychological responses; second, they related individual differences in physiology (i.e., vagal tone) to individual differences in behavior (i.e., neuroses); third, they recognized the pharmacological sensitivity of the vagal system to cholinergic agents; and fourth, they brought to the attention of the medical community the commonality of the vagal innervation of various peripheral organs.
In our model of vagal regulation, we adopt a stimulus-organism-response approach. The expression and regulation of emotions are the responses, which are dually determined by the stimulus and the organism. It is not just the stimuli that elicit a response; rather, the response is determined by a complex system of behavioral-physiological responses that involve perception of the stimulus, afferent feedback, and the regulation of approach-withdrawal behaviors via the vagal system. Because of the link between the right vagus and the processes of motion, emotion, and communication, individuals with low vagal tone and/or poor vagal regulation would be expected to exhibit difficulties in regulating emotion state, in appropriately attending to social cues and gestures, and in expressing contingent and appropriate emotions. Thus, the possibility exists that the vagal system may provide a physiological metaphor for the regulation of emotion states. Individual differences in vagal tone may index organismic factors related to the competency of the individual to react physiologically and to self-regulate.
VAGAL TONE RESEARCH
In testing our model of vagal regulation, we have been assessing empirically whether individuals with low vagal tone and/or difficulties in regulating vagal tone have problems with the expression and/or regulation of motion, emotion, and communication. The potential to move, express emotion, and communicate enables an individual to maneuver along a continuum of approach–withdrawal with the environment. Several behaviors are critical to this function, including behavioral reactivity, facial expressivity, and emotion regulation.
If vagal tone mediates the expression and regulation of emotion, developmental shifts in vagal tone might contribute to the observed developmental shifts in affective expression. Research has demonstrated that vagal control of the autonomic nervous system increases developmentally as the nervous system matures. We have reported a relation between gestational age and vagal tone in premature neonates (Porges, 1983) and a monotonic increase in vagal tone from birth through the first 18 days postpartum in rats (Larson & Porges, 1982). In the rat pups, these changes were paralleled by increased organization of behavior, including enhanced state regulation, exploration, and attention. Current longitudinal research with human infants has demonstrated that vagal tone increases monotonically from 3 to 13 months (Izard et al., 1991).
To evaluate whether vagal tone as a construct has properties that may be useful in explaining the expression of emotion and the regulation of affective state, the following sections review research on the relation between vagal tone and variables in the domains of reactivity, the expression of emotion, and self-regulation.
Reactivity
Here we provide theoretical justification and empirical support for the hypothesis that individual differences in vagal tone are related to heart rate and behavioral reactivity in young infants. The core proposition is that vagal tone indexes a dimension of central nervous system organization that disposes an individual to be hypo- or hyperreactive. Thus, subjects with higher levels of vagal tone should have more organized (i.e., consistent) autonomic responses with shorter latency and greater magnitude autonomic responses.
Before 1970, heart rate responses were defined as rapid increases and decreases to discrete stimuli. These response patterns were interpreted as an autonomic correlate of an orienting response (see Graham & Clifton, 1966). Research was not directed at the physiological mechanisms that may mediate individual differences in autonomic reactivity. Observed variations in heart rate response characteristics were assumed to be dependent on both the physical parameters of the stimulus and the subject's previous history with the stimulus. Individual differences that could not be attributed to these two sources were treated as experimental (i.e., measurement) error.
In the early 1970s, our research demonstrated that individual differences in spontaneous base-level heart rate variability were related to heart rate reactivity. These studies stimulated our interest in the vagal mechanisms mediating heart rate variability and in the development of methods to quantify vagal influences on the heart. The first studies (Porges, 1972, 1973) demonstrated, in a sample of college students, that individual differences in heart rate variability assessed during baseline conditions were related to heart rate responses and reaction time performance. These studies were followed by experiments with newborn infants that demonstrated a relation between baseline heart rate variability and the magnitude of heart rate responses to simple visual and auditory stimuli. Newborn infants with higher baseline heart rate variability reacted with larger heart rate responses to the onset and offset of auditory stimuli (Porges, Arnold, & Forbes, 1973) and with shorter latency responses to the onset of an increase in illumination (Porges, Stamps, & Walter, 1974). When the illumination was decreased, only the high heart rate variability subjects responded. Consistent with these findings, only the neonates with higher heart rate variability exhibited a conditioned heart rate response (Stamps & Porges, 1975).
Reports from other studies measuring RSA are consistent with the theme that vagal tone is an index of reactivity. Porter, Porges, and Marshall (1988) demonstrated in a sample of normal newborns that individual differences in RSA were correlated with heart rate reactivity to circumcision. Neonates with higher amplitude RSA exhibited not only larger heart rate accelerations but also lower fundamental cry frequencies in response to the surgical procedures, the latter having been hypothesized to be associated with greater vagal influences (see Lester & Zeskind, 1982). Consistent with these findings, Porter and Porges (1988) demonstrated in premature infants that individual differences in RSA tone are related to the heart rate response during lumbar punctures.
Behavioral reactivity and irritability in response to environmental stimuli assessed with the Neonatal Behavioral Assessment Scale (Brazelton, 1984) are also associated with vagal tone. In a sample of full-term healthy neonates, DiPietro, Larson, and Porges (1987) found that neonates who were breast-fed had higher amplitude RSA, were more reactive, and required more effort to test. DiPietro and Porges (1991) evaluated the relation between RSA and behavioral reactivity to gavage with a sample of preterm neonates. Gavage is a commonly used method to feed premature infants by passing food through a tube inserted into the stomach via the nasal or oral passages. Individual differences in RSA were significantly correlated with behavioral reactivity to the gavage method of feeding.
Similar relations between RSA and reactivity have been reported for older infants. Linnemeyer and Porges (1986) found that 6-month-old infants with higher amplitude RSA were more likely to look longer at novel stimuli, and only those with high-amplitude RSA exhibited significant heart rate reactivity to the visual stimuli. Richards (1985, 1987) reported convergent findings that infants with higher amplitude RSA were less distractible and had larger decelerative heart rate responses to visual stimuli. Huffman et al. (1998) observed that high-amplitude RSA at 3 months of age habituated to novel visual stimuli more rapidly than those with low-amplitude RSA; the former were more likely to suppress RSA during attention-demanding tasks and received a better attention score than the latter.
In summary, vagal tone mediates the behavioral and emotional response of the organism, and the RSA, as an index of vagal tone, provides a measure of behavioral and emotional reactivity. Neonates, infants, children, and adults with higher amplitude RSA exhibit appropriate autonomic reactivity and, in turn, appropriate behavioral, autonomic, and emotional responses to stimulation.
Expression of Emotion
Few studies have investigated individual differences in vagal tone as a mediating variable indexing individual differences in facial expressivity. There are two important reasons for posing this research question. First, both autonomic and facial responses have been theoretically associated with the expression of emotions. Second, measurement of RSA may provide an index of the neural organization necessary for facial expressions—a hypothesis suggested by the nature of the neurophysiological mechanisms that mediate facial expressions and autonomic reactions. Facial expressions and autonomic reactions associated with emotion states are controlled by brainstem structures that are in close proximity (i.e., the source nuclei of the facial nerve and the vagus). Quite often the facial nerve is included as the part of the "vagus complex." Therefore, if expressivity is assumed to be an individual difference determined by the neural tone of the facial nerve, measurement of the neural tone of the vagus might be related to the expressivity of the infant. Thus, vagal tone, monitored during a nonstressed period, might index a neural propensity to produce facial expressions.
Support for this hypothesis comes from studies that have related resting levels of heart rate variability to expressivity. Field, Woodson, Greenberg, and Cohen (1982) reported that newborn infants exhibiting greater resting heart rate variability were more expressive, and Fox and Gelles (1984) found that 3-month-old infants with higher resting heart rate variability displayed a longer duration of interest expressions. Consistent with these findings, Stifter, Fox, and Porges (1989) evaluated the relation between RSA and expressivity in 5-month-old infants and found that infants with higher amplitude RSA displayed more interest, more joy, and more look-away behaviors toward the stranger.
Self-Regulation
Self-regulation is a difficult process to operationalize. Behaviors as diverse as sustained attention, facial expressions, and latency to soothe can be interpreted as regulatory. Several studies have demonstrated the relation between measures of cardiac vagal tone and attention (for a review, see chapter 4). In general, higher vagal tone and proper suppression of vagal tone during an attention-demanding task are related to better performance. More important for the discussion of vagal tone and emotion self-regulation, indices of cardiac vagal tone (e.g., RSA) have also been shown to be related to the ability to self-soothe.
In both full-term and premature newborns, the ability to self-soothe is inversely related to vagal tone as measured by the amplitude of RSA. The higher vagal tone neonates are more irritable and exhibit greater difficulty in self-soothing. However, a subsequent increasing capacity to self-soothe is clearly seen in the high vagal tone neonates. One might speculate that the high vagal tone neonate's reactivity elicits more caregiving from the mother and that, once such an infant becomes physiologically stable, the capacity for self-soothing is consequently enhanced. Thus, the self-regulatory demands might be different for the neonate and for the older infant, and vagal tone might index this propensity to self-regulate under changing developmental demands. Support for this hypothesis comes from a study of 3-month-old infants that found significant relations between amplitude of RSA and soothability (Huffman et al., 1998); high base-level amplitude RSA was correlated with a low soothing score (i.e., little soothing was required) and a high Rothbart soothability score (i.e., distress was easily reduced).
The studies just summarized support the hypothesis that base-level vagal tone measured by the amplitude of RSA is an important determinant of self-regulatory autonomic and behavioral responses. Unfortunately, the relation is more complex, and there are infants with high vagal tone who do not suppress vagal tone under regulatory demands and who show poor emotion regulation (DeGangi, DiPietro, Greenspan, & Porges, 1991; Doussard-Roosevelt, Walker, Portales, Greenspan, & Porges, 1990). According to Greenspan (1991), infants older than 6 months of age who exhibit fussiness, irritability, poor self-calming, intolerance to change, and/or a hyperalert state of arousal are best described as being regulatory disordered.
Preliminary data suggest two important points about infants with regulatory disorders. First, these infants tend to have high vagal tone. Second, these infants tend to exhibit a deficit in the ability to suppress vagal tone during attention-demanding situations. Assessed at 9 months of age, this inability to suppress vagal tone predicts behavior problems at 3 years (see chapter 7) and at 54 months (Dale et al., in press). It appears that these "fussy babies" are hyperreactive not only to environmental stimuli but also to visceral feedback. The relation between higher vagal tone and greater reactivity is supported, but the relation between vagal tone and the ability to self-regulate, assessed via behavior and the suppression of vagal tone during tasks, is not consistent with that observed with normal infants.
CONCLUSIONS
What does vagal tone convey about an individual's ability to regulate and express emotion? To answer this question, we have proposed a model that integrates information regarding lateral brain function with the regulation of the peripheral autonomic nervous system. The model is based on the following observations.
1. The peripheral autonomic nervous system is asymmetrical.
2. The medullary regulation of the autonomic nervous system is also asymmetrical, with structures on the right side exhibiting greater control of physiological responses associated with emotion.
3. The right nucleus ambiguus is a source nucleus of the right vagus, which provides control of the larynx and SA node of the heart and controls vocal intonation and cardiac vagal tone.
4. The right central nucleus of the amygdala has direct influences on the right nucleus ambiguus to promote the laryngeal and cardiovascular responses associated with emotion (e.g., increased pitch of vocalization and increased heart rate).
5. Stimuli that are processed primarily by the right hemisphere produce greater cardiovascular responses than stimuli processed by the left hemisphere.
6. Damage to the right hemisphere blunts facial expression, vocal intonation, and autonomic reactivity.
Although each of these points has been documented, only recently has an experiment tested the proposed model linking vagal tone to right hemispheric regulation of emotion (Dufey, Hrtado, Fernandez, Manes, & Ibanez, in press). Since RSA is a sensitive measure of the input to the SA node from the right nucleus ambiguus, it provides a noninvasive measure of right hemisphere capacity to process emotion stimuli and to regulate emotional responses. To test this model adequately, it will be necessary to conduct experiments to evaluate measures of vagal tone and vagal reactivity of individuals with known right hemisphere disorders and evaluate covariations between individual differences in vagal tone and vagal reactivity and the expression and interpretation of emotions in non–brain-damaged subjects.
Providing evidence in support of this model, previous studies have addressed the relation between RSA and three dimensions related to the expression and regulation of emotion: reactivity, expressivity, and self-regulation. The literature and our ongoing research permit the following generalizations.
First, independent of developmental stage, RSA as an index of cardiac vagal tone is highly correlated with autonomic reactivity; individuals with higher amplitude RSA consistently exhibit larger and more reliable autonomic responses. Second, the relation between RSA and emotion expressivity appears to be dependent on development. A preliminary study has demonstrated that higher amplitude RSA was associated with greater facial expressivity in 5-month-old infants but failed to establish any such relation in 10-month-old infants. These data suggest that there is a developmental shift in the neurophysiological control of fa-cial expressivity: as infants become older, facial expressivity may become more dependent on higher brain control and less related to individual differences in brainstem function, manifest in the tonic outflow of cranial nerves (see chapter 8).
Third, independent of developmental stage, RSA is correlated with self-regulation. Individuals with high-amplitude RSA consistently suppress RSA or heart rate variability to enhance the intake of information from the environment. Fourth, there is a subset of individuals who have high vagal tone and who do not suppress RSA or heart rate variability during information processing. These individuals appear to have a regulatory disorder that is displayed on both behavioral and physiological levels; regulatory disordered infants are often labeled as fussy because of their continuous crying and disorganized behaviors, and they have difficulty self-soothing and maintaining a calm state (Dale et al., in press).
Finally, as the infant matures, the range of expressivity increases, the self-regulation of emotion is enhanced, and vagal tone increases; in the course of normal development, the increased myelination and regulation of autonomic function associated with enhanced vagal tone parallels the range and control of emotion states (see chapter 8). Thus, on both developmental and individual difference levels, vagal tone is clearly related to the processes of reactivity, expressivity, and self-regulation.
We introduce vagal tone as a physiological construct that is useful in explaining individual and developmental differences in the expression and regulation of emotion. As an organizing construct, vagal tone is useful in integrating central, autonomic, and psychological components of emotion. Vagal tone, as measured by RSA, may index individual differences in the homeostatic capacity of the autonomic nervous system to foster rapid expression and attenuation of sympathetic reactions. This function is dependent on neural regulation of the reciprocal relation between the antagonistic branches of the autonomic nervous system.
During emotion states, normal homeostatic function is perturbed to express emotions. Initially, sympathetic activity is expressed owing primarily to the withdrawal of the antagonistic vagal tone. Even without discrete sympathetic excitation, the vagal withdrawal will enhance the expression of sympathetic activity when the two systems have antagonistic influences on specific organs. The vagal withdrawal triggers the autonomic correlates of emotions.
If the emotion state is prolonged, the physiological state will be maintained by activation of sympathetic and endocrine systems. Excessive sympathetic activity reflects a deviation from normal homeostatic autonomic function, which then elicits vagal activity to self-regulate and return the autonomic state to homeostasis. In individuals with high vagal tone and appropriate vagal regulation capacities, the autonomic nervous system has the capacity to react (i.e., appropriate reactivity and expressivity) and to return rapidly to homeostasis (i.e., self-regulation and self-soothing).
The relation among the right hemisphere, the right vagus, and the processes involved in the expression and regulation of motion, emotion, and communication makes apparent the relevance of examining RSA in studies of emotion regulation. RSA and RSA regulation in the context of the vagal circuit of emotion regulation that we proposed may provide the physiological measures of the individual's ability to regulate motion, emotion, and communication.
SUMMARY
On the basis of current knowledge of neuroanatomy and our previous research with cardiac vagal tone, we have proposed the vagal circuit of emotion regulation. The vagal circuit of emotion regulation incorporates lateral brain function with the regulation of the peripheral autonomic nervous system in the expression of emotion. The vagus and the vagal circuit do not function independently of other neurophysiological and neuroendocrine systems. Research on brain activity (see Dawson, 1994; Fox, 1994) and research on adrenocortical activity (see Stansbury & Gunnar, 1994) demonstrate that cortisol levels are related to emotion states and to individual differences similar to those that we have investigated.
The vagal circuit emphasizes not only the vagus but also the lateralization of specific brain structures in emotion regulation. The emphasis of the vagal circuit on right-brainstem structures stimulates several testable hypotheses regarding the function of specific structures in the right brain in emotion regulation. These speculations are consistent with other reports (see Dawson, 1994; Fox, 1994) describing asymmetrical EEG activity during expressed emotions. Moreover, the vagal circuit does not exist independently of the brain structures and peptides systems regulating cortisol (see Stansbury & Gunnar, 1994). Areas in the brainstem (see chapters 12 and 19) regulating vagal activity are also sensitive to the peptides that regulate cortisol (e.g., oxytocin, vasopressin, and corticotropin-releasing hormone).
In this chapter, we have provided information regarding the relation between vagal tone and emotion regulation. A review of research indicates that baseline levels of RSA and RSA reactivity are associated with behavioral measures of reactivity, the expression of emotion, and self-regulation skills. Thus, we propose that cardiac vagal tone (i.e., RSA) can serve as an index of emotion regulation.
Historically, the vagus and other components of the parasympathetic nervous system have not been incorporated in theories of emotion. Recent developments in methodology have enabled us to define and accurately quantify cardiac vagal tone. Theories relating the parasympathetic nervous system to the expression and regulation of emotion are now being tested in several laboratories.
Co-authors for this chapter were J. A. Doussard-Roosevelt and A. K. Maiti.
CHAPTER 10
Emotion: An Evolutionary By-Product of the Neural Regulation of the Autonomic Nervous System
A new theory, the polyvagal theory of emotion, is presented which links the evolution of the autonomic nervous system to affective experience, emotional expression, vocal communication, and contingent social behavior. The polyvagal theory is derived from the well-documented phylogenetic shift in the neural regulation of the autonomic nervous system that expands the capacity of the organism to control metabolic output. The theory emphasizes the phylogenetic dependence of the structure and function of the vagus, the primary nerve of the parasympathetic nervous system. Three phylogenetic stages of neural development are described. The first stage is characterized by a primitive unmyelinated vegetative vagal system that fosters digestion and responds to novelty or threat by reducing cardiac output to protect metabolic resources. Behaviorally, this first stage is associated with immobilization behaviors. The second stage is characterized by a spinal sympathetic nervous system that can increase metabolic output and inhibit the primitive vagal system's influence on the gut to foster mobilization behaviors necessary for "fight or flight." The third stage, which is unique to mammals, is characterized by a myelinated vagal system that can rapidly regulate cardiac output to foster engagement and disengagement with the environment. The myelinated vagus originates in a brainstem area that evolved from the primitive gill arches and in mammals controls facial expression, sucking, swallowing, breathing, and vocalization. It is hypothesized that the mammalian vagal system fosters early mother–infant interactions and serves as the substrate for the development of complex social behaviors. In addition, the mammalian vagal system has an inhibitory effect on sympathetic pathways to the heart and thus promotes calm behavior and prosocial behavior.
The polyvagal theory of emotion proposes that the evolution of the autonomic nervous system provides the organizing principle to interpret the adaptive significance of affective processes. The theory proposes that the evolution of the mammalian autonomic nervous system, specifically the brainstem regulatory centers of the vagus and other related cranial nerves, provides substrates for emotional experiences and affective processes that are necessary for social behavior in mammals. In this context, the evolution of the nervous system limits or expands the ability to express emotions, which in turn may determine proximity, social contact, and the quality of communication. The polyvagal construct has been previously introduced (see chapter 2) to document the neurophysiological and neuroanatomical distinction between the two vagal branches and to propose their unique relation with behavioral strategies. This chapter elaborates on the polyvagal construct and proposes that affective strategies are derivative of the evolutionary process that produced the polyvagal regulation.
There is a consensus that affect is expressed in facial muscles and in organs regulated by the autonomic nervous system. However, with the exception of work by Cannon (1927, 1928), which focused on the sympathetic-adrenal system as the physiological substrate of emotion, the presumed neural regulation of affective state has not been investigated. Even contemporary researchers investigating affective signatures in the autonomic nervous system (Ax, 1953; Ekman, Levenson, & Friesen, 1983; Levenson, Ekman, & Friesen, 1990; Schachter, 1957) have tacitly accepted Cannon's assumption that emotions reflect responses of the sympathetic nervous system.
Unlike the architectural dictum that form (i.e., structure) follows function, the function of the nervous system is derivative of structure. The flexibility or variability of autonomic nervous system function is totally dependent on the structure. By mapping the phylogenetic development of the structures regulating autonomic function, it is possible to observe the dependence of autonomic reactivity on the evolution of the underlying structure of the nervous system. The phylogenetic approach highlights a shift in brainstem and cranial nerve morphology and function from an oxygen-sensitive system (i.e., the primitive gill arches) to a system that regulates facial muscles, cardiac output, and the vocal apparatus for affective communication.
CANNON'S BLUNDER
Cannon emphasized the idea that emotions were expressions of sympathetic-adrenal excitation. In limiting emotional experiences solely to the mobilization responses associated with sympathetic-adrenal activity, Cannon denied the importance of visceral feelings and neglected the contribution of the parasympathetic nervous system. Cannon's views were not compatible with earlier statements on the importance of visceral feedback and the parasympathetic nervous system. For example, in The Expression of Emotions in Man and Animals, Darwin (1872) acknowledged the importance of the bidirectional neural communication between the heart and the brain via the "pneumogastric" nerve. This, the Xth cranial nerve, is now called the vagus nerve and is the major component of the parasympathetic nervous system.
For Darwin, emotional state represented a covariation between facial expression and autonomic tone. However, he did not elucidate the specific neurophysiological mechanisms. Our current knowledge of the neuroanatomy, embryology, and phylogeny of the nervous system was not available to Darwin. At that time, it was not known that vagal fibers originated in several medullary nuclei, that branches of the vagus exerted control over the periphery through different feedback systems, and that the function of the branches of the vagus followed a phylogenetic principle. However, Darwin's statement is important, because it emphasizes afferent feedback from the heart to the brain, independent of the spinal cord and the sympathetic nervous system, as well as the regulatory role of the vagus in the expression of emotions.
The autonomic nervous system is related to visceral state regulation and the regulation of behaviors associated with mobilization or immobilization. For example, sympathetic excitation is clearly linked to mobilization. In vertebrates, the sympathetic nervous system is characterized by a trunk or column of ganglia paralleling the segmentation of the spinal cord. Skeletal motor pathways to the limbs are paralleled by sympathetic fibers to facilitate the metabolically demanding behaviors related to fight or flight. In fact, from Cannon's perspective and to many who followed, the sympathetic nervous system, due to its mobilizing capacity, was the component of the autonomic nervous system associated with emotion. This, however, neglected the autonomic components of affective experiences that were metabolically conservative, including processes such as signaling via facial expressions and vocalizations or specific immobilization responses.
AUTONOMIC DETERMINANTS OF EMOTION
Over the past 100 years we have learned much about the autonomic nervous system, its evolutionary origins, and how it relates to emotion. Initially, we can distinguish among three components of the autonomic nervous system (visceral afferents, sympathetic nervous system, and parasympathetic nervous system) and speculate how each might be related to affective experiences. First, the visceral afferents may be assumed to play a major role in determining "feelings." These mechanisms, which provide us with knowledge of hunger, also may convey a sense of nausea during emotional distress. We frequently hear subjective reports of individuals feeling "sick to their stomach" during periods of severe emotional strain associated with profound negative experiences. Similarly, negative states have been associated with reports of breathlessness or feelings that the heart has stopped. Second, the sympathetic nervous system and adrenal activity are associated with mobilization. Activation of the sympathetic nervous system is usually linked to increased skeletal movement of the major limbs. Thus, consistent with Cannon, the sympathetic nervous system provides the metabolic resources required for fight-or-flight behaviors. The sympathetic nervous system enhances mobilization by increasing cardiac output and decreasing the metabolic demands of the digestive tract by actively inhibiting gastric motility. Third, as proposed by Darwin and Bernard, the parasympathetic nervous system and specifically the vagus are related to emotional state. Few researchers have investigated the link between parasympathetic activity and affective state. However, over the last decade my laboratory has focused on this issue. We documented that cardiac vagal tone, a component of parasympathetic control, is related to affect and affect regulation (see chapter 9; Porges, 1991; Porges & Doussard-Roosevelt, 1997). We presented theoretical models explaining the importance of vagal regulation in the development of appropriate social behavior (see chapter 7). The parasympathetic nervous system is generally associated with fostering growth and restoration (Porges, 1992, 1995). Moreover, knowledge of the polyvagal system allows an appreciation of the importance of the brainstem origin of the specific vagal fibers in the determination of affective and behavioral response strategies (see chapter 7; Porges, Doussard-Roosevelt, Portales, & Suess, 1994).
Researchers and clinicians have had difficulties in the organization or categorization of intensive affective states that appear to have totally different etiologies or behavioral expressions. For example, intense feelings of terror might result in total immobilization or freezing. By contrast, intense feelings of anger or anxiety might be associated with massive mobilization activity. This problem exists, in part, because of a bias toward explanations of affective states defined in terms of either overt behavior such as facial expression (i.e., following Darwin) or sympathetic activity (i.e., following Cannon). The emphasis on sympathetic activity is based on three historical factors. First, theories regarding emotions have minimized or totally neglected the parasympathetic nervous system. Second, Cannon's focus on the sympathetic efferents and mobilization responses associated with fight or flight as the sole domain of autonomic reactivity during emotional states has not been challenged. Third, the database of autonomic correlates of affect, collected to identify autonomic "signatures" of specific affective states, is dominated by measures assumed to be related to sympathetic function (Ax, 1953; Ekman et al., 1983; Levenson et al., 1990; Schachter, 1957).
EVOLUTION OF THE AUTONOMIC NERVOUS SYSTEM: EMERGENT STRUCTURES FOR THE EXPRESSION OF EMOTIONS IN MAN AND ANIMALS
Although there is an acceptance that the autonomic nervous system and the face play a role in emotional expression, there is great uncertainty regarding the autonomic "signature" of specific or discrete emotions. Most researchers evaluating autonomic responses during affective experiences assumed, as Cannon did, that the sympathetic nervous system was the determinant of emotion or at least the primary physiological covariate of emotion. This, of course, neglects the potential role of the parasympathetic nervous system and its neurophysiological affinity to facial structures, including facial muscles, eye movements, pupil dilation, salivation, swallowing, vocalizing, hearing, and breathing. By investigating the evolution of the autonomic nervous system, we may gain insight into the interface between autonomic function and facial expression. In the following sections, the phylogenetic development of the autonomic nervous system will be used as an organizing principle to categorize affective experiences.
The polyvagal theory of emotion is derived from investigations of the evolution of the autonomic nervous system. The theory includes several rules and assumptions.
1. Emotion depends on the communication between the autonomic nervous system and the brain; visceral afferents convey information on physiological state to the brain and are critical to the sensory or psychological experience of emotion, and cranial nerves and the sympathetic nervous system are outputs from the brain that provide somatomotor and visceromotor control of the expression of emotion.
2. Evolution has modified the structures of the autonomic nervous system.
3. Emotional experience and expression are functional derivatives of structural changes in the autonomic nervous system due to evolutionary processes.
4. The mammalian autonomic nervous system retains vestiges of phylogenetically older autonomic nervous systems.
5. The phylogenetic "level" of the autonomic nervous system determines affective states and the range of social behavior.
6. In mammals, the autonomic nervous system response strategy to challenge follows a phylogenetic hierarchy, starting with the newest structures and, when all else fails, reverting to the most primitive structural system.
This chapter focuses on the phylogenetic shift in the neural regulation of the vertebrate heart. The heart has been selected because, in response to environmental challenge, cardiac output must be regulated to mobilize for fight-or-flight behaviors or to immobilize for death-feigning or hiding behaviors. To regulate cardiac output, several efferent structures have evolved. These structures represent two opposing systems: (1) a sympathetic-catecholamine system including chromaffin tissue and spinal sympathetics; and (2) a vagal system (a component of the parasympathetic nervous system) with branches originating in medullary source nuclei (i.e., dorsal motor nucleus of the vagus and nucleus ambiguus). In addition, vertebrates have chromaffin tissue containing high concentrations of catecholamines. Chromaffin tissue is defined as having morphological and histochemical properties similar to those of the adrenal medulla. Classes of vertebrates that do not have an adrenal medulla have relatively more chromaffin tissue, which regulates circulating catecholamines.
Table 10.1 lists the regulatory structures that influence the heart in vertebrates (Morris & Nilsson, 1994; Santer, 1994; Taylor, 1992). Two phylogenetic principles can be extracted from Table 10.1. First, there is a phylogenetic pattern in the regulation of the heart from endocrine communication, to unmyelinated nerves, and finally to myelinated nerves. Second, there is a development of opposing neural mechanisms of excitation and inhibition to provide rapid regulation of graded metabolic output.
TABLE 10.1. Method of Cardiac Control as a Function of Vertebrate Phylogeny
In the most primitive fish, the cyclostomes, the neural control of the heart is very primitive. Some cyclostomes such as the myxinoids (hagfish) use circulating catecholamines from chromaffin tissue to provide the sole excitatory influences on the heart. Other cyclostomes such as the lampetroids (lampreys) have a cardiac vagus. However, in contrast to all other vertebrates that have a cardioinhibitory vagus that acts via postganglionic muscarinic cholinoceptors, the cyclostome vagal innervation is excitatory and acts via nicotinic cholinoceptors. One striking feature of the cyclostome heart is the location of chromaffin tissue within the heart that stores large quantities of epinephrine and norepinephrine. As in other vertebrates, the circulating catecholamines produced by the chromaffin tissue stimulate beta-adrenergic receptors in the heart. Thus, the cyclostomes appear to have only excitatory mechanisms to regulate the heart.
The elasmobranchs (cartilaginous fish) are the first vertebrates to have a cardioinhibitory vagus. The vagus in these fish is inhibitory and the cholinoceptors on the heart are muscarinic as they are in other vertebrates. The cardioinhibitory vagus is functional in the elasmobranchs as a response to hypoxia. In conditions of hypoxia, metabolic output is adjusted by reducing heart rate. This modification of neural regulation may provide a mechanism to enable the elasmobranchs to increase their territorial range, by providing a neural mechanism that adjusts metabolic output to deal with changes in water temperature and oxygen availability. However, unlike more evolutionarily advanced fish or tetrapods, elasmobranchs do not have direct sympathetic input to the heart. Instead, cardiac acceleration and increases in contractility are mediated via beta-adrenergic receptors stimulated by circulating catecholamines released from chromaffin tissue. Thus, because activation of metabolic output is driven by circulating catecholamines and not by direct neural innervation, once the excitatory system is triggered, the ability to self-soothe or calm is limited.
In vertebrates with sympathetic and vagal neural innervation, vagal influences to the sinoatrial node inhibit or dampen the sympathetic influence and promote rapid decreases in metabolic output (Vanhoutte & Levy, 1979) that enable almost instantaneous shifts in behavioral state. As a whole, the teleosts may be considered phylogenetically the first class of vertebrates with both sympathetic and parasympathetic neural control of the heart, with innervation similar to that found in tetrapods. This enables rapid transitory changes in metabolic output, permitting changes from mobilization to immobilization. These are observed as "darting" and "freezing" behaviors. Amphibians, similar to the teleosts, have dual innervation of the heart via systems with direct neural components from the spinal cord via the sympathetic chain, producing increases in heart rate and contractility, and direct neural pathways from the brainstem via the vagus, producing cardioinhibitory actions.
True adrenal glands, in which a distinct medulla is formed of chromaffin tissue, are only present in birds, reptiles, and mammals (Santer, 1994). Neural regulation by the spinal sympathetics of the adrenal medulla provides a neural mechanism for rapid and controlled release of epinephrine and norepinephrine to stimulate cardiovascular function. In teleosts, chromaffin tissue is primarily related to parts of the cardiovascular system, but chromaffin tissue is also associated with the kidney. However, in amphibians, chromaffin tissue is primarily associated with the kidney, and substantial aggregations of chromaffin cells are located along the sympathetic chain ganglia. Thus, we can observe a phylogenetic shift in the location of chromaffin tissue and the concurrent evolution of a distinct adrenal medulla near the kidney.
In mammals, the morphology of the vagus changes (see chapter 2). Unlike that of all other vertebrates with cardioinhibitory vagi, the mammalian vagus contains two branches. One branch originates in the dorsal motor nucleus of the vagus and provides primary neural regulation of subdiaphragmatic organs, such as the digestive tract. However, at the level of the heart, the dorsal motor nucleus of the vagus does not play a major role in normal dynamic regulation of cardiac output. Rather, during embryological development in mammals, cells from the dorsal motor nucleus of the vagus migrate ventrally and laterally to the nucleus ambiguus (Schwaber, 1986) where they form the cell bodies for visceromotor myelinated axons that provide potent inhibition of the sinoatrial node, the pacemaker for the heart.
By transitory down-regulation of the cardioinhibitory vagal tone to the heart (i.e., removal of the vagal brake), mammals are capable of rapid increases in cardiac output without activating the sympathetic-adrenal system. By engaging this system rather than the sympathetic-adrenal system, mammals have the opportunity to rapidly increase metabolic output for immediate mobilization. Under prolonged challenge, the sympathetic system also may be activated. However, by rapidly reengaging the vagal system, mammals can inhibit sympathetic input on the heart (Vanhoutte & Levy, 1979) and rapidly decrease metabolic output to self-soothe and calm.
PHYLOGENETIC DEVELOPMENT OF THE AUTONOMIC NERVOUS SYSTEM: AN ORGANIZING PRINCIPLE FOR HUMAN EMOTION
Inspection of Table 10.1, which summarizes the primary regulatory structures of the heart in vertebrates, provides a basis for speculation on the behavioral repertoire of various classes of vertebrates. These speculations support the premise that the phylogenetic development of the autonomic nervous system provides an organizing principle for affective experiences and determines the limits on social behavior and, therefore, the possibility of affiliation. Phylogenetic development generally results in increased neural control of the heart via mechanisms that can rapidly increase or decrease metabolic output. This phylogenetic course results in greater central nervous system regulation of behavior, especially behaviors to engage and disengage with environmental challenges.
To further focus on the impact of phylogenetic development of the neural regulation of the autonomic nervous system, we can observe five phyloge-netically dependent response systems: (1) a chemical excitatory system via the catecholamine-rich chromaffin tissue to increase cardiac output and to support mobilization; (2) an inhibitory vagal system via the dorsal motor nucleus of the vagus to reduce cardiac output when metabolic resources are scarce and to support immobilization in response to danger; (3) a spinal sympathetic nervous system to provide neural excitation to promote rapid mobilization for behaviors associated with fight or flight; (4) a neurally regulated adrenal medulla system to provide more direct control over the release of circulating catecholamines to support mobilization for the prolonged metabolic requirements of fight-or-flight behaviors; and (5) the specialization of the mammalian vagal system into a "tonic" inhibitory system that allows graded withdrawal of the vagal brake, which can promote transitory mobilization and the expression of sympathetic tone without requiring sympathetic or adrenal activation. With this new vagal system, transitory incursions into the environment can be initiated without the severe biological price of either metabolic shutdown, via primitive vagal inhibition, or metabolic excitation, via sympathetic-adrenal activation.
The five phylogenetically dependent response systems are associated with three neuroanatomical constructs related to affective experience and expression: (1) dorsal vagal complex (DVC), (2) sympathetic nervous system (SNS), and (3) ventral vagal complex (VVC). Each of these three neural constructs is linked to a specific emotion subsystem observable in humans. Each emotion subsystem is manifested via differentiated motor output from the central nervous system to perform specific adaptive functions: to immobilize and to conserve metabolic resources, to mobilize in order to obtain metabolic resources, or to signal with minimal energy expense. The constituent responses associated with each subsystem are listed in Table 10.2.
TABLE 10.2. Physiological Functions Associated With Each Subsystem of the Autonomic Nervous System
THE DORSAL VAGAL COMPLEX: A VESTIGIAL IMMOBILIZATION SYSTEM
The DVC is primarily associated with digestive, taste, and hypoxic responses in mammals. It includes the nucleus tractus solitarius (NTS) and the interneuronal communication between the NTS and the dorsal motor nucleus of the vagus (DMX). The efferents for the DVC originate in the DMX, and primary vagal afferents terminate in the NTS. The DVC provides primary neural control of subdiaphragmatic visceral organs. It provides low tonic influences on the heart and bronchi. This low tonic influence is the vestige from the reptilian vagal control of the heart and lung. In contrast to reptiles, mammals have a great demand for oxygen and are vulnerable to any depletion in oxygen resources. The metabolic demand for mammals is approximately five times greater than that for reptiles of equivalent body weight (Else & Hulbert, 1981). Thus, reptilian dependence on this system provides a shutdown of metabolic activity to conserve resources during diving or death feigning. The DVC provides inhibitory input to the sinoatrial node of the heart via unmyelinated fibers and thus is less tightly controlled than the myelinated fibers from the VVC. Hypoxia or perceived loss of oxygen resources appears to be the main stimulus that triggers the DVC. Once triggered, severe bradycardia and apnea are observed, often in the presence of defecation. This response strategy is observed in the hypoxic human fetus. Although adaptive for the reptile, hypoxic triggering of this system may be lethal for mammals. In addition, it is important to note that the DVC has beneficial functions in humans. Under most normal conditions, the DVC maintains tone to the gut and promotes digestive processes. However, if up-regulated, the DVC contributes to pathophysiological conditions, including the formation of ulcers via excess gastric secretion and colitis. Research supports the importance of the unmyelinated vagal fibers in bradycardia (Daly, 1991) and suggests the possibility that massive bradycardia may be determined by the unmyelinated vagal fibers associated with the DVC recruiting myelinated vagal fibers to maximize the final vagal surge on the heart (Jones, Wang, & Jordan, 1995).
THE SYMPATHETIC NERVOUS SYSTEM: ADAPTIVE MOBILIZATION SYSTEM FOR FIGHT-OR-FLIGHT BEHAVIORS
The SNS is primarily a system of mobilization. It prepares the body for emergency by increasing cardiac output, stimulating sweat glands to protect and lubricate the skin, and inhibiting the metabolically costly gastrointestinal tract. The evolution of the sympathetic nervous system follows the segmentation of the spinal cord, with cell bodies of the preganglionic sympathetic motor neurons located in the lateral horn of the spinal cord. The sympathetic nervous system has long been associated with emotion. The label "sympathetic" reflects the historical identity of this system as a nervous system "with feelings" and contrasts it with the parasympathetic nervous system, a label that reflects a nervous system that "guards against feelings."
THE VENTRAL VAGAL COMPLEX: THE MAMMALIAN SIGNALING SYSTEM FOR MOTION, EMOTION, AND COMMUNICATION
The primary efferent fibers of the VVC originate in the nucleus ambiguus. The primary afferent fibers of the VVC terminate in the source nuclei of the facial and trigeminal nerves. The VVC has primary control of supradiaphragmatic visceral organs including the larynx, pharynx, bronchi, esophagus, and heart. Motor pathways from the VVC to visceromotor organs (e.g., heart and bronchi) and somatomotor structures (e.g., larynx, pharynx, and esophagus) are myelinated to provide tight control and speed in responding. In mammals, visceromotor fibers to the heart express high levels of tonic control and are capable of rapid shifts in cardioinhibitory tone to provide dynamic changes in metabolic output to match environmental challenges. This rapid regulation characterizes the qualities of the mammalian vagal brake that enable rapid engagement and disengagement in the environment without mobilizing the SNS.
A major characteristic of the VVC is that the neural fibers regulating somatomotor structures are derived from the branchial or primitive gill arches that evolved to form cranial nerves V, VII, IX, X, and XI. Somatomotor fibers originating in these cranial nerves control the branchiomeric muscles including facial muscles, muscles of mastication, neck muscles, larynx, pharynx, esophagus, and middle ear muscles. Visceromotor efferent fibers control salivary and lacrimal glands as well as the heart and bronchi. The primary afferents to the VVC come from facial and oral afferents traveling through the facial and trigeminal nerves and the visceral afferents, terminating in the NTS. The VVC is involved in the control and coordination of sucking, swallowing, and vocalizing with breathing.
EVOLUTION AND DISSOLUTION: HIERARCHICAL RESPONSE STRATEGY
The evolution of the autonomic nervous system provides substrates for the emergence of three emotion systems. This phylogenetic adjustment of the autonomic nervous system represents an exaptation (see Crews, 1997) of structures to express emotions that initially evolved in primitive vertebrates to extract oxygen from water, to oxygenate and transport blood, and to adjust metabolic output to match resources. The polyvagal theory of emotion is based on a phylogenetic model. The polyvagal theory of emotion proposes a hierarchical response strategy to challenge, with the most recent modifications employed first and the most primitive last. This phylogenetic strategy can be observed in our day-to-day interactions. Our social behavior follows a strategy that focuses initially on communication via facial expressions and vocalizations. The strategy has low metabolic demand and, if appropriately interpreted, results in contingent social interactions via verbal–facial mechanisms. Often, hand gestures and head movements contribute to increase the mammalian repertoire of communication-related behavior. An important characteristic of these prosocial behaviors is their low metabolic demand and the rapid contingent "switching" of transitory engagement to transitory disengagement strategies (i.e., speaking then switching to listening).
This phylogenetically based hierarchical response strategy is consistent with the concept of dissolution proposed by Jackson (1958) to explain diseases of the nervous system. Jackson proposed that "the higher nervous arrangements inhibit (or control) the lower, and thus, when the higher are suddenly rendered functionless, the lower rise in activity." This is observed in the polyvagal theory of emotion, not in terms of disease but in terms of response strategies to differential challenges to survival. The VVC with its mechanisms of "signaling" and "communication" provides the initial response to the environment. The VVC inhibits, at the level of the heart, the strong mobilization responses of the SNS. Withdrawal of VVC, consistent with Jackson's model, results in a "disinhibition" of the sympathetic control of the heart. Similarly, withdrawal of sympathetic tone results in a "disinhibition" of the DVC control of the gastrointestinal tract and a vulnerability of the bronchi and heart. There are several clinical consequences to unopposed DVC control, including defecation, due to relaxation of the sphincter muscles and increased motility of the digestive tract; apnea, due to constriction of the bronchi; and bradycardia, due to stimulation of the sinoatrial node. Thus, when all else fails, the nervous system elects a metabolically conservative course that is adaptive for primitive vertebrates but lethal to mammals. Consistent with the Jacksonian principle of dissolution, specific psychopathologies defined by affective dysfunction may be associated with autonomic correlates consistent with the three phylogenetic levels of autonomic regulation. The three levels do not function in an all-or-none fashion; rather, they exhibit gradations of control determined by both visceral feedback and higher brain structures.
UNVEILING DARWIN
Contemporary research and theory on emotion owes much to Darwin and his volume, The Expression of Emotions in Man and Animals (1872). Through careful and astute observations of facial expressions, Darwin insightfully interpreted emotional expressions within an evolutionary model of adaptation and natural selection. However, Darwin's knowledge of neurophysiology and neuroanatomy was limited. In contrast to Darwin's creative insights into the adaptive function of facial expression, his understanding of underlying physiological mechanisms and the linkage between facial muscles and emotion was synthetic and derivative. He repeatedly referenced the 1844 edition of Anatomy and Philosophy of Expressions, written by Sir Charles Bell, for physiological explanations of facial expression. As further support for the importance of facial muscles in emotional expressions, Darwin incorporated the work of Duchenne in his text. Duchenne conducted experiments by electrically stimulating the face of humans. Electrical stimulation of selected facial muscles provided expressions that were readily perceived as different emotional states.
In contrast to the polyvagal theory of emotion, which uses evolution of the autonomic nervous system as the primary organizing principle for the expression and experience of affect, Darwin's writings did not emphasize the importance of the nervous system as a structure involved in the evolution of emotion. Rather, he focused on affect as a functional system that responded to the determinants of evolution to produce the facial and vocal expressions of human emotion. Darwin neglected the importance of treating the nervous system as a structure that is vulnerable to the pressures of evolution. A choice between investigating affect as a functional behavioral system or investigating the structural determinants of affect (i.e., nervous system) was clearly made by researchers who followed Darwin. This research tradition followed the observational approach of organizing facial expression into affective categories. Although the physiological correlates of affect and facial expression were investigated (Ekman et al., 1983; Levenson et al., 1990; Stifter, Fox, & Porges, 1989). these investigations were made on a psychophysiological or correlative level and did not emphasize specific neural regulatory processes.
Consistent with the observational approach, Tomkins (1962, 1963) developed a theory of affect that emphasized the importance of the face not only as a structure of communication but also as a structure of self-feedback. Following Tomkins (1962, 1963), Ekman (1978) and Izard (1979) developed detailed coding systems for facial affect and have used these methods to study individual differences, developmental shifts, and the cross-cultural consistency of human facial expression,
Several contemporary theories of emotion have focused on facial expressions in a manner similar to that initially presented by Darwin. Rather than incorporating knowledge of neural regulation of the face or the evolution of neural regulation of autonomic function, researchers and theorists have attempted to organize information in terms of the functional significance of sequences or patterns of facial expressions. This difficult task, modeled on Darwin, often becomes bogged down in semantics, philosophical inconsistencies, and circularity. Darwin in his descriptions of emotions speculated and provided hypothetical examples of natural selection contributing to the uniqueness of species-specific affective response patterns. However, the terms selected to characterize specific emotions often vary from culture to culture. Tomkins and later Ekman and Izard promoted the description of affective experiences in terms of the specific facial muscles or groups of muscles involved in the facial expression. However, they then used subjective reports to label these facial expressions.
We may "unveil" Darwin by investigating the neural regulation that underlies facial expression. Facial expressions are controlled by cranial nerves. Motor pathways from the trigeminal nerve (V) control the muscles of mastication with branches to the temporalis, masseter, medial, and lateral pterygoid muscles. Motor pathways from the facial nerve (VII) control the muscles of facial expression, including zygomaticus, frontalis, orbicularis oculi, elevators, orbicularis oris, depressors, and platysma. The nucleus ambiguus serves as the source of cell bodies for motor pathways traveling through several cranial nerves, including the glossopharyngeal (IX), vagus (X), and accessory nerves (XI). Pathways from the glossopharyngeal nerve regulate pharyngeal muscles. Pathways from the vagus regulate the muscles of the pharynx and larynx, and pathways of the accessory nerve control the neck muscles, allowing rotation and tilting of the head. These cranial nerves are derivative from primitive gill arches (Gibbins, 1994; Langley, 1921) and may be collectively described as the ventral vagal complex. Thus, the evolutionary origins (i.e., primitive gill arches) of the somatomotor pathways traveling through these cranial nerves provide us with an organizing principle to understand affective expressions. In addition to the described neural regulation of somatomotor structures, these branchiomeric (i.e., derived from the primitive arches) cranial nerves also regulate the visceromotor processes associated with salivation, tearing, breathing, and heart rate.
Other cranial nerves contribute to the expression of emotions. The hypoglossal nerve (XII) innervates the muscles of the tongue. The trochlear (IV), abducens (VI), and oculomotor (III) nerves innervate muscles to provide movements of the eyes and eyelids. Thus, the facial expressions observed by Darwin, detailed by Tomkins, and coded by Ekman and Izard are a direct reflection of the regulation of the face by the cranial nerves.
VOODOO OR VAGUS DEATH? THE TEST OF THE POLYVAGAL THEORY
The polyvagal theory of emotion provides a theoretical framework to interpret the phenomenon of voodoo or fright death described by Cannon (1957) and Richter (1957). Cannon believed that extreme emotional stress, regardless of the specific behavioral manifestation, could be explained in terms of degree of sympathetic-adrenal excitation. In 1942, Cannon described a phenomenon known as voodoo death. Voodoo death was assumed to be directly attributable to emotional stress. Being wed to a sympathicoadrenal model of emotional experience (as just described), Cannon assumed that voodoo death would be the consequence of the state of shock produced by the continuous outpouring of epinephrine via excitation of the sympathetic nervous system. According to the Cannon model, the victim would be expected to breathe very rapidly and have a rapid pulse. The heart would beat fast and gradually lead to a state of constant contraction and, ultimately, to death in systole. Because his speculations were not empirically based, he offered the following challenge to test his model of voodoo death: "If in the future, however, any observer has opportunity to see an instance of 'voodoo death,' it is to be hoped that he will conduct the simpler tests before the victim's last gasp."
Richter responded to Cannon's challenge with an animal model. Rats were prestressed and placed in a closed turbulent water tank, and the latency to drowning was recorded. Most domestic laboratory rats lasted for several hours, whereas unexpectedly all of the wild rats died within 15 minutes. In fact, several wild rats dove to the bottom and, without coming to the surface, died. To test Cannon's hypothesis that stress-induced sudden death was sympathetic, Richter monitored heart rate and determined whether the heart was in systole or diastole after death. He assumed, on the basis of Cannon's speculations, that tachycardia would precede death and that at death the heart would be in a state of systole, reflecting the potent effects of sympathetic excitation on the pacemaker and the myocardium. However, Richter's data contradicted the Cannon model. Heart rate slowed prior to death, and at death the heart was engorged with blood, reflecting a state of diastole. Richter interpreted the data as demonstrating that the rats died a "vagus" death, the result of overstimulation of the parasympathetic system rather than the sympathicoadrenal system. However, Richter provided no physiological explanation except the speculation that the lethal vagal effect was related to a psychological state of "hopelessness."
The immediate and reliable death of the wild rats in Richter's experiment may represent a more global immobilization strategy. Sudden prolonged immobility or feigned death is an adaptive response exhibited by many mammalian species. Hofer (1970) demonstrated that several rodent species, when threatened, exhibited prolonged immobility accompanied by very slow heart rate. For some of the rodents, heart rate during immobility was less than 50% of the basal rate. During prolonged immobility, respiration became so shallow that it was difficult to observe, although the rate greatly accelerated. Although physiologically similar, Hofer distinguished between prolonged immobility and feigned death. The onset of feigned death was sudden with an apparent motor collapse during active struggling. Similar to Richter, Hofer interpreted this fear-induced slowing of heart rate as a vagal phenomenon. In support of this interpretation, he noted that of the four species that exhibited prolonged immobility, 71% of the subjects had cardiac arrhythmias of vagal origin; in contrast, in the two species that did not exhibit immobility behaviors, only 17% exhibited cardiac arrhythmias of vagal origin.
The polyvagal theory of emotion places Richter's and Hofer's observations in perspective. Following the Jacksonian principle of dissolution, the rodents would exhibit the following sequence of response strategies: (1) removal of VVC tone, (2) increase in sympathetic tone, and (3) a surge in DVC tone. The more docile domestic rats in Richter's experiment apparently progressed from removal of VVC tone, to increased sympathetic tone, and then death from exhaustion. However, the profile of the wild rats was different. Being totally unaccustomed to enclosure, handling, and also having their vibrissae cut, a mobilization strategy driven by increased sympathetic tone was not functional. Instead, these rats reverted to their most primitive system to conserve metabolic resources via DVC. This strategy promoted an immobilization response characterized by reduced motor activity, apnea, and bradycardia. Unfortunately, this mode of responding, although adaptive for reptiles, is lethal for mammals. Similarly, the onset of feigned death, as described by Hofer, illustrates the sudden and rapid transition from an unsuccessful strategy of struggling requiring massive sympathetic activation to the metabolically conservative immobilized state mimicking death associated with the DVC.
These data suggest that the vagus contributes to severe emotional states and may be related to emotional states of "immobilization" such as extreme terror. Application of the polyvagal approach enables the dissection of vagal processes into three strategic programs: (1) when tone of the VVC is high, the ability to communicate via facial expressions, vocalizations, and gestures exists; (2) when tone of the VVC is low, the sympathetic nervous system is unopposed and easily expressed to support mobilization such as fight-or-flight behaviors; and (3) when tone from DVC is high, immobilization and potentially life-threatening bradycardia, apnea, and cardiac arrhythmias occur.
CONCLUSION
Three important scientific propositions provide the basis for this theory. First, Darwin provided the concept of evolution and the processes that contribute to phylogenetic variation. Second, Jackson provided the concept of dissolution as a viable explanation for diseases of brain function. And, third, MacLean (1990) provided the concept that the human brain retains structures associated with phylogenetically more primitive organisms.
The polyvagal theory of emotion focuses on the evolution of the neural and neurochemical regulation of structures involved in the expression and experience of emotion as a theme to organize emotional experience and to understand the role of emotion in social behavior. Over 100 years ago Jackson, intrigued with Darwin's model of evolution, elaborated on how evolution in reverse, termed "dissolution," might be related to disease. According to Jackson, higher nervous system structures inhibit or control lower structures or systems and "thus, when the higher are suddenly rendered functionless, the lower rise in activity." The polyvagal theory of emotion follows this Jacksonian principle.
CHAPTER 11
Love: An Emergent Property of the Mammalian Autonomic Nervous System
There is no fear in love; but perfect love casteth out fear. —1 John 4:18
Love has had a variety of expressions. Foremost in our culture is the love between individuals of different genders. The products of this love are observed in terms of children, of cooperative and shared responsibilities to survive, of the transmitting of culture, and of pleasure and ecstasy. Although we assume that love is a unique human emotion, several neurobiological processes involved in the experience and expression of love are shared with other mammals. The phylogenetic origins of these processes reflect their antecedent adaptive function. In mammals, these processes have evolved into an integrated neurobehavioral system, which promotes proximity, reproduction, and physical safety. Central to the neural mediation of these processes is the autonomic nervous system. The focus of this chapter is to describe how the autonomic nervous system is involved in the processes associated with feelings of love and behaviors linked to reproduction. The chapter proposes a hypothetical model, which speculates that the phylogenetic changes in the autonomic nervous system are related to the emergence of two components of love: an appetitive phase associated with courting and seductive behaviors and a consummatory phase associate with passionate sexual behaviors and the establishment of enduring pair-bonds. According to this model, courting and seduction are dependent on phylogenetically newer structures. For example, the cortex, via corticobulbar pathways, regulates facial expressions and vocalizations to express availability to a prospective mate. In contrast, passionate visceral feelings are dependent on phylogenetically older structures, such as the hypothalamus and medulla, which involve phylogenetically more recent neuropeptides (oxytocin and vasopressin).
EVOLUTION AND DISSOLUTION: A HIERARCHICAL RESPONSE STRATEGY DURING MATE SELECTION
The evolution of the autonomic nervous system provides substrates for the emergence of the three emotion subsystems described above. Although reminiscent of the triune brain proposed by MacLean (1990), the polyvagal theory emphasizes that even the phylogenetically more primitive structures have changed in structure and function. This phylogenetic adjustment of the autonomic nervous system represents an exaptation (i.e. a shift in the function) of structures to express emotions. The ancient gill arches that characterize primitive vertebrates evolved into structures that convey emotional state via facial expressions, gestures and vocal communication.
The polyvagal theory emphasizes the phylogenetic changes in the autonomic nervous system. Specifically, as mammals evolved, the vagal regulation of the viscera was maintained by two vagal pathways. The second vagal circuit has myelinated efferent pathways and originates in the nucleus ambiguus, a nucleus ventral to the dorsal motor nucleus of the vagus. The ventral vagal circuit is described as the ventral vagal complex (VVC). The dorsal vagal circuit is shared with most other vertebrates and is described as the dorsal vagal complex (DVC). These two vagal circuits in concert with the sympathetic nervous system (SNS) form a hierarchical system regulating visceral organs. The polyvagal theory proposes a hierarchical response strategy to environmental challenges, with the most recent modifications employed first (i.e., VVC) and the most primitive (i.e., DVC) last. However, the response strategy is not all-or-none and may include transitional blends between the boundaries of the three emotion subsystems. These transitional blends may be determined by both visceral feedback and higher brain structures (including vasopressinergic and oxytocinergic pathways that communicate between the hypothalamus and the medullary source nuclei of the vagus). Thus, the neurophysiological substrate of specific states and behaviors may incorporate activation of more than one emotion subsystem. For example, sexual arousal, with features of facial and vocal expressiveness in concert with facial flush, sweating and tachycardia, may reflect a blend defined by the activation of the VVC and the SNS.
The proposed hierarchical response strategy provides a model for the exploration of human social behavior. This phylogenetic strategy can be observed in human mating strategies. Our mating behavior is usually initiated by communication via facial expressions and vocalizations, a strategy with low metabolic cost. If appropriately used, communication will determine availability, induce proximity and promote reproductive behavior. Or, the strategy will determine unavailability, induce social distance, and promote a search for another prospective mate. An important feature of this mate selection strategy is that it limits vulnerability and risk by allowing participants to rapidly switch between engagement and disengagement behaviors (i.e., speaking then switching to listening; moving toward then rapidly retreating).
This phylogenetically based hierarchical response strategy is consistent with the concept of dissolution proposed by John Hughlings Jackson (1958) to explain diseases of the nervous system. Jackson proposed that "the higher nervous arrangements inhibit (or control) the lower, and thus, when the higher are suddenly rendered functionless, the lower rise in activity." The polyvagal theory (see chapter 10) proposed dissolution, not in response to disease or brain trauma, but as a response strategy to differential challenges. The VVC with its mechanisms for signaling and communication provides the initial response to the environment. The VVC inhibits, at the level of the heart, the strong mobilization responses of the SNS. Withdrawal of the VVC, consistent with Jacksonian principles, results in a disinhibition of the sympathetic control of the heart. Similarly, withdrawal of sympathetic tone results in a disinhibition of the DVC control of the gastrointestinal tract and a vulnerability of the bronchi and heart. There are several clinical consequences to unopposed DVC control including defecation, due to a relaxation of the sphincter muscles and increased motility of the digestive tract; apnea, due to constriction of the bronchi; and bradycardia, due to stimulation of the sinoatrial node. Thus, when all else fails, the nervous system elects a metabolically conservative course that is adaptive for primitive vertebrates, but may be lethal for mammals. Consistent with the Jacksonian principle of dissolution, specific psychopathologies defined by affective dysfunction may be associated with autonomic correlates consistent with the three phylogenetic levels of autonomic regulation.
THE SOCIAL ENGAGEMENT SYSTEM: AN EMERGENT PROPERTY OF THE VENTRAL VAGAL COMPLEX
Phylogenetically, the VVC is the most recent neurophysiological affect system. The VVC is composed of a somatomotor component consisting of the special visceral efferents and a visceromotor component consisting of the myelinated vagal pathways from the nucleus ambiguus to the sinoatrial node of the heart and the bronchi. As illustrated in Figure 11.1, the special visceral efferents and the vagal brake collectively constitute an emergent social engagement system. The somatomotor components of the VVC contribute to the regulation of behaviors involved in exploration of the social environment (e.g., looking, listening, ingesting) and behaviors involved in acknowledging social contact (e.g., facial expressions, head gestures, and vocalizations). More specifically, the somatomotor components of the VVC are involved in head turning (via cranial nerve XI), vocalizations (IX, X), facial expression (VII, V), the filtering of low frequency sounds via the middle ear muscles to extract human voice from backgrounds sounds (VII) and mastication (V). The visceromotor components of the VVC contribute to the rapid modulation of vagal (X) control of the heart and the bronchi (X), which provides metabolic resources to engage and disengage in a social setting.
Three important features define the social engagement system. First, the efferent pathways that regulate the social engagement system originate in medullary structures (i.e., nucleus of cranial nerve V, nucleus of cranial nerve VII, nucleus ambiguus). Second, corticobulbar pathways, which originate in frontal cortex (i.e., upper motor neurons), enable the possibility of efficient cortical regulation of these medullary source nuclei (i.e., lower motor neurons). Third, on the medullary level, the structures that regulate the efferent regulation of social communication behaviors neuroanatomically communicate with structures that regulate ingestion (e.g., sucking, swallowing, salivation) and cardiac output. Thus, modulation of the vagal brake may either promote calming and self-soothing states (i.e., attenuate the influence of the sympathetic influence on the heart) or support mobilization (i.e., potentiate the sympathetic influence on the heart).
FIGURE 11.1. The social engagement system: social communication is determined by the cortical regulation of medullary nuclei via corticobulbar pathways. The social engagement system consists of a somatomotor component (special visceral efferent pathways that regulate the muscles of head) and a visceromotor component (the vagal brake that regulates the heart and bronchi).
Mammals have a unique vagal system that includes myelinated fibers that regulate heart rate. The mammalian vagus functions as an active vagal brake (see chapter 7) in which rapid inhibition and disinhibition of the vagal tone to the heart can change cardiac output to promote immediate engagement and disengagement with objects and individuals. Thus, the autonomic components, coincident with social interactions, may be mediated by changes in vagal tone to the heart, rather than the assumed changes in sympathetic arousal. Consistent with the polyvagal theory, difficulties in regulating the vagal brake may result in the phylogenetically older systems (i.e., neural regulation of the adrenal and the SNS) being recruited to regulate metabolic output to deal with environmental challenges. Consistent with the polyvagal theory, during states of mobilization, characterized by classic "fight–flight" behaviors and sympathetic excitation, both the vagal brake and the behavioral components of the social engagement system are not easily accessible.
The functional impact of the mammalian vagus on the heart produces a heart rate pattern known as respiratory sinus arrhythmia (RSA). RSA is the rhythmic increase and decrease in heart rate observed at the frequency of spontaneous breathing. Because the brainstem nuclei that regulate the mammalian vagus are neuroanatomically and neurophysiologically linked to the brainstem source nuclei of the special visceral efferents that regulate facial expression, monitoring dynamic changes in RSA and heart rate (i.e., the vagal brake) provides an efficient and noninvasive method of assessing the status of the social engagement system.
COURTING AND SEDUCTION: SPECIALIZED FUNCTIONS OF THE SOCIAL ENGAGEMENT SYSTEM
Seduction and courting behaviors convey an invitation to reduce physical and psychological distance. Primates express this invitation via cortical regulation of the brainstem source nuclei of the VVC that control facial expressions, head movements, and vocalizations. Mammalian social behaviors appear to be an emergent property of the cortical regulation of the VVC and the peripheral structures identified in Figure 11.1. Courting and other behaviors of social engagement require direct cortical modulation of the medullary source nuclei of the VVC. This cortical regulation of the brainstem inhibits protective and defensive response strategies that are dependent on subcortical structures (e.g., amygdala, hypothalamus). However, courting and other expressions of social behavior are not independent of neurophysiological state. When the brainstem is under subcortical control, such as in states associated with fight-or-flight behaviors, cardiac output is increased by activation of the SNS to support the mobilization behaviors necessary to avoid or to respond to potential conflict. During these states, cortical modulation of the visceromotor control of the heart (i.e., the vagal brake) would substantially reduce cardiac output and compromise the effectiveness of the adaptive fight-or-flight behaviors. Thus, cortical regulation of the VVC requires the setting to be perceived as safe. The perception of safety, or at least the lack of fight-or-flight responses, would provide a neurophysiological state in which cortical regulation of medullary nuclei could promote proximity and increase the probability of reproductive behaviors.
Love and the Polyvagal Theory
Based on the polyvagal theory, one would expect a neurobehavioral model of love to contain three phases, one representing each of the three emotion subsystems already described in chapter 10. The first two phases are easily identified. The first phase would be associated with the social engagement system, which would systematically signal and engage a prospective mate, and the regulation of vagal brake, which would modulate the metabolic resources necessary to carry out the behaviors. The first system would function only during periods of perceived safety. The second phase would be associated with mobilization and would provide the energy, via sympathetic excitation, to defend and to facilitate proximity for reproductive behaviors when separated. However, a third phase associated with immobilization is more difficult to conceptualize within the polyvagal theory. Although sexual behaviors often incorporate a state of immobilization, immobilization during coitus is not consistent with the state of fear predicted by the polyvagal theory. In addition, the initial description of the polyvagal theory made no statement regarding experiences of pleasure and ecstasy associated with sexual behavior. Therefore, to provide a neurophysiological explanation of love, the configuration of the polyvagal theory requires modification.
Love Without Fear: Hypothalamic Regulation of the DVC
The third emotional subsystem of the polyvagal theory assumes that immobilization is adaptive only in response to danger. A state of fear would be inconsistent with the acknowledged behavioral states associated with seduction and passion. However, for many mammals, behavioral immobilization of the female is required for intromission. This active inhibition of motor activity optimally occurs in humans, not in a state of terror or fear but in a state defined by safety and trust of the mate. If trust and safety do not characterize the period of copulation, intercourse may be painful and produce tissue damage. Alternatively, if a faulty sense of safety is perceived, both male and female may be vulnerable to predation.
How do mammals immobilize without fear? And what are the physiological mechanisms underlying the mating rituals that allow behavioral immobilization without the physiological consequences of fear-induced shutdown responses?
Although the polyvagal theory has emphasized the potentially lethal shutdown behaviors associated with massive surges from the dorsal motor nucleus of the vagus, the DVC is involved in other functions. The DVC, with motor fibers originating in the dorsal motor nucleus of the vagus and afferent fibers terminating in the nucleus of the solitary tract and area postrema, has been assumed to be involved primarily in homeostatic functions (Leslie, 1985). The DVC promotes anabolic activities related to the restoration and conservation of bodily energy and the resting of vital organs. The DVC regulates digestion by modulating digestive polypeptides and gastric motility (Rogers & Hermann, 1992). In addition, Uvnas-Moberg (1989, 1994) has proposed a parallel between DVC regulation of gastrointestinal hormones and the regulation of visceral states including stress, hunger, and satiety. Without external challenges, the DVC optimizes the function of the internal viscera. In contrast, by increasing metabolic output to deal directly with external challenges, the SNS attempts to optimize the organism's relationship with the environment. Thus, increases in ambient temperature, noise, pain, and pyrogenic agents produce not only increased sympathetic activity but an active inhibition of DVC actions on the gut (Uvnas-Moberg, 1987).
Paraventricular Nucleus and the DVC
The paraventricular nucleus of the hypothalamus is an important regulator of the DVC. Neural communication between the paraventricular nucleus and the DVC is involved in responses that are not only homeostatic but protective and defensive (e.g., nausea and vomiting, conditioned taste aversion, behavioral defense) (Lawes, 1990). Communication between the paraventricular nucleus and the DVC changes with experience and thus may exhibit a type of learning or memory. Associations may be established rapidly between environmental features or experiences and visceromotor responses. Perhaps, as in conditioned taste aversion, this memory is expressed as a learned association between a specific environmental feature and nausea. Once the association is made, subsequent exposure to the environmental feature may result in immediate nausea and defensive avoidance behaviors. These speculations regarding the changing communication between the paraventricular nucleus and the DVC with experience are consistent with general theories of aversion learning (Garcia, Lasiter, Bermudez-Rattoni, & Deems, 1985).
The paraventricular nucleus regulation of the DVC evolved in phylogenetically older vertebrates in which escape and avoidance behaviors contributed to the maintenance of visceral homeostasis (Lawes, 1990). Because the early vertebrates lacked an elaborate nervous system to control their viscera, behavior was a primary mechanism for the maintenance of homeostasis (e.g., moving to regulate thermoregulatory and oxygen requirements). As the nervous system evolved, an autonomic nervous system and neuroendocrine mechanisms emerged and displaced the need to use behavior to regulate internal state. The neural and neuroendocrine regulation of internal state allowed behavioral processes to be directed toward environmental challenges. However, the brain structures, specifically the paraventricular nucleus, that governed the homeostatically driven behaviors in the phylogenetically older species, evolved into the structures responsible for regulating internal homeostatic functions in the phylogenetically newer species (Leslie, Reynolds, & Lawes, 1992).
The role of the paraventricular nucleus in the regulation of the DVC in modern vertebrates retains phylogenetically older functions and continues to respond to threatening situations by contributing to visceral and endocrine responses. However, this phylogenetic organization results in vulnerabilities, because perceived challenges to survival, whether or not truly life-threatening, may elicit visceral and endocrine reactions that compromise normal physiological function.
The phylogenetic emphasis of the polyvagal theory emphasizes that defense and avoidance behaviors have a vagal component manifested through the DVC. For example, a physiological shutdown mediated by the DVC would support avoidance behaviors such as death feigning or freezing. However, the evolution of hypothalamic regulation of the DVC provides response alternatives. Specifically, in mammals, the paraventricular nucleus produces two neuropeptides, oxytocin and vasopressin, that differentially communicate with the sensory and motor portions of the DVC. Using the push-pull perfusion technique, Landgraf et al. (1990) demonstrated that both oxytocin and vasopressin are released in the DVC. Binding sites for vasopressin are prevalent in the sensory component, but are not represented in the motor component (Fuxe et al., 1994). In contrast, oxytocin appears to provide a primary pathway from the paraventricular nucleus to the dorsal motor nucleus of the vagus with oxytocin injections into the DVC mimicking the vagal responses normally observed immediately following feeding (Rogers & Hermann, 1992). Direct pathways from the nucleus of the solitary tract to the paraventricular nucleus provide a potential source of feedback for hypothalamic influences on visceromotor functions (Sawchenko & Swanson, 1982). The communication between the DVC and the paraventricular nucleus appears to modulate specific visceromotor reflexes involving cardiovascular (Nissen, Cunningham, & Renaud, 1993) and gastrointestinal systems (Bray, 1985).
Oxytocin and Vasopressin
Oxytocin and vasopressin are synthesized primarily in the paraventricular and supraoptic nuclei of the hypothalamus and released centrally via parvocellular neurons and systemically via magnocellular neurons (Swanson & Sawchenko, 1977). The central and systemic effects of these neuropeptides are different. Central release of oxytocin regulates the output of the dorsal motor nucleus of the vagus, usually maintaining output within levels optimal to support homeostasis. Peripheral release of oxytocin is related to milk ejection, uterine contractions, and ejaculation (Arletti, Benelli, & Bertolini, 1992; Wakerley, Clarke, & Summerlee, 1994). Central release of vasopressin appears to modulate afferent feedback from the viscera and to shift set points, independent of sensitivity, for vagal reflexes such as the baroreceptor reflex (Michelini, 1994). The raising of the baroreceptor set point would, by increasing cardiac output, potentiate fight-or-flight behaviors and allow sympathetic excitation of the heart to be unopposed by homeostatic vagal reflexes. Thus, central levels of oxytocin have been assumed to be associated with vagal processes and central levels of vasopressin have been assumed to be associated with sympathetic processes (Uvnas-Moberg, 1997).
Because the peripheral influences of oxytocin and vasopressin function through feedback, primarily via the sensory component of the DVC, the effects are less clear and may be level-dependent. For example, it is possible that peripheral vasopressin, by stimulating vagal afferents, may trigger massive vagal responses via the dorsal motor nucleus of the vagus. In support of this speculation, it is known that in humans, peripheral vasopressin, and not oxytocin, is related to the nausea experienced during motion sickness (Koch, Summy-Long, Bingaman, Sperry, & Stern, 1990). In addition, systemic vasopressin may induce a baroreceptor-mediated withdrawal of sympathetic tone, which is observed in increases in baroreceptor-elicited bradycardia and a fall in plasma concentration of norepinephrine (Buwalda, Koolhaas, & Bohus, 1992; Michelini, 1994).
Under certain conditions, such as during periods of perceived safety, small increases in peripheral vasopressin might trigger parvocellular release of both oxytocin and vasopressin. Potentially, this could occur via stimulation of either peripheral vagal afferents or vasopressinergic receptors in the area postrema. This stimulation would initiate communication between the paraventricular nucleus and both the sensory and motor nuclei of the DVC. The simultaneous central release of oxytocin and vasopressin would activate both vagal and sympathetic activity. This unique physiological state might characterize sexual arousal and would support intimate behaviors. Since vestibular stimulation elicits systemic vasopressin release (Koch et al., 1990), the perception of motion or environments that move might elicit visceral states vulnerable for intimacy. This could explain the selection of porch swings, trains, boats, planes, water beds, or even roller-coasters as preferred arenas for eliciting and experiencing passionate love.
Oxytocin may be part of a complex response profile related to the perception of the environment as safe. Consistent with this view, Uvnas-Moberg (1997) and Carter and Altemus (1997) propose that oxytocin promotes states resistant to stress (i.e., antistress). In contrast, vasopressin may be part of a complex response profile related to the perception that the environment is challenging or dangerous. In fact, central vasopressin could potentiate mobilization responses via sympathetic excitation, while high levels of systemic vasopressin may potentiate a physiological shutdown associated with fear (e.g., bradycardia) via feedback to the dorsal motor nucleus and inhibition of sympathetic outflow (Ferguson & Lowes, 1994). In addition, lesions of vagal afferents, which functionally block the visceral input to the sensory component of the DVC (areas sensitive to vasopressin), attenuate or abolish specific conditioned taste aversions (Andrews & Lawes, 1992).
Based on the polyvagal theory, the mammalian or smart vagus, with myelinated motor fibers originating in the nucleus ambiguus, provides a system for voluntary engagement with the environment with special features associated with the prosocial behaviors of communication. Paralleling this evolutionary shift in the vagus is a mammalian modification of the hypothalamic regulation of the DVC via both oxytocin and vasopressin. The advent of specific receptors for oxytocin and vasopressin increases the range of adaptive functions involving the DVC. In mammals, the dorsal motor nucleus of the vagus, the motor component of the DVC, is sensitive to oxytocin and insensitive to vasopressin. In contrast, the sensory components of the DVC, the nucleus of the solitary tract and area postrema, are most sensitive to vasopressin. Although the nucleus of the solitary tract has receptors for oxytocin (Landgraf et al. 1990), the area postrema may not be directly influenced by oxytocin (Carpenter, 1990). The differential sensitivity of specific components of the DVC to these two neuropeptides (the differential effects of central and systemic release on visceral function and a potential level dependency) results in a wider range of response options and the co-opting of the primitive vagal system to support avoidance (death feigning, vomiting), engagement (e.g., nursing, feeding) and copulation.
Conditioned Love: Physiological Mechanisms Involved in the Learning and Memory of Intimacy
Classical conditioning provides a potential neurophysiological process to associate gastrointestinal responses with specific sensory events. Classical conditioning may incorporate oxytocinergic and vasopressinergic pathways connecting the paraventricular nucleus with the dorsal motor vagal complex. Garcia et al. (1985), in presenting a general theory of aversion learning, speculated that two specialized coping systems evolved in mammals. The first coping system includes behaviors to protect oneself from predatory attack. This system employs instrumental mobilization behaviors, including active approach and avoidance behaviors. The second coping system deals with protecting the gut from toxic foods and includes the hedonic appraisal of visceral stimulation during eating and copulation. The second system produces rapidly conditioned and difficult-to-extinguish gustatory–visceral associations. Garcia and colleagues speculated that to approach a receptive mate is a product of the first system and to find a sexually satisfying mate more desirable is a product of the second system. This distinction between approach behaviors and conditioned visceral feelings is convergent with the proposed neurobiological theory of love, which distinguishes between seduction and conditioned or passionate love.
Consistent with the hypothesis that a conditioned association between positive visceral feelings and the mate are a product of copulation, Carter and colleagues (Carter, Devries, & Getz, 1995; Carter et al., 1997) have shown that sexual interactions, probably mediated through oxytocin and/or vasopressin, can facilitate pair-bonding. Oxytocin has been associated with positive states, such as physical proximity, touching, prosocial behavior, and the ingestion of food (Carter et al., 1997; Uvnas-Moberg, 1997). Oxytocin also has been implicated in the cephalic phase of digestion. The cephalic phase, the initial digestive phase, is stimulated by psychological factors such as sight, smell, taste, or associations with food before food enters the stomach. The cephalic phase is characterized by increased gastric secretion and reduced gastric motility, providing a receptive environment for the passage of food and allowing the gastric secretions to more efficiently aid in digestion of food in the stomach (Rogers & Hermann, 1992).
Oxytocin and vasopressin are related to other learned responses. For example, intracerebroventricular injection of oxytocin attenuates passive avoidance (Kovacs & Telegdy, 1982), whereas vasopressin enhances passive avoidance (De Wied, 1971). Although these findings may be paradigm-dependent, there is consistent evidence that central levels of both neuropeptides are involved in learning social cues and in the development of partner preferences (Carter, 1998; Engelmann, Wotjak, Neumann, Ludwig, & Landgraf, 1996). In addition, it is well known that the oxytocin released during milk ejection can be conditioned (Wakerley et al., 1994). Likewise, it is plausible that the oxytocin released during coitus could be conditioned and associated with specific social cues. Thus, engagements between pair-bonded mates would trigger a release of oxytocin, which might decrease the latency for subsequent sexual encounters.
Communication Between the Paraventricular Nucleus and the DVC: A Mammalian System for Love and Fear
The neural and neuropeptide communication between the paraventricular nucleus and the DVC, with its involvement in both emotional and learned associations, may provide the physiological mechanism that enables mammals to respond reliably to both fear-related and safety-related environmental cues. Thus, this communication, through the partitioned roles of two related peptides, vasopressin and oxytocin, may promote several classes of behavior. First, the oxytocinergic pathways from the paraventricular nucleus to the dorsal motor nucleus of the vagus appear to co-opt the ancient immobilization fear system that characterizes reptiles. By blunting the shutdown fear response mediated by the dorsal motor nucleus of the vagus, oxytocin modulates vagal function to promote homeostasis and shifts the function of the visceral organs to support progenitive behavior and experiences of passion. Second, in the absence of central oxytocin communication with the dorsal motor nucleus of the vagus, increases in systemic vasopressin would facilitate a fear-induced avoidance, consistent with phylogenetic origins, as a shutdown response system. Third, central vasopressin would facilitate mobilization via sympathetic excitation. Fourth, small increases in systemic vasopressin may trigger a coexcitation of central oxytocin and vasopressin coincident with the peripheral coexcitation of vagal and sympathetic activity characteristic of sexual arousal.
Consistent with the literature on visceral (i.e., vagal) conditioning and pair-bonding (Carter et al., 1997), the oxytocinergic communication between the paraventricular nucleus and the dorsal motor nucleus of the vagus, may provide a neurophysiological mechanism to explain how specific progenitive behaviors including proximity with a mate would be linked with positive visceral feelings. The conditioning process would be facilitated by both systemic vasopressin, which could trigger sexual arousal, and central oxytocin, which could modulate vagal responses. The conditioning process may provide a plausible mechanism to explain other classes of behavior including parent–child bonding, friendships, and the visceral reactions to the loss of a loved one either through death or violation of vows of love. For example, grief and unrequited love, often characterized by potent unpleasant visceral responses, may be mediated by increases in systemic vasopressin, which can trigger vagal responses (e.g., nausea and syncope), which are no longer protected or modulated within the homeostatic range by oxytocin. Vasopressin and oxytocin may work in concert in other behavioral states. For example, during sexual or nursing behaviors, peripheral oxytocin release may enable the visceral organs to be responsive and pliable, while central oxytocin and/or vasopressin release may result in a modulation or an attenuation of painful tactile stimulation, such as that reported following electrical stimulation of the solitary tract (Ren, Randich, & Gebhart, 1990; Uvnas-Moberg, 1998).
Additional neuroanatomical structures may be involved in the establishment of intimacy. The amygdala appears to play a major role in the retention of fear-related or aversive associations (Davis, 1992; LaBar & LeDoux, 1996; LeDoux, Iwata, Cicchetti, & Reis, 1988). For example, lesions of the central nucleus of the amygdala attenuate the conditioned bradycardia, independent of the conditioned corneoretinal potential (Gentile, Jarrell, Teich, McCabe, & Schneiderman, 1986) or the magnitude of the heart rate orienting response (Kapp, Frysinger, Gallagher, & Haselton, 1979). This research demonstrates a role for the amygdala in the retention of negative affective states. However, the role of the amygdala in the retention of positive affective states with prosocial consequences, such as a hypothesized conditioned love, has not been investigated.
Immobilization Without Fear: The Importance of Perceived Safety
By incorporating the influence of the neuropeptides oxytocin and vasopressin on the DVC, the polyvagal theory can be used to explain two classes of immobilization behaviors: one associated with fear and the other with passion. Mammals require a perception of safety to digest food efficiently, to sleep, and to reproduce. During perceived threat or fear, these processes are inhibited. The paraventricular regulation of the DVC provides a plausible mechanism for a central switching circuit that determines whether specific DVC processes are fostered or inhibited. Thus, the neuropeptide modulation of the DVC may contribute to two important processes: first, the determination of whether immobilization is due to fear or security; second, a specific conditioned association with each behavioral state. Similar to other conditioned vagal responses (e.g., taste aversion), the learned associations with either fear or security may be easily established and very difficult to extinguish.
The oxytocinergic pathways from the paraventricular nucleus to the dorsal motor nucleus of the vagus modulate the neural stimulation of organs of fear, digestion, and elimination to foster reproduction, feelings of safety, visceral sensations of pleasure and ecstasy and the conditioning of visceral associations with the mate. In contrast, the vasopressinergic pathways from the paraventricular nucleus to the nucleus of the solitary tract and area postrema may inhibit processes associated with digestion, elimination, and reproduction, thereby facilitating fight-or-flight (mobilization) behaviors. Other neural pathways from the hypothalamus to the DVC may promote either primitive avoidance behaviors (such as freezing and death feigning) or the more phylogenetically advanced flight-or-fight behaviors associated with the SNS. In addition, pathways from the amygdala may modulate the communication between the hypothalamus and the DVC (Lawes, 1990) and contribute to specific fear-associated behaviors (LeDoux et al., 1988; Rosen, Hamerman, Sitcoske, Glowa, & Schulkin, 1996).
Mammalian neuropeptides modulate autonomic functions during love-related behaviors. The polyvagal theory emphasizes the phylogenetically more recent vagal pathways that originate in the nucleus ambiguus. The nucleus ambiguus vagal pathways are involved in the voluntary behaviors required for social engagement, including the general category of seduction. However, the visceral experience of ecstasy, diffuse visceral pleasure, and analgesia are related to the phylogenetically older DVC. The DVC, according to the polyvagal theory is associated with an immobilization fear system. However, the mammalian neuropeptide oxytocin, which is released from the paraventricular nucleus of the hypothalamus, provides a neurophysiological mechanism to co-opt the function of the DVC. Oxytocin may modify the function of the DVC from an immobilization fear system to an immobilization passion or love system. Oxytocin, although stimulating vagal activity (e.g., DVC), appears to limit vagal activity to a functional range that protects the organism from experiencing massive vagal surges that would shut down physiological homeostasis.
Seduction or Rape?
The autonomic nervous system is involved in several aspects of human mating behavior. First, as discussed, the autonomic nervous system can support seduction (behaviors of engagement). The mammalian or smart vagus with its somatomotor regulation of facial expressions and vocalizations provides the neural regulatory structures for this phase of social engagement. Second, by an inhibition of the vagal system and an excitation of the SNS, cardiac output can be increased to support behavioral mobilization including active withdrawal from unwelcomed engagements, fighting behavior to protect the mate, and approach behaviors to reduce distance from the mate. Third, the phylogenetically older DVC (visceral or vegetative vagus) contributes to behavioral immobilization by initiating a primitive shutdown of physiological systems. This form of immobilization is usually in response to fear when there is no option to mobilize. Fourth, by co-opting the DVC, the neuropeptides from the paraventricular nucleus can promote sexual arousal, immobilization without fear, copulatory behavior, positive visceral experiences, and conditioned associations with the mating partner.
A physiological shutdown response profile may characterize the female during rape, an unwelcomed and physiologically dangerous event. This physiological shutdown, due to a massive surge from the dorsal motor nucleus of the vagus, may be associated with or conditioned to specific events or individuals. This classically conditioned response may require only a single trial to be learned and may exhibit great resistance to extinction. For example, following the rape, sexual encounters, even with a desired partner, may elicit a vagal syncope. Or the raped woman may become anxious about sexual encounters and physiologically mobilized via sympathetic excitation to escape. An important aspect of the immobilization phase is that it may follow the well-documented laws of conditioning visceral responses mediated by the DVC (e.g., nausea and vomiting) used to explain conditioned taste aversion (Garcia et al., 1985). A similar response profile may characterize individuals who fear death and believe that they are unable to escape, such as the description of hopelessness (Richter, 1957) and perhaps, clinical disorders such as post-traumatic stress disorder.
Seduction provides a prosocial vehicle for mate selection. Appropriate mate selection, in turn, changes the visceral and psychological experiences associated with female immobilization from fear to passion, which is required for intromission. Seduction allows intimacy to occur without trauma. When the female perceives the male as providing security, an immobilization love system is initiated. This immobilized love system is physiologically and psychologically incompatible with the immobilized fear system. Although both immobilization response patterns share common physiological substrates, the response profiles are different. The immobilized fear system results in a physiological shutdown and a functional inhibition of social behavior and sexual receptivity and responsivity of the genitalia. The immobilization fear system, when initiated, attempts to turn off behavior and consciousness by lowering heart rate and blood pressure, which may produce syncope. In contrast, the immobilization love system heightens sexual arousal in response to genital stimulation, lubricates genital tissue, maintains blood pressure, and raises pain thresholds. The immobilization love system co-opts the paraventricular communication with the DVC and modifies immobilization from a fear-related psychologically dissociative and physiologically compromised state to a love-related psychologically ecstatic and reproductively available state.
By co-opting the communication between the paraventricular nucleus and the DVC, an immobilized love circuit fosters reproductive behaviors. Reproductive behaviors occurring during states of immobilized love promote enduring associations between the mate and the ecstatic experiences. The development of these associations appears to follow the laws of visceral conditioning, since they are easy to establish and result in relatively permanent bonds (Carter et al., 1997). Love may be a classically conditioned response with enduring resistance to extinction. Perhaps species differences in monogamy (Carter et al., 1995; Dewsbury, 1987) might be related to differences in the capacity to recruit the paraventricular control over the DVC in prosocial associative learning paradigms. If this is true, the selective nature of the seduction phase and the fight-or-flight behaviors of the mobilization phase may become a functional barrier to this associative vulnerability. Thus, when we are careful about whom we allow ourselves and our children to be physically close to and to have sexual activities with, we are respecting our vulnerability to the conditioning that characterizes passionate love.
How does our nervous system organize these unique and important behaviors? As illustrated in Figure 11.1, the social engagement system and the vagal brake provide neurobiologically based constructs to describe mechanisms of seduction. Modulation of the social engagement system enables symbolic approach behaviors (e.g., facial expressions, head tilt, and vocalizations), and modulation of the vagal brake provides metabolic support for the behavioral mobilization necessary to engage a prospective mate physically. However, if there is a mismatch between the expectations of the prospective mating partners, mobilization may occur to ensure an increase in physical distance. This mobilization response requires sympathetic excitation and a withdrawal of the vagal brake (the smart vagus). If physical escape is not available, the vegetative vagus (DVC) may become activated to provide a primitive avoidance strategy characterized by a physiological shutdown and a possible loss of consciousness due to compromised homeostatic regulation (e.g., decreased blood pressure).
Successful mating and bonding are the product of a different sequence. In this sequence, seduction is successful and proximity between the prospective mating partners is reduced. Mobilization is restricted to the physical activities associated with the preparation to copulate and copulation. Finally, to ease intromission for the female and the postcoitus recovery for the male, an immobilization system is stimulated. This immobilization system is confined to situations of perceived security. Because states of immobilization for mammals are periods of vulnerability, this type of immobilization occurs only in a safe environment with mutual trust as a defining feature. Thus, an important feature of the love experience is not mediated by physical attractiveness, but is driven by trust and security.
As illustrated in Figure 11.2, hypothalamic–DVC regulation is flexible. Higher brain structures determine whether the hypothalamic–DVC communication results in an immobilized fear response or an immobilized love response. The amygdala may play a major role in determining which hypothalamic–DVC circuit is recruited. If Pavlovian conditioning provides a plausible metaphor for the enduring nature of a conditioned love (Garcia et al., 1985), then the amygdala may be involved in maintaining these learned associations.
The proposed model attempts to integrate the role of neuropeptides (i.e., oxytocin and vasopressin) with the autonomic nervous system. As illustrated in Figure 11.2, the perceptual mechanisms of higher brain structures determine if a situation is dangerous or safe. During perceived fear, there are two options. The organism may mobilize and express fight-or-flight behaviors. Or, if the first option is not available, the organism may immobilize. Immobilization results in a behavioral shutdown, death feigning, and a loss of consciousness. The effect of vasopressin released centrally via the parvocellular neurons, communicating between the hypothalamus and the sensory portion of the DVC (nucleus of solitary tract and area postrema), inhibits feedback from the viscera and promotes sympathetic activation and increased mobilization. However, vasopressin released systemically via the magnocellular neurons originating in the hypothalamus stimulates visceral afferents and promotes feedback to the DVC, which may result in the massive vagal surge associated with physiological shutdown.
FIGURE 11.2. Neural and neuropeptide regulation of the dorsal motor nucleus of the vagus: fear or love? Higher brain structures including the amygdala and cortex influence hypothalamic–dorsal vagal complex (DVC) communication. The DVC includes sensory nuclei in the nucleus of the solitary tract (NTS) and area postrema (AP) and motor nuclei in the dorsal motor nucleus of the vagus (DMX). During perceived danger, when mobilization is adaptive, central vasopressinergic pathways (AVP) communicate between the hypothalamus and both NTS and AP to change the set-point of vagal reflexes to facilitate sympathetic excitation. Immobilized fear occurs when fight-or-flight behaviors are not an option. Immobilized fear is fostered by vagal surges from DMX to visceral organs, which are potentiated by systemic AVP. Systemic AVP triggers increased DMX output by stimulating visceral afferents via NTS and AP. During perceived safety, oxytocin (OXT) is released centrally and systemically to foster an immobilized love response pattern. Central OXT limits DMX output to a functional range protecting homeostasis and systemic OXT stimulates visceral organs. Small increases in systemic AVP might, either via vagal afferents or direct stimulation of vasopressinergic receptors in AP, trigger central AVP and OXT and promote sexual arousal.
In contrast, perceived security changes the immobilization system, which originally evolved to cope with fear, to a system that promotes reproduction and provides heightened sensory experiences that are psychologically reinforcing. Perceived security enables the hypothalamic release of oxytocin centrally to modulate the vagal discharge that can stimulate the viscera and systemically to foster reproduction and heightened sensory feedback. Enhanced sensory feedback provides a positive visceral state that acts as a reinforcer to enhance the conditioned association between the mate and reproductive availability. In addition, sensory feedback acts as a modulator of pain (Komisaruk & Whipple, 1995), which allows areas of the body to experience sensations that would have been perceived as painful in other situations. The positive visceral feelings lead to experiences of pleasure, ecstasy, and the less well-defined emotional states that we associate with love. Thus, via the influence of oxytocin on the autonomic nervous system, organs of fear and vigilance become organs of pleasure, nurturance, and reproduction. Oxytocin, a hormone unique to mammals, co-opts ancient structures changing a fear-induced shutdown system in to a receptive and ecstatic reproductive system. In addition, as stated, there is the possibility that low levels of peripheral vasopressin, perhaps elicited by gentle vestibular stimulation in an environment perceived as safe (e.g., the porch swing or a sailboat) triggers central release of oxytocin.
Of course, the violation of trust changes the context, and a mate that once conveyed a sense of safety would now convey cues of danger. Thus, violation of trust might result in mobilized fight-or-flight behaviors, or the shutdown behavior associated with immobilized fear. In addition, unrequited love or the loss of a loved one might result in syncope or visceral feelings associated with nausea, characteristic of vagal activation due to systemic vasopressin release, without the protective or modulated influence of oxytocin.
The Monogamy Switch and the Biological Prenuptial
Love, as an emotional and motivational process, may have evolved to maximize the adaptive benefits associated with reproduction and safety. However, to achieve these benefits, individuals need to negotiate the relative costs of coping with two potent risks: (1) the vulnerability to predation; and (2) the vulnerability of the nervous system to develop enduring social bonds following the mating sequence. The first risk is obvious. The environment is competitive and often hostile. To survive, coalitions must be developed. Prominent among these coalitions is the social bond between mating partners. The second risk acknowledges that our nervous system may be stimulated to form enduring social bonds with an inappropriate mate. The articulation of the second risk is new to science. Rather, the second risk has been the focus of gender-specific myths and cultural expectations regarding chastity, promiscuity, and marriage. In contemporary culture, monogamy is the focal point around which seduction and sexual encounters are expected to revolve. However, not all sexual and love experiences lead to monogamous relationships. Although the partitioning of the love experience into two sequential components (seduction and a conditioned or enduring passionate love) assumes a monogamous end point, many individuals may focus on seduction and opt for relationships that are not monogamous.
Conditioned love with its enduring social bond might require a prerequisite neurophysiological state that might be conceptualized as a monogamy switch. Once a decision is made to become monogamous with a selected mate, the individual may immobilize without fear and the nervous system becomes vulnerable to conditioned love. Alternatively, to protect oneself from monogamy, the monogamy switch may be disabled by mobilization strategies, even during sexual encounters. Mobilization strategies engage SNS mechanisms, which are inhibitory of the conditioning processes associated with DVC mechanisms. For example, promiscuous sexual activity need not lead to enduring bonds, if the sexual activity were physically active and both sexual partners limited periods of immobilization. This strategy would limit oxytocin release, while experiencing the activation of sexual arousal. Illicit affairs fit this model, especially if the sexual activity is brief, intense, and under the threat of discovery.
The risks or vulnerabilities to states of conditioned love have been described in gender-specific myths, which promote female chastity and female vulnerability to first love in contrast to male promiscuity. Underlying these myths may be an implicit understanding of the monogamy switch, a neurophysiological mechanism that promotes pair-bonding or conditioned love. For example, female immobilization without fear might heighten a vulnerability to a conditioned or learned love. Thus, the male, who conquers the female's fears and gives her a sense of safety and security, not only is allowed to copulate with her, but in return she may be permanently bonded to him. Possibly, this gender bias might have evolved because copulatory behavior in mammals requires only the female to immobilize, but to immobilize outside the realm of fear. In contrast, the male tends to be more mobilized in the sexual act; only following ejaculation does the male become immobilized and at physical risk from the environment. It is this mobilization that may protect the male from a conditioned love. Perhaps, if the male, following copulation, remained immobilized or slept in the presence of the female, he would be as vulnerable as the female to conditioned love. Cultural and self-imposed prohibitions of spending the night together, even following sex, might reflect an implicit biological awareness of this phenomenon.
The potential gender differences in conditioned love vulnerability may result in an unstated but assumed biological prenuptial. The biological prenuptial reflects the interactive negotiations between prospective mating partners in which the male requests exclusive copulatory rights of the female's reproductive organs and the female requests that the male ensure her security and safety needs before activating the monogamy switch. Violation of the biological prenuptial occurs when the male physically abuses the female or when the female copulates with another male. The valence of these two violations appears to be gender specific. In support of this hypothesis, Buss, Larsen, Westen, and Semmelroth (1992) reported that men are more distressed by their mate's sexual infidelity, while women are more distressed by their mate's emotional infidelity. Violations of the biological prenuptial are destructive to the love bond and result in a lack of trust and security for both genders. This gender-biased prenuptial is so rooted in our history that it finds its way into our marriage vows and religious tenants, which have been used to support monogamy.
CONCLUSION
The evolution of the neural and hormonal regulation of the autonomic nervous system provides a framework to interpret mammalian love as an adaptive process that facilitates reproduction in a rapidly changing and challenging environment. The development of love and intimacy consists of several sequential processes with adaptive functions that promote safety and progenitive behavior. Love, as a neurophysiological construct, not only promotes reproduction but also provides a pair-bond to promote safety in the challenging environment. Within this adaptive context, love may have evolved functionally as a temporal shortcut to bypass the slow, often tedious, and potentially unsuccessful processes of communication and social engagement to foster physical proximity and to promote intimacy and reproductive behaviors.
CHAPTER 12
Social Engagement and Attachment: A Phylogenetic Perspective
As the scientific knowledge of neuroanatomy and neurophysiology expands, there is a growing interest in the role neural processes play in the development of normal social behavior and in the expression of the atypical social behaviors that may provide the roots of mental illness in children. Recent advances in neuroscience have enabled researchers to study nervous system function and structure in the intact living individual. Now neuronal function can be studied, and the structural hypotheses derived from animal models and postmortem histology can be challenged and explained. These new methods of assaying neural structure and function, coupled with the breakthroughs in molecular genetics, are providing new tools and models, which can be integrated with existing strategies that effectively monitor dynamic neural function by time sampling neuroendocrine and autonomic parameters.
DEFINING SOCIAL BEHAVIOR: THE GREAT CONCEPTUAL DIVIDE
An objective of this chapter is to build bridges among researchers who study the development of social behavior with both animal models and clinical populations. As a preliminary premise, it is assumed that both cohorts share the same objective of generating knowledge related to the mechanisms of normal and atypical social behavior that could be translated into clinical practice. The contrasts between the research strategies and methods of the two cohorts may force a reevaluation of this assumption.
Animal models often emphasize the role of a specific neural system, neurotransmitter, neuropeptide, hormone, or brain structure as a regulator of social behavior. In contrast, clinical research often focuses on studying aberrant psychological processes in clinical populations. When neurophysiological systems are studied with clinical populations, the research designs focus on establishing correlations with the disorders and, in general, preclude the possibility of distinguishing whether the physiological correlates are causes or effects of the disorder.
Although the two research strategies often use similar terms, the terms may reflect different domains of social behavior. Animal models tend to focus on the establishment of pair-bonds and generate paradigms to evaluate the strength of these bonds. In contrast, research with children, investigating normal and atypical social behavior, tends to focus on the behaviors that reduce social and physical distance between individuals. For example, the terminology associated with measuring and defining social behavior differs when contrasting the compromised social engagement strategies expressed by an institutionalized child with the ability to establish pair-bonds by a vole.
A final perplexing part of the conceptual divide relates to the translation of neuroscience principles and research findings into clinical practice. The clinician is the third limb of this triad. Paradoxically, although the link between social behavior and mental illness in children emerged directly from clinical observations, the features and dimensions of social behavior studied in both animal models and in laboratory studies of normal and atypical children often deviates from the features that clinicians use to define the pathology. Clinical researchers who conduct studies of social behavior are interested in either how outlier behaviors overlap with features of clinical diagnoses or how behavioral, psychological, and physiological parameters differentiate the clinical population from normal subjects. Often the parameters of interest or, at least, those that distinguish the clinical group from normal subjects focus on processes that do not have an obvious relation to the behaviors observed in clinical settings or used to define the pathology (e.g., cortisol).
Most research in psychopathology accepts the validity of clinical assessment and diagnostic systems (e.g., DSM-IV) as inclusion criteria and then attempts to demonstrate that deficits in psychological processes and/or atypical neurophysiological response patterns underlie the disorder. The research on processes and mechanisms, whether obtained from clinical populations or by studying animal models assumed to express behaviors similar to the clinical populations, does not easily enter the clinical realm and inform clinical assessment. Similarly, other than global diagnoses and quantitative information from standardized assessment instruments, little information from clinical observations regarding the specific features of behavior that have triggered the clinician's concern easily enters the research environment. Thus, the construct of social behavior is treated differently by researchers testing animal models, researchers studying normal social behavior, researchers studying the psychological and neurophysiological mechanisms and processes underlying a clinical diagnosis, and clinicians who diagnose and treat social behavior problems in children. Missing in this mix of metaphors, worldviews, paradigms, and diagnostic models is a shared agenda to translate research findings into practice (i.e., assessment and treatment) and to use clinical information to inform the theoretical models being tested.
SOCIAL BEHAVIOR AND ATTACHMENT
Several researchers who study the development of social behavior in children have focused on the construct of attachment. Several of these researchers conduct studies derived from the observations of Bowlby (1982) and the paradigm building research of Ainsworth et al. (1972) Much of the current research on human attachment is based on the Ainsworth typology, which applies a paradigm assessing infant responses to separation. Clinicians and researchers in developmental psychopathology assume that the Ainsworth classification system and recent derivatives (Cassidy & Shaver, 1999) will provide insights into the psychological mechanisms of specific disorders. In fact, diagnostic categories now include disorders such as reactive attachment disorder (RAD).
The traditional attachment schema derived from the Bowlby theory constitutes only a small part of social behavior. Moreover, traditional attachment theory by focusing on mother–infant relations does not include other putative attachment behaviors that are observed in the enduring bonds between peers, siblings, and mates. Missing from the traditional attachment theories is an articulation of the mechanisms mediating engagement between the individuals bonding or forming attachments.
SOCIAL ENGAGEMENT: THE PREAMBLE OF A SOCIAL BOND
To develop a social bond, individuals have to be in close proximity. This is true for the models focusing on both mother–infant attachment and the strong bonds associated with social monogamy. Both models test the strength and features of the relationship through separation paradigms. There are, of course, major differences between the contexts in which mother–infant attachment and the social bonds of reproductive partners are established and tested. One specific difference is the contrast in mobility between the mother–infant and reproductive partner dyads. In the mother–infant dyad there is an imbalance, with the infant having limited abilities to move toward or away from the mother. However, in the reproductive partner dyad, there is a balance between the behavioral repertoires of the two adults.
Although proximity is critical to the establishment of social bonds, proximity is totally caused by the ability to navigate across physical distance via voluntary behavior. If social bonds were dependent on voluntary motor behaviors, then the newborn infant would be greatly disadvantaged because the neural regulation of the spinal motor pathways are immature at birth and take several years to fully develop. However, in mammals not all muscles are driven by corticospinal pathways. Unlike the striated muscles of trunk and limbs, corticobulbar pathways regulate the striated muscles of the face and head. The corticobulbar pathways are sufficiently developed at birth to be available to the full-term infant to signal caregiver (e.g., vocalizations, grimace) and to engage the social (e.g., gaze, smile) and nutrient (e.g., sucking) aspects of the world. These motor pathways originate in the brainstem and regulate muscles through the branches of five cranial nerves (V, VII, IX, X, XI). Thus, the neural regulation of muscles that provide important elements of social cueing are available to facilitate the social interaction with the caregiver and function collectively as an integrated social engagement system (Porges, 2001a).
The muscles of the face and head influence both the expression and receptivity of social cues and can effectively reduce or increase social distance. Behaviorally this is observed as facial expressions, eye gaze, vocalizations, and head orientation. Neural regulation of these muscles can reduce social distance by making eye contact, expressing prosody in voice, displaying contingent facial expressions, and modulating the middle ear muscles to improve the extraction of human voice from background sounds. Alternatively, by reducing the muscle tone to these muscles, the eyelids droop, prosody is lost, positive and contingent facial expressions are diminished, the ability to extract human voice from background sounds is compromised, and the awareness of the social engagement behaviors of others may be lost. Thus, the neural regulation of the striated muscles of the face and head function both as an active social engagement system that reduces psychological distance and as a filter that can influence the perception of the engagement behaviors of others.
Special visceral efferent pathways mediate the neural regulation of the striated muscles of the face and head. Special visceral pathways emerge from three nuclei in the brainstem (nucleus of the trigeminal nerve, nucleus of the facial nerve, and nucleus ambiguus) and provide motor pathways that are contained within five cranial nerves (i.e., trigeminal, facial, hypoglossal, vagus, accessory). These pathways regulate structures that evolved from the ancient gill arches. From both clinical and research perspectives, the striated muscles of the face and head provide potent information regarding the behavioral dimensions used to express as well as to evaluate the strength of attachment or the stress to the social bond. For example, facial expressivity and prosody of vocalizations have been used as clinical indicators as well as quantifiable responses of separation distress (Newman, 1988).
THE SOCIAL ENGAGEMENT SYSTEM: PHYLOGENIC ORIGINS OF BEHAVIORAL AND AUTONOMIC COMPONENTS
The phylogenic origin of the behaviors associated with the social engagement system is intertwined with the phylogeny of the autonomic nervous system. As the striated muscles, via special visceral efferent pathways, evolved into a behavioral system that regulated social engagement behaviors, there was a profound shift in neural regulation of the autonomic nervous system. Phylogenetically, these changes in both somatomotor and visceromotor regulation are observed in the transition from reptiles to mammals. As the muscles of the face and head evolved into an ingestion (i.e., nursing) and social engagement system, a new component of the autonomic nervous system (i.e., a myelinated vagus) evolved that was regulated by a brainstem nucleus, which was also involved in the regulation of the striated muscles of the face and head (i.e., nucleus ambiguus). This convergence of neural mechanisms resulted in an integrated social engagement system with a synergism between behavioral and visceral features of social engagement. Thus, activation of the somatomotor component would trigger visceral changes that would support social engagement, while modulation of visceral state would either promote or impede social engagement behaviors. For example, stimulation of visceral states that would promote mobilization (i.e., fight-or-flight behaviors) would impede the ability to express social engagement behaviors, whereas increased activity through the myelinated vagus would promote the social engagement behaviors associated with a calm visceral state.
We can infer the specific neural mechanisms related to the effectiveness that feeding and rocking have on promoting calm behavioral and visceral states. Specifically, both the ingestive behaviors associated with feeding and the passive rocking of an infant promote calmness by influencing the myelinated vagus. Feeding activates the muscles of mastication via trigeminal efferent pathways, which, in turn, provide afferent feedback input to the nucleus ambiguus (i.e., the source nucleus of the myelinated vagus). Rocking provides an efficient and direct influence on the vagus by stimulating vagal afferent pathways via the baroreceptors. Moreover, activation of the social engagement system dampens the neural circuits including the limbic structures that support fight, flight, or freeze behaviors.
POLYVAGAL THEORY: THREE NEURAL CIRCUITS REGULATING REACTIVITY
Evolutionary forces have molded both human physiology and behavior. Via evolutionary processes, the mammalian nervous system has emerged with specific neural and behavioral features that react to challenge to maintain visceral homeostasis. These reactions change physiological state and, in mammals, limit sensory awareness, motor behaviors, and cognitive activity. To survive, mammals must determine friend from foe, evaluate whether the environment is safe, and communicate with their social unit. These survival-related behaviors are associated with specific neurobehavioral states that limit the extent to which a mammal can be physically approached and whether the mammal can communicate or establish new coalitions.
Through stages of phylogeny, mammals and especially primates have evolved a functional neural organization that regulates visceral state to support social behavior. The polyvagal theory (see chapters 2, 10, 11; Porges, 2001a) emphasizes the phylogenetic origins of brain structures that regulate social and defensive behaviors, domains compromised in individuals with autism and several psychiatric disorders. The polyvagal theory proposes that the evolution of the mammalian autonomic nervous system provides the neurophysiological substrates for the emotional experiences and affective processes that are major components of social behavior. The theory proposes that physiological state limits the range of behavior and psychological experience. In this context, the evolution of the nervous system determines the range of emotional expression, quality of communication, and the ability to regulate bodily and behavioral state. The polyvagal theory links the evolution of the autonomic nervous system to affective experience, emotional expression, facial gestures, vocal communication, and contingent social behavior. Thus, the theory provides a plausible explanation of several social, emotional, and communication behaviors and disorders.
The Social Engagement System
The polyvagal theory provides an explicit neurobiological model of how difficulties in spontaneous social behavior are linked to both facial expressivity and the regulation of visceral state, and, alternatively, how social behavior may serve as a regulator of physiological activity. The theory proposes a possible mechanism to explain how these difficulties might form a core domain of several psychiatric profiles. Relevant to this focus on psychiatric disorders are the specific deficits associated with several diagnoses in both the somatomotor (e.g., poor gaze, low facial affect, lack of prosody, difficulties in mastication) and visceromotor (difficulties in autonomic regulation resulting in cardiopulmonary and digestive problems) of the social engagement system. For example, clinicians and researchers have documented these deficits in individuals with autistic spectrum disorders. Deficits in the social engagement system would compromise spontaneous social behavior and social awareness and affect expressivity, prosody, and language development. In contrast, interventions that improve the neural regulation of the social engagement system hypothetically would enhance spontaneous social behavior, state and affect regulation, reduce stereotypical behaviors, and improve language skills.
Embryologically, components of several cranial nerves known as special visceral efferent pathways develop together to form the neural substrate of a social engagement system (see chapter 11). This system, as illustrated in Figure 3.1, p. 56, provides the neural structures involved in social and emotional behaviors. The social engagement system has a control component in the cortex (i.e., upper motor neurons) that regulates brainstem nuclei (i.e., lower motor neurons) to control eyelid opening (e.g., looking), facial muscles (e.g., emotional expression), middle ear muscles (e.g., extracting human voice from background noise), muscles of mastication (e.g., ingestion), laryngeal and pharyngeal muscles (e.g., vocalization and language), and head-turning muscles (e.g., social gesture and orientation). Collectively, these muscles function as filters that limit social stimuli (e.g., observing facial features and listening to human voice) and determinants of engagement with the social environment. The neural control of these muscles determines social experiences. In addition, the source nuclei (i.e., lower motor neurons) of these nerves, which are located in the brainstem, communicate directly with an inhibitory neural system that slows heart rate, lowers blood pressure, and actively reduces arousal to promote calm states consistent with the metabolic demands of growth and restoration of our neurophysiological systems. Direct corticobulbar pathways reflect the influence of frontal areas of the cortex (i.e., upper motor neurons) on the regulation of this system. Moreover, afferent feedback through the vagus to medullary areas (e.g., nucleus of the solitary tract) influences forebrain areas that are assumed to be involved in several psychiatric disorders. In addition, the anatomical structures involved in the social engagement system have neurophysiological interactions with the hypothalamic-pituitary-adrenal (HPA) axis, the neuropeptides of oxytocin and vasopressin, and the immune system (Porges, 2001a).
The study of comparative anatomy, evolutionary biology, and embryology may provide important hints regarding the functional relation between the neural control of facial muscles and emergent psychological experiences and behavior. The nerves that control the muscles of the face and head share several common features. Pathways from five cranial nerves control the muscles of the face and head. Collectively, these pathways are labeled as special visceral efferent. Special visceral efferent nerves innervate striated muscles, which regulate the structures derived during embryology from the ancient gill arches (Truex & Carpenter, 1969). The special visceral efferent pathways regulate the muscles of mastication (e.g., ingestion), muscles of the middle ear (e.g., listening to human voice), muscles of the face (e.g., emotional expression), muscles of larynx and pharynx (e.g., prosody and intonation), and muscles controlling head tilt and turning (e.g., gesture). In fact, a neural pathway involved in the regulation of the eyelids also tenses the stapedius muscle in the middle ear, which facilitates hearing human voice. Thus, the neural mechanisms for making eye contact are shared with those needed to listen to human voice. As a cluster, the difficulties in gaze, extraction of human voice, facial expression, head gesture, and prosody are common features of individuals with autism.
Disorders of the Social Engagement System: Maladaptive or Adaptive Behavioral Strategies?
Individuals with several psychiatric and behavioral disorders have difficulties in establishing and maintaining relations. Several clinical diagnostic categories include features associated with difficulties both in expressing social behavior and in reading social cues (i.e., social awareness). These features are observed in individuals with a variety of primary psychiatric diagnoses including autism, social anxiety, post-traumatic stress disorder, and RAD.
Although a compromised social engagement system results in "maladaptive" social behavior, do these asocial behavioral strategies have "adaptive" features? The phylogeny of the vertebrate autonomic nervous system serves as a guide (i.e., polyvagal theory) to understand these adaptive features. Phylogenetically, the vertebrate autonomic nervous system follows three general stages of development. Each stage supports a different category of behavior with only the phylogenetically most recent innovation (i.e., myelinated vagus) supporting social engagement behaviors. Because the neural regulation of the myelinated vagus is integrated into the social engagement system, when the social engagement system is compromised the effects are both behavioral and autonomic. The resultant changes in autonomic state support a range of adaptive defensive behaviors. Specifically, the compromised social engagement system is associated, neurophysiologically, with a change in autonomic regulation characterized by a reduction in the influence of the myelinated vagus (i.e., ventral vagal complex including nucleus ambiguus) on the heart. The removal of the regulatory influence of the ventral vagal complex on the heart potentiates the expression of the two phylogenetically older neural systems (i.e., sympathetic nervous system, dorsal vagal complex including dorsal nucleus of the vagus). These two older neural systems foster mobilization behaviors of fight or flight via the sympathetic nervous system or immobilization behaviors of death feigning, freezing, and behavioral shutdown via the dorsal vagal complex.
Neuroception: A Nervous System Evaluation of Risk
Regardless of the model of attachment or its dependence on cognitive, affective, behavioral, or biological constructs, the critical features that determine the valence of the interaction are related to perceived safety. Thus, the perception of safety is the turning point in the development of relationships for most mammals. The perception of safety determines whether the behavior will be prosocial (i.e., social engagement) or defensive. If the context and the other individual are perceived as safe, then the candidates for the social bond may inhibit the adaptive primitive neurobiological reactions of defense to allow the expression of social engagement. However, how are the adaptive neurobiological systems for defense functionally subdued to insure that attachment and the formation of social bonds will be the products of appropriate social engagement?
Before a social bond can occur, both individuals have to perceive each other as safe. What mediates the individual's ability to engage? Why would an infant look and coo at a caregiver, but avert gaze and cry as a stranger approached? Why would a gentle embrace be experienced as pleasurable when expressed by a lover and be experienced as assault when expressed by a stranger? Mammals have adaptive neurobehavioral systems for both defensive and social engagement behaviors. However, what enables engagement behaviors to occur, while disenabling the mechanisms of defense? The polyvagal theory with its focus on the phylogeny of the vertebrate autonomic nervous system provides a perspective to identify and to understand the plausible mechanisms that enable mammals to functionally switch between positive social engagement and defensive behavioral strategies. To effectively switch from defensive to social engagement strategies, the mammalian nervous system needs to perform two important processes: (1) assess risk, and (2) if the environment is perceived as safe, inhibit the more primitive limbic structures that control fight, flight, or freeze behaviors.
The nervous system, through the processing of sensory information from the environment, continuously evaluates risk. Because the neural evaluation of risk does not require conscious awareness, the term neuroception (see chapter 1) is introduced to emphasize the neural circuits that function as a safety-threat detection system capable of distinguishing among situations that are safe, dangerous, or life-threatening. Because of the phylogenetic heritage of mammals, neuroception can operate without cognitive awareness via relatively primitive mechanisms that are dependent on subcortical structures (e.g., limbic). As a product of evolution, new neural systems evolved in mammals that involved cortical regulation of subcortical structures and, in many instances, co-opted the defense functions of the primitive structures to support other functions including those related to reproductive behavior and pair bonding (see chapter 11).
Based on the relative risk of the environment, both social engagement and defense behaviors may be interpreted as either adaptive or maladaptive. For example, the inhibition of defense systems by the social engagement system would be adaptive and appropriate only in a safe environment. From a clinical perspective, it would be the inability to inhibit defense systems in safe environments (e.g., anxiety disorders, RAD) or the inability to activate defense systems in risk environments (e.g., Williams syndrome) that might contribute to the defining features of psychopathology. Thus, an invalid neuroception of safety or danger might contribute to maladaptive physiological reactivity and the expression of the defensive behaviors associated with specific psychiatric disorders.
There is a common feature between the invalid neuroception that identifies risk when no risk is there and McEwen's concept of "allostatic load" (McEwen & Wingfield, 2003). The physiological reaction to a valid risk, although metabolically costly, is adaptive. Thus, the increased metabolic activity necessary to support the mobilization behaviors of fight or flight are adaptive in the short term, but costly to the organism if maintained. The duration of the response is an important feature that distinguishes between adaptive and maladaptive reactions. The complex mammalian nervous system evolved with a great dependence on oxygen and, unlike the reptile, can survive only for short periods without oxygen. Thus, breath holding for mammals is adaptive only for short periods. In contrast, apnea is adaptive for reptiles, who because of their limited needs for oxygen can inhibit breathing for long periods, whereas apnea is potentially lethal for mammals (Porges et al., 2003). Similarly, temporal features, in part, determine the construct of allostatic load. McEwen describes chronic stress or allostatic state as a physiological response that, although having adaptive functions in the short term, can be damaging if used for long periods when it is no longer needed (i.e., invalid neuroception). This cost of adaptation or "maladaptation" McEwen refers to as allostatic load.
Safety Trumps Fear
In safe environments, autonomic state is adaptively regulated to dampen sympathetic activation and to protect the oxygen-dependent central nervous system from the metabolically conservative reactions of the dorsal vagal complex. However, how does the nervous system know when the environment is safe, dangerous, or life-threatening and what neural mechanisms evaluate risk in the environment?
New technologies, such as functional magnetic resonance imaging, have identified specific neural structures that are involved in detecting risk. The temporal lobe is of particular interest in expanding the construct of neuroception and in identifying neural mechanisms that modulate the expression of adaptive defensive behaviors and autonomic states. Functional imaging techniques document that areas of the temporal cortex, fusiform gyrus (FG), and superior temporal sulcus (STS) are involved in detecting features such as movements, vocalizations, and faces, which contribute to an individual being perceived as safe or trustworthy (Adolphs, 2002; Winston, Strange, O'Doherty, & Dolan, 2002). Slight changes in these stimuli can pose threat or signal endearment. Connectivity between these areas of the temporal cortex and the amygdala suggests a top-down control in the processing of facial features that could actively inhibit activity of the structures involved in the expression of defensive strategies (Pessoa, McKenna, Gutierrez, & Ungerleider, 2002).
Neuroanatomical and neurophysiological research with animals provides additional information regarding the modulation and inhibition of defensive behaviors via well-defined connections between the amygdala and the periaqueductal gray (PAG). The PAG is a heterogenous midbrain structure that consists of gray matter surrounding the cerebral aqueduct that connects the third and fourth ventricles. Studies have identified areas of the PAG that are organized to regulate flight, fight, or freeze behaviors and the autonomic states that support these behaviors (Keay & Bandler, 2001). Stimulating rostrally within the lateral and dorsolateral PAG produces confrontational defensive behaviors (i.e., fight), while stimulating caudally within the lateral PAG and dorsolateral PAG produces escape behaviors (i.e., flight). Autonomic shifts such as increases in heart rate and blood pressure parallel these behaviors. In contrast, stimulation in the region of the PAG ventrolateral to the aqueduct (vlPAG) evokes a passive reaction of immobility, a decrease in blood pressure, and a slowing of heart rate. Interestingly, excitation of the vlPAG evokes an opioid-mediated analgesia that might adaptively raise pain thresholds. In addition, there is evidence of a functional connection between the central nucleus of the amygdala and the vlPAG that modulates both antinociception and immobilization (Leite-Panissi, Coimbra, & Menescal-De-Oliveira, 2003). Consistent with the polyvagal theory, the vlPAG communicates with dorsal vagal complex, whereas the lPAG and dlPAG communicate with the sympathetic nervous system.
In the absence of threat, inhibitory projections from the FG and STS to the amygdala would be available to actively inhibit the limbic defense systems. This inhibition would provide an opportunity for social behavior to occur. Thus, the appearance of a friend or mate would subdue the limbic activation with the biobehavioral consequences of allowing proximity, physical contact, and other social engagement behaviors. In contrast, during situations in which the appraisal of risk is high, the amygdala and various areas of the PAG are activated. The amygdala and PAG only share connections through the central nucleus (Rizvi, Ennis, Behbehani, & Shipley, 1991).
The detection of safety subdues the adaptive defensive systems dependent on limbic structures. Thus, providing a plausible model of how a neural detection of environmental risk (i.e., neuroception) would modulate behavior and physiological state to support adaptive behaviors in response to safe, dangerous, and life-threatening environments. Conceptually, the process of detecting safety is inclusive of the detection of risk. Thus, the neural circuits that mediate the more primitive defense systems have through the processes of evolution been co-opted to support the social behavior necessary for mammalian survival. These behaviors include social engagement and the behaviors associated with social bonding (e.g., reproductive behaviors and nursing).
Co-opting the Immobilization Defense System for Reproductive Behaviors, Nursing, and the Formation of Social Bonds
Immobilization as a defense system is phylogenetically old and is associated with reduced metabolic demands and increased pain threshold. In reptiles, because of their limited need for oxygen, immobilization is a very effective defense strategy. In contrast, because mammals have a great need for oxygen, the inhibition of movement coupled with a shift in autonomic state to support the immobilization behavior (i.e., apnea and bradycardia) can be lethal (Hofer, 1970; Richter, 1957). However, several aspects of mammalian social behavior require immobilization, but immobilization without fear. Immobilization without fear is accomplished by co-opting the structures that regulate immobilization and pain thresholds to serve a broad range of social needs, including reproduction, nursing, and pair-bonding. By focusing on the area of the PAG that coordinates freezing behavior, we can see how a primitive immobilization defense system has been modified through evolution to serve the intimate social needs of mammals. In addition, when we study the vlPAG we find that it is rich in receptors for oxytocin, a neuropeptide associated with parturition, nursing, and the establishment of pair-bonds (Carter, 1998; Insel & Young, 2001).
Overlapping with the area of the PAG that organizes immobility (i.e., vlPAG) are areas that when stimulated produce lordosis and kyphosis. The lordosis reflex is a hormone-dependent behavior displayed by female rodents and other mammalian species during mating. In most mammals, lordosis involves the female immobilizing in a crouching posture with her hind end available to the male for copulation. Neural tracing studies have demonstrated that the vlPAG is part of the neural circuit involved in regulating lordosis (Daniels, Miselis, & Flanagan, 1999). Kyphosis is an upright crouching posture that is accompanied by inhibition of limb movements. This posture is stimulated by nipple attachment and provides an opportunity for the dam to feed simultaneously a large litter. When dams initiate a nursing bout, behavioral state shifts immediately from high activity to immobility (Stern, 1997). When the caudal portion of the vlPAG is lesioned, there are important consequences: (1) kyphotic nursing decreases, (2) litter weight gains decrease, and (3) the lesioned rats are more aggressive and more frequently attack strange males (Lonstein & Stern, 1998).
Test of the Model
The processes of attachment and the formation of social bonds require appropriate social engagement strategies. In the preceding sections, elements of a preliminary model that links social engagement to attachment and the formation of social bonds are presented. The model is expanded from the polyvagal theory and emphasizes the following points: (1) there are well-defined neural circuits to support social engagement behaviors and the defensive strategies of fight, flight, or freeze; (2) without being dependent on conscious awareness, the nervous system evaluates risk in the environment and regulates the expression of adaptive behavior to match the neuroception of a safe, dangerous, or life-threatening environment; (3) social engagement behaviors and the benefits of the physiological states associated with social support require a neuroception of safety; (4) social behaviors associated with nursing, reproduction, and the formation of strong pair bonds require immobilization without fear; and (5) immobilization without fear is mediated by a co-opting of the neural circuit regulating defensive freezing behaviors through the involvement of oxytocin, a neuropeptide involved in the formation of social bonds (Carter & Keverne, 2002; Winslow & Insel, 2002).
Figures 12.1, 12.2, and 12.3 illustrate the role that neuroception plays in determining the neural circuits recruited to regulate social engagement and fight, flight, or freeze behaviors. Each figure illustrates a different environment context (i.e., safe, dangerous, life threat). Figure 12.1 illustrates the assumed neural circuits involved in promoting social engagement behaviors in a safe context. The detection of safe or trustworthy features derived from face, voice, and movement activate a neural circuit that projects from the temporal cortex (i.e., FG, STS) to the central nucleus of the amygdala to inhibit defensive limbic functions (see Figures 12.2 and 12.3). This circuit disenables the limbic defense systems that organize and regulate fight, flight, or freeze behaviors and enables the corticobulbar pathways that regulate the social engagement behaviors (see Figure 3.1, p. 56). Figure 12.2 illustrates the neural circuits involved in a response to a neuroception of danger. In response to danger, the limbic defense circuits function to adaptively protect the individual. The specificity of the defense strategy, whether confrontational or avoidant (i.e., fight or flight), is regulated by the PAG. To support these mobilization behaviors, the sympathetic nervous system is activated and dominates autonomic state. Figure 12.3 illustrates the neural circuits involved in response to life threat. In response to life threat, the mammalian nervous system promotes immobilization or freezing behavior. Freezing, as a defense strategy, is coordinated by the PAG. To inhibit metabolic activity during immobilization, autonomic state is under the control of the dorsal vagal complex. As proposed by the polyvagal theory, the autonomic reactions during each adaptive behavioral strategy is hierarchically organized after the phylogeny of both the changes in the vertebrate autonomic nervous system and changes in the behavioral repertoire from immobilization to mobilization to social engagement.
FIGURE 12.1. Neural structures and pathways involved in a neuroception of safety.
The ability to evaluate whether the environment is safe or if a person is trustworthy is difficult for individuals with a variety of psychiatric diagnoses. Current research suggests that the areas in the temporal cortex (i.e., FG, STS), which are assumed to inhibit limbic defense reactions, are not activated in clinical populations that have difficulties with social engagement behaviors (e.g., autism, schizophrenia). Moreover, individuals with other psychiatric disorders, such as anxiety disorders and depression, which have as diagnostic features compromised social behavior, have difficulties in regulating visceral state (e.g., less vagal regulation of the heart) and supporting social engagement behaviors (e.g., reduced facial expressiveness and motor control of the striated muscles of the face and head). Thus, from a theoretical perspective, a potential root of several psychiatric disorders might be linked to an inability to detect safety in the environment and trustworthiness from interactions and, thus, the inability to express appropriate social engagement behaviors.
FIGURE 12.2. Neural structures and pathways involved in a neuroception of danger.
The study of attachment disorders such as RAD provides an intriguing test of the critical role of neuroception in mediating appropriate attachment and social behavior. RAD is described in both the DSM-IV (American Psychiatric Association, 1994) and the ICD-10 (World Health Organization, 1992) psychiatric diagnostic manuals. RAD comprises two clinical patterns (i.e., inhibited and uninhibited subtypes). The inhibited subtype is characterized by an emotionally withdrawn, unresponsive pattern in which there is an absence of attachment behaviors. The disinhibited subtype is characterized by indiscriminate attachment, which often is directed at strangers. These patterns have been described in institutionalized and maltreated children (Zeanah, 2000). From a neuroception perspective, in both subtypes, the evaluation of the risk in the environment is not accurate.
FIGURE 12.3. Neural structures and pathways involved in a neuroception of life threat.
Recent research on the outcomes of children raised in institutions in Romania has stimulated interest in RAD and in developing intervention strategies to remediate these devastating disturbances in social development. If an accurate neuroception of the environment is necessary for normal social behavior, then what features in the environment might potentiate normal social development? A recent study of Romanian toddlers (Smyke, Dumitrescu, & Zeanah, 2002) provides insight into the process. In this study, indices of RAD were evaluated in children as a function of the number of different caregivers. Two groups of institutionalized children were evaluated and contrasted with children who were never institutionalized. One group consisted of the standard institution unit in which 20 different caregivers worked rotating shifts with approximately 3 caregivers for 30 children on each shift. A second group consisted of a pilot unit in which the number of children were reduced to approximately 10, and the pool of caregivers was reduced to 4. If neuroception of safety is necessary to promote appropriate social behavior, then familiarity of caregiver would be critical. By having familiar caregivers, the child's detection of the caregiver's face, voice, and movements (the features that define a safe and trustworthy person) should trigger the inhibitory pathways to disenable the limbic defense system and foster the behaviors expressed by the social engagement system. In support of this model, the study demonstrated a monotonic relation between the number of different caregivers that a child had contact with and the indices of RAD. On all measures, the standard unit children were more likely to have higher indices of RAD, and on some measures the pilot group did not differ from the children who were never institutionalized. Thus, once we understand the contextual and social features that inhibit the neural circuits that mediate defensive behavioral strategies, we can optimize the developmental consequences of the neural circuits that promotes social engagement behaviors.
CHAPTER 13
The Polyvagal Hypothesis: Common Mechanisms Mediating Autonomic Regulation, Vocalizations, and Listening
Vocalizations are an intricate component of the complex biobehavioral repertoire of mammals. Since vocalizations are involved in coordinating behaviors for mammals that are both prosocial and survival-oriented, the predominance of research on mammalian vocalizations has focused on describing the adaptive functions of specific vocal features. In contrast, there is a paucity of research investigating the neurophysiological linkage between autonomic regulation and both the production of vocalizations and the processing of acoustic information. This chapter applies hypotheses and principles derived from the polyvagal theory (see chapter 2; Porges, 2001a, 2007a) to interpret mammalian vocalizations. The theory emphasizes the parallel phylogenetic shift in both the neural regulation of the autonomic nervous system and the evolutionary emergence of an integrated social engagement system that includes features optimizing conspecific vocal communication. The chapter presents the polyvagal hypothesis as a new way to interpret the adaptive functions of mammalian vocalizations. The chapter emphasizes neural mechanisms involved in social communication, including the reciprocal relationship between production and reception of vocalizations.
THE POLYVAGAL THEORY: THREE PHYLOGENETIC RESPONSE SYSTEMS
The polyvagal theory (see chapter 2; Porges, 2001a, 2007a) links parallel phylogenetic changes in the neural regulation of the autonomic nervous system with adaptive behaviors. The theory emphasizes the phylogenetic shift in the features of the vagus, the primary parasympathetic cranial nerve involved in regulating visceral state. The theory is named polyvagal, since only in mammals does the vagus contain two distinct efferent pathways. In addition to an unmyelinated pathway originating in the dorsal motor nucleus of the vagus shared with other vertebrates (i.e., reptilians, amphibians, teleosts, and elasmobrachs), mammals have a myelinated pathway originating in the nucleus ambiguus, and this circuit communicates in the brainstem with source nuclei regulating the striated muscles of the face and head that are involved in several of the features of social communication (e.g., facial expression, vocalizations, listening).
The polyvagal theory provides a physiological basis to link uniquely mammalian anatomical and physiological circuits involved in visceral state regulation to both the expressive and receptive domains of social communication. The development of these features reflects the phylogenetic distinction between reptiles and mammals, and includes a cluster of behaviors dependent on the function of structures that phylogenetically emerged with mammals. These changes include detached middle ear bones, the advent of a diaphragm, and a myelinated vagal system regulating supradiaphragmatic organs that is distinct from an unmyelinated vagal system regulating subdiaphragmatic organs. The classic definition of mammals focuses on the observation of mammary glands and hair. However, the fossil record, since it cannot be explored through these features, relies on the identification of detached middle ear bones as the defining feature of mammals. Coincident with the separation of the middle ear bones, other phylogenetic transitions resulted in brainstem areas regulating the vagus becoming intertwined with the areas regulating the striated muscles of the face and head. The result of this transition was a dynamic social engagement system with social communication features (e.g., facial expression, head movements, vocalizations, and listening) interacting with visceral state regulation.
The polyvagal theory articulates how each of three phylogenetic stages of the development of the vertebrate autonomic nervous system is associated with a distinct autonomic subsystem that is retained and expressed in mammals. These autonomic subsystems are phylogenetically ordered and behaviorally linked to social communication (e.g., facial expression, vocalization, listening), mobilization (e.g., fight-or-flight behaviors), and immobilization (e.g., feigning death, vasovagal syncope, and behavioral suppression). Social communication (i.e., social engagement system, see later discussion) involves the myelinated vagus, which serves to foster calm behavioral states by inhibiting the sympathetic influences to the heart and dampening the hypothalamic–pituitary axis (HPA) (e.g., Bueno et al., 1989). The mobilization system is dependent on the functioning of the sympathetic nervous system. The most phylogenetically primitive component, the immobilization system, is dependent on the unmyelinated vagus, which is shared with most vertebrates. With increased neural complexity due to phylogenetic development, the organism's behavioral and affective repertoire is enriched. The three circuits can be conceptualized as dynamically adjusting to provide adaptive responses to safe, dangerous, and life-threatening events and contexts.
Unlike reptiles, the mammalian nervous system did not evolve solely to survive in dangerous and life-threatening contexts, but it evolved to promote social interactions and social bonds in safe environments. To accomplish this adaptive flexibility, the mammalian nervous system evolved a new neural strategy for safe environments, while retaining two more primitive neural circuits to regulate defensive strategies (i.e., fight-or-flight and death-feigning behaviors). It is important to note that social behavior, social communication, and visceral homeostasis are incompatible with the neurophysiological states and behaviors promoted by the two neural circuits that support defense strategies. Thus, via evolution, the mammalian nervous system retains three neural circuits, which are in a phylogenetically organized hierarchy. In this hierarchy of adaptive responses, the newest circuit associated with social communication is used first, and if that circuit fails to provide safety, the older survival-oriented circuits are recruited sequentially.
By investigating the phylogeny of the regulation of the vertebrate heart (see chapters 2 and 10; Morris & Nilsson, 1994; Taylor, Jordan, & Coote, 1999), four principles can be extracted that provide a basis for speculation regarding emergent behaviors and social communication. These principles lead to testable hypotheses relating the specific neural mechanisms supporting social engagement, fight-or-flight, and death-feigning behaviors.
1. There is a phylogenetic shift in the regulation of the heart from endocrine communication, to unmyelinated nerves, and finally to myelinated nerves.
2. There is a development of opposing neural mechanisms of excitation and inhibition to provide rapid regulation of graded metabolic output.
3. A face–heart connection evolved as the brainstem source nuclei of vagal pathways shifted ventrally from the older dorsal motor nucleus to the nucleus ambiguus. This resulted in an anatomical and neurophysiological linkage between the neural regulation of the heart via the myelinated vagus and the special visceral efferent pathways that regulate the striated muscles of the face and head, forming an integrated social engagement system (see Figure 3.1, p. 56) (for more details see chapter 12; Porges, 2001a, 2007a).
4. With increased cortical development, the cortex exhibits greater control over the brainstem via direct (e.g., corticobulbar) and indirect (e.g., corticoreticular) neural pathways originating in motor cortex and terminating in the source nuclei of the myelinated motor nerves emerging from the brainstem (e.g., specific neural pathways embedded within cranial nerves V, VII, IX, X, and XI), controlling visceromotor structures (i.e., heart, bronchi) and somatomotor structures (muscles of the face and head).
THE SOCIAL ENGAGEMENT SYSTEM
The study of comparative anatomy, evolutionary biology, and embryology provide important hints regarding the functional relationship between the neural control of the striated muscles of the face and head and emergent behaviors, including facial expressions, head movements and vocalizations. The nerves that control the muscles of the face and head share several common features. Pathways from five cranial nerves control the muscles of the face and head. Collectively, these pathways are labeled as special visceral efferent. Special visceral efferent nerves innervate striated muscles, which regulate the structures derived during embryology from the ancient gill arches (Truex & Carpenter, 1969). The special visceral efferent pathways regulate the muscles of mastication (e.g., ingestion), muscles of the middle ear (e.g., listening to conspecific vocalizations), muscles of the face (e.g., emotional expression), muscles of the larynx and pharynx (e.g., prosody and intonation), and muscles controlling head tilt and turning (e.g., gesture). In fact, the neural pathway that enables the eyelids to be closed in a graded manner (i.e., winking) in humans is also involved in tensing the stapedius muscle in the middle ear to facilitate hearing the human voice (Djupesland, 1976).
The source nuclei of the circuits regulating the striated muscles of the face and head interact in the brainstem with the source nucleus of the myelinated vagus, forming an integrated social engagement system. This system (as illustrated in Figure 3.1, p. 56) provides the neural structures involved in social and emotional behaviors. The social engagement system has a control component in the cortex (i.e., upper motor neurons) that regulates brainstem nuclei (i.e., lower motor neurons) to control eyelid opening (e.g., looking), facial muscles (e.g., emotional expression), middle ear muscles (e.g., extracting vocalizations from background noise), muscles of mastication (e.g., ingestion), laryngeal and pharyngeal muscles (e.g., vocalization), and head-turning muscles (e.g., social gesture and orientation). Collectively, these muscles function as filters that limit social stimuli (e.g., observing facial features and listening to vocalizations) and determine engagement with the social environment. The neural control of these muscles determines social experiences by changing facial features (especially in humans and other primates), modulating laryngeal and pharyngeal muscles to regulate intonation of vocalizations, and coordinating both facial and vocal motor tone with respiratory actions (see also Smotherman, Schwartz, & Metzner, 2010). In addition, the frequency of breathing is encoded into the phrasing of vocalizations and may convey urgency by expressing short phrases associated with short exhalations (i.e., rapid breathing) or convey calmness by expressing long phrases associated with long exhalations (i.e., slow breathing).
The source nuclei (i.e., lower motor neurons) of the special visceral efferent pathways are located in the brainstem and communicate with the source nuclei of the mammalian myelinated vagus, which functions as a neural pathway inhibiting visceral functions (e.g., slowing heart rate, lowering blood pressure, etc.) to actively reduce arousal to promote calm states consistent with the metabolic demands of growth and restoration. Direct corticobulbar pathways reflect the influence of frontal areas of the cortex (i.e., upper motor neurons) on the regulation of this system. Moreover, afferent feedback, through the sensory vagus to medullary areas (e.g., nucleus of the solitary tract) influences forebrain areas and regulates states of arousal and alertness. In addition, the anatomical structures involved in the social engagement system have neurophysiological interactions with the HPA, the neuropeptides of oxytocin and vasopressin, and the immune system (for overview, see Porges, 2001a).
PHYLOGENY OF THE MIDDLE EAR
As vertebrates evolved from reptiles to mammals, the structures at the end of the mandible (i.e., jaw bone) that define components in the middle ear became detached (Luo, 2007; Luo, Crompton, & Sun, 2001; Rowe, 1996; Wang, Hu, Meng, & Li, 2001) and formed the small bones (auditory ossicles) of the middle ear. For humans and other mammals, sound in the environment impinges on the eardrum and is transduced from the eardrum to the inner ear via the ossicles in the middle ear. Bárány (1938) argued that the main purpose of the ossicular chain in land mammals was to reduce the effect of low-frequency sounds via bone conduction. In addition to the filtering imposed by bone separation, further attenuation can be achieved when the stapedius muscle (stabilizing stapes and innervated via a branch of the facial nerve) and the tensor tympani muscle (innervated via a branch of the trigeminal nerve) are contracted. Tension reduces the compliance of the ossicular chain and dampens the amplitude of the low-frequency acoustic stimulation from the environment reaching the inner ear. This process is similar to tightening the skin on a kettledrum. When the skin is tightened, the pitch of the drum is higher. When the ossicular chain is tightened, similar to the stretched skin, the movement of the eardrum is reduced and only higher frequencies bouncing against the eardrum are transmitted to the inner ear and to the auditory processing areas of the brain. The impact of these muscles on the perceived acoustic environment is to markedly attenuate low-frequency sounds, which facilitates the extraction of sounds in a higher frequency band associated with human voice and other mammalian vocalizations.
The detachment of the middle ear bones from the mandible was paralleled by two phylogenetic shifts: (1) the jawbone no longer restricted the cranium to expand and fostered the cortical development that characterizes modern mammals (Rowe, 1996); and (2) the mammalian middle ear enabled low-amplitude relatively high-frequency airborne sounds (i.e., sounds in the frequency of vocalizations) to be heard, even when the acoustic environment was dominated by low-frequency sounds.
The evolution of the mammalian middle ear enabled mammals to communicate in a frequency band that could not be detected by reptiles that, due to a dependence on bone conduction, were able to hear predominantly lower frequencies. This ability to hear low-amplitude, high-frequency airborne sounds in an acoustic environment dominated by loud low-frequency sounds is accomplished when the middle ear muscles are tensed to create rigidity along the ossicular chain. This mechanism unmasks the high-frequency sounds associated with mammalian vocalizations from background sounds. Without stiffening the ossicular chain, mammals would lose this advantage, and the soft airborne sounds of vocalizations would be easily lost in the acoustic background (see Borg & Counter, 1989). In fact, individuals who can voluntarily contract middle ear muscles exhibit an attenuation of approximately 30 dB at frequencies below 500 Hz, while there is no or minimal attenuation at frequencies above 1,000 Hz (see Kryter, 1985).
The earliest mammals were small, and vocal communication outside the acoustic range of their predominant predators (i.e., reptiles) was crucial to survival. The physics of the middle ear resulted in a specific range of frequencies requiring a lower sound pressure level to be detected. In small mammals, this frequency band was noticeably higher than the frequency band that large reptiles, due to a dependency on bone conduction, could easily detect. However, as mammals evolved, selective pressure resulted in larger mammals surviving. The size of the middle ear's structures increased with increased body size and the resonant frequency of the middle ear became lower. Thus, with large mammals (e.g., elephants and whales) the frequencies enhanced by middle ear structures foster infrasound communication with wavelengths that can travel over long distances and overlap with the lower frequencies that reptiles detect via bone conduction.
IMPACT OF MIDDLE EAR STRUCTURES ON SENSITIVITY TO CONSPECIFIC VOCALIZATIONS
The perception of sound is not equal at all frequencies. We hear sounds at low frequencies as if they were softer than their actual physical energy. In contrast, humans are relatively accurate in estimating the acoustic energy of frequencies associated with voice. This phenomenon initially reported as the Fletcher– Munson equal loudness contours (Fletcher & Munson, 1933), illustrated how human perception attenuated the "loudness" of low-frequency sounds. As measurement technologies improved, researchers refined the perceived loudness contours, and sound meters were modified to include a scale known as dB(A), which adjusted for the perceived differences in loudness as a function of frequency (i.e., the acoustic energy of lower frequencies had to be greatly increased to be perceived at the equivalent loudness of higher frequencies). This contrasts to sound pressure level, which describes the physical energy of the signal and does not apply any perceptually based weighting to the frequencies that constitute the acoustic stimulation.
The perceptual process of detecting conspecific vocalizations in background noise, illustrates the antimasking function of the middle ear muscles (attenuating the sounds at low frequencies). In addition to the antimasking function of the middle ear muscles, the middle ear structures act as a natural amplifier and contribute to the advantage in detecting conspecific vocalizations. Amplification occurs when the acoustic energy of vocalizations is characterized by a frequency band that overlaps with the resonance frequencies of the middle ear structures. Thus, due to the selective active antimasking by the middle ear muscles and the passive amplification by the middle ear structures, there are species-specific equal loudness contours. As a general rule, conspecific vocalizations occupy this frequency band of perceptual advantage.
Dammeijer, Dijk, Chenault, Manni, and Mameren (2007) evaluated the effect of noise exposure on the stapedius muscle in the rat. Their data suggest that even in the absence of loud noise the stapedius is active, with contraction observed at sound pressure levels much lower than those needed to elicit the acoustic reflex threshold (Pilz, Ostwald, Kreiter, & Schnitzler, 1997). In addition, the data were consistent with the assumed purpose of the stapedius in unmasking high-frequency signals by attenuating low-frequency, low-level constant noise in everyday life (Pang & Guinan, 1997). The middle ear muscles are composed primarily of relatively small fast-twitch fibers (deJong, Kingma, Wirtz, Berge, & Marres, 1988). Since the muscles are characterized, at least in the rat, by relatively high anaerobic, glycolytic, and aerobic oxidative enzyme activity, there is a natural resistance to fatigue. In addition, the numerous motor end plates, in combination with the numerous axon bundles, are indicative of small motor units and support the assumption that the middle ear muscles are able to perform finely graded contractions. In our laboratory, we are demonstrating a similar intensity-tuned function of the middle ear muscles in humans.
The antimasking role of the middle ear muscles in auditory processing is especially relevant when considering the impact of low-frequency background sounds on cochlear mechanisms. Standing waves from pure tone stimuli occupy larger regions of the basilar membrane as the intensity increases, reducing the sensitivity of cochlear filtering for intense stimuli. This process is observed in the flattening of the equal loudness profile. Thus, by attenuating low-frequency sound waves, even below the level of the acoustic reflex threshold, the graded contractions of the middle ear muscles would improve the frequency sensitivity and selectivity mediated by the outer hair cells in the frequency band of mammalian vocalizations.
The physics of the middle ear structures impose other filter characteristics. Although the stiffening of the ossicular chain functions as a high-pass filter by contracting the middle ear muscles and dampening the influence of low-frequency sounds on the inner ear, the physical characteristics of the ossicular chain also influence the acoustic energy reaching the inner ear. Ossicle inertia determines the highest frequencies that can pass through the middle ear (Hemilä et al., 1995). The high-frequency limit is inversely proportional to the ossicular mass. Although, in general, larger mammals have greater ossicular mass, there are instances of larger mammals being able to detect higher frequencies due to adaptive shifts in ossicle mass to facilitate the detection of sounds associated with predator, prey, and cohort. For example, cats have excellent sensitivity at very high frequencies and may benefit from hearing high-frequency sounds produced by small rodents (Forsman & Malmquist, 1988; Rosenzweig & Amon, 1955).
THE FREQUENCY BAND OF PERCEPTUAL ADVANTAGE
In very small mammals, the middle ear and inner ear structures can convey acoustic information in a range well above the audible sounds that humans can reliably detect. The upper frequency limit for audible sounds in humans is approximately 20,000 Hz. Thus, 20,000 Hz is often used to delineate audible from ultrasound in the acoustic spectrum. However, the terminology is misleading, since acoustic stimuli characterized as ultrasonic are within the "audible" range of several mammalian species.
When audiograms of mammals are plotted on a logarithmic x-axis (for review see Fay, 1988), the lowest threshold is observed in a band of frequencies functionally defined by the high pass of the middle ear muscles and the low-pass features of the combined effect of olivary–cochlear mechanisms and the inertia of the ossicular chain. This frequency band of perceptual advantage (i.e., lower thresholds to hear) is specific to each mammalian species with the smaller mammals, in general, having an advantage to hear higher frequencies. However, since low-frequency sounds dominate most acoustic environments, this frequency band of perceptual advantage is optimized only when lower frequencies do not overwhelm the acoustic apparatus. Thus, the importance of contracting the middle ear muscles is to reduce acoustic energy in the lower frequencies. In general, it is within this frequency band of perceptual advantage that mammalian species produce most of their relevant conspecific vocalizations. For example, with the human, although the generally accepted frequencies for human hearing are between 20 and 20,000 Hz, the human frequency band of perceptual advantage includes a band of frequencies from approximately 500 Hz to about 4,000 Hz. Within these frequencies, the second and third formant in both male and female human speech always occur, and in many cases so does the first formant. This select band, which conveys the information of human voice, is functionally amplified by the antimasking mechanisms of the middle ear muscles that dampen low-frequency activity and olivary–cochlear mechanisms that dampen high-frequency activity. Similar mechanisms in the rat provide a frequency band of perceptual advantage from approximately 5 kHz to 50 kHz (see Bjork, Nevalainen, Hakumaki, & Voipio, 1999).
Auditory information can be weighted within this frequency band to improve the extraction of human speech. Two such methods are known as the "index of articulation" (Kryter, 1962) and the more recent "speech intelligibility index" (American National Standards Institute, 1997). These indices emphasize the relative importance of specific frequencies in conveying speech-related information embedded in human vocalizations. In the normal ear, acoustic energy within the primary frequencies of these indices is not attenuated, as it passes through the middle ear structures to the inner ear. The frequency band defining the index of articulation is similar to the frequency band that composers have historically selected to express melodies. It is also the frequency band that mothers have used to calm their infants by singing lullabies. Modulation of the acoustic energy within the frequencies of human voice that characterize music, similar to vocal prosody, will recruit and modulate the neural regulation of the middle ear muscles, functionally calm the behavioral and physiological state by increasing vagal regulation of the heart, and promote more spontaneous social engagement behaviors. Vocal music duplicates the effect of vocal prosody and triggers neural mechanisms that regulate the entire social engagement system with the resultant changes in facial affect and autonomic state. Basically, we start to look and feel better when we listen to melodies.
The auditory system is capable of compressing the range of acoustic stimuli. Much of this compression occurs at the periphery. For example, stapedius muscle contraction reduces sound transmission at low frequencies. Zwislocki (2002), stated that: "placement of stimulus compression in the auditory periphery must have been an important evolutionary adaptation through which the remaining system can operate within a biologically more easily achievable range" (p. 14601). As Zwislocki suggests, by compressing the energy of loud low-frequency sounds, the inner ear and higher brain structures can process the higher frequency content of the acoustic signal in the range of vocalizations. From an engineering point of view, placing a mechanism to compress the nonlinearity at the periphery of a system reduces the dynamic range required of the remaining parts of the system. This compression mechanism would function as an automatic gain control to filter the acoustic energy at frequencies lower than the vocalizations and would allow higher brain structures to extract meaning and syntax by processing the acoustic energy in this narrower frequency range.
Although humans and other mammals can vocalize outside the frequency band of perceptual advantage, the within-species social communication is usually characterized by frequency modulated vocalizations within this frequency band. In contrast, danger and pain signals may be shrill cries (i.e., high pitch with diminished frequency modulation) at the upper edge of this frequency band. In addition, aggressive signaling may push vocalizations to lower frequencies outside this band (e.g., the roar of a lion). This preference to vocalize in a social context within the frequencies most easily detected by conspecifics has clear adaptive features, but it also creates challenges. In particular, the frequencies of these vocalizations are dependent on the processing of airborne acoustic energy and are above the frequencies easily conveyed through bone conduction. The higher frequencies of airborne mammalian vocalizations (audible and ultrasound) are characterized by very short wavelengths that dissipate rapidly with distance from the source. In contrast, low frequencies have long wavelengths that travel over long distances.
The short wavelength of mammalian vocalizations evolved with convergent mechanisms to aid in adaptive social engagement behaviors (see Porges, 2007a). For many mammals, including humans, facial expressivity and behavioral gestures (e.g., use of the hands by primates) are coordinated with the shifts in prosody (intonation) to reduce ambiguity of the acoustic message (Corballis, 2003). Thus, the signals of distress and danger often require concordant facial cues and hand gestures (see also Eberl, 2010). Areas in the temporal cortex are sensitive to this cross-modal binding of auditory–visual inputs during vocalizations. Congruent speech-related visual input activates supra-additive multisensory neurons in the superior temporal cortex. In contrast, reduced activation in these areas has been reported during tasks requiring the integration of auditory and visual language inputs in schizophrenics, a disorder frequently associated with auditory hallucinations (Surguladze et al., 2001). Functionally, the simultaneous observation of facial and head movements, while listening to human vocalizations, improves speech intelligibility (e.g., McGurk & MacDonald, 1976; Munhall, Jones, Callan, Kuratate, & Vatikiotis-Bateson, 2004) and has been reported to increase the ability to extract speech from background sounds by approximately 10–20 dB (Chen & Rao, 1998; Sumby & Polack, 1954).
One of the consequences of depending on high frequencies for social communication is that infants cannot stray far from the protection of their mother. In many small mammalian species (e.g., rats, mice) the predominance of ultrasound vocalizations by infants further restricts the distance that the caregiver can move from the litter. In rats, the frequencies used to communicate change developmentally. As rat pups develop and express exploratory behaviors, vocalizations shift from infantile ultrasound to adult type communication, which may also include audible sounds (Takahashi, 1992). As the rat pups mature, well-organized mobilization behaviors support exploration, and the pups extend the distance they explore from the mother. Paralleling this shift toward adult vocalizations are increases in the neural regulation of the larynx and pharynx, structures involved in the production and articulation of vocalizations, with a parallel increase in the neural regulation of the heart via the myelinated vagus (Larson & Porges, 1982).
ADAPTIVE COST FOR ACTIVE LISTENING TO VOCALIZATIONS
There may be a cost for actively dampening the sensitivity to low-frequency sounds and engaging the neural mechanisms involved in listening to the frequency band of perceptual advantage. Listening to the frequency band of perceptual advantage requires the neural implementation of an active filter that reduces the acoustic information at low frequencies that can reach the brain. Since the sounds associated with predators, especially the movements of larger animals, are characterized by low-frequency sounds, engaging in this active "listening" process has maladaptive consequences by reducing the ability to detect predators. Thus, the advantage of listening to conspecific vocalization also comes at a cost. In the "wild" the potential cost of social communication is reduction of predator detection. The adaptive consequence of this vulnerability is to restrict or to limit listening to vocalizations within the frequency band of perceptual advantage (an important component of social engagement and social communication) primarily in safe environments such as nests and burrows.
THE SOCIAL ENGAGEMENT SYSTEM AND THE POLYVAGAL "VOCALIZATION" HYPOTHESIS
As proposed by the polyvagal theory, the functioning and development of the striated muscles of the face and head involved in listening and in production of vocalizations parallel the maturation of the myelinated vagus (see Larson & Porges, 1982). This developmental covergence, among several neural circuits that constitute an integrated functional social engagement system (see Figure 3.1, p. 56), results in facilitating several adaptive behaviors, including: (1) an improved ability to regulate physiological state via myelinated vagal pathways (i.e., vagal brake) to both self-soothe and maintain calm states, as well as to mobilize by withdrawing the vagal brake to explore, forage, and defend; (2) increased neural regulation of the larynx and pharynx to promote conspecific vocalizations in the frequency band of perceptual advantage for the species to selectively signal peers and caregivers with a vocal mechanism that produces vocalizations with the highest adaptive value; and (3) improved thermoregulatory activity that reduces the need for the caregiver, as the autonomic nervous system matures.
The polyvagal theory emphasizes a phylogenetic parallel in the changing neural regulation of the autonomic nervous system and the neural regulation of the striated muscles of the face and head. This point is relevant to the study of mammalian vocalizations, since the striated muscles of the face and head are involved in both the detection of vocalizations (during listening) and in the production of these sounds through the coordination of the laryngeal and pharyngeal muscles with respiratory mechanisms.
The convergent phylogenetic changes in the neural regulation of the structures involved in the production and detection of mammalian vocalizations leads to the polyvagal hypothesis. Specifically, only mammals have a diaphragm to coordinate vocalizations with respiratory effort and volume. Consistent with the dependence on the mammalian diaphragm is the uniquely mammalian distinction between the two branches of the vagus: one dealing with supradiaphragmatic organs and the other dealing with subdiaphragmatic organs. The neural regulation of the subdiaphragmatic vagus is involved in abdominal breathing, while the neural regulation of supradiaphragmatic vagus is coordinated with the laryngeal and pharyngeal muscles that shape the acoustic features and provide facial expressions consistent with prosodic features of vocalizations. Moreover, slow exhalation, the respiratory process associated with expressive social vocalizations, enhances the impact of the myelinated vagus on the heart, promoting calm states.
The polyvagal hypothesis proposes that acoustic characteristics of vocalization not only serve to communicate to conspecifics relevant features in the environment, but also reflect the physiological state of the producer of the vocalizations. Mammals, but not reptiles or other phylogenetic antecedents to mammals, have a myelinated vagus, a diaphragm, detached middle ears, and neural circuits in the brainstem linking and coordinating the regulation of the myelinated vagus with the regulation of the striated muscles of the face and head. Specifically, engagement of this circuit conveys and expresses states of calmness and safety and would be associated with greater vagal influences to the heart and lungs to promote calmer physiological states, increased neural tone to the middle ear muscles to optimize listening in the frequency band of perceptual advantage, and increased neural tone to the laryngeal and pharyngeal muscles to shift the acoustic energy of vocalizations to lower frequencies and increased frequency modulation within the band of perceptual advantage. In contrast, retraction of this circuit conveys and expresses states of danger and distress, and would be associated with faster heart rate and breathing and higher pitched vocalizations. Thus, in humans the features of vocal prosody are expressed during social interactions and reduced during both mental and physical illness. Similarly, the vocalizations of small mammals may have a parallel to human prosody. Rats, while playing or experiencing more positive affective states, modulate their ultrasonic vocalizations across a range of frequencies within the rat's band of perceptual advantage, while vocalizations communicating negative states such as danger are characterized by vocalizations at a relatively constant frequency through a different neural mechanism (e.g., Brudzynski, 2007).
Consistent with the polyvagal hypothesis, medically compromised human infants have a high-pitched cry with little frequency modulation that is articulated in short bursts (Lester & Zeskin, 1982; Porter, Porges, & Marshall, 1988). Intonations of infant cries are regulated by neural tone, via source nuclei in the nucleus ambiguus, to the laryngeal muscles and the heart. In the physiologically stressed infant, decreased neural tone theoretically reduces the inhibitory effect on the heart and bronchi and the contraction of laryngeal muscles, thus producing dramatic increases in both heart rate and respiration rate and the fundamental frequency of the cry.
Porter and colleagues (1988) reported a convergence between withdrawal of cardiac vagal tone (i.e., measured by respiratory sinus arrhythmia and mediated via the myelinated vagal pathways originating in the nucleus ambiguus) and the shift in fundamental frequency of newborn pain cries in response to circumcision. Cardiac vagal tone was significantly reduced during the severe stress of circumcision, and these reductions were paralleled by significant increases in the pitch of the infants' cries. These results document the important role that the social engagement system has in signaling and responding to "stress" and pain, and provide a demonstration that the vocalizations convey information regarding visceral and emotional state.
SUMMARY
The polyvagal theory emphasizes the phylogenetic shifts in the neural regulation of the autonomic nervous system and how this evolutionary shift in neural regulation converged with the regulation of the middle ear muscles to facilitate mammalian vocal communication. The theory emphasizes the different neural circuits that support defensive behaviors (i.e., fight-or-flight and freeze) and social interactions. According to the theory, during defensive states, when the middle ear muscles are not contracted, acoustic stimuli are prioritized by intensity and during safe social engagement states, acoustic stimuli are prioritized by pitch. During safe states, hearing of the frequencies associated with conspecific vocalizations is selectively being amplified, while other frequencies are attenuated. During the defensive states, the loud, low-frequency sounds signaling a predator could be more easily detected, and the soft, higher frequencies of conspecific vocalizations are lost in background sounds. During social engagement behaviors, an integrated social engagement system regulates a shift in autonomic state to dampen sympathetic activity and to increase parasympathetic tone, while simultaneously increasing the neural tone to the striated muscles of the face and head (i.e., facial expressions, increased "emotional" cueing of the eyes associated with increased eye contact, increased prosody, and enhanced listening by contracting the middle ear muscles). During social interactions, the stiffening of the ossicular chain actively changes the transfer function of the middle ear, and functionally dampens low-frequency sounds and improves the ability to extract conspecific vocalizations. However, the selectivity to listen to conspecific vocalizations comes at a cost, and the detection of lower acoustic frequencies generated by predators becomes more difficult. Thus, the identification and construction of safe contexts (e.g., burrows, nests, or houses) plays an important role in enabling the social engagement system to promote prosocial behavior.
Co-author for this chapter was G. F. Lewis.
PART IV
THERAPEUTIC AND CLINICAL PERSPECTIVES
CHAPTER 14
The Vagus: A Mediator of Behavioral and Physiological Features Associated With Autism
The vagus nerve, as a system, provides a rich organizing principle to investigate several of the behavioral, psychological, and physiologic features associated with a diagnosis of autism. The vagus is not only a cranial nerve meandering through the periphery, but also an important bidirectional conduit carrying specialized motor and sensory signals that are part of a larger integrated feedback system that includes brain structures involved in the regulation of visceral state and affect. The premise of this chapter is that several features of autism become more understandable if a more integrated model of the nervous system is applied in which the vagus is a critical component.
THE VAGUS AND AFFECT REGULATION
The relation between affect and vagal afferent activity is not a recent idea. Darwin (1872) noted in The Expression of Emotions in Man and Animals the importance of the bidirectional neural communication between the heart and the brain via the "pneumogastric" nerve, now known as the vagus nerve.
Current research emphasizes the importance of the vagal afferents in the regulation of visceral state, mood, and affect. Studies have demonstrated that stimulation of vagal afferents regulate brain structures involved in epilepsy (Boon et al., 2001), depression (George et al., 2000), and even repetitive self-destructive behaviors often associated with autism (Murphy, Wheless, & Schmoll, 2000).
POLYVAGAL THEORY: THREE NEURAL CIRCUITS REGULATING REACTIVITY
Through stages of phylogeny, mammals, and especially primates, have evolved a functional neural organization that regulates visceral state to support social behavior. The polyvagal theory (see chapters 2, 10, and 11; Porges, 2001a) emphasizes the phylogenetic origins of brain structures that regulate social and defensive behaviors, domains compromised in individuals with autism. The polyvagal theory proposes that the evolution of the mammalian autonomic nervous system provides the neurophysiologic substrates for the emotional experiences and affective processes that are major components of social behavior. The theory proposes that physiologic state limits the range of behavior and psychological experience. In this context, the evolution of the nervous system determines the range of emotional expression, quality of communication, and the ability to regulate bodily and behavioral state. The polyvagal theory links the evolution of the autonomic nervous system to affective experience, emotional expression, facial gestures, vocal communication, and contingent social behavior. Thus, the theory provides a plausible explanation of several social, emotional, and communication behaviors and disorders associated with autism.
The polyvagal construct was introduced to emphasize and document the neurophysiologic and neuroanatomical distinction between two branches of the vagus and to propose that each vagal branch is associated with a different adaptive behavioral strategy. The theory proposes that the different branches are related to unique, adaptive behavioral strategies and articulates three phylogenetic stages of the development of the mammalian autonomic nervous system. These stages reflect the emergence of three distinct subsystems, which are phylogenetically ordered and behaviorally linked to social communication (e.g., facial expression, vocalization, listening), mobilization (e.g., fight-or-flight behaviors), and immobilization (e.g., feigning death, vasovagal syncope, and behavioral shutdown). The mobilization system is dependent on the functioning of the sympathetic nervous system. The most phylogenetically primitive component, the immobilization system, is dependent on the unmyelinated or "vegetative" vagus, which is shared with most vertebrates. With increased neural complexity due to phylogenetic development, the organism's behavioral and affective repertoire is enriched (see Table 14.1).
The theory emphasizes the functional aspect of neural control of both the striated muscles of the face and the smooth muscles of the viscera, because their functions rely on common brainstem structures. It does not make any assumptions regarding structural damage to either the vagal systems or the brain structures that regulate brainstem structures associated with the vagal systems. Thus, although the compromised brainstem features described by Rodier and colleagues (1996) are consistent with the predictions of the polyvagal theory, the theory emphasizes functional deficits and does not necessarily assume structural damage.
By investigating the phylogeny of the regulation of the vertebrate heart (Morris & Nilsson, 1994), three principles can be extracted. First, there is a phylogenetic shift in the regulation of the heart from endocrine communication, to unmyelinated nerves, and finally to myelinated nerves. Second, there is a development of opposing neural mechanisms of excitation and inhibition to provide rapid regulation of graded metabolic output. Third, with increased cortical development, the cortex exhibits greater control over the brainstem via direct (e.g., corticobulbar) and indirect (e.g., corticoreticular) neural pathways originating in the motor cortex and terminating in the source nuclei of the myelinated motor nerves emerging from the brainstem (e.g., specific neural pathways embedded within cranial nerves V, VII, IX, X, and XI), controlling visceromotor structures (i.e., heart, bronchi, thymus) and somatomotor structures (muscles of the face and head).
TABLE 14.1. The Three Phylogenetic Stages of the Neural Control of the Heart Proposed by the Polyvagal Theory
These phylogenetic principles provide a basis for speculations regarding the behavioral and physiologic responses associated with autism. With this new vagal system, transitory incursions into the environment or withdrawals from a potential predator can be initiated without the severe biologic cost associated with sympathetic-adrenal activation. Paralleling this change in neural control of the heart is an enhanced neural control of the face, larynx, and pharynx that enables complex facial gestures and vocalizations associated with social communication. This phylogenetic course results in greater central nervous system regulation of physiologic state that supports behaviors needed to engage and disengage with environmental challenges.
THE VAGAL BRAKE
Due to the tonic vagal influences to the sinoatrial node (i.e., the heart's pacemaker), resting heart rate is substantially lower than the intrinsic rate of the pacemaker. When the vagal tone to the pacemaker is high, the vagus acts as a brake on the rate at which the heart is beating (see chapter 7). When vagal tone to the pacemaker is low, there is little or no inhibition of the pacemaker. Thus, neurophysiologically, the vagal brake provides a mechanism to rapidly switch between physiologic states that either support social communication or mobilization. Functionally, the vagal brake, by modulating visceral state, enables the individual to rapidly engage and disengage with objects and other individuals and to promote self-soothing behaviors and calm behavioral states. These behaviors are obviously compromised in autism. Is it possible that autism is associated with a deficit in the vagal brake and an inability to switch between neurobiologic states that foster either defensive or social behaviors?
The polyvagal theory provides an explicit neurobiologic model of how difficulties in spontaneous social behavior are linked to both facial expressivity and the regulation of visceral state. The theory proposes a possible mechanism to explain how these difficulties might form a core domain of several psychiatric profiles. Relevant to autism are the specific deficits in both the somatomotor (e.g., poor gaze, low facial affect, lack of prosody, difficulties in mastication) and visceromotor (e.g., difficulties in autonomic regulation resulting in cardiopulmonary and digestive problems) areas of the social engagement system. Deficits in the social engagement system compromise spontaneous social behavior, social awareness, affect expressivity, prosody, and language development. In contrast, interventions that improve the neural regulation of the social engagement system hypothetically should enhance spontaneous social behavior and state and affect regulation, reduce stereotypical behaviors, and improve language skills.
During the development of the human embryo, components of several cranial nerves, known as special visceral efferent pathways, develop together to form the neural substrate of a social engagement system (see chapter 11). This system, as illustrated in Figure 3.1, p. 56, provides the neural structures involved in social and emotional behaviors. The social engagement system has a control component in the cortex (i.e., upper motor neurons) that regulates brainstem nuclei (i.e., lower motor neurons) to control eyelid opening (e.g., looking), facial muscles (e.g., emotional expression), middle ear muscles (e.g., extracting the human voice from background noise), muscles of mastication (e.g., ingestion), laryngeal and pharyngeal muscles (e.g., vocalization and language), and head-turning muscles (e.g., social gesture, orientation). Collectively, these muscles function as filters that limit social stimuli (e.g., observing facial features, listening to the human voice) and determinants of engagement with the social environment. The neural control of these muscles determines social experiences. In addition, the source nuclei (i.e., lower motor neurons) of these nerves, which are located in the brainstem, communicate directly with an inhibitory neural system that slows heart rate, lowers blood pressure, and actively reduces arousal to promote calm states consistent with the metabolic demands of growth and restoration of human neurophysiologic systems.
Direct corticobulbar pathways reflect the influence of frontal areas of the cortex (i.e., upper motor neurons) on the regulation of this system. Moreover, afferent feedback through the vagus to medullary areas (e.g., the nucleus of the solitary tract, which is the source nucleus of the afferent vagus) influences forebrain areas that are assumed to be involved in several psychiatric disorders. In addition, the anatomical structures involved in the social engagement system have neurophysiologic interactions with the hypothalamic-pituitary-adrenal (HPA) axis, the neuropeptides of oxytocin and vasopressin, and the immune system (Porges, 2001a). As a cluster, the difficulties with gaze, extraction of the human voice, facial expression, head gesture, prosody, and state regulation are common features of individuals with autism. For example, the neural pathway that raises the eyelids also tenses the stapedius muscle in the middle ear, which facilitates hearing the human voice (Borg & Counter, 1989). Thus, the neural mechanisms for making eye contact are shared with those needed to listen to the human voice.
Studies have demonstrated that the neural regulation of middle ear muscles, a necessary mechanism to extract human voice from loud low-frequency background noise, is defective in individuals with language delays, learning disabilities, and autistic spectrum disorders (Smith et al., 1988; Thomas et al., 1985). Middle ear infection (i.e., otitis media) may result in a total inability to elicit the "reflexive" contraction of the stapedius muscles (Yagi & Nakatani, 1987). Disorders that influence the neural function of the facial nerve (i.e., Bell's palsy) not only influence the stapedius reflex (Ardic, Topaloglu, Oncel, Ardic, & Uguz, 1997) but also affect the patient's ability to discriminate speech (Wormald et al., 1995). Thus, the observed difficulties that many autistic individuals have in extracting the human voice from background sounds may be dependent on the same neural system that is involved in facial expression.
PREDICTIONS BASED ON POLYVAGAL THEORY
Observations of the behaviors and physiologic response of autistic individuals suggest that they have great difficulties in recruiting the neural circuit that regulates the social engagement system. It appears that autism is associated with autonomic states that remove the individual from direct social contact by supporting the adaptive defensive strategies of mobilization (i.e., fight-or-flight behaviors) or immobilization (i.e., shutdown). Behaviorally, the retraction of the neural regulation of the social engagement system is expressed as limited use and regulation of the muscles of the face and head. The functional consequences limit facial expressions and head gestures, compromise the ability to extract the human voice from background sounds, and reduce prosody.
Neurophysiologically, because the vagus is integrated into several feedback systems involving both peripheral and central structures, depression or dysregulation of the vagus might be manifested on several levels. First, it may compromise the regulation of visceral organs, such as the gut, heart, and pancreas. Second, because the vagus is involved in the modulation of pain and the regulation of cytokines and the HPA axis, there may be regulational disorders in those systems. Third, because the brainstem areas regulating the myelinated vagal system provide both output and input to feedback systems involving other brain structures, the vagal system may provide a portal to assess and stimulate higher neural processes. Although there is a limited scientific literature evaluating the role of the vagus in autism, the plausibility of these predictions are reviewed in this chapter and discussed within the context of the current literature, which includes studies with other clinical populations and animal preparations.
VAGAL REGULATION OF HEART RATE AND HEART RATE VARIABILITY
Because vagal efferent pathways to the heart are cardioinhibitory, changes in vagal tone can influence the metrics used to monitor heart rate and heart rate variability. In general, greater cardiac vagal tone produces slower heart rate and regulates the transitory changes in heart rate in response to stimulation. The myelinated vagal efferents that synapse on the sinoatrial node have a respiratory rhythm. This rhythmic increase and decrease in cardioinhibitory activity through the vagus produces a cardiac rate rhythm known as respiratory sinus arrhythmia (RSA). The greater the cardioinhibitory influence through the vagus, the greater the rhythmic increases and decreases in this heart rate pattern. Thus, the amplitude of RSA provides a sensitive index of the influence of the myelinated vagus has on the heart. The rapid changes in heart rate in response to specific stimuli are primarily under vagal control. The characteristic heart rate pattern to stimulus changes—an immediate deceleration followed by either a continued deceleration or an acceleration—is primarily due to dynamic increases or decreases in cardioinhibitory activity through the myelinated vagus. The literature suggests that autism is associated with reliable differences in the amplitude of RSA and the transitory heart rate response pattern to various stimuli and task demands.
An early publication by Hutt, Rorresst, and Richer (1975) reported that normal children suppressed RSA more than autistic children did. Similarly, Althaus et al. (1999) found that children with a pervasive developmental disorder not otherwise specified (PDD-NOS) did not suppress RSA. Consistent with these findings, an early study of children diagnosed with schizophrenia (Piggott et al., 1973) identified significant differences in respiration and in the covariation between respiration and heart rate. The schizophrenic children had significantly faster and more shallow breathing patterns, a pattern consistent with reduced vagal efferent activity.
Other studies report that autistic children have dampened transitory heart rate responses to a variety of stimulation. Zahn, Rumsey, and Van Kammen (1987) reported unusually small deceleratory heart rate responses to auditory stimulation. Palkovitz and Wiesenfeld (1980) reported dampened heart rate responses to socially relevant speech, nonsense phrases, and a 500 Hz tone. Corona et al. (1998) reported that the heart rate of children with autism did not change across conditions.
VAGAL NERVE STIMULATION
Although not currently being used to treat autism, vagal nerve stimulation has been effective in treating epilepsy and depression. Vagal nerve stimulation is based on the assumption that stimulation of vagal afferents has a direct effect on the regulation of higher brain structures. The source nucleus of the vagal afferents is the nucleus of the solitary tract. This medullary nucleus plays an important role in the regulation of behavioral state, respiration, and blood pressure, and in conveying information to higher brain structures. The nucleus of the solitary tract relays the incoming sensory information via three primary pathways: (1) feedback to regulate the periphery, (2) direct projections to the reticular formation in the medulla, and (3) ascending projections to the forebrain, primarily through the parabrachial nucleus and the locus ceruleus. The parabrachial nucleus and the locus ceruleus send direct connections to all levels of the forebrain (e.g., hypothalamus, amygdala, the thalamic regions that control the insula and orbitofrontal and prefrontal cortices), areas that have been implicated in neuropsychiatric disorders. Thus, vagal afferent stimulation has direct input to both the lower motor neurons in the brainstem and the upper motor neurons in the cortex that regulate the social engagement system. Recent reviews provide a detailed description of the neurophysiologic basis for the intervention (George et al., 2000) and provide an explanation of the neural mechanisms involved in treating depression with vagal nerve stimulation (Marangell et al., 2002). Missing from these explanations is an acknowledgment of the communication between vagal afferents and the source nuclei of the nerves that regulate striated muscles of the face and head (i.e., special visceral efferent pathways), which collectively form the motor part of the social engagement system. It is this interaction that is emphasized in the polyvagal theory (Porges, 2001a).
Extrapolating from the vagal nerve stimulation model, one might speculate that other forms of vagal stimulation might have beneficial effects. Behaviorally, one of the most potent strategies for vagal stimulation is to stimulate the peripheral baroreceptors that regulate blood pressure. Rocking and swinging, in which the position of the head is changed relative to the position of the heart, will stimulate the baroreceptors and engage this feedback loop. This suggests that the frequently observed rocking and swinging behaviors in autistic individuals may reflect a naturally occurring biobehavioral strategy to stimulate and to regulate a vagal system that is not efficiently functioning.
One publication reported that vagal nerve stimulation reduced autistic-like behaviors (Murphy et al., 2000). In this study, vagal stimulation was administered to six patients with hypothalamic hamartoma, a congenital brain malformation that is associated with medically refractory epilepsy and injurious autistic behavior. Four of the six patients had autistic behaviors that included poor communication, ritualisms, compulsions, no social skills, and injury to self and others. The authors report that during vagal nerve stimulation, all four showed impressive improvements in behavior. In one subject, the behavioral improvements were immediately reversed when the vagal nerve stimulation was temporarily discontinued without worsening of seizure frequency.
VAGAL REGULATION OF THE GUT
Due to the high prevalence of gastrointestinal symptoms in individuals with autism (Horvath & Perman, 2002; Wakefield et al., 2002), there has been an interest in a possible link between gut and brain as a determinant of autism. This interest was stimulated by reports from parents who indicated that the administration of intravenous secretin reduced autistic symptoms. However, there has been no evidence for the efficacy of secretin when it was administered in a randomized, placebo-controlled, double-blind clinical trial (Owley et al., 2001).
Current research suggests that the prevalence of gastrointestinal symptoms represents an unsolved problem in autism. However, if we conceptualize the problem from a "vagal" perspective, we can identify the vagus as a primary regulator of the gut, with vagal afferents providing important information to brain structures. Support for this argument comes from animal studies in which it has been demonstrated that the vagus is involved in the regulation of secretin (Li, Chang, & Chey, 1998; Lu & Owyang, 1995). Thus given the compromised behavioral components of the social engagement system, it is not surprising to find that the vagal regulation of gastrointestinal processes is also compromised in autistic individuals.
Additional information regarding the role that vagal afferents from the gut have in modulating sensory experiences comes from research on eating disorders. Research suggests that vagal afferents are involved not only in regulating satiety via vagovagal reflexes but also in regulating nociceptive sensations via solitary-spinal pathways. Faris et al. (2000) and Raymond, Eckert, et al. (1999) have proposed that vagal afferent activation by binge eating and vomiting also activates the descending pain inhibitory pathway, resulting in an elevated pain threshold. Similarly they have reported elevated pain thresholds in anorexia nervosa subjects (Raymond, Faris, et al., 1999). Their research has led to administering ondansetron as an intervention for bulimia nervosa (Faris et al., 2000). Ondansetron is marketed for the prevention of vagally mediated emesis caused by cancer chemotherapeutic agents.
THE VAGUS AND THE IMMUNE SYSTEM
The subdiaphragmatic vagal afferents may be conceptualized as providing a targeted signal to the central structures that regulate immune function. Other researchers have linked the vagal efferent pathways to immune function. Bulloch and Pomerantz (1984) described motor pathways via the vagus to the thymus. The link between the vagal regulation of immune function and the polyvagal theory is not clear. However, it might be plausible to speculate that the neural mediation of the myelinated vagus may, via direct influence on thymus and direct inhibition of the sympathetic nervous system, trigger a physiologic state that would promote immune function. Likewise, mobilization strategies, resulting in a withdrawal of vagal tone to the heart, increased sympathetic tone, and the release of cortisol, have been associated with suppressed immune function. More relevant to the expression of psychiatric disturbances is the finding that the afferent vagus mediates behavioral depression, but not fever, in response to peripheral immune signals following abdominal inflammation (Konsman et al., 2000). Consistent with this model, it has been reported that autism spectrum disorder patients with developmental regression express excessive innate immune responses (Jyonouchi, Sun, & Le, 2001).
VAGAL REGULATION OF THE HPA AXIS
The vagus is involved in the regulation of the HPA axis. Vagal afferents exhibit an inhibitory influence on HPA axis and reduce cortisol secretion (e.g., Bueno et al., 1989; Miao et al., 1997). Studies (Cacioppo et al., 1995; Gunnar et al., 1995) have demonstrated a covariation between increases in cortisol and decreases in cardiac vagal tone (i.e., the amplitude of RSA). Thus, there appears to be a coordinated response that functions to promote metabolic activity and mobilization behaviors by withdrawing the vagal tone through the myelinated vagus and increasing both sympathetic activity and activation of the HPA axis.
Several studies have reported that the regulation of the HPA axis is dysfunctional in autistic children. Poorly developing autistic children were more likely to have an abnormal diurnal rhythm and an abnormal response on the dexamethasone suppression test than less severe cases (Hoshino et al., 1987). The results suggest that the negative feedback mechanism of the HPA axis may be disturbed in autistic children, especially in poorly developing individuals. Similarly, Jensen, Realmuto, and Garfinkel (1985) reported that most of the autistic patients studied failed to suppress cortisol with the dexamethasone test. Consistent with these reports, Jansen et al. (2000) reported the PDD-NOS children had a diminished cortisol response to physical stress.
THE VAGAL SYSTEM AS AN ORGANIZING PRINCIPLE
In this chapter, I have illustrated how the vagus is involved in the expression of several disparate symptoms associated with autism. Consistent with the polyvagal theory, the symptom clusters are associated with components of the vagal system. First, there are the behavioral characteristics linked to the neural regulation of the striated muscles of the face via special visceral efferent pathways (i.e., the somatomotor component of the social engagement system). Second, autism is associated with dysfunctional regulation of target organs (e.g., heart, gut) regulated by vagal efferent pathways (i.e., the visceromotor component of the social engagement system). Third, the vagal afferents exert a powerful regulatory influence on several systems—including visceral and tactile pain thresholds, the HPA axis, and the immune system—that are dysfunctional in autism. Fourth, the nucleus of the solitary tract (the source nucleus of the afferent vagus) influences areas of the forebrain that have been speculated to be compromised in autism.
CHAPTER 15
Borderline Personality Disorder and Emotion Regulation
The concept of a "borderline personality disorder" dates back to the early 1800s, when clinicians were unsure of the diagnosis of patients who displayed a combination of neurotic and psychotic symptoms. Since clinicians viewed these patients as being on the "border" between neurotic and psychotic, the borderline personality disorder (BPD) evolved as a diagnostic category and was listed as an Axis II diagnosis in 1980, with the publication of DSM-III (Hodges, 2003). The current DSM-IV-TR emphasizes that patients with BPD express symptoms that include affective instability, intense and tumultuous relationships, difficulty controlling anger, impulsivity, suicidal tendencies, and self-mutilation (American Psychiatric Association [APA], 2000; Rothschild, Haslam, Cleland, & Zimmerman, 2003). This cluster of symptoms indicates that BPD is associated with difficulty in regulating emotions, behavioral states, and relationships. BPD is a severe mental disorder that is more prevalent in women and is believed to impact approximately 2% of the population (APA, 2000; Hodges, 2003; Swartz, Blazer, George, & Winfield, 1990; Torgersen, Kringlen, & Cramer, 2001) and approximately 20% of the hospitalized psychiatric patients (Zanarini & Grankenbrug, 2001).
Because BPD is associated with problems in regulating emotions and responding appropriately to daily life events, BPD has been linked to a wide variety of poor outcomes including job-related problems (Zweig-Frank & Paris, 2002), dysfunction in developing strong personal relationships (Daley, Burge, & Hammen, 2000), social maladjustment, and reduced academic achievement (Bagge et al., 2004). Due to the breadth and severity of these frequently observed problems, BPD has been difficult to treat effectively (Hoffman, Buteau, Hooley, Fruzzetti, & Bruce, 2003).
The high correlation reported between BPD and past sexual abuse and family dysfunction (Weaver & Clum, 1993) has led to the developmental hypothesis that BPD may develop as a result of traumatic experiences early in life. Other adverse events, such as abandonment or fear of abandonment and lack of a secure emotional attachment with a caregiver, often accompany BPD (Benjamin, 1996; Gunderson, 1996). Furthermore, BPD is highly comorbid with other mood and anxiety disorders (Hodges, 2003; Skodol, Gunderson, et al., 2002; Weaver & Clum, 1993).
Despite the prevalence and severity of BPD, few studies have investigated the underlying neurological and physiological mechanisms of the disorder (e.g., Schmahl et al., 2004). Coccaro and Kavoussi (1991) suggested that an improved understanding of the neurological and physiological mechanisms mediating the clinical symptoms of BPD might lead to the development of more effective treatments. During the past decade, research has begun to identify specific neurobiological features that differentiate between BPD and controls. These features might provide clues to the mechanisms mediating the difficulties in emotion regulation experienced by individuals diagnosed with BPD.
Since impulse control is a characteristic deficit associated with BPD, dysfunction of the prefrontal cortex has been hypothesized to be a mediator of BPD. This speculation is based on observations of increased impulsivity following brain damage in prefrontal areas (Blair & Cipolott, 2000). Consistent with this hypothesis, individuals with BPD perform poorer on a go/no-go task, a test of impulse control assumed to evaluate prefrontal function (Völlm et al., 2004). In addition, volumetric studies applying imaging techniques have found smaller frontal lobes in BPD participants (Lyoo, Han, & Cho, 1998).
Imaging has also identified in individuals with BPD anomalies in limbic structures implicated in emotion regulation, such as smaller hippocampal and amygdala volumes (Tebartz van Elst et al., 2003). Volumetric reductions, especially in the hippocampus, are thought to be caused by the excessive stress that BPD patients experience (Schmahl, Vermetten, Elzinga, & Bremner, 2003). Because the hippocampus and the amygdala are involved in the processing of and responding to emotional stimuli (Anderson & Phelps, 2000; Nolte, 1993), a consequence of volumetric reductions might be related to the difficulties in emotion regulation that BPD individuals experience.
Other neurophysiological systems mediating processes such as emotion regulation, impulsivity, and aggressive behavior have been studied. Abnormalities in serotonin, a neurotransmitter linked to aggression, impulsivity, and suicidal behavior (Coccaro, 1989) have been reported in individuals diagnosed with BPD (Hansenne et al., 2002; New & Siever, 2002; Paris et al., 2004; Skodol, Siever, et al., 2002). BPD may be associated with a hyperresponsiveness of the hypothalamic-pituitary-adrenal (HPA) system (Rinne et al., 2002), a system implicated in stress responses, anxiety, and emotional reactivity. These findings provide limited evidence that a dysfunction in systems involved in controlling reactivity and emotion accompanies BPD.
Because several features of BPD are related to difficulties in regulating behavioral state and emotional reactivity, measurement of the autonomic nervous system might provide a portal into understanding the neural mechanisms of this disorder. Thus, it might be hypothesized that: (a) the sympathetic component of the autonomic nervous system, which supports fight-or-flight behaviors, would be hyperaroused; and (b) the parasympathetic component, which supports calm visceral states and social engagement behaviors, would be depressed. Previous research (for detailed review, see Herpertz, Kunert, Schwenger, & Sass, 1999; Schmahl et al., 2004) contrasted physiological responses regulated by the sympathetic nervous system in individuals with BPD and controls. Herpertz et al. (1999) monitored heart rate, skin conductance, and startle responses in a paradigm varying the emotional valence (pleasant, neutral, and unpleasant) and intensity of visual stimuli. Schmahl et al. (2004) measured heart rate, skin conductance, and blood pressure in response to reminders of personal experiences of severe stress (i.e., abandonment, trauma). Neither study found evidence of sympathetic hyperarousal associated with a diagnosis of BPD. However, both studies did not monitor the parasympathetic component of the autonomic nervous system or expose BPD participants to dynamically changing emotional stimuli (e.g., film clips).
The phylogenetic model of the autonomic nervous system described in the polyvagal theory (see chapters 2, 10, and 12; Porges, 2001a), provides an innovative theoretical framework to study the potential involvement of the parasympathetic nervous system in BPD. The theory focuses on the role that autonomic state plays in mediating both prosocial and defensive behaviors. The theory emphasizes an integrated social engagement system that regulates the muscles of the face and head involved in social engagement behaviors (e.g., gaze, expression, prosody, and gesture) and a component of the parasympathetic nervous system, the myelinated vagal pathways to the heart that calm visceral state and dampen sympathetic and HPA activity. The polyvagal theory emphasizes how neural circuits involved in the regulation of autonomic state evolved to support various adaptive biobehavioral responses to challenges. The theory proposes that autonomic reactions to challenges follow a phylogenetically ordered hierarchy with three distinct adaptive biobehavioral strategies. Each biobehavioral strategy reflects a specialized neurophysiological substrate that evolved to maximize adaptive strategies in safe, dangerous, or life-threatening contexts. Within this model the nervous system, through a process of "neuroception," is continuously evaluating risk and safety in the environment. Neuroception is not a conscious process; rather, it occurs via unconscious subcortical systems that functionally trigger one of these three adaptive neural circuits. Therefore, based on the polyvagal theory, difficulties in emotional regulation that are associated with a diagnosis of BPD could be interpreted as a behavioral expression of a physiological state that has evolved to support defensive strategies in dangerous and life-threatening situations. According to the polyvagal theory, the myelinated vagus, which phylogenetically evolved with mammals, is critical for two rea-sons: to inhibit defensive limbic circuits, and to establish social bonds (see chapter 12).
Phylogenetically, as mammals expressed special visceral efferent pathways to regulate the striated muscles of the face and head (e.g., facial expressions, head gesture), there was a parallel shift in the neural regulation of the heart from an unmyelinated to a myelinated vagus. This new myelinated (i.e., mammalian) vagus actively inhibits the sympathetic nervous system's influence on the heart and dampens HPA axis activity (Porges, 2001a). The mammalian vagus functions as an active vagal brake (see chapter 7) to maintain calm states in social contexts. However, when risk is detected, the vagal brake can be rapidly withdrawn to support defensive mobilization behaviors. Thus, BPD might be associated with difficulties in regulating the vagal brake in social settings.
The mammalian heart is characterized by a relatively strong vagal influence, via the myelinated pathways, on the heart's pacemaker (i.e., sinoatrial node). Functionally, the impact of the vagal brake produces a baseline or resting heart rate substantially lower than the intrinsic rate of the pacemaker. When the vagal brake is removed, heart rate can approximate the intrinsic rate of the pacemaker without recruiting sympathetic influences. When cardiac vagal tone via the myelinated vagus is high, the vagus acts as a restraint or brake limiting the rate the heart is beating. When vagal tone to the pacemaker is low, there is little or no inhibition of the pacemaker. Thus, the vagal brake may be used as a construct to describe the functional modulation of heart rate by the myelinated vagal efferent pathways.
The vagal brake provides a neural mechanism to change visceral states by slowing or speeding heart rate. Neurophysiologically, the influence of the vagal brake is reduced or removed to support the metabolic requirements for mobilization (e.g., fight-or-flight behaviors) and maintained or increased to support social engagement behaviors. The amplitude of respiratory sinus arrhythmia (RSA) indexes the state of the vagal brake. RSA is a natural rhythm in the heart rate pattern at approximately the frequency of spontaneous breathing. The amplitude of RSA provides a sensitive index of the impact that the myelinated vagus has on the heart (see chapter 2). By quantifying RSA during various challenges, it is possible to measure the dynamic regulation of the vagal brake.
It was hypothesized that the BPD participants, unlike typically behaving individuals, will have difficulties maintaining the vagal brake in social contexts. Thus, in response to social stimuli, BPD participants should rapidly shift from a calm state (i.e., high-amplitude RSA) to a state of agitation (i.e., low-amplitude RSA). To test this hypothesis, we contrasted the regulation of the vagal brake by measuring the amplitude of RSA in participants with BPD and controls during the presentation of film clips assumed to reflect emotional content. In addition, the experiment, by requiring an interaction between the participant and the experimenter, provides a secondary context related to social interactions.
METHOD
Participants
Participants (all female) consisted of 9 borderline personality disorder (BPD) patients and 11 controls between the ages of 18 and 45. Only females were recruited, since females represent the majority of individuals diagnosed with BPD and to remove gender as a possible source of variance in the neurophysiological regulation of autonomic state. The groups were equivalent in education level and age. BPD participants were referred to the study by clinicians in the Washington, DC, area. The BPD participants were identified and screened to eliminate comorbid diagnoses by National Institute of Mental Health clinical researchers at St. Elizabeth's Hospital (Washington, DC). The control group consisted of volunteers who were recruited from lists maintained by the National Institutes of Health. Both groups were free of drug and alcohol abuse and were not current users of prescription or illicit drugs. The diagnosis of BPD, based on DSM-IV criteria (APA, 1994), was confirmed by the Structured Clinical Interview for DSM-III-R (Spitzer, Williams, Gibbon, & First, 1990, 1992) and the Diagnostic Interview for Borderlines (Gunderson, Kolb, & Austin, 1981). The BPD participants were tested at St. Elizabeth's Hospital. The BPD participants, while participating in the study, were off medication and as a precautionary procedure were in residence at the hospital. Controls were free of psychiatric or neurological disorders. The control participants were not in residence at the hospital and were tested either at the hospital or at the Developmental Assessment Laboratory (University of Maryland, College Park) in similar testing environments.
Procedure
Following informed consent, participants were seated in a quiet room facing a television screen. To monitor electrocardiography (ECG), from which the heart period and RSA were derived, three disposable Ag–AgCl electrodes were placed on the participant's chest, connected to an ECG amplifier, and output to a Vagal Tone Monitor-II (Delta-Biometrics). After the initial baseline period, participants were instructed to watch three 10-min film clips. Each film clip was followed by questions specific to the film clip just viewed. Physiological data were collected during each of the four 10-min segments (i.e., baseline, film clip 1, film clip 2, film clip 3). To facilitate the measurement of a stable indicator of autonomic state in response to the specific emotional content of each film clip and to allow for a transition in physiological state due to the onset of each film clip, the last 5 minutes of data within each of the four conditions were analyzed. The experiment lasted approximately 1 hour. Heart period and RSA were collected for the baseline and film clip conditions. Participants were instructed to minimize their movements during the experiment. The experimenter remained in the room with the participant during the experimental procedures, since BPD is associated with an inability to tolerate being alone (see Gunderson, 1996). While in the experimental room, the experimenter ran the equipment and asked questions about each film clip.
Film clips 1 and 3 were selected to elicit a strong emotional response. Film clip 1 was a conflict scene with the mother in Frances, and film clip 3 was a conflict scene with the father in The Great Santini. In contrast to these two conflict scenes, film clip 2 was selected to be a neutral scene from A Handful of Dust. Participants rated the film clips on a Likert scale ranging from 0 (not arousing) to 10 (extremely arousing). Participants' ratings confirmed the assumed emotional content of each film sequence. The first conflict scene had a mean rating of 7.25 and the second conflict scene had a mean rating of 9.15. In contrast, the neutral scene had a mean rating of 1.75. The conflict scenes were rated as significantly more arousing than the neutral scene (p < .001). There were no significant differences in ratings between groups.
Data Quantification
The Vagal Tone Monitor detected the peak of the R-wave with 1-msec accuracy, timed sequential heart periods to the nearest ms (Riniolo & Porges, 1997), and stored the heart periods in files for offline analyses of RSA and heart period. The data files of sequential heart periods (i.e., R–R intervals in msec) were input into MXedit (version 2.21) software (Delta-Biometrics) in order to edit outlier data produced by movement and digitizing error. Editing consisted of integer addition or division of sequential values.
RSA estimates were calculated using the following procedures (Porges, 1985): (a) the heart period time series were converted to time-based data by resampling at successive 500-msec intervals; (b) a 21-point moving cubic polynomial filter was stepped through the time-sampled series to produce a smoothed template series; (c) the template series was subtracted from the original series to produce a residual time series; (d) the residual time series was processed by a digital bandpass filter with 25 coefficients to extract the variance in the frequency band of 0.12–0.40 Hz (i.e., the frequency of spontaneous breathing for adults); and (e) the bandpassed variance was transformed to its natural logarithm and used to quantify RSA. Data from our laboratory (Denver, Reed, & Porges, 2007) demonstrate that this methodology accurately captures the heart rate variability associated with spontaneous breathing. Denver et al. (2007) report a correlation of 0.99 between the frequency observed in the heart rate and respiration spectra. Moreover, the amplitude of the heart rate periodicity derived from spectral analysis was correlated 0.99 with the values derived from the MXedit analysis. These procedures result in a sensitive, noninvasive marker of the influence of the myelinated vagal fibers on the heart (see chapter 2; Porges, 2001a).
Analyses
Group (Borderline, Control) by Condition (Baseline, Film 1, Film 2, Film 3) analyses of variance (ANOVA) were used to identify statistical effects for RSA and heart period. Data from one participant during the baseline condition were lost due to technical error. Data from this participant were analyzed during the experimental conditions. Heart period was calculated as the time interval in msec between successive R-waves of the ECG. The metric for the current analyses represents the average R–R interval in msec for each condition. To evaluate the vagal contribution to heart period changes during each of the film clips, correlations between the changes from baseline in heart period and RSA were calculated. High correlations would illustrate that the changes in both variables were mediated by a common mechanism (i.e., vagal regulation of the heart). If the heart period change is totally dependent on vagal regulation, the correlation with RSA should approach 1.0. In contrast, low correlations would suggest that heart period is not tightly regulated by the vagus and would be mediated by other mechanisms.
RESULTS
Across all conditions there was a significant group effect for RSA, F(1,77) = 7.16, p < .05. The control group had significantly higher RSA than the BPD group. This main effect was functionally determined by the group trajectories during the experiment and statistically represented in the group × condition interaction, F(3,51) = 3.62, p < .05. As illustrated in Figure 15.1a, the trajectory of RSA during the experimental session differed between the groups. The values for RSA were similar for the two groups during baseline. However, over the course of the experiment, the control participants exhibited an increase in RSA, and the BPD participants exhibited a decrease in RSA. These distinct patterns reflect different neural strategies. The BPD participants exhibited a vagal withdrawal, which would support the increased metabolic demands of fight-or-flight behaviors. The control participants exhibited an increase in vagal influences to heart, which would support social engagement behaviors. As illustrated in Figure 15.1b, there was a significant group × condition interaction for heart period, F(3,51) = 6.49, p < .05. The heart period response pattern during the experiment confirms the observed group specific shifts in cardiac vagal tone. The BPD group progressively exhibited shorter heart periods (i.e., faster heart rate), while the control group exhibited longer heart periods (i.e., slower heart rate). Simple effects tests (see Table 15.1) confirm that at the end of the experiment, group differences were pronounced for both variables. Consistent with the RSA data, across conditions there was a group effect for heart period, F(1,77) = 14.2, p < .05.
The vagal contribution to heart period changes during the experiment was estimated by correlating the change from baseline in heart period with change from baseline in RSA. Correlations were calculated within each group to evaluate whether the observed heart period change from baseline to a specific film clip were under vagal control. As illustrated in Figure 15.2, the relations between changes in RSA and heart period are significantly correlated only for the control group. Thus, the changes in heart period are strongly linked to an increase in the vagal control of the heart, but only in the control group. In the BPD group, although there is a consistent vagal withdrawal, the changes in RSA are not sufficient to account for the changes in heart period. These findings demonstrate that BPD participants have poor vagal regulation and the changes in heart period may be due, in addition, to other influences such as sympathetic activation.
FIGURE 15.1. Means (± SE) for respiratory sinus arrhythmia in ln msec2 units (a) and heart period in msec (b) across test conditions.
TABLE 15.1. Simple Effects for ANOVA
DISCUSSION
Previous investigations of the autonomic nervous system of BPD participants focused on indices of the sympathetic nervous system. These studies did not identify differences between BPD and controls (Herpertz et al., 1999; Schmahl et al., 2004). In contrast, by focusing on the parasympathetic limb of the autonomic nervous system, the current study provides the first report of unique characteristics of autonomic regulation associated with a diagnosis of BPD.
Based on the limited literature studying the psychophysiology of post-traumatic stress disorder (e.g., Sahar, Shalev, & Porges, 2001), we assumed that BPD patients and controls would have similar levels of RSA during baseline. This assumption was confirmed. By challenging the participants with film clips of both high (clips 1 and 3) and low (clip 2) emotional content, the experiment was designed to elicit a withdrawal of the vagal brake (i.e., lower RSA and shorter heart period) in both groups during high-emotion stimuli with an expectation that the vagal brake would be reinstated (i.e., increases in RSA and heart period) during the low-emotion stimulus. However, we anticipated that the BPD participants, due to their sensitivity to affective stimuli and difficulty in state regulation, would exhibit an exaggerated withdrawal of the vagal brake (i.e., lower RSA and shorter heart period). Interestingly, we found that neither group exhibited a film clip–related response. Instead as illustrated in Figure 15.1a and b, the control group exhibited a trajectory of increasing vagal influences on the heart across the entire duration of the experiment, while the BPD group exhibited decreasing vagal influences on the heart.
At the start of the experiment, vagal influences on the heart were similar in both groups. However, as the experiment progressed, there was an intriguing divergence between the groups. The polyvagal theory provides two insights into the adaptive nature of the physiological state for each group as the experiment progressed. First, the polyvagal theory emphasizes that the physiological state that characterizes each group at the end of the experiment supports different classes of behavior. For the BPD group, the physiological state, characterized by a vagal withdrawal, would support the mobilization behaviors of fight or flight. In contrast, for the control group, the physiological state, characterized by increased vagal influence on the heart, would support spontaneous social engagement behaviors. Interestingly, embedded in the diagnosis of BPD are features related to compromised social engagement behaviors, including a hyperreactivity or emotional dysregulation during social interactions. Thus, although the experimental conditions provided the same context and task demands for all participants, the BPD group reacted with a visceral state to promote defensive behaviors, while the control group reacted with a visceral state to promote increased spontaneous social engagement behaviors. Second, the polyvagal theory (see chapter 1) proposes a mechanism (i.e., neuroception) that triggers defense strategies. Based on the clinical features of BPD, social interactions with a stranger (i.e., the experimenter) might provide a social stimulus that would trigger the nervous system and elicit a physiological state supporting fight-or-flight behaviors. However, the film clips in this experiment, with their depictions of social interactions, might have contributed to the need to mobilize. It would be interesting to see whether in the absence of a person in the experimental room, the physiological states of the two groups would still diverge during the film clips. Perhaps the physiological states of both groups would have been more similar and more stable throughout the experiment in the absence of the experimenter. Under these conditions, the BPD participants might not detect risk and elicit a defensive strategy and the control participants would not detect the cues from the experimenter to trigger spontaneous social engagement behaviors.
FIGURE 15.2. Correlations of change from baseline in heart period and change from baseline in RSA from film clips 1, 2, and 3; *p < .05.
The correlations between RSA and heart period provide additional support for the hypothesis that in individuals without psychiatric disturbances, there is a strong link between vagal regulation and the control of heart period. In previous research, we have reported that this covariation is compromised in other psychiatric disorders that have difficulties in regulating behavior (Sahar et al., 2001; Umhau et al., 2002). The lack of correlation between RSA and heart period for the BPD group is consistent with these studies.
It is possible that the neuroception of the environment provides an invalid indicator of risk for individuals with BPD. Thus, rather than being calm in the presence of another "nonthreatening" human, the autonomic nervous system is regulated to a state that supports fight or flight and not a state that would support spontaneous social engagement. If invalid neuroception related to social stimuli accurately describes individuals with BPD, it may partially explain why BPD patients have a variety of poor outcomes, especially related to social relationships and emotional instability in real-world situations.
This study has provided the first documented evidence of autonomic nervous system differences between controls and individuals diagnosed with BPD. Thus, it offers a theoretical framework to explain the emotional reactivity that is linked to BPD. The study, however, has several limitations, including small sample size, isolated experimental manipulation, and no measures of test–retest reliability. However, this investigation into the autonomic response profile of BPD patients may lead to additional research to confirm our hypotheses relating autonomic state and neuroception in a variety of other social challenges. Thus, monitoring RSA during social challenges may provide a measurable index of neuroception in BPD and provide a method to delineate the features in the social environment that would trigger autonomic states that support and promote defensive behaviors.
Co-authors for this chapter were M. A. Austin and T. C. Riniolo.
CHAPTER 16
Abuse History Is Related to Autonomic Regulation
Many individuals experience abuse, either in childhood or adulthood, and those who experience childhood abuse are more likely to experience adult abuse (e.g., Desai, Arias, Thompson, & Basile, 2002). Women are extremely vulnerable to victimization (Bremner & Vermetten, 2001), and the abuse is also more likely to occur by those closest to the victim. As in child abuse, the various types of adult abuse (e.g., domestic violence, sexual abuse, and emotional abuse) may occur in combination. Given the great overlap, it is difficult to investigate the effects of a particular type of child or adult abuse. Research suggests that history of childhood sexual abuse, an experience which occurs in about 16% of women, is the most common cause of post-traumatic stress disorder (PTSD) in women (Bremner & Vermetten, 2001).
It is important to note that women may experience abuse, including sexual abuse, without the severe debilitating features of PTSD. These women may present with symptoms that may or may not be consistent with other psychiatric diagnoses. Victims of abuse may develop mood disorders (e.g., Zavaschi et al., 2006), including depression (e.g., Schuck & Widom, 2001). The depressive symptoms may be related to self-esteem issues common in victims of abuse (e.g., Arata, Langhinrichsen-Rohling, Bowers, & O'Farrill Swails, 2005). Victims of abuse may also employ maladaptive coping mechanisms, which may be related to paying inordinate attention to internal stimuli and distorting environmental cues (Rothschild, 2000). These maladaptive coping strategies may initially be functional but, in the long term, result in detrimental consequences such as substance abuse, eating disorders, and suicide (Doyle, 2001).
Little is known about the effects of abuse on the victim's bodily responses to stressful events. Studies investigating the influence of abuse on the autonomic nervous system have generally focused on participants with chronic PTSD. The results of these studies have been inconsistent (Buckley & Kaloupek, 2001). For example, reported differences in baseline heart rate might not reflect a reliable difference in autonomic state but may be influenced by other psychologically related factors, including the laboratory setting, the experimental paradigm, and the psychological state (e.g., anticipatory anxiety) in which the participant was tested (Prims, Kaloupek, & Keane, 1995). In addition, supporting the idea that PTSD is associated with a sympathetic overdrive (Blanchard, 1990) studies have reported heightened physiological reactivity to stimuli related to traumatic events (e.g., Elsesser, Sartoy, & Tackenberg, 2004). Thus, assessment of cardiovascular regulation variables during both baseline and following a stressor may provide useful information in determining whether trauma history impacts on the physiological response strategies available to react and to calm in normal nonthreatening situations.
Even if the abuse does not result in PTSD, it may impact on day-to-day experiences and influence the development of social relationships. A history of abuse may "tune" the nervous system to be cautious and prepared for defensive fight-or-flight behaviors, even when real danger does not exist. Since the metabolic demands of exercise require a physiological adjustment that mimics physiological states associated with flight-or-fight behaviors, it is possible that abuse history may influence the neural circuits involved in calming following exercise. By monitoring heart rate patterns in response to mild exercise, it may be possible to evaluate an individual's ability to self-regulate and to promote a calm state following exercise.
The quantification of respiratory sinus arrhythmia (RSA) provides an opportunity to dynamically monitor the changing vagal regulation of the heart (see chapter 2) during recovery from exercise. Vagal regulation of the heart is associated with self-regulated behaviors. Optimally, during challenges demanding mobilization there is a rapid withdrawal of the vagal inhibition on the heart to raise heart rate to support the increased metabolic demands, and immediately following exercise there is a recovery of vagal tone to foster calm states. This rapid regulation of the vagus in support of social engagement behaviors has been labeled the vagal brake (see chapter 7).
Deficits in vagal regulation are present in perpetrators of violent abuse (Umhau et al., 2002) and are related to a variety of psychiatric disorders such as PTSD (Sack, Hopper, & Lamprecht, 2004; Sahar, Shalev, & Porges, 2001), generalized anxiety disorder (McLeod, Hoehn-Saric, Porges, Kowalski, & Clark, 2000), and depression (Rottenberg, Solomon, Gross, & Gotlib, 2005). There is an extensive literature demonstrating that vagal regulation of the heart, both in terms of level of cardiac vagal tone and the efficient regulation of the vagal brake, is related to processes such as emotional reactivity, social engagement, and reactions to stress.
Victims of abuse have state regulation difficulties with a bias toward behavioral states that are self-protective. This potential vulnerability to be defensive may result in difficulties in feeling safe with others and in developing trusting social relationships. A self-awareness of difficulties in regulating state, especially staying calm in the presence of others, may lead individuals to seek alternative strategies, such as yoga. Yoga exercises may help reduce symptoms of depression and anxiety, increase a sense of self-efficacy (Lee, Mancuso, & Charlson, 2004), and improve regulation of the autonomic nervous system (Sovik, 2000). Thus, individuals, who practice yoga, may be exercising their autonomic nervous system in an attempt to normalize an abuse-related damage to their ability to self-regulate.
Practitioners of yoga classes served as a forum to recruit participants for the current study. We hypothesized that women with a history of abuse, but without a diagnosis of PTSD, would exhibit greater physiological coping difficulties characterized by lower RSA and less RSA recovery following mild exercise. Additionally, we hypothesized that abuse history would be related to a greater use of dysfunctional coping strategies, increased mood disturbances, and lower self-concept.
METHOD
The analyses are based on 49 female participants, who were recruited from a local yoga studio. Forty-five participants described themselves as Caucasian, one as Hispanic, and three as multiracial. The majority (69.4%) were in a committed relationship, and approximately half (46%) reported having children. Most (96%) had college and graduate education. These participants were between 17 and 66 years of age. No participant reported PTSD, although diagnoses of depression (n = 7) and anxiety disorders (n = 4) were present.
Heart rate data were collected while the participant sat quietly for 5 minutes prior to and for 5 minutes following riding a stationary bike at her own pace for 1 mile.
RESULTS
Reported Abuse History
Reported abuse history is displayed in Table 16.1. Twenty-two participants did not report a history of abuse, 27 participants reported child abuse and/or adult abuse. Chi-square analysis indicated a significantly higher incidence of a mental health diagnosis in the abuse group, χ2(1, N = 49) = 6.23, p < .05. Specifically, 9 of the 11 participants who reported a current mental health diagnosis also reported a history of abuse. Frequency of yoga practice was not related to the physiological response pattern or abuse history.
TABLE 16.1. Reported History of Abuse: Number and Percent
Impact of Abuse History on Autonomic Regulation and Psychological Well-Being
To examine the potential impact of abuse history on autonomic state regulation, repeated measures analyses of variance were used to evaluate the pattern from pre- to post exercise for both RSA and heart rate. For these analyses, the participants were separated into two groups according to abuse history. A no-abuse-history group consisted of 22 participants and an abuse-history group consisted of 27 participants.
Heart rate was sensitive to the exercise challenge, but not related to abuse history. Heart rate significantly increased post exercise, regardless of abuse history, F(1, 47) = 76.17, p < .001. In contrast to the heart rate analyses, RSA was related to abuse history. Since age was significantly correlated with the pre-exercise level of RSA, r(49) = − .53, p < .001, an analysis of covariance was conducted to remove the influence of age on RSA level and RSA reactivity. When age was covaried, abuse history influenced level of RSA, F(1, 46) = 4.63, p < .05, and interacted with the reactivity to exercise, F(1, 46) = 4.06, p = .05. As illustrated in Figure 16.1 participants who reported abuse history had lower RSA and following exercise RSA did not recovery to pre-exercise levels.
In addition, abuse history was related to dysfunctional coping, F(1, 47) = 4.67, p < .05, and total self-concept, F(1, 47) = 9.24, p < .01. Women with an abuse history reported using more dysfunctional coping strategies. Women with an abuse history also reported having significantly lower self-concept.
To determine whether recency of abuse influenced the response profiles, participants were partitioned into three groups: no abuse (n = 22), child abuse only (n = 9), and adult abuse with or without child abuse (n = 18). Analyses with age being covaried indicated that pre-exercise level of RSA was significantly related to abuse group with the group reporting adult abuse having the lowest RSA, F(2, 45) = 3.33, p = .05. Moreover, the group experiencing adult abuse not only had lower levels of RSA but also exhibited the least recovery.
A cumulative abuse index was generated by assigning a 0 to the no-abuse group (n = 22), a 1 to the child-abuse-only group (n = 9), a 2 to the adult-only group (n = 6), and a 3 to those reporting child and adult abuse (n = 12). Although the categorization is only via self-report, the higher the self-reported index of abuse, the greater the decrease in RSA, but not heart rate, following exercise, r(49) = .34, p < .05. This relationship remained when partial correlations removed the influence of baseline RSA on the change scores, r(49) = .41, p < .01. Thus the two indices of autonomic regulation, baseline RSA and RSA reactivity, were independently related to abuse history.
FIGURE 16.1. RSA reactivity to exercise as a function of abuse history.
DISCUSSION
The current study investigated the potential impact of abuse history in women without a diagnosis of PTSD. Although more than half of the participants reported abuse either in childhood or adulthood, no participant reported a diagnosis of PTSD. This high percentage of reported abuse is larger than would be expected based on normative data (e.g., Administration on Children Youth and Families Children's Bureau, 2006). Several of the women reporting a history of abuse were experienced practitioners in yoga. It is possible that the yoga training might have helped improve their ability to self-regulate, to dampen physiological reactivity, and to feel more comfortable in their bodies.
We hypothesized that a history of abuse would "tune" the autonomic nervous system to facilitate defensive mobilization strategies of fight-or-flight behaviors and would compromise physiological recovery following mild exercise (i.e., low RSA and poor RSA recovery). This hypothesis was supported. Women who experienced abuse had lower RSA prior to mild exercise and poorer RSA recovery following mild exercise. These findings suggest that women with an abuse history are less able to recruit the calming influence of the vagal "brake" after exercise. Although these findings are consistent with the findings of Sack et al. (2004), who observed decreased RSA in response to traumatic reminders, our findings are unique since our experimental manipulation was not related to trauma or abuse.
The human autonomic nervous system evolved to rapidly and efficiently shift between physiological states that facilitate either social engagement or mobilization. The myelinated vagus provides a mechanism to rapidly adjust physiological state to behavioral demands. This vagal circuit is uniquely mammalian and evolved to dampen defensive systems. The myelinated vagal circuit allows us (and other mammals) selectively to down-regulate defensive systems when engaged by "safe" individuals and to remove this "brake" to promote the necessary fight-or-flight behaviors when engaged by "dangerous" individuals. Functionally, the vagal input to the heart results in the base-level heart rate being substantially lower than the intrinsic rhythm of the heart's pacemaker. When vagal influences are reduced to the pacemaker, the heart immediately beats faster. Thus, it appears that abuse history, even in individuals who appear to be functioning well, may lower the tonic influence of the vagal brake and limit the ability of the myelinated vagal circuit to rapidly recover and to support a calm physiological state.
The relationship between abuse history and an inability to rapidly engage neural regulation of the vagus to support calm physiological states is consistent with clinical impressions that abused individuals are more likely to exhibit asocial reactions such as fight-or-flight behaviors in response to stress, and to have difficulties shifting from a mobilized state to a state of calmness. These findings are consistent with research indicating that poor vagal modulation may be related to psychological dysfunction, including greater social anxiety (Movius & Allen, 2005) and poorer recovery from depression for depressed individuals (Rottenberg et al., 2005).
The impact of abuse history on mood disturbances and self-concept was also investigated. Women who participate in yoga may self-select due to a self-awareness of their difficulties in regulating state, especially staying calm in the presence of others, and may use yoga as a coping skill to regulate state. Consistent with these predictions, individuals with an abuse history had lower self-concept. Moreover, when considering total abuse, there appeared to be a graded relation between cumulative abuse experiences and both self-concept and mood disturbances. These findings are consistent with research suggesting that victims of abuse may also develop self-esteem issues (e.g., Arata et al., 2005) and mood disorders (e.g., Zavaschi et al., 2006) including depression (e.g., Schuck & Widom, 2001).
The findings that abuse history is related to autonomic regulation are consistent with the polyvagal theory (see chapter 2; Porges, 2001a, 2007a). The theory focuses on the adaptive functions of autonomic reactivity and describes how each of three phylogenetic stages in the development of the vertebrate autonomic nervous system is associated with a distinct autonomic subsystem that is retained and expressed in humans.
Abuse experiences challenge adaptive survival mechanisms. According to the polyvagal theory, the adaptive consequence is to react to features in the environment conservatively, by becoming hypervigilant and more sensitive to predators. Hypothetically, abuse experiences would "retune" the autonomic nervous system. Functionally, abuse history would dampen the ability to express socially "trusting" behaviors and facilitate more vigilant states to prepare for fight-or-flight behaviors. By measuring RSA in this study, an index of the functioning of the vagal brake was dynamically evaluated. Since the vagal brake supports trusting interactions and social engagement behaviors, it was hypothesized that abuse history would compromise the efficiency of this circuit. The data support the hypothesis and demonstrate that the individuals with abuse history who were tested in the current study had depressed cardiac vagal tone and a compromised ability to recruit and reinstate the vagal brake to self-soothe following the transitory physiological state disruption due to mild exercise. Thus, what appears to be a maladaptive inability to re-engage the vagal brake may have emerged from an adaptive response to potential abusive threats by promoting hypervigilance for danger and facilitating an immediate execution of fight-or-flight behaviors.
It has been speculated that PTSD may be a consequence of triggering the unmyelinated vagus as a primitive defense system, often in inescapable contexts, when mobilization defensive strategies cannot be employed (see Porges, 2007a). In this state, a lower brainstem system, more frequently employed by reptiles, is regulating peripheral physiology. This system reduces oxygenated blood flow to the brain and leads to fainting and experiences of dissociation. It is possible that a lower threshold to mobilize and a hypervigilance for danger might have potential survival consequences in this situation. Thus, from an adaptive perspective, the lower threshold to mobilize would protect the individual from recruiting this primitive shutdown circuit.
In the current study, we did not evaluate individuals with PTSD. However, it is possible that individuals with a history of PTSD, may react to the exercise challenge similar to our individuals with an abuse history. Thus, it is possible that individuals with a history of PTSD may have a lower threshold to remove the vagal brake and to employ fight-or-flight defensive strategies.
The study provides a plausible explanation of the mechanisms mediating fight-or-flight reactions clinically observed in clients with histories of abuse. It underscores the importance of demystifying personal experiences and understanding that the consequences of abuse are multidimensional and impacting both psychological and physiological domains. Trauma not only may retune perceptions of others, making it more difficult to feel safe with others, but also may retune the neural circuits by lowering the threshold to react defensively. These changes affect are ability to socially engage, which may lead to dysfunctional coping strategies, lower self-concept, and greater mood disturbance symptoms.
Therefore, assessments and interventions may need to incorporate a multidimensional conceptualization of trauma and abuse in which a retuning of autonomic regulation is a key feature. Within this model, treatments might apply clinical strategies that would enable the client to move from a fight-or-flight state to a physiological state associated with safety and social engagement (e.g., Ogden, Minton, & Pain, 2006). If treatments are unable to shift the client to a calmer physiological state, then access to the psychological mechanisms and processes that have been the bases of psychotherapy may not be efficiently available (see chapter 3).
Co-authors for this chapter were L. P. Dale, L. E. Carroll, G. Galen, J. A. Hayes, and K. W. Webb.
CHAPTER 17
Music Therapy, Trauma, and the Polyvagal Theory
Music is a documented feature of civilizations. Cultures have incorporated music into the educational process, religious and tribal rituals, and patriotic expressions. Vocal music, through both lyrics and melody, has been used both as a contemporary vehicle and an archival mechanism to transmit important cultural, moral, spiritual, and historical events and values. Music has been used to calm, to enable feelings of safety, to build a sense of community, and to reduce the social distance between people.
Music is an important component of the human experience. Music is intertwined with emotion, affect regulation, interpersonal social behavior, and other psychological processes related to personal responses to environmental, interpersonal, and intrapersonal challenges. Types of music have been uniquely associated with distinct feelings, experiences, and social interactions. These psychological processes shape our sense of self, contribute to our abilities to form relationships, and determine whether we feel safe in various contexts or with specific people. Although these processes can be objectively observed and subjectively described, they represent a complex interplay between our psychological experience and our physiology.
This chapter provides a novel insight into the traditions of music as a therapy aiding physical and mental health. More than listening to music, singing, or playing a musical instrument, music therapy involves the dynamic interactions among three features: (1) therapist, (2) client, and (3) music. In the following pages, the polyvagal theory is used to present a plausible model to explain how and why music therapy is helpful in supporting physical health and in enhancing function during compromised states associated with mental and physical illness, including the consequences of trauma. The polyvagal theory provides a strategy to understand the mechanisms and processes that enable music and music therapy to improve social engagement behaviors and to facilitate the regulation of bodily and behavioral states. The theory further provides insights that bridge music therapy to the nervous system and to health outcomes, and is used to deconstruct music therapy into two components: (1) the interpersonal relationship between therapist and client, and (2) the acoustic features of music used in the therapeutic setting as it pertains to clinical treatment.
THE POLYVAGAL THEORY
Our nervous system functions as a sentry by continuously evaluating risk in the environment. Through neural surveillance mechanisms (i.e., neuroception), our brain identifies features of risk or safety (see chapter 1). Many of the features of risk and safety are not learned, but are hard-wired into our nervous system and reflect adaptive strategies associated with our phylogenetic history. The way we react to the specific acoustic frequency bands that constitute music is determined by the same neural circuits that we use to evaluate risk in our environment. For example, low-frequency sounds elicit a sense of danger associated with an approaching predator. In "Peter and the Wolf," Prokofiev exploits this biological feature by conveying impending and predictable danger with the low-frequency sounds of the kettledrums.
Due to our phylogenetic history, the rumble of low-frequency sounds shifts our attention from social interactions to potential dangers in the environment. This reaction is shared with other vertebrates, including reptiles and amphibians. In contrast, high-pitched screams from another mammal (not just our children, but also our dogs and cats) elicit a sense of urgent concern or empathy as a response to the perceived pain or injury of a targeted other. In humans, high-frequency screams shift our attention from the social group or object of focus to the specific individual who is screaming. Music in the frequency band of human voice, however, elicits visceral and emotional states that are associated with neither impending doom nor a sense of urgency. As such, music in the frequency band of human voice is often used melodically in compositions to convey the functional and metaphorical "voice" of the composer. Thus, the orchestral instruments, which duplicate the human vocal range (e.g., violins, flutes, clarinets, trumpets, oboes, and French horns), are used to express the composer's emotional narrative.
Specific acoustic frequency bands in the environment elicit a variety of emotional experiences that are paralleled by adaptive physiological states. Each of these physiological states is functionally an adaptive state that influences affect regulation, social engagement behaviors, and our ability to communicate. We experience these states with feelings of safety, danger, or ultimate demise (i.e., life threat). Physiological state is an implicit component of the subjective experience of listening to or producing music. Music not only changes our emotive state but also elicits changes in our physiology that parallel the feelings of anxiety, fear, panic, and pain. For example, while listening to certain melodies, we relax, slow our heart rate, and smile. However, while listening to other music we may start to imagine danger and visualize marching off to war or protecting our loved ones. The feelings of danger will change our facial expression and increase our heart rate.
As Oliver Sacks (2007) discussed in Musicophilia, music appears to be part of the human experience, yet no brain area or circuit has been identified to explain or represent music. From a polyvagal perspective, this question is viewed differently. Rather than seeking specificity in the neural regulation required to process and to express music, this perspective points to the convergence and similarity between the neural mechanisms required to process music and the neural mechanisms required to process features of social engagement behaviors and risk in the environment. Furthermore, this convergence between physiological state and music-related emotional experience is neurophysiologically determined and explained by the polyvagal theory (see chapters 2, 10, 11, and 12; Porges, 2001a, 2007a). As this discussion progresses, the polyvagal theory (see chapter 2) is used as an organizing principle to explain how music, especially music expressed via music therapy, can recruit the neural mechanisms that integrate facial muscles and visceral state, and consequently promote restorative affective states and prosocial behavior.
In order to illustrate this link fully, it is important to briefly review the origins of the polyvagal theory. The polyvagal theory emerged from the study of the evolution of the vertebrate autonomic nervous system (ANS), and is based on the functions of the ANS that automatically regulate several major organs, including the heart, the lungs, and the gut. Circuits in the brain dynamically regulate the ANS. This regulation is bidirectional; the brain and its neural sentries continuously monitor body state, and body state dynamically influences brain function. Moreover, the neural regulation of the ANS is linked to the neural regulation of the muscles of the face and head, which signal to others our emotional state. These muscles of the face and head are involved both in actively listening to (i.e., the modulation of our middle ear muscles) and in producing of music whether by singing (i.e., the modulation of the laryngeal and pharyngeal muscles) or by playing a wind instrument (i.e., the modulation of the facial and oral muscles supporting an embouchure). Polyvagal theory assumes that many of our social behaviors and vulnerabilities to emotional disorders are "hard-wired" into our nervous system. Based on this theory, it is possible to understand various aspects of mental health, including trauma responses, and to develop treatment techniques that can help individuals communicate better and relate better to others.
This theoretical perspective is particularly important in understanding the mechanisms underlying music therapy, which require both the processing of acoustic stimuli and face-to-face social interactions within the therapeutic setting. Thus, the polyvagal theory provides insights into the beneficial effects of music therapy, since it provides an understanding of the neural control of structures involved in the two features of music therapy: (1) social interactions between the client and the therapist, and (2) listening and expressing music. The neural mechanisms recruited by these features of music therapy are described in further detail later in this chapter.
The Biobehavioral Quest for Safety, Survival, and a Painless Death
The polyvagal theory proposes that the evolution of the mammalian ANS provides the neurophysiological substrates for affective processes and stress responses. The theory proposes that physiological state limits the range of adaptive behaviors and psychological experiences. Thus, the evolution of the nervous system determines the range of emotional expression, quality of communication, and the ability to regulate body and behavioral state, including the expression and recovery of stress-related responses. Relevant to adaptive social and emotional behaviors, these phylogenetic principles illustrate the emergence of a brain-face-heart circuit and provide a basis not only for investigating the relation between several features of mental health and autonomic regulation, but also for deconstructing how music and music therapy can support mental and physical health.
The investigation of human phylogenetic history identifies changes in neural regulation that occurred as vertebrates evolved from jawless fish to humans and other mammals. Phylogenetic development resulted in humans having an increased neural control of the heart via a myelinated vagal system. The evolution of the myelinated vagus was paralleled by an enhanced neural control of the face, larynx, and pharynx. This integrated face–heart system enabled complex facial gestures and vocalizations associated with social communication to influence physiological states. The face–heart system can cue others of safety and danger via facial expressions or vocalization, while promoting transitory mobilization by increasing heart rate. Removing the myelinated vagal inhibition from the heart physiologically supports this biobehavioral process of mobilization. These mechanisms provide us with an understanding of how a warm smile simultaneously reflects a calm state and triggers calmness and a sense of safety and benevolence in the observer. In contrast, an angry face reflects a mobilized state and triggers a matching defensive state in the observer. Vocalizations, in addition to facial expression, can reflect and trigger bodily states. Similar to the relationship between smiling and calmness just described, the melodic patterns of vocalizations, which are not typically shrill or booming, provide convergent cues to the observer that indicate a relative sense of safety. However, the lower pitch associated with a booming voice will often startle and/or frighten the observer, while the high pitch of a shrill voice will generally evoke anxiety and fear.
THE SOCIAL ENGAGEMENT SYSTEM
The social engagement system regulates facial muscles, including the sphincter muscles around the eyes that promote social gaze and emotional expressivity; middle ear muscles that enable human voice to be extracted from background sounds; muscles of mastication that are involved in ingestion; laryngeal and pharyngeal muscles that are used in sucking, swallowing, vocalizing, and breathing; and muscles of head that enhance social gesture and orientation, such as turning and tilting. Collectively, these muscles act as filters of social stimuli and determinants of engagement with the social environment. It is interesting to note that the neural pathways regulating the orbicularis oculi, a sphincter muscle around the eye involved in expressive displays, also are involved in the dynamic regulation of the stapedius muscle in the middle ear (Djupesland, 1976). Thus, the neural mechanisms for emotional cueing via eye contact are shared with those needed to listen to human voice.
As a cluster, difficulties in behaviors associated with the social engagement system (e.g., avoidant gaze, nonresponsiveness to human voice, reduced facial affect and diminished vocal prosody, and atypical or lack of head gesture) are common features of individuals with autism, post-traumatic stress disorder, and other psychiatric disorders. This information leads the astute clinician to infer difficulties in both social engagement behaviors and physiological state regulation from facial expressions and vocal prosody. Several psychiatric disorders have deficits in both the somatomotor (e.g., poor gaze, low facial affect, lack of prosody, difficulties in mastication) and visceromotor (difficulties in autonomic regulation including cardiopulmonary and digestive problems) components of the social engagement system. Deficits in the social engagement system compromise spontaneous social behavior, social awareness, affect expressivity, prosody, and language development.
Human responses to trauma are devastating and compromise subsequent social behavior and emotion regulation. Understanding the mechanisms underlying the mammalian hard-wired response to life threat may demystify these debilitating consequences. From this neurophysiological perspective, a variety of clinical features, including severely compromised social behavior and difficulties in emotion regulation, are predictable. An understanding of the mechanisms mediating these atypical behaviors in response to trauma is helpful to the client, the family, and the therapist in developing supportive and restorative contexts and treatments.
Functionally, our nervous system is continuously evaluating risk in the environment through an unconscious process of neuroception (see chapter 1). Specific features in the environment trigger physiological states associated with feelings of safety, danger, or ultimate demise. The human nervous system evolved efficiently to shift between conditions of safety and danger. We easily adjust and calm following situations requiring fight-or-flight maneuvers. We use social interactions with attuned and contingent facial expressions, intonation of our voice (i.e., prosody), and gaze to calm and be calmed. However, in contrast to challenges of danger, reactions to life threat are not easily remediated. Attempts to socially engage a traumatized individual, rather than calming, may result in defensive strategies of rage and anger. Life threat triggers a very ancient neural circuit that severely limits social engagement behaviors and may distort neuroception, resulting in a false detection of risk. Thus, treatment of trauma requires a new model distinct from the traditional psychotherapeutic strategies of face-to-face dialog in order to trigger the calm states associated with the social engagement system. Music and music therapy strategies may provide an alternate portal to the social engagement system and avoid the initial face-to-face interactions that may be misinterpreted as threat by a traumatized individual.
HOW MUSIC AND PROSODIC VOCALIZATIONS TRIGGER THE SOCIAL ENGAGEMENT SYSTEM
As vertebrates evolved from reptiles to mammals, the structures at the end of the mandible (i.e., jaw bone) that define the middle ear bones became detached (Luo, Crompton, & Sun, 2001; Rowe, 1996; Wang, Hu, Meng, & Li, 2001). For humans and other mammals, sound in the environment impinges on the eardrum and is transduced from the eardrum to the inner ear via the small bones in the middle ear known as ossicles. When the stapedius (regulated by a branch of the facial nerve) and the tensor tympani (regulated by a branch of the trigeminal nerve) muscles are innervated, the ossicular chain becomes more rigid and dampens the amplitude of the low-frequency acoustic stimulation from the environment reaching the inner ear. This process is similar to tightening the skin on a kettledrum. When the skin is tightened, the pitch of the drum is higher. When the ossicular chain is tightened, similar to the stretched skin, only higher frequencies bouncing against the eardrum are transmitted to the inner ear and to the auditory processing areas of the brain. This functional relation is described and illustrated in a Scientific American article by Borg and Counter (1989).
The evolution of the human middle ear enabled low-amplitude, relatively high-frequency airborne sounds matching the frequency of the human voice to be heard, even when the acoustic environment was dominated by low-frequency sounds, such as sounds made by large predators. Detached middle ear bones were a phylogenetic innovation that enabled mammals to communicate in a frequency band that could not be detected by reptiles. Reptiles have difficulties in hearing higher frequencies, since their hearing is dependent on bone conduction.
Studies have demonstrated that this critical neural regulation of middle ear muscles, a necessary mechanism to extract the soft sounds of human voice from the loud sounds of low-frequency background noise, is defective in individuals with language delays, learning disabilities, and autistic spectrum disorders (Thomas, McMurry, & Pillsbury, 1985). Additionally, middle ear infection (i.e., otitis media) may result in a total inability to elicit the reflexive contraction of the stapedius muscles (Yagi & Nakatani, 1987). Disorders that influence the neural function of the facial nerve (i.e., Bell's palsy) not only influence the stapedius reflex (Ardic, Topaloglu, Oncel, Ardic, & Uguz, 1997), but may also affect the patient's ability to discriminate speech (Wormald, Rogers, & Gatehouse, 1995). The observed difficulties that individuals with a variety of physical and mental disorders have in extracting human voice from background sounds may be dependent on the same neural system that regulates facial expression. Thus, deficits in the social engagement system would compromise the expression of emotion, as well as social awareness and language development.
MUSIC THERAPY, TRAUMA, AND THE SOCIAL ENGAGEMENT SYSTEM
The frequency content of melodies in most musical compositions duplicates the frequency band of human voice. Functionally, acoustic properties of melodies, typically encompassing middle C and the two octaves above middle C, easily pass through the middle ear structures regardless of the neural tone to the middle ear muscles. Once the frequencies pass through the middle ear, they trigger a neural feedback mechanism to tense the ossicle chain. Vocal music duplicates the effect of vocal prosody and triggers neural mechanisms that regulate the entire social engagement system with the resultant changes in facial affect and autonomic state. Basically, we start to look and feel better when we listen to melodies. Thus, while social engagement is a common goal in music therapy practice, incorporating the polyvagal theory provides a plausible scientific justification for how music therapy may provide opportunities to exercise the social engagement system.
Consistent with the parallel between music and social communication, the same frequency band that characterizes melodies defines, in human voice, the frequency band in which all information (i.e., verbal content) is communicated. When this frequency band is weighted to enhance the understanding of voice, it is known as the "index of articulation" (Kryter, 1962) and more recently as the "speech intelligibility index" (American National Standards Institute, 1997). These indices emphasize the relative importance of specific frequencies in conveying the information embedded in human speech. In the normal ear, acoustic energy within the primary frequencies of these indices is not attenuated as it passes through the middle ear structures to the inner ear. The frequency band defining the index of articulation is similar to the frequency band that composers have historically selected to express melodies. It is also the frequency band that mothers have used to calm their infants by singing lullabies.
Trauma can turn off the social engagement system. Attempts to engage a person with a trauma history, rather that eliciting spontaneous social behavior, may trigger defensive and aggressive behaviors. From a clinical perspective, traumatized individuals often present features of gaze aversion and flat facial affect. If we were to monitor the physiological state of these individuals, we would observe an ANS that is poised to fight or flee (i.e., high heart rate and low vagal regulation of the heart). The traumatic experience functionally retunes neuroception to conservatively detect risk when there is no risk. Most therapeutic strategies attempt to engage with direct face-to-face eye contact. Working with traumatized individuals creates a great challenge to therapists, since the normal social engagement behaviors of the therapist may trigger fear and reactive defensive strategies. Music therapy provides a special portal to reengage the social engagement system that does not require an initial face-to-face interaction. Music can be used to stimulate the social engagement system without requiring face-to-face reciprocity. Since melodic music contains acoustic properties similar to vocal prosody, music may be used to recruit the social engagement system by challenging and modulating the neural regulation of the middle ear muscles. If the social engagement system is effectively recruited, positive facial expressions will emerge, eye gaze will spontaneously be directed at the therapist, and the traumatized individual will shift to a more calm and positive physiological state.
CONCLUSION
Based on the polyvagal theory, elements of music therapy can be deconstructed into biobehavioral processes that stimulate the social engagement system. When the social engagement system is stimulated, the client responds both behaviorally and physiologically. First, the observable features of social engagement become more spontaneous and contingent as the face and voice become more expressive. Second, there is a change in physiological state regulation that is expressed through increased behavioral regulation and calmness. The improved state regulation is mediated by the myelinated vagus, which directly promotes health, growth, and restoration. However for some clients, especially those who have been traumatized, face-to-face interactions can be threatening and may not elicit a neuroception of safety. In these circumstances, the social engagement system can potentially be activated through vocal prosody or music while minimizing direct face-to-face interactions.
From a polyvagal perspective, music therapy can be deconstructed into two integrated processes. As illustrated in Figure 17.1, the therapeutic environment often invites face-to-face interaction between the therapist and the client. This face-to-face interaction, if effective, will trigger the client's neuroception of safety. Second, the frequency band associated with melodies functionally duplicates the frequency band conveying information in the human voice. The human nervous system evolved to be very selective of these frequencies. Music, especially vocal music, produces melodies by modulating these frequencies. This process engages and exercises the neural regulation of the social engagement system with the positive effects of improved socioemotional behaviors and enhanced physiological state. Interestingly, the phrasing of music is also an important component of this process. The phrasing of music, especially when singing or playing a wind instrument, results in short inhalations and extended durations of exhalations. Physiologically, breathing "gates" the influence of the myelinated vagus on the heart. Functionally, when we inhale, the influence of the vagus is attenuated and heart rate increases. In contrast, when we exhale, the influence of the vagus is increased and heart rate decreases. This simple mechanical change in breathing increases the calming impact and health benefits of the myelinated vagus on our body. Thus, music therapy by engaging and exercising the social engagement system may promote positive outcomes improving several features related to quality of life.
FIGURE 17.1. Deconstructing music therapy: A polyvagal perspective.
PART V
SOCIAL BEHAVIOR AND HEALTH
CHAPTER 18
Reciprocal Influences Between Body and Brain in the Perception and Expression of Affect
Emotions, affect regulation, and interpersonal social behavior are psychological processes that describe basic human experiences in response to events, environmental challenges, and people. These processes shape our sense of self, contribute to our abilities to form relationships, and determine whether we feel safe in various contexts or with specific people. Although these processes can be objectively observed and subjectively described, they represent a complex interplay between our psychological experience and our physiological regulation. These psychological–physiological interactions are dependent on the dynamic bidirectional communication between peripheral organs and the central nervous system connecting the brain with these organs. For example, the neural circuits, providing a bidirectional communication between the brain and heart, can trigger either a rapid increase in heart rate to support protective fight-or-flight behaviors or a rapid decrease in heart rate to support social interactions. Peripheral physiological reactions can be initiated by the brain detecting features of danger in the environment, and alternatively, changes in peripheral physiological state can influence the brain and alter our perceptions of the world. Thus, affect and interpersonal social behavior are more accurately described as biobehavioral than psychological processes, since our physiological state can profoundly influence the quality of these psychological processes and our feelings can, in turn, determine dynamic changes in our physiology.
Our nervous system functions as a sentry by continuously evaluating risk in the environment. Through neural surveillance mechanisms (i.e., neuroception), our brain identifies features of risk or safety. Many of the features of risk and safety are not learned, but are hard-wired into our nervous system and reflect adaptive strategies associated with our phylogenetic history. For example, low-frequency sounds elicit in mammals a sense of danger associated with an approaching predator. This reaction is shared with other vertebrates, including reptiles and amphibians. Due to our phylogenetic history, the rumble of low-frequency sounds shifts our attention from social interactions to potential dangers in the environment. In contrast, high-pitched screams from another mammal (not just our children, but also our dogs and cats) elicit a sense of urgent concern or empathy for another who may be feeling pain or being injured. With humans, high-frequency screams shift our attention to the specific individual who is screaming. Through exposure and associative learning, we can link these features with other events. Specific features in the environment recruit physiological states differentially associated with feelings of safety, danger, or ultimate demise (i.e., life threat). Each of these states is characterized by a specific set of capacities for affect regulation and social engagement and communication (see chapter 12).
Current research in affective neuroscience focuses on brain structures and neural circuits related to specific motivational and emotional processes (e.g., Panksepp, 1998). These important discoveries emphasize cortical and subcortical structures in the emergence of the complex affective repertoire of humans and their contribution to social relationships (e.g., Schore, 1994, 2003; Siegel, 2007). However, underlying these contributions are details of an important and often overlooked neurobiological substrate, the neural circuits mediating the reciprocal communication between body states and brainstem structures, which impact on the availability of these affective circuits. These underlying circuits not only promote feelings (e.g., Damasio, 1999), but also form a bidirectional circuit (e.g., Darwin, 1872) that enables mental and psychological processes to influence body state and body state to color and, at times, to distort our perception of the world. Thus, the study of affective processes, especially in their prosocial and healing roles, requires an understanding of the neural circuits both between higher brain structures and the brainstem and between the brainstem and the visceral organs (e.g., the heart) mediated through the autonomic nervous system. All affective or emotional states are dependent on lower brain regulation of the visceral state and the important visceral, tactile, and nociceptive cues that travel to the brain from the periphery. Moreover, there are distinct visceral regulatory states that foster different domains of behavior. These states do not preclude the important bidirectional information from higher brain structures.
This chapter emphasizes the neural regulation of observable facial movements and concurrent subjective visceral experiences that characterize the expressions, feelings, and perceptions of emotion and affective state. The chapter uses the polyvagal theory (see chapters 2, 10, 11, and 12; Porges, 2001a, 2007a) as an organizing principle to explain the role of visceral state in the accessibility of prosocial emotions and restorative affective states. The polyvagal theory is an attempt to reorganize our conceptualization of the autonomic nervous system with a focus on the specific neural circuits involved in regulating visceral organs for specific adaptive functions related to affect, emotions, and social communication behaviors.
The polyvagal theory interprets social interactions and emotion as biobehavioral processes. Thus, the theory is particularly important for psychotherapists, who focus on the social interaction within the therapeutic setting and forgo pharmacological interventions. By treating the social interaction as a biobehavioral process, it is possible to conceptualize a therapeutic treatment that relies not on pharmacological manipulations but on the profound positive impact of social interactions and interpersonal behaviors on the neural regulation of body state and behavior. By exploring these bidirectional biobehavioral processes, psychotherapeutic treatments may change the neural regulation of physiological state, which in turn will support further benefits from interpersonal interactions.
EMOTION, MOTION, AND VISCERAL STATE: FEATURES OF MENTAL HEALTH
Regardless of the operational, and often arbitrary, distinction between emotion and affect or between emotional expressions and feelings, the measurement of physiological state (e.g., autonomic, endocrine, and muscle activity) needs to be embraced in affective neuroscience, particularly if there is to be a functional dialogue with experiential clinicians. In most cases physiological state has been conceptualized as a correlate or a consequence of higher brain structures (e.g., cortex) presumed to be driving emotion and affect. However, it would be naive not to explore the connections and potential bidirectional influences between peripheral physiological state and the brain circuits related to affective processes.
Physiological state is an implicit component of the subjective experiences associated with specific psychological constructs such as anxiety, fear, panic, and pain. The convergence between physiological state and emotional experience is neurophysiologically determined, since the metabolic requirements necessary to modulate the muscles of the face and body require supporting changes in autonomic state. All emotional and affective states require specific physiological shifts to facilitate their expression and to reach their implicit goals (e.g., fight, flight, freeze, proximity, etc.).
Through the study of phylogenetic shifts in the vertebrate autonomic nervous system, it is possible to link the different expressive features of emotion in humans with the phylogenetic transitions in visceral regulation observed in vertebrates. Physiological monitoring provides an important portal to monitor these reactions, since some affective responses are often not observable in overt behavior. For example, the convergence between the neural mechanism mediating autonomic state and facial expressions phylogenetically occurs in the transition from reptiles to mammals (see chapter 2; Porges, 2007a).
There is a rich history of research linking the neural regulation of face and viscera (e.g., heart) with brain circuits. Gellhorn (1964) elaborated on how proprioceptive discharges from facial muscles influence brain function and promote changes in visceral state. Thus, providing an example of the bidirectionality between peripheral and central structures and providing a neurophysiological basis for the assumed relation between facial expression and body feelings. Even earlier, Darwin (1872) acknowledged the important and often neglected bidirectional relation between the brain and the heart.
Although Hess was awarded the Nobel Prize in Physiology or Medicine in 1949 for his work emphasizing the importance of the central regulation of visceral state,* journals in contemporary affective neuroscience (e.g., Nature Neuroscience) and psychiatry (e.g., Biological Psychiatry) express a disconnect between subjective affective experience and visceral state regulation. Contemporary affective neuroscience with the aids of both imaging techniques and neurochemistry has focused on brain structures contributing to various neural circuits involved in adaptive behaviors with apparent motivational objectives.
Panksepp (1998) organizes affective experiences into seven neural-based motivational systems that include seeking, rage, fear, lust, care, panic, and play. However, missing from these functionally adaptive motivational circuits is the role that neural regulation of visceral state plays in potentiating or dampening these circuits. For example, if an individual is in a physiological state characterized by vagal withdrawal and high sympathetic excitation, body feelings of a fast pounding heartbeat are experienced and the threshold to react aggressively is low. In contrast, when in a physiological state characterized by an engaged myelinated vagus, sympathetic and hypothalamic-pituitary-adrenal axis reactivity are dampened. The physiological state is experienced as "calm." Intrusive stimuli that previously would have triggered aggressive behaviors when the vagal activity is withdrawn will now result in a dampened reaction. Accompanying this change in physiological state are options to further dampen reactivity through social interactions.
Most proponents of affective neuroscience embrace a science of parallelism that links either observable emotional expressions or subjective experiences with a "neural" specificity that is concretized and assumed to be validated by imaging studies that identify activation of brain areas or blockade studies interfering with appropriate functioning of these circuits. Thus, to many neuroscientists, affect resides solely in the brain and does not require inputs or outputs linking the body to the brain. Missing from this research agenda and theoretical explanation is an appreciation of the necessary contributions of both the sensory inputs from the periphery influencing central circuits and the motor outputs in the periphery driven by the central circuits. Focusing on the central circuits, without studying the sensory and motor contributions from the periphery, is like studying the behavior of a thermostat independent of information regarding both ambient temperature and the capacities of the heating, ventilation, and air conditioning components.
Hess in his Nobel speech articulates an awareness of the complex features of a system. From his perspective, although the components of a feedback circuit might be identified and studied independently, the functioning of independent parts will not explain how the system, as a whole, functions dynamically during moment-to-moment challenges. This limitation was, in part, dependent on the methodologies available to study neurophysiology that required pharmacological, surgical, or electrical manipulations to block or stimulate "global" branches of the autonomic nervous system that either shared a specific neurotransmitter (e.g., acetylcholine, epinephrine) or an easily identifiable nerve (e.g., vagus) that could be cut or stimulated.
Within the field of mental health, there is a similar acceptance of a disease model without a focus on the intervening feedback circuits that mediate the features of the disorder. Within the medical discipline of psychiatry, anxiety and depression are defined by clinical features and not by a measurable physiological substrate. The prevalent strategies in mental health research that use neurophysiological variables (e.g., imaging, autonomic measures) are not directed at defining anxiety or depression, but use neurophysiological variables as correlates of a clinical diagnosis.
The value of taking a different perspective can be illustrated with the construct of anxiety. If anxiety were viewed as dependent on a shift in autonomic state in which an individual's physiological state is dominated by the sympathetic nervous system, new clinical research strategies might emerge that focus on characterizing how states of anxiety and a vulnerability to being anxious would be potentiated or dampened by different autonomic states. Treatments would then be developed either to (1) dampen sympathetic tone or (2) enable the individual to move into environments or shift contexts that are less likely to trigger the increased reactivity associated with higher sympathetic excitation. Unfortunately, most researchers in psychiatry and psychology express little interest in the mapping autonomic regulation as a "vulnerability" dimension for various psychiatric disorders and behavioral problems, although visceral features are often symptoms of the disorders they are treating.
Clinical disciplines rarely acknowledge the proximal functions of visceral state. Clinicians seldom monitor the expression of vagal withdrawal or sympathetic excitation in their patients. Such a shift in autonomic state would be manifested in several physical and psychiatric symptoms including flat affect, difficulties in auditory processing, hyperacusis, tachycardia, and constipation. In addition, conventional models of mental disorders neglect the role of neurophysiological mechanisms dynamically interacting with contextual cues in the environment. In contrast, these disciplines have embraced distal constructs related to the functions of receptors within the brain that lead almost reflexively to drug treatment, while generally failing to recognize the important role of visceral state and visceral afferent feedback on the global functioning of the brain. This strategy is far from parsimonious and does not take into account either the phylogeny of the mammalian nervous system or the intervening neurophysiological and biobehavioral systems along a continuum from genes to behavior. Rather, these disciplines have assumed that clusters of observable behaviors or subjective experiences are linked parsimoniously and directly to neurochemical levels in specific brain circuits. Thus, this strategy misses the important potential of psychological and behavioral interventions (including changes in environment) that would be therapeutic by directly influencing physiological state without necessitating pharmacological treatments.
STATE REGULATION AND THE AUTONOMIC NERVOUS SYSTEM: A HISTORICAL PERSPECTIVE
Researchers for more than a century have measured autonomic variables (e.g., heart rate, palmar sweat gland activity) as indicators of emotional state related to perceived stress (e.g., fear, mental effort, workload, and anxiety). Historically, arousal theories (e.g., Berlyne, 1960; Darrow, 1943; Gray, 1971) provided scientists who study brain–behavior relations with a model that assumed that activation of peripheral physiological measures regulated by the sympathetic branch of the autonomic nervous system were sensitive indicators of brain "arousal" or "activation." This view was based on a rudimentary understanding of the autonomic nervous system in which changes in easily measured peripheral organs (e.g., sweat glands, heart) were assumed to be accurate indicators of how the brain was processing emotional stimuli. Usually, emotional states were associated with fight-or-flight behaviors and the sympathetic-adrenal system (e.g., increases in heart rate, sweat gland activity, and circulating catecholamines) as initially described by Cannon (1929b). Based on Selye (1936, 1956), emotional states were also associated with increased activity of the HPA axis (e.g., increases in cortisol). From a psychological level, arousal theories emphasized fight-or-flight behaviors and neglected or minimized the importance of both prosocial affective states that facilitated social interaction and also the defensive strategy of immobilization (e.g., fainting, death feigning).
An acceptance of a unitary arousal system is assumed in several research domains, including investigations of sleep, deception, sexual behavior, and anxiety. Moreover, it led to research on cortical "arousal" and the use of electroencephalography, single photon emission computed tomography, functional magnetic resonance imaging, and other imaging technologies that accepted the arousal construct with little interest in the distinction between activation of neural pathways that were excitatory or inhibitory. This resulted in difficulties in establishing whether "activation" represents the turning on or the turning off of a specific neural structure. From a physiological level, arousal theories emphasize an assumed continuity between central cortical activation and peripheral arousal marked by increases in the activity of the sympathetic nervous system and the adrenal hormones. However, arousal theories have neglected both the importance of the parasympathetic branch of the autonomic nervous system and the bidirectional communication between brain structures and visceral organs.
The continuity between brain and peripheral arousal created a research environment that neglected several important factors, including an understanding of the brain structures that regulate autonomic function; how these structures evolved from the most primitive vertebrates to mammals; how the autonomic nervous system interacts with the immune system, the hypothalamic-pituitary-adrenal axis, and the neuropeptides oxytocin and vasopressin; and the coevolution of stress and coping strategies with the increasing complexity of the autonomic nervous system. Missing from this dialog is a discussion of the role of the parasympathetic nervous system and especially the vagus (the Xth cranial nerve) with its bidirectional portal between the brain and specific visceral organs such as the heart.
THE POLYVAGAL THEORY: A PRIMER
The polyvagal theory (see chapter 2) emerged from the study of the evolution of the vertebrate autonomic nervous system. The theory assumes that many of our social behaviors and vulnerabilities to emotional disorders are "hard-wired" into our nervous system. Based on the theory, it is possible to understand various aspects of mental health and to develop treatment techniques that can help people communicate better and relate better to others. The term "polyvagal" combines "poly," meaning "many," and "vagal," which refers to the important nerve called the "vagus." To understand the theory, we need to investigate features of the vagus nerve, a primary component of the autonomic nervous system. The vagus nerve exits the brainstem and has branches that regulate several organs, including the heart. The theory proposes that there are two branches of the vagus that are related to different behavioral strategies, one related to social interactions in safe environments and the other related to adaptive responses to life threat.
Historically, the autonomic nervous system has been broken into two opposing components, one labeled sympathetic and the other parasympathetic. This organizational model was used to describe the function of the autonomic nervous system in the late 1800s and the early 1900s. In the 1920s this paired-antagonism model was formalized (Langley, 1921). This model characterized the function of the autonomic nervous system as a constant battle between the sympathetic nervous system associated with fight-or-flight behaviors and the parasympathetic nervous system associated with growth, health, and restoration. Because most organs of the body, such as the heart, the lungs, and the gut, have innervations from both sympathetic and parasympathetic components, the paired-antagonism model evolved into "balance theories." Balance theories attempted to link "tonic" imbalances to both physical and mental health. For example, a sympathetic dominance might be related to symptoms of anxiety, hyperactivity, or impulsivity, while a parasympathetic dominance might be related to symptoms of depression or lethargy. In addition to the tonic features of autonomic state, the paired-antagonism model also was assumed to explain the reactive features of the autonomic nervous system. This dependence on the construct of "autonomic balance" is still prevalent in textbooks, although there has been an intervening century in which neurophysiology has documented a second vagal pathway involved in regulating autonomic function. Unfortunately, this new knowledge of the second vagal pathway has not permeated the teaching of physiology, which still is dominated by descriptions of the paired antagonism between the sympathetic and parasympathetic components of the autonomic nervous system.
The primary parasympathetic influence to peripheral organs is conveyed through the vagus, a cranial nerve that exits the brain and innervates the gastrointestinal tract, respiratory tract, heart, and abdominal viscera. The vagus can be conceptualized as a tube or conduit containing several sensory and motor fibers originating or terminating in different areas of the brainstem. For example, the primary vagal motor pathways that regulate the lower gut originate in the dorsal nucleus of the vagus, the primary vagal motor pathways that regulate the heart and the lungs originate in the nucleus ambiguus, and the vagal pathways sending sensory information from the gut terminate in the nucleus of the solitary tract.
The polyvagal theory proposes that the autonomic nervous system reacts to real-world challenges in a predictable hierarchical manner that parallels, in reverse, the phylogenetic history of the autonomic nervous system in vertebrates. In other words, if we study the evolutionary path of how the autonomic nervous system unfolded in vertebrates (i.e., from ancient jawless fish to bony fish, amphibians, reptiles, and mammals), we learn not only that there is an increase in the growth and complexity of the cortex (the outer layer of the cerebrum), but also that there is a change in composition and function of the autonomic nervous system. In mammals, the autonomic nervous system functions as a hierarchical system that parallels phylogenetic states in reverse and not as the balance between sympathetic and parasympathetic components.
The phylogenetic changes in the autonomic nervous system (including changes in neural pathways and brainstem areas regulating the peripheral organs) determine how the autonomic nervous system reacts to challenges. In humans and other mammals, the hierarchy is composed of three neural circuits with the newer circuits having the capacity to override the older circuits. Under most challenges in our environment, we initially react with our newest system (i.e., myelinated vagus). If that circuit does not satisfy our biobehavioral quest for safety, an older circuit spontaneously reacts (i.e., sympathetic nervous system). Finally, if the former strategies are unsuccessful, as our last option we reflexively trigger the oldest circuit (i.e., unmyelinated vagus). Functionally, in humans the older vagal circuit is involved in adaptive reactions characterized by immobilization and decrease in metabolic resources, while the newer vagal circuit is involved in regulating calm states that promote both spontaneous social engagement and health, growth, and restoration. Along the phylogenetic hierarchy, between the two vagal circuits is the sympathetic nervous system that supports fight-or-flight behaviors.
THE POLYVAGAL THEORY: A BIOBEHAVIORAL QUEST FOR SAFETY
To survive, mammals must determine friend from foe, when an environment is safe, and communicate to their social unit. These survival-related behaviors limit the extent to which a mammal can be physically approached, whether vocalizations will be understood, and whether coalitions can be established. Moreover, these behavioral strategies, which are used to navigate through the "stress of life," form the bedrock on which social behaviors and higher cognitive processes can be developed and expressed. Thus, learning and other expansive mental processes must be structured, manipulated, and studied within the context of how the environment fosters or ameliorates stress-related physiological states.
The polyvagal theory proposes that the evolution of the mammalian autonomic nervous system provides the neurophysiological substrates for affective processes and stress responses. The theory proposes that physiological state limits the range of adaptive behaviors and psychological experiences. Thus, the evolution of the nervous system determines the range of emotional expression, quality of communication, and the ability to regulate body and behavioral state, including the expression and recovery of stress-related responses. Relevant to adaptive social and emotional behaviors, these phylogenetic principles illustrate the emergence of a brain-face-heart circuit and provide a basis for investigating the relation between several features of mental health and autonomic regulation.
Via evolutionary processes, the mammalian nervous system has emerged with specific features that react to challenge to maintain visceral homeostasis. In general, the domains of homeostasis, which have been monitored, have focused on the visceral systems involved in cardiovascular, digestive, reproductive, and immune functions. For example, studies have evaluated how long it takes heart rate to recover following a challenge to a prestress level. Adaptive coping requires minimizing the magnitude and duration of this deviation, whether the deviation is observed in raising heart rate, blood pressure, cortisol, or disrupting digestion.
By investigating the phylogeny of the regulation of the vertebrate heart (e.g., Morris & Nilsson, 1994), three principles can be extracted. First, there is a phylogenetic shift in the regulation of the heart from endocrine communication, to unmyelinated nerves, and finally to myelinated nerves. Second, there is a development of opposing neural mechanisms of excitation and inhibition to provide rapid regulation of graded metabolic output. Third, with increased cortical development, the cortex exhibits greater control over the brainstem via direct (e.g., corticobulbar) and indirect (e.g., corticoreticular) neural pathways originating in motor cortex and terminating in the source nuclei of the myelinated motor nerves emerging from the brainstem (e.g., specific neural pathways embedded within cranial nerves V, VII, IX, X, XI), controlling visceromotor structures (i.e., heart, bronchi, thymus) and somatomotor structures (muscles of the face and head) that results in a neural circuit that functions to facilitate social behavior and to maintain calm behavioral states.
These phylogenetic principles illustrate the emergence of a brain-face-heart circuit and provide a basis for investigating the relation between several features of mental health and autonomic regulation. In general, phylogenetic development results in increased neural control of the heart via the myelinated mammalian vagal system that is paralleled by an increase in the neural regulation of the facial muscles. This integrated system can "cue" others of safety and danger, while promoting transitory mobilization and the expression of sympathetic tone without requiring sympathetic or adrenal activation (i.e., raising heart rate by removing the myelinated vagal inhibition from the heart). Functionally, this phylogenetic progression provides a system that can respond rapidly (i.e., via myelinated pathways), selectively regulate the magnitude (i.e., via opposing inhibitory and excitatory circuits) and specificity of the features (e.g., via regulating the linkage between autonomic reactivity with facial muscles) of the reaction. With this new vagal system, transitory incursions into the environment or withdrawals from a potential predator can be initiated without the severe biological cost of the metabolic excitation associated with sympathetic-adrenal activation. Paralleling this change in neural control of the heart is an enhanced neural control of the face, larynx, and pharynx that enables complex facial gestures and vocalizations associated with social communication. This phylogenetic course results in greater central nervous system regulation of behavior, especially behaviors needed to engage and to disengage rapidly with environmental challenges. These phylogenetic shifts, which promote a greater bidirectional communication between brain and viscera, provide opportunities for mental processes, including voluntary behavior, to impact on body state. Thus, a greater understanding of the circuit mediating these interactions might lead to functional models of intervention that would both calm visceral state and promote more prosocial interactions. Consistent with this trend, new research and clinical programs are emerging. For example, Cleveland Clinic has created the Bakken Heart-Brain Institute and hosts an annual Heart-Brain Institute Summit to bring "together researchers, clinicians and others to stimulate greater collaboration and understanding of the heart–brain link and to positively impact research, education, and patient care."
THREE PHYLOGENETICALLY DEFINED AUTONOMIC CIRCUITS SUPPORT ADAPTIVE BEHAVIORS
The polyvagal theory (see chapters 2, 3, 10, 11, and 12; Porges, 2001a, 2007a) emphasizes and documents the neurophysiological and neuroanatomical distinction between the two branches of the vagus (i.e., Xth cranial nerve) and proposes that each vagal branch is associated with a different adaptive behavioral and physiological response strategy to stressful events. The theory describes three phylogenetic stages of the development of the mammalian autonomic nervous system. These stages reflect the emergence of three distinct subsystems, which are phylogenetically ordered and behaviorally linked to social engagement, mobilization, and immobilization. The phylogenetic orientation focuses our interest on the parasympathetic neural structures and neurobehavioral systems that we share with or have adapted from our phylogenetic ancestry. With increased neural complexity, due to phylogenetic development, the organism's behavioral and affective repertoire is enriched.
The polyvagal theory emphasizes the phylogenetic origins of brain structures that regulate social and defensive behaviors. For example, prosocial behaviors cue others that the environment is safe. Safe environments signal the individual to dispense with the hypervigilance required to detect danger and allows this precautionary strategy to be replaced with social interactions that further calm and lead to close proximity and physical contact. The prototypical prosocial behaviors in mammals are related to nursing, reproduction, interactive play, and being able to be calm in the presence of another. In contrast, defensive behaviors could be categorized into two domains: one related to mobilization including fight-or-flight behaviors and the other related to immobilization and death feigning that might be associated with dissociative psychological states. Within this dichotomy of defensive strategies, freezing behavior that requires increased muscle tension in the absence of movement, such as stalking or vigilance behaviors, is categorized within mobilization. In contrast, immobilization is associated with a decrease in muscle tension and often with fainting and other features of decreased metabolic activity. From a health perspective, the prosocial behaviors trigger neurophysiological circuits that not only support affect regulation and social interactions but also promote health, growth, and restoration.
Relevant to adaptive social and emotional behaviors, the polyvagal theory makes the following assumptions.
1. Evolution has modified the structures of the autonomic nervous system.
2. The mammalian autonomic nervous system retains vestiges of phylogenetically older autonomic nervous systems.
3. Emotional regulation and social behavior are functional derivatives of structural changes in the autonomic nervous system due to evolutionary processes.
4. In mammals, the autonomic nervous system response strategy to challenge follows a phylogenetic hierarchy, starting with the newest structures and, when all else fails, reverting to the most primitive structural system.
5. The phylogenetic stage of the autonomic nervous system determines the behavioral, physiological, and affective features of reactivity to people and objects in the environment.
The phylogenetic orientation focuses our interest on the parasympathetic neural structures and neurobehavioral systems that we share with or have adapted from our phylogenetic ancestry. First, there are three response systems proposed in the polyvagal theory: (1) cranial nerves to regulate the face and to mediate calm autonomic and behavioral states, (2) sympathetic-adrenal system to increase metabolic output, and (3) an inhibitory vagal system to decrease metabolic output and promote immobilization and defecation. These three response strategies are the products of distinct neurophysiological systems. Second, these distinct neurophysiological systems represent a phylogenetically dependent hierarchy with the use of cranial nerves to regulate facial expression emerging in mammals (well developed in primates), the sympathetic-adrenal system shared with other vertebrates including reptiles, and the inhibitory vagal system shared with more primitive vertebrates, including amphibians, bony fish, and cartilaginous fish (see chapters 10 and 11). The three systems represent different phylogenetic stages of neural development. This phylogenetic development starts with a primitive behavioral inhibition system, progresses to a fight-or-flight system, and, in humans (and other primates), culminates in a complex facial gesture and vocalization system. Thus, from a phylogenetic perspective, the nervous system of vertebrates evolved to support a greater range of behaviors and physiological states, including states that we often associate with social engagement behaviors.
HOW THE "MAMMALIAN" AUTONOMIC NERVOUS SYSTEM FOSTERS PROSOCIAL BEHAVIORS VIA A VAGAL BRAKE
The mammalian vagus (i.e., myelinated efferent pathways) functions as an active vagal brake (see chapter 7) in which rapid inhibition and disinhibition of vagal tone to the heart can support behavioral mobilization or self-soothe and calm an individual. When the vagal tone to the pacemaker is high, the vagus acts as a restraint or brake limiting heart rate. When vagal tone to the pacemaker is low, there is little or no inhibition of the pacemaker. Due to vagal influences to the sinoatrial node (i.e., the heart's pacemaker), resting heart rate is substantially lower than the intrinsic rate of the pacemaker. Neurophysiologically the vagal brake provides a mechanism to support the metabolic requirements for mobilization and communication behaviors; functionally, the vagal brake, by modulating visceral state, enables the individual to rapidly engage and disengage with objects and other individuals and to promote self-soothing behaviors and calm behavioral states. Thus, withdrawal of the vagal brake is associated with adaptive states of mobilization and a reinstatement of the vagal brake with calm behavioral recovery. In mammals, the primary vagal inhibitory pathways occur through the myelinated vagus originating in the nucleus ambiguus.
By transitory down-regulation of the cardioinhibitory vagal tone to the heart (i.e., removing the vagal brake), the mammal is capable of rapid increases in cardiac output without activating the sympathetic-adrenal system. This enables the ability to rapidly shift states from calm engagement to precautionary states of vagal withdrawal that rapidly increase cardiac output to support movements. But unlike the sympathetic-adrenal strategy, which is slow to initiate and slower to dampen, re-engaging the vagal brake instantaneously down-regulates cardiac output to produce a calm physiological state (Vanhoutte & Levy, 1979). By withdrawing the vagal brake, rather than stimulating the sympathetic-adrenal system, mammals have an opportunity to rapidly increase metabolic output for immediate, but limited mobilization. If the duration and intensity of mobilization is increased, the sympathetic nervous system is activated.
A withdrawal of the vagal brake will facilitate the recruitment of other neural mechanisms (e.g., excitation of sympathetic or the unmyelinated vagal path-ways) and neural chemical mechanisms (e.g., stimulation of the hypothalamic-pituitary-adrenal axis) to regulate physiological state. Thus, consistent with the polyvagal theory, if the vagal brake is not functioning or will not serve the survival needs of the organism, then the phylogenetically "older" systems (e.g., the sympathetic-adrenal system or unmyelinated vagus originating in the dorsal motor nucleus of the vagus) will be recruited to regulate metabolic output to deal with environmental challenges. For example, if the vagal brake is not functioning, there is the potential for greater dependence on the sympathetic excitation of the cardiovascular system. This dependence on sympathetic excitation to regulate cardiac output may create health risks (e.g., hypertension) and lead to difficulties in modulating behavioral state (i.e., rage, panic, aggression). Consistent with assumptions of the polyvagal theory, the vagal brake contributes to the modulation of cardiac output by decreasing or increasing the inhibitory vagal control of the heart to influence rate and thereby adjust metabolic resources to support either mobilization or social engagement behaviors.
THE SOCIAL ENGAGEMENT SYSTEM
As mammals evolved from more primitive vertebrates, a new circuit emerged to detect and to express signals of safety in the environment (e.g., to distinguish and to emit facial expressions and intonation of vocalizations) and to rapidly calm and turn off the defensive systems (i.e., via the myelinated vagus) to foster proximity and social behavior. This recent neural circuit can be conceptualized as a social engagement system. The Social Engagement System involves pathways traveling through several cranial nerves (i.e., V, VII, IX, X, and XI) that regulate the expression, detection, and subjective experiences of affect and emotion. Neuroanatomically, this includes special visceral efferent pathways regulating the striated muscles of the face and head (i.e., special visceral efferent) and the myelinated vagal fibers regulating the heart and lungs (see chapters 11 and 12; Porges, 2001a).
The social engagement system is an integrated system with both a somatomotor component regulating the striated muscles of the face and a visceromotor component regulating the heart via a myelinated vagus. The system is capable of dampening activation of the sympathetic nervous system and hypothalamic-pituitary-adrenal axis activity. By calming the viscera and regulating facial muscles, this system enables and promotes positive social interactions in safe contexts.
The somatomotor component includes the neural structures involved in social and emotional behaviors. Special visceral efferent nerves innervate striated muscles, which regulate the structures derived during embryology from the ancient gill arches (Truex & Carpenter, 1969). The social engagement system has a control component in the cortex (i.e., upper motor neurons) that regulates brainstem nuclei (i.e., lower motor neurons) to control eyelid opening (e.g., looking), facial muscles (e.g., emotional expression), middle ear muscles (e.g., extracting human voice from background noise), muscles of mastication (e.g., ingestion), laryngeal and pharyngeal muscles (e.g., prosody of vocalizations), and head-turning muscles (e.g., social gesture and orientation). Collectively, these muscles function as neural gatekeepers detecting and expressing features of safety (e.g., prosody, facial expression, head gestures, eye gaze) that cue others of intention and control social engagement with the environment.
The phylogenic origin of the behaviors associated with the social engagement system is intertwined with the phylogeny of the autonomic nervous system. As the muscles of the face and head emerged as social engagement structures, a new component of the autonomic nervous system (i.e., a myelinated vagus) evolved that was regulated by nucleus ambiguus, a medullary nucleus ventral to the dorsal motor nucleus of the vagus. This convergence of neural mechanisms produced an integrated social engagement system with synergistic behavioral and visceral components as well as interactions among ingestion, state regulation, and social engagement processes. As a cluster, difficulties in gaze, extraction of human voice, facial expression, head gesture, and prosody are common features of individuals with autism and other psychiatric disorders in which the social engagement system is compromised. Thus, we infer from the functioning of the face and the prosody of the voice, difficulties in both social engagement behaviors and physiological state regulation.
There are interneuronal connections between the source nuclei (i.e., lower motor neurons) of special visceral efferent pathways and the source nucleus of the myelinated vagus. These neurophysiological circuits provide an inhibitory pathway to slow heart rate and lower blood pressure, which, by actively reducing autonomic arousal, promote the calm states necessary to express social engagement behaviors and to support health, growth, and restoration. The brainstem source nuclei of this system are influenced by higher brain structures and by visceral afferents. Direct corticobulbar pathways reflect the influence of frontal areas of the cortex (i.e., upper motor neurons) on the medullary source nuclei of this system. Moreover, feedback through the afferent vagus (e.g., tractus solitarius) to medullary areas (e.g., nucleus of the solitary tract) influences both the source nuclei of this system and the forebrain areas that are assumed to be involved in several psychiatric disorders (e.g., Craig, 2005; Thayer & Lane, 2000). In addition, the anatomical structures involved in the social engagement system have neurophysiological interactions with the hypothalamic-pituitary-adrenal axis, the social neuropeptides (e.g., oxytocin and vasopressin), and the immune system (see chapter 19; Carter, 1998; Porges 2001b).
Afferents from the target organs of the social engagement system, including the muscles of the face and head, provide potent afferent input to the source nuclei regulating both the visceral and somatic components of the social engagement system. Thus, activation of the behavioral component (e.g., listening, ingesting, looking) could trigger visceral changes that would support social engagement, while modulation of visceral state, depending on whether there is an increase or decrease in the influence of the myelinated vagal efferents on the sinoatrial node (i.e., increasing or decreasing the influence of the vagal brake) would either promote or impede social engagement behaviors. For example, stimulation of visceral states that would promote mobilization (i.e., fight-or-flight behaviors) would impede the ability to express social engagement behaviors.
Relevant to psychiatric disorders are the specific deficits in both the somatomotor (e.g., poor gaze, low facial affect, lack of prosody, difficulties in mastication) and visceromotor (difficulties in autonomic regulation resulting in cardiopulmonary and digestive problems) components of the social engagement system. For example, clinicians and researchers have documented these deficits in individuals with autism. Thus, deficits in the social engagement system would compromise spontaneous social behavior, social awareness, affect expressivity, prosody, and language development. In contrast, interventions that improve the neural regulation of the social engagement system, hypothetically would enhance spontaneous social behavior, state and affect regulation, reduce stereotypical behaviors, and improve vocal communication (i.e., including enhancing both prosody in expressive speech and the ability extract human voice from background sounds). This is more than a plausible hypothesis. We not only have demonstrated relations between vagal regulation of the heart and social engagement behaviors, but have demonstrated in preliminary studies that is possible to improve social engagement behaviors in autistic individuals by engaging the neural regulation of the social engagement system (i.e., stimulating the neural regulation of the middle ear muscles with exaggerated prosodic acoustic stimulation), thus providing an empirical basis to understand the interpersonal social features, such as prosody and facial expressivity, that characterize individuals who effectively calm and sooth others.
DISORDERS OF THE SOCIAL ENGAGEMENT SYSTEM: MALADAPTIVE OR ADAPTIVE BEHAVIORAL STRATEGIES?
Several psychiatric and behavioral disorders are characterized as having difficulties in establishing and maintaining relationships. Diagnostic features often include features associated with difficulties both in expressing social behavior and in reading social cues (i.e., social awareness). These features are observed in a variety of psychiatric diagnoses, including autism, social anxiety, and post-traumatic stress disorder. From a psychopathology orientation, these clinical disorders have different etiologies and features. However, from a "polyvagal" perspective, they share a core component. This core component is characterized by a depressed social engagement system with the consequences of poor affect regulation, poor affect recognition, and poor physiological state regulation. Although a compromised social engagement system results in "maladaptive" social behavior, do these asocial behavioral strategies have "adaptive" features? The phylogeny of the vertebrate autonomic nervous system serves as a guide to understand these adaptive features.
Through the lens of the polyvagal theory, the vertebrate autonomic nervous system follows three general stages of phylogenetic development. In the mammalian autonomic nervous system the structures and circuits representing each of the stages remain, but have been co-opted for various adaptive functions. The neural circuit associated with each stage supports a different category of behavior with the phylogenetically most recent innovation (i.e., the myelinated vagus) capable of supporting high levels of social engagement behavior. Since the neural regulation of the "new" mammalian myelinated vagus (i.e., ventral vagus) is integrated into the social engagement system, when the social engagement system is compromised, the effects are both behavioral and autonomic. The resultant changes in autonomic state compromise spontaneous social engagement behaviors and minimize states of calmness, but support a range of adaptive defensive behaviors. Specifically, the compromised social engagement system (see Figure 3.1, p. 56) is associated, neurophysiologically, with a change in autonomic regulation characterized by a reduction in the influence of the myelinated vagus on the heart resulting in difficulties in behavioral state regulation and with a loss of neural regulation to the muscles of the face mediating the flat affective expression often observed in several clinical disorders. The removal of the regulatory influence of the myelinated vagus on the heart potentiates (i.e., disinhibits) the expression of the two phylogenetically older neural systems (i.e., sympathetic nervous system, unmyelinated vagus). These two older neural systems foster mobilization behaviors of fight or flight, via the sympathetic nervous system, or immobilization behaviors of death feigning, freezing, and behavioral shutdown via the unmyelinated vagus. Thus, withdrawal of the myelinated vagal circuit provides access to the more primitive adaptive defensive systems at a cost. If the removal is prolonged, there is an increased risk for both physical (e.g., risk for cardiovascular disorders) and mental (e.g., anxiety disorders, depression) illness as the protective antistress and self-soothing features of the myelinated vagus and the associated prosocial features of the social engagement system are lost.
NEUROCEPTION: CONTEXTUAL CUEING OF ADAPTIVE AND MALADAPTIVE PHYSIOLOGICAL STATES
To effectively switch from defensive to social engagement strategies, the mammalian nervous system needs to perform two important adaptive tasks: (1) assess risk, and (2) if the environment is perceived as safe, inhibit the more primitive limbic structures that control fight, flight, or freeze behaviors. In other words, any intervention that has the potential for increasing an organism's experience of safety has the potential of recruiting the evolutionarily more advanced neural circuits that support the prosocial behaviors of the social engagement system.
The nervous system, through the processing of sensory information from the environment and from the viscera, continuously evaluates risk. Since the neural evaluation of risk does not require conscious awareness and may involve subcortical limbic structures (e.g., Morris, Ohman, & Dolan, 1999), the term neuroception (see chapter 1) was introduced to emphasize a neural process, distinct from perception, that is capable of distinguishing environmental (and visceral) features that are safe, dangerous, or life-threatening. In safe environments, autonomic state is adaptively regulated to dampen sympathetic activation and to protect the oxygen-dependent central nervous system, and especially the cortex, from the metabolically conservative reactions of the dorsal vagal complex. However, how does the nervous system know when the environment is safe, dangerous, or life-threatening and what neural mechanisms evaluate this risk?
Neuroception might involve feature detectors involving the temporal cortex (see later discussion), since the temporal cortex responds to familiar voice and faces and hand movements and can influence limbic reactivity. Thus, the neuroception of familiar individuals and individuals with appropriately prosodic voices and warm expressive faces translates into a social interaction promoting a sense of safety. In most individuals (i.e., without a psychiatric disorder or neuropathology) the nervous system evaluates risk and matches neurophysiological state with the actual risk of the environment. When the environment is appraised as being safe, the defensive limbic structures are inhibited enabling social engagement and calm visceral states to emerge. In contrast, some individuals experience a mismatch and the nervous system appraises the environment as being dangerous, even when it is safe. This mismatch results in physiological states that support fight, flight, or freeze behaviors, but not social engagement behaviors. According to the theory, social communication can be expressed efficiently through the social engagement system, only when these defensive circuits are inhibited. Neuroception represents a neural process that enables humans to engage in social behaviors by distinguishing safe from dangerous contexts. Neuroception is proposed as a plausible mechanism mediating both the expression and the disruption of positive social behavior, emotion regulation, and visceral homeostasis.
New technologies, such as functional magnetic resonance imaging, have identified specific neural structures that are involved in detecting risk. The temporal lobe is of particular interest in expanding the construct of neuroception and in identifying neural mechanisms that, by detecting and evaluating risk, modulate the expression of adaptive defensive behaviors and autonomic states. Functional imaging techniques document the involvement of the temporal cortex, fusiform gyrus, and superior temporal sulcus in the evaluation of biological movement and intention including the detection of features such as movements, vocalizations, and faces, which contribute to an individual being perceived as safe or trustworthy (Adolphs, 2002; Winston, Strange, O'Doherty, & Dolan, 2002). Slight changes in these stimuli can be appraised as posing threat or alternatively signally endearment. Connectivity between these areas of the temporal cortex and the amygdala suggests a top-down control in the processing of facial features that could inhibit activity of the structures involved in the expression of defensive strategies (Pessoa, McKenna, Gutierrez, & Ungerleider, 2002).
Based on the relative risk of the environment, both social engagement and defense behaviors may be interpreted as either adaptive or maladaptive. For example, the inhibition of defense systems by the social engagement system would be adaptive and appropriate only in a safe environment. From a clinical perspective it would be the inability to inhibit defense systems in safe environments (e.g., anxiety disorders, post-traumatic stress disorder, reactive attachment disorder) or the inability to activate defense systems in risk environments (e.g., Williams syndrome, a genetic disorder with a behavioral repertoire characterized by engaging without detecting or respecting the emotional state of others) that might contribute to the defining features of psychopathology. Thus, an invalid neuroception of safety or danger might contribute to maladaptive physiological reactivity and the expression of the defensive behaviors associated with specific psychiatric disorders that include in their diagnostic criteria a social deficit (e.g., autism, social anxiety, Williams syndrome) or fear (e.g., various phobias, obsessive-compulsive disorder) (Leckman et al., 1997). However, in most individuals neuroception accurately reflects risk, and there is a consistency between the cognitive awareness of risk and the visceral response to risk.
The features of risk in the environment do not solely drive neuroception. Afferent feedback from the viscera provides a major mediator of the accessibility of prosocial circuits associated with social engagement behaviors. For example, the polyvagal theory predicts that states of mobilization would compromise our ability to detect positive social cues. Functionally, visceral states color our perception of objects and others. Thus, the same features of a person engaging another may result in range of outcomes, depending on the physiological state of the target individual. If the person being engaged is in a state in which the social engagement system is easily accessible, the reciprocal prosocial interactions are likely to occur. However, if the individual is in a state of mobilization, the same engaging response might be responded to with the asocial features of withdrawal or aggression. In such a state, it might be very difficult to dampen the mobilization circuit and enable the social engagement system to come back online.
The insula may be involved in the mediation of neuroception, since it has been proposed as a brain structure involved in conveying the diffuse feedback from the viscera into cognitive awareness. Functional imaging experiments have demonstrated that the insula has an important role in pain experience and the experience of several emotions, including anger, fear, disgust, happiness, and sadness. Critchley, Wiens, Rothstein, Ohman, and Dolan (2004) propose that internal body states are represented in the insula and contribute to subjective feeling states and have demonstrated that activity in the insula correlated with interoceptive accuracy.
CO-OPTING THE IMMOBILIZATION DEFENSE SYSTEM FOR REPRODUCTIVE BEHAVIORS, NURSING, AND THE FORMATION OF SOCIAL BONDS
Immobilization as a defense system is phylogenetically old and is associated with reduced metabolic demands and increased pain threshold. In reptiles, with a larger tolerance for reductions in oxygen, immobilization is a very effective defense strategy. In contrast, since mammals have a great need for oxygen, the inhibition of movement coupled with a shift in autonomic state to support the immobilization behavior (i.e., apnea and bradycardia) can be lethal (Hofer, 1970; Richter, 1957) such that death feigning can lead to death. In humans, fainting or dissociating in anticipation of death or painful injury reflects a less extreme form of this response.
However, several aspects of mammalian social behavior require immobilization, but do so in the absence of life threat. In these contexts an immobilization without fear is required. Immobilization without fear is accomplished by co-opting the structures that regulate immobilization in response to life threat to serve a broad range of social needs, including reproduction, nursing, and pair-bonding. The area of the periaqueductal gray that coordinates immobility as a primitive defense system has been modified in mammals to serve their intimate social needs. In addition, it has been reported that the ventral lateral portion of the periaqueductal gray is rich in receptors for oxytocin, a neuropeptide associated with parturition, nursing, and the establishment of pair-bonds (Carter, 1998; Insel & Young 2001).
CO-OPTING THE MOBILIZATION DEFENSE SYSTEM FOR PLAY
Often the playful "rough and tumble" behaviors observed in mammals are interpreted as preliminary exercises to develop adaptive defensive and aggressive behaviors. However, play is also inherently motivating and provides a unique and positive experience (Panksepp, 1998). Play, at least rough and tumble play, is characterized by mobilization. Thus, play shares with the defensive fight-or-flight behaviors a neurophysiological substrate that functionally increases metabolic output by increasing sympathetic excitation. Concurrent with the sympathetic excitation is a withdrawal of the myelinated vagal pathways that characterize the vagal brake. Just as the primitive mechanisms mediating immobilization in response to life threat can be co-opted to support loving and nutrient processes, so can mobilization mechanisms be involved occur to facilitate both defensive flight-or-fight behaviors and pleasurable "play."
How is play distinguished from aggressive behavior? More important, are there "neuroceptive" processes that either dampen or potentiate aggressive retaliation? If we observe play, we can reliably observe cues lead to either aggression or calming. Frequently play leads to acts that are painful and potentially aggressive. For example, often a playmate is injured. This may occur with various mammalian species. When dogs play, they may bite too hard and elicit a painful cry in the playmate. When a human is playing with a dog, the dog might accidentally hit in a vulnerable and tender place like the nose. When humans play a sport such as basketball, an individual may be hit with an elbow to the face. How are these situations diffused? What processes enable anger to be contained and play to be resumed?
Access to the social engagement system can transform potential aggression to play. The social engagement system cues others that the "intentionality" of the behavior is benign. For example, a fight is likely to occur if the individual who accidentally hits another in the face while playing basketball walks away without diffusing the tension through a face-to-face expression of concern. Similarly, play will not continue if dogs playing do not make face-to-face engagements after an accidental, but hurtful bite. Consistent with the importance of the social engagement system in the process of play, autism is associated with noninteractive (i.e., parallel) play. Thus, access to the social engagement system is critical in defining mobilization as play and not aggression. Team sports, which are prevalent in our culture, involve mobilization strategies that require face-to-face interactions to signal intentionality and share a common feature of integrating the social engagement system with mobilization.
Jogging and other forms of exercise also result in a physiological state similar to team sports or rough and tumble play. However, unlike exercise, a "polyvagal" definition of play requires reciprocal interactions and a constant awareness of the actions of others. Play is different than fight-or-flight behaviors. Although fight-or-flight behaviors often require an awareness of others, they do not require reciprocal interactions and an ability to restrain mobilizations. Play recruits another circuit that enables aggressive and defensive behaviors to be contained. The rapid recruitment of the social engagement system results in an immediate face-to-face evaluation of whether there is intentionality in the event that provoked the painful response. Areas of the cortex, such as superior temporal sulcus, provide a plausible location for this neuroceptive process. The superior temporal sulcus has been proposed to be an area of the brain that evaluates biological movement and intentionality. Thus, familiar voices, calming gestures, and appropriate facial expression can rapidly diffuse a possible physical conflict. Even the dogs that whimpers after being hit on the nose or bit on the leg while playing will rapidly make a face-to-face engagement and wait for a gesture that will provide the reassurance that the event was not intentional.
How does the social engagement system calm us down and keep us from expressing inappropriate aggressive acts? First, there are inhibitory pathways from the temporal cortex that dampen the limbic reactivity associated with defensive behaviors. Second, as Gellhorn (1964) noted almost 50 years ago, facial muscle activity influences the brain structures that regulate visceral state. This is frequently observed in humans of all ages from the very young infants who use sucking behaviors to calm to older individuals who use conversation, listening, smiling, and ingesting to calm. Consistent with and in contrast to these strategies of defusing conflict, walking away or turning the head away from the conflict can trigger a violent reaction.
By investigating the unique physiological mechanisms involved in play, we uncover the unique properties of reciprocal interactions that may define play and distinguish it from exercise and other solitary behaviors. Play requires turn taking in expressive motor movements and reciprocal receptive inhibition of activity. This is also observed in talking and listening, in throwing and catching, and in hiding and seeking. When there is mutual activity and contact, such as in rough and tumble play, there are more opportunities for cues to be mistaken and aggressive behaviors to unfold. However, if face-to-face engagement occurs rapidly with the appropriate features of concern and empathy, then the physiological state that was driven by the physical contact is evaluated for intentionality and diffused with the appropriate cues exchanged between two social engagement systems involved in the face-to-face exchange. Although play may share some of the neural mechanisms involved in fight-or-flight behaviors, unlike solitary exercise, play requires dynamic neural regulation of state to ensure safe interactions. Thus, both sympathetic activation to increase metabolic output to support motor activity and the vagal brake to restrain mobilization and to support the function of the social engagement system are recruited to maintain a mutual playful activity. Another adaptive process involves the coactivation of sympathetic excitatory and vagal inhibitory processes. This process is associated with sexual arousal, another vulnerable state that evolutionarily requires face-to-face interactions to evaluate intentionality of physical contact to determine whether the behaviors are caring or hurtful.
SUMMARY STATEMENTS
The polyvagal theory is an attempt to reorganize our conceptualization of the autonomic nervous system with a focus on the specific neural circuits involved in regulating visceral organs for specific adaptive functions, including the domains of affect, emotions, and goal-directed behaviors. The theory identifies specific variables that can be used to dynamically evaluate the changing neural regulation of specific adaptive circuits. Implicit in the theoretical model are four prominent features that impact directly on the development of testable hypotheses: (1) the role specific brain structures and neural circuits have in regulating autonomic state, (2) the justification of developing methods that can distinguish and track the dynamic vagal output to target organs through the myelinated vagus originating in the nucleus ambiguus and the unmyelinated vagus originating in the dorsal motor nucleus, (3) the role visceral afferents and sensory feature detectors have on the switching among the neural circuits regulating autonomic state, and (4) the relation between the regulation of visceral organs and the regulation of the striated muscles of the face and head involved in social engagement behaviors including affect recognition and emotional expression.
The polyvagal theory suggests that affective and emotional states are dependent on lower brain regulation of the visceral state and the important visceral, tactile, and nocioceptive cues that travel between the brain and the periphery. Through the lens of the polyvagal theory, specific bodily states foster different domains of behavior. Specifically, the neural regulation of five physiological states has been described and each state has been linked with a specific biologically based behavioral repertoire.
1. Social engagement: a state dependent on a well-defined social engagement system. This system promotes positive social interactions, reduces psychological distance, and promotes a sense of safety between people.
2. Mobilization—fight-or-flight: This state supports fight-or-flight behaviors and requires an increase in metabolic output.
3. Play: a blend of the above. Play is a hybrid state requiring features from both states of mobilization and social engagement.
4. Immobilization—life threat: This state is associated with life threat and is characterized by a reduction of metabolic output and shutdown behaviors. This primitive neural circuit works fine for reptiles but is potentially lethal in mammals.
5. Immobilization without fear: This state is associated with prosocial and positive states that require a reduction of movement without the massive reduction of metabolic resources. This circuit recruits pathways from the immobilization circuit and is used during nursing, childbirth, and reproductive behaviors and digestive and restorative processes.
Functionally, these five states color our perception of objects and others. Thus, the same features of a person engaging another may result in range of outcomes, if the target individual is in a different physiological state. If the person being engaged is in a state in which the social engagement system is easily accessible, the reciprocal prosocial interactions are likely to occur. If the individual is in a state of mobilization, the same engaging response might be responded to with the asocial features of withdrawal or aggression. This stimulus-organism-response (S-O-R) model is reminiscent of Woodworth (1928), who postulated an S-O-R model with an active organism intervening between stimulus and response. In the Woodworth model, processes internal to the organism mediated the effects of stimuli on behavior. Within the polyvagal theory, neuroception is an S-O-R model. Within this context, autonomic state is an intervening process that contributes to the transformation from the external physical stimulus to the complex internal cognitive-affective processes that determine the quality of the interpersonal interaction.
The five states described provide a good fit with the underlying physiological states required to successfully express the seven neural-based systems described by Panksepp (1998). Moreover, the polyvagal perspective, with its emphasis on phylogenetic shifts in visceral regulation, provides a unique insight into the use of psychological constructs. For example, the polyvagal theory will lead to three different visceral phenotypes for the emotion of fear. One type is characterized by mobilization strategies consistent with the features of fight-or-flight behaviors. A second type is characterized by immobilization (e.g., death feigning), a biobehavioral state that, due to metabolic depression, can potentially be lethal for a mammal. In humans this might be observed as fainting, defecating, and/or dissociating. A third type is more cognitive and involves a transitory depression of the social engagement system as a precautionary response to evaluate intentionality behaviors. If behavior is detected as dangerous, then the sympathetic nervous system is activated to support the fight-or-flight mobilization behaviors. All three are "fear" responses, but they have different behavioral topographies and different underlying neurophysiological substrates. Thus, the understanding of affective experiences and the strategy of organizing of these experiences into psychological constructs such as "emotions" may be informed by understanding the covariation between the specific phylogenetic shifts in the neural regulation of the viscera and the adaptive nature of these various affective states in phylogenetically older vertebrates
CONCLUDING COMMENTS
To optimize strategies studying the bridge between nervous system function and both clinical disorders and affective experiences, affective neuroscience will need to incorporate methodologies and to test hypotheses dependent on our expanding knowledge of neurophysiology and the central structures involved in both appraisal of context (i.e., neuroception) and neural regulation of visceral state. These questions have motivated previous (e.g., Cannon, Darwin, James, Gellhorn, Hess) and contemporary researchers (e.g., Critchley, 2005; Ekman, Levenson, & Friesen, 1983; Thayer & Lane, 2000) attempting to bridge the gap between visceral states and the subjective labels of affective experiences (i.e., emotions). To close this gap, new methodologies are necessary that are capable of evaluating dynamic changes in and interactions among various physiological (e.g., respiration, heart rate, blood pressure, vasomotor tone, and motor activity) variables in a changing context. In response to these needs, the polyvagal theory was developed.
The polyvagal theory provides a perspective to demystify features of clinical disorders. The theory provides principles to organize previously assumed disparate symptoms observed in several psychiatric disorders (i.e., a compromise in the function of the social engagement system). Moreover, by explaining features of disorders from an adaptive perspective, interventions may be designed that trigger the neural circuits that will promote spontaneous social engagement behaviors and dampen the expression of defensive strategies that disrupt social interactions.
* * *
*In 1949 W. Hess was awarded the Nobel Prize in Physiology or Medicine (http://nobelprize.org/ nobel_prizes/medicine/laureates/1949/hess-lecture.html). His Nobel lecture was titled "The Central Control of Activity of Internal Organs." In his lecture, he acknowledged the importance of the prevailing model of the autonomic nervous system, which emphasized the paired-antagonistic innervations of the internal organs and the definition of sympathetic and parasympathetic functions. However, he went well beyond this conceptualization to emphasize the importance of central structures in the regulation of visceral state by describing his studies that demonstrated the influence of the hypothalamus on the autonomic nervous system. By emphasizing the central mechanisms that mediate the dynamic regulation of peripheral organs, Hess anticipated the need for methodologies and technologies to continuously monitor the neural circuits involving both defined brain structures and peripheral nerves in the regulation of visceral function and state and to move from the prevailing conceptualization of the autonomic nervous system as a peripheral system. Hess's lecture (1) emphasized the importance of feedback circuits linking peripheral organs to brain structures and the bidirectionality of these feedback circuits, and (2) acknowledged that although much can be learned about neural structures and functions via traditional experimental paradigms (e.g., neural blockade, surgery, electrical stimulation), dynamic feedback circuits, that is, the moment-to-moment dynamic shifts in the system, cannot be adequately studied through these paradigms.
CHAPTER 19
Neurobiology and Evolution: Mechanisms, Mediators, and Adaptive Consequences of Caregiving
DEFINING FEATURES OF CAREGIVING
Caregiving includes providing food, protection, or other resources. However, caregiving also may extend beyond these physical elements to include social support to fulfill needs for affiliation and perceived safety. Most mammals, including humans and rodents, are altricial at birth and caregiving is necessary to compensate for the infant's undeveloped motor and autonomic nervous systems. Due to an immature corticospinal motor system, the infant is incapable of independently obtaining food or protecting from a predator. Due to an immature autonomic nervous system, the infant is incapable of independently thermoregulating to maintain a necessary body temperature to survive. Thus, the mature nervous system of the caregiver becomes intertwined with the undeveloped nervous system of the infant to create a model of "symbiotic regulation." The caregiver becomes part of a complex feedback system supporting the biological and behavioral needs of the infant. Within this model of symbiotic regulation, the caregiver is not solely giving to the infant. The behaviors of the infant also trigger specific physiological processes (e.g., neural and endocrine feedback circuits) that help establish strong bonds, provide emotional comfort for the caretaker, stimulate neural pathways, and support the health of the caregiver.
Through the process of maturation, motor and autonomic systems change. As the infant matures, there is a transition from a dependence on the caregiver for the regulation of biobehavioral processes to a greater degree of self-regulation. However, throughout the life span most mammals will continue to be dependent on others to maintain optimal well-being and state regulation (Hrdy, 2008). In some (but not all) mammalian species, caregiving is based on or induces selective and reinforcing emotional social relationships and bonds (see chapter 11; Carter. 1998). When enduring bonds are present, devastating reactions to separation or loss are to be expected (Bowlby, 1988).
Caregiving may or may not be reciprocal. However, reciprocity and the spontaneous reversal of the roles of giving and receiving are positive features of strong relationships and are the optimal features of symbiotic regulation. Conversely, a lack of reciprocity often signals distressed and vulnerable relationships. The inability of an individual to enter and to maintain reciprocal social relations is a feature of several psychiatric disorders (Teicher et al., 2003).
When a mammalian mother initially interacts with her offspring, usually she has just given birth and must provide milk to nurture the newborn. The onset of maternal caregiving is normally closely associated with birth and lactation. The physical events of birth and lactation provide endocrine windows of opportunity for the establishment of strong social bonds. Thus, the hormones of birth and lactation are plausible candidates to explain the beneficial effects of caregiving (Carter, 1998; Numan, 2007).
THE EVOLUTION OF A CAREGIVING SYSTEM: THE TRANSITION FROM "SELF-REGULATION" TO "OTHER REGULATION"
Evolutionary theories attempting to explain between- and within-species variation in social behavior tend to focus on ultimate causes and assumed selection pressures. These theories are based on ancient historical events and are limited to the fossil record. Thus, it is difficult to test evolutionary theories within the context of the expressed behavior or physiology of contemporary animals. However, a phylogenetic perspective that investigates the biological and behavioral shifts from reptiles to mammals illustrates several neurobiological features underlying sociality. For example, most behaviors associated with caregiving and prosociality in humans are uniquely mammalian and are not evident in reptiles. Differences among mammalian species, individual variation, and developmental changes in social contact and caregiving are common. Analyses of these variations provide experiments of nature, which cast a new light on the neurobiology of sociality.
In part, the phylogenetic transition from reptiles to mammals appears to be a shift from an organism capable of "self-regulation" to an organism that is dependent at certain points in development on "other regulation." It is within this phylogenetic transition, in which regulation by "other" becomes adaptive, that the neurobiology of sociality emerges. The defining features of the "other" in the mammalian model of regulation often have survival consequences that may involve various dimensions of support, including warmth, food, and protection. For most mammals, and especially humans, a developmental increase in self-regulation capacity parallels the development of specific features of the nervous system. With physical maturation, neural pathways from the cortex to the brainstem exhibit a greater efficiency in regulating the autonomic nervous system and enable the maintenance of physiological homeostasis in both safe and dangerous situations (Porges, 2001a). These maturational changes provide greater abilities to self-regulate and to reduce dependence on others.
THE PHYLOGENY OF MAMMALIAN SOCIAL ENGAGEMENT AND COMMUNICATION SYSTEMS
The mammalian autonomic nervous system retains three neural circuits, which are expressed in a phylogenetically organized hierarchy (Table 19.1). In this hierarchy of adaptive responses, the newest circuit, associated with social communication, is used first. If the newest circuit fails to provide safety, older survival-oriented circuits are recruited sequentially. It is important to note that social behavior, social communication, and visceral homeostasis are largely incompatible with neurophysiological states and behaviors that are regulated by circuits that support the defense strategies of both flight or fight and immobilization. Inhibition of systems, which are in general defensive or protective, is necessary to initiate social engagement and to allow positive social behaviors. Conversely, positive social behaviors may be inhibited during prolonged periods of adversity. However, systems that support sociality also may be protective against the costly or destructive effects of chronic fear or stress (Porges, 2001a, 2007b).
TABLE 19.1. Polyvagal Theory: Phylogenetic Stages of Neural Control
THE POLYVAGAL THEORY
Often overlooked, but especially critical to social and emotional behavior, is the autonomic nervous system (see chapter 2). The central nervous system cannot function without the support of visceral organs supplying oxygen and energy. The autonomic nervous system, via bidirectional pathways, regulates the viscera and conveys information upward to the hypothalamus, amygdala, and neocortex. Sensory information from the viscera contributes to what humans experience as "emotion" or "emotional states." These emotional states, in turn, are components of a "motivational" system that stimulates social engagement and allows sociality to be experienced as reinforcing. Often these emotional and motivational states involve other brain systems, including those that rely on dopamine and endogenous opioids.
Of particular importance to mammalian social behavior is the parasympathetic component of the autonomic nervous system. The evolution of this system in mammals permitted the emergence of complex social interactions and social communication. Embedded in the brainstem and associated cranial nerves are conserved and evolved elements that are essential for modern mammalian physiology and behavior (see chapters 2, 11, and 12; Porges, 2001a, 2007b). The vagus (Xth cranial) nerve is of special relevance, because it transmits and integrates complex bidirectional communication between the brain and peripheral organs involved in cardiovascular, respiratory, digestive, and immune functions. Critical to understanding mammalian sociality is knowledge of the origins of the neuranatomical pathways of the social engagement system, linking visceral state regulation and emotion to both the expressive and receptive domains of social communication.
In mammals, but not reptiles, the vagus nerve has two distinct efferent (motor) pathways with separate brainstem source nuclei (Table 19.1; see also Figure 3.1, p. 56). In addition, approximately 80% of vagal fibers are afferent pathways, transmitting sensory input from the viscera to the brainstem.
The evolution of the polyvagal system parallels the phylogenetic distinction between reptiles and mammals and includes a cluster of behaviors dependent on the function of structures that phylogenetically emerged first in mammals. These changes include detached middle ear bones, the advent of a diaphragm, and the myelinated vagal system regulating supradiaphragmatic organs. The latter vagal system is distinct from the unmyelinated vagus, which primarily regulates subdiaphragmatic organs. However, branches of both myelinated and unmyelinated vagus do reach the heart, where they coordinate the need for oxygen with the behavioral demands of a rapidly changing physical and social environment.
The phylogeny of the mammalian nervous system offers important clues to social behavior. In the transition from aquatic to terrestrial life, ancient gill (branchial) arches were co-opted to form the face and head, resulting in the modern versions of these structures, which included new or additional functions. Changes also were necessary in the muscles and nerves associated with the branchial arches. Taken together in modern mammals, including humans, these systems permit social engagement and social communication, including sucking, swallowing, facial expressions, and the production and receipt of airborne vocalizations.
Brainstem structures that regulate the efferent (motor) branch of the vagus have two source nuclei. The unmyelinated component of the vagus, which permits slowing of heart rate, originates in the dorsal motor nucleus of the vagus and provides the efferent component of the dorsal vagal complex. The unmyelinated vagus is shared by mammals with other vertebrates (i.e., reptiles, amphibians, teleosts, and elasmobrachs). The dorsal motor nucleus of the vagus is the source nucleus for the unmyelinated vagus, which carries neural impulses to and is capable of slowing heart rate. The more modern branch originates in the nucleus ambiguus of the ventral vagal complex and is myelinated, allowing rapid interaction between the brain and viscera. The myelinated vagus stabilizes cardiovascular function and is responsible for respiratory sinus arrhythmia (RSA), a rhythmic component of the heart rate pattern with a periodicity similar to spontaneous breathing. RSA is an index of the dynamic influence of the myelinated vagus on the heart. The ventral vagal complex also contains source nuclei for other facial nerves as well as autonomic functions. When RSA is withdrawn (reduced myelinated vagal tone), heart rate quickly accelerates. Thus, emotional states and social communication can be coordinated with visceral demands (see chapter 2).
Information from the myelinated vagus, measured by quantifying RSA, is cardioprotective and directly implicated in cortical oxygenation. RSA, due to its sensitivity to the influence of ventral vagal fibers on the heart is often used as an index of health and resilience in humans. The comparatively modern processes that supplied oxygen to the large primate cortex allowed the emergence of higher levels of cognitive functions. Such processes may be at the core of the uniqueness of human cognition. Furthermore, the myelinated vagus is associated in the brainstem with cranial nerves that innervate the face and head (see chapter 3). Thus, the myelinated vagal functions are coordinated with the neural regulation of the larynx and pharynx to coordinate sucking, swallowing, and breathing with vocalizations. The muscles of the human face, especially of the upper face involved in subtle emotional expressions, have projections from this system, which may be particularly important in social communication (see chapter 13).
THE NEUROANATOMY OF SOCIAL COGNITION
The expanding mammalian cortex set the stage for human cognition, speech, and more elaborate forms of caregiving beyond the maternal–infant interaction. Several features of anatomy that emerged concurrently with the expanding cortex provided the mechanisms necessary for mammalian social communication.
The classic definition of Mammalia focuses on the occurrence of mammary glands and hair. However, the fossil record relies on the identification of middle ear bones detached from the jawbone as the defining feature of Mammalia. The tiny middle ear bones form an ossicular chain that transmits sound vibrations from the ear drum to the inner ear. As the middle ear muscles tense or relax the middle ear structures change the frequency band that is transmitted to the inner ear. In safe environments, when there is no risk and defensive systems are down regulated, the ossicular chain is relatively tense. This state of stiffness in the ossicular chain enables mammals to process sounds in a frequency range that is higher than that available to reptiles. Thus, mammalian acoustic communication usually occurs at frequencies that cannot be heard by reptiles. The anatomical separation of the middle ear bones from the jawbone allows airborne auditory communication (see chapter 13). Concurrently, phylogenetic transitions in brainstem areas regulating the vagus are physically and functionally intertwined with the brainstem areas regulating the striated muscles of the face and head. The result of this transition is the emergence of a dynamic social engagement system with social communication features (e.g., head movements, production of vocalizations, and a selective ability to hear conspecific vocal communication).
In support of this new mammalian anatomy, a myelinated vagus emerged. The myelinated vagus could inhibit the sympathetic nervous system and the hypothalamic-pituitary-adrenal axis. This feature of the autonomic nervous system allowed mammals to inhibit mobilization and experience calm behavioral states. In this context it was possible for animals to engage in high levels of social interaction.
The phylogenetic transition from reptiles to mammals also resulted in a face–heart connection in which the striated muscles of the face and head were regulated in the same brainstem areas that evoked the calming influence of the myelinated vagus. The striated muscles of the face and head are involved in social cueing (e.g., facial expressions, vocalizations, listening, head gesture, etc.). These systems serve as "trigger" stimuli to the feature detectors in the nervous system that detect risk and safety in the environment (see discussion on neuroception). However, the expanded mammalian cortex also demands high levels of oxygen. Oxygenation of the cortex in mammals is accomplished in part through the same adaptations of the autonomic nervous system that permit elaborate forms of reciprocal sociality. These systems, including terrestrial lungs and a four-chamber heart, which support the oxygenation of the neocortex are also regulated in part by the myelinated branch of the vagus nerve.
This synergism of neural mechanisms in mammals allows for symbiotic regulation of behavior and the elaboration of reciprocal caregiving. These same systems provide setting conditions during which social behaviors can have a significant impact on cognition and health. In the human nervous system specific features of person-to-person interactions are innate triggers of adaptive biobehavioral systems, which in turn can support health and healing. In the absence of social interactions or under conditions of social adversity various forms of maladaptive behaviors and illness may be expressed.
NEUROCEPTION AND THE SOCIAL MANAGEMENT OF THREAT AND DANGER
The integrated functions of the myelinated vagus permit the expression of positive emotions and social communication. However, the nervous system also is constantly assessing risk in the environment through a process of "neuroception" (see chapter 1), during which specific neural circuits are triggered that may support defensive strategies of fight-or-flight or alternatively shutdown.
The mammalian nervous system has retained two primitive defensive neural circuits for either active or passive coping that can be called on in the presence of threat or danger. The fight-or-flight system allows mobilization and active coping. This system is supported by the sympathoadrenal systems, including the release of catecholamines and glucocorticoids, which increase available energy. However, active avoidance is not the only mechanism through which mammals may deal with threat or danger. Under some conditions, such as inescapable danger or other forms of extreme stress, mobilization strategies may be inhibited. This alternate defensive strategy is characterized by passive coping and immobility. Under more severe conditions many systems may be shut down, including those dependent on the neocortex. In these circumstances animals may show death-feigning and "helpless" behaviors.
The unmyelinated vagus tends to slow the heart, consistent with a reptilian adaptive strategy of freezing and conserving energy in the face of danger. However, mammals, with their large cortex, cannot maintain alertness without relatively high concentrations of oxygen. Thus, a prolonged slowing of heart rate could lead to unconsciousness and eventually death. Neurophysiological mechanisms exist for protecting the heart and brain from shutting down. As described shortly, the release of peptide hormones are among these neural mechanisms.
SOCIAL ENGAGEMENT SYSTEM
Critical to human social behavior are pathways from the five cranial nerves that control the muscles of the face and head. Collectively, these motor pathways are labeled as special visceral efferents. The special visceral efferent pathways regulate the muscles of mastication (e.g., ingestion), muscles of the middle ear (e.g., listening to vocalizations), muscles of the face (e.g., emotional expression), muscles of larynx and pharynx (e.g., vocal prosody and intonation), and muscles controlling head tilt and turning (e.g., gesture). The source nuclei of the circuits regulating the striated muscles of the face and head interact in the brainstem with the source nucleus of the myelinated vagus (i.e., nucleus ambiguus); together these form an integrated social engagement system. This system provides the neural structures involved in the expression of social and emotional behaviors and the feelings associated with these behaviors.
MAMMALIAN SOCIAL COMMUNICATION AND THE EVOLUTION OF SOCIAL COGNITION
Positive forms of communication, usually including speech and other vocalizations, are typically components of successful caregiving. Vocalizations also convey information regarding physiological state. For example, infant cries are indicators of health state and also can elicit caregiving. The coordinated regulation of social communication and visceral systems helps explain the relationship between positive social experiences and health.
Shared neural pathways underlie social communication and visceral functions, such as the regulation of the cardiovascular, digestive, and immune systems. For example, through the myelinated vagus, the brainstem regulates vocal communication (i.e., pathways controlling breath and the muscles of the larynx and pharynx) as well as heart rate. Thus, both heart rate, expressed in RSA, and the acoustic features of vocalizations, expressed in prosody, are parallel outputs of the integrated social engagement system.
A depressed social engagement system is characterized by low variability in both heart rate (i.e., low-amplitude RSA through the myelinated vagus) and vocal intonations (i.e., lack of prosody). Human voices that lack prosody fail to attract or interest others and are perceived as reflecting an emotionally detached or boring individual. In contrast, an optimally functioning social engagement system will have features of high variability in both heart rate (i.e., high-amplitude RSA) and vocal intonations (i.e., high prosody). Lack of prosody is a risk factor similar to low amplitude RSA and both may be used as indications of health risk.
Attention to features of acoustic vocalizations could provide insights into vagal regulation of the heart. For example, the positive features of vocal prosody in social interactions may be synchronized with RSA. The origins of these coordinated functions are most readily appreciated in the context of the evolution of neuroanatomical mechanisms for social engagement and social communication.
Collectively, the muscles of the face and head function as filters that limit social stimuli (e.g., observing facial features and listening to vocalizations) and determinants of engagement with the social environment. The neural control of these muscles determines social experiences by changing facial features (especially in humans and other primates), modulating laryngeal and pharyngeal muscles to regulate intonation of vocalizations (prosody), and coordinating both facial and vocal motor tone with respiratory actions. In addition, the frequency of breathing is encoded into the phrasing of vocalizations, which—independent of the content of speech—may express meaning. For example, urgency may be conveyed by short phrases associated with short exhalations (i.e., rapid breathing), while calmness would be conveyed by long phrases associated with long exhalations (i.e., slow breathing).
NEUROCHEMISTRY AND THE SOCIAL NERVOUS SYSTEM
Neuropeptides regulate sociality, emotion, and the autonomic nervous system. Social behaviors are supported and coordinated by both endocrine and autonomic processes (Grippo, Trahanas, Zimmerman, Porges, & Carter, 2009). The complex networks of biochemical systems necessary for reproduction and homeostasis also are implicated in social behavior. Given the energetic demands of social interactions, it is not surprising that the same neurotransmitters that are involved in social behavior also regulate the autonomic nervous system. Two mammalian hormones/neuromodulators, oxytocin and vasopressin, have been shown to be of particular importance to mammalian sociality. There is increasing evidence that the functions of these same molecules, especially oxytocin, are central to the causes and consequences of positive social behaviors, including sensitivity to social cues in others and constructs such as trust and caregiving (Heinrichs, von Dawans, & Domes, 2009).
Oxytocin and vasopressin are small neuropeptides that differ from each other in only two of nine amino acids (Landgraf & Neumann, 2004). Oxytocin is produced primarily in hypothalamic nuclei, including the supraoptic and paraventricular nuclei. Vasopressin is synthesized in the supraoptic and paraventricular nuclei as well as other brain regions implicated in the regulation of emotional behaviors, as well as circadian rhythms. In addition, and especially in males, vasopressin also is abundant in brain regions (e.g., amygdala, bed nucleus of the stria terminalis, and lateral septum) of particular importance to social and emotional regulation and self-defense (De Vries & Panzica, 2006).
Oxytocin and vasopressin are transported from the hypothalamus (i.e., supraoptic and paraventricular nuclei) to the mammalian posterior pituitary where they are released into the bloodstream and act as hormones on peripheral target tissues, such as the uterus or mammary tissue. Within the brain, these same chemicals also serve as neuromodulators, affecting a broad range of neural processes. Both oxytocin and vasopressin are capable of moving throughout the central nervous system, probably by passive diffusion (Landgraf & Neumann, 2004). Receptors for these molecules are found in various brain areas implicated in social behavior (Gimpl & Fahrenholz, 2001). In contrast to most biologically active compounds, oxytocin has only one receptor. Vasopressin has at least three distinct receptor subtypes with separable functions. However, the oxytocin peptide also may affect the vasopressin receptors and vice versa.
The neuroanatomy of the oxytocin system allows a coordinated effect on behavior, autonomic functions, and peripheral tissues. In some, but not all cases, vasopressin and oxytocin have opposite functions, possibly because they are capable of acting as antagonists to each other's receptors, while in other cases these peptides have similar effects. Dynamic interactions between oxytocin and vasopressin may in turn regulate physiology and behavior, allowing shifts between positive social behaviors and defensive states (Viviani & Stoop, 2008).
EVOLUTION OF OXYTOCIN AND VASOPRESSIN
It is likely that the essential elements on which sociality are based arose from physiological processes fundamental to the need to conserve water and minerals. Among these are adaptations allowing the transition from aquatic to terrestrial life, including internal fertilization and eventually pregnancy and placental reproduction. Although vasopressin is also known as the "antidiuretic hormone," both oxytocin and vasopressin influence kidney function in adults to conserve water and minerals. The capacity to maintain or reabsorb water was a critical element in the evolution of terrestrial mammals. The capacity for internal fertilization and the development of the placenta and lactation required a well-developed water regulation system. This shift also provided a protective environment for offspring before and after birth, and the emergence of contemporary versions of the neocortex and cognition.
Genes for the synthesis of oxytocin and vasopressin are very ancient, estimated to be over 700 million years old (Donaldson & Young, 2008). These genes existed before the split between vertebrates and invertebrates. The original molecular structure from which the peptides evolved is believed to be vasotocin. Vasotocin differs from oxytocin and vasopressin by only one amino acid. Vasotocin is found in mammalian fetuses, although its expression is reduced at the time of birth.
The specific coding sequences that define oxytocin and vasopressin may have evolved more than once, but the current form probably evolved around the time that mammals first emerged. Oxytocin, through its functions in birth and lactation, assists in maternal nurturing of a comparatively immature infant (Brunton & Russell, 2008; Numan, 2007). The capacity of oxytocin to induce uterine contractions may have allowed the expansion of the human skull and cortex, and eventually cognition. These changes contribute to the elaboration of human vocalizations into speech and other forms of social communication that rely on cognitive function and cortical structures (Hrdy, 2008).
NEUROPEPTIDES INFLUENCE AUTONOMIC FUNCTIONS THROUGH EFFECTS ON THE BRAINSTEM
The paraventricular nucleus of the hypothalamus (including cells that synthesize oxytocin and vasopressin) is an important site of convergence for neural communication coordinating endocrine and cardiovascular responses to various forms of challenge (Michelini, Marcelo, Amico, & Morris, 2003). At the level of the paraventricular nucleus, oxytocin may influence both the hypothalamic-pituitary-adrenal axis and autonomic functions. The presence of oxytocin receptors in the dorsal vagal complex has been verified by autoradiography in rodents (Gimpl & Fahrenholz, 2001). The amygdala, with connections to cortex, hypothalamus, and lower brainstem, integrates sensory, cognitive, and emotional information. The central nucleus of the amygdala also contains oxytocin and vasopressin as well as their receptors, and projections to and from the central nucleus of the amygdala may be critical determinants of emotional reactivity. Thus, the central nucleus of the amygdala is one site (among several) where shifts from positive to negative emotions may be managed. Receptors for oxytocin and vasopressin are found in the amygdala, providing an opportunity for these peptides to integrate social and emotional functions.
Oxytocin and vasopressin also can influence emotions and behavior through their effects on the autonomic nervous system. A growing literature implicates oxytocin and vasopressin and their interactions with vagal circuits. For example, there is evidence for interactions between oxytocin and vasopressin at the level of the central nucleus of the amygdala, which has inputs to lower brainstem nuclei controlling vagal circuits. These processes explain in part the capacity of oxytocin to down-regulate activity in the amygdala (measured by functional magnetic resonance imaging), especially under conditions of fear or emotional dysregulation (Meyer-Lindenberg, 2008). Vasopressin plays a complex role in behavior through effects on blood pressure and heart rate, as well as on the sympathetic-adrenal axis and parasympathetic functions. Both the vasopressin peptide and the vasopressin receptor (V1aR) have been identified in the central nucleus of the amygdala, and implicated in the regulation of brainstem areas including the myelinated vagus, with source nuclei in the ventral vagal complex, where oxytocin-containing processes have been observed.
Receptors for both oxytocin and vasopressin are found in pathways regulat-ing the myelinated vagus. However, oxytocin receptors are particularly abundant in the dorsal vagal complex, which regulates the unmyelinated vagus. As described, the unmyelinated vagus can slow the heart and, as part of a primitive defense system, could even trigger massive drops in blood pressure and fainting. However, under normal conditions the myelinated vagal system restrains the unmyelinated vagus, protecting this system from stopping the heart (Porges, 2007b). Under extremely stressful conditions, such as birth, oxytocin may act (on neural targets including the dorsal vagal complex) to protect the autonomic nervous system from reverting to this more primitive vagal system, which could lead to shutting down and reduced emotional, social, and cognitive function. Evolutionary changes in functions of neuropeptides, including the emergence of oxytocin and vasopressin as separate neuropeptides, facilitated the mammalian birth process (Brunton & Russell, 2008), and, in turn, these peptides fostered the reciprocal regulatory features of mammalian social behavior (Carter, 1998; Numan, 2007). A summary of the above model is illustrated in Figure 19.1.
FIGURE 19.1. The polyvagal theory: Hierarchal organization of neuroendocrine and autonomic processes implicated in social behavior and the adaptive management of stressful experiences. Neuropeptides, including oxytocin (OT), vasopressin (AVP), corticotropin-releasing factor (CRF), and endogenous opioids, as well as neurotransmitters such as serotonin (5-HT) and norepinephrine (NE) influence behavior and emotions through direct actions on the brain, as well as indirect effects on different components of the autonomic nervous system, including the ventral vagal complex (VVC), the dorsal vagal complex (DVC), and the hypothalamic-pituitary-adrenal (HPA) axis. Oxytocin, serotonin, and endogenous opioids acting in the brainstem may be protective during or against shutting down and immobilization.
NEUROPEPTIDES AND THE MANAGEMENT OF STRESSFUL EXPERIENCES
The hypothalamus, especially the paraventricular nucleus, is an important site of convergence for neural communication relating stress, affect, and cardiovascular regulation to social behavior. Thus, it is not surprising that oxytocin influences the hypothalamic-pituitary-adrenal axis and autonomic function (Carter, 1998; Viviani & Stoop, 2008). Oxytocin-deficient mice show disruptions in sympathetic-vagal balance and are impaired in their ability to manage stress (Michelini et al., 2003). In addition, oxytocin generally suppresses the activity of the hypothalamic-pituitary-adrenal axis (Neumann, 2008).
Oxytocinergic projections from the paraventricular nucleus to key brainstem regions are important in cardiovascular control. Oxytocin binding sites are found in the dorsal vagal complex, and oxytocin increases the excitability of vagal neurons (Viviani & Stoop, 2008). In addition, oxytocin receptors in the brainstem have been shown to modulate baroreflex control of heart rate by facilitating the bradycardic response to pressor challenges. Thus, under optimal conditions, systems that rely on oxytocin may modulate and constrain overarousal, which would allow optimal management of challenges and also be permissive for social engagement and caregiving (Porges, 2007b).
Peripheral oxytocin administration is able to reduce heart rate and blood pressure (Michelini et al., 2003). The protective effects of oxytocin or its absence may be most readily observed in the face of adversity or a stressful environment. For example, in highly social prairie voles (Carter, DeVries, & Getz, 1995) exogenous oxytocin ameliorated isolation-induced changes in behavior and heart rate (Grippo et al., 2009). This suggests that while endogenous increases in oxytocin are not sufficient to ameliorate isolation-induced changes in autonomic function, additional supplementation with exogenous oxytocin may have measurable effects. It is important to note that additional supplementation of oxytocin did not lower heart rate in prairie voles that were not isolated. Thus, at least some of the beneficial actions of oxytocin may only become apparent under conditions of stress and adversity.
Centrally released oxytocin can counter the defensive behavioral strategies associated with stressful experiences. Oxytocin also may inhibit the central effects of vasopressin and other adaptive peptides, such as corticotropin-releasing factor, which plays a major role in the hypothalamic-pituitary-adrenal axis (Neumann, 2008). Generally, but not always, the effects of endogenous oxytocin are neuroprotective. We have found, in studies done with the prairie vole, that intense stressors (such as restraint and exposure to a social intruder) can release both oxytocin and vasopressin. Milder stressors, such as handling, increase blood levels of vasopressin, but not usually oxytocin. Experiences, such as exposure to an infant vole, also may transiently release oxytocin, especially in reproductively naive males. Infant exposure concurrently blocks stress-induced increases in adrenal steroids and has the capacity to facilitate subsequent pair-bond formation (Carter, Grippo, Pournajafi-Nazarloo, Ruscio, & Porges, 2008). These and other examples support the general hypothesis that oxytocin plays a critical role in the management of stressful experiences, while also facilitating social behavior.
Animal research suggests that oxytocin affects the immune system, acting during development to "educate" the thymus. Oxytocin can also be a powerful anti-inflammatory agent, with the capacity to reduce inflammatory processes both in vivo and in vitro. For example, oxytocin can restore tissue following exposure to burns, protect against sepsis, and reduce the response to pathogens. At comparatively high levels endogenous oxytocin may promote wound healing, even in humans (Gouin et al., 2010). These functions of oxytocin could provide another set of mechanisms through which caregiving, under conditions that allow the release of oxytocin, might protect and heal both those who give and receive nurture.
SEX DIFFERENCES IN CAREGIVING AND STRESS MANAGEMENT
Sex differences in either the capacity to nurture or be nurtured are frequently observed and often debated. Women are more likely to give direct nurture, sometimes through their role as a spouse or parent, or in a professional capacity, such as nursing. In males, nurturance may be expressed through less direct caregiving behaviors, such as defense or protection of the family or resources.
Culture and experience play an important role in the development, expression, and maintenance of sex differences. However, it is also likely that male and female differences in the capacity to nurture are based in part on biology, which may in turn influence sexually dimorphic behavioral traits and states (Carter, Boone, Pournajafi-Nazarloo, & Bales, 2009).
Gonadal steroid hormones and their receptors can affect sex differences, especially in early development. In addition, sex differences in endogenous neuropeptides, such as oxytocin and vasopressin, or their receptors could influence sexually dimorphic social behaviors. For example, differential exposure to estrogen across the life span might be expected to enhance the availability of oxytocin. However, remarkably little research has actually addressed these questions and, at least in blood, oxytocin often does not differ between males and females. In addition, both the effects of estrogen and oxytocin are context-dependent (Grippo et al., 2009). Thus, other systems, including the hormones of the hypothalamic-pituitary-adrenal and gonadal axes, and autonomic states associated with activation or mobilization may influence the consequences of oxytocin of both endogenous and exogenous origins.
Vasopressin levels in blood also do not reliably differ between the sexes. However, sex differences do exist in central vasopressin, especially in a neural axis that includes the amygdala, bed nucleus of the stria terminalis, and lateral septum (De Vries & Panzica, 2006). This system is important to determining reactions to negative and positive stimuli, and may help explain select behavioral features of various disorders including autism, which are highly sexually dimorphic and also characterized by differences in social behavior and emotional reactions to stressful experiences (Carter, 2007). The effects of vasopressin may be dynamically influenced by oxytocin and vice versa (Viviani & Stroop, 2008).
Social experiences are likely to be major factors regulating both the synthesis of oxytocin and vasopressin and their receptors. Thus, the social history of an individual may be translated into an ontogenetic recalibration of neuropeptidergic functions with potential consequences for both physiology and behavior. Developmental exposure to exogenous oxytocin or vasopressin also can have lifelong consequences (Carter et al., 2009). These developmental influences interact with sex, because males and females often have differential responses to exogenous peptides, and have the potential to modify behavioral patterns across the life span. It seems likely that epigenetic changes in peptides and their receptors are one of the mechanisms through which social experiences are converted into long-lasting individual differences. Social behaviors, including alloparenting, are especially sensitive to social and hormonal experiences in early life. Thus, the fundamental components of caregiving may have evolved with the capacity to be modified by an interaction between early experience and hormones, such as oxytocin and vasopressin, that are capable of regulating sociality and emotional reactivity in later life.
CLINICAL IMPLICATIONS OF POSITIVE SOCIAL EXPERIENCES
Contemporary medicine, especially over the past century, has focused on mechanisms of disease. Medical advances have been largely technical. The natural mechanisms underlying health and healing remain remarkably poorly understood. Sophisticated neural mechanisms, based on ancient bodily systems that are unique to mammals, allow the human body to constantly monitor, manage, and restore its own health. However, humans are highly social creatures. Our capacity to heal ourselves is physically linked to our relationships with other people. When social bonds are absent or disrupted by the loss of a loved one, our health also is at risk.
In the context of caregiving, the quality of the person-to-person interactions between a caretaker and those being cared for is critical for survival. Often this involves contingent and "appropriate" gesture, facial expression, prosody, proximity, and touch. In addition to specific clinical treatments, social support and social engagement behaviors by friends and relatives may also be capable of reversing illness and maintaining health (Harris, 2009). It is likely that oxytocin is important to the positive consequences of social support, possibly through effects on the autonomic nervous system and immune system. For example, it has been shown in humans that immune responses to an endobacterial challenge (lipopolysaccaride) can be significantly blocked by concurrent treatment with oxytocin (Clodi et al., 2008).
Person-to-person interactions that trigger neural circuits promoting calm physiological states can contribute to health, healing, and growth processes. Alternatively, threatening interactions trigger defensive strategies associated with physiological states supporting mobilization (e.g., flight-or-fight behaviors) or immobilization (e.g., behavioral shutdown, syncope, death feigning). As described, the nervous system is constantly assessing the environment as being safe, dangerous, or life-threatening (see chapter 1). Through this process of neuroception, neural circuits are triggered that will either support health and healing or support defensive strategies of fight-or-flight or shutdown. Neuroception involves brain structures, including the amygdala, that can be modulated by neuropeptides including oxytocin and vasopressin. Under optimal conditions, person-to-person interactions can be innate triggers within the human nervous system for adaptive biobehavioral systems that support health and healing. Both the giving and receiving of caregiving or love has the capacity to protect, heal, and restore. The mechanisms underlying these processes are only now becoming apparent.
SUMMARY
As we look at social behavior across the life span, we notice long periods in which self-regulated behaviors, including spontaneous social engagement behaviors, are readily expressed. In contrast, we note that this timeline is anchored on both sides by a dependence on caregiver. This dependence on caregiver is paralleled by limitations in the neural regulation of autonomic state via the myelinated vagus. Whether the periods of caregiver dependence are modulated by neuropeptides is a question for future research. Perhaps the regulation of oxytocin during these periods may enable the infant or the elderly to be less selective of caregiver and to be able to be soothed and comforted by a variety of caregivers. As the neural control circuits develop and there are more opportunities to engage socially, then perhaps oxytocin and vasopressin play a more important role in modulating state to promote the establishment of strong social bonds.
As a species, humans are highly social mammals, dependent on others for survival and reproduction. Under optimal conditions, this dependency is both symbiotic and reciprocal. The evolved neural, autonomic, and endocrine underpinnings of sociality are shared with other species, permitting a cross-species analysis of the processes responsible for sociality. Awareness of the neurobiology of social engagement and social bonding also offers insights into human concepts, such as social support and caregiving, which in turn can be associated with good health and recovery from illness. These systems are integrated throughout the body, including at the level of the brainstem, where hormones, such as oxytocin and vasopressin, influence behavior, the autonomic nervous system, and immune system. Projections to and from these ancient systems are experienced by more modern brain structures, including the cortex, as diffuse and some-times powerful feelings or emotions. The same neuroendocrine and autonomic systems that permit high levels of social behavior and social bonds regulate the management of stressful experiences and the capacity of the mammalian body to heal itself. However, the activities of brainstem and autonomic systems are context-dependent. In a context of safety or comparatively mild or acute stressors, the release of oxytocin can promote health and restoration. In the context of chronic stress or fear, the actions of these same adaptive systems might have consequences that appear detrimental or destructive. Knowledge of the evolutionary origins and neurobiology of sociality provides a contextual perspective for understanding both the causes and consequences of mammalian caregiving behaviors.
Co-author for this chapter was C. S. Carter.
Epilogue
With increasing knowledge from the neurosciences and continuous feedback from clinicians, the polyvagal theory continues to evolve. In the future, the polyvagal theory (Porges, 2007a, 2007b) will expand into an integrated theory of neurovisceral regulation that will incorporate the role specific brain structures play in the neural regulation of the immune, endocrine, and autonomic systems. This integration will enable a more succinct understanding of the influence of both mental processes on bodily functions and health and bodily functions on mental processes. On a neuroanatomical level the integration of these systems is obvious and long overdue. Although these systems share common neuroanatomical structures, research communities have independently emerged with paradigms designed to maximize select response systems (i.e., immune, endocrine, or autonomic) as if they were independent. By proposing an expanded polyvagal theory, immune and endocrine responses will be interpreted within a phylogenetic hierarchy, similar to the three circuits outlined in the initial statement of the polyvagal theory (see chapter 2). Moreover, rather than interpreting immune and endocrine responses as being stressful or injurious, the responses will be interpreted within a phylogenetic framework and the adaptive function of these responses emphasized. The expanded theory will provide a better understanding of the bidirectional mind–body and brain–body responses that would explain important processes, such as the body's own ability to heal and the relation between physical disease and mental health.
Clinicians, especially traumatologists, have found the polyvagal theory useful in understanding mental health symptoms and in developing interventions and treatment models respecting the client's quest for safety. The theory also leads to a better understanding of the strategies an individual uses to regulate physiological state and feel safe. Within the realm of mental health, the polyvagal theory has already directed clinicians to look at the core features related to the social engagement system that are depressed in virtually all psychiatric illnesses. In the future, clinicians will be trained to be more sensitive to these features and will be skilled to attend to prosody of voice, facial expressivity, gaze, and auditory hypersensitivities as both diagnostic and prognostic indicators.
With an expanded polyvagal theory, easily monitored measures (e.g., heart rate, facial muscle activity, acoustic features of vocalization) may provide variables that reflect the same integrated systems involving endocrine and immune regulation as well as the function of brain structures and compromises due to white matter disease and other aspects of brain dysfunction. Moreover, in the future it may be possible to monitor these variables with noncontact technologies such as high-resolution infrared cameras. In our laboratory we are moving to the next generation of technologies and are able to measure with infrared cameras beat-to-beat heart rate and breathing. Our goal is to incorporate these noncontact measures into a polyvagal monitor that could be used to indicate when a client shifts from a physiological state that either supports defensive strategies (i.e., fear-induced mobilization or immobilization behaviors) or supports calm states associated with safety and social engagement. The technology would be helpful in increasing the awareness of therapists to the rapidly changing and often vulnerable physiological states of their clients.
The polyvagal theory, by centering its organizing constructs on current knowledge of neuroanatomy and evolution of the vertebrate autonomic nervous system, has led to a reinterpretation of autonomic reactions as a neural platform on which specific behaviors and psychological processes may occur. Within this model, many behaviors and psychological processes are neither learned nor correlates of neurophysiological processes. Rather, the model emphasizes that many behaviors and psychological processes are contingent emergent properties of well-defined neurophysiological states. Thus, neurophysiological state is a necessary but not sufficient condition for specific spontaneous behaviors and psychological processes to occur, including the symbiotic neurobehavioral regulation between two people that is often labeled as caring and loving (see chapters 18 and 19). Given this model, new therapies for mental and physical health would focus on manipulating neurophysiological state to provide an appropriate platform for mental and physical health and social behavior.
The polyvagal theory challenges the scientist to think in terms of both bidirectional and hierarchical neural feedback circuits involving the communication between peripheral organs and various structures in the brain. The theory challenges the clinician to interpret atypical behaviors and physiological reactions as adaptive. With these organizing principles, the polyvagal theory will be helpful in understanding the features that facilitate and optimize human social behavior and health.
Credits
PART I: THEORETICAL PRINCIPLES
Chapter 1. Porges, S. W. (2004). Neuroception: A subconscious system for detecting threat and safety. Zero to Three Journal, 24(5), 9–24. Copyright © 2004 ZERO TO THREE. Reprinted by permission of Zero to Three.
Chapter 2. Porges, S. W. (1995). Orienting in a defensive world: Mammalian modifications of our evolutionary heritage. A polyvagal theory. Psychophysiology, 32, 301–318. Reprinted by permission of John Wiley and Sons.
Chapter 3. Porges, S. W. (2009). The polyvagal theory: New insights into adaptive reactions of the autonomic nervous system. Cleveland Clinic Journal of Medicine, 76(Suppl 2), S86–S90. Reprinted with permission. Copyright © 2009 Cleveland Clinic Foundation. All rights reserved.
PART II: BIOBEHAVIORAL REGULATION DURING EARLY DEVELOPMENT
Chapter 4. Porges, S. W. (1992). Vagal tone: A physiological marker of stress vulnerability. Pediatrics, 90, 498–504. Reprinted by permission of the American Academy of Pediatrics.
The preparation of this chapter and much of the research described have been supported, in part, by grant HD 22628 from the National Institute of Child Health and Human Development. The construct of cardiac vagal tone described in this chapter has been measured with patented methods. These methods have been developed with support from National Institutes of Health grants HD-15968 and HD-05951 and National Institute of Mental Health grants MH-00054 and MH-18909 awarded to Dr. Porges. The methods have been incorporated in a vagal tone monitor that can evaluate vagal tone in real time. (Details regarding the vagal tone monitor can be obtained from Delta-Biometrics, Inc., 9411 Locust Hill Road, Bethesda, MD 20814-3960.)
Chapter 5. Porges, S. W. (1993). The infant's sixth sense: Awareness and regulation of bodily processes. Zero to Three Journal, 14, 12–16. Copyright © 1993 ZERO TO THREE. Reprinted by permission of Zero to Three.
Chapter 6. Porges, S. W. (1996). Physiological regulation in high-risk infants: A model for assessment and potential intervention. Development and Psychopathology, 8, 43–58. Reprinted by permission of Cambridge University Press.
Chapter 7. Porges, S. W., Doussard-Roosevelt, J. A., Portales, A. L., & Greenspan, S. I.. (1996). Infant regulation of the vagal "brake" predicts child behavior problems: A psychobiological model of social behavior. Developmental Psychobiology, 29, 697–712. Reprinted by permission of John Wiley and Sons.
Chapter 8. Porges, S. W., & Furman, S. A. (2010). The early development of the autonomic nervous system provides a neural platform for social behavior: A polyvagal perspective. Infant and Child Development. [Article first published online April 22, 2010.] Reprinted by permission of John Wiley and Sons.
The research described in this chapter was supported, in part, by NIH Grant R01 HD053570 from the National Institute of Child Health and Human Development and NIH Grant T32 MH18882 from the American Psychological Association Diversity Program in Neuroscience.
PART III: SOCIAL COMMUNICATION AND RELATIONSHIPS
Chapter 9. Porges, S. W., Doussard-Roosevelt, J. A., & Maiti, A. K. (1994). Vagal tone and the physiological regulation of emotion. In N. A. Fox (Ed.), The development of emotion regulation: Behavioral and biological considerations, Monographs of the Society for Research in Child Development, 59(2–3, Serial No. 240), 167–186. Reprinted by permission of John Wiley and Sons.
Chapter 10. Porges, S. W. (1997). Emotion: An evolutionary by-product of the neural regulation of the autonomic nervous system. Annals of the New York Academy of Sciences, 807, 62–77. Reprinted by permission of John Wiley and Sons.
Special thanks are extended to Sue Carter for encouraging me to formalize the ideas presented in this chapter. In addition, I would like to thank Jane Doussard-Roosevelt for commenting on earlier drafts and the students in my graduate seminar who provided a forum for the discussion of the concepts described in the polyvagal theory of emotion.
Chapter 11. Porges, S. W. (1998). Love: An emergent property of the mammalian autonomic nervous system. Psychoneuroendocrinology, 23, 837–861. Reprinted by permission of Elsevier.
The preparation of this chapter was supported in part by grant HD 22628 from the National Institute of Child Health and Human Development and by grant MCJ 240622 from the Maternal and Child Health Bureau. Special thanks are extended to Sue Carter for encouraging me to formalize the ideas presented here. In addition, I would like to thank Jack Clark, Jane Doussard-Roosevelt, Jaak Panksepp, and Kerstin Uvnas-Moberg for commenting on earlier drafts.
Chapter 12. Porges, S. W. (2003). Social engagement and attachment: A phylogenetic perspective. Annals of the New York Academy of Sciences, 1008, 31–47. Reprinted by permission of John Wiley and Sons.
This study was supported in part by a grant from the National Institutes of Health (MH60625). Several of the ideas presented in this article are the product of discussions with C. Sue Carter.
Chapter 13. Porges, S. W., & Lewis, G. F. (2010). The polyvagal hypothesis: Common mechanisms mediating autonomic regulation, vocalizations, and listening. In S. M. Brudzynsk (Ed.), Handbook of mammalian vocalizations: An integrative neuroscience approach (pp. 255–264). Amsterdam: Academic Press. Reprinted by permission of Academic Press.
PART IV: THERAPEUTIC AND CLINICAL PERSPECTIVES
Chapter 14. Porges, S. W. The vagus: A mediator of behavioral and physiologic features associated with autism. In M. L. Bauman & T. L. Kemper (Eds.), The neurobiology of autism (2nd ed.), (pp. 65–78). © 1994, 2005 The Johns Hopkins University Press. Reprinted with permission of The Johns Hopkins University Press.
The preparation of this chapter was supported in part by grant MH60625 from the National Institutes of Health. The author gratefully acknowledges the assistance of George Nijmeh in the preparation of this manuscript.
Chapter 15. Austin, M. A., Riniolo, T. C., & Porges, S. W. (2007). Borderline personality disorder and emotion regulation: Insights from the polyvagal theory. Brain and Cognition, 65, 69–76. Reprinted by permission of Elsevier.
A special thanks to Katherine C. Johnson for her input with this chapter, and to Janice Laben. The preparation of this manuscript was supported in part by a grant from the National Institutes of Health (MH60625).
Chapter 16. Dale, L. P., Carroll, L. E., Galen, G., Hayes, J. A., Webb, K. W., & Porges, S. W. (2009). Abuse history is related to autonomic regulation to mild exercise and psychological wellbeing. Applied Psychophysiology and Biofeedback, 34, 299–308. Reprinted by permission of Springer.
The authors wish to acknowledge the support of the administrators and participants at the local yoga studio. We are also grateful to Amanda Bliss, Allison M. Mattison, Lorinn M. Inserra, and Rebekah Jackson who helped with data collection; Jordana Klein and James DiLoretto who helped with data scoring; Rachel Schein who helped oversee the data scoring and entry; and Drs. Keri Heilman and John Denver for help with the physiological data.
Chapter 17. Porges, S. W (2010). Music therapy and trauma: Insights from the polyvagal theory. In K. Stewart (Ed.), Music therapy & trauma: Bridging theory and clinical practice (pp. 3–15). New York: Satchnote Press. Reprinted by permission of Satchnote Press.
PART V: SOCIAL BEHAVIOR AND HEALTH
Chapter 18. Porges, S. (2009). Reciprocal influences between body and brain in the perception and expression of affect: A polyvagal perspective. In D. Fosha, D. Siegel, & M. Solomon (Eds.), The healing power of emotion: Affective neuroscience, development, and clinical practice (pp. 27–54). New York: Norton. Reprinted by permission of W. W. Norton & Company.
Chapter 19. Porges, S. W., & Carter, C. S. (in press). Neurobiology and evolution: Mechanisms, mediators, and adaptive consequences of caregiving. In S. Brown, R. Brown, & L. Penner (Eds.), Self interest and beyond: Toward a new understanding of human caregiving. New York: Oxford University Press. Reprinted by permission of Oxford University Press.
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Acknowledgments
The polyvagal theory described in this volume evolved from four decades of research. The chapters provide a glimpse into the broad implications of the theory and how the development and expansion of the theory involved many collaborators. The ideas, conceptualized as the polyvagal theory, did not emerge in an intellectual vacuum but were a product of discussions and debates with colleagues and students. The theory is built on solid research and the extensive body of literature on the neural regulation of the autonomic nervous system. The polyvagal theory is not a doctrine but a work in progress. As we learn more, both from our research program and from our colleagues, the theory is updated and expanded. The chapters in this volume illustrate the changing features of the theory as new knowledge is discovered and relevant clinical problems are used to inform the theory.
During my scientific career, I have had the good fortune to meet and to interact with several individuals who have provided important scientific and intellectual guidance. Through these interactions, I became informed by clinical material and learned comparative neurophysiology, evolutionary biology, and time-series analyses. This exposure to perspectives from disparate areas of science provided a rare opportunity to integrate ideas, facts, and methods that led to the discovery of the bold organizing principles on which the polyvagal theory is based.
Several people played important roles in helping me navigate through this uncharted intellectual arena. Publishing this book provides an opportunity to acknowledge their important contributions. First, I want to acknowledge Sue Carter. For 40 years, Sue, as both my wife and intellectual partner, has supported my intellectual curiosity. She has provided me with the confidence to venture into unexplored scientific territories and to successfully find the voice to articulate my scientific and humanitarian vision, both of which are embedded in the theory.
During this process of discovery, there are specific individuals who have functionally "tilted" my thinking and enabled me to expand my perspectives. As a graduate student, David C. Raskin introduced me to the exciting world of psychophysiology with its potential to understand physiological mediators of behavior. Robert E. Bohrer tutored me in time-series statistics and provided me with the confidence to explore my intuitive mathematics. Bob unselfishly worked with me to translate my conceptualizations of the dynamic neural regulation of the autonomic nervous system into the mathematically robust time-series measures that are still the basic metrics used in my research. Stanley I. Greenspan stimulated my interest in psychiatric disorders and provided me with opportunities to deconstruct clinical features into the neurobiological constructs that I study. Peter A. Levine introduced me to the world of trauma research and somatic therapies. His insights into the somatic manifestations of trauma stimulated my passion to understand the neurobiological mechanisms mediating these debilitating effects. Ajit Maiti mentored me in neurophysiology and neuroanatomy. Ajit provided an intellectual bridge between the ancient wisdom of Eastern philosophy and modern Western science. Neil Schneiderman challenged my conceptualization of autonomic measures as psychophysiological indicators by encouraging me to investigate the neural mechanisms mediating heart rate. Hiram E. Fitzgerald shifted my career trajectory toward developmental questions and stimulated my intellectual curiosity to study the autonomic nervous system in young infants. David Crews encouraged me to seek an understanding of the adaptive function of physiological responses from a phylogenetic perspective. Evgeny Sokolov provided a model as a mentor and stimulated me to generate an integrative theory. These insightful and dedicated scientists and clinicians have all been instrumental in framing the intellectual strategies that I used to conceptualize the polyvagal theory. Understanding their contributions unlocks the mystery of the breadth of the theory, which encompasses methodology, mathematics, neuroscience, development, evolution, psychology, and clinical disease. In addition, I want to thank Theo Kierdorf, who encouraged me to integrate several of my papers into a volume on the polyvagal theory. Theo translated and edited several of my papers to create a "polyvagal reader," which was published in German by Junfermann and served as the core of the current volume. I also want to thank Allan Schore, who as editor welcomed this volume into the Norton series on interpersonal neurobiology.
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Foreword reprinted with permission from Bessel van der Kolk.
Copyright © 2011 by Stephen W. Porges
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Library of Congress Cataloging-in-Publication Data
Porges, Stephen W.
The polyvagal theory : neurophysiological foundations of emotions, attachment, communication, and self-regulation / Stephen W. Porges. — 1st ed.
p. cm. — (The Norton Series on Interpersonal Neurobiology) (A Norton Professional Book)
Includes bibliographical references and index.
ISBN 978-0-393-70700-7 (hardcover)
ISBN 978-0-393-70906-3 (e-book)
1. Affective neuroscience. 2. Emotions—Physiological aspects. 3. Vagus nerve.
I. Title.
QP401.P67 2011
612.8—dc22 2010044408
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Lord God, in your special love for children, you chose to raise up Mary Virginia Merrick to be the servant of the poor children. In laboring to serve the young and those without hope because of the crushing weight of poverty, she proclaimed the love of your Son. She made her life's work a demonstration that "nothing is ever too much to do for a child."
Grant that her example of selfless charity and courageous virtue in the face of her own suffering will inspire us to be as generous in the service of others. We humbly ask that your servant Mary Virginia Merrick be numbered among the Church's canonized saints for the edification of your holy people, in accord with your most holy will. Through her intercession, please hear and answer my request (state intention).
Download a Mary Virginia Merrick prayer card here.
Learn about guidelines for intercessory prayer for the canonization process.
Submit a favor received through Mary Virginia's intercession here. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,898 |
Cover-Up? Why Have the Media and Obama Administration Gone Silent on MH17?
By Niles Williamson
Region: Russia and FSU
The deafening silence of the US media and government about the investigation into the downing of Malaysian Airlines Flight MH17 one month ago reeks of a cover-up.
In the hours and days immediately after the crash, without a single shred of evidence, US officials alleged that the passenger jet was shot down by an SA-11 ground-to-air missile fired from pro-Russian separatist-held territory in eastern Ukraine. They launched a political campaign to obtain harsh economic sanctions against Russia and strengthen NATO's military posture in Eastern Europe.
Picking up on the scent, the CIA attack dogs in the US and European media blamed the crash squarely on Russian President Vladimir Putin. The cover of the July 28 print edition of German news magazine Der Spiegel showed the images of MH17 victims surrounding bold red text reading "Stoppt Putin Jetzt!" (Stop Putin Now!). A July 26 editorial in the Economist declared Putin to be the author of MH17's destruction, while the magazine ghoulishly superimposed Putin's face over a spider web on its front cover, denouncing Putin's "web of lies."
Anyone comparing the media's demonization of Putin with their treatment of Saddam Hussein or Muammar Gaddafi had to conclude that Washington was launching a campaign for regime change in Russia like those it carried out in Libya and Iraq—this time, recklessly pushing the United States towards war with a nuclear-armed power, Russia.
Having built up the crash into a casus belli against Russia, however, the US media suddenly dropped the matter completely. The New York Times has not found it fit to print a word on the MH17 crash since August 7.
There is no innocent explanation for the sudden disappearance of MH17 from the media and political spotlight. The plane's black box has been held in Britain for examination for weeks, and US and Russian spy satellites and military radar were intensively scanning east Ukraine at the time of the crash. The claim that Washington does not have detailed knowledge of the circumstances of the crash and the various forces involved is not credible.
If the evidence that is in Washington's hands incriminated only Russia and the Russian-backed forces, it would have been released to feed the media frenzy against Putin. If it has not been released, this is because the evidence points to the involvement of the Ukrainian regime in Kiev and its backers in Washington and the European capitals.
From the outset, the Obama administration presented no evidence to back up the incendiary charges that Putin was responsible for the MH17 crash. In his press briefing on July 18, the day after the crash, President Obama stated that it was still "too early for us to be able to guess what the intentions of those who might have launched this surface-to-air missile might have had."
While cynically exploiting the crash to pressure and threaten Russia, Obama warned that "there will likely be misinformation" in the coverage of the crash. In a backhanded acknowledgment that he had no evidence to support his claims, he said: "In terms of identifying specifically what individual or group of individuals or personnel ordered the strike, how it came about those are things that I think are still going to be subject to additional information that we're going to be gathering."
In the event, the misinformation on the MH17 crash came from the Obama administration itself. Secretary of State John Kerry went on a media blitz on July 20, arguing that the pro-Russian separatists and the Russian government were responsible for the shoot-down.
The sole evidence he presented were a few, dubious "social media records" posted to the Internet. He presented unauthenticated audio recordings of separatists speaking of a plane crash, edited and released by Ukraine's SBU intelligence agency, which works closely with the CIA; YouTube video clips showing a truck moving unidentified military equipment along a road; and a retracted social media statement claiming responsibility for shooting down a plane attributed to separatist leader Igor Strelkov.
Very quickly, the US government's story line on MH17 began to collapse. At a press briefing on July 21, State Department spokesperson and former CIA Middle East analyst Marie Harf declared that the Obama administration's conclusions regarding the downing of the plane were "based on open information which is basically common sense." Challenged by reporters to provide the evidence, she admitted that she could not: "I know it's frustrating. Believe me, we try to get as much out there are possible. And for some reason, sometimes we can't."
After a month during which Washington has failed to release evidence to support its charges against Putin, it is clear that the political offensive of the NATO governments and the media frenzy against Putin were based on lies.
If pro-Russian separatists had fired a ground-to-air missile, as the US government claims, the Air Force would have imagery in their possession confirming it beyond a shadow of a doubt. The US Air Force's Defense Support Program utilizes satellites with infrared sensors to detect missile launches anywhere on the planet, and US radar posts in Europe would have tracked the missile as it shot through the sky. These satellite and radar data have not been released, because whatever they show does not fit the storyline concocted by the US government and media.
What has emerged, instead, is a drumbeat of evidence pointing to the US-backed regime in Kiev's role in the MH17 shoot-down. The day after Kerry made his remarks, the Russian military presented radar and satellite data indicating that a Ukrainian SU-25 fighter jet was in the immediate vicinity and ascending towards MH17 as it was shot down. This claim has not been addressed, let alone refuted by the American government.
NSA whistle-blower William Binney and other retired American intelligence agents issued a statement at the end of July calling into question the social media data presented by Kerry, and demanding the publication of satellite imagery of the missile launch. They added, "We are hearing indirectly from some of our former colleagues that what Secretary Kerry is peddling does not square with the real intelligence."
On August 9, the Malaysian New Straits Times published an article charging the Kiev regime with shooting down MH17. It stated that evidence from the crash site indicated that the plane was shot down by a Ukrainian fighter with a missile followed by heavy machine gun fire.
While it is too early to say conclusively how MH17 was shot down, the preponderance of the evidence points directly at the Ukrainian regime and, behind them, the American government and the European powers. They created the conditions for the destruction of MH17, backing the fascist-led coup in Kiev this February that brought the current pro-Western regime to power. The Western media then supported the Kiev regime's war to suppress opposition to the putsch in east Ukraine, turning the region into a war zone in which MH17 was then shot down.
After the murder of the 298 people aboard MH17, in which they played an important if as-yet unexplained role, Western governments and intelligence agencies seized upon the tragedy in a reckless and sinister maneuver to escalate war threats against the Putin regime. Silence denotes consent, and the deafening silence of the Western media on the issue of Kiev's involvement in the MH17 crash testifies to the criminalization not only of the foreign policy establishment, but also of its media lackeys and the entire ruling class.
The original source of this article is World Socialist Web Site
Copyright © Niles Williamson, World Socialist Web Site, 2014
Articles by: Niles Williamson | {
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{"url":"http:\/\/mathoverflow.net\/revisions\/28846\/list","text":"4 Subscripts and braces\n\nThis is a comment to Sunil's answer on Euclid's proof of the infinitude of primes but I don't have enough point to leave a comment.\n\nThere is a generalization of Euclid's proof which should be well known but I seldom see it mentioned. If $a_1,...,a_r$ are pairwise relatively prime integers, then for any subset $I \\subset {1,...,r}$, \\{1,...,r\\}$,$a(r+1)=\\prod_{i a_{r+1}=\\prod_{i \\in I}a_i +\\prod_{i \\not\\in I}a_i$, is relatively prime to all the$a1,...,an$.a_1,...,a_r$.\n\n3 Fixed notation\n\nThis is a comment to Sunil's answer on Euclid's proof of the infinitude of primes but I don't have enough point to leave a comment.\n\nThere is a generalization of Euclid's proof which should be well known but I seldom see it mentioned. If $a_1,...,a_r$ are pairwise relatively prime integers, then for any subset $I \\subset {1,...,r}$, $a(r+1)=\\prod_{i \\in I}a_i +\\prod_{i \\not in I}ai$not\\in I}a_i$, is relatively prime to all the$a1,...,an$. 2 deleted 559 characters in body This is a comment to Sundil's Sunil's answer on Euclid's proof of the infinitude of primes but I don't have enough points point to leave a comment. There is a generalization of Euclid's proof which should be well known but I seldom see it mentioned. If$p1,...,pr$are distinct primes, then for any subset$I \\subset {1,...,r}$, and$P=\\prod_{i \\in I}pi + \\prod_{i \\not in I}pi$, (where empty product is 1) then$P$is relatively prime to all the pi and hence must contain a prime divisor different from all the$pi$. Sunil's addition of a multiple is new to me and one can include this as$P=m\\prod_{i \\in I}pi + n\\prod_{i \\not in I}pi$, where m is relatively prime to pi for i \\not in I and n is relatively prime to pi for i \\in I. The same argument also implies by looking at prime divisor that if$a_1,...,a_r$are pairwise relatively prime integers, then for any subset$I \\subset {1,...,r}$,$a(r+1)=\\prod_{i \\in I}a_i +\\prod_{i \\not in I}ai$, is relatively prime to all the$a1,...,an\\$.\n\n1 [made Community Wiki]","date":"2013-05-22 09:11:17","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9991602897644043, \"perplexity\": 934.0082868328614}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368701543416\/warc\/CC-MAIN-20130516105223-00093-ip-10-60-113-184.ec2.internal.warc.gz\"}"} | null | null |
Labskovs eller skipperlabskovs er en egnsret fra Nordtyskland. Det er en betegnelse for kogt eller stegt kød (eller fisk), skåret i terninger, brunet med løg, overhældt med suppe og kogt med ituskårne (eller mosede) kartofler og krydderier, men med flere varianter.
Ordet kendes også i plattysk og tysk, Labskaus, og engelsk lobscouse, der er blevet tolket som måltid for stærke; men i virkeligheden har etymologien været regnet for usikker. Madskribenten Petra Foede henviser til, at labs på lettisk betyder god, og kaušis betyder skål, krus, opøseske, og da Letland er en af søfartsnationerne omkring Østersøen, er forklaringen ikke usandsynlig.
Det var oprindeligt sømænd, som spiste labskovs. I ældre dansk sprog kaldtes det også lobescoves eller lobescowes.
I Danmark er retten kendt som skipperlabskovs (evt. med brød). Det er typisk husmandskost.
Litteratur
Petra Foede: Wie Bismarck auf den Hering kam. Kulinarische Legenden. Zürich 2009. (Side 131-134)
Referencer
Eksterne henvisninger
Ordbog over det Danske Sprog
Madretter
Mad fra Tyskland
Mad fra Danmark | {
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Nct Resonance Pre Order (Sep 2020) Get Assisted By Them.
Abigail Hazel September 22, 2020 September 23, 2020 no commentNo tags
Nct Resonance Pre Order (Sep 2020) Get Assisted By Them. >> The article, as mentioned earlier, is for the NCT resonance preorder.
K-pop has fans all over the world. People are crazy about the new Nct Resonance Pre Order. We have seen people getting excited about the recent launch all over the United States, Philippines, etc.
NCT's New Album
What will be the items in the preorder in the NCT Resonance box?
What is the estimated time of arrival?
What are the standard terms and conditions?
NCT is coming up with a new album soon. People are rooting for Nct Resonance Pre Order.
It will be a sequel to the NCT 2018 Empathy that was shot two years ago. A new album has been tentatively titled as NCT 2020: Resonance Pt. 1. There will also be a music video released for 'Make a Wish' on the same day. There will be 18 group members who will be a part of the NCT's subunits. The subunits are named as NCT Dream, NCT 127, NCT U, and Way. The first appearance of all the members was in the 'Black on Black.' One of the fourteen tracks include some record-breakers like Touch, Baby Don't stop, and Boss.
AS per the news, it is speculated that two more members will be added. The names of these two members are Sungchan and Shotaro, as is buzzed in the story. So, now, it would mean that there will be 23 people, though one cannot find sufficient information regarding this. There will be a teaser for 'Make a Wish' released on the 10th of October. Also, more details will be released on the 2nd of October. There will be first press along with the new Nct Resonance Pre Order.
There are two versions for that, including the past performance and future understanding. There are also no additional charges.
There will be a complete folded poster—one random 1p out of 3p.
There will be 1 p lyrics.
There will be a photocard: One will be a random 1p out of 23p.
There will be a book card: Random 1p out of 23
Also, there will be a special card available in the box.
There are All goods are official unless stated otherwise.
There is a fixed price for everything.
The estimated arrival time for the NCT Resonance will be in November. The last date for the payment will be on the 10th of November.
The terms and conditions for this are:
The user can claim for cancellation in 24 hours from the date when the submission has happened. After completing the order period, the user can cancel the order. Though, there will be a cancellation amount of P100 that the user will have to bear.
Thus, a lot of hype is generated in the market for the new Nct Resonance Pre Order. The K-pop fans are excited about the NCT resonance preorder. We think that it is going to be a big hit.
Ohshecomfy Reviews [Sep] – Is It a Fake Scam or Legit?
Microsoft Xbox One Series X Pre Order {Sep} Get Review! | {
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{"url":"https:\/\/www.dannyadam.com\/blog\/page\/8\/","text":"Categories\n\n## Installing Your Own Personal WeBWork Server, Part IV\n\nPart IPart IIPart III | Part IV\n\nThis series of posts covers the installation of\u00a0WeBWork. This post, Part IV of the series, will\u00a0cover how to configure your WeBWork server so that you can access the 20,000 freely available problems.\n\n\u2026 Continued from Part III\n\nThe last post of this series, Part III, concluded with an image showing the virtual machine and the WeBWork site loaded on my computer. Let\u2019s pick up where we left off, by loading the virtual machine and pointing our browser to http:\/\/localhost:14627\/webwork2. It is not necessary to log in to the wwadmin account on the Ubuntu machine, as Apache will serve the web pages without logging in. However, it is necessary that the virtual machine is turned on. The window can be minimized to free desktop space. I should mention at this point that I logged into the wwadmin account a few days ago and updated the machine\u2019s software using Ubuntu\u2019s update manager. I am not exactly sure why, but this caused problems with the WeBWork installation (i.e., I received errors when trying to use the site), so I reverted to an earlier snapshot of the virtual machine. I wanted to mention this in case you were considering upgrading the software. An upgrade of the WeBWork software would possibly solve the issues that I was having, but I have not tried this yet.\n\nCategories\n\n## Predicting Y When the Dependent Variable is a Transformation of Y\n\nI am going to try to start posting more frequently. This post covers topics that I\u2019ve been thinking about lately, including model estimation with ordinary least squares (OLS) and forecasting when OLS is used to fit a statistical model with a dependent variable that is a transformation of some variable we wish to forecast.\n\nSuppose we run a regression with the following specification:\n\n$Y=\\beta_{0}+\\beta_{1}X_{1}+\\beta_{2}X_{2}+\\ldots+\\beta_{n}X_{n}+\\varepsilon$\n\nLet\u2019s assume that the error term is distributed normally, and let\u2019s use OLS to solve for the coefficients in the model. Using a superscript to denote the m observations in the dataset, our m-by-n+1 design matrix $\\mathbf{X}$ is\n\n$\\left[\\begin{array}{ccccc} 1 & X_{1}^{(1)} & X_{2}^{(1)} & \\cdots & X_{n}^{(1)}\\\\ 1 & X_{1}^{(2)} & X_{2}^{(2)} & \\cdots & X_{n}^{(2)}\\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots\\\\ 1 & X_{1}^{(m)} & X_{2}^{(m)} & \\cdots & X_{n}^{(m)}\\end{array}\\right]$\n\nIf the coefficient vector is labeled\u00a0$\\overrightarrow{\\beta}$ and the vector containing the Y variable\u2019s values is labeled\u00a0$\\overrightarrow{y}$, then the OLS estimation for the coefficients can be calculated by solving $\\mathbf{X}^{\\top}\\mathbf{X}\\overrightarrow{\\beta}=\\mathbf{X}^{\\top}\\overrightarrow{y}$ for\u00a0$\\overrightarrow{\\beta}$.\n\n$\\overrightarrow{\\beta}=\\left(\\mathbf{X}^{\\top}\\mathbf{X}\\right)^{-1}\\mathbf{X}^{\\top}\\overrightarrow{y}$\n\nNow we have a set of coefficients that we can use to predict values of Y when we receive additional observations that have values for our independent variables X1, X2, \u2026, Xn, and no observed values of Y. Everything is fine.\n\nCategories\n\n## Probability Distributions Reference Table\n\nTable Description\n\nI have uploaded a reference table that I created with common probability distributions. Although the same formulas and similar descriptions can be obtained on Wikipedia, I find it useful to have the information contained on one table. The table does not have charts that plot probability density or probability mass functions (PDFs and PMFs) or cumulative distribution functions (CDFs). These charts can be found on Wikipedia or plotted with graphing software using the equations on the table. I find it helpful to look at PDF and CDF plots when learning about the distributions.\u00a0There might be some closed form CDFs, moment generating functions, or other equations that I omitted from the table. Also, some distributions have various formulations. For example, the four distributions defined below are all very similar, and I imagine that each might be referred to as a negative binomial, although certain specifications are more conventional than others.\n\n1. A distribution of the number of failures in a sequence of Bernoulli trials, before a specified number of successes.\n2. A distribution of the number of successes in a sequence of Bernoulli trials, before a specified number of failures.\n3. A distribution of the number of successes and failures in a sequence of Bernoulli trials, until a specified number of successes.\n4. A distribution of the number of successes and failures in a sequence of Bernoulli trials, until a specified number of failures.\n\nIn cases where there are multiple ways to define a distribution, I either used the definition that I prefer, or gave multiple formulations of the distribution. I am including a link to both a PDF of the table and an Excel file that has the table and a VBA macro that centers images across a column or multiple columns. Also included\u00a0is a link to the MathML markup file that was used to generate the equations on the table. Excel\u2019s equation editor was giving me problems, and I did not want to type $\\LaTeX$ code manually, so I searched for a what you see is what you get (WYSIWYG) equation editor on the internet. I ended up finding an open source project, Formulator MathML Editor\u00a0(http:\/\/code.google.com\/p\/formulator-mathml\/), which I think is great. After getting started, I also found a $\\LaTeX$ editor, LyX (http:\/\/www.lyx.org\/), which probably would have worked just as well, with the advantage that I\u2019m more familiar with\u00a0$\\LaTeX$ than MathML. I have spotted and corrected a few errors that I made when creating the table (e.g., I originally had the continuous uniform distribution\u2019s variance equal to $\\frac{1}{12}\\left(b+a\\right)^{2}$, which was corrected to $\\frac{1}{12}\\left(b-a\\right)^{2}$), but there still may be some errors. Please let me know if you find any.\n\nCategories\n\n## Installing Your Own Personal WeBWork Server, Part III\n\nPart IPart II | Part III | Part IV\n\nThis series of posts covers the installation of\u00a0WeBWork. This post, Part III of the series, has instructions for installing Ubuntu and WeBWork within VirtualBox.\n\nCreating a New Virtual Machine\n\nThe first step requires the configuration of a new virtual machine in VirtualBox. This new virtual machine will be used to install a copy of Ubuntu with WeBWork. To configure a new virtual machine, you must click New from the VirtualBox Manager. The images below show the progression of steps and my selected settings when configuring a new virtual machine. Please customize these settings according to your preferences and system. For example, if you are using a 32-bit system, please make sure to select Ubuntu (32 bit) when prompted, as opposed to the 64-bit option that I selected.\n\nCategories\n\n## Installing Your Own Personal WeBWork Server, Part II\n\nPart I | Part II | Part III | Part IV\n\nThis series of posts covers the installation of WeBWork. This post, Part II of the series, has instructions for installing VirtualBox. Although I am using Windows 7, the instructions should apply to other operating systems that VirtualBox can be installed on.\n\nWhy Install VirtualBox?\n\nWeBWork is designed to be installed alongside a web server on a Unix based computer. VirtualBox is a software program that allows you to install additional operating systems within the operating system you are currently running. Another\u00a0option would be to install WeBWork on a separate computer that is dedicated to web hosting. While this option may be appealing to colleges, it did not seem practical for the purpose of installing my own personal version of WeBWork. Additionally, the option presented here has a lower cost, as VirtualBox is available for free. For additional details on VirtualBox, please check out its\u00a0homepage and\/or the corresponding Wikipedia page.","date":"2021-03-04 05:50:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 13, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.49324092268943787, \"perplexity\": 697.5924863780162}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178368608.66\/warc\/CC-MAIN-20210304051942-20210304081942-00066.warc.gz\"}"} | null | null |
Q: PHP Web service connect with mssql sqlite and mysql I getting difficulty proceeding with this project if anyone can help.
i am coding a web service at will connect 3 database at the same time.
please check the picture so that you have an idea what am talking about.
http://i57.tinypic.com/j5w1sk.png
So there three data base will be exchanging information. input will obviously come from the android app where info will be entered. info from android app will be send xml file to webservice for processing. part of the info will go to mssql and part will go to mysql. sorting of info will be done by the webservice.
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layout: page
title: Livingston 55th Anniversary
date: 2016-05-24
author: Marilyn Bryant
tags: weekly links, java
status: published
summary: Maecenas suscipit non enim et aliquet. Pellentesque.
banner: images/banner/leisure-05.jpg
booking:
startDate: 09/26/2017
endDate: 09/29/2017
ctyhocn: BHMMBHX
groupCode: L5A
published: true
---
Phasellus dignissim nibh quis libero aliquet placerat. Mauris venenatis tempus est id tincidunt. Fusce porttitor eros risus, ut finibus nisl mattis finibus. Morbi tempus euismod vulputate. Maecenas tortor ante, luctus a faucibus et, blandit id nunc. Morbi scelerisque nibh est, eget rutrum enim scelerisque ut. Maecenas sit amet scelerisque justo, non consequat metus. Aliquam gravida interdum lectus quis scelerisque. Quisque pretium, eros eu ultricies viverra, est est hendrerit ligula, in pellentesque dolor leo vel nibh. Etiam ac ligula nisi. Nunc pellentesque ante a mi vulputate, id bibendum neque viverra.
* Fusce eu mi sit amet dolor blandit sagittis
* Vestibulum sed felis ultrices, sagittis mauris eu, lobortis augue
* Etiam tincidunt orci finibus urna pretium, quis sagittis dui commodo
* Nam vel tortor at odio bibendum commodo
* Nunc non nisl at augue dictum scelerisque in condimentum magna.
Praesent in aliquam arcu, non fermentum diam. In vitae placerat eros, et vehicula leo. Donec interdum quis sapien vel feugiat. Vivamus nec turpis a arcu hendrerit cursus. Phasellus condimentum vulputate leo, eget luctus enim. Donec aliquet quam quam, vitae viverra ligula maximus at. Donec tempor risus vitae urna consectetur cursus. Phasellus sit amet mi lectus. Curabitur at mi eleifend, varius ipsum sit amet, elementum tortor. Aliquam sed commodo ante. Proin sed lectus vel diam imperdiet aliquet ut et libero. Proin at tempus purus. Duis ultrices neque lacus, vehicula hendrerit elit vestibulum vel. Nulla eleifend, risus in sodales vulputate, elit orci sagittis leo, tempus elementum augue ante bibendum odio. Maecenas at sollicitudin turpis.
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Het jaar 1400 is het 100e jaar in de 14e eeuw volgens de christelijke jaartelling.
Gebeurtenissen
januari
16 - De graaf van Holland, Albrecht, verleent de stad Amsterdam het privilege om zelf haar burgemeesters te kiezen. Dit zal jaarlijks op de 1e februari worden gedaan door de aftredende burgemeesters en schepenen alsmede de in leven zijnde oud-burgemeesters en schepenen. Deze "Oudraad" kiest drie burgemeesters, die daarna een vierde kiezen uit de leden van de Oudraad.
juni
14 - Bij een stadsbrand in Den Bosch worden 36 huizen verwoest.
augustus
20 - Keurvorst Ruprecht III van de Palts wordt gekozen tot Rooms-koning als opvolger van de afgezette Wenceslaus.
september
16 - Owain Glyndŵr wordt uitgeroepen tot prins van Wales. Daarmee begint een Welshe revolte tegen de Engelse overheersers.
zonder datum
Wenceslaus van Luxemburg wordt afgezet als koning van Duitsland.
Timoer Lenk gaat op veldtocht door Anatolië en Syrië. Hij verovert, plundert en verwoest onder meer Sivas en Aleppo en slaat het beleg op voor Damascus.
Vanwege de aanval van Timoer Lenk moet Bayezid I zijn beleg van Constantinopel afbreken.
Regent Ho Quy Ly van Vietnam zet keizer Tran Thieu De af en kroont zichzelf tot keizer.
Het Vrouwenklooster Diepenveen wordt gesticht door Johannes Brinckerinck.
De Orde van Lijden van Christus wordt gesticht.
1400 was een Heilig Jaar.
Stadsbrand van Rhenen
oudst bekende vermelding: Neermoor
Kunst en literatuur
Taddeo di Bartolo: Madonna met kind (jaartal bij benadering)
Coluccio Salutati: De tyranno
Opvolging
Brunswijk-Wolfenbüttel - Frederik I opgevolgd door zijn broers Hendrik de Milde en Bernhard I
Duitsland - Wenceslaus van Luxemburg opgevolgd door Ruprecht van de Palts
Étampes - Lodewijk II van Évreux opgevolgd door zijn neef Jan van Berry
Korea - Jeongjong opgevolgd door zijn broer Taejong
Moldavië - Iuga opgevolgd door Alexander de Goede
Venetië (doge) - Antonio Venier opgevolgd door Michele Sten
Vietnam - Tran Thieu De opgevolgd door Ho Quy Ly
Geboren
29 maart - Bernard van Armagnac, Frans edelman
25 december - John Sutton, Engels edelman
Andreas Grego van Piscara, Italiaans monnik
Didacus, Spaans kloosterling
Otto II van Schaumburg, Duits edelman
Richard Neville, Engels edelman
Wenceslaus III van Oława, Silezisch edelman
Andries I Keldermans, Zuid-Nederlands architect (jaartal bij benadering)
Anton van Vaudémont, Frans edelman (jaartal bij benadering)
Bolko V de Hussiet, Silezisch edelman (jaartal bij benadering)
Filarete, Italiaans kunstenaar (jaartal bij benadering)
Gilles Binchois, Zuid-Nederlands componist (jaartal bij benadering)
Hendrik van Aragon, Aragonees prins (jaartal bij benadering)
Isabella, hertogin van Lotharingen (jaartal bij benadering)
Jacopo Bellini, Venetiaans schilder (jaartal bij benadering)
Jakob Kaschauer, Oostenrijks kunstenaar (jaartal bij benadering)
Jan van Luxemburg, Bourgondisch edelman (jaartal bij benadering)
Johannes Brugman, Nederlands geestelijke (jaartal bij benadering)
Juan Afonso Pimentel, Castiliaans edelman (jaartal bij benadering)
Luca della Robbia, Italiaans beeldhouwer (jaartal bij benadering)
Owen Tudor, Welsh soldaat (jaartal bij benadering)
Wartislaw IX, Pommeraans edelman (jaartal bij benadering)
Overleden
7 januari - John Montagu, Engels edelman
7 januari - Thomas Holland (~25), Engels edelman
13 januari - Thomas le Despenser (26), Engels edelman
16 januari - John Holland (~47), Engels edelman
14 februari - Richard II (33), koning van Engeland (1377-1399) (vermoedelijke datum)
24 maart - Florens Radewijns, Nederlands kerkhervormer
24 maart - Margaretha van Norfolk, Engels edelvrouw (of 1399)
april - Zweder van Abcoude, Nederlands edelman
6 mei - Lodewijk II van Évreux, Frans edelman
5 juni - Frederik I van Brunswijk-Wolfenbüttel (~42), Duits edelman
21 mei - John Bourchier, Engels edelman
25 oktober - Geoffrey Chaucer (~57), Engels schrijver (vermoedelijke datum)
november - Catharina van Beieren, Hollands edelvrouw
december - Archibald Douglas (~72), Schots edelman
Baldus de Ubaldis (~73), Italiaans jurist
Willem I van den Bergh, Nederlands edelman (jaartal bij benadering)
Jaar | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,897 |
108-й окремий гірсько-штурмовий батальйон — підрозділ гірської піхоти в складі 10-ї окремої гірсько-штурмової бригади.
Історія
5 грудня 2016 року сформовано 108 окремий гірсько-штурмовий батальйон, з дислокацією у смт Делятин Надвірнянського району, Івано-Франківської області та підпорядкуванням 10 окремій гірсько-штурмовій бригаді.
З 01.03.2017 по 11.04.2017 70 військовослужбовців батальйону у складі зведеної посиленої роти від 108-го та 109-го окремих гірсько-штурмових батальйонів 10 окремої гірсько-штурмової бригади в оперативному підпорядкуванні командира 72 окремої механізованої бригади в складі оперативно-тактичного угрупування «Донецьк» виконували завдання в районі промислової зони м. Авдіївка.
5 липня 2017 року на бригадних тактичних навчаннях з бойовою стрільбою був присутній Начальник генерального штабу — головнокомандувач Збройних Сил України генерал армії України В. М. Муженко.
З 20 липня 2017 року по 26 квітня 2018 року батальйон у повному складі виконував завдання в зоні бойових дій у складі оперативно-тактичного угрупування «Луганськ».
У грудні 2017 року підрозділами батальйону було зайнято та зачищено від бойовиків населений пункт Новоолександрівка Попаснянського району Луганської області.
24 серпня 2018 року Міністр оборони України Генерал армії України Полторак Степан Тимофійович вручив бойовий прапор 108 окремого гірсько-штурмового батальйону.
Станом на грудень 2018 року батальйон з 10 жовтня 2018 виконував завдання в зоні бойових дій у Луганській області у складі оперативно-тактичного угрупування «Північ».
24 серпня 2018 року на День Незалежності України, Президент Петро Порошенко вручив бойові прапори 8-му, 108-му, 109-му окремим гірсько-штурмовим батальйонам. Таким чином 10 окрема гірсько-штурмова бригада єдина в Збройних силах України, в якій всі окремі частини мають бойові прапори.
Із листопада 2019 року по ливень 2022 року батальйон виконував бойові завдання в районі н.п. Широкіне поблизу Маріуполя Донецької області.
З лютого 2021 року по листопад 2021 року батальйон знаходився біля н.п. Зайцево та н.п. Майорськ Донецької області.
Із 24 лютого 2022 батальйон знаходиться на передових позиціях б'ючи ворога спершу на Київському напрямку в подальшому на Луганщині та Донеччині.
За час існування батальйону 15 військовослужбовців нагороджено орденом Богадана Хмельницького різних ступенів, більше 30 військослужбоців орденом За мужність різних ступенів. Більше 300 військослужбоців відомчими нагородами МОУ та ГШ.
Виходцем батальйону являється Герой України майор Дронов Богдан Миколайович, командир 17-го окремого мотопіхотного батальйону 57 ОМПБр
Структура
Управління (включаючи штаб)
1 гірсько-штурмова рота
Взвод вогневої підтримки
2 гірсько-штурмова рота
3 гірсько-штурмова рота
Інженерно-саперний взвод
Взвод матеріального забезпечення
Розвідувальний взвод
Відділення безпілотних авіаційних комплексів
Медичний пункт
Командування
Осінь 2016—2017 — полковник Думанський Василь Степанович.
2017—2021 — підполковник Супрун Максим Вікторович.
2021 — майор Болехан Володимир Михайлович.
Втрати
Сержант Атакішиєв Сахіб Магеррам огли; 13 березня 2017 — бої за Авдіївку.
Старший солдат Довгий Роман Сергійович; 23 вересня 2017 — бої за Бахмутку.
Старший солдат Голубєв Микола Олегович, загинув 1 січня 2019 року, Бої за 29 блокпост.
Солдат Бурлака Артур Олегович загинув 22 березня 2019, під час виконання завдання біля Новотошківського. 15 березня 2019 йому виповнилося 19.
Головний сержант Кознюк Михайло загинув 12 лютого 2023 внаслідок артилерійського обстрілу. Найвідоміший вівчар України
Див. також
109-й окремий гірсько-штурмовий батальйон (Україна)
10-та окрема гірсько-штурмова бригада (Україна)
Гірська піхота України
Примітки
Посилання
10-та Гірська штурмова бригада
Гуманітарний вантаж для воїнів 108 батальйону 10 ОГШБ
Вони кріплять військову міць Держави
Надвірнянські свободівці передали домашніх смаколиків для воїнів гірсько-штурмової бригади
Керівництво району та міста зустрілось з командиром 108-го окремого гірсько-штурмового батальйону
Гірська піхота України
Батальйони України
Військові формування, засновані 2016
засновані в Україні 2016
2016 в Івано-Франківській області
Грудень 2016
Делятин | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,633 |
\section{Introduction}
\label{sec:intro}
Graph structured representations are one of the most commonly encountered data structure that naturally arises in nearly all scientific and engineering application~\cite{gama2018convolutional}. With the widespread use of networks neuroscience, molecular chemistry and other fields, it is not surprising that machine learning on graph data has become a key learning tool. Specifically, Graph neural networks (GNNs) have made significant advances on several fundamental tasks ranging from node classification to graph classification~\cite{kipf2016semi,xu2018how}. Despite GNNs' impressive performance on inferring from graph data, their susceptibility to test-time adversarial examples (i.e., carefully crafted perturbations to fool these models) is a major hurdle in a universal acceptance of GNN solutions in several high-regret applications. These results are consistent with the adversarial attacks on images \cite{liu2018zeroth,hogan2018universal,zhao2019design,xu2019interpreting,xu2018structured,chen2019zo,ye2019adversarial,zhao2019admm}.
Recently, a few attempts have been carried out in the robust machine learning community to devise robust GNN training methods, e.g., adversarial training.
Existing robust training methods for graph data \cite{zugner2019certifiable,xu2019topology} utilize convex relaxation to make the search domain continuous so that first-order methods can be used. Unfortunately, these approaches are highly inefficient (or time consuming) due to additional computations involved in the relaxation procedure and are infeasible to large-scale graphs. Furthermore, in practice the gradient information may not even be available, e.g., either due to inaccessible of the model parameters or model itself is discrete and non-continuous. In such scenarios, existing robust training methods fall short. To overcome these drawbacks, this paper proposes a general framework which leverages the greedy search algorithms and zeroth-order methods to obtain robust GNNs. Specifically, to address the inefficiency/scalability issue, we propose Greedy Topology Attacking method so that gradient-based adversarial training becomes plausible for massive size graphs. Next, to address the unavailability of gradient information issue, we propose Zeroth-order based Greedy Attack for gradient-free adversarial training.
Benchmarking on node classification tasks using GNNs, our Greedy Topology Attack method can achieve similar performance with current state-of-the-art attacks with significant speed up subject to the same topology perturbation budget. This demonstrates the effectiveness of our attack generation method through the lens of greedy search algorithm. Next, by leveraging our proposed greedy topology attack, we benchmark the robustness of our adversarial training technique for GNNs under different attacks and applications. We show that the proposed adversarial training technique is significantly less computationally expensive and, in some cases, more robust than the state-of-the-art methods making them suitable to large-scale problems.
\section{Related Works}
Some recent attentions have been paid to the vulnerability of GNNs. The authors in \cite{dai2018adversarial} investigated test-time non-targeted adversarial attacks on both graph classification and node classification problem by adding or deleting edges from a graph. The authors in \cite{zugner2018adversarial} considered both test-time (evasion) and training-time (data poisoning) attacks on node classification task. Different with~\cite{dai2018adversarial}, the node attributes were allowed to modify in~\cite{zugner2018adversarial}. Their algorithm is for targeted attacks on single node. It was shown that small perturbations on the graph structure and node features are able to misclassify the target node.
In \cite{zugner2019adversarial}, training-time attacks on GNNs were also studied for node classification by perturbing the graph structure. The authors solved a min-max problem in training-time attacks using meta-gradients and treated the graph topology as a hyper-parameter to optimize.
On the defense side, very recent work~\cite{xu2019topology} proposed adversarial training on GNNs, however, this method needs to perform convex relaxation, probabilistic sampling and bisection which pay extra computational cost and is not feasible to large graph datasets. Different from the adversarial training, \cite{zugner2019certifiable} considered the problem of certifiable robustness and also achieved robust model under a pre-specific bound.
\section{Problem Statement}
\label{sec: GNN}
A GNN predicts the class of an unlabeled node under the graph topology.
The $k$-th layer computation in a GNN can be formulated as
\begin{align}
\ H^{(k)} = \sigma \left ( \tilde{ A} H^{(k-1)} ( W^{(k-1)} )
\right ),
\end{align}%
where $H^{(1)} = X$ is the node feature matrix.
Here $\tilde { A}$ is defined as a normalized adjacency matrix $\tilde{ A} = \hat{ D}^{-1/2} \hat{ A} \hat{ D}^{-1/2}$, where $\hat{ A} = A + I$ and $A \in \{0,1\}^{N\times N}$ refers to the adjacency matrix.
We refer readers to \cite{kipf2016semi} for more details about GNN.
$\sigma$ is the ReLU activation function and $W$ is weight matrices.
\subsection{Adversarial Training for Graphs}
Following \cite{madry2018towards}, we consider an adversarial variant of standard Empirical Risk Minimization (ERM), where we minimize the risk over adversarial examples:
\begin{align}\label{eq: robust_train}
\begin{array}{cc}
\displaystyle \minimize_{W} \maximize_{ A^\prime \in \mathcal{C}} \, \mathcal{L}(A^\prime, W; X, y_L).
\end{array}
\end{align}
The training loss $\mathcal{L}$ is cross-entropy error over all training data $(X, y_L)$. $A^\prime$ is the perturbed adjacency matrix by the adversarial attack.
Here $\mathcal{C}$ are the constrains such as the maximum number of edges that can be perturbed, $A^\prime$ should be symmetric and most importantly, $A^\prime \in \{0,1\}^{N\times N}$.
Intuitively, adversarial training injects adversarial examples into training data via inner maximization problem to increase robustness of outer risk minimization (or training) problem.
The performance of adversarial training heavily depends on (a) the quality of the adversarial examples, and (b) the efficiency/scalability of the attack algorithms. Note that discrete constraint $\mathcal{C}$ in problem~\eqref{eq: attack_loss} make the gradient descent inapplicable. As mentioned before, current approaches to handle these constraints are computationally inefficient in solving inner optimization problem. Therefore, we first propose efficient algorithms to generate adversarial attacks via greedy algorithms.
\subsection{Efficient Adversarial Example Generation}
In this section, we propose two approaches to generate adversarial examples based on greedy algorithms and zeroth-order methods.
\subsubsection{Greedy Topology Attack} \label{sec:attack}
We fix node attributes $X$ and only consider edge perturbations as the adversarial attack. We find the perturbed adjacency matrix $A^\prime$ to minimize the negative of training loss
\begin{align}\label{eq: attack_loss}
\begin{array}{ll}
\displaystyle \minimize_{A^\prime} & -\mathcal{L}(A^\prime ; W, X, y_L)\\
\st & A^\prime \in \mathcal{C}.
\end{array}
\end{align}
Next, we introduce our Greedy Topology Attack (GTA) in Algorithm~\ref{alg: greedy_attack} which is able to handle discrete constraints.
\begin{algorithm}
\caption{Greedy topology attack (GTA)}
\begin{algorithmic}[1]
\State \textbf{Input:} graph $G = (A,X)$, number of maximum edges can be changed $M$, greedy search step $n$, label $y_L$
\State \textbf{Output:} Modified adversarial adjacency matrix $A^\prime$
\State $A^\prime \leftarrow A$\;
\State $R \leftarrow \emptyset$
\While{{$\|A^\prime - A\|_0 < 2M$}}
\LineComment{Compute gradient of Eq.~\eqref{eq: attack_loss}}
\State $g_{A^\prime} \leftarrow \nabla_{A^\prime} \mathcal{L}(A^\prime ; W, X, y_L)$
\State $S\leftarrow \nabla_{A^\prime}\odot(-2 A^\prime + 1)$
\State $e \leftarrow$ top $n$ elements in $S$ if they are not in $R$
\State $A^\prime \leftarrow$ remove or insert edge $e$ from/to $A^\prime$
\State $R \leftarrow$ add $e$ in record $R$
\EndWhile
\State \textbf{return:} $A^\prime$
\end{algorithmic}\label{alg: greedy_attack}
\end{algorithm}
To satisfy the symmetric attribute of $A$, we only perturb the upper triangular part of the matrix $A$ and replicate it to the lower triangular part. This is the reason that in Line 4 we have $2M$. In Line 6, we compute the gradient over $A^\prime$ of Eq.~\eqref{eq: attack_loss}. In Line 7, we conduct element-wise product between the gradient $g_{A^\prime} $ and a flipped adjacency matrix (0 becomes 1, and 1 becomes -1). This step actually help us express the edge that can be removed or inserted from/to $A^\prime$ also with the gradient value. In Line 8, we speed up the greedy search process. By calculating the $g_{A^\prime} $ once, we select at most $n$ edges to be changed in $A^\prime$, this improvement accelerate the whole attacking method around $n$ times. The sensitivity between the attacking performance and choice of $n$ will also be shown in experiment part. Line 9 records all the modified edges to avoid sometimes change repeatedly.
Overall, our attack method utilize the gradient information but also satisfies the strict constraints compared to the conventional gradient descent methods. The proposed greedy search step also helps in terms of the convergence speed.
\subsubsection{Zeroth-Order Greedy Topology Attack}\label{sec:zo_gta}
Note although during training stage, we usually use sparse matrix to store $A$, when optimizing graph over $A^\prime$, $g_{A^\prime}$ may be very dense, so that even efficient method like GTA may not be feasible to extremely large graph. Thus, to further improve the efficiency of GTA, we also address the expensiveness of obtaining gradient issue via zeroth-order methods.
As an extension to our GTA method, we propose Zeroth-order Greedy Topology Attack (ZO-GTA) to improve the feasibility on large graph. We summarize our ZO-GTA in Algorithm~\ref{alg: zo_greedy_attack}.
\begin{algorithm}
\caption{Zeroth-order Greedy topology attack (ZO-GTA)}
\begin{algorithmic}[1]
\State \textbf{Input:} graph $G = (A,X)$, number of maximum edges can be changed $M$, greedy search step $n$, label $y_L$
\State \textbf{Output:} Modified adversarial adjacency matrix $A^\prime$
\State $A^\prime \leftarrow A$
\State $L^{(0)} \leftarrow$ training loss $ \mathcal{L}(A^\prime; W, X, y_L)$
\State $t \leftarrow 1$
\While{{$\|A^\prime - A\|_0 < 2M$}}
\State $A^\prime \leftarrow$ random choice $n$ elements in $A^\prime$ and flip them.
\State $L^t \leftarrow$ training loss $ \mathcal{L}(A^\prime; W, X, y_L)$
\If{$L^t > L^{t-1} $}
\State continue
\Else
\State $A^\prime \leftarrow$ flip again the $n$ elements in $A^\prime$
\EndIf
\EndWhile
\State \textbf{return:} $A^\prime$
\end{algorithmic}\label{alg: zo_greedy_attack}
\end{algorithm}
Intuitively, we continuously check randomly flipped $n$ nodes to see whether they can help us maximize $ \mathcal{L}$ or not. If yes, we keep the $n$ nodes flipped, otherwise, we try this procedure again. The ZO-GTA evade to calculate the gradients in problem~\ref{eq: attack_loss} and preserve the discrete nature of $A^\prime$ as well.
\subsection{Algorithm for GNN Adversarial Training}
With the aid of our effective attack methods, the robust training for GNNs (see \eqref{eq: robust_train}) will be solved in this section.
Note that the inner maximization problem in \eqref{eq: robust_train} is exactly same as Eq.~\eqref{eq: attack_loss}. Thus, our GTA methods can approximate this inner optimization. For the outer minimize function, we follow~\cite{kipf2016semi} by using gradient descent to update $W$. This formulation aims to minimize the training loss for the worst case topology perturbations. We summarize our robust training method for GNNs in Algorithm~\ref{alg: adv_training}.
\begin{algorithm}
\caption{Robust training for solving problem \eqref{eq: robust_train}}
\begin{algorithmic}[1]
\State \textbf{Input:} graph $G = (A,X)$, number of maximum edges can be changed $N$, greedy search step $n$, learning rates $\beta$, and iteration numbers $T$, label $y_L$
\State \textbf{Output:} $W$
\State randomly initialize $W^{(0)}$
\For{$t = 1,2,\ldots, T$}
\State inner minimization over $A^\prime$: given $ W^{(t-1)}$, running \hspace*{0.18in}
GTA~\eqref{eq: attack_loss} and obtain $A^\prime$
\State outer maximization over $ W$: given $A^\prime$, obtain
{\small \begin{align*}
W^{t} = W^{t-1} - \beta \nabla_{ W} \mathcal{L}(A^\prime, W^{t-1}; X, y_L)
\end{align*}}%
\EndFor
\State return $W$
\end{algorithmic}\label{alg: adv_training}
\end{algorithm}
Note that this robust training algorithm is general enough to accommodate any existing attack method as long as constraint $ A^\prime \in \mathcal{C}$ in the inner minimization step is satisfied. For example, one can replace GTA with ZO-GTA or CE-PGD attack~\cite{xu2019topology} in Algorithm~\ref{alg: adv_training} to get different variants of robust models. Different attack methods will results in robust models with different degrees of robustness.
\section{Experiments}
\subsection{Experimental Setup}
This section presents our experimental results for both GTA and robust training on graph convolutional networks (GCN) \cite{kipf2016semi} on three popular datasets: Cora, Citeseer and Pubmed~\cite{sen2008collective}. The datasets statistics are summarized in Table~\ref{table:dataset}.
\vspace{-3mm}
\begin{table}[htb]
\centering
\caption{Dataset statistics summary~\cite{yang2016revisiting}}
\vspace{1mm}
\begin{tabular}{c|cccccc}
\toprule[1pt]
Dataset & Nodes ($N$) & Edges & Classes & Features \\
\midrule[1pt]
Cora & 2,708 & 5,429 & 7 & 1,433 \\
Citeseer & 3,327 & 4,732 & 6 & 3,703 \\
Pubmed & 19,717 & 44,338 & 3 & 500 \\
\bottomrule[1pt]
\end{tabular}
\label{table:dataset}
\end{table}
\vspace{-2mm}
We demonstrate the misclassification rate (namely, 1-prediction accuracy on unseen test nodes) of the proposed GTA and ZO-GTA method.
To provide reliable results, we repeat these experiments $5$ times based on different splits of training/testing nodes and report \textit{mean} $\pm$ \textit{standard deviation} of the performance.
We follow~\cite{zugner2019adversarial,xu2019topology} where for test nodes' predicted labels (not their ground-truth label, generate by an independent pre-trained model) can be used during the attack.
\subsection{Attack Performance}
As we mentioned in Section~\ref{sec:attack} and~\ref{sec:zo_gta}, the hyper parameter $n$ affects the speed and the performance of our final model. First we fix $M=0.05 N$ in each dataset.
Next, we conduct the experiment by searching $n \in [0.01N, 0.2N]$ which means we attack a graph by around 5 to 100 iterations. The results are shown in Figure~\ref{fig: ncheck}. As we can see, the misclassification rate drop as $n$ increases. However, the smaller the $n$ is, more iterations we need to execute the attack. Also, we can conclude that GTA performs better than ZO-GTA.
This is intuitive as ZO-GTA is an approximate scheme as compared to GTA which use gradient information.
Therefore, in the following experiments we set $n = 0.05N$.
We compare our GTA and ZO-GTA methods with DICE (`delete edges internally, connect externally')~\cite{waniek2018hiding}, CE-PGD and CW-PGD~\cite{xu2019topology}. We follow the hyper parameters and experimental settings as given in in~\cite{xu2019topology} for a fair comparison. The attack results and run time are reported in Table~\ref{table: attack_performance} and Table~\ref{table: attack_time}, respectively.
\begin{figure}[tb]
\centering
\begin{tabular}{c}
\includegraphics[width=.4\textwidth]{Figure/id_0_ncheck.pdf}
\end{tabular}
\caption {Misclassification rate (in \%) of GTA/ZO-GTA on Citeseer/Cora datasets vs $n$( $n$ is varied between $0.01N$ to $0.2N$).}
\label{fig: ncheck}
\end{figure}
\begin{table}[htb]
\centering
\caption{Misclassification rates (in $\%$) of GTA, ZO-GTA, DICE, CE-PGD and CW-PGD over Cora Citeseer and Pubmed datasets. No result means that the method is not feasible for this dataset.)
}
\vspace{1mm}
\begin{tabular}{c|ccc}
\toprule[1pt]
& Cora & Citeseer & Pubmed\\
\midrule[1pt]
Clean & $18.2\pm 0.1$ & $28.9\pm 0.3$ & $16.9\pm 0.5$ \\
\midrule
GTA & $25.8\pm 0.1$ & $35.2\pm 0.1$ & $\mathbf{19.3\pm 0.4}$ \\
ZO-GTA & $24.9\pm 0.3$ & $33.1\pm 0.3$ & $18.8\pm 0.5$ \\
\midrule
DICE & $18.9\pm 0.3$ & $29.8\pm 0.4$ & $17.4\pm 0.6$ \\
CE-PGD & $\mathbf{28.0\pm 0.1}$ & $36.0\pm 0.2$ &- \\
CW-PGD & $27.8\pm 0.4$ & $\mathbf{37.1\pm 0.5}$ &- \\
\bottomrule[1pt]
\end{tabular}
\label{table: attack_performance}
\end{table}
It can be clearly seen that GTA and ZO-GTA perform competitively with gradient methods, i.e., CE-PGD and CW-PGD and yield significantly better computational efficiency.
\begin{table}[htb]
\centering
\caption{Running time (in seconds) of GTA, ZO-GTA, DICE, CE-PGD and CW-PGD over Cora Citeseer and Pubmed datasets. No result means that the method is not feasible for this dataset.
}
\vspace{1mm}
\begin{tabular}{c|ccc}
\toprule[1pt]
& Cora & Citeseer & Pubmed\\
\midrule[1pt]
GTA & $37\pm 1$ & $36\pm 1$ & $109\pm 5$ \\
ZO-GTA & $\mathbf{19\pm 3}$ & $\mathbf{21\pm 3}$ & $43\pm 5$ \\
\midrule
CE-PGD & $147\pm 4$ & $144\pm 4$ &- \\
CW-PGD & $151\pm 5$ & $142\pm 2$ &- \\
\bottomrule[1pt]
\end{tabular}
\label{table: attack_time}
\end{table}
\vspace{-4mm}
\subsection{Defense Performance}
We next show the improved robustness of GCN by leveraging our proposed robust training algorithm against different topology attacks. We set $T = 1000$, $\beta = 0.01$ and $M = 0.05N$. We evaluate Algorithm~\ref{alg: adv_training} on Cora dataset to compare it with~\cite{xu2019topology}. GTA is used to solve the inner maximization problem since it yields better misclassification rate comapred to ZO-GTA. To test the robustness of the models, we use both GTA and CE-PGD attacking methods to attack them. Results in Table~\ref{table: robust_training} show that the GTA based robust training is competitive with CE-PGD based robust training.
However, recall the results in Table~\ref{table: attack_time}, GTA is much more computationally efficient than CE-PGD which helps our GTA method to handle time sensitive tasks and massive graphs.
\begin{table}[htb]
\centering
\caption{Misclassification rates (in $\%$) of two robust training methods against GTA and CE-PGD attack on Cora dataset. `clean-' means test without attack.
}
\vspace{1mm}
\begin{tabular}{c|cc}
\toprule[1pt]
& GTA training & CE-PGD training \\
\midrule[1pt]
clean-nature & \multicolumn{2}{c}{ $18.2\pm 0.1$ } \\
\midrule
clean-robust & $18.1 \pm 0.2$ & $18.0 \pm 0.3$ \\
\midrule
GTA attack & $20.4\pm 0.3$ & $20.8\pm 0.4$ \\
CE-PGD attack & $22.7\pm 3$ & $22.0\pm 0.2$ \\
\bottomrule[1pt]
\end{tabular}
\label{table: robust_training}
\end{table}
\section{Conclusion}
In this paper, we first introduce two GNN attacking method GTA and ZO-GTA based on the greedy search and zeroth-order algorithms. The proposed approaches are shown to be competitive with state-of-the-art attacks. Next a general and efficient framework of robust training on graph neural networks is proposed. Our experimental results show that the proposed robust training method is significantly computationally less expensive while achieving high robustness to various adversarial attacks. This makes them a potentially viable candidate to handle large-scale and time sensitive problems.
\clearpage
\bibliographystyle{IEEEbib}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,306 |
Q: Formatting output csv files Could I please get some help on the following problem. I can't seem to spot where I have gone wrong in my code. I have 2 output csv files from my code. The first produces the right format but the second does not:
First output file (fileB in my code)
A,B,C
D,E,F
Second output file (fileC in my code)
A,B,
C
D,E,
F
Here is my code:
file1 = open ('fileA.csv', 'rt', newline = '')
shore_upstream = open('fileB.csv', 'wt', newline = '')
shore_downstream = open('fileC.csv', 'wt', newline = '')
for line in file1:
first_comma = line.find(',')
second_comma = line.find(',', first_comma + 1)
start_coordinate = line [first_comma +1 : second_comma]
start_coordinate_number = int(start_coordinate)
end_coordinte = line [second_comma +1 :]
end_coordinate_number = int (end_coordinte)
upstream_start = start_coordinate_number - 2000
downstream_end = end_coordinate_number + 2000
upstream_start_string = str(upstream_start)
downstring_end_string = str(downstream_end)
upstream_shore = line[:first_comma]+','+ upstream_start_string + ',' + start_coordinate
shore_upstream.write(upstream_shore + '\n')
downstream_shore = line[:first_comma]+ ','+ end_coordinte + ',' + downstring_end_string
shore_downstream.write(downstream_shore + '\n')
file1.close()
shore_upstream.close()
shore_downstream.close()
By the way, I am using python 3.3.
A: Your variable end_coordinte may contain non-decimal characters in it, and probably contains a \n\t at the end, resulting in that output.
The simplest solution might be to evaluate those strings as a number, and printing them back as strings.
Replace:
upstream_shore = line[:first_comma]+','+ upstream_start_string + ',' + start_coordinate
downstream_shore = line[:first_comma]+ ','+ end_coordinte + ',' + downstring_end_string
by:
upstream_shore = line[:first_comma]+','+ upstream_start_string + ',' + str(start_coordinate_number)
downstream_shore = line[:first_comma]+ ','+ str(end_coordinate_number) + ',' + downstring_end_string
And pay attention to the line[:first_comma] output, as it may also contain characters you are not expecting.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,471 |
Q: jQuery call plugin with own init function I started trying to create my own plugin using jQuery and the prototype function.
My code looks like the following:
var Slider = function(element, controls, interval) {
this.element = element;
this.controls = controls;
this.interval = interval || 5000;
this.init();
};
Slider.prototype = {
init: function() {
if (this.controls)
this.setControls();
this.element.hide();
this.slides[0].show();
this.fadeIt();
}
}
And inside prototype are more functions.
Currently I'm creating a slider by calling the following:
new Slider($(".slide"), $(".slider").find(".controls").find("i"), 6000);
But how I want to do is is something like this:
Slider.init({
element: $(".slide"),
controls: $(".slider").find(".controls").find("i"),
interval: 6000,
});
I tried searching for a solutions but I only could find how to make something like this:
$(".slider").Slider();
Anyone who can help me out or knows what to search for?
Thanks instead!
A: I was looking up "jQuery Plugin init" when I found your question. I also found this great article with a jQuery Plugin boilerplate that include an init() function.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,253 |
CBD Guide
CBD Store Reviews
Cannabis Reviews
Seed Reviews
Cannabis Industry Pushes For Funding From Covid-19 Outbreak | CBDNerds.com
Authored: May 3, 2020
Cannabis Industry Pushes for Federal Funding During COVID-19 Outbreak
As COVID-19 continues to spread across the globe, the world economy is taking a hit harder than we've seen in modern history. With that, many governments are offering relief packages to both stimulate and maintain the economy. The United States has paid particular attention to small businesses most affected by these shutdowns with their upcoming relief package.
However, none of this relief yet pertains to the cannabis industry. Since cannabis remains a Schedule I substance on a federal level, cannabis businesses are, in many regards, illegitimate for such support. Luckily, a group of organizations seeks to change that.
Twenty-four state-level cannabis trade associations along with four financial institutions have recently sent a letter to Congress, urging for economic relief for the cannabis industry.
"Today's cannabis industry is a mature one, employing nearly 300,000 workers nationwide in direct and indirect jobs," the letter reads. "The industry is a job creator and enjoys bipartisan public support, but without federal funding eligibility to help offset emergency costs related to COVID on top of already extensive state and federal taxes, many cannabis employers and ancillary companies will have no option but to cut jobs or cease to operate."
The letter continues to explain that Congress calls to action with a relief package that allows funding for the cannabis industry. It also promotes the idea of decreasing taxes and promoting more business opportunities (i.e. bank loans) within the industry.
How Has COVID-19 Effected Cannabis?
When the nationwide quarantine began, it made headlines that cannabis was considered essential. Alongside these headlines were discussions as to how sales had spiked due to people hoarding a couple of month's supply of the plant.
However, since this time, cannabis sales have dropped back down to a stable level. The industry has had the benefit of remaining open in most states, but not without the costs of this epidemic.
When the world economy falls as much as we've seen over the last few months - when people lose their jobs and can barely afford to pay rent - cannabis, in many regards, becomes a luxury. The industry, just like any other industry, depends very much on the overall well-being of the financial institution. Therefore, it has been affected.
"In my view, there are businesses that are doing well and businesses that are not doing as well," claims Michael Bronstein, president of the American Trade Association for Cannabis and Hemp (ATACH). "The pandemic has been a place that's accelerated the normal marketplace of choosing winners and losers. And I think what we're finding is those who have run solid businesses will be able to weather the storm."
Though it took nearly a decade of advocacy, the cannabis industry now operates nearly identical to other industries (with a few exceptions that we'll get to later). Big corporations within the industry that have developed a stable marketplace for themselves have got a solid foundation for this financial collapse. However, the little guys that have been getting by on a month-to-month basis are the ones who are going to be most affected.
"As I see it, there's a lot of opportunity within the cannabis market that didn't exist prior to this pandemic," Bronstein informs us. "For some companies, it's created a place where they're assessing the markets they're in, they're assessing their business model, they're assessing what it is that they can do in this marketplace. For others, it's been intensely damaging.
"I think there's a mythology out there that since cannabis has been deemed an essential business, they haven't felt an economic impact from the response to COVID-19. And that's just not true."
Just like much of the United States economy, much of the cannabis industry is made up of smaller businesses. As we've discussed, these Mom and Pop operations are the ones struggling most through this economic shutdown.
"Even big cannabis businesses are struggling," Bronstein continues. "The capital market has frozen and cannabis businesses do not have access to traditional financing. And this has caused a number of hard issues hitting the industry all at once."
What Does the Letter to Congress Propose?
Congress is already in a heap of troubles due to this financial struggle. Much of their efforts are currently to maintain the economy in one way or another to what we knew just a few months ago. The big issue is, nobody knows when all this will end.
News has come out that states are reopening and this is a promising sign, in terms of the economy. However, there are a lot of factors to consider. For example, who's to say we won't see another spike in coronavirus infections once the colder weather comes back later in the year? Or, if people remain fearful of infection when the country re-opens, are they going to go out and fuel the economy the way they did before?
Everything is uncertain at this point and that's another worry when it comes to the cannabis industry. Nobody can predict the financial trends months down the line. More so, nobody can tell where this virus will be when autumn and winter rolls around. With that said, this letter to Congress is to build a foundation for cannabis as a vital part of the economic system.
"The number one thing is providing financial relief for cannabis businesses," proclaims Fred Niehaus, Chairman of the Policy Center for Public Health and Safety. "And that would be part of the next package, in whatever form it is. The second thing is state banking. While we have some banking, we don't really have full financial services."
This has always been one of the most difficult aspects of trying to make it in the cannabis industry. Unlike other industries, you can't walk into a bank and receive a loan for your cannabis business. Many businesses currently in operation were funded out of pocket.
And though this is something nearly impossible for a young entrepreneur to handle him/herself, we can't forget the fact that there are a number of taxes and strict regulations surrounding this industry. For example, most states require a license to grow cannabis and, in some of these states, licenses can easily enter into the thousands of dollars annually.
Making a Cannabis Industry for Everyone
Ideally, you want regulations to be on a comfortable middle ground. The Hemp and CBD industry is a solid example of this. Since hemp was legalized under the 2018 Farm Bill, little regulations have been set into place to help stimulate this industry. CBD itself still remains unapproved by the Federal Food & Drug Administration (FDA).
There are a number of problems when it comes to lacking regulations. For example, plenty of shady businesses have been able to enter the CBD marketplace, selling illegitimate products to people and local businesses. When this practice continues, we become vulnerable to damages such as that outbreak of vaping lung disease just last year. That outbreak, by the way, was linked to cannabis products on a number of occasions.
"What we're really looking for is standardization of the market," Bronstein explains. "and not to operate in a regulatory grey area."
Though the proposal of this letter to Congress was made in order to provide relief for the cannabis industry now, the true goal is to offer a framework for how the industry should operate alongside the federal government once this is all over.
"We have had conversations with some members of Congress that have been intimately involved with these issues," Niehaus tells us. "and with some of the states that have been involved with this issue. And the response has been extremely positive."
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Rated 4.6/5 Based on 36 Verified Ratings | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 965 |
What does X mean in guitar strumming?
Little X above the arrow means that it's muted. Big X in place of an arrow stands for the palm mute. > above the arrow means the accent, i.e. the note is stressed. You can also see empty spaces, they mean that no strums are made, you leave the strings ring for a number of beats.
What does the X above a chord mean?
An X above one of the strings means that you should mute or just not play that string. If you see a black dot or just a circle above one of the strings, that means that you should be playing that string open. Thick black lines or arcs that go over multiple strings represent bars.
What is a dead note on guitar?
Dead notes, also known as false notes and similar to ghost notes, are notes that have been muted to the point that there is no discernable pitch to the note. In guitar music, these are created via fret hand muting. Dead notes are often used to add rhythm and texture to spice up a part.
How do you read strumming symbols?
How to read & write strumming patterns – YouTube
What does the horizontal line on a guitar chord mean?
The horizontal lines on a chord diagram represent the frets on your guitar — the metal bars that run across the neck. The highest fret, closest to the nut, is the first fret. The one below it is the second fret, and so on.
What is a dead note guitar?
Dead notes, also known as false notes and similar to ghost notes, are notes that have been muted to the point that there is no discernable pitch to the note. In guitar music, these are created via fret hand muting.
What is Ghost Note music?
Ghost Note is defined by Wikipedia as 'a musical note with a rhythmic value, but no discernible pitch when played'. Drummers know it as the notes that are played very softly between the main notes. Though seemingly unnoticed to the listener, they fill out the beat and add greater dimension and depth to the music.
Is ghost notes same as dead notes?
How do you read strumming chords?
How to Read Strumming Notation – YouTube
How do you read guitar chords?
Reading Guitar Chord Charts
the right vertical line represents the 1st string.
the left vertical line represents the 6th string.
the horizontal lines represent the fret bars.
the space between the horizontal lines represent the frets.
the dots tell you where to put your fingers.
What do chord symbols mean?
Chord Symbol: A symbolic representation of the chord quality, such as major, minor, or diminished. If this is omitted it implies a major tonality. Chord Number: A number indicates additional notes, or extensions to the base chord.
How do I read guitar chords?
How do you hammer-on a guitar?
Hammer-Ons and Pull-Offs Guitar Lesson – YouTube
What are the 7 musical notes?
Most musicians use a standard called the chromatic scale. In the chromatic scale there are 7 main musical notes called A, B, C, D, E, F, and G. They each represent a different frequency or pitch.
What is the point of ghost notes?
The purpose of the ghost notes in drumming is to create color and texture in the groove. Drummers use them to create more complex drum beats and contrast in drumming. Ghost notes are particularly popular in funk and jazz music because they add diversity and create the funky sound of the drums.
How do you know when to strum up or down on guitar?
For most basic strumming patterns, consider down-strumming every quarter note and up-strumming on the eighth notes in-between. So, there is no strumming patterns or rules to follow, everybody can play the same song with different strumming patterns?
How do you read guitar rhythm?
How To Read Guitar Rhythms – Guitar Lesson – YouTube
How do you read chord symbols?
Chord symbols are written above the top staff of the written music. A chord symbol has two basic parts to it — the chord's root note followed by the chord quality. The root note is the main note on which the chord is built. The quality indicates the type of chord (i.e. major, minor, dominant, diminished, etc.).
What are the 5 main guitar chords?
The 5 chords we'll look at are the C major, A major, G major, E major, and D major. The reason we use all major chords is that the minor versions of any of these chords just require tiny adjustments.
What does C+ chord mean?
The C+ chord is made up of three notes – C, E, and G#. If you're new to chords, the '+' means 'augmented' and the chord is also referred to as the 'Caug chord', 'Caug triad', 'C augmented chord', or 'C augmented triad'. And you may notice that it's the same as the C Chord, but with a G# instead of G.
How do you read guitar chord symbols?
What is a ghost note on guitar?
Put simply, a ghost note is a percussive hit on the guitar with no pitch. So, when you mute the strings with your fretting hand, and strike them with the plectrum, that is a ghost note.
How do you slap a guitar?
Marc Seal Acoustic Guitar Lesson – How to Slap and Pop | ELIXIR Strings
Are there really only 12 notes?
Western music typically uses 12 notes – C, D, E, F, G, A and B, plus five flats and equivalent sharps in between, which are: C sharp/D flat (they're the same note, just named differently depending on what key signature is being used), D sharp/E flat, F sharp/G flat, G sharp/A flat and A sharp/B flat.
What is the most used note?
There were more than 8.2 billion Ben Franklins out in the world at the end of last year, making it the most popular bank note, by value, among the world's major currencies.
Does TennCare cover contact lenses?
How long does it take to cook a beef brisket in a smoker? | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,517 |
Q: How USB 3.1 power is managed by motherboard? There are several USB 3.1 ports in my mother board. According to specification USB 3.1 can handle up to 100W. What actual power PC motherboard can output? What if I connect 8 USB 3.1 with 100W (800W in total) power while power supple can output only 400W?
Is it safe to use high consuming device with week power supply?
A: USB can only handle 100W if the port is a USB Power Delivery Port. Power Delivery is an "extension" to USB, it does not state that every port supports it.
These ports are typically only set up on USB Chargers that are suitable for charging laptops or other devices. It is not supported by default by every port.
The maximum output of a "normal" USB port as would be fitted to a computer is 5V @ 900mA. See Wikipedia USB_3.0: Power and charging
USB 3.0 ports may implement other USB specifications for increased power, including the USB Battery Charging Specification for up to 1.5 A or 7.5 W, or, in the case of USB 3.1, the USB Power Delivery Specification for charging the host device up to 100 W
You will almost definitely not be able to use a computer to charge a laptop or other high current device.
A: Either your motherboard will block an unsafe condition, or your power supply will suffer. If your PC was bought in a store and you didn't upgrade it afterwards then the system is designed NOT to over load the power supply.
The motherboard manufacturer programs the USB system to have maximums that fall within your motherboards ability to handle it.
We would have to either contact your motherboards manufacturer or read the manual to see what is allowed.
It is likely that only 1 or maybe 2 ports will actually support 100w.
Further the motherboard manufacturer has a stated recommended power supply size, and not having that could melt the wires on the power supply.
So if you have a 400w power supply something approx 1/2 of that isn't for 5v. So maybe 200w of that is suitable for 5v. The other is 12v, now if your USB device is 20v at 5a then the motherboard up converts the 12v, and draws even more amps in the process. This assumes the motherboard even has the circuity to do this. At 12v to 20v at 5amps will result in a 9-11 amp draw from the 12v system.
Power is not magically generated. You can not draw more power from the power supply that it's providing without breaking it.
If your motherboard did allow this, which it probably doesn't, the minimum recommend power supply for it would be more like 800w or more.
A: There is a somewhat complex means in which USB devices draw power, and two ways in which a computer protects USB ports from damage due to devices drawing too much power.
The chances of damage to your computer or ports because of too much power draw are small because your computer has circuit breakers or fuses, much like you see in a home's 120 volt AC distribution panel only much smaller. Unlike a circuit breaker in a 120 volt distribution panel the circuit breakers inside your computer are made to reset themselves, they don't have a handle to flip but instead removing and restoring power will reset them. Like any circuit breaker they can sometimes fail to act swiftly enough to prevent damage, but they are better than nothing.
The circuit breakers are a brute force means of managing power, they limit power by just turning everything off. That's a good way to prevent damage but also a good way to damage valuable data.
The primary means USB power is managed is with "bookkeeping" and each device asking nicely for power before taking it.
USB devices are allowed 5 volts and about 100 milliamps of current, a minimum allowance that has changed over time. Low power devices like a mouse or keyboard stop there. Devices that want more power can use USB 2.0, USB-BC, USB-PD, or some other protocol to ask for more power. If there is not enough power available then the request is denied and, presumably, the device will respect that denial. This is safe because the USB host will know to keep a reserve in the bookkeeping for new devices to make these requests. There will not be enough ports to run out of power by plugging in devices that take this small amount of power without asking first.
The USB ports on a computer must be able to supply some minimum amount of power to meet the USB spec. USB 3.1 requires at least 7.5 watts, allows up to 15 watts, and anything beyond 15 watts requires use of some other protocol like USB-PD. If the ports don't meet this spec then they can't claim to be USB compatible. Failure to comply means the manufacturer gets drowned in lawsuits. There will be wailing, gnashing of teeth, rending of garments, really biblical proportions of legal troubles that leave all that bear witness to be left in madness to the point of being unable to speak. Well, maybe not exactly like that.
One thing that prevents too much power from flowing also is that there's only 5 volts at the port unless a request to raise the voltage is granted. This voltage is too low to do much harm even if there is a short circuit. The USB spec allows 100 watts supplied only at 20 volts, if there's not 20 volts then there's not 100 watts. If something tries to take more power anyway then the circuit breakers open and there's no voltage. No voltage means no power.
If you plug in a laptop that would charge from a 100 watt USB-C power supply into the USB-C port on another computer they will "chat" with each other for a split second, the request for power is denied, and you might get a note on the screen that no power is flowing. Or there's a problem, lawyers get involved, and masses of people are left naked, mute, and catatonic in their wake. Or something.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,516 |
SYNONYM
#### According to
The Catalogue of Life, 3rd January 2011
#### Published in
Just's Bot. Jahresber. , Abt. 2 56: 362 (1937)
#### Original name
Coniothyrium zygophylli Syd. & P. Syd., 1913
### Remarks
null | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,983 |
Q: PowerShell Hard Disks verifier I have a PowerShell script to get hard disks information (like hard disk type, Serial number, Size).
My idea is that I will get DeviceID and if the name of the model in DeviceID has "SSD". It will show this is SSD and vice versa.
However, my script is so long and there are a lot of repeat codes.
##PhysicalDisk_Slot 00
$Disk00 = Get-PhysicalDisk | Where-Object {$_.DeviceID -match '0' -and $_.model -match ("SSD")}
if ($Disk00.FriendlyName -like "*SSD*")
{
$DiskMD00 = Get-PhysicalDisk | where { $_.DeviceID -match '0'} | Select-object -ExpandProperty FriendlyName
$DiskSN00 = Get-PhysicalDisk | where { $_.DeviceID -match '0'} | Select-object -ExpandProperty SerialNumber
$DiskSize00 = Get-PhysicalDisk | where { $_.DeviceID -match '0'} | Select-object -ExpandProperty Size
$DiskSize0 = $DiskSize00 / 1024MB
Write-Host -fore Green `n `t "Specs: DeviceID: 0"
Write-Host -fore Green `t "Specs: SSD Model" `t`t`t ":"$DiskMD00
Write-Host -fore Green `t "Specs: SSD Serial Number" `t ":"$DiskSN00
if($DiskSize0 -ge 210)
{
Write-Host -fore Green `t "Specs: SSD Size" `t`t`t ": 250GB"
}
else
{
Write-Host -fore Green `t "Specs: SSD Size" `t ":"([Math]::Round($DiskSize0, 2))"GB"
}
}
else {
$DiskMD00 = Get-PhysicalDisk | where { $_.DeviceID -match '0'} | Select-object -ExpandProperty FriendlyName
$DiskSN00 = Get-PhysicalDisk | where { $_.DeviceID -match '0'} | Select-object -ExpandProperty SerialNumber
$DiskSize00 = Get-PhysicalDisk | where { $_.DeviceID -match '0'} | Select-object -ExpandProperty Size
$DiskSize0 = $DiskSize00 / 1024MB
Write-Host -fore Green `n `t "Specs: DeviceID: 0"
Write-Host -fore Green `t "Specs: HDD Model" `t`t`t ":"$DiskMD00
Write-Host -fore Green `t "Specs: HDD Serial Number" `t ":"$DiskSN00
if($DiskSize0 -ge 910)
{
Write-Host -fore Green `t "Specs: HDD Size" t`t`t ": 1TB"
}
else
{
Write-Host -fore Green `t "Specs: HDD Size" `t ":"([Math]::Round($DiskSize0, 2))"GB"
}
}
##PhysicalDisk_Slot 01
$Disk01 = Get-PhysicalDisk | Where-Object {$_.DeviceID -match '1' -and $_.model -match ("SSD")}
if ($Disk01.FriendlyName -like "*SSD*")
{
$DiskMD01 = Get-PhysicalDisk | where { $_.DeviceID -match '1'} | Select-object -ExpandProperty FriendlyName
$DiskSN01 = Get-PhysicalDisk | where { $_.DeviceID -match '1'} | Select-object -ExpandProperty SerialNumber
$DiskSize01 = Get-PhysicalDisk | where { $_.DeviceID -match '1'} | Select-object -ExpandProperty Size
$DiskSize1 = $DiskSize01 / 1024MB
Write-Host -fore Green `n `t "Specs: DeviceID: 1"
Write-Host -fore Green `t "Specs: SSD Model" `t`t`t ":"$DiskMD01
Write-Host -fore Green `t "Specs: SSD Serial Number" `t ":"$DiskSN01
if($DiskSize1 -ge 210)
{
Write-Host -fore Green `t "Specs: SSD Size" `t`t`t ": 250GB"
}
else
{
Write-Host -fore Green `t "Specs: SSD Size" `t ":"([Math]::Round($DiskSize1, 2))"GB"
}
}
else {
$DiskMD01 = Get-PhysicalDisk | where { $_.DeviceID -match '1'} | Select-object -ExpandProperty FriendlyName
$DiskSN01 = Get-PhysicalDisk | where { $_.DeviceID -match '1'} | Select-object -ExpandProperty SerialNumber
$DiskSize01 = Get-PhysicalDisk | where { $_.DeviceID -match '1'} | Select-object -ExpandProperty Size
$DiskSize1 = $DiskSize01 / 1024MB
Write-Host -fore Green `n `t "Specs: DeviceID: 1"
Write-Host -fore Green `t "Specs: HDD Model" `t`t`t ":"$DiskMD01
Write-Host -fore Green `t "Specs: HDD Serial Number" `t ":"$DiskSN01
if($DiskSize1 -ge 910)
{
Write-Host -fore Green `t "Specs: HDD Size" `t`t`t ": 1TB"
}
else
{
Write-Host -fore Green `t "Specs: HDD Size" `t ":"([Math]::Round($DiskSize1, 2))"GB"
}
}
##PhysicalDisk_Slot 02
$Disk02 = Get-PhysicalDisk | Where-Object {$_.DeviceID -match '2' -and $_.model -match ("SSD")}
if ($Disk02.FriendlyName -like "*SSD*")
{
$DiskMD02 = Get-PhysicalDisk | where { $_.DeviceID -match '2'} | Select-object -ExpandProperty FriendlyName
$DiskSN02 = Get-PhysicalDisk | where { $_.DeviceID -match '2'} | Select-object -ExpandProperty SerialNumber
$DiskSize02 = Get-PhysicalDisk | where { $_.DeviceID -match '2'} | Select-object -ExpandProperty Size
$DiskSize2 = $DiskSize02 / 1024MB
Write-Host ""
Write-Host -fore Green `n `t "Specs: DeviceID: 2"
Write-Host -fore Green `t "Specs: SSD Model" `t`t`t ":"$DiskMD02
Write-Host -fore Green `t "Specs: SSD Serial Number" `t ":"$DiskSN02
if($DiskSize2 -ge 210)
{
Write-Host -fore Green `t "Specs: SSD Size" `t`t`t ": 250GB"
}
else
{
Write-Host -fore Green `t "Specs: SSD Size" `t ":"([Math]::Round($DiskSize2, 2))"GB"
}
}
else {
$DiskMD02 = Get-PhysicalDisk | where { $_.DeviceID -match '2'} | Select-object -ExpandProperty FriendlyName
$DiskSN02 = Get-PhysicalDisk | where { $_.DeviceID -match '2'} | Select-object -ExpandProperty SerialNumber
$DiskSize02 = Get-PhysicalDisk | where { $_.DeviceID -match '2'} | Select-object -ExpandProperty Size
$DiskSize2 = $DiskSize02 / 1024MB
Write-Host -fore Green `n `t "Specs: DeviceID: 2"
Write-Host -fore Green `t "Specs: HDD Model" `t`t`t ":"$DiskMD02
Write-Host -fore Green `t "Specs: HDD Serial Number" `t ":"$DiskSN02
if($DiskSize1 -ge 910)
{
Write-Host -fore Green `t "Specs: HDD Size" `t`t`t ": 1TB"
}
else
{
Write-Host -fore Green `t "Specs: HDD Size" `t ":"([Math]::Round($DiskSize2, 2))"GB"
}
}
and the result below:
Specs: DeviceID: 0
Specs: SSD Model : Samsung SSD 860 EVO 250GB
Specs: SSD Serial Number :
Specs: SSD Size : 250GB
Specs: DeviceID: 1
Specs: HDD Model : ST1000DM010-2EP102
Specs: HDD Serial Number :
Specs: HDD Size : 1TB
Specs: DeviceID: 2
Specs: HDD Model : ST1000DM 003-1ER162
Specs: HDD Serial Number :
Specs: HDD Size : 1TB
In my script, how I can compact my script?
A: Here is one way to do it:
Get-PhysicalDisk | ForEach-Object -Begin { $i = 0 } -Process {
Write-Host -Object "`n `t Specs: DeviceID: $i" -ForegroundColor Green
Write-Host -ForegroundColor Green "`t Specs: SSD Model `t`t`t : $( $_.FriendlyName )"
Write-Host -ForegroundColor Green "`t Specs: SSD Serial Number `t`t`t : $( $_.SerialNumber )"
if ( ( $DiskSize = $_.Size ) / 1GB -ge 210 ) { $Size = "250 GB" }
else { $Size = "$( [Math]::Round( $DiskSize, 2 ) ) GB" }
Write-Host -ForegroundColor Green "`t Specs: SSD Size `t`t`t : $( $Size )"
$i += 1
}
A: I would use a dedicated function to make the code a lot more concise.
function Get-DiskSpecs {
# create and output an object with the wanted properties
Get-PhysicalDisk | ForEach-Object {
# calculate the size. (this does not match the real sizes ???)
$tempSize = $_.Size / 1024MB
$size = '{0}GB' -f ([Math]::Round($tempSize, 2))
if ($_.MediaType -eq 'SSD') {
if ($tempSize -ge 210) { $size = '250GB'}
}
elseif ($tempSize -ge 910) { $size = '1TB'}
[PsCustomObject]@{
DeviceId = $_.DeviceId
Type = $_.MediaType # instead of: if ($_.FriendlyName -like "*SSD*") {'SSD'} else {'HDD'}
Model = $_.FriendlyName
SerialNumber = $_.SerialNumber
Size = $size
}
}
}
Now, all you have to do is call
Get-DiskSpecs
and it returns a collection of objects like
DeviceId : 0
Type : HDD
Model : ST3500418AS
SerialNumber : 9VMRAT50
Size : 465,76GB
DeviceId : 1
Type : SSD
Model : Samsung SSD 750 EVO 250GB
SerialNumber : S33SNWDH513210X
Size : 250GB
However, I think you are calculating the size in a really funny way, and I would opt for another helper function to format the given size in bytes:
function Format-ByteSize {
[CmdletBinding()]
param(
[Parameter(Mandatory = $true, ValueFromPipeline = $true)]
[ValidateRange(0, [double]::MaxValue)]
[double]$SizeInBytes
)
$units = "Bytes", "KB", "MB", "GB", "TB", "PB"
$index = 0
while ($SizeInBytes -gt 1024 -and $index -le $units.length) {
$SizeInBytes /= 1024
$index++
}
if ($index) {
return '{0:N2} {1}' -f $SizeInBytes, $units[$index]
}
"$SizeInBytes Bytes"
}
If you put that on top of your script and use that, the function Get-DiskSpecs can be shortened to:
function Get-DiskSpecs {
# create and output an object with the wanted properties
Get-PhysicalDisk | ForEach-Object {
[PsCustomObject]@{
DeviceId = $_.DeviceId
Type = $_.MediaType # instead of: if ($_.FriendlyName -like "*SSD*") {'SSD'} else {'HDD'}
Model = $_.FriendlyName
SerialNumber = $_.SerialNumber
Size = Format-ByteSize $_.Size
}
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 765 |
// Copyright (c) Microsoft Corporation. All rights reserved.
// Licensed under the MIT License.
// Code generated by Microsoft (R) AutoRest Code Generator.
package com.azure.resourcemanager.devtestlabs.models;
import com.azure.resourcemanager.devtestlabs.fluent.models.ApplicableScheduleInner;
import java.util.Map;
/** An immutable client-side representation of ApplicableSchedule. */
public interface ApplicableSchedule {
/**
* Gets the id property: Fully qualified resource Id for the resource.
*
* @return the id value.
*/
String id();
/**
* Gets the name property: The name of the resource.
*
* @return the name value.
*/
String name();
/**
* Gets the type property: The type of the resource.
*
* @return the type value.
*/
String type();
/**
* Gets the location property: The geo-location where the resource lives.
*
* @return the location value.
*/
String location();
/**
* Gets the tags property: Resource tags.
*
* @return the tags value.
*/
Map<String, String> tags();
/**
* Gets the labVmsShutdown property: The auto-shutdown schedule, if one has been set at the lab or lab resource
* level.
*
* @return the labVmsShutdown value.
*/
Schedule labVmsShutdown();
/**
* Gets the labVmsStartup property: The auto-startup schedule, if one has been set at the lab or lab resource level.
*
* @return the labVmsStartup value.
*/
Schedule labVmsStartup();
/**
* Gets the inner com.azure.resourcemanager.devtestlabs.fluent.models.ApplicableScheduleInner object.
*
* @return the inner object.
*/
ApplicableScheduleInner innerModel();
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,933 |
Editor's note: Samuel E. Poole III, a Fort Lauderdale lawyer who served as executive director of the South Florida Water Management District from 1994 to 1999, responds to the Nov. 15 column "New reservoir needed to stop algae blooms'' by Erik Eikenberg, CEO of the Everglades Foundation.
I agree with everything in Erik Eikenberg's column concerning Everglades restoration except his priority of treating Lake Okeechobee's nutrient problems after they enter the lake. Our first priority must be stopping the nutrients from entering the lake.
Lake Okeechobee receives about five times the 105 tons of phosphorus per year limit, and nearly all of that phosphorus enters the lake with stormwater from suburban and agricultural development as far north as Orlando. My more than 40 years of experience addressing environmental consequences of development in Florida has taught me that complicated engineering projects are seldom the best response to the unanticipated problems created by complicated engineering projects. Filling in the channel dug through the Kissimmee River is the most successful environmental restoration project in the United States. The takeaway is that undoing mistakes is often more effective than building and maintaining another project to treat the mistake.
For Lake Okeechobee, this means going upstream as close to the source of the stormwater runoff as possible. This also means working with the owners of roughly 1 million acres of undeveloped land remaining in the watershed to undo the drainage ditches and canals sending water to Lake Okeechobee and to hold and clean up stormwater on their land. Using a market-based system known as a fee for environmental services, landowners are compensated based on their effectiveness in holding and treating stormwater.
Regulation alone has not prevented damage to our lakes, rivers, bays and aquifers. Supplementing our regulatory system with a fee for environmental services approach is a rational alternative to more big engineering solutions.
This approach is not new to Florida. The Florida Ranchlands Environmental Services Project proved the concept in the Lake Okeechobee watershed. Although the program lost leadership and focus with cuts in the South Florida Water Management District budgets, it still exists.
Cost comparisons between government purchasing thousands of acres of land; designing, permitting and constructing; and then maintaining, operating and making corrective modifications to big engineering systems in perpetuity, and paying private landowners to produce clean water crops continue to evolve. It is clear that service payments to landowners will need to make the business of dispersed water management competitive with other uses of land, including agriculture and development. In its mature form, fee for services could allow a mix of uses on small and light footprints, with the majority of the landscape available for water storage and treatment.
• Dispersed storage and treatment can be scaled up and have immediate impacts.
• Design, permitting and construction of a 60,000-acre storage and treatment system will take years before it's fully operational.
• Holding and treating stormwater north of the lake will help the estuaries and allow the lake to begin recovery, including cycling out the legacy nutrients in lake sediments.
• Storing and treating stormwater south of the lake will help the estuaries but allow Lake Okeechobee to continue to receive very high nutrient loads.
• If the storage and treatment capacity of the lake's watershed is restored, the important reconnection of Lake Okeechobee with the sawgrass Everglades would be a different project — more like a natural flow way and less like a new lake south of Lake Okeechobee.
Finally, holding stormwater for aquifer recharge instead of dumping it into the estuaries could ease Central Florida's concern about future water supplies. I have experienced too many big engineering fixes to believe that "this time we will get it right."
Samuel E. Poole III is a Fort Lauderdale lawyer and served as executive director of the South Florida Water Management District from 1994 to 1999. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,048 |
{"url":"https:\/\/cran.r-project.org\/web\/packages\/nhdR\/vignettes\/network.html","text":"Pull stream network associated with a waterbody catchment\n\n2019-05-01\n\nThe goal of this vignette is to demonstrate how to pull the stream network associated with a waterbody catchment. We will use the extract_network function to accomplish this task using the built-in mendota dataset.\n\nlibrary(nhdR)\nlibrary(dplyr)\nlibrary(ggplot2)\nlibrary(sf)\n\nLet\u2019s start by pulling the coordinates associated with Lake Mendota, which will be the largest object (by area) in the waterbody object.\n\ndata(mendota)\n\nlargest_waterbody <- which.max(st_area(mendota$sp$NHDWaterbody))\nmendota_lake <- mendota$sp$NHDWaterbody[largest_waterbody,]\nmendota_lake <- st_transform(mendota_lake, crs = 4326)\nmendota_centroid <- st_coordinates(st_centroid(mendota_lake))\n\nNext, we can use the extract_network function to pull the stream network associated with Lake Mendota. Notice that we have set the maxsteps parameter to Inf. If the network is anticipated to be very large it can be a good idea to set this to a discrete (lower) number to avoid returning very large lines objects.\n\nmendota_network <- extract_network(lon = mendota_centroid[1],\nlat = mendota_centroid[2],\nmaxsteps = Inf)\n\nFinally, we compare the stream network from a geometric buffer around Lake Mendota against the output of extract_network to make sure everything is working properly.\n\nggplot() +\ngeom_sf(data = mendota$sp$NHDFlowLine) +\ngeom_sf(data = mendota_lake, fill = \"cyan\") +\ngeom_sf(data = mendota_network, color = \"blue\")","date":"2019-05-25 13:47:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3150758743286133, \"perplexity\": 4396.603760295127}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-22\/segments\/1558232258058.61\/warc\/CC-MAIN-20190525124751-20190525150751-00366.warc.gz\"}"} | null | null |
\section{Introduction}
\label{sec-intro}
\subsection{Overview}
\label{sec-overview}
Fix $\gamma \in (0,2)$, suppose that $U \subseteq \mathbbm C$ is a domain, and let $h$ be an instance of (some form of) the Gaussian free field (GFF) on $U$. The $\gamma$-\emph{Liouville quantum gravity} (LQG) surface described by $h$ formally corresponds to
\begin{equation}
\label{eqn:lqg_metric}
e^{\gamma h(z)} (dx^2 + dy^2), \quad z=x+iy
\end{equation}
where $dx^2+dy^2$ denotes the Euclidean metric on $U$. This expression does not make literal sense since $h$ is a distribution and not a function so does not take values at points. Previously, the volume form associated with~\eqref{eqn:lqg_metric} was constructed by Duplantier and Sheffield in \cite{shef-kpz} using a regularization procedure. Namely, for each $\epsilon > 0$ and $z \in U$ so that $B_\epsilon(z) \subseteq U$ let $h_\epsilon(z)$ denote the average of $h$ on $\partial B_\epsilon(z)$. Then the volume form $\mu_h$ is given by the limit as $\epsilon \to 0$ of
\begin{equation}
\label{eqn:lqg_measure}
\epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)} dx dy, \quad z=x+iy
\end{equation}
where $dx dy$ denotes Lebesgue measure on $U$. The factor $\epsilon^{\gamma^2/2}$ is necessary for the limit to exist and be non-trivial. It is also possible to use a similar procedure to make sense of the lengths of certain types of curves \cite{shef-kpz,shef-zipper}. See~\cite{kahane,rhodes-vargas-review} for a more general theory of random measures of this type.
The LQG measure satisfies a certain change of coordinates formula~\cite[Proposition 2.1]{shef-kpz}. Suppose that $\widetilde{U} \subseteq \mathbbm C$ is another domain, $\phi \colon U \to \widetilde{U}$ is a conformal transformation, and
\begin{equation}
\label{eqn:lqg_coord_change}
\widetilde{h} = h \circ \phi ^{-1} + Q\log |(\phi ^{-1})'|, \quad Q = \frac{2}{\gamma} + \frac{\gamma}{2} .
\end{equation}
Then a.s.\ $\mu_h(A) = \mu_{\widetilde{h}}(\phi(A))$ for all Borel sets $A \subseteq U$. Two domain/field pairs $(U,h)$, $(\widetilde{U},\widetilde{h})$ are said to be \emph{equivalent as LQG surfaces} if they are related as in~\eqref{eqn:lqg_coord_change}. An \emph{LQG surface} is an equivalence class of domain/field pairs with respect to this equivalence relation. We think of two equivalent pairs as being two embeddings of the same surface.
In a previous series of papers~\cite{lqg-tbm1,lqg-tbm2,lqg-tbm3}, a metric (distance function) associated with a $\sqrt{8/3}$-LQG surface was constructed in the special case when $\gamma=\sqrt{8/3}$. These works also showed that a certain special $\sqrt{8/3}$-LQG surface is equivalent, as a metric measure space, to the Brownian map of Le Gall \cite{legall-uniqueness} and Miermont \cite{miermont-brownian-map}.
This work is part of a larger project which is focused on constructing for all $\gamma\in (0,2)$ the metric space structure of $\gamma$-LQG, i.e., the Riemannian distance function associated with~\eqref{eqn:lqg_metric}, and proving its basic properties.
We now explain the construction of the metric, which was carried out in the previous works~\cite{dddf-lfpp,local-metrics,lqg-metric-estimates,gm-confluence,gm-uniqueness}.
It is shown in~\cite{dg-lqg-dim,dzz-heat-kernel} that for each $\gamma \in (0,2)$, there is an exponent $d_\gamma > 2$ which can be defined in several equivalent ways, e.g., as the ball volume growth exponent for certain random planar maps in the $\gamma$-LQG universality class.
It is shown in~\cite{gp-kpz} that $d_\gamma$ is the Hausdorff dimension of a $\gamma$-LQG surface, viewed as a metric space.
The value of $d_\gamma$ is not known explicitly except that $d_{\sqrt{8/3}}=4$, but see~\cite{dg-lqg-dim,gp-lfpp-bounds,ang-discrete-lfpp} for reasonably sharp bounds on $d_\gamma$.
We define
\begin{equation} \label{eqn-xi-def}
\xi = \xi_\gamma := \frac{\gamma}{d_\gamma} .
\end{equation}
The significance of the parameter $\xi$ is as follows: for a smooth function $f$, the Riemannian distance function associated with the metric tensor $e^f (dx^2 + dy^2)$ is obtained by integrating $e^{f/2}$ with respect to the Euclidean length measure on smooth paths. This makes it so that scaling the volume form by a factor of $C$ corresponds to scaling distances by a factor of $C^{1/2}$. The Hausdorff dimension of the $\gamma$-LQG metric is $d_\gamma$, rather than 2, so scaling the volume form (i.e., the LQG area measure) by a factor of $C$ should correspond to scaling distances by a factor of $C^{1/d_\gamma}$. This is achieved by defining the distance function using a regularization of $e^{\xi h}$ instead of $e^{\gamma h / 2}$.
Suppose for simplicity that $h$ is a whole-plane GFF.
In light of the preceding paragraph, a natural way to approximate the distance function associated with~\eqref{eqn:lqg_metric} is via the random metrics
\begin{equation} \label{eqn-lfpp}
D_h^\varepsilon(z,w) := \inf_{P : z\rightarrow w} \int_0^1 e^{\xi h_\varepsilon(P(t))} |P'(t)| \, dt ,
\end{equation}
where the infimum is over all piecewise continuously differentiable paths from $z$ to $w$ and $\{h_\varepsilon\}_{\varepsilon > 0}$ is a certain family of continuous functions which approximate the GFF as $\varepsilon\rightarrow 0$ (for technical reasons convergence has only been shown when we take $h_\varepsilon$ to be the convolution of $h$ with the heat kernel).
It is shown in~\cite{dddf-lfpp} that the family of random metrics~\eqref{eqn-lfpp} (suitably re-scaled) is tight w.r.t.\ the local uniform topology on $\mathbbm C \times \mathbbm C$, and every possible subsequential limit is a metric which induces the Euclidean topology.
See also~\cite{ding-dunlap-lqg-fpp,df-lqg-metric,ding-dunlap-lgd} for earlier tightness results for approximations of the LQG metric, preceding~\cite{dddf-lfpp}.
Subsequently, it was shown in~\cite{gm-uniqueness}, building on~\cite{local-metrics,lqg-metric-estimates,gm-confluence}, that the subsequential limit is unique, and in fact the metrics $D_h^\varepsilon$, suitably re-scaled, converge in probability as $\varepsilon\rightarrow 0$ to a metric $D_h$ on $\mathbbm C$. This $D_h$ is defined to be the $\gamma$-LQG metric.
The metric $D_h$ is characterized by a list of axioms including the metric version of the coordinate change formula~\eqref{eqn:lqg_coord_change} for all complex affine functions. These conditions are listed just below. In particular, the metric for $\gamma=\sqrt{8/3}$ is the same as the one in~\cite{lqg-tbm1,lqg-tbm2,lqg-tbm3}. By the local dependence of $D_h$ on $h$, it follows that one can measurably associate an LQG metric for $\gamma \in (0,2)$ with the GFF on any planar domain (see~\cite[Remark 1.5]{gm-uniqueness}).
The purpose of this work is to show that the resulting metric satisfies the metric analog of~\eqref{eqn:lqg_coord_change} for general conformal maps. Consequently, the metric constructed in~\cite{gm-uniqueness} is intrinsic to the quantum surface structure of an LQG surface, i.e., the particular choice of embedding does not change the definition of the metric. As we will see, establishing~\eqref{eqn:lqg_coord_change} for general conformal maps from the case of just complex affine maps is trickier than one might expect.
Although this work builds on~\cite{dddf-lfpp,local-metrics,lqg-metric-estimates,gm-confluence,gm-uniqueness}, it can be read without any knowledge of these works, or even any knowledge about LQG beyond basic properties of the GFF.
The reason for this is that we take the axiomatic definition of the whole-plane $\gamma$-LQG metric from~\cite{gm-uniqueness} as our starting point, and deduce our results from these axioms.
\bigskip
\noindent\textbf{Acknowledgements.} We thank two anonymous referees for helpful comments on an earlier version of this paper. We thank Jian Ding, Julien Dub\'edat, Alex Dunlap, Hugo Falconet, Josh Pfeffer, Scott Sheffield, and Xin Sun for helpful discussions.
EG was supported by a Herchel Smith fellowship and a Trinity College junior research fellowship.
JM was supported by ERC Starting Grant 804166.
\subsection{Main results}
\label{sec-main-results}
We will now define a notion of a $\gamma$-LQG metric for arbitrary open domains $U\subset \mathbbm C$.
The definition of the $\gamma$-LQG metric in~\cite{gm-uniqueness} is the special case when $U = \mathbbm C$.
We first need some preliminary definitions. Throughout, $(X,D)$ denotes a metric space.
\medskip
\noindent
For a continuous curve $P : [a,b] \rightarrow X$ (here $[a,b]$ is equipped with the Euclidean metric and $X$ is equipped with the metric $D$), the \emph{$D$-length} of $P$ is defined by
\begin{equation*}
\operatorname{len}\left( P ; D \right) := \sup_{T} \sum_{i=1}^{\# T} D(P(t_i) , P(t_{i-1}))
\end{equation*}
where the supremum is over all partitions $T : a= t_0 < \dots < t_{\# T} = b$ of $[a,b]$. Note that the $D$-length of a curve may be infinite.
\medskip
\noindent
For $Y\subset X$, the \emph{internal metric of $D$ on $Y$} is defined by
\begin{equation} \label{eqn-internal-def}
D(x,y ; Y) := \inf_{P \subset Y} \operatorname{len}\left(P ; D \right) ,\quad \forall x,y\in Y
\end{equation}
where the infimum is over all paths $P$ in $Y$ from $x$ to $y$.
Then $D(\cdot,\cdot ; Y)$ is a metric on $Y$, except that it is allowed to take the value $\infty$.
\medskip
\noindent
We say that $(X,D)$ is a \emph{length space} if for each $x,y\in X$ and each $\varepsilon > 0$, there exists a curve of $D$-length at most $D(x,y) + \varepsilon$ from $x$ to $y$.
\medskip
\noindent
A \emph{continuous metric} on an open domain $U\subset\mathbbm C$ is a metric $D$ on $U$ which induces the Euclidean topology on $U$.
We equip the space of such metrics with the local uniform topology for functions from $U\times U$ to $[0,\infty)$.
We allow a continuous metric to satisfy $D(u,v) = \infty$ if $u$ and $v$ are in different connected components of $U$.
In this case, in order to have $D^n\rightarrow D$ w.r.t.\ the local uniform topology we require that for large enough $n$, $D^n(u,v) = \infty$ if and only if $D(u,v)=\infty$.
\medskip
\noindent
A \emph{GFF plus a continuous function} on an open domain $U\subset \mathbbm C$ is a random distribution $h$ on $U$ which can be coupled with a random continuous function $f$ in such a way that $h-f$ has the law of the (zero-boundary or whole-plane, as appropriate) GFF on $U$. We emphasize that $f$ is not required to extend continuously to $\overline U$.
\medskip
\noindent
For $U\subset \mathbbm C$, let $\mathcal D'(U)$ be the space of distributions (in the sense of Schwartz) on $\mathbbm C$, equipped with the usual weak topology.
\begin{defn} \label{def-lqg-metric}
A \emph{$\gamma$-LQG metric} is a collection of functions $h\mapsto D_h$, one for each open set $U\subset\mathbbm C$, from $\mathcal D'(U)$ to the space of continuous metrics on $U$ with the following properties.
Let $U\subset \mathbbm C$ and let $h$ be a GFF plus a continuous function on $U$.\footnote{Our axioms for a $\gamma$-LQG metric only concern a.s.\ properties of $D_h$ when $h$ is a GFF plus a continuous function. So, once we have defined $D_h$ a.s.\ when $h$ is a GFF plus a continuous function, we can take $D$ to be any measurable mapping $\mathcal D'(U) \rightarrow \{\text{continuous metrics on $U$}\}$ which is a.s.\ consistent with our given definition when $h$ is a GFF plus a continuous function.
In fact, the construction of the metric in~\cite{dddf-lfpp,lqg-metric-estimates,gm-confluence,gm-uniqueness} only gives an explicit definition of $D_h$ in the case when $h$ is a GFF plus a continuous function. }
Then the associated metric $D_h$ satisfies the following axioms.
\begin{enumerate}[I.]
\item \textbf{Length space.} Almost surely, $(U,D_h)$ is a length space, i.e., the $D_h$-distance between any two points of $U$ is the infimum of the $D_h$-lengths of $D_h$-continuous paths (equivalently, Euclidean continuous paths) in $U$ between the two points. \label{item-metric-length}
\item \textbf{Locality.} Let $V \subset U$ be a deterministic open set.
The $D_h$-internal metric $D_h(\cdot,\cdot ; V)$ is a.s.\ equal to $D_{h|_V}$ (so in particular it is a.s.\ determined by $h|_V$). \label{item-metric-local}
\item \textbf{Weyl scaling.} Let $\xi$ be as in~\eqref{eqn-xi-def}. For a continuous function $f : U\rightarrow \mathbbm R$, define \label{item-metric-f}
\begin{equation} \label{eqn-metric-f}
(e^{\xi f} \cdot D_h) (z,w) := \inf_{P : z\rightarrow w} \int_0^{\operatorname{len}(P ; D_h)} e^{\xi f(P(t))} \,dt , \quad \forall z,w\in U,
\end{equation}
where the infimum is over all continuous paths from $z$ to $w$ in $U$ parameterized by $D_h$-length.
Then a.s.\ $ e^{\xi f} \cdot D_h = D_{h+f}$ for every bounded continuous function $f: U\rightarrow \mathbbm R$.
\item \textbf{Conformal coordinate change.} Let $\widetilde U\subset \mathbbm C$ and let $\phi : U \rightarrow \widetilde U$ be a deterministic conformal map. Then with $Q$ as in~\eqref{eqn:lqg_coord_change}, a.s.\ \label{item-metric-coord0}
\begin{equation} \label{eqn-metric-coord0}
D_h \left( z,w \right) = D_{h\circ\phi^{-1} + Q\log |(\phi^{-1})'|}\left(\phi(z) , \phi(w) \right) ,\quad \forall z,w \in U.
\end{equation}
\end{enumerate}
\end{defn}
It is shown in~\cite[Theorem 1.2 and Corollary 1.3]{gm-uniqueness} (building on~\cite{dddf-lfpp,local-metrics,lqg-metric-estimates,gm-confluence}) that there is a unique LQG metric in the special case when $U=\mathbbm C$, i.e., one has the following statement.
\begin{thm}[\!\cite{gm-uniqueness}] \label{thm-uniqueness}
There is a measurable function $h\mapsto D_h$ from $\mathcal H'(\mathbbm C)$ to the space of continuous metrics on $\mathbbm C$ which satisfies the above axioms with $U = \widetilde U = \mathbbm C$. In this restricted setting, Axiom~\ref{item-metric-local} is replaced by the requirement that $D_h(\cdot,\cdot;V)$ is a.s.\ determined by $h|_V$ and Axiom~\ref{item-metric-coord0} reads as follows.
\begin{enumerate}[I$'$.]
\setcounter{enumi}{3}
\item \textbf{Coordinate change for complex affine maps.} For each fixed deterministic $a,b \in\mathbbm C$, $a\not=0$, a.s.\ \label{item-metric-coord}
\begin{equation}
D_h \left( a z + b , a w + b \right) = D_{h(a\cdot + b) +Q\log |a| }(z,w) , \quad \forall z,w \in \mathbbm C.
\end{equation}
\end{enumerate}
Furthermore, if $h\mapsto D_h$ and $h\mapsto \widetilde D_h$ are two such measurable functions, then there is a deterministic constant $C>0$ such that a.s.\ $D_h = C \widetilde D_h$ whenever $h$ is a GFF plus a continuous function.
\end{thm}
We call $D_h$ from Theorem~\ref{thm-uniqueness} the \emph{$\gamma$-LQG metric associated with $h$}.
Following~\cite[Remark 1.2]{gm-confluence}, it is not hard to extend the definition of $D_h$ from Theorem~\ref{thm-uniqueness} to GFF-type distributions on proper subdomains of $\mathbbm C$, as we now explain.
Suppose that $h$ is a whole-plane GFF. For each deterministic open set $U\subset \mathbbm C$, the metric $D_h(\cdot,\cdot;U)$ is a.s.\ determined by $h|_U$ so we can simply define $D_{h|_U} := D_h(\cdot,\cdot ;U)$.
We can write $h|_U = \mathring h^U + \mathfrak h^U$, where $\mathring h^U$ is a zero-boundary GFF on $U$ and $\mathfrak h^U$ is a random harmonic function on $U$ independent from $\mathring h^U$.
In the notation~\eqref{eqn-metric-f}, we define
\begin{equation}
D_{\mathring h^U} := e^{-\xi \mathfrak h^U} \cdot D_{h|_U} .
\end{equation}
It is easily seen from Axioms~\ref{item-metric-local} and~\ref{item-metric-f} that $D_{\mathring h^U}$ is a measurable function of $\mathring h^U$ (see~\cite[Remark 1.2]{gm-confluence}).
In light of Axiom~\ref{item-metric-f}, we can then define $D_{\mathring h^U + f}$ as a measurable function of $\mathring h^U + f$ for any random continuous function $f : U\rightarrow\mathbbm R$.
By inspection, this function from distributions to metrics satisfies Axioms~\ref{item-metric-length} through~\ref{item-metric-f} above.
The main result of this paper is the following theorem which verifies that the above metric satisfies Axiom~\ref{item-metric-coord0}. This completes the program to define the $\gamma$-LQG metric for all $\gamma \in (0,2)$ on an arbitrary planar domain.
\begin{thm}
\label{thm-coord}
Let $U\subset\mathbbm C$ be an open domain and let $\phi : \mathbbm C\rightarrow U$ be a conformal map.
Also let $h^U$ be a GFF on $U$ plus a continuous function.
Almost surely, the $\gamma$-LQG metric satisfies the coordinate change formula
\begin{equation} \label{eqn-coord}
D_{h^U}(z,w) = D_{h^U\circ\phi^{-1} + Q\log|(\phi^{-1})'|}\left(\phi(z),\phi(w)\right) ,\quad\forall z,w \in U .
\end{equation}
That is, the mapping $h^U \mapsto D_{h^U}$ constructed in~\cite{gm-uniqueness} is a $\gamma$-LQG metric in the sense of Definition~\ref{def-lqg-metric}.
\end{thm}
As noted above, Theorem~\ref{thm-coord} says that the LQG metric depends intrinsically on the $\gamma$-LQG surface $(U,D_h)$, i.e., it does not depend on the particular choice of parameterization for this surface. Hence a $\gamma$-LQG surface with any choice of underlying conformal structure makes sense as a metric space.
\begin{comment}
\begin{remark} \label{remark-dependence}
The only external input used in the proof of Theorem~\ref{thm-coord} is some general theorems about local metrics of the Gaussian free field from~\cite{local-metrics}, which we re-state as they are used.
This paper depends on~\cite{dddf-lfpp,lqg-metric-estimates,gm-confluence,gm-uniqueness} only to establish the existence of a $\gamma$-LQG metric satisfying the above axioms.
\end{remark}
\end{comment}
\subsection{Outline}
\label{sec-outline}
Throughout most of the proof of Theorem~\ref{thm-coord}, we will work with a whole-plane GFF $h$ restricted to a domain in $\mathbbm C$. We will transfer to other variants of the GFF at the very end of the argument using Axiom~\ref{item-metric-f}.
For an open set $U\subset\mathbbm C$ and a conformal map $\phi : U \rightarrow \phi(U)$, we define
\begin{equation} \label{eqn-coord-field-def}
h^\phi := h\circ\phi^{-1} +Q\log|(\phi^{-1})'| \quad\text{and} \quad
D_h^\phi(z,w) := D_{h^\phi}(\phi(z),\phi(w)),\quad\forall z,w\in U .
\end{equation}
By the conformal invariance of the GFF, $h^\phi$ is the sum of a zero-boundary GFF and a harmonic function on $\phi(U)$. Therefore $D_{h^\phi}$ is defined as explained before the statement of Theorem~\ref{thm-coord}.
Furthermore, from the locality of $D_h$ it is easily seen that $D_h^\phi$ is a local metric for $h|_U$ and is a.s.\ determined by $h|_U$.
We want to show that a.s.\ $D_h^\phi = D_{h|_U}$.
\begin{comment}
Let $V\subset U$. Then $D_h(\cdot,\cdot; V)$ is determined by $h|_V$, so $D_{h^\phi}(\cdot,\cdot ;\phi(V))$ is determined by $h|_V$. We have $D_h^\phi(\cdot,\cdot;V) = D_{h^\phi}(\phi(\cdot),\phi(\cdot) ; \phi(V))$.
\end{comment}
As one might expect, the basic idea of the proof is to use that the conformal map $\phi$ looks approximately like a complex affine map in a small neighborhood of a typical point, then apply Axiom~{\hyperref[item-metric-coord]{IV$'$}}. However, there are a number of complications in making this argument work which make the proof of Theorem~\ref{thm-coord} more difficult than one might expect at first glance.
The first main step of the proof, which is carried out in \textbf{Section~\ref{sec-bilip}}, is to show that if $z\in U$ and $r >0$ is small, then $D_h$ and $D_h^\phi$ are close on $B_r(z)$, in the sense that with high probability $\sup_{u,v\in B_r(z)} |D_h^\phi(u,v) - D_h(u,v)|$ is of smaller order than the $D_h$-diameter of $B_r(z)$ (which by Axioms~\ref{item-metric-f} and~{{\hyperref[item-metric-coord]{IV$'$}}} is typically of order $r^{\xi Q} e^{\xi h_r(z)}$).
See Proposition~\ref{prop-scaled-metric-uniform} for a precise statement.
Here we note that when $r > 0$ is small, the $D_h$-diameter of $B_r(z)$ is smaller than its $D_h$-distance to $\partial U$, so the restrictions to $B_r(z)$ of $D_{h|_U} = D_h(\cdot,\cdot ; U)$ and $D_h$ agree.
The main difficulty in this step is that we do not know a priori that $\phi\mapsto D_h^\phi$ depends continuously on the conformal map $\phi$ in the almost sure sense.
This is because we do not know that $\phi \mapsto D_{h\circ\phi^{-1}}$ is continuous. Rather, we only know that if $\phi^{-1}$ is uniformly close to the linear map $z\mapsto\alpha z$ (which will be the case if we start with an arbitrary conformal map $\phi$ and zoom in on a sufficiently small neighborhood of any given point) then the law of $h\circ\phi^{-1}$ is close to the law of $h(\alpha\cdot)$ in the total variation sense (Lemma~\ref{lem-tv-conv}). This tells us that the \emph{marginal laws} of $D_{h\circ\phi^{-1}}$ and $D_{h(\alpha\cdot)}$ are close.
We will show in Lemma~\ref{lem-field-metric-converge} that \emph{joint law} of $D_{h\circ\phi^{-1}}$ and $D_{h(\alpha\cdot)}$ is close to the joint law of two copies of the same instance of $D_{h(\alpha\cdot)}$. The basic idea of the argument is as follows. If $\{\phi_n\}_{n\in\mathbbm N}$ is a sequence of conformal maps such that $\phi_n^{-1}$ converges uniformly on compact subsets of $\mathbbm C$ to $z\mapsto \alpha z$, then using basic facts about the GFF we can establish the convergence of joint laws
\begin{equation}
(h \circ\phi_n^{-1} , D_{h\circ\phi_n^{-1}}) \rightarrow (h(\alpha\cdot) , D_{h(\alpha\cdot)} ) \quad\text{and} \quad (h , h\circ\phi_n^{-1}) \rightarrow (h , h(\alpha\cdot)).
\end{equation}
This implies that the joint laws of the 4-tuples $(h , D_h , h\circ\phi_n^{-1} , D_{h\circ\phi_n^{-1}})$ are tight, and moreover allows us to show that any possible subsequential limit is of the form $(h,D_h,h(\alpha\cdot) , D_{h(\alpha\cdot)})$.
By re-scaling and applying Axiom~{\hyperref[item-metric-coord]{IV$'$}}, the preceding paragraph allows us to show that if $\phi : U\rightarrow \phi(U)$ is a conformal map, then the metric $D_{h|_U}$ and the metric $D_h^\phi $ appearing in~\eqref{eqn-coord} are close at small scales in the desired sense.
In \textbf{Section~\ref{sec-coord-proof}}, we upgrade from the statement that $D_{h|_U}$ and $D_h^\phi$ are close with high probability in a small neighborhood of any point to the statement that $D_{h|_U}$ and $D_h^\phi$ are close with high probability everywhere.
This will be carried out in two steps. In Section~\ref{sec-coord-bilip}, we show that $D_{h|_U}$ and $D_h^\phi$ are a.s.\ bi-Lipschitz equivalent using a general criterion for bi-Lipschitz equivalence of two local metrics for the same GFF (Theorem~\ref{thm-bilip}). We then show that the optimal bi-Lipschitz constant is 1 in Section~\ref{sec-attained} using a ``good annulus covering" argument similar to the one in~\cite[Section 3]{gm-uniqueness}.
The reason why we need to use a two-step argument of this form is as follows. Even though we know that $D_{h|_U}$ and $D_h^\phi$ are close at small scales, our estimates are not sharp enough to say directly that a quantity of the form $\sum_{j=1}^N |D_{h|_U}(P(t_{j-1}) , P(t_j)) - D_h^\phi(P(t_{j-1}) , P(t_j))|$ is small when $P : [0,T] \rightarrow U$ is a $D_h$-rectifiable path and $0 = t_0 < t_1 < \dots < t_N = T$ is a fine partition of $[0,T]$.
The arguments of Section~\ref{sec-coord-proof} allow us to restrict attention to ``good" scales where we can say that the ratios of certain $D_{h|_U}$-distances and $D_h^\phi$-distances are close to 1.
\subsection{Basic notation}
\label{sec-notation}
\noindent
We write $\mathbbm N = \{1,2,3,\dots\}$ and $\mathbbm N_0 = \mathbbm N \cup \{0\}$.
\medskip
\noindent
For $a < b$, we define the discrete interval $[a,b]_{\mathbbm Z}:= [a,b]\cap\mathbbm Z$.
\medskip
\noindent
If $f :(0,\infty) \rightarrow \mathbbm R$ and $g : (0,\infty) \rightarrow (0,\infty)$, we say that $f(\varepsilon) = O_\varepsilon(g(\varepsilon))$ (resp.\ $f(\varepsilon) = o_\varepsilon(g(\varepsilon))$) as $\varepsilon\rightarrow 0$ if $f(\varepsilon)/g(\varepsilon)$ remains bounded (resp.\ tends to zero) as $\varepsilon\rightarrow 0$.
We similarly define $O(\cdot)$ and $o(\cdot)$ errors as a parameter goes to infinity.
We will often specify any requirements on the dependencies on rates of convergence in $O(\cdot)$ and $o(\cdot)$ errors in the statements of lemmas/propositions/theorems, in which case we implicitly require that errors, implicit constants, etc., appearing in the proof satisfy the same dependencies.
\medskip
\noindent
For $z\in\mathbbm C$ and $r>0$, we write $B_r(z)$ for the Euclidean ball of radius $r$ centered at $z$. We also define the open annulus
\begin{equation} \label{eqn-annulus-def}
\mathbbm A_{r_1,r_2}(z) := B_{r_2}(z) \setminus \overline{B_{r_1}(z)} ,\quad\forall 0 < r_r < r_2 < \infty .
\end{equation}
\medskip
\section{Comparison of $D_h$ and $D_h^\phi$ at small scales}
\label{sec-bilip}
The goal of this section is to show that in the notation~\eqref{eqn-coord-field-def}, the metrics $D_{h^\phi}$ and $D_{h|_U}$ are close with high probability at small scales (see Proposition~\ref{prop-scaled-metric-uniform} just below).
We will be working with conformal maps, so since circles are not preserved under conformal maps it is sometimes convenient to use a slightly different normalization for the GFF than the usual $h_1(0) = 0$. In particular, we fix a smooth compactly supported, radially symmetric bump function $\mathbbm f : \mathbbm C\rightarrow [0,1]$ with $\int_{\mathbbm C} f(z)\,dz = 1$ and for $z\in\mathbbm C$ and $r>0$ we define
\begin{equation} \label{eqn-smooth-normalize}
h_{\mathbbm f , r}(z) := (h(r\cdot+z) , \mathbbm f) = (h , r^{-2} \mathbbm f(r^{-1}(\cdot-z))) .
\end{equation}
We will often normalize $h$ by requiring $h_{\mathbbm f , 1}(0) = 0$ instead of $h_1(0) = 0$.
The advantage of this is that, since $\mathbbm f$ is smooth, the smoothed average $h_{\mathbbm f , 1}(0)$ depends continuously on $h$ in the distributional topology. This fact is needed in the proof of Lemma~\ref{lem-field-metric-converge0} below.
The main result of this section is the following proposition.
\begin{prop} \label{prop-scaled-metric-uniform}
Let $h$ be a whole-plane GFF normalized so that $h_{\mathbbm f,1}(0) = 0$.
For each fixed $\delta > 0$ and compact set $K\subset U$,
\begin{equation} \label{eqn-scaled-metric-uniform}
\lim_{r\rightarrow 0} \inf_{z\in K} \mathbbm P\left[ \sup_{u,v\in B_r(z)} | D_h^\phi(u,v) - D_h(u,v) | \leq \delta r^{\xi Q} e^{\xi h_{\mathbbm f , r}(z)} \right] =1 .
\end{equation}
\end{prop}
Proposition~\ref{prop-scaled-metric-uniform} involves the smooth averages $h_{\mathbbm f , r}(z)$ instead of circle averages, but it is easy to convert to statements which do not depend on the choice of normalization for the field. For example, we have the following consequence of Proposition~\ref{prop-scaled-metric-uniform} which will be used in Section~\ref{sec-attained}.
\begin{lem} \label{lem-scaled-metric-ratio}
Let $h$ be a whole-plane GFF, with any choice of normalization.
Fix $\delta , b \in (0,1)$. For each fixed compact set $K\subset U$,
\begin{equation} \label{eqn-scaled-metric-ratio}
\lim_{r\rightarrow 0} \inf_{z\in K} \mathbbm P\left[ 1-\delta \leq \frac{D_h^\phi(u,v)}{D_h(u,v)} \leq 1+\delta ,\: \forall u,v\in B_r(z) \: \text{with}\: |u-v|\geq b r \right] = 1 .
\end{equation}
\end{lem}
\begin{proof}
By Axiom~\ref{item-metric-f}, changing the normalization of $h$ (i.e., adding a constant to $h$) does not affect the value of $D_h^\phi(u,v) / D_h(u,v)$.
Therefore, we can assume without loss of generality that $h$ is normalized so that $h_{\mathbbm f , 1}(0) = 0$.
By Axioms~\ref{item-metric-f} and~{\hyperref[item-metric-coord]{IV$'$}}\, and the scale and translation invariance of the law of $h$, modulo additive constant, for each $\varepsilon > 0$ we can find $c > 0$ depending only on $b$ such that with probability at least $1-\varepsilon$,
\begin{equation*}
D_h(u,v) \geq c r^{\xi Q} e^{\xi h_{\mathbbm f ,r}(z)} ,\: \forall u,v\in B_r(z) \: \text{with}\: |u-v|\geq b r .
\end{equation*}
The lemma statement follows by combining this with Proposition~\ref{prop-scaled-metric-uniform} with $c\delta$ in place of $\delta$, then sending $\varepsilon\rightarrow 0$.
\end{proof}
\subsection{$D_h^\phi$ converges to $D_h$ as $\phi$ converges to a linear map}
\label{sec-field-metric-converge}
Throughout this section we let $h$ be a whole-plane GFF normalized so that $h_{\mathbbm f , 1}(0) = 0$, with $\mathbbm f$ as in~\eqref{eqn-smooth-normalize}.
The main step in the proof of Proposition~\ref{prop-scaled-metric-uniform} is the following lemma, which we will prove in this section.
\begin{lem} \label{lem-field-metric-converge}
Let $\phi_n : U_n\rightarrow\phi_n(U_n)$ be a sequence of conformal maps such that $\phi_n(0) = 0$, $\phi_n'(0) \rightarrow 1/\alpha \in\mathbbm C \setminus \{0\}$ as $n\rightarrow\infty$, and each fixed compact subset of $\mathbbm C$ is contained in $U_n$ for large enough $n$.
Then
\begin{equation} \label{eqn-field-metric}
\left( h , D_h , h^{\phi_n} , D_h^{\phi_n} \right)
\rightarrow \left( h , D_h , h(\alpha \cdot) , D_h \right)
\end{equation}
in law with respect to the distributional topology and the local uniform topology on $\mathbbm C\times\mathbbm C$, as appropriate.
\end{lem}
The main difficulty in the proof of Lemma~\ref{lem-field-metric-converge} is comparing the metrics $D_{h\circ\phi_n^{-1}}$ and $D_{h(\alpha\cdot)}$.
We will accomplish this using the outline discussed in Section~\ref{sec-outline}.
We first need the following elementary lemma for the GFF.
\begin{lem} \label{lem-tv-conv}
Let $\phi_n : U_n\rightarrow\phi_n(U_n)$ be as in Lemma~\ref{lem-field-metric-converge}.
Then for each compact set $K\subset\mathbbm C$,
\begin{equation} \label{eqn-as-conv}
h\circ\phi_n^{-1} |_K \rightarrow h(\alpha\cdot)|_K,\quad\text{almost surely in the distributional sense}
\end{equation}
and
\begin{equation} \label{eqn-tv-conv}
\left( h\circ\phi_n^{-1} - (h\circ\phi_n^{-1})_{\mathbbm f , |\alpha| }(0) \right)|_K \rightarrow h(\alpha \cdot)|_K ,\quad \text{in the total variation sense} .
\end{equation}
\end{lem}
\begin{proof}
By the Koebe distortion theorem, $\phi_n'(z) \rightarrow 1/\alpha$ uniformly on compact subsets of $\mathbbm C$.
\begin{comment}
Let $\phi_{R,n} := \alpha R^{-1} \phi_n(R \cdot)$. Then $\phi_{R,n}$ is conformal on $\mathbbm D$ for large enough $n$ and satisfies $\phi_{R,n}'(0) = 1$.
By the Koebe distortion theorem,
\begin{equation*}
\frac{1-|z|}{(1+|z|)^3} \leq |\phi_{R,n}'(z)| \leq \frac{1-|z|}{(1-|z|)^3} .
\end{equation*}
Setting $z = R^{-1} w$ gives
\begin{equation*}
\frac{1-|w|/R}{(1+|w|/R)^3} \leq |\alpha \phi_n'(w)| \leq \frac{1-|w|/R}{(1-|w|/R)^3} .
\end{equation*}
Hence $|\alpha \phi_n'(w)|$ is uniformly close to 1 when $n$ is large.
\end{comment}
It follows that $\phi_n(z) \rightarrow z/\alpha$ uniformly on compact subsets of $\mathbbm C$.
By the Cauchy integral formula, all of the higher-order derivatives of $\phi_n$ converge to zero uniformly on compact subsets of $\mathbbm C$.
Furthermore, $\phi_n^{-1} \rightarrow \alpha z$, $(\phi_n^{-1})'(z) \rightarrow \alpha $, and all of the higher-order derivatives of $\phi_n^{-1}$ converge to zero uniformly on compact subsets of $\mathbbm C$.
Consequently, if $f : \mathbbm C\rightarrow\mathbbm R$ is a smooth, compactly supported function then $f\circ \phi_n$ and all of its derivatives of all orders converge uniformly to $f(\alpha\cdot)$ and its corresponding derivatives as $n\rightarrow\infty$.
Therefore,
\begin{equation}
( h\circ\phi_n^{-1} , f) = (h , |\phi_n'|^2 (f \circ\phi_n) ) \rightarrow (h , |\alpha|^{-2} f(\alpha^{-1} \cdot) ) = (h(\alpha \cdot) , f) .
\end{equation}
This gives~\eqref{eqn-as-conv}.
To prove~\eqref{eqn-tv-conv}, write $h|_{U_n} = \mathring h_n + \mathfrak h_n$, where $\mathring h_n$ is a zero-boundary GFF on $U_n$ and $\mathfrak h_n$ is an independent random harmonic function on $U_n$. Then $\mathring h_n \circ\phi_n^{-1}$ is a zero-boundary GFF on $\phi(U_n)$.
By~\cite[Proposition 2.10]{ig4}, we have $\mathring h_n\circ\phi_n^{-1}|_K \rightarrow h|_K$ in the total variation sense if we view both $\mathring h_n$ and $h$ as being defined modulo a global additive constant.
The field $h(\alpha \cdot)$ is normalized so that $(h(\alpha \cdot))_{\mathbbm f , |\alpha| }(0)$ is zero.
Therefore,
\begin{equation} \label{eqn-tv-conv0}
\left(\mathring h_n\circ\phi_n^{-1} - (\mathring h_n\circ\phi_n^{-1})_{\mathbbm f , |\alpha|}(0) \right)|_K \rightarrow h(\alpha \cdot)|_K
\end{equation}
in total variation, \emph{without} having to view the distributions as being defined modulo additive constant.
On the other hand, basic estimates for the harmonic part of the GFF (see the proof of~\cite[Proposition 2.10]{ig4}) combined with the aforementioned convergence of $\phi_n^{-1}$ to $z\mapsto \alpha z$ shows that for any fixed compact set $K'\subset\mathbbm C$, the Dirichlet energy of $\mathfrak h_n|_{K'}$ tends to zero in probability as $n\rightarrow\infty$.
Combining this with the convergence of $\phi_n^{-1}$ and all of its derivatives mentioned above, we get that the same is true with $\mathfrak h_n\circ\phi_n^{-1}$ in place of $\mathfrak h_n$.
Recall that if $f$ is a smooth compactly supported bump function on $U_n$, then the laws of $\mathring h_n \circ\phi_n^{-1}$ and $\mathring h_n \circ\phi_n^{-1} + f$ are mutually absolutely continuous, and the Radon-Nikodym derivative of the latter with respect to the former is $\exp\left( (\mathring h_n \circ\phi_n^{-1} ,f)_\nabla - \frac12 (f,f)_\nabla \right)$, where $(g_1,g_2)_\nabla := \frac{1}{2\pi} \int_{U_n} \nabla g_1(z) \cdot\nabla g_2(z) \, d^2 z$ denotes the Dirichlet inner product.
By applying this formula with $f$ equal to a smooth, compactly supported bump function times $\mathfrak h_n\circ\phi_n^{-1} - (\mathfrak h_n\circ\phi_n^{-1})_{\mathbbm f , |\alpha|} $, we obtain~\eqref{eqn-tv-conv} from~\eqref{eqn-tv-conv0} and the preceding paragraph.
\end{proof}
We can now establish the convergence of the second two coordinates in Lemma~\ref{lem-field-metric-converge}.
\begin{lem} \label{lem-field-metric-converge0}
Let $\phi_n : U_n\rightarrow\phi_n(U_n)$ be as in Lemma~\ref{lem-field-metric-converge}.
Then
\begin{equation} \label{eqn-field-metric0}
\left( h^{\phi_n} , D_h^{\phi_n} \right)
\rightarrow \left( h(\alpha \cdot) + Q\log|\alpha| , D_h \right)
\end{equation}
in law with respect to the distributional topology on the first coordinate and the local uniform topology on $\mathbbm C\times\mathbbm C$ on the second coordinate.
\end{lem}
\begin{proof}
Consider a large bounded open set $V\subset \mathbbm C$. Since $D_h(\cdot,\cdot; V)$ is a deterministic functional of $h|_V$ (Axiom~\ref{item-metric-local}), the total variation convergence in Lemma~\ref{lem-tv-conv} implies that
\begin{equation} \label{eqn-field-metric-tv}
\left( \left( h\circ\phi_n^{-1} - (h\circ\phi_n^{-1})_{\mathbbm f , |\alpha| }(0) \right) |_V , D_{ h\circ\phi_n^{-1} - (h\circ\phi_n^{-1})_{\mathbbm f , |\alpha| }(0) }(\cdot,\cdot ; V) \right)
\rightarrow \left( h(\alpha \cdot)|_V , D_{h(\alpha \cdot)}(\cdot,\cdot; V) \right)
\end{equation}
in the total variation sense. Since the function $\mathbbm f$ of~\eqref{eqn-smooth-normalize} is smooth and compactly supported, we can apply~\eqref{eqn-as-conv} of Lemma~\ref{lem-tv-conv} to get that
\begin{equation} \label{eqn-normalization-to-0}
(h\circ\phi_n^{-1})_{\mathbbm f , |\alpha| }(0)
\rightarrow (h(\alpha \cdot))_{\mathbbm f , |\alpha| }(0)
= h_{\mathbbm f , 1}(0) = 0
\end{equation}
in law as $n\rightarrow\infty$.
By combining~\eqref{eqn-field-metric-tv} and~\eqref{eqn-normalization-to-0} (and using Axiom~\ref{item-metric-f} to get that the map $c\mapsto D_{h+c}$ is continuous), then letting $V$ increase to all of $\mathbbm C$, we obtain
\begin{equation} \label{eqn-field-metric-law}
\left( h\circ\phi_n^{-1} , D_{ h\circ\phi_n^{-1} } \right)
\rightarrow \left( h(\alpha \cdot) , D_{h(\alpha \cdot)} \right)
\end{equation}
in law.
Recall that $h^{\phi_n} = h\circ\phi_n^{-1} + Q\log|(\phi_n^{-1})'|$ and $D_h^{\phi_n} =D_{h^{\phi_n}}(\phi_n(\cdot),\phi_n(\cdot))$.
We have $\phi_n(z) \rightarrow \alpha^{-1} z$ and $Q\log|(\phi_n^{-1})'| \rightarrow Q\log|\alpha |$ uniformly on compact subsets of $\mathbbm C$ (see the beginning of the proof of Lemma~\ref{lem-tv-conv}).
Combining this with~\eqref{eqn-field-metric-law} and using Axiom~\ref{item-metric-f} to deal with the convergence of the metrics shows that
\begin{equation} \label{eqn-field-metric-law'}
\left( h^{\phi_n} , D_h^{\phi_n} \right)
\rightarrow \left( h(\alpha \cdot) + Q\log|\alpha| , D_{h(\alpha \cdot) + Q\log|\alpha |}(\alpha^{-1} \cdot,\alpha^{-1} \cdot) \right)
\end{equation}
in law. The right side of~\eqref{eqn-field-metric-law'} equals $\left( h(\alpha \cdot) + Q\log|\alpha| , D_h \right) $ by Axiom~{\hyperref[item-metric-coord]{IV$'$}}.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem-field-metric-converge}]
By Lemma~\ref{lem-field-metric-converge0} and the Prokhorov theorem, for any sequence of $n$'s tending to $\infty$, there is a subsequence $\mathcal N$ and a coupling $\left( h , D_h ,h' , D_{h'} \right)$ of two whole-plane GFF's and their associated metrics such that as $\mathcal N \ni n\rightarrow\infty$,
\begin{equation*}
\left( h , D_h , h^{\phi_n} , D_h^{\phi_n} \right)
\rightarrow \left( h , D_h , h'(\alpha \cdot) + Q\log|\alpha| , D_{h'} \right) ,
\end{equation*}
in law. By the a.s.\ convergence part of Lemma~\ref{lem-tv-conv}, we have $(h , h^{\phi_n}) \rightarrow (h, h(\alpha \cdot) + Q\log|\alpha| ) $ in law.
Hence $h'=h$ a.s., so also $D_{h'} =D_h$ a.s.
Therefore our subsequential limit is given by the right side of~\eqref{eqn-field-metric}.
Since our initial choice of subsequence was arbitrary, we obtain the statement of the lemma.
\end{proof}
\subsection{Uniform comparison of $D_h$ and $D_h^\phi$}
\label{sec-scaled-metric-uniform}
Continue to assume that $h$ is a whole-plane GFF normalized so that $h_{\mathbbm f , 1}(0) = 0$. To deduce Proposition~\ref{prop-scaled-metric-uniform} from Lemma~\ref{lem-field-metric-converge}, we need to re-scale to convert from a statement about conformal maps at small scales to a statement about conformal maps which are close to linear at constant-order scales; and we need to ensure that the estimate we obtain is uniform over all $z\in U$.
The re-scaling will be accomplished by means of the following basic calculation.
\begin{lem} \label{lem-coord-scale}
Fix $r > 0$ and $z\in\mathbbm C$ and let $\widetilde h := h(r\cdot + z) - h_{\mathbbm f ,r}(z)$, so that $\widetilde h \overset{d}{=} h$.
Also let $\widetilde\phi : r^{-1}(U-z) \rightarrow r^{-1}(\phi(U) -z)$ be defined by $\widetilde\phi(w) = r^{-1}(\phi(r w + z) - z)$.
Then
\begin{equation}
D_{\widetilde h}^{\widetilde\phi}( u , v) = r^{-\xi Q} e^{-\xi h_{\mathbbm f , r}(z)} D_h^\phi(r u + z , r v + z) ,\quad\forall u ,v\in r^{-1}(U-z) .
\end{equation}
\end{lem}
\begin{proof}
Recall the definition of $h^\phi$ from~\eqref{eqn-coord-field-def}.
We apply Axiom~\ref{item-metric-f} and then Axiom~{\hyperref[item-metric-coord]{IV$'$}}\, to $h^\phi$ to get that for $u,v \in r^{-1}(U-z)$,
\alb
D_{\widetilde h}^{\widetilde\phi}(u,v)
&= D_{h( \phi^{-1}(r\cdot + z) ) + Q\log|(\phi^{-1})'(r\cdot + z)| - h_{\mathbbm f , r}(z) }\left(r^{-1} (\phi(r u + z) - z) , r^{-1} ( \phi(r v + z) -z) \right) \notag\\
&= e^{-\xi h_{\mathbbm f , r}(z)} D_{h^\phi(r\cdot + z)}\left(r^{-1} (\phi(r u + z) - z) , r^{-1} ( \phi(r v + z) -z) \right) \quad \text{(Axiom~\ref{item-metric-f})} \notag\\
&= r^{-\xi Q} e^{-\xi h_{\mathbbm f , r}(z)} D_{h^\phi } \left( \phi(r u + z) , \phi(r v + z) \right) \quad \text{(Axiom~{\hyperref[item-metric-coord]{IV$'$}})} \notag \\
&= r^{-\xi Q} e^{-\xi h_{\mathbbm f , r}(z)} D_h^\phi(r u + z , r v + z) \quad \text{(by the definition~\eqref{eqn-coord-field-def} of $D_h^\phi$)}.
\ale
\end{proof}
In what follows, we fix $\delta > 0$ and for $z\in U$ and $r> 0$ such that $B_r(z)\subset U$, we let
\begin{equation} \label{eqn-scaled-metric-event}
F_r(z) := \left\{ \sup_{u,v\in B_r(z)} | D_h^\phi(u,v) - D_h(u,v) | \leq \delta r^{\xi Q} e^{\xi h_{\mathbbm f , r}(z)} \right\}
\end{equation}
be the event of Proposition~\ref{prop-scaled-metric-uniform}.
\begin{lem} \label{lem-scaled-metric-conv}
Consider a sequence of points $\{z_n\}_{n\in\mathbbm N} \subset U$ and radii $\{r_n\}_{n\in\mathbbm N}$ such that $z_n\rightarrow z\in U$ and $r_n\rightarrow 0$.
In the notation of Proposition~\ref{prop-scaled-metric-uniform}, we have $\lim_{n\rightarrow\infty} \mathbbm P[F_{r_n}(z_n)] = 1$.
\end{lem}
\begin{proof}
For $n\in\mathbbm N$, define the conformal map
\begin{equation*}
\phi_n : r_n^{-1} (U_n - z_n) \rightarrow r_n^{-1}(\phi(U_n) - z_n) \quad \text{by} \quad \phi_n(w) = r_n^{-1} \left( \phi (r_n w + z_n) - z_n \right)
\end{equation*}
Then $\phi_n(0) = 0$, $\phi_n'(0) = \phi'(z_n) \rightarrow \phi'(z)$, and (since $r_n\rightarrow 0$ and $z$ lies at positive distance from $\partial U$) every compact subset of $\mathbbm C$ is contained in $r_n^{-1} (U_n - z_n)$ for large enough $n$.
Define the field $h_n := h(r_n \cdot+z_n) - h_{\mathbbm f , r_n}(z_n) \overset{d}{=} h$.
By Lemma~\ref{lem-field-metric-converge} applied with $h_n$ in place of $h$ and $\alpha = 1/\phi_n'(z)$, we have the convergence of joint laws
\begin{equation}
\left( h_n , D_{h_n} , h_n^{\phi_n} , D_{h_n}^{\phi_n} \right) \rightarrow \left( h,D_h,h(\cdot/\phi'(z)) + Q\log |1/\phi'(z)| ,D_h \right) .
\end{equation}
In particular, it holds with probability tending to 1 as $n\rightarrow\infty$ that
\begin{equation} \label{eqn-scaled-metric-close}
| D_{h_n}^{\phi_n}(u,v) - D_{h_n}(u,v) | \leq \delta ,\quad \forall u,v\in \mathbbm D .
\end{equation}
By Lemma~\ref{lem-coord-scale} along with Axioms~\ref{item-metric-f} and~{\hyperref[item-metric-coord]{IV$'$}}\, for $h$, if $r_n$ is sufficiently large that $\mathbbm D\subset r_n^{-1}(U-z_n)$, then
\allb \label{eqn-scaled-metric-compare}
&D_{h_n}(u,v) = r_n^{-\xi Q} e^{-\xi h_{\mathbbm f , r_n}(z_n)} D_h(r_n u + z_n , r_n v + z_n) \quad \text{and} \notag \\
&\qquad D_{h_n}^{\phi_n}(u,v) = r_n^{-\xi Q} e^{-\xi h_{\mathbbm f , r_n}(z_n)} D_h^\phi(r_n u + z_n , r_n v + z_n) ,
\quad \forall u ,v \in \mathbbm D .
\alle
Combining~\eqref{eqn-scaled-metric-close} with~\eqref{eqn-scaled-metric-compare} gives the statement of the lemma.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop-scaled-metric-uniform}]
Assume by way of contradiction that for some compact set $K\subset U$, the relation~\eqref{eqn-scaled-metric-uniform} fails.
Then there is an $\varepsilon > 0$, a sequence $r_n\rightarrow 0$, and a sequence of points $z_n \in K$ such that $\mathbbm P[F_{r_n}(z_n)]\leq 1-\varepsilon$ for every $n\in\mathbbm N$.
By possibly passing to a subsequence, we can assume without loss of generality that $z_n\rightarrow z\in K$.
Then Lemma~\ref{lem-scaled-metric-conv} shows that $\lim_{n\rightarrow\infty} \mathbbm P[F_{r_n}(z_n)] = 1$, which is a contradiction.
\end{proof}
\section{Proof of Theorem~\ref{thm-coord}}
\label{sec-coord-proof}
Recall the notation $D_h^\phi$ from~\eqref{eqn-coord-field-def}.
To prove Theorem~\ref{thm-coord}, we want to upgrade from the statement that $D_{h|_U}$ and $D_h^\phi$ are close with high probability at small scales (Proposition~\ref{prop-scaled-metric-uniform}) to the statement that these two metrics are a.s.\ close globally. This will be done using various local independence properties of the GFF.
In this section we will mostly use circle averages rather than the smoothed average $ h_{\mathbbm f , r}(z)$ of Section~\ref{sec-bilip}.
\subsection{Bi-Lipschitz equivalence of $D_h$ and $D_h^\phi$}
\label{sec-coord-bilip}
Before establishing that $D_{h|_U} =D_h^\phi$, we will show that the two metrics are bi-Lipschitz equivalent.
\begin{prop} \label{prop-coord-bilip}
Let $h$ be a whole-plane GFF, with any choice of additive constant.
For each conformal map $\phi : U\rightarrow\phi(U)$, there is a constant $C \geq 1$ such that a.s.\
\begin{equation}
C^{-1} D_{h|_U}(z,w) \leq D_h^\phi(z,w) \leq C D_{h|_U}(z,w) ,\quad\forall z,w\in U .
\end{equation}
\end{prop}
Proposition~\ref{prop-coord-bilip} will be a consequence of Proposition~\ref{prop-scaled-metric-uniform} together with a general criterion for metrics coupled with the same GFF to be bi-Lipschitz equivalent which is proven in~\cite{local-metrics}. To state the criterion, we need a couple of preliminary definitions.
\begin{defn}[Jointly local metrics] \label{def-jointly-local}
Let $U\subset \mathbbm C$ be a connected open set and let $(h,D_1,\dots,D_n)$ be a coupling of a GFF on $U$ and $n$ random continuous length metrics.
We say that $D_1,\dots,D_n$ are \emph{jointly local metrics} for $h$ if for any open set $V\subset U$, the collection of internal metrics $\{ D_j(\cdot,\cdot; V) \}_{j = 1,\dots,n}$ is conditionally independent from $(h|_{U\setminus V} , \{ D_j(\cdot,\cdot; U\setminus \overline V) \}_{j = 1,\dots,n} )$ given $h|_V$.
\end{defn}
We note that if each of $D_1,\dots,D_n$ is a local metric for $h$ and is determined by $h$, then $D_1,\dots,D_n$ are jointly local for $h$.
This is a consequence of~\cite[Lemma 1.4]{local-metrics}.
In particular, $D_{h|_U}$ and $D_h^\phi$ are jointly local for $h|_U$.
\begin{defn}[Additive local metrics] \label{def-additive-local}
Let $U\subset \mathbbm C$ be a connected open set and let $(h^U ,D_1,\dots,D_n)$ be a coupling of a GFF on $U$ and $n$ random continuous length metrics which are jointly local for $h$.
For $\xi \in\mathbbm R$, we say that $ D_1,\dots,D_n $ are \emph{$\xi$-additive} for $h^U$ if for each $z\in U$ and each $r> 0$ such that $B_r(z) \subset U$, the metrics $(e^{-\xi h_r^U(z)} D_1,\dots, e^{-\xi h_r^U(z)} D_n)$ are jointly local metrics for $h^U - h_r^U(z)$ (where, as per usual, $h^U_r(z)$ denotes the circle average).
\end{defn}
It is clear from Axiom~\ref{item-metric-f} that the metrics $D_{h|_U}$ and $D_h^\phi$ are $\xi$-additive for $h|_U$.
The following theorem is~\cite[Theorem 1.6]{local-metrics}.
For the statement, we recall the notation for Euclidean annuli from~\eqref{eqn-annulus-def}.
\begin{thm} \label{thm-bilip}
Let $\xi \in\mathbbm R$, let $h$ be a whole-plane GFF normalized so that $h_1(0) = 0$, let $U\subset\mathbbm C$, and let $(h,D,\widetilde D)$ be a coupling of $h$ with two random continuous metrics on $U$ which are jointly local and $\xi$-additive for $h|_U$.
There is a universal constant $p \in (0,1)$ such that the following is true.
Suppose there is a constant $C>0$ such that for each compact set $K\subset U$, there exists $r_K > 0$ such that
\begin{equation} \label{eqn-bilip}
\mathbbm P\left[ \sup_{u,v \in \partial B_r(z)} \widetilde D\left(u,v; \mathbbm A_{r/2,2r}(z) \right) \leq C D(\partial B_{r/2}(z) , \partial B_r(z) ) \right] \geq p ,\quad \forall z\in K, \quad \forall r \in (0,r_K] .
\end{equation}
Then a.s.\ $\widetilde D(z,w) \leq C D(z,w)$ for all $z,w\in\mathbbm C$.
\end{thm}
Let us now check the condition~\eqref{eqn-bilip} for the metrics $D_{h|_U}$ and $D_{h^\phi}$ using Proposition~\ref{prop-scaled-metric-uniform}.
\begin{lem} \label{lem-coord-ratio}
Let $h$ be a whole-plane GFF, with any choice of additive constant.
For each $p\in (0,1)$, there exists a constant $C = C(p,\gamma) > 0$ such that for each choice of conformal map $\phi : U \rightarrow \phi(U)$ and each compact set $K\subset U$, there exists $r_K = r_K(\phi) > 0$ such that
\begin{equation}
\mathbbm P\left[ \sup_{u,v \in \partial B_r(z)} D_h^\phi\left(u,v; \mathbbm A_{r/2,2r}(z) \right) \leq C D_{h|_U}(\partial B_{r/2}(z) , \partial B_r(z) ) \right] \geq p ,\quad \forall z\in K, \quad \forall r \in (0,r_K] ;
\end{equation}
and the same is true with $D_h^\phi$ and $D_h$ interchanged.
\end{lem}
\begin{proof}
As in the proof of Lemma~\ref{lem-scaled-metric-ratio}, due to Axiom~\ref{item-metric-f} we can assume without loss of generality that $h$ is normalized so that $h_{\mathbbm f ,1}(0) = 0$.
By Axioms~\ref{item-metric-f} and~{\hyperref[item-metric-coord]{IV$'$}}, the fact that $D_h$ induces the Euclidean topology, and the scale and translation invariance of the law of $h$, modulo additive constant, we can find small constants $a,b >0$ such that for each $z\in\mathbbm C$ and each $r>0$, it holds with probability at least $1-(1-p)/2$ that the following is true.
\begin{enumerate}
\item $D_h(\partial B_r(z) , \partial \mathbbm A_{r/2,2r}(z)) \geq a r^{\xi Q} e^{\xi h_{\mathbbm f , r}(z)} $. \label{item-coord-ratio-across}
\item For each $u,v\in \partial B_r(z)$ with $|u-v| \leq b r$, we have $D_h(u,v) \leq (a/100) r^{\xi Q} e^{\xi h_{\mathbbm f , r}(z)}$. \label{item-coord-ratio-around}
\end{enumerate}
By Proposition~\ref{prop-scaled-metric-uniform}, for each compact set $K\subset U$ there exists $r_K = r_K(\phi) > 0$ such that for each $z\in K$ and each $r\in (0,r_K]$, it holds with probability at least $1-(1-p)/2$ that
\begin{equation}
\sup_{u,v\in B_{2r}(z)} | D_h^\phi(u,v) - D_h(u,v) | \leq \frac{a}{100} r^{\xi Q} e^{\xi h_{\mathbbm f , r}(z)} .
\end{equation}
Combining these estimates shows that for each $z\in K$ and each $r\in (0,r_K]$, it holds with probability at least $p$ that
\begin{enumerate}
\item $D_h^\phi\left(\partial B_r(z) , \partial \mathbbm A_{r/2,2r}(z) \right) \geq (a/2) r^{\xi Q} e^{\xi h_{\mathbbm f , r}(z)} $.\label{item-coord-ratio-across'}
\item For each $u,v\in \partial B_r(z)$ with $|u-v| \leq b r$, we have $D_h^\phi(u,v) \leq (a/2) r^{\xi Q} e^{\xi h_{\mathbbm f , r}(z)}$. \label{item-coord-ratio-around'}
\end{enumerate}
If $u,v\in \partial B_r(z)$ such that $D_h(u,v) \leq D_h(\partial B_r(z) , \partial \mathbbm A_{r/2,2r}(z))$, then $D_h(u,v) = D_h(u,v; \mathbbm A_{r/2,2r}(z))$.
By applying condition~\ref{item-coord-ratio-around} for $D_h$ and the triangle inequality at most $2\pi/b$ times, we therefore have that
\begin{equation}
\sup_{u,v \in \partial B_r(z)} D_h \left(u,v; \mathbbm A_{r/2,2r}(z) \right) \leq \frac{a \pi}{50 b} r^{\xi Q} e^{\xi h_{\mathbbm f , r}(z)} .
\end{equation}
Similarly, by condition~\ref{item-coord-ratio-around'} for $D_h^\phi$ and applying the triangle inequality at most $2\pi/b$ times, we have that
\begin{equation}
\sup_{u,v \in \partial B_r(z)} D_h^\phi\left(u,v; \mathbbm A_{r/2,2r}(z) \right) \leq \frac{a \pi}{ b} r^{\xi Q} e^{\xi h_{\mathbbm f , r}(z)}.
\end{equation}
Combining the preceding two estimates with condition~\ref{item-coord-ratio-across} for each of $D_h$ and $D_h^\phi$ gives the statement of the lemma with $C = \pi/b$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop-coord-bilip}]
The metrics $D_h$ and $D_h^\phi$ are each local metrics for $h|_U$. Moreover, these metrics are determined by $h|_U$ so they are jointly local for $h|_U$. By Axiom~\ref{item-metric-f}, for $z\in \mathbbm C$ and $r>0$, the metrics $e^{-\xi h_r(z)} D_h$ and $e^{-\xi h_r(z)} D_h^\phi$ are each local for $h|_U - h_r(z)$, so in particular these metrics are $\xi$-additive for $h|_U$.
Therefore, the proposition statement follows from Lemma~\ref{lem-coord-ratio} and Theorem~\ref{thm-bilip}.
\end{proof}
\subsection{The bi-Lipschitz constant is 1}
\label{sec-attained}
We will now show that in fact the constant $C$ in Proposition~\ref{prop-coord-bilip} can be taken to be one.
\begin{prop} \label{prop-attained}
Let $h$ be a whole-plane GFF (with any choice of normalization). Let $U\subset\mathbbm C$ be an open domain and let $\phi : U\rightarrow \phi(U)$ be a conformal map.
Almost surely, we have $D_h^\phi(z,w) \leq D_{h|_U}(z,w)$ for each $z,w\in U$.
\end{prop}
Before proving Proposition~\ref{prop-attained}, we explain why it implies our main result.
\begin{proof}[Proof of Theorem~\ref{thm-coord}, assuming Proposition~\ref{prop-attained}]
For a whole-plane GFF $h$ and a domain $U\subset\mathbbm C$, we can write $h|_U$ as the sum of a zero-boundary GFF on $U$ and an independent random harmonic function on $U$.
Therefore, if $h^U$ is a random distribution on $U$ as in Theorem~\ref{thm-coord}, then we can couple $h^U$ with $h$ in such a way that $g := h|_U -h^U$ is a random continuous function on $U$.
By Proposition~\ref{prop-attained} and Axiom~\ref{item-metric-f}, it follows that a.s.\
\alb
D_{h^U}(z,w)
= (e^{\xi g}\cdot D_{h|_U})(z,w)
&\geq (e^{\xi g} \cdot D_h^\phi)(z,w) \notag \\
&= (e^{\xi (g\circ\phi^{-1} ) }\cdot D_{h^\phi})\left(\phi(z) , \phi(w) \right) \notag\\
&= D_{h^U\circ\phi^{-1} + Q\log|(\phi^{-1})'|}\left(\phi(z),\phi(w)\right) ,\quad\forall z,w \in U .
\ale
This is a one-sided version of~\eqref{eqn-coord}.
By the conformal invariance of the law of the zero-boundary GFF, we can apply this one-sided statement with $h^U\circ\phi^{-1} + Q\log|(\phi^{-1})'|$ in place of $h$ and $\phi^{-1}$ in place of $\phi$ to get the opposite inequality in~\eqref{eqn-coord}.
\end{proof}
The proof of Proposition~\ref{prop-attained} is similar to the proof of~\cite[Proposition 3.6]{gm-uniqueness}.
Fix a small $\delta \in (0,1)$ (which we will eventually send to zero) and a parameter $\alpha\in (1/2,1)$, a little bit less than 1.
We will use Lemma~\ref{lem-scaled-metric-ratio} together with a general independence result for events for the GFF restricted to annuli (Lemma~\ref{lem-annulus-iterate} just below) to cover a given compact set $K\subset U$ by Euclidean balls of the form $B_{r/2}(z)$ such that $D_h^\phi(u,v) \leq (1+\delta) D_{h|_U}(u,v)$ for each $u \in \partial B_{\alpha r}(z)$ and each $v\in \partial B_{r}(z)$ which can be joined by a $D_{h|_U}$-geodesic contained in $\overline{\mathbbm A_{\alpha r , r}(z)}$.
If we assume that $D_h^\phi \leq C D_{h|_U}$ for some $C>1$, then by considering the times when a $D_{h|_U}$-geodesic between two fixed points $\mathbbm z , \mathbbm w \in \mathbbm C$ crosses the annulus $\mathbbm A_{\alpha r , r}(z)$ for such a $z$ and $r$, we will be able to show that $ D_h^\phi(\mathbbm z , \mathbbm w) \leq C_\delta D_{h|_U}(\mathbbm z,\mathbbm w)$ for a constant $C_\delta$ which is strictly smaller than $C$ if $\delta$ is chosen to be sufficiently small.
This shows that one has to have $D_h^\phi \leq C D_{h|_U}$ for $C=1$, since if the optimal constant for which this holds is strictly bigger than 1, then this optimal constant can be improved, which is a contradiction.
The following annulus iteration lemma, which is \cite[Lemma~3.1]{local-metrics}, a generalization of a result from \cite{mq-geodesics}, will be used to produce the desired covering by balls of the form $B_{r/2}(z)$.
\begin{lem} \label{lem-annulus-iterate}
Fix $0 < s_1<s_2 < 1$. Let $\{r_k\}_{k\in\mathbbm N}$ be a decreasing sequence of positive real numbers such that $r_{k+1} / r_k \leq s_1$ for each $k\in\mathbbm N$ and let $\{E_{r_k} \}_{k\in\mathbbm N}$ be events such that $E_{r_k} \in \sigma\left( (h-h_{r_k}(0)) |_{\mathbbm A_{s_1 r_k , s_2 r_k}(0) } \right)$ for each $k\in\mathbbm N$.
For $K\in\mathbbm N$, let $N(K)$ be the number of $k\in [1,K]_{\mathbbm Z}$ for which $E_{r_k}$ occurs.
\item For each $a > 0$ and each $b\in (0,1)$, there exists $p = p(a,b,s_1,s_2) \in (0,1)$ and $c = c(a,b,s_1,s_2) > 0$ such that if \label{item-annulus-iterate-high}
\begin{equation} \label{eqn-annulus-iterate-prob}
\mathbbm P\left[ E_{r_k} \right] \geq p , \quad \forall k \in \mathbbm N,
\end{equation}
then
\begin{equation} \label{eqn-annulus-iterate}
\mathbbm P\left[ N(K) < b K\right] \leq c e^{-a K} ,\quad\forall K \in \mathbbm N.
\end{equation}
\end{lem}
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=1]{figures/fig-attained-long.pdf}
\vspace{-0.01\textheight}
\caption{
Illustration of condition~\ref{item-attained-long} in the definition of $E_r(z)$. Suppose that $u \in \partial B_{\alpha r}(z)$ and $v \in \partial B_{r}(z)$ are ``far apart" in the sense that $v$ is further from $u$ than the boundary of the large blue annulus $\mathbbm A_{r/2,r}(z)$, w.r.t.\ either $D_h$ or $D_h^\phi$. Then condition~\ref{item-attained-long} says that any path from $u$ to $v$ which is contained in $\overline{\mathbbm A_{\alpha r , r}(z)}$ (such as the red path in the figure) has to have $D_h$-length strictly larger than $D_h(u,v;\mathbbm A_{r/2,2r}(z) )$, so in particular such a path cannot be a $D_h$-geodesic. To obtain that this condition holds with high probability, we will choose $\alpha$ to be close to 1 and use the fact paths which are ``close" to a circular arc have large $D_h$-lengths (see Lemma~\ref{lem-attained-long}).
}\label{fig-attained-long}
\end{center}
\vspace{-1em}
\end{figure}
Let us now define the events to which we will apply Lemma~\ref{lem-annulus-iterate}.
For $z \in U$, $r > 0$ such that $B_r(z) \subset U$, and parameters $\alpha \in (1/2,1)$, $ A > 1$, and $\delta \in (0,1)$ and let $E_r(z) = E_r(z; \alpha,A,\delta)$ be the event that the following is true.
\begin{enumerate}
\item For each $u \in \partial B_{\alpha r}(z)$ and each $v\in \partial B_{r}(z)$ such that there is a $D_h$-geodesic from $u$ to $v$ which is contained in $\overline{\mathbbm A_{\alpha r , r}(z)}$, we have $D^\phi_h(u,v) \leq (1+\delta) D_h(u,v)$. \label{item-attained-dist}
\item If $u \in \partial B_{\alpha r}(z)$ and $v\in \partial B_r(z)$ such that either $D_h(u,v) > D_h(u , \partial\mathbbm A_{r/2,2r}(z))$ or $ D^\phi_h(u,v) > D^\phi_h(u , \partial\mathbbm A_{r/2,2r}(z))$, then each path from $u$ to $v$ which stays in $\overline{\mathbbm A_{\alpha r , r}(z)}$ has $D_h$-length strictly larger than $D_h\left( u , v ; \mathbbm A_{r/2,2r}(z) \right)$ (see Figure~\ref{fig-attained-long} for an illustration). \label{item-attained-long}
\item There is a path in $\mathbbm A_{\alpha r , r}(z)$ which disconnects the inner and outer boundaries of $\mathbbm A_{\alpha r , r}(z)$ and has $D_h$-length at most $A D_h\left(\partial B_{\alpha r}(z) , \partial B_{ r}(z) \right)$. \label{item-attained-around}
\end{enumerate}
Condition~\ref{item-attained-dist} is the main point of the event $E_r(z)$, as discussed just above.
The purpose of condition~\ref{item-attained-long} is to ensure that $E_r(z)$ is determined by $h|_{\mathbbm A_{r/2,2r}(z)}$.
Indeed, as we will see in the proof of Lemma~\ref{lem-attained-msrble} just below, on the event that this condition is satisfied $h|_{\mathbbm A_{r/2,2r}(z)}$ determines which paths in $\overline{\mathbbm A_{\alpha r ,r}(z)}$ are $D_h$-geodesics.
The purpose of condition~\ref{item-attained-around} is to ensure that the annuli $\mathbbm A_{\alpha r , r}(z)$ for which $E_r(z)$ occurs (rather than just the balls $B_r(z)$ for which $E_r(z)$ occurs) cover a positive fraction of the $D_h$-length of a $D_h$-geodesic.
Indeed, if a $D_h$-geodesic between two points outside of $B_r(z)$ enters $B_{\alpha r}(z)$, then it must cross the path from condition~\ref{item-attained-around} twice.
Since geodesics are length minimizing, this means that it can spend at most $A D_h\left(\partial B_{\alpha r}(z) , \partial B_{ r}(z) \right)$ units of time in $B_{\alpha r}(z)$: otherwise the path from condition~\ref{item-attained-around} would provide a shortcut.
We want to use Lemma~\ref{lem-annulus-iterate} to argue that with high probability we can cover any given compact subset of $U$ by balls $B_{r/2}(z)$ for which $E_r(z)$ occurs.
We first check the measurability condition in Lemma~\ref{lem-annulus-iterate}
\begin{lem} \label{lem-attained-msrble}
For each $z\in\mathbbm C$ and $r>0$,
\begin{equation} \label{eqn-attained-msrble}
E_r(z) \in \sigma\left( (h-h_{4r}(z)) |_{ \mathbbm A_{r/2,2r}(z) } \right) .
\end{equation}
\end{lem}
\begin{proof}
By Axiom~\ref{item-metric-f} subtracting $h_{4r}(0)$ from $h$ results in scaling each of $D_h$ and $D^\phi_h$ by $e^{\xi h_{4r}(0)}$, so does not affect the occurrence of $E_r(z)$.
Hence it suffices to show that $E_r(z) \in \sigma( h|_{\mathbbm A_{r/2,2r}(z)})$.
By Axiom~\ref{item-metric-local}, condition~\ref{item-attained-around} in the definition of $E_r(z)$ is determined by $h |_{ \mathbbm A_{r/2,2r}(z) }$.
For $u \in \partial B_{\alpha r}(z)$ and $v\in \partial B_{r}(z)$, we can determine whether $D_h(u,v) > D_h(u , \partial\mathbbm A_{r/2,2r}(z))$ from the internal metric $D_h\left(\cdot,\cdot ;\mathbbm A_{r/2,2r}(z) \right)$: indeed, $D_h(u , \partial\mathbbm A_{r/2,2r}(z))$ is clearly determined by this internal metric and $D_h(u,v) \leq D_h(u , \partial\mathbbm A_{r/2,2r}(z))$ if and only if $v$ is contained in the $D_h$-ball of radius $ D_h(u , \partial\mathbbm A_{r/2,2r}(z))$ centered at $v$, which is contained in $\overline{\mathbbm A_{r/2,2r}(z)}$.
Similar considerations hold with $D^\phi_h$ in place of $D_h$.
By the locality of the metrics $D_h$ and $D_h^\phi$, it follows that condition~\ref{item-attained-long} in the definition of $E_r(z)$ is determined by $h|_{\mathbbm A_{r/2,2r}(z)}$.
If $P$ is a path from $ u \in \partial B_{\alpha r}(z)$ to $v\in \partial B_r(z)$ which stays in $\overline{\mathbbm A_{\alpha r ,r}(z)}$, then $P$ is a $D_h$-geodesic if and only if $\operatorname{len}(P ; D_h) = D_h(u,v)$.
Therefore, if condition~\ref{item-attained-long} holds, then in order for $P$ to be a $D_h$-geodesic we must have $D_h(u,v) \leq D_h(u , \partial\mathbbm A_{r/2,2r}(z) )$ and $ D^\phi_h(u,v) \leq D^\phi_h(u , \partial\mathbbm A_{r/2,2r}(z) )$ (note that $D_h(u,v ; \mathbbm A_{r/2,2r}(z)) \geq D_h(u,v)$).
If this is the case, then we can tell whether $P$ is a $D_h$-geodesic from the restriction of $h$ to the $D_h$-metric ball of radius $D_h(u , \partial\mathbbm A_{r/2,2r}(z) )$ centered at $u$.
We know this restriction is determined by $h|_{\mathbbm A_{r/2,2r}(z)}$ by Axiom~\ref{item-metric-local}.
On the event that $D_h(u,v) \leq D_h(u , \partial\mathbbm A_{r/2,2r}(z) )$ and $ D^\phi_h(u,v) \leq D^\phi_h(u , \partial\mathbbm A_{r/2,2r}(z) )$, both $D_h(u,v)$ and $D^\phi_h(u,v)$ are determined by $h|_{\mathbbm A_{r/2,2r}(z)}$.
Therefore, the intersection of conditions~\ref{item-attained-dist} and~\ref{item-attained-long} in the definition of $E_r(z)$ is determined by $h|_{\mathbbm A_{r/2,2r}(z)}$.
Hence we have proven~\eqref{eqn-attained-msrble}.
\end{proof}
We now use Lemma~\ref{lem-scaled-metric-ratio} to prove a lower bound for the probability that $E_r(z)$ occurs for at least one small value of $r$.
\begin{lem} \label{lem-shorter-annulus}
For each $q>1$, there exist parameters $\alpha \in (1/2,1)$ and $A>1$, depending only on $q$, such that for each compact set $K\subset U$ and each $\delta\in(0,1)$, we have
\begin{equation} \label{eqn-shorter-annulus}
\inf_{z\in K} \mathbbm P\left[ \text{$E_r(z)$ occurs for at least one $r\in [\varepsilon^2 , \varepsilon ] \cap \{2^{-k} : k\in\mathbbm N\}$} \right] \geq 1 - O_\varepsilon(\varepsilon^q ) .
\end{equation}
\end{lem}
To deal with condition~\ref{item-attained-long} in the definition of $E_r(z)$, we will use the following lower bound for the $D_h$-lengths of paths in a narrow Euclidean annulus, which is~\cite[Lemma 2.11]{gm-uniqueness} (note that the number $\mathfrak c_r$ from~\cite{gm-uniqueness} is equal to $r^{\xi Q}$, see~\cite[Section 1.4]{gm-uniqueness}).
\begin{lem}[\cite{gm-uniqueness}] \label{lem-attained-long}
For each $S > s > 0$ and each $p\in (0,1)$, there exists $\alpha_* = \alpha_*(s,S,p) \in (1/2,1)$ such that for each $\alpha\in [\alpha_*,1)$, each $z\in\mathbbm C$, and each $r > 0$,
\begin{equation} \label{eqn-attained-long}
\mathbbm P\left[ \inf\left\{ D_h\left( u , v ; \mathbbm A_{\alpha r , r}(z) \right) : u , v \in \mathbbm A_{\alpha r , r }(z) , D_h(u,v) \geq s r^{\xi Q} e^{\xi h_{\mathbbm r}(z)} \right\} \geq S r^{\xi Q} e^{\xi h_r(z)} \right] \geq p .
\end{equation}
\end{lem}
\begin{proof}[Proof of Lemma~\ref{lem-shorter-annulus}]
By Lemma~\ref{lem-attained-msrble}, we can apply Lemma~\ref{lem-annulus-iterate} to find that there exists $p = p(q) \in (0,1)$ such that if~\eqref{eqn-shorter-annulus-show} just below holds, then~\eqref{eqn-shorter-annulus} holds:
\begin{equation} \label{eqn-shorter-annulus-show}
\text{$\exists r_0 = r_0(K,\delta) > 0$ such that} \: \mathbbm P[E_r(z)] \geq p ,\quad \forall z\in K , \quad \forall r \in (0,r_0] .
\end{equation}
It therefore suffices to choose $\alpha$ and $A$ in a manner depending only on $p$ in such a way that~\eqref{eqn-shorter-annulus-show} holds.
We first deal with condition~\ref{item-attained-long}.
By Axioms~\ref{item-metric-f} and~{\hyperref[item-metric-coord]{IV$'$}}, the fact that $D_h$ induces the Euclidean topology, and the scale and translation invariance of the law of $h$, modulo additive constant, we can find $S > s > 0$ depending on $ p$ such that for each sufficiently small $r>0$ (depending only on $K$), for each $z\in K$ it holds with probability at least $1 - (1- p)/4$ that
\begin{equation}
D_h\left( \partial \mathbbm A_{3r/4,r}(z) , \partial \mathbbm A_{r/2 , 2r}(z) \right) \geq s r^{\xi Q} e^{\xi h_r(z)} \quad \text{and} \quad
\sup_{u,v\in \mathbbm A_{3 r /4 , r}(z)} D_h\left( u ,v ; \mathbbm A_{r/2,2r}(z)\right) \leq S r^{\xi Q} e^{\xi h_r(z)} .
\end{equation}
Since $\phi$ is nearly linear at small scales, after possibly decreasing $s$ and increasing $S$ we can arrange that the same is true with $D^\phi_h$ in place of $D_h$.
Since $ \mathbbm A_{\alpha r , r}(z) \subset \mathbbm A_{3 r /4 , r}(z) $ for any choice of $\alpha \in [3/4,1)$, Lemma~\ref{lem-attained-long} with the above choice of $s$ and $S$ gives an $\alpha \in [3/4,1)$ depending on $ p$ such that for each sufficiently small $r > 0$, it holds for each $z\in K$ that the probability of condition~\ref{item-attained-long} in the definition of $E_r(z)$ is at least $1 - (1-p)/3$.
By applying Axioms~\ref{item-metric-f} and~{\hyperref[item-metric-coord]{IV$'$}}\, as above, we can find $A > 1$ depending on $p$ such that for each sufficiently small $r > 0$, it holds for each $z\in K$ that the probability of condition~\ref{item-attained-around} in the definition of $E_r(z)$ is at least $1 - (1 - p)/3$.
By Lemma~\ref{lem-scaled-metric-ratio} applied with $b = 1-\alpha$, for each sufficiently small $r > 0$, it holds for each $z\in K$ that the probability of condition~\ref{item-attained-dist} in the definition of $E_r(z)$ is at least $1-(1-p)/3$. Combining the three preceding paragraphs shows that~\eqref{eqn-shorter-annulus-show} holds.
\end{proof}
\begin{lem} \label{lem-shorter-annulus-all}
There is a universal constant $q>1$ such that if $\alpha$ and $A$ are chosen as in Lemma~\ref{lem-shorter-annulus} for this choice of $q$, then for each compact set $K\subset U$ and each $\delta\in(0,1)$, it holds with probability tending to 1 as $\varepsilon\rightarrow 0$ that the following is true. For each $z\in K$, there exists $r \in [\varepsilon^2,\varepsilon] \cap \{2^{-k} : k\in\mathbbm N\}$ and $w\in \left(\frac{\varepsilon^2 }{4} \mathbbm Z^2 \right) \cap B_\varepsilon(K)$ such that $z\in B_{r/2}(w)$ and $E_r(w) $ occurs.
\end{lem}
\begin{proof}
Upon choosing $q$ sufficiently large, this follows from Lemma~\ref{lem-shorter-annulus} and a union bound over all $w \in \left(\frac{\varepsilon^2}{4} \mathbbm Z^2 \right) \cap B_\varepsilon(K)$.
\end{proof}
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=1]{figures/fig-attained.pdf}
\vspace{-0.01\textheight}
\caption{Illustration of the proof of Proposition~\ref{prop-attained}. The $D_h$-geodesic $P$ from $\mathbbm z$ to $\mathbbm w$ along with one of the balls $B_{r_j}(w_j)$ hit by $P$ for which $E_{r_j}(w_j)$ occurs are shown.
The time $t_j$ is the first time after $t_{j-1}$ at which $P$ exits $B_{r_j}(w_j)$ and the time $s_j$ is the last time before $t_j$ at which $P$ hits $\partial B_{\alpha r_j}(w_j)$. Condition~\ref{item-attained-dist} in the definition of $E_{r_j}(w_j)$ tells us that $D_h^\phi(P(s_j) , P(t_j)) \leq (1+\delta) (t_j - s_j)$.
The orange path comes from condition~\ref{item-attained-around} in the definition of $E_{r_j}(w_j)$.
It has $D_h$-length is at most $A D_h(\partial B_{\alpha r_j}(w_j) , \partial B_{r_j}(w_j)) \leq A (t_j - s_j)$.
Since $P$ is a $D_h$-geodesic and $P$ crosses this orange path both before time $t_{j-1}$ and after time $s_j$, it follows that $s_j - t_{j-1} \leq A (t_j - s_j)$. This allows us to show that the ``good" intervals $[s_j , t_j]$ occupy a uniformly positive fraction of the total $D_h$-length of $P$. This then allows us to show that $D_h^\phi(\mathbbm z , \mathbbm w) \leq C_\delta D_{h|_U}(\mathbbm z ,\mathbbm w)$ for a constant $C_\delta >0$ which is strictly smaller than $C_*$ if we assume that $C_*>1$ and $\delta$ is chosen to be sufficiently small.
}\label{fig-attained}
\end{center}
\vspace{-1em}
\end{figure}
\begin{proof}[Proof of Proposition~\ref{prop-attained}]
See Figure~\ref{fig-attained} for an illustration of the proof.
\medskip
\noindent\textit{Step 1: setup.}
Let $\alpha$ and $A$ be chosen as in Lemma~\ref{lem-shorter-annulus-all}.
Also let
\begin{equation} \label{eqn-max-def}
C_* := \inf\left\{ C > 1 : \mathbbm P\left[ \sup_{z,w\in U, z\not=w} \frac{D_h^\phi(z,w)}{D_{h|_U}(z,w)} \leq C \right] = 1 \right\} .
\end{equation}
Proposition~\ref{prop-coord-bilip} implies that $C_* <\infty$. We want to show that $C_* \leq 1$.
To this end, we will show that a.s.\
\begin{equation} \label{eqn-lower-ratio}
D_h^\phi(\mathbbm z , \mathbbm w) \leq C_\delta D_{h|_U}(\mathbbm z , \mathbbm w) ,
\quad\forall \mathbbm z,\mathbbm w \in U , \quad
\text{where} \quad C_\delta := 1 + \delta + \frac{A}{ A+1 }(C_* - 1 - \delta) .
\end{equation}
If $C_* > 1$ and $\delta > 0 $ is chosen sufficiently small (depending on $C_*$ and $A$), then $C_\delta < C_*$. This contradicts the definition of $C_*$, so we infer that $C_* \leq 1$.
It remains only to prove~\eqref{eqn-lower-ratio}.
\medskip
\noindent\textit{Step 2: regularity event.}
The idea of the proof of~\eqref{eqn-lower-ratio} is to use Lemma~\ref{lem-shorter-annulus-all} and conditions~\ref{item-attained-dist} and~\ref{item-attained-around} in the definition of $E_r(z)$ to show that if $P : [0,D_h(\mathbbm z,\mathbbm w)] \rightarrow \mathbbm C$ is a $D_h$-geodesic, then with high probability the following is true. We can cover a $1-\frac{A}{A+1}$-fraction of the interval $[0,D_h(\mathbbm z,\mathbbm w)]$ by intervals of the form $[s,t]$ such that $D_h^\phi(P(s) ,P(t)) \leq (1+\delta) (t-s)$.
In order for the comparison of $D_h$-lengths and $D_h^\phi$-lengths to make sense, we need to make sure that our $D_h$-geodesic $P$ stays in $U$. We now introduce a regularity event on which we can force certain $D_h$-geodesics to stay in $U$.
Fix a compact set $K\subset U$ and let $\zeta\in (0,1)$ be a small parameter which we will eventually send to zero.
By the continuity of $D_h$, we can find a small parameter $\rho \in (0,1)$ and a compact set $K'$ satisfying $K\subset K' \subset U$, depending on $K$ and $\zeta$, such that with probability at least $1-\zeta$, we have
\begin{equation} \label{eqn-attained-dist-reg}
D_h(\mathbbm z , \mathbbm w) \leq D_h(\mathbbm z , \mathbbm C\setminus K') ,\quad \forall \mathbbm z , \mathbbm w \in K \quad \text{with} \quad |\mathbbm z-\mathbbm w| \leq \rho .
\end{equation}
If~\eqref{eqn-attained-dist-reg} holds, then $D_h(\mathbbm z , \mathbbm w) = D_{h|_U}(\mathbbm z,\mathbbm w)$ for each pair of points $\mathbbm z , \mathbbm w \in K $ with $|\mathbbm z-\mathbbm w| \leq \rho $ and moreover every $D_h$-geodesic between two such points is contained in $K'$, so in particular is also a $D_{h|_U}$-geodesic.
For $\varepsilon > 0$, let $F_{K'}^\varepsilon$ be the event that~\eqref{eqn-attained-dist-reg} holds and the event of Lemma~\ref{lem-shorter-annulus-all} occurs with the above choices of $\alpha,A,\delta $ and with $K'$ in place of $K$, so that $\mathbbm P[F_{K'}^\varepsilon] \geq 1 -\zeta - o_\varepsilon(1)$.
\medskip
\noindent\textit{Step 3: reducing to an estimate for nearby points.}
We claim that on $F_{K'}^\varepsilon$, it is a.s.\ the case that
\begin{equation} \label{eqn-attained-show}
D_h^\phi(\mathbbm z , \mathbbm w) \leq C_\delta D_{h|_U}(\mathbbm z , \mathbbm w) + o_\varepsilon(1) ,\quad \forall \mathbbm z , \mathbbm w \in K \quad \text{with} \quad |\mathbbm z-\mathbbm w| \leq \rho ,
\end{equation}
where the $o_\varepsilon(1)$ is a random error which tends to zero in probability as $\varepsilon\rightarrow 0$, uniformly over all $\mathbbm z,\mathbbm w \in K$.
Before proving~\eqref{eqn-attained-show}, we explain why it implies~\eqref{eqn-lower-ratio}.
Applying~\eqref{eqn-attained-show} and sending $\varepsilon\rightarrow 0$ shows that with probability at least $1-\zeta$, we have $D_h^\phi(\mathbbm z , \mathbbm w) \leq C_\delta D_{h|_U}(\mathbbm z , \mathbbm w)$
for each $ \mathbbm z , \mathbbm w \in K$ with $|\mathbbm z-\mathbbm w| \leq \rho $. This implies that with probability least $1-\zeta$, the $D_h^\phi$-length of any path contained in $K$ is at most $C_\delta$ times its $D_{h|_U}$-length. Since $D_{h|_U}$ and $D_h^\phi$ are length metrics, sending $\zeta \rightarrow 0$ and letting $K$ increase to all of $U$ gives~\eqref{eqn-lower-ratio}.
\medskip
\noindent\textit{Step 4: decomposition of a $D_h$-geodesic into segments. }
Assume that $F_{K'}^\varepsilon$ occurs, let $\mathbbm z,\mathbbm w \in K$ with $|\mathbbm z-\mathbbm w| \leq \rho$, and let $P$ be a $D_h$-geodesic from $\mathbbm z$ to $\mathbbm w$. As noted after~\eqref{eqn-attained-dist-reg}, we have $P\subset K'$.
We will define several objects which depend on $P$ and $\varepsilon$, but to lighten notation we will not make $P$ and $\varepsilon$ explicit in the notation. See Figure~\ref{fig-attained} for an illustration of the definitions.
Let $t_0 = 0$ and inductively let $t_j$ for $j\in\mathbbm N$ be the smallest time $t \geq t_{j-1}$ at which $P$ exits a Euclidean ball of the form $B_{r}(w)$ for $w\in \left(\frac{\varepsilon^2 }{4} \mathbbm Z^2 \right) \cap B_\varepsilon(K)$ and $r\in [\varepsilon^2 , \varepsilon] \cap \{2^{-k} : k\in\mathbbm N\}$ such that $P(t_{j-1}) \in B_{r/2}(w)$ and $E_r(w)$ occurs; or let $t_j = D_h(\mathbbm z , \mathbbm w )$ if no such $t$ exists.
If $t_j < D_h(\mathbbm z , \mathbbm w )$, let $w_j$ and $r_j$ be the corresponding values of $w$ and $r$.
Also let $s_j$ be the last time before $t_j$ at which $P$ exits $B_{\alpha r_j}(w)$.
Note that $s_j \in [t_{j-1} , t_j]$ and $P([s_j , t_j]) \subset \overline{ \mathbbm A_{\alpha r_j , r_j}(w_j)}$.
Define the indices
\begin{equation}
\underline J := \max\left\{ j\in \mathbbm N : |\mathbbm z - P(t_{j-1} )| < 2 \varepsilon \right\}
\quad \text{and} \quad
\overline J := \min\left\{j\in\mathbbm N : |\mathbbm w - P(t_{j+1} )| < 2 \varepsilon \right\} .
\end{equation}
Since $r_j \leq \varepsilon $ and $P(t_j) \in B_{r_j}(w_j)$ for each $j$, we have $\mathbbm z , \mathbbm w \notin B_{ r_j}(w_j)$ for $j \in [\underline J , \overline J]_{\mathbbm Z}$.
By the definition of $F_{K'}^\varepsilon$, on this event we have $t_j < D_h(\mathbbm z,\mathbbm w)$ and $|P(t_{j-1}) - P(t_j)| \leq 2 \varepsilon $ whenever $|\mathbbm w - P(t_{j-1})| \geq \varepsilon $.
Therefore, on $F_{K'}^\varepsilon$,
\begin{equation} \label{eqn-endpoint-contain}
P(t_{\underline J}) \in B_{4\varepsilon }(\mathbbm z) \quad \text{and} \quad P(t_{\overline J}) \in B_{4\varepsilon }(\mathbbm w) .
\end{equation}
Since $P$ is a $D_h$-geodesic, for $j\in [\underline J , \overline J]_{\mathbbm Z}$ also $P|_{[s_j , t_j]}$ is a $D_h$-geodesic from $P(s_j) \in \partial B_{\alpha r_j}(w_j)$ to $P(t_j) \in \partial B_{r_j}(w_j)$.
By definition, this $D_h$-geodesic stays in $\overline{\mathbbm A_{\alpha r_j , r_j}(w_j)}$.
Combining this with condition~\ref{item-attained-dist} in the definition of $E_{r_j}(w_j)$ (applied with $u = P(s_j)$ and $v = P(t_j)$) and the definition~\eqref{eqn-max-def} of $C_*$, we obtain
\begin{equation} \label{eqn-increment-tilde-D}
D^\phi_h\left( P(s_j) , P(t_j) \right) \leq (1+\delta) (t_j - s_j)
\quad \text{and} \quad
D^\phi_h\left( P(t_{j-1}) , P(s_j) \right) \leq C_* (s_j - t_{j-1} ) ,\quad \forall j \in [\underline J ,\overline J]_{\mathbbm Z}.
\end{equation}
\medskip
\noindent\textit{Step 5: comparing $s_j - t_{j-1}$ to $t_j - s_j$.}
In order to extract a non-trivial upper bound for $D_h^\phi(\mathbbm z,\mathbbm w)$ from~\eqref{eqn-increment-tilde-D}, we need to show that the ``good" intervals $[s_j,t_j]$ occupy a positive fraction of the total length of the time interval $[0,D_h(\mathbbm z,\mathbbm w)]$. To this end, we will now argue that $ s_j - t_{j-1}$ is not too much larger than $t_j - s_j$.
If $j \in [\underline J , \overline J]_{\mathbbm Z}$, then since $r_j \leq \varepsilon $ and $|P(t_j) -\mathbbm z| \wedge |P(t_j) - \mathbbm w| \geq 2\varepsilon $, the geodesic $P$ must cross the annulus $\mathbbm A_{\alpha r_j,r_j}(w_j)$ at least once before time $t_{j-1}$ and at least once after time $s_j$.
By the definition of $E_{r_j}(w_j)$, there is a path disconnecting the inner and outer boundaries of this annulus with $D_h$-length at most $A D_h\left(\partial B_{\alpha r_j}(w_j) , \partial B_{ r_j}(w_j) \right)$.
The geodesic $P$ must hit this path at least once before time $t_{j-1}$ and at least once after time $s_j$.
Since $P$ is a geodesic and $P(s_j) \in \partial B_{\alpha r_j}(w_j)$, $P(t_j) \in \partial B_{ r_j}(w_j)$, it follows that
\begin{equation*}
s_j - t_{j-1} \leq A D_h\left(\partial B_{\alpha r_j}(w_j) , \partial B_{ r_j}(w_j) \right) \leq A (t_j - s_j) .
\end{equation*}
Adding $A (s_j - t_{j-1})$ to both sides of this inequality, then dividing by $A+1$, gives
\begin{equation} \label{eqn-increment-compare}
s_j - t_{j-1} \leq \frac{A}{A+1} (t_j - t_{j-1}) .
\end{equation}
\medskip
\noindent\textit{Step 6: upper bound for $D^\phi_h$. }
As above, we assume that $F_{K'}^\varepsilon$ occurs, we let $\mathbbm z,\mathbbm w\in K$ with $|\mathbbm z-\mathbbm w|\leq \rho$ and we let $P$ be the $D_h$-geodesic from $\mathbbm z$ to $\mathbbm w$. In the notation above, it holds for each $j\in [\underline J +1 , \overline J]_{\mathbbm Z}$ that
\allb \label{eqn-attained-increment}
D_h^\phi\left(P(t_{j-1}) , P(t_j) \right)
&\leq D^\phi_h\left( P(t_{j-1}) , P(s_j) \right) + D^\phi_h\left( P(s_j) , P(t_j) \right) \quad \text{(triangle inequality)} \notag \\
&\leq C_*( s_j - t_{j-1}) + (1+\delta) (t_j - s_j) \quad \text{(by \eqref{eqn-increment-tilde-D})} \notag \\
&= (1+\delta) (t_j - t_{j-1} ) + (C_* - 1-\delta ) (s_j - t_{j-1}) \notag \\
&= C_\delta (t_j - t_{j-1} ) \quad \text{(by \eqref{eqn-increment-compare} and the definition of $C_\delta$)} .
\alle
We now apply~\eqref{eqn-endpoint-contain} and sum the estimate~\eqref{eqn-attained-increment} to get
\allb \label{eqn-attained-sum}
D^\phi_h\left( B_{4 \varepsilon \mathbbm r}(\mathbbm z) , B_{4 \varepsilon \mathbbm r}(\mathbbm w) \right)
&\leq D_h^\phi\left( P(t_{\underline J}) , P(t_{\overline J}) \right) \quad \text{(by~\eqref{eqn-endpoint-contain})} \notag \\
&\leq \sum_{j=\underline J+1}^{\overline J} D_h^\phi\left(P(t_{j-1}) , P(t_j) \right) \notag\\
&\leq C_\delta (t_{\overline J} - t_{\underline J}) \quad \text{(by~\eqref{eqn-attained-increment})} \notag\\
&\leq C_\delta D_{h|_U}(\mathbbm z,\mathbbm w) \quad \text{(since $P$ is a $D_{h|_U}$-geodesic)}.
\alle
By the continuity of $(z,w) \mapsto D_h^\phi(z,w)$ and the triangle inequality, a.s.\
\begin{equation} \label{eqn-sup-distance}
D^\phi_h\left( \mathbbm z , \mathbbm w \right) \leq D^\phi_h\left( B_{4 \varepsilon \mathbbm r}(\mathbbm z) , B_{4 \varepsilon \mathbbm r}(\mathbbm w) \right) + o_\varepsilon(1)
\end{equation}
where the $o_\varepsilon(1)$ tends to 0 in probability as $\varepsilon\rightarrow 0$, uniformly over all $\mathbbm z,\mathbbm w \in K $. Combining this with~\eqref{eqn-attained-sum} gives~\eqref{eqn-attained-show}.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,828 |
Петер Слотердайк () е германски философ и общественик.
Биография и творчество
Петер Слотердайк е роден на 26 юни 1947 г. в Карлсруе, Германия.
Професор по естетика, понастоящем той е ректор на Hochschule für Gestaltung в родния си град. Бил е преподавател в Ню Йорк, Париж, Цюрих и Виена. Получил е десетки престижни награди за своето творчество, между които "Фридрих Меркер" за есеистика (1999).
Книгата му "Критика на циничния разум" (1983 г.) е преведена на повече от 30 езика. Слотердайк е автор на десетки книги, монографии и есета, като няколко от тях предизвикват дискусии не само в Германия, но и по света. Особено се коментира издадената през 1999 г.
Правила за човешкия парк, която прави аналогии между практиките на хуманизма и животновъдството. Юрген Хабермас, който окуражава Слотердайк в ранните години от кариерата му, се оказва между именитите критици на книгата.
В годините между 1998 и 2004 излизат томовете от негова трилогия Сфери, мащабно (2565 стр.) философско изследване на самосъздаващото се човечество.
Избрани трудове
Literatur und Organisation von Lebenserfahrung. Autobiographien der Zwanziger Jahre. Hanser, München 1978, ISBN 3-446-12627-9.
Kritik der zynischen Vernunft. 2 Bände. Suhrkamp, Frankfurt am Main 1983, ISBN 3-518-11099-3.
Der Zauberbaum. Die Entstehung der Psychoanalyse im Jahr 1785. Ein epischer Versuch zur Philosophie der Psychologie. Suhrkamp, Frankfurt am Main 1985, ISBN 3-518-37945-3.
Der Denker auf der Bühne. Nietzsche's Materialismus. Suhrkamp (es 1353), Frankfurt am Main 1986, ISBN 3-518-11353-4.
Kopernikanische Mobilmachung und ptolemäische Abrüstung. Ästhetischer Versuch. Suhrkamp (es 1375), Frankfurt am Main 1987, ISBN 3-518-11375-5.
Eurotaoismus. Zur Kritik der politischen Kinetik. Suhrkamp (es 1450), Frankfurt am Main 1989, ISBN 3-518-11450-6.
Im selben Boot. Versuch über die Hyperpolitik. Suhrkamp, Frankfurt am Main 1993, ISBN 3-518-38947-5.
Medienzeit. Drei gegenwartsdiagnostische Versuche. Schriftenreihe der Staatlichen Hochschule für Gestaltung Karlsruhe, Band 1, hrsg. v. Heinrich Klotz, Cantz, Stuttgart 1993, ISBN 3-89322-586-2.
Falls Europa erwacht. Gedanken zum Programm einer Weltmacht am Ende des Zeitalters ihrer politischen Absence. Suhrkamp, Frankfurt am Main 1994, ISBN 3-518-39902-0.
Regeln für den Menschenpark. Ein Antwortschreiben zu Heideggers Brief über den Humanismus. Suhrkamp, Frankfurt am Main 1999, ISBN 3-518-06582-3.
Tau von den Bermudas. Versuch über das Verlangen nach Neuzeit. Suhrkamp (es), Frankfurt am Main 2001, ISBN 3-518-06632-3.
Nicht gerettet. Versuche nach Martin Heidegger. Suhrkamp, Frankfurt am Main 2001, ISBN 3-518-41279-5.
Luftbeben. An den Wurzeln des Terrors. Suhrkamp, Frankfurt am Main 2002, ISBN 3-518-12286-X.
Im Weltinnenraum des Kapitals. Zu einer philosophischen Geschichte der terrestrischen Globalisierung. Suhrkamp, Frankfurt am Main 2005, ISBN 3-518-45814-0.
Zorn und Zeit. Politisch-psychologischer Versuch. Suhrkamp, Frankfurt am Main 2006, ISBN 3-518-41840-8.
Der ästhetische Imperativ. Schriften zur Kunst. Hrsg. v. Peter Weibel, PHILO & PhiloFineArts, Hamburg 2007, ISBN 978-3-86572-629-2.
Derrida – Ein Ägypter: über das Problem der jüdischen Pyramide. Suhrkamp, Frankfurt 2007, ISBN 978-3-518-12502-1.
Gottes Eifer. Vom Kampf der drei Monotheismen. Verlag der Weltreligionen im Insel Verlag, 2007, ISBN 978-3-458-71004-2.
Theorie der Nachkriegszeiten: Bemerkungen zu den deutsch-französischen Beziehungen seit 1945. Suhrkamp Verlag, 2008.
Du mußt dein Leben ändern Über Anthropotechnik. Suhrkamp, Frankfurt am Main 2009, ISBN 978-3-518-41995-3.
Philosophische Temperamente: Von Platon bis Foucault. Diederichs, München 2009, ISBN 978-3-424-35016-6.
Die nehmende Hand und die gebende Seite: Beiträge zu einer Debatte über die demokratische Neubegründung von Steuern. Suhrkamp, Berlin 2010, ISBN 978-3-518-06141-1.
Von der Domestikation des Menschen zur Zivilisierung der Kulturen. Zur Beantwortung der Frage, ob die Menschheit zur Selbstzähmung fähig ist. In: Die Neugier des Glücklichen. Hrsg. v. B.-Christoph Streckhardt. Verlag der Bauhaus Universität Weimar 2012, ISBN 978-3-86068-474-0.
Zeilen und Tage. Notizen 2008–2011. Suhrkamp, Berlin 2012, ISBN 978-3-518-42342-4.
Ausgewählte Übertreibungen: Gespräche und Interviews 1993–2012. Suhrkamp, Berlin 2013, ISBN 978-3-518-42200-7.
Reflexionen eines nicht mehr Unpolitischen. Dankrede von Peter Sloterdijk zur Entgegennahme des Ludwig-Börne-Preises 2013. Laudatio: Hans Ulrich Gumbrecht: Wachheit. Von Ludwig Börne zu Peter Sloterdijk. Suhrkamp, Berlin 2013, ISBN 978-3-518-06070-4.
Gespräche über Gott, Geist und Geld. Mit Thomas Macho, Herder, Freiburg im Breisgau 2014, ISBN 978-3-451-30928-1.
Die schrecklichen Kinder der Neuzeit. Suhrkamp, Berlin 2014, ISBN 978-3-518-42435-3.
Was geschah im 20. Jahrhundert? Unterwegs zu einer Kritik der extremistischen Vernunft. Suhrkamp, Berlin 2016, ISBN 978-3-518-42507-7.
Das Schelling-Projekt. Ein Bericht. Suhrkamp, Berlin 2016, ISBN 978-3-518-42524-4.
Nach Gott: Glaubens- und Unglaubensversuche. Suhrkamp, Berlin 2017, ISBN 978-3-518-42632-6 bzw. ISBN 3-518-42632-X.
Neue Zeilen und Tage. Notizen 2011-2013. Suhrkamp, Berlin 2018, ISBN 978-3-518-42844-3.
Polyloquien. Ein Brevier. Hrsg. v. Raimund Fellinger, Suhrkamp, Berlin 2018, ISBN 978-3-518-42775-0.
Den Himmel zum Sprechen bringen: Elemente der Theopoesie. Suhrkamp, Berlin 2020, ISBN 978-3-518-42933-4.
Награди и отличия
1993: "Награда Ернст Роберт Курциус" за есеистика
1999: "Награда Фридрих Меркер" за есеистика
2001: Christian-Kellerer-Preis für die Zukunft philosophischer Gedanken
2005: Wirtschaftsbuchpreis der Financial Times Deutschland
2005: "Награда Зигмунд Фройд"
2005: Österreichisches Ehrenzeichen für Wissenschaft und Kunst
2006: Кавалер на "Ordre des Arts et des Lettres"
2008: "Награда Лесинг за критика"
2008: Cicero Rednerpreis
2008: Internationaler Mendelssohn-Preis zu Leipzig (Kategorie Gesellschaftliches Engagement)
2008: "Prix européen de l'essai Charles Veillon" für sein Werk Zorn und Zeit aus dem Jahr 2006.
2009: BDA-Preis für Architekturkritik
2011: Ehrendoktorwürde Universität Nijmegen, Niederlande
2013: "Награда Лудвиг Бьорне"
2017: Helmuth-Plessner-Preis
Бележки
Външни препратки
Офциален сайт
Петер Слотердайк и Славой Жижек, Цивилизация на кредита?, в-к Култура, бр. 27 (2645), 15 юли 2011
Петер Слотердайк, Правила за човешкия парк, сп. Социологически проблеми, 2006, кн. 1-2, с.20-45
Германски философи
Философи от 20 век
Философи от 21 век
Феноменология
Германски есеисти
Мюнхенски университет
Доктор хонорис кауза
Членове на Европейската академия на науките и изкуствата
Родени в Баден-Вюртемберг | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,344 |
Q: Translating Nebular Calendar component I'm trying Nebular UI library for Angular (https://akveo.github.io/nebular) to see its capabilities for future projects and it has almost everything I need usually but i18n support. In particular, I'd like to translate the strings in the calendar and calendar range components (NbCalendarComponent and NbCalendarRangeComponent) but I don't see any options I can modify.
Any suggestion on this? Thanks!
A: As suggested above in the Github thread by @Rich, I've changed the LOCALE_ID of the application to match my language and now the month/day names in the Calendar component are shown correctly. There's still a minor problem, which is that the 'Today' text is not translated, but I've removed the calendar header. Until the component features multilingual support, I'll use this solution for the application.
Cheers,
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,997 |
*New to React DnD? [Read the overview](docs-overview.html) before jumping into the docs.*
Testing
===================
React DnD is test-friendly. The whole drag and drop interaction, including the rendering of your components, as well as the side effects you perform in response to the drag and drop events, can be tested.
There are several different approaches to testing React components. React DnD is not opinionated and lets you use either of them. **Neither of those approaches require the browser event system to be available.**
A few test examples are included with the React DnD inside its `examples` folder. Run `npm install` and `npm test` inside the `react-dnd` root folder to start them.
### Testing the Components in Isolation
If you are only interested in testing the *rendering* of your components in isolation, and not their interaction, you may use the `DecoratedComponent` static property available on any class wrapped with React DnD to access the original class. You may then test it with the different props without any dependency on React DnD, supplying an identity function to stub the connector methods.
-------------------
```js
var React = require('react');
var TestUtils = require('react/lib/ReactTestUtils');
var expect = require('expect');
it('can be tested independently', function () {
// Obtain the reference to the component before React DnD wrapping
var OriginalBox = Box.DecoratedComponent;
// Stub the React DnD connector functions with an identity function
var identity = function (el) { return el; };
// Render with one set of props and test
var root = TestUtils.renderIntoDocument(
<OriginalBox name='test'
connectDragSource={identity} />
);
var div = TestUtils.findRenderedDOMComponentWithTag(root, 'div');
expect(div.props.style.opacity).toEqual(1);
// Render with another set of props and test
root = TestUtils.renderIntoDocument(
<OriginalBox name='test'
connectDragSource={identity}
isDragging />
);
div = TestUtils.findRenderedDOMComponentWithTag(root, 'div');
expect(div.props.style.opacity).toEqual(0.4);
});
```
-------------------
```js
import React from 'react';
import TestUtils from 'react/lib/ReactTestUtils';
import expect from 'expect';
it('can be tested independently', () => {
// Obtain the reference to the component before React DnD wrapping
const OriginalBox = Box.DecoratedComponent;
// Stub the React DnD connector functions with an identity function
const identity = el => el;
// Render with one set of props and test
let root = TestUtils.renderIntoDocument(
<OriginalBox name='test'
connectDragSource={identity} />
);
let div = TestUtils.findRenderedDOMComponentWithTag(root, 'div');
expect(div.props.style.opacity).toEqual(1);
// Render with another set of props and test
root = TestUtils.renderIntoDocument(
<OriginalBox name='test'
connectDragSource={identity}
isDragging />
);
div = TestUtils.findRenderedDOMComponentWithTag(root, 'div');
expect(div.props.style.opacity).toEqual(0.4);
});
```
-------------------
-------------------
### Testing the Drag and Drop Interaction
If you want to test the whole interaction, and not just the individual component rendering, you should use the [test backend](docs-test-backend.html) that ships with React DnD. **The test backend does not require the DOM** so you can also use it with [`ReactShallowRenderer`](https://facebook.github.io/react/docs/test-utils.html#shallow-rendering) just fine.
This is currently the least documented part of React DnD because it exposes the underlying concepts from the [DnD Core](https://github.com/gaearon/dnd-core), the library powering React DnD inside. You can learn more about the test backend and its methods from the [DnD Core tests](https://github.com/gaearon/dnd-core/tree/v1.1.0/src/__tests__).
Here is an example to get you started:
-------------------
```js
var React = require('react');
var Component = React.Component;
var TestBackend = require('react-dnd/modules/backends/Test');
var DragDropContext = require('react-dnd').DragDropContext;
var TestUtils = require('react/lib/ReactTestUtils');
var expect = require('expect');
/**
* Wraps a component into a DragDropContext that uses the TestBackend.
*/
function wrapInTestContext(DecoratedComponent) {
return DragDropContext(TestBackend)(
React.createClass({
render: function () {
return <DecoratedComponent {...this.props} />;
}
})
);
}
it('can be tested with the testing backend', function () {
// Render with the test context that uses the test backend
var BoxContext = wrapInTestContext(Box);
var root = TestUtils.renderIntoDocument(<BoxContext name='test' />);
// Obtain a reference to the backend
var backend = root.getManager().getBackend();
// Test that the opacity is 1
let div = TestUtils.findRenderedDOMComponentWithTag(root, 'div');
expect(div.props.style.opacity).toEqual(1);
// Find the drag source ID and use it to simulate the dragging operation
var box = TestUtils.findRenderedComponentWithType(root, Box);
backend.simulateBeginDrag([box.getHandlerId()]);
// Optionally you can pass in a clientOffset for testing operations that
// depend on mouse movements.
// backend.simulateBeginDrag([box.getHandlerId()], {
// clientOffset: { x: 0, y: 0 },
// getSourceClientOffset: () => { x: 0, y: 0 },
// });
// Verify that the div changed its opacity
div = TestUtils.findRenderedDOMComponentWithTag(root, 'div');
expect(div.props.style.opacity).toEqual(0.4);
// See other backend.simulate* methods for more!
});
```
-------------------
```js
import React, { Component } from 'react';
import TestBackend from 'react-dnd/modules/backends/Test';
import { DragDropContext } from 'react-dnd';
import TestUtils from 'react/lib/ReactTestUtils';
import expect from 'expect';
/**
* Wraps a component into a DragDropContext that uses the TestBackend.
*/
function wrapInTestContext(DecoratedComponent) {
return DragDropContext(TestBackend)(
class TestContextContainer extends Component {
render() {
return <DecoratedComponent {...this.props} />;
}
}
);
}
it('can be tested with the testing backend', () => {
// Render with the test context that uses the test backend
const BoxContext = wrapInTestContext(Box);
const root = TestUtils.renderIntoDocument(<BoxContext name='test' />);
// Obtain a reference to the backend
const backend = root.getManager().getBackend();
// Test that the opacity is 1
let div = TestUtils.findRenderedDOMComponentWithTag(root, 'div');
expect(div.props.style.opacity).toEqual(1);
// Find the drag source ID and use it to simulate the dragging operation
const box = TestUtils.findRenderedComponentWithType(root, Box);
backend.simulateBeginDrag([box.getHandlerId()]);
// Verify that the div changed its opacity
div = TestUtils.findRenderedDOMComponentWithTag(root, 'div');
expect(div.props.style.opacity).toEqual(0.4);
// See other backend.simulate* methods for more!
});
```
-------------------
```js
import React, { Component } from 'react';
import TestBackend from 'react-dnd/modules/backends/Test';
import { DragDropContext } from 'react-dnd';
import TestUtils from 'react/lib/ReactTestUtils';
import expect from 'expect';
/**
* Wraps a component into a DragDropContext that uses the TestBackend.
*/
function wrapInTestContext(DecoratedComponent) {
@DragDropContext(TestBackend)
class TestContextContainer extends Component {
render() {
return <DecoratedComponent {...this.props} />;
}
}
return TestContextContainer;
}
it('can be tested with the testing backend', () => {
// Render with the test context that uses the test backend
const BoxContext = wrapInTestContext(Box);
const root = TestUtils.renderIntoDocument(<BoxContext name='test' />);
// Obtain a reference to the backend
const backend = root.getManager().getBackend();
// Test that the opacity is 1
let div = TestUtils.findRenderedDOMComponentWithTag(root, 'div');
expect(div.props.style.opacity).toEqual(1);
// Find the drag source ID and use it to simulate the dragging operation
const box = TestUtils.findRenderedComponentWithType(root, Box);
backend.simulateBeginDrag([box.getHandlerId()]);
// Verify that the div changed its opacity
div = TestUtils.findRenderedDOMComponentWithTag(root, 'div');
expect(div.props.style.opacity).toEqual(0.4);
// See other backend.simulate* methods for more!
});
```
-------------------
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,684 |
I have recently fallen in love with Dieter Ram's 606 Universal Shelving System, or shall I say, I have fallen in love again. Growing up in Germany in the early 1980s, I remember these in the houses of most of my parents' friends. Now, I am finding that I am including these shelves in many projects on which I am currently working.
One of the things I love about Dieter Ram's shelving (apart from the sleek elegant aesthetics) is that they are incredibly practical. They are the preultimate "kit" so you can totally tailor it to your needs and space and they can be easily reconfigured. They work just as well in kitchens, living rooms as they do in bedrooms and offices and even retail spaces as shelves, workstations, display cabinets, sideboards and storage.
The only draw back is their cost. Like anything which is superbly designed and made, they are expensive. As I'll be insist on taking them with me to my nursing home I'm planning on getting a lot of use out of them.
In the meantime, I am installing some in my office combining a workstation, drawers, filing unit and shelving and would love to do the same in my husband's office, but he isn't convinced. Growing up in Australian suburbia in the 1970s, he says the shelves feel too institutional.
Is furniture something you have for a life time or is that way too much of a commitment? Do you like the simple post-war aesthetics and practicality of Dieter Ram's design or is it too austere for your tastes? | {
"redpajama_set_name": "RedPajamaC4"
} | 2,030 |
\section{Introduction}\label{Sec:Intro}
Throughout this paper, we write permutations as words in one-line notation. Let $S_n$ denote the set of permutations of $[n]$. A \emph{descent} of a permutation $\pi=\pi_1\cdots\pi_n\in S_n$ is an index $i\in[n-1]$ such that $\pi_i>\pi_{i+1}$. A \emph{peak} of $\pi$ is an index $i\in\{2,\ldots,n-1\}$ such that $\pi_{i-1}<\pi_i>\pi_{i+1}$.
\begin{definition}\label{Def1}
We say the permutation $\sigma=\sigma_1\cdots\sigma_n$ \emph{contains the pattern} $\tau=\tau_1\cdots\tau_m$ if there are indices $i_1<\cdots<i_m$ such that $\sigma_{i_1}\cdots\sigma_{i_m}$ has the same relative order as $\tau$. Otherwise, we say $\sigma$ \emph{avoids} $\tau$. Denote by $\operatorname{Av}(\tau^{(1)},\ldots,\tau^{(r)})$ the set of permutations that avoid the patterns $\tau^{(1)},\ldots,\tau^{(r)}$. Let $\operatorname{Av}_n(\tau^{(1)},\ldots,\tau^{(r)})=\operatorname{Av}(\tau^{(1)}\ldots,\tau^{(r)})\cap S_n$. Let $\operatorname{Av}_{n,k}(\tau^{(1)},\ldots,\tau^{(r)})$ be the set of permutations in $\operatorname{Av}_n(\tau^{(1)},\ldots,\tau^{(r)})$ with exactly $k$ descents.
\end{definition}
A set of permutations is called a \emph{permutation class} if it is the set of permutations avoiding some (possibly infinite) collection of patterns. Equivalently, a permutation class is a downset in the poset of all permutations ordered by containment. The \emph{basis} of a class is the minimal set of permutations not in the class.
The notion of pattern avoidance in permutations, which has blossomed into an enormous area of research and which plays a lead role in the present article, began in Knuth's book \emph{The Art of Computer Programming} \cite{Knuth}. In this book, Knuth described a so-called \emph{stack-sorting algorithm}; it was the study of the combinatorial properties of this algorithm that led him to introduce the idea of pattern avoidance. In his 1990 Ph.D. thesis \cite{West}, West defined a deterministic version of Knuth's stack-sorting algorithm, which we call the \emph{stack-sorting map} and denote by $s$. The stack-sorting map is a function defined by the following procedure.
Suppose we are given an input permutation $\pi\in S_n$. Place this permutation on the right side of a vertical ``stack." Throughout this process, if the next entry in the input permutation is larger than the entry at the top of the stack or if the stack is empty, the next entry in the input permutation is placed at the top of the stack. Otherwise, the entry at the top of the stack is annexed to the end of the growing output permutation. This procedure stops when the output permutation has length $n$. We then define $s(\pi)$ to be this output permutation. Figure \ref{Fig1} illustrates this procedure and shows that $s(3142)=1324$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1\linewidth]{SortingGRWCPIC}
\end{center}
\caption{The stack-sorting map $s$ sends $3142$ to $1324$.}
\end{figure}\label{Fig1}
If $\pi\in S_n$, we can write $\pi=LnR$, where $L$ (respectively, $R$) is the (possibly empty) substring of $\pi$ to the left (respectively, right) of the entry $n$. West observed that the stack-sorting map can be defined recursively by $s(\pi)=s(L)s(R)n$ (here, we also have to allow $s$ to take permutations of arbitrary finite sets of positive integers as arguments). There is also a natural definition of the stack-sorting map in terms of tree traversals of decreasing binary plane trees (see \cite{Bona,Defant,Defant2}).
The ``purpose" of the stack-sorting map is to sort the input permutation into increasing order. Hence, we say a permutation $\pi\in S_n$ is \emph{sortable} if $s(\pi)=123\cdots n$. The above example illustrates that the stack-sorting map does not always do its job. In other words, not all permutations are sortable. In fact, the following characterization of sortable permutations follows from Knuth's work.
\begin{theorem}[\!\!\cite{Knuth}]\label{Thm1}
A permutation $\pi$ is sortable if and only if it avoids the pattern $231$.
\end{theorem}
Even if a permutation is not sortable, we can still try to sort it via iterated use of the stack-sorting map. In what follows, $s^t$ denotes the composition of $s$ with itself $t$ times.
\begin{definition}
A permutation $\pi\in S_n$ is called $t$-\emph{stack-sortable} if $s^t(\pi)=123\cdots n$. Let $\mathcal W_t(n)$ denote the set of $t$-stack-sortable permutations in $S_n$. Let $W_t(n)=|\mathcal W_t(n)|$.
\end{definition}
Theorem \ref{Thm1} states that $\mathcal W_1(n)=\operatorname{Av}_n(231)$, so it follows from the well-known enumeration of $231$-avoiding permutations that $W_1(n)=C_n=\frac{1}{n+1}{2n\choose n}$ is the $n^\text{th}$ Catalan number. In his thesis, West proved \cite{West} that a permutation is $2$-stack-sortable if and only if it avoids the pattern $2341$ and also avoids any $3241$ pattern that is not part of a $35241$ pattern. He also conjectured the following theorem, which Zeilberger proved in 1992.
\begin{theorem}[\!\!\cite{Zeilberger}]\label{Thm2}
We have
\begin{equation}\label{Eq1}
W_2(n)=\frac{2}{(n+1)(2n+1)}{3n\choose n}.
\end{equation}
\end{theorem}
Combinatorial proofs of this theorem arose later in \cite{Cori,Dulucq2,Dulucq,Goulden}. Some authors have studied the enumerations of $2$-stack-sortable permutations according to certain statistics \cite{BonaSimplicial,Bousquet98,Bouvel,Dulucq}. Recently, the authors of \cite{Duchi} introduced new combinatorial objects known as \emph{fighting fish} and showed that they are counted by the numbers $W_2(n)$. Fang has now given a bijection between fighting fish and $2$-stack-sortable permutations \cite{Fang}. The authors of \cite{Bevan} study what they call $n$-\emph{point dominoes}, and they have made the fascinating discovery that the number of these objects is $W_2(n+1)$. In \cite{Bona}, B\'ona draws attention to the fact that a certain class of lattice paths are counted by the numbers $2^{2n-1}W_2(n)$ (see \cite{Bousquet02}), and he asks about the possibility of using this result to obtain a simple combinatorial proof of \eqref{Eq1}.
The primary purpose of this article is to study preimages of permutation classes under the stack-sorting map. This is a natural generalization of the study of sortable and $2$-stack-sortable permutations since $\mathcal W_1(n)=s^{-1}(\operatorname{Av}_n(21))$ and $\mathcal W_2(n)=s^{-1}(\operatorname{Av}_n(231))$. In fact, Bouvel and Guibert \cite{Bouvel} have already considered stack-sorting preimages of certain classes in their study of permutations that are sortable via multiple stacks and $D_8$ symmetries (we state some of their results in Section \ref{Sec:Back}). Claesson and \'Ulfarsson \cite{Claesson} have also studied this problem in relation to a generalization of classical permutation patterns known as \emph{mesh patterns}, which were introduced\footnote{The idea of a mesh pattern appears earlier under different names. For example, it appears in \cite{Bousquet06,Dulucq,West}. Mesh patterns are sometimes called \emph{barred patterns}.} in \cite{Branden}. They showed that each set of the form $s^{-1}(\operatorname{Av}(\tau^{(1)},\ldots,\tau^{(r)}))$ can be described as the set of permutations avoiding a specific collection of mesh patterns, and they provided an algorithm for computing this collection. From this point of view, the current paper provides a new method for counting permutations that avoid certain mesh patterns. In specific cases, $s^{-1}(\operatorname{Av}(\tau^{(1)},\ldots,\tau^{(r)}))$ is a bona fide permutation class. For example, $s^{-1}(\operatorname{Av}(m(m-1)\cdots 321))$ is a permutation class. Hence, we give a new method for enumerating (or at least estimating) some permutation classes.
The idea to count the preimages of a permutation under the stack-sorting map dates back to West, who called $|s^{-1}(\pi)|$ the \emph{fertility} of the permutation $\pi$ and went to great lengths to compute the fertilities of the permutations of the forms \[23\cdots k1(k+1)\cdots n,\quad 12\cdots(k-2)k(k-1)(k+1)\cdots n,\quad\text{and}\quad k12\cdots(k-1)(k+1)\cdots n.\] The very specific forms of these permutations indicates the initial difficulty of computing fertilities. We define the fertility of a set of permutations to be the sum of the fertilities of the permutations in that set. With this terminology, our main goal in this paper is to compute the fertilities of sets of the form $\operatorname{Av}_n(\tau^{(1)},\ldots,\tau^{(r)})$.
Bousquet-M\'elou \cite{Bousquet} studied permutations with positive fertilities, which she termed \emph{sorted} permutations. She mentioned that it would be interesting to find a method for computing the fertility of any given permutation. This was achieved in much greater generality in \cite{Defant} using new combinatorial objects called ``valid hook configurations." The theory of valid hook configurations was the key ingredient used in \cite{Defant2} in order to improve the best-known upper bounds for $W_3(n)$ and $W_4(n)$, and it will be one of our primary tools in subsequent sections.
It is important to note that we do not have a thorough understanding of valid hook configurations. This is why an explicit formula for $W_3(n)$ remains out of reach, and it is why enumerating the permutations in $s^{-1}(\operatorname{Av}(\tau^{(1)},\ldots,\tau^{(r)}))$ is still highly nontrivial for many choices of patterns $\tau^{(1)},\ldots,\tau^{(r)}$. However, valid hook configurations still provide a powerful tool for computing fertilities. Recently, the authors of \cite{Defant3} used valid hook configurations to establish connections among permutations with fertility $1$, certain weighted set partitions, and cumulants arising in free probability theory. The current author has also investigated which numbers arise as the fertilities of permutations \cite{Defant4}.
In Section \ref{Sec:VHCs}, we review the definitions and necessary theorems concerning valid hook configurations. We also prove a theorem that ameliorates the computation of fertilities in many cases. This theorem was stated in \cite{Defant2}, but the proof was omitted because the result was not needed in that paper. We have decided to prove the result here because we will make use of it in our computations. This result is also used in \cite{Defant4}, so it is important that a proof appears in the literature.
Section \ref{Sec:Back} reviews some facts about generalized patterns and stack-sorting preimages of permutation classes. In Section \ref{Sec:4patterns}, we study the set $s^{-1}(\operatorname{Av}(132,231,312,321))$, which is a permutation class. In Section \ref{Sec5}, we study $s^{-1}(\operatorname{Av}(132,231,321))$ and $s^{-1}(\operatorname{Av}(132,312,321))$, the latter of which is a permutation class. We show that these sets are both enumerated by central binomial coefficients. A corollary of the results in this section actually settles a problem of Bruner \cite{Bruner}. In Section \ref{Sec6}, we consider $s^{-1}(\operatorname{Av}(231,312,321))$, which turns out to be a permutation class. We enumerate this class both directly and by using valid hook configurations, leading to a new identity involving well-studied orderings on integer compositions and integer partitions. Section \ref{Sec7} considers the set $s^{-1}(\operatorname{Av}(132,231,312))$. Finding the fertilities of permutations in $\operatorname{Av}(132,231,312)$ allows us to prove that some of the estimates used in \cite{Defant2} are sharp. In addition, we will find that the permutations in $s^{-1}(\operatorname{Av}(132,231,312))$ are enumerated by the Fine numbers, giving a new interpretation for this well-studied sequence. Section \ref{Sec:312,321} is brief and is merely intended to state that $s^{-1}(\operatorname{Av}(312,321))$ is the permutation class $\operatorname{Av}(3412,3421)$, which Kremer \cite{Kremer} has proven is enumerated by the large Schr\"oder numbers. Section \ref{Sec:132,321} enumerates the permutations in $s^{-1}(\operatorname{Av}(132,321))$. In Section \ref{Sec:Pair}, we prove that $|s^{-1}(\operatorname{Av}_n(132,312))|=|s^{-1}(\operatorname{Av}_n(231,312))|$. Finally, we prove that \[8.4199\leq\lim_{n\to\infty}|s^{-1}(\operatorname{Av}_n(321))|^{1/n}\leq 11.6569\] in Section \ref{Sec:321}. This is notable because $s^{-1}(\operatorname{Av}(321))$ is a permutation class; it is not clear how to obtain such estimates using standard methods for enumerating permutation classes. In most of these sections, we actually refine our counts by enumerating stack-sorting preimages of permutation classes according to the number of descents and according to the number of peaks.
Our results lead to several fascinating open problems and conjectures. In fact, we believe that this paper opens the door to a vast new collection of enumerative problems. We accumulate these problems and conjectures in Section 11.
\section{Valid Hook Configurations and Valid Compositions}\label{Sec:VHCs}
In this section, we review some of the theory of valid hook configurations. Our presentation is almost identical to that given in \cite{Defant4}, but we include it here for completeness. Note that the valid hook configurations defined below are, strictly speaking, different from those defined in \cite{Defant} and \cite{Defant2}. For a lengthier discussion of this distinction, see \cite{Defant3}.
To construct a valid hook configuration, begin by choosing a permutation $\pi=\pi_1\cdots\pi_n\in S_n$. Recall that a descent of $\pi$ is an index $i$ such that $\pi_i>\pi_{i+1}$. Let $d_1<\cdots<d_k$ be the descents of $\pi$. We use the example permutation $3142567$ to illustrate the construction. The \emph{plot} of $\pi$ is the graph displaying the points $(i,\pi_i)$ for $1\leq i\leq n$. The left image in Figure \ref{Fig2} shows the plot of our example permutation. A point $(i,\pi_i)$ is a \emph{descent top} if $i$ is a descent. The descent tops in our example are $(1,3)$ and $(3,4)$.
\begin{figure}[t]
\centering
\subfloat[]{\includegraphics[width=0.2\textwidth]{FertilityPIC2}}
\hspace{1.5cm}
\subfloat[]{\includegraphics[width=0.2\textwidth]{FertilityPIC3}}
\caption{The left image is the plot of $3142567$. The right images shows this plot along with a single hook.}\label{Fig2}
\end{figure}
A \emph{hook} of $\pi$ is drawn by starting at a point $(i,\pi_i)$ in the plot of $\pi$, moving vertically upward, and then moving to the right until reaching another point $(j,\pi_j)$. In order for this to make sense, we must have $i<j$ and $\pi_i<\pi_j$. The point $(i,\pi_i)$ is called the \emph{southwest endpoint} of the hook, while $(j,\pi_j)$ is called the \emph{northeast endpoint}. The right image in Figure \ref{Fig2} shows our example permutation with a hook that has southwest endpoint $(3,4)$ and northeast endpoint $(6,6)$.
\begin{definition}\label{Def2}
A \emph{valid hook configuration} of $\pi$ is a configuration of hooks drawn on the plot of $\pi$ subject to the following constraints:
\begin{enumerate}[1.]
\item The southwest endpoints of the hooks are precisely the descent tops of the permutation.
\item A point in the plot cannot lie directly above a hook.
\item Hooks cannot intersect each other except in the case that the northeast endpoint of one hook is the southwest endpoint of the other.
\end{enumerate}
\end{definition}
\begin{figure}[t]
\begin{center}
\includegraphics[width=.7\linewidth]{FertilityPIC6}
\caption{Four configurations of hooks that are forbidden in a valid hook configuration.}
\label{Fig3}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=.7\linewidth]{FertilityPIC4}
\caption{All of the valid hook configurations of $3142567$.}
\label{Fig4}
\end{center}
\end{figure}
Figure \ref{Fig3} shows four placements of hooks that are forbidden by Conditions 2 and 3.
Figure \ref{Fig4} shows all of the valid hook configurations of $3142567$.
Observe that the total number of hooks in a valid hook configuration of $\pi$ is exactly $k$, the number of descents of $\pi$. Because the southwest endpoints of the hooks are the points $(d_i,\pi_{d_i})$, there a natural ordering of the hooks. Namely, the $i^\text{th}$ hook is the hook whose southwest endpoint is $(d_i,\pi_{d_i})$. We can write a valid hook configuration of $\pi$ concisely as a $k$-tuple $\mathcal H=(H_1,\ldots,H_k)$, where $H_i$ is the $i^\text{th}$ hook.
A valid hook configuration of $\pi$ induces a coloring of the plot of $\pi$. To color the plot, draw a ``sky" over the entire diagram and color the sky blue. Assign arbitrary distinct colors other than blue to the $k$ hooks in the valid hook configuration. There are $k$ northeast endpoints of hooks, and these points remain uncolored. However, all of the other $n-k$ points will be colored. In order to decide how to color a point $(i,\pi_i)$ that is not a northeast endpoint, imagine that this point looks directly upward. If this point sees a hook when looking upward, it receives the same color as the hook that it sees. If the point does not see a hook, it must see the sky, so it receives the color blue. However, if $(i,\pi_i)$ is the southwest endpoint of a hook, then it must look around (on the left side of) the vertical part of that hook. See Figure \ref{Fig5} for the colorings induced by the valid hook configurations in Figure \ref{Fig4}. Note that the leftmost point $(1,3)$ is blue in each of these colorings because this point looks around the first (red) hook and sees the sky.
To summarize, we started with a permutation $\pi$ with exactly $k$ descents. We chose a valid hook configuration of $\pi$ by drawing $k$ hooks according to Conditions 1, 2, and 3 in Definition \ref{Def2}. This valid hook configuration then induced a coloring of the plot of $\pi$. Specifically, $n-k$ points were colored, and $k+1$ colors were used (one for each hook and one for the sky). Let $q_i$ be the number of points colored the same color as the $i^\text{th}$ hook, and let $q_0$ be the number of points colored blue (the color of the sky). Then $(q_0,\ldots,q_k)$ is a composition\footnote{Throughout this paper, a \emph{composition of }$b$ \emph{into} $a$ \emph{parts} is an $a$-tuple of \emph{positive} integers whose sum is $b$.} of $n-k$ into $k+1$ parts; we say the valid hook configuration \emph{induces} this composition. Let $\mathcal V(\pi)$ be the set of compositions induced by valid hook configurations of $\pi$. We call the elements of $\mathcal V(\pi)$ the \emph{valid compositions} of $\pi$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=.7\linewidth]{FertilityPIC7}
\caption{The different colorings induced by the valid hook configurations of $3142567$.}
\label{Fig5}
\end{center}
\end{figure}
Let $C_j=\frac{1}{j+1}{2j\choose j}$ denote the $j^\text{th}$ Catalan number. Given a composition $(q_0,\ldots,q_k)$, let \[C_{(q_0,\ldots,q_k)}=\prod_{t=0}^kC_{q_t}.\] The following theorem explains why valid hook configurations are so useful when studying the stack-sorting map.
\begin{theorem}[\!\!\cite{Defant}]\label{Thm5}
If $\pi\in S_n$ has exactly $k$ descents, then the fertility of $\pi$ is given by the formula \[|s^{-1}(\pi)|=\sum_{(q_0,\ldots,q_k)\in\mathcal V(\pi)}C_{(q_0,\ldots,q_k)}.\]
\end{theorem}
\begin{example}\label{Exam1}
The permutation $3142567$ has six valid hook configurations, which are shown in Figure \ref{Fig4}. The colorings induced by these valid hook configurations are portrayed in Figure \ref{Fig5}. The valid compositions induced by these valid hook configurations are (reading the first row before the second row, each from left to right) \[(3,1,1),\quad (2,2,1),\quad(1,3,1),\quad(2,1,2),\quad(1,2,2),\quad(1,1,3).\] It follows from Theorem \ref{Thm5} that \[|s^{-1}(3142567)|=C_{(3,1,1)}+C_{(2,2,1)}+C_{(1,3,1)}+C_{(2,1,2)}+C_{(1,2,2)}+C_{(1,1,3)}=27.\]
\end{example}
It is also possible to refine Theorem \ref{Thm5} according to certain permutation statistics such as the number of descents and the number of peaks\footnote{Theorem \ref{Thm7} was originally stated in \cite{Defant} in terms of ``valleys" instead of peaks, but the formulation we give here is equivalent.}. Recall that a peak of a permutation $\pi=\pi_1\cdots\pi_n\in S_n$ is an index $i$ such that $\pi_{i-1}<\pi_i>\pi_{i+1}$. In what follows, we consider the \emph{Narayana numbers} $N(i,j)=\frac 1i{i\choose j}{i\choose j-1}$. Let us also define \[V(i,j)=2^{i-2j+1}{i-1\choose 2j-2}C_{j-1}.\] It is known\footnote{See sequence A091894 in the Online Encyclopedia of Integer Sequences \cite{OEIS}.} that $V(i,j)$ is the number of decreasing binary plane trees with $i$ vertices and $j$ leaves, and this is actually why these numbers arise in this context.
\begin{theorem}[\!\!\cite{Defant}]\label{Thm7}
If $\pi\in S_n$ has exactly $k$ descents, then the number of permutations in $s^{-1}(\pi)$ with exactly $m$ descents is \[\sum_{(q_0,\ldots,q_k)\in\mathcal V(\pi)}\sum_{j_0+\cdots+j_k=m+1}\prod_{t=0}^kN(q_t,j_t),\] where the second sum ranges over positive integers $j_0,\ldots,j_k$ that sum to $m+1$.
The number of permutations in $s^{-1}(\pi)$ with exactly $m$ peaks is
\[\sum_{(q_0,\ldots,q_k)\in\mathcal V(\pi)}\sum_{j_0+\cdots+j_k=m+1}\prod_{t=0}^k V(q_t,j_t),\] where the second sum ranges over positive integers $j_0,\ldots,j_k$ that sum to $m+1$.
\end{theorem}
We will often make implicit use of the following result, which is Lemma 3.1 in \cite{Defant2}.
\begin{theorem}[\!\!\cite{Defant2}]\label{Thm6}
Each valid composition of a permutation $\pi\in S_n$ is induced by a unique valid hook configuration of $\pi$.
\end{theorem}
In her study of sorted permutations (permutations with positive fertilities), Bousquet-M\'elou introduced the notion of the \emph{canonical tree} of a permutation and showed that the shape of a permutation's canonical tree determines that permutation's fertility \cite{Bousquet}. She asked for an explicit method for computing the fertility of a permutation from its canonical tree. The current author reformulated the notion of a canonical tree in the language of valid hook configurations, defining the \emph{canonical valid hook configuration} of a permutation \cite{Defant2}. Here, we describe a method for computing a permutation's fertility from its canonical valid hook configuration. Specifically, we show how to describe all valid compositions of $\pi$ from the canonical valid hook configuration. This method was stated in \cite{Defant}, but the proof was omitted. We include the proof here because we will make use of this method. In addition, one wishing to program a computer to compute fertilities will likely find that this method is easier to implement than a method that involves finding all valid hook configurations of a permutation directly.
As before, let $d_1<\cdots<d_k$ be the descents of $\pi$. We will construct the canonical valid hook configuration of $\pi$, which we denote by $\mathcal H^*=(H_1^*,\ldots,H_k^*)$. That is, $H_i^*$ is the hook in $\mathcal H^*$ whose southwest endpoint is $(d_i,\pi_{d_i})$. In order to define $\mathcal H^*$, we need to choose the northeast endpoints of the hooks $H_1^*,\ldots,H_k^*$. To start, consider all possible points that could be northeast endpoints of $H_k^*$; these are precisely the points above and to the right of $(d_k,\pi_{d_k})$. Among these points, choose the leftmost one (equivalently, the lowest one) to be the northeast endpoint of $H_k^*$. Next, consider all possible points that could be northeast endpoints of $H_{k-1}^*$ (given that we already know $H_k^*$ and that we need to satisfy the conditions in Definition \ref{Def2}). Among these points, choose the leftmost one to be the northeast endpoint of $H_{k-1}^*$. Continue in this fashion, always choosing the leftmost possible point as the northeast endpoint of $H_\ell^*$ given that $H_{\ell+1}^*,\ldots,H_k^*$ have already been chosen. If it is ever impossible to find a northeast endpoint for $H_\ell^*$, then $\pi$ has no valid hook configurations (meaning $\pi$ is not sorted). Otherwise, we obtain the canonical valid hook configuration of $\pi$ from this process. Figure \ref{Fig7} shows the canonical valid hook configuration of a permutation.
\begin{figure}[t]
\begin{center}
\includegraphics[height=6.0cm]{SortingPIC10}
\end{center}
\captionof{figure}{The canonical valid hook configuration of \\$2\,\,7\,\,3\,\,5\,\,9\,\,10\,\,11\,\,4\,\,8\,\,1\,\,6\,\,12\,\,13\,\,14\,\,15\,\,
16$. } \label{Fig7}
\end{figure}
Let us assume $\pi$ is sorted so that it has a canonical valid hook configuration $\mathcal H^*=(H_1^*,\ldots,H_k^*)$. Recall that $d_1<\cdots<d_k$ are the descents of $\pi$. We make the conventions $d_0=0$ and $d_{k+1}=n$. For $1\leq i\leq k+1$, the $i^\text{th}$ ascending run of $\pi$ is the string $\pi_{d_{i-1}+1}\cdots\pi_{d_i}$. We use $\mathcal H^*$ to define certain parameters as follows.
\begin{itemize}
\item Let $(b_i^*,\pi_{b_i^*})$ be the northeast endpoint of $H_i^*$.
\item Let $(q_0^*,\ldots,q_k^*)$ be the valid composition of $\pi$ induced from $\mathcal H^*$.
\item For $1\leq i\leq k$, define $e_i$ by requiring that $\pi_{b_i^*}$ is in the $e_i^\text{th}$ ascending run of $\pi$. In other words, $d_{e_i-1}<b_i^*\leq d_{e_i}$. Furthermore, put $e_0=k+1$.
\item Let $\alpha_j=\vert\{i\in\{1,\ldots,k\}\colon e_i=j\}\vert$ denote the number of northeast endpoints $(b_i^*,\pi_{b_i^*})$ such that $\pi_{b_i^*}$ is in the $j^{\text{th}}$ ascending run of $\pi$.
\end{itemize}
\begin{example}\label{Exam}
Let $\pi=2\,\,7\,\,3\,\,5\,\,9\,\,10\,\,11\,\,4\,\,8\,\,1\,\,6\,\,12\,\,13\,\,14\,\,15\,\,
16$ be the permutation whose canonical valid hook configuration appears in Figure \ref{Fig7}. We have $d_0=0$, $d_1=2$, $d_2=7$, $d_3=9$, and $d_4=16$. Furthermore,
\begin{itemize}
\item $(b_1^*,b_2^*,b_3^*)=(5,13,12)$;
\item $(q_0^*,q_1^*,q_2^*,q_3^*)=(7,2,2,2)$;
\item $(e_0,e_1,e_2,e_3)=(4,2,4,4)$;
\item $(\alpha_1,\alpha_2,\alpha_3,\alpha_4)=(0,1,0,2)$.
\end{itemize}
\end{example}
We are now in a position to state and prove the theorem that allows one to combine the above pieces of data in order to describe the valid compositions of $\pi$. The reader who is interested only in the enumerative results of Section \ref{Sec:Back} can safely bypass the proof of the following theorem.
\begin{theorem}\label{Thm4}
Let $\pi\in S_n$ be a sorted permutation, and preserve the notation from above. A composition $(q_0,\ldots,q_k)$ of $n-k$ into $k+1$ parts is a valid composition of $\pi$ if and only if the following two conditions hold:
\begin{enumerate}[(a)]
\item For every $m\in\{0,\ldots,k\}$, \[\sum_{j=m}^{e_m-1}q_j\geq\sum_{j=m}^{e_m-1}q_j^*.\]
\item For all $m,p\in\{0,1,\ldots,k\}$ with $m\leq p\leq e_m-2$, we have \[\sum_{j=m}^pq_j\geq d_{p+1}-d_m-\sum_{j=m+1}^{p+1}\alpha_j.\]
\end{enumerate}
\end{theorem}
\begin{proof}
To ease notation, let $P(i)=(i,\pi_i)$. Suppose $(q_0,\ldots,q_k)\in\mathcal V(\pi)$, and let $\mathcal H=(H_1,\ldots,H_k)$ be the valid hook configuration inducing $(q_0,\ldots,q_k)$. Let $P(b_i)$ be the northeast endpoint of $H_i$. Put $\pi_0=n+1$, $\pi_{n+1}=n+2$, and $b_0=b_0^*=n+1$. It will be convenient to view the ``sky" as a hook $H_0=H_0^*$ with southwest endpoint $(d_0,\pi_{d_0})=(0,n+1)$ and northeast endpoint $(b_0,\pi_{b_0})=(n+1,n+2)$ (this conflicts with our definition of a hook since these points are not in the plot of $\pi$, but we will ignore this technicality).\footnote{The term ``skyhook" refers to, among other things, a recovery system employed by the United States CIA and Air Force in the $20^{\text{th}}$ century. The system allows an airplane to attach to a cable while in flight. The cable is, in turn, attached to a person, who is then dragged behind the plane. One might find it helpful (or not) to picture an airplane at the point $(n+1,n+2)$, a person at the point $(0,n+1)$, and a blue cable connecting the plane and the person.} Put $\mathcal B^*=\{b_0^*,\ldots,b_k^*\}$ and $\mathcal B=\{b_0,\ldots,b_k\}$.
If we build $\mathcal H$ by choosing the northeast endpoints $P(b_k),\ldots,P(b_1)$ in this order, then every possible choice for $P(b_m)$ was also a choice for $P(b_m^*)$ when we built $\mathcal H^*$. It follows from our choice of $P(b_m^*)$ that $b_m^*\leq b_m$ and $\pi_{b_m^*}\leq \pi_{b_m}$. This implies that $H_m$ lies above $H_m^*$ or is equal to $H_m^*$ for every $m\in\{0,\ldots,k\}$.
Suppose $m,p\in\{0,\ldots,k\}$ and $m\leq p\leq e_m-1$ (recall that $\pi_{b_m^*}$ is in the $e_m^\text{th}$ ascending run of $\pi$). Let $X=\{d_m+1,d_m+2,\ldots,\min\{b_m^*,d_{p+1}\}\}$. Suppose $b_\gamma\in X\cap\mathcal B$, where $\gamma\neq m$. Because $b_m^*\leq b_m$, we must have $d_m<b_\gamma<b_m$. This means that $b_\gamma$ lies below the hook $H_m$, so $H_\gamma$ lies below $H_m$. Deducing that $m+1\leq\gamma$, we find that $d_m<b_\gamma^*$. We also know that $b_\gamma^*\leq b_\gamma$, so $d_m<b_\gamma^*\leq \min\{b_m^*,d_{p+1}\}$ (because $b_\gamma\in X$). This proves the implication $b_\gamma\in X\cap\mathcal B\Longrightarrow b_\gamma^*\in X\cap\mathcal B^*$. The map $X\cap\mathcal B\to X\cap\mathcal B^*$ given by $b_j\mapsto b_j^*$ is an injection, so
\begin{equation}\label{Eq10}
\vert X\cap\mathcal B\vert\leq\vert X\cap\mathcal B^*\vert.
\end{equation}
Choose $x\in X\setminus\mathcal B$. Recall that $\mathcal H$ induces a coloring of the plot of $\pi$. The point $P(x)$ lies below the hook $H_m$. None of the hooks $H_0,H_1,\ldots,H_{m-1}$ lie below $H_m$, and all of the hooks $H_{p+1},H_{p+2},\ldots,H_k$ appear to the right of $P(x)$. Therefore, if $P(x)$ looks directly upward, it sees one of the hooks $H_m,H_{m+1},\ldots,H_p$. Letting $\mathcal A_{m,p}$ be the set of points that are given the same color as one of the hooks $H_m,H_{m+1},\ldots,H_p$, we see that $x\in \mathcal A_{m,p}$. This shows that $(X\setminus\mathcal B)\subseteq\mathcal A_{m,p}$. Hence,
\begin{equation}\label{Eq11}
\sum_{j=m}^pq_j=|\mathcal A_{m,p}|\geq\vert X\setminus\mathcal B\vert.
\end{equation}
Suppose $p\leq e_m-2$ (meaning $\min\{b_m^*,d_{p+1}\}=d_{p+1}$). We have $\vert X\cap\mathcal B^*\vert=\displaystyle{\sum_{j=m+1}^{p+1}\alpha_j}$ and $\vert X\vert=d_{p+1}-d_m$, so we can use \eqref{Eq10} to find that
\[\vert X\setminus\mathcal B\vert=|X|-|X\cap\mathcal B|\geq\vert X\vert-\vert X\cap\mathcal B^*\vert=d_{p+1}-d_m-\sum_{j=m+1}^{p+1}\alpha_j.\]
Combining this last inequality with \eqref{Eq11} yields $(b)$.
Next, suppose $p=e_m-1$ (meaning $\min\{b_m^*,d_{p+1}\}=b_m^*$). In this case, the elements of $X\setminus\mathcal B^*$ are the indices $x$ such that $P(x)$ lies below $H_m^*$ and is not a northeast endpoint of a hook in $\mathcal H^*$. These points are precisely those that are given the same color as one of the hooks $H_m^*,\ldots,H_{e_m-1}^*$ in the coloring induced by $\mathcal H^*$. The number of such points is $\displaystyle \sum_{j=m}^{e_m-1}q_j^*$, so we deduce from \eqref{Eq10} and \eqref{Eq11} that \begin{equation}\label{Eq3}
\sum_{j=m}^{e_m-1}q_j\geq|X\setminus\mathcal B|\geq|X\setminus\mathcal B^*|=\sum_{j=m}^{e_m-1}q_j^*.
\end{equation}
This proves $(a)$.
To prove the converse, suppose we are given a composition $(q_0,\ldots, q_k)$ of $n-k$ into $k+1$ parts that satisfies $(a)$ and $(b)$. We wish to construct a valid hook configuration $\mathcal H=(H_1,\ldots,H_k)$ of $\pi$ that induces the composition $(q_0,\ldots, q_k)$. To do so, it suffices to specify the northeast endpoints of the hooks. Letting $P(b_i)$ denote the northeast endpoint of $H_i$ as before, we see that we need only choose the indices $b_i$. We will choose them in the order $b_k,\ldots,b_1$.
Let $\ell\in\{1,\ldots,k\}$. Suppose that we have already chosen $b_k,\ldots,b_{\ell+1}$ and that we are now ready to choose $b_\ell$. Let $Z_\ell$ be the set of indices $z\in\{d_\ell+1,\ldots,n\}\setminus\{b_{\ell+1},\ldots,b_k\}$ such that $P(z)$ does not lie below any of the hooks $H_{\ell+1},\ldots,H_k$. Let us write $Z_\ell=\{z_\ell(1),\ldots,z_\ell(\theta_\ell)\}$, where $\theta_\ell=\vert Z_\ell\vert$ and $z_\ell(1)<\cdots<z_\ell(\theta_\ell)$. Put $b_\ell=z_\ell(q_\ell+1)$.
This choice of $b_\ell$ is actually forced upon us. Indeed, we \emph{must} put $b_\ell=z_\ell(w)$ for some $w$. The points $P(z_\ell(1)),\ldots,P(z_\ell(w-1))$ are precisely the points that see the hook $H_\ell$ when they look directly upward. Therefore, if we can show that this construction actually produces a valid hook configuration $\mathcal H$, we will know that $(q_0,\ldots,q_k)$ is the valid composition of $\pi$ induced by $\mathcal H$. Furthermore, we will know that $\mathcal H$ is the unique valid hook configuration of $\pi$ inducing $(q_0,\ldots,q_k)$ (this is essentially the proof of Theorem \ref{Thm6} given in \cite{Defant2}).
We first need to verify that $\theta_\ell\geq q_\ell+1$ so that $z_\ell(q_\ell+1)$ actually exists. There are $n-d_\ell-(k-\ell)$ indices $z\in\{d_\ell+1,\ldots,n\}\setminus\{b_{\ell+1},\ldots,b_k\}$, and $\displaystyle{\sum_{j=\ell+1}^{k+1}q_j}$ of them are such that $P(z)$ lies below one of the hooks $H_{\ell+1},\ldots,H_k$. Consequently,
\begin{equation}\label{Eq7}
\theta_\ell=\vert Z_\ell\vert=n-d_\ell-(k-\ell)-\sum_{j=\ell+1}^kq_j.
\end{equation}
We know that $b_i^*>d_i$ for all $i$. Therefore, among the numbers $b_1^*,\ldots,b_k^*$, only $b_1^*,\ldots,b_{\ell-1}^*$ could possibly lie in the first $\ell$ ascending runs of $\pi$. This shows that $\displaystyle \sum_{j=1}^\ell\alpha_j\leq\ell-1$.
Combining this with \eqref{Eq7} and the fact that $\displaystyle{\sum_{j=0}^\ell q_j=n-k-\sum_{j=\ell+1}^kq_j}$ gives
\[\theta_\ell=\ell-d_\ell+\sum_{j=0}^\ell q_j=\ell-1-d_\ell+\sum_{j=0}^{\ell-1} q_j+q_\ell+1\geq\sum_{j=0}^{\ell-1}q_j-\left(d_\ell-\sum_{j=1}^\ell\alpha_j\right)+q_\ell+1.\]
Setting $m=0$ and $p=\ell-1$ in condition $(b)$ yields \[\sum_{j=0}^{\ell-1}q_j\geq d_\ell-\sum_{j=1}^\ell\alpha_j,\] so $\theta_\ell\geq q_\ell+1$.
Now that we have defined the indices $b_\ell$, we can construct the hooks $H_\ell$. Specifically, $H_\ell$ is the hook with southwest endpoint $P(d_\ell)$ and northeast endpoint $P(b_\ell)$. To check that this is in fact a hook, we must verify that $d_\ell<b_\ell$ and $\pi_{d_\ell}<\pi_{b_\ell}$ for all $\ell$. We constructed $b_\ell$ so that $d_\ell<b_\ell$. We also know that $\pi_{d_\ell}<\pi_{b_\ell^*}$ because $H_\ell^*$ is a hook with southwest endpoint $P(d_\ell)$ and northeast endpoint $P(b_\ell^*)$. Hence, it suffices to show that $\pi_{b_\ell^*}\leq\pi_{b_\ell}$.
Observe that $b_k^*=d_k+q_k^*+1$ because the points lying below $H_k^*$ are precisely $P(d_k+1),\ldots,$ $P(d_k+q_k)$. Likewise, $b_k=d_k+q_k+1$. Setting $m=k$ in condition $(a)$ yields $q_k\geq q_k^*$ because $e_k=k+1$. This shows that $b_k^*\leq b_k$. This also forces the inequality $\pi_{b_k^*}\leq \pi_{b_k}$ since $\pi_{b_k^*}$ and $\pi_{b_k}$ both lie in the $(k+1)^\text{th}$ ascending run of $\pi$. It follows that $H_k$ lies above $H_k^*$ or is equal to $H_k^*$. Now, choose some $\ell\in\{1,\ldots,k-1\}$, and suppose that $b_m^*\leq b_m$ and $\pi_{b_m^*}\leq\pi_{b_m}$ for all $m\in\{\ell+1,\ldots,k\}$. For each such $m$, this means that $H_m$ lies above or is equal to $H_m^*$. Recall the definition of $Z_\ell$ from above. It is straightforward to check that the entries $\pi_{z_\ell(1)},\ldots,\pi_{z_\ell(\theta_\ell)}$ are left-to-right maxima of the string $\pi_{d_\ell+1}\cdots\pi_n$ (a left-to-right maximum of a string of positive integers $w_1\cdots w_m$ is an entry $w_j$ such that $w_j>w_i$ for all $i\in\{1,\ldots,j-1\}$).
We know from our definition of $e_\ell$ that $b_\ell^*\geq d_{e_\ell-1}+1$. We wish to show that $b_\ell\geq b_\ell^*$, which will of course imply that $b_\ell\geq d_{e_\ell-1}+1$. By way of contradiction, let us first assume $b_\ell\leq d_{e_\ell-1}$. We have $d_p+1\leq b_\ell\leq d_{p+1}$ for some $p\in\{\ell,\ell+1,\ldots,e_\ell-2\}$. Construct the hook $H_\ell$ with southwest endpoint $P(d_\ell)$ and northeast endpoint $P(b_\ell)$ (we do not yet know that $\pi_{d_\ell}<\pi_{b_\ell}$, but we can still connect $P(d_\ell)$ and $P(b_\ell)$ with line segments and call the resulting shape a ``hook"). Because $\pi_{b_\ell}=\pi_{z_\ell(q_\ell+1)}$ is a left-to-right maximum of $\pi_{d_\ell+1}\cdots\pi_n$, every point $P(x)$ with $d_\ell+1\leq x\leq d_p$ lies below $H_\ell$. Furthermore, each of the hooks $H_{\ell+1},\ldots,H_p$ must lie (entirely) below $H_\ell$ because $P(b_\ell)$ cannot lie above any of these hooks (one can verify that our construction guarantees that no point in the plot of $\pi$ lies above any of these hooks). It follows from our construction that there are precisely $\displaystyle{\sum_{j=\ell}^pq_j}$ points that lie below the hook $H_\ell$ and are not in $\{P(b_1),\ldots,P(b_k)\}$. Each of the points $P(b_{\ell+1}),\ldots,P(b_p)$ lies below $H_\ell$ because the hooks $H_{\ell+1},\ldots,H_p$ lie below $H_\ell$. This means that the total number of points lying below $H_\ell$ is at least $p-\ell+\displaystyle{\sum_{j=\ell}^pq_j}$. For each such point $P(z)$, we have $d_\ell+1\leq z\leq d_{p+1}$, so
\begin{equation}\label{Eq9}
p-\ell+\sum_{j=\ell}^pq_j<d_{p+1}-d_\ell
\end{equation}
Note that the inequality here is strict because $b_\ell$ is an element of $\{d_\ell+1,\ldots,d_{p+1}\}$ and $P(b_\ell)$ does not lie below $H_\ell$.
If $\delta$ is an index such that $d_\ell+1\leq b_\delta^*\leq d_{p+1}$, then $\delta\leq p$. The point $P(b_\delta^*)$ must lie below $H_\ell^*$ because $d_\ell<b_\delta^*\leq d_{p+1}<b_\ell^*$. Hence, $\ell+1\leq \delta\leq p$. This shows that there are at most $p-\ell$ possible choices for $\delta$, so \[\sum_{j=\ell+1}^{p+1}\alpha_j\leq p-\ell.\] It now follows from \eqref{Eq9} that \[\sum_{j=\ell+1}^{p+1}\alpha_j+\sum_{j=\ell}^pq_j<d_{p+1}-d_\ell,\] which we can see is a contradiction by setting $m=\ell$ in condition $(b)$. We conclude that $b_\ell\geq d_{e_\ell-1}+1$.
Note that $H_\ell$ lies above the hooks $H_{\ell+1},\ldots,H_{e_\ell-1}$, so the number of points not in the set $\{P(b_1),\ldots,P(b_k)\}$ that lie below $H_\ell$ is at least $\displaystyle{\sum_{j=\ell}^{e_\ell-1}q_j}$. By condition $(a)$, this is at least $\displaystyle{\sum_{j=\ell}^{e_\ell-1}q_j^*}$. Moreover, each of the points $P(b_{\ell+1}),\ldots,P(b_{e_\ell-1})$ lies below $H_\ell$. This shows that there are at least \[e_\ell-1-\ell+\sum_{j=\ell}^{e_\ell-1}q_j^*\] points below $H_\ell$.
If $\eta$ is an index such that $P(b_\eta^*)$ lies below $H_\ell^*$, then $d_\ell<d_\eta<b_\eta^*<b_\ell^*\leq d_{e_\ell}$. This guarantees that $\ell<\eta\leq e_\ell-1$, so there are at most $e_\ell-1-\ell$ such indices $\eta$. The number of points below $H_\ell^*$ that are not of the form $P(b_\eta^*)$ is $\displaystyle{\sum_{j=\ell}^{e_\ell-1}q_j^*}$, so the total number of points below $H_\ell^*$ is at most \[e_\ell-1-\ell+\sum_{j=\ell}^{e_\ell-1}q_j^*.\] According to the previous paragraph, the number of points below $H_\ell$ is at least the number of points below $H_\ell^*$. Therefore, $H_\ell$ lies above $H_\ell^*$ or is equal to $H_\ell^*$. In other words, $b_\ell^*\leq b_\ell$ and $\pi_{b_\ell^*}\leq \pi_{b_\ell}$. It follows by induction that $b_i^*\leq b_i$ and $\pi_{b_i^*}\leq\pi_{b_i}$ for all $i\in\{1,\ldots,k\}$.
We have shown that the hooks $H_1,\ldots,H_k$ are in fact bona fide hooks. It is straightforward to check that our construction guarantees that $\mathcal H=(H_1,\ldots,H_k)$ is a valid hook configuration of $\pi$, so the proof is complete.
\end{proof}
\section{Background on Preimages of Permutation Classes}\label{Sec:Back}
In this section, we review some known results concerning preimages of permutation classes. We also establish some notation for subsequent sections.
First, suppose $\pi\in S_n$ for some $n\geq 4$, and write $\pi=LnR$ so that $s(\pi)=s(L)s(R)n$. Either $L$ or $R$ has length at least $2$, and the permutations $s(L)$ and $s(R)$ each must end in their last entries. It follows that $s(\pi)$ contains the pattern $123$, so \[|s^{-1}(\operatorname{Av}_n(123))|=0\quad\text{whenever}\quad n\geq 4.\] It is easy to see that \[|s^{-1}(\operatorname{Av}_n(213))|=C_n.\] Indeed, suppose $\pi\in s^{-1}(\operatorname{Av}_n(213))$. Since $s(\pi)$ avoids $213$ and has last entry $n$, $s(\pi)$ must be the identity permutation of length $n$. In other words, $s^{-1}(\operatorname{Av}(213))=s^{-1}(\operatorname{Av}(21))=\operatorname{Av}(231)$ by Theorem \ref{Thm1}. Theorem \ref{Thm2} tells us that \[|s^{-1}(\operatorname{Av}_n(231))|=\frac{2}{(n+1)(2n+1)}{3n\choose n}.\] Theorem 3.2 in \cite{Bouvel} states that we also have \[|s^{-1}(\operatorname{Av}_n(132))|=\frac{2}{(n+1)(2n+1)}{3n\choose n}.\] Part of Theorem 3.4 in \cite{Bouvel} states that \[|s^{-1}(\operatorname{Av}_n(312))|=\frac{2}{n(n+1)^2}\sum_{k=1}^n{n+1\choose k-1}{n+1\choose k}{n+1\choose k+1}.\] This last expression is also the number of so-called \emph{Baxter permutations} of length $n$, and it produces the sequence A001181 in the Online Encyclopedia of Integer Sequences \cite{OEIS}. The only length-$3$ pattern $\tau$ for which $|s^{-1}(\operatorname{Av}_n(\tau))|$ is not known is $321$. The sequence $(|s^{-1}(\operatorname{Av}_n(321))|)_{n\geq 1}$ appears to be new. In Section \ref{Sec:321}, we use valid hook configurations to derive estimates for the exponential growth rate of this sequence.
The above remarks show that the sets $s^{-1}(\operatorname{Av}(\tau^{(1)},\ldots,\tau^{(r)}))$ are relatively uninteresting when one of the patterns $\tau^{(i)}$ is an element of $\{123,213\}$. Hence, we will focus our attention on permutation classes whose bases are subsets of $\{132,231,312,321\}$.
As mentioned in the introduction, it is always possible to describe $s^{-1}(\operatorname{Av}(\tau^{(1)},\ldots,\tau^{(r)}))$ as the set of permutations avoiding some finite collection of \emph{mesh patterns}. We find it simpler to describe our stack-sorting preimage sets in terms of barred patterns or vincular patterns. A \emph{barred pattern} is a permutation pattern in which some entries are overlined. Saying a permutation contains a barred pattern means that it contains a copy of the pattern formed by the unbarred entries that is not part of a pattern that has the same relative order as the full barred pattern. For example, saying a permutation contains the barred pattern $3\overline{5}241$ means that it contains a $3241$ pattern that is not part of a $35241$ pattern. In fact, West \cite{West} introduced barred patterns in order to describe $2$-stack-sortable permutations, showing that a permutation is $2$-stack-sortable if and only if it avoids the classical pattern $2341$ and the barred pattern $3\overline{5}241$.
A \emph{vincular pattern} is a permutation pattern in which some consecutive entries can be underlined. We say a permutation \emph{contains} a vincular pattern if it contains an occurrence of the permutation pattern in which underlined entries are consecutive. For example, saying that a permutation $\sigma=\sigma_1\cdots\sigma_n$ contains the vincular pattern $\underline{32}41$ means that there are indices $i_1<i_2<i_3<i_4$ such that $\sigma_{i_4}<\sigma_{i_2}<\sigma_{i_1}<\sigma_{i_3}$ and $i_2=i_1+1$. Vincular patterns appeared first in \cite{Babson} and have received a large amount of attention ever since \cite{Steingrimsson}.
If $\tau$ is a classical, barred, or vincular pattern, then we say a permutation \emph{avoids} $\tau$ if it does not contain $\tau$. Let $\operatorname{Av}(\tau^{(1)},\ldots,\tau^{(r)})$ be the set of permutations avoiding $\tau^{(1)},\ldots,\tau^{(r)}$, and let $\operatorname{Av}_n(\tau^{(1)},\ldots,\tau^{(r)})=\operatorname{Av}(\tau^{(1)},\ldots,\tau^{(r)})\cap S_n$.
Recall the notation from Theorem \ref{Thm5} and Theorem \ref{Thm7}. When counting preimages of permutations according to numbers of descents and peaks, we will make use of the generating functions \[F(x,y)=\sum_{n\geq 1}\sum_{m\geq 1}N(n,m)x^ny^{m-1}\quad\text{and}\quad G(x,y)=\sum_{n\geq 1}\sum_{m\geq 1}V(n,m)x^ny^{m-1}.\] It is known that
\begin{equation}\label{Eq2}
F(x,y)=\frac{1-x(y+1)-\sqrt{1-2x(y+1)+x^2(y-1)^2}}{2xy}
\end{equation}
and
\begin{equation}\label{Eq4}
G(x,y)=\frac{1-2x-\sqrt{(1-2x)^2-4x^2y}}{2xy}.
\end{equation}
We let $[z_1^{n_1}\cdots z_r^{n_r}]A(z_1,\ldots,z_r)$ denote the coefficient of $z_1^{n_1}\cdots z_r^{n_r}$ in the generating function \linebreak $A(z_1,\ldots,z_r)$.
\section{$s^{-1}(\operatorname{Av}(132,231,312,321))$}\label{Sec:4patterns}
It turns out that $s^{-1}(\operatorname{Av}(132,231,312,321))$ is an actual permutation class; we have \[s^{-1}(\operatorname{Av}(132,231,312,321))=\operatorname{Av}(1342,2341,3142,3241,3412,3421).\] One could probably enumerate this class directly, but we will use valid hook configurations in order to illustrate this uniform method for finding fertilities.
\begin{theorem}\label{Thm3}
For $n\geq 2$, we have \[|s^{-1}(\operatorname{Av}_n(132,231,312,321))|=2C_n-2C_{n-1}.\] The number of elements of $s^{-1}(\operatorname{Av}_n(132,231,312,321))$ with $m$ descents is \[N(n,m+1)+\sum_{i=1}^{n-2}\sum_{j=1}^mN(n-i-1,j)N(i,m-j+1).\] The number of elements of $s^{-1}(\operatorname{Av}_n(132,231,312,321))$ with $m$ peaks is \[V(n,m+1)+\sum_{i=1}^{n-2}\sum_{j=1}^mV(n-i-1,j)V(i,m-j+1).\]
\end{theorem}
\begin{proof}
The only elements of $\operatorname{Av}_n(132,231,312,321)$ are $123\cdots n$ and $2134\cdots n$. The only valid composition of $123\cdots n$ is $(n)$. Each valid hook configuration of $2134\cdots n$ has exactly one hook. This hook has southwest endpoint $(1,2)$ and has northeast endpoint $(n+1-i,n+1-i)$ for some $i\in\{1,\ldots,n-2\}$. This valid hook configuration induces the valid composition $(i,n-i-1)$. It follows from Theorem \ref{Thm5} and the standard Catalan number recurrence relation that \[|s^{-1}(\operatorname{Av}_n(132,231,312,321))|=C_n+\sum_{i=1}^{n-2}C_iC_{n-i-1}=C_n+\sum_{i=0}^{n-1}C_iC_{n-i-1}-2C_{n-1}=2C_n-2C_{n-1}.\] The second and third statements of the theorem follow immediately from Theorem \ref{Thm7}.
\end{proof}
\section{$s^{-1}(\operatorname{Av}(132,231,321))$ and $s^{-1}(\operatorname{Av}(132,312,321))$}\label{Sec5}
It turns out that \[s^{-1}(\operatorname{Av}(132,231,321))=\operatorname{Av}(1342,2341,3241,45231,3\overline{5}142).\] The set $s^{-1}(\operatorname{Av}(132,312,321))$ is actually equal to the permutation class $\operatorname{Av}(1342,3142,3412,3421)$.
Let \[\sigma_{n,\ell}=\ell 12\cdots(\ell-1)(\ell+1)\cdots n\quad\text{and}\quad\gamma_{n,\ell}=23\cdots(\ell-1)1(\ell+1)\cdots n.\] It is straightforward to check that $\operatorname{Av}_n(132,231,321)=\{\sigma_{n,1},\ldots,\sigma_{n,n}\}$ and $\operatorname{Av}_n(132,312,321)=\{\gamma_{n,1},\ldots,\gamma_{n,n}\}$. For example, $\operatorname{Av}_4(132,231,321)=\{1234,2134,3124,4123\}$.
West \cite{West} found formulas for the fertilities of $\sigma_{n,\ell}$ and $\gamma_{n,\ell}$ and found that they are equal. It follows that \[|s^{-1}(\operatorname{Av}_n(132,231,321))|=|s^{-1}(\operatorname{Av}_n(132,312,321))|.\]
This equality is easy to verify with the theory of valid hook configurations. Indeed, for $2\leq \ell\leq n$, we have \[\mathcal V(\sigma_{n,\ell})=\{(n-\ell-i+1,\ell+i-2):1\leq i\leq n-\ell\}\hspace{.29cm}\text{and}\hspace{.29cm}\mathcal V(\gamma_{n,\ell})=\{(\ell+i-2,n-\ell-i+1):1\leq i\leq n-\ell\}.\]
That is, the valid compositions of $\sigma_{n,\ell}$ are obtained by interchanging the two parts in the valid compositions of $\gamma_{n,\ell}$. Along with Theorem \ref{Thm7}, this also implies the following.
\begin{theorem}\label{Thm15}
The number of elements of $s^{-1}(\operatorname{Av}_n(132,231,321))$ with $m$ descents is equal to the number of elements of $s^{-1}(\operatorname{Av}_n(132,312,321))$ with $m$ descents. The number of elements of $s^{-1}(\operatorname{Av}_n(132,231,321))$ with $m$ peaks is equal to the number of elements of $s^{-1}(\operatorname{Av}_n(132,312,321))$ with $m$ peaks.
\end{theorem}
Note that West did not prove these refined equalities.
\begin{theorem}\label{Thm16}
We have \[|s^{-1}(\operatorname{Av}_n(132,231,321))|=|s^{-1}(\operatorname{Av}_n(132,312,321))|={2n-2\choose n-1}.\] The number of elements of $s^{-1}(\operatorname{Av}_n(132,231,321))$ (equivalently, $s^{-1}(\operatorname{Av}_n(132,312,321))$) with $m$ descents is \[{n-1\choose m}^2.\] The number of elements of $s^{-1}(\operatorname{Av}_n(132,231,321))$ (equivalently, $s^{-1}(\operatorname{Av}_n(132,312,321))$) with $m$ peaks is \[2^{n-2m-2}{n\choose 2m+2}{2m+2\choose m+1}.\]
\end{theorem}
\begin{proof}
By Theorem \ref{Thm15}, we need only consider the preimage sets $s^{-1}(\operatorname{Av}_n(132,231,321))$. We will prove the second and third statements; the first statement will then follow from the second and the well-known identity $\sum_{m=0}^{n-1}{n-1\choose m}^2={2n-2\choose n-1}$. The only valid composition of $\sigma_{n,1}=123\cdots n$ is $(n)$. For $2\leq \ell\leq n$, the valid compositions of $\sigma_{n,\ell}$ are $(n-\ell-i+1,\ell+i-2)$ for $1\leq i\leq n-\ell$. In particular, we can ignore $\sigma_{n,n}$ because it has no valid compositions (that is, $\sigma_{n,n}$ is not sorted).
Using the first part of Theorem \ref{Thm7}, we find that the number of elements of
\linebreak $s^{-1}(\operatorname{Av}_n(132,231,321))$ with $m$ descents is \[N(n,m+1)+\sum_{\ell=2}^{n-1}\sum_{(q_0,q_1)\in\mathcal V(\sigma_\ell)}\sum_{j_0+j_1=m+1}N(q_0,j_0)N(q_1,j_1)\] \[=N(n,m+1)+\sum_{\ell=2}^{n-1}\sum_{i=1}^{n-\ell}\sum_{j=1}^m N(n-\ell-i+1,j)N(\ell+i-2,m-j+1).\] Letting $r=\ell+i$, this becomes
\[N(n,m+1)+\sum_{\ell=2}^{n-1}\sum_{r=\ell+1}^n\sum_{j=1}^mN(n-r+1,j)N(r-2,m-j+1)\] \[=N(n,m+1)+\sum_{\ell=1}^{n-2}\sum_{r=\ell}^{n-2}\sum_{j=1}^m N(n-r-1,j)N(r,m-j+1)\] \[=N(n,m+1)+\sum_{r=1}^{n-2}r\sum_{j=1}^m N(n-r-1,j)N(r,m-j+1)\] \[=[x^ny^m]\left(F(x,y)+x^2y\,F(x,y)\cdot\frac{\partial}{\partial x}F(x,y)\right).\] Applying \eqref{Eq2} and some algebraic manipulations, we find that this is equal to \[[x^ny^m]\left(\frac{x}{\sqrt{1+x^2(y-1)^2-2x(y+1)}}\right).\] Standard methods now allow us to see that this expression is equal to ${n-1\choose m}^2$.
The proof of the third statement in the theorem proceeds exactly as in the proof of the second statement. In this case, we find that the number of elements of $s^{-1}(\operatorname{Av}_n(132,231,321))$ with $m$ peaks is \[[x^ny^m]\left(G(x,y)+x^2y\,G(x,y)\cdot\frac{\partial}{\partial x}G(x,y)\right).\]
Applying \eqref{Eq4} and some algebraic manipulations, we find that this is equal to \[[x^ny^m]\left(\frac{x}{\sqrt{1-4x-4x^2(y-1)}}\right).\] Standard methods now allow us to see that this expression is equal to $2^{n-2m-2}{n\choose 2m+2}{2m+2\choose m+1}$.
\end{proof}
Recently, Bruner proved that \[|\operatorname{Av}_n(2431, 4231, 1432, 4132)|={2n-2\choose n-1}.\] She also listed several other permutation classes that appear to be enumerated by central binomial coefficients but did not prove that this is the case. One of these classes is $\operatorname{Av}(1243,2143,2413,2431)$. Of course, a permutation is in this class if and only if its reverse is in the class \[\operatorname{Av}(1342,3142,3412,3421)=s^{-1}(\operatorname{Av}(132,312,321)).\] Therefore, the following corollary, which follows immediately from Theorem \ref{Thm16} and the discussion immediately preceding that theorem, settles one of the enumerative problems that Bruner listed.
\begin{corollary}
We have \[|\operatorname{Av}_n(1342,3142,3412,3421)|={2n-2\choose n-1}.\] The number of elements of $\operatorname{Av}_n(1342,3142,3412,3421)$ with $m$ descents is \[{n-1\choose m}^2.\] The number of elements of $\operatorname{Av}_n(1342,3142,3412,3421)$ with $m$ peaks is \[2^{n-2m-2}{n\choose 2m+2}{2m+2\choose m+1}.\]
\end{corollary}
\section{$s^{-1}(\operatorname{Av}(231,312,321))$}\label{Sec6}
We will find it convenient to identify a permutation with a configuration of points in the plane via its plot. Doing so, we can build permutations by placing the plots of smaller permutations in various configurations. For example, if $\lambda=\lambda_1\cdots\lambda_\ell\in S_\ell$ and $\mu=\mu_1\ldots\mu_m\in S_m$, then the \emph{sum} of $\lambda$ and $\mu$, denoted $\lambda\oplus\mu$, is obtained by placing the plot of $\mu$ above and to the right of the plot of $\lambda$. More formally, the $i^\text{th}$ entry of $\lambda\oplus\mu$ is \[(\lambda\oplus\mu)_i=\begin{cases} \lambda_i & \mbox{if } 1\leq i\leq \ell; \\ \mu_{i-\ell}+\ell & \mbox{if } \ell+1\leq i\leq \ell+m. \end{cases}\]
Let $\text{Dec}_a=a(a-1)\cdots 1\in S_a$ denote the decreasing permutation of length $a$. The permutations in $\operatorname{Av}(231,312)$ are called \emph{layered}; each is of the form $\text{Dec}_{a_1}\oplus\cdots\oplus \text{Dec}_{a_t}$ for some composition $(a_1,\ldots,a_t)$. For example, $32154687=321\oplus 21\oplus 1\oplus 21$ is the layered permutation corresponding to the composition $(3,2,1,2)$. Under this correspondence between layered permutations and compositions, the permutations in $\operatorname{Av}(231,312,321)$ correspond to compositions whose parts are all at most $2$. It follows that these permutations are counted by the Fibonacci numbers.
The set $s^{-1}(\operatorname{Av}(231,312,321))$ is actually just the permutation class $\operatorname{Av}(2341,3241,3412,3421)$. A permutation $\sigma$ is in this class if and only if the first and third entries in every $231$ pattern in $\sigma$ are consecutive integers. For example, $236541$ is not in this class because the entries $3,6,1$ form a $231$ pattern while $3$ and $1$ are not consecutive integers. In this section, we use valid hook configurations to derive a formula for $|s^{-1}(\operatorname{Av}_n(231,312,321))|$. We then enumerate this class directly; showing that these permutations are counted by the terms in sequence A049124 in the Online Encyclopedia of Integer Sequences \cite{OEIS}. Together, these results give a new formula and a new combinatorial interpretation for the terms in this sequence.
Let $\operatorname{Comp}_a(b)$ denote the set of all compositions of $b$ into $a$ parts (that is, $a$-tuples of positive integers that sum to $b$). Define a partial order $\preceq$ on $\operatorname{Comp}_a(b)$ by declaring that $(x_1,\ldots,x_a)\preceq(y_1,\ldots,y_a)$ if $\sum_{i=1}^\ell x_i\leq\sum_{i=1}^\ell y_i$ for all $\ell\in\{1,\ldots,a\}$. A \emph{partition} is a composition whose parts are nonincreasing. Following \cite{Stanley}, we let $L(u,v)$ denote the set of all partitions (including the empty partition) with at most $u$ parts and with largest part at most $v$. Endow $L(u,v)$ with a partial order $\leq$ by declaring that $(\lambda_1,\ldots,\lambda_\ell)\leq(\mu_1,\ldots,\mu_m)$ if $\ell\leq m$ and $\lambda_i\leq\mu_i$ for all $i\in\{1,\ldots,\ell\}$. Geometrically, $L(u,v)$ is the set of all partitions whose Young diagrams fit inside a $u\times v$ rectangle, and $\lambda\leq\mu$ if and only if the Young diagram of $\lambda$ fits inside of the Young diagram of $\mu$.
Given $x=(x_1,\ldots,x_a)\in\operatorname{Comp}_a(b)$, let $\psi(x)\in L(b-a,a-1)$ be the partition that has exactly $x_i-1$ parts of length $a-i$ for all $i\in\{1,\ldots,a-1\}$. The map $\psi:\operatorname{Comp}_a(b)\to L(b-a,a-1)$ is an isomorphism of posets. For $x\in\operatorname{Comp}_a(b)$, let \[D_x=|\{y\in\operatorname{Comp}_a(b):y\preceq x\}|.\] Equivalently, $D_{\psi^{-1}(\lambda)}$ is the number of partitions (including the empty partition) whose Young diagrams fit inside of the Young diagram of the partition $\lambda$. Recall the notation $C_{(x_0,\ldots,x_k)}=\prod_{t=0}^k C_{x_t}$, where $C_j$ is the $j^\text{th}$ Catalan number.
\begin{theorem}\label{Thm8}
Preserving the notation of the preceding paragraph with $a=k+1$ and $b=n-k$, we have \[|s^{-1}(\operatorname{Av}_n(231,312,321))|=\sum_{k=0}^{n-1}\sum_{q\,\in\,\operatorname{Comp}_{k+1}(n-k)}C_qD_q=\sum_{k=0}^{n-1}\sum_{\lambda\in L(n-2k-1,k)}C_{\psi^{-1}(\lambda)}D_{\psi^{-1}(\lambda)}.\]
\end{theorem}
\begin{proof}
Recall that each permutation in $\operatorname{Av}(231,312,321)$ is of the form $\text{Dec}_{a_1}\oplus\cdots\oplus\text{Dec}_{a_t}$, where $a_i\in\{1,2\}$ for all $i$. Suppose $\pi\in\operatorname{Av}_n(231,312,321)$ has descents $d_1<\cdots<d_k$. For $p\in\{0,\ldots,k-1\}$, let $u_p=d_{p+1}-p$. Defining $y_0=u_0$, $y_i=u_i-u_{i-1}$ for $1\leq i\leq k-1$, and $y_k=n-k-u_{k-1}$, we obtain a composition $y=(y_0,\ldots,y_k)\in\operatorname{Comp}_{k+1}(n-k)$. Given this composition $y$, we can easily reconstruct the permutation $\pi$. Thus, there is a bijective correspondence between compositions in $\operatorname{Comp}_{k+1}(n-k)$ and permutations in $\operatorname{Av}_n(231,312,321)$ with $k$ descents.
We are going to use Theorem \ref{Thm4} to describe all of the valid compositions of $\pi$. Preserve the notation from that theorem and the discussion immediately preceding it. We must compute the canonical valid hook configuration $\mathcal H^*=(H_1^*,\ldots,H_k^*)$ of $\pi$. This is fairly simple to do: the hook $H_i^*$ has southwest endpoint $(d_i,\pi_{d_i})$ and northeast endpoint $(d_i+2,\pi_{d_i+2})$. Thus, $b_i^*=d_i+2$. We also have $(q_0^*,\ldots,q_k^*)=(n-2k,1,\ldots,1)$, $e_0=k+1$, and $e_i=i+1$ for all $i\in\{1,\ldots,k\}$. Finally, $\alpha_1=0$, and $\alpha_i=1$ for all $i\in\{2,\ldots,k+1\}$.
Every composition $(q_0,\ldots,q_k)\in\operatorname{Comp}_{k+1}(n-k)$ satisfies condition $(a)$ in Theorem \ref{Thm4}. In condition $(b)$, the inequality $m\leq p\leq e_m-2$ is only satisfied when $m=0$. When $m=0$, \[d_{p+1}-d_m-\sum_{j=m+1}^{p+1}\alpha_j=d_{p+1}-p=u_p.\] Hence, Theorem \ref{Thm4} tells us that a composition $q=(q_0,\ldots,q_k)\in\operatorname{Comp}_{k+1}(n-k)$ is a valid composition of $\pi$ if and only if \[\sum_{j=0}^pq_j\geq u_p\] for all $p\in\{0,\ldots,k-1\}$. This occurs if and only if $y\preceq q$.
Combining these observations with Theorem \ref{Thm5}, we find that (recall the definition of
\linebreak $\operatorname{Av}_{n,k}(\tau^{(1)},\ldots,\tau^{(r)})$ from Definition \ref{Def1}) \[|s^{-1}(\operatorname{Av}_n(231,312,321))|=\sum_{k=0}^{n-1}\,\sum_{\pi\in\operatorname{Av}_{n,k}(231,312,321)}\,\sum_{q\in\mathcal V(\pi)}C_q=\sum_{k=0}^{n-1}\sum_{y\in\operatorname{Comp}_{k+1}(n-k)}\sum_{\substack{q\in\operatorname{Comp}_{k+1}(n-k)\\y\preceq q}}C_q\]
\[=\sum_{k=0}^{n-1}\sum_{q\,\in\,\operatorname{Comp}_{k+1}(n-k)}C_qD_q.\] The identity \[\sum_{k=0}^{n-1}\sum_{q\,\in\,\operatorname{Comp}_{k+1}(n-k)}C_qD_q=\sum_{k=0}^{n-1}\sum_{\lambda\in L(n-2k-1,k)}C_{\psi^{-1}(\lambda)}D_{\psi^{-1}(\lambda)}\] follows from the discussion immediately preceding this theorem (with $a=k+1$ and $b=n-k$).
\end{proof}
\begin{theorem}\label{Thm9}
For $n\geq 1$, we have \[|s^{-1}(\operatorname{Av}_n(231,312,321))|=\sum_{k=0}^{n-1}\frac{1}{n+1}{n-k-1\choose k}{2n-2k\choose n}.\]
\end{theorem}
\begin{proof}
Let $h(n)=|s^{-1}(\operatorname{Av}_n(231,312,321))|=|\operatorname{Av}_n(2341,3241,3412,3421)|$ (with $h(0)=1$), and put \[H(x)=\sum_{n\geq 0}h(n)x^n.\] We will prove that
\begin{equation}\label{Eq6}
H(x)=1+\frac{xH(x)^2}{1-x^2H(x)^2}.
\end{equation}
Lagrange inversion (the details of which we omit) then allow one to extract the desired formula from this functional equation for $H(x)$. In fact, \[\sum_{k=0}^{n-1}\frac{1}{n+1}{n-k-1\choose k}{2n-2k\choose n}\] is the $(n+1)^\text{th}$ term of the sequence A049124 in the Online Encyclopedia of Integers Sequences \cite{OEIS}, and the generating function $A(x)$ of that sequence satisfies $A(x)=x+\dfrac{A(x)^2}{1-A(x)^2}$ (implying that $A(x)=xH(x)$).
We can rewrite \eqref{Eq6} as
\begin{equation}\label{Eq5}
H(x)=1+xH(x)^2+x^2H(x)^2(H(x)-1);
\end{equation}
it is this form of the equation that we will prove. For convenience, say a permutation is \emph{special} if it is an element of $s^{-1}(\operatorname{Av}(231,312,321))$ (including the empty permutation). As mentioned above, a permutation $\sigma=\sigma_1\cdots\sigma_n$ is special if and only if the first and third entries in every $231$ pattern in $\sigma$ are consecutive integers. Note that the empty permutation is special and accounts for the term $1$ on the right-hand side of \eqref{Eq5}. We now show how to construct nonempty special permutations. It is convenient to split these permutations into two types.
The first type of nonempty special permutation $\sigma$ is that in which the first entry $\sigma_1$ is not part of a $231$ pattern. As mentioned above, we identify permutations with their plots so that we can build large permutations by arranging plots of smaller permutations. Each nonempty special permutation of the first type is formed by choosing two special permutations $\eta$ and $\zeta$ and arranging them as follows:
\begin{center}
\includegraphics[width=.15\linewidth]{ClassPreimagesPIC1}.
\end{center}
Accordingly, the nonempty special permutations of the first type contribute the term $xH(x)^2$ to the right-hand side of \eqref{Eq5}.
The second type of nonempty special permutation $\sigma$ is that in which $\sigma_1$ is part of a $231$ pattern. We describe how to build such a permutation, leaving the reader to check that every nonempty special permutation of the second type is built uniquely via this procedure.
Begin by choosing special permutations $\lambda=\lambda_1\cdots\lambda_\ell$, $\tau=\tau_1\cdots\tau_t$, and $\mu=\mu_1\cdots\mu_m$ such that $\mu$ is nonempty. Write $\mu'=\mu_2\cdots\mu_m$ (so $\mu'$ is empty if $m=1$). Write $\lambda=\lambda'\lambda''$, where $\lambda'$ is the first descending run of $\lambda$. In other words, $\lambda'=\lambda_1\cdots\lambda_j$, where $j$ is the smallest index that is not a descent of $\lambda$. If $\lambda$ is empty, then $\lambda'$ and $\lambda''$ are also empty. The plot of $\sigma$ is formed by arranging the plots of $\lambda,\tau,$ and $\mu$ as follows:
\begin{center}
\includegraphics[width=.35\linewidth]{ClassPreimagesPIC2}.
\end{center}
The points in the section labeled $\mu$ are arranged vertically so that they form a permutation that is order isomorphic to $\mu$. In other words, the point labeled $\mu_1$ should be placed higher than exactly $\mu_1-1$ of the points in the box labeled $\mu'$. Similarly, the points in the section labeled $\lambda$ should form a permutation that is order isomorphic to $\lambda$.
This construction shows that the nonempty special permutations of the second type contribute the term $x^2H(x)^2(H(x)-1)$ to the right-hand side of \eqref{Eq5}.
\end{proof}
\begin{example}\label{Exam2}
Let us illustrate the part of the proof of Theorem \ref{Thm9} that describes the construction of a nonempty special permutation of the second type. Choose $\tau=21$, $\lambda=31542$, and $\mu=231$. The resulting permutation is $4\,\,2\,\,1\,\,10\,\,8\,\,6\,\,3\,\,7\,\,5\,\,12\,\,11\,\,9$, whose plot is
\begin{center}
\includegraphics[width=.29\linewidth]{ClassPreimagesPIC3}.
\end{center}
The extra boxes drawn in this plot are intended to highlight the appearances of $\tau,\lambda$, and $\mu$ in the plot.
\end{example}
Combining Theorem \ref{Thm8} and Theorem \ref{Thm9} yields the identity \[\sum_{k=0}^{n-1}\sum_{q\,\in\,\operatorname{Comp}_{k+1}(n-k)}C_qD_q=\sum_{k=0}^{n-1}\frac{1}{n+1}{n-k-1\choose k}{2n-2k\choose n}.\] Numerical evidence suggests the following conjecture.
\begin{conjecture}\label{Conj1}
In the notation of Theorem \ref{Thm8}, we have \[\sum_{q\,\in\,\operatorname{Comp}_{k+1}(n-k)}C_qD_q=\frac{1}{n+1}{n-k-1\choose k}{2n-2k\choose n}\] for all nonnegative integers $n$ and $k$.
\end{conjecture}
When combined with Theorem \ref{Thm8}, a proof of Conjecture \ref{Conj1} would yield an alternative proof of Theorem \ref{Thm9}. We know from the proof of Theorem \ref{Thm8} that \[|s^{-1}(\operatorname{Av}_{n,k}(231,312,321))|=\sum_{q\,\in\,\operatorname{Comp}_{k+1}(n-k)}C_qD_q,\] so Conjecture \ref{Conj1} is equivalent to the identity \[|s^{-1}(\operatorname{Av}_{n,k}(231,312,321))|=\frac{1}{n+1}{n-k-1\choose k}{2n-2k\choose n}.\]
\section{$s^{-1}(\operatorname{Av}(132,231,312))$}\label{Sec7}
It turns out that \[s^{-1}(\operatorname{Av}(132,231,312))=\operatorname{Av}(2341,3412,1342,3142,34\underline{21},\underline{32}41).\]
In this subsection, we make use of the \emph{generalized Narayana numbers} \[N_k(n,r)=\frac{k+1}{n}{n\choose r+k}{n\choose r-1}.\] Note that the standard Narayana numbers are simply $N(n,r)=N_0(n,r)$.
Finding the fertilities of permutations in $\operatorname{Av}(132,231,312)$ is interesting because it proves the tightness of certain estimates that were used in \cite{Defant2} in order to obtain upper bounds for $W_3(n)$ and $W_4(n)$. More specifically, the following theorem is an immediate consequence of Corollary 3.1 and Lemma 4.1 in \cite{Defant2}.
\begin{theorem}[\!\!\cite{Defant2}]\label{Thm14}
Suppose $\pi\in S_n$ has $k$ descents. We have \[|s^{-1}(\pi)|\leq\frac{2k+2}{n+1}{2n-2k-1\choose n}.\] Furthermore, the number of elements of $s^{-1}(\pi)$ with exactly $m$ descents is at most \[N_k(n-k,m-k+1)=\frac{k+1}{n-k}{n-k\choose m+1}{n-k\choose m-k}.\]
\end{theorem}
It turns out that these estimates are sharp when $\pi\in\text{Av}_n(132,231,312)$. In fact, it is straightforward to check that the only permutation in $\text{Av}_n(132,231,312)$ with exactly $k$ descents is \[\theta_{n,k}=(k+1)k(k-1)\cdots 321(k+2)(k+3)\cdots n\] (this permutation is the sum of a decreasing permutation of length $k+1$ and an increasing permutation of length $n-k-1$). For example, $\theta_{7,2}=3214567$ is the only permutation of length $7$ that has $2$ descents and avoids the patterns $132,231,312$.
\begin{theorem}\label{Thm10}
With notation as above, \[|s^{-1}(\theta_{n,k})|=\frac{2k+2}{n+1}{2n-2k-1\choose n}.\] Furthermore, the number of permutations in $s^{-1}(\theta_{n,k})$ with exactly $m$ descents is \[N_k(n-k,m-k+1).\]
\end{theorem}
\begin{proof}
As in Section \ref{Sec6}, let $\operatorname{Comp}_a(b)$ denote the set of compositions of $b$ into $a$ parts. Corollary 3.1 in \cite{Defant} states that \[\sum_{(q_0,\ldots,q_k)\in\operatorname{Comp}_{k+1}(n-k)}C_{(q_0,\ldots,q_k)}=\frac{2k+2}{n+1}{2n-2k-1\choose n}.\] Lemma 4.1 in that same paper tells us that \[\sum_{(q_0,\ldots,q_k)\in\operatorname{Comp}_{k+1}(n-k)}\,\,\sum_{(j_0,\ldots,j_k)\in\operatorname{Comp}_{k+1}(m+1)}\,\,\prod_{t=0}^kN(q_t,j_t)=N_k(n-k,m-k+1).\] Invoking Theorems \ref{Thm5} and \ref{Thm7}, we see that is suffices to show that $\mathcal V(\theta_{n,k})=\operatorname{Comp}_{k+1}(n-k)$. We know that $\mathcal V(\theta_{n,k})\subseteq\operatorname{Comp}_{k+1}(n-k)$, so it remains to prove the reverse containment. This is actually quite simple to do if we apply Theorem \ref{Thm4} with the permutation $\theta_{n,k}$ in place of $\pi$. Preserve the notation from that theorem and the discussion immediately preceding it.
First, we have $d_i=i$ for $0\leq i\leq k$. It is straightforward to compute the canonical valid hook configuration $\mathcal H^*$ of $\theta_{n,k}$. Namely, $\mathcal H^*=(H_1^*,\ldots,H_k^*)$, where $H_i^*$ is the hook with southwest endpoint $(i,k-i+2)$ and northeast endpoint $(2k-i+2,2k-i+2)$. The valid composition induced by $\mathcal H^*$ is \[(q_0^*,\ldots,q_k^*)=(n-2k,1,\ldots,1).\] We have $e_i=k+1$ for all $i\in\{0,\ldots,k\}$. Finally, $\alpha_i=0$ for $i\in\{1,\ldots,k\}$, and $\alpha_{k+1}=k$.
Choose $(q_0,\ldots,q_k)\in\operatorname{Comp}_{k+1}(n-k)$. If $m\in\{1,\ldots,k\}$, then \[\sum_{j=m}^{e_m-1}q_j\geq\sum_{j=m}^{e_m-1}1=\sum_{j=m}^{e_m-1}q_j^*.\] If $m=0$, then \[\sum_{j=m}^{e_m-1}q_j=n-k=\sum_{j=m}^{e_m-1}q_j^*.\] Hence, $(q_0,\ldots,q_k)$ satisfies condition $(a)$ in Theorem \ref{Thm4}. If $m,p\in\{0,1,\ldots,k\}$ and $m\leq p\leq e_m-2=k-1$, then \[\sum_{j=m}^p q_j\geq p-m+1=d_{p+1}-d_m-\sum_{j=m+1}^{p+1}\alpha_j.\] This shows that $(q_0,\ldots,q_k)$ satisfies condition $(b)$ in Theorem \ref{Thm4}, so it follows from that theorem that $(q_0,\ldots,q_k)\in\mathcal V(\theta_{n,k})$.
\end{proof}
In the proof of Theorem \ref{Thm10}, we showed that $\mathcal V(\theta_{n,k})=\operatorname{Comp}_{k+1}(n-k)$. According to Theorem \ref{Thm7}, the number of permutations in $s^{-1}(\theta_{n,k})$ with $m$ peaks is \[\sum_{(q_0,\ldots,q_k)\in\operatorname{Comp}_{k+1}(n-k)}\sum_{(j_0,\ldots,j_k)\in\operatorname{Comp}_{k+1}(m+1)}\prod_{t=0}^k V(q_t,j_t).\] We leave it as an open problem to find a simple closed-form expression for these numbers.
The following theorem gives a new combinatorial interpretation for the Fine numbers $F_n$, which are defined by \[\sum_{n\geq 0}F_nx^n=\frac{1}{x}\frac{1-\sqrt{1-4x}}{3-\sqrt{1-4x}}.\] The sequence of Fine numbers, which is sequence A000957 in the Online Encyclopedia of Integer Sequences \cite{OEIS}, has many combinatorial connections with Catalan numbers. See the survey \cite{Deutsch} for more on this ubiquitous sequence. The following theorem also involves two reFinements of the Fine numbers. These are the numbers \[g_{n,m}=\sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor}N_k(n-k,m-k+1)\] and \[h_{n,m}=\frac{2^{n-2m-1}}{n+2}{n+2\choose m+1}{n-m-1\choose m}.\] These numbers, which have combinatorial interpretations in terms of Dyck paths, appear as sequences A100754 and A114593 in \cite{OEIS}. It is known that\footnote{These identities are stated without proof in the Online Encyclopedia of Integer Sequences, but they can be proven by standard (yet somewhat tedious) arguments involving generating functions.}
\begin{equation}\label{Eq8}
\sum_{m=0}^{n-1}g_{n,m}=\sum_{m=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor}h_{n,m}=F_{n+1}.
\end{equation}
\begin{theorem}\label{Thm11}
In the notation of the preceding paragraph, we have \[|s^{-1}(\operatorname{Av}_n(132,231,312))|=F_{n+1}.\] Moreover, the number of permutation in $s^{-1}(\operatorname{Av}_n(132,231,312))$ with exactly $m$ descents is $g_{n,m}$. The number of permutation in $s^{-1}(\operatorname{Av}_n(132,231,312))$ with exactly $m$ peaks is $h_{n,m}$.
\end{theorem}
\begin{proof}
Recall that $\operatorname{Av}_n(132,231,312)=\{\theta_{n,0},\theta_{n,1},\ldots,\theta_{n,n-1}\}$. It follows from Theorem \ref{Thm10} that $s^{-1}(\theta_{n,k})$ is empty if $k>\left\lfloor\frac{n-1}{2}\right\rfloor$. It now follows immediately from Theorem \ref{Thm5} that the number of permutations in $s^{-1}(\operatorname{Av}_n(132,231,312))$ with exactly $m$ descents is $g_{n,m}$. Along with \eqref{Eq8}, this implies that $|s^{-1}(\operatorname{Av}_n(132,231,312))|=F_{n+1}$.
Recall the generating function $G(x,y)$ from \eqref{Eq4}. According to the preceding paragraph and Theorem \ref{Thm7}, the number of permutations in $s^{-1}(\theta_{n,k})$ with $m$ peaks is \[\sum_{(q_0,\ldots,q_k)\in\operatorname{Comp}_{k+1}(n-k)}\,\,\sum_{(j_0,\ldots,j_k)\in\operatorname{Comp}_{k+1}(m+1)}\,\,\prod_{t=0}^kV(q_t,j_t).\] This is nothing more than the coefficient of $x^ny^m$ in $x^ky^k\,G(x,y)^{k+1}$. Consequently, the number of permutations in $s^{-1}(\operatorname{Av}_n(132,231,312))$ with $m$ peaks is \[[x^ny^m]\left(\sum_{k=0}^{n-1}x^ky^k\,G(x,y)^{k+1}\right)=[x^ny^m]\left(\sum_{k=0}^\infty x^ky^k\,G(x,y)^{k+1}\right)=[x^ny^m]\left(\frac{G(x,y)}{1-xy\,G(x,y)}\right).\]
Straightforward algebraic manipulations show that \[\frac{G(x,y)}{1-xy\,G(x,y)}=\frac{1-2x-\sqrt{1-4x+4x^2-4x^2y}}{xy(1+2x+\sqrt{1-4x+4x^2-4x^2y})}.\] This last expression is equal to the generating function $\displaystyle\sum_{n\geq 1}\sum_{m=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor}h(n,m)x^ny^m$.
\end{proof}
\section{$s^{-1}(\operatorname{Av}(312,321))$}\label{Sec:312,321}
Applying the algorithm of Claesson and \'Ulfarsson shows that $s^{-1}(\operatorname{Av}(312,321))$ is equal to the set of permutations avoiding certain mesh patterns. However, Chetak Hossain has drawn the current author's attention to the fact that this set is actually a permutation class. Specifically, $s^{-1}(\operatorname{Av}(312,321))=\operatorname{Av}(3412,3421)$. We leave the proof of this statement to the reader; it requires nothing more than the definition of the stack-sorting map and the definition of permutation pattern avoidance. Hossain has also pointed the current author to the paper \cite{Kremer}. In this paper, Kremer proves that $|\operatorname{Av}_n(3412,3421)|$ is the $(n-1)^\text{th}$ large Schr\"oder number. In other words, we have the following theorem.\footnote{Kremer did not mention the stack-sorting map in her theorem.}
\begin{theorem}[\!\!\cite{Kremer}]\label{Thm17}
We have \[\sum_{n\geq 1}|s^{-1}(\operatorname{Av}_n(312,321))|x^n=\sum_{n\geq 1}|\operatorname{Av}_n(3412,3421)|x^n=\frac{1-x-\sqrt{1-6x+x^2}}{2}.\]
\end{theorem}
\section{$s^{-1}(\operatorname{Av}(132,321))$}\label{Sec:132,321}
We have \[s^{-1}(\operatorname{Av}(132,321))=\operatorname{Av}(1342, 34251, 35241, 45231, \underline{31}42).\]
It appears as though the sequence enumerating these permutations has not been studied before, but we will see that its generating function is fairly simple. Let $C(x)=\sum_{n\geq 0}C_nx^n=\dfrac{1-\sqrt{1-4x}}{2x}$ be the generating function of the sequence of Catalan numbers. Recall the generating functions $F(x,y)$ and $G(x,y)$ from \eqref{Eq2} and \eqref{Eq4}. Let $\mathfrak a(n,m)$ denote the number of elements of $s^{-1}(\operatorname{Av}_n(132,321))$ with exactly $m$ descents, and let $\mathfrak b(n,m)$ denote the number of elements of $s^{-1}(\operatorname{Av}_n(132,321))$ with exactly $m$ peaks.
\begin{theorem}
In the notation of the preceding paragraph, we have \[\sum_{n\geq 1}|s^{-1}(\operatorname{Av}_n(132,321))|\,x^n=C(x)-1+x^3(C'(x))^2.\] Furthermore, \begin{equation}\label{Eq12}
\sum_{n\geq 1}\sum_{m\geq 0}\mathfrak a(n,m)x^ny^m=F(x,y)+x^3y\left(\frac{\partial}{\partial x}F(x,y)\right)^2,
\end{equation} and
\begin{equation}\label{Eq13}
\sum_{n\geq 1}\sum_{m\geq 0}\mathfrak b(n,m)x^ny^m=G(x,y)+x^3y\left(\frac{\partial}{\partial x}G(x,y)\right)^2.
\end{equation}
\end{theorem}
\begin{proof}
We will prove \eqref{Eq12}; the proof of \eqref{Eq13} is similar. In addition, the first statement of the theorem follows from \eqref{Eq12} and the fact that $F(x,1)=C(x)-1$.
For $h,i,t\geq 1$, let \[\delta_{h,i,t}=(h+1)(h+2)\cdots(h+i)12\cdots h(h+i+1)(h+i+2)\cdots(h+i+t)\in S_{h+i+t}.\] For example, $\delta_{1,3,2}=234156$. It is straightforward to check that \[\operatorname{Av}_n(132,321)=\{123\cdots n\}\cup\{\delta_{h,i,t}:h,i,t\geq 1,\,\,h+i+t=n\}.\] Moreover, the set of valid compositions of $\delta_{h,i,t}$ is \[\mathcal V(\delta_{h,i,t})=\{(i+j-\ell,h+\ell-1):1\leq \ell\leq t\}.\] Invoking Theorem \ref{Thm7}, we find that \[\mathfrak a(n,m)=N(n,m+1)+\sum_{\substack{h,i,t\geq 1\\ h+i+t=n}}\sum_{\ell=1}^t\sum_{j=1}^mN(i+t-\ell,j)N(h+\ell-1,m+1-j)\] \[=N(n,m+1)+\sum_{h=1}^{n-2}\sum_{i=1}^{n-h}\sum_{\ell=1}^{n-h-i}\sum_{j=1}^mN(n-h-\ell,j)N(h+\ell-1,m+1-j)\] \[=N(n,m+1)+\sum_{h=1}^{n-2}\sum_{\ell=1}^{n-h}\sum_{i=1}^{n-h-\ell}\sum_{j=1}^mN(n-h-\ell,j)N(h+\ell-1,m+1-j)\] \[=N(n,m+1)+\sum_{h=1}^{n-2}\sum_{\ell=1}^{n-h}\sum_{j=1}^m(n-h-\ell)N(n-h-\ell,j)N(h+\ell-1,m+1-j).\] The substitution $r=n-h-\ell$ gives \[\mathfrak a(n,m)=N(n,m+1)+\sum_{h=1}^{n-2}\sum_{r=1}^{n-h-1}\sum_{j=1}^mr\,N(r,j)N(n-r-1,m+1-j)\] \[=N(n,m+1)+\sum_{r=1}^{n-2}\sum_{j=1}^mr(n-r-1)\,N(r,j)N(n-r-1,m+1-j).\] It is now routine to verify that this last expression is the coefficient of $x^ny^m$ in \[F(x,y)+x^3y\left(\frac{\partial}{\partial x}F(x,y)\right)^2.\qedhere\]
\end{proof}
\section{$s^{-1}(\operatorname{Av}(132,312))$ and $s^{-1}(\operatorname{Av}(231,312))$}\label{Sec:Pair}
We have \[s^{-1}(\operatorname{Av}(132,312))=\operatorname{Av}(1342,3142,3412,34\underline{21})\] and \[s^{-1}(\operatorname{Av}(231,312))=\operatorname{Av}(2341,3412,34\underline{21},3\overline{5}241).\]
From these descriptions of these sets, there is no obvious reason to expect that $|s^{-1}(\operatorname{Av}_n(132,312))|$
$=|s^{-1}(\operatorname{Av}_n(231,312))|$. However, this is indeed the case; valid hook configurations make the proof quite painless.
\begin{theorem}\label{Thm12}
For all positive integers $n$, we have \[|s^{-1}(\operatorname{Av}_n(132,312))|=|s^{-1}(\operatorname{Av}_n(231,312))|.\] In fact, the number of permutations in $s^{-1}(\operatorname{Av}_n(132,312))$ with $m$ descents is the same as the number of permutations in $s^{-1}(\operatorname{Av}_n(231,312))$ with $m$ descents. Moreover, the number of permutations in $s^{-1}(\operatorname{Av}_n(132,312))$ with $m$ peaks is the same as the number of permutations in $s^{-1}(\operatorname{Av}_n(231,312))$ with $m$ peaks.
\end{theorem}
\begin{proof}
Invoking Theorem \ref{Thm5} and Theorem \ref{Thm7}, we see that it suffices to exhibit a bijection $\varphi:\operatorname{Av}_n(132,312)\to \operatorname{Av}_n(231,312)$ with the property that $\mathcal V(\pi)=\mathcal V(\varphi(\pi))$ for all $\pi\in\operatorname{Av}_n(132,312)$.
Suppose we are given a permutation $\pi\in\operatorname{Av}_n(132,312)$. Recall that a \emph{left-to-right maximum} of a permutation is an entry of the permutation that is larger than all entries to its left. Because $\pi$ avoids $312$, we can write \[\pi=b_1(1)b_1(2)\cdots b_1(c_1)\,\,b_2(1)b_2(2)\cdots b_2(c_2)\,\,\cdots\,\, b_\ell(1)b_\ell(2)\cdots b_\ell(c_\ell),\] where the entries $b_i(1)$ are the left-to-right maxima of $\pi$ and the strings $b_i(1)b_i(2)\cdots b_i(c_i)$ are decreasing. In other words, the strings $b_i(1)b_i(2)\cdots b_i(c_i)$ are the descending runs of $\pi$. It turns out that $\pi$ is uniquely determined by the numbers $c_1,\ldots,c_\ell$ and the assumption that $\pi\in\operatorname{Av}_n(132,312)$. Indeed, the entries of $\pi$ that are not left-to-right maxima must appear in decreasing order lest $\pi$ contain a $312$ pattern or a $132$ pattern. These entries must also all be less than the first entry of the permutation. Each permutation in $\operatorname{Av}_n(231,312)$ is also uniquely determined by the lengths of its descending runs. This is because the entries in each descending run of the permutation must be consecutive integers. Indeed, the permutations in $\operatorname{Av}(231,312)$ are called \emph{layered permutations}; as described at the beginning of Section \ref{Sec6}, a permutation is in $\operatorname{Av}(231,312)$ if and only if it can be written as a sum of decreasing permutations.
For example, the unique permutation in $\operatorname{Av}_{10}(132,312)$ whose descending runs have lengths $2,3,1,2,1,1$ is $5\,\,4\,\,6\,\,3\,\,2\,\,7\,\,8\,\,1\,\,9\,\,10$ (whose plot is shown on the left in Figure \ref{Fig6}). The unique permutation in $\operatorname{Av}_{10}(231,312)$ whose descending runs have lengths $2,3,1,2,1,1$ is $2\,\,1\,\,5\,\,4\,\,3\,\,6\,\,8\,\,7\,\,9\,\,10$ (whose plot is shown on the right in Figure \ref{Fig6}). We now define $\varphi:\operatorname{Av}_n(132,312)\to \operatorname{Av}_n(231,312)$ by declaring $\varphi(\pi)$ to be the unique permutation in $\operatorname{Av}_n(231,312)$ whose $i^\text{th}$ descending run has the same length as the $i^\text{th}$ descending run of $\pi$ for all $i$.
Figure \ref{Fig6} illustrates the map $\varphi$. In this figure, we have drawn a valid hook configuration on each of the plots. In general, the valid hook configurations of $\pi\in\operatorname{Av}_n(132,312)$ correspond bijectively to the valid hook configurations of $\varphi(\pi)$. Specifically, each hook of a valid hook configuration of $\varphi(\pi)$ is obtained from a hook in the corresponding valid hook configuration of $\pi$ by keeping fixed the horizontal coordinates of the endpoints of the hooks. In other words, we can obtain the plot of $\varphi(\pi)$ by vertically sliding some of the points in the plot of $\pi$; we keep the hooks attached to their endpoints throughout this sliding motion. Corresponding valid hook configurations induce the same valid compositions, so we have $\mathcal V(\pi)=\mathcal V(\varphi(\pi))$ for all $\pi\in\operatorname{Av}_n(132,312)$.
\end{proof}
\begin{figure}[t]
\begin{center}
\includegraphics[width=.6\linewidth]{FertilityPIC20}
\end{center}
\caption{The map $\varphi$ from the proof of Theorem \ref{Thm12} sends the permutation $5\,\,4\,\,6\,\,3\,\,2\,\,7\,\,8\,\,1\,\,9\,\,10$ to the permutation $2\,\,1\,\,5\,\,4\,\,3\,\,6\,\,8\,\,7\,\,9\,\,10$. The valid hook configuration drawn on the plot of $5\,\,4\,\,6\,\,3\,\,2\,\,7\,\,8\,\,1\,\,9\,\,10$ corresponds to the one drawn on the plot of $2\,\,1\,\,5\,\,4\,\,3\,\,6\,\,8\,\,7\,\,9\,\,10$. Both valid hook configurations induce the valid composition $(1,1,2,1,1)$.}\label{Fig6}
\end{figure}
We have seen that the sequences $(|s^{-1}(\operatorname{Av}_n(132,312))|)_{n\geq 1}$ and $(|s^{-1}(\operatorname{Av}_n(231,312))|)_{n\geq 1}$ are identical. Numerical evidence suggests that this sequence is, up to reindexing, the same as the sequence A071356 in the Online Encyclopedia of Integer Sequences \cite{OEIS}. The latter sequence is defined as the expansion of a relatively simple generating function, but it also has some combinatorial interpretations. In addition, it appears as though $(|s^{-1}(\operatorname{Av}_n(132,231))|)_{n\geq 1}$ is the same sequence. We state these observations formally in the following conjecture.
\begin{conjecture}\label{Conj2}
We have \[\sum_{n\geq 1}|s^{-1}(\operatorname{Av}_n(132,312))|x^n=\sum_{n\geq 1}|s^{-1}(\operatorname{Av}_n(132,231))|x^n=\frac{1-2x-\sqrt{1-4x-4x^2}}{4x}.\]
\end{conjecture}
\section{$s^{-1}(\operatorname{Av}(321))$}\label{Sec:321}
As discussed in the introduction, there are known formulas for $|s^{-1}(\operatorname{Av}_n(\tau))|$ whenever $\tau$ is a permutation pattern of length $3$ other than $321$. By contrast, the sequence $(|s^{-1}(\operatorname{Av}_n(321))|)_{n\geq 1}$ appears to be new.\footnote{We have added it as sequence A319027 in the Online Encyclopedia of Integer Sequences.} This sequence is of interest because $s^{-1}(\operatorname{Av}(321))$ is equal to the permutation class $\operatorname{Av}(34251,35241,45231)$. We will use valid hook configurations to establish nontrivial estimates for the growth rate of this sequence. Note that the trivial estimates for this growth rate\footnote{The nitpicky reader might beg for a proof of the existence of the limit defining this growth rate. We say a sequence of real numbers $(a_m)_{m=1}^\infty$ is \emph{supermultiplicative} if $a_ma_n\leq a_{m+n}$ for all positive integers $m,n$. The multiplicative version of Fekete's lemma \cite{Fekete} states that if $(a_m)_{m=1}^\infty$ is a supermultiplicative sequence, then $\displaystyle{\lim_{m\to\infty}\sqrt[m]{a_m}}$ exists. It is straightforward to show (in the notation of Section \ref{Sec6}) that $s(\sigma\oplus\mu)=s(\sigma)\oplus s(\mu)$ for any $\sigma\in S_m$ and $\mu\in S_n$. It follows that there is an injective map $s^{-1}(\operatorname{Av}_m(321))\times s^{-1}(\operatorname{Av}_n(321))\to s^{-1}(\operatorname{Av}_{m+n}(321))$ given by $(\sigma,\mu)\mapsto\sigma\oplus\mu$. Hence, $(|s^{-1}(\operatorname{Av}_n(321))|)_{n\geq 1}$ is supermultiplicative.} are given by \[4\leq\lim_{n\to\infty}|s^{-1}(\operatorname{Av}_n(321))|^{1/n}\leq 16.\] The lower bound follows from the fact that $|s^{-1}(123\cdots n)|=C_n$ has growth rate $4$. We know the upper bound because $|\operatorname{Av}_n(321)|=C_n$ has growth rate $4$ and because each permutation of length $n$ has fertility at most $4^n$.\footnote{From a permutation $\sigma\in S_n$, we obtain a word $w_\sigma$ of length $2n$ over the alphabet $\{A,B\}$ as follows. Send $\sigma$ through the stack as described in the introduction. Each time an entry is pushed into the stack, write an $A$. Each time an entry is popped out of the stack, write a $B$. For example, the permutation $3142$ from Figure \ref{Fig1} gives rise to the word $w_{3142}=AABBAABB$. It is straightforward to check that, for each $\pi\in S_n$, the map $\sigma\mapsto w_\sigma$ is injective on $s^{-1}(\pi)$. Therefore, $|s^{-1}(\pi)|\leq 2^{2n}=4^n$.}
\begin{theorem}\label{Thm13}
We have \[8.4199\leq\lim_{n\to\infty}|s^{-1}(\operatorname{Av}_n(321))|^{1/n}\leq 11.6569.\]
\end{theorem}
\begin{proof}
By reversing permutations, we see that $|\operatorname{Av}_{n,k}(321)|=|\operatorname{Av}_{n,n-1-k}(123)|$. The authors of \cite{Barnabei} have computed $|\operatorname{Av}_{n,k}(123)|$; one can easily use their results to see that the values of $|\operatorname{Av}_{n,k}(321)|$ are given by sequence A091156 in the Online Encyclopedia of Integer Sequences \cite{OEIS} (this interpretation of the sequence A091156 is not new to this article). A precise formula is given by
\begin{equation}\label{Eq14}
|\operatorname{Av}_{n,k}(321)|=\frac{1}{n+1}{n+1\choose k}\sum_{j=0}^{n+1-2k}{k+j-1\choose k-1}{n+1-k\choose n-2k-j}.
\end{equation}
To prove the desired upper bound, we combine Theorem \ref{Thm14} with \eqref{Eq14} to find that
\[|s^{-1}(\operatorname{Av}_n(321))|=\sum_{k=0}^{n-1}|s^{-1}(\operatorname{Av}_{n,k}(321))|\leq\sum_{k=0}^{n-1}\frac{2k+2}{n+1}{2n-2k-1\choose n}|\operatorname{Av}_{n,k}(321)|\] \[=\sum_{k=0}^{n-1}\frac{2k+2}{n+1}{2n-2k-1\choose n}\frac{1}{n+1}{n+1\choose k}\sum_{j=0}^{n+1-2k}{k+j-1\choose k-1}{n+1-k\choose n-2k-j}.\] Up to a subexponential factor, this upper bound is \[\sum_{k=0}^{n-1}{2n-2k\choose n}{n\choose k}\sum_{j=0}^{n-2k}{k+j\choose k}{n-k\choose n-2k-j}.\] The sum over $j$ in the last expression is maximized when $j\sim n/2-k$, so our upper bound is (again, up to a subexponential factor) \[\sum_{k=0}^{n-1}{2n-2k\choose n}{n\choose k}\sum_{j=0}^{n-2k}{n/2\choose k}{n-k\choose n/2-k}=\sum_{k=0}^{n-1}(n-2k+1){2n-2k\choose n}{n\choose k}{n/2\choose k}{n-k\choose n/2-k}.\] Let $K(n)$ denote the value of $k$ for which the term in this last summation is maximized, and put $c(n)=K(n)/n$. Note that $c(n)\in(0,1/2)$. A straightforward application of Stirling's formula shows that, up to a subexponential factor, this last upper bound is at most \[\frac{(2-2c(n))^{2-2c(n)}}{(1-2c(n))^{1-2c(n)}}\frac{1}{c(n)^{c(n)}(1-c(n))^{1-c(n)}}\frac{(1/2)^{1/2}}{c(n)^{c(n)}(1/2-c(n))^{1/2-c(n)}}\frac{(1-c(n))^{1-c(n)}}{(1/2-c(n))^{1/2-c(n)}(1/2)^{1/2}}\] \[=f(c(n)),\] where \[f(x)=\frac{(2-2x)^{2-2x}}{(1-2x)^{1-2x}x^{2x}(1/2-x)^{1-2x}}.\] One can easily verify that $f(x)\leq 11.6569$ whenever $x\in(0,1/2)$. This proves the desired upper bound.
The \emph{reverse complement} of a permutation $\pi_1\cdots\pi_n\in S_n$ is the permutation whose $i^\text{th}$ entry is $n+1-\pi_{n+1-i}$. The proof of the desired lower bound requires two crucial observations. The first is that the set $\operatorname{Av}_{n,k}(321)$ is closed under taking reverse complements. If $\pi\in S_n$ has $\ell$ left-to-right maxima, then the reverse complement of $\pi$ has $n-\ell$ left-to-right maxima. It follows that at least half of the permutations in $\operatorname{Av}_{n,k}(321)$ have at least $n/2$ left-to-right maxima.
The second observation is that if $\pi\in \operatorname{Av}_{n,k}(321)$ has last entry $\pi_n=n$, then $\pi$ has a valid hook configuration (i.e., $\pi$ is sorted). Indeed, let $d_1<\cdots<d_k$ denote the descents of $\pi$. For each $i\in\{1,\ldots,k\}$, let $\pi_{b_i}$ be the leftmost left-to-right maximum of $\pi$ that lies to the right of $\pi_{d_i}$. The condition $\pi_n=n$ guarantees that this left-to-right maximum exists for all $i$. The canonical valid hook configuration $(H_1^*,\ldots,H_k^*)$ of $\pi$ (described in Section \ref{Sec:VHCs}) is formed by declaring that $H_i^*$ has southwest endpoint $(d_i,\pi_{d_i})$ and northeast endpoint $(b_i,\pi_{b_i})$ for all $i$.
There are exactly $|\operatorname{Av}_{n-1,k}(321)|$ permutations in $\operatorname{Av}_{n,k}(321)$ with last entry $n$ (we obtain a bijection by adding the entry $n$ to the end of a permutation in $\operatorname{Av}_{n-1,k}(321)$). According to the discussion above, there are at least $\frac 12|\operatorname{Av}_{n-1,k}(321)|$ permutations in $\operatorname{Av}_{n,k}(321)$ with at least $n/2$ left-to-right maxima. Choose one such permutation $\pi$, and let $\ell$ be the number of left-to-right maxima in $\pi$. The canonical valid hook configuration of $\pi$ (described in the previous paragraph) induces a valid composition $(q_0^*,\ldots,q_k^*)\in\mathcal V(\pi)$. In the coloring of the plot of $\pi$ induced by the canonical valid hook configuration, there are exactly $\ell-k$ points colored blue (sky-colored). Indeed, these points are precisely the left-to-right maxima of $\pi$ that are not northeast endpoints of hooks. This tells us that $q_0^*=\ell-k$. According to Theorem \ref{Thm5},
\begin{equation}\label{Eq17}
|s^{-1}(\pi)|\geq C_{(q_0^*,\ldots,q_k^*)}=C_{\ell-k}C_{q_1^*}\cdots C_{q_k^*}.
\end{equation}
For convenience, we define $C_x=\dfrac{\Gamma(2x+1)}{\Gamma(x+2)\Gamma(x+1)}$, where $\Gamma$ denotes the Gamma function. When $x$ is a positive integer, $C_x$ is simply the $x^\text{th}$ Catalan number. One can show that
\begin{equation}\label{Eq15}
C_{x+\varepsilon}C_{y-\varepsilon}<C_xC_y\quad\text{whenever}\quad 0<x<y\quad\text{and}\quad 0<\varepsilon\leq\frac{y-x}{2}.
\end{equation} In other words, a product of (generalized) Catalan numbers decreases when we make the indices closer while preserving the sum of the indices. Let us assume that $n$ is sufficiently large, that $5<k<0.4n$, and that $\pi$ is chosen as in the previous paragraph. By the properties of valid compositions, we know that
\begin{equation}\label{Eq16}
\ell-k+q_1^*+\cdots+q_k^*=n-k.
\end{equation} It follows from \eqref{Eq15} and \eqref{Eq16} that
\begin{equation}\label{Eq18}
C_{\ell-k}C_{q_1^*}\cdots C_{q_k^*}\geq C_{\ell-k}C_{(n-\ell)/k}^k.
\end{equation} The assumption $5<k<0.4n$ and the fact that $\ell\geq n/2$ guarantee that \[\frac{n-\ell}{k}\leq \frac{n}{2k}<\frac n2-k\leq\ell-k.\] By \eqref{Eq15}, we have
\begin{equation}\label{Eq19}
C_{\ell-k}C_{(n-\ell)/k}^k\geq C_{n/2-k}C_{n/(2k)}^k.
\end{equation}
When we combine \eqref{Eq15}, \eqref{Eq18}, and \eqref{Eq19} with the discussion above, we find that there are at least $\frac 12|\operatorname{Av}_{n-1,k}(321)|$ permutations in $\operatorname{Av}_{n,k}(321)$ that each have at least $C_{n/2-k}C_{n/(2k)}^k$ preimages under $s$ (for $5<k<0.4n$). We now use \eqref{Eq14} to see that \[|s^{-1}(\operatorname{Av}_n(321))|\geq \frac 12|\operatorname{Av}_{n-1,k}(321)|C_{n/2-k}C_{n/(2k)}^k\] \[\geq\frac{1}{2n}{n\choose k}\sum_{j=0}^{n-2k}{k+j-1\choose k-1}{n-k\choose n-2k-j-1}C_{n/2-k}C_{n/(2k)}^k\] \[\geq\frac{1}{2n}{n\choose k}{k+(\left\lfloor n/2\right\rfloor-k)-1\choose k-1}{n-k\choose n-2k-(\left\lfloor n/2\right\rfloor-k)-1}C_{n/2-k}C_{n/(2k)}^k\] \[=\frac{1}{2n}{n\choose k}{\left\lfloor n/2\right\rfloor-1\choose k-1}{n-k\choose \left\lceil n/2\right\rceil-k-1}C_{n/2-k}C_{n/(2k)}^k.\] This holds whenever $5<k<0.4n$. In particular, we can put $k=\left\lfloor 0.06582n\right\rfloor$ (this value is chosen to maximize the lower bound). With this choice of $k$, we can use Stirling's formula to see that our lower bound is at least $8.4199^n$ for sufficiently large $n$.
\end{proof}
\section{Concluding Remarks and Further Directions}
Let us collect some open problems and conjectures arising from and related to the topics studied in this article.
Recall that a sequence $a_1,\ldots,a_m$ is called \emph{unimodal} if there exists $j\in\{1,\ldots,m\}$ such that $a_1\leq\cdots\leq a_{j-1}\leq a_j\geq a_{j+1}\geq\cdots\geq a_m$ and is called \emph{log-concave} if $a_j^2\geq a_{j-1}a_{j+1}$ for all $j\in\{2,\ldots,m-1\}$ \cite{Branden2}. This sequence is called \emph{real-rooted} if all of the complex roots of the polynomial $\sum_{k=1}^ma_kx^k$ are real. It is well-known that a real-rooted sequence of nonnegative numbers is log-concave and that a log-concave sequence of nonnegative numbers is unimodal.
The notions of unimodality, log-concavity, and real-rootedness are prominent in the study of the stack-sorting map. For example, let $f_k(\pi)$ denote the number of elements of $s^{-1}(\pi)$ with $k$ descents. B\'ona proved \cite{BonaSymmetry} that the sequence $f_0(\pi),\ldots,f_{n-1}(\pi)$ is symmetric and unimodal for each $\pi\in S_n$. We conjecture the following much stronger result, which we have verified for all permutations of length at most $8$.
\begin{conjecture}\label{Conj3}
For each permutation $\pi\in S_n$, the sequence $f_0(\pi),\ldots,f_{n-1}(\pi)$ is real-rooted.
\end{conjecture}
Of course, even if it is too difficult to prove that $f_0(\pi),\ldots,f_{n-1}(\pi)$ is always real-rooted, it would be very interesting to prove the weaker statement that this sequence is always log-concave. One could also attempt to find large classes of permutations for which Conjecture \ref{Conj3} holds.
A consequence of B\'ona's result is that
\begin{equation}\label{Eq20}
W_t(n,0),\ldots,W_t(n,n-1)
\end{equation} is symmetric and unimodal for all $t,n\geq 1$, where $W_t(n,k)$ is the number of $t$-stack-sortable permutations in $S_n$ with $k$ descents. Knowing that the sequence in \eqref{Eq20} is real-rooted when $t=1$ and when $t=n$, B\'ona \cite{BonaSymmetry} conjectured that the sequence is real-rooted in general. Br\"and\'en \cite{Branden3} later proved this conjecture in the cases $t=2$ and $t=n-2$. This leads us to the following much more general problem.
\begin{question}\label{Quest1}
Given a set $\mathcal U$ of permutations, let $f_k(\mathcal U\cap S_n)$ denote the number of permutations in $s^{-1}(\mathcal U\cap S_n)$ with exactly $k$ descents. Can we find interesting examples of sets $\mathcal U$ (such as permutation classes) with the property that $f_0(\mathcal U\cap S_n),\ldots,f_{n-1}(\mathcal U\cap S_n)$ is a real-rooted sequence for every $n\geq 1$? Is this sequence always real-rooted?
\end{question}
Recall from Section \ref{Sec:Back} that $s^{-1}(\operatorname{Av}_n(123))$ is empty when $n\geq 4$. In general, $s^{-1}(\operatorname{Av}_n(123\cdots m))$ is empty if $n\geq 2^{m-1}$. This is certainly true for $m\leq 3$. To see that this is true for $m\geq 4$, suppose $n\geq 2^{m-1}$ and $\pi\in S_n$. Write $\pi=LnR$ so that $s(\pi)=s(L)s(R)n$. One of $L$ and $R$ has length at least $2^{m-2}$, so it follows by induction on $m$ that either $s(L)$ or $s(R)$ contains an increasing subsequence of length $m-1$. Therefore, $s(\pi)$ contains the pattern $123\cdots m$.
Despite the uninteresting behavior of the sequence $(|s^{-1}(\operatorname{Av}_n(123\cdots m))|)_{n\geq 1}$ for large values of $n$, it could still be interesting to study the initial terms in this sequence. For example, the nonzero terms of the sequence are $1,2,6,10,13,10,3$ when $m=4$.
\begin{conjecture}\label{Conj4}
For each integer $m\geq 2$, the sequence $(|s^{-1}(\operatorname{Av}_n(123\cdots m))|)_{n=1}^{2^{m-1}-1}$ is unimodal.
\end{conjecture}
We next consider some natural questions that we have not attempted to answer.
\begin{question}\label{Quest6}
Can we obtain interesting results by enumerating sets of the form
\linebreak $s^{-1}(\operatorname{Av}(\tau^{(1)},\ldots,\tau^{(r)}))$ when the patterns $\tau^{(1)},\ldots,\tau^{(r)}$ are not all of length $3$?
\end{question}
\begin{question}\label{Quest2}
Let $S_{n,k}$ denote the set of permutations in $S_n$ with exactly $k$ descents. Can we find formulas for $|s^{-1}(S_{n,1})|$, $|s^{-1}(S_{n,2})|$, or $|s^{-1}(S_{n,3})|$?
\end{question}
Throughout this article, we often enumerated permutations in sets of the form \linebreak
$s^{-1}(\operatorname{Av}_n(\tau^{(1)},\ldots,\tau^{(r)}))$ according to their number of descents or number of peaks. This is because the techniques in \cite{Defant} allow us to use valid hook configurations to count preimages of permutations according to these statistics. It is likely that there are other permutation statistics that can be treated similarly using valid hook configurations. It would be interesting to obtain results for these other statistics analogous to those derived above for descents and peaks.
The article \cite{Defant} provides a general method for computing the number of decreasing plane trees of various types that have a specified permutation as their postorder reading (see the article for the relevant definitions). In the special case in which the trees are decreasing binary plane trees, this is equivalent to computing fertilities of permutations. This suggests that one could obtain enumerative results analogous to those from this paper by replacing decreasing binary plane trees with other types of trees. This provides a very general new collection of enumerative problems. Namely, we want to count the decreasing plane trees of a certain type whose postorders lie in some permutation class. Two very specific examples of this type of problem are the following. Preserve the notation from the article \cite{Defant}.
\begin{question}\label{Quest3}
How many decreasing $\mathbb N$-trees have postorders that lie in the set $\operatorname{Av}_n(132,231,312)$? How many unary-binary trees have postorders that lie in the set $\operatorname{Av}_n(132,231,312)$?
\end{question}
We now collect the open problems and conjectures that arose throughout Sections 4--10. First, recall the following conjecture from Section \ref{Sec6}.
\begin{reptheorem}{Conj1}
In the notation of Theorem \ref{Thm8}, we have \[\sum_{q\,\in\,\operatorname{Comp}_{k+1}(n-k)}C_qD_q=\frac{1}{n+1}{n-k-1\choose k}{2n-2k\choose n}\] for all nonnegative integers $n$ and $k$.
\end{reptheorem}
In Section \ref{Sec7}, we defined \[\theta_{n,k}=(k+1)k(k-1)\cdots 321(k+2)(k+3)\cdots n\] and mentioned that the number of permutations in $s^{-1}(\theta_{n,k})$ with $m$ peaks is
\begin{equation}\label{Eq21}
\sum_{(q_0,\ldots,q_k)\in\operatorname{Comp}_{k+1}(n-k)}\sum_{(j_0,\ldots,j_k)\in\operatorname{Comp}_{k+1}(m+1)}\prod_{t=0}^k V(q_t,j_t).
\end{equation}
\begin{question}\label{Quest4}
Can we find a simple closed form for the expression in \eqref{Eq21}?
\end{question}
We stated the following intriguing conjecture in Section \ref{Sec:Pair}.
\begin{reptheorem}{Conj2}
We have \[\sum_{n\geq 1}|s^{-1}(\operatorname{Av}_n(132,312))|x^n=\sum_{n\geq 1}|s^{-1}(\operatorname{Av}_n(132,231))|x^n=\frac{1-2x-\sqrt{1-4x-4x^2}}{4x}.\]
\end{reptheorem}
Of course, our results in Section \ref{Sec:321} are far from perfect.
\begin{question}\label{Quest5}
Can we enumerate the permutations in $s^{-1}(\operatorname{Av}(321))$ exactly? Can we at least improve the estimates from Theorem \ref{Thm13}?
\end{question}
We have focused primarily on preimage sets of the form $s^{-1}(\operatorname{Av}(\tau^{(1)},\ldots,\tau^{(r)}))$ when
$\emptyset\neq\{\tau^{(1)},\ldots,\tau^{(r)}\}\subseteq\{132,231,312,321\}$. The astute reader may have realized that the only preimage set of this form that we have not mentioned is $s^{-1}(\operatorname{Av}(231,321))$. This set appears to be enumerated by the OEIS sequence A165543 \cite{OEIS}. More precisely, we have the following conjecture.
\begin{conjecture}\label{Conj5}
We have \[\sum_{n\geq 0}|s^{-1}(\operatorname{Av}_n(231,321))|x^n=\frac{1}{1-xC(xC(x))},\] where $C(x)=\dfrac{1-\sqrt{1-4x}}{2x}$ is the generating function of the sequence of Catalan numbers.
\end{conjecture}
\section{Acknowledgments}
The author thanks Amanda Burcroff for a useful conversation about skyhooks. He also thanks Michael Engen for valuable conversations and thanks Steve Butler for providing data that was used to formulate several theorems and conjectures. He thanks Chetak Hossain for the contributions mentioned in Section \ref{Sec:312,321}. The author was supported by a Fannie and John Hertz Foundation Fellowship and an NSF Graduate Research Fellowship.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,084 |
31-й выпуск Париж — Рубе — шоссейной однодневной велогонки по дорогам Франции. Гонка прошла 20 апреля 1930 года.
Участники
Стартовало 86 велогонщиков, финишировало - 35.
Маршрут
Ход гонки
Результаты
Гонка изначально выиграна французом Жаном Марешалем, показавшим время 8 часов 10 минут 50 секунд. Сразу после финиша ставший вторым Жюльен Верваке и его спортивный директор Людовик Фёйет подали протест и потребовали дисквалифицировать Марешаля, обвинив его в том, что он спровоцировал падение Верваке. Комиссары велогонки, выслушав объяснения обеих сторон, приняли решение отдать победу Верваке, а второе место Марешалю.
Примечания
Ссылки
Официальный сайт
Париж — Рубе 1930 на FirstCycling
Париж — Рубе 1930 на CycleBase
Париж — Рубе 1930 на Bikeraceinfo
Страницы истории велоспорта: Париж-Рубэ-1930 на Velolive | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,584 |
Cloud computing is a model for enabling ubiquitous, convenient, on-demand network access to shared pool of configurable computing resources. Now, that might include network, servers, storage, applications and even services. The goal here is to be able to rapidly provision and release these instances, these virtual machines, for e.g. as needed, without requiring any kind of provision assistance from a service provider. This cloud consumer defines which computing resources that they want to be use for their instances. This is an abstract overview of what cloud computing is and how they provide __ as a Service.
which is completely defined into above definition.
If we compare traditional workload with cloud we are looking at needing to be able to scale-up. As in database server you have back in your data center. As you want to satisfy additional demand as your business is growing, what do you need to do? You need to make that database server, or that data center virtualized database server, as in it becomes bigger. You need to add RAM, storage, CPU capacity, etc. If you see, cloud workloads are fundamentally different. The goal there is the idea of being able to scale-out. That is nothing but elastic nature. Being able to add additional instances of a web server. for e.g. let's say that you are designing a web environment that is serving up content for some major sporting events. So, as 11 months out of a year, people would care less about that sporting event. They'll just occasionally connect, so you might need only one or two web servers. But during the event itself, we are all looking to gain access to that information on-line, and so the number of web servers that you need is going to be huge, so we need to be able to scale out by adding those additional virtual instances to meet that demand.
In terms of deployment, we actually have a few different models, that we can work from.
Private cloud — This is a cloud used by single organization. In fact, for many, they have the hardware residing in their own data center, but the applications are cloud workload applications, and they look to be able to scale out their resources.
Public cloud — This is a cloud that is available to the general public, and is run by a cloud provider.
Hybrid cloud — Perhaps provide the most interesting deployment model. In this case we are using the private cloud, for the core of our resources, the standard amount of resources, that, let's say we need for that 11 months(as above example) that the sporting event is not happening. And that in that 12 months during the sporting event, we are able to, well, some call this a cloudburst, we expand into the public cloud. same application, same type of resources.
IaaS is a term that is used to talk about providing infrastructure services. Things like those virtual machines, instances, servers, maybe some network resources, additional storage resources. Those things we are building out in our data center. That's infrastructure.
PaaS, application development, being able to streamline and speed up the delivery of those applications, those development platform is provided by PaaS. it helps us to automate the deployment of those applications and is also providing platforms that are more cloud aware for those cloud type workloads.
SaaS, services allow users to access specific application over the web i.e. internet. The cloud service provider own the application and customers use the service on subscription basis.
DaaS(Desktop as a Service), allows a customers to host their entire computing environment through cloud. This we called as "everything as a service".
Again, many giant companies and start-up's started providing other service through the cloud as, SECaas(Security as a Service), DBaaS(Database as a Service), MaaS(Monitoring as a Service), CaaS(Communication as a Service). Note: CaaS even refers to Container as a Service which is emerging in the cloud era. I'll write about it in upcoming blogs. XaaS(Anything as a Service), etc. You can choose your cloud service provider based on how you are going to use, your requirements and optimal strategy which fulfills the needs.
So, as the term The Cloud finally emerging into the masses and evolving of it benefits not just big giant but even an early-stage startup or students, researchers, blogger, etc. The next decade will be promising to provide even better service, one click cloud solutions, mature the current problems and security flaws, and unable the consumers to fully understand the benefits. From Enterprise to SMB to end users, Cloud Services will inevitably bring immense benefits and cost savings. All that is now required is for consumers to know what all those unnecessarily complicated acronyms mean! | {
"redpajama_set_name": "RedPajamaC4"
} | 3,055 |
//
// URLParse.m
// QC Utilities
//
// Created by Randall Maas on 5/19/14.
//
#import "URLParse.h"
@implementation URLParse
/* We need to declare the input / output properties as dynamic as Quartz Composer will handle their implementation */
@dynamic inputURL, outputStructure, outputIsFileURL, outputStandardizedURL, outputError;
/// Holds the attributes for this plugin
static NSDictionary* portAttributes;
+ (void) initialize
{
RegisterExceptionHandler();
portAttributes =
@{
@"inputURL":
@{
QCPortAttributeNameKey : @"File path or URL",
QCPortAttributeDefaultValueKey: @"",
QCPortAttributeTypeKey : QCPortTypeString
},
@"outputStructure":
@{
QCPortAttributeNameKey: @"output",
QCPortAttributeTypeKey: QCPortTypeStructure
},
@"outputIsFileURL":
@{
QCPortAttributeNameKey: @"is file",
QCPortAttributeTypeKey: QCPortTypeBoolean
},
@"outputStandardizedURL":
@{
QCPortAttributeNameKey: @"standardized URL",
QCPortAttributeTypeKey: QCPortTypeString
},
@"outputError":
@{
QCPortAttributeNameKey: @"error",
QCPortAttributeTypeKey: QCPortTypeStructure
},
};
}
+ (NSDictionary*) attributes
{
/* Return the attributes of this plug-in */
return @{
QCPlugInAttributeNameKey : @"URL Parser",
QCPlugInAttributeCopyrightKey : @"Randall Maas (c) 2014",
QCPlugInAttributeCategoriesKey : @[@"Utility/Structure", @"Utility/String", @"Utility/Network"],
QCPlugInAttributeDescriptionKey: @"Parses a URL string.\n\n"
};
}
+ (NSDictionary*) attributesForPropertyPortWithKey:(NSString*)key
{
/* Return the attributes for the plug-in property ports */
return portAttributes[key];
}
+ (QCPlugInExecutionMode) executionMode
{
/* This plug-in is a processor (it just processes and may change with time) */
return kQCPlugInExecutionModeProcessor;
//
}
+ (QCPlugInTimeMode) timeMode
{
// Either idle or time base. I'm going with timeBase
return kQCPlugInTimeModeNone;
}
/** @brief This method is called by Quartz Composer whenever the plug-in needs to recompute its result
@param context
@param time
@param arguments
*/
- (BOOL) execute:(id<QCPlugInContext>)context
atTime:(NSTimeInterval)time
withArguments:(NSDictionary*)arguments
{
// Check for any changes
if (![self didValueForInputKeyChange:@"inputURL"] && time )
return YES;
// create a URL
NSURL* url = [NSURL URLWithString:self.inputURL];
// If it is just a file name, we have to try a backup method
if (!url)
url = [NSURL fileURLWithPath:self.inputURL];
// Update our results
NSURL* tmp = [url standardizedURL];
self . outputStandardizedURL = tmp?[tmp absoluteString] :@"";
// If it is a file do something different than if it is a full item)
if ([url isFileURL])
{
self . outputIsFileURL = true;
self . outputStructure =
@{
@"absolute" : _n([url absoluteString]),
// The relative portion of a URL. If baseURL is nil this is the same as absolute
@"relative" : _n([url relativeString]),
// The same as path if baseURL is nil
@"relativePath" : _n([url relativePath]),
@"base" : _n([url baseURL]),
@"absolute" : _n([url absoluteString]),
@"extension" : _n([url pathExtension])
};
// Check for error's accessingit
NSError* e= nil;
[url checkResourceIsReachableAndReturnError: &e];
self . outputError= NSError2Struct(e);
}
else
{
self . outputError = @[];
self . outputStructure =
@{
@"absolute" : _n([url absoluteString]),
// The relative portion of a URL. If baseURL is nil this is the same as absolute
@"relative" : _n([url relativeString]),
// The same as path if baseURL is nil
@"relativePath" : _n([url relativePath]),
@"base" : _n([[url baseURL] absoluteString]),
@"extension" : _n([url pathExtension]),
@"scheme" : _n([url scheme]),
@"resourceSpecifier": _n([url resourceSpecifier]),
/* If the URL conforms to rfc 1808 (the most common form of URL), the following accessors will return the
various components; otherwise they return nil. The litmus test for conformance is as recommended in
RFC 1808 - whether the first two characters of resourceSpecifier is @"//". In all cases, they return the
component's value after resolving the receiver against its base URL.
*/
@"host" : _n([url host]),
@"port" : _n([url port]),
@"user" : _n([url user]),
@"password" : _n([url password]),
@"path" : _n([url path]),
@"fragment" : _n([url fragment]),
@"parameters" : _n([url parameterString]),
@"query" : _n([url query]),
};
}
return YES;
}
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,832 |
\section{Introduction}
Recent years have seen the emergence of unexpected mathematical structures in scattering amplitudes,
especially so in $\mathcal{N}=4$ supersymmetric Yang Mills (sYM).
Planar loop integrands in this theory are in principle known to all loop orders \cite{ArkaniHamed:2010kv}, with the on-shell data entering the former being described by a positive Grassmannian \cite{ArkaniHamed:2009dn}. Furthermore the integrand has a dual geometric description \cite{Arkani-Hamed:2013jha}.
There is considerable evidence that cluster algebras play an important role for the amplitudes, both at the level of the loop integrand \cite{Arkani-Hamed:2016byb}, as well as for the functions obtained after integration \cite{Golden:2013xva}.
For example, planar six- and seven-gluon scattering amplitudes appear to be governed by the finite $A_3$ and $E_6$ cluster algebras, respectively.
This suggests that their function space is a certain set of generalized polylogarithms, which is the starting point for the bootstrap program \cite{Dixon:2011pw,Drummond:2014ffa}.
Further constraints
come from the absence of discontinuities in overlapping channels, the (extended) Steinmann relations \cite{Caron-Huot:2016owq,Caron-Huot:2018dsv,Caron-Huot:2019bsq}, which are closely related to cluster adjacency properties \cite{Drummond:2017ssj,Drummond:2018dfd}.
These findings have been instrumental for bootstrapping amplitudes to very high loop orders, see e.g. \cite{Dixon:2016nkn,Drummond:2018caf,Caron-Huot:2019vjl,Dixon:2020cnr}, and the review \cite{Caron-Huot:2020bkp}.
How general is the appearance of cluster algebras in quantum field theory?
On the one hand, all known cases are related to planarity, and concern finite parts of amplitudes in $\mathcal{N}=4$ sYM, which have additional symmetries \cite{Drummond:2008vq,Berkovits:2008ic,Drummond:2009fd}.
On the other hand, it motivates us that
several
structures initially found in $\mathcal{N}=4$ sYM, such as insights into the transcendental structure of Feynman integrals \cite{ArkaniHamed:2010gh}, their evaluation \cite{Henn:2013pwa} and dealing with their analytic properties and identities between them \cite{Goncharov:2010jf}, are by now common tools in generic quantum field theories, as reviewed in \cite{Duhr:2019wtr,Henn:2020omi}.
In this Letter we initiate a study of cluster algebras in Feynman integrals in $D=4- 2 \epsilon$ dimensions, without relying on extra symmetries.
\section{Cluster algebras and associated function spaces}
Cluster algebras \cite{1021.16017,1054.17024,CAIII,CAIV} (see \cite{2008arXiv0807.1960K,Lampe13,Fomin2016,Fomin2017} for introductory articles) are commutative algebras equipped with a distinguished set of generators $a_i$, the \emph{cluster $\mathcal{A}$-coordinates}, grouped into overlapping subsets $\boldsymbol{a}\equiv\{a_1,\ldots,a_d\}$ of \emph{rank} $d$, the \emph{clusters}. Starting from an initial cluster, they may be constructed recursively by a \emph{mutation} operation on the cluster coordinates. They may also be generalized to contain \emph{frozen coordinates} or \emph{coefficients} $\{a_{d+1},\ldots,a_{d+m}\}$, whose main difference from the cluster coordinates is that they do not mutate.
How the cluster coordinates transform under a mutation is encoded in an integer $(d+m)\times d$ \emph{exchange matrix} $B$, whose components we will denote as $b_{ij}$. Restricting to the components with $i,j\le d$ corresponds to the \emph{principal part} of $B$, which must be skew-symmetrizable.
Then, mutating a cluster $(\boldsymbol{a},B)$ along the $k$-th variable,
with $1 \le k\le d$, we obtain the new cluster $(\boldsymbol{a}',B')$, whose exchange matrix $B'$ is related to the previous one by
\begin{align}\label{eq:Bmutation}
b'_{ij} =
\begin{cases}
-b_{ij} \; &\text{for}\,i=k\,\text{or}\,j=k\\
b_{ij} + \left[-b_{ik}\right]_+b_{kj} + b_{ik}\left[b_{kj}\right]_+\; &\text{otherwise}
\end{cases}\,,
\end{align}
where $\left[x\right]_+ = \max\left(0,x\right)$. The cluster coordinates $a_{i}$ are unchanged for $i\neq k$ and $a_{k}$ is mutated according to
\begin{equation}
\label{equ:clusterMutation}
a'_k = a_k^{-1}\left(\prod_{i=1}^{d+m}a_{i}^{\left[b_{ik}\right]_+} + \prod_{i=1}^{d+m}a_{i}^{\left[-b_{ik}\right]_+}\right) \,.
\end{equation}
The $C_2$ cluster algebra will play a prominent role in this Letter. For the coefficient-free case $m=0$ it is defined by
\bar\eta
B=\left(
\begin{array}{cc}
0&1\\
-2&0
\end{array}\right)\,.
\end{equation}
Then, it is easy to show that under a mutation
\eqref{eq:Bmutation} $B$ only changes by a sign, and thus the corresponding cluster transformation~\eqref{equ:clusterMutation} simplifies to
\bar\eta\label{simplifiedmutationrule}
a_{m+1}a_{m-1}=
\begin{cases}
1+a_m&\text{if $m$ is odd}\,,\\
1+a_m^2&\text{if $m$ is even}\,,\\
\end{cases}
\end{equation}
where $\{a_1,a_2\}$ are the $\mathcal{A}$-coordinates of the initial cluster, $a_3=a'_1$ and so on. The coordinates obtained in this manner are shown in the \emph{exchange graph} of Fig.~\ref{B2C2ExchangeGraph}, where clusters $\{a_i,a_{i+1}\}$ are represented by vertices, and the mutations relating them by edges. The circle topology indicates that mutating six times takes us back to where we started, $a_{i+6}=a_i$.
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=\textwidth]{c2clusterv2.pdf}
\end{subfigure}
\caption{The exchange graph of the $C_2$ cluster algebra, with cluster coordinates ordered as $a_i,a_{i+1}$.
}
\label{B2C2ExchangeGraph}
\end{figure}
Below we will also find it useful to consider an alternative set of coordinates,
called {\it cluster $\mathcal{X}$-coordinates} \cite{FG03b}, which are defined as
\begin{equation}
\label{eq:xtoa}
x_{i} \equiv \prod_{l=1}^{d+m}a_{l}^{b_{li}}\,,\quad i=1,\ldots d\,.
\end{equation}
The $\mathcal{A}$-coordinate mutation \eqref{equ:clusterMutation} and eq.~\eqref{eq:xtoa} imply a relation between the associated $\mathcal{X}$-coordinates, which may be written directly as
\begin{equation}\label{eq:xMutation}
x_i' = \begin{cases}
1/x_i & k = i\,,\\
x_i\bigl(1+x_k^{-{\rm sgn}(b_{ki})}\bigr)^{-b_{ki}} & k \neq i\,.
\end{cases}
\end{equation}
In fact the latter provide another way of defining cluster algebras (more precisely, cluster Poisson varieties): one may start with the $\mathcal{X}$-coordinates and the principal part of $B$ of the initial cluster, and obtain all other clusters by virtue of the $\mathcal{X}$-coordinate mutation \eqref{eq:xMutation}.
Let us now associate a natural function space to a cluster algebra, given a set $\{a_i\}$ of $\mathcal{A}$- (or similarly, $\mathcal{X}$-) coordinates.
A {\it cluster (polylogarithm) function} $f$ \cite{Parker:2015cia} of (transcendental) weight $w$ has the defining property that its differential has the form
\begin{align}\label{definition_cluster_function}
d \,f^{(w)} = \sum_{i} f^{(w-1)}_{i} d \log a_{i} \,,
\end{align}
where the $f_{i}$ are again cluster functions, of weight ${(w-1)}$. The iterative definition starts with the weight zero function, which is a constant.
From this it follows that cluster functions of weight $w$ can be expressed as $w$-fold Chen iterated integrals \cite{chen1977}.
The definition (\ref{definition_cluster_function}) is very similar to {\it{canonical differential equations}} satisfied by certain classes of Feynman integrals \cite{Henn:2013pwa},
\begin{align}\label{canonicalDE}
d\, {\bf f}(\vec{z};\epsilon) = \epsilon \left[ \sum_i {\bf A}_{i} d \log \alpha_i(\vec{z}) \right] {\bf f}(\vec{z};\epsilon) \,,
\end{align}
where ${\bf f}$ is a basis of Feynman integrals under consideration, ${\bf A}_{i}$ are constant matrices, and the $\alpha_i$ are algebraic functions of the kinematic variables $\vec{z}$, and $d= \sum_j dz_j \partial_{z_j}$. Solving eq. (\ref{canonicalDE}) as a series in $\epsilon$ yields Chen iterated integrals, with the weight corresponding to the order in $\epsilon$.
Moreover, the {\it symbol} \cite{Goncharov:2010jf} of the answer, which amounts to the solution modulo integration constants, can be read off from eq. (\ref{canonicalDE}).
The set of $\alpha$'s is called the {\it{alphabet}}, and its elements are called {\it letters}.
Knowing the alphabet of a given Feynman integral or scattering amplitude is an important piece of information, because it is essential for bootstrapping the answer.
In the following, we wish to study whether the alphabets of certain Feynman integrals coincide with the $\mathcal{A}$- or $\mathcal{X}$-coordinates of some cluster algebra (up to multiplicative redefinitions).
\section{$C_{2}$ cluster algebra and four-particle scattering with one off-shell leg}
\begin{figure}
\centering
\begin{subfigure}{.15\textwidth}
\includegraphics[width=\textwidth]{4ptdiag1.pdf}
\end{subfigure}%
\;\;
\,\,\begin{subfigure}{.15\textwidth}
\includegraphics[width=\textwidth]{4ptdiag2.pdf}
\end{subfigure}%
\,\,\begin{subfigure}{.15\textwidth}
\includegraphics[width=\textwidth]{4ptdiag4.pdf}
\end{subfigure}%
\caption{Examples of known two- and three-loop four-point integrals with one off-shell leg, $P^2 \neq 0$.}
\label{fig:2Loop1MassBoxes}
\end{figure}
We consider four-particle scattering processes with one off-shell leg and massless internal lines, cf. Fig.~\ref{fig:2Loop1MassBoxes}.
This kinematic situation has many important physical applications.
It applies to helicity amplitudes of a Higgs and three partons \cite{Gehrmann:2011aa,Duhr:2012fh} (in the limit of infinite top quark mass),
or of a vector boson and three partons \cite{Gehrmann:2013vga}, that are relevant for collider physics.
Interestingly, it has been observed that the Higgs amplitudes bear close resemblance to form factors of a composite operator inserted into three on-shell states in $\mathcal{N}=4$ sYM \cite{Brandhuber:2017bkg}.
What is more, very recently, an integrability description for form factors in $\mathcal{N}=4$ sYM was derived \cite{Sever:2020jjx}, which provides valuable all-loop and even non-perturbative information.
Denoting the Lorentz invariants by $z_1 =s / P^2$ and $ z_2= t/ P^2$, with $s = 2 p_1 \cdot p_2$ and $t=2 p_2 \cdot p_3$,
we note that all known Feynman integrals (i.e. all planar and non-planar two-loop integrals \cite{Gehrmann:2000zt,Gehrmann:2001ck}, and certain planar three-loop integrals \cite{DiVita:2014pza}, see Fig.~\ref{fig:2Loop1MassBoxes}) in this kinematics can be expressed in terms of the following alphabet (to all orders in $\epsilon$),
\begin{align}\label{eq:2dHPL}
\Phi_{\text{2dHPL}}=\{z_1,z_2, z_3, 1-z_1,1-z_2,1-z_3\}\,,
\end{align}
with $z_1 + z_2 + z_3 =1$.
The associated class of functions, dubbed two-dimensional harmonic polylogarithms (2dHPL), is well-studied in the physics literature \cite{Gehrmann:2001jv,Duhr:2012fh}
\footnote{At one loop, the five-letter subalphabet $\{z_1,z_2,1-z_1,1-z_2,z_1+z_2\}$ is sufficient. It is well-known that it corresponds to the $A_{2}$ cluster algebra.
Closely related function spaces have also appeared in off-shell form factors, four-point CFT correlation functions, and in the soft anomalous dimension matrix.}.
\renewcommand{\arraystretch}{1.25}
\begin{table}[tp]
\caption{Dimension of the $C_2$ cluster functions space (modulo transcendental constants), after constraints.}
\begin{tabular}{|c| c c c c c c c c |}
\hline
weight & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
{{First entry condition}}
& 3 & 12 & 45 & 165 & 597 & 2143& 7653& 27241\\
Adjacency constraint & 3 & 12 & 42 & 138 & 438 & 1362 & 4182& 12738 \\
\hline
\end{tabular}
\label{tableB2integrability}
\end{table}%
We now show that surprisingly, the 2dHPL alphabet (\ref{eq:2dHPL}) is equivalent to the $C_{2}$ alphabet given in Fig.~\ref{B2C2ExchangeGraph}.
To see this one applies the mapping
\begin{equation}
z_1 = - \frac{a_2^2}{1+a_1} \,, \quad z_2= - \frac{1+a_1+a_2^2}{a_1 (1+a_1)} \,. \label{z1z2a1a2}
\end{equation}
This identification allows us to make a number of interesting observations:
1. The cluster $\mathcal{X}$-coordinates are useful for choosing a particular polylogarithmic representation of the functions and for writing down functional identities \cite{Gehrmann:2001jv,Duhr:2012fh,Parker:2015cia,Golden:2014xqa}. For example, if $x$ is an $\mathcal{X}$-coordinate, then ${\rm Li}_{n}(-x)$ is a cluster function, as can be understood from Eq.(\ref{eq:xMutation}). In \cite{Parker:2015cia} an algorithm for constructing a basis of $A_n$ cluster functions at any weight is given.
Moreover, we note that the $C_{2}$ cluster mutations generate a subset of the automorphisms of the 2dHPL alphabet.
2. The cluster algebra suggests to group the alphabet letters according to the clusters.
In certain functions in $\mathcal{N}=4$ sYM the interesting property of {\it{cluster adjacency}} \cite{Drummond:2017ssj} has been observed, which in the language of differential equations \eqref{canonicalDE} translates to the statement that ${\bf A}_{i}.{\bf A}_{j} = 0$ if $\alpha_i,\alpha_j$ do not appear together in any cluster.
Note that this proves the adjacency property to all orders in $\epsilon$.
Inspecting the differential equations for the two- and three-loop planar integrals of \cite{Gehrmann:2000zt,DiVita:2014pza}, we observe that they obey a subset of $C_{2}$ adjacency conditions, namely
\bar\eta\label{eq:ObservedAdjacency}
{\bf A}_{i}.{\bf A}_{j} = 0\,, \;\; {\rm for} \; i,j \in \{1,3,5\} \; {\rm with} \; i \neq j\,.
\end{equation}
{Eqs. \eqref{eq:ObservedAdjacency} also hold for the non-planar integrals of \cite{Gehrmann:2001ck}, as was first noticed in \cite{Dixon:2020bbt}.}
In terms of the equivalent alphabet (\ref{eq:2dHPL}), this implies that the letters $1-z_i$ and $1-z_j$ for $i\neq j$ never appear next to each other in a symbol.
The adjacency conditions we observe significantly reduce the dimension of the space containing the aforementioned integrals, as shown in table \ref{tableB2integrability}.
Furthermore, the adjacency property extendeds to rational functions \cite{Drummond:2018dfd},
as can be seen by inspecting for example the two-loop amplitudes of \cite{Brandhuber:2017bkg,Jin:2018fak}.
This is obvious because the latter only contain poles at $z_{i}=0$.
\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=0.35]
\definecolor{bg}{RGB}{246,202,203}
\coordinate (A) at (1,0.8);
\coordinate (B) at (9,0.8);
\coordinate (C) at (13,-0.8);
\coordinate (D) at (-3,-0.8);
\draw[thick, color=white, fill=bg,fill opacity=0.6] (A) -- (B) -- (C) -- (D) -- cycle;
\draw[very thick,white] (0,0) -- (2.75,0.25) ;
\draw[very thick,white] (7.25,0.25) -- (10,0);
\draw[very thick,white] (2,-0.25) -- (8,-0.25);
\draw[very thick,white] (0,0) -- (2,-0.25);
\draw[very thick,white] (8,-0.25) -- (10,0);
\draw[very thick,white] (2.75,0.25) -- (7.25,0.25);
\draw[very thick,white] (0,0-0.05) -- (2.75,0.25-0.05) ;
\draw[very thick,white] (7.25,0.25-0.05) -- (10,0-0.05);
\draw[very thick,white] (2,-0.25-0.1) -- (8,-0.25-0.1);
\draw[very thick,white] (0,0-0.05) -- (2,-0.25-0.05);
\draw[very thick,white] (8,-0.25-0.05) -- (10,0-0.05);
\draw[very thick,white] (2.75,0.25-0.05) -- (7.25,0.25-0.05);
\draw[thick,fill=none] (0,0) -- (1,3) -- (2,-0.25) -- (1,-3.5) -- cycle;
\draw[thick,fill=none] (2,-0.25) -- (8,-0.25);
\draw[thick,fill=none] (10,0) -- (9,3) -- (8,-0.25) -- (9,-3.5) -- cycle;
\draw[thick,fill=none] (1,3) -- (5,5) -- (9,3);
\draw[thick,fill=none] (1,-3.5) -- (5,-5.25) -- (9,-3.5);
\draw[thick,gray,dashed] (0,0.0) -- (2.75,0.25) ;
\draw[thick,gray,dashed] (7.25,0.25) -- (10,0.0);
\draw[thick,gray,dashed] (2.75,0.25) -- (5,3.25) -- (7.25,0.25) -- (5,-2.75) -- cycle;
\draw[thick,gray,dashed] (5,3.25) -- (5,5);
\draw[thick,gray,dashed] (5,-2.75) -- (5,-5.25);
\end{tikzpicture}
\end{center}
\caption{
The intersection of the exchange graph of the $A_3$ cluster algebra with the parity invariant plane (in pink) is shown in white. It can be identified with the $C_2$ cluster exchange graph shown in Fig.~\ref{B2C2ExchangeGraph}.}
\label{Stasheff_triang}
\end{figure}
3.
We can interpret the observed adjacency in the following way: As shown in Fig.~\ref{Stasheff_triang}, $C_2$ is the parity-invariant surface of the $A_{3}$ cluster algebra, relevant for six-gluon amplitudes in $\mathcal{N}=4$ sYM
\footnote{More generally, we find that $C_{n}$ can be identified as the parity-invariant surface inside $A_{2n-1}$, and that the same holds true for $F_4$ inside $E_{6}$. Similarly, we have obtained $G_{2}$ and $B_{n}$ from $D_{4}$ and $D_{2n-1}$, respectively. See also \cite{Arkani-Hamed:2020tuz}.}.
On the latter surface, corresponding to $\Delta\equiv(u+v+w-1)^2-4 u v w=0$, where $u,v,w$ are the dual conformal cross-ratios parametrizing the kinematics, only parity-even combinations of the $A_3$ ${\cal A}$-coordinates are relevant. Choosing them as
\bar\eta
\begin{aligned}\label{eq:B2toA3Letters}
a_1&=\sqrt \frac{u}{v w} \,,\;\;\;\;\, a_3=\sqrt \frac{w}{u v} \,,\;\;\;\;\; a_5=\sqrt \frac{v}{u w} \,,\\
a_2&=\sqrt \frac{1-v}{v} \,,\; a_4=\sqrt \frac{1-u}{u} \,,\; a_6=\sqrt \frac{1-w}{w} \,,
\end{aligned}
\end{equation}
where $\{a_1,a_2\}$ are associated to the leftmost equatorial cluster in Fig.~\ref{Stasheff_triang},
$\{a_6,a_1\}$ to the left front cluster, and so on, we find that they satisfy exactly the $C_{2}$ mutation rule, eq. (\ref{simplifiedmutationrule}), on the $\Delta=0$ surface.
By virtue of eq. (\ref{eq:B2toA3Letters}), we recognize that the adjacency restrictions~(\ref{eq:ObservedAdjacency}) precisely correspond to the extended Steinmann relations~\cite{Caron-Huot:2018dsv,Caron-Huot:2019bsq} (i.e. applying to any consecutive entries in the symbol) for six-particle
massless scattering!\footnote{Note however that the first entries are $z_{i}$ with $i=1,2,3$, while the adjacency conditions apply to the $1-z_{i}$ entries, i.e. the relations found cannot be interpreted as Steinmann relations in the kinematic space of four-particle scattering with one off-shell leg.}\footnote{It is very interesting to note that the $A_{3}$ mutations that lie on the parity-invariant surface relate
$a_{2i-1}\leftrightarrow a_{2i+1}$, i.e. exactly the pairs appearing in the adjacency relations~\eqref{eq:ObservedAdjacency} we observe.}
4. Finally, let us comment on the massless limit $P^2 \to 0$. Denoting $z=t/s$, Eq. (\ref{eq:2dHPL}) reduces to the alphabet $\{z, 1+z \}$ in the limit.
This is consistent with the fact that all currently known Feynman integrals in this kinematics (up to three loops, planar and non-planar) satisfy differential equations (\ref{canonicalDE}) with this alphabet \cite{Henn:2013fah,Henn:2013nsa,Henn:2020lye}.
It is interesting to note that one can obtain this on-shell four-particle alphabet from an $A_{1}$ cluster algebra with one frozen variable.
\section{Matching function spaces and cluster algebras, and applications}
\begin{table}
\caption{One-loop integrals related to cluster algebras.}
\begin{tabular}{|c|c|c||c|}
\hline
\multirow{2}{*}{ Integral family } & {\# } & {\# } & \multirow{2}{*}{ cluster algebra } \\ & var. & letters & \\
\hline
\multirow{2}{*}{ 1-loop Bhabha scattering \cite{Henn:2013woa} } & \multirow{2}{*}{2} & \multirow{2}{*}{8} & $\subset A_3$ \\ & & & $\lim C_3$ \\
\hline
2-mass-easy box & 3 & 7 (9) & $ \subset A_3$ \\
\hline
\multirow{2}{*}{2-mass-hard box} & \multirow{2}{*}{3} & \multirow{2}{*}{10} & $\subset C_3$ \\ & & & $\lim D_4$ \\
\hline
3-mass box & 4 & 16 & $ \subset C_4$ \\
\hline
\end{tabular}
\label{tableidentifications}
\end{table}
There are many other cases where Feynman integrals evaluate to generalized polylogarithmic functions that are characterized by an alphabet.
Are any of them associated to cluster algebras? If so, how can we find this relation?
We use the following procedure to find {{embeddings of alphabets into cluster algebras}}.
Let $\vec{\alpha}$ be an alphabet depending on $d$ variables $z_1,\ldots,z_d$ in a rational way.
We would like to express all letters $\vec \alpha$ in terms of the letters $\vec\beta$ of a candidate cluster alphabet (with more or equal number of letters),
\begin{align}\label{transformation}
d\log(\alpha_i) = \sum_j n_{ij} d\log(\beta_{j}) \,,
\end{align}
where $n_{ij}$ are integers.
We proceed in the following steps. Usually it is possible to simplify the letters $\vec{\alpha}$ by a birational change of variables, such that $\{ z_i, 1+z_i\}_{i=1}^d$ are among the new set of letters. Next, we order the letters according to their `complexity', namely the number of $z$-variables they depend on.
Parsing though the transformations with small integer coefficients $n_{ij}$, we identify those which are consistent with the one-variable letters.
Then we further restrict the set of allowed transformations demanding that two-variable letters
also factorize into the cluster letters, and so on for letters of higher complexity.
This yields a host of novel matchings, which are indicated in table~\ref{tableidentifications} with the $\subset$ sign.
Let us illustrate this for the two-mass easy box integral family with massive legs $p_1$ and $p_3$.
The corresponding 10-letter alphabet is given by \footnote{In fact, only 8 independent combinations of the letters appear in the one-loop differential equations.}
\begin{align}
\Phi_{2me} = & \{ s,t,p_1^2,p_3^2, s-p_1^2 , s-p_3^2, t-p_1^2 , t-p_3^2, \notag\\
& \quad s t - p_1^2 p_3^2 , s+t-p_1^2 - p_3^2 \} \,.
\end{align}
The change of variables
\begin{align}
z_1 = - \frac{p_1^2}{s} ,\;
z_2 = - \frac{t}{p_3^2} ,\;
z_3 = \frac{t-p_1^2}{s-p_3^2} \,,
\end{align}
brings it to a simpler form,
\begin{align}
\Phi_{2me} \approx & \{ z_1 , z_2, z_3, 1+z_1 , 1+z_2, 1+z_3 , \notag\\
& \quad z_1 - z_2 , z_1 - z_3, z_2 - z_3 \}\,,
\end{align}
where we dropped a trivial mass scale. This 9-letter alphabet is equivalent to the cluster $A_3$ alphabet.
We can find further identifications by considering {\it degenerations} of cluster algebra alphabets with fewer variables.
What we mean by this is to consider the subalphabet obtained at a hypersurface $\alpha_{i}=0$ (for some $i$).
We have already seen an example of this, when going from $C_{2}$ to $A_{1}$.
We note that another degeneration of $C_{2}$ yields the important alphabet $\{z, 1+z,1-z \}$ corresponding to harmonic polylogarithms \cite{Remiddi:1999ew}.
The results of our heuristic search with this method are denoted in table~\ref{tableidentifications} by `lim'.
\begin{table}
\caption{Six-dimensional integrals and cluster algebras.}
\begin{tabular}{|c|c|c||c|}
\hline
\multirow{2}{*}{ Integral } & {\# } & {\# } & \multirow{2}{*}{ cluster algebra } \\ & var. & letters & \\
\hline
1-mass hexagon & 4 & 16 & $D_4$\\
\hline
2-mass-easy hexagon & 5 & 24 & $\subset D_5$\\
\hline
{ 2-mass-hard hexagon } & \multirow{2}{*}{5} & \multirow{2}{*}{27} & \multirow{2}{*}{$\lim G(4,8)$} \\ pentagon with one off-shell leg & & & \\
\hline
\end{tabular}
\label{tableidentifications2}
\end{table}
The identifications of alphabets discussed so far are valid to all orders in the dimensional regularization parameter $\epsilon$.
However, when computing finite physical quantities, often simplifications occur, and a reduced alphabet is sufficient to describe the answer.
Moreover, it could be that cluster algebras appear in general only for integer dimensions.
For these reasons we find it is interesting to look for cluster algebras for finite Feynman integrals, or when truncating the $\epsilon$ expansion.
It is known that the (dual conformal) six-dimensional hexagon integral corresponds to the $A_{3}$ cluster algebra. Here find that its one-mass version \cite{DelDuca:2011jm,DelDuca:2011wh}, which has $20$ letters and $4$ dimensionless variables, corresponds to the $D_{4}$ cluster algebra.
Likewise, we find that a `two-mass-easy' hexagon \cite{DelDuca:2011wh} can be embedded into the $D_{5}$ cluster algebra.
So far we have discussed finite cluster algebras, however in $\mathcal{N}=4$ sYM also infinite cluster algebras are seen to play a role.
For example, the infinite Grassmannian $G(4,8)$ algebra is expected to govern dual conformal eight-particle scattering, generalizing the six- and seven-particle case, with $A_{3} \sim G(4,6)$ and $E_{6} \sim G(4,7)$, respectively. Very recently, a natural way of identifying a finite set of letters from $G(4,8)$ has been proposed in \cite{Arkani-Hamed:2019rds,Henke:2019hve,Drummond:2019cxm}.
Starting from the eight-particle alphabet thus obtained in the latter paper, here we will establish a connection with five-particle scattering in generic gauge theory. The eight-particle alphabet of \cite{Drummond:2019cxm} consists of 272 rational and 18 square-root letters, and contains both the alphabets of the two-loop MHV~\cite{CaronHuot:2011ky} and NMHV amplitudes~\cite{Zhang:2019vnm} (see also \cite{He:2020vob,Mago:2020kmp,He:2020uhb}).
In a first step, we specify the generic eight momentum twistors \cite{Hodges:2009hk} such that they describe two adjacent massive legs, cf. Fig.~\ref{two-mass-hard-hexagon}.
We find that this reduces the $G(4,8)$ alphabet to 30+5 rational and algebraic letters, consistent with the symbol of the six-dimensional two-mass-hard hexagon, as obtained from \cite{Spradlin:2011wp}.
The alphabet we find also applies to non dual-conformal integrals: by interpreting the bi-twistor $Z_{7} \wedge Z_{8}$ as the infinity twistor, the kinematics is equivalent to five-particle scattering with one massive leg, cf. Fig.~\ref{two-mass-hard-hexagon}.
Indeed we reproduce all letters appearing in the one-loop integrals of \cite{Abreu:2020jxa}, except for $W_{48}, W_{49}$.
Taking in addition the {$P^2\to 0$} limit, we find $22$ letters from the planar pentagon alphabet \cite{Gehrmann:2015bfy}.
These include all one-loop letters except for $W_{31}$ (using the notation of \cite{Chicherin:2017dob}).
Moreover, the reduction of the octagon alphabet also captures some of the additional two-loop letters; upon cyclic symmetrization, we obtain all planar two-loop letters, apart from $W_{31}$.
This is very interesting, because $W_{31}$ has been observed to drop out of appropriately defined finite quantities, such as the hard part (after infrared subtraction) of the two-loop $\mathcal{N}=4$ sYM \cite{Abreu:2018aqd,Chicherin:2018yne} and $\mathcal{N}=8$ supergravity \cite{Chicherin:2019xeg,Abreu:2019rpt} amplitudes.
The same is true for the hard part of the two-loop $q \bar{q} \to \gamma\gamma\gamma$ amplitude \cite{Abreu:2020cwb,Vasilycomm} and the two-loop five-gluon amplitudes \cite{Badger:2018enw,Abreu:2019odu,Simonecomm} in QCD.
\begin{figure}[t]
\begin{align*}
\begin{array}{c}
\includegraphics[width=3.5cm]{twistors8b.pdf}
\end{array} \qquad
\begin{array}{c}
\includegraphics[width=3.5cm]{pentagon_kinematics_offshell4.pdf}
\end{array}
\end{align*}
\caption{Momentum twistors in a dual conformal two-mass-hard hexagon configuration.
This is equivalent to five-particle kinematics with one off-shell leg $P$.}
\label{two-mass-hard-hexagon}
\end{figure}
Interested experts may find the technical details of the identifications mentioned above in the Supplemental Material to this Letter.
\section{Discussion and outlook}
In this Letter, we have provided for the first time evidence that cluster algebras are relevant to Feynman integrals beyond $\mathcal{N}=4$ sYM.
In particular, we have uncovered that the important class of two-dimensional harmonic polylogarithms corresponds to the $C_{2}$ cluster algebra.
This suggests that a number of physical processes, including Higgs plus jet amplitudes in the heavy top quark limit, may be amenable to bootstrap methods \cite{Brandhuber:2012vm}.
Moreover, we found unexpected adjacency relations that further constrain the function space. By identifying $C_2$ as a subspace of the $A_3$ cluster algebra, we have shown that these adjacencies are equivalent to the extended Steinmann relations of planar six-particle scattering in $\mathcal{N}=4$ sYM~\cite{Caron-Huot:2018dsv,Caron-Huot:2019bsq}.
Such relations have been used for six- and higher-particle amplitudes (and five-particle scattering with one off-shell leg \cite{Abreu:2020jxa}),
but their implications are not yet known for massless five-particle scattering. This makes the adjacency relations we found in a five-particle kinematics all the more interesting.
We have demonstrated that the constraints we found significantly reduce the relevant functions space, cf. Table~\ref{tableB2integrability}. This should prove valuable in bootstrap approaches.
This is very timely in view of the recent integrability results for form factors in $\mathcal{N}=4$ sYM \cite{Sever:2020jjx}.
We have outlined heuristic procedures for matching Feynman integrals with cluster algebras, and found a host of one-loop identifications. This suggests a number of questions.
What other processes are related to cluster algebras? What cluster algebras describe the higher-loop versions of the cases discussed here?
We leave these questions to future work, and hope that our results motivate a systematic taxonomy of the relationship between Feynman integrals and cluster algebras.
In planar $\mathcal{N}=4$ sYM, there is a direct link between the kinematics and geometry of the scattering process, and cluster algebras.
This leads us to the most pressing open question: is there a similar story for generic Feynman integrals?
\acknowledgments
We thank S.~Badger, L. Dixon, P.~Mastrolia, A. McLeod, V.~Sotnikov, M. Wilhelm and S.~Zoia
for useful correspondence, N.~Henke for discussions, and M.~Spradlin for comments on the manuscript. JMH and GP thank the Higgs Centre for Theoretical Physics for hospitality during the workshop {\it Cluster Algebras and the Geometry of Scattering Amplitudes} in the early stages of this project. This research received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 725110), {\it Novel structures in scattering amplitudes}. GP acknowledges support from the Deutsche Forschungsgemeinschaft under Germany's Excellence Strategy EXC 2121 {\it Quantum Universe} 390833306.
\section{Supplemental material}
\subsection{Cluster algebras and one-loop alphabets}
Here we present the technical details of the identifications of one-loop alphabets with cluster algebras mentioned in the main part of the Letter.
In the following we deal with the cluster alphabets $\Phi$ of the $A_n, \, C_n, \, D_n$ series in their ${\cal X}$-coordinate formulation \p{eq:xMutation}, which contain ${n(n+3)}/{2}$, $n(n+1)$, and $n^2$ letters, respectively. We find convenient to make a birational change of variables to simplify the cluster letters \cite{Arkani-Hamed:2020tuz} as follows,
\begin{widetext}
\begin{align}
& \Phi_{A_n}= \bigcup\limits_{i=1}^n \{ z_i , 1+ z_{i} \} \, \cup \, \bigcup\limits_{1\leq i<j \leq n} \{ z_i - z_j \} \; ,\qquad
\Phi_{C_n} = \Phi_{A_n} \cup \bigcup_{1\leq i \le j \leq n -1 } \{z_i z_j + z_n\} \label{Cn} \\
& \Phi_{D_n} = \bigcup\limits_{\substack{1\leq i< \\ j \leq n-2}} \{ z_i - z_j , z_i (z_{n-1}+z_n) - z_{n-1} z_n - z_i z_j + z_i - z_j \} \, \cup \bigcup\limits_{i=1}^{n} \{ z_i , 1+ z_{i} \} \, \cup \bigcup\limits_{i=1}^{n-2} \{ z_i - z_{n-1}, z_i - z_{n}, z_i + z_{n-1} z_n \} \label{Dn}
\end{align}
\end{widetext}
{\it Two-mass hard box.} We consider the family of box integrals with two adjacent massive legs $p_1, \,p_2$. Parametrizing $p_1^2 = s y_1 y_2$ and $p_2^2 = s (1-y_1)(1-y_2)$, we find the following rational 11-letter alphabet,
\begin{align}
& \Phi_{2mh} = \{ s ,\, t ,\, y_1 ,\, 1 - y_1 ,\, y_2 ,\, 1 - y_2 ,\, y_1 - y_2 ,\, s y_1 y_2 - t , \notag \\
& s y_1(1-y_2) + t ,\, s y_2(1-y_1) + t ,\, s(1-y_1)(1-y_2) - t \} \,, \notag
\end{align}
where $s = (p_1 + p_2)^2$ and $t = (p_2+ p_3)^2$.
After the birational change of variables
\begin{align}
&t = -\frac{s z_1 z_3}{(1+z_2)(z_1 - z_3)} \notag\\
&y_1 = \frac{z_1}{z_1 - z_3} , \quad y_2 = \frac{1}{1+z_2}
\end{align}
the alphabet becomes
\begin{align}
\Phi_{2mh} \approx \{ & s,z_1,z_2,z_3,1+z_1,1+z_2 ,1+z_3,\notag\\
& z_1 - z_2, z_1 - z_3, z_2 - z_3, z_1 z_2 + z_3\} \,. \label{2mh_z}
\end{align}
Ignoring the trivial mass scale we establish the embedding in the $C_3$ cluster alphabet formulated as in eq. \p{Cn}, $\Phi_{2mh} \subset \Phi_{C_3}$.
Alternatively, one could start with the rank-4 cluster alphabet $D_4$, eq. \p{Dn}, and perform the degeneration $z_1 \to -1$,
\begin{align}
\lim_{z_1 \to -1} D_4 \approx \{ & z_2, z_3,z_4, 1+z_2 , 1+ z_3 , 1+z_4, \notag\\
& z_2 - z_3, z_2 - z_4, 1 - z_3 z_4 , z_2 + z_3 z_4 \}\,.
\end{align}
The resulting 10-letter alphabet is equivalent to \p{2mh_z} after a simple variable change,
\begin{align}
z_2 \to \frac{1}{z_1} ,\, z_3 \to \frac{1}{z_2} ,\, z_4 \to - \frac{z_3}{z_2} \,.
\end{align}
{\it Three-mass box.}
The family of box integrals with three massive legs $p_1,p_2,p_3$ is described by the 18-letter alphabet
\begin{widetext}
\begin{align}
& \Phi_{3m} = \{ y_1 , y_2,y_3, y_4, 1- y_1 , 1-y_2 , 1- y_3 , 1-y_4 , y_1 - y_2 , y_3 - y_4 , 1-y_1 -y_3, 1-y_1 - y_4 , 1-y_2 - y_3, 1-y_2 - y_4, \notag\\
& (1- y_1)(1- y_2) - y_3 y_4 , y_1 y_2 - (1-y_3)(1-y_4) , (1 - y_1 -y_2) (1-y_3)(1-y_4) + y_1 y_2 (1-y_3 - y_4), \, p_1^2 \}\,
\end{align}
\end{widetext}
in the rational parametrization of the kinematics
\begin{align}
& s = (1-y_1)(1-y_2) p_1^2 , \;\;
p_2^2 = y_1 y_2 p_1^2 , \notag\\
& t = (1-y_3)(1-y_4) p_1^2 , \;\;
p_3^2 = y_3 y_4 p_1^2 \,.
\end{align}
In fact, only 16 multiplicative combinations of the specified 17 dimensionless letters appear in the one-loop system of differential equations \p{canonicalDE}. After the change of variables
\begin{align}
& y_1 = \frac{z_1-z_2}{1+z_1},\quad
y_2 = -\frac{z_4 + z_1 z_2}{z_1 - z_4},\notag\\
& y_3 = \frac{1+z_2}{1+z_3},\quad
y_4 = \frac{z_3 (1+z_2)}{z_3 - z_4}
\end{align}
and putting aside the trivial mass scale we find that $ \Phi_{3m} \subset \Phi_{C_4}$, cf. eq. \p{Cn}.
{\it One-loop Bhabha scattering.}
Bhabha scattering is a $2 \to 2$ process of massive electron scattering. The kinematics is specified by $s,t$ Mandelstam invariants and mass $m$. It is convenient to introduce dimensionless variables $x,y$: $-{s}/{m^2} = {(1-x)^2}/{x}$, $-{t}/{m^2} = {(1-y)^2}/{y}$. Then the relevant family of one-loop master integrals is described by the 8-letter alphabet \cite{Henn:2013woa}
\begin{align}
\Phi_{\rm Bhabha} = \{ x, 1-x,1+x, y , 1-y,1+y, x+y,1+xy\} \,. \label{Bhabha}
\end{align}
We can reproduce it by degenerating $z_3 \to -1$ the cluster alphabet $C_3$, eq. \p{Cn}, and identifying $x = z_1, \, y=z_2$.
The same alphabet can be obtained from the $A_3$ cluster alphabet via non-singular degeneration.
In order to see this it is more convenient to use the cluster ${\cal X}$-coordinates \p{eq:xtoa},
\begin{align}
\Phi_{A_3} \approx \{ & x_1, x_2, x_3, 1+ x_1 , 1+ x_2, 1+x_3 ,\notag\\
& 1+ x_2 + x_1 x_2, 1+ x_3 + x_2 x_3,
,\notag\\
& 1+ x_3 + x_2 x_3 + x_1 x_2 x_3\}\,,
\end{align}
where $(x_1,x_2,x_3)$ is the initial seed with the exchange matrix $B$,
\begin{align}
B = \begin{pmatrix}
0 & -1 & 0 \\
1 & 0 & -1 \\
0 & 1 & 0
\end{pmatrix} .
\end{align}
Then the non-singular limit $x_1 \to - {1}/{x_3}$ and the following change of variables,
\begin{align}
x_3 = x , \quad x_2 = - \frac{x + y}{1-x}
\end{align}
bring the cluster alphabet $\Phi_{A_3}$ to the form of the one-loop Bhabha scattering alphabet \p{Bhabha}. Let us recall that the $A_3$ cluster alphabet describes the planar hexagon scattering in ${\cal N} = 4$ sYM theory. The imposed constraint $x_1 x_3 = -1$ on the ${\cal X}$-variables corresponds to $u+v+w=1$ in the space of dual-conformal cross-ratios.
{\it Six-dimensional hexagons in easy-mass configurations.}
The six-dimensional hexagon integrals with up to three massive legs in the easy-mass configuration (where at least one massless leg separates two massive legs) are pure dual-conformal invariant functions of uniform weight three \cite{DelDuca:2011wh}. They depend on the cross-ratios $u_i$ in the dual momenta coordinates, with five cross-ratios in the two-mass-easy configuration and four-cross ratios in the one-mass configuration. The letters of the alphabets are algebraic in the cross-ratios, and in order to eliminate square roots it is very convenient to use a momentum-twistor parametrization of the cross-ratios. In order to make contact with notations in \cite{DelDuca:2011wh} we reproduce the rational parametrization of the cross-ratios $u_i$ by $y_1,\ldots,y_5$ in the two-mass-easy configuration (as compared to eq. (44) in \cite{DelDuca:2011wh}, we made one of the massive legs massless by putting there $y_8 \to 0$, i.e. $u_6 \to 0$, and we relabelled $x_5,x_8,y_2,z_2,z_5$ with $y_1,\ldots,y_5$, respectively),
\begin{align}
&u_1 = \frac{y_4-y_5}{(1-y_3)(1-y_5)}\, , \,
u_2 = \frac{y_1 - y_2}{(1-y_2)(1-y_5)} \,, \notag\\
& u_3 = -\frac{y_3}{(1-y_2)(1-y_3)} \,,\,
u_4 = \frac{y_4 (1-y_5)}{y_4-y_5} \,, \notag\\
& u_5 = \frac{y_1 (1-y_2)}{y_1 - y_2} \,.
\end{align}
We obtain the one-mass configuration from the two-mass-easy configuration by putting $y_1 \to 0$, which corresponds to $u_5 \to 0$.
Then the one-mass and two-mass-easy hexagon alphabets extracted from the one-loop calculation of \cite{DelDuca:2011wh} contain $16$ and $24$ letters, respectively, and they take the following form
\begin{widetext}
\begin{align}
\Phi_{\rm hex_{1m}} = \{ & y_2, y_3 , y_4 , y_5 , 1-y_2, 1-y_3 , 1-y_4 , 1-y_5, y_4 - y_5, 1-y_3 - y_4 , 1-y_2 + y_2 y_3 , 1-y_5 + y_2 y_5 , y_4 - y_5 + y_2 y_5 , \notag\\
& 1- y_3 - y_4 + y_3 y_5 , y_4 - y_5 + y_2 y_5 - y_2 y_3 y_5 , 1-y_2- y_4 + y_2 y_3 + y_2 y_4 \}\,, \notag \\[0.2cm]
\Phi_{\rm hex_{2me}} = \{ & y_1, y_2, y_3, y_4, y_5, 1-y_1, 1-y_2, 1-y_3, 1-y_4, 1-y_5, y_1 - y_2, y_4 - y_5 , 1 -y_3 - y_4 , 1 - y_1 - y_5 , 1 -y_2 + y_2 y_3 , \notag\\
& y_1 - y_2 + y_2 y_3 , y_4 - y_5 - y_1 y_4 , 1- y_1 - y_5 + y_2 y_5 , 1- y_3 - y_4 + y_3 y_5 , y_4 - y_5 -y_1 y_4 + y_2 y_5, \notag\\
& 1-y_2 - y_4 + y_2 y_3 + y_2 y_4 , 1- y_1 - y_3 - y_4 + y_1 y_3 + y_1 y_4 + y_3 y_5 , y_4 - y_5 - y_1 y_4 + y_2 y_5 - y_2 y_3 y_5 , \notag\\
& y_1 - y_2 - y_1 y_4 + y_2 y_3 + y_2 y_4 - y_2 y_3 y_5 \}\,.
\end{align}
\end{widetext}
The birational change of variables
\begin{align}
& y_1 = \frac{z_3 - z_5}{z_2 - z_5} \,,\,
y_2 = -\frac{z_3 - z_5}{(1+z_4)z_5} \,, \notag\\
&
y_3 = \frac{z_1 (1+z_5)}{z_1-z_5} \,,\,
y_4 = \frac{(z_1 - z_3)z_5}{z_3(z_1 - z_5)} \,,\notag\\
&
y_5 = \frac{(z_2 - z_3)z_5}{z_3(z_2 - z_5)} \,, \label{2mHexChange}
\end{align}
shows that $\Phi_{\rm hex_{2me}}$ is embedded in the 25-letter cluster alphabet $D_5$ in notations of eq. \p{Dn}, i.e. $\Phi_{\rm hex_{2me}} \subset \Phi_{D_5}$. More precisely, the extra letter of the cluster alphabet is $z_2 (z_4 + z_5) - z_4 z_5 - z_2 z_3 + z_2 - z_3$.
The change of the variables \p{2mHexChange} relating the two-mass-easy hexagon alphabet and the cluster alphabet is compatible with the degenerating $y_1 \to 0$ into the one-mass hexagon that demands $z_2 \to \infty$. Indeed, the resulting birational transformation
\begin{align}
& y_2 = -\frac{z_3 - z_5}{(1+z_4)z_5} \,, \,
y_3 = \frac{z_1 (1+z_5)}{z_1-z_5} \,, \notag\\
&
y_4 = \frac{(z_1 - z_3)z_5}{z_3(z_1 - z_5)} \,, \, y_5 = \frac{(z_2 - z_3)z_5}{z_3(z_2 - z_5)} \,, \label{1mHexChange}
\end{align}
establishes the equivalence of the one-mass hexagon with the $D_4$ cluster alphabet $\Phi_{\rm hex_{1m}} \approx \Phi_{D_4}$.
\subsection{Reduction of the octagon alphabet to pentagon functions}
\begin{figure}[t]
\includegraphics[width=3.5cm]{hexagon.pdf}
\caption{Two-mass-hard hexagon configuration. The dual coordinates $x_{i}$ denote region momenta.}
\label{two-mass-hard-hexagon2}
\end{figure}
We consider an alphabet describing the dual-conformal invariant finite part part of the eight-particle amplitude in ${\cal N} = 4$ sYM theory, and we show how to reduce it to the pentagon scattering with one massive leg. There are several alphabets relevant for eight-particle scattering in ${\cal N} = 4$ sYM. In order to be specific, we consider the dual-conformal invariant alphabet \cite{Zhang:2019vnm} containing 172 rational letters and 18 algebraic letters in the momentum twistor variables. The latter involve two different square roots. We refer to it as the octagon alphabet $\Phi_{oct}$. The kinematics and the alphabet letters can be parametrized by 9 independent variables.
We introduce the dual momenta $x_i$ such that $x_{i+1} - x_{i} = p_{i+1}$ to describe the light-like kinematics $p_i^2 = 0$, $i=1,\ldots,8$, which automatically take into account momentum conservation. It is convenient to specify the kinematics by 8 unconstrained momentum twistors $Z_i$, which are points on $\mathbb{P}^3$, such that a pair of twistors (a line in $\mathbb{P}^3$) represents a dual momentum, $x_i \sim Z_i \wedge Z_{i+1}$. Intersecting lines $Z_i \wedge Z_{i+1}$ and $Z_{i+1}\wedge Z_{i+2}$ correspond to a pair of null-separated dual momenta, $(x_i-x_{i+1})^2 = 0$, and the intersection point $Z_{i+1}$ corresponds to the light-like momentum $p_{i+1}$. We would like that pairs $p_1$, $p_8$ and $p_6$, $p_7$ of the light-like momenta form massive corners (two-mass-hard dual-conformal hexagon), see Fig.~\ref{two-mass-hard-hexagon2}. It corresponds to the configuration of the momentum twistors depicted in Fig.~\ref{two-mass-hard-hexagon}. Twistor translations which do not alter the line configuration are allowed. We are interested in those multiplicative combinations of $\Phi_{oct}$ letters which are completely specified by this kinematic configuration, i.e. they are annihilated by four differential operators
\begin{align}
Z_5 \cdot \partial_{Z_6} ,\,
Z_2 \cdot \partial_{Z_1} ,\,
Z_7 \cdot \partial_{Z_8} , \,
Z_8 \cdot \partial_{Z_7} \,.
\end{align}
Resolving these constraints we obtain the 27-letter alphabet $\Phi_{\rm hex_{2m}}$ with 22 rational letters and 5 algebraic letters with one square root (in the momentum twistor variables). The letters of $\Phi_{\rm hex_{2m}}$ are functions of 5 independent variables.
Switching to the frame $x_7 \to \infty$ in the dual-conformal letters of $\Phi_{\rm hex_{2m}}$ we obtain the one-mass pentagon configuration without any extra symmetry restrictions, cf. Fig.~\ref{two-mass-hard-hexagon}, where $P = p_6+p_7 + p_8 + p_1$, such that $P^2 \neq 0$. The latter is equivalent to identifying $Z_7 \wedge Z_8$ as an infinity bi-twistor that allows us to express `distance' $x_{ij}^2 = (x_i - x_j)^2$ in terms of the twistor 4-brackets $\vev{abcd} \equiv \det(Z_a Z_b Z_c Z_d)$, and eventually, transform the alphabet to the Mandelstam variables $s_{ij} = (p_i + p_j)^2$,
\begin{align}
& s_{12} = x_{25}^2 = \frac{\vev{2356}}{\vev{2378}\vev{5678}} \,,\,
s_{23} = x_{13}^2 = \frac{\vev{1234}}{\vev{1278}\vev{3478}} \,,\notag\\
& s_{34} = x_{24}^2 = \frac{\vev{2345}}{\vev{2378}\vev{4578}} \,,\,
s_{45} = x_{35}^2 = \frac{\vev{3456}}{\vev{3478}\vev{5678}} \,,\notag\\
& s_{15} = x_{14}^2 = \frac{\vev{1245}}{\vev{1278}\vev{4578}} \,,\,
p_1^2 = x_{15}^2 = \frac{\vev{1256}}{\vev{1278}\vev{5678}} \,.
\end{align}
Thus, comparing the 28-letter alphabet (after restoring an overall scale) with the letters $\{ W_{i} \}_{i = 1}^{58}$ of the planar one-mass pentagon alphabet from \cite{Abreu:2020jxa}, we identify among the latter
\begin{align}
&
W_{1}, \ldots, W_{9}, W_{12}, \ldots, W_{15}, W_{18}, W_{19}, W_{22}, W_{23},W_{24},
\notag\\
& W_{33} , W_{34}, W_{37}, W_{38}, W_{40}, W_{43}, \ldots , W_{47}\,, \label{one-loop-pent-1m}
\end{align}
and the square-root inherited from the algebraic letters of $\Phi_{\rm hex_{2m}}$ equals to $\sqrt{\Delta_3}$ of \cite{Abreu:2020jxa}, which is the typical square root of the three-mass-triangle Feynman integral with massive corners $P, p_2+p_3,p_4+p_5$. This alphabet is enough to express the finite one-loop one-mass-pentagon Feynman integrals as well as finite parts of the one-loop scattering amplitudes (hard functions).
Larger octagon alphabets are also discussed in the literature. In particular, the parity restricted tropical ${\rm G}(4,8)$ alphabet presented in \cite{Drummond:2019cxm} contains 272 rational and the same 18 algebraic letters as $\Phi_{oct}$. The previous reduction procedure in this case leads to a bigger set of the planar one-mass pentagon letters \cite{Abreu:2020jxa}. In addition to the one-loop letters \p{one-loop-pent-1m}, it also captures genuine two-loop letters
\begin{align}
W_{16},W_{17},W_{29},W_{30},W_{52},W_{53},W_{55},W_{56\,}.
\end{align}
Let us note that after taking the soft limit $P^2 \to 0$ and doing cyclic symmetrization (or equivalently, doing cyclically shifted reductions of the octagon alphabet) we reproduce all letters of the planar two-loop massless pentagon alphabet \cite{Chicherin:2017dob} which are needed to express the finite part of the scattering amplitudes (hard functions).
\bibliographystyle{apsrev4-1}
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Q: Create a SCOM Monitor/Alert with a threshold relative to a performance counter? Is it possible to have a SCOM monitor that is related to a performance counter. For instance, we monitor a WCF Service Operation that gives us an indicator of our system's workload ("Calls Per Second"), can we create a Monitor of errors that is related to this figure, a percentage say?
Surely this is a pretty standard requirement but I can't seem to see how this is done in SCOM.
Here's how see it:
Volume Monitor: (Based on 'Calls Per Second')
Error Monitor: (Based on 'Errors Per Second' / 'Calls Per Second')
The we could generate an Alert on the Error Monitor say if it goes above 0.5% or some arbitrary value that we could tweak.
A: Unfortunately the standard SCOM modules don't provide a way of calculating the ratio of two performance counters, or using a performance counter as a threshold. Here's an old TechNet thread on the topic - nothing has changed in SCOM 2012.
One option is to author a custom monitor that uses a script module to perform the calculation. (Note: if you use the performance data mapper to map the resultant ratio to performance data, you can at least use the standard modules for threshold comparison etc)
If you have control over the source application, it may be easier to modify it to provide the error ratio as a performance counter.
| {
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Q: Given a circle with cente $O$, radius $R=6$ and diameter $AB$. On the radius $OB$ we take $C$, so that $OC=2$. Find the length of $EZ$. Given a circle with cente $O$, radius $R=6$ and diameter $AB$. On the radius $OB$ we take $C$, so that $OC=2$. $EZ$ is a chord of the circle which intersects $AB$ at $C$, so that $\angle ECB=60^o$. Find the length of $EZ$.
After drawing it accurately, I believe that $EZ=10$. I tried to work this out but didn't manage. My thoughts are the following:
$\angle ACZ=\angle ECB=60^o$
$OC=2,CB=4,AO=6$
If $E'$ is Es intidiametric point, then $E'E^2+E'Z^2=EZ^2$. We also have that $AC*CB=EC*CZ$ and $AEBZ$ is inscribed. This is all I could think of. Could you please explain to me how to solve this question?
A: Connect points $E$ and $Z$ to $O$ and draw a perp from $O$ to chord $EZ$. Say it meets $EZ$ at $H$ then $OH = OC \sin 60^0$.
Now use Pythagoras in $\triangle OEH$ to find $EH$ and that leads to $EZ$.
EDIT: If you want to use intersecting chords theorem,
$ \small EC \cdot CZ = (EH-1) \cdot (HZ+1) = 8 \cdot 4$ (as $\small CH = 1$). Now you know $\small CH = HZ$.
| {
"redpajama_set_name": "RedPajamaStackExchange"
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Q: Why asp:HyperLink.NagigateUrl in user control, placed in sub-folder, contains this sub-folder? ~/Folder1/UserControl1.ascx:
<%@ Control Language="C#" CodeBehind="WebUserControl1.ascx.cs" Inherits="WebApplication1.WebUserControl1" %>
<asp:HyperLink runat="server" NavigateUrl="?foo=bar">HyperLink1</asp:HyperLink>
~/UserControl2.ascx:
<%@ Control Language="C#"CodeBehind="WebUserControl2.ascx.cs" Inherits="WebApplication1.WebUserControl2" %>
<asp:HyperLink runat="server" NavigateUrl="?foo=bar">HyperLink2</asp:HyperLink>
Result:
http://localhost/Folder1/?foo=bar
http://localhost/?foo=bar
Why does it happen?
A: WHen you do not have a / or ~ at the beginning of the path, it is considered relative to the CURRENT position.
From a user control, current position is the position of the control.
A: The links are resolve relative to the directory of the user control, they actually call ResolveClientUrl(); internally, so you see this same behavior.
Description from MSDN:
The URL returned by this method is relative to the folder containing the source file in which the control is instantiated. Controls that inherit this property, such as UserControl and MasterPage, will return a fully qualified URL relative to the control.
Resolving the urls with Page.ResolveClientUrl() in the code-behind will solve the problem.
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Comparison of oxygen-15 PET and transcranial Doppler CO2-reactivity measurements in identifying haemodynamic compromise in patients with symptomatic occlusion of the internal carotid artery
S. Persoon, L.J. Kappelle, B.N.M. van Berckel, R. Boellaard, C.H. Ferrier, A.A. Lammertsma, C.J.M. Klijn
Radiology and nuclear medicine
EJNMMI Research
https://doi.org/10.1186/2191-219X-2-30
10.1186/2191-219X-2-30
300415Final published version, 595 KB
Persoon, S., Kappelle, L. J., van Berckel, B. N. M., Boellaard, R., Ferrier, C. H., Lammertsma, A. A., & Klijn, C. J. M. (2012). Comparison of oxygen-15 PET and transcranial Doppler CO2-reactivity measurements in identifying haemodynamic compromise in patients with symptomatic occlusion of the internal carotid artery. EJNMMI Research, 2(1), 30. [30]. https://doi.org/10.1186/2191-219X-2-30
Persoon, S. ; Kappelle, L.J. ; van Berckel, B.N.M. et al. / Comparison of oxygen-15 PET and transcranial Doppler CO2-reactivity measurements in identifying haemodynamic compromise in patients with symptomatic occlusion of the internal carotid artery. In: EJNMMI Research. 2012 ; Vol. 2, No. 1. pp. 30.
@article{9d1975a9091e42c0be5ad2fe4b17dff2,
title = "Comparison of oxygen-15 PET and transcranial Doppler CO2-reactivity measurements in identifying haemodynamic compromise in patients with symptomatic occlusion of the internal carotid artery",
author = "S. Persoon and L.J. Kappelle and {van Berckel}, B.N.M. and R. Boellaard and C.H. Ferrier and A.A. Lammertsma and C.J.M. Klijn",
doi = "10.1186/2191-219X-2-30",
pages = "30",
journal = "EJNMMI Research",
publisher = "Springer Berlin",
Persoon, S, Kappelle, LJ, van Berckel, BNM, Boellaard, R, Ferrier, CH, Lammertsma, AA & Klijn, CJM 2012, 'Comparison of oxygen-15 PET and transcranial Doppler CO2-reactivity measurements in identifying haemodynamic compromise in patients with symptomatic occlusion of the internal carotid artery', EJNMMI Research, vol. 2, no. 1, 30, pp. 30. https://doi.org/10.1186/2191-219X-2-30
Comparison of oxygen-15 PET and transcranial Doppler CO2-reactivity measurements in identifying haemodynamic compromise in patients with symptomatic occlusion of the internal carotid artery. / Persoon, S.; Kappelle, L.J.; van Berckel, B.N.M. et al.
In: EJNMMI Research, Vol. 2, No. 1, 30, 2012, p. 30.
T1 - Comparison of oxygen-15 PET and transcranial Doppler CO2-reactivity measurements in identifying haemodynamic compromise in patients with symptomatic occlusion of the internal carotid artery
AU - Persoon, S.
AU - Kappelle, L.J.
AU - van Berckel, B.N.M.
AU - Boellaard, R.
AU - Ferrier, C.H.
AU - Lammertsma, A.A.
AU - Klijn, C.J.M.
U2 - 10.1186/2191-219X-2-30
DO - 10.1186/2191-219X-2-30
JO - EJNMMI Research
JF - EJNMMI Research
Persoon S, Kappelle LJ, van Berckel BNM, Boellaard R, Ferrier CH, Lammertsma AA et al. Comparison of oxygen-15 PET and transcranial Doppler CO2-reactivity measurements in identifying haemodynamic compromise in patients with symptomatic occlusion of the internal carotid artery. EJNMMI Research. 2012;2(1):30. 30. doi: 10.1186/2191-219X-2-30 | {
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{"url":"http:\/\/math.stackexchange.com\/questions\/836708\/generalization-of-liouvilles-theorem","text":"Generalization of Liouville's theorem\n\nAs proposed in this answer, I wonder if the answer to following question is known.\n\nLet $E = E_0$ be the set of elementary functions. For each $i > 0$, inductively define $E_i$ to be the closure of the set of functions whose derivative lies in $E_{i-1}$ with respect to multiplication, inversion, and composition. Does there exist an integer $n$ such that $E_n = E_{n+1}$?\n\nThis seems like such a natural generalization of Liouville's theorem, it has to have been asked before. After a couple of quick internet searches, I can't seem to find anything.\n\n-\nI feel like you need to throw some kind of closure operation in there -- something like \"Define $E_i$ to be the set of all functions that can be formed by a composition of functions whose derivatives lie in $E_{i-1}$.\" But maybe that is unnecessary. \u2013\u00a0mweiss Jun 17 '14 at 0:01\nIndeed, it is necessary. Thanks. \u2013\u00a0RghtHndSd Jun 17 '14 at 0:07\nSimilar question has been asked before math.stackexchange.com\/questions\/686445\/\u2026 but alas no answers. \u2013\u00a0Conifold Jun 17 '14 at 1:03\nAnd how would this relate to notions of computability (e.g Turing Machine), since an elementary functin can be a symbol along with the other (finite) symbols of \"addition\", \"nth power\", \"division\" etc.. This may indeed be related.. And provide sth analogous to an \"algebraic closure\", but if taken in a computable sense, this would be non-decidable.. \u2013\u00a0Nikos M. Jun 17 '14 at 20:36\n@NikosM.: I don't understand how \"notions of computability\" can be related to my question. What does this mean? Perhaps you are asking \"do the E_i's consist of computable functions\"? Also regarding the last sentence, precise what is it that \"would be non-decidable\"? \u2013\u00a0RghtHndSd Jun 17 '14 at 21:04","date":"2016-05-03 09:14:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8382852673530579, \"perplexity\": 318.5381130822715}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-18\/segments\/1461860121090.75\/warc\/CC-MAIN-20160428161521-00037-ip-10-239-7-51.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction.}
The notion of a self--consistent ``gravitational entropy'', distinct from (though possibly related with) the entropy of the sources (thermal sources or black holes) is an open problem with interesting theoretical ramifications in General Relativity. This notion comes originally from Penrose's old idea \cite{arrow1} of the ``arrow of time'', associated with the ratio of scalars contractions of the Weyl and Ricci tensors. This idea was further developed and modified by different authors~\cite{arrow1,arrow2,arrow3,arrow4,arrow5,arrow6}. A more recent approach by Clifton, Ellis and Tavakol (CET) \cite{CET} no longer relies on invariant curvature scalars, but on an ``effective'' energy--momentum tensor associated with the ``free gravitational field'' and obtained from the Bell--Robinson tensor. An alternative approach is based on an entropy functional from the Kullback-Leibler divergence of Information Theory \cite{Info}, applied by Hosoya and Buchert (HB) \cite{HB1,HB2,HB3} to a cosmological context, and in a modified form (the ``HBq'' proposal denoting the original HB proposal by ``HBp'') to spherically symmetric Lema\^{\i}tre--Tolman--Bondi (LTB) dust models in \cite{part1} by means of the quasi--local weighted average instead of Buchert's average.
In the present article we use generic LTB models to probe the CET, HBp and HBq gravitational entropy proposals, as a first step to test the theoretical solidity of their predictions and properties. While LTB models \cite{LTB} are highly idealized toy models, they are particularly well suited to understand and study a host of non--linear non--perturbative relativistic effects of cosmological and astrophysical self--gravitating systems by means of mathematically tractable methods (see the comprehensive reviews in \cite{kras1,kras2,BKHC2009,celerier,focus}). They have been used to describe a wide variety of phenomena: structure formation and late time cosmological inhomogeneities \cite{KH,ltbstuff}, fitting cosmological observations without resorting to dark energy (see \cite{kras2,BKHC2009,marranot} for a review), testing averaging formalisms \cite{LTBave1,LTBave2,sussBR,sussIU,suss2011}, cosmic censorship \cite{lemos,joshi} and even describing some effects in quantum gravity \cite{quantum}.
In order to use LTB models to examine and compare the gravitational entropy proposals, we describe their dynamics in terms of an initial value formulation based on an alternative representation of coordinate independent scalars (the ``quasi--local'' or ``q--scalars'') that follow from a weighted proper volume average on comoving domains (see \cite{part1,part2} for a comprehensive study). These scalar variables have been very useful to look at the models under a dynamical systems approach \cite{sussDS1,sussDS2}, to examine their asymptotic behaviour in the radial direction \cite{RadAs}, the evolution of radial profiles and void formation \cite{RadProfs}, the existence of back--reaction and ``effective'' acceleration in the context of Buchert's formalism \cite{sussBR,sussIU,suss2011}, and even to study dark energy sources compatible with the LTB metric \cite{sussQL,suss2009}. As shown in \cite{part2}, the q--scalars and their fluctuations and perturbations lead to a covariant and gauge invariant formalism of ``exact perturbations'' on a Friedmann-Lema\^itre-Robertson-Walker (FLRW) abstract background defined by the q--scalars themselves (which satisfy FLRW time evolution laws). Hence, we find it useful to express these exact perturbations in terms of an exact covariant generalization of the growing and decaying density modes of linear perturbations of dust sources (see comprehensive discussion in \cite{sussmodes}). We must also mention the dynamical studies of LTB models (with zero and nonzero $\Lambda$) by Wainwright and Andrews \cite{WainAndr} in terms of growing and decaying density modes (see the review of this article in Appendix B of \cite{sussmodes}) and in terms of frame variables by Coley {\it et al} \cite{coley}.
It is known that the condition for positive entropy production for the HBp and HBq proposals is a negative statistical correlation between the fluctuations of the energy density and Hubble expansion scalar (see \cite{HB1,HB2,HB3,part1}). We prove in this paper that the entropy production condition for the CET proposal is also a negative correlation between density and Hubble fluctuations. This is a new result that enhances the marginal discussion in \cite{CET} of the application of the CET proposal to LTB models. However, there are subtle but important differences between the various proposals: the condition for positive entropy production from the CET proposal is a local, necessary and sufficient, condition, while the conditions for the HBp and HBq proposals are domain dependent, and thus non-local, though they are expressible in terms of local fluctuations as sufficient (not necessary) conditions.
While the conditions for entropy production are appealing and elegant, it is necessary to verify their actual fulfilment on the models. We undertake this task by means of analytic relations that hold at various asymptotic limits characteristic of the models: near the Big Bang, asymptotic time range, maximal expansion, near collapse and radial asymptotic regime. By collecting all this information we are able to provide a roughly consistent qualitative description of the full time behaviour of the entropy and entropy production valid for the three proposals. This description reveals that entropy production for the three entropies is non--negative for all domains and regions of generic LTB models in which the density growing mode is dominant. As a consequence, entropy production is necessarily negative in the early time evolution of all models with a non-zero decaying mode (which is always dominant near a non--simulteneous Big Bang). These results can be connected with those of theoretical studies by Goode and Wainwright \cite{GooWain} and Lim {\it et al} \cite{Limetal} of the early time behavior and initial singularities in generic inhomogeneous models, and also with those obtained in a recent article by Bolejko and Stoeger \cite{bolstoeg}, who undertook a numerical study of entropy production (from various conceptual proposals) for spherically symmetric spacetimes with non-zero pressure and viscosity (we compare our results with those of these articles in section 10).
The content of the article is given as follows. In section 2, we review the definitions of the CET and the original HB entropy proposals. Section 3 is devoted to the presentation of LTB spacetimes in terms of the q--scalars, their fluctuations and perturbations. We provide a full derivation in section 4 of the conditions for positive entropy production for all proposals. In section 5 we comment on the subtle differences between these entropy production conditions: their `local' vs `non--local' and `necessary and sufficient' vs `only sufficient' nature. We examine in section 6 the fulfilment of these conditions in the asymptotic ranges of the time evolution of the models, looking at the general case (both density growing and decaying modes are non-zero), as well as the cases when one either one of these modes is suppressed. We use in section 7 the information obtained in section 6 to describe qualitatively the time evolution of the entropies, and present in section 8 three numerical examples that fully corroborate this qualitative description for the CET entropy. In section 9 we examine the integrability conditions of the CET entropy (which is defined through a Gibbs one--form), as well the radial asymptotic behaviour of the three entropies. We present a full discussion and summary of our results in section 10. The relation between the q--scalars and the traditional LTB variables is discussed in Appendix A, analytic solutions of the Friedman equation are summarized in Appendix B, Appendix C provides a brief discussion of the evolution equations used for the numerical examples. Formal results on the convergence of the HBp and HBq entropy functionals used in section 7 are proven in Appendix D.
\section{The gravitational entropy.}
In this section, we will briefly present the two entropy proposals studied in this paper. We refer the reader to the original papers for more details.
\subsection{The Clifton, Ellis and Tavakol (CET) entropy}
In the CET entropy proposal \cite{CET} a gravitational entropy $s_{\tiny{\textrm{gr}}}$ is constructed from the ``free'' gravitational field by demanding that it complies with basic consistency criteria, namely: that $s_{\tiny{\textrm{gr}}}$ and its associated entropy production are non--negative and that it is compatible with the Beckenstein--Hawking area formula when applied to black holes. For this purpose CET consider the Bell--Robinson tensor, $T_{abcd}$, which is the only totally symmetric traceless tensor that can be constructed with the conformal Weyl tensor $C_{abcd}$.
However, since $T_{abcd}$ is fourth order (and thus its dimensions are $1/\hbox{cm}^4$), CET consider a ``square root'' procedure expressing it as an irreducible algebraic decomposition in terms of a symmetric traceless second order tensor $t_{ab}$, which allows for a derivation of an ``effective'' or ``super'' energy--momentum tensor ${\cal T}_{ab}$ associated with the free gravitational field, with ``gravitational'' energy--momentum fluxes (energy density $\rho_{\tiny{\textrm{gr}}}$, pressure $p_{\tiny{\textrm{gr}}}$, anisotropic stresses $\Pi^{ab}_{\tiny{\textrm{gr}}}$ and heat flux $q^a_{\tiny{\textrm{gr}}}$) constructed by invariant contractions with the matter 4--velocity $u^a$ and projector $h_{ab}=u_au_b+g_{ab}$. A self consistent form for the ``gravitational entropy'' emerges by analogy with standard laws of fluid thermodynamics applied to the quantities associated with ${\cal T}_{ab}$. CET consider two paradigmatic types of gravitational fields, for which ${\cal T}_{ab}$ takes simple forms in terms of the Newman--Penrose conformal invariants $\Psi_2$ and $\Psi_4$: the ``Coulomb--like'' Petrov type D and the ``wave--like'' Petrov type N fields (see \cite{CET} for further details).
For Coulomb--like fields CET obtain the following effective tensor and associated fluxes:
\begin{eqnarray} \fl \frac{{\cal T}^{ab}}{8\pi}=\alpha|\Psi_2| \left[x^ax^b+y^ay^b-2\left(z^az^b-u^au^b\right)\right]=\rho_{\tiny{\textrm{gr}}} u^au^b + p_{\tiny{\textrm{gr}}} h^{ab}+2q^{(a}_{\tiny{\textrm{grav}}} u^{b)}+\Pi^{ab}_{\tiny{\textrm{gr}}},\nonumber\\
\fl 8\pi\rho_{\tiny{\textrm{gr}}}=2\alpha|\Psi_2|,\quad p_{\tiny{\textrm{gr}}}=q^a_{\tiny{\textrm{gr}}}=0,\quad 8\pi\Pi^{ab}_{\tiny{\textrm{gr}}}=\frac{\alpha|\Psi_2|}{2}(x^ax^b+y^ay^b-z^az^b+u^au^b),\nonumber\label{effective}\\
\end{eqnarray}
where $\alpha$ is a positive constant to provide the appropriate physical units and $\{x^a,\,y^a,\,z^a,\,u^a\}$ is an orthonormal tetrad. By analogy with the off--equilibrium Gibbs equation (see \cite{rund}) CET obtain the following expression for the entropy production:
\begin{equation}T_{\tiny{\textrm{gr}}}\dots_{\tiny{\textrm{gr}}} = (\rho_{\tiny{\textrm{gr}}} V)\dot{}=-V\sigma_{ab}\left[\Pi^{ab}_{\tiny{\textrm{gr}}}+\frac{4\pi(\rho+p)}{3\alpha|\Psi_2|}E^{ab}\right],\label{gibbs}\end{equation}
where $V$ is a suitable local volume, $E^{ab}=u_au_b C^{acbd}$ is the electric Weyl tensor and the ``gravitational'' temperature $T_{\tiny{\textrm{gr}}}$ is given by
\begin{equation} T_{\tiny{\textrm{gr}}} = \frac{\left|\dot u_az^a+\theta/3+\sigma_{ab}z^az^b\right|}{2\pi},\label{Tgr} \end{equation}
where $\dot u_a =u^b\nabla_a u_b$ is the 4--acceleration and $\theta\equiv \tilde\nabla_cu^c=h_c^b\nabla_b u^c$ is the isotropic expansion scalar. As commented by CET, the terms inside the brackets in the right hand side of (\ref{gibbs}) play the role of ``effective'' relativistic dissipation terms in analogy with dissipative matter sources, though this is merely an analogy, and since the actual sources are conserved, the Gibbs equation (\ref{gibbs}) does not imply that they exchange energy or momentum with the free gravitational fields associated with (\ref{effective}). On the other hand, CET justify $T_{\tiny{\textrm{gr}}}$ in (\ref{Tgr}) as a local temperature that reduces to the semi--classical Unruh and Hawking temperatures in the appropriate limits (see \cite{CET}).
\subsection{The Hosoya--Buchert (HB) entropy}
The HB entropy is originally inspired by the relative information entropy, also known as the Kullback-Leibler divergence \cite{Info}, defined as:
$$
s_{\textrm{\tiny{KL}}}=\sum_{i}p_{i}\ln\left[\frac{p_{i}}{{\cal P}_{i}}\right],\label{KL}
$$
where $p_{i}$ is the actual probability density of the random variable $i$, and ${\cal P}_{i}$ the expected one. If the two probability densities coincide for all the variables (i.e. if our information on the system is complete), then $s_{\textrm{\tiny{KL}}}=0$. On the other hand, if, for at least one of the variables, $p_{i}\neq {\cal P}_{i}$, then in general, $s_{\textrm{\tiny{KL}}}\neq 0$. Hosoya and Buchert constructed in \cite{HB1,HB2,HB3} an entropy based on the Kullback-Leibler divergence, suitable to study the emergence of inhomogeneities in cosmological models. They defined what we shall denote by the Hosoya-Buchert (HB) entropy:
\begin{equation}
s_{\textrm{\tiny{HB}}}=\int_{{\cal D}}\rho\ln\left[\frac{\rho}{\langle\rho\rangle_{\cal D}}\right]{\rm{d}}\mu,\label{sHB}
\end{equation}
where $\rho$ is the relativistic energy density and $d\mu$ is the Riemannian measure on the spacelike hypersurfaces on which the Buchert's average $\langle\,.\,\rangle_{\cal D}$ of $\rho$:
\begin{equation}\langle\rho\rangle_{{\cal D}}=\frac{\int_{{\cal D}}{\rho\,{\rm{d}}\mu}}{\int_{{\cal D}}{{\rm{d}}\mu}},\label{rhoav}\end{equation}
is computed over an averaging domain ${\cal D}$. Evidently, an entropy measure like that of HB can also be defined with any self--consistent energy density average . The entropy measure (\ref{sHB}) can also be defined with a scalar weighed average (q--average) (see \cite{part1}). We provide rigorous proof in Appendix D.1 that $s_{\textrm{\tiny{HB}}}$ is non--negative for every scalar averaging formalism and for every domain as long as $\rho\geq 0$.
\section{LTB dust models in the q--scalar representation.}
We shall describe LTB dust models in the following useful FLRW--like metric parametrization
\footnote{The relation between this metric parametrization and the standard metric form and variables of the models is discussed in detail in Appendix A. See \cite{part1,part2} for a comprehensive discussion on the q--scalar representation introduced in this section.}
\begin{eqnarray} \fl {\rm{d}} s^2 =-{\rm{d}} t^2+ a^2\left[\frac{\Gamma^2}{1-{\cal{K}}_{q0}r^2}{\rm{d}} r^2+r^2\left({\rm{d}}\vartheta^2+\sin^2\theta{\rm{d}}\varphi^2\right)\right],\label{ltb2}\\
\fl a=a(t,r),\qquad \dot a^2 =\left[\frac{\partial a}{\partial t}\right]^2=\frac{8\pi\rho_{q0}}{3a}-{\cal{K}}_{q0},\qquad \Gamma=1+\frac{ra'}{a},\qquad a' =\frac{\partial a}{\partial r},\label{aGdef}\end{eqnarray}
where ${\cal{K}}_{q0}={\cal{K}}_q(t_0,r)$ and $\rho_{q0}=\rho_q(t_0,r)$ are defined further ahead (see equation (\ref{rhoKK})) and the subindex ${}_0$ will denote henceforth evaluation at an arbitrary fiducial hypersurface $t=t_0$. We remark that $a_0=\Gamma_0=1$ (see Appendix A).
It is useful to describe the dynamics of the models by means of their covariant objects given in terms of the representation of ``q--scalars'' and their perturbations (see \cite{part1,part2,sussmodes} for a comprehensive discussion on the definition and properties of these perturbations). For every LTB scalar $A$, the associated q--scalar $A_q$, perturbation $\delta^{(A)}_q$ and fluctuation ${\textrm{\bf{D}}}_q(A)$ are defined by the correspondence rules
\begin{eqnarray} A_q =\frac{\int_0^r{A\,a^3\,\Gamma\,\bar r^2\,{\rm{d}}\bar r}}{\int_0^r{a^3\,\Gamma\,\bar r^2\,{\rm{d}}\bar r}}=\frac{3\int_0^r{A\,a^3\,\Gamma\,\bar r^2\,{\rm{d}}\bar r}}{a^3r^3},\label{Aqdef}\\
\delta^{(A)}_q =\frac{A-A_q}{A_q} = \frac{rA'_q/A_q}{3\Gamma}=\frac{1}{r^3 a^3 A_q}\int_0^r{A'\,\bar r^3\, a^3{\rm{d}} \bar r},\label{Dadef}\\
{\textrm{\bf{D}}}_q(A)\equiv A-A_q=A_q\delta^{(A)}_q=\frac{r A'_q}{3\Gamma}=\frac{1}{r^3\,a^3}\int_0^r{A'\, \bar r^3\,a^3{\rm{d}} \bar r}.\label{Deldef}
\end{eqnarray}
where (\ref{Dadef}) and (\ref{Deldef}) follow directly by differentiation and integration by parts of (\ref{Aqdef}) and allow, in general, to compute $\delta^{(A)}_q$ and ${\textrm{\bf{D}}}_q(A)$ in terms of the gradients $A'_q$ and the scale factor $\Gamma$. The q--scalars are directly related to proper volume averages with weight factor ${\cal{F}}=\sqrt{1-{\cal{K}}_{q0}r^2}$ (see \cite{part1,part2} for further details and explanations).
The basic LTB covariant scalars are: (i) the rest--mass density $\rho$, (ii) the Hubble scalar ${\cal{H}}\equiv \theta/3$ and (iii) the spatial curvature scalar ${\cal{K}}\equiv {\cal{R}}^{(3)}/6$ (with ${\cal{R}}^{(3)}$ the Ricci scalar of hypersurfaces $t=$ const.). In the q--scalar representation these scalars take the forms of exact perturbations \cite{part1,part2}:
\begin{equation}\fl \rho=\rho_q(1+\delta^{(\rho)}_q),\qquad {\cal{H}}={\cal{H}}_q(1+\delta^{(\HH)}_q),\qquad {\cal{K}}={\cal{K}}_q(1+\delta^{(\KK)}_q),\label{rhoHHKK}\end{equation}
with their associated q--scalars and perturbations given by \cite{sussmodes}:
\begin{eqnarray} \fl \frac{8\pi}{3}\rho_q =\frac{8\pi}{3}\frac{\rho_{q0}}{a^3}=\frac{\Omega_{q0}{\cal{H}}_{q0}^2}{a^3},\qquad {\cal{K}}_q=\frac{{\cal{K}}_{q0}}{a^2}=\frac{(\Omega_{q0}-1){\cal{H}}_{q0}^2}{a^2},\label{rhoKK}\\
\fl {\cal{H}}_q =\frac{\dot a}{a}=\frac{\left[\frac{8\pi}{3}\rho_{q0}-{\cal{K}}_{q0}a\right]^{1/2}}{a^{3/2}}=\frac{{\cal{H}}_{q0}\left[\Omega_{q0}-(\Omega_{q0}-1)a\right]^{1/2}}{a^{3/2}},\label{HHq}\\
\fl \delta^{(\rho)} _q= \frac{\JJ_{\tiny{\textrm{(g)}}}+\JJ_{\tiny{\textrm{(d)}}}}{1-\JJ_{\tiny{\textrm{(g)}}}-\JJ_{\tiny{\textrm{(d)}}}}=\frac{1+\delta^{(\rho)}_0}{\Gamma}-1,\label{Drho}\\
\fl \delta^{(\KK)}_q = \frac{2\,(\JJ_{\tiny{\textrm{(g)}}}+\JJ_{\tiny{\textrm{(d)}}}-\Delta_0^{\tiny{\textrm{(g)}}})}{3(1-\JJ_{\tiny{\textrm{(g)}}}-\JJ_{\tiny{\textrm{(d)}}})}=\frac{2/3+\delta^{(\KK)}_0}{\Gamma}-\frac{2}{3},\label{DKK}\\
\fl \delta^{(\HH)}_q= \frac{(2+\Omega_q)(\JJ_{\tiny{\textrm{(g)}}}+\JJ_{\tiny{\textrm{(d)}}})-2(1-\Omega_q)\Delta_0^{\tiny{\textrm{(g)}}}}{6(1-\JJ_{\tiny{\textrm{(g)}}}-\JJ_{\tiny{\textrm{(d)}}})}=\frac{\Omega_q}{2}\delta^{(\rho)}-\frac{\Omega_q-1}{2}\delta^{(\KK)},\label{Dh}\end{eqnarray}
where the q--scalar $\Omega_q$ and its perturbation are given by
\begin{equation}\fl \Omega_q \equiv \frac{8\pi\rho_q}{3{\cal{H}}_q^2}=\frac{\Omega_{q0}}{\Omega_{q0}+(1-\Omega_{q0})a},\qquad \delta^{(\Omega)}_q = \frac{(1-\Omega_q)\,\left(\JJ_{\tiny{\textrm{(g)}}}+\JJ_{\tiny{\textrm{(d)}}}+2\Delta_0^{\tiny{\textrm{(g)}}}\right)}{3(1-\JJ_{\tiny{\textrm{(g)}}}-\JJ_{\tiny{\textrm{(d)}}})},\label{OmDOm}\end{equation}
and $\JJ_{\tiny{\textrm{(g)}}},\,\JJ_{\tiny{\textrm{(d)}}}$ are the exact generalizations of the growing and decaying density modes of linear perturbation theory (see \cite{sussmodes}):
\begin{eqnarray} \JJ_{\tiny{\textrm{(g)}}} = 3\Delta_0^{\tiny{\textrm{(g)}}}\left[{\cal{H}}_q (t-t_{\textrm{\tiny{bb}}})-\frac{2}{3}\right],\qquad \hbox{density growing mode},\label{gmode}\\
\JJ_{\tiny{\textrm{(d)}}} = 3\Delta_0^{\tiny{\textrm{(d)}}}\,{\cal{H}}_q,\qquad\qquad\qquad \hbox{density decaying mode}, \label{dmode}\end{eqnarray}
where $t_{\textrm{\tiny{bb}}}=t_{\textrm{\tiny{bb}}}( r)$ is the Big Bang time, $\Delta_0^{\tiny{\textrm{(g)}}}$ and $\Delta_0^{\tiny{\textrm{(d)}}}$ (both assumed non-zero unless stated otherwise) are the ``amplitudes'' of the modes:
\begin{eqnarray} \fl \Delta_0^{\tiny{\textrm{(g)}}} \equiv \frac{\delta^{(\rho)}_{q0}-\frac{3}{2}\delta^{(\KK)}_{q0}}{1+\delta^{(\rho)}_{q0}},\label{Dig}\\ \fl \Delta_0^{\tiny{\textrm{(d)}}} \equiv -\frac{{\cal{H}}_{q0}(t_0-t_{\textrm{\tiny{bb}}})\left[\delta^{(\rho)}_{q0}-\frac{3}{2}\delta^{(\KK)}_{q0}\right]+\delta^{(\KK)}_{q0}-\delta^{(\rho)}_{q0}}{{\cal{H}}_{q0}(1+\delta^{(\rho)}_{q0})}=\frac{r\,t_{\textrm{\tiny{bb}}}'}{3(1+\delta^{(\rho)}_{q0})} \label{Did}\end{eqnarray}
which can also be given in terms of $\delta^{(\HH)}_{q0},\,\delta^{(\Omega)}_{q0}$ by the relations
\begin{equation} \delta^{(\rho)}_{q0}=\delta^{(\Omega)}_{q0}+2\delta^{(\HH)}_{q0},\qquad \delta^{(\KK)}_{q0}=2\delta^{(\HH)}_{q0}-\frac{\Omega_{q0}\delta^{(\Omega)}_{q0}}{1-\Omega_{q0}}.\label{ivpconstr} \end{equation}
Given the constraints among the basic initial q--scalars, $A_{q0}=\rho_{a0},\,{\cal{H}}_{q0},\,{\cal{K}}_{q0},\,\Omega_{q0}$, and the relation between their gradients and the perturbations $\delta^{(A)}_{q0}=(r/3)A'_{q0}/A_{q0}$ that follows from (\ref{Dadef}), any LTB model can be uniquely specified by selecting as free parameters (initial conditions) any two of the four $A_{q0}$. The dynamics of the models becomes fully determined, either analytically through the scaling laws (\ref{rhoHHKK})--(\ref{ivpconstr}) once we have add the analytic expression relating ${\cal{H}}_q(t-t_{\textrm{\tiny{bb}}})$ with $a$ (see (\ref{HTq})--(\ref{H32}) in Appendix B), or numerically by solving the appropriate systems of evolution equations (see Appendix C and examples in \cite{part2,sussmodes}). We shall use for various calculations and graphs in the remaining of this paper both the numerical and analytic approaches.
The main LTB proper tensors are the shear tensor ($\sigma_{ab}=\tilde\nabla_{(a}u_{b)}-{\cal{H}} h_{ab}$), the electric Weyl tensor ($E_{ab}=u_au_b C^{acbd}$) and Weyl tensor ($C^{acbd}$). These tensors and their eigenvalues take the form:
\numparts\begin{eqnarray} \fl \sigma_{ab}=\Sigma\,\hbox{\bf{e}}_{ab},\qquad E_{ab}=\Psi_2\,\hbox{\bf{e}}_{ab},\qquad C^{ab}_{cd}=\Psi_2\,\left(h^{[a}_{[c}-3u_{[c}u^{[a}\right)\,\hbox{\bf{e}}^{b]}_{d]},\label{sigEC}\\
\fl \Sigma =\frac{1}{6}\hbox{\bf{e}}_{ab}\sigma^{ab}=-\frac{\dot\Gamma}{3\Gamma}=-{\textrm{\bf{D}}}_q({\cal{H}}),\qquad \Psi_2 =\frac{1}{6}\hbox{\bf{e}}_{ab}E^{ab}=\frac{4\pi}{3}\,{\textrm{\bf{D}}}_q(\rho),\label{SigPsi2}\end{eqnarray}\endnumparts
where $\Psi_2$ is the Petrov type D conformal invariant in (\ref{effective}) and $\hbox{\bf{e}}^a_b=h^a_b-3n^a n_b$, with $n_a=\sqrt{g_{rr}}\delta^r_a$, is the common covariantly constant tensor of Petrov type D LRS spacetimes \cite{LRS}.
\section{Gravitational entropy in LTB dust models.}
\subsection{The CET gravitational entropy.}
From the expressions associated with the effective energy--momentum tensor (\ref{effective}) and the entropy production law (\ref{gibbs}) and gravitational temperature (\ref{Tgr}), CET obtain the following entropy production law for LTB models:
\begin{equation} T_{\tiny{\textrm{gr}}}\dots_{\tiny{\textrm{gr}}}=\partial_{t}(\rho_{\tiny{\textrm{gr}}} V),\label{CET2} \end{equation}
where $V=\ell^3 = a^3\Gamma$ is the local volume defined by the condition ${\cal{H}} =\dot\ell/\ell=(a^3\Gamma)\dot{}/(3a^3\Gamma)$ and $\rho_{\tiny{\textrm{gr}}},\,T_{\tiny{\textrm{gr}}}$ are given by
\begin{eqnarray}
\fl 8\pi\rho_{\tiny{\textrm{gr}}} &=&2\alpha\frac{|M-4\pi R^{3}\rho/3|}{R^{3}}=2\alpha|\Psi_2|=\frac{8\pi\alpha}{3}|{\textrm{\bf{D}}}_q(\rho)|=\frac{8\pi\alpha}{3}\rho_q|\delta^{(\rho)}_q|,\label{rhograv2}\\
\flT_{\tiny{\textrm{gr}}}&=&\frac{1}{2\pi}\left|\frac{\dot{R}^{'}}{R^{'}}\right|=\frac{|{\cal{H}}_q|\,|\, 1+3\delta^{(\HH)}_q\,|}{2\pi}.\label{Tgrav2}
\end{eqnarray}
where $R=a r$, and we used (\ref{ivf1}), (\ref{basic}), (\ref{qbasic}) and (\ref{SigPsi22}) to express $\rho,\,M/R^3$ and $\dot R'/R'$ in terms of q--scalars and their fluctuations. Inserting (\ref{rhograv2}) and (\ref{Tgrav2}) into (\ref{CET2}) we obtain:
\begin{equation}
\fl \dots_{\tiny{\textrm{gr}}}=\frac{2\pi\alpha}{3}\frac{\partial_{t}\left(\rho_{q}a^3\Gamma\left|\delta^{(\rho)}\right|\right)}{|{\cal{H}}_{q}||1+3\delta^{(\HH)}_q|}= \frac{2\pi\alpha\rho_{q0}}{3}\frac{\partial_{t}\left(\Gamma\left|\delta^{(\rho)}\right|\right)}{|{\cal{H}}_{q}||1+3\delta^{(\HH)}_q|}\label{CETc12}.
\end{equation}
which determines the sign of $\dots_{\tiny{\textrm{gr}}}$:
\begin{equation}
\dots_{\tiny{\textrm{gr}}}\geq 0 \quad \Leftrightarrow \quad \partial_{t}\left[\Gamma|\delta^{(\rho)}_q|\right]\geq 0.\label{CETc22}
\end{equation}
From (\ref{rhoHHKK}) and (\ref{SigPsi2}) we obtain after some calculations:
\begin{eqnarray} \fl \partial _t\left(\Gamma\,|\delta^{(\rho)}_q|\right) = \dot\Gamma|\delta^{(\rho)}|+\Gamma \partial_t(|\delta^{(\rho)}|)=\left[\partial_t(\,|\delta^{(\rho)}_q|\,)+3{\cal{H}}_q|\delta^{(\rho)}_q|\delta^{(\HH)}_q\right]\Gamma,\end{eqnarray}
where we used $\dot \Gamma =3\Gamma {\textrm{\bf{D}}}_q({\cal{H}})=3\Gamma{\cal{H}}_q\delta^{(\HH)}$ (from (\ref{SigPsi2})) and we assume henceforth that $\Gamma>0$ holds to avoid shell crossing singularities \cite{RadAs,RadProfs,sussmodes}. Considering that
\begin{eqnarray}\fl |\delta^{(\rho)}_q| = \left\{\begin{array}{c}
\delta^{(\rho)}_q\quad\hbox{if}\quad \delta^{(\rho)}_q> 0,\\
0 \quad\hbox{if}\quad \delta^{(\rho)}_q= 0,\\
-\delta^{(\rho)}_q \quad\hbox{if}\quad \delta^{(\rho)}_q< 0,
\end{array}\right.\quad \partial_t(\,|\delta^{(\rho)}_q|\,) = \left\{\begin{array}{c}
\partial_t(\,\delta^{(\rho)}_q\,)=\dot\delta^{(\rho)}_q\quad\hbox{if}\quad \delta^{(\rho)}_q> 0,\\
0 \qquad\qquad\quad\hbox{if}\quad \delta^{(\rho)}_q= 0,\\
\partial_t(-\delta^{(\rho)}_q)=-\dot\delta^{(\rho)}_q \quad\hbox{if}\quad \delta^{(\rho)}_q< 0,
\end{array}\right.\nonumber\\
\end{eqnarray}
and using the evolution equation (19a) of \cite{part2} to eliminate $\dot\delta^{(\rho)}_q$, we obtain
\begin{equation}\fl \partial _t\left(\Gamma\,|\delta^{(\rho)}|\right) = \left\{\begin{array}{c}
-3{\cal{H}}_q\delta^{(\HH)}_q=-3{\cal{H}}_q\delta^{(\HH)}_q\delta^{(\rho)}_q/|\delta^{(\rho)}_q| \quad\hbox{if}\quad \delta^{(\rho)}_q> 0,\\
0 \qquad\qquad\qquad\qquad\qquad\qquad \hbox{if}\quad \delta^{(\rho)}_q= 0,\\
3{\cal{H}}_q\delta^{(\HH)}_q=3{\cal{H}}_q\delta^{(\HH)}_q\delta^{(\rho)}_q/|\delta^{(\rho)}_q| \quad\hbox{if}\quad \delta^{(\rho)}_q< 0.
\end{array}\right.\label{CETc122}\end{equation}
where we used the fact that $\delta^{(\rho)}_q/|\delta^{(\rho)}_q|=\hbox{signum}(\delta^{(\rho)})$. From (\ref{Deldef}), and since $\rho_q\geq 0$ implies that $\delta^{(\rho)}_q$ and ${\textrm{\bf{D}}}_q(\rho)$ have the same sign, we can now express condition (\ref{CETc22}) in terms of the fluctuations ${\textrm{\bf{D}}}_q({\cal{H}})$ and ${\textrm{\bf{D}}}_q(\rho)$
\begin{equation}
\dots_{\tiny{\textrm{gr}}}\geq 0 \quad \Leftrightarrow \quad \left\{\begin{array}{c}
{\textrm{\bf{D}}}_q({\cal{H}})<0\quad\hbox{if}\quad {\textrm{\bf{D}}}_q(\rho)> 0,\\
{\textrm{\bf{D}}}_q({\cal{H}})=0 \quad\hbox{if}\quad {\textrm{\bf{D}}}_q(\rho)= 0,\\
{\textrm{\bf{D}}}_q({\cal{H}})>0 \quad\hbox{if}\quad {\textrm{\bf{D}}}_q(\rho)< 0,
\end{array}\right..\label{CETc23}
\end{equation}
or equivalently from (\ref{Deldef}):
\begin{equation} \fl \dots_{\tiny{\textrm{gr}}}\geq 0 \quad \Leftrightarrow \quad {\textrm{\bf{D}}}_q(\rho){\textrm{\bf{D}}}_q({\cal{H}})=\frac{\rho'_q{\cal{H}}'_q}{(3\Gamma/r)^2}=\frac{\int_0^r{\rho'\,(a\bar r)^3{\rm{d}}\bar r}\int_0^r{{\cal{H}}'\,(a\bar r)^3{\rm{d}}\bar r}}{R^6}\leq 0,\label{CETcond}\end{equation}
which is reminiscent to a condition of negative correlation between fluctuations of the energy density and Hubble scalar.
We remark, from (\ref{SigPsi2}), that the non--negative entropy production condition (\ref{CETcond}) can also be written as:
\begin{equation} \fl \dots_{\tiny{\textrm{gr}}}\geq 0 \quad \Leftrightarrow \quad \Sigma\,\Psi_2 \geq 0\qquad \hbox{or}\qquad \dots_{\tiny{\textrm{gr}}}\geq 0 \quad \Leftrightarrow \quad \sigma_{ab}E^{ab}\geq 0\label{CETcond2}\end{equation}
where $\Sigma$ and the conformal invariant $\Psi_2$ are, respectively, the eigenvalues of the shear and electric Weyl tensors. Another alternative form for (\ref{CETcond}) is
\begin{equation} \dots_{\tiny{\textrm{gr}}}\geq 0 \quad \Leftrightarrow \quad {\textrm{\bf{D}}}_q({\cal{R}})\,{\textrm{\bf{D}}}_q({\cal{H}}) \leq 0,\end{equation}
where ${\cal{R}}$ is the (4--dimensional) Ricci scalar and we used the fact that ${\cal{R}}=8\pi\rho$, and thus ${\textrm{\bf{D}}}({\cal{R}})=8\pi{\textrm{\bf{D}}}(\rho)$, hold for LTB models.
\subsection{The HBp and HBq gravitational entropies.}
For a fixed arbitrary spherical domain ${\cal {D}}[r_b]$ centered on $r=0$, whose proper volume is
\begin{equation} \fl {\cal{V}}_p[r_b]= \int_{{\cal {D}}[r]}{{\rm{d}}{\cal{V}}_p}=4\pi\int_0^{r_b}{{\cal{F}}^{-1}a^3\Gamma \bar r^2{\rm{d}}\bar r},\qquad {\cal{F}}\equiv \sqrt{1-{\cal{K}}_{q0}r^2},\label{propvol}\end{equation}
the original HB entropy functional (\ref{sHB}) for Buchert's average (\ref{rhoav}) (to be denoted henceforth as the ``HBp'' entropy) applied to LTB models takes the form:
\footnote{We will use the subindex ${}_p$ to emphasize the connection to Buchert's proper volume average, using the notation ``$\langle A\rangle_p[r_b]$'' as domain indicator instead of the usual form $\langle A\rangle_{\cal {D}}$. The subindex ${}_p$ will be attached to all quantities related to this average, such as the local and non--local fluctuations ${\textrm{\bf{D}}}_p(A)$ and ${\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_p(A)$ \cite{part2}, and their associated local functions, $A_p$, to distinguish them from the analogous objects constructed with the correspondence rule (\ref{Aqdef}): the q--scalars, the q--average and their local and non--local fluctuations and perturbations, all of which carry the subindex ${}_q$.}
\begin{eqnarray} \fl s_{\textrm{\tiny{HB}}}{}_p[r_b] -s_{\textrm{\tiny{HB}}}{}_p^{(\textrm{\tiny{eq}})}= \gamma_0\int_{{\cal {D}}[r_b]}{\rho\ln\left[\frac{\rho}{\langle \rho\rangle_p[r_b]}\right]{\rm{d}}{\cal{V}}_p}=\gamma_0{\cal{V}}_p[r_b]\left\langle \rho\ln\left[\frac{\rho}{\langle \rho\rangle_p[r_b]}\right]\right\rangle_p[r_b],\nonumber\\\label{sHB2}\end{eqnarray}
where $s_{\textrm{\tiny{HB}}}{}_p^{(\textrm{\tiny{eq}})}>0$ is the ``equilibrium'' entropy, $\gamma_0$ is a constant so that the left hand side of (\ref{sHB2}) has units of entropy, while the density average is
\begin{equation} \langle \rho\rangle_p[r_b]=\frac{\int_{{\cal {D}}[r]}{\rho\,{\rm{d}}{\cal{V}}_p}}{\int_{{\cal {D}}[r]}{{\rm{d}}{\cal{V}}_p}} =\frac{M_p[r_b]}{{\cal{V}}_p[r_b]}\label{rhoave2}\end{equation}
where $M_p$ is the domain's proper mass--energy function
\begin{equation} M_p[r_b]=\int_{{\cal {D}}[r]}{\rho\,{\rm{d}}{\cal{V}}_p}=\int_0^{r_b}{\rho {\cal{V}}'_p {\rm{d}} r}=4\pi\int_0^{r_b}{\rho\, {\cal{F}}^{-1}a^3\Gamma \bar r^2{\rm{d}}\bar r},\label{Mp}\end{equation}
which is independent of $t$ (since $\rho a^3\Gamma=\rho_0$ follows from (\ref{rhoHHKK}), (\ref{rhoKK}) and (\ref{Drho})). Following \cite{HB1,HB2,HB3} we evaluate $\dot s_{\textrm{\tiny{HB}}}{}_p$ by applying to (\ref{sHB2}) the time derivative commutation rule for any averaged scalar (we omit the domain indicator $[r_b]$ to simplify notation):
\begin{eqnarray} \fl \langle A\rangle\dot{}_p-\langle\dot A\rangle_p =3\left[\langle A{\cal{H}}\rangle_p-\langle A\rangle_p\langle \HH\rangle_p\right]=3\langle (A-\langle A\rangle_p)({\cal{H}}-\langle \HH\rangle_p)\rangle_p=\hbox{{\bf Cov}}_p(A,{\cal{H}}),\nonumber\\
\label{tder}\end{eqnarray}
which yields after some algebraic manipulation:
\begin{eqnarray} \fl \frac{\dot s_{\textrm{\tiny{HB}}}{}_p}{\gamma_0{\cal{V}}_p}&=&-\langle \dot\rho\rangle_p+\langle \rho\rangle\dot{}_p=-3\left[\langle \rho{\cal{H}}\rangle_p-\langle \rho\rangle_p\langle \HH\rangle_p\right]=- 3\langle {\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_p(\rho){\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_p({\cal{H}})\rangle_p\nonumber\\
\fl &=&-\hbox{{\bf Cov}}_p(\rho,{\cal{H}}),\label{Sdot1}\end{eqnarray}
where
\begin{equation}{\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_p(\rho)=\rho( r)-\langle \rho\rangle_p[r_b],\qquad {\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_p({\cal{H}})={\cal{H}}( r)-\langle \HH\rangle_p[r_b]\end{equation}
are the non--local fluctuations
\footnote{The ${\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_p(A)$ and their analogues ${\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_q(A)$ are non--local fluctuations because they depend on both inner points $r<r_b$ and on the boundary $r=r_b$ of ${\cal {D}}[r_b]$ (notice that $0\leq r\leq r_b$, see \cite{part1,part2}). In contrast, the fluctuations ${\textrm{\bf{D}}}_q(A)=A-A_q$ defined by (\ref{Deldef}) and (\ref{denlq}) are local because both $A$ and $A_q$ are evaluated for the same value of $r$ (the same holds for the analogous local fluctuations ${\textrm{\bf{D}}}_p(A)=A-A_p$).}
of $\rho$ and ${\cal{H}}$, while $\hbox{{\bf Cov}}_p$ denotes the covariance statistical moment (correlation) with respect to the involved Buchert's averages (we have removed the domain indicator $[r_b]$ to simplify notation). The necessary and sufficient condition for a positive HBp entropy production is then
\begin{equation}\dot s_{\textrm{\tiny{HB}}}{}_p[r_b]\geq 0 \quad \Leftrightarrow\quad \hbox{{\bf Cov}}_p[r_b](\rho,{\cal{H}})\leq 0,\label{HBcond}\end{equation}
which directly relates this entropy production to the negative statistical correlation of $\rho$ and ${\cal{H}}$ in an arbitrary domain ${\cal {D}}[r_b]$.
Instead of using Buchert's density average (\ref{rhoave2}) to define the gravitational entropy (\ref{sHB}) for LTB models, we may consider the quasi--local weighted average (q--average) for a domain ${\cal {D}}[r_b]$ defined by the correspondence rule (\ref{Aqdef})
\footnote{Notice that $A_q( r)$ and $\langle A\rangle_q[r_b]$ are different objects even if both follow from the same correspondence rule (\ref{Aqdef}). The q--average $\langle A\rangle_q[r_b]$ is a functional and $A_q( r)$ is the function constructed from this functional by considering a varying domain boundary. Hence, $\langle A\rangle_q[r_b]$ is effectively a constant for inner points $r<r_b$ of ${\cal {D}}[r_b]$, while $A_q( r)$ is locally varying for these points and both coincide at the boundary $r=r_b$ for every $r_b$ (see figure 1 of \cite{sussBR} and equation (\ref{AqAavq})). Analogous local functions $A_p$ are also defined for Buchert's average functional $\langle A\rangle[r_b]$ in (\ref{Apdef}) (see \cite{part1} and \cite{sussBR} for a comprehensive discussion).}
\begin{equation} \langle \rho\rangle_q[r_b]=\frac{\int_{{\cal {D}}[r_b]}{\rho\,{\cal{F}}\,{\rm{d}}{\cal{V}}_p}}{\int_{{\cal {D}}[r_b]}{{\cal{F}}\,{\rm{d}}{\cal{V}}_p}}=\frac{M_q[r_b]}{{\cal{V}}_q[r_b]},\label{rhoaveq2}\end{equation}
where ${\rm{d}} {\cal{V}}_q={\cal{F}}{\rm{d}}{\cal{V}}_p$ defines the quasi--local volume of the domain
\begin{equation} {\cal{V}}_q[r_b]=\int_{{\cal {D}}[r_b]}{{\rm{d}}{\cal{V}}_q}=4\pi\int_0^{r_b}{a^3\,\Gamma\,\bar r^2\,{\rm{d}}\bar r}=\frac{4\pi}{3}a^3(r_b)r_b^3.\label{qvol}\end{equation}
and $M_q$ is the quasi--local mass--energy function of the domain
\begin{equation} M_q[r_b]=\int_{\cal {D}}{\rho\,{\rm{d}}{\cal{V}}_q}=4\pi\int_0^{r_b}{\rho\,{\cal{V}}'_q{\rm{d}} \bar r}=4\pi\int_0^{r_b}{\rho\,a^3\,\Gamma\,\bar r^2{\rm{d}}\bar r},\label{Mq}\end{equation}
which is (like $M_p$) independent of $t$ by virtue of (\ref{rhoHHKK}), (\ref{rhoKK}) and (\ref{Drho}). Following \cite{part1}, we define the ``HBq'' gravitational entropy for the domain ${\cal {D}}[r_b]$ along the lines of the HBp entropy defined before:
\begin{eqnarray} \fl s_{\textrm{\tiny{HB}}}{}_q[r_b] -s_{\textrm{\tiny{HB}}}{}_q^{(\textrm{\tiny{eq}})}= \gamma_0\int_{{\cal {D}}[r_b]}{\rho\ln\left[\frac{\rho}{\langle \rho\rangle_q[r_b]}\right]{\cal{F}}{\rm{d}}{\cal{V}}_p}=\gamma_0{\cal{V}}_q[r_b]\left\langle \rho\ln\left[\frac{\rho}{\langle \rho\rangle_q[r_b]}\right]\right\rangle_q[r_b],\nonumber\\\label{sHBq}\end{eqnarray}
As shown in \cite{part1}, the definition (\ref{sHBq}) yields results that are equivalent to those obtained from (\ref{sHB2}):
\begin{eqnarray} \fl \frac{\dot s_{\textrm{\tiny{HB}}}{}_q[r_b]}{\gamma_0{\cal{V}}_q[r_b]}=-3\left[\langle \rho{\cal{H}}\rangle_q-\langle \rho\rangle_q\langle \HH\rangle_q\right]=- 3\langle {\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_q(\rho){\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_q({\cal{H}})\rangle_q=-\hbox{{\bf Cov}}_q(\rho,{\cal{H}}),\nonumber\\
\label{Sdot1q}\end{eqnarray}
so that
\begin{equation}\fl \dot s_{\textrm{\tiny{HB}}}{}_q[r_b]\geq 0 \quad \Leftrightarrow\quad \hbox{{\bf Cov}}_q[r_b](\rho,{\cal{H}})= \langle{\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_q({\cal{H}}){\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_q(\rho)\rangle_q[r_b]\leq 0,\label{HBcondq}\end{equation}
where
\begin{equation}\fl {\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_q(\rho)=\rho( r)-\langle \rho\rangle_q[r_b],\qquad {\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_q({\cal{H}})={\cal{H}}( r)-\langle \HH\rangle_q[r_b]\label{denlq}\end{equation}
are the non--local fluctuations with respect to the q--averages $\langle \rho\rangle_q[r_b]$ and $\langle \HH\rangle_q[r_b]$.
\section{Local vs non--local entropies.}
It is quite interesting that the condition for a positive entropy production from the CET proposal (\ref{CETcond}) resembles that obtained from the HBp proposal (\ref{HBcond}) and its quasi--local version HBq in (\ref{HBcondq}). In fact, the resemblance is more striking if we consider the HBq entropy because q--scalars and q--averages coincide at the domain boundary (see \cite{part1} and figure 1 of \cite{sussBR} for a comprehensive discussion):
\begin{equation}A_q( r_b) = \langle A\rangle_q[r_b]\quad \Rightarrow\quad {\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_q(A)|_{r=r_b}={\textrm{\bf{D}}}_q(A)|_{r=r_b}.\label{AqAavq}\end{equation}
Nevertheless, whether we consider Buchert's average or the q--average, it is important to remark that there are important but subtle differences between (\ref{CETcond}) and either one of (\ref{HBcond}) or (\ref{HBcondq}): the CET entropy is defined for a local volume $V$, which is consistent with the fact that the ${\textrm{\bf{D}}}_q$ fluctuations are local, and thus (\ref{CETcond}) is a local condition evaluated in a point--wise manner, whereas the ${\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_p$ and ${\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_q$ fluctuations are non--local, and thus the HB and HBq entropies in (\ref{HBcond}) and (\ref{HBcondq}) must be evaluated though proper volume averaging over a domain ${\cal {D}}[r_b]$.
This difference is important, since (\ref{CETcond}) is necessary and sufficient by definition, as it follows directly from the original CET article \cite{CET}) and its fulfillment can be tested by local evaluation of the involved quantities (we only need to evaluate both fluctuations ${\textrm{\bf{D}}}_q(\rho)$ and ${\textrm{\bf{D}}}_q({\cal{H}})$ at each point). Conditions (\ref{HBcond}) and (\ref{HBcondq}) are also necessary and sufficient by definition (see references \cite{HB1,HB2,HB3,part1}), but their necessary and sufficient nature is strictly domain dependent, that is: it only applies after the integrals in the involved averages have been evaluated for any given domain. As a consequence, we cannot rule out that either one of the latter conditions (say (\ref{HBcond})) may hold ({\it i.e.} $\hbox{{\bf Cov}}_p(\rho,{\cal{H}})\leq 0$) even if ${\textrm{\bf{D}}}_p(\rho){\textrm{\bf{D}}}_p({\cal{H}})\leq 0$ fails to hold in inner points $r<r_b$ of ${\cal {D}}[r_b]$ (the same situation occurs for $\hbox{{\bf Cov}}_q(\rho,{\cal{H}})$).
However, we can obtain weaker conditions that are only sufficient (and not necessary) by looking for sign conditions in the integrands {\it before} the evaluation of the integrals: if these conditions are fulfilled the integrals (once evaluated) will have the desired sign, but the converse statement is false. These weaker conditions follow from the fact that if ${\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_p(A){\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_p(B)\leq 0$ holds for every $0\leq r\leq r_b$, it implies $\hbox{{\bf Cov}}_p(A,B)\leq 0$ for every pair of scalars $A,\,B$ (the same occurs with the ${\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_q$), which leads to the following sufficient but not necessary conditions:
\numparts\begin{eqnarray} {\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_p(\rho){\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_p({\cal{H}})\leq 0 \quad \forall\;\; 0\leq r\leq r_b\quad \Rightarrow\quad \dots_{\textrm{\tiny{HB}}}{}_p(r_b) \geq 0,\\
{\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_q(\rho){\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_q({\cal{H}})\leq 0 \quad\forall\;\; 0\leq r\leq r_b\quad \Rightarrow\quad \dots_{\textrm{\tiny{HB}}}{}_q(r_b) \geq 0, \label{NLconds}\end{eqnarray}\endnumparts
which must still be evaluated at every point in ${\cal {D}}[r_b]$. More useful sufficient (not necessary) conditions on $\dots_{\textrm{\tiny{HB}}}{}_p$ and $\dot s_{\textrm{\tiny{HB}}}{}_q$ can be obtained that only involve evaluating local fluctuations at the boundary of each domain. For this purpose, we define the local fluctuations equivalent to the quasi--local fluctuations (\ref{Deldef}):
\begin{equation}{\textrm{\bf{D}}}_p(A)=A-A_p = \frac{A'_p}{{\cal{V}}'_p/{\cal{V}}_p}=\frac{1}{{\cal{V}}_p}\int_0^r{A'\,{\cal{V}}_p\,{\rm{d}}\bar r},\label{Apdef}\end{equation}
where $A_p$ are the local functions whose correspondence rule is the same as that of Buchert's average ({\it i.e.} equation (\ref{rhoave2})). The following results on quadratic fluctuations proven in \cite{part1} and \cite{sussBR}
\numparts\begin{eqnarray} \langle {\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_p(\rho){\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_p({\cal{H}})\rangle_p[r_b]=\langle{\textrm{\bf{D}}}_p(\rho){\textrm{\bf{D}}}_p({\cal{H}})\rangle_p[r_b],\label{result1}\\
\langle {\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_q(\rho){\textrm{\bf{D}}}^{(\tiny{\textrm{NL}})}_q({\cal{H}})\rangle_q[r_b]=\langle{\textrm{\bf{D}}}_q(\rho){\textrm{\bf{D}}}_q({\cal{H}})\rangle_q[r_b],\label{result2}\end{eqnarray}\endnumparts
yield the following sufficient (not necessary) conditions
\numparts\begin{eqnarray} \left[\,{\textrm{\bf{D}}}_p(\rho){\textrm{\bf{D}}}_p({\cal{H}})\,\right]|_{r=r_b}\leq 0 \quad \Rightarrow\quad \dots_{\textrm{\tiny{HB}}}{}_p(r_b) \geq 0,\label{HBcond01}\\
\left[\,{\textrm{\bf{D}}}_q(\rho){\textrm{\bf{D}}}_q({\cal{H}})\,\right]|_{r=r_b}\leq 0 \quad \Rightarrow\quad \dots_{\textrm{\tiny{HB}}}{}_q(r_b) \geq 0, \label{HBcond01}\end{eqnarray}\endnumparts
which are valid for every domain, and since $r=r_b$ is arbitrary, they can be evaluated locally for all the range of $r$ and can be stated simply as local sufficient conditions:
\numparts\begin{eqnarray} {\textrm{\bf{D}}}_p(\rho){\textrm{\bf{D}}}_p({\cal{H}})\leq 0 \quad \Rightarrow\quad \dots_{\textrm{\tiny{HB}}}{}_p \geq 0,\label{HBcond1}\\
{\textrm{\bf{D}}}_q(\rho){\textrm{\bf{D}}}_q({\cal{H}})\leq 0 \quad \Rightarrow\quad \dots_{\textrm{\tiny{HB}}}{}_q \geq 0. \label{HBcond2}\end{eqnarray}\endnumparts
This is an important result, as it shows that the conditions for entropy production that emerge from the CET proposal and the HBq proposal are essentially the same, as both are based on q--scalars: compare (\ref{CETcond}) and (\ref{HBcond2}), the only difference being that they are necessary and sufficient for the CET proposal and sufficient but not necessary for the HBq proposal.
The conditions from the HBp proposal (with Buchert's average) are also the same as those from CET, but the fluctuations in the HBp case must be evaluated from the local functions $A_p$ in (\ref{Apdef}) associated with Buchert's average and that are analogous to $A_q$. However, we can readily obtain sufficient (not necessary) conditions for (\ref{CETcond}) and (\ref{HBcond2}) that are valid for both averages under certain restrictions: if both $\rho$ and ${\cal{H}}$ are monotonous in a given domain ${\cal {D}}[r_b]$, then
\begin{equation}\fl \hbox{if}\quad\rho'{\cal{H}}'\leq 0\quad\hbox{holds}\quad \Rightarrow\quad \dots_{\textrm{\tiny{HB}}}{}_p[r_b] \geq 0\quad\hbox{{\bf and}}\quad\dots_{\textrm{\tiny{HB}}}{}_q[r_b] \geq 0\quad \hbox{hold},\label{HBcond3}\end{equation}
where we used the fact that the integrals in (\ref{Deldef}) and (\ref{Apdef}) for $A=\rho,\,{\cal{H}}$ involve $\rho'$ and ${\cal{H}}'$ as integrands. Hence, if these gradients have opposite signs in all points of a domain ${\cal {D}}[r_b]$ the products of local fluctuations ${\textrm{\bf{D}}}_q(\rho){\textrm{\bf{D}}}_q({\cal{H}})$ and ${\textrm{\bf{D}}}_p(\rho){\textrm{\bf{D}}}_p({\cal{H}})$ will be negative (but the converse is false).
\section{Probing the conditions for entropy production in the asymptotic limits.}
While the conditions for positive entropy production from the CET and HB proposals are elegant and plausible, it is necessary to verify their actual fulfilment in generic regular LTB models, at least in the asymptotic ranges of their time evolution. We assume henceforth that shell crossings are absent ($\Gamma>0$ holds for $a>0$, see the conditions for this in \cite{sussmodes}). We examine first the case of the CET entropy in (\ref{CETcond}), which is a sufficient but not necessary condition for the HBq case in (\ref{HBcondq}) and (under certain restrictions) for the HB case in (\ref{HBcond}).
In order to examine qualitatively the fulfilment of (\ref{CETcond}) and (\ref{HBcondq}) in various evolution ranges we remark that
\begin{equation} {\textrm{\bf{D}}}_q(\rho){\textrm{\bf{D}}}_q({\cal{H}})=\rho_q\,{\cal{H}}_q\,\delta^{(\rho)}_q\,\delta^{(\HH)}_q.\label{deltacond}\end{equation}
where, following \cite{sussmodes}, we will use the expressions for the perturbations $\delta^{(\rho)}_q,\,\delta^{(\HH)}_q$ given by (\ref{Drho}) and (\ref{Dh}) in terms of the exact generalizations of the growing/decaying density modes (\ref{gmode})--(\ref{ivpconstr}) and the exact analytic forms for $t-t_{\textrm{\tiny{bb}}},\,{\cal{H}}_q$ and ${\cal{H}}_q(t-t_{\textrm{\tiny{bb}}})$ given by (\ref{hypsol})--(\ref{ellsol}) and (\ref{HTq})--(\ref{H32}) in Appendix B.
\subsection{The general case: $\JJ_{\tiny{\textrm{(g)}}}$ and $\JJ_{\tiny{\textrm{(d)}}}$ non-zero.}
\subsubsection{Near the (non--simultaneous) big bang singularity.}
At very early times it is safe to assume that ${\cal{H}}_q>0$ and $\rho_q>0$ hold. Hence, conditions (\ref{CETcond}) and (\ref{HBcondq}) are equivalent to
\begin{equation} \delta^{(\rho)}_q\,\delta^{(\HH)}_q\leq 0\quad \Leftrightarrow\quad \dots_{\tiny{\textrm{gr}}}\geq 0.\label{condpos}\end{equation}
We have for $t\approx t_{\textrm{\tiny{bb}}}$ (or $0<a\ll 1$):
\begin{eqnarray} \fl \JJ_{\tiny{\textrm{(g)}}} \approx \frac{2(1-\Omega_{q0})\Delta_0^{\tiny{\textrm{(g)}}} a}{3\Omega_{q0}}\to 0,\quad \JJ_{\tiny{\textrm{(d)}}}\approx \frac{3\Delta_0^{\tiny{\textrm{(d)}}}{\cal{H}}_{q0}\Omega_{q0}^{1/2}}{a^{3/2}}\to-\infty,\quad \Omega_q\approx 1+O(a)\to 1,\nonumber\label{JgJd1}\\
\end{eqnarray}
where we used (\ref{HTq}) and assumed that $\Delta_0^{\tiny{\textrm{(d)}}}\leq 0$ ($t_{\textrm{\tiny{bb}}}'\leq 0$) to comply with absence of shell crossings \cite{sussmodes}. These expressions lead to
\begin{equation}\fl \delta^{(\rho)}_q\approx \frac{\JJ_{\tiny{\textrm{(d)}}}}{1-\JJ_{\tiny{\textrm{(d)}}}}\approx -1+\frac{a^{3/2}}{3|\Delta_0^{\tiny{\textrm{(d)}}}|{\cal{H}}_{q0}\Omega_{q0}^{1/2}},\qquad \delta^{(\HH)}_q \approx 2\delta^{(\rho)}_q\approx -\frac{1}{2}+O(a^{3/2}), \label{deltasbb} \end{equation}
where we have assumed that $\Delta_0^{\tiny{\textrm{(d)}}}<0$ holds to comply with absence of shell crossings \cite{sussmodes}. Evidently, $\delta^{(\rho)}_q$ and $\delta^{(\HH)}_q$ have the same sign and so (\ref{condpos}) is violated.
\subsubsection{Asymptotic late times.}
For ever-expanding hyperbolic models we can also assume ${\cal{H}}_q>0$ and $\rho_q>0$ and test the fulfilment of (\ref{CETcond}) and (\ref{HBcondq}) through (\ref{condpos}). In the asymptotic time range ($t\to\infty$ or $a\to\infty$) we have $\Omega_q\sim O(a^{-1})\to 0$, and from (\ref{HTq})
\begin{eqnarray} \fl \JJ_{\tiny{\textrm{(d)}}}\approx \frac{3\Delta_0^{\tiny{\textrm{(d)}}}\sqrt{1-\Omega{q0}}}{a}\to 0,\quad \JJ_{\tiny{\textrm{(g)}}}\approx \Delta_0^{\tiny{\textrm{(g)}}}\left[1+3\Omega_q\left(1+\ln\left(\frac{\sqrt{\Omega_q}}{2}\right)\right)\right]\to\Delta_0^{\tiny{\textrm{(g)}}},\nonumber\label{JgJd2}\\
\end{eqnarray}
hence, from (\ref{Drho}) and (\ref{Dh}), we find that condition (\ref{condpos}) holds:
\begin{equation} \delta^{(\rho)}_q\delta^{(\HH)}_q \approx \left(\Delta_0^{\tiny{\textrm{(g)}}}\right)^2\Omega_q\left[1+\ln\left(\frac{\sqrt{\Omega_q}}{2}\right)\right]\leq 0,\end{equation}
since $\Omega_q\ll 1$ and so the logarithmic term inside the square brackets necessarily takes large negative values. The fact that (\ref{condpos}) holds irrespective of the sign of $\Delta_0^{\tiny{\textrm{(g)}}}$ is an important result, because this sign determines the void ($\Delta_0^{\tiny{\textrm{(g)}}}\geq 0$) or clump ($\Delta_0^{\tiny{\textrm{(g)}}}\leq 0$) nature of the time asymptotic radial density profile \cite{sussmodes}.
For elliptic models in their expanding stage (${\cal{H}}_q>0$) a late time regime is given by layers approaching the maximal expansion when ${\cal{H}}_q\to 0$, which corresponds to $t\approx t_{\textrm{\tiny{max}}}$ with $t<t_{\textrm{\tiny{max}}}$, where $t_{\textrm{\tiny{max}}}$ given by (\ref{tmax}), or equivalently $a\toa_{\textrm{\tiny{max}}}=\Omega_{q0}/(\Omega_{q0}-1)$ and $\Omega_q\sim (a-a_{\textrm{\tiny{max}}})^{-1}\to\infty$. From (\ref{HTq}) we have in this limit
\begin{eqnarray}\fl \JJ_{\tiny{\textrm{(d)}}} \approx \frac{3\Delta_0^{\tiny{\textrm{(d)}}}{\cal{H}}_{q0}(\Omega_{q0}-1)^2}{\Omega_{q0}^{3/2}}(a-a_{\textrm{\tiny{max}}})\to 0,\quad \JJ_{\tiny{\textrm{(g)}}}\approx \Delta_0^{\tiny{\textrm{(g)}}}\left[-2+\frac{3\pi}{2\sqrt{\Omega_q}}\right]\to -2\Delta_0^{\tiny{\textrm{(g)}}},\nonumber\label{JgJd3}\\
\end{eqnarray}
hence, from (\ref{Drho}) and (\ref{Dh}), we find that (\ref{condpos}) holds:
\begin{equation} \delta^{(\rho)}_q\delta^{(\HH)}_q \approx -\frac{\pi\,(\Delta_0^{\tiny{\textrm{(g)}}})^2}{2(1+2\Delta_0^{\tiny{\textrm{(g)}}})^2}\sqrt{\Omega_q}\leq 0.\label{condposmax1}\end{equation}
If we approach the maximal expansion limit from the collapsing stage (${\cal{H}}_q\to 0$ with ${\cal{H}}_q<0$), then we need to use (\ref{deltacond}) instead of (\ref{condpos}):
\begin{equation} \rho_q{\cal{H}}_q\delta^{(\rho)}_q\delta^{(\HH)}_q=-|{\cal{H}}_q|\rho_q\delta^{(\rho)}_q\delta^{(\HH)}_q\leq 0\quad \Rightarrow\quad \delta^{(\rho)}_q\delta^{(\HH)}_q\geq 0,\label{condposmax2}\end{equation}
with the forms of $\JJ_{\tiny{\textrm{(g)}}}$ and $\JJ_{\tiny{\textrm{(d)}}}$ in the maximal expansion limit obtained from (\ref{HTqc}). This yields the same signs for each one of $\delta^{(\rho)}_q$ and $\delta^{(\HH)}_q$. Hence, (\ref{condposmax2}) is fulfilled. As a consequence, (\ref{CETcond}) and (\ref{HBcondq}) hold in the maximal expansion limit.
\subsubsection{Collapsing regime.}
As $t\tot_{\textrm{\tiny{coll}}}$ we have $\rho_q\to\infty$ and ${\cal{H}}_q\to-\infty$ as $a\to 0$ and $\Omega_q\to 1$. Hence, conditions (\ref{CETcond}) and (\ref{HBcondq}) are equivalent to (\ref{condposmax2}). From (\ref{HTqc}) and (\ref{H32}) (with ${\cal{H}}_q<0$) we have in this regime ($\Omega_q-1\approx 0$):
\begin{equation} \fl \JJ_{\tiny{\textrm{(g)}}}\approx -\frac{3\pi\Delta_0^{\tiny{\textrm{(g)}}}}{(\Omega_q-1)^{3/2}}\to-\infty,\qquad \JJ_{\tiny{\textrm{(d)}}}\approx -\frac{3\Delta_0^{\tiny{\textrm{(d)}}}(\Omega_{q0}-1)^{3/2}}{\Omega_{q0}(\Omega_q-1)^{3/2}}\to\infty,\label{JgJd4}\end{equation}
where we assumed that $\Delta_0^{\tiny{\textrm{(g)}}}\geq 0$ and $\Delta_0^{\tiny{\textrm{(d)}}}\leq 0$ hold to comply with absence of shell crossings \cite{sussmodes}. From the forms above and (\ref{Drho}) and (\ref{Dh}) we obtain
\numparts\begin{eqnarray} \delta^{(\rho)}_q \approx -1 +\frac{\Omega_{q0}(\Omega_q-1)^{3/2}}{3[\Delta_0^{\tiny{\textrm{(d)}}}(\Omega_{q0}-1)^{3/2}+\pi\Delta_0^{\tiny{\textrm{(g)}}}\Omega_{q0}]},\label{Drhocoll}\\
\delta^{(\HH)}_q \approx -\frac{1}{2}+\frac{\Omega_{q0}(\Omega_q-1)^{3/2}}{6[\Delta_0^{\tiny{\textrm{(d)}}}(\Omega_{q0}-1)^{3/2}+\pi\Delta_0^{\tiny{\textrm{(g)}}}\Omega_{q0}]},\label{DHcoll}\end{eqnarray}\endnumparts
which imply that (\ref{condposmax2}) is fulfilled.
\subsection{Models with suppressed decaying mode.}
From (\ref{Did}), the suppression of the decaying mode, $\Delta_0^{\tiny{\textrm{(d)}}}=0$, is equivalent to a simultaneous Big Bang: $t_{\textrm{\tiny{bb}}}=t_{\textrm{\tiny{bb(0)}}}=$ constant. Following (\ref{tbb}), models of this type (which can be hyperbolic or elliptic) are characterized by initial conditions based on a single free function ($\Omega_{q0}$):
\begin{equation}{\cal{H}}_{q0}=\frac{Y_{q0}}{t_0-t_{\textrm{\tiny{bb(0)}}}},\qquad \delta^{(\HH)}_0 = \frac{\Omega_{q0}}{Y_{q0}}\frac{{\rm{d}} Y_{q0}}{{\rm{d}}\Omega_{q0}}\delta^{(\Omega)}_0,\label{ivfSDM}\end{equation}
where $Y_{q0}=Y_q(\Omega_{q0})$ follows from (\ref{Y}) for a given choice of $\Omega_{q0}$ (while $\delta^{(\Omega)}_0=(r/3)\Omega'_{q0}/\Omega_{q0}$). As shown by the expansions (\ref{JgJd1}), (\ref{JgJd2}), (\ref{JgJd3}) and (\ref{JgJd4}), the decaying mode is only relevant in the very early evolution times $t\approx t_{\textrm{\tiny{bb}}}$ and near the collapsing singularity (see \cite{sussmodes} for a comprehensive discussion). Therefore, suppression of this mode does not alter the asymptotic forms of perturbations $\delta^{(\rho)}_q,\,\delta^{(\HH)}_q$, either in the asymptotic time range ($t\to\infty$ of hyperbolic models and $t\tot_{\textrm{\tiny{max}}}$ in elliptic models) or near collapse ($t\tot_{\textrm{\tiny{coll}}}$). As a consequence, condition (\ref{condpos}) holds when ${\cal{H}}_q>0$ or (\ref{condposmax2}) (for $t\tot_{\textrm{\tiny{max}}}$ with $t>t_{\textrm{\tiny{max}}}$ and ${\cal{H}}_q<0$), while near collapse the growing mode has the same form as in (\ref{JgJd4}) and the perturbations take the forms (\ref{Drhocoll}) and (\ref{DHcoll}) with $\Delta_0^{\tiny{\textrm{(d)}}}=0$, hence condition (\ref{condposmax2}) is fulfilled. On the other hand, the suppression of the decaying mode makes an important difference in (\ref{condpos}) in the limit $t\to t_{\textrm{\tiny{bb(0)}}}$. The growing mode $\JJ_{\tiny{\textrm{(g)}}}$ takes the same form as in (\ref{JgJd1}), hence the perturbations now take the following forms
\begin{equation} \delta^{(\rho)}_q\approx -\frac{2}{5}\Delta_0^{\tiny{\textrm{(g)}}}(\Omega_q-1)\to 0,\qquad \delta^{(\HH)}_q\approx -\frac{1}{3}\delta^{(\rho)}_q\to 0,\label{DrhoDh0}\end{equation}
which imply that (\ref{condpos}) is fulfilled as $t\tot_{\textrm{\tiny{bb(0)}}}$. This is the opposite result from that when $t\tot_{\textrm{\tiny{bb}}}$ with a non-zero decaying mode, which indicates a strong relation between early times homogeneity ({\it i.e.} the fact that $\delta^{(\rho)}_q,\,\delta^{(\HH)}_q\to 0$) and early times positive entropy production (see Figures 1, 3, 5 and 6).
\subsection{Models with suppressed growing mode.}
Since the decaying mode is only dominant at very early times, suppression of the growing mode does not affect the perturbations in the range $t\approx t_{\textrm{\tiny{bb}}}$, in which (\ref{condpos}) is violated if $\Delta_0^{\tiny{\textrm{(d)}}}\ne 0$. However, the suppression of this mode does affect the late time forms of the perturbations. Demanding absence of shell crossings \cite{sussmodes}, we have two types of models with a suppressed growing mode: hyperbolic models complying with $\Delta_0^{\tiny{\textrm{(g)}}}=0$ and ${\cal{H}}_q(t-t_{\textrm{\tiny{bb}}})>2/3$ (elliptic models with $\Delta_0^{\tiny{\textrm{(g)}}}=0$ exhibit shell crossings), and parabolic models for which $\Delta_0^{\tiny{\textrm{(g)}}}\ne 0$ but ${\cal{H}}_q(t-t_{\textrm{\tiny{bb}}})=2/3$. For the hyperbolic models the decaying mode has the same asymptotic time form as in (\ref{JgJd2}), with the perturbations taking the following forms in this limit ($\Omega_q\approx 0$):
\begin{equation}\delta^{(\rho)}_q \approx -\frac{3(1-\Omega_{q0})^{3/2}\Delta_0^{\tiny{\textrm{(d)}}}}{\Omega_{q0}}\Omega_q\to 0,\qquad \delta^{(\HH)}\approx \frac{1}{3}\delta^{(\rho)}_q\to 0,\label{ivfSGM}\end{equation}
which imply that (\ref{condpos}) is violated. For parabolic models (${\cal{K}}_q=0$) we have ${\cal{H}}_q^2=(8\pi/3)\rho_q$, hence we obtain from (\ref{Dadef}): $2\delta^{(\HH)}_q=\delta^{(\rho)}_q$. Since this is an exact (not asymptotic) relation, then (\ref{condpos}) is violated throughout the whole time evolution and radial domains of parabolic models.
\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{Fig1new_entropy_v2.pdf}
\caption{{\bf CET entropy time evolution.} The panels display $s_{\tiny{\textrm{gr}}},\,\dots_{\tiny{\textrm{gr}}}$ and $\ddots_{\tiny{\textrm{gr}}}$ as functions of time of a representative dust layer for hyperbolic models (general case in panel (a) and suppressed decaying mode in panel (b)) and elliptic models (general case in panel (c) and suppressed decaying mode in panel (d)). The forms of the graphs were obtained qualitatively from the results of section 6 and also apply to the HBp and HBq entropies (see detailed discussion in section 7). Notice how the decaying mode (dominant for early times) forces $s_{\tiny{\textrm{gr}}}$ to diverge as $t\to t_{\textrm{\tiny{bb}}}$ with $\dots_{\tiny{\textrm{gr}}}\to-\infty$, whereas for models with a suppressed decaying mode ((b) and (d)) $\dots_{\tiny{\textrm{gr}}}\geq 0$ holds for all times.}
\label{fig1}
\end{center}
\end{figure}
\section{A qualitative look at the time evolution of the entropies.}
We have examined in previous sections the conditions for $\dots_{\tiny{\textrm{gr}}}\geq 0,\,\dots_{\textrm{\tiny{HB}}}{}_p\geq 0$ and $\dots_{\textrm{\tiny{HB}}}{}_q\geq 0$ and their fulfilment for various asymptotic time ranges. A qualitatively robust description of the full time evolution of these entropies follows by putting together this information. We look at the CET and HB cases separately below (see also Figures 1 and 3).
\subsection{The CET entropy.}
We obtain directly from (\ref{rhograv2})--(\ref{CETc12}) and (\ref{CETc122}) the exact form of $\dot s_{\tiny{\textrm{gr}}}$:
\begin{equation} \dot s_{\tiny{\textrm{gr}}}= -\frac{2\pi\, \alpha\, \rho_{q0}\, \Gamma\, \delta^{(\HH)}_q}{|1+3\delta^{(\HH)}_q|}\frac{{\cal{H}}_q\,\delta^{(\rho)}_q}{|{\cal{H}}_q|\,|\delta^{(\rho)}_q|},\label{sgrt}\end{equation}
which allows us to examine all relevant sub--cases below:
\begin{itemize}
\item {\underline {General case}} $\JJ_{\tiny{\textrm{(g)}}},\,\JJ_{\tiny{\textrm{(d)}}}$ non-zero. Considering from (\ref{Drho}) that
\begin{equation} \Gamma =(1+\delta^{(\rho)}_{q0})\,(1-\JJ_{\tiny{\textrm{(g)}}}-\JJ_{\tiny{\textrm{(d)}}}),\end{equation}
and using (\ref{JgJd1}) and (\ref{deltasbb}), we have for hyperbolic and elliptic models:
\begin{equation} \fl \dots_{\tiny{\textrm{gr}}}\approx \frac{3(1+\delta^{(\rho)}_{q0})\Delta_0^{\tiny{\textrm{(d)}}}{\cal{H}}_{q0}\Omega_{q0}^{1/2}}{a^{3/2}}\to -\infty,\label{CETc12bb}\quad\hbox{as}\quad t\tot_{\textrm{\tiny{bb}}},\end{equation}
where we used the fact that $(1+\delta^{(\rho)}_{q0})\Delta_0^{\tiny{\textrm{(d)}}}\leq 0$ (to avoid shell crossings) and ${\cal{H}}_q>0$, so ${\cal{H}}_q/|{\cal{H}}_q|=1$ holds in this limit. The limit (\ref{CETc12bb}) implies that $s_{\tiny{\textrm{gr}}}$ must decrease (for both hyperbolic and elliptic models) from infinite values, but there must always exists an extremum of $s_{\tiny{\textrm{gr}}}$ since $\dots_{\tiny{\textrm{gr}}}$ eventually becomes positive for all models as the evolution proceeds (and $\JJ_{\tiny{\textrm{(g)}}}$ dominates over $\JJ_{\tiny{\textrm{(d)}}}$). The qualitative late time behaviour of $s_{\tiny{\textrm{gr}}}$ is different for hyperbolic and elliptic models:
\begin{itemize}
\item Hyperbolic models (see qualitative plot in Figure 1a and numeric plots in Figures 5b and 6). In the asymptotic time regime we have ${\cal{H}}_q\to 0$ (with ${\cal{H}}_q>0$) so ${\cal{H}}_q/|{\cal{H}}_q|=1$ holds, as well as $\delta^{(\rho)}_q\to \Delta_0^{\tiny{\textrm{(g)}}}/(1-\Delta_0^{\tiny{\textrm{(g)}}})$ and $\delta^{(\HH)}_q\to 0$ (see also \cite{sussmodes}), then $\dot s_{\tiny{\textrm{gr}}}\to 0$ holds in the this limit. As a consequence, the extremum of $s_{\tiny{\textrm{gr}}}$ must be a minimum with $\ddots_{\tiny{\textrm{gr}}}>0$, but the curve of $s_{\tiny{\textrm{gr}}}$ becomes convex as $\ddots_{\tiny{\textrm{gr}}}$ becomes positive with $\dots_{\tiny{\textrm{gr}}}\to 0$ and $s_{\tiny{\textrm{gr}}}$ necessarily reaching a finite (position dependent) asymptotic value.
\item Elliptic models (see qualitative plot in Figure 1c and numeric plot in Figure 6). In this case $s_{\tiny{\textrm{gr}}}$ also has an extremum (minimum) as $\dots_{\tiny{\textrm{gr}}}$ becomes positive and grows as the maximal expansion is reached (see equations (\ref{condposmax1}) and (\ref{condposmax2})). However, from the fact that ${\cal{H}}_q<0$ and thus ${\cal{H}}_q=-|{\cal{H}}_q|$, the form of $\dots_{\tiny{\textrm{gr}}}$ near the collapse singularity has the same form as in (\ref{CETc12bb}) but with a positive sign, which leads to $\dots_{\tiny{\textrm{gr}}}\to\infty$ as $t\tot_{\textrm{\tiny{coll}}}$. Hence, the form of $s_{\tiny{\textrm{gr}}}$ as a function of $t$ is convex with $\ddots_{\tiny{\textrm{gr}}}>0$ for all the evolution.
\end{itemize}
\item {\underline {Suppressed decaying mode}}. We have $\dots_{\tiny{\textrm{gr}}}\to\infty$ as $t\tot_{\textrm{\tiny{bb(0)}}}$ for both hyperbolic and elliptic models, hence $s_{\tiny{\textrm{gr}}}$ must be finite at $t_{\textrm{\tiny{bb(0)}}}$ and display an initial growth with infinite slope. For the late time evolution we have:
\begin{itemize}
\item Hyperbolic models (see qualitative plot in Figure 1b and numeric plot in Figures 4 and 5a). We have $\dots_{\tiny{\textrm{gr}}}\geq 0$ for the full time evolution, but as the evolution proceeds $\dots_{\tiny{\textrm{gr}}}$ decreases with $\dots_{\tiny{\textrm{gr}}}\to 0$ as $t\to\infty$. As a consequence, for all the evolution $s_{\tiny{\textrm{gr}}}$ increases towards a the same position dependent terminal value as the general case, but with $\ddots_{\tiny{\textrm{gr}}}<0$ for all $t$ (concave curves).
\item Elliptic models (see qualitative plot in figure 1d). Since $\dots_{\tiny{\textrm{gr}}}> 0$ as $t\tot_{\textrm{\tiny{max}}}$, the change of sign of $\dots_{\tiny{\textrm{gr}}}$ must occur in the expanding stage, with $s_{\tiny{\textrm{gr}}}$ reaching a position dependent minimal value for some $t<t_{\textrm{\tiny{max}}}$ and growing afterwards with $\dots_{\tiny{\textrm{gr}}}\to\infty$ and $s_{\tiny{\textrm{gr}}}\to\infty$ at the collapse. Hence $\ddots_{\tiny{\textrm{gr}}}>0$ holds for the full evolution.
\end{itemize}
\item {\underline {Suppressed growing mode}}. For both hyperbolic and parabolic models we have $\dots_{\tiny{\textrm{gr}}}\to\-\infty$ as $t\tot_{\textrm{\tiny{bb}}}$ and $\dots_{\tiny{\textrm{gr}}}< 0$ with $\dots_{\tiny{\textrm{gr}}}\to 0$ as $t\to\infty$, hence $s_{\tiny{\textrm{gr}}}$ decreases for all the time evolution from infinite to a finite terminal value.
\end{itemize}
\subsection{The HBp and HBq entropies.}
In models with a non-zero decaying mode the integrals in (\ref{sHB2}) and (\ref{sHBq}) that define $s_{\textrm{\tiny{HB}}}{}_p$ and $s_{\textrm{\tiny{HB}}}{}_q$, for any domain ${\cal {D}}[r_b]$, must be evaluated along some time slices that are not complete and everywhere regular: $\rho$ and its average diverge as the slices ``intersect'' the non--simultaneous Big Bang singularity. The same phenomenon occurs for any domain of elliptic models in which some slices necessarily intersect the non--simultaneous Big Crunch collapsing singularity (see Figure 2). For the regular slices the radial integration range of (\ref{sHB2}) and (\ref{sHBq}) is complete ({\it i.e.} $0\leq r\leq r_b$), but for slices hitting singularities it is restricted by the conditions $t_{\textrm{\tiny{bb}}}'\leq 0$ and $t_{\textrm{\tiny{coll}}}'\geq 0$ that follow from demanding absence of shell crossings. For any domain bounded by a finite $r=r_b$, the effects of the non--simultaneity of the singularities on (\ref{propvol})--(\ref{Mp}) and (\ref{rhoaveq2})--(\ref{sHBq}) are illustrated by Figure 2 and summarized below:
\begin{itemize}
\item Near the Big Bang in hyperbolic and elliptic models or regions (see Figure 2a): for time slices $t_{\textrm{\tiny{bb}}}(0)>t_s>t_{(-)}=t_{\textrm{\tiny{bb}}}(r_b)$ we have $\rho\to \infty$ and ${\cal{H}}\to \infty$ and $a\to 0$ as $r\to r_s$. Hence, all integrals in (\ref{sHB2}) and (\ref{sHBq}) and in (\ref{Sdot1}) and (\ref{Sdot1q}) are improper integrals whose convergence in this limit must be verified in the restricted integration range $r_s<r\leq r_b$, where $t_s=t_{\textrm{\tiny{bb}}}(r_s)$. As the slices approach $t=t_{(-)}$, the spherical domains ${\cal {D}}[r_b]$ ``shrink'' to very thin shells around the singular point $[r_s,t_{\textrm{\tiny{bb}}}(r_s)]$ corresponding to a ``narrowing'' of the radial range as $r_b\approx r_s$. The limit $t_s\to t_{(-)}$ is equivalent to both, the convergence limit $r\to r_s$ and the limit $r_s\to r_b$ of ``narrow'' domains (see Figure 2a). As proven in Appendix D.2, for every fixed arbitrary $r_b$ we have
\numparts\begin{eqnarray} \fl \lim_{r_s \to r_b} s_{\textrm{\tiny{HB}}}{}_p[r_b]-s_{\textrm{\tiny{HB}}}{}_p^{(\textrm{\tiny{eq}})}=0,\qquad \lim_{r_s \to r_b} s_{\textrm{\tiny{HB}}}{}_q[r_b]-s_{\textrm{\tiny{HB}}}{}_q^{(\textrm{\tiny{eq}})}=0,\label{shblim}\\
\fl \lim_{r_s \to r_b} \dots_{\textrm{\tiny{HB}}}{}_p[r_b]=\infty,\qquad \lim_{r_s \to r_b} \dots_{\textrm{\tiny{HB}}}{}_q[r_b]=\infty,\label{dotshblim}\end{eqnarray}\endnumparts
which imply that the very early time behaviour of $s_{\textrm{\tiny{HB}}}{}_p$ and $s_{\textrm{\tiny{HB}}}{}_p$ is radically different from that of $s_{\tiny{\textrm{gr}}}$: for every domain (since $r_b$ is arbitrary) their earliest time value at some $t=t_{(-)}$ is finite and given by the ``equilibrium'' entropies (these can be a different integration constant for different domains), with the initially instantaneous infinite entropy growth depicted by Figures 3a and 3c for $t$ close to $t=t_{(-)}$ (this behaviour is analogous to the very early time evolution of $s_{\tiny{\textrm{gr}}}$ depicted by Figures 1b and 1d for models with a suppressed decaying mode).
\item Near the Big Crunch in elliptic models or regions (see Figures 2b, 3c and 3d): in slices $t_{\textrm{\tiny{coll}}}(0)<t_s<t_{(+)}=t_{\textrm{\tiny{coll}}}(r_b)$ the integration range is also $r_s<r\leq r_b$, but now with $t_s=t_{\textrm{\tiny{coll}}}(r_s)$ and with the lower bound $r_s$ of the integral in (\ref{sHB2}) also approaching $r_b$ as $t_s\to t_{(+)}$. The limit (\ref{shblim}) also holds for $s_{\textrm{\tiny{HB}}}{}_p[r_b]$ and $s_{\textrm{\tiny{HB}}}{}_q[r_b]$, but not (\ref{dotshblim}). Instead, we have (see Appendix D.2):
\begin{equation} \lim_{r_s \to r_b} \dots_{\textrm{\tiny{HB}}}{}_p[r_b]=-\infty,\qquad \lim_{r_s \to r_b} \dots_{\textrm{\tiny{HB}}}{}_q[r_b]=-\infty,\label{dotshblimc}\end{equation}
As a consequence, the values of $s_{\textrm{\tiny{HB}}}{}_p[r_b]$ and $s_{\textrm{\tiny{HB}}}{}_q[r_b]$ plummet down with a final infinite slope to their final equilibrium value (the same as in the Big Bang) for every domain $r_b$.
\end{itemize}
Because of (\ref{dotshblim}), the values of $s_{\textrm{\tiny{HB}}}{}_p$ and $s_{\textrm{\tiny{HB}}}{}_q$ must increase from their equilibrium values at least for domains bounded by $r_b$ that intersect the non--simultaneous Big Bang in slices $t_s$ close to $t_{(-)}$.
While for every domain the terminal collapse value of the HB entropies is the same as their initial Big Bang value in elliptic models, the terminal time asymptotic values as $t\to\infty$ in hyperbolic models is necessarily larger than their initial Big Bang values at $t=t_{(-)}$ (see proof in Appendix D.3).
This information on the early and late asymptotic time limits, together with the sufficient conditions for the production of entropy, are sufficient to obtain a qualitative picture of the time evolution of $s_{\textrm{\tiny{HB}}}{}_p[r_b]$ and $s_{\textrm{\tiny{HB}}}{}_q[r_b]$ when both density modes are nonzero (this is depicted by Figures 3a and 3c). The CET and HB entropies clearly exhibit a different behaviour for slices intersecting non--simultaneous singularities, as the integration domains of the integrals that define $\dots_{\textrm{\tiny{HB}}}{}_p$ and $s_{\textrm{\tiny{HB}}}{}_q$ in (\ref{HBcond}) and (\ref{HBcondq}) are incomplete (this follows from (\ref{shblim})--(\ref{dotshblim}) and (\ref{dotshblimc})). However, for slices $t>t_{\textrm{\tiny{bb}}}(0)$ (see figure 2a) and $t_{\textrm{\tiny{bb}}}(0)<t<t_{\textrm{\tiny{coll}}}(0)$ (elliptic models, figure 2b) these integrals are evaluated for arbitrary complete domains $0\leq r\leq r_b$. Since their integrands are similar to those of the CET entropy in (\ref{CETcond}), the HBp and HBq entropies should exhibit for these complete slices a qualitatively similar evolution (see Figures 3a and 3c) to that displayed by Figures 1a and 1c for the CET entropy. This assessment follows from the fact that the fulfillment of $\dots_{\tiny{\textrm{gr}}}( r)\geq 0$ for every $0\leq r\leq r_b$ implies for complete slices (through the sufficient conditions (\ref{HBcond1})--(\ref{HBcond2}) and (\ref{HBcond3})) the fulfillment of $\dots_{\textrm{\tiny{HB}}}{}_p\geq 0$ and $\dots_{\textrm{\tiny{HB}}}{}_q\geq 0$ for $r=r_b$. Moreover, these sufficient conditions may place strong restrictions on the radial profiles of $\rho$ and ${\cal{H}}$, therefore we cannot exclude the possibility that well behaved models exist in which $\dots_{\textrm{\tiny{HB}}}{}_p< 0$ and/or $\dots_{\textrm{\tiny{HB}}}{}_q<0$ could hold in some subdomains $r<r_b$ of ${\cal {D}}[r_b]$, or in restricted time ranges of complete slices, even if $\dots_{\tiny{\textrm{gr}}}(r_b)>0$ holds for these times.
The evaluation (and early time evolution) of $s_{\textrm{\tiny{HB}}}{}_p$ and $s_{\textrm{\tiny{HB}}}{}_q$ is completely different for models (hyperbolic and elliptic) with a suppressed decaying mode, as in this case the Big Bang is simultaneous ($t_{\textrm{\tiny{bb}}}=t_{\textrm{\tiny{bb(0)}}}$), hence all slices $t>t_{\textrm{\tiny{bb(0)}}}$ are regular and the radial integration range of $s_{\textrm{\tiny{HB}}}{}_p$ and $s_{\textrm{\tiny{HB}}}{}_q$ is complete for all domains (there is no need to consider improper integrals). As we prove in Appendix D.4, the integrals in the right hand sides of (\ref{sHB2}) and (\ref{sHBq}) vanish and $\dots_{\textrm{\tiny{HB}}}{}_p\to\infty,\,\dots_{\textrm{\tiny{HB}}}{}_q\to\infty$ in the limit $t\tot_{\textrm{\tiny{bb(0)}}}$, which are the same limits (\ref{shblim})--(\ref{dotshblim}) but valid for all domains. Hence, as depicted by figures 3b and 3d, $s_{\textrm{\tiny{HB}}}{}_p$ and $s_{\textrm{\tiny{HB}}}{}_q$ increase from their initial equilibrium value at $t=t_{\textrm{\tiny{bb(0)}}}$ for all domains and follow similar evolution patterns as those displayed in Figures 1b and 1d (though the behaviour of $s_{\textrm{\tiny{HB}}}{}_p$ and $s_{\textrm{\tiny{HB}}}{}_q$ near the Big Crunch is the same as in the general case: compare the curves of Figures 3c and 3d as $t\to t_{(+)}$).
\begin{figure}
\begin{center}
\includegraphics[scale=0.4]{Fig2new_entropy.pdf}
\caption{{\bf HB entropies near non--simultaneous singularities.} The panels display the time evolution of a generic domain bounded by $r=r_b$ (shaded region) in time slices close to non--simultaneous singularities: the Big Bang ($t=t_{\textrm{\tiny{bb}}}( r)$ in panel (a)) and the collapsing Big Crunch ($t=t_{\textrm{\tiny{coll}}}( r)$ in panel (b)). Notice that for a typical time slice $t=t_s$ that ``intersects'' these singularities $\rho$ and ${\cal{H}}$ diverge as $r\to r_s$. Therefore, the integrals in (\ref{sHB2}) and (\ref{sHBq}) for any domain $r=r_b$ must be evaluated in the restricted range to $r_s<r\leq r_b$. Since these integrals are improper, we need to verify their convergence as $r\to r_s$ (see Appendix D.2).}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{Fig3new_entropy.pdf}
\caption{{\bf Time evolution of the HB entropies.} The panels display $s_{\textrm{\tiny{HB}}}[r_b]$ (which can be either one of $s_{\textrm{\tiny{HB}}}{}_p[r_b]$ or $s_{\textrm{\tiny{HB}}}{}_q[r_b]$) as a function of time for a representative generic domain ${\cal {D}}[r_b]$ with $r_b$ finite. Hyperbolic and elliptic models respectively correspond to panels (a)--(b) and ( c)--(d). Panels (a) and ( c) depict the cases with non-zero decaying mode, while the cases with zero decaying mode (simultaneous Big Bang $t=t_{\textrm{\tiny{bb(0)}}}$) are depicted by panels (b) and (d). The forms of the graphs were obtained qualitatively from the discussion in section 7.2 and the proofs of convergence of the two HB entropies in Appendix D.2 and D.4 (see the text of these sections for further detail).}
\end{center}
\end{figure}
\section{Numerical examples.}
We complement the qualitative study of the previous section by the following three numeric examples:
\footnote{We only examine the CET entropy. The second and third examples are meant to illustrate the behaviour of entropy production $\dots_{\tiny{\textrm{gr}}}$. They are not meant to be ``realistic'' or to comply with observational constraints.}
\subsection{Cosmological void with suppressed decaying mode.}
Figure 4 displays $\log |{\textrm{\bf{D}}}_q(\rho){\textrm{\bf{D}}}_q({\cal{H}})|$ as a function of $t$ and $r$ for the LTB void model studied in \cite{February:2009pv} (model \# 2), which is radially asymptotic to an Einstein--de Sitter FLRW model. The free parameter is a present day ($a=a_0=1$) matter density profile (denoted by `` $\Omega_m$'' in \cite{February:2009pv}) that exactly corresponds to the initial value function:
\begin{equation}\Omega_{q0}( r)=\Omega_{\textrm{\tiny{out}}}-\left(\Omega_{\textrm{\tiny{out}}}-\Omega_{\textrm{\tiny{in}}}\right)\exp\left(-\frac{r^{2}}{\sigma^{2}}\right),\label{OmSDM}\end{equation}
where the parameters were selected from compliance with best-fit values to SN1a and age data (more details in \cite{February:2009pv}): $H_{0}=64.33\mbox{ km s}^{-1}\mbox{Mpc}^{-1}$, $\Omega_{\textrm{\tiny{in}}}=0.120$, $\Omega_{\textrm{\tiny{out}}}=1$ and $\sigma=3.77\mbox{ Mpc}$, with cosmic age $t_0 =13.46$ Gyr (the remaining initial value functions ${\cal{H}}_{q0}, \delta^{(\HH)}_{q0},\,\delta^{(\Omega)}_{q0}$ follow from (\ref{ivfSDM}) with $t_{\textrm{\tiny{bb(0)}}}=0$). We see that condition (\ref{CETcond}) is fulfilled: ${\textrm{\bf{D}}}_q(\rho){\textrm{\bf{D}}}_q({\cal{H}})<0$ holds all across spacetime, thus showing that entropy production is positive for the whole time evolution of all observers.
\begin{figure}
\begin{center}
\includegraphics[scale=0.4]{Fig4new_entropy.pdf}
\caption{$\ln |{\textrm{\bf{D}}}_q(\rho){\textrm{\bf{D}}}_q({\cal{H}})|$ as a function of $t$ and $r$ for model \# 2 of \cite{February:2009pv} described in the text. ${\textrm{\bf{D}}}_q(\rho){\textrm{\bf{D}}}_q({\cal{H}})$ is negative throughout spacetime, which implies a positive entropy production everywhere in spacetime.}
\end{center}
\end{figure}
\subsection{Cosmological void model with non-zero decaying mode.}
As a second example we consider a hyperbolic void model whose decaying mode is non-zero, but otherwise it is almost identical to the previous one: it is also asymptotic (in the radial direction) to an Einstein--de Sitter background. This model follows from the same form of $\Omega_{q0}( r)$ in (\ref{OmSDM}), with its second initial value function ${\cal{H}}_{q0}( r)$ not given by (\ref{ivfSDM}), but by
\begin{equation}\fl {\cal{H}}_{q0}=\frac{Y_{q0}(\Omega_{q0})}{t_0-t_{\textrm{\tiny{bb}}}( r)},\qquad t_{\textrm{\tiny{bb}}}=0.01\left[\exp\left(-\frac{r^2}{\sigma_0^2}\right)-1\right],\qquad \sigma_0=1\,\hbox{Mpc},\label{OmDM} \end{equation}
which, evidently, introduces a small decaying mode via a non--simultaneity of the Big Bang (marked $t_{\textrm{\tiny{bb}}}=t_{\textrm{\tiny{bb(0)}}}=0$ in the previous example). This yields a position dependent cosmic age that goes from $t_0=13.46$ Gyr for central observers to an asymptotic value 1\% larger ($\sim 10^8$ years) for observers in the Einstein--de Sitter background. We plot in Figure 5 the product $-\delta^{(\rho)}_q\delta^{(\HH)}_q$ obtained from the numerical solution of the system (21a)--(21d) in \cite{part2}, which is proportional to the sign of $\dots_{\tiny{\textrm{gr}}}$ (from (\ref{condpos})). Comparison of panels (a) and (b) of this figure reveals that $\dots_{\tiny{\textrm{gr}}}>0$ behaves as depicted in the qualitative plots in Figures 1a and 1b: it has almost identical form for both models, save for early very times $a<10^{-3}$ in which $\dots_{\tiny{\textrm{gr}}}$ in panel (b) becomes negative and tends to $-\infty$ as $t\tot_{\textrm{\tiny{bb}}}$ (or $a\to 0$). Since the present day time is taken as $a=a_0=1$, these early times can be identified with times before the last scattering surface $z\sim 1000$ where the dust source is no longer a valid model of cosmic matter.
\begin{figure}
\begin{center}
\includegraphics[scale=0.4]{Fig5new_entropy.pdf}
\caption{{\bf Entropy production at early times in a void model with zero and non-zero decaying mode.} The figure depicts the plot of $-\delta^{(\rho)}_q\delta^{(\HH)}_q \propto \dots_{\tiny{\textrm{gr}}}$ as a function of $\log a$ and $r$ marking the present day time as $a=a_0=1$. Panel (a) corresponds to the model of Figure 2 with suppressed decaying mode and panel (b) corresponds to the closely related model with a non-zero decaying mode whose initial value functions are (\ref{OmSDM}) and (\ref{OmDM}). Notice that the curves have the forms of $\dots_{\tiny{\textrm{gr}}}$ depicted qualitatively in figures 1a and 1b and that the effects of the decaying mode (change of sign of $\dots_{\tiny{\textrm{gr}}}$) are only significant for very early times $a< 10^{-3}$ in panel (b). }
\label{fig3}
\end{center}
\end{figure}
\subsection{Dust gravitational collapse.}
The third example is furnished by a ``spherical collapse model'' defined by a ``mixed'' elliptic/hyperbolic configuration, so that ``inner'' dust layers near the symmetry centre (elliptic region) collapse to a non--simultaneous Big Crunch, while ``external'' layers perpetually expand (hyperbolic region). The initial value functions are given by
\begin{equation}\fl \frac{4\pi\rho_{q0}}{3H_0^2}=\frac{\Omega_0\,(1+\epsilon_1 + x^2)}{2(1+x^2)},\qquad \frac{{\cal{K}}_{q0}}{H_0^2}=\frac{(\Omega_0-1)(1+\epsilon_2+x^{3/2})}{1+x^{3/2}},\end{equation}
with the same value of $H_0$ as in the first example, $\Omega_0=0.8,\,\epsilon_1=2.0,\,\epsilon_2=-1.25,\,\,x=r/\sigma_0$ with $\sigma_0$ an arbitrary length scale and $t$ is normalized with the Hubble factor at the last scattering surface $t_{\textrm{\tiny{LS}}}$ (so that present day is $t_0\sim 10^4$). We have a collapsing elliptic region (${\cal{K}}_{q0}>0$) for $0\leq x<0.397$ and an expanding hyperbolic region (${\cal{K}}_{q0}<0$) for $r>0.397$. The curves of Figure 6, which were obtained by solving numerically the system of evolution equations (\ref{EVq21})--(\ref{EVq24}), clearly reveal how $\dots_{\tiny{\textrm{gr}}}\to\infty$ as $t\tot_{\textrm{\tiny{coll}}}$ in the collapsing layers of the elliptic region (like the late time form of $\dots_{\tiny{\textrm{gr}}}$ in Figure 1c), whereas $\dots_{\tiny{\textrm{gr}}}\to 0$ as $t\to\infty$ in the expanding layers of the hyperbolic region (as in its late time form in figure 1a). The early time evolution is not displayed by Figure 6, but since $\JJ_{\tiny{\textrm{(d)}}}\ne 0$ it is qualitatively analogous to that depicted by figures 1a, 1c and 5b ($\dots_{\tiny{\textrm{gr}}}\to-\infty$ as $t\tot_{\textrm{\tiny{bb}}}$) .
\begin{figure}
\begin{center}
\includegraphics[scale=0.4]{Fig6new_entropy.pdf}
\caption{{\bf Entropy production in a collapse/expansion evolution.} The figure displays $\dots_{\tiny{\textrm{gr}}}$ given by (\ref{sgrt}) as a function of $\log t$ and $r$ for a mixed configuration made of an ``inner'' elliptic region surrounded by an ``outer'' hyperbolic region that expands perpetually. Notice that $\dots_{\tiny{\textrm{gr}}}\to\infty$ for collapsing layers while $\dots_{\tiny{\textrm{gr}}}\to 0$ for asymptotic layers as $t\to\infty$. }
\label{fig1}
\end{center}
\end{figure}
\section{Radial scaling and asymptotic behaviour.}
\subsection{Integrability conditions for the CET entropy.}
As opposed to the HBp and HBq entropies, the CET proposal defines entropy through the convective derivative $\dots_{\tiny{\textrm{gr}}}$ in the Gibbs equation (\ref{CET2}), which is a relation between entropy and energy one--forms with integrating factor $T_{\tiny{\textrm{gr}}}$:
\begin{equation}{\rm{d}} s_{\tiny{\textrm{gr}}} =\frac{{\rm{d}} (\rho_{\tiny{\textrm{gr}}} V)}{T_{\tiny{\textrm{gr}}}}.\end{equation}
For the spherically symmetric LTB models in coordinates $(t,r,\vartheta,\phi)$ we have ${\rm{d}} s_{\tiny{\textrm{gr}}}=[\dots_{\tiny{\textrm{gr}}},\,s_{\tiny{\textrm{gr}}}',0,0]$, and so the components of the Gibbs one form yield an integrability condition
\begin{equation}\fl \dot s_{\tiny{\textrm{gr}}} =\frac{(\rho_{\tiny{\textrm{gr}}} V)\,\dot{}}{T_{\tiny{\textrm{gr}}}},\quad s_{\tiny{\textrm{gr}}}' =\frac{(\rho_{\tiny{\textrm{gr}}} V)'}{T_{\tiny{\textrm{gr}}}} \quad \Rightarrow\quad T_{\tiny{\textrm{gr}}}'\dots_{\tiny{\textrm{gr}}} - \dotT_{\tiny{\textrm{gr}}}s_{\tiny{\textrm{gr}}}'=0,\end{equation}
which leads to the following fully general expression for $s_{\tiny{\textrm{gr}}}'$:
\begin{equation} s_{\tiny{\textrm{gr}}}' =-\frac{\partial_r[F(T_{\tiny{\textrm{gr}}})]}{\partial_t[F(T_{\tiny{\textrm{gr}}})]}\,\dots_{\tiny{\textrm{gr}}},\label{sgrrdef}\end{equation}
where $F(T_{\tiny{\textrm{gr}}})$ is an arbitrary smooth function. Considering for simplicity the particular case $F(T_{\tiny{\textrm{gr}}})=T_{\tiny{\textrm{gr}}}^2$, and substituting the forms of $T_{\tiny{\textrm{gr}}}$ and $\dot s_{\tiny{\textrm{gr}}}$ in (\ref{Tgrav2}), (\ref{CETc12}), (\ref{CETc122}) and (\ref{sgrt}) into (\ref{sgrrdef}), yields after some algebraic manipulation
\begin{equation} \fl s_{\tiny{\textrm{gr}}}' = -\frac{3\alpha\,\Omega_q\,a^3}{4\,H_{q0}^2}\,\frac{\Gamma\,\delta^{(\HH)}_q\,\delta^{(\rho)}_q\,{\cal{H}}_q}{|1+3\delta^{(\HH)}_q|\,|\delta^{(\rho)}_q|\,|{\cal{H}}_q|}\,\frac{{\cal{H}}'_q\,(1+3\delta^{(\HH)}_q)+3{\cal{H}}_q(\delta^{(\HH)}_q)'}{[1+3\delta^{(\HH)}_q+\frac{1}{2}\Omega_q\,(1+3\delta^{(\rho)}_q)]},\label{sgrr}\end{equation}
where $\Omega_q$ is given by (\ref{OmDOm}) and we eliminated $2\pi\rho_{q0}/{\cal{H}}_q^2,\,\dot{\cal{H}}_q,\,\dot\delta^{(\HH)}_q$ from (\ref{HHq}) and the evolution equations (19b) and (19d) of \cite{part2}.
\subsection{Radial asymptotic convergence of LTB models.}
In order to examine the radial asymptotic behaviour of the CET and the two HB entropies we will rely on the results of the comprehensive study of the radial asymptotics of LTB models undertaken in \cite{RadAs}. Assuming absence of shell crossing singularities and a well behaved radial coordinate (since radial rays are spacelike geodesics, the proper radial length along them must be a monotonous function of $r$), we characterize the radial asymptotic behaviour of the covariant parameters of the models by means of the definition of radial asymptotic convergence ``$A\sim \tilde A$\,'' given in section 7.1 of \cite{RadAs}, considering the polynomial asymptotic forms for the following initial value functions:
\begin{equation} \fl \rho_{q0}\sim m_0+m_1r^{-\alpha},\qquad {\cal{K}}_{q0}\sim k_0+k_1 r^{-\beta},\qquad {\cal{H}}_{q0}\sim H_0+H_1\,r^{-\nu},\label{asrhok}\end{equation}
where $m_0\geq 0,\,m_1,\,k_0,\,k_1,\,H_0,\,H_1,\,\alpha,\,\beta$ and $\nu$ are real constants, whose values and restrictions (given in Table 1) correspond to the various classes of radial asymptotic convergence studied in \cite{RadAs}. Using (\ref{asrhok}) yields the following asymptotic forms for the scale factors:
\begin{equation} \\
a\sim 1+ (H_0+H_1 r^{-\nu})(t-t_0),\qquad \Gamma \sim 1-\frac{\nu H_1 r^{-\nu}\,(t-t_0)}{1+H_0 (t-t_0)},\label{asaG}\end{equation}
obtained by looking at the radial asymptotic behaviour of the exact solutions (\ref{hypsol})--(\ref{ellsol}) (expressed in terms of $a,\,\rho_{q0}$ and ${\cal{K}}_{q0}$) for the various classes of radial asymptotic convergence listed in Table 1.
From (\ref{asrhok}) and (\ref{asaG}) and considering the values of the asymptotic parameters given in Table 1, the asymptotic radial limits of the q--scalars and their perturbations for the various classes of models are given below (see \cite{RadAs} for a comprehensive discussion):
\begin{itemize}
\item Models asymptotic to FLRW: Einstein de Sitter, open FLRW and Milne ($H_0>0$)
\begin{equation} \fl (a,\,\rho_q,\,{\cal{H}}_q,\,\Omega_q)\to (\tilde a,\,\tilde\rho,\,\tilde{\cal{H}},\,\tilde\Omega),\quad (\delta^{(\rho)}_q,\,\delta^{(\HH)}_q)\to 0,\quad \Gamma\to 1,\label{asympt1}\end{equation}
where $\tilde a,\,\tilde\rho,\,\tilde{\cal{H}},\,\tilde\Omega$ are the scale factor, density, Hubble scalar and Omega factor of the FLRW model ($\tilde\rho=0$ for models asymptotic to Milne).
\item Models asymptotic to Minkowski: MD, VD and G ($H_0=0$, see Table 1)
\numparts\begin{eqnarray} \fl (\rho_q,\,{\cal{H}}_q)\to 0,\quad (\delta^{(\rho)}_q,\,\delta^{(\HH)}_q)\to (\delta^{(\rho)}_{q_\infty},\,\delta^{(\HH)}_{q_\infty}),\quad (a,\,\Gamma)\to 1,\label{asympt2}\\
\fl \Omega_q\to 0\,\, \hbox{(VD)},\qquad \Omega_q\to 1\,\, \hbox{(MD)},\qquad \Omega_q\to \frac{8\pi m_1}{8\pi m_1+3k_1}<1\,\, \hbox{(G)},\label{asympt3}\end{eqnarray}\endnumparts
where $\delta^{(\rho)}_{q_\infty}=-\alpha/3$ and $\delta^{(\HH)}_{q_\infty}=-\nu/3$ are the finite non-zero asymptotic values of the perturbations.
\end{itemize}
\subsection{The CET entropy.}
It is evident from (\ref{sgrt}), (\ref{asympt1}) and (\ref{asympt2}) that $\dots_{\tiny{\textrm{gr}}}\to 0$ as $r\to\infty$ for all convergence classes, as for models converging to FLRW we have in this limit $\rho_{q0}\to m_0>0$ but $\delta^{(\HH)}\to 0$, while for models converging to Milne or Minkowski we have $\delta^{(\HH)}\to -\nu/3<0$ but $\rho_{q0}\to 0$.
Regarding $s_{\tiny{\textrm{gr}}}'$ in (\ref{sgrr}), the asymptotic forms for $\Omega_q,\,a$ and $\Gamma$ in (\ref{asympt1}) and (\ref{asympt2})--(\ref{asympt3}) are non-zero for all models (save $\Omega_q$ for the VD hyperbolic models). However, we have for all convergence classes the following asymptotic form
\begin{equation}\fl {\cal{H}}'_q (1+3\delta^{(\HH)}_q) + 3{\cal{H}}_q (\delta^{(\HH)}_q)' \sim \frac{\nu(\nu-1)\,H_1\,r^{-\nu-1}}{[1+H_0(t-t_0)]^2}+O(r^{-2\nu-1})\to 0,\end{equation}
where we used (\ref{Dadef}), (\ref{HHq}) and the asymptotic forms (\ref{asaG}). Therefore, we have $s_{\tiny{\textrm{gr}}}'\to 0$ for all models compatible with a radial asymptotic range. Since both $\dots_{\tiny{\textrm{gr}}}$ and $s_{\tiny{\textrm{gr}}}'$ tend to zero as $r\to\infty$, then $s_{\tiny{\textrm{gr}}}$ (which is defined up to an additive constant) must reach a finite constant asymptotic ``equilibrium'' value in the radial direction for all time slices.
\begin{table}
\begin{center}
\begin{tabular}{|c| c| c|}
\hline
\hline
\hline
\hline
\multicolumn{3}{|c|}{{\bf Parameters of radial asymptotic convergence.}}
\\
\hline
\hline
\hline
\hline
\multicolumn{3}{|c|}{Hyperbolic models, ${\cal{K}}_{q0}<0$ or $0<\Omega_{q0}<1$.}
\\
\hline
\hline
\hline
\hline
{Asymptotic class} &{$m_0,\,k_0,\,\alpha,\,\beta$} &{$H_0,\,\nu$}
\\
\hline
{{\bf open FLRW}} &{$m_0>0,\,k_0<0$} &{$H_0=[2m_0+|k_0|]^{1/2}$ }
\\
{} &{$\alpha>0,\,\beta>0 $} &{$\nu =\hbox{min}(\alpha,\beta)$}
\\
\hline
{{\bf Milne}} &{$m_0=0,\,k_0<0$} &{$H_0=|k_0|^{1/2}$}
\\
{} &{$0<\alpha\leq 3,\,\,\beta>0$} &{$\nu =\hbox{min}(\alpha,\beta)$}
\\
\hline
{{\bf Einstein de Sitter}} &{$m_0>0,\,k_0=0$} &{$H_0=[2m_0]^{1/2}$}
\\
{} &{$\alpha>0,\,\,0<\beta\leq 2$} &{$\nu =\hbox{min}(\alpha,\beta)$}
\\
\hline
{{\bf Minkowski}} &{$m_0=k_0=0$} &{$H_0=0,\,H_1>0$}
\\
{} &{$0<\alpha\leq 3,\,\,0<\beta\leq 2$} &{$\nu =\alpha/2$ (MD: $\beta>\alpha$)}
\\
{} &{} &{$\nu =\beta/2$ (VD: $\beta<\alpha$)}
\\
{} &{} &{$\nu =\gamma/2$ (G: $\gamma=\beta=\alpha$)}
\\
\hline
\hline
\hline
\hline
\multicolumn{3}{|c|}{Elliptic models, ${\cal{K}}_{q0}>0$ or $\Omega_{q0}>1$. }
\\
\hline
\hline
\hline
\hline
{Asymptotic class} &{$m_0,\,k_0,\,\alpha,\,\beta$} &{$H_0,\,\nu$}
\\
\hline
{{\bf Einstein de Sitter}} &{$m_0>0,\,k_0=0$} &{$H_0=[2m_0]^{1/2}$}
\\
{} &{$\alpha>0,\,\,\beta\geq 2$} &{$\nu =\hbox{min}(\alpha,\beta)$}
\\
\hline
{{\bf Minkowski}} &{$m_0=k_0=0$} &{$H_0=0,\,H_1>0$}
\\
{} &{$0<\alpha\leq 3,\,\,\beta\geq 2$} &{$\nu =\alpha/2$ (MD $\beta>\alpha$)}
\\
\hline
\hline
\hline
\hline
\end{tabular}
\end{center}
\caption{{\bf{Parameters in the classification of radial asymptotic convergence}}. The values for the parameters describe all the classes of radial asymptotic convergence of LTB models discussed in \cite{RadAs} for polynomial trial functions (\ref{asrhok}). Models converging to Minkowski are subdivided in the following classes according to the limit of $\alpha_{q0}\propto {\cal{K}}_{q0}/\rho_{q0}$ as $r\to\infty$: ``MD'' (matter dominated, $\alpha<\beta$) if $\alpha_{q0}\to 0$, ``VD'' (vacuum or curvature dominated, $\alpha>\beta$) if $\alpha_{q0}\to \infty$ and ``G'' (generic, $\alpha=\beta$) if $\alpha_{q0}\to \alpha_0=$ constant. The restrictions on the range of $\alpha$ and $\beta$ in models converging to Milne, Einstein de Sitter and Minkowski strictly follow from the conditions to avoid shell crossings examined in \cite{RadAs}. The relation between $\alpha,\,\beta$ and $\nu$ follows from the asymptotic behavior of ${\cal{H}}_{q0}^2=(8\pi/3)\rho_{q0}-{\cal{K}}_{q0}$ for the different classes of convergence. The values $\alpha=3$ and $\alpha=\beta=2$ respectively correspond to models asymptotic to Schwarzschild and to self similar dust solutions (see further detail in \cite{RadAs}).}
\label{tabla3}
\end{table}
\subsection{The HBp and HBq entropies.}
The radial scaling of these entropies depends on the behaviour of the integrals (\ref{sHB2}) and (\ref{sHBq}) as functions of an increasing domain boundary $r_b$, up the asymptotic limit $r_b\to\infty$ that would correspond to domains that encompass whole time slices ($t$ constant hypersurfaces). While (\ref{sHB2}) and (\ref{sHBq}) are analogous to the proper volume mass--energy integrals $M_p$ and $M_q$ in (\ref{Mp}) and (\ref{Mq}), with $\rho\ln[\rho/\langle \rho\rangle_p]$ and $\rho\ln[\rho/\langle \rho\rangle_q]$ playing the role of entropy ``densities'', the asymptotic convergence of these ``densities'' does not imply the asymptotic convergence of the proper volume integrals (\ref{sHB2}) and (\ref{sHBq}), just as a converging $\rho$ does not prevent the mass--energy functionals $M_p$ and $M_q$ in (\ref{Mp}) and (\ref{Mq}) from diverging in the radial asymptotic range.
In order to explore the asymptotic convergence of $s_{\textrm{\tiny{HB}}}{}_p$ and $s_{\textrm{\tiny{HB}}}{}_q$ we use (\ref{propvol}) and (\ref{qvol}) to rewrite (\ref{sHB2}) and (\ref{sHBq}) as
\numparts\begin{eqnarray} \fl s_{\textrm{\tiny{HB}}}{}_p[r_b] -s_{\textrm{\tiny{HB}}}{}_p^{(\textrm{\tiny{eq}})}\nonumber\\
\fl = 4\pi\gamma_0\left[\int_0^{r_b}{\frac{\rho_{q0}(1+\delta^{(\rho)}_0)\,\bar r^2}{\sqrt{1-{\cal{K}}_{q0} \bar r^2}}\ln\rho\,{\rm{d}} \bar r}-\ln\langle \rho\rangle_p[r_b]\int_0^{r_b}{\frac{\rho_{q0}(1+\delta^{(\rho)}_0)\,\bar r^2}{\sqrt{1-{\cal{K}}_{q0} \bar r^2}}{\rm{d}} \bar r}\right],\label{sHBpp}\\
\fl s_{\textrm{\tiny{HB}}}{}_q[r_b] -s_{\textrm{\tiny{HB}}}{}_q^{(\textrm{\tiny{eq}})} \nonumber\\
\fl =4\pi\gamma_0\left[\int_0^{r_b}{\rho_{q0}(1+\delta^{(\rho)}_0)\,\bar r^2\,\ln\rho\,{\rm{d}} \bar r}-\ln\langle \rho\rangle_q[r_b]\int_0^{r_b}{\rho_{q0}(1+\delta^{(\rho)}_0)\,\bar r^2 \,{\rm{d}} \bar r}\right],\label{sHBqq}
\end{eqnarray}\endnumparts
where we assume integration over complete time slices (no intersection with $t=t_{\textrm{\tiny{bb}}}( r)$ as in section 7.2) and we used (\ref{rhoHHKK}), (\ref{rhoKK}) and (\ref{Drho}).
We obtain the asymptotic forms of $\rho,\,\langle \rho\rangle_p[r_b],\,\langle \rho\rangle_q[r_b]$ by substitution of (\ref{asrhok}) and (\ref{asaG}) in (\ref{rhoHHKK}), (\ref{rhoKK}), (\ref{Drho}), (\ref{rhoave2}) and (\ref{rhoaveq2}). Following the convergence of integrals described in Appendix B of \cite{RadAs}, we substitute these asymptotic forms together with (\ref{asrhok}) into the integrals (\ref{sHBpp})--(\ref{sHBqq}) and evaluate their asymptotic forms. After tedious algebraic manipulations we finally obtain the following results:
\begin{itemize}
\item Hyperbolic and elliptic models converging to Minkowski: both $s_{\textrm{\tiny{HB}}}{}_p[r_b]$ and $s_{\textrm{\tiny{HB}}}{}_q[r_b]$ diverge as $r_b\to\infty$.
\item Models converging to a FLRW state: the convergence of $s_{\textrm{\tiny{HB}}}{}_p[r_b]$ and $s_{\textrm{\tiny{HB}}}{}_q[r_b]$ strongly depends on the details of the convergence of $\rho$ and ${\cal{K}}$, which is determined by the convergence of $\rho_{q0}$ and ${\cal{K}}_{q0}$ (see \cite{RadAs}). Considering the case $\alpha=\beta=\nu$ (see Table 1) we obtain
\begin{itemize}
\item Hyperbolic models converging to open FLRW ($\nu>0$)
\numparts\begin{eqnarray}\fl s_{\textrm{\tiny{HB}}}{}_p[r_b] -s_{\textrm{\tiny{HB}}}{}_p^{(\textrm{\tiny{eq}})}\sim r_b^{2(1-\nu)},\qquad s_{\textrm{\tiny{HB}}}{}_p[r_b]\,\,\hbox{converges for}\,\,\nu\geq 1,\\ \fl s_{\textrm{\tiny{HB}}}{}_q[r_b] -s_{\textrm{\tiny{HB}}}{}_q^{(\textrm{\tiny{eq}})}\sim r_b^{3-2\nu},\qquad s_{\textrm{\tiny{HB}}}{}_q[r_b]\,\,\hbox{converges for}\,\,\nu\geq 3/2, \end{eqnarray}\endnumparts
\item Hyperbolic models converging to Milne ($0<\nu\leq 3$)
\numparts\begin{eqnarray}\fl s_{\textrm{\tiny{HB}}}{}_p[r_b] -s_{\textrm{\tiny{HB}}}{}_p^{(\textrm{\tiny{eq}})}\sim r_b^{2-\nu/2},\qquad s_{\textrm{\tiny{HB}}}{}_p[r_b]\,\,\hbox{diverges},\\ \fl s_{\textrm{\tiny{HB}}}{}_q[r_b] -s_{\textrm{\tiny{HB}}}{}_q^{(\textrm{\tiny{eq}})}\sim r_b,\qquad\qquad s_{\textrm{\tiny{HB}}}{}_q[r_b]\,\,\hbox{diverges}, \end{eqnarray}\endnumparts
\item Hyperbolic and elliptic models converging to Einstein de Sitter ($0<\nu\leq 2$)
\numparts\begin{eqnarray}\fl s_{\textrm{\tiny{HB}}}{}_p[r_b] -s_{\textrm{\tiny{HB}}}{}_p^{(\textrm{\tiny{eq}})}\sim r_b^{2-3\nu/2},\qquad s_{\textrm{\tiny{HB}}}{}_p[r_b]\,\,\hbox{converges for}\,\,\nu\geq 4/3,\\ \fl s_{\textrm{\tiny{HB}}}{}_q[r_b] -s_{\textrm{\tiny{HB}}}{}_q^{(\textrm{\tiny{eq}})}\sim r_b^{3-2\nu},\qquad s_{\textrm{\tiny{HB}}}{}_q[r_b]\,\,\hbox{converges for}\,\,\nu\geq 3/2, \end{eqnarray}\endnumparts
\end{itemize}
\end{itemize}
Evidently, the exact value of $\nu$ that distinguishes convergence form divergence may change if we relax the condition $\nu=\alpha=\beta$ (see Table 1). However, for whatever values of these exponents allowed by regularity conditions, the following results emerge:
\begin{itemize}
\item $s_{\textrm{\tiny{HB}}}{}_p[r_b]$ and $s_{\textrm{\tiny{HB}}}{}_q[r_b]$ diverge, and thus scale with volume ({\it i.e} they are ``extensive'' entropies) for models that radially converge to vacuum states (Milne and Minkowski).
\item $s_{\textrm{\tiny{HB}}}{}_p[r_b]$ and $s_{\textrm{\tiny{HB}}}{}_q[r_b]$ converge (and thus do not scale with volume and are ``non--extensive'') when $\rho$ converges sufficiently fast to the density of a non--vacuum FLRW asymptotic state (open FLRW or Einstein--de Sitter).
\end{itemize}
The relation between the convergence of these entropies and the radial asymptotic behaviour of the density modes is evident: both entropies converge for models radially converging to non--vacuum FLRW for which both modes $\JJ_{\tiny{\textrm{(g)}}}$ and $\JJ_{\tiny{\textrm{(d)}}}$ vanish in the radial asymptotic limit (as all perturbations and fluctuations vanish in this limit for these models \cite{RadAs}). On the other hand, assuming (\ref{asrhok}) for models converging to Minkowski (see Table 1), we have $\delta^{(\rho)}_{q0}\sim -\alpha/3,\,\delta^{(\KK)}_{q0}\sim-\beta/3$, thus (\ref{gmode})--(\ref{ivpconstr}), (\ref{asaG}) and (\ref{HTq})--(\ref{H32}) lead to:
\begin{itemize}
\item MD (matter dominated models, see Table 1): $\Omega_q\sim 1+O(\Omega_{q0}-1)\to 1$, and thus
\begin{equation}\fl \JJ_{\tiny{\textrm{(g)}}} \sim -\frac{\alpha-\frac{3}{2}\beta}{3-\alpha}\,O(\Omega_{q0}-1),\qquad \JJ_{\tiny{\textrm{(d)}}}\sim -\frac{\beta}{2(3-\alpha)}+O(\Omega_{q0}-1),\end{equation}
\item VD (vacuum or curvature dominated models, see Table 1): $\Omega_q\sim O(\Omega_{q0})\to 0$, and thus
\begin{equation}\fl \JJ_{\tiny{\textrm{(g)}}} \sim -\frac{\alpha-\frac{3}{2}\beta}{3(3-\alpha)}+O(\Omega_{q0}),\qquad \JJ_{\tiny{\textrm{(d)}}}\sim -\frac{\beta}{2(3-\alpha)}+O(\Omega_{q0}).\end{equation}
\end{itemize}
As a consequence, for $\alpha<3$ we have for MD models $s_{\textrm{\tiny{HB}}}{}_p$ and $s_{\textrm{\tiny{HB}}}{}_q$ diverging under conditions that are similar to those of early times for the general case: $\JJ_{\tiny{\textrm{(g)}}}\to 0$ and $\JJ_{\tiny{\textrm{(d)}}}$ finite so that $|\JJ_{\tiny{\textrm{(d)}}}|\gg |\JJ_{\tiny{\textrm{(g)}}}|$, while for VD models the entropies diverge under different asymptotic conditions: both modes tend to finite values with (in general) $|\JJ_{\tiny{\textrm{(g)}}}|> |\JJ_{\tiny{\textrm{(d)}}}|$ (notice that $t_{\textrm{\tiny{bb}}}'\to-\infty$ holds in all models radially asymptotic to Minkowski \cite{RadAs}, but $t_{\textrm{\tiny{bb}}}'$ is not the only factor involved in the decaying mode). In the case $\alpha=3$ (so that $\rho_{q0}\sim r^{-3}$, asymptotically Schwarzschild models) both entropies and both modes diverge for MD and VD models and thus further examination is required to verify which mode is dominant.
\section{Summary and conclusion.}
We have undertaken in this paper a comprehensive study of the application to generic LTB dust models of two different definitions of a gravitational entropy (section 2): the CET (Clifton, Ellis and Tavakol) proposal \cite{CET}, and two variants of the HB (Hosoya and Buchert) proposal: the original one (denoted by HBp) based on Buchert's average \cite{HB1,HB2,HB3} and one (denoted by HBq) constructed with a weighted average (the q--average) and specially suited for LTB models \cite{part1}. In order to probe these entropy proposals on LTB models, we described (section 3) their dynamical and geometric properties by means of an initial value parametrization of their metric, together with a covariant representation of q--scalars, their fluctuations and perturbations \cite{part1,part2} expressed in terms of exact generalizations of the density growing and decaying modes of linear perturbation theory \cite{sussmodes}. We summarize below our main results:
\begin{description}
\item[Conditions for entropy growth.] The usage of q--scalars and their perturbations allowed for a unifying description of the necessary and sufficient conditions for non--negative entropy production for the three proposals described above (conditions (\ref{CETcond}), (\ref{HBcond}) and (\ref{HBcondq})):
\begin{eqnarray} \fl \dot s_{\textrm{\tiny{HB}}}{}_p \geq 0 \quad \Leftrightarrow\quad \left\langle\, (\rho-\langle \rho\rangle_p)({\cal{H}}-\langle \HH\rangle_p)\,\right\rangle_p\leq 0,\qquad\hbox{HBp proposal},\nonumber\\
\fl \dot s_{\textrm{\tiny{HB}}}{}_q \geq 0 \quad \Leftrightarrow\quad \left\langle\, (\rho-\langle \rho\rangle_q)({\cal{H}}-\langle \HH\rangle_q)\,\right\rangle_q\leq 0,\qquad\hbox{HBq proposal},\nonumber\\
\fl \dots_{\tiny{\textrm{gr}}} \geq 0 \quad \Leftrightarrow\quad (\rho-\rho_q)({\cal{H}}-{\cal{H}}_q)\leq 0,\qquad\qquad\quad\hbox{CET proposal},\nonumber\end{eqnarray}
where we remark that the condition above for the CET proposal is a new result not obtained in the original CET paper \cite{CET}, while the conditions for the HBp and HBq proposals were previously known \cite{HB1,HB2,HB3,part1}. Entropy growth for the three proposals is directly related to a non--positive correlation of analogous (though strictly different) fluctuations of $\rho$ and ${\cal{H}}$. We regard this finding as an appealing and important result. However, as discussed in section 5, there are subtle differences between these correlations:
\begin{itemize}
\item Entropy growth in the HBp and HBq proposals involves, strictly speaking, statistical correlations (statistical covariance moment) involving average functionals $\langle \rho\rangle_p,\,\langle \HH\rangle_p$ and $\langle \rho\rangle_q,\,\langle \HH\rangle_q$.
\item Entropy growth in the CET proposal involves local correlations that are not statistical, as they involve the local functions (q--scalars) $\rho_q,\,{\cal{H}}_q$ whose correspondence rule is the same as $\langle \rho\rangle_q,\,\langle \HH\rangle_q$.
\end{itemize}
As a consequence, the CET condition only needs to be evaluated locally at each point, whereas the HBp and HBq entropy production conditions are domain dependent, and thus are necessary and sufficient {\it only if} the integrals in the involved averages are evaluated for each given domain. Because of the integral non--local nature of these entropies, we can have entropy growth for a given domain ${\cal {D}}[r_b]$, even if it decreases in local regions inside the domain (or in a smaller domain ${\cal {D}}[r_c]\subset {\cal {D}}[r_b]$ with $r_c\ll r_b$). However, it is possible to obtain weaker (sufficient but not necessary) entropy production conditions for the HB entropies in terms of a uniform behavior of local fluctuations (see conditions (\ref{HBcond1})--(\ref{HBcond2}) and (\ref{HBcond3})). In particular, since $\rho_q,\,{\cal{H}}_q$ coincide with the functionals $\langle \rho\rangle_q,\,\langle \HH\rangle_q$ at the boundary of any domain, the entropy production condition for the CET entropy above is also an entropy production condition for the HBq entropy, but in the latter it is only sufficient and in the former it is necessary and sufficient.
\item[The CET entropy in the asymptotic evolution time ranges.] We proved analytically (section 6) that the CET entropy grows in all asymptotic evolution ranges in which the decaying mode is subdominant or is suppressed: asymptotic time range of expanding hyperbolic models, maximal expansion and collapse of elliptic models and near a simultaneous Big Bang (which follows by suppressing the decaying mode). The CET entropy decreases only in models (such as parabolic models) in which the growing mode is (artificially) suppressed and (for general models) near the non--simultaneous Big Bang where the decaying is always dominant.
\item[The HB entropies near singularities.] Because of their domain dependent nature the HBp/HBq entropies exhibit a different behavior from that of the CET entropy near the non--simultaneous Big Bang and Big Crunch (see figure 2) in models with non-zero decaying mode: while the CET entropy diverges as dust layers reach these singularities, the HBp/HBq entropies are bounded in ``shrunk'' domains intersecting the singularities, though their time derivatives (entropy production) diverge for these domains (all this was proven in Appendix D.2).
\item[Qualitative time evolution and numerical examples.] The asymptotic time behavior of the CET entropy production condition (section 7) yields sufficient information to put together a complete qualitative description of its full time evolution. This evolution is depicted in Figures 1 for a typical dust layer. Considering dust layers in models for which the condition (\ref{CETcond}) for $\dots_{\tiny{\textrm{gr}}}\geq 0$ holds (depicted by figures 1), and using the sufficient conditions (\ref{HBcond1})--(\ref{HBcond2}) and (\ref{HBcond3}), we can infer qualitatively the time evolution of both HB entropies in domains bounded by these dust layers in the non--asymptotic time range (see section 7). This evolution is depicted by figures 3 for a typical domain. We added (section 8) four numerical examples that fully corroborate the qualitative results on the CET entropy: a void model with suppressed decaying mode that fits supernovae and age constraints \cite{February:2009pv} (Figure 4), a similar void model but with non-zero decaying mode to compare with the previous example (Figure 5), and a ``spherical collapse model'' made of an elliptic collapsing region in a hyperbolic expanding exterior (Figure 6).
\item[Terminal entropy and net entropy gain.] The time asymptotic behaviour of the HBp/HBq entropies for ever-expanding models reveals (see proof in Appendix D.3) that there is a net entropy gain for the full time evolution of these models (Figures 3a and 3b), as the asymptotic value is a domain dependent ``equilibrium'' terminal value necessarily larger than the initial ``equilibrium'' value at the Big Bang. There is also an analogous behaviour (a position dependent terminal equilibrium value) in the CET entropy applied to these models and locally evaluated along each local dust layer in the asymptotic time range (Figures 1a and 1b), though in this case the net entropy gain does not hold for the full time evolution because $s_{\tiny{\textrm{gr}}}\to\infty$ as $t\tot_{\textrm{\tiny{bb}}}$ (moreover, such entropy net gain occurs in the late time evolution: see Figures 1a and 1c, and if the decaying mode is suppressed: see Figures 1b and 1d). These entropy gains are a consistent result, since the CET and HBp/HBq entropy productions are positive as the evolution proceeds and the growing mode becomes dominant. The HBp/HBq and CET entropies are more similar to each other in models with suppressed decaying mode (convergence results for these models were proven in Appendix D.4).
\item[Radial scaling of the CET entropy.] Since the CET entropy is defined by its entropy production law ($\dots_{\tiny{\textrm{gr}}}$) through a Gibbs one--form, we solved the corresponding integrability condition to obtain its radial gradient $s_{\tiny{\textrm{gr}}}'$ (section 9). From this result we were able to provide analytic proof of the convergence of the CET entropy in the radial asymptotic range for all the different classes of radial asymptotic convergence of LTB models studied and classified in \cite{RadAs} (see Table 1). Since both $\dots_{\tiny{\textrm{gr}}}$ and $s_{\tiny{\textrm{gr}}}'$ vanish as $r\to\infty$, the radial asymptotic ``equilibrium'' state must be characterized by a time independent constant $s_{\tiny{\textrm{gr}}}$ (as $s_{\tiny{\textrm{gr}}}$ is defined up to an additive constant) for all models. This result is consistent with the construction of the CET entropy with the Weyl and Bell--Robinson tensors, as LTB models converge asymptotically in the radial direction to spacetimes (FLRW or Minkowski) for which these tensors vanish.
\item[Radial scaling of the HB entropies.] Probing the radial asymptotic convergence of the HB entropies involved looking at these functionals for increasingly large domains in a given (complete) time slice. Using polynomial asymptotic trial functions derived in \cite{RadAs} we obtained the following results: the HBp/HBq entropies tend to an asymptotic time dependent ``equilibrium'' value in models radially converging to a FLRW background, but {\it only} if $\rho$ and ${\cal{H}}$ converge sufficiently fast to their asymptotic FLRW values. For a slow convergence to FLRW and for models converging to a vacuum state (Minkowski or Milne), these entropies diverge in the asymptotic radial range.
\end{description}
\subsection{The CET gravitational entropy and isotropic cosmological singularities.}
We have shown from the asymptotic and qualitative study of the CET entropy that:
\begin{itemize}
\item $s_{\tiny{\textrm{gr}}}$ decreases in time ranges in which the decaying mode $\JJ_{\tiny{\textrm{(d)}}}$ is dominant, either when the growing mode $\JJ_{\tiny{\textrm{(g)}}}$ is suppressed (section 6.3) or if $\JJ_{\tiny{\textrm{(g)}}}\ne 0$ as dust layers emerge from a non--simultaneous Big Bang.
\item $s_{\tiny{\textrm{gr}}}$ increases near the Big Bang singularity and throughout the full time evolution {\it only} in models examined in section 6.2 in which the decaying mode is totally suppressed: $\JJ_{\tiny{\textrm{(d)}}}=0$ (this is also true for the HBp/HBq entropies).
\end{itemize}
However, as shown by Wainwright and Andrews \cite{WainAndr}, LTB models with a suppressed decaying mode emerge from an isotropic Big Bang and converge in earlier times to a spatially flat EdS model. This suggests an important theoretical connection between the early time behavior of the CET entropy and basic geometric features of the initial singularity and early time evolution of the models. In particular, the comprehensive studies by Goode and Wainwright \cite{GooWain} and Lim {\it et al} \cite{Limetal} may further suggest that an ever increasing CET entropy may also be a characteristic robust property of generic inhomogeneous perfect fluid models ($\Lambda\ne 0$ is assumed in \cite{Limetal}) admitting an isotropic initial singularity associated with an early times convergence to a spatially flat FLRW model (EdS model). Lim {\it et al} characterize this class of models by only three free parameters, which for the case of pure dust sources ($p=\Lambda=0$) reduce to a single free function. Evidently, LTB models with $\JJ_{\tiny{\textrm{(d)}}}=0$ are the spherically symmetric dust sub--case (with $\Lambda=0$) of this class of generic models defined by Lim {\it et al}, as they are fully specified by a single free function (see \cite{sussmodes}) and (as shown by \cite{WainAndr}) they satisfy the asymptotic conditions that \cite{GooWain} and \cite{Limetal} use to define the isotropic Big Bang and early time EdS behavior. Since a full detailed comparison with the formalism of \cite{GooWain} and \cite{Limetal} is outside the scope of this paper, we examine only their asymptotic condition $\hat\Omega=1$ near the initial singularity, where $\hat\Omega$ is given by:
\begin{equation} \hat \Omega \equiv \frac{8\pi\,\rho}{3{\cal{H}}^2} = \Omega_q \frac{1+\delta^{(\rho)}_q}{(1+\delta^{(\HH)}_q)^2},\qquad \Omega_q = \frac{8\pi\,\rho_q}{3{\cal{H}}_q^2},\end{equation}
and must not be confused with the q--scalar $\Omega_q\ne \hat\Omega$ defined by (\ref{OmDOm}) (Lim {\it et al} denote $\hat\Omega$ by ``$\Omega$'' and this can be confusing with our notation). Since $\Omega_q\to 1$ holds for all LTB models as $t\tot_{\textrm{\tiny{bb}}}$ (irrespective of whether the decaying mode is suppressed or not), then the early time EdS behavior $\hat\Omega\to \Omega_q\to 1$ as $t\to t_{\textrm{\tiny{bb(0)}}}$ occurs only if $\JJ_{\tiny{\textrm{(d)}}}=0$, as for these models we have $\delta^{(\rho)},\,\delta^{(\HH)}\to 0$ as $t\tot_{\textrm{\tiny{bb(0)}}}$ (see equation (\ref{DrhoDh0})). However, LTB models with a nonzero decaying mode ($\JJ_{\tiny{\textrm{(d)}}}\ne 0$) do not comply with the conditions of \cite{GooWain} and \cite{Limetal} for an isotropic Big Bang and EdS behavior, as we have for these models $\delta^{(\rho)}_q\to -1$ and $\delta^{(\HH)}_q\to-1/2$ (see (\ref{JgJd1}) and (\ref{deltasbb})), so that $\hat\Omega\to 0$ holds as $t\tot_{\textrm{\tiny{bb}}}$.
Lim {\it et al} proved that the isotropic Big Bang and associated EdS behavior are robust geometric features preserved by a change of frame (``time gauge'' in the sense of a 4--velocity boost \cite{coley}), hence the early time increasing behavior of $s_{\tiny{\textrm{gr}}}$ that we have proved for the CET entropy in models with $\JJ_{\tiny{\textrm{(d)}}}=0$ is also robust in this sense. However, the CET and HBp/q entropies in the late time evolution and/or models with $\JJ_{\tiny{\textrm{(d)}}}\ne 0$ are likely to be affected by such a change of frame, since the entropy growth conditions (\ref{CETcond}), (\ref{HBcond}) and (\ref{HBcondq}) also depend on fluctuations of the Hubble scalar ${\cal{H}}$ related to the eigenvalue of the shear tensor (see (\ref{sigEC})--(\ref{SigPsi2})), which is sensitive to the choice of 4--velocity. Looking at this issue in detail is beyond the scope of the present paper.
It is also important to emphasize that an isotropic singularity is not a necessary condition for a non--negative CET entropy production, as we have $\dots_{\tiny{\textrm{gr}}}\to\infty$ as dust layers approach the collapse singularity, which is not an isotropic singularity in the sense of \cite{GooWain,Limetal} (notice that $\hat\Omega\to 0$ as $t\tot_{\textrm{\tiny{coll}}}$ in elliptic models). However, entropy production from the the HBp/q entropies become negative as $t\tot_{\textrm{\tiny{coll}}}$.
\subsection{Gravitational entropy vs cosmological ``homogenization''.}
Our results (summarized before) are in excellent agreement with those of a recent numerical study by Bolejko and Stoeger \cite{bolstoeg}, who considered various entropy proposals (including the old ``arrow of time'' notion) in spherically symmetric models endowed with a rather general matter content: general perfect fluids and anisotropic fluids with non-zero viscosity. These authors showed that there is always a period in the evolution of their models in which entropy (in its various definitions) decreases, an effect they associate with a process in which the Universe "homogenizes" when the decaying mode is present and dominant. Evidently, this result is identical to our analytical findings on the early time behaviour of the CET and HBp/HBq entropies in models with a non-zero decaying mode (see section 7 and Figures 1a, 1c, 3a, 3c, 5b and 6). Furthermore, they show that, after this ``homogenization'' stage, their models evolve with increasing entropy towards an asymptotic terminal inhomogeneous state (they did not examine collapsing configurations). Again, this numerical result is identical to our analytic and qualitative result that the initially decreasing entropies begin to grow until reaching a (position or domain dependent) terminal profile in the asymptotic time range of hyperbolic models (see section 7, Appendix D.3 and Figures 1a, 1b, 3a, 3b). While Bolejko and Stoeger examined more general spherically symmetric models, we were only concerned with LTB dust models. Hence, they had to rely on a numerical treatment in which the identification of the (necessarily coupled) growing or decaying density modes of fully non--linear sources is practically impossible, whereas we were able to undertake a fully analytic treatment of the CET and the two HB entropies using the analytic (exact) non--linear forms for these coupled modes obtained for LTB models in \cite{sussmodes}.
As a further note: Bolejko and Stoeger identify the ``homogenization'' phase of their models with a sort of Einstein--de Sitter unstable saddle point: this result was obtained rigorously for LTB models in the dynamical systems study undertaken in \cite{sussmodes}.
\subsection{The gravitational entropy and the cosmological constant.}
While we have only considered gravitational entropy for LTB models with $\Lambda=0$, important qualitative information on the entropy growth, at least for the CET entropy, follows from previous work on the dynamics of LTB models with $\Lambda>0$ \cite{sussDS2}. Since the invariant scalar $\Psi_2$ (and thus $\rho_{\tiny{\textrm{gr}}}$) is the same when we consider a $\Lambda>0$ term, the condition for the CET entropy growth is also given by (\ref{CETcond}) in this case. Hence, figures 2, 3, 4, 5 and 8 of \cite{sussDS2} provide numerical examples in which $\delta^{(\rho)}\delta^{(\HH)}<0$, and thus $\dots_{\tiny{\textrm{gr}}}>0$, hold in the asymptotic time range of ever--expanding LTB models with $\Lambda>0$. However, a proper examination of the gravitational entropies for generic LTB models with $\Lambda> 0$ requires a fully separate study, which is relevant given the fact that these models are an inhomogeneous generalization of the $\Lambda$CDM model.
\subsection{The gravitational entropy and the need to suppress the decaying mode.}
The full time non--negative entropy production when the decaying mode is totally suppressed ($\JJ_{\tiny{\textrm{(d)}}}=0$) seem to lend further weight to the preference of using models with this feature in cosmological and astrophysical applications, in particular in the effort to fit cosmological observations without assuming the existence of dark energy \cite{Zibin:2011ma,Bull:2011wi} (though see \cite{CBK,clarkreg} for a critical approach to this line of thought). However, we believe that our results do not justify the total suppression of the decaying mode, which is an excessively extreme and not strictly necessary option, as the early times negative entropy production associated with a decaying mode can be regarded merely as a signal that LTB models are no longer viable for these radiation dominated cosmic times. In fact, a non--relativistic dust source with zero pressure gradients is not expected to be a physically plausible matter model near any singularity, not even an isotropic singularity (in the sense of \cite{GooWain,Limetal}) associated with a positive entropy production like the simultaneous Big Bang of models with $\JJ_{\tiny{\textrm{(d)}}}=0$. While CMB constraints and compatibility with the inflationary paradigm require nearly homogeneous conditions for sufficiently early times at the onset of (non-relativistic) matter dominated era, this requirement can be met by LTB models (such as our second numerical example in section 8) with a decaying mode that has become sufficiently subdominant (though not totally suppressed) at the required times, discarding the evolution of the model for previous times when radiation is dominant.
\subsection{Stability and extensivity.}
Considering the relation between the type of the extrema of the entropy ({\it i.e.} concavity of its time profile) and the stability of equilibrium states, we can associate the perpetual expansion of hyperbolic models with a time asymptotic entropy maximum for all proposals (see the convex time profile of the curves of Figures 1a, 1b, 3a and 3b for large times), which corresponds to a stable asymptotic terminal equilibrium state. On the other hand, the concave time profiles of the three entropies, either in regions where decaying modes are dominant (early times in Figures 1a, 1c, 2a and 2c), or in elliptic models (Figures 1c, 1d, 3c, 3d), suggest that decaying modes and collapsing configurations should be characterized by unstable equilibrium states of the gravitational entropy. This unstable equilibrium state is reminiscent of the unstable equilibria that characterize the Boltzmann--Gibbs entropy in non--collisional Newtonian systems subjected to relaxation processes and Antonov's instability \cite{Newtonian}. However, the gravitational entropy is a different concept from the Boltzmann--Gibbs entropy and these Newtonian systems evolve towards terminal stationary states, while a stationary state in elliptic LTB models only arises at the instant of maximal expansion. Nevertheless, despite these differences, this correspondence between the unstable equilibria of self gravitating systems in all these entropies is worth exploring, specially by looking in future research at the gravitational entropy proposals for general relativistic stationary (or asymptotically stationary) systems.
Another point worth commenting upon is the issue of the ``extensive'' or ``non--extensive'' nature of the CET and the two HB entropies. If we define an extensive quantity as scaling with volume, then the CET entropy is clearly non--extensive for all LTB models, while the HB entropies are only non--extensive for models converging sufficiently fast to a FLRW background. The HB entropies for slow convergence to FLRW or convergence to vacuum states are then extensive. On the other hand, if we regard large decaying modes as unphysical, then the extensive nature of the two HB entropies in LTB models converging to vacuum states seems to be consistent with the viability of a non--extensive entropy, as some of these models exhibit large decaying modes in the asymptotic radial range (though further study is needed to find out which one of the decaying or growing mode is dominant). These facts point out to a possible theoretical connection with studies of thermodynamical properties of Newtonian self--gravitating systems, which show that the long range nature of gravity blurs the simple distinction between intensive and extensive thermodynamical variables and may lead to non--extensive energy and entropy \cite{Newtonian}. A possible theoretical connection to Tsallis' non--extensive entropy \cite{Tsallis} may also be worth exploring in future research.
\subsection{Final comments.}
We would like to highlight the utility of describing LTB models with the q--scalar representation, as the original LTB variables would not have allowed us to link the CET entropy production to fluctuations of $\rho$ and ${\cal{H}}$, which resemble the fluctuations that result from the the two HB entropies, and thus provide the key theoretical unifying connection among all these entropies. This fact also provides a strong motivation to study possible extensions of the q-scalars to spacetimes more general than LTB (q--scalars have been used for the study of LTB models with $\Lambda>0$ in \cite{sussDS2}), and even non-spherical spacetimes, in order to examine general theoretical properties of the inhomogeneous gravitational field (including the study of the CET and HB entropies in less symmetrical contexts). We are currently elaborating a follow up paper to deal with the gravitational entropies in the case $\Lambda>0$ and for the non--spherical Szekeres dust models, as formal results valid for LTB models can be easily generalized for the latter models along the lines of \cite{sussbol}. Spacetimes with perfect fluid and dissipative sources, as well as the ``wave--like'' Petrov type N spacetimes examined by \cite{CET}, are also important candidates for further investigation.
\section*{Acknowledgments:} RAS acknowledges support from Mellon Foundation during a visit to Rhodes University, when this work was initiated. RAS also acknowledges financial support from grant CONACYT 132132. The authors thank Timothy Clifton, George Ellis, Thomas Buchert and Krzysztof Bolejko for useful comments and discussions.
\begin{appendix}
\section{LTB models in their standard variables.}
LTB dust models are usually given by the following traditional metric:
\begin{equation} {\rm{d}} s^2 =-{\rm{d}} t^2+\frac{R'{}^2}{1+2E}\,{\rm{d}} r^2 +R^2\left({\rm{d}}\vartheta^2+\sin^2\vartheta\,{\rm{d}}\varphi^2\right),\label{ltb1}\end{equation}
where $R=R(t,r),\,E=E( r),\,R'=\partial R/\partial r$ and $R$ satisfies the Friedman--like equation
\begin{equation}\dot R^2 = \frac{2M}{R}+2E,\label{Friedeq1}\end{equation}
with $M=M_q( r)$ (the quasi--local mass--energy functional in (\ref{Mq}) but treated as a function). The Friedman--like LTB metric (\ref{ltb2}) the Friedman--like equation (\ref{HHq}) follow from (\ref{ltb1}) and (\ref{Friedeq1}) by selecting the radial coordinate such that $R_0=R(t_0,r)=r$ for an arbitrary fiducial hypersurface $t=t_0$ and defining
\begin{equation} a\equiv \frac{R}{r},\qquad \Gamma =\frac{rR'}{R}=1+\frac{ra'}{a},\end{equation}
so that $a_0=\Gamma_0=1$. The relation between the free functions $M$ and $E$ in (\ref{ltb1})--(\ref{Friedeq1}) and the basic initial value q--scalars $\rho_{q0},\,{\cal{K}}_{q0},\,{\cal{H}}_{q0},\,\Omega_{q0}$ is given by
\footnote{Under this initial value parametrization the ``Big Bang time'' $t_{\textrm{\tiny{bb}}}$ follows as a function of any two basic initial q--scalars. See equation (\ref{tbb}). }
\begin{eqnarray} \fl \frac{8\pi}{3}\rho_{q0}=\frac{2M}{r^3}=\Omega_{q0}{\cal{H}}_{q0}^2,\qquad {\cal{K}}_{q0}=-\frac{2E}{r^2}=(\Omega_{q0}-1){\cal{H}}_{q0}^2,\label{ivf1}\\
\fl {\cal{H}}_{q0}=\left[(8\pi/3)\rho_{q0}-{\cal{K}}_{q0}\right]^{1/2}=\frac{[2M+2Er]^{1/2}}{r^{3/2}},\quad \Omega_{q0}=\frac{8\pi\rho_{q0}}{3{\cal{H}}_{q0}^2}=\frac{M}{M+Er},\label{ivf2}\end{eqnarray}
with their initial value perturbations $\delta^{(A)}_{q0}$ obtained from (\ref{Dadef}) with $\Gamma=1$. The main covariant scalars in (\ref{rhoHHKK}) and their corresponding q--scalars take the following form in terms of the standard variables:
\begin{eqnarray}\fl 4\pi\rho =\frac{M'}{R^2R'},\qquad {\cal{H}} = \frac{\partial_t(R^2 R')}{3R^2 R'},\qquad
{\cal{K}} = -\frac{4(ER)'}{R^2 R'},\label{basic}\\
\fl \frac{4\pi}{3}\rho_q =\frac{M}{R^3},\qquad {\cal{H}}_q = \frac{\dot R}{R},\qquad
{\cal{K}}_q = -\frac{2E}{R^2}.\label{qbasic}\end{eqnarray}
The eigenvalues of the shear, electric Weyl and Weyl tensors in the standard variables are
\begin{equation} \Sigma = -\frac{\partial_t(R'/R)}{3\,R'/R},\qquad \Psi_2=\frac{M}{R^3}-\frac{4\pi}{3}\rho.\label{SigPsi22}\end{equation}
It is straightforward to show that these forms are identical to those in (\ref{SigPsi2}).
\section{Analytic solutions.}
The analytic solutions of the Friedman equation (\ref{HHq}) (equivalent to (\ref{Friedeq1}))
\begin{equation} {\cal{H}}_q =\frac{\dot a}{a}={\cal{H}}_{q0}\left[\frac{\Omega_{q0}}{a^3}-\frac{\Omega_{q0}-1}{a^2}\right]^{1/2},\end{equation}
can be given in terms of $\Omega_q$ for initial conditions ${\cal{H}}_{q0},\,\Omega_{q0}$ as
\footnote{The ``parabolic'' case follows from the elliptic and hyperbolic cases as the limit $\Omega_{q0}\to 1$ (or $E=0$ in (\ref{Friedeq1})). For these models ${\cal{H}}_q(t-t_{\textrm{\tiny{bb}}})=2/3$ holds exactly, hence $\JJ_{\tiny{\textrm{(g)}}}=0$ even if $\Delta_0^{\tiny{\textrm{(g)}}}\ne 0$ (see \cite{sussmodes}).}
\begin{eqnarray}
\fl {\underline{\hbox{hyperbolic models:}}}\,\,0<\Omega_q<1\,\,({\cal{K}}_q<0\,\,\,\hbox{or}\,\, E>0),\qquad , \nonumber\\
t-t_{\textrm{\tiny{bb}}} = \frac{Y_q(\Omega_q)}{{\cal{H}}_q},\label{hypsol}\\
\fl {\underline{\hbox{elliptic models:}}}\,\,\Omega_q>1\,\,({\cal{K}}_q>0\,\,\,\hbox{or}\,\, -1<E<0),\qquad ,\nonumber\\
\fl t-t_{\textrm{\tiny{bb}}}= \left\{ \begin{array}{l}
Y_q(\Omega_q)/{\cal{H}}_{q},\qquad\qquad\qquad
{\hbox{expanding phase}}\quad {\cal{H}}_q>0,\\
2\pi\beta_q - Y_q(\Omega_q)/{\cal{H}}_q, \qquad
{\hbox{collapsing phase}}\quad {\cal{H}}_q<0,\\
\end{array} \right.\label{ellsol}\end{eqnarray}
with the functions $\beta_{q}$ and $Y_q$ given by
\begin{eqnarray}
\beta_q = \frac{4\pi\rho_q}{3|{\cal{K}}_q|^{3/2}}=\frac{\Omega_q}{2|1-\Omega_q|^{3/2}{\cal{H}}_q}=\beta_{q0}\qquad\left(\Rightarrow \dot\beta_q=0\right),\label{beta}\\
Y_q(\Omega_q) = \frac{\epsilon}{|1-\Omega_q|}\left[1-\frac{\Omega_q}{2|1-\Omega_q|^{1/2}}{\cal{A}}\left(\frac{2}{\Omega_q}-1\right)\right], \label{Y}
\end{eqnarray}
where $\epsilon =1,\, {\cal{A}}=$ arccosh correspond to the hyperbolic case and $\epsilon =-1,\, {\cal{A}}=$ arccos to the elliptic case. Notice that these solutions can be given in terms of the scale factor $a$ by substituting the form of $\Omega_q$ in (\ref{OmDOm}) into the right hand sides of (\ref{hypsol}) and (\ref{ellsol}) (see Appendix A2 of \cite{sussBR}).
The Big Bang time $t_{\textrm{\tiny{bb}}}=t_{\textrm{\tiny{bb}}}(r )$ and its gradient are expressible in terms of primary initial value functions and perturbations:
\begin{equation} t_{\textrm{\tiny{bb}}}=t_0-\frac{Y_q(\Omega_{q0})}{{\cal{H}}_{q0}},\qquad rt_{\textrm{\tiny{bb}}}' = (1+\delta^{(\rho)}_{q0})\Delta_0^{\tiny{\textrm{(d)}}},\label{tbb}\end{equation}
where we substituted $t=t_0$ in (\ref{hypsol}) and (\ref{ellsol}), while the form of $t_{\textrm{\tiny{bb}}}'$ follows from (\ref{Did}). Besides $t_{\textrm{\tiny{bb}}}$, elliptic models have the following characteristic times:
\begin{eqnarray}
\fl \hbox{elliptic expanding:}\quad 0< t-t_{\textrm{\tiny{bb}}} \leq t_{\textrm{\tiny{max}}}-t_{\textrm{\tiny{bb}}}=\pi\beta_{q0},\label{tmax}\\
\fl \hbox{elliptic collapsing:}\quad t_{\textrm{\tiny{max}}}-t_{\textrm{\tiny{bb}}} < \tau_q<t_{\textrm{\tiny{coll}}}-t_{\textrm{\tiny{bb}}}=2\pi\beta_{q0},\label{tcoll}
\end{eqnarray}
where $t=t_{\textrm{\tiny{max}}}$ and $t=t_{\textrm{\tiny{coll}}}$ mark the times of maximal expansion (${\cal{H}}_q=0$) and the collapse singularity (``Big Crunch'' ${\cal{H}}_q\to-\infty$).
The perturbations and fluctuations become fully determined once we compute $\JJ_{\tiny{\textrm{(g)}}}$ and $\JJ_{\tiny{\textrm{(d)}}}$. For this purpose, we need the following expressions:
\begin{eqnarray}
\fl {\cal{H}}_q(t-t_{\textrm{\tiny{bb}}})=Y_q,\qquad \hbox{(hyperbolic \& elliptic expanding)},
\label{HTq}\\
\fl {\cal{H}}_q(t-t_{\textrm{\tiny{bb}}})=Y_q-\frac{\pi\Omega_q}{(\Omega_q-1)^{3/2}},\quad\hbox{(elliptic collapsing)}, \label{HTqc}\end{eqnarray}
\begin{equation}\fl {\cal{H}}_q =\pm{\cal{H}}_{q0}\frac{\Omega_q}{\Omega_{q0}}\,\left[\frac{1-\Omega_{q0}}{1-\Omega_q}\right]^{3/2},\label{H32}\end{equation}
where $Y_q=Y_q(\Omega_q)$ is given by (\ref{Y}), $\epsilon = 1,-1$ correspond to hyperbolic and elliptic cases and we can use the scaling law (\ref{OmDOm}) to express ${\cal{H}}_q(t-t_{\textrm{\tiny{bb}}})$ above in terms of the scale factor $a$ and initial value functions.
\section{Evolution equations.}
While the dynamics of LTB models is fully determined analytically, they can also be studied numerically by solving the evolution equations for the q--scalars and their perturbations given by (19a)--(19d) or (21a)--(21d) of \cite{part2}. However, these evolution equations can be problematic for collapsing configurations, since $\delta^{(\HH)}_q,\,\Omega_q$ and $\delta^{(\Omega)}$ diverge as ${\cal{H}}_q\to 0$ (when $t\tot_{\textrm{\tiny{max}}}$), and thus the expanding (${\cal{H}}_q>0$) and collapsing (${\cal{H}}_q<0$) stages of elliptic models must be treated separately. For a unified numerical treatment of collapsing configurations (as the third example of section 8) we used instead the following evolution equations for the q--scalars and their fluctuations ${\textrm{\bf{D}}}_q(\rho),\,{\textrm{\bf{D}}}_q({\cal{H}})$ which are bounded at $t=t_{\textrm{\tiny{max}}}$:
\begin{eqnarray} \dot \rho_q &=& -3 \rho_q{\cal{H}}_q,\label{EVq21}\\
\dot {\cal{H}}_q &=& -{\cal{H}}_q^2-\frac{4\pi}{3}\rho_q, \label{EVq22}\\
\left[{\textrm{\bf{D}}}_q(\rho)\right]\dot{} &=& -3[\rho+{\textrm{\bf{D}}}_q(\rho)]\,{\textrm{\bf{D}}}_q({\cal{H}})-3{\cal{H}}_q{\textrm{\bf{D}}}_q(\rho),\label{EVq23}\\
\left[{\textrm{\bf{D}}}_q({\cal{H}})\right]\dot{} &=& -\frac{4\pi}{3}{\textrm{\bf{D}}}_q(\rho) - [2{\cal{H}}_q+3{\textrm{\bf{D}}}_q({\cal{H}}_q)]{\textrm{\bf{D}}}({\cal{H}}_q).\label{EVq24}\end{eqnarray}
\section{Formal results on the HB entropies.}
\subsection{Non--negativity of the HB functionals.}
The HB entropy functional (\ref{sHB}) is strictly non--negative for $\rho\geq 0$ and any associated scalar average of the form (\ref{rhoav}) (which includes the quasi--local average (\ref{sHBq})). The proof (private communication from T. Buchert) follows by remarking that the inequality
\begin{equation}\ln x \leq x-1 \qquad x = \frac{\langle \rho\rangle_{\cal {D}}}{\rho}\end{equation}
holds for all non--negative $x$. Multiplying both sides by $-\rho\leq 0$ and using the property $-\ln x=\ln x^{-1}$ leads to the desired result:
\begin{eqnarray} \fl \rho\,\ln \left[\frac{\rho}{\langle \rho\rangle_{\cal {D}}}\right]\geq \rho-\langle \rho\rangle_{\cal {D}}\quad \Rightarrow\quad s_{\textrm{\tiny{HB}}} =\left\langle \rho\,\ln \left[\frac{\rho}{\langle \rho\rangle_{\cal {D}}}\right]\right\rangle_{\cal {D}} {\cal{V}}_{\cal {D}}\geq 0,\end{eqnarray}
since $\langle\, \rho-\langle \rho\rangle_{\cal {D}}\,\rangle_{\cal {D}} =0$ holds for any scalar average (\ref{rhoav}).
\subsection{Proof of the limits (\ref{shblim}) and (\ref{dotshblim}).}
We elaborate the convergence proofs only for $s_{\textrm{\tiny{HB}}}{}_q$, as the proofs for $s_{\textrm{\tiny{HB}}}{}_p$ are analogous. We look first at the case of slices intersecting the Big Bang, considering domains shown in Figure 2a in a slice $t=t_s\approx t_{(-)}$, such that $r_b>r_s$ (with $r_b\approx r_s$), where $t_s=t_{\textrm{\tiny{bb}}}(r_s)$, hence $t_s-t_{\textrm{\tiny{bb}}}( r)\approx -t_{\textrm{\tiny{bb}}}'{}_s(r-r_s)$ holds with $t_{\textrm{\tiny{bb}}}'{}_s=t_{\textrm{\tiny{bb}}}'(r_s)<0$ (from demanding absence of shell crossings). Since $\rho\to\infty$ as $r\to r_s$, the convergence test of (\ref{sHBq}) follows by rewriting this integral as the limit
\begin{eqnarray} \fl s_{\textrm{\tiny{HB}}}{}_q[r_b] -s_{\textrm{\tiny{HB}}}{}_q^{(\textrm{\tiny{eq}})}=\gamma_0 \lim_{\epsilon\to 0}\int_{r_s+\epsilon}^{r_b}{M'_q(\bar r) \ln\left[\frac{\rho}{\langle \rho\rangle_q[r_b]}\right]{\rm{d}}\bar r},\qquad M'_q=4\pi\rho_{q0}(1+\delta^{(\rho)}_{q0})r^2,\nonumber\\
\label{sHBq2}\end{eqnarray}
where $\epsilon>0$. Since $a\ll 1$ holds for the domains we are interested in, we use (\ref{deltasbb}) and expand the solutions (\ref{hypsol}) and (\ref{ellsol}) in this limit \cite{sussBR} to obtain at first order in $r-r_s$
\begin{eqnarray}\fl a^{3/2}(t_s,r)\approx \left[\frac{3}{2}\sqrt{\Omega_{q0}}{\cal{H}}_{q0}|t_{\textrm{\tiny{bb}}}'|\right]_s(r-r_s),\qquad 1+\delta^{(\rho)}_{q}(t_s,r) \approx \frac{3(1+\delta^{(\rho)}_{q0s})}{2r_s}(r-r_s),\nonumber\\ \label{approx1}\end{eqnarray}
where the subscript ${}_s$ will denote henceforth evaluation at $r=r_s$. While $\rho$ along $t=t_s$ can be directly computed by inserting (\ref{approx1}) in (\ref{rhoHHKK}) and (\ref{rhoKK}), the density average in this slice is no longer given by (\ref{rhoaveq2}), but by $\langle \rho\rangle_q[r_b]=(M_{qb}-M_{qs})/{\cal{V}}_q[r_b]$, with $M_{qb}=M_q(r_b)$ and $M_{qs}=M_q(r_s)$. Applying (\ref{approx1}) to the appropriate forms for $\rho$ and $\langle \rho\rangle_a[r_b]$ we obtain
\begin{eqnarray}\fl \rho\approx \frac{M'_{qs}}{6\pi\,\Omega_{q0s}{\cal{H}}_{q0s}^2|t_{\textrm{\tiny{bb}}}'{}_s|^2\,r_s(r-r_s)},\quad \langle \rho\rangle_q[r_b]\approx \frac{M'_{qs}}{3\pi\, \Omega_{q0s}{\cal{H}}_{q0s}^2 |t_{\textrm{\tiny{bb}}}'{}_s|^2\,r_s(r_b-r_s)},\nonumber\\\label{approx2}\end{eqnarray}
where we used $M_{qb}-M_{qs}\approx M'_{qs}(r_b-r_s)$. By inserting these forms into (\ref{sHBq2}) and taking the limit as $\epsilon\to 0$ we obtain:
\begin{eqnarray} s_{\textrm{\tiny{HB}}}{}_q[r_b] -s_{\textrm{\tiny{HB}}}{}_q^{(\textrm{\tiny{eq}})}= (1-\ln 2)\,M'_{qs}(r_b-r_s)>0,\label{approx3}
\end{eqnarray}
which implies the convergence of the integral (\ref{sHBq2}) for the domains under consideration. The second limit in (\ref{shblim}) follows readily as $r_s\to r_b$. For the HBp entropy (first limit in (\ref{shblim})) we obtain the same result as (\ref{approx3}), but with $M'_{qs}$ replaced by $M'_{ps}$. The collapse case yields the same result as (\ref{approx3}) for the HBp and HBq entropies, since for slices intersecting $t=t_{\textrm{\tiny{coll}}}$ we have $a^{3/2},\,\rho$ and $\langle \rho\rangle_q[r_b]$ taking the same forms (\ref{approx1}), but proportional to $t_{\textrm{\tiny{coll}}}-t_s\approx t_{\textrm{\tiny{coll}}}'(r_s)(r-r_s)$, hence the ratio $\rho/\langle \rho\rangle_q$ has the same form as that obtained with (\ref{approx2}).
In order to verify the convergence of $\dots_{\textrm{\tiny{HB}}}{}_q[r_b]$ in a slice $t=t_s$ intersecting the Big Bang, we re--write (\ref{Sdot1q}) as
\begin{equation} \dots_{\textrm{\tiny{HB}}}{}_q[r_b]=\gamma_0\left[\lim_{\epsilon \to 0}\int_{r_s+\epsilon}^{r_b}{M'_q\,{\cal{H}}\,{\rm{d}}\bar r}-\langle \HH\rangle_q[r_b]\,(M_{qb}-M_{qs})\right],\label{dotSHB2}\end{equation}
where we used the identities $\rho {\cal{V}}'_q =M'_q$ and $\langle \rho\rangle_q[r_b]\,{\cal{V}}_q[r_b]=M_{qb}-M_{qs}$ and we are taking into consideration that ${\cal{H}}\to\infty$ as $r\to r_s$. Considering that $M'_q\approx M'_{qs}+M''_{qs}(r-r_s)$ and expanding ${\cal{H}}$ and $\langle \HH\rangle_q[r_b]=\dot a(t_s,r_b)/a(t_s,r_b)$ along $t=t_s$ at first order in $r-r_s$ we obtain
\begin{equation} {\cal{H}}\approx \frac{1}{3\,|t_{\textrm{\tiny{bb}}}'{}_s|(r-r_s)},\qquad \langle \HH\rangle_q[r_b]\approx\frac{2}{3\,|t_{\textrm{\tiny{bb}}}'{}_s|(r_b-r_s)}, \label{approx4}\end{equation}
all of which inserted into (\ref{dotSHB2}) yields:
\begin{equation} \dots_{\textrm{\tiny{HB}}}{}_q[r_b]\approx \frac{\gamma_0 M'_{qs}}{3\,|t_{\textrm{\tiny{bb}}}'{}_s|}\left[\,\lim_{\epsilon \to 0}\int_{r_s+\epsilon}^{r_b}{\frac{{\rm{d}} \bar r}{\bar r-r_s}}-2\right]\to \infty,\label{approx5}\end{equation}
where we used $M_{qb}-M_{qs}\approx M'_{qs}(r_b-r_s)$. We obtain the same result for $\dots_{\textrm{\tiny{HB}}}{}_p[r_b]$, with $M'_{qs}$ replaced by $M'_{ps}$. Since we keep $r_b>r_s$ fixed, the lack of convergence of these integrals implies the limits (\ref{dotshblim}).
Near the collapsing singularity we obtain similar forms as in (\ref{approx4}) with $|t_{\textrm{\tiny{bb}}}'{}_s|$ replaced by $|t_{\textrm{\tiny{coll}}}'{}_s|$, but now we have ${\cal{H}}<0$ and $\langle \HH\rangle_a[r_b]<0$. Hence the right hand side of (\ref{approx5}) has the opposite sign, leading to $\dots_{\textrm{\tiny{HB}}}{}_q[r_b]\to -\infty$ and $\dots_{\textrm{\tiny{HB}}}{}_p[r_b]\to -\infty$ as $r\to r_s$, and thus the limits (\ref{dotshblimc}) follow.
\subsection{The HB entropies in the asymptotic time range of hyperbolic models.}
To examine the asymptotic time range $t\gg t_0$ of hyperbolic models we expand (\ref{hypsol}) for $a\gg 1$ (or $\Omega_q\ll 1$) and use (\ref{JgJd2}) to obtain
\begin{equation} a \approx |{\cal{K}}_{q0}|^{1/2}\Delta,\qquad 1+\delta^{(\rho)}\approx \frac{1}{1-\Delta_0^{\tiny{\textrm{(g)}}}}=\frac{1+\delta^{(\rho)}_{q0}}{1+\frac{3}{2}\delta^{(\KK)}_{q0}},\end{equation}
where $\Delta=t-t_0$. This yields
\begin{equation}\fl \rho\approx \frac{\rho_{q0}(1+\delta^{(\rho)}_{q0})}{\Delta^3\,|{\cal{K}}_{q0}|^{3/2}\left(1+\frac{3}{2}\delta^{(\KK)}_{q0}\right) },\quad \langle \rho\rangle_q[r_b]\approx \frac{\int_0^{r_b}{\rho_{q0}(1+\delta^{(\rho)}_{q0}) \bar r^2{\rm{d}}\bar r}}{\Delta^3\,\int_0^{r_b}{|{\cal{K}}_{q0}|^{3/2}\left(1+\frac{3}{2}\delta^{(\KK)}_{q0}\right)\bar r^2{\rm{d}}\bar r} }, \end{equation}
which inserted into (\ref{sHBq2}) (with $r_s=0$) leads to the following asymptotic terminal expression as $t\to\infty$:
\begin{eqnarray} \fl s_{\textrm{\tiny{HB}}}{}_q(r_b) -s_{\textrm{\tiny{HB}}}{}_q^{(\textrm{\tiny{eq}})}\approx\nonumber\\
\fl \int_0^{r_b}{\rho_{q0}(1+\delta^{(\rho)}_{q0})\,\bar r^2\,\ln\left[\frac{\rho_{q0}(1+\delta^{(\rho)}_{q0})}{\langle\rho_{q0}(1+\delta^{(\rho)}_{q0})\rangle_q[r_b]}\,\frac{\langle |{\cal{K}}_{q0}|^{3/2}(1+\frac{3}{2}\delta^{(\KK)}_{q0})\rangle_q[r_b]}{|{\cal{K}}_{q0}|^{3/2}(1+\frac{3}{2}\delta^{(\KK)}_{q0})}\right]{\rm{d}}\bar r},
\end{eqnarray}
whose right hand side is obviously not zero, hence it must be positive (see proof of non--negativity in Appendix D.1). Therefore, the terminal value of $s_{\textrm{\tiny{HB}}}{}_q[r_b]$ for all domains with $r_b>0$ is necessarily larger that the equilibrium initial value (either at $t=t_{(-)}$ when the decaying mode is non-zero or at $t=t_{\textrm{\tiny{bb(0)}}}$ when this mode is suppressed). The same results hold for $s_{\textrm{\tiny{HB}}}{}_p[r_b]$.
\subsection{Models with a suppressed decaying mode.}
For these models we evaluate the integral in (\ref{sHBq}) for slices $t\approx t_{\textrm{\tiny{bb(0)}}}$. Expanding (\ref{hypsol}) and (\ref{ellsol}) with $t_{\textrm{\tiny{bb}}}=t_{\textrm{\tiny{bb(0)}}}$ for $a\ll 1$ and using (\ref{DrhoDh0}) we obtain
\begin{equation}\fl a^{3/2}\approx \frac{3}{2}\sqrt{\Omega_{q0}}{\cal{H}}_{q0}\Delta,\qquad 1+\delta^{(\rho)}_{q} \approx 1-\frac{2(\Omega_{q0}-1)\Delta_0^{\tiny{\textrm{(g)}}}}{5\Omega_{q0}}\,a, \label{approx6}\end{equation}
where now $\Delta\equiv t-t_{\textrm{\tiny{bb(0)}}}$. The density and its q--average take the form
\begin{eqnarray} \fl \rho \approx \frac{4}{9\Delta^2}\left[1-\frac{18^{1/3}\Delta_0^{\tiny{\textrm{(g)}}} (\Omega_{q0}-1)({\cal{H}}_{q0})^{2/3}}{5(\Omega_{q0})^{2/3}\Delta^{2/3}}\right],\quad
\langle \rho\rangle_q[r_b]\approx \frac{4}{9\Delta^2}\left[1-\frac{M'_{qb}}{5M_{qb}\Delta^{2/3}}\right],\nonumber\\\end{eqnarray}
so that $\rho( r)/\rho_q[r_b]\approx 1+F(r,r_b)\Delta^{2/3}$, where $F(r,r_b)$ follows from the initial value functions $\Omega_{q0},\,{\cal{H}}_{q0},\,\Delta_0^{\tiny{\textrm{(g)}}}$ and the domain dependent constants $M_{qb},\,M'_{qb}=M'_q(r_b)$. Since $\ln[\rho( r)/\langle \rho\rangle_q[r_b]]\approx F(r,r_b)\Delta^{2/3}$, the HBq entropy becomes
\begin{equation}s_{\textrm{\tiny{HB}}}{}_q(r_b) -s_{\textrm{\tiny{HB}}}{}_q^{(\textrm{\tiny{eq}})}\approx \frac{\Delta}{4\pi}\int_0^{r_b}{M'_q(\bar r) F(\bar r,r_b){\rm{d}} \bar r},\end{equation}
and thus the right hand side vanishes for all domains in the limit $\Delta\to 0$. Regarding the behaviour of $\dots_{\textrm{\tiny{HB}}}{}_q[r_b]$, we use the form (\ref{dotSHB2}) but for the whole integration range $0\leq r \leq r_b$, hence $M_{qs}=0$. Since ${\cal{H}}$ and $\langle \HH\rangle_q[r_s]$ are both proportional to $1/\Delta$ for slices $t\approx t_{\textrm{\tiny{bb(0)}}}$, then $\dots_{\textrm{\tiny{HB}}}{}_q[r_b]\to\infty$ as $t-t_{\textrm{\tiny{bb(0)}}}$. The same result follows readily for $\dots_{\textrm{\tiny{HB}}}{}_p[r_b]$.
\end{appendix}
\section*{References}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,595 |
Lauren Ogilvie Bosworth (często po prostu Lo) (ur. 29 września 1986) – amerykańska osobowość telewizyjna, znana polskim widzom z serii programów Laguna Beach, który tworzyła razem ze swoją przyjaciółką Lauren Conrad.
Życiorys
Bosworth urodziła się w Laguna Beach w Kalifornii. Jest dzieckiem Richarda i Ellen Bosworth. Ma starszą siostrę Erin (1983) i młodszego brada Christiana (1990). Bosworth jest absolwentką Uniwersytetu Kalifornijskiego w Los Angeles.
Kariera
Lauren Bosworth pojawiła się u boku najlepszych przyjaciół: Lauren Conrad, Stephena Coletti i Kristin Cavallari w programie MTV Laguna Beach. Występuje również gościnnie w reality show Wzgórza Hollywood (The Hills).
Linki zewnętrzne
Urodzeni w 1986
Amerykańskie aktorki filmowe
Uczestnicy programów typu reality show | {
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If you are looking for a business ready compact solution PC, the HP ProDesk 400 & 600 Desktop Mini PC's is your ideal selection!
Powerful, reliable, secure and manageable!
With technologies like HP Sure Start Gen4 you are protected against BIOS attacks and with HP BIOSphere Gen4 you can help prevent data breaches. In additional you can easily manage devices through Microsoft System Center Configuration Manager with HP Manageability Integration Kit.
Ideal for call centers, small office spaces, coworking establishments and educational environments.
The perfect solution for the modern working environment! | {
"redpajama_set_name": "RedPajamaC4"
} | 8,532 |
about Oliver Brown
Oliver or Ollie Brown may refer to: Oliver Brown (plaintiff) (1918–1961), plaintiff in the case Brown v. Board of Education Oliver Brown (baseball) (1849–1932), baseball player Oliver Brown (footballer) (1908–1953), English footballer, also known as Buster Brown, active in the 1930s Oliver Brown (snooker player), English snooker player Oliver C. Brown (born 1946), American percussionist Ollie Brown (baseball) (1944–2015), Major League Baseball player Ollie E. Brown (born 1953), American drummer and record producer, half of duo Ollie & Jerry Oli Brown (born 1990), British blues musician Oliver Brown, son of abolitionist John Brown William Oliver Brown (1903–1976), Scottish nationalist activist
Name : Oliver Brown
Place of Birth : 0 E 10, 51 N 29
Current Vimshottari Dasa : Mon-Sun
Sign Libra Sagittarius Leo Libra Scorpio Capricorn Sagittarius Scorpio Aries Libra
Lord Ven Jup Sun Ven Mar Sat Jup Mar Mar Ven
Oliver Brown is born in Libra lagn. Libra is the 7th sign of the zodiac and is ruled by venus. Oliver Brown will have an impressive personality. Oliver Brown will be helpful to others. Oliver Brown is fond of travelling to religious places. Oliver Brown have a very prominent nose. Oliver Brown is a competent business person. Oliver Brown is fond of astrology. His voice is very sweet and impressive. Oliver Brown is not greedy. Oliver Brown will travel to many places. Oliver Brown may live far from His family. Oliver Brown would have had some unhappiness in His childhood but will be happy later in life. Oliver Brown will have a normal life. Oliver Brown will rise in life after 31 32 years of age. Oliver Brown make many friends. Oliver Brown have a dignified personality. Oliver Brown is succeptible to cough. Oliver Brown is wise and is always smiling. Oliver Brown have very attractive and impressive eyes. Oliver Brown is very able and dutiful. Oliver Brown can read other peoples's thoughts. Oliver Brown is kind hearted. Oliver Brown is famous for His judgements. Oliver Brown is a competent administrator. Oliver Brown is peace loving. Oliver Brown is fond of art and music. Oliver Brown will succeed in big ventures. Oliver Brown is fond of physical pleasures,luxury and perfumes. Though Oliver Brown come from an ordinary family Oliver Brown will rise high. Oliver Brown will acquire a lot of success in government and social affairs. Oliver Brown will head projects. Oliver Brown have a broad face. Oliver Brown is attractive,good looking and have an impressive personality. Oliver Brown is very ambitious. Oliver Brown is skilled in many arts. Oliver Brown will travel to many countries. Oliver Brown is truth loving. Oliver Brown take interest in social work. As Oliver Brown were born in this nakshatra Oliver Brown will be rich, intelligent, helpful and kind. Oliver Brown will be successful in business and will lead a life of comforts. Oliver Brown will have numerous servants and will be very religious minded. Oliver Brown will have a harmonious relationship with His spouse and will be devoted to His parents but Oliver Brown will be over strict and will have a violent nature. Oliver Brown will be clear minded and will be clear in His actions. His liver will be weak. Oliver Brown must perform pooja as Oliver Brown is born in the gund mool nakshatra. Oliver Brown will gain after 25 years of age.
Sun Prediction of Oliver Brown
Moon Prediction of Oliver Brown
Mars Prediction of Oliver Brown
Mercury Prediction of Oliver Brown
Jupiter Prediction of Oliver Brown
Venus Prediction of Oliver Brown
Saturn Prediction of Oliver Brown
Rahu Prediction of Oliver Brown
Ketu Prediction of Oliver Brown
Vimshottari Dasha Analysis of Oliver Brown
Numerology of Oliver Brown
Sadesati of Oliver Brown
Yearly Prediction (Varshphal) of Oliver Brown
Gem Remedies of Oliver Brown
Lal Kitab Remedies of Oliver Brown
Sadesati Remedies of Oliver Brown | {
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I am also protein c deficienc and it is almost unable to be detected.
Anyone else have a qualitative platelet disorder and factor 1 hemophilia?
My son was diagnosed at age 6 with a qualitative platelet disorder. His platelets are missing the mechanism that causes them to stack. I recently decided his medical records and noticed that when they were doing his blood work for the diagnosis his fibrinogen level was too low also but they never told me that. Could this mean that he is also a factor 1 hemophiliac... or am I over reacting?
Has anyone been told they have microbleeding in the joints?
I'm trying to find out if anyone with vW type 2b has constantly low platelet counts that doctors cannot explain.
got a packet from the non-profit chapter about this company and their product. was just wanting to see the reviews from people that i have connected to or a support group that i am in.
its derived from human based cells, from what i am gathering. just not sure on it as last i remember when they had tried something along that line the AIDS and Hep C outbreak occurred.
I am interested in finding out if anyone with vW type 2b ever developed low platelet counts over time?
Does anyone else have side effects from Amicar?
I have been on Amicar most of my life and dealt with the side effects, but they seem to be affecting me more lately. I've had nausea and vomiting, and i get really dizzy and ligt headed to the point of fainting. The worst though is along with that, I get super overheated. Does anyone else have these side effects? It's gotten so bad I have even considered trying to switch to Lysteda in hopes it won't have the same effects. Anyone have experiences with switching?
Call the manufacturer they can help. | {
"redpajama_set_name": "RedPajamaC4"
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Q: Listen to port from docker I have a php app running inside a docker container. The app uses SMTP for mailing. Pre-docker we used a tool called FakeSMTP that allowed us to intercept outgoing e-mails.
We entered the port 2525 into the FakeSMTP tool, and it would listen on this port for e-mails. Whenever our app would send mails on port 2525 we could intercept them. I want to do the same, but with docker.
I tried mapping the port (-p 2525:2525) from docker to host, but that means that the port can not be bound to by any other tools.
Is expose the way to go?
Or is this even possible?
A: Why not map the port to something else on the container, then have FakeSMTP listen to that other port?
-p 41950:2525
Then have FakeSMTP listen to port 41950 and send the intercepted mail out on port 2525.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,128 |
Dear Readers, The selection for Probationary Officers in State Bank of India(SBI) 2018 will be done through a three tiered process.
Preliminary Examination consisting of Objective Test for 100 marks will be conducted online. The test will have of 3 Sections (with separate timings for each section). | {
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var fs = require('fs');
var os = require('os');
var child_process = require('child_process');
var path = require('path');
var jscs = require('./jscs_handler');
var _ = require('lodash');
function ensurePathExists(req, res, next) {
var path = req.param('path');
if (!fs.existsSync(path)) {
res.json(400, { error: 'No such path: ' + path, errorCode: 'no-such-path' });
} else {
next();
}
}
var reviewCode = function(req, res) {
var repoPath = req.param('path');
var files = _.clone(req.body.changedFiles);
for(var i in files) {
files[i] = repoPath + '/' + files[i];
}
console.log('files', files);
console.log(jscs(files));
jscs(req.body.changedFiles).then( function (errorList) {
res.json(errorList);
});
};
exports.install = function(env) {
var app = env.app;
var ensureAuthenticated = env.ensureAuthenticated;
var git = env.git;
app.get(env.httpPath + '/review-code', ensureAuthenticated, ensurePathExists, function(req, res) {
res.json({ exists: false });
});
app.post(env.httpPath + '/review-code', ensureAuthenticated, ensurePathExists, reviewCode);
app.get(env.httpPath + '/changes', ensureAuthenticated, ensurePathExists, function(req, res) {
var repoPath = req.param('path');
git.getRemoteAddress(repoPath, 'origin')
.fail(function(err) {
res.json(400, err);
})
.done(function(remote) {
console.log('remote', remote);
res.json(200, 'dupcia!')
});
});
}
console.log(reviewCode({
param: function() {
return "."
},
body: {
changedFiles: [
"backend.js",
"jscs_handler.js"
]
}
}));
| {
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'Noble deeds and hot baths are the best cures for depression,' wrote author Dodie Smith in I Capture The Castle. 'I am sure there are things that can't be cured by a good bath but I can't think of one,' Sylvia Plath agreed in The Bell Jar. A long, hot soak has long been seen as the remedy for many of life's challenges – and now a new study has proved that the theory may actually have a grounding in science.
During the research, which was reported by New Scientist earlier this week, scientists at the University of Freiburg, Germany, asked half of a group of 45 people with depression to take a 30 minute warm bath every day for eight weeks. This was followed by 20 minutes of wrapping up in blankets and relaxing with a hot water bottle.
They then asked the other half of the test group to take 40 to 45 minutes of aerobic exercise twice a week. After eight weeks, those in the bathing group scored six points lower on a commonly used depression scale, while those in the exercise group scored an average of three points lower.
Although the trial was small, and more extensive research would be required to prove the connection conclusively, the results do seem to suggest that taking a daily bath may help to relieve symptoms of depression in a more effective way than regular exercise.
As for why the results were slanted this way? It's thought to be associated with circadian rhythms, the cycles which control the sleep patterns and behaviour of our bodies, and are connected to temperature. People with depression tend to have a 'flatter' circadian rhythm, but taking a bath would increase their core temperature, thereby 'strengthening' their rhythm and likely improving the quality of their sleep.
Previous research has indicated that increases in body temperature can stimulate the release of 'happy' hormone serotonin, so the process of bathing also could lead to an overall improvement in mood.
The German study is a promising start, and although there are no guarantees that you'll feel the benefit, a soothing soak is usually a pleasant experience regardless – so it certainly can't hurt to give it a go. | {
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{"url":"http:\/\/digitalhaunt.net\/West-Virginia\/calculus-error-function.html","text":"WVVA.net Inc opened for business in 1997 as C & H Computers, a computer\/networking retail and service company. In 2001, C & H expanded to offer dialup internet access in the Giles and Monroe County areas. In 2004, C & H Computers became WVVA.net Inc and began offering high speed wireless internet as well as dialup and computer services. WVVA.net Inc now has offices in Rich Creek, Virginia and Bluefield, Virginia. The offices are located at: 405 Virginia Avenue 601 Virginia Avenue, VA 24147 CreekBluefield24605 540-726-2317 (276) -322-2696 toll free 1-866-WVVA-NET fax 540-726-9522\n\nAddress 601 Virginia Ave, Bluefield, VA 24605 (540) 726-2317 http:\/\/www.wvva.net\n\n# calculus error function Elkhorn, West Virginia\n\nSimilarly, (8) (OEIS A103979 and A103980). Loading... Algebra [Notes] [Practice Problems] [Assignment Problems] Calculus I [Notes] [Practice Problems] [Assignment Problems] Calculus II [Notes] [Practice Problems] [Assignment Problems] Calculus III [Notes] [Practice Problems] [Assignment Problems] Differential Equations [Notes] Extras FAQ - A few frequently asked questions.\n\nNew York: Dover, pp.179-182, 1967. and Stegun, I.A. (Eds.). \"Error Function and Fresnel Integrals.\" Ch.7 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. Midpoint Trapezoid Simpson\u00e2\u20ac\u2122s n Approx. These bounds will give the largest possible error in the estimate, but it should also be pointed out that the actual error may be significantly smaller than the bound.\u00c2\u00a0 The bound\n\nSign in 9 Loading... For , (5) where is the incomplete gamma function. Error function Select the second example, showing the error function, This is another function that is defined using an integral. Linked 112 Proving $\\int_{0}^{\\infty} \\mathrm{e}^{-x^2} dx = \\dfrac{\\sqrt \\pi}{2}$ Related 1Calculus questions involving intermediate theorem?1Trigonometric Functions Involving Calculus2Find my error?\n\nGiven a function \u00c2\u00a0we call dy and dx differentials and the relationship between them is given by, Note that if we are just given \u00c2\u00a0then the differentials are MrYouMath 39,130 views 4:44 Beta Function, Gamma Function and their Properties - Duration: 5:01. This equation clearly shows which error sources are predominant, and which are negligible. Princeton, NJ: Princeton University Press, p.105, 2003.\n\nI also have quite a few duties in my department that keep me quite busy at times. In it, you'll get: The week's top questions and answers Important community announcements Questions that need answers see an example newsletter By subscribing, you agree to the privacy policy and terms But even with special functions I can't understand how to do it... Where are the answers\/solutions to the Assignment Problems?\n\nConsider the multiplication of two quantities, one having an error of 10%, the other having an error of 1%. Error 8 15.9056767 0.5469511 17.5650858 1.1124580 16.5385947 0.0859669 16 16.3118539 0.1407739 16.7353812 0.2827535 16.4588131 0.0061853 32 16.4171709 0.0354568 16.5236176 0.0709898 16.4530297 0.0004019 64 16.4437469 0.0088809 16.4703942 0.0177665 16.4526531 0.0000254 128 16.4504065 Integrals and Series, Vol.2: Special Functions. This will present you with another menu in which you can select the specific page you wish to download pdfs for.\n\nThen all you need to do is click the \"Add\" button and you will have put the browser in Compatibility View for my site and the equations should display properly.\n\nCan Show Answer Answer\/solutions to the assignment problems do not exist. Class Notes Each class has notes available. 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Proof: The mean of n values of x is: The average deviation of the mean is: The average deviation of the mean is obtained from the propagation rule appropriate to average In such cases the experimenter should consider whether experiment redesign, or a different method, or better procedure, might improve the results. Calculus I (Notes) \/ Applications of Derivatives \/ Differentials [Notes] [Practice Problems] [Assignment Problems] Calculus I - Notes Derivatives Previous Chapter Next Chapter Integrals Linear Approximations Previous Section Next\n\nAlternatively, you can view the pages in Chrome or Firefox as they should display properly in the latest versions of those browsers without any additional steps on your part. Not the answer you're looking for? This equation has as many terms as there are variables.\n\nThen, if the fractional errors are small, the differentials dR, dx, dy and dz may be replaced by the absolute errors What are the benefits of a 'cranked arrow' delta wing?\n\nWe are using the word \"average\" as a verb to describe a process. DrJamesTanton 13,324 views 13:45 Normal distribution's probability density function derived in 5min - Duration: 4:50. Also, when I first started this site I did try to help as many as I could and quickly found that for a small group of people I was becoming a Sign in Share More Report Need to report the video?","date":"2018-11-14 07:22:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3579619824886322, \"perplexity\": 2782.659899709425}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-47\/segments\/1542039741660.40\/warc\/CC-MAIN-20181114062005-20181114083304-00010.warc.gz\"}"} | null | null |
\section{Introduction}
Knowledge graphs, such as DBpedia~\cite{Lehmann15} or
Wikidata~\cite{Kroetzsch14}, are the state of the art structure for
storing information and to draw knowledge from. They are knowledge
bases represented as graphs and consist essentially of \emph{items}
which are related through \emph{properties} and \emph{values}. This
enables them to fulfill the task of giving exact answers to exact
questions. However, they are limited when it comes to provide a
concise overview of the contained metric data and give characteristic
insights. For example, the number of such metric data sets in
Wikidata is tremendous: since (presumably) the set of all cities of
the world, including their geographic coordinates, is included in
Wikidata, this constitutes a metric data set. Further examples are
chemical compounds and their physical properties like mass and size or
celestial bodies and their trajectories.
One possibility to enhance the understanding of the metric data is to
identify outstanding elements, i.e., outstanding items. Based on such
elements it is possible to compose or enhance item recommendations to
users. For example, such recommendations could provide a set of the
most relevant cities in the world with respect to being outstanding in
their local surroundings. However, it is a challenging task to
identify outstanding items in metric data sets. In cases where the
metric space is equipped with an additional valuation function, this
task becomes more feasible. Such functions, often called \emph{scores}
or \emph{height} function, are frequently naturally provided: cities
may be ranked by their population; the importance of scientific
publications maybe ranked by the $h$-index \cite{Hirsch05} of their
corresponding authors. A naïve approach for recommending relevant
items in such settings would be based on the claim: items with higher
scores are more relevant items. As this method seems reasonable for
many applications, some obstacles may arise if the ``highest'' items
of the topic may be concentrated into a specific region of the
underlying metric data space. For example, returning the twenty most
populated cities in the world as an overview for the city landscape
would return no European city\footnote{\url{https://en.wikipedia.org/wiki/List_of_largest_cities} on
2019-06-16}, recommending the hundred highest mountain peaks of the
world would not lead to any knowledge about the mountainscapes
outsides of Asia\footnote{\url{https://en.wikipedia.org/wiki/List_of_highest_mountains_on_Earth}
on 2019-06-16}.
To overcome this problem we propose a novel approach: we combine the
valuation measure (e.g., ``height'') and distances drawn from the
metric in order to provide new valuation functions on the set of
items, called \emph{prominence} and \emph{isolation}. In contrast to
the naïve approach, those functions do value an item based on its
height in relation to the valuations of the surrounding items. This
results in a valuation function on the set of items that reflects the
extend to which an item is locally outstanding. The basic idea behind
the novel valuation functions is the following. The prominence
function values an item based on the minimal descent (with respect to
the height function) that is needed to get to another point of at
least same height. Furthermore, the isolation function, sometimes also
called \emph{dominance radius}, values the distance to the next higher
point with respect to the given metric and height function. These
measures are adapted from the field of topography where topographic
isolation and topographic prominence are used in order to identify
outstanding mountain peaks. Our approach is based on~\cite{Schmidt18},
where the authors Schmidt \& Stumme proposed prominence and dominance
for networks. We will transfer and adapt these through generalization
to the realm of bounded metric space.
To give a first insight to the potential of the novel valuation
functions in knowledge graphs, we will empirically verify their
ability to identify relevant items for a given topic. For this we
employ a supervised machine learning task. We evaluate if isolation
and prominence functions can contribute to the task of identifying
relevant items in the sets of French and German cities.
The contributions of this paper are as follows:
\begin{inparaitem}
\item We propose prominence and isolation for bounded metric
spaces. For this we generalize the results in~\cite{Schmidt18} which
were limited to finite, undirected graphs.
\item We demonstrate an artificial machine learning task for
evaluating novel valuation functions in metric data.
\item We introduce a general approach for using prominence and
isolation to enrich metric data in knowledge graphs. We show
empirically that this information helps to identify a set of
representative items.
\end{inparaitem}
The remainder of this paper is organized as follows. In~\cref{sec:rel}
we give a short overview over related work. This is followed
by~\cref{sec:math} were the necessary mathematical foundation is laid
out. \cref{sec:appl} gives a first insight in how the novel valuation
functions can be employed in a possible recommendation process. We
evaluate this in~\cref{sec:exp} and conclude our work within~\cref{sec:conc}.
\section{Related Work}
\label{sec:rel}
Item recommendations for knowledge graphs is a contemporary topic of high
interest in research. Investigations cover for example music recommendation
using content and collaborative information~\cite{Oramas16} or movie
recommendations using PageRank like methods~\cite{Catherine16}. The former is
based on the common notion of embedding, i.e., embedding of the graph structure
into $d$-dimensional $\mathbb{R}$ vector spaces. The latter operates on the
relational structure itself. Our approach differs from those as it is based on
combining a valuation measure with the metric of the data space. Nonetheless,
given an embedding into an finite dimensional $\mathbb{R}$ vector space, one
could apply isolation and prominence in those as well.
The novel valuation functions prominence and isolation are inspired by
topographic measures, which have their origin in the classification of mountain
peaks. The idea of ranking peaks solely by their absolute height was already
deprecated in 1978 by Fry in his work~\cite{Fry87}. The author introduced
prominence for geographic mountains, a function still investigated in this
realm, e.g., in Torres et. Al.~\cite{Torres18}, where the authors use deep
learning methods to identify prominent mountain peaks. Another
recent step for this was made in~\cite{Kirmse17}, where the authors investigated
methods for discovering new ultra-prominent mountains. Isolation and more
valuations functions motivated in the orometric realm are collected
in~\cite{Helman05}.
Recently the idea of transferring orometric functions to different realms of
research gained attention: The authors of \cite{Nelson19} used topographic
prominence to identify population areas in several U.S.\ States.
In~\cite{Schmidt18} the authors Schmidt \& Stumme transferred prominence and
dominance, i.e., isolation, to co-author graphs in order to evaluate their
potential of identifying ACM Fellows. We build on this for proposing our
valuation functions on bounded metric data. This generalization results in a
wide range of applications.
\section{Mathematical Modeling}
\label{sec:math}
Let us consider the following scenario: We have a data set $M$, consisting of a
set of items, in the following called \emph{points}, equipped with a metric
$d$ and a valuation function $h$, in the following called \emph{height
function}. The goal of the orometric (topographic) measures prominence and
isolation is, to provide measures that reflect the extend to which a point is
locally outstanding in its neighborhood.
Let $M$ be a non-empty set and $d: M \times M \to \mathbb{R}_{\geq 0}$. We call
$d$ a \emph{metric} on the set $M$ iff
\begin{inparaenum}
\item $\forall x,y \in M:d(x,y)=0\iff x=y$, and
\item $d(x,y)=d(y,x)$ for all $x,y \in M$, called symmetry, and
\item $\forall x,y,z \in M: d(x,z) \leq d(x,y) + d(y,z)$, called triangle inequality.
\end{inparaenum}
If $d$ is a metric on $M$, we call $(M,d)$ a \emph{metric space} and if $M$ is
finite we call $(M,d)$ a \emph{finite metric space}. If there exists a
$C\in\mathbb{R}_{\geq0}$ such that we have $d(m,n)\leq C$ for all $m,n \in M$ ,
we call $(M,d)$ \emph{bounded}. For the rest of our work we assume that $|M|>1$
and $(M,d)$ is a bounded metric space. Additionally, we have that $M$ is equipped with a height
function (valuation / score function) $h:M \to \mathbb{R}_{\geq 0}, m\mapsto h(m)$.
\begin{definition}[Isolation]\label{def:isolation}
Let $(M,d)$ be a bounded metric space and let $h: M \to \mathbb{R}_{\geq 0}$ be
a height function on M. The \emph{isolation of a point} $x \in M$ is then
defined as follows:
\begin{itemize}
\item If there is no point with at least equal height to $m$,
than $\iso(m)\coloneqq\sup\{d(m,n)\mid n \in M\}$. The boundedness of $M$ guarantees
the existence of this suprenum.
\item If there is at least one other point in $M$
with at least equal height to $m$, we define its isolation by:
\begin{equation*}
\iso(m)\coloneqq\inf\{d(m,n)\mid n \in M \setminus \lbrace m \rbrace \wedge h(n) \geq h(m)\}.
\end{equation*}
\end{itemize}
\end{definition}
The isolation of a mountain peek is often called the \emph{dominance radius} or
sometimes the \emph{dominance}. Since the term \emph{orometric dominance} of a
mountain sometimes refers to the quotient of prominence and height, we
will stick to the term \emph{isolation} to avoid confusion.
While the isolation can be defined within the given setup, we have to equip our
metric space with some more structure in order to transfer the notion of
prominence. Informally, the prominence of a point is given by the minimal
vertical distance one has to descend to get to a point of at least the same
height. To adapt this measure to our given setup in metric spaces with a height
function, we have to define what a path is. Structures that provide paths in a
natural way are graph structures. For a given graph $G=(V,E)$ with vertex set
$V$ and edge set $E\subseteq{V\choose{2}}$, \emph{walks} are defined as
sequences of nodes $\{v_i\}_{i=0}^n$ which satisfy $\{v_{i-1},v_i\} \in E$ for
all $i \in \lbrace 1,...,n \rbrace$. If we also have $v_i \neq v_j$ for
$i \neq j$, we call such a sequence a \emph{path}. For $v,w\in V$ we say $v$ and
$w$ are \emph{connected} iff there exists path connecting them. Furthermore, we
denote by $G(v)$ the \emph{connected component} of $G$ containing $v$, i.e.,
$G(v)\coloneqq\{w\in V\mid v\ \text{is connected with}\ w\}$.
To use the prominence measure as introduced by Schmidt
\& Stumme in~\cite{Schmidt18}, which is indeed defined on graphs, we have to
derive an appropriate graph structure from our metric space.
The topic of graphs embedded in finite dimensional vector spaces, so called
spatial networks \cite{Barthelmy11}, is a topic of current interest. These
networks appear in real world scenarios frequently, for example in the modeling
of urban street networks \cite{Jiang04}. Note that our setting, in contrast to
the afore mentioned, is not based on a priori given graph structure. In our
scenario the graph structure must be derived from the structure of the given
metric space.
Our approach is, to construct a \emph{step size graph} or
\emph{threshold graph}, where we consider points in the metric space as nodes and
connect two points through an edge, iff their distance is smaller then a given
threshold $\delta$.
\begin{definition}($\delta$-Step Graph)\label{def:stepgraph}
Let $(M,d)$ be a metric space and $\delta > 0$. We define the
\emph{$\delta$-step graph} or \emph{$\delta$-threshold graph}, denoted by
$G_{\delta}$, as the tuple $\left( M,E_\delta \right)$ via
\begin{equation}
E_{\delta}\coloneq \{ \{ m, n\} \in {M\choose{2}} \mid d(m,n) \leq \delta \}\}
\end{equation}
\end{definition}
This approach is similar to the one found in the realm of random
geometric graphs, where it is common sense to define random graphs by
placing points uniformly in the plane and connect them via edges if
their distance is less than a given threshold~\cite{Penrose2003}.
Since we introduced a possibility to derive a graph that just depends
on the metric space, we use a slight modification of the definition of
prominence compared to~\cite{Schmidt18} for networks.
\begin{definition}[Prominence in Networks]\label{def:prom}
Let $G=(V,E)$ be a graph and let $h:V \to \mathbb{R}_{\geq 0}$ be a
height function. The \emph{prominence} $\prom_G(v)$ of $v\in V$ is
defined by
\begin{equation}
\label{eq:prom}
\prom_{G}(v)\coloneqq\min\{h(v),\mindesc_{G}(v)\}
\end{equation}
where
$\mindesc_{G}(v)\coloneqq \inf\{\max\{h(v)-h(u)\mid u\in p\}\mid p
\in P_v\}$. The set $P_{v}$ contains all paths to vertices $w$ with
$h(w)\geq h(v)$, i.e.,
$P_{v}\coloneqq\{\{v_i\}_{i=0}^n \in P\mid v_0=v\wedge v_n\neq v \wedge
h(v_n)\geq h(v)\}$,
where $P$ denotes the set of all paths of the graph $G$.
\end{definition}
Informally, $\mindesc_{G}(v)$ reflects on the minimal descent in order
to get to a vertex in $G$ which has a height of at least $h(v)$. For
this the definition makes use of the fact that $\inf\emptyset=\infty$
in cases where no such point exists. This case results in $\prom_G(v)$
being the height of $v$. An essential distinction to the prior
definition in~\cite{Schmidt18} is, that we now consider all paths and
not just shortest paths. Based on this we are able to transfer the
notions above to metric spaces.
\begin{definition}[$\delta$-Prominence in Metric Spaces]\label{def:prominences}
Let $(M,d)$ be a bounded metric space and
$h:M \to \mathbb{R}_{ \geq 0 }$ be a height function. We define the
$\delta$-prominence $\prom_\delta (m)$ of $m \in M$ as
$\prom_{G_{\delta}}(v)$, i.e, the prominence of $m$ in the step
graph $G_{\delta}$ from~\cref{def:stepgraph}.
\end{definition}
We now have a prominence term for all metric spaces that depends on a
parameter $\delta$ to choose. For all knowledge procedures, choosing
such a parameter is a demanding task. Hence, we want to provide in
the following a natural choice for $\delta$. The ideas for this is
informally the following: We consider only those values for $\delta$
such that corresponding $G_{\delta}$ does not exhibit noise, i.e.,
there is no element without a neighbor. In other words, we allow only
those values of $\delta$ such that
$\forall m\in M\exists e\in E_\delta:m\in e$.
\begin{definition}[Minimal Threshold]\label{def:minimal-delta}
For $(M,d)$ a bounded metric space with $|M|>1$ we define the
\emph{minimal threshold} $\delta_M$ of $M$ as
\begin{equation*}
\delta_{M}\coloneqq \sup\{\inf \{d(m,n)\mid n \in M \setminus\{m\}\}\mid m \in M\}.
\end{equation*}
\end{definition}
Based on this definition a natural notion of prominence for metric
spaces (equipped with a height function) emerges via a limit process.
\begin{lemma}
Let $M$ be a bounded metric and $\delta_M$ as
in~\cref{def:minimal-delta}. For $m \in M$ the descending limit
\begin{equation}
\lim_{\delta \searrow \delta_M} \prom_\delta (m)
\end{equation}
exists.
\end{lemma}
\begin{proof}
Fix any $\hat{\delta} > \delta_M$ and consider on the open interval
from $\delta_M$ to $\hat{\delta}$ the function that maps $\delta$ to
$\prom_\delta(m)$:
\[\prom_{(.)}(m) :\ ]\delta_M, \hat{\delta}[ \to \mathbb{R}, \delta
\mapsto \prom_\delta (m).\] It is well known that it is sufficient
to show that $\prom_{(.)}(m)$ is monotone decreasing and bounded
from above. Since we have for any $\delta$ that
$\prom_\delta (m) \leq h(m)$ holds, we need to show the monotony.
Let $\delta_1, \delta_2$ be in $] \delta_M, \hat\delta [$ with
$\delta_1 \leq \delta_2$. If we consider the corresponding graphs
$(M,E_{\delta_1})$ and $(M, E_{\delta_2})$, it easy to see
$E_{\delta_1} \subseteq E_{\delta_2}$. Hence, we have to consider
more paths in~\cref{eq:prom} for $E_{\delta_2}$, resulting in a not
larger value for the infimum. We obtain
$\prom_{\delta_1}(m) \geq \prom_{\delta_2}(m)$, as required.
\end{proof}
This leads in a natural way directly to the following definition.
\begin{definition}[Prominence in Metric Spaces]\label{def:metric-prom}
If $M$ is a bounded metric space with $|M|>1$ and a height function $h$, the
prominence $\prom(m)$ of $m$ is defined as:
\begin{equation}
\prom(m) \coloneqq \lim_{\delta \searrow \delta_M} \prom_\delta (m).
\end{equation}
\end{definition}
Note, if we want to compute prominence on a real world finite metric data set,
it is possible to directly compute the prominence values: in that case the
supremum in \cref{def:minimal-delta} can be replaced by a maximum and the
infimum by a minimum, which leads to $\prom(m)$ being equal to
$\prom_{\delta_M}(m)$. Hence, we can compute prominence and isolation for every
point in the finite data set. There are results for efficiently creating such
threshold graphs~\cite{Bentley75}. However, for our needs in this work, in
particular in the experiment section, a quadratic brute force approach for
generating all edges is sufficient.
We want to show that our prominence definition for bounded metric spaces is a
natural generalization of~\cref{def:prom}.
\begin{lemma}
Let $G=(V,E)$ be a finite, connected graph with $ \left| V \right| \geq
2$. Consider $V$ equipped with the shortest path metric as a metric space. Then
the prominence $\prom_{G}(\cdot)$ from~\Cref{def:prom} and $\prom(\cdot)$
from~\cref{def:metric-prom} coincide.
\end{lemma}
\begin{proof}
Let $M\coloneqq V$ be equipped with the shortest path metric $d$ on $G$. As $G$
is connected and has more than one node, we have $\delta _M=1$. This yields
that $(M,E_{\delta_M})$ from~\Cref{def:stepgraph} and $G$ are equal. Hence,
the prominence terms coincide.
\end{proof}
\section{Application}
\label{sec:appl}
\subsection{Score based item recommending}
As an application of our valuation functions, we envisage a general approach for
a score based item recommending process. The task of item recommending in
knowledge graphs is a current research topic. However, most approaches are
solely based on knowledge about preferences of the given user and graph
structural properties, often accessed through knowledge graph embeddings. The
idea of the recommendation process we imagine differs from those. We stipulate
on a procedure that is based on the information entailed in the connection of
the metric aspects of the data together with some (often present) height
function. Of course, we are aware that this limits our approach to metric data
in knowledge graphs, only. Nonetheless, given the large amounts of metric item
sets in prominent knowledge graphs, we claim the existence of a plenitude of
applications. For example, while considering sets of cities, such a system could
recommend a \emph{relevant} subset, based on a height function, like population,
and a metric, like geographical distances. By doing so, we introduce a source of
information for recommending metric data in relational structures, like
knowledge graphs. A common approach for analyzing and learning in knowledge
graphs is knowledge graph embedding. There is an extensive amount of research
about that, see for example~\cite{Wang14,Bordes11}. Since our novel methods rely
solely on bounded metric spaces and some valuation function, one may apply those
after the embedding step as well. In particular, one may use isolation and
prominence for investigating or completing knowledge graph embeddings. This
constitutes our second envisioned application. Finally, common item recommending
scores/ranks can also be used as height functions in our sense. Hence, computing
prominence and isolation for already setup recommendation systems is another
possibility. Here, our valuation functions have the potential to enrich the
recommendation process with additional information. In such a way our measures
can provide a novel additional aspect to existing approaches.
The realization and evaluation of our proposed recommendation approach is out of
scope of this paper. Nonetheless, we want to provide some first insights for the
applicability of valuation functions for item sets based on empirical
experiments. As a first experiment, we will evaluate if isolation and prominence
help to separate important and unimportant items in specific item sets in
Wikidata. More specifically, we will evaluate if the valuation functions help to
differentiate important and unimportant municipalities in the countries of
France and Germany, solely based on their geographic metric properties and their
population as height function.
\subsection{Enriching metric item sets in Wikidata}
In this section we depict an universal approach for enriching finite metric item
sets in Wikidata using the introduced functions isolation and prominence. In
order to enhance the grasp for the reader, we accompany every step with a
running example. To which extend do municipalities stand out with respect to
their local surroundings, based on population (height)? The particular
steps are as follows:
\begin{enumerate}
\item \textbf{Identify a metric item set in the knowledge graph:} For this we
need to identify the metric space of all items in some considered set. One may
pre-compute their pairwise distances, if applicable.
For our experiments we identify the set of German municipalities and French
municipalities with their geographic coordinates in longitude and latitude and
compute as well their pairwise (approximated) distances.
\item \textbf{Identify height function:} Since we want to compute the prominence
and isolation of the items, we also have to identify a height function. Hence,
we need to identify a valued property shared by all items identified in the
step above which is also relevant to the enriching task.
In our running example we identify the population of the municipalities as
such a relevant shared valued property.
\item \textbf{Compute isolation:} Based on the steps before we are now
abled to compute the isolation for all items in the item sets.
For our running example, we compute the isolation for all municipalities for
the item sets of Germany and France.
\item \textbf{Compute the threshold graph:} For computing the prominence values
for all items in the item sets, we need to compute the threshold graph and the threshold $\delta_{M}$
using~\cref{def:stepgraph} and ~\cref{def:minimal-delta}.
In our running examples, for Germany we compute the value $\delta_{M}\approx 32$
kilometers. This value is necessary in order to preserve a connection between
Borkum (Q25082) and Krummhörn (Q559432). For the French item set we compute
$\delta_{M}\approx 54$ kilometers in order to preserve the connection between
Mende (Q191772) and La Grand-Combe (Q239967).
\item \textbf{Compute prominences:} Equipped with the threshold graph we are now
able to compute the prominence values for all items
using~\cref{def:prominences}.
\end{enumerate}
\subsection{Resulting Questions}
The sections above raise the natural question for an objective evaluation of the
functions prominence and isolation. In this section we present such an
evaluation scheme by means of two qualitative questions connected to this task.
Assume we have given a bounded metric space $M$ representing our data set and a
given height function $h$. The aim of the research questions we propose in the
following is to evaluate if our functions isolation and prominence provide
useful information about the relevance of given points in the metric space. If
$(M,d,h)$ is a metric space equipped with an additional height function, let the
map $c:M \to \lbrace 0,1 \rbrace$ be a binary function that classifies the
points in the data set as relevant (1) or not (0). We want to answer the
following question to evaluate if there is a connection between the extent to
which a data point is local outstanding (i.e., has high isolation and
prominence) and relevance. We connect this to our running example using the
classification function that classifies municipalities having a university (1)
and municipalities that do not have an university (0). We admit that the
underlying classification is not meaningful in itself. However, since this setup
is essentially a benchmark framework (in which we assume cities with
universities to be more relevant) we refrain from employing a more meaningful
classification task in favor of a controllable classification scenario.
\begin{enumerate}
\item \textbf{Are prominence and isolation alone characteristical for
relevance?} \newline We use isolation and/or prominence for a given set of
data points as features. To which extend do these features improve learning a
classification function for relevance?
This question manifests in our running example as follows: are prominence and
isolation useful features to classify the university locations of France and
Germany?
\item \textbf{Do prominence and isolation provide additional information, not
catered by the absolute height?} \newline Do prominence and isolation
improve the prediction performance of relevance compared to just
using the absolute height? Does a classifier that uses prominence and
isolation as additional features produce better results than a classifier
that just uses the absolute height?
In the context of our running example: Do prominence and isolation of
municipalities add information to the population feature, that help to
characterize the university locations, compared to using the plain population
value?
\end{enumerate}
We will evaluate the proposed setup in the realm a knowledge graph and take on
the questions stated above in the following section and present some
experimental evidence.
\section{Experiments}
\label{sec:exp}
\subsection{Dataset}
We extract information about municipalities in the countries of Germany and
France from the Wikidata knowledge graph. This knowledge graph is a structure
that stores knowledge via \emph{statements}, linking \emph{entities} via
\emph{properties} to \emph{values}. A detailed description can be found in
\cite{Kroetzsch14}, while \cite{Hanika19} gives an explicit mathematical
structure to the Wikidata graph and shows how to use the graph for extracting
implicational knowledge from Wikidata subsets. We investigate in the following
if prominence and isolation of a given municipality can be used as features to
predict university locations in a classification setup. We use the query
service of Wikidata\footnote{\url{https://query.wikidata.org/}} to extract
points in the country maps from Germany and France and to extract all their
universities. For every relevant municipality we extract the coordinates and the
population. The necessary SPAQRL queries we employed for all the followings
tasks are documented in our GitHub
repository\footnote{\url{https://github.com/mstubbemann/Orometric-Methods-in-Bounded-Metric-Data}}
for our paper project. While constructing the needed metric space, we have to
overcome some obstacles.
\begin{itemize}
\item Wikidata provides different relations for extracting items that are
instances of the notion city. The most obvious choice is to employ the
\emph{instance of} (P31) property for the item \emph{city} (Q515). Using this,
including \emph{subclass of} (P279), we find insufficient results for
generating our data sets. More specific, we find only 102 French cities and
2215 German cities.\footnote{Queried on 07-08-19} For Germany, there exists a
more commonly used item \emph{urban municipality of Germany} (Q42744322) for
extracting all cities, while to the best of our knowledge, a counterpart for
France is not provided.
\item The preliminary investigation led us to use not cities but
\emph{municipality} (Q15284), again including the \emph{subclass of} (P279)
property, with more than 5000 inhabitants.
\item Since there are multiple french municipalities that are not located in the
mainland of France, we encounter problems for constructing the metric
space. To cope with that we draw a basic approximating square around the
mainland of France and consider only those municipalities inside.
\item We find the class of every municipality, i.e, university location or
non-university location, through the following approach. We use the
properties \emph{located in the administrative territorial entity} (P131) and
\emph{headquarters location} (P159) on the set of all universities and checked
if these are set in Germany or France. An example of a German University that
has not set P131 is \emph{TU Dortmund} (Q685557).\footnote{last checked on
19-06-25}
\item Using a Python script we then matched the list of municipalities with the
indicated properties of the universities. This method was necessary for the
following reason. Some universities are not related to municipalities through
property P131. For example, the item \emph{Hochschule Niederrhein} (Q1318081)
is located in the administrative location \emph{North Rhine-Westphalie}
(Q1198), which is a federal state containing multiple municipalities. For these
cases we checked the university locations manually. Some basic statistics on
our dataset can be found in~\cref{table:basics}, a graphic overview of the
municipality and university distribution is depicted in~\cref{fig:countries}.
\item During the construction of the data set we encounter universities that are
associated to a country having neither \emph{located in the administrative
territorial entity} (P131) nor \emph{headquarters location} (P159). There
are ten German and twelve French universities for this case. We checked them
manually and were able to discard them all for different reasons, for example,
items that were wrongly related to the university item.
\end{itemize}
\begin{table}
\caption{Basic statistics of the country datasets extracted from
wikidata.}
\label{table:basics}
\begin{center}
\begin{tabular}{lrr}
\toprule
{} & Municipalities & University Locations \\
\midrule
France & 2063 & 92 \\
Germany & 2863 & 164 \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\begin{figure*}[htbp]
\includegraphics[trim=9em 0 0 0, width=0.8\columnwidth,
clip]{mixed_g.png}
\includegraphics[trim=0 0 9em 0, width=0.8\columnwidth,
clip]{mixed_f.png}
\caption{Municipalities in Germany (left) and France (right) having an
university (Uni) and not having an university (No Uni).}
\label{fig:countries}
\end{figure*}
\subsection{ Binary Classification Task }
\subsubsection*{Setup}
For both France and Germany, we compute the prominence and isolation of all
data points. We then normalize the population, isolation and prominence values
to be in the range from $0$ to $1$. Since our data set is highly imbalanced,
most of the common classifiers would tend to simply predict the majority
class. A variety of methods were proposed in the past to deal with such
problems. An overview can be found in~\cite{Kotsiantis06}. Sampling approaches
like undersampling or oversampling via the creation of synthetic
examples~\cite{Chawla02} are an established method for dealing with such
imbalances. We want to stress out again that the goal for the to be introduced
classification task is not to identify the best classifier. Rather we want to
produce evidence for the applicability of employing isolation and prominence as
(more suitable) features for learning a classification function. Since we need a
classification algorithm that provides useful predictions on single features, we
decide to use logistic regression with $L^2$ regularization and Support Vector
Machines~\cite{Cortes95} with a radial kernel. To overcome the imbalance, we use
inverse penalty weights with respect to the class distribution.
For our experiments, we use the algorithms for Support Vector Machines
(\texttt{SVC}) and \texttt{LogisticRegression} that are provided by the Python library
Scikit-Learn~\cite{Pedregosa12}. To solve the resulting minimization problem,
our setup of Scikit-Learn uses the LIBLINEAR library, see \cite{fan08}. As
penalty factor for the \texttt{SVC} we set $C=1$, however, we also experiment
with $C\in\{0.5,1,2,5,10,100\}$. As in~\cite{Akbani04}, where the authors
compared multiply methods to use support vector machines for imbalanced data
sets, we choose $\gamma=1$ for our radial kernel. For all possible combinations of
population, isolation and prominence we use hundred iterations of five
cross-validation. We analyze to which extent the novel valuation functions help to
classify university municipalities in Germany and France.
\subsubsection*{Evaluation}
We use the g-mean (i.e., geometric mean) as evaluation function. Consider the confusion
matrix depicted in~\cref{tabdingohneschlaege}.
\begin{table}[b]
\caption{Confusion Matrix}
\begin{tabular}{l|c|c}
\toprule
{} & Predicted Negative & Predicted Positive \\
\midrule
Actual Negative &TN (True Negative) &FP (False Positive)\\
Actual Positive &FN (False Negative) &TP (Tue Positive)\\
\bottomrule
\end{tabular}
\label{tabdingohneschlaege}
\end{table}
Overall accuracy (i.e., how many test examples are classified correctly) is
highly misleading in the context of heavily imbalanced data. It is obvious that
for any classifier function predicting the majority would lead to an excellent
accuracy~\cite{Chawla10}. Therefore, we will evaluate the classification
decisions by using the geometric mean of the accuracy on the positive instances,
$acc_+ := \frac{TP}{TP+FN}$, often called sensitivity, and the accuracy on the
negative instances $acc_- :=\frac{TN}{TN+FP}$, often called specificity. Hence,
the g-mean score is then defined by the formula $g_{mean} :=\sqrt{acc_+ \cdot
acc_-}$. The evaluation function g-mean is established in the topic of
imbalanced data mining. It is mentioned in \cite{He09} and used
for evaluation in \cite{Akbani04}.
In our setup, the university locations are the positive class, meaning
that $acc_+$ corresponds to the classification results on the
university locations, and $acc_-$ corresponds to the accuracy on non
university locations.
For our experiments we now compare the values for g-mean for the following
cases. First, we train a classifier function purely on the features population,
prominence or isolation. Secondly, we also try combinations of them for the
training process. We consider in all those experiments the classifier solely
trained using the population feature as baseline, since this classification
function does not incorporate any metric aspects of the data set.
Then, an increase in g-mean when using prominence or isolation together with the
population function is evidence for the utility of the introduced valuation
functions. Furthermore, when directly comparing a classifier function that is
trained on isolation/prominence with a version trained on population, an
increase in g-mean strongly indicates the importance of the novel features.
In our experiments, we are not expecting high values for g-mean, since the
placement of university locations depends on many additional features, including
historical evolution of the country and political decisions. However, we claim
that the evaluation setup above is sufficient to show that the novel features
are potentially helpful for identifying interesting and useful items in
different tasks.
\subsubsection*{Results}
\begin{table*}[htbp]
\begin{center}
\caption{Results of the binary classification task. The results
for every combination of features and every classifier. The best
value for every combination of features is printed in bold.}
po=population, pr=prominence, is=isolation \\
SVM= Support Vector Machine, LR = Logistic Regression
\label{tab:binary}
\begin{tabular}{ll||rr|rr||rr|rr}
\toprule
Country & & \multicolumn{4}{l}{France} & \multicolumn{4}{l}{Germany} \\
Classifier & & \multicolumn{2}{l}{SVM} & \multicolumn{2}{l}{LR} & \multicolumn{2}{l}{SVM} & \multicolumn{2}{l}{LR} \\
& Score & mean & std & mean & std & mean & std & mean & std \\
\midrule
iso & acc+ & 0.5700 & 0.0075 & 0.6185 & 0.0036 & 0.5407 & 0.0044 & 0.6201 & 0.0043 \\
& acc- & 0.9595 & 0.0010 & 0.9468 & 0.0010 & 0.9751 & 0.0003 & 0.9564 & 0.0009 \\
& g-mean & 0.7395 & 0.0048 & 0.7652 & 0.0024 & 0.7261 & 0.0030 & 0.7701 & 0.0027 \\
\midrule
pr & acc+ & 0.2273 & 0.0041 & 0.3967 & 0.0065 & 0.1643 & 0.0035 & 0.3380 & 0.0075 \\
& acc- & 1.0000 & 0.0000 & 0.9968 & 0.0004 & 1.0000 & 0.0000 & 0.9990 & 0.0002 \\
& g-mean & 0.4767 & 0.0043 & 0.6288 & 0.0051 & 0.4054 & 0.0044 & 0.5811 & 0.0065 \\
\midrule
po & acc+ & 0.4684 & 0.0035 & 0.5815 & 0.0139 & 0.3370 & 0.0065 & 0.4949 & 0.0057 \\
& acc- & 0.9932 & 0.0004 & 0.9834 & 0.0008 & 0.9970 & 0.0003 & 0.9886 & 0.0005 \\
& g-mean & 0.6820 & 0.0025 & 0.7562 & 0.0092 & 0.5796 & 0.0056 & 0.6994 & 0.0041 \\
\midrule
iso+pr & acc+ & 0.5577 & 0.0100 & 0.6114 & 0.0088 & 0.5109 & 0.0075 & 0.5915 & 0.0061 \\
& acc- & 0.9616 & 0.0011 & 0.9499 & 0.0008 & 0.9782 & 0.0006 & 0.9648 & 0.0009 \\
& g-mean & 0.7323 & 0.0065 & 0.7621 & 0.0055 & 0.7069 & 0.0052 & 0.7554 & 0.0040 \\
\midrule
iso+po& acc+ & 0.6038 & 0.0131 & 0.6273 & 0.0050 & 0.6012 & 0.0055 & 0.6549 & 0.0061 \\
& acc- & 0.9691 & 0.0011 & 0.9611 & 0.0007 & 0.9809 & 0.0007 & 0.9721 & 0.0005 \\
& g-mean &\textbf{0.7649} & 0.0083 & \textbf{0.7764} & 0.0031 & \textbf{0.7680} & 0.0035 & \textbf{0.7979} & 0.0037 \\
\midrule
pr+po & acc+ & 0.4770 & 0.0060 & 0.5543 & 0.0091 & 0.3524 & 0.0029 & 0.4966 & 0.0086 \\
& acc- & 0.9960 & 0.0004 & 0.9895 & 0.0007 & 0.9978 & 0.0001 & 0.9927 & 0.0005 \\
& g-mean & 0.6892 & 0.0044 & 0.7406 & 0.0061 & 0.5930 & 0.0024 & 0.7021 & 0.0061 \\
\midrule
iso+pr+po & acc+ & 0.5992 & 0.0115 & 0.6233 & 0.0066 & 0.5945 & 0.0053 & 0.6410 & 0.0075 \\
& acc- & 0.9694 & 0.0011 & 0.9629 & 0.0008 & 0.9817 & 0.0006 & 0.9744 & 0.0006 \\
& g-mean & 0.7622 & 0.0073 & 0.7747 & 0.0041 & 0.7640 & 0.0034 & 0.7903 & 0.0046 \\
\bottomrule
\end{tabular}
\end{center}
\end{table*}
The results of our evaluation can be found in Table \ref{tab:binary}. In the
following we collect the observations drawn from this table.
\begin{inparaitem}
\item \textbf{Isolation is a good indicator for structural relevance.}
Considering the results for both countries we notice that using isolation as
the only feature leads to a solid prediction of university and non-university
locations. For both countries and classifiers, it outperforms population.
\item \textbf{Combining absolute height with our valuation functions leads to
better results.} Combining our orometric functions with population leads to
better performance compared to solely the population feature.
\item \textbf{Prominence is not useful as a solo indicator.} Our result raises
confidence that prominence alone is not an useful indicator for finding
university locations. We may propose the following explanation. Prominence is
a very strict valuation function: recall that we constructed the graphs by
using distance margins as indicators for edges, leading to a dense graph
structure in more dense parts of the metric space. It follows that a point in
a more dense part has many neighbors and thus many potential paths that may
lead to a very low prominence value. Observing
definition~\cref{def:prom}, one can see that having a higher neighbor,
with respect to the height function, always leads to a prominence value of
zero. As mentioned earlier, the threshold is about 32 kilometers for Germany
and 54 kilometers for France. Hence, a municipality has a not vanishing
prominence if it is the most populated point in a radius of over 32
kilometers, respectively 54 km. Only 75 municipalities of France have non zero
prominence, with 41 of them being university locations. Germany has 124
municipalities with positive prominence with 78 of them being university
locations. Thus, prominence alone as a feature is insufficient for the
prediction of university locations. As indicated in~\Cref{tab:binary}, the low
g-mean score results from bad accuracy on the positive instances. Overall, it
is an useful feature for identifying outstanding ``peaks''.
\item \textbf{The results for Germany differ from the results for France.} The
margin in which isolation outperforms population as solely feature is for
Germany greater than for France. The same holds for the score improvement if
we add prominence and isolation as features to population. We assume that this
observation is based on the difference in the geographic population
distribution in France and in Germany: Having another look
at~\Cref{fig:countries}, one may observe a tendency of clustering of
university locations in some French areas. For example, looking at the area
around Paris, one may observe a variety of universities located in the
regional surrounding. The represented municipalities are all dominated by the
nearby city Paris. As a consequence, they have a low isolation and prominence
value.
\item \textbf{Support vector machine and logistic regression lead to similar
results.} To the question, whether our valuation functions improve the
classification compared with the population feature, support vector machines and
logistic regressions provide the same answer: isolation always outperforms
population, a combination of all features is always better then using just the
plain population feature.
\item\textbf{Support vector machine penalty parameter.} Finally, for our last
test we check the different results for support vector machines using the
penalty parameters $C\in\{0.5,1,2,5,10,100\}$. We observe that increasing the
penalty results in better performance using the population feature. However,
for lower values of $C$, i.e., less overfitting models, we see better
performance in using the isolation feature. In short, the more the model
overfits due to $C$, the less useful are the novel valuation functions we
introduced in this paper.
\end{inparaitem}
\section{Conclusion and Outlook}
\label{sec:conc}
In this work, we presented a novel approach to identify outstanding elements in
item sets. For this we employed orometric valuation functions, namely prominence
and isolation. We investigated a computationally reasonable transfer to the
realm of bounded metric spaces. In particular, we generalized previously known
results that were researched in the field of finite networks.
The theoretical work was motivated by the observation that knowledge graphs,
like Wikidata, do contain huge amounts of metric data. These are often equipped
with some kind of height functions in a natural way. Based on this we proposed
in this work the groundwork for an item recommending scheme. This envisioned
system would be capable of enriching conventional setups.
To evaluate the capabilities for identifying outstanding items we selected an
artificial classification task. We identified all French and German
municipalities from Wikidata and evaluated if a classifier can learn a
meaningful connection between our valuation functions and the relevance of a
municipality. To gain a binary classification task and to have a benchmark, we
assumed that universities are primarily located at relevant municipalities. In
consequence, we evaluated if a classifier can use prominence and isolation as
features to predict university locations. Our results showed that isolation and
prominence are indeed helpful for identifying relevant items.
For future work we propose to develop the conceptualized item recommender system
and to investigate its practical usability in an empirical user
study. Furthermore, we urge to research the transferability of other orometric
based valuation functions. Finally, we acknowledge that our results about
valuation functions in metric spaces are surely already present in mathematical
theory. To identify the related mathematical notions and therefore to nourish
from advanced mathematical results would be the next theoretical goal.
\begin{acks}
The authors would like to express thanks to Dominik Dürrschnabel for fruitful
discussions. This work was funded by the German Federal Ministry of Education
and Research (BMBF) in its program ``Quantitative Wissenschaftsforschung'' as
part of the REGIO project under grant 01PU17012.
\end{acks}
\citestyle{acmnumeric}
\bibliographystyle{ACM-Reference-Format}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,030 |
Q: htaccess rewrite rules conflict www.example.com/category/sub_category/Having problem with the rewririte rules in .htaccess file.
my current .htaccess file looks like this.
RewriteRule ^([0-9a-zA-Z_-]+)/([0-9]+)$ products.php?cat=$1&id=$2 [NC,L]
RewriteRule ^([^/]*)/([0-9a-zA-Z_-]+)/([0-9]+)$ product_categories.php?cat=$2&id=$3 [NC,L]
RewriteRule ^(.*)/(.*)/([0-9a-zA-Z_-]+)/([0-9]+)$ product_details.php?cat=$3&id=$4 [NC,L]
RewriteRule ^([0-9]+)$ gallery.php?id=$1 [NC,L]
I'm trying create urls like the following.
www.example.com/product_name/1
www.example.com/category/sub_category/21
www.example.com/category/sub_category/product_name/233
www.example.com/gallery/872
www.example.com/gallery/872 is redirecting to www.example.com/category/sub_category/872 instead of gallery.php?id=872
edit:corrected url from www.example.com/gallery/872 to www.example.com/category/sub_category/872.
A: Your issue is that the first rule matches, the last one can never get applied...
RewriteEngine on
RewriteRule ^gallery/([0-9]+)/?$ gallery.php?id=$1 [NC,L]
RewriteRule ^([0-9a-zA-Z_-]+)/([0-9]+)$ products.php?cat=$1&id=$2 [NC,L]
RewriteRule ^([^/]*)/([0-9a-zA-Z_-]+)/([0-9]+)$ product_categories.php?cat=$2&id=$3 [NC,L]
RewriteRule ^(.*)/(.*)/([0-9a-zA-Z_-]+)/([0-9]+)$ product_details.php?cat=$3&id=$4 [NC,L]
Rule of thumb: first the specific exceptions, then the more general rules.
The NC flag does not make sense if you also specify both, lower and upper case letters in your regex patterns. It is either/or, not and.
(note: I also included the correction @anubhava posted in his answer)
A: Your last rule will need regex modification since you're matching /gallery/872 and your pattern is only matching 1 or more digits.
RewriteRule ^gallery/([0-9]+)/?$ gallery.php?id=$1 [NC,L,QSA]
RewriteRule ^([0-9a-zA-Z_-]+)/([0-9]+)$ products.php?cat=$1&id=$2 [QSA,L]
RewriteRule ^([^/]*)/([0-9a-zA-Z_-]+)/([0-9]+)$ product_categories.php?cat=$2&id=$3 [QSA,L]
RewriteRule ^(.*)/(.*)/([0-9a-zA-Z_-]+)/([0-9]+)$ product_details.php?cat=$3&id=$4 [QSA,L]
Also you need to recorder your rules like I showed above.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,534 |
Myriaporidae is a family of bryozoans belonging to the order Cheilostomatida.
Genera:
Leieschara M.Sars, 1863
Myriapora de Blainville, 1830
Myriozoella Levinsen, 1909
Myriozoum Donati, 1750
References
Cheilostomatida | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,689 |
With a bamboo stick, skewer one piece of candy. Scoop out 1 heaping tablespoon from the cheesecake. Use your hands to mold the cheesecake into a ball around the candy of your choice on the skewer. Repeat with remaining candy and cheesecake. Skewer with raspberries, if desired. Roll in chips or candy of choice and place in the freezer to mold. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,741 |
"till they have seen the kingdom of God come" (Mark 9:1).
John and Jesus taught that the kingdom was "at hand" (Matthew 3:2; 4:17). They obviously believed it was forthcoming.
Some followers were so convinced of it, they attempted to force Jesus to lead a coupe d'état to declare Israel's independence from Rome (which, by the way, he rejected, cf. John 6:15). Nevertheless, Jesus continued preaching about the kingdom's imminent arrival.
Did it ever come? Was it only metaphor? Is it yet future?
Truly I say to you, there are some of those who are standing here who will not taste death until they see the kingdom of God after it has come with power" (Mark 9:1, NASB).
Some suggest that Jesus' transfiguration, which was witnessed only by Peter, James and John, and which took place in relative proximity to these words, was the fulfillment of them (Matthew 17:1-13).
Is there any reason to assume that Jesus' glorious transfiguration was the coming of the kingdom about which he spoke?
The evidence is lacking. The saying and the transfiguration are relatively close in proximity, just six days (Mark 9:2). Beyond that, there is little else to connect the transfiguration to this saying. One would have to assert that Jesus' transfiguration is the kingdom; or, at the best, a foreshadowing of the kingdom – a sort of, kingdom-in-earnest presentation, as it were. That is quite a stretch, and we believe it an unnecessary one.
Some who propose this theory also suggest that the kingdom has yet to come at all, that we are yet waiting for it. Which is it? Did the kingdom come metaphorically when Jesus was glorified before Peter, James and John, or is it yet to come? It can't be both, but it could be neither.
The meaning evidently is, "till they shall see my kingdom," i.e., my church…All this was accomplished. All these apostles, except Judas, lived to see the wonders of the day of Pentecost; some of them, John particularly, saw the Jewish nation scattered, the temple destroyed, the gospel established in Asia, Rome, Greece, and in a large part of the known world.
This is the best argument, that the kingdom Jesus promised to usher in, and the church Jesus promised to build – which came into being on the first Pentecost following his resurrection – are precisely the same entity. They are not complementary entities. They are not consecutive entities (the church first, then the kingdom). They are the one and the same.
The only reason to interpret this prophetic promise of Jesus as being fulfilled in his transfiguration is to avoid the fact that the church and the kingdom are the same entity.
And, behold, I send the promise of my father upon you: but tarry ye in the city of Jerusalem, until ye be endued with power from on high (Luke 24:49).
Some disciples then living saw the Lord's kingdom come with power. That power was the Holy Spirit descending upon the apostles and giving them the ability to miraculously communicate with all dialects the gospel of Christ on the Day of Pentecost.
When people obeyed the gospel that day, and since, they entered into that kingdom (Colossians 1:14), which is the church (Acts 2:47).
Jesus told his disciples that when the power would come, the kingdom would, too (Acts 1:6-8).
You are right, John! And I originally had this as part of the article, but it was getting lengthy so I cropped it, focusing on the fact that even Barnes, the Presbyterian, understood the interpretation. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,472 |
- [Awesome php](https://github.com/ziadoz/awesome-php)
- [A fractal of bad design](http://eev.ee/blog/2012/04/09/php-a-fractal-of-bad-design/)
- [Hacker guide](http://php.net/manual/en/internals2.php)
- [Demystifying regex](http://www.sitepoint.com/demystifying-regex-with-practical-examples/)
- [A php course](http://www.hackingwithphp.com)
## Doc
- [PHP.net](http://www.php.net/)
- [An zend exam example](http://www.zendexam.com/)
## Ecosystem
- [PHP Weekly](http://www.phpweekly.com/)
## Tools
- [Regex](https://regex101.com/) | {
"redpajama_set_name": "RedPajamaGithub"
} | 9,785 |
CONTENTS
_TITLE PAGE_
_DEDICATION_
_EPIGRAPH_
CHAPTER 1
CHAPTER 2
CHAPTER 3
CHAPTER 4
CHAPTER 5
CHAPTER 6
CHAPTER 7
CHAPTER 8
CHAPTER 9
CHAPTER 10
CHAPTER 11
CHAPTER 12
CHAPTER 13
CHAPTER 14
CHAPTER 15
CHAPTER 16
CHAPTER 17
_ACKNOWLEDGMENTS_
_ABOUT THE AUTHOR_
_COPYRIGHT_
**For William Hanson,
1977–2005**
Scars have the strange power to remind us that our past is real.
**—C ORMAC MCCARTHY,
_All the Pretty Horses_**
**1**
**T** he two boys walked the high ridge at the center of the wood road, avoiding the muddy ruts along the sides. Loggers had powered their hulking machines along the makeshift pathways—the huge skidder tires clawing deep cuts into the soft earth. The men had taken the timber of any value and only the undesirable trees remained: the young, the mangled and twisted, the rotten and sick. The boys made their way through the difficult clutter of leftover branches that now thatched the forest floor. The sun broke the sparse canopy and beat on their sweating necks.
Terry caught a toe on the cut end of an exposed root and stumbled into several lurching steps. His backpack rattled. The other boy sidestepped the splintered butt of wood and quickly tiptoed around a small birch stump. Terry stood a head taller than the boy and he was half again as broad, but he wore his body like an oversize suit. The boy was still small and nimble, but he wasn't happy about it. He looked at Terry's body and he wanted one of his own. Terry's neck and arms were thick like a man's. The backpack looked like a child's toy, dangling between his broad shoulders.
Terry tripped again. "Cocksucker," he said. He hopped the rut at the side of the path and took a seat on a broad stump. The cut wood was still pale and creamy. White sawdust clung to the dead leaves on the ground like early snow. Terry bent over his knees and clutched the laces of his work boot. He wore them untied and loose, as was the fashion in their school for boots and high-tops. Now he straightened his leg and leaned back, pulling the boot tight. He bow-tied the laces and set to his other foot.
The boy eyed Terry's hands and forearms as he pulled. They were covered in coarse red hair that matched the color of his closely shorn scalp. The boy's arms were undefined. What hair he had on his body was blond and thin.
Terry grunted when he stood. He hopped back on the trail. The boy was six months his senior, but Terry's size earned him the lead through the maze of skid roads. When Terry wondered which way to proceed, the boy pointed knowingly from behind. He'd grown up hunting the Darling land with his father and uncle. But several years back, Mr. Darling had died and his children had sold the property to a developer. Within weeks, NO TRESPASSING signs surrounded the four hundred acres. Within months, the land had been subdivided and the town's zoning board confronted with plans for a handful of upscale housing developments.
In effect, the boys were trespassing, but there was no one around to catch them. When the economy had gone bad and stayed bad, the development stopped. The groaning cement trucks quit their runs in and out of the new neighborhoods. The swarms of subcontractors disappeared and the developer's Mercedes no longer made its rounds about town. It was rumored that the money from the recent logging contract was all he had left to fend off foreclosure.
The boys walked out into the clearing of Woodbury Heights, the last of the developer's projects. He'd pushed the road into the woods, paved it, and even managed to cut several of the prospective house lots before the recession settled in. Piles of soil and unearthed boulders now cluttered the landscape. Leafless trees lay prone, their roots reaching elliptically into the air. The deep black of the new pavement stood out from the mess of the rest of the scene.
The boys made their way to the culvert at the end of the road. The August sun hung heavily on the two and came back at them from the hot blacktop.
"You sure?" the boy said.
Terry nodded. He slid his arms out of the backpack and pulled out three glass bottles.
"How you know?"
"My brother," Terry told him. "Two parts gas, one part oil." He took out three socks and tied knots in them. He soaked the socks with the mixture in the bottles and stuffed a knot through each open bottleneck. Then he went to the side of the road and wiped his hands on the tall grass.
When he returned, he took up one of the bottles, held a lighter to the sock, and heaved the cocktail. It crashed and set a good portion of pavement afire.
"See," Terry said. "Told you."
The boy smiled. "No shit," he said.
They watched the fire slowly subside.
Terry lit and tossed a second. Again the pavement burned.
"Let me," the boy said.
Terry handed him the last of the three bottles and the boy held it, his arm cocked and ready. Terry thumbed the lighter and touched it to the sock. The boy waited for the flame to catch, crow-hopped a quick three steps, and overhanded the bottle. It reminded him of some second-rate firework, the trajectory neither high nor fast. When it crashed down, the flames spilled across the tarmac and waved in the air. The boy stared at the fire, a dumb smile on his soft face.
A jab in the ribs brought him around quickly. Terry pointed a thumb down the road. His head was cocked, an ear in the direction of his hand. His eyes looked at the sky. The boy heard it too, an engine in low gear, climbing the hill. Terry turned and sprinted. The boy chased after him but couldn't keep up. With the sound of the engine growing closer, Terry didn't try to make it to the trail they'd come on. Instead he bolted over the shoulder of the road, through the underbrush, and into the woods. The boy followed.
With the broad hardwoods gone, the hiding wasn't good. Terry sprawled behind a fir sapling and the boy crouched behind a good-size stump. He panted, catching his breath. When he saw the police cruiser, his chest froze and he could hardly get more than a quick gasp. He looked back at Terry.
"Think it's burning?" the boy said.
Terry shrugged. "Come on," he said, once the cruiser had passed. He jumped to his feet and waved for the boy to follow.
"I don't think we should move."
"No way," Terry said, as he turned and lumbered into the woods. The boy looked back at the road. He heard Terry crashing through the brush and dead leaves behind him. He turned and ran after his friend. He didn't want to be alone.
Though Terry was a more powerful sprinter, his size worked against him over a longer distance and the boy overtook him.
"Where you going?" said the boy.
Terry pointed in the direction he was running.
The boy shook his head and motioned off to his right.
Terry nodded and followed.
When the two could run no farther, they stopped and rested, their torsos bent, hands heavy on their knees.
"Duncan?" the boy said.
Terry shrugged.
"I hope it was Duncan."
"Me too."
"Don't tell."
"Don't tell me not to tell. Christ," Terry said.
The boy looked down and then away.
"Besides," Terry told him, "I'm the one reeks of gas." He reached down and wiped his hands on the leaves of a small tree.
They walked until they reached Sandy Creek, the first development to go up on the Darling property. Prior to the building and landscaping, it had been a sandpit where teenagers rallied dirt bikes and hopped-up pickups. Where teenagers and twenty-somethings gathered around bonfires and drank beer until the police chased them out. Terry and the boy had hunted bullfrogs at the water holes there. When they couldn't catch them, they threw stones.
Now it was Sandy Creek, but in the heat of summer, the creek was more of a bog, and it plagued the new neighborhood with mosquitoes. The local carpenters and handymen had turned a good profit screening in the expansive porches of the new homes. On the road, the boy couldn't help noticing how different the subdivision was from the rest of the town. The uniform homes rose like tiered gunships from the ground, sitting nearly on top of one another—their grand picture windows looked out on other grand picture windows. The lawns were flat and cropped like crew cuts. The trees were planted. The gardens didn't bear produce.
At the end of the Sandy Creek road, the two boys stopped. Terry lived in one direction and the boy in the other.
"Want to come over?" Terry said.
The boy shook his head, unwilling to chance another encounter with Terry's two older brothers. They had a routine called the Daily Beating, and although the name implied a schedule, they simply pounced on Terry when the feeling struck them. During the boy's last visit, they included him in a new game they called Help Him—He's Drowning. One brother grabbed the boy by the back of the neck and dunked his head up and down in their above-ground pool. The other stood on the patio, pointing and shouting, "Help him—he's drowning! For the love of God, someone help!"
After hacking out the chlorinated water, hunkered on his hands and knees, the boy walked all the way home in his dripping swimsuit.
"No thanks," he told Terry.
Terry paused for a moment but the boy didn't extend an invitation.
"See you," Terry said.
"Yeah," said the boy.
When Terry was out of sight, the boy went back into Sandy Creek. He walked to the elaborate entryway of one of the large homes. It was hardly dusk, but the chandelier over the door was already lit. He'd ditched Terry because he knew his chances of getting invited inside were greater without him. Terry had fought Kevin Dennison earlier in the summer and fattened his lip. The boy pushed the doorbell and heard the elegant chime inside. He heard footsteps approaching and the door opened.
"Theodore," Mrs. Dennison said. She smiled and her eyes squinted. "How are you?"
The boy hated being called Theodore but he smiled back. "Fine," he said. "You?"
"Not bad," she said. Mrs. Dennison was young. Young to have kids his age, the boy thought. She was slender and her short dark hair looked shiny and soft to the touch. "Ready for school?"
He shook his head.
She smiled. "High school already. I can hardly believe it," she said. "Are you nervous?"
"Nah," he said.
"Don't tell him I said so, but I think Kevin is."
He smiled. "Kevin and Bobby here?"
"They're with their father," she told him.
He nodded. He knew the Dennisons were separated. He knew Mr. Dennison already had a girlfriend.
"Why don't you come by tomorrow," she said. "I'll tell them you were here."
"All right—have a good night."
"You too, Theodore." She smiled and retreated back inside the house.
He spun about, hopped down the stairs, and walked across the brick sidewalk to the road. When he turned, he balked at the sight of Terry, standing at the corner where they had parted. Terry had a cigarette. The boy knew the way he tried to hide it—his hand cupped around the butt, arm hanging casually at his side. Terry looked at the boy, took a drag, and left for the second time. An ill feeling settled inside the boy. He'd been caught in the midst of his defection, and worse, Terry had predicted the betrayal.
The boy scuffed his shoe hard across the pavement. He picked up a rock and hurled it at a real estate sign in front of a home. It struck loud and metallic. He winced and quickly looked around for any witnesses. The flash of fear overwhelmed the feeling he'd had upon seeing Terry—upon seeing Terry see him. He jogged to the end of the Sandy Creek road, but Terry was gone. The boy stood for a moment at the intersection before he turned and headed home.
As he walked, the din of evening crickets poured in from the surrounding woods. The pavement was old and cracked at the edges. The sand the town spread for traction in winter collected in small dunes in the ditch. Trees grew close at the sides and reached over the road. Some bore scars from accidents and run-ins with snowplows. Here and there a beer can littered the ditch, sometimes a hubcap or paper coffee cup.
Before the boy got to the Humphreys' house, he bent over and fisted two good-size stones. The Humphreys always had one mongrel dog or another that came to snarl at the foot traffic that passed. Every couple years the dogs were run down by cars and replaced. The boy passed in front of their home, but the dog didn't show. A barn stood on the back corner of their property and an old pony, round-bellied and sway-backed, wandered a corral back there. The boy remembered sneaking through the woods to throw rocks at the horse and watch it twitch and buck. He passed out of sight of the Humphreys' and dropped the two stones.
When he came around the bend to his house, he saw the real estate sign on his lawn. It was only two months old and it still caught his eye. His father had sold life insurance until the recession whittled away his commissions.
Earlier that summer, he found better work, but far off—in Pennsylvania. The boy's mother had fought the decision but her salary as a schoolteacher wouldn't cover the bills and there was little she could do but concede. The father moved into an efficiency, eight hours south, while the boy and his mother stayed home—the father hopeful for its sale, the mother for a change in the economy.
The boy walked across the mouth of the driveway and down the edge of the lawn. He eyed the front of the house and listened for coming cars. When all seemed clear he bent over and wrenched the sign back and forth, loosening the soil's grip on it. After that bit of handiwork, the sign came out easily. Clods of dirt still clung to its legs. With a hand on the top two corners, he jogged to the opposite side of the road, raised the sign above his head, and heaved it into the ditch for the third time. On each occasion, the real estate agent would track it down, skewer it back in the lawn, and curse the local teenage hooligans. It was a small delight for the boy, and he knew he would keep at it until he was caught.
**2**
**D** espite the Dennisons' air-conditioning and swimming pool, the three boys panted away the following afternoon at the boy's home. Kevin and Bobby wanted soda, but there was none. They wanted to watch cable television, but the only box on the top of the television had two rabbit-ear antennae protruding from it.
"You don't have crap to do," Bobby said. He was short and squat with a thick neck, like his older brother and father.
The boy shrugged. Bobby always complained that there wasn't anything to do.
"He has a gun," Kevin said.
"Bull," Bobby said.
"He does."
"Show us," Bobby said. "It's probably just a BB gun."
"I have two," the boy told them.
"Prove it," Bobby said.
"What kinds?" said Kevin.
"A twelve-gauge," the boy told them. "And a twenty-two."
"Let's see the twelve-gauge," Kevin said.
The boy shook his head. The shotgun was old, a hand-me-down from his uncle. He was afraid they would focus on the gun's nicks and scratches, or the way the pump action was worn and loose.
"He's bull," Bobby said. "That's why he won't."
The boy looked out the window and saw his mother working in the backyard. "I'll show you my twenty-two," he said.
"That's a stupid gun," Bobby said.
"Shut up," Kevin told his brother.
The boy shook his head. The gun wasn't stupid. His .22 was the only new firearm in the house—the others had been in the family for years: the 30-06 under the couch in the living room, the .300 Savage and the bolt-action 20-gauge in a closet in the basement, the shotgun under his father's bed.
He led the brothers to the china cabinet in the dining room. He reached underneath.
"Why here?" Kevin asked him.
"For the crows," he said.
Kevin looked confused.
"They wake him up some mornings," the boy said. "He lets me shoot them. Out the window." He glanced over his shoulder at the window that looked out on the backyard.
"That's awesome," Kevin said.
The boy didn't mention that he had yet to fire a single shot. Every time he opened the window, the crows scattered in the air. And he knew better than to go firing off a volley at the airborne birds—there were houses back there beyond the woods, and though a small caliber, a .22 could send a bullet over a mile.
He drew the gun out by the stock and cradled it.
"It's a Remington," he said. He pointed the barrel at the floor and walked the Dennisons through the parts he knew. The brothers stared, and for a moment, they listened patiently.
"Let me hold it," Bobby said.
The boy pointed the barrel at the ceiling and handed him the gun. Bobby seemed surprised at how heavy it was. He smiled and leveled it at his brother.
"Bam," he said. "You're dead."
The boy grabbed the barrel and wrenched the gun away from him.
"Never do that," he said.
"What?"
"Point it."
"It's not loaded," Bobby said. "That's stupid."
"You're stupid," the boy said. He pulled out a drawer in the cabinet and removed a small box. "These are the bullets." He fingered open the box and removed one for them to see. Kevin held out his hand and the boy dropped it into his palm. The two brothers bent their necks to study the round.
"It's puny," Bobby said.
"Show us how to put it in." Kevin held the bullet back in the boy's direction.
"There," he told Kevin, pointing to the chamber. "And close this." He slid the bolt closed and locked it down.
"No," Kevin said. "For real."
The boy shook his head.
"Why?" Bobby said.
"We're inside."
"Come on," Kevin said. "Do it."
"Show us," Bobby said.
The boy looked at them. Their eyes seemed so eager, so captivated. He held out his hand and Kevin returned the bullet to his palm. He took a breath and drew back the action. He looked at Kevin and Bobby as if to say, _Like this,_ and let the bullet into chamber. He fisted the ball on the end of the bolt, slid it forward, and locked it down. It felt beautiful—the slide and clack of steel coupling with steel. He exhaled and looked at them. They smiled. Bobby rocked back and forth from one foot to another.
A noise outside jarred the boy and he walked to the window. His mother wasn't in the backyard. He looked nervously from the .22 to the Dennisons. He knew that showing it off would land him in a shit storm of trouble. He bent over and slid it back under the cabinet.
He moved quickly to the window in the adjacent room and was relieved to find her in the side yard. She was sweeping up the sand on the side of the road. With a snow shovel she lifted piles of it into a wheelbarrow. His mother was small but strong. When she got ahold of him, he was helpless in her hard hands. She could shake him like a rag doll. His breathing calmed at the sight of her, but he felt guilty, watching her work. He knew he should be helping.
As he turned from the window, he flinched at the concussion of the loud clap in the house. He was puzzled only for an instant. It was different inside, but still he knew the noise. His fists clenched and his jaw bore down on his teeth until it felt as though they might give. He was sure his father would find out, would take his gun and get furious and silent for days. He wondered if he could hide it, some small cavity in the Sheetrock or molding. Maybe the shot went through the floor, the hole hidden by the nap of the carpet, the bullet lost in the dark basement.
As the boy breached the doorway to the room and saw Bobby on his back, the anger left him. His lungs stammered, unable to commit to the next breath that his body so needed. He looked at Kevin. Kevin was crying.
"My ears," Kevin said.
"What?"
"My ears," he said. "I can't hear."
"What happened?"
"He wanted it and I wasn't done. He pulled it." Kevin shoved the gun back at the boy and left the room. The boy walked toward Bobby. A small bubble of spit rose from Bobby's lips and burst. His eyes blinked, but other than that he didn't move. The boy moved closer and Bobby's eyes found him. There was a confusion in his face, like a dog hearing a new sound. The boy nearly jumped when Bobby moved. His hand came up and brushed at the small bloody spot on his chest like it was an itch or some crumbs. Then his arm stopped and lay still again. The spot on Bobby's chest was close to his left nipple. With his hand over it, he looked ready for the national anthem or the Pledge of Allegiance.
The boy looked down at the .22. He drew back the bolt and held out his hand to catch the spent shell. He held it to his nose and smelled the sulfurous mouth. He pocketed the casing, closed the action, and put the gun back under the cabinet. He turned and crouched, both feet under him, his elbows on his knees. He watched Bobby closely, the small movements here and there. Bobby stared at the ceiling.
The boy turned when he heard a noise behind him, and he was startled to see his mother. His body seized with the fear of a forthcoming punishment, but she didn't come for him. She didn't scream. Without so much as acknowledging the boy's presence, she walked quickly over and fell to her knees beside Bobby.
"Call an ambulance," she said. He stood alone behind her and watched. She took Bobby's wrist in her hand. She put her ear to his mouth. She pinched Bobby's nose and put her lips to his. She exhaled, paused, and exhaled again. She kneeled upright, put one hand atop the other, and pumped on Bobby's chest. She turned her hands over and looked at them. One had a thick sheen of blood. She looked from her hand to the boy behind her. "Go, goddammit."
He went to the next room and dialed the three numbers. He did his best to explain what had happened.
"Is he breathing?" the voice on the other end asked.
"A little, maybe," he said. "I don't know."
"I have some people on the way now, but I need to ask you a few more questions."
"Fine," he said.
The dispatcher proceeded and he answered the best he could but he was distracted by the sounds from the neighboring room. He stretched the cord and took the receiver away from his ear so that he could peer around the doorjamb. His mother sat up and wiped her hands on the legs of her jeans. She slowly shook her head. Her shoulders jumped twice and she brought a hand to her face. After a moment, she leaned over Bobby and pushed his bangs from his forehead. She ran her fingers through his hair and parted it to the side. She whispered something but the boy could hear only the soft outline of her voice. She patted his hair in place and slowly drew her hands back along his cheeks. She bent again to press her mouth to Bobby's and the boy felt a pang as witness to this. He felt separate. He felt alone and he quickly looked away.
When the ambulance arrived, he put the phone down on the counter and went to the door. Two men and a woman moved quickly up the sidewalk carrying heavy shoulder bags. The boy held the door open for them and pointed to the dining room but they stopped on the front stairs. The man in front asked him what had happened. The boy told him that Bobby had been shot.
"Who did it?" the man asked him.
"His brother."
"Where is he now?"
The boy shrugged. "Bobby's right in here. He needs help." He couldn't understand their hesitation.
"Where's the kid with the gun?" the man said. But before the boy could answer, the woman suddenly stepped forward and shouldered the man aside.
"Where is he?" she said to the boy, her head bent down to him. "The hurt one?"
The boy pointed and she walked past him.
"Wait," the man shouted after her. "There's a kid with a gun."
"He doesn't have the gun," the boy told him. "He left it."
The man shook his head and stepped through the doorway, the second man on his heels. The boy followed them in but stopped at the doorway of the dining room. The woman pushed his mother back from Bobby and handed her something for the blood on her hands. After feeling Bobby's wrist and neck and listening to his mouth, the first man placed a plastic mask over Bobby's face. He gave Bobby two blows and the second man set to pumping on his chest—much harder than his mother had. Bobby's head bucked. It looked terribly painful.
His mother took him hard by the wrist and pulled him out of the room. She had sweat on her brow. She took him to the couch in the other room and they both sat.
"What did you do?" she said.
"Mum, it wasn't me."
She didn't say anything. She dipped her chin and looked hard at him.
"It wasn't," he told her. "It was Kevin."
"Tell me," she said. "Exactly."
He told her how he had showed them the gun, loaded it, and stuffed it back under the cabinet.
"You loaded it?" she said.
He nodded.
"Goddammit," she said. "You know better than that."
He started to speak, but he heard more sirens approaching. He lifted himself onto the arm of the couch and looked over his shoulder, out the window. There was a quick squeal of tires as the cruiser turned into the driveway. Almost immediately another cop car appeared and parked behind the first. The boy's breathing got away from him.
"Watch what you say," his mother said. He shook and began to cry. His mother locked a hard hand around his biceps and squeezed. "Theodore," she said. "You didn't load that gun. Understand?"
He looked at his mother.
"Do you understand me?"
He dropped his chin twice in agreement.
The officers banged at the door but they didn't wait. Duncan walked immediately to the boy and his mother, the other officer helped one of the EMTs wheel a gurney into the room where Bobby was. Duncan was tall with short gray hair and a matching mustache.
"Who can give me an idea of what happened here?" he said.
The boy slowly raised his hand as if he were in school, as if he had an answer he wasn't sure was right. He started his story again, and again he was interrupted.
"Wait," Duncan said, holding up a hand. "Where's Kevin now?"
"I don't know," the boy said.
"Shit," said Duncan. He picked up the radio that clung to his lapel and began talking into it. As he mumbled into the radio, the boy watched them wheel Bobby by. He wanted Bobby to smile and give him the thumbs-up, but Bobby didn't move. Two men maneuvered the gurney while a woman pumped a plastic ball that attached to Bobby's face with a clear mask.
"How long ago did he leave?" Duncan said.
The boy didn't know. It felt like days and seconds at the same time.
"He can't be far," the mother said. "It couldn't have been more than fifteen minutes."
"I need a description," Duncan said.
The boy and the mother pieced together some details.
"Varney," Duncan shouted.
"Talk to me," the other officer said as he walked into the room.
"We got a missing shooter, or possible shooter," Duncan said. "Take the unit and see if you can find him. Early teens, brown hair, five foot or so. Kevin Dennison. Lives on Sandy Creek. Head out there."
"Got you," Varney said. He left the room.
The boy heard the screen door bang shut.
"Possible shooter?" the mother said.
"I'm not here to make any decisions, Donna."
"It was an accident," she said. "Teddy just told you."
"It's not my job to determine that," Duncan told her. "I'm here to secure the scene and separate the witnesses, which seems to have already been done for me."
"You're going to separate us?"
"He's a minor so you have the right to stay with him."
"We can have a lawyer, right?" she said.
Duncan shook his head. "You don't get Miranda until we arrest you."
"What are you going to arrest him for?"
"The troopers and the attorney general will determine the charges," Duncan said.
"I don't think all this is necessary," she told him.
"In case you haven't noticed," Duncan said. "We got a shooting where the accused has fled the scene. Not to mention a missing weapon."
"It's under the cabinet," the boy said.
"Huh?" Duncan looked quickly at him.
The boy pointed toward the other room.
"Why is it there?"
"That's where it belongs."
"I see," Duncan said.
A voice began to squawk from the radio on his lapel and he stepped out of the room. The boy tried to listen to the exchange between Duncan and his radio, but the officer mumbled in a low tone and the voice from the radio was distorted and static-ridden.
"We got Kevin," Duncan said as he reentered the room. "And we've got an investigator on the way. You expecting Pete?"
"He's out of town," the mother said. "I'll call him when you leave."
Duncan shook his head. "Not if you want to stay with your boy. He's headed to the station."
"Do I have a minute?" she said.
"What for?"
She motioned at the dark stains on the legs of her pants.
"Go on," he said. After she left the room, he leaned in to the boy and spoke quietly: "This all has to go a certain way, no matter what happened. Understand?"
The boy nodded. He wanted to smile but he didn't. Something in him had changed. He couldn't say exactly what it was, but he felt different. It didn't feel like he was sitting on the couch, there in the room. It felt like he was somewhere inside his skull, watching the room through the windows of his eyes.
**3**
**W** aiting in the holding room reminded the boy of being in the doctor's office. Any minute the door could open, but the minutes kept passing and no one came. The fluorescent lights above them hummed and flickered. His chair squeaked whenever he shifted his weight.
"I'll handle this," his mother said.
"Ma," he whined.
"You have no idea what all's at stake here," she hissed in a whisper. "You let me do what I can." Her tone stopped his protest. He nodded and looked down at his clasped hands in his lap.
When the door finally opened, he flinched and his mother stood to greet the officer.
"Hello, Mrs. LeClare," he said, holding out a hand. "Trooper Thompson, Department of Major Crime." The two shook hands.
"Pleased to meet you," the mother said. "Well, not really." She faked a smile.
"I understand," he said. He reached toward the boy and they shook.
The boy was intimidated by the trooper. He looked young and he looked like a bruiser. His jaw was thick and square and covered in dark stubble. His left eyebrow was cut in two by a scar.
"I'll be working to determine what happened today," he said. "I apologize for my appearance." He rubbed his face. "I've been on all night—we're a bit short-staffed at the moment." He took a seat across from them. "I got most of the basics from Duncan," he said. "But there's a lot more to cover. Can I ask you this, Teddy—have you ever been in trouble before?"
"Of course not," his mother said.
"Mrs. LeClare, I need answers from Teddy right now. You'll have your chance later."
"Me?" she said.
"Of course. He's your son, it was your home, and you were the first on the scene," he said. "Now, Teddy, any trouble?"
"What kind?"
"Like this. With the law."
"Duncan picked us up one time, on the highway, on our bikes."
"Us?"
"Me and Terry," the boy said. "Terry Duvall. We were taking the shortcut to the Waterhouse, the convenience store."
"Anything else?" the trooper said.
The boy looked at his mother. Then he looked at the floor. "One time he picked us up hitching on the bypass."
"What?" the mother said.
The trooper held a hand up. "You can hide him for it later," he told her. "Let's just get through this." He let his hand down and looked back at the boy. He was fighting a smile. "You and Terry?"
The boy nodded.
"Where were you headed this time?"
The boy shrugged.
"How far you get?"
"Duncan was the only one to stop," the boy said.
"What did he do?"
"Took us home."
"Nice of him," the trooper said.
The boy nodded. "He even let us off about a hundred yards from our houses. So our folks wouldn't see us get out of the car."
The trooper smiled and shook his head. "I have to find a job in a town like this."
"Better hurry up," the mother said. "It isn't going to be small for long."
The trooper nodded. "A lot of changes, I'm sure," he said.
"You wouldn't believe it," she said.
"I'm a transplant myself," the trooper told her.
"I know it. The way you talk. You're from Boston. Moved here to raise your kids, I bet."
"Nail on the head," he said. "I'm impressed."
"I've been here a little while."
"Teddy, let's start yesterday, around this time," the trooper said. "What were you doing?"
"What's yesterday got to do with anything?"
"Mrs. LeClare, it's my job to collect a twenty-four-hour history of everyone involved. Now, will you let me do my job?"
"Please, go ahead."
He was about to start when she interrupted him again.
"You can call me Donna."
He nodded. "Teddy, yesterday, about this time—talk to me."
"Me and Terry," the boy said. "We were out in the woods."
"The dynamic duo strikes again."
"We weren't doing nothing wrong."
"I didn't say you were," he said. "In the woods. Anywhere in particular?"
The boy shrugged.
"What were you doing?" the trooper said.
"Nothing."
"Nowhere doing nothing," the trooper said. "Sounds a bit vague, doesn't it?"
"I said we were walking the trails in the woods."
"You know when I get vague answers?" the trooper said. "When someone's trying to hide something. What are you trying to hide, Teddy?"
"Nothing."
"You guys out hitching on the bypass again? Or worse? You two fighting with the Dennison boys somewhere?"
The boy realized he was only causing more trouble. "We were on the Darling place," he said. "Or what used to be."
"That doesn't mean anything to me," the trooper said.
"They were trespassing," the mother told him. "The land's posted."
"Why were you trespassing?" the trooper asked him.
"They just logged it. We wanted to see what it looked like."
"I can accept that," the trooper said. "See how much easier this is when you tell the truth?"
The boy nodded.
"There's no reason for all that land to be posted," the mother said.
"That's something you can take up with the owner," the trooper said. "Let's keep it rolling, Teddy."
He told the trooper about ditching Terry to see the Dennisons. Then he had to explain the fight that Terry had had with Kevin at the beginning of the summer. The trooper didn't seem to care much about it. He did want to know about the Dennisons.
"How would you describe them, as a family?" the trooper asked.
"Fine," said the boy. "Nice."
"Nothing you ever seen or heard made you question that?"
The boy shook his head.
"Nothing besides them being separated and him having a girlfriend half his age," the mother said.
Thompson nodded at her. "Violence?" he said to the boy. "Drugs or alcohol?"
"I hardly know them," the boy said. "They're pretty new here."
"Kevin?" said the trooper. "Ever hear him say anything about smoking a little pot or drinking a few beers?"
The boy shook his head.
"How about Kevin and Bobby? How did they get along?"
"Fine," said the boy.
"They fight much?"
"They argued some but I never saw them really fight."
"What did they argue about?"
"Stupid stuff. Brother stuff."
The trooper seemed satisfied and continued. The boy told him about the night after his stop at the Dennisons' and the following morning. The trooper seemed uninterested.
"Tell me about what happened after Kevin and Bobby arrived," the trooper said.
"They wanted to see my guns."
"Why?" the trooper said.
"They were bored."
"How did they go about asking to see the guns?"
"They said I didn't have anything to do. They said my house was boring. Mostly Bobby. Then Kevin told him about my guns and they wouldn't quit after that."
"How did Kevin know you had a gun?"
The boy shrugged. "It wasn't a big secret. I probably said it sometime."
"Then what?"
"Then I showed them."
"Can you explain that for me?"
"I took them to the dining room and pulled it out from under the cabinet," he said. "We kept it there because the crows bothered my dad sometimes."
"There's a season for crows," the mother said. "They both got a small-game license."
"I don't care about the crows," the trooper said. "Was the gun loaded when you took it from under the cabinet?"
"Of course not," the mother said. "What kind of people you think we are?"
The trooper waited until she finished. "Teddy?"
"It wasn't loaded," he said.
"How did you know it wasn't?"
"We never keep them loaded. But I knew it wasn't because the action was open and I could see the chamber was empty."
"Then what?"
"Bobby wanted to hold it so I handed it to him," the boy said. "But he pointed it at his brother so I took it back. I told him he was stupid."
"Why?"
"Because you're supposed to treat every gun like it's loaded. No matter what."
"How would you describe your experience with firearms?"
"My experience?"
"How much have you been around them? How much instruction have you had?"
"Dad taught me to shoot at what, Ma—six?"
She nodded.
"He and my uncle always taught me to be safe," he said. "I had to take the hunters' safety course to get my deer tags. I got the card at the house."
The trooper nodded. "And how would you describe the Dennisons' experience, before today?"
The boy shrugged. "Probably none."
"And you were aware of this?"
"I guess."
"So you took the gun back from Bobby," the trooper said. "Then what?"
"I showed them the bullets."
"Where were they?"
"In a drawer, in their box."
"Who handled the bullets?"
"I took them out and handed one to Kevin."
"How did you hold it?" the trooper said.
The boy shrugged. "What do you mean?"
"By the casing? By the tip?"
He shrugged again. "I don't know," he said.
The trooper reached down and pulled a round from his belt. He held it out toward the boy. "Could you show me?"
The boy hesitated but then reached up and took the bullet. "Like this, I guess." The pads of his thumb and index finger held the brass casing.
"Did you touch any other parts of it?" the trooper asked him. "The primer?"
"There's no primer on a twenty-two," he told the trooper. "It's rimfire."
"Did you touch any other part?"
"Maybe," he said. "When I put it back in the box."
"You're sounding a little vague again, Ted."
"How is he supposed to remember that?" the mother interrupted. "A boy was just shot."
The trooper held up an open hand in her direction. "You don't have to remind me what happened, Mrs. LeClare. Then what?" he said to the boy.
The boy shrugged.
"How did the firearm get loaded?"
"He knows better than to load a gun in the house," the mother said.
"Mrs. LeClare, I won't ask you again. Teddy, how did the firearm get loaded?"
The boy looked at his mother. He looked back to the trooper. He shrugged again.
"Teddy?"
"I don't know," he said. "I heard something. I thought it was my mom so I put the gun back under the cabinet. I couldn't see her in the backyard and I went to another window. Then I heard the gun go off and I ran back. Bobby was on the ground and Kevin had it. He said Bobby tried to take it from him. Then he made me take it. I took the casing out and put the gun back under the cabinet. That's when my mom came."
"So how did the gun get loaded?"
"I don't know," he said. "I guess one of them."
"I thought you said they didn't have any experience with guns," Thompson said. "How would they know how to load it?"
"I showed them, before."
"How did you show them?"
"I pointed to the chamber and told them that's where a bullet goes," he said. "Then I closed the bolt."
"Where's the empty casing now?"
The boy stood and reached into his front pocket. He came out with the empty shell and handed it over to the trooper. The trooper held up his hand for him to wait. He went into his pocket and came out with a handkerchief. He unfolded the fabric and held it open for the casing. The boy dropped it in and the trooper carefully set it to rest on the table before him.
"How come you took it?"
"Seemed important," he said.
"Can't argue with that," the trooper said. "I'm going to step out to get this bagged. Give me a minute." He took the handkerchief and left the room.
When the door closed, the boy spun to face his mother. "Ma?" he said.
She looked at the floor.
"Ma?" He was more insistent, near pleading.
"I don't know," she said. "Good Christ, I don't."
He was shaken by this change in her and he began to cry. When she saw him, she sat upright. She seemed to gather herself. She slid her chair closer and pulled him into an embrace. Her hand was warm and heavy on the back of his neck.
"Easy," she whispered. "Just breathe. You're crazy if you think I'd let them lay a hand on you."
He made sucking sounds as he tried to stifle the heaving in his chest. He remembered when two older neighborhood boys had chased him home, threatening to beat him up after an argument turned ugly. He sprinted to his yard and found his mother at work in the garden on her knees. Embarrassed as he was, he took shelter behind her and, without asking, she stood and faced off with the two teenagers. "There's two of you," she said. "Seems only fair there be two of us." She gripped a small spade like she meant to use it and the two boys departed, cursing under their breath.
Just as the boy was getting his breathing back under control, the trooper and Duncan stepped through the door.
"Could we have a moment with Theodore?" the trooper asked his mother.
"Alone?" she said.
He nodded. "Just a moment."
"No," she said. She shook her head.
"Donna?" Duncan said.
"We know this was an accident," the trooper told her. "Kevin's story and Teddy's corroborate that much."
"So what's the problem?" the mother asked him.
"There's some differences between their stories."
"Maybe Kevin's lying," the mother said.
The trooper paused. "Maybe," he said.
"Wouldn't you?" she asked him.
"Excuse me?" he said.
"If it was your brother?"
He looked at her for a moment. "I don't know, Mrs. LeClare."
"Donna," Duncan said. "It'll just be a minute."
"It's either me or a lawyer," she said. "Take your pick."
"Will you both sign statements?" Duncan asked her. "At-testing to what you've said?"
She nodded.
"All right," said the trooper. "Let's get that started. We're also going to need prints done and residue tests on your hands. And we're going to need your clothes, Teddy."
The boy looked at him.
"You didn't tell him to bring a change of clothes?" The trooper looked at Duncan.
Duncan shrugged.
"What kind of operation is this?" Thompson smiled. "They'll also need to be tested for gunshot residue."
"I got some sweats he can borrow," Duncan said.
"You can change in here while we question your mother down the hall," the trooper said.
"How do I know this isn't some scheme to get him alone," the mother said.
"Why are you so worried about him being alone?" the trooper said.
"I know the games you guys can play with a boy like him," she said. "I expect your mother would do the same for you."
"I expect you're right," the trooper said. "But I assure you we're not playing games. You have my word we won't question him without you."
"Then let's get it over with." She stood up from her chair.
"We have to wait a minute," Duncan said. "Mrs. Dennison's in the hall."
"I'd like to talk to her," the mother said.
"No you wouldn't," Duncan told her.
"How is Bobby?" she said.
"They're doing everything they can," the trooper told her.
"How is he?" she said. She looked to Duncan.
Duncan looked at the floor and shook his head.
"Please," she said. "Can't I talk to her?"
Duncan shook his head again. "I know you mean well, Donna, but you're about the last person I'd put in front of her right now."
At the car, Duncan opened the front door for the mother and the back for the boy. The sweatpants Duncan had given him didn't have a string, so he held a fistful of the waistband to keep them from falling. He cuffed the legs and he pushed the sleeves up over his elbows to keep them put. The boy slid onto the plastic seat in the rear of the car and pulled the door closed behind him. He felt the small tilt in the car as Duncan climbed in. The car shivered when the engine started.
"Now what?" the mother asked Duncan.
"Now we wait," he said. "The evidence, the tests got to come back."
The mother looked at him.
"So they can substantiate a story," he said. "Teddy claims Kevin loaded the gun but Kevin says the opposite. Unfortunately Bobby won't be around to say which is so. And neither me or Thompson is much in the mood to go bullying these kids at the moment. Not with what they seen today."
"But you said you knew it was an accident."
"We can't rightly say that," he said. "That was a mistake. Say Kevin is lying about loading the gun. Then we got to ask ourselves why he's not telling the truth. Did he get upset with his brother and whammo?"
"I don't think it was like that," the mother said.
"And even if it was an accident, that doesn't mean there won't be charges," he said. "Say it comes out that the gun was loaded before the boys even got there."
"No," she said. "Not a chance."
"I'm saying hypothetical, Donna. If that was the case they could bring charges of negligence on you and Pete, resulting in a loss of life. And if it was Teddy loaded it, he could get hit with the same—even reckless behavior. Same goes for Kevin."
"How could Teddy be guilty of anything if he wasn't even in the room?"
"If he left a loaded gun with two kids he knew had no experience with firearms?" Duncan said. "Some might see that as negligent or reckless behavior."
"This doesn't make any sense to me," she said.
"Well, so you know," Duncan said, "we got a mother with a dead boy who thinks that even owning a gun should be a crime."
She nodded. "How about that trooper—Thompson?"
"What about him?"
"Is he a good man?"
"I believe he is," Duncan told her.
"I know he isn't from here," she said. "But is he like us?"
"Donna? What does that mean?"
"You know exactly," she said. "Does he live in some condo and make his kids wear helmets when they walk down the sidewalk? Does he think it's a crime if a dog shits on the wrong lawn? Don't tell me you don't know what I mean, Dick."
"I think he's fine," Duncan said.
She nodded. "How long for the evidence?"
"Hard to say. This isn't the TV. The recession's hit the troopers hard. Thompson was telling me he had a murder case where he waited so long for a set of prints to come back that he finally sent them to his old precinct in Boston. And that was just one set of prints."
"How long?" she said.
"Couple months anyway," Duncan said. "Probably a safe bet."
They drove the rest of the way to the house in silence. In the driveway, Duncan stopped the mother before she could leave the car. "If either of you has any trouble, please call. We're always around and we know some good professionals."
"Excuse me?" the mother said.
"Counseling," he said.
"Oh, no." She batted a hand in the air. "That's okay."
"You know where to find us," he said.
"Thank you, but we'll be fine." She got out and opened the rear door for the boy, then headed up the walk, throwing a quick wave at Duncan as he backed out. She pulled open the spring-loaded screen door and the boy followed her inside. He kicked off his shoes, and climbed the stairs to the quiet privacy of his room. He sprawled on his bed and forced his face into the pillow, wishing he could crawl into some small corner of his room and hide. He wanted to slip behind the Sheetrock and stand there between the wall studs, waiting for everything to pass. He lay there for some time, breathing through his pillow, willing himself to disappear, wishing to be like Bobby—invisible and gone and blameless.
There was a soft knock at his door. He didn't respond, and it came again. The door opened and his mother softly called his name, but he didn't move. He tried to breathe as if he were sleeping. She called again and placed a hand on his leg, but still he didn't budge. After a moment, he heard the footsteps withdraw and the door close again.
When he lifted his head, there was a wet spot where his breath had soaked the cotton of the pillowcase. He sat up on the edge of his bed, his elbows on his knees and his head in his hands. He pushed on his eyes until it hurt. He pulled at handfuls of his hair.
"Stupid," he hissed. "Stupid. Stupid."
He drew his hand out away from his head and made a fist. He hesitated for a moment and then dashed himself with his knuckles. Each time he swung, there was less hesitation and the pop of his fist against his skull grew louder. When his knuckles grew sore, too sore, he started in with his other hand.
When it left him, after the feeling had burned out, he rested his head back in his hands and ran his fingers over the lumps on his scalp. He knew to do it above the hairline—he'd answered enough questions for the day.
Later in the night, after he heard his mother climb the stairs to her bedroom, after he gave her time to fall asleep, he made his way back down to the first floor of the house. Sleep seemed impossible. His mind raced with all of the things he could have done, and all the things that still might happen. His stomach ached and turned with hunger. He walked to the refrigerator and took out a plate of lasagna, covered in plastic wrap. He slowly pulled a drawer, eased a fork out, and headed for his room.
When he passed the dining room, he stopped. He left the plate and fork on the telephone stand in the corner and pushed the round button of the dimmer switch. He spun the knob to dull the glare of the small chandelier and walked to where Bobby had fallen. A rectangular patch of carpet was missing—cut out and taken as evidence by the troopers who had examined the room while they were at the station. The removed swath wasn't nearly as big as death would seem to demand.
The round had entered between two ribs, punctured Bobby's heart, and come to rest in his scapula. Since the bullet never left Bobby's body, there wasn't anywhere for the blood to go. It slowly pooled inside him.
The boy crouched and ran his hand over the carpet around the cutout. It didn't feel any different. He slowly dropped his nose—it smelled stale and a bit musty. He turned his face and looked under the cabinet, but the .22 was gone. He looked back to the vacant rectangle where Bobby had been and for a moment it struck him: Bobby Dennison was dead, and he would never be anything but that again.
**4**
**L** ate that night, the father arrived from Pennsylvania. The boy heard the car pull in well after two in the morning and he quickly scrambled to kill his bedroom light. He stripped off his pants, threw them across the room, and crawled into bed. He heard the car door thump closed in the driveway and moments later he heard the front door of the house whine open. He heard the father's heavy footsteps in the house and then the dull, empty sounds of his parents greeting each other. As he listened, he practiced several poses of feigned sleep—on his back, mouth open, on his side, mouth slightly open, on his stomach, face concealed—but they never came for him. When they finally climbed the stairs, he heard them turn and head for their own bedroom.
The three ate a quiet breakfast together the following morning. The mother, still in her robe, cleared the kitchen table after the meal and left the room at the father's request. The boy and father sat kitty-corner to each other, the boy hunched, his head hanging. The father was barrel-chested and gray-haired. His elbows dug into the table, his eyes scrambled behind thick bifocals. As a child, the father had had a cyst removed from his left eyelid and now it sagged slightly. It gave him an inquisitive, untrusting look.
The father looked out from behind his brass-rimmed glasses and rubbed his graying stubble. A stiff, coarse sound came from between his hand and face. On weekends he sometimes waited until noon before making his pilgrimage to the bathroom where he showered and scraped the stubble from his heavy cheeks. He had trouble with the neck. The boy knew this from childhood, when he stood on the toilet seat and watched his father work the razor delicately around the peach pit of his Adam's apple.
Now the father sat at the head of the table, quiet, patient. This was the father's method: silence, silence for some time, and then a question, a question generally rhetorical in nature. It set the boy to floundering every time.
"So what are we going to do now?" the father said.
The boy shrugged.
"I know what you been through. With the law fixing to come down on you, I can't see throwing another log on the pyre."
The boy squirmed in his chair as a knot formed in his throat. This was another of the father's tactics: unexpected kindness. It worked to choke him up and send him reeling further.
The father shook his head and paused, perhaps to regain his composure. "I've heard what the police and your mother had to say. But I still can't figure how two kids got their hands on a gun in my house."
The boy looked down at the table.
"Well?" the father said.
"I showed them."
"Yeah?"
The boy shrugged. "They wanted to see it."
"So you do whatever any fool wants you to? You know what that makes you?"
The boy had trouble breathing. His throat started to close and his face grew tight.
"Who loaded that gun?"
The boy shrugged.
"Don't give me that."
"I don't know." The boy coughed it out.
"You realize that what you're saying makes no sense at all?"
"It must've been one of them," the boy said. "I didn't do it." He couldn't hold it back any longer and he went full-blown. His sniffling and gurgling were the only sounds in the room for some time. He used his shirt to wipe the tears and snot from his face.
"Quit the crybaby act," the father said. "It ain't going to work on me."
The boy didn't say anything.
"I hope you're not crying for yourself."
The boy shook his head.
"You best not be," the father said. "There's a dead kid out of this mess and a family who's got to live with that. Your mother's got to go teach in a school next week where every Tom, Dick, and Harry knows some kid just got himself killed in her dining room. And me, Ted, I'm a salesman. All I got is a reputation."
"I know," the boy nearly shouted. "Jesus, you think I ain't thought of all that?"
The father flinched. "Good," he said. "So this is what we're going to do. We're not going to say a word about it. Someone asks, you just shrug it off. Pretend like you don't know what they're talking about, like nothing ever happened. This rumor mill of a town doesn't need its fire stoked any more than it is. You get me?"
"Yes," the boy said.
"Good," said the father. "We're lucky we can leave. I don't know how we could stay here after something like this." He slid his chair back with a grunt. He stood and pushed the chair back under the table before he left the kitchen. Shortly after his departure, the mother returned from her seclusion in the living room. She rubbed the boy's back.
"You did good," she whispered.
"Ma. We can't do this."
"Yes we can."
"They're going to find out."
"It's your word against Kevin's," she said. "They can't prove anything."
He shook his head.
"If you don't play your cards right, they're going to take you away from me," she said. "And if that isn't enough, they've got something called a civil suit. They find us guilty of negligence or recklessness or God knows what else, those Dennisons will sue us and they'll win. You want to lose the house?" she asked him. "The cars? Everything?"
"No," he said and shook his head.
"Then you keep saying what you're saying. Understand?"
He didn't answer.
"Theodore," she said. "We can't take the gun out of Kevin's hands and we can't give that boy back to his poor mother. It's not going to happen. All we're doing is keeping you out of trouble. You get it?"
He nodded. His arms were folded on the table before him and he put his forehead in the crook of an elbow, hiding his face in the darkness his body created.
The father, son, and mother did what they could to duck one another's company for the next five days. They stuck to the father's policy—they didn't speak of the shooting, or Bobby's death, or what was to come. They came to the table for meals and they stared wide-eyed at the television in the evenings, but little was said among the three. The father busied himself with paperwork at a desk in the basement. The mother had two weeks' worth of the father's laundry to work through. The boy stayed in his bedroom and slept as much as he could. When he woke, he lay in bed, staring at the ceiling for as long as he could stand. He inventoried the contents of his bureau and closet, twice. He read random selections from a collection of encyclopedias he had on a bookshelf. Given the chance, his mind ground through the events around Bobby's death and his imagination ran off into the future, where he saw himself locked up in huge, clamoring institutions.
When he managed to get his mind away from the shooting, it took off with his fears of school. It started in the coming week. His small town was one of six that sent their students to a large regional high school—he was going from an eighth grade of ninety-five to a freshman class of over six hundred.
The day finally came when the father had to return to Pennsylvania and the house was busy preparing for his departure. The boy sat out of sight at the top of the stairs. He could see some of the interaction on the floor below in a mirror on the downstairs wall. His mother's brother, John, and his wife, Margaret, had come to see the father off. The father held the local paper that his brother-in-law had handed him.
"What?" the father said.
"Article on the editorial page," John said. "Look."
The father glanced quickly at the paper. "I don't have time for this." He handed the paper back.
"Ahem," John said, shaking the paper to attention. "'I think it is an outrage that a firearm was left accessible to young people.'" John paused and looked at the father. "This woman is trying to screw you."
"John," shouted the mother from another room.
"'I think it is time that gun owners are held responsible for any and all events that involve their firearms, even accidents.'" He let the paper down.
"I don't need this right now," the father said.
"She's trying to make it your fault," John said.
"John," the mother said, walking into the room. "Let it go."
"She has the right to say who can and can't go to the goddamn services," John said. "But this is too much."
"It doesn't concern you," she told him.
"How can you say that?" he said. "It's bullshit."
"Watch your language," the mother said. "I got a boy in the house."
The boy loved listening to his uncle cuss. He tried to swear with the same authority, but it always sounded cheap and phony.
"You think if you ignore it, it'll all go away," John said.
"One more word," she told him. "One more and you're gone."
"Donna," the father said. "Take it easy." There was a pause in the conversation. "John, we appreciate the concern, but I think it's just easier to let this lady do her thing. If and when they file charges, we'll do what we have to."
"This is your town," John said. "Both of you. You can't let her chase you out."
"Don't get dramatic on me, John—it doesn't suit you," the father said.
John didn't reply.
"There isn't work here," the father told him. "I can't live where there isn't work."
"It'll come back," John said. "Everyone's saying it."
"When it comes back, I'll come back."
"With women like this," John shook the newspaper. "There won't be anything worth coming back to. It'll be one big suburb. Nothing but Volvos and Saabs."
"I guess the only thing left to do is get drunk and cry about it," the father said.
"Piss off," John said.
"Nice talk," said the father.
The boy heard the front door open and slam shut.
"Ted," the father hollered.
The boy waited a moment to give the impression that he'd been in his room. "Yeah?"
"Get down here."
The boy rumbled down the stairs.
"What?"
"Give me a hand." The father had a gun case in each hand. He pointed at two other cases on the floor. The boy took them up and followed him into the yard. The trunk of his Pontiac was open and half full, but the father went to Margaret's station wagon instead.
"What are you doing?"
"They're going to John's," his father said. "He's going to keep them for us."
"Why?" said the boy.
"You know why." The father took the two guns from him and slid them into the station wagon. "Come on," he said. They walked back into the house. The father made his way up the stairs to the boy's bedroom. He pulled a chair from the corner and sat down. He pointed at the bed and the boy sat, facing him.
"You all right?"
"Yes," the boy said.
"You're the man here now."
"I know."
"No more fooling around," his father said. "Time to batten down the hatches and pull it together, yeah?"
"Yeah," said the boy.
"You got to help your mother."
"I know."
"You ready for school?"
He shrugged.
"It ain't going to be easy."
"I know."
"Remember, just keep your head down, keep quiet, and let this thing blow over. I'll be back again in another week or two. If you have any trouble when I'm not here, you see John. You can't reach him, you call Dave Benson."
"Mr. Benson?" the boy said, surprised at the mention of an almost forgotten fishing partner of the father's.
"I know—it's been years. But he knows folks."
The boy played with the fringe at the edge of his bedspread. "How long you think?" he said. "Till the house sells?"
"It doesn't look good," the father said, "but you never know."
The boy nodded. He'd seen the real estate signs around town, the developments full of new, empty homes. He'd heard of people forced to sell their places for half of what they'd paid only a year prior. He'd heard worse. Notices of bank foreclosures and property auctions were showing up in the local paper with increasing frequency.
"You know, some places in Pennsylvania, kids get the first day of deer season off," the father said.
"Yeah?"
"Some schools even give the whole first week."
The boy smiled.
"Honest to God," said the father. "No kidding. It'll be just like the old days. Me and you, in the woods." He smiled at the boy and gave him a slap on the side of the shoulder with his heavy hand. The boy was afraid to speak. He thought his voice might crack and he would break into tears. He always felt overwhelmed when taken back into the father's good favor.
"Come on," the father said. "Let's get this show on the road. If I don't leave soon, I'll never get there."
In the driveway the father rearranged the contents of the trunk and slammed it shut. The mother and Margaret stood before the car and John appeared from the backyard and stood behind them. The boy sat on the front steps.
"No baloney now," the father said.
"Don't flatter yourself," John told him. The father smiled and the two men shook hands. Margaret stepped to the father next and kissed him on the cheek.
"Drive safe," she said.
"Always," said the father. He approached his wife and kissed her.
"Work hard," she said, softly rubbing his arm.
The father nodded, turned, and approached the boy. He reached out his hand and the boy shook it.
"Be good," the father said, clapping him hard on the shoulder.
The boy nodded.
"Two weeks," said the father. "Two weeks is nothing." He climbed into the driver's seat and slammed the door. The engine turned over and started and the transmission clunked into reverse. The father smiled and waved through the windshield before he looked over his shoulder and backed into the road. The horn blasted twice and they waved at the tail end of the Pontiac.
"Finally got rid of him," John said.
The boy smiled back but the women didn't.
"Don't worry about dinner," Margaret said to the mother. "I've got a shepherd's pie in the freezer."
The mother nodded. "Thank you," she said, and the two moved into an embrace. The boy watched as his mother pushed her face into Margaret's shoulder and Margaret slowly rubbed her back in circles. John skirted the two and approached him.
"Come on," he said. "Let's get out of here." John pointed at the passenger side of the station wagon and the boy walked to the side of the car. "You'll be over later?" John said to the women. They nodded and John got in the driver's seat. He started the car and drove the short distance to his house.
An oak bar stood in the back corner of the basement room. The head and antlers of two white-tailed deer hung from the back wall, next to a bearskin and a few mounted fish. Against the side walls stood well-crafted glass-faced gun cabinets, each lit from within, making them glow in the dull light. John had spent his tax refunds on the room, and while the rest of the basement lacked even Sheetrock, this room had hardwood molding. Margaret complained that the money could have paid for a vacation, and given the chance, John always replied, "That room _is_ my vacation."
"You know where all these come from?" the boy said. He ran his finger around half the room, encompassing a good portion of John's gun collection in a single pass.
"I'd imagine," John said.
The boy walked over to a cabinet and opened it. "This one?" he said.
"A friend needed the money," John said. "Hit a kid in a crosswalk. Drunk."
The boy pointed at another.
"Estate auction."
He pointed again.
"Some fool didn't know what he had. I bought that for fifty bucks out of the want ads."
He stepped to the next cabinet down. "This one?"
"Me and your mother used to shoot that," John said. "We went to this dump as kids and went through the rubbish with sticks, for bottles and jars. We stood them on the side of an old icebox and I made up charges against them: disorderly conduct, drunk and disorderly, driving while drunk. We took turns until they were all shot up."
"Yeah?" said the boy.
"Don't tell her I told you, but we shot it at an oak door in the basement once. Had a bet whether or not it would go through."
"Who won?" said the boy. "The bet."
John shrugged. "It didn't go through. I know that. And deaf? My ears rang for a week."
The boy smiled and went on to the next cabinet. He opened the door and looked closely at an old pump-action shotgun.
"That was your great-grandfather's."
"I know. I never met him."
"Go ahead." He motioned for the boy to take up the firearm.
He hesitated.
"It's all right," John said. "It won't bite."
"That's okay." He tried to smile.
"Pick it up, Ted."
The boy reached into the cabinet and took it by the barrel. The action was open and he looked quickly to check the chamber.
"There you go," John said. "That's about as close as you're going to get to him. Without stepping in front of a bus anyway."
He pulled the stock to his shoulder. The wood was worn and smooth against his cheek. He let the gun down and looked it over again. It wouldn't have been the gem of anyone else's collection—the wood was nicked here and there and the blue steel had worn in places and shone silver. The boy used his shirttail to wipe it down before returning it. He looked at the next gun in the cabinet.
"Dad gave that to your mother for confirmation."
"What did you get?" the boy asked.
John shrugged. "I got that, when she got tired of it. She was his favorite."
"How come?"
"She went along with his stories," John said.
"What were his stories about?"
John squinted and looked at the ceiling as if to remember. "About how nice he was, how happy we all were. I tended to call his bluff."
"Why?"
"I couldn't help it," John said. "I guess I wasn't very good at telling stories."
The boy nodded as if he understood. He motioned to the next firearm in the row.
"Came back from World War Two with your great-uncle Dale's," John said.
The next cabinet down had just been stocked with the guns from the boy's home. He stood for a moment, staring through the glass.
"I know you've probably been asked this a hundred times," his uncle said. "But how you holding up through all this?"
"Fine."
"You sure?"
"Yeah."
"You want to tell me what happened?"
"Dad told me not to talk any more about it."
"Yeah, well, he's probably right."
"Besides," said the boy. "What's there to tell? Some kid killed his brother—you know that."
"I do," John said. "But that's a lot to carry. Whether you know it or not."
"You ever seen anyone die?"
John nodded. "Car accident," he said. "A woman got pitched from a truck when it flipped and it rolled right on over the top of her. She was terrible, head split open, arms and legs all twisted up. Gave me nightmares for weeks. Couldn't drive in the car for a while without getting these little spells where my heart raced and my breathing got all crazy."
"Bobby wasn't anything like that," said the boy. "You could hardly tell he was hurt. And I ain't had no nightmares."
"Well, just be careful," John said. "When it comes, it comes."
The boy nodded absentmindedly. His eyes were on the back wall. He walked over to one of the mounted deer. Its eyes were wide and glassy and the nose shone like it was still wet. He reached up and ran a hand down its thick neck, which felt stiff and hollow, the fur dry and brittle. He remembered the day his uncle had shot the buck. The boy had arrived just after John finished gutting it. The hind legs were splayed and steam rose from the empty carcass. Its eyes had already clouded over and the tongue that stuck out from between its clenched teeth was caked with dirt.
John cut the heart and the liver from the gut pile and he cut two feet of a branch with a wide fork at one end. He skewered the organs and slid them down until they came to rest on the _Y_ where the branch split in two. He gave the stick a quick shake and when the makeshift handle proved suitable, he laid it to rest inside the deer and grabbed fistfuls of downed leaves to wipe the blood from his hands. Together, they noosed a rope around the animal's neck and front legs and dragged it back to John's house, where they hung it in a small shed. They went inside and John gave the boy his first beer. They said, "Cheers," and they drank.
**5**
**T** he boy stepped down the three rubber-clad stairs and stood for a moment, looking out at the crowd, before he took the last step from the bus to the pavement. The driveway ran in a crescent in front of the school. Hordes of kids poured from their buses and crowds waited on the walk for friends to arrive. The red-brick bell tower stood high over them, the clock on its face off by hours. The boy saw a few people he recognized, but for the most part he stood surrounded by strangers.
He tried to stop and survey his surroundings, but the crowd behind him surged and forced him into the mob on the sidewalk. He bumped and squirmed through the maze of bodies and book bags. He saw Karen Hatch. She smiled, waved, and took quick little steps toward him.
"Hey, Teddy," she said. She leaned in and hugged him quick.
"Hey." He smiled. They weren't particularly close but he had known her since elementary school.
"There's so many new people," she said. "It's so exciting."
He nodded.
She eyed everything around her except him.
"You seen Terry Duvall?"
She shook her head and he nodded.
"Well," she said. "See you."
"Yeah," he said.
"Good luck," she told him. She smiled and waved again before she walked away.
When he stumbled into some free space, he pulled a folded sheet of paper from his pocket. It was a map of the school he'd received in the mail several weeks back. The silhouette of each building was sketched on the paper with the names written inside. A star marked the location of his homeroom. He oriented himself, quickly stuffed the map back into his pocket, and set off for the star.
He rounded a corner and saw Darren Bell coming toward him. He smiled and tried to catch his eye, but Darren quickly looked at the pavement and passed without speaking.
The boy wondered how much they knew. He wondered if Karen's "good luck" was intended for the day of school or the investigation. He wondered if that was why Darren ignored him. Since he was a minor, his name hadn't been released to the public, but that didn't mean much in their small town.
Two students sat in his homeroom when he arrived. He looked at the clock on the wall and saw that he still had another fifteen minutes before he had to be there. There was a 5-×-7 card on each desk. He found his name, sat, and pulled out his schedule and his map and plotted his course through the day. Then he rested his head on his folded arms and closed his eyes.
Before long the bell rang and students came pouring in. He recognized a few and nodded quickly at them. A huge kid with a shaved head took the seat behind him. He seemed older than the rest of the freshmen. After a few minutes he tapped a hard finger on the boy's shoulder. The boy turned. Without saying a word, the kid took hold of each side of his bottom lip and turned it out for the boy to see. On the inside of his lip, dark capital letters spelled out SKINS.
"It's killing me," he said once he had let his lip go. "Just got it done."
The boy nodded. Skinheads. He'd heard of them.
When the kid didn't say anything else, the boy turned around and put his head back on the desk. He tried to close his eyes, but they kept springing open.
He made his way to and from his first four classes without trouble, but a bottleneck in the hallway and the resulting crowd kept him from the lunchroom. The doorway had to accommodate both the inbound and the outbound traffic. He tried to edge his way into the single file that passed through the door, but no one would let him in. Just as a girl stopped and let the line move forward ahead of her, making room for him, he saw Kevin Dennison coming from the opposite direction. The boy froze and looked at the floor.
"You going?" said the girl who'd made room for him.
He shook his head.
"Fine," she snapped.
He turned and walked into the bathroom behind him. He went across the tile floor to one of the stalls and swung the flimsy door. It didn't quite fit in the frame, so he shouldered it shut. He sat on the toilet without pulling his pants down. The sounds of other students in the bathroom echoed around the walls of the stall.
Kevin Dennison had looked different. The boy couldn't place it exactly, but it was there. Maybe it was his face. His expression was stern, stoic. Perhaps it was the black T-shirt. Kevin was a prep in junior high and his outfits had included colorful collared tops and khakis.
The boy heard footsteps approach. Someone took the toilet next to him. He heard the flick of a lighter and soon he smelled smoke. He leaned forward and saw a familiar pair of work boots.
"Terry?" he said.
"Ted? The hell you doing in here?"
"Nothing," said the boy.
"You were pulling it, weren't you?"
He looked up and saw Terry, standing on the toilet, resting his elbows on the wall of the stall.
The boy shook his head. "You got an extra one of those?"
"An after-sex cigarette?"
"Come on." The boy waved his hand for one.
"What are you doing in here?" Terry reached down behind the wall and came up with his pack of Marlboros. He flicked the top open with his thumb and held it out to the boy. "You ain't taking a shit with your pants like that. Least I hope."
"Just taking a breather," the boy said.
"That's a good one," Terry said. "I'm going to steal that. Taking a breather." He inhaled hard on his cigarette and held the smoke in his lungs. "Take another one for later." Smoke poured out of his mouth as he spoke.
The boy stood and leaned against the wall opposite Terry. He put one cigarette behind his ear and held one between his lips.
"Suppose you need a light too." Terry reached down behind the wall again and his shoulder bobbed as he rummaged in his pocket. He leaned over the wall and lit the boy's cigarette. "Need one?" Terry held out the lighter to him. "I got another in my bag."
The boy nodded and took it. He focused on inhaling the way Terry had taught him. The smoke burned all the way down. He felt like coughing, but he held it back. The graffiti on the walls around him began to wobble and go crooked. He felt dizzy, so he sat back down. He liked the feeling. For a moment the two smoked in silence.
"So what happened?" Terry asked him.
"I can't talk about it."
"Come on."
The boy shook his head.
"That trooper came by my house," Terry said. "My parents loved that. I been meaning to say thanks."
The boy took another cautious drag.
"You told him we were on Darling's place."
"I had to."
"I sounded like a jackass, making some other story up," Terry said. "Then he told me you already told him."
"I had to say it."
"How long you figure until they know it was us setting the street on fire?"
"I don't think they care much about that."
"We'll see," Terry said. He stepped down from the toilet, out of the boy's sight. "I guess I should've known."
"What?" said the boy.
"You know. You can't ever keep your mouth shut."
"You don't know what you're talking about."
"Well," Terry said, "I know what I know."
The boy heard the quick hiss of Terry's cigarette hitting the water in the toilet. He heard the door of his stall whine open and bang shut.
"See you," he said.
"Yeah," Terry called back, the heels of his boots pounding the tile.
He stood and tossed what was left of his cigarette into the toilet. He flushed it with his foot and watched the butt circle and circle and eventually spin down the drain. He slid the second cigarette and the lighter into a small pocket of his book bag.
The crowd in the doorway to the lunchroom was gone when he got there. He saw Terry's orange hair above a table on the far side of the room. Terry sat with some older guys—the boy figured they were friends of Terry's brothers. He scanned the rest of the room and finally took a corner seat at a half-empty table. The nausea from the cigarette had stifled his appetite. He ate what he could of his lunch and then put his head on his folded arms and waited for the bell to ring.
His last class was in one of the portable buildings. They were long and narrow and looked like extended trailer homes. They sat between the tall bell tower and the four-story rectangle of the vocational building. In the late seventies the school had hauled the portables in as temporary classrooms for a sudden boom in the student body. More than a decade later they still housed history and social studies. He walked half the length of the hallway in the second portable to his American government class. The floor had a hollow, bouncy feel. When the door of the classroom closed, the whole wall seemed to shudder.
He knew his teacher from town. Mrs. Kimball had a son a year ahead of him and a daughter the year behind. She stood at the front of the class straightening papers on her desk. She put a folder in the drawer and called attendance. When his name came, he slowly raised his hand. He knew that every time he did so, he gave someone else the chance to put the name to his face, the chance to put his face in the stories they had heard.
Mrs. Kimball passed out her syllabus and gave an overview of what the class would entail. Students raised their hands and asked questions and she answered them. When the questions ceased, she began to outline their first assignment.
"Think about that old cliché assignment What I Did on My Summer Vacation," she said. "It's a little bit like that." She smiled and a few students laughed. "But I want it to be an analysis of something political that occurred over the summer. I want you to pursue your own interests. Doug, for instance, since he is interested in business, might want to focus on a specific interaction between government and the business world that occurred this summer. Some of you might be able to use some personal experience. Someone in this class might even be able to shed some light for us on the criminal-justice system." She looked directly at the boy when she said it, and he looked at his desk.
"Yes?" Mrs. Kimball said, acknowledging a hand somewhere behind him.
"Can I write about the gun debate?" a boy asked.
"In what way?" she asked him.
"Well, to start with, the ridiculous concept of blaming an inanimate object for our country's woes, and how that figures into the larger theme of liberal America's inability to accept responsibility for their own actions?"
Mrs. Kimball stood silently for a moment. "Do you have a specific event in mind?" she said.
"Of course," he said.
"What might that be?"
"I think you know what I'm talking about."
"I'm not a mind reader, Mr....?"
"Jackson," he said. "Jeffrey. Or J.J. But most people call me Peckerhead."
The class giggled. The boy tried to look over his shoulder but he couldn't get a glimpse of him.
"I don't think I'll be calling you that, Mr. Jackson."
"I'm just saying if you want to, I don't mind."
"I don't want this assignment to simply be a platform for your own political views, Mr. Jackson," Mrs. Kimball said. "It's an analysis of a specific event, with as much objectivity as possible."
"I'm sorry if I angered you," he said. "I meant no disrespect."
"You haven't angered me, Mr. Jackson."
"I come from a family where healthy debate is encouraged—that's all."
"I'm curious," Mrs. Kimball said. "Are your parents conservatives too?"
"I can only wish that was the case," Peckerhead said. "Bleeding hearts, the both of them."
The boy stood when the bell rang. He quickly packed his things and turned for the door. He looked at the ground. He feared that Mrs. Kimball would pull him aside and want to discuss what had happened with the Dennisons. He joined the herd at the doorway, waited his turn to leave, and hit a quick stride in the hallway. He didn't want to miss his bus.
"Theodore," someone called from behind him. It was the kid's voice from class.
He kept up his pace.
"Hey," the kid said. "Hey, man."
The boy looked when the kid grabbed his shoulder. Peckerhead Jackson was a skinny kid. Short brown hair, some acne. Black suspenders stood out from his white T-shirt. His khakis had long sharp creases.
"Don't call me that," the boy said. "It's Ted."
"Ted," he said. "Peckerhead." He held out his hand.
The boy shook it quickly. "Peckerhead?"
"Nobody forgets a name like Peckerhead." He smiled at the boy. "I didn't pick it."
The boy nodded.
"We had third-period bio together too. I sat three rows over."
"Yeah?"
"I was a little more reserved in bio. That was bullshit, what she pulled on you." Peckerhead pointed a thumb in the direction of the classroom that they had just left.
The boy kept walking.
"What are you doing after school?" Peckerhead said. "I got some friends. We all get together—you should come."
"So I can get a nickname like Peckerhead?"
"Funny. You're a ball-breaker—they'll like that."
"I got to run. I can't miss my bus." The boy quickened his pace, but Peckerhead kept up.
"We'll give you a ride. A few of the guys got cars."
"That's all right," the boy said.
"You like guns?" Peckerhead smiled. "We got a lot of guns."
The boy shook his head. He turned and jogged toward the drive where the buses idled.
"See you," Peckerhead called out.
The boy threw a hand up in a quick wave.
He had a dull evening of chores and homework and a quiet dinner with his mother. When he finished eating, he excused himself. He said he had homework and went to his room. He changed out of his school clothes, sat, and pulled a photo album out of his desk drawer. The leather cover was dry and cracked. The brown finish had worn off the edges and the corners were curled over. He liked to look at the pictures of his father in adolescence. He was pale, skinny, almost fragile-looking. On football teams where his father had insisted he was a standout, the boy saw that he was one of the smallest players. He always thought the realization should have been disheartening, but each time it actually let him breathe. The boy was a late bloomer. At fourteen, his armpits were bald and he still weighed less than most of the girls he wanted to date. The pictures made it easier. Sure he was skinny and hairless, but at least he was no worse than his father had been.
He turned the page to reveal four photos of several figures, heaped in winter clothes, standing on the vast, white expanse of a frozen lake. Their faces were so concealed by hoods and scarves that if it had not been for the father's stories, the boy would not have known who they were. The father and his father and brothers took to the ice when winter came, cut holes, and jigged for walleye and perch. But unlike the other fishermen—who drove trucks on the frozen lake and pulled shanties and sat around fires drinking schnapps—they stood some distance from one another and jigged away in silence. After they departed from the shore together, a toboggan loaded with gear in tow, the father's father would periodically auger a hole and leave a son, auger a hole and leave a son. They stood alone, weighed down in layer upon layer of wool, shifting from foot to foot, shivering. Rotating about the holes to keep their backs to the wind, they were sometimes envious of the houses on the shore, sometimes of the shanties, strewn like black pox on the ice.
When fish were caught, their skulls were dashed. Before they froze stiff, their bellies were slit and guts pulled free. The father spoke of reluctantly removing his mittens and with cold-blunted fingers opening the blade of his pocketknife. On good days, beside each hole grew a pile of fish and beside them grew a pile of their innards.
When the boy asked the father why they never heard from his brothers, the father told him that they weren't much different at home than they had been on the ice. Even in closer proximity, communication was functional and sparse at best. With the passing of their folks, there was little left to draw them together.
The boy clapped the album closed, put it away, and stood up from his desk. He opened the door and looked down the hallway. His mother was in her room, watching television. He closed the door, went to the window, and slowly eased it open. The screen proved a little difficult, but he got it off and out of the way. He rifled through his bag and came up with the cigarette and lighter. He killed the light in his room, stuffed pillows under his blankets and stepped out the window to the roof of the garage. The pitch was perfect to lie back on and look up at the sky. He thumbed the lighter, touched the flame to the cigarette, and drew in the smoke.
He inhaled slowly and methodically on the cigarette. He tried exhaling through his nose, but it burned much too badly. He thumbed the lighter again and stared into the flame. He started counting and got past fifty. His thumb began to burn and he stopped. He lifted his left arm and looked at his biceps. He pushed the searing hot metal into the soft pale flesh inside his arm. His whole body lurched and went tight at the feeling. He closed his eyes and white light exploded on the backs of his lids. It took everything he had to keep it there. Once the burning went dull, he pulled the lighter back and slid it into the top of his sock. Without the burning he felt slow and heavy. The backs of his eyes went back to black. He let his head fall on the stone-dust shingles and he took a long slow breath.
**6**
**P** eckerhead Jackson approached the boy every day after school. He even sought the boy out before and after their third-period biology class. The boy tried to act uninterested, but it was difficult, since Peckerhead was the only person who tried to talk to him in the course of the school day. After a week of walking the halls alone, he started looking forward to Peckerhead's company. On the Tuesday after Labor Day weekend, Peckerhead approached him.
"You been thinking about us," Peckerhead said.
The boy shrugged.
"I'm not going to keep following you 'round like some whipped dog," he said.
"I never asked you to."
"Can't you see I'm trying to do you a favor?"
"I never asked for nothing from you."
"Would you quit being so stubborn and come on?"
"Yeah?" the boy said.
"Hell, yes," Peckerhead said. "Quit being so pigheaded." He smiled and the boy smiled back at him. He followed Peckerhead out to the school parking lot, where a group of guys stood in a circle. Many of them had on the same outfit as Peckerhead: black suspenders, white T-shirts, pressed khakis. The group parted as Peckerhead and the boy approached. A handful of the closest boys stood staring.
"Guys, this is Ted LeClare. He's a freshman."
Most of the group smiled and a handful called out a greeting to him. The two closest to him held out a hand and he shook them. An older guy hopped off the trunk of a car and walked toward him. The crowd parted.
"Theodore LeClare, nice to finally meet you," he said. "I'm George Haney."
"Ted," the boy said. He reached out to shake George's hand. "Just Ted."
George smiled and they shook. The boy noticed that his hand was small for his age, his handshake a bit weak. George was also somewhat pear-shaped. He had narrow shoulders and wider hips. "How is school treating you?" George asked.
"Fine."
"Anyone giving you trouble?"
The boy shook his head. There was something to the way George spoke. He looked the boy in the eye and patiently waited at the end of each question. The boy felt as though his answer mattered.
"He likes guns," Peckerhead called out from the other side of the group. "I told him you'd show him."
"I didn't say that," the boy said.
"You don't have to be ashamed." George said it slowly and shook his head. His demeanor was calming to the boy. "We're all red-blooded Americans here. We like guns too." He smiled.
The boy smiled back.
"I have to work for a couple hours at my mother's shop," George said. "There won't be many customers. If you'd like to come along, we can chat and I can give you a ride home when I get off."
The boy shrugged.
"You'll be home for dinner," George said. "Come on." He was easy to listen to and the boy followed him through the crowd to his Ford sedan.
The store was small, just an addition off their single-story ranch house. The sign out front read VIRA'S PANTRY. GUNS, AMMO, KNIVES, HUNTING ACCES., HOME PROTECTION. He followed George across the dirt parking lot. It was uneven and rutted out. An electric chime rang as George opened the front door. His mother sat behind the register reading a magazine. Behind her stood a vertical row of at least thirty rifles and shotguns. Above them on the wall hung what looked to be an M16, an AK-47, and two other assault rifles the boy didn't recognize. In front of her was a long glass display case full of handguns and knives. When they got close, she looked down at her watch and shook her head. She was a big woman with short gray hair, but when she shook her head, the boy saw that it was long in the back.
"Would it kill you to get here on time?" she said.
"I have school, Mother," George said.
"Don't give me that."
"You can go now," he told her.
"I know what I can and can't do," she said. "Don't get lippy just 'cause one of your little peckerwood friends is here."
"I'm sorry, Mother."
"I told you not to give me that tone."
"Fine," he said. "Just leave."
The boy was startled by the jump in George's voice. He had walked down one of the aisles, toward the back of the store, to get away from the argument.
George's mother shook her head. The floor creaked as she walked across the room and he heard the door that led into their house open and shut.
"I apologize for that," George said.
"It's nothing."
"I find it very embarrassing."
"Don't," said the boy.
"She doesn't agree with my attempts to better myself. She says it's bad for business, if you can believe that," George said. "It kills her that we're going under. I'm sure you've noticed our lack of inventory."
The boy had noted the sparsely stocked shelves, but he shrugged as if he didn't.
"You're kind, but the truth is that we've lost our credit with most of our suppliers."
"My dad had to move to find work," the boy said. He walked over to the glass case and peered in at the selection of handguns. George went behind the counter and stood across from him. The boy was fascinated by the revolvers. He loved the exposure of their firing mechanisms. The newer semiautos tucked away most of their moving parts, but with the revolvers, it was all there to be seen.
George unlocked the case and reached inside. "Hold this one," he said. George pushed a pistol into the boy's hand. "That's German design, German manufacture."
"It's heavy," the boy said.
"That's partly the age of the piece, the lack of materials technology that we have today, but it's also due to the exceptional caliber of the round." George set a large bullet on the glass counter. "That, Ted, would do considerable—" He was interrupted by the electronic chime of the front door. Another teenager stepped into the shop. The first thing the boy noticed was his shaved head.
"What's up," the other boy said, throwing a casual, salute-like greeting at George.
"Jason," George said. "I was just showing this young man the Luger."
"He's a tease," Jason said to the boy. "He ain't let me shoot it yet."
"You can shoot it when you buy it," George told him.
"And just give me a minute to sell my car and whore-out my mother," Jason said. He laughed with George. The boy smiled.
"Jason Becker, Ted," George said.
Jason Becker reached out to shake his hand.
"Good to meet you, Ted," said Jason. "Sorry I can't hang—my mother's got me by the short ones. Tonight?"
"Of course," George told him.
"I'll be there," Jason said. "See you around."
"Sure," the boy said. After the door closed he asked George if Jason was a skinhead.
"No," said George. "He's Youth."
The boy shrugged.
"American Youth," George said. "It's a small group I coordinate. We get together and discuss politics, activism, that kind of thing. You should come."
"Maybe," the boy said.
"Definitely," George told him. "Tonight—think you can?"
"I can call my mom and ask."
"Good," George said. "The phone's back here behind the counter."
The boy was nervous. He sat in the center of a circular booth, surrounded on all sides by George's cohorts. They held meetings at the local Denny's, a twenty-four-hour chain diner. On his right sat Jason Becker, the largest and most intimidating of the bunch. If George was the leader, Jason seemed a second-in-command of sorts. He was the brawn to George's brains. To the right of Becker sat Peckerhead, who kept opening his mouth to speak but never said much more than a few words before he was interrupted. On the boy's left sat Dan; when George introduced him he made it clear that Dan was captain of the varsity wrestling team. Beyond Dan sat Birch, a tall, lanky guy with an enormous pompadour. George pulled a chair up to the open spot at the table and sat. Behind him a group of young cadets moved their chairs and crooked their necks to try to get closer to the action in the booth.
"What do you think?" Jason Becker asked the boy.
The boy shrugged.
"You know it's their welfare spending killing the economy, keeping your dad out of work," Becker said.
"My dad has a job," the boy said.
"Simply put, Ted, I'm an American," George interrupted from across the table. "I believe in freedom." He took a breath. "I personally don't think that people in New Hampshire should have to live like people in Rhode Island or Alabama. Or them like us, for that matter. I believe in America the republic, where the power is equally dispersed among the states." Taking a sip of coffee, George shushed someone else at the table with an open palm raised in the air. "But if you're a Federalist—they call themselves Democrats but they have nothing to do with democracy—if you're a Federalist, Ted, you think that the national government has the right to force the same laws over everybody, with no concern for local culture or community."
"Dude, it goes way back," Peckerhead said. "Alexander Hamilton and Thomas Jefferson were having the same argument."
"Civil War," Becker said, holding up a finger to make a point. "All that slave stuff was secondary to the real conflict. The North wanted to force a trade tariff on all the states. The South wanted the right to make laws on a state-by-state basis."
"These days all they preach is diversity, but it's just the new hegemony," George said. "If they're going to embrace the Hollywood whores and homosexuals, they also have to accept those of us who don't greet them with open arms." George's voice rose in energy. "They can have the whole of California, for all I care. Massachusetts, Connecticut—keep them." He paused and raised an index finger in the air above the table. "Just as long as we can keep what's ours." George brought a fist down on the table, rattling the plates and silverware there.
"Hell, yeah," Becker said. He reached up and gave George a high-five.
"I'm not here to tell anyone else how to go about their business," George said. "But I'll be damned if anyone comes into my home and tries to tell me how I got to live."
High-fives went around the table. They smiled and punched fists. The boy found himself consumed by the energy in the booth. Goose bumps grew on his forearms and when Dan held up his hand, the boy reached up and slapped it hard. Becker threw a jab at the boy's shoulder and slapped him hard on the back. The boy didn't understand it entirely—George had lost him somewhere around _hegemony—_ but whatever it was, it felt good.
The Youth wardrobe was influenced by both the professional golf tour and the local skinheads. Loafers and argyle socks were a mainstay and their khakis were hiked up to show them off. Narrow black suspenders stood out against their white T-shirts.
Much of the community adored the Youth. In school they acted as a sort of underground vigilante force. They ratted out drug dealers and finked on folks who boozed before the pep rallies. They went to church and took part in volunteer organizations to support their view that social welfare was the obligation of the community and not the federal government. They organized protests against anything that defied what they thought to be good, wholesome, and true. They organized protests against protests.
After they left Denny's, it was clear the Youth members had another destination in mind, but they made a detour and dropped the boy off at home before he could discover where they were headed.
"Think about it, Ted," George said.
The boy nodded. He stood in his driveway, outside George's car window.
"You're with us, man, I know you are," George told him. "But you have to make up your mind."
The boy nodded again.
"Have a good night, Ted," George said. There were other farewells from within the car.
"See you," the boy said. The car backed out and they were off.
The boy stood for a moment facing the empty road. He turned and looked at his house. His mother was home, but the only evidence of her presence was the flickering light of the television in the living room. He walked to the mouth of the driveway and looked again at the house. When he didn't see her, he walked across the yard and took hold of the real estate sign. He wrenched it out of the ground and jogged across the street. He took the sign by the legs, spun several times, and let it fly. It wobbled in the air like an unbalanced Frisbee, clanged into a good-size pine, and crashed into the underbrush. He loped back across the street and down his driveway.
Inside the house, he kicked off his shoes and walked into the living room. His mother sat on the couch with an afghan over her legs.
"Where you been?" she said.
"I told you," he said. "Out."
She looked at him.
"Denny's, with those guys, the ones I told you about."
"Tell me again."
"They're a group. Like an organization."
"They got a name?"
"American Youth," he said. "They're political."
"Political?" she said. "What the heck's a kid your age got to be political about?"
"Plenty, Ma. They're against drugs and alcohol. They're for American values."
"That's political, huh?"
"Why wouldn't it be?"
"Just seems strange for teenage boys to get excited about something like that."
"That's why it's political, Ma."
"Take it easy," she told him. "If it's going to keep you out of drugs and liquor, go ahead and be political then."
"They don't like the new people in the developments either."
"Why not?"
"Because they're trying to change everything here."
She nodded. "Not troublemakers or anything?"
He shook his head.
"You can't get in trouble," she said.
"I know."
"I'm not kidding, Theodore. You get in trouble again and it could ruin everything. You hear me?"
He nodded. "You heard anything?"
She shook her head. "Dick Duncan said if I called anymore he'd file harassment charges. You?" she said. "Anything going around school?"
"I'm sure they talk," he said. "They don't say it to me, though."
She nodded. Her eyes stayed on the television. "I left dinner on the stove."
He turned to walk into the kitchen.
"Hey," she called.
He stopped and turned.
"Would you quit throwing that sign?"
He looked at her.
"I'm not getting it this time." She still looked at the television.
"It doesn't matter whether it's there or not," he said. "People don't buy houses where that happened." He pointed in the direction of the dining room.
"Stop that," she told him. "Stop it now."
"Why?" he said. "It's the truth."
"Does it make you feel any better to think that the house will never sell and we'll be away from your father for good?"
He shook his head.
"So why think that way?" she asked him. "The truth doesn't matter."
He nodded and looked at the television. He turned and walked into the kitchen, stopping for a moment at the entrance of the dining room to look in. It was dark and he could just make out the outlines of the table and chairs, the china cabinet. His mother had thrown a decorative rug over the missing rectangle of carpet, but it was hard for him not to think about it. It was hard not to think about how different everything could be.
**7**
**F** ollowing another day of school, an afternoon at the Haneys' shop, and an evening spent at Denny's, the boy loaded up with the Youth and headed out again. He sat with two others in the back of George's Ford sedan. Peckerhead was in the front, next to George. Four other guys were packed into Jason Becker's Volkswagen Rabbit—the car was covered with such a skin of stickers and graffiti that at first glance it looked like something from the circus. But a closer look at the slogans made it clear that a crowd of clowns wasn't about to pile out: BURN MY FLAG AND I'LL BURN YOUR ASS; KILL A COMMIE FOR MOMMY; ABORTION = HOMICIDE; A FIRESTORM TO PURIFY.
George turned off the bypass and plunged into darkness, leaving the lights of the strip malls and gas stations behind them. It was a rainy night and the wipers cut back and forth across the windshield. The boy was excited to see that they weren't heading in the direction of his home. He wondered where they were off to. It surprised him when he looked over his shoulder and saw that Becker's Rabbit no longer followed them. George pushed a tape into the car stereo and some loud music spilled from the speakers. The boy tried to listen to the lyrics but the singer was indecipherable. Peckerhead pounded his fists on the dashboard of the car to the beat of the music.
George pulled into one of the new housing developments and the boys glared out the windows of the car.
"Here, here, here," Peckerhead said. "Stop!" he shouted. He pushed open the door of the car, jumped to the pavement, and ran toward the mouth of a driveway. Through the rain-speckled rear window of the car, the boy watched him squat and get hold of a good-size ceramic flowerpot. He raised it over his head and dashed it to the pavement. The two adjacent flowerpots suffered a similar fate. The boys in the backseat of the car cheered him on. George had killed the headlights and he scanned their surroundings for any potential witnesses. Finished with the flowerpots, Peckerhead got both hands on a small cedar and ripped it from the ground. He shook it victoriously over his head and tossed it to the far side of the street. Then he attempted to tear down the mailbox. When it wouldn't budge, he ran back to the car.
"Go! Come on, go!" he yelled as he jumped into the front seat. He was wet from being out in the rain and his breathing was heavy. The boy looked back again to see the broken pottery, the black soil spilled out over the flowers. The tree lay limp and alone where it had been thrown. It could probably be replanted, the boy thought.
"It's like this," George said after he'd turned the headlights back on and accelerated to a suitable speed. "Vandalism is a form of protest."
"Hell, yeah," Peckerhead said. "The German tribe shit, tell him the German tribe shit..."
"Peckerhead," George said, holding up an index finger. "I'm getting there." George turned onto another road of the development. "The Vandals tore up the Roman temples to protest the encroachment into their territory." George looked to see if he was satisfied and Peckerhead nodded. "These people are encroaching on us."
"They don't belong here," Peckerhead said.
George navigated the car through the maze of the development and the boy could tell that they were on the lookout for another display to assault.
"How do you know?" the boy said.
"What?" George said.
"How do you know?" the boy said again.
"How do we know what?" George said.
"That they're Federalists."
"Look," George said. "Do these houses look like our houses?"
The boy looked at George's reflection in the rearview mirror.
"Damn," Peckerhead suddenly burst out. "This is my favorite shirt." George hit the interior light and Peckerhead held the fabric up to show everyone that the rain had caused his black suspenders to bleed onto his white T-shirt. Earlier in the night the boy had noticed that the shirt had a simple depiction of Reagan's face on the front. The two other boys in the backseat now lifted their own suspenders to see if their shirts had suffered a similar fate. They'd been spared but the boy could tell that Peckerhead's experience had seriously stifled anyone's interest in further action. There were no more requests for George to stop, even though the boy saw several appealing targets: a mailbox in the shape of a swordfish, a water fountain that glowed the color of a healthy pumpkin, a false well house with bucket, rope, and hand crank.
A large illuminated sign stood at one of the exits of the development. It was wooden, the letters ornately carved and painted: WESTCHESTER ESTATES. The car stood at the intersection as the boys bent their necks down to see it atop a small knoll. Cars passed more frequently on the road the development spilled out onto.
"Westchester, my ass," Peckerhead hissed. He kicked open the door and jumped out of the car.
"Peckerhead," George yelled. He rolled down his window. "Peckerhead," he said again. "The goddamn road."
Undaunted, Peckerhead marched up the hill to the sign. George panicked a little and finally backed the car up, away from the main road, with the door still open. The boy reached forward and closed it, turning out the interior light. It was quiet in the car as they watched Peckerhead do everything he could to try to bring the sign down. He karate-kicked it twice, then got a running start and threw his shoulder into it. The sign held its ground and Peckerhead stumbled backward, rubbing his arm. He ran at it again and clambered awkwardly atop the sign. He shifted his weight back and forth in an attempt to loosen it from its footing. The sign didn't budge and Peckerhead eventually lost his balance and fell to the ground. The boys in the car flinched at his landing. Peckerhead scrambled back to his feet, picked up a rock, and smashed the lights that illuminated the wooden facade. When he got back to the road, he turned and hurled the rock at the sign, missing it by a foot or so.
George drove the car out onto the bypass and eased it up to the fifty-five-mile-per-hour speed limit. In response to the general feeling of defeat that lingered in the car, George began a lecture on the new local economy: "They want their Egg McMuffins and coffee on their commute—"
"What the hell is that?" Peckerhead cut in. He looked at George and then the boys in the backseat. He held out his hand and it shook in the air. George looked around the dashboard for some sort of sign. The shimmy quickly progressed, and soon the entire car shuddered.
"Pull over," the boy said, but George kept driving. The thumping began, getting louder, quickly louder, and the car shook more violently.
"I messed up my car," George said with a sad confusion in his voice.
"Pull over," the boy said again, and George finally let it coast into the breakdown lane.
Outside they stood in a semicircle, staring at the smoking rear tire. There was an awe among them, an awkward silence that confused the boy. They were frozen.
"We're sitting ducks," Peckerhead said.
"The cops can't prove anything," George told him.
"I'm more worried about the guy whose place I just tore up," Peckerhead said. "Getting vigilante on us." He rubbed the top of his head and looked back and forth in either direction on the dark road.
It suddenly became clear to the boy what the problem was: These guys didn't know how to change a tire. "Pop the trunk," he said.
Even though he didn't destroy a single Federalist lawn ornament, the boy still ended up the hero of the night. There were pats on his back and high-fives, and when they met up with the others back at Denny's, they all began calling him Teddy the Mechanic. By the end of the night it had been shortened to Teddy the Wrench, and the boy liked this very much.
He liked it all very much. He liked the handshakes, the smiles, the older guys telling him he'd do just fine with them. He started to see that everything that was good in the world was a result of honest American values. Anything bad was a result of a departure from those core principles.
Those kids in seventh grade, the ones that made fun of him because his sneakers only _looked_ like Nikes: fucking Federalist assholes.
George didn't want to drive far on the spare tire, so Jason agreed to take the boy home. Some angry music spilled from the tape deck and the two spoke little. The boy was surprised when the car slowed and Becker turned into the Sandy Creek development. For a moment he thought Becker had taken a wrong turn. He was about to say something when Becker looked over and said, "This is for you, Teddy." He throttled the car. "Hold on," he said.
The car quickly accelerated and the boy reached up and grabbed the handle above the door. With his left hand he reached down and got a good grip on the bottom of the seat. The Volkswagen barreled onto the fresh-cut lawn, mowed with the precision of a fresh flattop. Becker spun the wheel sharp and wrenched up on the emergency brake. The car went into a wild power slide and the tail end came almost all the way around. When the car came to a rest, Becker threw it into reverse, floored it, and cut the wheel. As they spun, the boy saw the lights of the house come and go, come and go from his field of vision. The car was filled with the noise of the racing engine and the clatter of soil and stone pummeling the wheel wells. They came to a stop and the boy heard the music again. Becker shifted and drove quickly back to the road. He looked over to the boy.
"We haven't forgotten," he said.
"What?"
"That they tried to blame you for what happened."
"How do you know where they live?"
"Mrs. Dennison?" Becker chuckled. "Man, she's high on our list. We've been sending her anonymous hate mail since she made her debut on the town council."
"Why?" the boy said.
"Don't give me that naïve crap," Becker said. "This isn't some little game. This is our town. This is everything."
The boy sat on the edge of the bed with his head in his hands. He didn't feel good about the Dennisons' lawn. He didn't feel good about the hate mail. He still didn't feel good about Bobby, not even okay. He saw Kevin around school and he could see that he didn't feel so good about it either. Kevin wore high combat boots and his hair was shorn unevenly. He wore a lot of black. He had the look of a boy with troubles.
The boy ran the fingers of his right hand under the sleeve of his T-shirt, along the inside of his left arm. A fingernail caught on the edge of scab and he picked at it. The most recent burn was swollen and tender to the touch—his fingers jumped away from the blistered flesh. Each time, he swore it was the last. For a spell of days and nights he could keep away from it, but when his mind got to racing, he always justified another.
He couldn't say why he did it, or why it felt so good. He just knew that when he did it, he became the burning, he became the pain, and when he was the pain, he didn't have to be anything else.
**8**
**A** fter school the next day, a small entourage of Youth members took the boy on a trip to the mall. After leaving one of the larger department stores, Becker motioned for the others to follow him. Once in the bathroom, Becker pulled a pair of black suspenders out of his pants. He saw the look of surprise on the boy and said, "Corporate chains rob us blind." The boy put on the suspenders. Before they left the bathroom, Becker said, "Cuff up your pants." In doing so the boy further exposed the old pair of loafers Becker had traded him for a Wrist-Rocket slingshot. Birch leaned back and kicked open the bathroom door. It resounded with a boom and the boom's echo, but the boy never flinched. The four of them hit their stride on the concourse. The boy's skin rose in goose bumps and static tingled in his scalp.
After the mall, Jason and the boy went to the gun shop to wait for George to get off. They sat around reading the magazines that collected in the shop. George's girlfriend walked through the door, setting off the electronic chime. The boy looked up from the most recent copy of _Soldier of Fortune._ He thought she was cute—Colleen Crenshaw—with her sad eyes and the nose ring that George hated. Colleen and George smiled and kissed over the counter.
"Mom's going to let me off early," George said.
"What do you want to do?" She bent over and rested her head in her hands. Her elbows sat atop the glass gun case. She wore a short skirt and the boy tried desperately not to stare. They were both in ninth grade, but Colleen was a year and a half older than he was. She had a driver's license and a blue Ford Escort.
George shrugged. "You?" he said.
"Let's go shopping."
"Yeah?"
She nodded.
When George's mother arrived, the four climbed into Becker's car. George insisted on riding in the front seat, leaving Colleen and the boy cramped together in the back. Colleen leaned in to speak to him, so close as to send soft washes of breath against his ear. This perpetuated flurries of what felt like electricity to the ends of his limbs. When the car turned, she reached down and took hold of the boy's thigh to steady herself. When they stopped to stroll through music shops and pawn shops and used-clothing stores, the boy couldn't wait to return to the cramped backseat of the Volkswagen.
They stopped at a barn that had been converted into a large indoor flea market. On summer weekends, stands and tables covered much of the old pasture, but during the week they vanished and the market retreated inside the musty barn. Jason said it was a great place to find switchblades and billy clubs. The barn was mostly filled with old odds and ends of furniture, but there were the occasional glass cases that held strange collections of coins, knives, thimbles, and sports cards. George and Colleen had taken off on their own as soon as they arrived, leaving Becker and the boy to stroll by themselves.
"I always wanted one of those," Becker said.
The boy stood at his side. "The racetrack?"
"Box looks brand new," Jason said.
The boy nodded.
"Ten bucks," Jason said as he fingered the stringed price tag. "Way too much." He walked around a corner. "Hey," he called out. "Come here a minute."
An old man scuffled over in a pair of slippers. He looked at Jason.
"What's the best you can do on this?" Jason said.
"What's it say?"
"Ten bucks."
"Well," the old man said. "You can read."
"Ten?" said Jason. "Come on. I got five." He held out a wad of bills.
"Get another five off your buddy and you got a deal." The old man smiled.
"Probably doesn't even work," Jason said.
"I took a couple laps on her myself. Just to be sure."
"I need a birthday present for my little brother," Jason said.
"Seven," the old man told him.
"Six," Jason said.
"Six?"
"Six," said Jason.
The old man finally nodded and the two carried out the transaction. The old man smiled, said thanks, and shuffled back to his post. Jason took up the package. "I'll take this out to the car," he told the boy. "You go and get them two."
The boy nodded and proceeded to wander about the barn. When he didn't find them on the first floor, he headed up into the loft. He was about halfway up the stairs when his head cleared the second floor. In a gap between two pieces of furniture, he caught sight of George and Colleen. He had moved past the opening and he had to lean back to see them again. Colleen sat high on a bureau and George stood between her legs. Her arms were wrapped around his neck and their heads moved back and forth as they kissed. The boy's breathing quickened. He knew he should call out or head to the car and wait, but he didn't move. Colleen took an arm from around George's neck and reached down to take hold of his wrist. She pulled his hand up to her breast and moved it in slow circles. George suddenly pulled his hand away and let it fall back to its place on the bureau. She tried to take hold of his wrist again but George held firm and then withdrew from their embrace.
"How many times have I told you?" he said.
She smiled at him.
"It's not funny," he said in a harsh whisper.
"I don't think feeling me up is going to get you sent to hell."
"One thing leads to another. Then what?"
"I don't know," she said slowly. "Then what?"
"Stop it," he told her.
She pulled him close and began whispering in his ear. George pushed her arms off of him and quickly stepped back. "Goddammit," he said. "Why do you have to be such a slut?" He turned and walked away from her. When he rounded the corner to the stairwell, he came face-to-face with the boy. Neither spoke, they stared, wide-eyed, at each other.
"I was just coming for you," the boy said finally. "Jason sent me. He's out at the car."
George rumbled past him on the stairs. "Let's get out of here," he said. Colleen came around the corner of the stairs and caught the boy's eye. He looked up and she wowed her eyes at him. She mouthed the word _Oops_ and put a hand on his shoulder as she passed him.
In the backseat of the car the boy slouched under the box of the electric racetrack. It protruded from the hatchback, just over his head, and cramped him further in the already small space.
"You're going to get a crick in your neck," Colleen said. She smiled. She reached up and massaged his shoulder, working her small fingers up to his neck and then to the back of his head. He held his breath. He looked up to the front of the car. Jason and George were having a loud conversation over the loud music from the stereo. "Don't worry," she told him, her hand still working on him. "They're busy."
He quickly looked away from her, out the window.
That evening, Jason Becker's mother went out on a date and a significant portion of the Youth congregated at his apartment. The boys crowded around the television in the living room, where they watched a videotape of a band whose political platform cohered with their own. The bald, bare-chested singer screamed out precisely what was wrong with the world. Heads rocked to the steely and static-ridden music that squawked from the television speaker. There were shots of the sweaty and seething crowd—fists shot into the air and the mob surged here and there as a whole, like a collection of small fish or birds.
"All right, all right," the singer said between songs. "Take care of each other out there. You might've heard this next one...'Take It Back.'"
George and Colleen sat at the back of the crowd, in the kitchen, the boy across the table from them. The racetrack wound in a figure eight on the floor and electric cars sped around under the command of two triggered controls in Jason Becker's hands.
"He's so cute," Colleen said quietly. She pointed at Jason.
The boy looked and smiled.
"Want to go for a walk?" she asked George.
George shook his head. His eyes were on the television across the room.
"I knew you wouldn't," she said. "Teddy?" She looked at him.
He shook his head.
"Come on," she said, standing. "Take me for a walk." She came around the table and took his hand.
He looked at George.
George made a motion with his head and the boy got up. He felt awkward walking through the crowd with his hand in Colleen's, so he pulled it free. He followed her outside and around the corner of the apartment complex. She rifled through her purse and came out with a cigarette and a lighter.
"What are you doing?" he said, louder than he had expected to.
"Want one?" she said.
"You're going to get me killed."
"I have gum."
He stayed silent for a moment, looking at her. Colleen's face didn't change.
"No thanks," he said. He walked in small circles while she smoked. He could see the lights of their high school in the distance, but he didn't want to think about school. Without his father around, he had little energy for his studies. He knew his grades were lagging already this quarter and he knew he would hear about it from his father, but he also knew he wouldn't have to hear about it for long. His father would be home to bark at him for two or three days and then be gone again for two or three weeks.
"What was it like?" Colleen suddenly said. She took a drag off her cigarette and exhaled. "When that boy died?"
The boy balked and for a moment he was speechless. "I'm not supposed to talk about it," he finally said.
"How come?" she said.
"There's an investigation."
"Could you go to jail?"
He shrugged. "I don't know," he told her. "How do you know about it?"
She chuckled. "Everyone knows about it, Ted. Why do you think they're so interested in you? You're like their cause."
The boy stared out into the darkness. He was about to ask for a cigarette when Youth members began streaming out the door behind them. Colleen dropped her cigarette, stepped on it, and exhaled before turning around. George approached them.
"You need to go home," he told Colleen. "Come on, Ted."
"What?" she said. "No way."
"You smell that, Teddy?" George said. "That smell like cigarettes to you?"
"Fine," she said. She walked over and hastily kissed him on the cheek.
"Good night," George told her.
At the parking lot, the Youth divided among several cars. The boy found himself in George's mother's sedan.
"I don't think it's cool," Peckerhead said.
"This is not about thinking," George replied. "This is principles."
"His parents," Peckerhead said. "They just divorced, for crap sake."
George didn't respond.
"I'm not doing this," Peckerhead said. "Screw this."
"You don't have to," George said. "I just wish you wouldn't make up excuses."
"What does that mean?" Peckerhead said.
"You don't have the stomach for this kind of thing," George said. "Fine, but don't make excuses for Dan when they're really for you."
"Fuck you," Peckerhead said.
"Fuck me?" George said. He paused for a moment. "I expect better from you, J.J."
No one spoke for the rest of the ride. They stopped behind a supermarket not far from their high school. No one else was there. They got out and the boy was glad when George pulled him aside and whispered, "Stay back—let the veterans take care of this."
Becker's car arrived shortly. He and Birch climbed out of the front seat and several of the younger guys weaseled out from the backseat. Another car rolled in with another couple of guys. The group mingled about until they had effectively isolated Dan.
In a very formal tone, George said, "You've been untrue to us, Daniel." He paused. "You were drinking at a jock party last weekend."
"No way," Dan said.
"Are you calling me a liar?"
"Not you," Dan said. "Whoever said it."
"I'm saying it," George said.
Dan didn't reply.
"If you come clean, we can begin to forgive you," George said. "But this, this gets us nowhere."
"Fuck this," Dan said.
There was a moment of silence.
George walked back to the crowd.
Two of the younger guys stepped forward and tried to get ahold of Dan's arms, but he shucked them both. They tried again, and again he shucked them, shoving one in the chest and one with a palm to the face. The two stepped back and Becker moved in. Becker swung and Dan ducked, lurched forward, and got ahold of Becker's upper body. He heaved Becker off the ground and toppled him to the pavement. With Becker lying on his back, Dan could've easily kicked him, but he didn't—he kept moving, not allowing the group to corner him against the building.
Birch stormed in and dived at Dan's hips, but Dan sprawled his legs back and forced Birch's face to the pavement. Dan quickly jumped off him and Birch came up holding his cheek, his pompadour in a mess.
"What the heck?" George said.
"He's a goddamn wrestler," Becker said. His palms were scraped and he was trying to brush the sand from the cuts.
"All go at once," George blurted.
When five of them moved in together, Dan turned and sprinted off. No one followed. The group stood around, speechless in their defeat. A few rubbed at sore spots.
They spent the rest of the night driving around the nearby roads, but Dan was not to be found. At a loss for what to do next, George sent a contingent to Dan's house. They broke down his mailbox and someone pissed on the door handle of his parents' car.
**9**
**T** he first time she called, she asked the boy if he had seen George. When he said no, she said okay and hung up. The second time Colleen called, she said, "You seen George?"
"No," he said. "I think he's at the shop."
"I thought so."
"Why don't you call?"
"I don't want to talk to him," she said. "What are you doing?"
"Homework. Algebra."
"I hate math," she said.
"What are you doing?" he asked.
"Trying not to smoke cigarettes."
"You trying to quit?"
"Don't you ever just want to say, 'Screw it,' and get wicked fucked up?"
"I don't know. Sometimes."
"We should do it," she told him. "Just me and you, sneak off and get totally cocked."
"Yeah. Maybe."
"You're afraid of them."
"The guys?" he said.
"You shouldn't be," she said. "I bet you could take any of them."
"One's not the problem."
"You shouldn't let them tell you what to do."
"You do."
"I do it so I can be with George."
"I do it so I can hang out with them."
"Why?" she said.
"The politics—I don't know."
"I hate that crap the worst," Colleen said. "They just sit and talk. It's so boring."
He laughed.
"What?" she said.
"I'm surprised to hear you say that."
"I don't believe in any of their crap," she said.
"Then why are you with George?"
"I don't know. The way he talks. He's smart. He really listens. Sometimes."
He nodded. "I know what you mean."
"Listen, I have to go because my mom needs the phone, but you should call me," she said. "We should do what we talked about."
He hesitated and she hung up before he could answer.
The third time she called, sometime later in the week, she asked if he'd seen George, he said no, and she asked if she could come over to his house.
"Here?" he said.
"Yeah, I'm bored."
He'd never invited a girl over and he didn't know how his mother would react to this. He certainly didn't know how he could entertain Colleen because what he did know for sure was that his mother would insist upon the two remaining in some supervised space such as the living room or kitchen.
"My mother's home," he said.
"We don't have to stay."
Her answer relieved him.
"We could go to the mall," she said. "Let's go there."
He agreed, as this solution settled the problem of the mother. But as he hung up the phone he thought of the altogether separate and still unsolved dilemma of George Haney. It didn't necessarily rest upon him yet, though, for surely if this was merely two friends going to the mall, there was no dilemma.
During the drive, Colleen continued her habit of touching the boy. Her hand reached for him when not busy shifting the manual transmission. She stalled the car at a traffic light just before the mall entrance. "Fucking piece of shit," she said. Her timing of clutch, stick, and accelerator had yet to be perfected, and often the car bucked in between gears. After restarting the car, she revved the engine and popped the clutch, making the tires spin weakly and scratch at the dry pavement.
Colleen drove around the outskirts of the mall and parked in the back, outside one of the smaller department stores. She looked at the boy, leaned over, and kissed him. It caught him off guard, but the quick kiss didn't require much—he'd managed to slightly purse his lips before she got to him. She leaned back to her side of the car, smiled, and got out. The dilemma of George Haney had arrived. Inside the mall there were other brief pecks here and there—slight taps of their lips together. Since it was a weekday evening, the mall was nearly empty.
Colleen reached down and squeezed his hand. He squeezed back, but he didn't look at her. She pulled him through a set of doors, into a back hallway that led to some restrooms. She leaned against a wall and pulled him close. She closed her eyes and cocked her head to the side. He leaned in and pressed his lips to hers. Her mouth opened and he felt her tongue slide between his lips. He opened his mouth and cautiously let his tongue touch hers. She opened and closed her mouth ever so slightly and he did his best to follow her lead. Just when he thought he was getting the hang of it, Colleen opened her eyes and pushed him back.
"You have to close your eyes," she said.
"Okay." He leaned in toward her, but her hand kept him at bay.
"I'm not kidding," she said. "It's important."
"All right—I'll do it."
They closed their eyes and kissed again. He pressed his body tightly to hers and Colleen did not shy from this friction. With her hands on the small of his back she pulled him tighter. He ran his hands over her breasts, but when he tried to slip his hand under her shirt, she stopped him.
"Nope," she said. "Sorry. Not here."
"Someplace else?" he said.
She smiled and shook her head. "You boys are all the same," she said, straightening her shirt. "Come on."
The comment confused him. Wasn't she the one who pulled him back there? And just how many boys was she basing this on? He knew he wasn't the first—with George and all—or even the second or third. He had heard what the guys said, that she and George often fought over the fact that she had slept with two guys from the punk-rock crowd before they got together.
As they strolled, he began to distance himself from her. Casually at first—he pulled his hand away from hers to yawn or stretch—then he moved out of range of her outstretched arm to look at something in a storefront window. Soon he ran out of polite maneuvers and his remove became obvious. Colleen didn't wait to respond. She stopped and grabbed his shirt.
"What's wrong?" She looked up at him with squinting, suspicious eyes. He shrugged at first and tried to turn away but Colleen held on to the handful of his shirt and turned him to face her.
"I don't know," he said. This was followed by another jolt to his shirt. "All right," he said, pulling her hand from its grip on him. "What about George?"
"What about him?"
"Yeah," he said. "What about him."
"I was with him—now I'm with you," she said.
The simplicity of it puzzled him. He doubted it could work out so neatly.
"They're going to kill me," he told her.
"It's not like you stole me," she said. "Me and George broke up two days ago."
This news was somewhat settling to the boy and momentarily it did seem as simple as she claimed: She had been with George, and now she was with him.
On the drive home, Colleen insisted on holding his hand, and when that became sweaty she placed her hand high on his thigh.
"Where do we go now?" she said. They were stopped at the traffic light at the end of the exit ramp into town.
"I live that way," he said, pointing east on the bypass.
"We're not done yet." Colleen smiled. She patted his thigh. "Where can we go?"
The only place he could think of was Woodbury Heights.
When the Escort pulled into the culvert at the end of the road, it parked amid charring and scars in the road. Black streaks of squealed tires and unskilled graffiti marred the pavement.
Colleen and the boy kissed and fondled each other for some time. He reached around her waist to the lever that controlled her seat. He pulled at it and leaned into her so that she was nearly flat. Before he could climb atop her, she stopped him, hands up, on his chest.
"We're not going to have sex," she said.
"Okay." He stopped and pulled away.
"Is that why you're here?" She paused. "Is it?" It was as if her affection had fled and turned to disgust.
"No," he said. He said nothing else, but seeing Colleen's unchanging face, he realized that nothing else would not suffice. "I care about you." He said it because it was the only thing he could think to say—the only thing he could think to say to make her like him again.
"Good," she said, suddenly smiling. She pulled him down to her so they could continue kissing. He felt a tinge of something, perhaps guilt—shame, maybe—but that feeling was lost when for the second time he slid his hand under her shirt and under her bra and this time she did not stop him. She let him undo the front of her pants and she let him touch her there. He shivered. He tried to ease her pants past her hips but she shook her head. She started to reach into his own trousers when a strange cramp gripped his abdomen. He winced and had to pull away.
"What?" she said.
"I don't know."
"What is it?"
"Nothing," he said. "My stomach's weird—I don't know."
"You're nervous," she told him.
He shook his head.
"It's cute," she said.
He bent back down to kiss her, but he heard something. Both of their heads jerked upright. A car was coming. He scrambled back toward his seat but he was only halfway there when the car passed. The engine raced and its tires squealed around the turn. He heard hollering over the rumble of the engine and recognized several of the local metalheads in the half-rusted muscle car. Terry Duvall sat in the front seat with his fist out the open window. Another hand protruded from the backseat, the middle finger raised at them. The boy ducked, fearing that if they saw him they might stop. When the car reached the far side of the culvert, he heard the engine thunder and the tires squeal. It was gone as fast as it had come.
"Jesus," Colleen said. Her head was crooked down, looking to button her pants. "Such assholes."
"They're all right," he said.
"They're idiots," she said.
He shook his head. "I grew up with them."
"If they're so great, why don't you hang out with them?"
The boy shrugged. He looked at her. One of her breasts, the one closer to him, was sandwiched between her raised shirt and her lowered bra. Once she finished with her pants, she dipped two fingers into her bra cup and pulled it out, allowing her breast to settle. Then she pulled her shirt down.
"Why not?" she said.
"I don't know," he said. "It's complicated."
Colleen sat and stared at him. "Tell me," she said.
After a moment, he started, "When I was getting questioned about the shooting, I told the cops some stuff that pissed Terry off."
"Did he get in trouble?"
He shook his head.
"So what's he mad about?"
"I don't know. It's complicated, I told you."
"It sounds stupid to me."
"Maybe I don't care what it sounds like to you."
"I didn't mean that you or him were stupid," she said. "I just meant the reason you guys are fighting seems stupid."
The boy sat silently, looking out the windshield of the car.
"What happened that day?" Colleen asked him.
"You know what happened. Kevin Dennison shot his brother by accident."
"I know I know that," she said. "But what happened?"
"I can't talk about it, I told you."
"Because of the investigation?"
He nodded.
"Will you tell me after it's over?"
"Why is it so important?"
Colleen shrugged. "It just seems important."
He shook his head. "Will you take me home?"
"Why?"
"I need to go," he said. "Take me home."
"Fine." Colleen started the car and the Escort bucked as she let out the clutch too quickly.
**10**
**T** he following day the boy walked cautiously to the group of Youth members congregated around Jason Becker's Volkswagen in the school parking lot. He shook hands with a few and exchanged high-fives with a few others. He didn't sense any apprehension in their gestures. He couldn't make out any fury in their faces. It seemed impossible for them to know of his drive with Colleen, but still he worried. Espionage was at the heart of their role in the school. They listened.
After making his way to the center of the group and jovially punching fists with Jason Becker, he felt sure his secret was safe.
"The Wrench," Becker said.
"Hey, Becker," the boy said, smiling.
"You a good little boy in school today?"
"Always," he said.
Becker nodded.
The boy heard some hurried whispering from behind him. Becker craned his neck to look over his shoulder, then eased the boy aside and stepped past. The boy walked to the edge of the crowd to see who was drumming up the attention and he saw Kevin Dennison skulking across the lot. His pants were cuffed up to show off his black leather boots. A multitude of patches and safety-pinned scraps of material broke up the black of his backpack. Day by day Kevin seemed to grow further away from the prep-school image he'd once had. The boy had passed Kevin one day at his locker and come close enough to read the patches on his backpack. The one that stuck with him said TOO DRUNK TO FUCK.
Before getting to his car, Kevin turned and looked at the mob of staring Youth. He squinted his eyes and bobbed a middle finger in the air between them.
"Fuck you, faggot!" someone in the group shouted.
"Come on over here, you pussy," someone else yelled.
"Hey, Dennison," Becker shouted. Kevin didn't look up from unlocking the door of his car. "Why don't you smoke a dick instead of all those joints?"
At that the group fell into laughter. One after another they reached high-fives out to Jason Becker. He smiled and clapped the hands above him in the air. Kevin got in his car and left without looking back.
"Think he knows?" Birch said.
"That burnout?" said Becker. "He doesn't know his head from his ass."
The excitement of the group waned and they began mulling about in smaller, separate conversations. The boy made his way back over to Becker's side.
"What's up?" the boy said.
"What's up with what, my man?"
"With Dennison." The boy pointed a thumb back to where Kevin had been parked.
"I thought George had filled you in." Becker put his arm around him and pulled him close. "You're especially going to love this. Thing is, we found out Mr. Dennison's been peddling a little grass at the Coffee House. And this is the sweet part: We know he keeps it on him, and that he keeps it all wrapped separately." He threw a jab at the boy and smiled.
The boy didn't smile back.
"That's 'intent to distribute,'" Jason said. "That's deep shit. We're going to keep an eye on the Coffee House. If he's there this weekend, George is going to call it in. It's going to be killer—we're going to videotape it from across the street."
"You going to the Haneys' shop?" the boy said.
"In a bit."
"Can I come?"
"Sure. What's up?"
"Nothing," the boy said. "I need to talk to George."
George stared at the boy from behind the counter and the boy stared back.
"This junkie tried to ruin your life and now you want to help him?" George said.
"You can't do it," said the boy.
"The fuck we can't," Becker said.
"He didn't try to ruin my life."
"I'd say shooting his brother in your home and trying to pin it on you is pretty close," George said.
"It was an accident."
"I don't give a damn," Becker said. "The kid's a drug dealer. Period."
"I'll talk to him," the boy said.
"The hell you will." Becker leaned in close to the boy.
"Easy, Jason," George said. "Back off."
The boy looked at Becker. "Yeah," he said.
"What?" Becker said. "Who is this tough guy?"
"Jason," George snapped. He pointed at the door. "Take a walk."
Becker stared at him.
"I can't have a conversation with you like this." George pointed again at the door. Jason glared at the boy but did as he was told. The door boomed behind him as he barged out.
"There," George said.
"Yeah," the boy said. "There."
"Ted, we're doing this for you."
The boy shook his head. "I don't want you to."
"I know," George said. "But I don't get it. This could turn the investigation in your favor."
"His brother died."
"So we should pardon every criminal with a good excuse?" George said. "Every thief who grew up poor? Every prostitute who was abused?"
"You guys break the law," the boy said. "Tearing up people's stuff."
"They deserve it."
"Just because you say so," said the boy. "Just because you make it up."
"I don't make anything up," George told him. "We're working for a greater good, for a greater law. We can't let sentimentality sway us, Ted. Mr. Dennison must be punished. Case closed."
"I don't think I can let you do it."
"Then I'm afraid that we'll have to deal with you too."
"Fine," the boy said. He turned to leave.
"Teddy."
The boy stopped.
"Please think about this. We make better friends than enemies."
"I know," the boy said. He paused. "I will." He walked out the door and across the lot to the side of the bypass. Jason Becker leaned against his car, leering at the boy. He didn't look at Jason. He walked down the side of the bypass toward home. He turned and stuck out his thumb each time a car passed. An old man in a Chevy pickup finally slowed and stopped. The boy jogged to catch up and hopped in the cab. They greeted each other but spoke little more than that.
The boy thought about Kevin. It made sense to let the Youth have their way. Kevin's arrest would reflect poorly upon him. Perhaps the investigators would look more favorably upon his own story after seeing what Kevin was capable of.
The old man stopped at the corner of his street. The boy thanked him, hopped out, and walked the rest of the way home. Shortly after he arrived, the phone rang. He hesitated to pick it up, but on the fourth ring he pulled the phone from its cradle.
"Ted," his Uncle John said. "How are you?"
The boy was relieved. "Fine. You?"
"Got any plans this evening?"
"Some homework."
"Got time to give me a hand with a deer? Skin and quarter it?" John said. "I'd let it hang but it's so warm I got to get it in the fridge."
"Season's open already?"
"Opened this week," John said.
"You want me to ride my bike over?"
"Nah," John said. "Get your father's good cutlery. I'll be there in ten minutes."
"Sure," the boy said. "See you."
He hung up the phone and ran upstairs to change. Then he went to the basement and rifled through a box of his father's hunting gear. He found the two knives he was looking for, pulled them from their sheaths, and ran a finger along their blades. He grabbed a stone from the box and small bottle of oil and went to the workbench. He laid out the stone and oiled it. He started with the skinning knife, slowly drawing the rounded blade along the stone, alternating from one side to the other. He touched the blade again and his thumb instinctively jumped back. Satisfied, he set to the longer, more slender blade of the boning knife.
He waited in the driveway for his uncle. Once at John's house, they walked around back to a small shed and John lifted a two-by-four from the yoke between the doors. One side swung open. A rope from a rafter was noosed around the deer's neck and held it upright in the shed.
"Nice deer," the boy said. "Six-point."
"Five," the uncle told him. "Other side's only got two."
"Healthy," the boy said. He ran a hand down the soft, shiny coat. The deer was thick in the midsection and good and wide across the hindquarters. "Where'd you get him?"
The uncle smiled and held a finger to his lips. He pointed a finger out in the direction behind the house. The Darling property extended from behind the boy's house all the way over the hill and abutted the uncle's land.
"Poacher," the boy said, smiling.
"Nobody knows any different," John said. He took a hacksaw from a nail on the side of the shed. "Want to get ahold of him for me?"
The boy nodded and grabbed the deer to keep it from swinging on the rope. John took a hoof and ran the saw back and forth through the leg at the knee. The sound wasn't pleasant, but the boy had grown accustomed to it. John threw each leg in a corner and hung the saw back on the wall. He took out his knife and ran it around the deer's neck, just behind the ears, just deep enough to cut the hide. Then he cut the hide down the throat and chest to the opening in the carcass at the bottom of the sternum. They each took a corner of the hide and began to pull it away from the body, carefully running a blade along the fissure between muscle and skin.
After they cut the skin away from the front legs and pulled it back behind the shoulders, they put their knives down. They each took ahold of the fur and put their weight into it. The hide pulled away from the carcass with a hiss. They skinned the hind legs and hacked off the bone that connected the tail. The deer hung naked, the muscle red and purple, the rump thick with white fat. John took the skin and hung it over the back of a sawhorse.
The boy took hold of the deer again as the uncle set to the tenderloins on the inside of the lower back. The muscle was still warm. Without the fur, the boy could see where the arrow had punctured the rib cage just behind the shoulder. It was a textbook shot, the broadhead probably puncturing the heart or lungs. Death came fast, as the blood quickly pooled inside.
The boy allowed his hand to slide down from the deer's shoulder. He ran his finger around the entrance wound. It was black at the edges, where the blood had clotted. He dipped the tip of his finger into the hole and felt the dried, hardened flesh, the sharp edge of bone. He closed his eyes. In the darkness he saw Bobby, noosed and naked, and the boy's finger dipped into the hole in Bobby's chest. His eyes popped open and his hands jumped back from the deer. The carcass swung away from his uncle.
"Hey," John said, looking around the deer.
"Sorry," he said. "I had an itch." He took hold of the deer again and John finished with the second tenderloin. He laid it on a small tray beside the first. He cut into the meat of the lower back, all the way to the spine on either side.
"You want to catch?" John said.
"Huh?"
"I'll cut, you catch?" John said. "You all right?"
The boy nodded.
"You sure?"
He nodded again and crouched next to the animal. He got a good grip on the hindquarters and John set to the spine with the hacksaw. The boy wobbled under the weight when the saw cleared the bone and John helped him carry it inside. They separated the hindquarters and set each side in the bottom of a refrigerator in the basement. They went to the shed and returned with the two front shoulders. They cut down the torso and cut out the chops, the neck, and the belly meat. They stacked the cuts in the refrigerator.
"That's all?" the boy said.
John nodded. "I want to age it some," he said. "I'll get to it this weekend."
He took the knives and the cutting board they had used to a large sink in the corner to begin the cleanup. The boy looked at the rib cage on the table. With the chops cut away, the spine ran like a tall, narrow mohawk down the center. He walked over and looked again at the entry wound. With the flesh cut away, it looked small and hardly seemed capable of bringing the animal down. The boy ran a finger back and forth over the opening. The heat had dissipated and it was cool to the touch.
"What the heck is wrong with you tonight?"
The boy flinched and saw that John was watching him from across the room. "Nothing," he told his uncle.
"Come on," John said. "You're stumbling around here like a zombie. What's up?"
The boy shrugged. "I got to decide something."
"What about?"
"Remember Kevin Dennison?" the boy said.
"Kind of a tough one to forget."
"He's selling pot. And some guys I know want to rat him out."
"What's the choice you got?"
"I could tell him or not," the boy said. "What they're going to do."
"Why would you?"
"It doesn't seem right," the boy said. "To get him in more trouble."
"More?" John said.
"You know, more than we're already in."
"I'm going to ask you something and I don't want you to take it the wrong way."
The boy looked at his uncle.
"You shoot that kid? That brother of his?"
"No," the boy said. "Hell no. Why?"
The uncle shrugged. "Something about it doesn't sit right with me."
"I didn't shoot him," the boy said. "I swear."
"All right," John told him. "Take it easy—I believe you."
The boy looked at the floor. "I loaded the gun," he said. He looked up at his uncle.
John nodded. "Why aren't you telling the truth?"
"Ma says we could get sued and lose everything," he said. "You can't tell her I told you. You can't."
"I won't say anything," John told him. "But I will tell you something about your mother."
"What?"
"Our father," he said. "When we were kids, a terrible drunk. Sober, he was straight as the Pope, but get a couple drinks in him and he was a twisted son of a bitch. Mean, I'm telling you, a real ball-breaker. And your mother," John said. "She was the type who could wake up the morning after one of his nights and go to school with a smile on her face like nothing happened. Hell, she'd have a story explaining the bruise on the side of my head before we got to the breakfast table. Me, I tried, believe me." He looked at the boy and the boy nodded. "I don't know what type you are," John said. "But I'll tell you, it ain't about the law. It's about how something sits with you. Your guts won't always say when you're doing right, but they know sure as shit when you're doing wrong. I believe that, Ted. That's why I ain't going to tell you what to do. I know you know."
The boy nodded. "You won't tell?" he said.
John shook his head.
"Thanks," the boy said.
"Long as you don't repeat what I told you."
"I won't," the boy said. He smiled and John smiled back.
"Come on," John said. "Let's get you home."
The following day the boy walked cautiously through school. He knew the Youth were keeping a close eye on him. He saw them about, eyeing him, waiting for him to make his move, waiting for some indication of treachery. But they kept their distance and he kept his. He realized he was running out of time. It was Friday and he knew the Youth would try to make their move over the weekend.
He saw Kevin twice in school and both times his heart started to race and his breathing quickened. He was afraid of the Youth but he was also terribly afraid of Kevin—he had been since the day he lied. It was because of that lie that the boy knew he couldn't let the Youth turn Kevin in. But it was also that lie that made approaching Kevin impossible. What could he possibly say to him? And if he called, what would he do if Mrs. Dennison picked up the phone?
In English class, he stared at the blank page in his notebook. Mr. O'Shea gave them the last five or ten minutes of class to free-write in their journals. The boy thought about reflecting on "Ode on a Grecian Urn," as O'Shea had suggested, but he didn't get the poem in the least. He listened to the scratch of pens against paper as the students around him scribbled away. He took up his pen and started in the top left corner of the page:
_Kevin,_
_They know you are selling drugs. They know you do it at the Coffee House. They are going to call the cops. They are going to videotape it from across the street. You know who they are. Please don't get caught._
_A Friend_
O'Shea gave him a dirty look when he tore the page out of his notebook. They were supposed to keep their journal writings together over the course of the quarter. The boy shrugged, folded the paper, and tucked it into his pocket. Neither O'Shea nor grades was high on his list of priorities.
After the bell rang, the boy made his way into the surge of students in the hall. He moved with the tide of bodies and kept an eye out for Youth members. He went to a water fountain and ducked his head for a drink. He stood and looked either way. His next class was in the science wing, but he took a quick turn toward the gym. He knew he was taking a chance even being seen near Kevin's locker; he was sure the Youth knew where it was. Before he got there he pulled the note out of his pocket, and as he passed, he slipped it through the vent at the top of the metal door. He walked to the end of the hallway and turned around, heading back in the direction of his next class.
When he turned the corner toward the science wing, he saw Colleen standing down the hall, leaning against the wall with her arms crossed. He wanted to take another turn, a quick detour around her, but he saw that she saw him.
"Hey," he said when he arrived before her.
"Why are you avoiding me?" she said.
"I'm not."
"I saw you turn back there, down the other hall."
He shook his head. "It's not what you think," he told her.
"I can talk to George," she said. "I'll make it okay."
"No." He shook his head. "No you won't."
"So you want to pretend that we aren't even dating?"
"We've only been on one date."
"Fine," she said.
"You want to go out this weekend?"
"I don't think so. I think I'm busy."
"Fine," said the boy.
The two turned and walked in opposite directions in the hallway.
**11**
**H** e spent an uneventful weekend by himself, anxious about what the coming week would bring. On Monday, he was relieved when the Youth members he saw smiled and acted warmly toward him. Birch even gave him a quick pat on the back on his way by in the hallway. But underneath the relief, he also felt a bit of remorse, perhaps even sadness. If the Youth were happy, their plot to bring down Kevin Dennison must have gone as planned. Maybe the note had been lost in Kevin's locker. Or maybe Kevin never returned to it on Friday afternoon.
Peckerhead was late coming into third-period biology and he was also quick to leave. The boy grew uneasy and his suspicions heightened when Peckerhead disappeared after their last-period American government class. He walked alone down the hallway of the portable and squinted as he stepped out into the bright fall sun. He headed across the cement courtyard to the front of the school, where the buses idled. It seemed safest to head home. Perhaps he could probe the Youth with a few phone calls that evening.
He was halfway across the courtyard when he saw Jason Becker and Birch coming his way. He was about to put his head down and make a jog for the bus, but they both smiled. Becker even waved. The boy slowed so that their paths intersected.
"Teddy," Birch said, holding his fist out.
The boy punched at it.
Becker gave him a quick, playful shove. "The Wrench," he said. "What are you doing?"
"Nothing," he said.
"Heading home?" Birch said, looking in the direction of the buses.
The boy nodded. "I got homework," he said.
"Nah," Jason said. "Come with us." He smiled and shoved him again.
"Yeah?" the boy said.
"Don't be a dumb-ass." Jason smiled and walked in the direction of the parking lot. The boy fell in behind them.
When they got to the parking lot, something in the boy told him to run. It felt wrong. The usual crowd wasn't there, no George, no Peckerhead, no mob of young cadets. He looked back over his shoulder and saw the last two buses groaning out of the drive.
"Where is everyone?" he said.
"At the shop," Birch told him. "We're all meeting up there."
Becker climbed into the driver's seat of the Volkswagen and Birch held the passenger door open for him. The front seat was folded forward, but the boy hesitated. Becker bent his neck down to look at him.
"Come on," he said.
The boy didn't move.
"What choice you got?" said Birch. "You going to walk?"
The boy got into the backseat.
"That a boy," Becker said, smiling. Once they were in the flow of traffic, Becker looked over his shoulder. "Everyone's at the shop to watch the video from this weekend."
The boy looked down at his lap. His hands played with a torn corner of nylon that hung from the seat in front of him.
"Cool," he said. He smiled quickly, but then he winced, thinking of how he had failed Kevin again.
When they arrived at the shop, he saw Colleen's car in the lot. He walked through the door, and it wasn't the mob of Youth members that surprised him. It wasn't the television on the counter or the ring of chairs set up around it. The quiet looks on their faces didn't catch his attention. What struck him when he entered the shop was the sight of Colleen Crenshaw standing at George's side, behind the glass case of handguns and knives. Her hand was flat on the counter and his was atop hers. George's eyes were upon him as soon as he entered the store, but Colleen looked down, away, anywhere but in his direction.
"Glad you could make it, Theodore," George said.
The boy nodded.
"Take a seat," George told him, but he didn't move.
"Please," George said, motioning to the chairs before the television.
When the boy still didn't budge, Becker took him by an elbow and walked him across the room.
"Sit," Becker said, and the boy sat.
"We have a little something we'd like you to see," George said. He reached over the top of the counter and turned on the television. It erupted in static. He pushed a tape into the VCR and the television went quiet, the snow on the screen replaced by a night shot of the local coffeehouse. A girl walked out holding a foam coffee cup. The boy heard whispering in the background of the video. The camera panned down the sidewalk and came to rest on three boys standing by a car in the parking lot. When the camera zoomed in, the boy recognized one of them as Kevin Dennison. The three leaned in to speak privately. Then they all leaned back and laughed. Kevin buckled over, holding his guts. One of his friends held a hand over his face and leaned back on a car. They seemed stoned out of their minds. The boy grew nauseous thinking where it was all headed.
One of Kevin's friends stepped behind a car for a moment and then he walked down the sidewalk toward the entrance. He had a black hooded sweatshirt on that hung past his waist.
Halfway to the entrance he bobbled his empty coffee cup and dropped it. When he bent over to grab it, his bare ass popped out from under his sweatshirt. It happened so quickly that the boy doubted what he'd seen. The kid got to the entrance, threw out his cup in a trash can, and returned to his friends in the parking lot. The three laughed wildly. There was silence in the room around the boy.
"What the fuck?" someone whispered on the videotape.
Kevin regained his composure and headed down the sidewalk. He was stiff, unnaturally upright. He stopped, looked down, and quickly bent over as if to tie his shoe. His pale ass blinked out from under his untucked shirt. He stood, but held his shirt up and slapped his ass. Then he pulled up his pants and turned to face the camera. He bobbed two middle fingers up and down and turned it into a kind of dance, gyrating, turning, flipping the camera off the entire time. When he finished his dance, he faced the camera again and pulled out the insides of his pockets. Then he smiled and held his hands up in a mocking shrug. The two other boys quickly ran over and began to moon the camera again. The boy was entranced by the footage. He wanted to laugh and cheer for them, but the screen went back to static.
George reached over the counter and turned off the television. No one spoke. The boy could hear the many Youth members shifting their weight on the creaky floor around him. He tried to steady his breathing and he wiped his hands along the legs of his pants.
"So," George said. "We have been humiliated." He walked out from behind the counter. "And I can only assume it was possible because of you, Theodore."
The boy looked at George. He thought through his options.
"But this isn't the end of the world," George said to him. "We all make mistakes. And with the proper amount of repentance, we can all be forgiven. At the end of the day, Theodore, we want you on our side."
The boy knew he could grovel and beg his way back into their favor. But he looked at the group around him—their khakis and loafers, their goddamn argyle socks. He looked at Colleen, behind the counter, picking at a fingernail. She looked up quickly and he caught her eye. She blinked slowly. She saw George watching her and she went back to working at her finger.
"Are you even going to try to deny it?" George asked him.
"Does it matter?" the boy said. He thought of Kevin and his two friends. He thought of Terry Duvall and Dan the wrestler. He looked quickly at the door and saw that he had a clear alley.
"Of course it does," George said. "Did you tell him, Theodore?"
"Don't call me that," he said. He reached up and scratched his head. Then he lurched out of the folding chair and sprinted for the exit. He felt the tug of someone grabbing his shirt, but the tension quickly gave way. He hit the horizontal bar of the door at full stride and blew it open. It slowed him a bit and gave them the time they needed. He felt a hard kick to his trailing leg, which sent his foot colliding into his opposite heel. Tripped up at the top of three stairs, he had nowhere to go but down. He got his hands out in front of him, but they hardly slowed his fall. He came to the ground with a grunt and the clash of his teeth. A blast of white light flashed inside his clenched eyes. He bounced and slid to a stop in the rocky dirt of the drive.
He rolled over and immediately clasped a hand over his mouth. His tongue had been between his teeth. He couldn't feel it and he feared he had bitten it off. He opened his mouth and touched the tip. It was still there, but his fingers came away thick with blood. He looked up at the group of awestruck boys. Peckerhead cringed and turned away.
The boy grabbed a baseball-size rock and clambered to his feet. He stayed low, the rock back and ready. He felt like an animal. He knew he would swing the stone with all his might at anyone who approached and the Youth seemed to sense this. The boy felt the warm fluid dripping off his chin. He took a couple of steps back and wiped at his face. There was more of his blood than he'd ever seen. He spat at his feet and it came out crimson and bubbly. He tried to threaten them. He tried to tell them he would kill anyone who so much as stepped a foot closer, but it came out garbled and meaningless. He continued to back up, step by step. He saw another good-size rock at his feet and he grabbed it with his empty hand. He shook it at them.
As he backed past a parked car, the tingling in his mouth began to turn into a searing pain. The superhuman strength he felt, the fearlessness, was melting from his limbs. He leaned against the car. He shook the rocks at the boys another time and then turned and sat, his back resting against the tire. He kept the rocks tight in his fists and listened carefully for footsteps. He spat again at his side and at the sight of the blood he realized the fear was coming fast. What if he'd done something they couldn't fix? What if he couldn't speak again?
"Teddy?" George called out.
The boy grunted. He jumped to his feet and hurled one of his stones. The group scattered and the rock bashed into the vinyl siding of the shop. The boy shook his second stone before turning and sitting again.
"Ted," George hollered. "What is wrong with you? Come on out of there so we can get you to the hospital."
The boy shook his rock above the fender of the car for everyone to see.
"This isn't at all what we wanted, Ted," George went on, but the boy was shrinking back inside his head. He was filled with a terrible fury. He wanted to beat them all to pulp. Even more, he wanted to dash in his own head—take the sharp end of the rock and go for his temple, over and over again, until the job was done. He was furious with himself for being so desperate, for needing their company, for believing Colleen. He hated this bleeding, whimpering thing he'd become, hiding like a child behind this car. Tears began to run down his cheeks, and they only added to his rage.
"Ted?" Colleen's voice drew him back. "Please don't throw a rock at me, okay? It's just me. I'm going to pull my car over there and we're going to go to the hospital. Okay?"
He fisted the rock until the edges began to cut into his hand. He wanted to kick her until she wept. He heard her car start and shift. He heard her back out and pull forward in his direction. He wiped fiercely at his face to hide the tears. The Escort pulled up beside him and she leaned over and threw open the door. He didn't move.
"Come on," she said. She waved him into the front seat.
He still didn't budge. He glared at her out of the corner of his eye.
"It's either me or them," she said.
He slowly got to his feet and walked to the car. He never looked back at the group of boys in the lot.
"How bad is it?" Colleen said.
He flinched when she reached across the car but she only went for the visor above the windshield and pulled it down.
"Do you want to look?" she said.
He saw himself in the mirror. Blood was smeared across his cheek and jawline. Fresh drops ran down from the corners of his mouth. His eyes were runny and bloodshot. He slowly opened his mouth. At first it didn't seem too bad. A cut ran halfway across his tongue. But when he stuck it out of his mouth, the weight of the hanging tip pulled the wound open. He saw the raw meat of the deep cut and drew it back in. He closed his eyes and cringed. He shook his head.
"Bad?" she said.
He nodded.
"I'm so sorry, Ted," she said.
He told her to fuck off.
She leaned closer. "Huh?" she said.
He shook his head. He batted a hand that told her to forget about it. He put his face in his hands and leaned over against the door.
"Back there," she said. "It wasn't what it looked like."
He didn't move.
"When I heard what they were going to do, I went to George to try to stop him," she said. "I thought if I was there they wouldn't hurt you."
He shook his head. "Fanks," he said.
"I'm serious," she said.
He held up a hand for her to stop talking. She did.
At the hospital they sat before the registry desk. When the woman returned, she had several packets of large gauze pads. She helped the boy open them and he used the first couple to wipe his face. He spit a mouthful of blood into the next handful and held a couple more over his mouth.
"Okay," she said. "Let's get some paperwork done so we can get you in to see someone."
He nodded.
"You can't talk?" she said.
He shook his head.
"This will do," she said. She took a prescription pad off her desk and a pen out of a coffee mug and handed them both to the boy. She asked him questions and he wrote out the answers.
"How did the injury occur?" she said.
He wrote, _Fell._
"He fell?" she said to Colleen.
Colleen nodded. "He tripped at the top of a couple stairs," she said.
The woman behind the desk nodded. "You're under eighteen? Do you have a number where we can reach your parents?"
He wrote down his home number and the woman dialed it. She held the phone to her head but eventually shook her head. "No answer," she said.
He scratched a second number on the pad and wrote, _Ask for Donna LeClare._
The woman dialed and asked for his mother. "It's an emergency," she said. After a long wait she introduced herself and explained the situation to his mother. "We'd like to get him in to a doctor as soon as possible. He can't talk at the moment. He has a friend here with him. Would you mind giving me consent before I do that? Yes, I think it's important that he's seen immediately. Okay. Yes. Thank you. Here's his friend." She handed the phone over to Colleen and came around the end of the desk. "Come with me, Teddy," she said.
He followed her through a set of swinging doors, down a hall, and into one of many small rooms.
"Climb right up there on the bed," she said. "Someone will be in shortly."
He sat with his head in his hands and then leaned back and pulled his legs up on the bed. He let an arm fall across his face to hide his eyes from the bright fluorescent lights. There was a quick knock on the door before it opened. A smiling Indian man in a white coat stepped through the door, a younger woman following.
"Mr. LeClare," he said. "I hear we had a run-in with some stairs."
The boy nodded and he began to sit up, but the doctor was already at his side with a warm, heavy hand on his chest. The woman stayed back, just behind him.
"You can stay," he told the boy. He had him rest his head on the pillow and he ran his hands over his jaw and neck, pressing and probing the bone and muscle. "Any pain when I do this?" he said.
The boy shook his head.
"Lose consciousness at all?"
He shook his head again.
"See any stars or bright flashes?"
He shrugged.
"Yes?" the doctor said. "Sounds like you took a pretty good hit."
The boy smiled.
"Okay," the doctor said. "Now let's get a look at that tongue." He went across the room and came back with a wooden tongue depressor. He had him open his mouth and he moved his tongue around with the wooden depressor and his gloved fingers. "Looks like the stairs won this round," he said. The woman looked over his shoulder. "But I think we'll live to fight another day." He listed off the things he would need to the woman and she wrote them down on a clipboard.
"I'll be right back," he said to the boy. "And we'll get you stitched up." He smiled and stepped through the door.
There was another knock, but this time Colleen poked her head through. She smiled when she saw him. She walked over and sat on the chair beside his bed. She looked down at him and ran a hand up his chest and along his cheek, dragging her fingernails across his scalp. He closed his eyes and focused on the feeling. She reached back and massaged his neck.
"Sorry I took so long," she said. "Your mother had a ton of questions. She'll be here soon."
The door opened and Colleen took her hand away. The boy opened his eyes and saw the woman who had come in with the doctor. She had a tray of odds and ends that she placed on the table beside the bed.
The doctor was close behind her. He adjusted the height of a wheeled stool and folded the railing of the bed down. He situated himself above the boy and arranged the things on the tray beside him.
"Do you mind if I stay?" Colleen said.
"Not at all," the doctor said. "I only ask that if you have a soft spot for blood or needles or anything of the sort that you look away and stay seated. I don't need another patient." He smiled and Colleen smiled back.
"Okay," she said.
"Now, Ted," the doctor said. "This is going to be a bit like the dentist. Ever had Novocain?"
The boy nodded.
"I'm going to give you a couple injections of something like it. And it's going to burn like a son of a gun. But after that you shouldn't feel much of anything. Sound all right?"
The boy shrugged. He felt Colleen take his hand in both of hers. When he saw the doctor reaching in with the needle, he opened his mouth and closed his eyes. A heavy hand fell on his chin and then it felt as if the doctor were pushing a red-hot needle into his tongue. He bore down on Colleen's hand and arched his back against the bed.
"A few more spots and we'll be done," the doctor said. "You're doing great."
The boy felt the needle withdraw and plunge again and again. Each time he had to fight the urge to clamp his jaw shut. His hand grew hot and sweaty in Colleen's.
"Now we're just going to give the medicine a chance to work," the doctor said.
The boy nodded. He opened his eyes and looked over at Colleen. She had her forehead down on the back of her hand that held his. He looked back to the doctor. The doctor smiled. The boy noticed that his tongue had begun to disappear.
After a moment the doctor was back in his mouth. The boy felt the pressure of his tongue being held in place. When he saw the doctor draw the needle up to pull the suture tight, he felt the tug, tug, tug of the string on his tongue.
The doctor prescribed an oral antibiotic and an antiseptic mouthwash. He told the boy that the stitches would dissolve on their own and that he didn't need to return unless something went wrong. He said the swelling should go down in a day or so and he told the boy that Popsicles would help. The doctor smiled when he said _Popsicles._
The boy shook his hand and left the room. He waited for Colleen, then headed down the hall toward the waiting room. He saw his mother at the opposite end, but she didn't see him. She walked quickly, her purse under her arm. She ducked her head as if to go faster. The boy stayed in her path. When she looked up to see the obstruction, she was startled to see it was him.
"Goodness," she said. She put an open hand on her chest. "Already done?"
The boy smiled, amused with himself for surprising her. He nodded.
"Let me see," she said. She reached and took him by the chin, pushing his mouth gently open. He stuck his tongue out. "Ouch," she said. "Hurt?"
He shrugged.
"You fell?" his mother said.
He nodded.
"He tripped at the top of a couple stairs," Colleen said.
"Thanks for being here with him, Colleen."
Colleen smiled and nodded.
"My boy must be growing," the mother said. "He's getting clumsy." She smiled at the both of them.
The boy held out the two prescriptions and his mother took them.
"We can pick these up on the way home," she said.
"The doctor also said Popsicles would help with the swelling," Colleen said.
"Popsicles it is, then," the mother said. They headed down the hall.
"Mrs. LeClare?" Colleen said. "Do you mind if I come over?"
"No," the mother said. "If your parents don't mind."
"No one's home," Colleen said. "I get creeped out by myself."
"I'm sure Teddy would like the company."
The boy looked at Colleen and shook his head. He wanted to ask her who the hell she thought she was, inviting herself over to his house. He wanted to tell her to go on back to George and his little study group. He wanted to explain to her just how upset seeing her with George had made him, but all that was impossible with his tongue as useless as it was. He could only frown and shake his head.
Colleen playfully slapped at his forearm and followed him out of the hospital.
Colleen and the mother ate bacon, lettuce, and tomato sandwiches on toasted white bread while the boy sucked on a milk-shake that his mother made. After dinner she went upstairs to grade student work while he and Colleen sat on the couch and watched television. After some time Colleen pulled his head down to her lap. He put his feet up and rested his head on her thigh. She ran her fingers through his hair and massaged his scalp.
At a commercial break he got up to use the bathroom. When he returned, he saw that Colleen was sprawled across the couch. Rather than sitting back up, she opened her arms for him. He paused for a moment and then lay beside her, letting his head rest against her warm, soft chest. She kissed his forehead and ran her hand up his back, under his shirt. He closed his eyes and felt his breathing go slow and heavy.
Sometime later he heard movement upstairs. He untangled himself from Colleen and sat up.
"Teddy?" his mother called down. "Ted?"
He grunted.
"I think it's time for Colleen to head home."
"Okay, Mrs. LeClare," Colleen said. "Thank you for dinner."
"Glad to have you," the mother said. "Have a good week at school."
"You too, Mrs. LeClare."
On the front steps outside the house, Colleen ran her hand up and down his chest. She kissed him on the lips. "Good night baby," she said.
He nodded and smiled.
After she'd backed out and headed down the road, he went inside. He turned off the television and the lights on the first floor, then walked up the stairs to say good night to his mother.
In the bathroom he stuck out his tongue in the mirror. It was swollen and discolored. The black stitches wound like a helix through either side of the cut. He brushed his teeth and rinsed with the antiseptic mouthwash. He winced at the pain. He killed the light in the bathroom and walked across the hall to his room.
Undressed and under the covers, he didn't think of Bobby or the investigation. He didn't think of George and the Youth. He thought of the warm feeling that had filled him in Colleen's arms. He thought of the searing pain that had filled his mouth earlier in the day. Sleep came fast.
The following day in school, he carried around a sheet of paper. It said, _I have stitches in my tongue. I can't talk. Thanks for understanding._ After he handed each teacher the note, he opened his mouth as proof. When the teacher asked how it happened, he flipped the paper over. There it said, _I fell._
By the end of the week the swelling was gone and the cut had begun to close. Still the boy chose not to talk. He liked the silence. It allowed him to stay back, behind his eyes, where it was quiet and less complicated.
The Youth stayed out of his way for a spell. Whether they were sympathetic to his injury or fearful of his rock-wielding rage, the boy did not know.
**12**
**A** t the end of the following week, Colleen's sister and her husband invited Colleen and the boy over for the evening, for drinks. The boy had never been anywhere for drinks. Sure, he'd gone places _to_ drink—the gravel pits and basements about town—but this was different. _For drinks_ sounded sophisticated. He knew he was breaking Youth doctrine, but hell, he was messing around with Colleen and he'd already ruined the Youth plot against Kevin Dennison. Really, the boy thought, what was a couple of drinks?
When they arrived, Colleen's nephew was hanging in a swinging chair and her sister was spooning food into his mouth. Jenny was a rounder, lighter-haired version of Colleen. She greeted them, placing the jar of baby food on the coffee table to shake the boy's hand. Jenny told them to make themselves comfortable. She said she'd be done in a minute. When her husband, Doug, came into the room, the boy was surprised at how young he looked. He gave them a weak smile.
"Get drinks," Jenny told Doug. And Doug went to get drinks. "Daiquiris," Jenny said. And Doug made daiquiris—strawberry, frozen, from a can in the freezer.
"You want some of this?" Doug asked him, holding up the blender pitcher. "I've got beers," Doug added. He nodded and Doug handed two mugs of daiquiris to Colleen and Jenny, went to the fridge, and came back with two cans of Budweiser. He handed one to the boy. "Only the best," Doug said.
"Cheers," said Jenny, and the four leaned in and clicked their drinks together.
It wasn't long until several rounds had gone down and Doug was sent back into the kitchen for another batch of frozen cocktails.
Jenny said, "Piña coladas," and Doug made piña coladas with another can of frozen mix. "I'm so glad you're not with that..." Jenny took a sip off the top of her new cocktail. "That Nazi." She looked at Colleen. The two of them were giddy with drink. "You're not into that no-drinking, no-drugs, no-sex thing, are you, Ted?" She reached out to her son's swing and gave it a light push.
The boy shook his head and held up his most recent Budweiser with a smile. He was feeling the alcohol as well.
"That's good," Jenny said. "Good for Colleen, anyway." She pointed at her sister and laughed out loud.
"Shut up," said Colleen.
"Serious, though." Jenny stopped laughing. "George and those guys are just weird."
The boy smiled and raised the beer to his lips. He finished the last quarter of the can.
"All right," Jenny said. "Part of the deal is that you guys have to watch Jimmy while me and Doug go and check on some friends." She gently rubbed her son's head with one hand and her other hand shot up with the empty glass in it. Doug retrieved the glass and filled it with what was left in the pitcher. With her drink full, Jenny took Doug's hand and pulled him toward the door.
"Adiós," Doug said.
"We'll knock," said Jenny through the closing gap of the door. "Take care of my baby," she said at the last second. Colleen chuckled and looked at the boy. He shrugged.
"They're going to smoke pot," Colleen said. "They have friends two floors up. They'll be a while."
He smiled.
"I do it sometimes," she said. "You?"
"I tried it," he said. "Didn't get anything off it, though."
"We'll do it," she said. "Me and you. We'll get totally stoned."
He leaned forward to put his beer on the coffee table. When he returned to the couch, he slid over, close to Colleen. He put his arm around her and she leaned into him. When she looked up at him, he kissed her.
"I couldn't wait for them to leave," she said.
She kissed him back, thrusting her tongue into his mouth. His own tongue was still a bit tender, but he managed to flop it about as the kissing required. After waiting for what he thought was an appropriate time, he ran a hand over one of her breasts. She rubbed the crotch of his jeans. When she leaned back, he followed her.
"Take off your shoes," she said.
He hesitated.
"It's a new sofa," she told him.
He sat up and slid off his loafers. Colleen smiled and kicked her feet up and down on the couch. He untied her sneakers and pulled them off, tossing them to the floor. She looked back, above her head, and gave the swing her nephew sat in a gentle push.
The boy lay upon her and they kissed. He pushed her shirt up to her armpits and fussed with her bra until she laughed at him, sat up, and undid the contraption herself. He fell greedily upon her breasts.
When he tired of them, he set to her pants. She raised her hips from the couch, allowing him to pull them free. She reached back and pushed the swing again. Her nephew gurgled and swatted at the rattle mounted before him. The boy finally realized that the swing mechanism was broken.
They were both down to underwear and he ground himself into her. It wasn't a particularly good feeling—something like the early stages of rug burn—but he couldn't bring himself to stop. When he came poking out the front of his boxers, he didn't reach to return it—the grinding felt much better against the satin of her underwear than it did against the cotton of his own.
"You want to?" he said.
Colleen shook her head.
He paused. "You sure?"
She nodded.
He had raised himself up off her to speak. Doing so exposed her breasts, and he fell upon them again with his hands and mouth, as if he had forgotten they existed. He kept up the grinding of pelvises and soon he found himself pulling her underwear aside. His aim always seemed skewed. He finally used a hand for direction, but as he slid his hips forward, Colleen stopped him.
"No," she said. She looked over her shoulder toward her nephew.
He pushed his hips farther forward, stopping only when their hips met. They remained motionless for a moment, staring into each other's eyes. She weakly shook her head. He nodded.
"At least put something on," she said.
He sat up and reached down for his pants, scuffling about for his wallet. When he set to putting the rubber on, it proved difficult, impossible, really.
"You've got it backward," she said.
He corrected his mistake and fell back upon her after he finished with the condom. She kept one hand on the swing, rocking it steadily. There was again the question of aim, but with some help from his hand, he corrected it. As he proceeded through the motions, he heard noises from Colleen. He wanted to believe that the sounds were evidence of pleasure, evidence that he was doing something right. As he continued, he realized she was humming. By the time he'd finished, he could tell that the tune was something of a lullaby, and not meant for him at all.
He sat up and stared at the television. There was a movie on. He tried to catch up with what he had missed. Colleen stood up quickly and grabbed her clothes. Before she left the room she glared back at him and said, "Are you happy?" She looked angry. "Now that you can tell your friends?"
He thought it strange. He didn't have friends. And if he told the friends he once had, they would most certainly kick his ass. And she knew this. When she came back into the room, she was clothed. She took her nephew and left again. The boy put his pants back on. She returned without the child and took a seat in a chair across the room from him.
"That was my first time," he said.
"What?"
"I've never done it," he said. He thought it was obvious.
She came slowly toward him. She took up a seat on the couch. "Really?"
"Yeah," he said. He grabbed his T-shirt off the floor. He turned it right side out and reached his hands above his head to pull it on.
"What are those?" Colleen said, touching the inside of his arm.
He flinched and quickly pulled the shirt down and over his head. He looked at the television and tried to let the moment pass.
"Hey," she said. She tried to lift his elbow, but he held his arm tightly to his side. "What happened?"
"It's nothing," he told her.
"It's not nothing," Colleen said.
"It's old," he said.
Colleen shook her head. "One's still scabbed."
He stared at the television.
"Hey," she said. She tugged at his hand.
He looked at her. He shook his head.
They both looked back to the television.
"When I was in middle school," Colleen said. "I rubbed a pencil eraser back and forth on my arm. I made a cross. The counselor kept saying it was because of my parents' divorce, but I was just bored."
"It's not like that," he said.
"Don't get mad," she said. She put her head on his shoulder and wrapped her arms around him. She held him quietly for some time. "Have you told anyone about Bobby?" she said.
"My mother."
"Any of your friends?"
He shook his head.
"Will you tell me now? What happened?"
"He died," the boy told her.
"What was it like?"
He thought for a moment. "Not wicked bloody or anything like that," he said. "He just stopped."
"Stopped what?" Colleen said.
"Stopped everything."
A commercial break came on the television and he slipped out of her embrace. He eased her down on the couch and slid his hand up under her shirt. He ran the tip of his index finger around her nipple. He pulled her shirt up and sucked at her breast. His pelvis started to gyrate against the outside of her thigh.
"Already?" she said.
He looked up and smiled.
After it was over, Colleen smiled and said, "You're not lying? I'm your first?"
He nodded. "You?" he said.
"Once before," she told him. "Twice now."
He raised his eyebrows at what he thought was a lie.
"I thought I was in love with him," she said.
"Punk-rock guy?"
"Yeah," she said. "How did you know?"
"Youth guys," he said.
"What else did they tell you?"
He shook his head.
"You shouldn't believe them," she said. "They know I don't believe in any of their shit."
He nodded. They put on their pants and watched what was left of the movie.
Jenny and Doug came back squinting and yawning. They said thank you and good-bye. Colleen and the boy walked through the drizzle to the car, and after they were back on the highway, he said, "You won't tell?"
"About the drinking?" she said.
"And us?"
Colleen paused for a moment. "No," she said.
It made the boy half happy. Half happy because it was one of the two things he had been fretting about. He was also concerned about returning to his house half drunk and late. It was Friday. He knew his father was due home and hoped he would be sleeping. If all went well, he could slip quietly into his room, tell his parents a half-truth the following morning, and everything would turn out half good.
They were about two hundred feet from his driveway when he said, "Stop here."
"What?" said Colleen.
"Stop," he said. "Stop, here."
"They know you're with me, right?"
"Don't pull in," he said. "Keep going, keep going." He grabbed the steering wheel and pulled the car back onto the road.
"Why?"
"Pull over up there, past the house," he said. Colleen coasted the car to the spot he directed her to. She stopped the car, but kept it running. She looked straight ahead.
"What?" he said.
She shook her head.
"Come on," he said. "I'm already late."
"Did you even say you were with me?"
"Yes."
"Then why don't you want me to drop you off?"
"It's not that," he said. "My dad's home. I don't want to wake him up."
"I thought your parents were divorced."
"I never said that."
"He's never here," she said. "I just thought—"
"He moved for work. My mom and I will too."
"Move?" she said. "When?"
"When the house sells."
"Why didn't you say?"
He shrugged.
She looked out the window.
"I'm late," he said. "I got to go."
The two leaned in and exchanged a quick peck. He left the car and tried to close the door as quietly as possible.
At the front porch he took his shoes off, to creep more quietly. The decking was wet but the porch was creaky as hell. He inserted the key and slowly turned the bolt. Once the door was free of the jamb, he swung it quickly to avoid the haunted-house groan that happened if it opened slowly. He held his breath to hear. The furnace kicked on with a rumble. He stepped inside and set his sneakers down.
As he made his way to the kitchen, he saw his father's slippered feet propped up on the coffee table in the living room. He stopped. He could see the hem of his father's blue bathrobe.
"What are you doing up?" he said. The stairway blocked any eye contact between the two.
"Can't sleep," the father said, in the hushed tone of a man with a wife asleep in the house. The boy walked around the front of the stairs and entered his line of sight. The father looked up over a paperback, the title declared in a daunting raised font on the front cover.
"A little late?" he said, looking back to his book.
The boy shrugged.
"What you been up to?"
"Hanging out."
"She says you got a girl. The ex of one of these political guys."
The boy nodded.
"You be careful."
"They were broke up," he said.
"You watch out for her too," the father said. "Don't go and get yourself hurt."
The boy nodded.
"Still no word on the charges," the father said.
He shook his head.
The father nodded. "I'd be calling it a night. We got a funeral in the morning."
"Who?" the boy said.
"Lawrence," the father said. "Your mother's uncle."
"How?" the boy said.
"Diabetes, mostly. Then he killed himself. Apparently."
"How?"
The father made the shape of a gun with his hand and put his index finger in his mouth. The cocked hammer of his thumb fell.
"Wow," the boy said.
The father reached for the fallen book with his open hand. The boy stooped and handed it to him. "Sleep tight," his father said.
The boy stood for a moment before he nodded and left.
In the darkness of his room he pulled the covers over his head, and in the confines of his sheets his body convulsed in several quick sobs. He didn't tell his father that he loved him, or that he was glad to see him. And he did. And he was. Something had grown up between the two.
When the boy's childhood dog had been struck down in the street, there had been such a draw to his father, that, once in his arms, the boy stayed there for hours. He buried his face in his father and hid there, crying and soaking his shirt. His father rubbed warm circles across his back.
But now it was as if the poles had reversed. The closer the boy and father became, the harder they pushed each other away. The boy felt broken up and confused. There was something that he couldn't get his head around. He wished he could still seek comfort in the father's arms, but his body chose instead to turn in on itself, milking the last warm remnants of the night's beer buzz.
He ran his fingers over the scars and scabs on the inside of his arm. Each was in the shape of a simple smiling face. The rollers of the lighter formed two eyes, the curved metal flame guard a deep smile. When the boy made a muscle in the bathroom mirror, they all stared back at him, laughing.
**13**
**I** t was a Catholic service and, like most, it did little for the boy. He was bored by all the repetition. And they were stuck with Father Thomas—he was in his seventies and his recent hip replacement had aged him considerably more. At one point in the service Father Thomas started saying _Clarence_ instead of _Lawrence._ The boy scarcely knew his great-uncle, but from what he gathered, his life had grown difficult. The diabetes he'd managed for most of his life had finally overrun him. It had sapped a good deal of his sight and worked on his legs—subtracting from them a joint at a time—leaving him confined to a wheelchair, half blind.
He did it with a .45 pistol, in his mouth. They found him behind the barn, as he told them they would. He had said, "You got yourself a lame horse, you know what you do: You take it out back, you take care of it." The story of him saying this went around and around at his service. Perhaps to help purge the guilt of those who had actually heard him say it and had done nothing, or maybe because some found it eerily noble, as some final expression of virility. It seemed like strength, control, some manifest destiny. In the action, Lawrence had proclaimed: _I cannot live my life as I would like; therefore I assume the responsibility of ending it._ Or perhaps it was more like: _This sucks—fuck it._
The ceremony at the cemetery had an altogether different effect on the boy. His great-uncle had served in Korea and he was buried with full military honors. The boy had never seen anything like it. He watched the soldiers in full dress march through the procedure. There were two things that broke him up: The first—while folding up the flag from his great-uncle's coffin, one of the soldiers was clearly in tears. The image of that sadness amid the formality was wrenching. The boy thought he must have been a relation, but at the end, he left in the ROTC van without a word to anyone. The second thing was the gunfire. The first shots were so sharp and so startling that the boy flinched, and in this motion something seemed to come loose in him. The man calling out the command for the shots had a southern accent and it came out, _Aiem, faher!_ As the shots continued, they grew less shocking, but for some reason he didn't want them to end. _Aiem, faher!_ There was something tremendously beautiful about it. _Aiem, faher!_ By the last volley the boy was choked up and wet in the eyes. It confused him. He'd never really known the guy.
After the services there was a get-together at the local Elks' Lodge. The family had congregated there time and again for wedding and funeral receptions, graduation parties and wedding anniversaries. It was an enormous sharp-white Victorian building of three floors. From the windows at the front there was a beautiful view of the lake. During the many events in the past the boy and several cousins had explored what they could of the building. In the basement they'd found a bar at the head of six bowling lanes. The boy was the first to get a running start and baseball-slide into the pins. On the third floor the kids thought they had found an illegal gambling operation when they discovered a room with a roulette wheel, tables for craps, poker, and blackjack. The father, hearing of the conspiracy, chuckled and said, "It's pretend—they use play money, for charity." And then more seriously, "The hell you doing up there? Stay where you belong."
The father, the mother, and the boy stood in the middle of the room that ran adjacent to the first-floor bar and looked out upon the lake.
"Donna?" the father said. "Whiskey and ginger?" The mother nodded. The grandfather and grandmother left another circle of folks and approached them.
"You all set?" the father said to his in-laws. He made a motion to the bar. The two held up their usual cocktails—the grandmother a Manhattan, the grandfather scotch and water.
"Come give me a hand," the father said. The boy followed.
The father rested on the bar, a twenty in his hand so that the bartender would know his intentions. It was only slightly past noon and it was a funeral reception, but still the bartender was rushed. It was this way with the family. The father made a motion for the boy to stand next to him at the bar.
"I been thinking," he said to the boy. "Seems like you've been having a hard go of it."
The boy nodded. His father handed him his mother's cocktail and grabbed his own after carrying out the transaction with the bartender.
"You want," said the father, "you can come down and stay with me."
The boy nodded. "Yeah," he said.
"That's it?" said the father. "'Yeah'?"
The boy shrugged.
The father shook his head and turned away. The boy stood for a moment and watched his father walk back to the family. He looked across the room and saw the lake through the wall of windows. He didn't know why, but he didn't want to go to Pennsylvania with his father.
The boy's mother was busy talking to his grandparents when he approached, so he waited at her elbow until there was a break in the conversation. He handed her the drink.
"Well, Teddy." His grandmother pulled him by the back of the neck to kiss his cheek.
"Hey, Grandma," the boy said.
His grandfather reached out his hand and the boy shook it. He was an old man, but his hand was still heavy and hard. He had broad forearms and his neck was thick like a linebacker's.
"Howdy, Chief," his grandfather said.
"Grandpa," the boy said. He nodded and stayed silent so as to allow their conversation to resume. In no time they were talking about Lawrence. They agreed that it was a shame, but his grandmother claimed that if he'd taken better care of himself he would've lasted longer. She said that a diabetic shouldn't have drunk like he did. When the boy felt that they had sufficiently forgotten his presence in their conversation, he left the circle.
He went to a corner and stood alone, watching other groups about the room. He felt uneasy in large gatherings of his family. They were strangers, after all, for the most part. Sure, they were allowed to hug and kiss you, but this made it worse. Most of them only saw one another on these ceremonial occasions—and that didn't seem to constitute an intimate relationship. Yet somehow it did. He looked for his uncle John, but he was nowhere to be found. John had arrived late to the funeral and remained standing at the rear of the church. At the burial he stayed out along the fringes of the crowd.
The boy crossed the room to the buffet on the far wall. Passing over the two tin pans of lasagna, he noticed that his grandfather was choosing from the several bowls of coleslaw. Before leaving the line, the boy split a dinner roll and stuffed it with roast beef from the cold-cut plate at the end of the table. He found a small bowl of shock-yellow mustard and doused the makeshift sandwich.
He chose a folding seat at a round folding table that was covered with a linen cloth. He was surprised when his grandfather took a seat just two chairs over. As a child the boy had felt a distinct dislike from his grandfather, but as he had grown he'd realized that the grandfather seemed annoyed by all children. They frustrated him with their dependence and their inadequacy.
"Say I shouldn't eat like I do," his grandfather said. "Say I'll end up like old Lawrence there. Well, not like him, I suppose."
"He was your younger brother?" the boy asked.
"Yup, old Lawrence," said the grandfather. "Weren't like brothers you'd think today. They didn't let kids stay 'round much."
The boy nodded.
"How's it sit with you?" the grandfather asked him. "The way he done it?"
The boy didn't know what answer his grandfather was looking for, so he used a mouthful of sandwich as an excuse to shrug.
"Women seem upset," the grandfather said. "Don't bother me none." The grandfather ate a forkful of coleslaw. "I take a man's word. You?"
The boy nodded.
"Some ways are just no way. You follow me?"
The boy nodded again.
"Weren't like that boy that got himself killed," the grandfather said. "What was his name?"
"Bobby," said the boy. "Bobby Dennison."
"Now that's a goddamn shame. At his age?"
The boy nodded.
"They still investigating?"
The boy's mouth was full with a bite of sandwich, so he didn't respond.
"Doesn't seem right, really. You didn't do nothing wrong," the grandfather said. "Leaving a couple fools with a gun ain't the brightest thing I ever heard, but how can you be held responsible for something they did on their own?"
"I don't know," the boy said.
"What you think will come of it?"
The boy shrugged.
The two ate in silence for some time.
"Old Lawrence," the grandfather said. "Now there was one miserable son of a bitch. Told me once, 'I wouldn't piss on you if you were on fire.'" He raised an eyebrow and nodded at the boy. "Just the way he was. Probably ain't right to talk like that but he was never one to try to tell you what a sweetheart he was."
After saying their good-byes, the father, son, and mother walked across the parking lot. The mother put her arm through the boy's and squeezed it tight.
"What'd you and your grandfather talk about?"
"Suicide."
"What'd he say?"
"Thinks it's fine," he said.
"I don't know what makes them think they have the right," she said.
"They're a different generation," the father said. He unlocked the car doors and they climbed in.
"Doesn't matter," she said. "You can't do that to your family."
"I'm sure he thought he was doing them a favor," the father said.
"If he even thought about them at all."
"I'm just saying he probably felt like he was a burden."
"It's selfish," she said.
"Can we talk about something else?" the father said. "Something decent?"
"Yes," said the mother. "You're right."
"Thank you."
A moment of silence passed before the boy blurted out, "I don't want to move to Pennsylvania."
"Did you hear your father?" the mother said. "He drove eight hours to spend the weekend with us—the least we can do is make it time well spent. Right?"
The boy remained silent.
"Well?" the mother asked again.
"This is fucked," he whispered.
"What?" she said.
"Nothing," said the boy.
"What did you say?"
"We're so screwed up."
The mother spun in her seat and shot a finger in the boy's face. "Don't you ever say that about my family. You don't have the right. Did you ever stop to think why we're in this mess?"
"Back off," the father hollered. "The both of you."
The mother turned and sat back in her seat. The car was silent until the boy heard the mother's pained and sporadic breathing.
"We're fine," she sobbed. "Everything is fine." Her voice was hollow and aching. "It's just the funeral," she said. "Everyone's a little emotional, that's all. A little overtired—we just need some rest. Everything's fine," she repeated. "We're fine."
The boy wanted to open the door and jump from the moving vehicle, haul ass into the woods, and hide there. Instead, he found himself pondering Lawrence's suicide. He tried to imagine what he might do in the position of Lawrence's children—if it had been his father to put the gun in his mouth and pull the trigger. What would he do with the pistol that had taken his father's life? Of course he would have to wait for the police to finish their investigation and rule out wrongdoing before he could even have access to it, but after that, what?
The boy imagined retrieving it—bagged as it had been found—and some night, after his mother had gone to sleep, he would take it to the kitchen sink and wash the sidearm free of his father's remains. Covering the kitchen table with newspaper, he would disassemble the gun and clean its intricacies to the best of his ability. Oiling the many moving parts, he would put it back together and wrap the gun in the light-blue felt of a Crown Royal bag. Then what? Sell it? The thought sickened him. Bury it? It seemed such a waste.
The boy leaned his face against the cool glass of the window and watched the passing world. They were nearly home and the car was quiet. He turned away from the window, where everything was dark and indistinguishable, and looked at the shadowed dome of his father's head. He was happy it was whole. He was glad he didn't have to see inside.
**14**
**I** n school the following Monday, Colleen held the boy's hand in front of people. She held her head up to be kissed when they parted, even when the hallway was busy with other students. The boy did his best at making minor, casual objections, but he finally blurted out, "We can't do this."
"Why?"
" _Why?_ " he said.
"We're together."
"I'm going to get my ass kicked."
"You're afraid of them," she said.
"Shut up." He stormed down the hallway to his next class.
Later in the day Colleen approached him at his locker. She tugged on his shirt.
"I'll talk to George," she said. "I'll make him understand."
"Are you kidding?"
"Teddy."
"That's retarded," he told her.
"Why are you doing this?"
He didn't respond.
"Hey." She tugged at the hem of his shirt again.
Still he didn't respond.
"You know what," she said. "You can fuck off."
He looked away from her.
"Fuck you," she said, and walked down the hall.
He didn't try to stop her.
He knew he wasn't handling it right. He couldn't understand it. Sex, intercourse—he had ached for it. At times he'd felt as though some vital inner organ would rupture if he remained without it any longer. But now he felt something else, and he didn't know exactly what or why. He thought he must be an idiot, maybe even gay, for here was Colleen—cute and nice and affectionate Colleen—and he couldn't even kiss her without wincing.
The kid and the couch and the broken swing—he wanted to forget it, and that was all the easier when she wasn't around.
When he walked into school the next day, the first Youth member he saw had a vicious sneer upon his face. A moment later, farther down the hall, two more members confronted him.
"You're so dead," Birch told him.
"After school, motherfucker, wicked dead," said a kid whose name he always forgot.
He and Peckerhead Jackson were lab partners in biology, and for the first half of the period Peckerhead managed to remain silent, but he was a talker and it was killing him.
Without looking up from his lab report, Peckerhead finally whispered, "What the heck, Ted?"
"What?" the boy said.
Peckerhead didn't reply.
"What, Peckerhead?"
When he finally said what was going around, the boy felt gut-shot. Point blank. The word passing through him, the concussion sending waves out from its place of impact somewhere in his torso. _Rape._ He felt himself sway. He wished his body would collapse. He wished he would fall and hit his head on something, lose consciousness, and wake up at home or in the hospital, anywhere other than third-period biology.
Peckerhead told him that Colleen had confessed the details of her night with the boy to Becky Stanton, another Youth member's girlfriend. Becky relayed the story to her own boyfriend and by the time it all got back to George Haney, it went like this: Ted LeClare got drunk and raped Colleen Crenshaw.
"She said she told you no," Peckerhead said.
"It wasn't like that."
"Did she say it?"
"It wasn't like that," the boy said.
"Did she say no, Ted?"
"I need to talk to them."
"They won't believe you," Peckerhead said.
"Because they want to kick my ass."
"No," Peckerhead said. "Because they want you back. If they can't have that, then they'll kick your ass."
"No way," the boy said.
Peckerhead shrugged. "Go up to George after school and tell him how sorry you are and see," he said. "You're his golden boy."
"Jesus," said the boy. "Why did you get me involved with them?"
"Because if I didn't, they could do this to me," Peckerhead said.
The boy shook his head.
He didn't follow his regular path to his next class. He walked straight to where he thought he'd find the only person who might help him—his childhood friend, and known bruiser, Terry Duvall. Terry wasn't in the bathroom in the vocational wing, where he was known to smoke between classes, but on his way back, the boy found him sitting on a railing outside the building. The boy explained his predicament.
"Fucking slut," Terry said.
"What should I do?"
"They're pussies and shit-talkers," Terry told him. "You can get on my bus after school if you want."
"Thanks," the boy said.
"What were you doing with those cocksuckers anyway?" Terry said.
"I don't know."
"You find me," Terry told him. "I ain't going to go looking for you."
The boy nodded.
He skulked from class to class, staying away from his standard route. Whenever he saw a Youth member, he did what he could to avoid him, even if that required stopping and walking in the opposite direction. He was late to his last two classes because of the tangents he took.
While the last bell meant an end to this miserable day, the boy also knew it marked an open season on himself. Once off school property, he was fair game. He searched the herd of students making their way to the buses. When he saw Terry's orange hair, he struggled through the crowd to get to him.
Chris Awdry walked with Terry. While Terry had come to be a known bruiser, Chris was a known psycho. He was not right. The boy knew this. It was not an act with Chris Awdry. Chris scared the boy. This was the kid whose left nipple had been disfigured by a knife his sister threw at him when he was eleven. This was the kid who had set a bonfire in his own attic.
They made their way to the bus without trouble and took the seats at the back. The boy felt relieved when the door squealed closed and the bus groaned forward.
A couple of miles from school Terry said, "There they are."
On the road behind them was Jason Becker's Volkswagen Rabbit. At first the Rabbit was two cars back, but soon it rode close to the bumper of the bus. The car was packed full with Youth. They made every kind of violent and vulgar gesture known in the vocabulary of hands.
"Should we tell the driver?" the boy asked.
Chris didn't acknowledge the comment.
Terry just looked at him and smiled. "That's how they would do it," he said. "Ain't it?"
While the boy sat shifting uneasily in the green vinyl seat, Terry and Chris prepared themselves. Terry tied tight the work boots he normally wore loose and open. Chris took off his jean jacket and stuffed it in the boy's backpack. Then he jumped into a crouch between the last two seats and threw two middle fingers at the car behind them.
"Get in your seat!" the bus driver shouted from the front. Chris lingered a moment before heeding the command.
"There's a lot of them," the boy said.
"There's three of us," Terry said.
Chris looked at the boy and said, "Get a rock or a stick if you can."
Terry leaned over and looked out the back again, "When they ratted out Devo for dealing, some of that weed he had was mine."
When they got off the bus, Jason Becker stood with one foot out of his car, one hand on the roof and the other on top of the open door.
"This has nothing to do with you, Terry," he said.
The boy realized that it was true—this had nothing to do with Terry or Chris, and yet here they were, more inclined to go through with this fight than the boy was himself.
"Fuck you, peckerwood," Chris shouted.
"What do you know about it?" Terry said.
Becker's car was holding up traffic. He tried to wave the cars by, but they wouldn't pull around on the blind corner. When the car behind him began honking, Becker finally got in and pulled away in search of a more suitable pull-off or turnaround. As they passed, the Youth in the car were screaming through the windows, holding up middle fingers and fists. Terry and Chris stood their ground. The boy remembered how those same hands had shaken his own and patted his back. He had seen those screaming, bitter faces smile, warm with affection.
"Come on," Terry said to the boy, once the car had passed around a corner, out of sight. When the boy hesitated, Terry said, "Hurry up dumb-ass or they'll see us." Chris and Terry crossed the ditch on the far side of the road.
They picked their way through the woods, eventually coming out in Chris's backyard. There was an empty above-ground pool and a deck that ran between the pool and a sliding glass door on the house. Chris took a key from his pocket and unlocked the back door. He flipped open a pizza box that sat on the kitchen table and took a slice. He motioned for Terry and the boy to do likewise. They took the last three pieces—they were dry and curled at the edges, the cheese white and the crust difficult to chew.
Back out on the porch they smoked some pot from a small metal pipe. The smoke scorched the boy's lungs. Terry and Chris laughed as he coughed and hacked. Chris rapped the ash out of the pipe. "You guys hungry?" he said.
Terry and the boy shrugged.
"I'm starving," Chris said. "Got any money?"
He collected several dollars each from Terry and the boy. He pulled another couple of singles from his pocket. He walked inside to the foot of the stairs. "Hey, you want dinner?" he shouted.
"Here," a voice hollered down. Chris ran upstairs and returned with a ten-dollar bill. A moment later he was on the phone ordering two large cheese pizzas.
The boy felt safe at Chris's house. He doubted that any of the Youth knew where Chris lived and he doubted even more that they would make any attempt to get at him there. Chris's fight with a senior the previous year was a part of the school's collective memory. There had only been about two dozen people present at the bout, but anyone could tell you what had happened. They said that Chris was half the senior's size. They said the senior asked for it, taunting Chris that way. They said that when it was over, the senior was not only missing his shirt but a front tooth and a good fistful of hair as well.
The three boys watched a movie on a pay station. The boy's body felt heavy and his mind slow, so slow. He felt like he was melting into the couch. He found himself stopping to remind himself exactly what was happening: _This is a movie, this is Chris's house, it is after school._ He dreaded the thought of ever having to leave—having to go home to his mother, stoned—having to leave the safety of Chris and Terry.
The boy jumped at the sound of the doorbell. Terry pointed at him and laughed. Chris got up and went to the door. When he returned, he put the two pizza boxes on the coffee table in front of them. He yelled at the ceiling, "Dinner! Hey!"
The boy leaned forward and took a paper plate and a slice of pizza. Behind him he heard someone descending the stairs. She had pajama bottoms on and a large T-shirt that hung nearly to her knees.
"You know Ted?" Chris said to her. She didn't look at the boy but she shook her head. "My sister, Shelly," Chris said to the boy.
"Hi," the boy said.
Shelly didn't respond. She took two slices of pizza and went back upstairs.
"She's just like that," Terry told the boy.
The boy had noticed that her nipples had protruded through her T-shirt. And the thought of her nipples led to thoughts of sex, which led the boy to thoughts of Colleen and the whole mess he was in. He did his best to think of something else. The pizza helped.
He waited until well after dark to leave and cautiously walked the side of the road, ducking into the woods when headlights appeared. The pot hadn't relaxed him. He heard cars approaching that were not cars approaching but wind in the woods. When he ducked off the road, he went much farther than was necessary. He found himself walking deeper still into the forest, even after the car had come and gone. He had to remind himself: _I'm walking home; home is that way._
He walked in the front door and kicked off his shoes. The house was warm and smelled of food. He heard the television.
"You with the boys?" his mother asked.
He hung up his jacket and dropped his backpack at the foot of the stairs. "Nah," he said.
"No?"
"Remember Terry?" he said.
"Duvall?" she said. "Of course. Is he friends with the boys now?"
"No," he said. He walked into the living room. He did his very best impersonation of a sober boy.
She had her legs pulled up on the couch and an afghan draped over them. A dirty plate and fork sat on the coffee table in front of her. The television was on. He smiled quickly at her and passed through to the kitchen. A casserole dish of chicken breasts cooked in cream-of-mushroom soup rested on the stove. A pot of rice pilaf sat next to it. He took a plate and a chicken breast, a scoop of rice. He ladled the sauce from the casserole dish over both.
"You okay?" she asked him.
"Yeah," he said.
"You sure?"
He shrugged.
"Something with the boys?"
He shrugged again.
The mother stayed silent.
"They don't like me anymore."
"No?"
"Nope." He was eager for the chicken and rice, despite the three slices of pizza at Chris's house.
"Was it Colleen?" the mother asked.
He shrugged.
"You know, I thought she was trouble," the mother said. "I should've done something."
He nodded. He ate quickly and rinsed the plate before putting it in the dishwasher.
"I'm going upstairs," he told her. "Do some homework."
"Okay," she said.
In his room he fell into a panic. He had seen the commercials, "No means no." And Colleen had said no. But it hadn't been like that. Or had it? By law he was guilty, wasn't he? Yes, but it hadn't been like that. It hadn't been. They'd had sex after that. After that she'd said yes. How did she explain that to George? How did they explain that?
The lethargy he'd experienced earlier had transformed into a terrible paranoia. He wasn't getting anywhere and he wished for some way to extract the marijuana chemistry from his mind, some antidote to slow down the ideas and quell the whirlwind in his head. For several hours he sat on his bed with his history book open, and in that time he never finished reading the section he'd started on. Finally there was a knock on his door.
"You still awake?" his mother asked.
"Yeah," he said.
She let herself in.
"I'm reading," he said.
"Any good?" She looked at the book splayed before him.
He shrugged.
"Sure that's the only thing keeping you up?"
"Yeah, Ma. I have to do it."
"You don't want to talk?"
"I'm fine, Ma. Just stop. Please."
"You haven't told anyone, have you?" she said.
He shook his head.
"What happened with Kevin and Bobby? Not even Colleen?"
"No, Ma. Jesus."
"You can't trust anyone," she said. "See how quickly they can turn on you?"
He nodded. "I know."
She bent over and hugged him. "Good night," she said.
"Good night."
He remained rigid against her embrace. Rather than look at his mother, he kept his eyes on the text that explained some event a hundred years prior. When she left, he gave up on the book and threw it to the floor. He pushed himself up and sat on the edge of the bed, both his elbows resting on his knees. He pushed up his sleeve on the left side. The inside of his upper arm was growing cluttered with the small, smiling burns. It hurt, somewhere inside, to see his body so abused. But there was something else in him that said he deserved it, that told him he deserved worse—it wanted to cover his whole body with the burns, it wanted worse. He winced and shook his head, but it wouldn't stop.
He held the lighter out and flicked the flame to life. He counted slowly. He moved his thumb back from the rollers, maintaining pressure on the plastic tab, keeping the flame alive. When he passed one hundred, he turned the lighter around in his hand and gripped it. He rested his opposite arm across his thigh, his palm facing the ceiling. He brought the hot metal down on the inside of his forearm. His arm bucked to get away, but he held it there as the heat of the metal dissipated into his flesh. He leaned back on his bed and let the lighter fall away.
His mind was clear for a moment, but soon the same circle of thoughts plagued him again. No matter how he went about it, he always came back to the fact that she had shaken her head. It hadn't been like that, he told himself. But it had. But it hadn't. But it had.
**15**
**F** or the rest of the week, the boy took Terry's bus and waited at his house for a couple of hours before heading home. To be safe, he walked in the woods and along an old railroad grade that paralleled one of the roads between their homes.
He was quickly developing a habit of smoking grass. Terry handed him small amounts in the twisted-off corners of sandwich bags and in turn the boy gave him a five-or ten-dollar bill. If it was early he went for a walk in the woods, and if it was late he turned off his bedroom lights, stuffed pillows under the covers, and proceeded out the window to the roof.
There were times that were quite beautiful: stars, the smells of wood—living, dying, and dead—the silhouettes of large pine boughs rocking in the wind. There were also times when he panicked over a sound in the house and everything went to shit.
While most of what had occurred in his life could account for his general ill feeling, it was another run-in with the Youth that had made one of the days notably worse than the rest.
He'd come face-to-face with Colleen in the school hallway between periods. There was a moment where neither of them spoke.
"I didn't say it like they said it," she said.
He nodded.
"I didn't," she said. "I swear."
"I figured."
"I'm sorry," she said.
"Me too."
"They can't stop talking about it." She pouted and he felt like he was falling for her all over again. "No matter what I say. It doesn't matter."
"I know what you mean," he said.
"I thought they'd be over it faster." Suddenly she looked over his shoulder. Her smile disappeared and she looked at the floor. Before the boy could look back, a hand grabbed his shoulder and spun him around.
"What are you thinking?" Jason Becker barked at him.
He froze, a frightened stare on his face.
"You think you can talk to her?" Becker said it loudly enough that the kids coursing down the hallway looked. Some even stopped to look.
The boy looked over his shoulder for Colleen, but she was gone.
"Don't think we've forgotten," Becker said, "you degenerate fuck." Becker reached up and put his hand on the boy's face. He gave him a hard shove and the boy stumbled backward, knocking into several people. The girl that took the brunt of it whined, "What the fuck?"
The boy turned and apologized.
"Asshole," she said.
When the boy looked back, Becker was leaving, but there was a crowd staring at him. When he walked away he heard someone say under his breath, "What a pussy."
The worst part was that he knew he was to blame for all of it. He couldn't complain. He couldn't bitch. But he could get stoned. Now he gave a quick glance in each direction of the railroad bed and pulled the joint out of his pocket. It was in an empty pack of gum for protection and it had a slight air of mint because of that. He stopped for a moment to get it lit, and once it was burning, he continued on.
He tried to imagine a train passing on this grade that once had held tracks. It seemed so hard to believe, a train where now there was only an overgrown path in the woods. There were elaborate stone bridges where the trail passed over creeks and culverts and there were other reminders around town of some other time. As kids they'd found stone walls in the woods that seemed absurdly obsolete—dividing only forest from forest.
On walks during hunting season, the grandfather had pointed out the ruins of the first Darling farmhouse. The stone foundation was six or seven feet deep, with small trees growing out of the basement floor. The boy and his friends had found their way back to it, daring one another into the small, dark root cellar. There was a narrow stone well that made them think of the stories of children falling into such small, tight places.
The grandfather had told him about his days working on the farm: hay season, jacking deer, the farmer beating him worse than his father ever could. It was the Depression and the great-grandparents sent the grandfather away. At the farm they could feed him, and the farmer occasionally brought food to his parents as payment for his labor.
The boy thought about his grandfather's situation and felt a flash of guilt—here he was, well fed and underworked, and still such a screwup. An ache grew in him, and he closed his eyes and crooked his neck to hit at the joint. He hardly coughed anymore, but he flinched when he looked up—Mr. Benson, in a headband and sweatsuit, jogged toward him on the path. He tried to let his arm fall casually to his side, and he flicked the joint over the edge of the trail. He turned his head to the side and let the smoke slowly out of his nose.
"Hello, Teddy," Mr. Benson said.
"Hey, Mr. Benson."
Mr. Benson kept jogging and disappeared as quickly and as quietly as he had come. "Fuck," the boy said under his breath. "Goddamn fuck-all."
He turned, and when he saw that Mr. Benson was long gone, he went back and retrieved the joint. He hit at it several more times, snuffed it out, and slid it back into the empty pack of gum.
"The luck I got," he said. He remembered the hot afternoons he and his father had spent fishing on Mr. Benson's boat. He wondered if Mr. Benson had recognized the smell of pot. Then he wondered if Mr. Benson would fink on him.
The boy arrived home to an empty house. He went first to the bathroom and brushed his teeth. Then he tipped his head back and held each eye open for a couple of drops of Visine. He wiped at the corners of his eyes with the sleeve of his shirt and left the bathroom. In the kitchen he filled a bowl with pretzels, pulled a pickle out of a jar in the fridge, grabbed one of his mother's diet sodas, and went to the living room to watch television. After several sit-com reruns, his mother arrived. She had stayed late at school for a PTA meeting and picked up a pizza on the way home. The two greeted each other and made their way to the kitchen to eat. The house was quiet without the television on.
"Did you talk to Mr. O'Rourke?" she said.
"Nah," he said, chewing on one of those little pieces of sausage that reminded him of rabbit turds.
"Theodore," she said.
"What?"
"He gave you a D on your last exam." She whispered it, like she didn't want anyone else in the empty house to hear about her son's grade.
"It's just French," he said. "What am I, going to France?"
She shook her head. "Should I tell your father?"
"He said pay attention to my _important_ classes." He took a bite of the crust that lay on his plate. "I got B's in algebra and technology." He said it through his half-chewed pizza and his mother shook her head some more.
"B minuses do not count as B's."
"They're not C's, are they?" he said. "Then they must be B's. Ma—"
"Quit it," the mother said. "Nobody likes a wiseass." He stopped. When she started cussing, he knew to listen. "The stove downstairs needs wood," she said. "I moved what was left upstairs before I left for school."
"Is there enough for tonight?" he asked.
His mother shook her head and he frowned. This meant he'd have to haul wood over the twenty-five yards that lay between their house and the woodshed. It wasn't a terrible chore, but he wondered why she couldn't have just left enough for the night. He also wondered why the hell they'd built the woodshed so far from the house in the first place. Or why they were so stingy that they couldn't let the oil furnace heat the place for a night.
The mother finished her usual two slices of pizza and conscientiously left two pieces for lunch the following day. She sat and watched her son devour his half of the pie. He stuffed down bite after bite and the orange grease of cheese and sausage collected at the corners of his mouth. He wiped his face with the back of his hand.
"Am I grossing you out?" he said.
"You need to eat if you want to grow," she said.
He shook his head. He wanted to disgust her.
"I want you to be big," she told him. "So when you get married and we dance, I can put my head right here," she patted the inside of her shoulder. "I want my hand to feel small in yours."
"That's gross," he said. He pushed his chair back and left the table.
As much as he resented having to get the wood, he couldn't deny that it was a beautiful night. The empty wheelbarrow bounced over the bumps and roots on the path to the woodshed. The cool evening air felt good on his still half-stoned head. He looked up and saw a handful of stars through the thick pine boughs. He smelled the cut wood in the shed some ten or twenty yards before he got there. He remembered scampering to his father's lap in the evenings and the way he smelled after a day of cutting firewood—the thick scent of his perspiration mixed with chainsaw exhaust and the smell of the sawdust that still clung to his damp T-shirt. The father's fragrance often contained a hint of cheap American pilsner. A great fondness for the man welled up in the boy, but it was quickly followed by a tremendous ache—his father was so, so far from him, standing in the backyard as he was, half stoned and torn up as hell. He stood for a moment and waited for it to pass. When it wouldn't, he set to his work anyway, wiping at his nose with the coarse collar of his work glove.
He was back inside, unloading his third wheelbarrow of wood, when he heard his mother call down to him.
"What?" he shouted at the unfinished ceiling of the basement.
"Get up here," she hollered.
"Goddammit," the boy said to himself. He threw off the heavy leather gloves and set to shedding his boots for the trek upstairs.
When he turned the corner at the top of the stairs, he saw Officer Duncan standing with his mother by the front door. The boy's mind raced at the sight. He thought of the half-smoked joint in his bedroom. He thought of Mr. Benson and wondered if he'd finked on him.
"Evening, Teddy," Duncan said.
The boy nodded and stopped where he was, halfway down the hall.
"He wants to talk to you," the mother said.
"Come here—I ain't going to bite," Duncan told him.
The boy closed the distance between them and jammed his hands in his pants pockets.
"I want to start by saying this is an unofficial visit."
"What's that mean?" the mother asked him.
"Means no one sent me. Means I'm here on my own."
Neither the boy nor his mother responded.
"I'm here as a friend. To tell you I think they've got a case against Theodore." Duncan shifted his weight from one foot to the other. "I can't say specifics but...do you mind if I come in and have a seat and we can talk about this?"
"I think it's best if you stay where you are," the mother said.
Duncan nodded. "I think it would look good if Teddy came clean. He could say he was in shock and afraid and not thinking right. I think we'd have a much better chance of getting him off."
"I think you should leave," the mother said.
"Donna? You've known me a long time, haven't you?"
"Yes I have," she said. "But I've also lived here all my life and I can see how things are." She shook her head and rubbed at the bridge of her nose. "You're going to scapegoat this kid so you all can come out looking shiny as hell and so you can keep selling off this town as a country club. You think they're going to start moving here again if one of these kids gets killed and no one gets punished for it? Damn right they're not. And more people eventually means more money for each of you, I know it. I know how it is, Dick."
Duncan stood speechless for a moment. "That's quite a scheme you got cooked up," he said. "But it ain't right. I'm trying to tell you the best way through this thing that I can see. I'm insulted. I've always been square with you two."
"Say what you want, Dick, but this is my boy." She pointed at him. "The only one I got in this world. Now, you want him, you can take him, but know that you're going to have come through me. Because without him, nothing in this world means a thing to me." She stopped and wiped at the outside corners of her eyes. "Now, _as a friend,_ " she said. "I'm going to have to ask you to leave, kindly."
"All right. But this is all I can do for you. This is the end of the road for me."
The mother nodded and closed the door behind him. She turned and looked at the boy. His eyes were wet. She stepped close and put her arms around him. He didn't take his hands out of his pockets, but he let his face fall to her shoulder. His torso jumped with quick sobs.
"Don't you think we should tell?" he said.
She shook her head. "They can lie, you know. They can say whatever they want to try and get you to talk. It's legal for them to do it. I bet you anything he's here because they hit a dead end. I bet they're saying the same thing to Kevin as we speak."
"You think?"
"Don't worry," she said. "Trust me." She stepped back from him. "Now go and finish the wood so you can get to sleep."
He nodded and headed back down to the basement.
The following morning the boy showered and dressed. When he got to the kitchen, he found his mother at the table with her head in her hands. When she lifted her head, she looked distraught.
"What?" he said.
She shook her head.
"Mom?"
"I don't know why someone would do this to us," she said. She pointed at the front of the house. He walked to a window.
There was a helix of tire tracks across the lawn—brown dirt billowed over the grass where the tires had slid and spun. The mailbox had been knocked over and dashed in with a large rock that still sat atop it. The driveway bore two broken flowerpots, their soil spread in halos around them.
"Son of a bitch," he said. He took a deep breath.
"Do you know anything about this?" The mother stood behind him. "Could it be Kevin?"
He shook his head. He felt ready to explode. Hadn't Becker seen Colleen, smiling and talking with him? Wasn't that enough?
He winced when he heard his mother calling in sick.
"Something just came over me," she said. "I feel terrible."
The boy took the bus and fumed through school. Youth members smiled and chuckled at him in the hallways. He wanted to make a move on them: sucker punch one in the throat, throw a brick through a windshield in the parking lot, lob a Molotov cocktail through the storefront of George's mother's shop.
He found Terry at lunch and told him what had happened and what he wanted to do.
"No way," Terry told him.
"What?" the boy said.
"We mess with them, they tell on us," Terry said. "They're tattletales. You know the type."
"Piss off," said the boy.
"You want to do something, go right ahead," Terry said. "Just don't count me in."
The boy left the lunch room. He couldn't take the glaring eyes of the Youth any longer. He went to the bathroom and sat in the stall. If he'd had a cigarette he would have smoked it. If he'd had a lighter he would have burned himself. Instead he sat with his head in his hands, his elbows on his knees. He clenched his eyes as tightly as he could, but several small tears managed to push their way through the corners. He cursed himself for being so soft.
He skulked through the rest of the school day and took his bus home. When he got off at his stop, he saw that the mailbox was gone and the driveway clear of debris. As he crossed the lawn, he stepped over the tire tracks. The billowed dirt had been raked flat and sprinkled with pale flecks of grass seed.
He entered the house and greeted his mother but the two spoke little. He poured himself a glass of water and went to the couch. He turned on the television and bounced through the channels.
The phone rang and his mother answered it. By the tone of her voice, he knew it was his father. She spoke with him for a moment and then held the receiver out to him.
"He wants to talk to you," she said.
He shook his head.
She nodded and shook the phone.
He stood and wiped his mouth with the sleeve of his shirt. He crossed the room and took the phone.
"Hey," he said.
"What is going on up there?" the father said.
"I don't know."
"I find it hard to believe you know nothing about this lawn business."
The boy didn't speak.
"You can lie to me all you want," said the father. "But you tell whoever did it, I ain't you. When I come home I'm not going to lie in bed, pretending to be asleep while some kids tear up my property."
"Dad," he said. "I don't know who did it. I swear I didn't even hear them."
"This really helps sell the house, you know," the father said. "Me and your mother are killing ourselves trying to keep this family together and you keep doing the opposite."
"Dad—"
"Ted, I know about the drugs," the father said. "I got a phone call. And I got a good feeling this is something to do with that."
The boy held his breath. He didn't speak.
"Don't you think you've put your mother through enough?" the father said. "Don't you think? You're going to quit that right now. And you're coming down here with me soon as we can do the paperwork for school. Understand?"
"What about Ma?"
"What part of this don't you understand?" the father said. "You're under investigation for manslaughter. You keep messing around and you're going to jail. Then what about your mother?"
"I don't want to go to jail." He choked it out and a silence came over the line. "Dad?" he said.
"Christ, Ted. I'll do everything in my power. Everything I can. But you got to quit doing this stuff that can get you there. Understand?"
"Yes," he said.
"Say it," the father said. "Tell me you understand."
"I understand."
"Good," the father told him. "We'll get this all taken care of."
There was a moment of silence on the line.
"I don't want to move down there," the boy said.
"You say it like you have a choice."
"I do," he said. "I do."
"You know what a fool you sound like?" the father said. "You don't understand—"
The boy pulled the phone away from his head and held it at arm's length. He listened to the hollow outline of his father's voice at that distance. He turned the phone over and gently put it back in the receiver.
"If he calls back," he told his mother, "I'm not talking to him."
Later that night the boy sat at the desk in his bedroom. He struggled with his homework, distracted as he was. He heard the phone ring through the closed door of his bedroom. The ringing stopped and he heard his mother call his name. He stood and went to the hall.
"I'm not talking to him," he said.
"It's Terry," his mother called out from her bedroom.
He rumbled down the stairs and picked up the phone in the kitchen. He waited for the click of his mother hanging up.
"What's up?" he said.
"What are you doing?" Terry asked him.
"Homework."
"Can you get out?"
"Why?"
"Can you?"
"After she goes to sleep," he said. "Why?"
"Be out in your driveway at eleven."
He heard the click of Terry hanging up and he dropped the receiver back in the cradle. He went back to his room and sat at his desk. He struggled with his homework for the next hour and a half.
At ten-thirty he stuffed pillows under the covers of his bed. He wrote a quick note to his mother:
_Mom,_
_Don't worry. I couldn't sleep. I went for a walk. I'll be back soon. I'm sorry about everything._
_Love, Ted_
He folded it and put it on the pillow, where she would easily find it, were she to turn on the lights. He pulled on a pair of jeans, grabbed a hooded sweatshirt from his closet, eased open the door of his room, and stepped out into the hallway. He listened. One cautious step at a time, he walked down the hall and down the stairs, stopping at the bottom to listen again. He went through the dark living room, carefully weaving around the coffee table and footstool. He turned and went down the stairs into the basement. At the bottom he listened, and when he heard nothing from the house above him, he walked quickly across the cement floor to the door outside. He eased it open and closed as quietly as possible and went around to the front of the house.
The night air was sharp. It cut at his bare skin and he pulled up the hood of his sweatshirt and slid his hands into the pocket at the front. Autumn was upon them and the crickets no longer sounded. The boy felt as though he could smell the fallen leaves melting back into the forest floor. That was the smell of fall to him, the smell of decay.
He sat on the short stone wall that ran halfway down the driveway. When his butt grew cold and stiff, he stood and walked back up the driveway and across the sidewalk at the front of the house. In the moonlight he could just make out the front yard. He could see the fresh soil of the tire tracks. The anger began to grow in him again.
He heard footsteps in the sand at the end of his driveway and he saw Terry's silhouette out there in the darkness. The boy made his way down the walk and out to meet him.
"Hey," the boy whispered.
Terry nodded. "Come on," he said, waving the boy to follow. Terry took a right turn at the street. The only sound was their feet scratching at the sand on the side of the road. They came upon a car parked in the ditch. When Terry opened the door, the interior light came on and the boy saw that it was Terry's mother's Ford. He stood for a moment, knowing that Terry didn't have a license, knowing that no good could come of this.
The boy knew he could shake his head. He could turn and walk back to his house and return to his bedroom. He could sit alone on the side of his bed and he could do the things he did to himself there. He flinched and shook his head. When Terry reached over and unlocked the passenger side door, the boy pulled it open and stepped in.
"You ready?" Terry said.
The boy shrugged. He could smell booze on Terry's breath. "You been drinking?" he said.
"You think I'd lift my mom's car sober?" Terry reached under the seat and came out with a bottle of liquor. It was flat and slightly rounded so as to fit conveniently in a back pocket of a pair of pants or the inside pocket of a coat. Terry unscrewed the cap, took a sip, and handed it to the boy. "Whiskey," he said, his eyes closed and his face winced in a sort of pucker. The boy took the bottle and pulled hard on it. The whiskey was hot and smoky in his mouth. Even hotter in this throat. He coughed quickly and took another hard sip. "Hey, go easy," Terry said.
"I got to catch up," the boy said.
"Case you ain't realized, I got a few pounds on you," Terry said.
The boy smiled and drank again from the bottle.
"Hey man, I'm serious," Terry said. He twisted the whiskey from the boy's hand. "We got shit to do."
"What you got in mind?" the boy said. He coughed into his hand.
"We're going to get back at those cocksuckers."
"The Youth?"
"Your mom..." Terry said. He took a sip from the bottle and screwed the cap back on. "Remember those multiplication tests? The timed ones?"
The boy nodded.
"Man, I just could not do those things. I'd break my pencil in the middle of the test to get out of it, you know? Your mom kept me in at recess for extra help and I must've broken five pencils one time. So obvious—but your mom never said a word. She just smiled and told me it was okay and let me try again."
"Yeah?" said the boy. He almost laughed at the thought of it.
"I'll kill those motherfuckers." Terry pounded a fist on the steering wheel. "Messing with your house like that." He pulled out a pack of cigarettes and rolled down the window. He flipped open the top and pulled one out between his lips. He held the pack out to the boy and the boy took one. "Roll down your window," Terry told him. "I can't have this thing stinking to hell." The boy did as he was told and Terry lit both of their cigarettes. He started the car, turned on the lights, and pulled out onto the road.
The boy's head started to spin from the nicotine and his body grew warm with the flush of alcohol. It felt wonderful. And as soon as he felt it, he was afraid the feeling would leave him.
"Give me some more," he said. He held out his hand. Terry reached under the seat for the bottle and handed it over.
"Go easy," he said. "I'm serious. We got work to do."
"You got a plan?" The boy took a long gulp and hissed between his teeth. "Yow," he said. He shook his head back and forth, back and forth. A flash of nausea rolled through him and he thought he would cool it on the booze for a bit.
"Yes," Terry said. "I got a plan."
The car rolled and wound through the wooded country roads. The boy was impressed with Terry's driving, although once he did drop two tires in the rocky ditch and the car bumped and rumbled until he jerked it back to the pavement.
The boy knew it was dangerous, driving with a drunk. He'd seen the commercials. He'd been subject to the campaigns in school. But to care about your physical well-being, you have to care about your physical well-being. The boy's drunk mind fantasized about crashing full speed into one of the broad pines on the side of the road—his body flying into the dashboard, through the windshield, headlong into the trees and small saplings. Pain was what his body craved. It pleaded to be burned and scalded and dashed to pieces. It longed for relief.
They took the back way out of town toward the high school. The boy wondered what Terry had in mind, but he didn't ask. Terry stopped at the intersection on the bypass. He looked both ways several times. He cautiously throttled the car out onto the two-lane and gently brought the car up to speed. He looked back and forth between the road and the speedometer, carefully working the throttle to keep the car at a steady speed.
The wind boomed through the windows, setting everything loose to fluttering. The car rolled over the uneven road like a ship on the water. Terry bent his neck and took another cigarette from the pack with his lips. He couldn't get his lighter to work in the wind so he reached down and pressed the lighter in the dash. When it popped out, he retrieved it and set his cigarette to smoldering.
The boy looked across the car at Terry, illuminated by the odd glow from the dash. The lights from the passing cars and street lamps came in and over them, providing quick flashes of clarity. The boy stared at Terry's hand on the wheel, the cigarette sprouting from between his fingers. Terry's left hand took hold of the bottom of the wheel as he brought the cigarette to his mouth. The red tip went bright as he inhaled. His hand returned to the wheel and he blew the smoke in the direction of the open window. His eyes squinted and his lips were tight as he forced the smoke out of his lungs. Looking at Terry, the boy felt warm with something.
"Thanks," he said.
"Huh?"
"Thank you," the boy said.
"For what?"
"For everything."
"Don't start that."
"I'm serious," the boy said.
"You're drunk is what you are."
The boy shook his head.
"Drunk or queer," Terry said. "Take your pick."
The boy smiled and looked back out the windshield at the tunnel the headlights created before them.
They passed the Haneys' gun shop. They passed the school. Terry finally slowed the car and turned into the parking lot of Jason Becker's apartment complex. He drove out to the far corner, where Jason's car sat alone. Terry smiled at the boy. He turned the car around and pulled back onto the bypass. He took the next left turn and after a couple hundred yards, he pulled over. He pointed at the woods.
"His car is that way," Terry told him. He took out the bottle and took a long drink. He handed it to the boy. "Finish it," he said. The boy was already reeling, but he took the bottle and did as he was told. He hardly flinched at the burn.
They got out of the car and Terry went to the trunk. He twisted the key in the lock and it popped. Terry bent over and came out with a small backpack. He threw it over his shoulder and it rattled.
The boy put his hand on Terry's arm. "What's in there?" he said.
"I'm doing this," Terry said. "You can come or not."
"Not his apartment," said the boy.
"Are you crazy?" Terry said. "That little piece-of-shit car of his."
"I don't know," the boy said. His tongue was thick with the whiskey and his voice sounded funny to him.
"You can come or not," Terry said. "This ain't for you anyway. This is for your mom." He turned and headed into the woods. The boy stood for moment but he quickly set off after Terry. He didn't want to be alone.
He had trouble walking in the woods. He stumbled on downed limbs and bumped into trees. The whiskey was working hard on him. He followed the sound of Terry's footsteps and the red glow of his cigarette. He was glad to see the woods open into the field above Jason's apartment complex. Before he stepped into the clearing, a briar patch got hold of his pants and drove thorns into his legs. He winced and set to pulling them free from his clothing. As he pulled them from his pants, they took hold of the arms of his sweatshirt. His predicament was getting worse. He grew frustrated and stood, tearing his arms up and away from the briars. He got balanced and forced his way step after step out of the patch. The vines ripped and popped out of his clothing and clawed at his skin. He stepped out into the grass of the field and it felt good to be free. Terry was halfway across the opening and the boy jogged to catch up, rubbing at the burning in his thighs where the thorns had got to him.
He caught up with Terry at a hedge just above the parking lot. Jason's car sat in the lot some twenty-five yards away. Terry had the backpack down on the ground and he fished two glass bottles out of it. He went back for some socks at the bottom and quickly assembled the Molotov cocktails.
"Whose socks you use?" the boy said.
"My brother's," Terry said, smiling. "I took the newest, cleanest pair I could find."
The two giggled for a moment.
Terry pushed one of the bottles at the boy, but he didn't take it.
"Come on," Terry said. "It'll be faster if we do it at the same time."
Still he didn't take it.
"Think about it," Terry said. "All they've done to you?"
The boy reached out and took the bottle in his fist. "Light it," he said.
Terry thumbed the lighter and held the flame to the socks. They both watched the flames climb the cotton. "Let's do it," Terry said. He smiled at the boy and ran around the corner of the hedge. The boy followed. Terry ran half the distance to the car and hurled the bottle, but the boy kept running. He watched Terry's bottle crash to the pavement just under the driver's-side door. The flames burst and waved up to the window. The boy was hardly fifteen feet from the car when he let his go. The bottle crashed into the rear bumper and set the hatchback on fire. He stood for a second, admiring his throw, and then turned and sprinted. Terry was already gone. When the boy neared the tree line, he heard Terry call his name. He followed Terry's voice and found him hunched over, his hands on his knees.
"Holy shit," Terry said. "I can't believe you hit it."
The boy turned and watched the car burn. Terry's was nearly out, but flames still throbbed across the back of the car. The boy held his breath, waiting for the whole vehicle to catch, waiting for some grand explosion or finale, but the gas gradually burned off, leaving only the smoldering edges of a few bumper stickers.
"That's it?" said the boy.
"That's it," Terry said.
"It's not enough."
"He'll get it."
"It's not enough," said the boy.
"It's enough for tonight," Terry said.
The boy shook his head. He wanted to run back down the hill and jump on Jason's hood. He wanted to kick his foot through the windshield and piss all over the interior.
Terry reached over and took him hard by the arm. He looked and Terry shook his head. He motioned back to the car with his head. "Come on, man," Terry said. "We should git."
He nodded.
Back in the car, on the way home, he looked over at Terry.
"My dad knows I been smoking dope," he said.
"How?"
"Mr. Benson caught me on the railroad bed."
"You ain't told on me?"
The boy shook his head.
"You better not," Terry told him.
"I won't," he said.
"What did he say?"
"I got to move down there with him."
"You going to?"
"Why not?" the boy said.
"Friends, I don't know."
"What friends I got?"
"You got me," Terry said.
"Yeah?"
Terry nodded. "You always had me."
The boy nodded. He set his jaw and tried to swallow the knot in his throat.
"Do I got you?" Terry said.
"Yeah," said the boy. "You do."
"Good," Terry told him.
**16**
**T** he boy woke in desperate need to piss. He rolled over and stood, only to immediately take a seated position on the side of the bed. It felt as though someone had driven a nail behind his left eye. His stomach was acidic and pukey. His mouth tasted as he imagined the soles of his shoes might if he had walked through an ashtray and a puddle of vomit.
"Oh, good Christ," he hissed. He hung his head and clenched his eyes.
He took slow, cautious steps to the bathroom. Once there, he sat on the toilet to urinate, his elbows on his knees, his head in his hands. He stood and retrieved his underwear from around his ankles. He went to the sink and brushed his teeth. He grabbed the bottle of mouthwash from below the counter and gargled to try to kill the whiskey he imagined was on his breath.
Back in the hallway, he leaned against the wall, looking down the stairs.
"Ma?" he hollered.
"Teddy?"
"Can you call in for me? I feel terrible."
"What's wrong?"
"I told you—I feel like crap."
"How so?" She came around the corner and looked up at him.
"I got a headache and I think I'm going to puke."
"Fever?" she said.
He shook his head.
"Come down and have something to eat," she said. "Maybe you'll feel better."
He shook his head. "I'm going back to bed. Just call in for me."
"Are you going to miss anything important?"
"Ma, come on."
"I think you should go."
"I think you should call."
She didn't say anything, just stared up at him.
"You owe me," he said.
"What?"
"Just do it, Ma."
"What do I owe you for, mister?"
He shook his head.
"Say it," she said.
"For keeping your secret."
She shook her head. "Ted," she said. She paused for a moment. "It's not my secret."
He leaned back and punched the wall. "Fine," he snapped. "Don't call in. I don't give a shit." He walked back into his room and slammed the door. He grabbed the chair from his desk and jammed it just under the knob. He tested the door and the chair held. He went to his bed and pulled the covers over him. He listened for his mother at the door, but the knob never moved. Sometime later he heard her car start in the drive and back out. After that he heard his bus come and go. He lay under the covers with his eyes clenched, but sleep would not come. The headache and the nausea and the fury kept him tossing.
Eventually he threw back the covers and got dressed. He went downstairs and poured himself a glass of water and fished a few aspirin from a bottle in the cabinet just beside the sink. He choked down the bitter, chalky tablets and went to the couch in the living room. He lay down and pulled the afghan from the back of the couch. He covered himself and reached out from under the blanket for the television remote. The tube sprang to life and he went through the channels until he found a station with morning cartoons. An episode of _Tom and Jerry_ was in full swing. A clothes iron fell from its perch and struck the cat in the face. The cat's face took on the flat shape of the iron.
The boy was startled awake by a loud knock at the door. He sat up quickly and rubbed at his eyes. A commercial squawked from the television and the knock came again. He stood and looked out on the driveway. His body grew rigid at the sight of the local police cruiser. He crouched back down, out of sight of the windows. The knock came again, louder and more insistent. He knew it was Duncan. It had to be.
"Ted," a voice hollered from the other side of the door. It said something else but he couldn't make it out. He knew he could stay hidden. He knew he could lie on the floor and wait for Duncan to leave. But he also knew that doing so would only prolong whatever was coming.
He went to the door, turned the bolt, and drew it open. Duncan stood at the top of the stairs.
"Ted," Duncan said.
"Yeah?"
"You look tired."
"I was sleeping on the couch," he told him. "I'm sick."
"That's what they said at school."
He nodded, realizing that his mother had called in before she left. "My mom's not here."
"I know," Duncan said. "Feel like going for a ride?"
He shook his head. "Not really. I'm sick."
"Come on. Your parents wouldn't want people seeing the cruiser in the driveway." He pointed his thumb at the car.
"Since when do you care what my parents want?"
"Don't try to make me the bad guy, Ted," Duncan told him. "I've always been square with you."
He looked at the floor, embarrassed, because he knew Duncan was right.
"Get your shoes on. Come on."
He nodded and went to the living room to turn off the television. He took his keys from the kitchen table and walked back to the front door. He stepped into his shoes and grabbed his sweatshirt off the back of the chair beside the door. It had a slight smell of cigarette smoke.
They passed the entrance to Sandy Creek in silence. The boy eyed all the buttons and switches on the dash. The radio squawked now and again. When they neared Woodbury Heights, Duncan slowed the car and they turned in. He drove to the culvert at the top of the road and stopped. He put the car in park and looked over at the boy. The boy was afraid to turn his head.
"Hand me my lunch?" Duncan pointed to the brown bag near his feet.
He bent over and returned with the bag. He handed it over.
"I like to come up here to eat," Duncan said. "It's quiet, out of the way." Duncan opened the bag and rummaged around inside. He came out with a cellophane-wrapped sandwich.
"You like egg salad?"
The boy shrugged.
"Here," Duncan said, holding out half the sandwich. "My wife makes great egg salad."
"That's all right," said the boy.
"Take it." Duncan thrust the half-sandwich at him. "I've got a whole other one in here."
He took the sandwich and the two ate.
"It's good," the boy said, holding up the sandwich.
"Told you," Duncan said. "Chips?" He held out an open sandwich bag full of crinkle-cut potato chips.
The boy reached in and took one.
"Take a handful. You're doing me a favor. If I ate everything my wife packed, I'd be a blimp."
The boy reached over and took a few more chips. A little later Duncan offered him a cookie and he took it.
"You know why we're here?" Duncan said.
The boy shrugged. "Something with the Dennison case?"
"Yes and no." Duncan chewed and swallowed his last bite of cookie and chased it down with a sip from a can of diet cola. "They had a little incident over by the high school last night and I thought you might know something about it."
"What happened?"
"Someone tried to set a car on fire."
"Huh," the boy said. "Sounds pretty crazy."
"Don't play stupid, Teddy."
"I don't know what you're talking about."
"I'm going to say this once," Duncan said. "You have no idea how much shit you are in at the moment. And you better get it through that head of yours that I'm trying to help you."
The boy shook his head.
"You don't get it, Ted—this is just the beginning. Look at your life since this happened. You're falling like a homesick rock. You think that getting off these charges is going to stop that? Two wrongs only make way for more wrongs. You screwed up—you loaded the gun. You can't change that, but you can tell the truth."
"It was Kevin and Bobby," the boy said.
Duncan shook his head. "Yours are the only prints on the bolt," he said. "Kevin's came up on the casing and the trigger and everywhere else, but yours are the only ones on the action of that gun, Ted. They're waiting for a few more test results, but they have what they need to file charges against you."
"No way," he said.
"They've got the physical evidence to corroborate Kevin's story," Duncan said. "What is there to back up yours?"
"My word."
"Pardon me for saying this, Ted, but your word is looking a little shoddy at the moment. And if this little firebomb incident sticks, you're not going to look so repentant."
The boy shook his head. He looked out the window at the tree line. "It was their fault," he said.
"This isn't about them. This is about letting yourself off the hook. You need to understand that. I've seen other boys like you, Ted—boys who never get it and they just keep screwing up and screwing up, just like you're doing now. If you don't come clean, you don't have a chance. I know it."
"I'm not screwing up," he said.
"The day before Bobby was killed you admitted that you and Terry were up here on the Darling property. That day I found the remains of three Molotov cocktails up here in the circle. Last night two very similar incendiary devices were used to light up a car belonging to one of these Young American kids and they're pointing the finger at you, Ted."
"They're just pissed because I stole one of their girlfriends."
"You don't even want to know what they're saying about you and her," Duncan told him.
"I know what they're saying and it's crap. Go ahead and ask Colleen."
"I hope you're right, but you're missing my point, Ted. I'm not trying to get you in more trouble—I'm trying to keep you out of it. If you just tell the truth, I think we can get you out of most of this mess."
"Take me home," he said.
"For Christ's sake, Ted, we know you loaded the gun. And if you get up there and lie to the judge, it's going to reflect very poorly on you. Lying in court—perjury, Ted. It's a crime. It can be added to whatever else they get you on."
"Take me home."
"I don't have to tell anyone about the broken bottles I found up here, Ted. We can get you off the arson charges. And if you tell us what happened with the Dennisons, I think it'll be clear to everyone that it was just an accident."
"You're lying," he said. "She said you'd do this. I say what you want and then you put me away and they sue us for everything."
"Your mother loves you very much. Too much, Ted. She's not seeing this clearly."
"Take me home now."
"You got to trust me," Duncan told him.
The boy opened the door of the cruiser and pushed himself out. He got his feet under him and sprinted for the tree line, not bothering to close the door of the car. He ran as far and as fast as he could, dodging trees and small saplings, hiking his legs up and over the downed limbs and rocks in his path. When he stopped, his lungs were raw from the fierce breathing. Terry's cigarettes the night before hadn't helped. With his hands on his knees and his head hanging, he heaved several times until the remnants of the egg salad, chips, and cookie poured out his mouth and fell to the dry leaves on the ground. He hacked and spit several times and wiped his mouth with the sleeve of his sweatshirt. He looked back in the direction of Woodbury Heights and was happy to see that Duncan hadn't followed him.
The floor of the forest was deep with downed leaves and the boy kicked and crashed through them. He jumped a couple of deer as he crested a small knoll. He heard them before he saw them, and then only caught a glimpse of their white tails as they bounded away. On the other side of the knoll, he found the old stone wall he was looking for. If he followed it to the right, he would come to the small creek that eventually crossed the road just down from his house. Instead he turned left on the wall and walked until the stones took a sharp right turn. He knew to bear slightly left from the corner of the wall and walk until he came to the swamp.
When he was young, when trouble had erupted in the house, when his parents had exploded at him or each other, he ran for the woods, where it was quiet and safe. He built small forts and dreamed of never returning. He thought that if he could build a warm enough place to sleep, he could sneak back into his parents' home while they were at work and steal the food and supplies he needed and stay away forever.
The feeling struck him again. He wished he never had to go back to any of it. But even as a child he knew the futility of his escape plans. The evening air poured through the flimsy walls of his small huts and lean-tos. No matter how tightly he thatched the branches that he broke from the surrounding trees, light from the rising moon and stars poured through the makeshift ceilings. The cold and the solitude sent him shivering home every time. Upon his return, he found his parents glued to nighttime television. They looked up, even greeted him, but never seemed to acknowledge his absence. He felt somehow robbed, unable to drum up their attention—they never even acknowledged the familial strife that had sent him running in the first place.
He came to the swamp and walked the edge of it for some time. After fighting through a small stand of young pines, he poked out into his uncle's backyard.
He crossed the small patch of grass and climbed the wooden stairs that led up to the porch. He held a hand to his brow and looked in a window. It wasn't much past noon and he figured his uncle and aunt wouldn't return until five or six. He reached above the light fixture at the top of the door and came down with the spare key. He unlocked the door and returned the key to its hiding place.
The house was quiet and smelled of food. He found some bread and deli meat and made a simple sandwich. He took a box of off-brand cheese-flavored crackers from the top of the refrigerator and poured a small pile of them on the side of his plate. He pulled a beer from the door of the fridge and went to the table to eat. When he finished, he washed and dried his plate, took another beer, drank half, and poured the rest down the drain.
He went into the basement and let himself into his uncle's room. The deer heads stared, wide-eyed and unblinking. The gun cabinets reflected the light that came in from the small cellar windows just below the ceiling. He walked around the room and stopped at the cabinet that held the guns from his house. He pulled at the door but it was locked, so he reached his hand up and ran it along the top of the cabinet. He found the small key and fingered it into the lock.
His father's shotgun was older than the boy but the father had kept it well and the age did not show. There was ornamental carving on the wooden stock and the blue steel receiver of the Remington 1100.
In the years before the Darling land was sold, the father pulled him from school for the first week of deer season—no matter how much his teachers objected. In the woods, on the planks of a tree stand, the father stood with his back to the tree, the boy with his back to the father, the shotgun standing before them both. When the cold finally overtook him, and the boy's body shivered uncontrollably, the father unbuttoned his heavy Woolrich coat, pulled the boy to the warmth inside, and closed the garment around them both—the three, father, son, and firearm, becoming some strange totem.
The boy reached to the floor of the cabinet and fingered open a yellow-and-green box of double-ought buckshot. He let a shell into the chamber. With one hand he released the action, and with the other he let it slowly close. He thumbed the rest of the shells from the box into the belly of the firearm.
He sat in his uncle's chair and stood the gun vertically, stock down, between his knees. He found it difficult to get his top teeth over the front sight. Once past his teeth, the sight dug into the roof of his mouth. When he exhaled, there was a new sound: the hollow call of his breath around the barrel. He closed his eyes and clicked the safety off with his index finger. It was all he could do to focus on the placement of his fingers. There wasn't room for the guilt, for the sick feelings that over-ran him. There was just enough space for the concentration it took to keep his fingers off the trigger.
He wondered what Lawrence had felt in his final moments behind the barn. For the boy, there was little feeling in it. The concentration it required seemed to quash that part of him. His grandfather's words ricocheted through his head, _Some ways are just no way, some ways are no way, some ways are no way at all._
The steel of the gun was cold in his damp hands. The smooth wooden stock was slick with his perspiration. He felt guilty for implicating the firearm in such an egregious act, for somehow tainting all the good memories.
He ran his finger along the rounded edge of the trigger guard. The pad of his finger stopped where the guard returned to the stock, where the trigger entered the receiver. His finger slid down in the direction of the floor and found the round button of the safety. He popped it back on.
"Some ways are no way," his whispered when the barrel was free from his mouth. He rested the butt of the stock on his inner thigh and drew back the action. He held out a hand and caught the ejected shell. He closed and opened the action four more times until the magazine was empty.
He took his shirtsleeve and gently wiped the barrel free of the moisture from his breath. He wrapped the cotton around his finger and twisted it inside the muzzle. After he finished with the barrel, he wiped down the receiver and the stock.
With the gun back in the cabinet, he thought how foolish he had been—loading more than one shell, as if one wouldn't have done the trick.
He twisted the lock and pulled the door to his uncle's home closed behind him. He took the porch stairs two at a time and loped across the lawn. Again he fought through the small pines and followed the edge of the swamp. He found the stone wall and walked the length of it. Instead of following the creek to his house, he jumped across it and fought his way up the small hill on the other side. The ground was slick with leaves and he grabbed at small trees to help him up the incline.
Darling's Rock hulked just past the crest of the small knoll. The granite boulder was an oddity in the landscape—it was the size of a small bungalow and sat perched atop the hill. The side he approached was pointed like a shark's snout on one end and slightly rounded on the other, making it look like a distorted map of the United States. He walked up close and rested a hand on the cool, coarse rock. He bent to look into the mouth of a small cavern beneath it—as a child he climbed inside and had immediately felt a wash of terror when he realized the immense weight that hung over him. He'd scampered out and shown his father a souvenir from the cave: a small rounded ball, brown and about the size of a grape or olive.
"It's shit," the father said.
The boy looked closer.
"Porcupine," the father said. "They like places like that."
The boy quickly flung away the turd and wiped his hand on the leg of his dungarees. His father laughed. The boy asked how the rock got there and the father explained glaciers to him—how the huge sheets of ice could move such things at random. The boy tried to imagine it.
"What about the trees?" he asked.
The father passed a flat hand horizontally across the landscape. "No trees."
The boy, now grown and leaning on the rock, tried again to imagine the terrain—nothing but ice, and when it melted, nothing but stone and mud and water in every direction. He walked around the rock, under the overhang of the shark snout, to the flat face of the far side. Names and dates were chiseled into the stone, the oldest going back to the 1700s, when the surrounding area was most likely nothing but pasture—having been logged barren again.
What little he knew of his family history told him they had not yet arrived in the area at that time. They were French-Canadians who had fought in the Civil War in exchange for American citizenship. Even after fighting for the Union, the early French immigrants were ostracized by the town. They were forbidden to attend the local Catholic church, and so built their own—St. Peter's, the parish the boy's family still attended when holidays brought them around to mass.
He turned from the rock and looked out into the forest. He realized that his family was not unlike the folks who now populated the new developments. They came, clung close to one another, and bore the brunt of the locals' condemnation. Their acceptance required time, and more than that, another new group to persecute. For the first time, he saw a fundamental flaw in the Youth doctrine—they fought to preserve a status quo where there had never been anything but change.
He left Darling's Rock and followed the ridge up and around until he saw the open backyards of Sandy Creek. He quickly jogged between two houses to the street. From the edge of the pavement, he followed the brick walkway up to the front steps of the house. He looked at the chandelier and walked to the door. He reached out and touched the glowing doorbell. He heard the chime inside and he heard footsteps approaching. The door opened and Mrs. Dennison stood in front of him. She didn't flinch. She didn't cry out or fly into a fit of rage. Mrs. Dennison stood in the open door and looked at him. She was dressed in a pair of jeans and a blue sweatshirt. She didn't look mad or sad or much of anything.
"Theodore," she said.
"Mrs. Dennison." He looked at her feet. She had on white leather sneakers.
"Kevin's at school," she told him.
"I know," he said.
"Why aren't you at school?"
"I didn't feel good."
"Are you feeling better now?"
He shook his head.
"Do you have a cold?" she said. "The flu?"
"I'm awful hung-over," he said. He looked up at her and she nodded.
"Kevin's been drinking and smoking marijuana something terrible," she said. "We don't know what to do."
He nodded.
"I can't say I blame him," she said. "If I thought it would help, I'd do it too."
"It doesn't," he said.
"I know. I'd ask you inside but the house is a mess. I haven't had the energy for that sort of thing. You want to sit?" She stepped out of the house and motioned to the steps. He turned and took a seat on the second step from the top. She sat on the step above his, on the opposite side. "Thank goodness we have someone to take care of the yard," she said. They looked out at the lawn, the shaped shrubs, the small mulched island planted with a variety of small flowers. They sat for a moment without speaking.
"I loaded the gun," he said.
"I know you did," she said.
He looked down at his hands. "I'm sorry."
"I know," she said. "There was a time when I hated you for it, but I'm through that."
He nodded.
"Did you ever hate me?" she said.
He shrugged.
"You can say it. It's okay."
"I did," he said.
"It felt good, didn't it? Better, anyway. It felt better."
"Anything felt better," he said.
"I thought I'd have to kill myself to get away from it. I had to see someone. I have to take pills."
He reached down and pulled back the sleeve of his sweatshirt. He pushed the cuff up past his forearm to his elbow. He rotated his arm so that the inside pointed toward the sky. A scabbed, smiling face stood out against the pale flesh. He looked from Mrs. Dennison down to his arm and pointed to the burn with his index finger.
"What's that?" she said.
"Burn."
"From what?"
"Top of a lighter."
"You did it?"
He nodded.
"Did it feel good?"
"It felt really good."
"You shouldn't," she said. "Not anymore. That wouldn't make Bobby happy. He wouldn't want that."
Tears poked out of the corners of his eyes and crawled down his cheeks. "I'm so goddamn sorry." His chin bounced as he spoke.
"I know, honey." Her voice cracked. She reached over and rubbed his back. She slid closer and put an arm around his shoulder. Their bodies shook together.
He wiped at his eyes with a sleeve. "I'm going to tell what happened," he said.
"That's good."
"Please don't sue us," he said.
"What?"
"Don't sue my parents, please."
"You don't have anything I want," she told him. "It wouldn't bring him back."
He looked out at the lawn.
"I don't think about him as much as I used to," she said. "Sometimes I feel like a bad mother because of that. I feel so terrible letting him go. But I try to think of what he would want. First I thought he wanted me to vindicate him or something. But that's what I wanted. He just wants us to be happy."
"I hope so," he said.
"He's forgiven you," she said. "I know he has."
"Kevin probably hates me," he said.
Mrs. Dennison shrugged. "At the moment it's easier for me to communicate with the dead than the living."
"I bet he's pissed," he said.
"He's a lot like us. He's so busy being mad at everyone else, he can't see how mad he is at himself."
"I'm not mad."
"Look at your arm again and tell me that," she said. "You've got to let it go."
"I don't know how."
"Yes you do." She smiled at him. "You do. You're doing it right now."
From Sandy Creek the boy walked back in the direction of his house. As he walked down the side of the road, his bus roared past. The wake of the vehicle gusted and seemed to push him off balance. He stepped into the sand on the side of the road and regained his footing. Before he got to the Humphreys', he kicked around in the downed leaves on the side of the road in search of a few suitable stones. He came up with one about the size of a baseball, worn smooth and oblong like an egg. A few feet over he found a smaller, more jagged one. He gripped the smoother piece of granite in his throwing hand and held the smaller one on the opposite side for backup. He crossed the road to put the tarmac between him and the Humphreys' yard.
He saw the dog lying at the top of the drive, next to the cargo van that hadn't moved in years. The dog's head rested on the pavement between its two outstretched paws. The boy was halfway past the yard when the dog's ears perked up and its head rose. It scrambled to its feet and trotted down the drive and across the yard. A low rumble came from the dog. Its long fur hung in patches of white, black, and several shades of brown. The boy knew he shouldn't make eye contact with the dog, but he did. He half hoped the dog would come for him, would cross the pavement and make a lunge. He wanted an excuse to club it with the rock. He held it up and ready. The dog came a few feet out onto the pavement and barked once, sharp and loud. It snarled at him and its jaws popped. The boy didn't turn his back to it, but he kept walking. He kept the rock ready. The dog made several false lunges at him, but it seemed to understand the threat of the raised stone. It didn't come within striking distance.
It followed him well past the Humphreys' property line. The boy grew tired of walking backward, but anytime he tried to turn around, the dog made for his heels.
"Go on," he shouted. "Git!" But the dog still slunk along on the opposite side of the street. He swapped the two stones in his hands. He got a good grip on the small jagged rock. He turned his back to the dog but kept an eye over his shoulder. The dog came for him. He waited until it was well into the road. When the dog was close, he turned, hopped twice quickly, and overhanded the stone. It caught the dog in the rib cage with a hollow, dull sound. The dog shuddered at the blow and let out a quick yelp. It cowered for a moment in the middle of the road before turning to skulk away. When he saw the dog turn, he burst into a sprint after it, letting his feet fall hard and loud on the tarmac. The dog shot a quick glance back at him and bolted into the woods. He watched until the dog was out of sight.
He heard a car coming. He saw that he was in the middle of the road and took several quick steps to the opposite ditch. The noise grew as the car approached. It came around the bend ahead of him and he was glad that he still had the larger of the two stones. Jason Becker's Volkswagen slowed and pulled off the road just ahead of him. He thought about ducking into the woods and making a run for it, but he was tired of all that. He was surprised to see that only Jason, George, and Peckerhead were in the car. They climbed out and walked toward him. Jason and George were in the front. Peckerhead lingered back, near the hood of the car.
"You can put the rock down, Theodore," George said. "We're not here for anything like that."
"It's not for you," he said.
"You can put it down," George told him.
He shook his head.
"You're a coward," Becker said.
"Jason," George said, "you said you could do this."
"This little punk sets my car on fire and you want me to be cordial? Fuck that," Becker said. "Don't even try to deny it."
"I won't," said the boy. "Long as you don't try to deny tearing up my yard."
"Only because you raped Colleen, you fuck," Jason said.
"You know it wasn't like that," the boy said. "Peckerhead, tell him, for Christ's sake."
Peckerhead shrugged. He stayed back by the car.
"Guys," George said. "We can go back and forth like this forever. And where will it get us? Nowhere. So just quit it for a minute. Please." He looked at both Jason and the boy. "This is exactly what they want. It's infighting. It distracts us and weakens our cause. I've been thinking, and as it stands I'd say we're all about even. I think we can call it water under the bridge and get on with business."
"You want him back?" Becker said.
"Not immediately," George said.
"You're kidding me," Becker said. "He rapes your girlfriend and torches my car and you want him back?"
"I didn't rape her, you idiot," the boy told Becker. "She wanted it and you know it."
"What did you call me?" Becker said.
"A dumb-ass," he said.
Becker came at him. The boy had the rock back and ready, but George jumped between the two and got a palm on both of their chests.
"What is wrong with you two?" George hollered.
Jason swatted at George's hand and tried to push him out of the way.
The boy pushed George's hand off of him, but rather than stepping away, he took a step closer. George was busy with Jason and he never saw the blow coming. The rock landed on the side of his head, just above his ear, just where the boy had aimed. It sounded like a stick coming down quickly on a wet sack of mud. George's head bucked at the impact and he went to the ground like liquid. His face landed in the sand and his arms sprawled at odd angles from his torso. His eyes stared blankly at the far side of the street. His pupils twitched back and forth so fast they seemed to vibrate. Sand clung to his cheek and his body shivered. His hair was already wet around the wound. For a moment, it was quiet. The boy couldn't say why he did it, but even in the act of protecting him, George seemed more threatening than Jason could ever have been.
"What the fuck," Jason said to the boy.
He stepped back and held the rock up and ready.
"What is wrong with you?" Jason said. He looked back to George and squatted down beside him. "George? Man, you all right?" George didn't respond. Peckerhead came close and looked down. Jason looked up at him. "Get help," he said. "Go."
Peckerhead looked at the boy.
"My house is closest," he said. "Come on." He broke into a run and he heard Peckerhead calling after him.
"I think you killed him," Peckerhead said.
He didn't answer. He turned into his driveway and then down the sidewalk. He ran up the stairs and found the door locked. He jabbed at the doorbell until he heard the lock turn in the door. He pushed through the door and by his mother and went to the telephone.
"You have got some explaining to do," she said.
He dialed the three numbers and waited. He told the dispatch that a boy was unconscious on the side of the road and he gave her a general location.
"Was there an accident?" the dispatcher said.
"No," he told her. "I hit him." There was a pause. "You should probably send the police too," he said.
His mother had him by the arm. "What is going on?"
He hung up the phone. "I hit George Haney," he said. He held up the stone for his mother to see. "He's out on the side of the road. He might be dead."
His mother stared at him.
"I'm going to tell them the truth," he said. "I already told Mrs. Dennison."
She reached back and slapped him across the face. It stung but he didn't move. He looked her in the eye. She took a breath like she was about to let him have it, but she stopped. "Fine," she said. "But you're on your own. All of it," she swung a hand toward the front door. "It's yours." She turned and walked down the hall. She grabbed the railing and began climbing the stairs. He heard the door of her bedroom close.
He walked back outside. He followed the sidewalk and the driveway back out to the street. Jason squatted on his haunches next to George. He was bare-chested and he held what must have been his shirt to the side of George's head. Peckerhead stood above the two with his hand on the hood of the Volkswagen. He looked at the boy for a moment but then looked back to George.
The boy walked back to his driveway and sat on the edge of the stone wall. He heard the sirens approaching and he watched the ambulance blow past. Shortly after, the local cruiser passed. He knew it wouldn't be long.
He stood when the cruiser pulled in. Duncan looked at him, leaned down, and said something into the radio. He opened the door and stepped out. He came around the front of the car and stood before the boy.
"Can't say I didn't warn you," Duncan said.
"Is he dead?"
Duncan smiled and shook his head.
The boy nodded. "How bad?"
"Pretty mean concussion," Duncan said. "Some stitches and rest and he'll be fine."
"I was afraid I killed him," he said.
Duncan shook his head and smiled. "Nah, just knocked the heck out of him."
The boy shrugged.
"So what's your story," Duncan said. "I got three guys over there saying you just blindsided him."
"That sounds about right."
"Was he threatening you?" Duncan asked. "Talking like he was going to hurt you or anything?"
"No," he said.
"Ted," Duncan said, "you're not giving me much to work with."
"You going to take me in?"
"I'm afraid I'm going to have to."
"Will you get that trooper?" he said. "Thompson?"
"How come?"
"'Cause I want to talk to him."
"Sure," Duncan said. "We can make that happen."
"You going to cuff me?"
"Nah. Just get in." Duncan opened the front door and the boy climbed inside. He pulled the door shut and drew the seat belt across his chest. Duncan stood outside for a moment, talking into his radio. He climbed in and they backed out.
As they drive, the boy stares out the window at the blur of the forest. The car feels like a bubble, the windows bulging out against the world. He knows that outside, it's all still happening. And once he steps out of the cruiser, it will begin again. He wishes for even a week, a month, or a year more, but there isn't any more time. Nothing will stop.
His face is reflected in the window at his side and he looks through himself to see the world outside. The next economic boom is already beginning out there, wherever that sort of thing happens. The forest outside the window will be cut for new house lots and condominium developments. The town will sing again with chainsaws—the pop of nail guns providing a disjointed rhythm for the sickly wail of power tools. One day he will look at the same road and hardly be able to see the world where all of this transpired. There will be glints of it here and there—a place where the old faded tarmac peeks out from under the new pavement, the house of an elderly couple that resists renovation—but for the most part, it will disappear, it will change, and all the while, his own memory of it will recede farther and farther from him.
**17**
**H** e sits on the corner of his bed and with the tip of his finger he traces the ornamental engraving on the wooden stock and steel receiver. He rolls the gun upright and with the butt on his thigh he reaches up and opens the action. The sound of sliding metal and the clack of the action locking in place break the silence in the room. Holding the lever firm, he thrusts the bolt forward and closed again. He pulls the gun to his shoulder and looks down the sights, taking aim on a model airplane that hangs from the ceiling. Then he quickly slings the barrel to the side, bearing down on a ceramic figurine.
It's a pink piggy bank, and it seems to point out a certain absurdity. The gun has gone the way of the dirt bikes and four-wheel-drive trucks—now that the roads have been paved and the sand pits landscaped. It's difficult for him to think that he will never again hunt the hill behind his house. He looks at the .22, and it's still difficult to think that he will never again see Bobby Dennison.
Telling the truth about loading the gun wasn't quite the cure-all that he'd hoped it would be. With the stories corroborated and clear on paper, Trooper Thompson and the attorney general did agree that the shooting was indeed accidental and that Kevin and Theodore had suffered enough. But that wasn't the end of their legal troubles. Kevin was recently arrested on a possession charge when he was pulled over with marijuana in his pocket. The boy was charged with assault for what he did to George. Duncan assured him that mandatory counseling would be included in his sentence.
The boy takes a slow breath and runs his fingers along the inside of his arm. He pulls up the sleeve of his shirt to look. He traces the wicked smiles with the tip of a finger. Some of the older scars have faded slightly, melting back into his skin, forming soft, pale outlines in his flesh. While he can't give it up, the burns blossom less frequently.
He lets the sleeve of his shirt fall and he takes the .22 and slides it back under his bed. He hopes to give it to some child of his at an appropriate moment, and though it feels strange, he hopes the child will in turn do the same. It isn't so much the desire to possess the gun itself, but rather the ill feeling that comes over him when he thinks of someone else taking it up, perhaps with no idea of what it has done.
**Acknowledgments**
The author wishes to thank the Syracuse University Creative Writing Program, the Western Michigan University Prague Summer Program in Creative Writing, and the Summer Literary Seminars in St. Petersburg for their support; Peter Straus, Melanie Jackson, Daniel Menaker, and the gang at Random House for their belief in this manuscript; Arthur Flowers and George Saunders for their kindness and wisdom from the beginning to the end of this project; Mary Karr for revealing an easier, softer way through it all; the following technical advisers: Chris Decker, firearms instructor and armorer for the New Hampshire State Police, Wendy Foley of the Windham, New Hampshire, Police Department, Kenneth R. Martell of the Bristol, New Hampshire, Police Department, Phillip Maurice LaMarche of the Plattsburgh, New York, Fire Department; and the following readers for their advice and encouragement: Christian TeBordo, Adam Levin, Nina Shope, Erin Brooks Worley, Rahul Mehta, Stephanie Carpenter, Brian Evenson, Tobias Wolff, Robert Eversz, and Michael Schellenberg.
The author would also like to thank his family, Patty and Todd Davis, for their understanding and support; Cheryl and Greg Doble for sharing their home and their two wonderful boys; his parents for all the slingshots, guns, fireworks, and dirt bikes, for sticking by him through the ensuing trips to the hospital, for the hunting and fishing, and for showing him all the fun and beauty there is to be had in the world. He would also like to express his gratitude to Caroline, his best friend, his partner in crime, his favorite.
**ABOUT THE AUTHOR**
PHIL LAMARCHE was a writing fellow in the Syracuse University graduate creative writing program. He was also awarded the Ivan Klíma Fellowship in fiction in Prague and a Summer Literary Seminars fellowship in St. Petersburg, Russia. His story "In the Tradition of My Family," published in the spring 2005 edition of _Ninth Letter_ and the _2005 Robert Olen Butler Prize Stories_ anthology, has been made into a film by or Later Productions. He lives in central New York State.
This is a work of fiction. Names, characters, places, and incidents are the products of the author's imagination or are used fictitiously. Any resemblance to actual events, locales, or persons, living or dead, is entirely coincidental.
Copyright © 2007 by Phil LaMarche
All rights reserved.
Published in the United States by Random House, an imprint of The Random House Publishing Group, a division of Random House, Inc., New York.
RANDOM HOUSE and colophon are registered trademarks of Random House, Inc.
LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA
LaMarche, Phil.
American youth : a novel / Phil LaMarche.
p. cm.
1. Teenage boys—Fiction. 2. Firearms accidents—Fiction. 3. New England—Fiction. 4. Psychological fiction. I. Title.
PS3612.A5435A8 2007
813'.6—dc22 2006042921
www.atrandom.com
eISBN: 978-1-58836-605-4
v3.0
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Урош Вељовић (Краљево, 5. новембра 1996) српски је фудбалер, који тренутно наступа за Слогу из Краљева.
Каријера
Вељовић је тренирао у школи фудбала Бамби у родном Краљеву. Своју сениорску каријеру почео је у чачанској Слободи, где је забележио 30 наступа уз један погодак у Српској лиги Запад. Одатле је прешао у омладинску селекцију Металца из Горњег Милановца, док је лета 2015. приступио екипи Полета из Љубића. Годину дана касније прешао је у ФАП из Прибоја и ту се задржао такође једну сезону. Током такмичарске 2017/18. био је члан Славије из Источног Сарајева са којом је наступао у Првој лиги Републике Српске.
Вељовић се након тога вратио у ФАП крајем августа 2018. и ту се задржао до краја исте календарске године. После тога је прешао у ужичку Слободу. Наступао је и за у . Лета 2020. се вратио у родно Краљево и приступио екипи Слоге, за коју је дебитовао у 4. колу Прве лиге Србије, против Жаркова.
Референце
Спољашње везе
Урош Вељовић на сајту Resultados
Урош Вељовић на сајту Macsonuclarim
Урош Вељовић на сајту Sortitoutsi
Рођени 1996.
Спортисти из Краљева
Српски фудбалери
Везни играчи у фудбалу
Фудбалери Слободе Чачак
Фудбалери Металца Горњег Милановца
Фудбалери Слободе Ужица
Фудбалери Слоге Краљево
Фудбалери Прве лиге Србије
Фудбалери Славије Источно Сарајево | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,358 |
Rita Eleanor Hauser
Rita EleanorHauser
by Ruth Gursky
One of the first women to graduate from Harvard Law School during the 1950s, Rita Eleanor Hauser's career has taken her from presidential campaigns to the United Nations and back to law once again. Along the way, she worked towards achieving peace in the Middle East, established the Hauser Foundation, and donated $13 million to her alma mater, the largest contribution ever to a law school.
Institution: Private collection
Rita E. Hauser is a woman of many accomplishments. She was a trailblazer for women in law, politics and foreign affairs at a time when few women entered the legal profession or achieved top-level positions in business and politics. She was instrumental in persuading Yasir Arafat and the Palestine Liberation Organization to renounce terrorism publicly and to recognize Israel. She has been involved in Republican presidential politics since Richard Nixon's presidential campaign, and she was invited to join a major Wall Street law firm as its first woman partner.
Born in New York City on July 12, 1934, to Nathan and Frieda (Litt) Abrams, Rita Eleanor (Abrams) Hauser was the elder of two daughters. She received her B.A. from Hunter College (1954). After college, she won a Fulbright scholarship for graduate work in France. She attended the University of Strasbourg and was awarded a doctorate in political economy. At a time when women represented only one percent of those graduating from law school, she attended Harvard Law School (1955–1956), received her LL.B. from New York University Law School (1959), and became one of the few Americans to obtain a License en droit [French LL.B.] from the University of Paris (1958). She is licensed to practice law in New York and the District of Columbia.
Hauser's first job as an attorney took her to Washington, D.C., to serve as counsel in the Justice Department's Appellate Tax Division under the Attorney General's Honors Program. While in Washington, she was invited to join Richard Nixon's first presidential campaign and distinguished herself as a speechwriter and campaign strategist. Raised in a Republican family, Hauser subscribes to the party's approach in governance. After Nixon lost the 1960 election to John F. Kennedy, she moved to New York and worked in the international law department of a New York law firm while continuing her association with Republican politics.
In Nixon's successful 1968 presidential campaign, Hauser served as cochair of New Yorkers for Nixon. She was next appointed the United States representative to the United Nations Commission on Human Rights, a position she held from 1969 to 1972. In addition, she served as a member of the United States delegation to the twenty-fourth UN General Assembly. It was during her tenure at the UN that she met the key players in Middle East politics and became committed to her pursuit of conflict resolution in the Middle East, human rights and humanitarian law. During her term at the UN, she helped Jewish immigrants leave Russia and visited Palestinian refugee camps throughout the Middle East.
Hauser identified Golda Meir, whom she met during her tenure at the UN, as her mentor and role model. Meir inspired Hauser's involvement in Middle East politics and encouraged her, a secular Jew, to learn more about Jewish history and her own Jewishness.
In 1972 Hauser was invited to join a major Wall Street law firm, Stroock, Stroock and Lavan, as a senior partner and the firm's first female partner. In that capacity, she was instrumental in developing its international law department, became mentor to a number of its female associates and advocated an increase in the number of women partners in the firm.
Throughout her career Hauser has served the United States in numerous high-level public service assignments. Of particular note, in her capacity as head of the American branch of the International Center for Peace in the Middle East (1984–1991), she participated in secret diplomatic negotiations, coordinated by the Swedish foreign minister, which culminated in Yasir Arafat's 1988 public recognition of the State of Israel and the Palestine Liberation Organization's renunciation of terrorism. As chair of the International Peace Academy, Hauser was invited by the head of the Palestine Elections Commission to be an official observer of the 1996 Palestinian elections.
In 1988 Rita Hauser and her husband, Gustave M. Hauser, created the Hauser Foundation, a private philanthropic organization, to "meet the challenge of bringing about the peaceful resolution of conflict and promoting democracy"—particularly in the Middle East. In 1996 the Hausers donated thirteen million dollars to Harvard Law School, the largest gift ever donated to a law school, to fund the building of Hauser Hall. They also founded Harvard's Hauser Center for Nonprofit Organizations in 1997. Previously, they gave five million dollars to New York University Law School to establish the Global Law School Program, which supports the Hauser Scholar Program for postgraduate foreign students and advances the integration of foreign law into the law school curriculum. In 2001 they gave an additional five million dollars to the program, which was then renamed in their honor. In 1999 Hauser proposed the endowment of the Edward Said Chair of Middle Eastern Studies at Columbia University. She also helped fund the position, which was established in 2003. In 1992 Hauser left the full-time practice of law to manage the work of the Hauser Foundation, of which she is president. In her current capacity of counsel to the Stroock law firm, she continues practicing international law, with a client base in Europe, the Middle East and Latin America.
Over the years Hauser has been the recipient of numerous awards and honors and currently serves on diverse boards of directors. In 2001 President George W. Bush appointed her to the President's Foreign Intelligence Advisory Board and the President's Intelligence Oversight Board. Hauser credits the "enormous support" she has received from her husband, whom she married in 1957, in helping her to handle her overlapping careers as wife, mother, attorney, United States representative to the UN and world traveler. Gustave Hauser, an attorney, business executive and pioneer of the modern cable television industry, is the chairman and CEO of Hauser Communications, Incorporated. They have two children, Ana Patricia (b. 1962) and Glenvil (b. 1963).
Hauser, Rita. Curriculum vitae, and interview by author, December 20, 1995; Waring, Nancy. "Gus and Rita Hauser's Lifetime Legal Merger." Harvard Law Bulletin (Summer 1995): 3–7.
More on Rita Eleanor Hauser
Profiles: Rita Eleanor Hauser
Gursky, Ruth. "Rita Eleanor Hauser." Jewish Women: A Comprehensive Historical Encyclopedia. 27 February 2009. Jewish Women's Archive. (Viewed on January 20, 2020) <https://qa.jwa.org/encyclopedia/article/hauser-rita-eleanor>. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,696 |
Grease 2 ist ein Musikfilm aus dem Jahr 1982 mit Maxwell Caulfield und Michelle Pfeiffer in den Hauptrollen. In der Fortsetzung des Films Grease von 1978 spielte Michelle Pfeiffer ihre erste Hauptrolle.
Handlung
Herbst 1961. Vorgestellt werden neue Schülergruppen: Mitglieder der T-Birds, angeführt von Johnny Nogerelli, und die Pink Ladies, angeführt von Stephanie Zinone. Er möchte sich mit ihr verabreden, sie hat aber keine Gefühle mehr für ihn.
Unterdessen kommt Michael, ein Austauschschüler aus England, an. Er ist der Cousin von Sandy Olsson (die Rolle, die Olivia Newton-John im ersten Film spielte). Michael versucht an Stephanie heranzukommen, aber sie möchte nicht mit einem normalen Schüler ausgehen. Stephanie träumt von einem "Cool Rider" auf einem Motorrad.
Michael erledigt gegen Bezahlung die Hausaufgaben der T-Birds, so dass er sich bald sein eigenes Motorrad kaufen kann. Damit will er dann Stephanie beeindrucken.
Nach einem Bowling-Spiel kommt es zu einer Konfrontation zwischen den T-Birds und ihren Rivalen, den Cycle Lords. Als einer der T-Birds in die Hände der Cycle Lords fällt und er kurz davor ist, von diesen verprügelt zu werden, erscheint ein mysteriöser Biker und geht dazwischen. Dies verärgert die Cycle Lords. Sie lassen von ihrem Opfer ab und verfolgen den Unbekannten. Dabei wird Stephanie auf ihn aufmerksam und schwärmt nun von ihm. Jetzt steckt Michael in der Zwickmühle: entweder er spielt weiter den mysteriösen Biker oder er erzählt Stephanie, dass er der Biker ist, der sie so sehr beeindruckt hat und geht das Risiko ein, sie möglicherweise für immer zu verlieren.
Johnny ist auf Stephanie und den mysteriösen Biker (Michael) eifersüchtig. Er schwört, sich an ihm zu rächen, und droht Stephanie, ihn zu verprügeln, wenn er ihn das nächste Mal sieht.
Beim June Moon Talent Contest (einer Probe für die Abschlussfeier 1962) verfolgen die T-Birds den mysteriösen Biker bis zu einer Baustelle an einem gefährlichen Straßenabschnitt ("Dead Man's Curve"), von wo er in die Nacht springt. Man sucht vergeblich nach ihm und glaubt, dass er an dieser gefährlichen Stelle in den Tod gesprungen ist. Stephanie ist am Boden zerstört.
Johnny und Stephanie werden bei der Abschlussfeier als König und Königin der Talentshow "gekrönt". Etwas später tauchen die Cycle Lords auf und drohen damit, alles zu zerstören. Aus dem Nichts taucht der mysteriöse Biker plötzlich wieder auf. Er gibt sich nun auch als Michael zu erkennen.
Stephanie ist schockiert, als sie erkennt, dass der Mann, in den sie sich verliebt hat, ihr Schulkollege Michael ist. Nach dem ersten Schock macht Johnny Michael zu einem Mitglied der T-Birds, indem er ihm eine T-Bird-Jacke überreicht.
Kritik
Grease 2 wurde bei Kritik und Publikum ein Flop. Der Film konnte nicht den Hype des Vorgängers auslösen. Kritiker bemängelten auch, dass die Lieder nicht zu der Handlungslinie passten.
Auszeichnungen
Michelle Pfeiffer und der Film als bester Musikfilm waren 1983 für den Young Artist Award nominiert.
Soundtrack
Back to School Again
Score Tonight
Do It For Our Country
Who's That Guy?
Brad
Cool Rider
Reproduction
Charades
Prowlin'
A Girl For All Seasons
(Love Will) Turn Back the Hands of Time
Rock-a-Hula Luau (Summer Is Coming)
We'll Be Together
Weblinks
Einzelnachweise
Filmtitel 1982
US-amerikanischer Film
Musicalfilm
Filmkomödie
Filmdrama | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,951 |
package io.schinzel.basicutils.str;
import java.util.Locale;
abstract class AbstractIStr<T extends IStr<T>> implements IStr<T> {
private StringBuilder sb = new StringBuilder();
@Override
public T a(String s) {
sb.append(s);
return this.getThis();
}
@Override
public Locale getLocale() {
return Locale.US;
}
@Override
public String asString() {
return sb.toString();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,372 |
'use strict';
const fs = require('fs');
const DEFAULT_AMOUNT = 0;
global.currencyName = 'Gyarabuck';
global.currencyPlural = 'Gyarabucks';
let Economy = global.Economy = {
/**
* Reads the specified user's money.
* If they have no money, DEFAULT_AMOUNT is returned.
*
* @param {String} userid
* @param {Function} callback
* @return {Function} callback
*/
readMoney: function (userid, callback) {
// In case someone forgot to turn `userid` into an actual ID...
userid = toId(userid);
let amount = Db.currency.get(userid, DEFAULT_AMOUNT);
if (callback && typeof callback === 'function') {
// If a callback is specified, return `amount` through the callback.
return callback(amount);
} else {
// If there is no callback, just return the amount.
return amount;
}
},
/**
* Writes the specified amount of money to the user's "bank."
* If a callback is specified, the amount is returned through the callback.
*
* @param {String} userid
* @param {Number} amount
* @param {Function} callback (optional)
* @return {Function} callback (optional)
*/
writeMoney: function (userid, amount, callback) {
// In case someone forgot to turn `userid` into an actual ID...
userid = toId(userid);
// In case someone forgot to make sure `amount` was a Number...
amount = Number(amount);
if (isNaN(amount)) {
throw new Error("Economy.writeMoney: Expected amount parameter to be a Number, instead received " + typeof amount);
}
let curTotal = Db.currency.get(userid, DEFAULT_AMOUNT);
Db.currency.set(userid, curTotal + amount);
let newTotal = Db.currency.get(userid);
if (callback && typeof callback === 'function') {
// If a callback is specified, return `newTotal` through the callback.
return callback(newTotal);
}
},
writeMoneyArr: function (users, amount) {
this.writeMoney(users[0], amount, () => {
users.splice(0, 1);
if (users.length > 0) this.writeMoneyArr(users, amount);
});
},
logTransaction: function (message) {
if (!message) return false;
fs.appendFile('logs/transactions.log', '[' + new Date().toUTCString() + '] ' + message + '\n');
},
logDice: function (message) {
if (!message) return false;
fs.appendFile('logs/dice.log', '[' + new Date().toUTCString() + '] ' + message + '\n');
},
};
function rankLadder(title, type, array, prop, group) {
let groupHeader = group || 'Username';
const ladderTitle = '<center><h4><u>' + title + '</u></h4></center>';
const thStyle = 'class="rankladder-headers default-td" style="background: -moz-linear-gradient(#576468, #323A3C); background: -webkit-linear-gradient(#576468, #323A3C); background: -o-linear-gradient(#576468, #323A3C); background: linear-gradient(#576468, #323A3C); box-shadow: -1px -1px 2px rgba(0, 0, 0, 0.3) inset, 1px 1px 1px rgba(255, 255, 255, 0.7) inset;"';
const tableTop = '<div style="max-height: 310px; overflow-y: scroll;">' +
'<table style="width: 100%; border-collapse: collapse;">' +
'<tr>' +
'<th ' + thStyle + '>Rank</th>' +
'<th ' + thStyle + '>' + groupHeader + '</th>' +
'<th ' + thStyle + '>' + type + '</th>' +
'</tr>';
const tableBottom = '</table></div>';
const tdStyle = 'class="rankladder-tds default-td" style="box-shadow: -1px -1px 2px rgba(0, 0, 0, 0.3) inset, 1px 1px 1px rgba(255, 255, 255, 0.7) inset;"';
const first = 'class="first default-td important" style="box-shadow: -1px -1px 2px rgba(0, 0, 0, 0.3) inset, 1px 1px 1px rgba(255, 255, 255, 0.7) inset;"';
const second = 'class="second default-td important" style="box-shadow: -1px -1px 2px rgba(0, 0, 0, 0.3) inset, 1px 1px 1px rgba(255, 255, 255, 0.7) inset;"';
const third = 'class="third default-td important" style="box-shadow: -1px -1px 2px rgba(0, 0, 0, 0.3) inset, 1px 1px 1px rgba(255, 255, 255, 0.7) inset;"';
let midColumn;
let tableRows = '';
for (let i = 0; i < array.length; i++) {
if (i === 0) {
midColumn = '</td><td ' + first + '>';
tableRows += '<tr><td ' + first + '>' + (i + 1) + midColumn + Tsunami.nameColor(array[i].name, true) + midColumn + array[i][prop] + '</td></tr>';
} else if (i === 1) {
midColumn = '</td><td ' + second + '>';
tableRows += '<tr><td ' + second + '>' + (i + 1) + midColumn + Tsunami.nameColor(array[i].name, true) + midColumn + array[i][prop] + '</td></tr>';
} else if (i === 2) {
midColumn = '</td><td ' + third + '>';
tableRows += '<tr><td ' + third + '>' + (i + 1) + midColumn + Tsunami.nameColor(array[i].name, true) + midColumn + array[i][prop] + '</td></tr>';
} else {
midColumn = '</td><td ' + tdStyle + '>';
tableRows += '<tr><td ' + tdStyle + '>' + (i + 1) + midColumn + Tsunami.nameColor(array[i].name, true) + midColumn + array[i][prop] + '</td></tr>';
}
}
return ladderTitle + tableTop + tableRows + tableBottom;
}
exports.commands = {
'!wallet': true,
atm: 'wallet',
wallet: function (target, room, user) {
if (!target) target = user.name;
if (!this.runBroadcast()) return;
let userid = toId(target);
if (userid.length < 1) return this.sendReply("/wallet - Please specify a user.");
if (userid.length > 19) return this.sendReply("/wallet - [user] can't be longer than 19 characters.");
Economy.readMoney(userid, money => {
this.sendReplyBox(Tsunami.nameColor(target, true) + " has " + money + ((money === 1) ? " " + currencyName + "." : " " + currencyPlural + "."));
//if (this.broadcasting) room.update();
});
},
gs: 'givecurrency', //You can change "gs" and "givestardust" to your currency name for an alias that applies to your currency Example: AwesomeBucks could be "ga" and "giveawesomebucks"
givestardust: 'givecurrency',
gc:'givecurrency',
givecurrency: function (target, room, user, connection, cmd) {
if (!this.can('forcewin')) return false;
if (!target) return this.sendReply("Usage: /" + cmd + " [user], [amount]");
let splitTarget = target.split(',');
if (!splitTarget[2]) return this.sendReply("Usage: /" + cmd + " [user], [amount], [reason]");
for (let u in splitTarget) splitTarget[u] = splitTarget[u].trim();
let targetUser = splitTarget[0];
if (toId(targetUser).length < 1) return this.sendReply("/" + cmd + " - [user] may not be blank.");
if (toId(targetUser).length > 19) return this.sendReply("/" + cmd + " - [user] can't be longer than 19 characters");
let amount = Math.round(Number(splitTarget[1]));
if (isNaN(amount)) return this.sendReply("/" + cmd + "- [amount] must be a number.");
if (amount > 1000) return this.sendReply("/" + cmd + " - You can't give more than 1000 " + currencyName + " at a time.");
if (amount < 1) return this.sendReply("/" + cmd + " - You can't give less than one " + currencyName + ".");
let reason = splitTarget[2];
if (reason.length > 100) return this.errorReply("Reason may not be longer than 100 characters.");
if (toId(reason).length < 1) return this.errorReply("Please specify a reason to give " + currencyName + ".");
Economy.writeMoney(targetUser, amount, () => {
Economy.readMoney(targetUser, newAmount => {
if (Users(targetUser) && Users(targetUser).connected) {
Users.get(targetUser).popup('|html|You have received ' + amount + ' ' + (amount === 1 ? currencyName : currencyPlural) +
' from ' + Tsunami.nameColor(user.userid, true) + '.');
}
this.sendReply(targetUser + " has received " + amount + ((amount === 1) ? " " + currencyName + "." : " " + currencyPlural + "."));
Economy.logTransaction(user.name + " has given " + amount + ((amount === 1) ? " " + currencyName + " " : " " + currencyPlural + " ") + " to " + targetUser + ". (Reason: " + reason + ") They now have " + newAmount + (newAmount === 1 ? " " + currencyName + "." : " " + currencyPlural + "."));
});
});
},
ts: 'takecurrency', //You can change "ts" and "takestardust" to your currency name for an alias that applies to your currency Example: AwesomeBucks could be "ta" and "takeawesomebucks"
takestardust: 'takecurrency',
tc:'takecurrency',
takecurrency: function (target, room, user, connection, cmd) {
if (!this.can('forcewin')) return false;
if (!target) return this.sendReply("Usage: /" + cmd + " [user], [amount]");
let splitTarget = target.split(',');
if (!splitTarget[2]) return this.sendReply("Usage: /" + cmd + " [user], [amount], [reason]");
for (let u in splitTarget) splitTarget[u] = splitTarget[u].trim();
let targetUser = splitTarget[0];
if (toId(targetUser) === user.userid) return this.errorReply("You cannot transfer stardust to yourself.");
if (toId(targetUser).length < 1) return this.sendReply("/" + cmd + " - [user] may not be blank.");
if (toId(targetUser).length > 19) return this.sendReply("/" + cmd + " - [user] can't be longer than 19 characters");
let amount = Math.round(Number(splitTarget[1]));
if (isNaN(amount)) return this.sendReply("/" + cmd + "- [amount] must be a number.");
if (amount > 1000) return this.sendReply("/" + cmd + " - You can't take more than 1000 " + currencyName + " at a time.");
if (amount < 1) return this.sendReply("/" + cmd + " - You can't take less than one " + currencyName + ".");
let reason = splitTarget[2];
if (reason.length > 100) return this.errorReply("Reason may not be longer than 100 characters.");
if (toId(reason).length < 1) return this.errorReply("Please specify a reason to give " + currencyName + ".");
Economy.writeMoney(targetUser, -amount, () => {
Economy.readMoney(targetUser, newAmount => {
if (Users(targetUser) && Users(targetUser).connected) {
Users.get(targetUser).popup('|html|' + Tsunami.nameColor(user.userid, true) + ' has removed ' + amount + ' ' + (amount === 1 ? currencyName : currencyPlural) +
' from you.<br />');
}
this.sendReply("You removed " + amount + ((amount === 1) ? " " + currencyName + " " : " " + currencyPlural + " ") + " from " + Chat.escapeHTML(targetUser));
Economy.logTransaction(user.name + " has taken " + amount + ((amount === 1) ? " " + currencyName + " " : " " + currencyPlural + " ") + " from " + targetUser + ". (Reason: " + reason + ") They now have " + newAmount + (newAmount === 1 ? " " + currencyName + "." : " " + currencyPlural + "."));
});
});
},
confirmtransferstardust: 'transfercurrency', //You can change "transferstardust" and "confirmtransferstardust" to your currency name for an alias that applies to your currency Example: AwesomeBucks could be "transferawesomebucks" and "confirmtransferawesomebucks"
transferstardust: 'transfercurrency',
confirmtransfercurrency: 'transfercurrency',
transfercurrency: function (target, room, user, connection, cmd) {
if (!target) return this.sendReply("Usage: /" + cmd + " [user], [amount]");
let splitTarget = target.split(',');
for (let u in splitTarget) splitTarget[u] = splitTarget[u].trim();
if (!splitTarget[1]) return this.sendReply("Usage: /" + cmd + " [user], [amount]");
let targetUser = (Users.getExact(splitTarget[0]) ? Users.getExact(splitTarget[0]).name : splitTarget[0]);
if (toId(targetUser).length < 1) return this.sendReply("/" + cmd + " - [user] may not be blank.");
if (toId(targetUser).length > 18) return this.sendReply("/" + cmd + " - [user] can't be longer than 18 characters.");
let amount = Math.round(Number(splitTarget[1]));
if (isNaN(amount)) return this.sendReply("/" + cmd + " - [amount] must be a number.");
if (amount > 1000) return this.sendReply("/" + cmd + " - You can't transfer more than 1000 " + currencyName + " at a time.");
if (amount < 1) return this.sendReply("/" + cmd + " - You can't transfer less than one " + currencyName + ".");
Economy.readMoney(user.userid, money => {
if (money < amount) return this.sendReply("/" + cmd + " - You can't transfer more " + currencyName + " than you have.");
if (cmd !== 'confirmtransfercurrency' && cmd !== 'confirmtransferstardust') {
return this.popupReply('|html|<center>' +
'<button class = "card-td button" name = "send" value = "/confirmtransfercurrency ' + toId(targetUser) + ', ' + amount + '"' +
'style = "outline: none; width: 200px; font-size: 11pt; padding: 10px; border-radius: 14px ; text-shadow: 1px 1px 2px rgba(0, 0, 0, 0.4); box-shadow: 0px 0px 7px rgba(0, 0, 0, 0.4) inset; transition: all 0.2s;">' +
'Confirm transfer to <br><b style = "color:' + Tsunami.hashColor(targetUser) + '; text-shadow: 1px 1px 1px rgba(0, 0, 0, 0.8)">' + Chat.escapeHTML(targetUser) + '</b></button></center>'
);
}
Economy.writeMoney(user.userid, -amount, () => {
Economy.writeMoney(targetUser, amount, () => {
Economy.readMoney(targetUser, firstAmount => {
Economy.readMoney(user.userid, secondAmount => {
this.popupReply("You sent " + amount + ((amount === 1) ? " " + currencyPlural : " " + currencyPlural) + " to " + targetUser);
Economy.logTransaction(
user.name + " has transfered " + amount + ((amount === 1) ? " " + currencyPlural : " " + currencyPlural) + " to " + targetUser + "\n" +
user.name + " now has " + secondAmount + " " + (secondAmount === 1 ? " " + currencyPlural : " " + currencyPlural) + " " +
targetUser + " now has " + firstAmount + " " + (firstAmount === 1 ? " " + currencyPlural : " " + currencyPlural)
);
if (Users.getExact(targetUser) && Users.getExact(targetUser).connected) {
Users.getExact(targetUser).send('|popup||html|' + Tsunami.nameColor(user.name, true) + " has sent you " + amount + ((amount === 1) ? " " + currencyPlural : " " + currencyPlural));
}
});
});
});
});
});
},
moneylog: function (target, room, user) {
if (!this.can('forcewin')) return false;
if (!target) return this.sendReply("Usage: /moneylog [number] to view the last x lines OR /moneylog [text] to search for text.");
let word = false;
if (isNaN(Number(target))) word = true;
let lines = fs.readFileSync('logs/transactions.log', 'utf8').split('\n').reverse();
let output = '';
let count = 0;
let regex = new RegExp(target.replace(/[-\/\\^$*+?.()|[\]{}]/g, '\\$&'), "gi"); // eslint-disable-line no-useless-escape
if (word) {
output += 'Displaying last 50 lines containing "' + target + '":\n';
for (let line in lines) {
if (count >= 50) break;
if (!~lines[line].search(regex)) continue;
output += lines[line] + '\n';
count++;
}
} else {
if (target > 100) target = 100;
output = lines.slice(0, (lines.length > target ? target : lines.length));
output.unshift("Displaying the last " + (lines.length > target ? target : lines.length) + " lines:");
output = output.join('\n');
}
user.popup("|wide|" + output);
},
'!richestuser': true,
richestusers: 'richestuser',
richestuser: function (target, room, user) {
if (!target) target = 100;
target = Number(target);
if (isNaN(target)) target = 100;
if (!this.runBroadcast()) return;
let keys = Db.currency.keys().map(name => {
return {name: name, money: Db.currency.get(name)};
});
if (!keys.length) return this.sendReplyBox("Money ladder is empty.");
keys.sort(function (a, b) { return b.money - a.money; });
this.sendReplyBox(rankLadder('Richest Users', currencyPlural, keys.slice(0, target), 'money') + '</div>');
},
resetstardust: 'resetmoney',
resetmoney: function (target, room, user) {
if (!this.can('roomowner')) return false;
if (!target) return this.parse('/help resetmoney');
target = toId(target);
Economy.writeMoney(target, 0);
this.sendReply(target + " now has 0 " + currencyName + ".");
},
resetmoneyhelp: ['/resetmoney [user] - Resets target user\'s currency to 0. Requires: &, ~'],
customsymbol: function (target, room, user) {
let bannedSymbols = ['!', '|', '‽', '\u2030', '\u534D', '\u5350', '\u223C'];
for (let u in Config.groups) if (Config.groups[u].symbol) bannedSymbols.push(Config.groups[u].symbol);
if (!user.canCustomSymbol && !user.can('vip')) return this.sendReply('You need to buy this item from the shop to use.');
if (!target || target.length > 1) return this.sendReply('/customsymbol [symbol] - changes your symbol (usergroup) to the specified symbol. The symbol can only be one character');
if (target.match(/([a-zA-Z ^0-9])/g) || bannedSymbols.indexOf(target) >= 0) {
return this.sendReply('This symbol is banned.');
}
user.customSymbol = target;
user.updateIdentity();
user.canCustomSymbol = false;
this.sendReply('Your symbol is now ' + target + '. It will be saved until you log off for more than an hour, or the server restarts. You can remove it with /resetsymbol');
},
removesymbol: 'resetsymbol',
resetsymbol: function (target, room, user) {
if (!user.customSymbol) return this.sendReply("You don't have a custom symbol!");
delete user.customSymbol;
user.updateIdentity();
this.sendReply('Your symbol has been removed.');
},
economy: 'economystats',
currency: 'economystats',
stardust: 'economystats',
economystats: function (target, room, user) {
if (!this.runBroadcast()) return;
const users = Db.currency.keys().map(curUser => ({amount: Db.currency.get(curUser)}));
const total = users.reduce((acc, cur) => acc + cur.amount, 0);
let average = Math.floor(total / users.length) || 0;
let output = "There " + (total > 1 ? "are " : "is ") + total + " " + (total > 1 ? currencyPlural : currencyName) + " circulating in the economy. ";
output += "The average user has " + average + " " + (average > 1 ? currencyPlural : currencyName) + ".";
this.sendReplyBox(output);
},
};
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,953 |
Louis Jacques Thénard (ur. 4 maja 1777 w La Louptière, zm. 21 czerwca 1857 w Paryżu) – francuski chemik, członek Francuskiej Akademii Nauk, profesor na uniwersytecie i politechnice w Paryżu.
Thénard w 1808 roku odkrył bor (w tym samym czasie dokonali tego także Humphry Davy i Joseph Louis Gay-Lussac), a w 1818 roku – nadtlenek wodoru. Zbadał wpływ światła na reakcję chloru z wodorem w 1809 roku. W tym samym roku stwierdził, że siarka jest pierwiastkiem chemicznym. W 1799 roku otrzymał pigment zwany od jego nazwiska "błękitem Thénarda".
Upamiętnienie
Jego nazwisko pojawiło się na liście 72 nazwisk na wieży Eiffla.
Przypisy
Bibliografia
Francuscy chemicy
Odkrywcy pierwiastków chemicznych
Członkowie Francuskiej Akademii Nauk
Wyróżnieni na liście 72 nazwisk na wieży Eiffla
Urodzeni w 1777
Zmarli w 1857 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 932 |
Sebastian Santin (born 15 June 1994) is an Austrian professional footballer who plays as a midfielder for Austrian Second League club FC Dornbirn.
Club career
He made his Austrian Football First League debut for WSG Wattens on 21 July 2017 in a game against TSV Hartberg.
References
External links
1994 births
Living people
Austrian footballers
Association football midfielders
WSG Tirol players
FC Vaduz players
FC Dornbirn 1913 players
2. Liga (Austria) players
Austrian Football Bundesliga players
Austrian Regionalliga players
Swiss Super League players
Austrian expatriate footballers
Austrian expatriate sportspeople in Liechtenstein
Expatriate footballers in Liechtenstein
People from Bregenz
Footballers from Vorarlberg | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,768 |
Q: KDE Recent Documents *only* (Not Locations) In KDE Plasma 5, the "Recent Documents" item in the Application Menu includes both Recent Documents, and Recent Locations (i.e. shows not only recent documents, but also all the folders I've browsed to in Dolphin). Is there a way to disable the "recent locations" behavior, so it shows only recent documents (i.e. similar to how it behaves on Windows)?
Thanks in advance
A: This isn't really a great solution, but after a month of trying & posting in half a dozen different places (here, official KDE forums, KDE bug tracker, etc), the best I can come up with is:
sqlite3 ~/.local/share/kactivitymanagerd/resources/database "DELETE FROM ResourceScoreCache WHERE initiatingAgent = 'dolphin';" ; pkill -f kactivitymanagerd ; /usr/lib/x86_64-linux-gnu/libexec/kactivitymanagerd &
If you create a launcher item with this command, you can run it to delete all the clutter from the menu prior to using it. The menu will therefore temporarily appear as though it were behaving properly (aka "Recent Documents" will actually show recent documents). It will of course continue to become cluttered again as you navigate around the filesystem, but at least it provides a one-off way to simulate a more sane behavior.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,931 |
People operating an 'adult gaming centre' must have a gaming machines general operating licence issued by the Gambling Commission and premises licence from us.
This licence will enable you to make available up to four Category B3 or B4 Gaming Machines and an unlimited number of Category C and D machines.
No one under the age of 18 years is permitted to enter an adult gaming centre.
If you want to operate an arcade that admits children, see our family entertainment centres page.
Will there be any conditions attached to my licence?
We may attach conditions to a premises licence to promote the licensing objectives.
You can see our Current list of gambling fees. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,884 |
The Denver Social Anxiety Group is for people in the Denver area who have a desire to improve the quality of their lives by learning and discussing various topics related to psychology. Many of our members suffer from shyness or social anxiety. Others have problems with public speaking anxiety, self-esteem, depression, or weight management.
We meet a few times a month. Our meetings consist of one member giving a brief presentation on a topic relevant to the group, and then we have a discussion. Anyone can give the presentation and you can choose to participate in the discussion or just listen. There's never any pressure to do anything. So far, we've discussed topics such as self-esteem, exposure therapy, mindfulness, and we've even gotten into the ancient teachings of the Buddha.
It's a fun and supportive group. It's purely a self-help group and it's by no means a substitute for therapy. None of us are professional therapists. What we can provide is a sense of normalcy, though, since all of us have issues we're trying to deal with. | {
"redpajama_set_name": "RedPajamaC4"
} | 765 |
Posts Tagged 'Robert Wood Johnson'
Poorer kids are also sicker kids
July 8, 2011 | General | No Comments
I've written before about how children from poor families have a higher chance of needing PICU care than do children from more affluent families. Eligibility for Medicaid is a good marker for this; nearly half the population of most urban PICUs is made up of children on Medicaid, even though the national average (it varies a little from state to state) for children on Medicaid is about 25%. So poor kids are more likely to become critically ill.
A report from the Robert Wood Johnson foundation, a renowned health policy organization, lays out how poverty correlates so closely to poor health. This chart is the most telling. It measures a somewhat vague quantity, something they call "children in less than very good health." They obtain this value by surveying parents, so you could quibble about the validity of whatever it is the term measures. That quibble would make sense to me if the numbers weren't so striking.
But they are striking. For example, among white, non-Hispanic children, 20% of poor children have "less than very good health," compared with 6% of well-off children. The differences among black and Hispanic children are much more dramatic. Nearly 50% of poor, Hispanic children are not in optimal health.
What this means to me is fairly obvious, and it has been obvious for a long time — health status is linked to socio-economic status. We shouldn't need a study to tell us that, but it is helpful to have such a graphic demonstration of the effect. I'm sure it's partly because poor families can't afford health insurance. But that isn't the whole story — all of these poorest children, the group with the most severe health problems, would qualify for Medicaid, even in the states with the most stringent requirements.
Thus whatever we do about healthcare reform will be closely linked to what is happening in the economy. Perhaps the best thing we can do for healthcare is reduce poverty.
Poorer kids are sicker kids Shares Facebook Twitter LinkedIn I've written before about how children...
Poor kids are not only sicker kids — they make sicker adults, too Shares Facebook Twitter LinkedIn I've written before (here and here)...
Children in the PICU without health insurance have higher mortality Shares Facebook Twitter LinkedIn I've written before about how poor...
Unequal burdens among the states in Medicaid financing Shares Facebook Twitter LinkedIn Medicaid is the joint state/federal program...
Uninsured kids in the PICU — more likely to die Shares Facebook Twitter LinkedIn I've written before about how poor...
Poorer kids are sicker kids
October 20, 2008 | General | No Comments
Now a new report from the Robert Wood Johnson foundation, a renowned health policy organization, lays out how poverty correlates so closely to poor health. This chart is the most telling. It measures a somewhat vague quantity, something they call "children in less than very good health." They obtain this value by surveying parents, so you could quibble about the validity of whatever it is the term measures. That quibble would make sense to me if the numbers weren't so striking.
Poorer kids are also sicker kids Shares Facebook Twitter LinkedIn I've written before about how children...
The PICU — canary in the healthcare coal mine Shares Facebook Twitter LinkedIn We have known for a long...
Poor children are far more likely to end up in the PICU than are affluent children Shares Facebook Twitter LinkedIn It's widely known socioeconomic status correlates...
How to read the medical news I: the nature of medical evidence 1 view | posted on May 23, 2011 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 439 |
\section{Introduction}
Cigarette smoking is responsible for over 480,000 deaths per year in the United States, with total yearly economic cost exceeding \$300 billion \citep{CDC2014}. Quitting smoking quickly and substantially lowers the risk of heart attack, stroke, and cancer \citep{CDC2010}, and most adult smokers are motivated to quit \citep{Jamal2018}. Unfortunately, fewer than one in three smokers who attempt to quit use evidence-based methods --- often due to barriers to access --- and there is a large divide between yearly quit attempts ($>$50\%) and successful quitting ($<$10\%) \citep{Babb2017}. Consequently there is an ongoing and urgent need to develop novel, effective, and accessible cessation interventions.
To more effectively support cessation, we can seek to better understand the immediate causes (or precursors) of smoking events. Currently, there is limited understanding of the factors leading to lapse (i.e. smoking during a quit attempt), and the process through which a single lapse becomes a relapse (i.e. failed quit attempt). If moments of high lapse risk can be identified, possible interventions include accessing self-help materials \citep{Siu2015}, telephone support/counseling \citep{Siu2015,Stead2013}, or use of short-acting pharmacologic treatment such as a nicotine lozenge \citep{Kotlyar2017}, oxytocin \citep{Miller2016}, or a nicotine film \citep{Du2015}. Smoking risk has been estimated based on passive physiologic measurements \citep{Chatterjee2016}, self-reported symptoms (e.g. urge, stress) \citep{Businelle2016}, and previously observed daily patterns \citep{Chandra2011}. Studies in controlled settings have also suggested that the surrounding environment plays an important role \citep{McClernon2016,Stevenson2017}. However, less emphasis has been placed on the underlying temporal dynamics of smoking events --- in other words, the natural rhythms and patterns of smoking within and between days. The goal of this work is to improve prediction and knowledge of smoking risk by utilizing a prediction model that can learn these dynamics.
Temporal patterns of smoking have been studied since the 1970s, when \citet{Frederiksen1977} and \citet{Robinson1980} independently observed consistent daily patterns of smoking in several small cohorts of smokers. These patterns were even more striking when considering individual participants, whose behaviors were more consistent than the cohorts as a whole. More recent work by \citet{Chandra2007} has strengthened and extended this line of research with much larger study (n=351) that identified four common daily smoking patterns (e.g. daily decline, morning high). Subsequently these patterns were confirmed and additionally tied to craving, including evidence that craving tends to precede smoking \citep{Chandra2011}. The connection between smoking patterns and craving --- a factor which also predicts failure to quit \citep{Killen1997} --- is strong motivation for predicting craving \citep{Chatterjee2016} as a proxy for smoking risk to inform an intervention. Nevertheless risk and craving are distinct, and their relative merits in predicting smoking and/or lapses are currently unknown.
Related to daily patterns, smoking is influenced by plasma and saliva nicotine concentrations, which peak within minutes of smoking and return to baseline over several hours ($\sim$2hr plasma half-life) \citep{Benowitz2009}. Plasma nicotine concentration and craving are negatively correlated, suggesting that craving is driven in part by nicotine seeking \citep{Jarvik2000}. \citet{Gomeni2001} directly studied the relative effects of nicotine and circadian patterns on craving with a blinded study design, finding that nicotine replacement can attenuate craving. Indeed, this effect is the motivation for nicotine-based cessation treatment, which has been shown to be effective \citep{Kotlyar2017,Du2015}. Given the strong relationship between craving and smoking itself \citep{Chandra2011}, these results suggest that cigarette smoking is a self-triggering (i.e. self-reinforcing) process in which the probability of smoking is modulated by previous smoking events.
To our knowledge, this self-triggering dynamic has not yet been incorporated in smoking prediction models. For example, \citet{rathbun2016mixed} model smoking events as a Poisson process with intensity function (i.e. rate of smoking) modulated by self-reports of affect, arousal, attention, and restlessness; but in their model, previously observed smoking events do not directly modify the current intensity. Here we propose to explore the self-triggering nature of smoking with a Hawkes process, wherein the intensity is influenced not only by exogenous factors --- such as demographics or self-reports --- but also the accumulating influence of previous smoking events.
Specifically, we develop the time-varying semi-parametric Hawkes process (TV-SPHP), then fit it to smoking events reported by smokers (n=42) over a $>$1 week period. This approach allows us to capture the effects of the following predictors on current smoking intensity (i.e. risk): (a) fixed covariates such as demographics; (b) time-varying covariates, including speed as measured by GPS; and (c) previous smoking events. The relationship between recent smoking and current risk is modeled with a semi-parametric \textit{impact function}, which is itself a sum of parametric basis functions.
From a clinical perspective, this approach offers two important advantages. First, it allows us to investigate the self-triggering nature of smoking in the real world by examining the learned impact function. We hypothesize that the impact function will exhibit daily periodicity and peak within 1-4 hours of smoking, which may reflect increased nicotine seeking. Second, it may improve prediction performance compared to alternative approaches by modeling the effect of previous events on current risk. We hypothesize that the TV-SPHP will improve prediction performance --- measured as the mean absolute error over periods ranging from 10 minutes to 1 hour --- compared to baseline and existing models.
\paragraph{Technical Significance}
This work develops the time-varying, semi-parametric Hawkes process (TV-SPHP) and its associated learning/inference algorithm. The TV-SPHP extends previous Hawkes process variants by modeling the intensity function as a sum of exogenous and endogenous factors. The exogenous factors include both time-invariant and time-varying participant features, while the endogenous factors capture the self-triggering nature of smoking using a semi-parametric impact function.
\paragraph{Clinical Relevance}
The clinical significance of this work is twofold. First, we improve on existing smoking prediction models and establish a new performance baseline. Smoking prediction models can identify high risk contexts either (a) in real-time to trigger a just-in-time adaptive intervention, or (b) in advance to help smokers avoid these contexts. Second, we connect the self-triggering nature of smoking to nicotine metabolism as a hypothesis to be explored further in future work.
\section{Related Work}
\subsection{Smoking Prediction}
Compared to the smoking detection literature --- which is intimately connected to the large, rapidly progressing activity recognition literature \citep{ordonez2016deep} --- less work has focused on smoking prediction. \citet{Dumortier2016} compared the performance of several machine learning classifiers (Naive Bayes, discriminant analysis, decision tree) to discriminate between high-urge and low-urge states based on self-reported contextual information. \citet{Chatterjee2016} also predicted high and low craving with an approach based on conditional random fields, but their system was based on physiologic measurements passively collected with wearable devices.
The work of \citet{rathbun2016mixed} is most closely related to the current work. A total of 304 smokers self-reported all smoking events along with contextual information with a hand-held device. These events were fitted to an inhomogeneous Poisson process with intensity modified by the time-varying contextual predictors. In contrast to our work, (a) the focus was model interpretation and not prediction; (b) their model did not include a self-triggering component; and (c) time-varying covariates were self-reported contextual information rather than GPS measurements.
\subsection{Temporal point processes and Hawkes processes}
Temporal point processes~\citep{daley2007introduction} have been widely used to model real-world event sequences.
In the simplest 1D case, given an individual's sequence of events $\bm{s}=\{t_1,t_2,...|t_i\in[0,T)\}$ and the corresponding counting process $N(t)=|\{t_i\in\bm{s}|t_i\leq t \}|$, a point process model characterizes the sequence by modeling the expected instantaneous event occurrence rate over time:
\begin{eqnarray}\label{intensity}
\begin{aligned}
\lambda(t) = \frac{\mathbb{E}[dN(t)|\mathcal{H}_t]}{dt},
\end{aligned}
\end{eqnarray}
where $\mathcal{H}_t=\{t_i\in\bm{s}|t_i\leq t \}$ contains all historical events happening before or at time $t$ and $\mathbb{E}[dN(t)|\mathcal{H}_t]$ computes the expected number of events in $[t, t+dt)$ conditioned on the history.
Generally, we call $\lambda(t)$ the \emph{intensity function}.
The key to the point process model is modeling and learning $\lambda(t)$ based on observed data.
A important kind of point process called the Hawkes process~\citep{hawkes1971point} is useful for explicitly modeling sequences whose events are self- and mutually-triggering. Mathematically, the intensity function of a 1D Hawkes process has the following form:
\begin{eqnarray}\label{hp}
\begin{aligned}
\lambda(t)= \mu(t) + \int_{0}^{T}\phi(t-s)dN(s)=\mu(t) + \sideset{}{_{t_i\in\mathcal{H}_t}}\sum\phi(t-t_i).
\end{aligned}
\end{eqnarray}
where $\mu(t)$ corresponds to the background intensity caused by exogenous factors~\citep{xu2017benefits}, which can be modeled as homogeneous or inhomogeneous Poisson process.
The term $\int_{0}^{T}\phi(t-s)dN(s)$ represents the accumulation of endogenous intensity caused by the history~\citep{farajtabar2014shaping}.
The nonnegative function $\phi(t-s)$, where $t\geq s$, is called the \emph{impact function}. It represents the influence of an historical event at time $s$ on the intensity at time $t$.
Recently, Hawkes processes have attracted many researchers, and a number of variants have been proposed. These include the mixture of Hawkes processes~\citep{xu2017dirichlet}, the nonlinear Hawkes process~\citep{xu2017patient}, and the locally-stationary Hawkes process~\citep{roueff2016locally,xu17b}.
Applications include network analysis~\citep{zhao2015seismic}, quantitative finance~\citep{da2014hawkes,bacry2015hawkes}, and e-health~\citep{xu2017patient}.
Maximum likelihood estimation (MLE) is one of the most popular approaches for learning Hawkes processes~\citep{lewis2011nonparametric,zhou2013learning}.
In addition to MLE, least squares methods~\citep{eichler2017graphical}, Wiener-Hopf equations~\citep{bacry2015hawkes}, and the Cumulants-based method~\citep{achab2016uncovering} have been applied.
However, stochastic optimization for Hawkes processes has not been studied systematically.
For nonparametric Hawkes processes, online learning methods were proposed in~\citep{hall2016tracking,yang2017online}, but they use time-consuming discretization or kernel-estimation when learning models, and thus have poor scalability.
The risk bound and the sample complexity of learning a single Hawkes process are investigated in~\citep{daneshmand2014estimating,eichler2017graphical,yang2017online}.
\section{Cohort}
\subsection{Participants and Study Design}
Adult smokers (N=60) were recruited from the Durham, NC area. All participants completed an IRB-approved informed consent form. Participant characteristics (age, sex, race, ethnicity, marital status, highest education completed, number of years smoked, and cigarettes smoked per day) were collected during a baseline study visit. After the visit, smokers carried a small, rechargeable GPS tracker for approximately one week. The trackers are smaller than a cigarette pack and fit easily in a pocket or purse. Participants were required to keep the tracker on their person at all times except when it could become wet (e.g. while bathing or swimming), and they received detailed instructions on tracker use and recharging.
A button on the front of the tracker can be pressed to indicate the times of events of interest. Participants were instructed to manually log all cigarettes smoked by pressing this button just before lighting the cigarette. \citet{shiffman2002immediate} showed that smoking events self-reported with a button press were highly correlated with cotinine and exhaled CO levels as well as participant recall on timeline follow-back assessment. Nevertheless, we took additional steps (see Section 3.3) to validate these data.
\subsection{Inclusion and exclusion criteria}
Participants were required to be generally healthy and between the ages of 18 and 55. They were also required to have smoked at least 5 cigarettes with normal nicotine content ($>$0.5mg) per day for at least a year, with an afternoon expired CO concentration $>$8 ppm.
Those with current or previous significant health problems (e.g. chronic hypertension, COPD, seizure disorder, liver or kidney disorder, coronary heart disease, myocardial infarction, arrhythmia) were excluded. Other exclusion criteria included use of psychoactive medications, use of smokeless tobacco, current alcohol or drug abuse (confirmed by urine drug screen), and use of nicotine replacement therapy or other smoking cessation treatment.
\subsection{Data collection and pre-processing}
Raw data downloaded from the GPS trackers included GPS coordinates, GPS performance level (coordinate accuracy), estimated speed, the time of measurement, and a button press indicator. These data were collected at 30 second intervals throughout the collection period with two exceptions. First, the trackers incorporate a ``smart tracking" functionality that prevents GPS collection when no movement is detected to conserve battery life. Second, each logged smoking event (button press) prompted an additional measurement at that time. Available features are summarized in Table 1.
\begin{table}[t]
\centering
\small{
\caption{The list of collected features}\label{tab1}
\begin{tabular}{l|l}\hline\hline
{Feature Domain} &
{Options} \\
\hline
Exhaled CO (ppm)&1. $<10$; 2. $[10, 20]$; 3. $[20, 30]$; 4. $>30$\\
\# Years Smoked &1. $<10$; 2. $[10, 20]$; 3. $>20$\\
Age &1. $<25$; 2. $[20, 25]$; 3. $>50$\\
Sex &1. Male; 2. Female\\
Race &1. White; 2. American India; 3. Asian; 4. Black; 5. Hispanic; 6. Other\\
Ethnicity &1. Non-Hispanic; 2. Hispanic; 3. Unknown\\
Education &1. Elementary; 2. Middle; 3. High; 4. Coll/Tech; 5. College Graduate\\
Marital Status &1. Single; 2. Separated; 3. Widowed; 4. Married; 5. Divorced\\
GPS Measurements &Longitude; Latitude; Speed (km/h); Distance (m)\\
Time of Day &1. Night; 2. Morning; 3. Afternoon; 4. Evening
\\ \hline\hline
\end{tabular}
\label{tab:example}
}
\end{table}
The raw data from all N=60 participants were first pre-processed to exclude participants for whom there was strong evidence of unreliable button presses. This was determined by partitioning each participant's data into individual days and comparing (a) the rate of daily button presses to the expected daily rate (based on self-reported cigarettes per day) by nonparametric two-sample proportion test; and (b) comparing the rate of button presses over the entire collection period to the expected rate. Participants were excluded when $p<0.01$ for two or more days --- an event expected to occur with approximately 0.002 probability given 7 days of collection (21 pairs of days) --- or when $p<0.01$ for the collection period as a whole.
As a final pre-processing step, repeated button presses ($<$60 seconds apart) were removed. Multiple rapid sequences of presses were observed in several participants. Since the tracker does not provide feedback/confirmation when the button is pressed, these sequences may reflect the participant's uncertainty as to whether their button presses have been recorded. The 60 second threshold was chosen was a reasonable lower bound on the time between successive cigarettes. We assume that these sequences do represent a smoking event, however, so the final button press in each such sequence was retained.
\section{Proposed method}
\subsection{Time-varying Semi-parametric Hawkes processes}
Consider $N$ individuals' smoking behaviors, $i.e.$, $\mathcal{S}=\{\bm{s}_{n}\}_{n=1}^{N}$, where $\bm{s}_n=\{t_1^n, t_2^n,...\}$ records the timestamps of the $n$-th individual's smoking behaviors.
Participant characteristics ($e.g.$, race, gender, age and education) are recorded as a time-invariant feature $\bm{f}_0^n\in\mathbb{R}^C$.
A participant's motion status ($e.g.$, location and speed) is also recorded as a time-varying feature vector $\bm{f}^n(t)\in\mathbb{R}^D$.
For each individual, the sequence of his/her smoking behaviors is an instantiation of a temporal point process system.
As previously mentioned, the dynamics of the system ($i.e.$, the occurrence of a smoking behavior at certain time) are influenced by three factors: an intrinsic propensity of that individual, the current environment and status, and the triggering from historical smoking events.
From the viewpoint of Hawkes processes, the first two factors are history-independent, and may be viewed as exogenous factors of the system; while the remaining factor depends on historical behaviors, and is the endogenous factor of the system.
As this analysis suggests, the Hawkes process is well-suited for capturing factors affecting smoking behavior in its intensity function.
In particular, we propose the following semi-parametric Hawkes process model for smoking behaviors.
For each individual $n$, the intensity function is written as:
\begin{eqnarray}\label{sphp}
\begin{aligned}
\lambda^n(t) = \underbrace{\bm{\mu}_{TI}^{\top}\bm{f}_0^n +\bm{\mu}_{TV}^{\top}\bm{f}^n(t)}_{\text{exogenous factors}}+\underbrace{\sideset{}{_{t_i^n<t}}\sum \phi(t-t_i)}_{\text{endogenous factor}},~\text{for}~n=1,...,N,
\end{aligned}
\end{eqnarray}
where $\bm{\mu}_{TI}\in\mathbb{R}^C$ and $\bm{\mu}_{TV}\in\mathbb{R}^D$ are parameters corresponding to time-invariant and time-varying features, respectively.
The $\bm{\mu}_{TI}$, $\bm{\mu}_{TV}$, and impact function $\phi(t)$ are the parameters of our model, which are shared across all subjects.
In contrast to traditional parametric or nonparametric Hawkes process models~\citep{zhou2013learning,zhou2013learning2,eichler2017graphical}, here we design a time-varying semi-parametric Hawkes process (TV-SPHP) model: the exogenous intensity $\mu(t)$ in (\ref{hp}) is parametrized as $\bm{\mu}_{TI}^{\top}\bm{f}_0^n +\bm{\mu}_{TV}^{\top}\bm{f}^n(t)$ to take advantage of all available features, while the impact function $\phi(t)$ is modeled nonparametrically to reduce the risk of model misspecification.
In particular, we use the basis representation method to model the impact function as
\begin{eqnarray}\label{impact}
\begin{aligned}
\phi(t) = \sideset{}{_{m=1}^{M}}\sum a_m\kappa_m(t),
\end{aligned}
\end{eqnarray}
where $\{\kappa_m(t)\}_{m=1}^{M}$ are predefined basis functions ($e.g.$, wavelets, Gabor, Gaussian kernel, etc.) and $\bm{a}=\{a_m\}_{m=1}^M$ are the corresponding coefficients.
Given our training event sequences, we apply the basis selection method proposed in~\citep{xu2016learning} to decide the number and bandwidth of the basis functions.
\subsection{Learning algorithm}
Given training event sequences $\mathcal{S}=\{\bm{s}^n\}$ from $N$ subjects and the proposed model mentioned above, we can learn the parameters of the model via maximum likelihood estimation (MLE), which is a common method used in many existing works~\citep{lewis2011nonparametric,zhou2013learning,luo2015multi}.
In particular, the learning problem of our model is the following optimization problem:
\begin{eqnarray}\label{loss}
\begin{aligned}
\sideset{}{_{\bm{\theta}\geq\bm{0}}}\min\mathcal{L}(\bm{\theta}; \mathcal{S})+\gamma\mathcal{R}(\bm{\theta}),
\end{aligned}
\end{eqnarray}
where $\bm{\theta}=[\bm{\mu}_{TI};\bm{\mu}_{TV};\bm{a}=[a_m]]\in\mathbb{R}^{C+D+M}$ are the parameters of our model.
According to~\citet{daley2007introduction}, the negative log-likelihood function $\mathcal{L}(\bm{\theta};\mathcal{S})$ is
\begin{eqnarray}\label{nll}
\begin{aligned}
\mathcal{L}(\bm{\theta};\mathcal{S})=\sideset{}{_{n=1}^{N}}\sum\Bigl\{ \int_{0}^{T}\lambda^n(s)ds - \sideset{}{_{i=1}^{I_n}}\sum \log\lambda^n(t_i^n) \Bigr\},
\end{aligned}
\end{eqnarray}
where $T$ is the length of training event sequence, and $I_n$ is the number of events of the $n$-th sequence.
Based on (\ref{sphp}) and (\ref{impact}), it is easy to prove that (\ref{nll}) is convex~\citep{zhou2013learning}.
$\mathcal{R}(\bm{\theta})$ is an arbitrary convex regularizer and its significance is controlled by the weight $\gamma$.
In this work, we only impose sparsity on the parameters, $i.e.$, $\mathcal{R}(\bm{\theta})=\|\bm{\theta}\|_1=\|\bm{\mu}_{TI}\|_1+\|\bm{\mu}_{TV}\|_1+\|\bm{a}\|_1$, where $\|\cdot\|_1$ denotes the $\ell_1$ norm.
Similar to the algorithms in~\citep{zhou2013learning}, we introduce an auxiliary variable and a dual one for $\bm{\theta}$, denoted as $\bm{y}$ and $\bm{z}$, and rewrite (\ref{loss}) as a Lagrangian form:
\begin{eqnarray}\label{lagrange}
\begin{aligned}
\sideset{}{_{\bm{\theta},\bm{y},\bm{z}\geq\bm{0}}}\min \widetilde{\mathcal{L}}(\bm{\theta},\bm{z},\bm{y};\mathcal{S})=\mathcal{L}(\bm{\theta}; \mathcal{S})+\gamma\|\bm{z}\|_1 + \rho\bm{y}^{\top}(\bm{\theta}-\bm{z}) + \frac{\rho}{2}\|\bm{\theta}-\bm{z}\|_2^2.
\end{aligned}
\end{eqnarray}
This problem can be solved iteratively by the alternating direction method of multipliers (ADMM)~\citep{zhou2013learning}.
In particular, the scheme of our learning algorithm is shown in Algorithm~\ref{alg1}, where the function $(\cdot)_{+}$ in line 5 sets all negative elements to zeros and the function $S_{\tau}(\cdot)$ in line 6 is the soft-thresholding function with threshold $\tau$.
The derivation details of the gradient of $\mathcal{L}(\bm{\theta};\mathcal{S})$ at $\bm{\theta}^{(j-1)}$, $i.e.$, $\frac{\partial \mathcal{L}(\bm{\theta};\mathcal{S})}{\partial \bm{\theta}}|_{\bm{\theta}^{(j-1)}}$, are shown in Appendix~A.
\begin{algorithm}[t]
\caption{Learning Time-varying Semi-parametric Hawkes Processes (TV-SPHP)}
\label{alg1}
\begin{algorithmic}[1]
\STATE \textbf{Input:} Event sequences $\mathcal{S}$. Weight $\gamma$. Learning rate $\delta$. Iteration number $J$.
\STATE \textbf{Output:} Parameters $\bm{\theta}=[\bm{\mu}_{TI};\bm{\mu}_{TV};\bm{a}]$.
\STATE Initialize $\bm{\theta}^{(0)}$ randomly from uniform distribution, $\bm{z}^{(0)}=\bm{\theta}^{(0)}$, $\bm{y}^{(0)}=\bm{0}$.
\FOR{$j=1,...,J$}
\STATE Gradient descent: $\bm{\theta}^{(j)}=(\bm{\theta}^{(j-1)}-\delta(\frac{\partial \mathcal{L}(\bm{\theta};\mathcal{S})}{\partial \bm{\theta}}|_{\bm{\theta}^{(j-1)}}+\rho(\bm{\theta}-\bm{z}^{(j-1)}+\bm{y}^{(j-1)})))_{+}$.
\STATE $\min\frac{\rho}{2}\|\bm{\theta}^{(j)}+\bm{y}^{(j-1)}-\bm{z}\|_2^2+\gamma\|\bm{z}\|_1$ by soft-thresholding: $\bm{z}^{(j)}=S_{\gamma/\rho}(\bm{\theta}^{(j)}+\bm{y}^{(j-1)})$.
\STATE Update dual variable: $\bm{y}^{(j)}=\bm{y}^{(j-1)}+\bm{\theta}^{(j)}-\bm{z}^{(j)}$.
\ENDFOR
\STATE $\bm{\theta}=\bm{\theta}^{(J)}$.
\end{algorithmic}
\end{algorithm}
\subsection{Simulation-based prediction}
Given the learned point process model and historical observations, we can predict the expected number of events in future time intervals.
In particular, for the target interval, we can simulate a set of synthetic event sequences based on Ogata's modified thinning method~\citep{ogata1981lewis}.
The expected number of events in this interval can be estimated by the mean of the numbers of the events in these synthetic event sequences.
It should be noted that when predicting future events based on our TV-SPHP model, we do not have access to the time-varying features for the future time interval.
Fortunately, for predictions in a short interval ($e.g.$, next 20 minutes or next one hour), we may assume that the time-varying features are relatively stable, allowing them to be approximated as the features corresponding to the last observed event.
In summary, the scheme of our prediction method is shown in Algorithm~\ref{alg2}, in which the supremum of the intensity function in $[t, t_0+T]$ is approximated by discretization: $\sup_{s\in [t, t_0+T]}\lambda(s)\approx \max_{i=0,1,...,\frac{t_0+T-t}{\Delta}}\lambda(t+i\Delta)$, where the step size $\Delta$ is predefined.
\begin{algorithm}[t]
\caption{Predicting the expected number of events in $[t_0, t_0+T]$}
\label{alg2}
\begin{algorithmic}[1]
\STATE \textbf{Input:} History $\mathcal{H}_{t_0}$. Learned model $\bm{\theta}$. Features $f_0$ and $f(t_0)$. Simulation trials $J$.
\STATE \textbf{Output:} The expected number of events $\widehat{N}$.
\STATE Initial $\widehat{N}=0$.
\FOR{$j=1,...,J$}
\STATE $t=t_0$.
\REPEAT
\STATE Compute $m(t)=\sup_{s\in [t, t_0+T]}\lambda(s)$. Generate $s\sim \exp(m(t))$, $u\sim \mbox{Uniform}([0, 1])$.
\STATE $t=t+s$. \textbf{If} $t<t_0+T$ \textbf{and} $u<\frac{\lambda(t)}{m(t)}$, \textbf{then} $\widehat{N}=\widehat{N}+1$.
\UNTIL{$t>t_0+T$}
\ENDFOR
\STATE $\widehat{N}=\widehat{N}/J$.
\end{algorithmic}
\end{algorithm}
\begin{table}[t]
\centering
\small{
\caption{Prediction performance of all models}\label{tab2}
\begin{tabular}{c|c|cccccc}\hline\hline
\multirow{2}{*}{Method} &
\multirow{2}{*}{LogLike.} &
10 mins & 20 mins & 30 mins & 40 mins & 50 mins & 1 hr\\
& & MAE & MAE & MAE & MAE & MAE & MAE \\
\hline
Poisson &-2132.7 &
0.080 & 0.173 &
0.259 & 0.381 &
0.570 & 0.802\\
Multi-task HP &-2093.9 &
0.075 & 0.169 &
0.246 & \textbf{0.356} &
0.530 & 0.751\\
TI-SPHP &-2095.6 &
\textbf{0.074} & 0.174 &
0.247 & 0.364 &
0.545 & 0.763\\
TV-SPHP &\textbf{-2058.4} &
\textbf{0.074} & \textbf{0.167} &
\textbf{0.243} & 0.365 &
\textbf{0.523} & \textbf{0.733}
\\ \hline\hline
\end{tabular}
\label{tab:example}
}
\end{table}
\section{Experimental Results}
There were 42 subjects with valid data who jointly recorded a total of 5483 smoking events. Individual participants logged between 37 and 389 smoking events (1.9-19.5 packs), with a median of 113 (5.7 packs) and an interquartile range of 79 to 152 (4.0-7.6 packs).
The dataset was divided into $54$ smoking event sequences.
Each sequence records the timestamps of a subject's smoking behaviors over a $4$ day period.
The first $3$ days' events are used for training and the remaining events are used for testing.
Based on different learned models, we predict the expected number of smoking events in the next time interval given the history of observations.
The length of the time interval ranges from $10$ minutes to $1$ hour in $10$ minute increments, as shown in Table 2, to explore how the different models are affected. We feel that different windows in this range may be appropriate depending on the specific intervention (see Discussion).
To demonstrate the feasibility and superiority of our time-varying semi-parametric Hawkes process (\textbf{TV-SPHP}) method, we test it on our dataset and compare its performance with its variants and other state-of-the-art methods, including the parametric inhomogeneous Poisson process (\textbf{Poisson})~\citep{rathbun2016mixed}, the multi-task Hawkes process (\textbf{Multi-task HP})~\citep{xu2017benefits}, and the time-invariant semi-parametric Hawkes process (\textbf{TI-SPHP}).
In particular, the Poisson method was designed for smoking event analysis, but only considers time-invariant and time-varying features while ignoring triggering patterns caused by previous smoking events.
The multi-task HP does not utilize features and relies solely on temporal information, so that the exogenous part for each subject $n$ is represented as a value $\mu^n$ and learned directly.
The TI-SPHP utilizes only the time-invariant features, $i.e.$ it ignores the term $\bm{\mu}_{TV}^{\top}\bm{f}^n(t)$ in (\ref{sphp}).
\begin{table}[t]
\centering
\small{
\caption{Standard deviation of prediction performance across simulation trials}\label{tab3}
\begin{tabular}{c|cccccc}\hline\hline
\multirow{2}{*}{Method} &
10 mins & 20 mins & 30 mins & 40 mins & 50 mins & 1 hr\\
& SD & SD & SD & SD & SD & SD \\
\hline
Poisson &
0.011 & 0.009 &
0.010 & 0.017 &
0.014 & 0.008\\
Multi-task HP &
0.007 & 0.008 &
0.010 & 0.008 &
0.010 & 0.013\\
TI-SPHP &
0.005 & 0.012 &
0.011 & 0.011 &
0.015 & 0.016\\
TV-SPHP &
0.005 & 0.005 &
0.007 & 0.011 &
0.012 & 0.011
\\ \hline\hline
\end{tabular}
\label{tab:example}
}
\end{table}
As previously mentioned, the time-invariant features include participant characteristics, which can be represented as $31$-dimensional ($C=31$) binary vectors.
Each time-varying feature is an $8$-dimensional ($D=8$) vector consisting of a $4$-dimensional real-valued vector containing the participant's location (latitude and longitude), their speed, and the distance from their location to their home; and a $4$-dimensional binary vector representing the time of day.
The basis $\{\kappa_m\}$ are the Gaussian basis, with their number and bandwidth set using the method in~\citep{xu2016learning}.
In all of the above models, we apply an iterative optimization method to learn the parameters.
For our learning algorithm (Algorithm~\ref{alg1}), the number of iterations is $J=30$ and the learning rate is $\delta=0.01$.
The weight $\gamma=0.2$ is decided by cross-validation.
For our prediction algorithm (Algorithm~\ref{alg2}), the number of simulation trials is $20$, the step size $\Delta$ is $10$ minutes.
\subsection{Model Performance}
For each testing sequence $k$, $k=1,...,K$, the true number of smoking events in a specific testing interval is $N_k$, and the expected number obtained from a certain learned model is $\widehat{N}_k$. Note that for the current dataset, $K$ is 54, the number of test sequences. We then calculate the following performance measures:
\begin{enumerate}
\item \textbf{MAE.} The mean absolute error of the estimated number:
\begin{eqnarray*}
\begin{aligned}
\mbox{MAE}=\frac{1}{K}\sideset{}{_{i=1}^{K}}\sum |N_k-\widehat{N}_k|.
\end{aligned}
\end{eqnarray*}
\item \textbf{Testing Log-Likelihood.} The log-likelihood values of the testing sequences are calculated for each model.
\end{enumerate}
Performance of all models is compared in Table 2. TV-SPHP performance was superior to the other models except for the 40-minute prediction window, and was superior to the Poisson model for all windows. Table 3 shows the standard deviation of the MAE across all simulation trials, demonstrating that results are consistent across testing sequences. In most cases, standard deviation is lowest for the TV-SPHP.
Binary classification accuracy (i.e. will smoke, will not smoke) was identical for all models -- values were 92.6\%, 85.2\%, 77.8\%, 64.8\%, and 53.7\% for 10-60 minute windows, respectively -- because smoking is a rare event with respect to these prediction windows, so all models heavily favor not smoking over smoking. For this reason, the MAE is the more appropriate measure of smoking risk in this application (see Discussion).
\subsection{Model Interpretation}
Model parameters are summarized and compared in Figure 1. Among the time-varying parameters, time of day was weighted much more heavily than the GPS measurements. The impact function, visualized in Figure 2, peaks within the first hour then decays over several hours, with marked daily periodicity.
\begin{figure}[h]
\centering
\includegraphics[width=.8\textwidth]{exogenous2.pdf}
\caption{Standardized coefficients for all exogenous factors}
\label{fig:coefficients}
\end{figure}
\begin{figure}[h]
\centering
\subfigure[Endogenous impact function]{
\includegraphics[width=.4\textwidth]{impact_days.pdf}
}
\subfigure[Enlarged endogenous impact function]{
\includegraphics[width=.4\textwidth]{impact_hours.pdf}
}
\caption{Visualization of (a) the learned impact function with length $3$ days, and (b) an enlarged view of the first $6$ hours. For better visual effect, we show the square root of the impact function, $i.e.$, $\phi^{0.5}(t)$, in the time domain.
}
\label{fig:impact}
\end{figure}
\section{Discussion}
To our knowledge, this study is the first to model smoking as a self-triggering process, and among the first to evaluate the performance of smoking event prediction. Moreover, we have developed the time-varying semi-parametric Hawkes process (TV-SPHP), an extension of previous Hawkes process variants that is well-suited to this application. These methods and performance statistics can serve as baselines for future prediction models, leading to improved prediction of smoking risk. Effective prediction can be used to identify moments of high lapse risk and their causes/precursors, ultimately supporting more effective cessation strategies that could reduce death from smoking.
As hypothesized, prediction performance favored the TV-SPHP over its variants and previous methods, validating the advantages of modeling smoking as a self-triggering process. The models with and without time-varying features were similar over shorter prediction windows, but the TV-SPHP was superior over longer windows, most notably the 1 hour window. This suggests that time-varying features --- such as speed and time of day -- offer greater predictive advantage as the time since the last known smoking event increases.
The most appropriate prediction window depends on the application or intervention. A 10-minute window might fit a nicotine lozenge intervention due to its fast effect, whereas a longer window might be more appropriate for a self-help or peer support intervention. We feel that prediction windows longer than an hour are less reasonable, because (a) the time-varying features may change substantially during the window, and (b) smoking events occurring during the window affect the expected number of subsequent events.
Smoking is a rare event with respect to our prediction windows, therefore all models predict that the participant will not smoke in a given window. For this reason, accuracy (and other binary prediction measures) is a misleading measure of performance. The TV-SPHP most closely estimates the true number of cigarettes smoked (i.e. lowest MAE), which may be used as a measure of smoking risk. When placing a threshold on model output to trigger an intervention, a low threshold would be needed to achieve high sensitivity. This would also result in a high false positive rate, which may be acceptable given the importance of avoiding a lapse. The appropriate trade-off between these measures is intervention-specific.
The Hawkes process impact function is capable of capturing both short-term temporal dynamics, which may relate to nicotine metabolism and/or nicotine seeking, as well as longer-term daily patterns. For example, the peak $\sim$30 minutes after smoking may reflect nicotine seeking as plasma nicotine levels decline. This hypothesis could be explored in future work by testing how the learned impact function differs between groups with/without nicotine replacement. The location of this peak, which we hypothesized would occur within 1-4 hours of smoking, occurred earlier than expected. On average, plasma nicotine levels remain above $\frac{2}{3}$ of their peak value when the impact function peaks, but importantly, they are declining sharply.
The peaks 1 and 2 days after smoking, on the other hand, more likely reflect daily patterns and preferences rather than direct effects of nicotine. The clear daily periodicity of the impact function is striking, and corroborates previous observations that individual smokers exhibit consistent daily patterns of smoking \citep{Frederiksen1977,Robinson1980,Chandra2007}. The nature of the factors underlying these patterns (e.g. physiologic versus environmental) remains to be explored.
\subsection{Limitations and Future Work}
A first and arguably most important limitation of this work is the uncertain reliability of self-reported smoking events. This problem is a well-known difficulty of detecting or predicting real-world smoking. Our pre-processing excluded participants with an unreasonable (high or low) number of button presses, but knowing the number is consistent with the participant's smoking habits does not guarantee that it is accurate. This step eliminated almost a third of participants (18/60), which underscores the difficulty of relying on self-report. Further, there is also no way to know whether the \textit{timing} of individual presses is accurate, which is particularly important given our focus on temporal dynamics. Research in smoking risk prediction will benefit from technologies that reliably detect smoking, bypassing the problems of self-report.
This pre-processing was essential to the goals of this study, but it also introduces a potential source of bias: participants with unpredictable smoking patterns may have been excluded because they appeared to be unreliable self-reporters. We have tried to mitigate this possibility by using a conservative exclusion threshold.
Finally, the measurements and participant populations in the parent studies were tailored to the objectives of those studies. We intend to extend this work with new data collection that includes a more comprehensive set of physiologic, self-report, and environmental measures. Many such factors are known to be associated with smoking risk and would be expected to improve prediction performance. A broader cohort of participants will be recruited, including those with significant health problems. These participants may be good candidates for intervention due to their more frequent interaction with the healthcare system. Follow-up work will also explore more meaningful incorporation of location information via a hierarchical model that converts raw GPS coordinates to personal activity space (e.g. distinct locations visited by the participant throughout their day).
Future work will also explore relationships between nicotine metabolism (e.g. nicotine metabolite ratio) and the self-triggering nature of smoking, as quantified by the Hawkes process impact function. Nicotine metabolism is known to affect the relative efficacy of varenicline versus nicotine patch in smoking cessation \citep{Lerman2015}, for example. With our approach, the effect of pharmacologics and/or lapses on subsequent smoking risk could be explored through simulation, which may have important implications for cessation interventions.
\section{Conclusion}
In this work, we have developed the time-varying semi-parametric Hawkes process and applied it to the important problem of smoking prediction using data collected by 42 smokers. Motivated by the known influence of daily patterns and plasma nicotine levels on smoking behaviors, our approach models smoking as a self-triggering process. Results illustrate the temporal dynamics of smoking and demonstrate improved prediction performance compared to previous methods. Future work will extend the approach to a more comprehensive dataset and explore the effects of interventions such as nicotine replacement on temporal smoking dynamics.
Effective smoking prediction models can pinpoint moments of high lapse risk in order to trigger a just-in-time intervention, for example with a mobile device. Modeling smoking as a self-triggering process may help to explain how a single lapse can progress to a failed quit attempt, thereby supporting cessation and reducing death from smoking.
| {
"redpajama_set_name": "RedPajamaArXiv"
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Feline Nuvo™ offers a variety of play and lounging options. Sisal-wrapped supports provide a tough, resilient scratching post for your cat's scratching needs while the plush dark fabric easily wipes clean from cats accidents.
Escapade: Bolstered Lounging Perch, Resting Bench, Lounging Lookout & Low Bolstered Bed with Sisal-Wrapped Support.
32" X 18" X 40" | {
"redpajama_set_name": "RedPajamaC4"
} | 4,872 |
var app = angular./**
* app Module
*
* Description
*/
module('app', []).config(function($provide) {
$provide.constant('loadUser', 'postRequests/profile/loadUser.php');
});
app.controller('controller', ['$scope', '$http', 'loadUser', function($scope, $http, loadUser){
$scope.data = {},
$scope.data.profile = {};
$scope.init = function(user_id) {
$scope.data.profile.user_id = user_id;
var promiseLoadBasicProfile = $http({
method: "post",
url: loadUser,
data: {
profile_id: $scope.profile.user_id
},
headers: { 'Content-Type': 'application/x-www-form-urlencoded' }
});
promiseLoadBasicProfile.then(function(successReponse) {
$scope.data.profile = successReponse.data;
}, function(errorResponse) {
console.log('Error fetching user profile');
});
}
}]); | {
"redpajama_set_name": "RedPajamaGithub"
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Mustafà ibn Mahmud () (Tunis, agost de 1786 - la Goleta, 10 d'octubre de 1837) fou bei de Tunis de la dinastia husaynita de Tunísia de 1835 a 1837.
Era fill de Mahmud ibn Muhammad i el germà petit d'al-Husayn II ibn Mahmud, i fou designat hereu pel seu germà el 1824 i el va succeir quan va morir el 20 de maig de 1835. El 15 de desembre següent fou nomenat general de l'exèrcit otomà.
El febrer de 1837, sota consell del gran visir Sidi Rashid al-Shakir Sahib al-Tabaa va fer un cens dels joves del àis entre 10 i 25 anys per reclutar-los per l'exèrcit; la població de la ciutat, que fins aleshores estava lliure de servir a l'exèrcit, es va enfurismar, i es va crear una gran tensió. Els notables van fer saber que el projecte violava els privilegis tradicionals dels ciutadans. El bey però mantenia el seu pla, i només el va abandonar per una revolta de les tribus de l'oest del país, fent necessari l'enviament a la zona de l'exèrcit; suprimida la rebel·lió el projecte fou arxivat. L'instigador, el gran visir, fou executat l'11 de setembre de 1837 al palau del Bardo; el bey el va acusar de conspirar contra la dinastia i contra la seva vida. El va substituir un altre mameluc de nom Mustafà Khaznadar, antic esclau d'origen grec, que tenia només 21 anys.
Va morir pocs dies després, el 10 d'octubre de 1837 i el va succeir el seu fill Ahmad I ibn Mustafà.
Bibliografia
El Mokhtar Bey, Les Beys de Tunis 1705-1957. Tunis, 2002.
Beis de Tunísia
Morts a Tunísia
Polítics tunisians | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,056 |
Aughton Park
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"redpajama_set_name": "RedPajamaCommonCrawl"
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<B>Uses of Class<br>com.netflix.astyanax.serializers.AbstractSerializer</B></H2>
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Packages that use <A HREF="../../../../../com/netflix/astyanax/serializers/AbstractSerializer.html" title="class in com.netflix.astyanax.serializers">AbstractSerializer</A></FONT></TH>
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<TR BGCOLOR="white" CLASS="TableRowColor">
<TD><A HREF="#com.netflix.astyanax.serializers"><B>com.netflix.astyanax.serializers</B></A></TD>
<TD> </TD>
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<A NAME="com.netflix.astyanax.serializers"><!-- --></A>
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Uses of <A HREF="../../../../../com/netflix/astyanax/serializers/AbstractSerializer.html" title="class in com.netflix.astyanax.serializers">AbstractSerializer</A> in <A HREF="../../../../../com/netflix/astyanax/serializers/package-summary.html">com.netflix.astyanax.serializers</A></FONT></TH>
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<TABLE BORDER="1" WIDTH="100%" CELLPADDING="3" CELLSPACING="0" SUMMARY="">
<TR BGCOLOR="#CCCCFF" CLASS="TableSubHeadingColor">
<TH ALIGN="left" COLSPAN="2">Subclasses of <A HREF="../../../../../com/netflix/astyanax/serializers/AbstractSerializer.html" title="class in com.netflix.astyanax.serializers">AbstractSerializer</A> in <A HREF="../../../../../com/netflix/astyanax/serializers/package-summary.html">com.netflix.astyanax.serializers</A></FONT></TH>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/AnnotatedCompositeSerializer.html" title="class in com.netflix.astyanax.serializers">AnnotatedCompositeSerializer<T></A></B></CODE>
<BR>
Serializer for a Pojo annotated with Component field annotations
Serialized data is formatted as a list of components with each component
having the format: <2 byte length><data><0></TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/AsciiSerializer.html" title="class in com.netflix.astyanax.serializers">AsciiSerializer</A></B></CODE>
<BR>
Almost identical to StringSerializer except we use the US-ASCII character set
code</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/BigIntegerSerializer.html" title="class in com.netflix.astyanax.serializers">BigIntegerSerializer</A></B></CODE>
<BR>
Serializer implementation for BigInteger</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/BooleanSerializer.html" title="class in com.netflix.astyanax.serializers">BooleanSerializer</A></B></CODE>
<BR>
Converts bytes to Boolean and vice versa</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/ByteBufferSerializer.html" title="class in com.netflix.astyanax.serializers">ByteBufferSerializer</A></B></CODE>
<BR>
The BytesExtractor is a simple identity function.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/BytesArraySerializer.html" title="class in com.netflix.astyanax.serializers">BytesArraySerializer</A></B></CODE>
<BR>
A BytesArraySerializer translates the byte[] to and from ByteBuffer.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/CharSerializer.html" title="class in com.netflix.astyanax.serializers">CharSerializer</A></B></CODE>
<BR>
Uses Char Serializer</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/CompositeSerializer.html" title="class in com.netflix.astyanax.serializers">CompositeSerializer</A></B></CODE>
<BR>
</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/DateSerializer.html" title="class in com.netflix.astyanax.serializers">DateSerializer</A></B></CODE>
<BR>
Converts bytes to Date and vice versa, by first converting the Date to or
from a long which represents the specified number of milliseconds since the
standard base time known as "the Unix epoch", that is January 1, 1970,
00:00:00 UTC.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/DoubleSerializer.html" title="class in com.netflix.astyanax.serializers">DoubleSerializer</A></B></CODE>
<BR>
Uses LongSerializer via translating Doubles to and from raw long bytes form.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/DynamicCompositeSerializer.html" title="class in com.netflix.astyanax.serializers">DynamicCompositeSerializer</A></B></CODE>
<BR>
</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/FloatSerializer.html" title="class in com.netflix.astyanax.serializers">FloatSerializer</A></B></CODE>
<BR>
Uses IntSerializer via translating Float objects to and from raw long bytes
form.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/IntegerSerializer.html" title="class in com.netflix.astyanax.serializers">IntegerSerializer</A></B></CODE>
<BR>
Converts bytes to Integer and vice versa</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/JaxbSerializer.html" title="class in com.netflix.astyanax.serializers">JaxbSerializer</A></B></CODE>
<BR>
Serializes Objects using Jaxb.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/LongSerializer.html" title="class in com.netflix.astyanax.serializers">LongSerializer</A></B></CODE>
<BR>
Converts bytes to Long and vise a versa</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/ObjectSerializer.html" title="class in com.netflix.astyanax.serializers">ObjectSerializer</A></B></CODE>
<BR>
The ObjectSerializer is used to turn objects into their binary
representations.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/PrefixedSerializer.html" title="class in com.netflix.astyanax.serializers">PrefixedSerializer<P,S></A></B></CODE>
<BR>
</TD>
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<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/ShortSerializer.html" title="class in com.netflix.astyanax.serializers">ShortSerializer</A></B></CODE>
<BR>
<CODE>Serializer</CODE> for <A HREF="http://download.oracle.com/javase/6/docs/api/java/lang/Short.html?is-external=true" title="class or interface in java.lang"><CODE>Short</CODE></A>s (no pun intended).</TD>
</TR>
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<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/SpecificCompositeSerializer.html" title="class in com.netflix.astyanax.serializers">SpecificCompositeSerializer</A></B></CODE>
<BR>
</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/StringSerializer.html" title="class in com.netflix.astyanax.serializers">StringSerializer</A></B></CODE>
<BR>
A StringSerializer translates the byte[] to and from string using utf-8
encoding.</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/TimeUUIDSerializer.html" title="class in com.netflix.astyanax.serializers">TimeUUIDSerializer</A></B></CODE>
<BR>
</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/TypeInferringSerializer.html" title="class in com.netflix.astyanax.serializers">TypeInferringSerializer<T></A></B></CODE>
<BR>
A serializer that dynamically delegates to a proper serializer based on the
value passed</TD>
</TR>
<TR BGCOLOR="white" CLASS="TableRowColor">
<TD ALIGN="right" VALIGN="top" WIDTH="1%"><FONT SIZE="-1">
<CODE> class</CODE></FONT></TD>
<TD><CODE><B><A HREF="../../../../../com/netflix/astyanax/serializers/UUIDSerializer.html" title="class in com.netflix.astyanax.serializers">UUIDSerializer</A></B></CODE>
<BR>
A UUIDSerializer translates the byte[] to and from UUID types.</TD>
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,128 |
from django.conf import settings
from django.core.management.base import BaseCommand, CommandError
from pathlib import Path
import logging, requests, json, base64, html, glob, os
logger=logging.getLogger('webframe.commands.install')
try:
import xml.etree.cElementTree as ET
except ImportError:
logger.warning('Using the Python ElementTree instead of CElementTree')
import xml.etree.ElementTree as ET
b64=lambda path: base64.b64encode(Path(path).read_text().encode('utf-8')).decode('ascii')
ns='http://jasperreports.sourceforge.net/jasperreports'
class Command(BaseCommand):
help = 'Install and prepare the reports into JasperServer'
def add_arguments(self, parser):
parser.add_argument('--host', dest='rpthost', type=str, help='The hostname of the report server; Default: http://rpthost:8080', default='http://rpthost:8080')
parser.add_argument('--admin', dest='adminuser', type=str, help='The admin username of the report server; Default: jasperadmin', default='jasperadmin')
parser.add_argument('--pass', dest='adminpass', type=str, help='The admin password of the report server; Default: jasperadmin', default='jasperadmin')
parser.add_argument('--prefix', dest='prefix', type=str, help='The prefix of installed reports URI; It should be starting and ending with a slash \'/\'; Default: /webframe/', default='/webframe/')
parser.add_argument('--reports', dest='reports', type=str, help='The path of the target report(s); Default: ./rpt/MyReports/*xml', default='./rpt/MyReports/*.jrxml')
parser.add_argument('--dbconn', dest='dbconn', type=str, help='Specify the database connection for each imported report; Default: ./rpt/MyReports/DBConn.xml', default='./rpt/MyReports/DBConn.xml')
parser.add_argument('--role', dest='role', type=str, help='The role to access the report repository', default='webframe')
def jasperserver(self, url, **kwargs):
''' Communicate/Query the JasperServer '''
myargs=self.kwargs.copy()
myargs.update(kwargs)
data=dict() if 'data' not in myargs else json.dumps(myargs['data']).encode('utf-8')
auth=(myargs['adminuser'], myargs['adminpass'])
headers={'Content-Type': myargs.get('contentType', 'application/json')}
if 'headers' in kwargs: headers.update(kwargs['headers'])
url=url.format(**myargs)
method=myargs.get('method','put').upper()
if method=='PUT':
logger.debug(' Putting resources<{0}>: {1}'.format(kwargs['contentType'], url))
rep=requests.put(url, auth=auth, data=data, headers=headers)
elif method=='POST':
logger.debug(' Posting resources<{0}>: {1}'.format(kwargs['contentType'], url))
rep=requests.post(url, auth=auth, data=data, headers=headers)
elif method=='GET':
logger.debug(' Getting resources<{0}>: {1}'.format(kwargs['contentType'], url))
rep=requests.get(url, auth=auth, data=data, headers=headers)
elif method=='DELETE':
logger.debug(' Deleting resources<{0}>: {1}'.format(kwargs['contentType'], url))
rep=requests.delete(url, auth=auth, data=data, headers=headers)
else:
raise TypeError('Unknow HTTP method: {0}'.format(method))
return rep
def init_role(self):
# Checking the role are existing
url='{rpthost}/rest_v2/roles/{role}'
rep=self.jasperserver(url, contentType='application/json', method='get')
if rep.status_code != 200:
#If NOT exists
logger.info(' Creating reporting role: %s ...'%self.kwargs['role'])
rep=self.jasperserver(url, data={}, contentType='application/json', method='put')
if 200<=rep.status_code<300:
logger.info(' Role<%s> has been created successfully!'%self.kwargs['role'])
else:
logger.info(' Role<%s> cannot been created, status-code: %s'%(self.kwargs['role'], rep.status_code))
logger.debug(rep.text)
def init_user(self):
# Make sure the user is created.
user=settings.REPORTING['username']
pawd=settings.REPORTING['password']
url='{rpthost}/rest_v2/users/{user}'
rep=self.jasperserver(url, contentType='application/json', method='get', user=user)
if rep.status_code!=200:
logger.info(' Creating/Updating report user: %s ...'%settings.REPORTING['username'])
rep=self.jasperserver(url, contentType='application/json', method='put', user=user, data={'fullName': 'Deploy by script', 'enabled': True, 'password': pawd, 'roles': [{'name': 'ROLE_USER'}, {'name': self.kwargs['role']}]})
if 200<=rep.status_code<300:
logger.info(' Role<%s> has been created successfully!'%self.kwargs['role'])
else:
logger.info(' Role<%s> cannot been created, status-code: %s'%(self.kwargs['role'], rep.status_code))
logger.debug(rep.text)
def import_report(self, filename, root):
#Import the jasper report (*.jrxml)
types=dict()
def gettype(rptname, types, tipe):
if tipe not in types:
uri='{rpthost}/rest_v2/resources{prefix}{rptname}_Data/{tipe}'
contentType='application/repository.dataType+json'
rep=self.jasperserver(uri, method='get', contentType=contentType, tipe=tipe, rptname=rptname, headers={'accept': 'application/json'})
if 200 <= rep.status_code < 300:
types[tipe]=rep.json()['uri']
else:
logger.debug(' Creating contentType for {0}'.format(tipe))
#Server support type: text|number|date|dateTime|time
if tipe in ['java.lang.Double', 'java.lang.Float', 'java.lang.Integer', 'java.lang.Long', 'java.lang.Short', 'java.math.BigDecimal']:
serverType='number'
elif tipe=='java.sql.Date':
serverType='date'
elif tipe=='java.sql.Time':
serverType='time'
elif tipe=='java.sql.Timestamp' or tipe=='java.util.Date':
serverType='datetime'
else:
serverType='text'
rep=self.jasperserver(uri, method='put', contentType=contentType, tipe=tipe, rptname=rptname, data={
'label': tipe,
'description': 'Data-Type for {0}'.format(tipe),
'permissionMask': '0',
'version': '1',
'type': serverType,
'strictMax': 'false',
'strictMin': 'false',
})
if 200 <= rep.status_code < 300:
logger.info(' Created/Updated data-type for {0}: {1}...'.format(rptname, tipe))
url=rep.url
url=url[url.index('rest_v2/resources')+17:]
types[tipe]=url
else:
logger.debug(' [{0}]: {1}'.format(rep.status_code, rep.text))
raise TypeError('Cannot create the data-type for report: {0}'.format(rptname))
return types[tipe]
def property(rptname, tag):
logger.debug(' Handling parameter<{0}> as {1}'.format(tag.attrib['name'], tag.attrib['class']))
tipe=gettype(rptname, types, tag.attrib['class'])
uri='{rpthost}/rest_v2/resources{prefix}{rptname}_Data/{name}'
contentType='application/repository.inputControl+json'
rep=self.jasperserver(uri, method='get', contentType=contentType, name=tag.attrib['name'], rptname=rptname, headers={'accept': 'application/json'})
if 200 <= rep.status_code < 300:
return rep.json()['uri']
else:
desc=tag.find('{{{0}}}parameterDescription'.format(ns))
rep=self.jasperserver(uri, contentType=contentType, name=tag.attrib['name'], rptname=rptname, data={
'name': tag.attrib['name'],
'label': 'InputControl.{0}'.format(tag.attrib['name']),
'description': None if not desc else desc.text,
'permissionMask': '0',
'version': '1',
'mandatory': True,
'type': '2', #2==Single Value
'dataType': {'dataTypeReference': {'uri': tipe}},
'usedFields': tag.attrib['name'],
'visible': True,
})
if 200 <= rep.status_code < 300:
logger.info(' Created/Updated parameter<{0}> for report<{1}>...'.format(tag.attrib['name'], rptname))
return rep.url[rep.url.index('rest_v2/resources')+17:]
else:
logger.debug(' [{0}]: {1}'.format(rep.status_code, rep.text))
raise TypeError('Cannot create the parameter<{0}> for report<{1}>'.format(tag.attrib['name'], rptname))
def subreport(cwd, rptname, tag, params):
path=tag.find('{{{0}}}subreportExpression'.format(ns)).text #$P{rptprefix}+"Customer"+$P{rptsuffix}
path=path[path.index('\"')+1:] #Customer"+$P{rptsuffix}
path=path[0:path.index('\"')] #Customer
path='{0}.jrxml'.format(path)
filename=os.path.join(cwd, "{0}_Data".format(rptname), path)
sr=ET.ElementTree(file=filename).getroot()
logger.debug(' Handling subreport<{0}>: {1}...'.format(sr.attrib['name'], filename))
uri='{rpthost}/rest_v2/resources{prefix}{rptname}_Data/{path}'.format(rpthost=self.kwargs['rpthost'], prefix=self.kwargs['prefix'], path=path, rptname=rptname)
desc=sr.find('{{{0}}}property[@name=\'com.jaspersoft.studio.report.description\']'.format(ns))
rep=self.jasperserver(uri, contentType='application/repository.file+json', method='get', headers={'accept': 'application/json'})
if 200 <= rep.status_code < 300:
pass
else:
rep=self.jasperserver(uri, contentType='application/repository.file+json', method='put', data={
'label': sr.attrib['name'],
'description': None if not desc else desc,
'permissionMask': '0',
'version': '1',
'type': 'jrxml',
'content': b64(filename),
})
if 200 <= rep.status_code < 300:
logger.info(' Created/Updated subreport<{0}>...'.format(sr.attrib['name']))
else:
logger.debug(' [{0}]: {1}'.format(rep.status_code, rep.text))
raise TypeError('Cannot create the subreport<{0}>...'.format(sr.attrib['name']))
name=root.attrib['name'].replace('-', '_')
logger.warning('Importing "{1}" JRXML: {0}...'.format(filename, name))
cwd=os.path.dirname(filename)
parameters=dict()
for p in root.findall('{{{0}}}parameter'.format(ns)):
parameters[p.attrib['name']]=property(name, p)
for r in root.iter('{{{0}}}subreport'.format(ns)):
subreport(cwd, name, r, parameters)
uri='{rpthost}/rest_v2/resources{prefix}{name}'
rep=self.jasperserver(uri, contentType='application/repository.reportUnit+json', method='get', name=name, headers={'accept': 'application/json'})
if not 200 <= rep.status_code < 300:
desc=root.find('{{{0}}}property[@name=\'com.jaspersoft.studio.report.description\']'.format(ns))
rep=self.jasperserver(uri, contentType='application/repository.reportUnit+json', method='put', name=name, data={
'label': root.attrib['name'],
'description': None if not desc else desc,
'permissionMask': '0',
'version': '1',
'controlsLayout': 'popupScreen',
'alwaysPromptControls': 'true',
'dataSource': {'dataSourceReference': {'uri': self.dbconn}},
'inputControls': [{'inputControlReference': {'uri': parameters[p]}} for p in parameters],
'jrxml': {
'jrxmlFile': {
'type': 'jrxml',
'label': 'jrxml',
'content': b64(filename)
}
}
})
if 200 <= rep.status_code < 300:
logger.info(' Created/Updated report<{0}>...'.format(name))
else:
logger.debug(' [{0}]: {1}'.format(rep.status_code, rep.text))
raise TypeError('Cannot create the report<{0}>...'.format(name))
rep=self.jasperserver('{rpthost}/rest_v2/permissions', method='post', name='permission', contentType='application/json', data={
'uri': '{prefix}{name}'.format(prefix=self.kwargs['prefix'], name=name),
'recipient': 'role:/{0}'.format(self.kwargs['role']),
'mask': '1',
})
if not 200<= rep.status_code < 300:
logger.info(' Failed to grant permission for role: {0} in {1}{2}: {3}{4}'.format(self.kwargs['role'], self.kwargs['prefix'], name, rep.status_code, rep.text))
def import_file(self, filename):
# Import the xml file
logger.debug(' Importing file: {0}'.format(filename))
root=ET.ElementTree(file=filename).getroot()
if root.tag=='jdbcDataAdapter':
#Importing the jdbc
self.import_ds(root)
elif root.tag=='{{{0}}}jasperReport'.format(ns):
#Importing the JasperReport
self.import_report(filename, root)
else:
logger.info(' Unknow ROOT Tag in the xml, skipping...')
def import_ds(self, root):
# Make sure the data-source created
url='{rpthost}/rest_v2/resources{prefix}'
contentType='application/repository.jdbcDataSource+json'
name=root.find('name').text
rep=self.jasperserver(url, name=name, contentType=contentType, method='get')
if rep.status_code==200:
self.jasperserver(url, name=name, contentType=contentType, method='delete')
rep=self.jasperserver(url, name=name, contentType=contentType, method='post', data={
'label': name,
'description': 'The database connection imported by deployreports.py',
'permissionMask': '0',
'version': '1',
'driverClass': root.find('driver').text,
'username': root.find('username').text, #Refer to rpt/createUser.sql
'password': root.find('password').text, #Refer to rpt/createUser.sql
'connectionUrl': root.find('url').text,
})
if 200 <= rep.status_code < 300:
logger.info(' Create/Update datatsource<{0}> successfully.'.format(name))
self.dbconn='{prefix}{name}'.format(prefix=self.kwargs['prefix'], name=name)
else:
logger.info(' Create/Update datasource<{0}> failed with status code {1}'.format(name, rep.status_code))
logger.warning(rep.text)
def handle(self, *args, **kwargs):
verbosity=int(kwargs['verbosity'])
if verbosity==3:
logger.setLevel(logging.DEBUG)
elif verbosity==2:
logger.setLevel(logging.INFO)
elif verbosity==1:
logger.setLevel(logging.WARNING)
else:
logger.setLevel(logging.ERROR)
self.kwargs=kwargs
logger.warning('Report server are located at "{host}" with admin-user: {user}'.format(host=self.kwargs['rpthost'], user=kwargs['adminuser']))
self.init_role()
self.init_user()
logger.warning('Importing datasource: {0}...'.format(self.kwargs['dbconn']))
self.import_file(self.kwargs['dbconn'])
for f in glob.glob(self.kwargs['reports']):
self.import_file(f)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,596 |
{"url":"http:\/\/gmatclub.com\/forum\/goizueta-emory-calling-all-2013-mba-applicants-141069-320.html","text":"Find all School-related info fast with the new School-Specific MBA Forum\n\n It is currently 29 May 2016, 11:08\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n# Events & Promotions\n\n###### Events & Promotions in June\nOpen Detailed Calendar\n\n# Goizueta (Emory) - Calling All 2013 MBA Applicants\n\nAuthor Message\nIntern\nJoined: 01 Mar 2013\nPosts: 42\nLocation: India\nGMAT 1: 750 Q51 V40\nGPA: 3.51\nWE: Analyst (Consulting)\nFollowers: 0\n\nKudos [?]: 6 [0], given: 7\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants\u00a0[#permalink]\n\n### Show Tags\n\n08 Mar 2013, 23:23\nchdheeraj1 wrote:\nLooks like only 1 indian in this forum received an invite so far. Given the decisions come out on Mar 15, the chances are looking bleak now\n\nThe best we can hope for is a WL. I am sure there wouldn't be any interview before decision date now. Damn!\nIntern\nJoined: 28 Nov 2010\nPosts: 36\nFollowers: 0\n\nKudos [?]: 6 [0], given: 2\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants\u00a0[#permalink]\n\n### Show Tags\n\n09 Mar 2013, 22:46\nAJUnited wrote:\nchdheeraj1 wrote:\nLooks like only 1 indian in this forum received an invite so far. Given the decisions come out on Mar 15, the chances are looking bleak now\n\nThe best we can hope for is a WL. I am sure there wouldn't be any interview before decision date now. Damn!\n\nDont hope for that as well..BTW I am on the same boat..so keep no hard feelings..\nManager\nJoined: 06 Mar 2013\nPosts: 61\nFollowers: 0\n\nKudos [?]: 6 [0], given: 5\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants\u00a0[#permalink]\n\n### Show Tags\n\n11 Mar 2013, 01:14\nall hi. i got an interview weeks ago. it went.. weird, well, it was really hard to descript..my feelings about the meeting. well, basically, i don't think the adcom showed her interest(great? normal? plain? i don't know) in me.. no request about my personal life, no detailed questions about my work experience or acedemic records.. so i tend to put it as a negative sign... waiting for the DING from goizueta...\nwell, anyone has any comment? to encourage me up or.. just let me sink down...\nthanks anyway!!\nbtw, good luck to all!!\nIntern\nJoined: 01 Mar 2013\nPosts: 42\nLocation: India\nGMAT 1: 750 Q51 V40\nGPA: 3.51\nWE: Analyst (Consulting)\nFollowers: 0\n\nKudos [?]: 6 [1] , given: 7\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants\u00a0[#permalink]\n\n### Show Tags\n\n11 Mar 2013, 01:34\n1\nKUDOS\nxiangxder wrote:\nall hi. i got an interview weeks ago. it went.. weird, well, it was really hard to descript..my feelings about the meeting. well, basically, i don't think the adcom showed her interest(great? normal? plain? i don't know) in me.. no request about my personal life, no detailed questions about my work experience or acedemic records.. so i tend to put it as a negative sign... waiting for the DING from goizueta...\nwell, anyone has any comment? to encourage me up or.. just let me sink down...\nthanks anyway!!\nbtw, good luck to all!!\n\nYou should just chill out. You had an interview which means that the Adcom liked you application in the first place. From what I understand the interview is not blind (the interviewer has read your file previously) and hence they may not ask detailed questions on work-ex or academia, unless you have something glaring in your profile that they may want clarify on. Plus a lot depends on the interviewer themselves, so you can't read a lot into that.\n\nThe results are 5 days away, I would suggest just stop thinking about it and worry on Friday, not before that as you can't really do anything about it in anycase. Maybe speak to some current students to show your interest in GBS, but other than that just sit tight and keep your fingers crossed.\nManager\nJoined: 01 Oct 2012\nPosts: 94\nLocation: United States\nGMAT 1: 720 Q V\nGMAT 2: Q V0\nGPA: 3.2\nWE: Operations (Investment Banking)\nFollowers: 2\n\nKudos [?]: 25 [0], given: 19\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants\u00a0[#permalink]\n\n### Show Tags\n\n11 Mar 2013, 06:53\nAJUnited wrote:\nFrom what I understand the interview is not blind (the interviewer has read your file previously) and hence they may not ask detailed questions on work-ex or academia, unless you have something glaring in your profile that they may want clarify on.\n\nI was told that your application\/essay review and interview are conducted by separate members of AdCom fwiw.\nIntern\nJoined: 20 Feb 2013\nPosts: 7\nLocation: China\nConcentration: Marketing, Entrepreneurship\nGMAT 1: 750 Q V0\nGPA: 3.4\nFollowers: 0\n\nKudos [?]: 7 [0], given: 0\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants\u00a0[#permalink]\n\n### Show Tags\n\n11 Mar 2013, 20:01\nxiangxder wrote:\nall hi. i got an interview weeks ago. it went.. weird, well, it was really hard to descript..my feelings about the meeting. well, basically, i don't think the adcom showed her interest(great? normal? plain? i don't know) in me.. no request about my personal life, no detailed questions about my work experience or acedemic records.. so i tend to put it as a negative sign... waiting for the DING from goizueta...\nwell, anyone has any comment? to encourage me up or.. just let me sink down...\nthanks anyway!!\nbtw, good luck to all!!\n\nHi Xiangxder, I won't see it as a negative sign. Are you also applying from China? I got my interview few weeks ago too, and it was quite some standarded plus few behavioral questions. The conversation went well, but I got no reply to my thank you email later on, so I guess it's hard to predict. Just chill out and relax. We will find out the result in just 4 days, hopefully with calls.\nManager\nJoined: 06 Mar 2013\nPosts: 61\nFollowers: 0\n\nKudos [?]: 6 [0], given: 5\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants\u00a0[#permalink]\n\n### Show Tags\n\n11 Mar 2013, 22:31\nTJ83 wrote:\nxiangxder wrote:\nall hi. i got an interview weeks ago. it went.. weird, well, it was really hard to descript..my feelings about the meeting. well, basically, i don't think the adcom showed her interest(great? normal? plain? i don't know) in me.. no request about my personal life, no detailed questions about my work experience or acedemic records.. so i tend to put it as a negative sign... waiting for the DING from goizueta...\nwell, anyone has any comment? to encourage me up or.. just let me sink down...\nthanks anyway!!\nbtw, good luck to all!!\n\nHi Xiangxder, I won't see it as a negative sign. Are you also applying from China? I got my interview few weeks ago too, and it was quite some standarded plus few behavioral questions. The conversation went well, but I got no reply to my thank you email later on, so I guess it's hard to predict. Just chill out and relax. We will find out the result in just 4 days, hopefully with calls.\n\nhi, TJ, nice to see you here and thank you for your reply. I think we are probably on the same boat, good luck to both of us!! btw, I had also applied Darden but haven't received a ding or interview invite yet till now. I saw your darden status is a D, so i guess you probably had received a definite notification from Darden, right?? so, well, do you think i should still waiting for the invite or just let it go?? personally, i tried to put it into a negative answer..again, haha!! well, thanks again and finger crossed for us!!\nManager\nJoined: 06 Mar 2013\nPosts: 61\nFollowers: 0\n\nKudos [?]: 6 [0], given: 5\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants\u00a0[#permalink]\n\n### Show Tags\n\n11 Mar 2013, 22:32\nAJUnited wrote:\nxiangxder wrote:\nall hi. i got an interview weeks ago. it went.. weird, well, it was really hard to descript..my feelings about the meeting. well, basically, i don't think the adcom showed her interest(great? normal? plain? i don't know) in me.. no request about my personal life, no detailed questions about my work experience or acedemic records.. so i tend to put it as a negative sign... waiting for the DING from goizueta...\nwell, anyone has any comment? to encourage me up or.. just let me sink down...\nthanks anyway!!\nbtw, good luck to all!!\n\nYou should just chill out. You had an interview which means that the Adcom liked you application in the first place. From what I understand the interview is not blind (the interviewer has read your file previously) and hence they may not ask detailed questions on work-ex or academia, unless you have something glaring in your profile that they may want clarify on. Plus a lot depends on the interviewer themselves, so you can't read a lot into that.\n\nThe results are 5 days away, I would suggest just stop thinking about it and worry on Friday, not before that as you can't really do anything about it in anycase. Maybe speak to some current students to show your interest in GBS, but other than that just sit tight and keep your fingers crossed.\n\nyeah, very stressful days, my friend in India, thanks for your words. and good luck to all of us!!\nManager\nJoined: 29 Nov 2012\nPosts: 123\nConcentration: Marketing, Strategy\nSchools: Kelley (Indiana) - Class of 2015\nGMAT 1: 720 Q49 V40\nGPA: 3.5\nWE: Operations (Non-Profit and Government)\nFollowers: 2\n\nKudos [?]: 38 [0], given: 23\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants\u00a0[#permalink]\n\n### Show Tags\n\n11 Mar 2013, 22:45\nxiangxder wrote:\nbtw, I had also applied Darden but haven't received a ding or interview invite yet till now. I saw your darden status is a D, so i guess you probably had received a definite notification from Darden, right?? so, well, do you think i should still waiting for the invite or just let it go?? personally, i tried to put it into a negative answer..again, haha!! well, thanks again and finger crossed for us!!\n\nYou should never give up, but you might want to think about casting your net a bit wider and applying to some other schools in the final round if you've only applied to Darden and Goizueta. Marshall, Kelley, McDonough and Mendoza are all pretty much Goizueta's peer schools and have deadlines on or after March 15. (I'm not sure if you might have visa issues coming from China and applying so late) I think Mendoza in particular is pretty low on international students so it could work in your favour.\nIntern\nJoined: 20 Feb 2013\nPosts: 7\nLocation: China\nConcentration: Marketing, Entrepreneurship\nGMAT 1: 750 Q V0\nGPA: 3.4\nFollowers: 0\n\nKudos [?]: 7 [0], given: 0\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants\u00a0[#permalink]\n\n### Show Tags\n\n11 Mar 2013, 23:56\nxiangxder wrote:\nTJ83 wrote:\nxiangxder wrote:\nall hi. i got an interview weeks ago. it went.. weird, well, it was really hard to descript..my feelings about the meeting. well, basically, i don't think the adcom showed her interest(great? normal? plain? i don't know) in me.. no request about my personal life, no detailed questions about my work experience or acedemic records.. so i tend to put it as a negative sign... waiting for the DING from goizueta...\nwell, anyone has any comment? to encourage me up or.. just let me sink down...\nthanks anyway!!\nbtw, good luck to all!!\n\nHi Xiangxder, I won't see it as a negative sign. Are you also applying from China? I got my interview few weeks ago too, and it was quite some standarded plus few behavioral questions. The conversation went well, but I got no reply to my thank you email later on, so I guess it's hard to predict. Just chill out and relax. We will find out the result in just 4 days, hopefully with calls.\n\nhi, TJ, nice to see you here and thank you for your reply. I think we are probably on the same boat, good luck to both of us!! btw, I had also applied Darden but haven't received a ding or interview invite yet till now. I saw your darden status is a D, so i guess you probably had received a definite notification from Darden, right?? so, well, do you think i should still waiting for the invite or just let it go?? personally, i tried to put it into a negative answer..again, haha!! well, thanks again and finger crossed for us!!\n\nHi Xiangxder, I applied Darde R1 in a rush, so my case may not be applicable to yours, but so far I would say no interview is definitely not a good sign.\nBack to Emory - I heard AO will start to call admitted students a bit earlier than the actually decision date. If lucky, we can expect a call or email by 15th morning China time, according to what happened to R1\/2 admitted.\nGood luck to all R3 international applicants!\nManager\nJoined: 06 Mar 2013\nPosts: 61\nFollowers: 0\n\nKudos [?]: 6 [0], given: 5\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants\u00a0[#permalink]\n\n### Show Tags\n\n13 Mar 2013, 02:29\nTo TJ83, applicant in the States has received R3 admission. the party is started!!!\nnone in mine...\ni have been so stressful recently that i just couldn't get focused on my work!!...\n\nLast edited by xiangxder on 13 Mar 2013, 21:11, edited 1 time in total.\nIntern\nJoined: 20 Feb 2013\nPosts: 7\nLocation: China\nConcentration: Marketing, Entrepreneurship\nGMAT 1: 750 Q V0\nGPA: 3.4\nFollowers: 0\n\nKudos [?]: 7 [0], given: 0\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants\u00a0[#permalink]\n\n### Show Tags\n\n13 Mar 2013, 06:27\nXiangder, I didn't get any update on OPUS either. I guess it didn't come to Chinese pool yet. Relax....\n\nPosted from my mobile device\nManager\nJoined: 01 Oct 2012\nPosts: 94\nLocation: United States\nGMAT 1: 720 Q V\nGMAT 2: Q V0\nGPA: 3.2\nWE: Operations (Investment Banking)\nFollowers: 2\n\nKudos [?]: 25 [0], given: 19\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants\u00a0[#permalink]\n\n### Show Tags\n\n13 Mar 2013, 06:29\nxiangxder wrote:\napplicant in the States has received R3 admission. the party is started!!!\n\nHi, out of curiosity where did you see this?\nIntern\nJoined: 03 Jan 2013\nPosts: 30\nFollowers: 0\n\nKudos [?]: 0 [0], given: 4\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants\u00a0[#permalink]\n\n### Show Tags\n\n14 Mar 2013, 08:51\nAJUnited wrote:\nchdheeraj1 wrote:\nLooks like only 1 indian in this forum received an invite so far. Given the decisions come out on Mar 15, the chances are looking bleak now\n\nThe best we can hope for is a WL. I am sure there wouldn't be any interview before decision date now. Damn!\n\nGuys did you get any update on the status??\nI am too from India, looks like many of us didn't get call, does that mean we can hope for an interview invite even after the DD??\nManager\nJoined: 01 Oct 2012\nPosts: 94\nLocation: United States\nGMAT 1: 720 Q V\nGMAT 2: Q V0\nGPA: 3.2\nWE: Operations (Investment Banking)\nFollowers: 2\n\nKudos [?]: 25 [0], given: 19\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants\u00a0[#permalink]\n\n### Show Tags\n\n14 Mar 2013, 09:04\nI've been refreshing my Opus every 20 minutes for the past 48 hours...\nIntern\nJoined: 18 Sep 2012\nPosts: 7\nGPA: 3.23\nWE: Consulting (Telecommunications)\nFollowers: 0\n\nKudos [?]: 5 [0], given: 0\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants\u00a0[#permalink]\n\n### Show Tags\n\n14 Mar 2013, 09:57\nI have been refreshing OPUS as well since Monday and I have not gotten a different message other than \"Your admission decision is not available at this time. Please check your Admission To Do list for remaining application requirements. If all application materials have been received, check your school's website for decision release dates or for decision timeframes. Thank you for your interest in Emory University.\"\n\nI also have not received an email or phone call. It looks like they might be waiting until tomorrow to release decisions.\nManager\nJoined: 01 Oct 2012\nPosts: 94\nLocation: United States\nGMAT 1: 720 Q V\nGMAT 2: Q V0\nGPA: 3.2\nWE: Operations (Investment Banking)\nFollowers: 2\n\nKudos [?]: 25 [0], given: 19\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants\u00a0[#permalink]\n\n### Show Tags\n\n14 Mar 2013, 11:28\nGot the e-mail from Goizueta. I'm in! $$, but amount not specified. 2-yr FullTime. Best of luck guys Last edited by DMNY31 on 14 Mar 2013, 12:24, edited 1 time in total. Intern Joined: 02 Jul 2012 Posts: 6 Followers: 0 Kudos [?]: 3 [1] , given: 1 Re: Goizueta (Emory) - Calling All 2013 MBA Applicants [#permalink] ### Show Tags 14 Mar 2013, 11:39 1 This post received KUDOS I am in! International student. Email - no call! So happy! best of luck to EVERYONE! Intern Joined: 23 Jan 2013 Posts: 15 Concentration: Entrepreneurship, General Management GMAT 1: 720 Q49 V39 GPA: 3.8 WE: General Management (Transportation) Followers: 0 Kudos [?]: 1 [0], given: 0 Re: Goizueta (Emory) - Calling All 2013 MBA Applicants [#permalink] ### Show Tags 14 Mar 2013, 12:07 In @ 1-year program! I applied to both (one year being the first choice) and havn't received the decision regarding the 2-year program yet. Email notification. I'm an international applicant residing in Goergia now. So does anyone receive the 2-year decision yet? Manager Status: Poor but happy Joined: 06 Mar 2012 Posts: 156 Concentration: Marketing, Healthcare Schools: Chicago Booth - Class of 2015 GMAT 1: 720 Q49 V40 GMAT 2: Q V WE: Engineering (Health Care) Followers: 5 Kudos [?]: 45 [1] , given: 27 Re: Goizueta (Emory) - Calling All 2013 MBA Applicants [#permalink] ### Show Tags 14 Mar 2013, 12:40 1 This post received KUDOS My OPUS account says I'm waitlisted and then I get an email saying I'm accepted. A quick check with the adcom confirmed that I have indeed been accepted into the 2 year program, with$$.\n\nBut oh man, I wish I hadn't logged onto OPUS before I saw that acceptance mail . Horrible way to get your first acceptance.\nAnyhoo, good luck to all!\n_________________\n\nMy MBA application journey, one step at a time https:\/\/spoonfulofmba.wordpress.com\n\nRe: Goizueta (Emory) - Calling All 2013 MBA Applicants \u00a0 [#permalink] 14 Mar 2013, 12:40\n\nGo to page \u00a0 Previous \u00a0 \u00a01\u00a0\u00a0...\u00a0\u00a09\u00a0\u00a0\u00a010\u00a0\u00a0\u00a011\u00a0\u00a0\u00a012\u00a0\u00a0\u00a013\u00a0\u00a0\u00a014\u00a0\u00a0\u00a015\u00a0\u00a0\u00a016\u00a0\u00a0\u00a017\u00a0\u00a0\u00a018\u00a0\u00a0\u00a019\u00a0\u00a0\u00a020\u00a0 \u00a0 Next \u00a0[ 396 posts ]\n\nSimilar topics Replies Last post\nSimilar\nTopics:\n3 Goizueta (Emory) 2016 Intake: Calling all Waitlisted Applicants!! 21 09 Dec 2015, 02:07\n100 Calling all Emory(Goizueta) Applicants: (2016 Intke) Class of 2018!! 527 20 Jun 2015, 09:26\n80 Calling all Goizueta(Emory) Applicants (2015 Intake) class of 2017 388 08 Jul 2014, 12:22\n75 Goizueta (Emory) Class of 2016 Calling all applicants! 482 25 Apr 2013, 08:41\nEmory One year MBA 2013- Calling all applicants 0 14 Feb 2013, 21:23\nDisplay posts from previous: Sort by","date":"2016-05-29 18:08:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3130677342414856, \"perplexity\": 7296.55547215161}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-22\/segments\/1464049281869.96\/warc\/CC-MAIN-20160524002121-00029-ip-10-185-217-139.ec2.internal.warc.gz\"}"} | null | null |
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