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Pokasin es una localidad de Croacia en el municipio de Gradec, condado de Zagreb.
Geografía
Se encuentra a una altitud de 151 msnm a 57,4 km de la capital nacional, Zagreb.
Demografía
En el censo 2011, el total de población de la localidad fue de 66 habitantes.
Según estimación 2013 contaba con una población de 62 habitantes.
Referencias
Enlaces externos
Este artículo contiene datos geográficos extraídos de Google Earth.
Localidades de Croacia | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,698 |
Q: Not getting response with my routes from remote server I have a node js app running okay locallyenter image description here but when deployed on remote server it is not giving response of the request but only giving response for home page
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,216 |
Berken est une commune suisse du canton de Berne, située dans l'arrondissement administratif de Haute-Argovie.
Histoire
Berken fait partie de la seigneurie, puis du bailliage d'Aarwangen jusqu'en 1798.
Références
Liens externes
Commune du canton de Berne | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,591 |
\section{INTRODUCTION}
According to the most popular progenitor model of long-duration Gamma-Ray
Bursts (GRBs), the collapsar model \citep{Woosley1993}, a GRB is the result of
ultra-relativistic jets ejected by an accreting black hole formed by the
core-collapse of a massive star (most probably a Wolf-Rayet star). This
predicts a physical link between GRBs and supernovae (SNe) that has been
spectroscopically confirmed in four cases so far: XRF 020903
\citep{Soderberg2005}, GRB 021211/SN2002lt \citep{DellaValle2003}, GRB
030329/SN2003dh \citep[e.g.,][]{Hjorth030329, Stanek030329} and GRB
031203/SN2003lw \citep[e.g.,][Mazzali et al. 2005, in preparation]
{Malesani2004}. Further evidence comes from the observation of late-time bumps
in GRB afterglows that can be modelled very well by an underlying SN component
\citep{Bloom1999, ZKH}, and which have led to the conclusion that in fact all
long-duration GRBs are physically related to SN explosions \citep[][Paper
I]{ZKH}. Furthermore, the collapsar model implies that the progenitors of
long-duration GRBs are associated with regions of high-mass star formation
\citep{Paczynski1998}, which might reveal themselves by a detectable
extinction in the GRB host galaxies along the lines of sight towards the
burster. This idea is further supported by the so-called dark bursts
\citep{Groot1998}, for which no optical afterglow has been discovered despite
rapid and deep searches in small error box regions. While most
none-discoveries are the result of too shallow searches and too large error
boxes \citep[e.g.,][]{Jakobsson2004b, Rol2005}, a small percentile remains
that require intrinsic extinction to dim the afterglow, e.g., GRB 970828
\citep{Groot1998, Djorgovski2001}, 990506 \citep{Taylor2000} and 020819
\citep{JakobssonIPR}, while others might have been intrinsically underluminous
\citep[for a discussion, see e.g.,][]{Fynbo2001b, Lazzati2002, Klose2003}.
Of particular interest within the dust extinction model is the statistical
distribution of the amount of visual extinction in the GRB host galaxies, as
well as the nature of the corresponding dust. Given the fact that GRBs and
their afterglows can be observed up to high redshift, they offer the
possibility to get insight into the nature of cosmic dust when the universe
was much younger and galaxies were much less evolved. Furthermore, since the
optical properties of the dust grains trace the environmental conditions in
the interstellar medium \citep{FitzpatrickMassa1986, Draine2000,
Bradley2005}, they are to some degree an indicator for the physical
conditions to make a GRB progenitor.
The present paper is the third in a series of papers where we employ a large
database of photometry gathered from the literature to reanalyze all optical
afterglow light curves of GRBs in the pre-\emph{Swift} era in a consistent
manner to derive a homogenous sample, which is then used to study afterglow
properties in a statistical way. In Paper I and an update \citep{ZKH05}, the
properties of the supernovae underlying nearby afterglow light curves were
explored. In Paper II \citep{ZKK} the optical light curve parameters were
derived for a large (and basically complete) sample of afterglows observed by
the end of 2004, up to the launch of the \emph{Swift} mission. In this paper,
we extend this systematic analysis to the spectral energy distribution (SED)
of GRB afterglows in the optical/near-infrared bands in order to search for
signals from extinction by dust in the GRB host galaxies.
In \kref{data}, we present the methods with which we analyzed the afterglow
SEDs. Among the 59 GRBs studied in Paper II, 30 had data quality sufficiently
high to be included in a sample for an investigation of the SEDs. We then
further reduce this sample to a Golden Sample of 19 GRB afterglows with
parameters derived from specific dust model fits. In \kref{RaD} we present the
results derived from our analysis and discuss our findings in the context of
the standard fireball model. The fitting process allows us to derive the host
galaxy extinction $\tn{\emph{A}}_{\tn{V}}$ along the line of sight and the intrinsic,
extinction-corrected spectral slope $\beta$ ($F_\nu\propto\nu^{-\beta}$) of
the afterglow light in the optical/near-infrared bands. The $\tn{\emph{A}}_{\tn{V}}$-$\beta$
sample is then further analyzed statistically to derive conclusions about the
environment of GRBs and the dust properties of high-redshift galaxies.
\section{DATA ANALYSIS}
\label{data}
\subsection{The fitting Procedure}
\label{analysis}
In Paper II we presented the analysis of a complete list of optical/NIR
afterglow light curves for which sufficient public data was available, up to
GRB 041006. One parameter derived in these light curve fits is the
normalization constant $m_k$, the magnitude of the afterglow at a certain
time, being one day after the GRB trigger in the case of a light curve fit
with a simple power-law ($F_\nu(t)\propto t^{-\alpha}$), or the jet break
time, $t_b$, in case of a fit with a smoothly broken power-law with a
pre-break decay slope $\alpha_1$ and a post-break decay slope $\alpha_2$
\citep {Beuermann1999}. All data used to fit these light curves is corrected
for Galactic extinction with $E(B-V)$ derived from the COBE maps \citep{SFD}
employing the Milky Way extinction curve of \cite{CCM}. The contemporaneous
afterglow brightness in different photometric bands is then transformed into a
spectral energy distribution, using zero fluxes and median wavelengths taken
from the literature. The SED for GRB 030329 is derived in an alternate way,
see Appendix \ref{LitComp}. The fits use a Levenberg-Marquardt $\chi^2$
minimization algorithm. Unless stated otherwise, all errors in this paper are
at the $1\sigma$ confidence level.
Initially, we assume no extinction and the SEDs are fit with a simple
power-law, $F_\nu\propto\nu^{-\beta}$. In the following, we label a slope
derived from such a fit $\beta_0$. A steep $\beta_0$ combined with a
non-linear (curved) $F_\nu$ is then indicative of extinction in the GRB host
galaxy.
In order to derive the visual extinction $\tn{\emph{A}}_{\tn{V}}$ in the GRB host galaxy along
the line of sight, we transform the wavelengths of the SED into the host
galaxy frame following \citep{Fynbo2001}, using redshifts taken from the
compilation of \cite{FB2005}. In two cases redshift estimates are used. For
GRB 980519, we use a redshift of $z=1.5$, following \cite{Jaunsen2001} who
state that $z\geq1.5$ due to the lack of a supernova bump. For XRF 030723, we
varied the redshift and fit the SED for each redshift, and find the best
results in a redshift range of $z=0.3$ to 0.4. Thus, we adopt
$z=0.35$. \cite{FynboXRF030723} find tentative evidence for a low redshift
(although the host galaxy would be very faint), and \cite{ButlerXRF} find a
preferred $z=0.4$ from the Amati relation \citep{Amati2002}. In both cases,
the SED is fit well by a power-law with an additional small amount of source
frame extinction. The influence of intergalactic Lyman absorption on the
photometry was accounted for by excluding SED data points lying beyond
$2.4\cdot 10^{15}$ Hz in the host frame from the fit\footnote{For GRB 030323
($z=3.37$), the V band also had to be excluded as the host galaxy is a DLA
with a very wide Lyman $\alpha$ absorption line \citep{Vreeswijk2004}.}. The
observed SED (corrected for Galactic extinction) is then modeled by the
function
\begin{equation}
F_{\nu}=F_0\cdot\nu^{-\beta}\cdot e^{-\tau(\nu_{\tn{host}})}
\end{equation}
with
\begin{equation}
\tau(\nu_{host})=\frac{1}{1.086}\cdot \tn{\emph{A}}_{\tn{V}}\cdot\eta(\nu_{host})\,.
\end{equation}
Here, $\beta$ is the intrinsic power-law slope of the SED, and $F_0$ a
normalization constant (we choose the unextinguished flux density at
$5\cdot10^{14}$ Hz in the host galaxy frame). The function
$\eta(\nu_{host})=A_{\lambda _{\tn{host}}}/\tn{\emph{A}}_{\tn{V}}$ is the extinction law assumed
for the interstellar medium (ISM) of the GRB host galaxy. We call this
extinction source frame extinction. It encompasses local extinction close to
the site of the GRB and host galaxy extinction further away in case the
afterglow passes through a significant part of the host galaxy along the line
of sight. The extinction law for the Milky Way (MW), the Large Magellanic
Cloud (LMC), and the Small Magellanic Cloud (SMC) was taken from
\cite{Pei1992}. These three dust types differ strongly in the wavelength
region we examine (host frame UV/optical/NIR), especially toward shorter
wavelengths. The sequence MW, LMC, SMC is given by a decreasing strength of
the 2175 {\AA} bump, an increasing FUV extinction, and a decreasing reddening
per H atom \citep{Draine2000}. The former implies the absence of very small
carbonaceous grains \citep{WD2001}, while the latter is consistent with the
trend observed for the metallicity in HII regions, with MW representing the
highest metallicity and SMC the lowest. Thus, SMC dust traces the dust
properties in low-metallicity environments. As the wavelengths of the SED are
shifted from the UV into the optical/NIR, a distinction between the three dust
types is easiest for GRBs at high redshift, whereas for nearby GRBs the three
extinction curves are almost identical in the considered wavelength regime.
Since the fit has three free parameters ($F_0$, $\beta$ and $\tn{\emph{A}}_{\tn{V}}$), the
observed SED has to have at least four data points. The one exception is GRB
000131 (Appendix \ref{LitComp}), which we include because of
its very high redshift \citep[$z=4.5$,][]{Andersen2000}. The fitting process is
very sensitive to slight variations in the SED, especially if the $m_k$ data
points have small errors, so we did a careful check for each SED and removed
outliers. Furthermore, we added a systematic error of 0.03 magnitudes to each
data point of the SED (with the exception of those small number of cases where
$m_k$ is based on only one data point, as these almost always have much larger
errors anyway) to encompass uncertainties that may derive from the process of
data reduction. This results in a strong decrease of $\chi^2$ for the fits and
reduces the importance of this measure for discerning among different
models for SED fits.
\subsection{Selection of a Golden Sample}
\label{Golden}
Our final sample (Table \ref{tabALL}) comprises 30 afterglows out of the 59
bursts included in the sample of Paper II, with fits to the three dust models
for each afterglow. Figure \ref{SEDs} shows all 30 SEDs and the associated
best fits for each of the three dust models. This figure contains all fit
results that we obtain, including those with unphysical results,
just in order to show the failure of certain
extinction laws to reproduce the observational data.
Table \ref{tabALL} reveals that many of the fits are unsatisfactory. First of
all, some fits have very large error bars $\Delta\tn{\emph{A}}_{\tn{V}}$ and $\Delta\beta$. Since
the SEDs are fit by allowing all three free parameters (incl. $F_0$) to vary
simultaneously, this creates a cross-sectional error, as the errors of $\beta$
and $\tn{\emph{A}}_{\tn{V}}$ are correlated. The steepest spectral slope ($\beta+\Delta\beta$)
results in the smallest extinction ($\tn{\emph{A}}_{\tn{V}}-\Delta\tn{\emph{A}}_{\tn{V}}$) and vice versa. Secondly,
many fits, especially when using the MW dust model, find \emph{negative}
extinction. This would imply bright emission features, for which we have no
evidence. Thirdly in some cases $\beta\leq0$ is found, while the standard
fireball model implies $\beta\geq0$ in the timespans (hours to days) and
wavelength bands that afterglows are typically observed in, and thus the
timespan where we derive $m_k$. These afterglow SEDs (e.g. GRB 971214) are
categorized by a relatively shallow unextinguished slope $\beta_0\approx0.5$
to 1 but a strong spectral curvature. Correction of this curvature results in
$\beta\leq0$. One solution would be dust with an increased FUV extinction, but
recent results on high redshift quasars \citep{Maiolino2004} do not support
this idea. We note that in these cases the data base is also sparse in
photometric bands other than $R_C$.
To derive a more homogenous sample, which we call the Golden Sample, we employ
the following criteria:
\begin{enumerate}
\item The 1$\sigma$ error in $\beta$ ($\Delta\beta$) and the 1$\sigma$ error in
$\tn{\emph{A}}_{\tn{V}}$ ($\Delta\tn{\emph{A}}_{\tn{V}}$) should both be $\leq0.5$.
\item $\tn{\emph{A}}_{\tn{V}}+\Delta\tn{\emph{A}}_{\tn{V}}\geq0$.
\item We do not consider GRBs where all fits (MW, LMC, and SMC) find $\tn{\emph{A}}_{\tn{V}}<0$,
even if the previous criterium is fulfilled.
\item $\beta>0$ (although we do not reject cases with
$\beta-\Delta\beta\leq0$).
\item A known redshift.
\end{enumerate}
As there is no case where LMC dust is clearly preferred compared to SMC
and MW dust (Table \ref{tabALL}), in the following we remove all LMC dust
cases from further consideration.
After applying these criteria, for the remaining 19 GRBs, in eight cases only
one dust model remains. For GRB 000131 no distinction can be made, as it is
not fit with extinction. The remaining ten GRBs give ambiguous results when
it comes to distinguishing between the two dust models. With the exception
of GRB 020813 and GRB 030328, where visual inspection of the fits shows that
the inclusion of SMC dust reproduces the observed SED better than MW dust, no
true distinction can be made at all. For these GRBs, we do not claim any dust
model preference, but for the sake of consistency, we choose the dust model
result which produces smaller errors in the derived parameters. This results
in 19 $\beta$-$\tn{\emph{A}}_{\tn{V}}$ pairs which we designate our Golden Sample (Table
\ref{tabGold}).
\subsection{Highly extinguished GRBs outside our sample}
\label{Others}
There are some GRBs for which high visual source frame extinction has been
deduced in the literature but which are not included in Table \ref{tabALL}.
For GRB 980329, no redshift is known, although it is assumed to be high.
\cite{Jaunsen2003} derive a Bayesian photometric redshift of $z=3.6$. The
very steep observed spectral slope (uncorrected for extinction) has been noted
by several authors. \cite{Yost2002} use their own data combined with data from
the literature and derive an $\tn{\emph{A}}_{\tn{V}}\approx1$ for $z=3$, corresponding to
$\tn{N}_{\tn{H}}\approx2 \cdot10^{21}\tn{cm}^{-2}$ assuming a Galactic dust-to-gas
ratio. This is concurrent with our result $\tn{\emph{A}}_{\tn{V}}=1.03\pm0.65$ for SMC dust and
under the assumption of a redshift of $z=3.6$. However, this result also has
$\beta\leq0$, implying that the true host extinction is lower than $\tn{\emph{A}}_{\tn{V}}=1$.
Two afterglows which have redshifts but viable data in only two colors, GRB
990705 and GRB 000418, could not be fitted with the procedure outlined in
\S \ref{analysis}. \cite{Masetti2000} find $\beta_0=2$, in agreement with our
result. While they do discuss reddening by intrinsic dust, their lack of a
redshift measurement does not allow them to find a definite result. The steep
initial decay suggests a wind environment with a cooling break blueward of the
optical bands. Using this case and SMC dust, we derive $\tn{\emph{A}}_{\tn{V}}=1.15$. For GRB
000418, assuming a wind (ISM) environment with the cooling break blueward of
the optical bands and the new derived light curve parameters from Paper II we
fix $\beta$ via the $\alpha$-$\beta$ relations and find $\tn{\emph{A}}_{\tn{V}}=1.01$ ($\tn{\emph{A}}_{\tn{V}}=
0.77$). The latter is slightly smaller than in \cite{Klose2000} (who used an ISM
environment), while \cite{Berger2001} find $\tn{\emph{A}}_{\tn{V}}=0.4$ for LMC dust. Finally,
there are indications of very high source frame extinction from the NIR
afterglows of GRB 030528 \citep{Rau2004} and GRB 040827 \citep{DeLuca040827},
but we were not able to perform any fits.
\section{RESULTS AND DISCUSSION}
\label{RaD}
\subsection{The prevalence of SMC-like dust in GRB host galaxies}
\label{SMCchapter}
In the selection of our Golden Sample, there were eight bursts where
application of our selection criteria yielded an unambiguous result.
SMC dust is by far the most preferred model, with seven out of the eight cases
(Table \ref{tabGold}), only GRB 970508 has evidence for MW dust. This result is
expected from studies of GRB host galaxies, which indicate that most hosts
are low-metallicity galaxies, as they are Lyman $\alpha$ emitters, blue, and
subluminous \citep[e.g.,][]{Fynbo2003, LeFloch2003, Jakobsson2005b}. Thus
we deduce that the ISM in these galaxies is probably metal-poor.
While the preference of SMC-like dust in GRB environments has already been
found by many groups for several bursts, e.g., GRB 000301C \citep{Jensen2001},
GRB 000926 \citep{Fynbo2001}, GRB 010222 \citep{Lee2001}, GRB 020813
\citep{Savaglio2004}, GRB 021004 \citep{Holland2003}, GRB 030226
\citep{Klose2004}, GRB 030429 \citep{Jakobsson2004}, XRF 030723
\citep{FynboXRF030723}, our study puts this conclusion on a statistical
basis. Given the sparse knowledge we have so far on the dust properties in
cosmologically remote galaxies we note however that this finding does not
necessarily imply the requirement of low-metallicity environments for the
creation of GRB progenitors, as it might be indicated by SMC-like dust. The
fact that at least some afterglows SEDs are fitted best assuming MW dust
\citep[e.g., the spectral feature found in the spectrum of GRB 991216,] [which
is mirrored in our SED result, and the strong preference of MW dust for GRB
030227, although the large errors make this case unsure]{Vreeswijk2005} raises
doubt on such a general requirement. Even though one can not rule out the
possibility that in such cases the line of sight passes through foreground
material with different grain properties and ISM metallicity. In any case,
it is clear from this study that GRB afterglows can be used as a tool in order
to explore the properties of cosmic dust in the cosmologically remote
universe, in star-forming regions in particular and in galaxies in
general. But high quality photometric data are essential for building larger
samples.
\subsection{The extinction within GRB host galaxies}
\label{AVchapter}
Among the 19 GRBs in our Golden Sample (Table \ref{tabGold}), we find eleven
with $\tn{\emph{A}}_{\tn{V}}-\Delta\tn{\emph{A}}_{\tn{V}}>0$. We note that almost all of these cases are also those
with the highest quality data. Basically, this is not surprising, as more SED
data reduces errors. It is more speculative to state that, given high quality
SED data, there will always be small amounts of source frame extinction
detected in GRB afterglows. In eight of these cases, the detection is
significant at the 2$\sigma$ level or higher, it is significant at more than
3$\sigma$ in two cases and at 4$\sigma$ in one case.
The distribution of derived intrinsic extinctions is given in Figure
\ref{A_VHist}. A strong clustering toward low extinctions ($\tn{\emph{A}}_{\tn{V}}\leq0.2$) is
evident. Zooming in with smaller bins shows that values between $\tn{\emph{A}}_{\tn{V}}=0.1$ and
$\tn{\emph{A}}_{\tn{V}}=0.2$ are preferred, with a non-error-weighted mean value of
$\overline{\tn{\emph{A}}_{\tn{V}}}=0.21\pm0.04$. No afterglow is extinguished by more than 0.8
magnitudes. As a comparison, the prime example of a dark burst is plotted, GRB
970828. For this burst, \cite{Djorgovski2001} derive $\tn{\emph{A}}_{\tn{V}}\gtrsim3.8$ by
interpolating between X-ray and radio data and comparing this with deep
optical observations that found no afterglow \cite{Groot1998}. The intervening
space is free of examples, creating a potential Dark Burst Desert. On the
other hand, there are two afterglows in Table \ref{tabALL} that have mean
$\tn{\emph{A}}_{\tn{V}}$ values lying in this Dark Burst Desert but are not included in the
Golden Sample due to the large errors of their fits (criterium (1) in
\S~\ref{Golden}). For GRB 010921, MW dust is preferred, as SMC dust finds
$\beta<0$. We derive $\tn{\emph{A}}_{\tn{V}}=0.91\pm0.82$, in agreement with
\cite{Price2003}. While this is a positive detection, the error encompasses
most other bursts of the Golden Sample. For GRB 980703, no dust model
preference can be derived, but even for the SMC result with smaller
extinction, we find $\tn{\emph{A}}_{\tn{V}}=1.32\pm0.59$.
How real is the non-existence of highly extinguished afterglows? A first
argument against the reality of the Dark Burst Desert is that our sample is
biased toward afterglows with low source frame extinction, as we require
viable photometric data in several wavebands and a known redshift. Intrinsic
extinction complicates the gathering of photometry (starting with the
discovery of the afterglow) and a spectroscopic redshift. In \kref{Others}, we
mentioned several probably highly extinguished afterglows that are not part of
our sample due to the sparsity of their photometric data. Highly extinguished
afterglows are not observed well enough to determine conclusively that they
are highly extinguished and not just intrinsically faint \citep{Klose2003}.
Figure \ref{A_Vtoz} shows the derived visual extinctions (Table \ref{tabGold})
plotted as a function of redshift $z$ \citep[values taken from][]{FB2005}. With
the exception of GRB 030429 the amount of extinction drops sharply toward high
redshifts. Most likely, this can be explained by an observational bias, at
least in the pre-\emph{Swift} era where only very few afterglows were already
localized within minutes after a burst. In particular, due to the wavelength
dependence of the dust opacity, the higher the redshift the more effective
dust can dim an afterglow in the optical bands. Furthermore, the redshift
measurement itself is biased toward bright and unextinguished afterglows, as
many redshifts are determined from absorption spectra taken when the afterglow
is much brighter than the underlying host galaxy. Finally, our data indicate
that, on average, low-$z$ GRBs have brighter afterglows (cf. Figure
\ref{BigPic1}), and thus have a greater chance of being detected through
significant extinction. Unfortunately, the chances of detecting a "Rosetta
Stone Dark Burst" at low redshift that is highly obscured by dust but still
bright enough to yield viable data and a redshift might be small if the GRB
frequency is coupled with the star-formation rate \citep[e.g.,][]{LR2002,
Firmani2004} and low metallicities \citep{MW1999}.
It is also visible that the the ability to discern between different dust
models is coupled to the redshift, as the strongest deviations (2175 {\AA}
bump, FUV extinction) appear in the UV region and the burst has to lie at a
certain redshift to shift these features into the optical bands.
\subsection{Host extinction versus star formation rate}
\label{submmchap}
Assuming that our database is not seriously affected by an observational bias
(dark bursts), Figure \ref{submm} shows that the standard GRB afterglow is
nearly unextinguished in its host galaxy. Since any long-lasting dust
destruction by the intense fireball radiation should represent itself in a
color evolution of the afterglow \citep{Waxman2000}, which we do not find in
our data, we conclude that on average there is not much evidence for dust in
GRB environments along the line of sight. Furthermore, any dust destruction
could only affect the dust in the immediate GRB environment, say, within 10 pc
around the burster \citep{Waxman2000, Draine2000, RhoadsFruchter2001,
DraineHao2002}. Dust at larger distances will still produce
extinction. Additionally, the \emph{Swift} satellite has made observations of
early GRB afterglows routine, and a major result is the almost complete
absence of reverse shock UV flashes \citep{Roming2005}. It seems that in most
cases the absence of large amounts of dust is not due to dust destruction but
due to the fact that the standard GRB progenitor (seen pole-on along its
rotational axis) is not enshrouded by dust, globally and locally. Thus, if in
those cases where significant extinction is found the extinction is not just
produced in the immediate GRB environment (local extinction) but prevails
within the host galaxy (global extinction), we would expect that GRBs for
which we find significant optical extinction are located in very dusty,
submm-bright hosts.
In order to explore this possibility, we have taken from the literature all
submm data for GRB hosts with a flux density $F_{850}>0$ at the 1$\sigma$
level that are also in our main sample of 30 afterglows \citep{Berger2003,
Tanvir2004} and calculated the corresponding star formation rate (SFR) via the
procedure developed by \cite{YC2002} \citep[cf.][]{Berger2003} to take the
different redshifts into account. Since the SFR is a direct measure of the
far-infrared luminosity of a galaxy \citep{K1998}, it traces its total amount
of radiating dust. For GRB 980703, we conservatively used the result for SMC
dust, which has the lowest extinction, and took the SFR from
\cite{Berger2001b}. We added GRB 000418 ($\tn{\emph{A}}_{\tn{V}}=1.01$, \kref{Others}),
arbitrarily assuming a 1~$\sigma$ error of 0.5 mag, and transformed the $\tn{\emph{A}}_{\tn{V}}$
result for GRB 011211 (SMC dust) into an upper limit, as we find $\beta\leq0$,
making $\tn{\emph{A}}_{\tn{V}}$ unsure. The resulting relation of SFR ($M_\odot$ per year)
versus $\tn{\emph{A}}_{\tn{V}}$ is shown in Figure \ref{submm}. A trend is visible, a rising SFR
is coupled to a rising intrinsic visual extinction along the line of sight,
with the exception of GRB 010222. A linear fit to the data (excepting GRB
010222\footnote{ The most remarkable outlier in the potential SFR-$\tn{\emph{A}}_{\tn{V}}$
relation is GRB 010222. A Hubble Space Telescope image of the GRB 010222 host
galaxy reveals the location of the GRB being offset by a small margin from
the center of the galaxy \citep{Fruchter2001, Galama2003}. We find a low
extinction value along the line of sight, whereas persistent submillimeter
flux indicates that this is a dusty starburst galaxy \citep{Frail2002}. The
discrepancy is resolved if the GRB happened toward the edge of the galaxy, but
our line of sight places it in front of the galaxy.}) gives
$SFR\;\mbox{$(M_\odot/$yr)}\;=220\pm130 + (500\pm180)\cdot\tn{\emph{A}}_{\tn{V}}$. We conclude
that, on average, GRB afterglows that show significant extinction along the
line of sight in their host are located in galaxies with a substantial star
formation rate and, hence, a globally acting extinction by large amounts of
dust. If this interpretation is correct, then the trend seen in the data
indicates that the line-of-sight can pass through a significant extent of the
host galaxy. Since the sample is still very small and the error bars on both
the SFR and the extinction $\tn{\emph{A}}_{\tn{V}}$ are very large, it is clear that more and
better data are required in order to verify this result.
\subsection{The intrinsic spectral slope and the power-law
index of the electron distribution function}
\label{betachapter}
Figure \ref{betaHist} displays the distribution of the intrinsic spectral
slopes, $\beta$, for the bursts from Table \ref{tabGold}. The distribution is
broad, ranging from 0.2 to 1.2, and features a peak around $\beta=0.7$. We
find a non-error-weighted mean value of $\overline{\beta}=0.57\pm0.05$. This
result of a broad $\beta$ distribution of the intrinsic SED of the afterglows
is probably robust, since (1) our light curve fits of the individual
afterglows usually include many data points covering several days, so that we
are not much sensitive to individual measurement errors and (2) these fits do
not include the very early phase of an afterglow when its spectral properties
might develop much faster than at later times. In fact, with the exception of
the early afterglows of GRB 021004 and GRB 030329, we have never found clear
evidence for color variations of the genuine afterglow light in our data.
While it is not the goal of the present paper to find reliable explanations
for the observed width of the distribution of spectral slopes $\beta$ of the
afterglows in the optical/NIR bands, we note that the mean of this
distribution gives a value for the power-law index of the electron
distribution ($N(E)dE \propto E^{-p} dE$) of $p=2.4$, assuming a wind
environment and the cooling frequency blueward of the optical bands
\cite[][and references therin]{ZhangMeszaros2004} and the peak of the
histogram (Figure \ref{betaHist}). This is in general agreement with
theoretical predictions for ultra-relativistic shocks \citep{Kirk2000,
Achterberg2001}. The maximum values we find for $\beta$ (Figure
\ref{betaHist}) could then be explained by those afterglows which had
the cooling frequency redward of the optical bands during the entire
time span when they were observed, provided that
$p$ is a universal number. On the other hand, a universality of $p$ seems to
be difficult to reconcile with those afterglows which have $\beta<0.5$.
Assuming the standard afterglow models \citep{ZhangMeszaros2004}, these
afterglows require $p<2$, provided that the basic model assumptions are indeed
fulfilled in these cases. Note that in Paper II we also found afterglows that
require $p<2$ based on their light curve shapes alone.
The fact that some afterglows require $p<2$ has already been reported and in
detail explored by others \citep[cf. ][]{PK2000, PK2001, P2005}. Based on our
SED fits three of 19 afterglows have $\beta\leq0.5$ within their 1$\sigma$
error bar (GRB 970508, GRB 991216 and GRB 030429) which implies $p<2$, while a
very shallow afterglow decay slope $\alpha_2$ brings GRB 990123, GRB 991216,
GRB 010222, GRB 030328 and GRB 041006 into this sample (Paper II). Obviously,
only a minority of afterglows require $p<2$. It is not obvious why these
afterglows are specific in some sense. The afterglow of GRB 991216 is one of
the brightest afterglow for redshifts $z\lesssim1$. On the other hand, the
afterglow of GRB 041006 is one of the less luminous afterglows
(\kref{redshifting}). Thus, $p<2$ is not a question of luminosity. Note that
both afterglows have only small intrinsic extinction ($\tn{\emph{A}}_{\tn{V}}\approx0.1$), so
these low values for $\beta$ are not just an artifact of the fitting
process. For GRB 991216, the unextinguished slope is $\beta_0=0.54\pm0.03$,
for GRB 041006 it is $\beta_0=0.49\pm0.05$. We also note that all these
afterglow light curves are sampled fairly (GRB 990123) to very well (GRB
010222), implying that the finding of a flat spectral slope is not a question
of data quality either.
\subsection{Dust-to-gas ratios in GRB host galaxies}
\label{NH}
X-ray observations or the modeling of Lyman $\alpha$ absorption allow the
determination of the hydrogen column density $\tn{N}_{\tn{H}}$ along the line of sight
to the GRB in its host galaxy after correcting for the column density
in our Galaxy \citep{DL1990}. In Figure \ref{NHtoA_V} we present a sample of
afterglows for which we have derived $\tn{\emph{A}}_{\tn{V}}$ and for which $\tn{N}_{\tn{H}}$ values are
reported in the literature (Table \ref{Xraydata}). This is an
update of the plot first presented in \cite{GalamaWijers2001} and expanded
in \cite{Stratta2004} \footnote{Note that we, unlike \cite{GalamaWijers2001}
and \cite{Stratta2004}, use a log-log plot to avoid crowding toward low
$\tn{\emph{A}}_{\tn{V}}$, leading to a linear depiction of the dust model curves.}. The results
reinforce our findings presented in \kref{SMCchapter}. With two exceptions,
all points in the plot lie on the theoretical prediction for SMC dust or even
above it. The two exceptions are GRB 021004 and 030329, where only upper
limits for $\tn{N}_{\tn{H}}$ are given in the literature, leading to dust-to-gas ratios
higher than even for the Milky Way. On the other hand, both afterglow SEDs
are best fit with SMC dust (although the preference is only weak for GRB
030329), implying that the three dust models we use are not applicable in all
cases. Incidentally, GRB 021004 and GRB 030329 have the two best observed
afterglow light curves with the most pronounced substructure (Paper II). The
strongest outlier in Figure \ref{Xraydata} is GRB 990123, for which we find a
very low source frame extinction. The extremely bright UV flash of this burst
\citep{Akerlof1999} may have burned significant amounts of dust along the
line of sight \citep[cf.][]{Waxman2000, GalamaWijers2001, RhoadsFruchter2001,
PLF2003}, reducing the dust-to-gas ratio. No early multicolor data exists to
probe the color variations that are expected in the early light curve.
\subsection{The luminosity distribution of the afterglows}
\label{redshifting}
Knowledge of the intrinsic spectral slope of the afterglows allows us to
determine their luminosity distribution (Appendix \ref{Math}).
At first, in Figure \ref{BigPic1}, we show the $R_C$-band light curves of all
30 afterglows of the SED sample (Table \ref{tabALL}) plotted with
smoothed splines connecting the data points to guide the eye (for reasons
of clarity, error bars on the photometry have been omitted). The data have
been corrected for Galactic extinction and host contribution. For the
light curve of GRB 030329, the supernova contribution has also been
subtracted, using the data from \cite{ZKH05}. A very large spread of
magnitudes is seen, the range at one day after the trigger is 7.5
magnitudes between the afterglows of GRB 021211 and GRB 030329. Further
bright afterglows are those of GRB 991208 and GRB 991216, other faint
afterglows are GRB 040924, GRB 030227 and GRB 971214.
Figure \ref{BigPic2} displays the afterglow light curves of the Golden
Sample (Table \ref{tabGold}) after applying the cosmological $k$-correction
(the second term of eq. \ref{delta1}) (Table \ref{tabzCorr}) and after the
time shifting to a common redshift of $z_1=1$ (the first term of eq.
\ref{delta1}). In other words, Figure \ref{BigPic2} is a measure of the
absolute $R_C$ band magnitudes of the afterglows up to a constant. Compared
to Figure \ref{BigPic1} the magnitude range has decreased, now being 5.7
magnitudes at one day after the burst. Nine of the 16 afterglows that have
data one day after the burst lie in a range only two magnitudes wide,
approximately clustered around the afterglow of GRB 030329. In other
words, this afterglow is now seen to be quite typical. The two afterglows
above this range are those of GRB 021004 and GRB 991208 (assuming for the
latter burst an extrapolation of the decay with $\alpha\approx2.5$ to one
day after the burst in the host rest frame, Paper II). The six afterglows
beneath this range are GRB 030226, GRB 020405, GRB 030328, GRB 011121, GRB
041006 and GRB 040924. Of these six afterglows, only two (GRB 030226 and
GRB 030328) lie at $z>1$. GRB 021004 is the most luminous afterglow at all
times, although it is possible that the afterglow of GRB 991208 was
brighter at earlier times when it was not yet discovered
\citep{Castro-Tirado2001}\footnote{We note that both \cite{Nardini2005}
and \cite{LZ2005} have reached similar conclusions using host galaxy
extinction values derived from the literature.}.
Knowledge of the unextinguished light curves allows us to determine the
luminosity distribution of our afterglow sample. Based on Fig.~\ref{BigPic2}
we derive the $R_C$ band magnitudes at one and four days (corresponding to
half a day and two days after the burst in the host frame at $z$=1) and
transform them into absolute magnitudes $\tn{M}_{\tn{B}}$. We do not extrapolate the
afterglow light curves, except for GRB 030429 and GRB 020813, which both have
their final data points close to 4 days and are post-break. Thus, not all
light curves are included (e.g., GRB 991208 is not yet detected at one day,
GRB 030328 is not detected any more at four days if at $z=1$). The sample then
contains 16 GRBs at both one day and at four days, the results are given in
Table \ref{tabMB}. Thereby, we computed the luminosity errors in a
conservative fashion. Given that the errors of $\beta$ and $\tn{\emph{A}}_{\tn{V}}$ are
correlated (the result $\tn{\emph{A}}_{\tn{V}}+\Delta\tn{\emph{A}}_{\tn{V}}$ is coupled to $\beta-\Delta\beta$ and
the other way around), we calculate three values of $\tn{M}_{\tn{B}}$ for the pairs
$(\beta,\;\tn{\emph{A}}_{\tn{V}})$, $(\beta+\Delta \beta,\;\tn{\emph{A}}_{\tn{V}}-\Delta\tn{\emph{A}}_{\tn{V}})$, and
$(\beta-\Delta\beta,\;\tn{\emph{A}}_{\tn{V}}+\Delta\tn{\emph{A}}_{\tn{V}})$. The pair
$(\beta+\Delta\beta,\;\tn{\emph{A}}_{\tn{V}}-\Delta\tn{\emph{A}}_{\tn{V}})$ results in a lower luminosity, and the
pair $(\beta-\Delta\beta,\;\tn{\emph{A}}_{\tn{V}}+\Delta\tn{\emph{A}}_{\tn{V}})$ in a higher luminosity. The
difference between these luminosities and the luminosity derived from the pair
($\beta,\;\tn{\emph{A}}_{\tn{V}}$) is a conservative upper limit for the error of $\tn{M}_{\tn{B}}$. In
addition, we impose limits $\beta-\Delta\beta\geq0$ and
$\tn{\emph{A}}_{\tn{V}}-\Delta\tn{\emph{A}}_{\tn{V}}\geq0$. If $\beta-\Delta\beta<0$ or $\tn{\emph{A}}_{\tn{V}}-\Delta\tn{\emph{A}}_{\tn{V}}<0$, we set
$\beta=0$ or $\tn{\emph{A}}_{\tn{V}}=0$, respectively, when computing $\tn{M}_{\tn{B}}$.
The resulting luminosity distributions are given in Figures \ref{MB1} and
\ref{MB4}. In addition, in order to search for a potential evolutionary effect
or an observational bias, we distinguish GRBs with $z<1.4$ and $z\geq1.4$, this
being the median of the redshift distribution of our sample. A bimodal
distribution of the afterglow luminosities is evident in Figures \ref{MB1z} and
\ref{MB4z}. Afterglows at $z<1.4$ tend to be less luminous, which might be
explainable by an observational bias: the chances to detect intrinsically
faint afterglows are higher for a lower $z$.
At one day after the GRB at $z=1$, the complete distribution has a two peaks,
the mean lying at $\overline{\tn{M}_{\tn{B}}}=-23.2\pm0.4$, with the most luminous
afterglow, GRB 021004, having $\tn{M}_{\tn{B}}=-25.58^{+0.04}_{-0.04}$, and the faintest,
GRB 040924, having $\tn{M}_{\tn{B}}=-19.84^{+0.17}_{-0.65}$. The means for the
distributions of the $z<1.4$ and $z\geq1.4$ GRBs are
$\overline{\tn{M}_{\tn{B}}}=-22.4\pm0.6$ and $\overline{\tn{M}_{\tn{B}}}=-24.1\pm0.5$,
respectively. The difference between the mean values is $1.7\pm0.7$ mag,
indicating a bimodality.
At four days after the burst at $z=1$, a single broad peak remains,
for the complete distribution, the mean value is $\overline{\tn{M}_{\tn{B}}}=-21.4
\pm0.4$. Now, the brightest afterglow (GRB 021004) has $\tn{M}_{\tn{B}}=-24.07^{+0.04}
_{-0.04}$ and the faintest afterglow (GRB 041006) has $\tn{M}_{\tn{B}}=-18.89^{+0.17}
_{-0.65}$. Once again, the bimodal distribution for nearby ($z<1.4$) and
distant ($z\geq1.4$) GRB afterglows is evident. The means for the
distributions of the $z<1.4$ and $z\geq1.4$ GRBs are $\overline{\tn{M}_{\tn{B}}}=
-21.2\pm0.5$ and $\overline{\tn{M}_{\tn{B}}}=-21.8\pm0.6$, respectively. The difference
between the mean values is $0.6\pm0.8$ mag, the significance of the
bimodality has been strongly reduced.
\cite{GB2005} analysed the X-ray afterglow light curves of GRBs and shifted
them to $z=1$ in an analog process, finding a bimodal flux distribution.
There are eight GRBs in their sample (GRB 970508, GRB 990123, GRB 991216,
GRB 000926, GRB 010222, GRB 011121, GRB 030226, and GRB 030329) that are
also among the afterglows we have plotted in Figure \ref{BigPic2}. GRB
990123, GRB 991216, GRB 000926, and GRB 010222 are in \emph{group I} of
\cite{GB2005}, while GRB 970508, GRB 011121, GRB 030226, and GRB 030329
are in their \emph{group II}. We note that for $z=1$, $t=1$ day lies at
the end of the initial plateau phase of the light curve of GRB 970508.
The afterglow brightens by 1.5 magnitudes shortly afterward. A comparison
reveals that while the two groups mix in the optical (the afterglow of
GRB 030329 being more luminous than those of GRB 990123 and GRB 010222
at one day after the burst at $z=1$), the mean absolute
magnitude of \emph{group I} afterglows is $\overline{\tn{M}_{\tn{B}}}=-24.2\pm
0.4$, while it is $\overline{\tn{M}_{\tn{B}}}=-22.7\pm0.5$ for \emph{group II}
afterglows, with the difference being $1.5\pm0.7$ mag. While this
finding is intriguing, the sample is too small to draw any conclusions.
Knowledge of the extinction corrected afterglow magnitudes allows us to
compare the luminosities of the afterglows at a common point in the evolution
of the jet, the jet break time. Ignoring any fine structure in the light
curves, for all bursts in our Golden Sample with a jet break (Paper II) we
computed the apparent $R_C$-band afterglow magnitudes at the time of the jet
break from the Beuermann equation \citep{Beuermann1999}. We included the fit
around the jet break of GRB 030329 but excluded GRB 021004, as the reality of
the late break we found in Paper II is unclear due to the many rebrightening
episodes. These magnitudes were then converted to luminosities and
normalized to the luminosity of the afterglow of GRB 990123 at the time of its
jet break.
We use these luminosity ratios to search for correlations between afterglow
luminosity and parameters of the prompt $\gamma$ emission of the GRBs. We take
the isotropic energy release $E_{\tn{iso}}$ and the source frame peak energy
$E_p$ from \cite{FB2005}, and the collimation corrected energies $E_\gamma$
from Paper II. We do not find any correlations between the afterglow
luminosity during the break time and these three parameters, with the absolute
value of the correlation coefficient being smaller than 0.25 in all cases.
\section{SUMMARY \& CONCLUSIONS}
\label{conclusions}
We have presented a sample of 30 GRB afterglow spectral energy distributions
in the optical/NIR bands which have been modeled with various dust extinction
curves (Milky Way, Large Magellanic Cloud and Small Magellanic Cloud) to
derive the source frame extinction, $\tn{\emph{A}}_{\tn{V}}$, intrinsic to the host galaxies and
the spectral slope, $\beta$, of the afterglows unaffected by any dust
extinction. As all afterglows have been analyzed in a systematic way, the
results are fully comparable, making this sample unique in terms of both size
and consistency. For the further statistical study, we selected 19 afterglows,
our Golden Sample, which have physically reasonable results and small error
bars.
The preferred dust models we find (\kref{SMCchapter}) as well as the deduced
source frame dust-to-gas ratios (\kref{NH}) based on the inclusion of data
taken from the literature, both indicate that the majority of GRBs we have
investigated, covering the redshift range from 0.1 to 4.5, occur in
low-metallicity environments. The $\tn{\emph{A}}_{\tn{V}}$ distribution that we have derived from
these data (\kref{AVchapter}) highlights a sparsity of strongly extinguished
afterglows, creating a Dark Burst Desert, even though it is unclear if the
preference of low extinctions is more than an observational and sample
selection bias. Our finding that most afterglows suffer from only low
extinction in their hosts could indicate that afterglows are usually not
obscured by dust close to the burster. One would then expect that the most
extinguished afterglows are in fact located in globally dusty hosts. Indeed,
we find weak evidence for a correlation between the submm flux of GRB host
galaxies and the source frame extinction $\tn{\emph{A}}_{\tn{V}}$. Although the statistical
significance is low due to the small sample size and the large errors, this
finding calls for a more thorough investigation.
Knowledge of $\beta$ and $\tn{\emph{A}}_{\tn{V}}$ allowed us then to correct the afterglow light
curves for intrinsic extinction and to derive the true luminosity distribution
of our afterglow sample at chosen times in the host galaxy frame
(\kref{redshifting}). We find that, on average, low-$z$ afterglows are less
luminous than high-$z$ afterglows. The most likely explanation we have at
hand for this finding is an observational bias against intrinsically faint
afterglows at high redshifts. A bimodal distribution found by \cite{GB2005} in
similarly corrected X-ray afterglows is not clearly seen in the optical
although, on average, GRBs with fainter X-ray afterglows also have fainter
optical afterglows. Unfortunately, the available sample size is still too
small to reach definite conclusions. A search for correlations between prompt
emission parameters and the luminosity of the optical afterglows at jet break
time has come up empty.
Since our sample is exclusively composed of GRBs from the pre-\emph{Swift}
era, a similar study in a few years time on \emph{Swift}-discovered GRB
afterglows will shed light on the Dark Burst Desert and the true afterglow
luminosity distribution by removing observational bias factors via rapid and
highly precise GRB localizations. Already, \emph{Swift} has lead to the
discovery of very faint afterglows \citep[e.g. GRB 050126, GRB 050607,]
[]{BergerSwift2005, Rhoads2005} including what may be the "darkest" burst
ever, GRB 050412 \citep{KosugiGCN, JakobssonIPR}. The recent discovery of
the first afterglows of short GRBs \citep{HjorthShort, FoxShort, CovinoShort,
BergerShort, SB051221} opens the possibility of finally making a comparison
of the environment of the two different classes of GRBs.
\acknowledgments We thank the anonymous referee for helpful comments that
improved this paper. D.A.K. and S.K. acknowledge financial support by DFG grant
Kl 766/13-2. A.Z. and S.K. acknowledge financial support by DFG grant Kl
766/11-1. We wish to thank S. Covino, J. Gorosabel, T. Kawabata, B. C. Lee,
K. Lindsay, E. Maiorano, N. Masetti, R. Sato, M. Uemura, P. M. Vreeswijk and
K. Wiersema for contributing unpublished or otherwise unavailable data to
the database, and S. Cortes (Clemson University) for reducing additional data.
D.A.K. wishes to thank N. Masetti and D. Malesani for enlightening
discussions. Furthermore, we wish to thank Scott Barthelmy, NASA for the
upkeep of the GCN Circulars and Jochen Greiner, Garching, for the "GRB Big
List".
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,199 |
Il ghiacciaio Coley è un ghiacciaio lungo circa 9 km situato sull'isola di James Ross, davanti alla costa orientale della penisola Trinity, l'estremità settentrionale della Terra di Graham, in Antartide. Assieme al ghiacciaio Coley Nord e al ghiacciaio Coley Sud, il ghiacciaio Coley fa parte di un sistema di tre ghiacciai tutti localizzati all'interno di un ripido circo glaciale, dove sono separati da speroni rocciosi, sito nella parte orientale dell'isola; qui i tre ghiacciai fluiscono rispettivamente verso sud-est, verso nord-est e verso est fino a entrare nel golfo Erebus e Terror, poco a ovest di punta Gage.
Storia
Così come l'intera isola di James Ross, il ghiacciaio Coley è stato cartografato per la prima volta nel corso della Spedizione Antartica Svedese, condotta dal 1901 al 1904 al comando di Otto Nordenskjöld, tuttavia esso è stato così battezzato solo in seguito dal Comitato britannico per i toponimi antartici in onore di John A. Coley, membro della squadra meteorologica del British Antarctic Survey, allora chiamato "Falklands Islands Dependencies Survey", di stanza presso la stazione di baia Speranza nel periodo 1952-53.
Note
Voci correlate
Ghiacciai dell'Antartide
Collegamenti esterni
Coley | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,265 |
MMi's Thomas Bridge in MediaPost: Succeed In Spot TV Advertising During Political Season - Media Management Inc.
In a new MediaPost column, MMi CEO Thomas Bridge reviews the fallacy of advertisers' TV schedules not posting "due to political advertising". As Thomas points out, political advertising does have a lot of impact on demand for the available supply of ad time, but it has nothing to do with whether a schedule delivers or not.
"Demand on commercial inventory has nothing to do with the audience for those commercials (and hence the ratings that they generate)," he writes.
Political ad cycles can certainly affect pricing and availability, but they have absolutely nothing to do with whether an individual spot achieves its estimated rating or not. Read the full article to find out what Thomas says clients can do to prepare for the upcoming political advertising window.
Read Thomas' full MediaPost column here. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,898 |
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Final Fantasy VII ( ファイナルファンタジーVII, Fainaru Fantajī Sebun? ) est un jeu vidéo de rôle développé par Square ( devenu depuis Square Enix) sous la direction de Yoshinori Kitase et sorti en 1997, constituant le septième opus de la série Final Fantasy. Premier jeu de la série à être produit pour la console Sony PlayStation ainsi que pour les ordinateurs dotés de Windows.
Final Fantasy VII ( яп.
ファイナルファンタジーVII файнару фантадзи: сэбун) — японская ролевая игра, разработанная Square Co Ltd. ( ныне Square Enix) и выпущенная Sony Computer Entertainment для игровой приставки Sony PlayStation в 1997 году как седьмая номерная часть. We' re nearing three years since Final Fantasy VII Remake' s reveal at Sony' s infamous E3 press conference, but as has become customary with anticipated Square Enix projects, we haven' t. | {
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Pelosi says right-wing DC rally is meant to 'praise the people who were out to kill' during Capitol riot
William September 14, 2021 3 min read
Democratic House Speaker Nancy Pelosi has slammed those who plan to attend a rally outside the Capitol building later this month, arguing that they're "coming back to praise the people who were out to kill" during the insurrection on 6 January.
According to an internal memo obtained by CNN, Capitol Police are preparing for the possibility of violence during a right-wing protest planned for 18 September outside Congress in Washington DC.
Officials have told The Associated Press that far-right extremists are expected to attend the rally, such as members of the Proud Boys and Oath Keepers.
The rally has been called "Justice for J6" and is aimed at voicing support for the Capitol riot and the suspected insurrectionists who have been criminally charged and sentenced for taking part in the siege.
The goal of the riot was to stop Congress from certifying President Joe Biden's electoral college victory.
Interest in the event was boosted when the identity of the officer who shot Trump supporter Ashli Babbitt was revealed, according to CNN. Ms Babbitt died of her injuries after being shot as she tried to climb through a broken window towards parts of the Capitol building where lawmakers were still present.
Ms Pelosi slammed those intent on protesting outside the Capitol on 18 September.
"These people are coming back to praise the people who were out to kill, out to kill members of Congress," she told reporters on Wednesday. She added that rioters "successfully [caused] the deaths… of our law enforcement".
"We intend to have the integrity of the Capitol be intact," she said when asked about security measures around the Capitol. "What happened on January 6th was such an assault on this beautiful Capitol, a Capitol under the dome that Lincoln built during the Civil War."
According to the memo obtained by CNN, there has been an increase in violent online rhetoric in connection to the 18 September event. The memo adds that some attendees may see the rally as a "Justice for Ashli Babbitt" event as well and not just an opportunity to support the criminally charged suspected rioters.
One online chat suggested Jewish centres and liberal churches be attacked while law enforcement is distracted by the rally.
The Capitol Police requested that temporary fencing be put back into place around the Capitol. The request is likely to be approved but the fencing is expected to be smaller and not interfere with traffic.
The former Deputy Director of the FBI, Andrew McCabe, told CNN that law enforcement should take the rally "very seriously. In fact, they should take it more seriously than they took the same sort of intelligence that they likely saw on January 5".
Lead rally organizer Matt Braynard told the outlet: "This is a completely peaceful protest. And we have told people that when they come, we don't want to see any messaging about the election, we don't want to see any messaging on T-shirts and flags or signs about candidates or anything like that."
Around 500 people have indicated that they are likely going to attend the rally.
Tags: Capitol kill meant Pelosi people praise rally rightwing riot
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Q: Reset / Reuse / Clone cURL Multi Handle I have a specific problem which requires me to reuse cURL multi handles. Is there a way to do that? I tried to use
curl_copy_handle()
And it did not work saying
curl_copy_handle(): supplied resource is not a valid cURL handle resource
Which is not totally unexpected. Is there a way to reuse or clone a cURL multi handle?
Edit: Calling
clone
Also does not work
Fatal error: __clone method called on non-object
A: I don't think this is possible with a single built-in function.
As you creating the original cURL multi handle, right before each curl_multi_add_handle() call, save a copy of each easy (regular) handle in an array A with curl_copy_handle().
Then, when you need to re-use the multi handle:
*
*Create a new empty one with curl_multi_init()
*Loop through each element of A
*In the loop, use curl_multi_add_handle() to add a copy (again, with curl_copy_handle()) of each easy handle to the new multi handle
| {
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Q: How to add weight of edges when calculate personallization Page Rank https://networkx.github.io/documentation/development/_modules/networkx/algorithms/link_analysis/pagerank_alg.html
I want to calculate weighted personalized page rank. How should I implement with specific start node?
Code:
import newtworkx as nx
G = nx.DiGraph()
[G.add_node(k) for k in [1,2,3,4]]
G.add_edge(2,1)
G.add_edge(3,1)
G.add_edge(4,1)
ppr1 = nx.pagerank(G,personalization={1:1, 2:1, 3:1, 4:1})
How to set the nstart and personalization ? Thanks!
| {
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Q: Cannot read property '706142720527171634' of undefined. discord music bot crashes My bot has a skip and stop commands it gives this error when I try that command
Cannot read property '706142720527171634' of undefined
I also re-wrote the code,
These are the two commands' codes
if(msg.content === 'r!' + 'skip'){
server.dispatcher = connection.play(ytdl(server.queue[0],{filter: "audioonly"}));
var server = servers[msg.guild.id];
if (server.dispatcher) server.dispatcher.end();
}
if(msg.content === 'r!' + 'stop'){
var server = server[msg.guild.id];
if(msg.guild.voice.connection){
for(var i = server.queue.length -1; i >=0; i--){
server.queue.splice(i, 1);
}
A: Here, you used server instead of servers: server[msg.guild.id];.
The following should work:
if(msg.content === 'r!' + 'skip'){
server.dispatcher = connection.play(ytdl(server.queue[0],{filter: "audioonly"}));
var server = servers[msg.guild.id];
if (server.dispatcher) server.dispatcher.end();
}
if(msg.content === 'r!' + 'stop'){
var server = servers[msg.guild.id];
if(msg.guild.voice.connection){
for(var i = server.queue.length -1; i >=0; i--){
server.queue.splice(i, 1);
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
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einestages war ein am 8. Oktober 2007 gestartetes Zeitgeschichte-Portal von Spiegel Online, das ab März 2014 als reguläres Ressort von Spiegel Online fortgeführt wurde und im Januar 2020 zugunsten eines neuen Geschichtsressorts eingestellt wurde.
Auf einestages wurde eine Mischung historischer Themen und Zeitzeugenberichte veröffentlicht. Leser konnten sich dabei als Mitglieder registrieren und Beiträge einreichen. Die Berichte wurden von einer Redaktion geprüft, redigiert und mit Fotos und Videos illustriert. Ressortleiter war der Journalist und Schriftsteller Benjamin Maack. Nach Angaben des Verlags hatten sich bereits in der ersten Woche über 2.000 Mitglieder registriert und die Seiten wurden insgesamt 8,2 Millionen Mal angeklickt.
Partner des Projekts waren unter anderem Progress Film-Verleih, das Deutsche Auswandererhaus, das Bundesarchiv, die Deutsche Fotothek, das Bildarchiv Preußischer Kulturbesitz, die Bundesstiftung zur Aufarbeitung der SED-Diktatur und das Militärgeschichtliche Forschungsamt der Bundeswehr.
Im März 2008 erhielt das Portal bei den LeadAwards die Auszeichnung "Webmagazin des Jahres".
Im September 2008 wurde die Testausgabe eines gleichnamigen Printmagazins veröffentlicht.
Einzelnachweise
Onlinemagazin
Der Spiegel (online)
Deutschsprachiges Medium
Ersterscheinung 2007
Erscheinen eingestellt 2020 | {
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She Was Generous. She Was Also Racist. Should This Ballpark Carry Her Name?
Published June 10, 2020 at 5:00 AM EDT
University of Cincinnati pitcher Nathan Moore considers the proposed renaming of Marge Schott Stadium "a really simple decision."
As nationwide protests against police brutality and racism demand change to current laws and institutions, the ripple effects are reaching historic symbols of white supremacy.
The effort to dismantle, relocate or rename symbols is happening in the sports world as well.
Athletes have gotten involved in a potential name change at the University of Cincinnati.
Former UC baseball player Jordan Ramey started a petition drive to change the name of Marge Schott Stadium.
Schott owned Major League Baseball's Cincinnati Reds from 1984 to 1999. To say she was controversial is an understatement. Her racial and ethnic slurs against African American, Jewish and Japanese people prompted a one-year ban from baseball in 1993. After publicly praising Adolf Hitler in a 1996 ESPN interview, Schott was forced to give up day-to-day control of the Reds until 1998.
Schott also was known as a philanthropist in Cincinnati — she gave money, through her foundation, to the city zoo, hospitals and the university. In 2006, the Bearcats named their baseball stadium after Schott, following a foundation gift of $2 million to the school's athletic programs.
University of Cincinnati pitcher Nathan Moore has played home games at Marge Schott Stadium for the past four years. But it wasn't until he heard about former teammate Ramey's petition that he learned about Schott's past.
Moore has since become an outspoken supporter of the petition drive, saying the call for a stadium name change should be "a really simple decision" for the university.
"I think it would set a great example for schools around the world that still commemorate people who clearly thought the wrong way and were racist," he said.
The University of Cincinnati released a statement from Athletic Director John Cunningham.
"We appreciate the willingness of our current and former student-athletes to have tough conversations and express their feelings about the name of our baseball stadium. The Department of Athletics is providing the University Administration any information or context they may need to better understand this issue from the perspectives of our student-athletes. We are One Team and I want to thank our student-athletes for their candor and let each and every one of them know I'm always available to them via phone or text if they want or need to talk."
Beyond the statement, the university says no one will comment further.
Moore says he was initially conflicted when he discovered the university named the stadium after Schott 14 years ago.
"I was honestly confused as to why nothing was done before," he told NPR. "But I didn't want to waste too much time on that. I knew I had a powerful voice and a platform to use, so, I was done waiting around for someone to do it I guess.
"Honestly, I mean no disrespect to Marge Schott and I'm sure she had good intentions, but just from my beliefs personally, money that comes from somebody who has that hatred in her heart for fellow human beings, is not money worth accepting in my opinion."
Rob Yowell wasn't part of the negotiations between UC and the Schott Foundation. But as president of Gemini Sports Group, an agency that specializes in stadium naming rights, he's been in similar negotiations and he has a pretty good idea what happened in 2006.
"[The university] could've said no," Yowell said, "but I imagine there wasn't anyone else in line with a $2 million check [for the school's athletic programs]. They made a business decision at that point."
As far as what UC will do now, Yowell said a decision could be made based on the current environment and the school's ability to get new money.
"At the end of the day," Yowell said, "[the university] wants a piece of inventory in naming rights to try to sell [to someone else] and generate money. And at the same time, distance itself from an individual who's been identified in the past as someone very insensitive to race."
He says the university wouldn't have to return the $2 million "because at this point, it's a gift. You're talking about 14-plus years. It's not like this happened in recent history. They gave a lump-sum donation."
There's precedent, however, for a similar controversy prompting a donation return. Although it happened in a much smaller window of time.
In 2014, UCLA returned nearly a half million dollars donated by the Donald Sterling Charitable Foundation, and canceled what would've been a $3 million, multiyear gift from the former NBA owner.
UCLA acted after racist comments Sterling made were revealed that year, which ultimately led to his lifetime ban from the NBA. He also was forced to sell his team, the Los Angeles Clippers.
Focus On Sport / Getty Images
Cincinnati Reds owner Marge Schott, at the 1990 World Series at which the Reds swept the Oakland Athletics, was known for using racial and ethnic slurs and even praising Adolf Hitler.
In this time of free-flowing anger after the killing of George Floyd in police custody, there's little tolerance for those whose words have contributed to the country's toxic racial environment. No doubt, Marge Schott was a contributor. But with her life back in public view, there's talk of a legacy that's more complex. That is certainly the case in Cincinnati, where she was born and where she died in 2004.
On that day 16 years ago, Cincinnati Enquirer sports columnist Paul Daugherty wrote this:
"The true measure of a person is whether he or she leaves a place better than he or she found it. Is Cincinnati better for having raised and rooted Marge Schott?
"Yeah. Probably.
"Good Marge competed with Bad Marge, daily."
Now the University of Cincinnati has to navigate this dual legacy, although thousands of petition signers see nothing complex about Marge Schott. They see a name on a stadium representing a person whose words directly contradict one of the school's stated goals: to foster a community that prioritizes inclusion.
A spokesman for the Schott Foundation declined a request to talk about the petition, but provided this statement:
"The Schott Foundation continues on its mission to support and find qualified charitable programs and organizations for the betterment of the greater Cincinnati community. The foundation will support its community partners in any decisions that will progress unity of purpose and stand against prejudice in any manner."
Meanwhile, there is a similar renaming effort at Clemson University in South Carolina.
Clemson University still honors the name of slave owner John C. Calhoun on its buildings, signs, and in the name of its honors program. I am joining the voices of the students and faculty to restarted this petition to rename the Calhoun Honors College. https://t.co/1198BZ8FeS pic.twitter.com/xQmXmBPUeW
— Deandre Hopkins (@DeAndreHopkins) June 8, 2020
NFL stars DeAndre Hopkins, a wide receiver with the Arizona Cardinals, and Houston Texans quarterback Deshaun Watson, both Clemson alums, are joining a petition drive calling for the university to change the name of its Calhoun Honors College. John C. Calhoun, who served as vice president of the United States in the 1820s and 30s, was a strong proponent of slavery. Clemson, according to the university's website, is built on Calhoun's Fort Hill Plantation, where he owned "some 70-80 enslaved African-Americans."
The Greenville News reports the honors college director sent an email to students that said:
"The decision on any change in the college's name does not rest with us; such decisions are the purview of University leadership, which is aware of the situation. We have heard your concerns and wishes, have shared those with others, and will continue to make sure those voices for change are heard." | {
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Home › What's New on Netflix › New This Week on Netflix (November 16th, 2018)
New This Week on Netflix (November 16th, 2018)
by Cheryl Greenway
Published on November 17th, 2018, 12:48 am EST
Joaquín Cosío, Diego Luna, Horacio Garcia Rojas. 'Narcos: Mexico.' Image: Carlos Somonte/Netflix
Happy Friday, Netflixers! You are in for a great couple of days because Netflix has a bountiful supply of streaming goodies for you this weekend. We took the time to round up all the new titles and highlights for you that have hit Netflix in the past week.
New week, new titles! And you're in for a great couple of days of bingeing because Netflix has added more than enough quality titles to keep you going through the weekend. From new Netflix Original movies to a grown-up reboot of She-Rah, there really is something for everyone this weekend.
Below are my picks for the week. I have seen them all and would personally recommend them.
For your convenience, a full list of the titles added this week is at the end of this article. To keep up with new additions be sure to follow our What's New page, updated daily.
'The Break-Up' is new this week on Netflix.
Like most Coen brothers movies this film is a bleak affair. Wrought with silliness wrapped into tales of morality, each chapter of this six-part Western anthology is more intriguing than the last. I was hooked from the beginning. Some are very surprising, others just plain sad. I tried to guess with each chapter where each story would lead and was never right. I love a movie that can surprise me. This film definitely deserves a spot on your list.
This latest installment in the Netflix library of drug soap-operas is very worthy of a watch. It stars Diego Luna (Rogue One: A Star Wars Story) and Michael Peña (Crash, End of Watch) and explores the origins of the modern drug war by going back to its roots, beginning at a time when the Mexican trafficking world was a loose and disorganized confederation of independent growers and dealers. Both Luna and Peña are excellent actors and they don't disappoint here. We see the rise of the Guadalajara Cartel in the 1980s and Luna shines as its leader.
https://youtu.be/VBLcYJ7C4F0
This Original series is brought to us by a man whose life was changed by a dog. A criminal prosecutor, he had no fulfillment in his life. He happened upon a stray dog one day and saved it from being euthanized. Inspired by the volunteers at the shelter, he became one himself and found a new calling. Each portion of this series is about how a dog changed a person's life. It is all the best parts of humanity wrapped into a series. If I had one complaint it would be that we don't see enough of the dogs themselves. But it's truly a heartwarming show that should be watched.
Complete List of New Titles Added This Week:
26 New Movies Added This Week
Aalorukkam (2017)
BuyBust
Cam (2018) Netflix Original
Christmas Wedding Planner (2017)
Follow Me (2017)
Green Room (2015)
Halkaa (2018)
Killer Elite (2011)
Loudon Wainwright III: Surviving Twin (2018) Netflix Original
Mala Kahich Problem Nahi (2017)
May the Devil Take You (2018) Netflix Original
Nothing to Hide (2018) Netflix Original
Odu Raja Odu (2018)
Only the Dead (2015)
Pimpal (2017)
Savita Damodar Paranjpe (2018)
Shopkins: Chef Club (2016)
Shopkins: Wild (2018)
Shopkins: World Vacation (2017)
The Ballad of Buster Scruggs (2018) Netflix Original
The Break-Up (2006)
The Crew (2000) Netflix Original
The Giant (2017)
The Princess Switch (2018) Netflix Original
The Workshop (2017)
Walt: The Man Behind the Myth (2001)
15 New TV Series Added This Week
Age of Tanks (Season 1)
Beyblade Burst Evolution (Season 1)
Birth of a Beauty (Season 1)
Dogs (Season 1) Netflix Original
Helix (Season 1)
My Little Pony: Friendship Is Magic (Renewed)
Narcos: Mexico (Season 1) Netflix Original
Ponysitters Club (Season 2) Netflix Original
Prince of Peoria (Part 1) Netflix Original
She-Ra and the Princesses of Power (Season 1) Netflix Original
Super Fan Builds (Season 1)
Tech Toys 360 (Season 1)
The Kominsky Method (Season 1) Netflix Original
Vai Anitta (Season 1) Netflix Original
Warrior (Season 1) Netflix Original
1 New Documentaries/Docuseries
Muzaffarnagar Baaqi Hai (2015)
Article by Cheryl Greenway
Cheryl has written for What's on Netflix for over three years. She's a confessed streaming addict and also runs a Netflix based community on Facebook with over 10,000 users. Cheryl specializes in documentaries and covers weekly additions for the US in addition to breaking news stories. She is the Weekend Editor for What's On Netflix. Cheryl resides in Virginia, USA.
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Coming Soon to Netflix / Netflix News
Manga Adaptations Coming to Netflix in 2023 and Beyond | {
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Ernst Eugen Karl August Bernhard Paul von Hessen-Philippsthal (* 20. Dezember 1846 in Philippsthal; † 22. Dezember 1925 in Eisenach) aus dem Haus Hessen war der letzte Titularlandgraf von Hessen-Philippsthal.
Leben
Ernst war der älteste Sohn des paragierten Landgrafen Karl II. von Hessen-Philippsthal (1803–1868) aus dessen Ehe mit Marie (1818–1888), Tochter des Herzogs Eugen von Württemberg.
Nachdem das Kurfürstentum Hessen einschließlich Hessen-Philippsthal 1866 von Preußen annektiert worden war, erhielt Ernst, der 1868 auf alle Nachfolgeansprüche in Hessen-Philippsthal verzichtet hatte, gemeinsam mit Titularlandgraf Alexis von Hessen-Philippsthal-Barchfeld 1880 aus dem kurhessischen Fideikommiss eine Rente von 300.000 Mark, sowie die Schlösser Hanau, Rotenburg und Schönfeld als Privatfideikommiss der Philippsthaler Linien.
Als Standesherr hatte er eine Virilstimme im Kommunallandtag Kassel und ab 1867 im Preußischen Herrenhaus.
Ernst starb 1925 unverheiratet und ohne Nachkommen. Da sein jüngerer Bruder Carl bereits 1916 verstorben war, erlosch mit ihm die Linie Hessen-Philippsthal. Vom Haus Hessen existierten damit nur noch die drei Linien Hessen-Philippsthal-Barchfeld, Hessen-Rumpenheim und Hessen-Darmstadt. Ernst wurde in der Fürstengruft von Schloss Philippsthal bestattet.
Literatur
Eckhart G. Franz (Hrsg.): Haus Hessen. Biografisches Lexikon. (=Arbeiten der Hessischen Historischen Kommission N.F., Bd. 34) Hessische Historische Kommission, Darmstadt 2012, ISBN 978-3-88443-411-6, Nr. HP 36, S. 258–259 (Andrea Pühringer).
Jochen Lengemann: MdL Hessen. 1808–1996. Biographischer Index (= Politische und parlamentarische Geschichte des Landes Hessen. Bd. 14 = Veröffentlichungen der Historischen Kommission für Hessen. Bd. 48, 7). Elwert, Marburg 1996, ISBN 3-7708-1071-6, S. 182.
Dieter Pelda: Die Abgeordneten des Preußischen Kommunallandtags in Kassel 1867–1933 (= Vorgeschichte und Geschichte des Parlamentarismus in Hessen. Bd. 22 = Veröffentlichungen der Historischen Kommission für Hessen. Bd. 48, 8). Elwert, Marburg 1999, ISBN 3-7708-1129-1, S. 84–85.
Christoph Carl Hoffmeister: Carl II. Landgraf zu Hessen-Philippsthal. Nach Leben, Wirken und Bedeutung, Marburg 1869.
Weblinks
Hessen-Philippsthal In: Meyers Konversations-Lexikon. 1888.
Einzelnachweis
Mitglied des Preußischen Herrenhauses
Mitglied des Kurhessischen Kommunallandtags
Familienmitglied des Hauses Hessen (Linie Philippsthal)
Geboren 1846
Gestorben 1925
Mann | {
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{"url":"https:\/\/mathoverflow.net\/questions\/272505\/fourier-series-of-e-cos-x","text":"# Fourier series of $e^{\\cos x}$\n\nI need to compute the fourier series of $f(t)=e^{\\cos(t)}, 0 \\leq t < 2\\pi$.\n\nThe fourier series are defined as $f(t) = \\sum_{n=-\\infty}^\\infty c_n e^{2\\pi int\/T}$ with $c_n = \\frac 1 T \\int_0^T e^{-2\\pi int\/T}f(t) \\, dt$.\n\nI have tried to do this by using the definition of the $c_n$, but i get stuck when with the integration by parts. I also have tried to use the approach used in of this question but i cannot go forward more than expanding $e^{\\cos(t)}$. Could any of you help me with a hint?\n\n\u2022 I think this is a question for math.stackexchange \u2013\u00a0Michael Freimann Jun 18 '17 at 20:59\n\u2022 That's not entirely fair; the coefficients are not elementary. As I recall the Fourier coefficients of $\\exp(\\cos t)$, and more generally $\\exp(c \\cos t)$, can be expressed in terms of Bessel functions. \u2013\u00a0Noam D. Elkies Jun 18 '17 at 21:33\n\n$$\\int_0^{2\\pi} \\exp(int) \\exp(\\cos(t))\\; dt = \\int_{-\\pi}^{\\pi} \\cos(n t) \\exp(\\cos(t))\\; dt = 2 \\pi I_n(1)$$ where $I_n$ is a modified Bessel function of the first kind and thus $$I_n(1)=\\frac12\\sum_{k\\geq0}\\frac1{4^kk!(n+k)!}.$$","date":"2021-01-24 13:49:01","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\": 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\": 0.98527592420578, \"perplexity\": 82.83325429442804}, \"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-2021-04\/segments\/1610703548716.53\/warc\/CC-MAIN-20210124111006-20210124141006-00243.warc.gz\"}"} | null | null |
\section{Introduction}
In \cite{I,II} we started a program to promote the HOMFLY
polynomials \cite{HOMFLY} to character expansions, representing them
as linear combinations of the Schur functions $S_Q\{p_k\}$ (i.e. the
characters of linear groups) \cite{Fulton}. Such an expansion
depends on the choice of a braid realization of the
knot, thus,
its coefficients by themselves are not knot invariants, instead they
are pure group theory quantities and possess many nice properties.
For an $m$-strand braid ${\cal B}$ the HOMFLY polynomial in
representation $R$ is expanded as\footnote{Our calculus is based on the approach by \cite{RT},
though that of \cite{inds,Rama00,ZR} is, by essence, also very close.
The first part of this formula is related to Chern-Simons theory in the
temporal gauge \cite{MorSm}. The new element is a special emphasis on the
character expansion, which allows one to extend the knot polynomials
to arbitrary time variables \cite{I} and provides
very simple {\it general} formulas for entire classes of knots.
However,
following this line, we
omit the additional {\it factor}
$(A^\alpha q^\gamma)^{w^{\cal B}}$
in front on the trace in (\ref{1}), where $w^{\cal B}$ is the writhe
number of the braid, while $\alpha$ and $\gamma$ depend
on normalization of the ${\cal R}$-matrix.
Throughout this paper we use a special normalization of
${\cal R}$-matrices,
though in our normalization $\alpha$ and $\gamma$ are
actually non-vanishing: for most purposes (in the standard framing)
$\alpha = -|R|$ and $\gamma = -4\varkappa_R$.
We discuss this issue in a separate
subsection \ref{ff}, and actually restore the factors
in the formulas of the Appendix.
To simplify the notations we do not put star on $H(A)$
in this paper,
as we do in \cite{I,II,IMMMfe},
because the extended polynomials are almost not mentioned here.
They can be, however, obtained from the formulas of the Appendix
simply by removing the stars from $\SS_Q$, thus, promoting
them from the quantum dimensions to the Schur functions. }
\be\label{1}
H_R^{\cal B} = \Tr_{R^{\otimes
m}}\Big((q^\rho)^{\otimes m}{\cal B}\Big) = \sum_{Q\vdash m|R|}
h_{RQ}^{\cal B}\,S_Q^*(A)
\ee
where
\be\label{tolo}
S_Q^*(A) = \Tr_{Q\in
R^{\otimes m}} (q^\rho)^{\otimes m} = S_Q\{p_k^*\}, \ \ \ \ \ \ \ \
\ \
p_k^* = {[kN]\over [k]} = \frac{A^k-A^{-k}}{q^k-q^{-k}}
\ee
are quantum dimensions of representations $Q$ of $SU(N)$,
where $A=q^N$ and $[x] = \frac{q^x-q^{-x}}{q-q^{-1}}$.
The coefficients $h_{RQ}^{\cal B}$ do not depend on $A$,
i.e. on $N$,
thus, they can be evaluated from analysis of an arbitrary group $SU_q(N)$.
Instead they can be represented as traces in auxiliary
spaces of intertwining operators ${\cal M}_{R^mQ}$, whose dimension
is the number ${\rm dim}\, {\cal M}_{R^mQ} = N_{R^mQ}$ of times
the irreducible representation $Q$ appears in the $m$-th tensor power
of representation $R$,
\be
R^{\otimes m} = \sum_{Q\vdash m|R|} {\cal M}_{R^mQ} \otimes Q
\label{decoRmQ}
\ee
The trace is taken of a product of the diagonal quantum ${\cal R}$-matrices
$\widehat{\cal R}$ acting in ${\cal M}_{R^mQ}$,
and the "mixing matrices" intertwining ${\cal R}$-matrices
that act on different pairs of adjacent strands in the braid.
These mixing matrices, in their turn, can be represented as products
of universal constituents, associated with a switch between two
"adjacent" trees describing various decompositions (\ref{decoRmQ}).
In \cite{II} we exhaustively described
such representations for the coefficients $h_{RQ}^{\cal B}$
for {\it arbitrary} $m=2,3,4$-strand braids
and for the simplest fundamental representation $R=[1]$:
\be\label{4}
\begin{array}{ccc}
m=2, & {\cal B} = {\cal R}^a: &
H_{[1]}^{(a)} = q^a S_2^*(A) + \left(-\frac{1}{q}\right)^a S_{11}^*(A)
= q^a S_2^*(A) \ +\ \left( q \longrightarrow -\frac{1}{q}\right)
\end{array}
\\
\\
\ee
\be
m=3, \ \ \ {\cal B} = ({\cal R}\otimes I)^{a_1}(I\otimes {\cal R})^{b_1}
({\cal R}\otimes I)^{a_2}(I\otimes {\cal R})^{b_2} \ldots \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ee
$$
H_{[1]}^{(a_1,b_1|a_2,b_2|\ldots)} =
q^{\sum_i (a_i+b_i)} S_3^*(A) + \left(-\frac{1}{q}\right)^{\sum_i(a_i+b_i)} S_{111}^*(A)
+ \Big(\Tr_{2\times 2} \widehat{\cal R}_2^{a_1}U_2 \widehat{\cal R}_2^{b_1}U_2^\dagger
\widehat{\cal R}_2^{a_2}U_2 \widehat{\cal R}_2^{b_2}U_2^\dagger \ldots\Big) S_{21}^*(A)
$$
\be\label{6}
\\
m=4, \ \ \ {\cal B} = ({\cal R}\otimes I\otimes I)^{a_1}(I\otimes {\cal R}\otimes I)^{b_1}
({\cal R}\otimes I\otimes I)^{c_1}
({\cal R}\otimes I\otimes I)^{a_2}(I\otimes {\cal R}\otimes I)^{b_2}
({\cal R}\otimes I\otimes I)^{c_2} \ldots \ \ \ \ \ \ \ \
\ee
$$
H_{[1]}^{(a_1,b_1,c_1|a_2,b_2,c_2|\ldots)} =
q^{\sum_i (a_i+b_i+c_i)} S_4^*(A) + \left(-\frac{1}{q}\right)^{\sum_i(a_i+b_i+c_i)} S_{1111}^*(A) +
$$
$$
+ \Big(\Tr_{2\times 2} \widehat{\cal R}_2^{a_1}U_2 \widehat{\cal R}_2^{b_1}U_2^\dagger
\widehat{\cal R}_2^{c_1+a_2}U_2 \widehat{\cal R}_2^{b_2}U_2^\dagger \widehat{\cal R}_2^{c_2+a_3}
\ldots\Big) S_{22}^*(A) +
$$
$$
+ \left\{\Big(\Tr_{3\times 3} \widehat{\cal R}_3^{a_1}U_3 \widehat{\cal R}_3^{b_1} V_3U_3\widehat{\cal R}_3^{c_1}
U_3^\dagger V_3^\dagger U_3^\dagger
\widehat{\cal R}_3^{a_2}U_3 \widehat{\cal R}_3^{b_2} V_3U_3\widehat{\cal R}_3^{c_2}
U_3^\dagger V_3^\dagger U_3^\dagger
\ldots\Big) S_{31}^*(A)\ + \ \left(q \longrightarrow -\frac{1}{q}\right)\right\}
$$
where
\be
\hat{\cal R}_2 = \left(\begin{array}{cc} q & \\ &-\frac{1}{q}\end{array}\right)\ \ \ \ \ \ \ \ \ \
\hat{\cal R}_3 = \left(\begin{array}{ccc} q & & \\ & q &\\ &&-\frac{1}{q}\end{array}\right)\nn
\ee
\be
U_2 = \left(\begin{array}{cc} c_2 & s_2\\ -s_2 &c_2\end{array}\right)\ \ \ \ \ \ \ \
U_3 = \left(\begin{array}{ccc} 1&& \\ &c_2 & s_2\\ &-s_2 &c_2\end{array}\right)\ \ \ \ \ \ \ \
V_3 = \left(\begin{array}{ccc} c_3 & s_3 & \\ -s_3 &c_3 & \\ && 1\end{array}\right)
\label{UVmatr}
\ee
Subscripts refer to the size of the matrices, the entries of rotation matrices $U$ and $V$
are given by
\be
c_k = \frac{1}{[k]}\,, \ \ \ \
s_k = \sqrt{\,1-c_k^2\,} = \frac{\sqrt{\,[k-1]^{\phantom {1^1}}\!\!\![k+1]\,}}{[k]}
\ee
These formulas provide a very transparent and convenient representation for infinitely
many HOMFLY polynomials
and seem to be very useful for any theoretical analysis of their general properties,
from integrability to linear Virasoro like relations
(including $A$-polynomials and spectral curves \cite{Apol,DiFu,3dAGT,Brini},
AMM/EO topological recursion \cite{AMM/EO,GS} etc).
Therefore, further insights are important about the structure of these formulas
and their generalizations
(in \cite{II} the $m=5$ case was also investigated,
and the general formula for the coefficients $h_{[1]}^{[m-1,1]}$ was suggested
for all $m$).
The limitations in \cite{II} are pure technical:
to make the paper readable and the main ideas understandable,
we considered only the implications of the theory of $SU_q(2)$ quantum group:
this allowed us to calculate only contributions of the Young diagrams $Q$
with one and two columns or rows.
Including the diagrams with $l$ columns or rows, which arise when the
number of strands in the braid is $m\geq 5$, requires the similar
use of the $SU_q(l)$ quantum group theory, which is tedious but straightforward,
and will be considered in further publications.
(We emphasize once again that $l$ has nothing to do with $N$ in $A=q^N$,
the relevant $l$ is related to the number $m$ of strands,
and for small $m\leq 4$ the smallest $l=2$ is sufficient to describe
{\it all} the HOMFLY polynomials with $R=[1]$.)
Another restriction in \cite{II} was to $R=[1]$.
It is also partly related to restriction to $SU_q(2)$, but not only.
It is the purpose of the present paper to make a first step
in the direction towards the {\it colored} HOMFLY polynomials with $|R|>1$,
that is, to the symmetric representation $R=[2]$.
{\bf Instead of performing this calculation directly, {\it a la} \cite{II}, we use a shortcut:
we determine the five parameters (angles) of the three orthogonal matrices $\hat U_{[51]}$, $\hat U_{[321]}$
and $\hat U_{[42]}$ by comparison with the answers for torus and composite knots and links
in eq.(\ref{H3n}) below.}
This can be also compared with the results of the long-lasting impressive work by the
Indian group \cite{inds} using the direct evaluation of the Racah coefficients.
Our goal is to find the necessary ingredients for formulas like (\ref{4})-(\ref{6}),
which provide an {\it exhaustive} description of {\it all} braids with a given number
of strands and in a given representation.
The final task would be to find general formulas that depend explicitly on
{\it all} the parameters: the number of strands $m$, the set $(a_{1i},\ldots,a_{mi})$,
specifying the $m$-strand braid (i.e. on the element of the braid group),
and on the Young diagram $R$ labeling the representation.
The formula is going to be for a coefficient in front of a particular character,
like the Schur function $S_Q\{p\}$ (or, alternatively, Hall-Littlewood \cite{HLexp}
or some other element of an appropriate expansion basis).
Now we perform a step of this program: find such {\it parametric} formulas
for given $m=3$ and $R=[2]$. For $R=[1]$ and $m=3,4$ (and partly $m=5$)
they were already derived in \cite{II}.
The formulas of this paper for the mixing matrices, which we obtain here indirectly, can be obtained directly
using the representation theory, like it was done in \cite{II}.
We consider the mixing matrices for a more generic case in
\cite{IV}.
\section{The case of 2 strands, $m=2$}
To determine emerging $Q$ in this case, one suffices to expand the product of two symmetric representations:
\be
\l[2]\times[2] = [4]+[31]+[22],
\ \ \ \
\l[11]\times [11] = [22] + [211] + [1111]
\ee
This decomposition can be easily obtained from the decomposition of the characters.
Indeed, given $S_{[2]} = \frac{1}{2}p_2 + \frac{1}{2}p_1^2$, $S_{[11]} = -\frac{1}{2}p_2 + \frac{1}{2}p_1^2$ and
\be
S_{[4]} = \frac{1}{4}p_4 + \frac{1}{3}p_3p_1 + \frac{1}{8}p_2^2 + \frac{1}{4}p_2p_1^2 + \frac{1}{24}p_1^4, \\
S_{[31]} = -\frac{1}{4}p_4 - \frac{1}{8}p_2^2 + \frac{1}{4}p_2p_1^2 + \frac{1}{8}p_1^4, \\
S_{[22]} = - \frac{1}{3}p_3p_1 + \frac{1}{4}p_2^2 + \frac{1}{12}p_1^4, \\
S_{[211]} = \frac{1}{4}p_4 - \frac{1}{8}p_2^2 - \frac{1}{4}p_2p_1^2 + \frac{1}{8}p_1^4, \\
S_{[1111]} = -\frac{1}{4}p_4 + \frac{1}{3}p_3p_1 + \frac{1}{8}p_2^2 - \frac{1}{4}p_2p_1^2 + \frac{1}{24}p_1^4,
\ee
it is easy to check that
\be
S_{[2]}^2 = S_{[4]} + S_{[31]} + S_{[22]}, \ \ \ \ \ \ S_{[11]}^2 = S_{[22]} + S_{[211]} + S_{[1111]}
\ee
In particular, for the ordinary dimensions $d_Q = S_Q\{p_k = N\}$ of representations $Q$ of
the $SU(N)$ algebra it reads as
$\left(\frac{N(N+1)}{2}\right)^2 = \frac{N(N+1)(N+2)(N+3)}{24} +
\frac{(N-1)N(N+1)(N+2)}{8} + \frac{(N^2-1)N^2}{12} $ etc.
and very similarly for the quantum dimensions $D_Q = S_Q^* = S_Q\{p_k = [N]_q\}$:
\be
\left(\frac{[N][N+1]}{[2]}\right)^2 = \frac{[N][N+1][N+2][N+3]}{[2][3][4]} +
\frac{[N-1][N][N+1][N+2]}{[2][4]} + \frac{[N-1][N]^2[N+1]}{[2]^2[3]}\\
\left({[N-1][N]\over [2]}\right)^2= \frac{[N-1][N]^2[N+1]}{[2]^2[3]}+{[N-2][N-1][N][N+1]\over [2][4]}+
{[N-3][N-2][N-1][N]\over [2][3][4]}
\ee
The HOMFLY polynomials are given by \cite{chi} (see also \cite{DMMSS} for
an extension to superpolynomials \cite{sup})
\be
H^{2,n}_{[1]} = \sum_{Q\in [1]\times [1]} \epsilon_Q^n q^{n\varkappa_Q} S_Q
= q^{n} S_{[2]} \mp q^{-n} S_{[11]} \\
H^{2,n}_{[2]} = \sum_{Q\in [2]\times [2]} \epsilon_Q^n q^{n\varkappa_Q} S_Q
= q^{6n} S_{[4]} \mp q^{2n} S_{[31]} + S_{[22]} \\
H^{2,n}_{[11]} = \sum_{Q\in [2]\times [2]} \epsilon_Q^n q^{n\varkappa_Q} S_Q
= q^{-6n} S_{[1111]} \mp q^{-2n} S_{[211]} + S_{[22]}
\label{H2n}
\ee
where\footnote{Similarly to \cite{I,II},
we use here $\varkappa_Q = -\nu_Q + \nu_{Q'}$ with the opposite sign as compared with \cite{DMMSS}.}
$\varkappa_Q = -\nu_Q + \nu_{Q'}$, $\nu_Q=\sum (i-1)Q_i$, $Q'$ is the transposed Young diagram and
the sign factors $\epsilon_Q$ are defined from the Adams rule \cite{chi}: "the initial condition"
$H^{m,0}_{[R]}\{p\} = \widehat{Ad}_mS_R\{p\}$ with
$\widehat{Ad}_m p_k = p_{mk}$ is imposed at $n=0$
\be
\widehat{Ad}_2 S_{[1]}\{p\} = \frac{p_2+p_1^2}{2} = S_{[2]} - S_{[11]} \\
\widehat{Ad}_2 S_{[2]}\{p\} = \frac{p_4+p_2^2}{2} = S_{[4]} - S_{[31]} + S_{[22]} \\
\widehat{Ad}_2 S_{[11]}\{p\} = \frac{p_4+p_2^2}{2} = S_{[1111]} - S_{[211]} + S_{[22]}
\label{Ad2knots}
\ee
It should not be mixed with the "physical" initial conditions for the $n$-evolution
of \cite{DMMSS},
\be
H^{m,n}_R =\ H^{n,m}_R
\ee
imposed at $0<n<m$.
For links one has instead of (\ref{Ad2knots}):
\be
\Big(\widehat{Ad}_1 S_{[1]}\{p\}\Big)^2 = p_1^2 = S_{[2]}+S_{[11]}, \\
\Big(\widehat{Ad}_1 S_{[2]}\{p\}\Big)^2 = \frac{(p_2+p_1^2)^2}{4} = S_{[4]} + S_{[31]} + S_{[22]}, \\
\Big(\widehat{Ad}_1 S_{[11]}\{p\}\Big)^2 = \frac{(-p_2+p_1^2)^2}{4} = S_{[1111]} + S_{[211]} + S_{[22]}
\label{Ad2links}
\ee
Accordingly, the signs $\mp$ at the r.h.s. of (\ref{H2n}) are minuses and pluses
for knots and links ($n$ odd or even) respectively.
In particular, for the unknot with $(m,n)=(2,1)$
\be
H_{[1]}^{2,1} = qS_{[2]}^* - q^{-1}S_{[11]}^* = qS_{[1]}^*\\
H_{[2]}^{2,1} = q^{6}S_{[4]}^* - q^{2}S_{[31]}^* + S_{[22]}^* = A^2q^4S_{[2]}^* \\
H_{[11]}^{2,1} = q^{-6}S_{[1111]}^* - q^{-2}S_{[211]}^* + S_{[22]}^* = A^2q^4S_{[11]}^*
\ee
for the Hopf link with $(m,n)=(2,2)$
\be
H_{[1]}^{2,2} = q^{2}S_{[2]}^* + q^{-2}S_{[11]}^* = \left((q^2-1+q^{-2})A-A^{-1}\right){S_{[1]}^*\over\{q\}} \\
H_{[2]}^{2,2} = q^{12}S_{[4]}^* + q^{4}S_{[31]}^* + S_{[22]}^* = \left((q^{12}-q^{10}-q^8+2q^6-q^2+1)A^2
-q^8 +q^4-q^2-1+q^2A^{-2}\right)q{S_{[2]}^*\over\{q\}\{q^2\}} \\
H_{[11]}^{2,2} = q^{-12}S_{[1111]}^* + q^{-4}S_{[211]}^* + S_{[22]}^* =\\= \left((1-q^{-2}+2q^{-6}-q^{-8}-q^{-10}+
q^{-12})A^2-1-q^{-2}+q^{-4}-q^{-8}+q^{-2}A^{-2}\right){S_{[11]}\over q\{q\}\{q^2\}}
\ee
and for the trefoil with $(m,n)=(2,3)$
\be
H_{[1]}^{2,3} = q^{3}S_{[2]}^* - q^{-3}S_{[11]}^* = \left((q^2+q^{-2})A-A^{-1}\right)S_{[1]}^* \\
H_{[2]}^{2,3} = q^{18}S_{[4]}^* - q^{6}S_{[31]}^* + S_{[22]}^* =\left( (q^{12}+q^6+q^4+1)A^2-q^8-q^6-q^2-1
+q^2A^{-2}\right)q^4S_{[2]}^* \\
H_{[11]}^{2,3} = q^{-18}S_{[1111]}^* - q^{-6}S_{[211]}^* + S_{[22]}^* = \left((q^{-12}+q^{-6}+q^{-4}+1)A^2-
q^{-8}-q^{-6}-q^{-2}-1+q^{-2}A^{-2}\right){S_{[11]}^*\over q^4}
\ee
\section{ 3 strands, $m=3$}
\subsection{Structure of the answer}
Now
\be
\l[2]\times[2]\times[2] = ([4]+[31]+[22])\times [2] = \\ =
([6] + [51] + [42]) + ([51] + [42] + [411] + [33] + [321]) + ([42]+[321]+[222])
\ee
For example, for the dimensions of $SU(2)$ representations one has
$3^3=27 = (7+5+3) + (5+3+0+1 + 0) + (3+0+0)$. Again, this decomposition is obtained as the decomposition of
the characters:
\be
S_{[6]} = \frac{1}{6}p_6+\frac{1}{5}p_5p_1+\frac{1}{8}p_4p_2+\frac{1}{8}p_4p_1^2
+\frac{1}{18}p_3^2+\frac{1}{6}p_3p_2p_1+\frac{1}{18}p_3p_1^3+\frac{1}{48}p_2^3+
\frac{1}{16}p_2^2p_1^2+\frac{1}{48}p_2p_1^4+\frac{1}{720}p_1^6, \\
S_{[51]} = -\frac{1}{6}p_6 - \frac{1}{8}p_4p_2+\frac{1}{8}p_4p_1^2
-\frac{1}{18}p_3^2 +\frac{1}{9}p_3p_1^3-\frac{1}{48}p_2^3+
\frac{1}{16}p_2^2p_1^2+\frac{1}{16}p_2p_1^4+\frac{1}{144}p_1^6, \\
S_{[42]} = -\frac{1}{5}p_5p_1+\frac{1}{8}p_4p_2-\frac{1}{8}p_4p_1^2 +\frac{1}{16}p_2^3+
\frac{1}{16}p_2^2p_1^2+\frac{1}{16}p_2p_1^4+\frac{1}{80}p_1^6, \\
S_{[411]} = \frac{1}{6}p_6
+\frac{1}{18}p_3^2-\frac{1}{6}p_3p_2p_1+\frac{1}{18}p_3p_1^3-\frac{1}{24}p_2^3
-\frac{1}{8}p_2^2p_1^2+\frac{1}{24}p_2p_1^4+\frac{1}{72}p_1^6, \\
S_{[33]} = -\frac{1}{8}p_4p_2-\frac{1}{8}p_4p_1^2
+\frac{1}{9}p_3^2+\frac{1}{6}p_3p_2p_1-\frac{1}{18}p_3p_1^3-\frac{1}{16}p_2^3+
\frac{1}{16}p_2^2p_1^2+\frac{1}{48}p_2p_1^4+\frac{1}{144}p_1^6, \\
S_{[321]} = \frac{1}{5}p_5p_1 - \frac{1}{9}p_3^2 - \frac{1}{9}p_3p_1^3 + \frac{1}{45}p_1^6, \\
\ldots \\
S_{[222]} = -\frac{1}{8}p_4p_2+\frac{1}{8}p_4p_1^2
+\frac{1}{9}p_3^2-\frac{1}{6}p_3p_2p_1-\frac{1}{18}p_3p_1^3+\frac{1}{16}p_2^3+
\frac{1}{16}p_2^2p_1^2-\frac{1}{48}p_2p_1^4+\frac{1}{144}p_1^6, \\
\ldots
\ee
Thus, one needs the $2\times 2$ mixing matrices for representations $[51]$ and $[321]$
and the $3\times 3$ mixing matrix for representation $[42]$.
The answer for the HOMFLY polynomial in the fundamental representation for
the generic $3$-strand knot $(a_1,b_1|a_2,b_2|\ldots)$
has the following form:
\be
H_{[1]}^{a_1,b_1|a_2,b_2|\ldots} =
q^{a_1+b_1+a_2+b_2+\ldots}\,S_{[3]}
+ \left(-\frac{1}{q}\right)^{a_1+b_1+a_2+b_2+\ldots} S_{[111]} + \\
+ \tr_{2\times 2} \Big\{\hat{\cal R}_{[21]}^{a_1} \hat U_{[21]} \hat{\cal R}_{[21]}^{b_1} \hat U_{[21]}^\dagger
\hat{\cal R}_{[21]}^{a_2} \hat U_{[21]} \hat{\cal R}_{[21]}^{b_2} \hat U_{[21]}^\dagger \ldots\Big\}S_{[21]}
\ee
with
\be
\hat{\cal R}_{[21]}
= \left(\begin{array}{cc} q^{\varkappa_{[2]}} & 0 \\ 0 & -q^{\varkappa_{[11]}} \end{array}\right)
= \left(\begin{array}{cc} q & 0 \\ 0 & -q^{-1} \end{array}\right), \ \ \ \ \ \
\hat U_{[21]} = \left(\begin{array}{cc} c_2 & s_2 \\ -s_2 & c_2 \end{array}\right)
\ee
Likewise, in the symmetric representation, it is going to be
\be
H_{[2]}^{a_1,b_1|a_2,b_2|\ldots} =
(q^6)^{a_1+b_1+a_2+b_2+\ldots}\,S_{[6]}
+ (-q^2)^{a_1+b_1+a_2+b_2+\ldots}\,\Big(S_{[411]}+S_{[33]}\Big) + S_{[222]} + \\
+ \tr_{2\times 2} \Big\{\hat{\cal R}_{[51]}^{a_1} \hat U_{[51]} \hat{\cal R}_{[51]}^{b_1} \hat U_{[51]}^\dagger
\hat{\cal R}_{[51]}^{a_2} \hat U_{[51]} \hat{\cal R}_{[51]}^{b_1} \hat U_{[51]}^\dagger \ldots\Big\}S_{[51]}
+ \\
+ \tr_{2\times 2} \Big\{\hat{\cal R}_{[321]}^{a_1} \hat U_{[321]} \hat{\cal R}_{[321]}^{b_1} \hat U_{[321]}^\dagger
\hat{\cal R}_{[321]}^{a_2} \hat U_{[321]} \hat{\cal R}_{[321]}^{b_1} \hat U_{[321]}^\dagger \ldots\Big\}S_{[321]}
+ \\
+ \tr_{3\times 3} \Big\{\hat{\cal R}_{[42]}^{a_1} \hat U_{[42]} \hat{\cal R}_{[42]}^{b_1} \hat U_{[42]}^\dagger
\hat{\cal R}_{[42]}^{a_2} \hat U_{[42]} \hat{\cal R}_{[42]}^{b_1} \hat U_{[42]}^\dagger \ldots\Big\}S_{[42]}
\label{H3n}
\ee
Here
\be
\hat{\cal R}_{[6]} = q^{\varkappa_{[4]}} = q^6,\ \ \ \ \ \ \ \
\hat{\cal R}_{[411]} = \hat{\cal R}_{[33]} = -q^{\varkappa_{[31]}} = -q^2, \ \ \ \ \ \ \ \
\hat{\cal R}_{[222]} = q^{\varkappa_{[22]}} = 1, \\ \\
\hat{\cal R}_{[51]}
= \left(\begin{array}{cc} q^{\varkappa_{[4]}} & 0 \\ 0 & -q^{\varkappa_{[31]}} \end{array}\right)
= \left(\begin{array}{cc} q^6 & 0 \\ 0 & -q^{2} \end{array}\right), \ \ \ \ \ \ \nn
\ee
\be
\hat{\cal R}_{[321]}
= \left(\begin{array}{cc} -q^{\varkappa_{[31]}} & 0 \\ 0 & -q^{\varkappa_{[22]}} \end{array}\right)
= \left(\begin{array}{cc} -q^2 & 0 \\ 0 & 1 \end{array}\right), \ \ \ \ \ \nn
\ee
\be
\hat{\cal R}_{[42]}
= \left(\begin{array}{ccc} q^{\varkappa_{[4]}} & 0 & 0\\ 0 & -q^{\varkappa_{[31]}} & 0 \\
0 & 0 & q^{\varkappa_{[22]}} \end{array}\right)
=\left(\begin{array}{ccc} q^6 & 0 & 0 \\ 0 & -q^{2} & 0 \\ 0 & 0 & 1 \end{array}\right)
\label{R42}
\ee
and the mixing matrices $\hat U_Q$ need to be calculated.
Two of them , those for the double-line diagrams $[51]$ and $[42]$, can be evaluated with the help
of representation theory of $SU_q(2)$, but in the $[321]$ sector at least $SU_q(3)$ would be needed.
Instead of performing this calculation directly, {\it a la} \cite{I,II}, we use a shortcut:
determine the five parameters (angles) of the three orthogonal matrices $\hat U_{[51]}$, $\hat U_{[321]}$
and $\hat U_{[42]}$ by comparison with the answers for torus and composite knots and links.
\subsection{Torus knots}
For torus knots $T[3,n]$ with $a_1=b_1=\ldots=a_n=b_n=1$ one has an alternative decomposition \cite{chi}:
\be
\underline{H}_{[2]}^{3,n} = \sum_{Q\vdash 6} q^{\frac{2n}{3}\varkappa_Q} C_{[2]}^QS_Q
\label{Hbar3[2]}
\ee
where the coefficients are defined from the Adams rule
\be
\widehat{Ad}_3 S_{[2]} = \frac{p_6+p_3^2}{2} = \sum_{Q\vdash 6} C_{[2]}^QS_Q =
S_{[6]} - S_{[5,1]} + \underline{0\cdot S_{[42]}} + S_{[411]}
+ S_{[33]} - S_{[321]} + S_{[222]}, \\
\Big(\widehat{Ad}_1 S_{[2]}\Big)^3 = \frac{(p_2+p_1^2)^3}{8} = \sum_{Q\vdash 6} C_{[2]}^QS_Q =
S_{[6]} + 2S_{[5,1]} + \underline{3\cdot S_{[42]}} + S_{[411]}
+ S_{[33]} + 2S_{[321]} + S_{[222]}
\ee
for knots and links, i.e. for $n=1,2\ ({\rm mod}\ 3)$ and $n = 0\ ({\rm mod}\ 3)$ respectively.
Thus for the knots, $n=1,2\ ({\rm mod}\ 3)$
\be
\underline{H}_{[2]}^{3,n} = q^{10n}S_{[6]} - q^{6n}S_{[5,1]} + \underline{0\cdot q^{10n/3}S_{[42]}} + q^{2n}S_{[411]}
+ q^{2n}S_{[33]} - S_{[321]} + q^{-2n}S_{[222]} = \\
= q^{-2n}\Big(q^{12n}S_{[6]} - q^{8n}S_{[5,1]} + \underline{0\cdot q^{16n/3}S_{[42]}} + q^{4n}S_{[411]}
+ q^{4n}S_{[33]} - q^{2n}S_{[321]} + S_{[222]}\Big)
\label{H3n[2]kn}
\ee
Note that the only would be contribution with non-integer value of $\frac{1}{3}\varkappa_Q$
(underlined) does not contribute in the case of torus knots: the Adams coefficient
$C_{[2]}^{[42]}=0$.
Looking at the coefficients in front of the fully known "singlet" terms
$S_{[6]}$, $S_{[411]}$, $S_{[33]}$, $S_{[222]}$,
which do not involve yet unknown mixing matrices,
we see that eq.(\ref{Hbar3[2]}) differs from the correct expression by a factor of
\be
\underline{H}_{[2]}^{3,n} = q^{-2n}{H}_{[2]}^{3,n}
\ee
For the generic single-line (symmetric) representations $[p]$ and arbitrary number $m$ of strands
one gets, comparing the coefficients in front of $S_{[pm]}$:
\be\label{chicorr}
\underline{H}_{[p]}^{m,n} = q^{\frac{2n}{m}\varkappa_{[mp]}-n(m-1)\varkappa_{[2p]}}{H}_{[p]}^{m,n}
= q^{-n(m-2)p(p-1)}{H}_{[p]}^{m,n}
\ee
so that there is no discrepancy for either the first fundamental representation $p=1$
or for the case of $m=2$ strands, when all the knots are torus.
For the links, $n=0\ ({\rm mod}\ 3)$
\be
\underline{H}_{[2]}^{3,n} = q^{10n}S_{[6]} +2 q^{6n}S_{[5,1]} + \underline{3\cdot q^{10n/3}S_{[42]}}
+ q^{2n}S_{[411]} + q^{2n}S_{[33]} + 2S_{[321]} + q^{-2n}S_{[222]} = \\
= q^{-2n}\Big(q^{12n}S_{[6]} + 2q^{8n}S_{[5,1]} + \underline{3\cdot q^{16n/3}S_{[42]}} + q^{4n}S_{[411]}
+ q^{4n}S_{[33]} + 2q^{2n}S_{[321]} + S_{[222]}\Big)
\label{H3n[2]ln}
\ee
This time the underlined terms are non-vanishing, but since for the links $n\vdots 3$,
the power is integer in this case.
Note that the coefficients are the same for knots and links in front of the
terms $S_{[6]}$, $S_{[411]}$, $S_{[33]}$ and $S_{[222]}$, in full accordance with
(\ref{H3n}), because for the torus knots and links $a_i+b_i$ is either $2$ or $0$,
i.e. always even, so that the corresponding signs $\epsilon_Q$ can not affect the answers
in the torus case (however, they affect the answers for the composite knots, see s.\ref{comp} below).
These formulas generalize those for the fundamental representation:
\be
H_{[1]}^{3,n} = q^{2n}S_{[3]} - S_{[21]} + q^{-2n}S_{[111]},\ \ \ \ n=1,2\ ({\rm mod}\ 3) \\
H_{[1]}^{3,n} = q^{2n}S_{[3]} + 2S_{[21]} + q^{-2n}S_{[111]},\ \ \ \ n=0\ ({\rm mod}\ 3)
\label{H3n[1]}
\ee
considered in \cite{II}.
\subsection{$2\times 2$ matrices $\hat U_{[51]}$ and $\hat U_{[321]}$ from the torus knots}
When mixing matrix is of the size $2\times 2$, it can be parameterized by a single parameter $s$,
sine of the mixing angle, cosine $c$ being related through $c^2+s^2=1$.
Then we have for an elementary building block
\be
\hat{\cal R}^{a} \hat U \hat{\cal R}^{b} \hat U^\dagger =
\left(\begin{array}{cc} \epsilon q^{\varkappa} & 0 \\ 0 & \tilde\epsilon q^{\tilde\varkappa}
\end{array}\right)^{a}
\left(\begin{array}{cc} c & s \\ -s & c
\end{array}\right)
\left(\begin{array}{cc} \epsilon q^{\varkappa} & 0 \\ 0 & \tilde\epsilon q^{\tilde\varkappa}
\end{array}\right)^{b}
\left(\begin{array}{cc} c & -s \\ s & c
\end{array}\right)
= \\
= \left(\begin{array}{cc}
\epsilon^{a+b} q^{\varkappa(a+b)}c^2 + \epsilon^a\tilde\epsilon^b q^{\varkappa a + \tilde\varkappa b}s^2
& \left( -\epsilon^{a+b} q^{\varkappa(a+b)}
+ \epsilon^a\tilde\epsilon^b q^{\varkappa a + \tilde\varkappa b}\right)cs \\
\left( \tilde\epsilon^{a+b} q^{\tilde\varkappa(a+b)}
- \epsilon^b\tilde\epsilon^a q^{\varkappa b + \tilde\varkappa a}\right)cs
& \tilde\epsilon^{a+b} q^{\tilde\varkappa(a+b)}c^2
+ \epsilon^b\tilde\epsilon^a q^{\varkappa b + \tilde\varkappa a}s^2
\end{array}\right)
\ee
In the case of torus knots and links $a=b=1$ and this reduces to
\be
\hat{\cal R} \hat U \hat{\cal R} \hat U^\dagger =
\left(\begin{array}{cc}
q^{2\varkappa}c^2 + (\epsilon\tilde\epsilon) q^{\varkappa + \tilde\varkappa}s^2
& \left( - q^{2\varkappa}
+ (\epsilon\tilde\epsilon) q^{\varkappa + \tilde\varkappa }\right)cs \\
\left( q^{2\tilde\varkappa}
- (\epsilon\tilde\epsilon) q^{\varkappa +\tilde\varkappa }\right)cs
& q^{2\tilde\varkappa }c^2
+ (\epsilon\tilde\epsilon) q^{\varkappa \tilde\varkappa }s^2
\end{array}\right)
\ee
and
\be
\Tr_{2\times 2} \hat{\cal R} \hat U \hat{\cal R} \hat U^\dagger =
(q^{2\varkappa}+q^{2\tilde\varkappa })c^2 + 2(\epsilon\tilde\epsilon) q^{\varkappa + \tilde\varkappa}s^2
= \Big(q^{2\varkappa}+q^{2\tilde\varkappa }\Big)
- \Big(q^\varkappa - (\epsilon\tilde\epsilon)q^{\tilde\varkappa}\Big)^2s^2
\label{Tr31}
\ee
Now it remains to substitute the relevant values of $\varkappa$, $\tilde\varkappa$,
$\epsilon$ and $\varepsilon$, and compare this trace with the relevant coefficient of the HOMFLY
polynomial for the torus knot $T[3,1]$
(it is essentially the unknot but realized by a non-simplest braid; since we do not need to restrict
ourselves to the topological locus here, this expression is not the same as $S_{[R]}$).
After that one can calculate the traces of powers of this matrix and check that with the same
value of $s$ they
reproduce the values of the coefficient for all other torus knots and links $T[3,n]$ with different $n$.
This is, in fact, not a problem, because one should just check that with the right value of $s$
the matrix $\hat{\cal R} \hat U \hat{\cal R} \hat U^\dagger$ has appropriate eigenvalues,
proportional to the roots of unity.
Finally, the same value of $s$ determines the coefficient for all other 3-strand braids
$(a_1,b_1,a_2,b_2,\ldots)$.
\bigskip
{\bf The case of $R=[1]$ and the term $S_{[21]}$.}
We start with this already known case, \cite{II} for illustrative purposes. One has to
substitute $\varkappa = \varkappa_{[2]} = 1$, $\tilde\varkappa_{[11]} = -1$,
$\epsilon=1$, $\tilde\epsilon = -1$ and compare (\ref{Tr31}) with the value of the coefficient
in front of $S_{21}$ in (\ref{H3n[1]}) with $n=1$, which is $-1$. This gives:
\be
q^2+q^{-2} - (q+q^{-1})^2s^2 = -1 \ \Longrightarrow \
s = \frac{\sqrt{q^2+1+q^{-2}}}{q+q^{-1}}=\frac{\sqrt{[3]}}{[2]}=s_2, \ \
c = \frac{1}{q+q^{-1}} = \frac{1}{[2]} = c_2
\ee
This reproduces the answer (\ref{UVmatr}) for $U_2$ from \cite{II}.
It is easy to check that, with this values of $s$ and $c$,
\be
{\det}_{2\times 2}\Big(\hat{\cal R} \hat U \hat{\cal R} \hat U^\dagger - \lambda I\Big)
= \frac{\lambda^3-1}{\lambda-1} = \lambda^2+\lambda+1
\ee
i.e. the two eigenvalues of $\hat{\cal R} \hat U \hat{\cal R} \hat U^\dagger$ are $e^{\pm \frac{2\pi i}{3}}$,
so that
\be
\Tr_{2\times 2} \Big(\hat{\cal R} \hat U \hat{\cal R} \hat U^\dagger\Big)^n
= \left\{\begin{array}{ccc} -1 & {\rm for} & n=1,2\ ({\rm mod}\ 3) \\
+2 & {\rm for} & n=0\ ({\rm mod}\ 3)\end{array}\right.
\ee
in full agreement with (\ref{H3n[1]}).
\bigskip
{\bf The case of $R=[2]$ and the term $S_{[51]}$.}
One has to substitute $\varkappa = \varkappa_{[4]} = 6$, $\tilde\varkappa_{[31]} = 2$,
$\epsilon=1$, $\tilde\epsilon = -1$ and compare (\ref{Tr31}) with the value of the coefficient
in front of $S_{[51]}$ in (\ref{H3n[2]kn}) with $n=1$, which is $-q^8$. This gives:
\be
q^{12}+q^{4} - (q^6+q^{2})^2s^2 = -q^8 \ \Longrightarrow \
s = \frac{\sqrt{q^4+1+q^{-4}}}{q^2+q^{-2}} = \frac{\sqrt{[3]_{q^2}}}{[2]_{q^2}}, \ \
c = \frac{1}{q^2+q^{-2}} = \frac{1}{[2]_{q^2}}
\ee
where $[x]_{q^2}\equiv {q^{2x}-q^{-2x}\over q^2-q^{-2}}$.
Again, it is a simple exercise to check that with these values of $s$ and $c$
\be
{\det}_{2\times 2}\Big(\hat{\cal R} \hat U \hat{\cal R} \hat U^\dagger - \lambda I\Big)
= \lambda^2 + q^8\lambda+q^{16}
\ee
i.e. the two eigenvalues of $\hat{\cal R} \hat U \hat{\cal R} \hat U^\dagger$ are
$q^8e^{\pm\frac{2\pi i}{3}}$ and
\be
\Tr_{2\times 2} \Big(\hat{\cal R} \hat U \hat{\cal R} \hat U^\dagger\Big)^n
= \left\{\begin{array}{ccc} -q^{8n} & {\rm for} & n=1,2\ ({\rm mod}\ 3) \\
+2q^{8n} & {\rm for} & n=0\ ({\rm mod}\ 3)\end{array}\right.
\ee
in full agreement with (\ref{H3n[2]kn}) and (\ref{H3n[2]ln}).
\bigskip
{\bf The case of $R=[2]$ and the term $S_{[321]}$.}
One has to substitute $\varkappa = \varkappa_{[31]} = 2$, $\tilde\varkappa_{[22]} = 0$,
$\epsilon=-1$, $\tilde\epsilon = 1$ and compare (\ref{Tr31}) with the value of the coefficient
in front of $S_{[321]}$ in (\ref{H3n[2]kn}) with $n=1$, which is $-q^2$. This gives:
\be
q^{4}+1 - (q^2+1)^2s^2 = -q^2 \ \Longrightarrow \
s = \frac{\sqrt{q^2+1+q^{-2}}}{q+q^{-1}} = \frac{\sqrt{[3]}}{[2]}, \ \
c = \frac{1}{q+q^{-1}} = \frac{1}{[2]}
\ee
With these values of $s$ and $c$
\be
{\det}_{2\times 2}\Big(\hat{\cal R} \hat U \hat{\cal R} \hat U^\dagger - \lambda I\Big)
= \lambda^2 + q^2\lambda+q^{4}
\ee
i.e. the two eigenvalues of $\hat{\cal R} \hat U \hat{\cal R} \hat U^\dagger$ are
$q^2e^{\pm\frac{2\pi i}{3}}$ and
\be
\Tr_{2\times 2} \Big(\hat{\cal R} \hat U \hat{\cal R} \hat U^\dagger\Big)^n
= \left\{\begin{array}{ccc} -q^{2n} & {\rm for} & n=1,2\ ({\rm mod}\ 3) \\
+2q^{2n} & {\rm for} & n=0\ ({\rm mod}\ 3)\end{array}\right.
\ee
again in excellent agreement with (\ref{H3n[2]kn}) and (\ref{H3n[2]ln}).
\subsection{Constraining the $3\times 3$ matrix $\hat U_{[42]}$ from the torus knots}
When orthogonal mixing matrix is of the size $3\times 3$, it can be parameterized by three
independent Euler angles, namely by their sines and cosines:
\be
\hat U = \left(\begin{array}{ccc} c_1 & 0 & s_1 \\ 0& 1 & 0\\ -s_1 & 0 & c_1 \end{array}\right)
\left(\begin{array}{ccc} 1&0&0\\0&c_2 & s_2 \\ 0&-s_2 & c_2 \end{array}\right)
\left(\begin{array}{ccc} c_3 & 0 & s_3\\ 0& 1 & 0\\ -s_3 & 0 & c_3 \end{array}\right)
\label{Eude}
\ee
One now needs to perform the same trick: to compare the traces of powers of
$\hat{\cal R} \hat U \hat{\cal R} \hat U^\dagger$, where $\hat{\cal R}$ is given in (\ref{R42})
with the known coefficients in front of $S_{[42]}$ in (\ref{H3n[2]kn}).
This comparison tells that
\be
\tr_{3\times 3} (\hat{\cal R} \hat U \hat{\cal R} \hat U^\dagger)^n =
\left\{\begin{array}{ccc} 0 & {\rm for} & n = 1,2\ ({\rm mod}\ 3) \\
3q^{16n/3} & {\rm for} & n = 0\ ({\rm mod}\ 3)
\end{array}\right.
\label{probtri}
\ee
for diagonal
\be
\hat{\cal R} = \left(\begin{array}{ccc} q^6 & 0 & 0 \\ 0 & -q^2 & 0 \\ 0 & 0 & 1\end{array}\right)
\ee
The choice of the Euler decomposition in (\ref{Eude}) is obviously adjusted to this form of the
matrix $\hat{\cal R}$.
At $q=1$ a solution is obvious:
\be
\hat{\cal R} = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{array}\right)
\ \Longrightarrow \
\hat{U} = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & c & s \\ 0 & -s & c\end{array}\right), \ \ \ \
\hat{\cal R} \hat U \hat{\cal R} \hat U^\dagger =
\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & c^2-s^2 & -2cs \\ 0 & 2cs & c^2-s^2 \end{array}\right)
\ee
i.e. one gets the rotation matrix with the doubled angle $\phi$, $s = \sin\phi$.
Then (\ref{probtri}) means that $6\phi = 2\pi k$ with any integer $k$,
i.e. $\phi = \frac{\pi k}{3}$, and $c = \pm \frac{1}{2}$, $s = \pm\frac{\sqrt{3}}{2}$.
Of course, at $q=1$ there is a huge degeneracy: any rotation involving only the first and the third
lines leaves $\hat{\cal R}(q=1)$ intact, and one can take many other $\hat U$, obtained by such a
rotation, for example,
$\hat{U} = \left(\begin{array}{ccc} c & s & 0 \\ -s & c & 0 \\ 0 & 0 & 1\end{array}\right)$
with the same $c$ and $s$. For $q=1$ only one of the three Euler angles in $U$ is fixed by
conditions (\ref{probtri}).
At $q\neq 1$ conditions (\ref{probtri}) imply that the three eigenvalues of
$\hat{\cal R} \hat U \hat{\cal R} \hat U^\dagger$ are three cubic roots of unity
times $q^{16/3}$, i.e. that
\be
{\det}_{3\times 3} \Big(\hat{\cal R} \hat U \hat{\cal R} \hat U^\dagger - \lambda I\Big)
= q^{16}-\lambda^3
\ee
Clearly, these are only two conditions, so that only two of the three Euler angles
will be fixed by (\ref{probtri}).
One extra condition, not just coming from the 3-strand torus knot and link polynomials, will be needed
to fix $\hat U_{[42]}$ unambiguously.
We impose this condition by making an "educated guess" that $c_3=c_1$ and $s_3=-s_1$.
Then
\be
c_1 = c_3 = \frac{1+q^4}{\sqrt{(2q^4-q^2+2)(1+q^2+q^4)}}\,,
\ \ \ \ \ \ c_2 = -\frac{q^4-q^2+1}{1+q^4} = -\frac{1+q^6}{(1+q^2)(1+q^4)}
\ee
(the sign in $c_2$ is essential).
One can use an alternative parametrization instead of (\ref{Eude}):
\be
\hat U = \left(\begin{array}{ccc} c_1' & 0 & s_1' \\ 0& 1 & 0\\ -s_1' & 0 & c_1' \end{array}\right)
\left(\begin{array}{ccc} c_2' & s_2' & 0 \\ -s_2' & c_2' &0\\0&0&1 \end{array}\right)
\left(\begin{array}{ccc} c_3' & 0 & s_3'\\ 0& 1 & 0\\ -s_3' & 0 & c_3' \end{array}\right)
\label{Eude'}
\ee
In this case the Euler angles are given by
\be
c_1' = c_3' = \sqrt{\frac{1-q^{10}}{(2q^4-q^2+2)(1-q^6)}}= -s_1=s_3,
\ \ \ \ \ \ c_2' = -\frac{q^4-q^2+1}{1+q^4} = -\frac{1+q^6}{(1+q^2)(1+q^4)}=c_2
\ee
i.e. are "dual" to those for (\ref{Eude}).
In both cases one obtains the same matrix $\hat U$, see eq.(\ref{U42}) below.
It remains an open question, whether a nicer decomposition exists for this
rather sophisticated mixing matrix.
\subsection{The final answer}
Substituting into (\ref{H3n}) the values of the mixing angles,
found in the previous subsections
one finally obtains for arbitrary $3$-strand braid:
\be
H_{[2]}^{a_1,b_1|a_2,b_2|\ldots} =
q^{6(a_1+b_1+a_2+b_2+\ldots)}\,S_{[6]}
+ (-q^2)^{a_1+b_1+a_2+b_2+\ldots}\,\Big(S_{[411]}+S_{[33]}\Big) + S_{[222]} + \\
+ \tr_{2\times 2}\left\{
\left(\begin{array}{cc} q^6 & 0 \\ 0 & -q^2\end{array}\right)^{a_1}
\left(\begin{array}{cc} -\frac{1}{[2]_{q^2}} & \frac{\sqrt{[3]_{q^2}}}{[2]_{q^2}} \\
-\frac{\sqrt{[3]_{q^2}}}{[2]_{q^2}} & -\frac{1}{[2]_{q^2}}\end{array}\right)
\left(\begin{array}{cc} q^6 & 0 \\ 0 & -q^2\end{array}\right)^{b_1}
\left(\begin{array}{cc} -\frac{1}{[2]_{q^2}} & -\frac{\sqrt{[3]_{q^2}}}{[2]_{q^2}} \\
\frac{\sqrt{[3]_{q^2}}}{[2]_{q^2}} & -\frac{1}{[2]_{q^2}}\end{array}\right)
\ldots\right\}S_{[51]} + \\
+ \tr_{2\times 2}
\left\{
\left(\begin{array}{cc} -q^2 & 0 \\ 0 & 1\end{array}\right)^{a_1}
\left(\begin{array}{cc} -\frac{1}{[2]_{q}} & \frac{\sqrt{[3]_{q}}}{[2]_{q}} \\
-\frac{\sqrt{[3]_{q}}}{[2]_{q}} & -\frac{1}{[2]_{q}}\end{array}\right)
\left(\begin{array}{cc} -q^2 & 0 \\ 0 & 1\end{array}\right)^{b_1}
\left(\begin{array}{cc} -\frac{1}{[2]_{q}} & -\frac{\sqrt{[3]_{q}}}{[2]_{q}} \\
\frac{\sqrt{[3]_{q}}}{[2]_{q}} & -\frac{1}{[2]_{q}}\end{array}\right)
\ldots\right\}S_{[321]}
+ \\
+ \tr_{3\times 3}\left\{
\left(\begin{array}{ccc} q^6 &&\\&-q^2 & \\ && 1 \end{array}\right)^{a_1}
U_{[42]}
\left(\begin{array}{ccc} q^6 &&\\&-q^2 & \\ && 1 \end{array}\right)^{b_1}
U_{[42]}^\dagger
\ldots\right\}S_{[42]}\ \ \ \ \ \ \ \ \ \ \
\label{ansH3n}
\ee
The matrix $U_{[42]}$ is equal to:
\be
\left(\begin{array}{ccc}
\frac{q^4}{(q^4+1)(q^4+q^2+1)}
& -\frac{q\sqrt{q^8+q^6+q^4+q^2+1}}{(q^4+1)\sqrt{q^4+q^2+1}}
& -\frac{\sqrt{q^8+q^6+q^4+q^2+1}}{q^4+q^2+1}\\
\frac{q\sqrt{q^8+q^6+q^4+q^2+1}}{(q^4+1)\sqrt{q^4+q^2+1}}&-\frac{q^4-q^2+1}{q^4+1}
&\frac{q}{\sqrt{q^4+q^2+1}}\\
-\frac{\sqrt{q^8+q^6+q^4+q^2+1}}{q^4+q^2+1}&-\frac{q}{\sqrt{q^4+q^2+1}}
&\frac{q^2}{q^4+q^2+1}
\end{array}\right)=\\
= \left(\begin{array}{ccc}
\frac{[2]}{[3][4]}& -\frac{[2]}{[4]}\sqrt{\frac{[5]}{[3]}} & -\frac{\sqrt{[5]}}{[3]} \\
\frac{[2]}{[4]}\sqrt{\frac{[5]}{[3]}} & -\frac{[6]}{[3][4]} & \frac{[1]}{\sqrt{[3]}} \\
-\frac{\sqrt{[5]}}{[3]} & -\frac{1}{\sqrt{[3]}} & \frac{1}{[3]}
\end{array}\right)
\label{U42}
\ee
This is the same matrix as the matrix of the Racah coefficients, (A.20) in \cite{Kaul93}.
\subsection{Composite knots and links: a check
\label{comp}}
In this section we perform further checks, making use of topological equivalence
between different braids, i.e. homotopic equivalence of the corresponding
knots and links.
Accordingly, in this section we can consider only the ordinary HOMFLY polynomials
$H$ reduced to the topological locus
\be
p_k = p_k^* = \frac{A^k-A^{-k}}{q^k-q^{-k}}
\label{tolo2}
\ee
\bigskip
{\bf The case of $b_1=a_2=b_2=\ldots =0$:}
In this simplest example, with only one non-vanishing parameter $a_1$,
the 3-strand knot/link splits into untied a 2-strand knot/link and the unknot.
Accordingly,
\be
H_R^{3,(a,0,0,0,\ldots)} =\ H_R^{2,a}\cdot {H_R^{0}}
\label{H3aaadeco}
\ee
At the same time, in this case (\ref{ansH3n}) is also drastically simplified:
all mixing matrices drop away from the formula and it reduces to just
\be
H_{[2]}^{3,(a,0,0,0,\ldots)} = q^{6a}S_{[6]} + (-q^2)^a\Big(S_{[411]}+S_{[33]}\Big) + S_{[222]} +\\
+ \Big(q^{6a}+(-q^2)^a\Big)S_{[51]} + \Big((-q^2)^a+1\Big)S_{[321]} +
\Big(q^{6a} + (-q^2)^a + 1\Big)S_{[42]}
\label{H3a000}
\ee
Note that, in variance with expressions for the 3-strand {\it torus} knots and links,
this formula is sensitive to the sign of the $R$-matrix eigenvalue $-q^2$.
It remains to reduce (\ref{H3a000}) to the topological locus (\ref{tolo2}), where the Schur functions
turn into the quantum dimensions, and check that this coincides with the r.h.s.
of (\ref{H3aaadeco}) with $R=[2]$, where the unknot polynomial is just $H_R^{0} = S_R^*$
and $H_{[2]}^{2,a}$ is given by the second line of (\ref{H2n}).
Of course, such a relation can {\it not} be lifted to the entire $p$-space: (\ref{H3a000})
does {\it not} coincide with $H_{[2]}^{(2,a)} S_{[2]}$ beyond the topological locus (\ref{tolo2}):
one suffices to note that the former depends on $p_6$, while the latter one does not.
\subsection{Results}
\subsubsection{The figure eight knot $4_1$}
This knot can be realized with the braid
\be
4_1:\ \ \ \ (a_1,b_1|a_2,b_2) = (1,-1|1,-1),
\ee
similar to a possible 3-strand realization of the trefoil,
which is a torus knot $T[2,3]=T[3,2]$
\be
3_1:\ \ \ \ (1,1|1,1)
\ee
In the fundamental representation one had
\be
H^{4_1}_{[1]} = S_{[3]}^* + \Big(q^4 - 2q^2 + 1 - 2q^{-2} + q^{-4}\Big)S^*_{[21]} +S_{[111]}^*
= \Big(A^2 -(q^2-1+q^{-2}) + A^{-2}\Big) S_{[1]}^*
\ee
while
\be
H^{3_1}_{[1]} = q^{4}S_{[3]}^* - S_{[21]}^* + q^{-4}S_{[111]}^* =
\Big((q^2+q^{-2})A^2 - 1 \Big) S_{[1]}^*
\ee
The second expression is highly asymmetric,
while the formula for $4_1$ is {\it very} symmetric even when expressed in terms of the $A$
variable: this is a specifics of $4_1$.
In the symmetric representation $R=[2]$ the answer is
\be
H^{4_1}_{[2]}
= \Big( q^4A^4
-(1+q^2)(1-q^2+q^6)q^{-2}A^2
+ (q^6-q^4+3-q^{-4}+q^{-6}) - \\
-(1+q^{-2})(1-q^{-2}+q^{-6})q^2A^{-2}
+q^{-4}A^{-4}\Big)S^*_{[2]}
\label{H241}
\ee
This can be compared with the asymmetric formula for the trefoil $(1,1,1,1)$
\be
H^{3_1}_{[2]}
= q^8\Big(A^4(1+q^4+q^6+q^{12}) - A^2(1+q^2)(1+q^6) + q^2\Big)S_{[2]}^*
\ee
Expression (\ref{H241}) certainly coincide with results presented in existing literature,
see e.g. \cite{Rama00}. Moreover, it turns out that in the case of $4_1$ one can get the result for {\it any}
symmetric $[p]$ and antisymmetric $[1^p]$ representation \cite{IMMMfe}.
The HOMFLY polynomials in the symmetric representation for other 3-strand knots with no more than 8 crossings
are collected in the Appendix.
\subsection{Antisymmetric representation}
In order to construct the HOMFLY polynomials in the antisymmetric representation $[1,1]$, one could repeat the
standard machinery of the mixing matrices etc we described above. However, the result can be obtained much simpler
using a symmetry of the HOMFLY polynomials.
Indeed, the character expansions of the HOMFLY polynomials possess a
$Z_2$-symmetry
\be\label{symas}
A,\ q,\ S_R^* \ \ \longleftrightarrow \ A,\ -\frac{1}{q}\,,\ S_{R'}^*
\ee
where $R'$ is a transposition of the Young diagram $R$. This symmetry can be easily understood, since
$S_{R'}\{p_k\} = S_R\big\{(-)^{k-1}p_k\big\}$ and $\kappa_R=-\kappa_{R'}$. At the same time, all the ($SU_q(N)$)
group representation quantities (in particular, the mixing matrices) are also possess this antipodal symmetry. Hence,
one can calculate the HOMFLY polynomials in the antisymmetric representation just making a substitution $q\to -1/q$
in the HOMFLY polynomials for the symmetric representation obtained in the previous sections.
\subsection{Ooguri-Vafa conjecture}
In the paper \cite{OV}, the authors conjectured a connection of the Chern-Simons theory with topological string on the
resolution of the conifold. In fact, they proposed that the generating function $Z$ of average of the Wilson loop
in different representations is associated with the topological string partition function $Z_{str}$.
In accordance with the Ooguri-Vafa result \cite{GV} $Z$ is given by the sum\footnote{
This sum can be obtained as the Chern-Simons average of the Ooguri-Vafa operator $\exp\sum_n {1\over n}
\Tr \left(\oint_K Adx\right)^n\Tr V^n$, where $A$ is the gauge field, $p_k=\Tr V^k$ are external sources and
the traces are taken over the fundamental representation.
}
\be
Z(q,A,K)=\sum_R\chi_R(p)H_R^K(q,A)
\ee
where the sum runs over all the irreducible representation of $SU(N)$ ($A=q^N$). Now the topological nature
of this object implies that the "connected" correlators $f_R(q,A)$ defined by the expansion
\be
\log Z=\sum_{n=0,R}{1\over n}f_R(q^n,A^n)\chi_R(p^{(n)})
\ee
where the set of variables $p^{(n)}_k\equiv p_{nk}$, has the generic structure
\be
f_R(q,A)=\sum_{n,k} \tilde N_{R,n,k}{A^nq^k\over q-q^{-1}}
\ee
$\tilde N_{R,n,k}$ are integer and the parity of $n$ in the sum coincides with the parity of $|R|$ while the parity of $k$ is inverse.
These numbers are related to the Gopakumar-Vafa
integers $n_{\Delta,n,k}$ \cite{GV} by the relation
$$
n_{\Delta,n,k}=\sum_R \Phi_R(\Delta)\tilde N_{R,n,k}
$$
where $\Phi_R(\Delta)$ is the character of the symmetric group $S_{|\Delta|}$. The integers $\tilde N_{R,n,k}$ are
more refined, since their integrality implies that $n_{\Delta,n,k}$ are integer but not vise verse.
In fact, one can consider even more refined integers \cite{LMV}
\be\label{numN}
f_R(q,A)=\sum_{n,k,R_1,R_2} C_{RR_1R_2}\Sigma_{R_1}(q) N_{R_2,n,k}A^n\Big(q^{-1}-q\Big)^{2k-1}
\ee
where
\be
C_{RR_1R_2}=\sum_{\Delta}{\Phi_R(\Delta)\Phi_{R_1}(\Delta)\Phi_{R_2}(\Delta)\over z_{\Delta}}
\ee
the Clebsh-Gordon coefficients of the symmetric group, $z_{\Delta}$ is the standard symmetric factor of the
Young diagram \cite{Fulton} and $\Sigma_R(q)$ is a monomial non-zero only for the corner Young diagrams
$R = [l-d,1^{d}]$ and is equal to
\be
\Sigma_R(q)=(-1)^dq^{2d-l+1}
\ee
First few terms for $f_R$ and $N_{R,n,k}$ are
\be
f_{[1]}(q,A)=H_{[1]}(q,A)\\
f_{[2]}(q,A)=H_{[2]}(q,A)-{1\over 2}\Big( H_{[1]}(q,A)^2+H_{[2]}(q^2,A^2)\Big)\\
f_{[1,1]}(q,A)=H_{[1,1]}(q,A)-{1\over 2}\Big( H_{[1]}(q,A)^2-H_{[2]}(q^2,A^2)\Big)\\
...
\ee
and
\be
f_{[1]}(q,A)=\sum_{n,k}N_{[1],n,k}\Big(q^{-1}-q\Big)^{2k-1}A^n\\
f_{[2]}(q,A)=\sum_{n,k}\Big(q^{-1}N_{[2],n,k}-qN_{[1,1],n,k}\Big)\Big(q^{-1}-q\Big)^{2k-1}A^n\\
f_{[1,1]}(q,A)=\sum_{n,k}\Big(-qN_{[2],n,k}+q^{-1}N_{[1,1],n,k}\Big)\Big(q^{-1}-q\Big)^{2k-1}A^n\\
...
\ee
We calculate both the Ooguri-Vafa polynomials $f_{[2]}(q,A)$ and the numbers $N_{[2],n,k}$
for all 3-strand knots with no more than 8 crossings in the Appendix. The integrality
of these numbers and their using in product formulas is discussed in \cite{Peng}.
\subsection{"Special" polynomials}
The "special" polynomials are defined \cite{DMMSS} as the limit of ratio of the HOMFLY polynomials and
the quantum dimensions as $q\to 1$:
\be
{\mathfrak{H}}_R^{\cal K}(A) = \lim_{q\rightarrow 1} \frac{H_R^{\cal
K}(q,A)}{S_R^*(q,A)}
\label{speHdef}
\ee
Note that the limit is taken with fixed $A$, and both the HOMFLY polynomial
$H_R$
and the quantum dimension $S_R^*$ are singular behaving as
$(q-q^{-1})^{-|R|}$.
Here $|R|$ is the number of boxes in the Young diagram $R$.
Note that in this limit
\be
\lim_{q\rightarrow 1} S_R(A) = d_RS_{[1]}(A)^{|R|}
\ee
where
\be
d_R=S_R\{p\}\Big|_{p_k=\delta_{k,1}}=\prod_{(i,j)\in R}{1\over h_{i,j}}
\ee
and $h_{i,j}$ is the "hook" length.
The conjectured property of the "special" polynomials reads as \cite{DMMSS,IMMMspe}
\be
{\mathfrak{H}}^{\cal K}_R(A) = \Big({\mathfrak{H}}_{[1]}^{\cal
K}(A)\Big)^{|R|}
\label{spepro}
\ee
and is presumably valid for arbitrary ${\cal K}$ and $R$.
For example,
\be
{\mathfrak{H}}_{[2]}^{3_1}(A) = (2A^2-1)^2, \\
{\mathfrak{H}}_{[1]}^{3_1}(A) = 2A^2-1,
\ee
\be
{\mathfrak{H}}_{[2]}^{4_1}(A) = \Big(A^2-1+A^{-2}\Big)^2, \\
{\mathfrak{H}}_{[1]}^{4_1}(A) = A^2-1+A^{-2}
\ee
etc.
This conjecture is an amusing "dual" of a somewhat similar conjecture
\be
{\mathfrak{A}}_{R}^{\cal K}(q) = {\mathfrak{A}}_{[1]}^{\cal
K}\left(q^{|R|}\right)
\label{alepro}
\ee
for the Alexander polynomial
\be
{\mathfrak{A}}_{R}^{\cal K}(q) = \lim_{A\rightarrow 1}
\frac{H_R^{\cal K}(q,A)}{S_R^*}
\ee
We check these two conjectures for the concrete knots in the Appendix.
Similarly, one can consider the "special" limit of $q\to 1$ for other polynomials, e.g. for the Ooguri-Vafa
polynomials $f_R(q,A)$. The special Ooguri-Vafa polynomials
${\mathfrak{f}}_{R}(A)\equiv\lim_{q\rightarrow 1}{f_R(q,A)\over
S_{[R]}^*}$, however, depend on the representation much less trivially than the "special" and Alexander polynomials
(see the Appendix for examples). Note that ${\mathfrak{f}}_{[2]}(A)=-
{\mathfrak{f}}_{[11]}(A)$.
\subsection{Framing factor\label{ff}}
In this text we assume that the ${\cal R}$-matrix is normalized so
that in the channel $Q\in R\otimes R$ its eigenvalue is equal to
$\pm q^{\varkappa_Q}$ and is independent of $R$. This simplifies our
formulas, and this is important for their extension beyond the
topological locus (\ref{tolo}). However, instead this breaks some
properties, important for the knot theory, including topological
(ambient isotopy) invariance. Still, this difference is very easy to
take into account by adding an overall
factor, which is simple, but depends on representation and even on
the rank of the group $SU(N)$. This factor is also important in the
definition of the Ooguri-Vafa polynomials\footnote{For the fully symmetric
knots like
the figure eight $4_1$ with the vanishing writhe number the
factor is unity and the Ooguri-Vafa polynomials can be easily extended
beyond the topological locus \cite{IMMMfe}. For generic knots this
extension needs a separate discussion. } and is ambiguously determined
due to the freedom in choosing the framing \cite{MV}. We choose
the standard, or canonical framing.
Then, the ${\cal R}$-matrix, which
is adequate for knot theory calculations is actually normalized
differently:
\be
{\cal R}_{R\otimes R}^{norm} = A^{-|R|}
q^{-4\varkappa_R} {\cal R}_{R\otimes R} \label{normcon}
\ee
This
means that all our answers for the HOMFLY polynomials should be
multiplied by the additional factor
\be
H_R^{\cal B}\
\longrightarrow\ H_R^{\cal K} =
\Big(A^{|R|}q^{4\varkappa_R}\Big)^{-w^{\cal B}} H^{\cal B}_R\{p^*_k\}
\label{normH}
\ee
where $w^{\cal B}$ is the algebraic number of
intersections in the braid ${\cal B}$ called {\it the writhe
number}. We illustrate the significance of this factor by three
examples. First of them concerns the topological invariance, the second
one the identity (\ref{alepro}) for the Alexander polynomials: in
both cases the additional factor is essential. The third
example demonstrates that (\ref{normH}) is consistent with existing
literature.
\paragraph{Example 1:} If the torus knot $3_1$ is represented
by the $2$-strand braid $[2,3]=(1)^3=(1,1,1)$, then one gets for the
HOMFLY polynomial
\be
\frac{q^3\ {^*\!S}_{[2]} - q^{-3}\
{^*\!S}_{[11]}} { {^*\!S}_{[1]}} = \frac{q^3(Aq-A^{-1}q^{-1}) -
q^{-3}(Aq^{-1}-A^{-1}q)}{q^2-q^{-2}} = A(q^2+q^{-2}) - A^{-1}
\ee
If
the same knot is represented by the $3$-strand braid $[3,2] =
(1,1)^2=(1,1|1,1)$, one gets instead
\be
\frac{q^4\ {^*\!S}_{[3]} -
\tr_{2\times 2}{\Big(\hat{\cal R}\hat U\hat{\cal R}\hat
U^{\dagger}\Big)^2} {^*\!S}_{[21]} + q^{-4}\ {^*\!S}_{[111]} }{
{^*\!S}_{[1]} } = A^2(q^2+q^{-2}) - 1
\ee
Clearly, these two
expressions do not coincide and differ by a factor of $A$, which is
exactly taken into account by the correction factor (\ref{normH}),
because $w^{[3,2]}=4$, while $w^{[2,3]}=3$, and in this example
$\varkappa_{[1]}=0$. For $p=2$ the two HOMFLY polynomials,
calculated in this paper, differ by a factor of $A^2q^4$, which is again
nicely eliminated by (\ref{normH}), because $\varkappa_{[2]}=1$. Note that in the Appendix
we choose the opposite orientation for the trefoil: $(-1,-1,-1)$ in order to better match formulas
from the standard knot tables.
\paragraph{Example 2:} In fact, the Alexander polynomials made from our
extended HOMFLY polynomials,
\be
{\mathfrak A}^{[2,n]}_{[p\,]}(q) =
\lim_{A\rightarrow 1} \frac{q^{p(2p-1)n}\ \SS_{[2p\,]} -
q^{p(2p-3)n}\ \SS_{[2p-1,1]}}{\SS_{[p]}}
\ee
(all other Young
diagrams from the decomposition of $[p\,]\otimes[p\,]$ do not
contribute at $A=1$), satisfy
\be
{\mathfrak{A}}^{[2,n]}_{[p\,]}(q)\
=\ q^{2p(p-1)n}\ {\mathfrak{A}}^{[2,n]}_{[1]}(q^p)\ = \
q^{4\varkappa_{[p\,]}\cdot w^{[2,n]}}\
{\mathfrak{A}}^{[2,n]}_{[1]}(q^p)
\ee
rather than (\ref{alepro}).
Unwanted factors in this relation are eliminated after the
factor (\ref{normH}) is taken into account.
\paragraph{Example 3:} As we
know from (\ref{chicorr}), the torus polynomial character expansion
of \cite{chi} based on use of the Adams operation, and also
suitable for continuation from the topological locus (\ref{tolo}) to the
entire space of time-variables, differs from ours by a factor of
$q^{-n(m-2)p(p-1)}$. The normalized HOMFLY obtained from ours by
the rule (\ref{normH}) should, therefore, differ from that one by a
factor of $q^{-2n(m-1)p(p-1)}\cdot q^{n(m-2)p(p-1)} = q^{-nmp(p-1)}$
(since $w^{[m,n]} = (m-1)n$ and $\varkappa_{[p\,]}=
\frac{p(p-1)}{2}$):
\be
H_{[p\,]}^{[m,n]} = A^{-(m-1)np}
q^{-mnp(p-1)} \underline{H}^{m,n}_{[p\,]} =
A^{-(m-1)n|R|}q^{-2mn\varkappa_{[p\,]}} \underline{H}^{m,n}_{[p\,]}
\ee
This is exactly the factor used in \cite{chi} for arbitrary
representation $R$. It can deserve noting that $mn$ is {\it not} the
writhe number of the braid associated with the torus knot, and
coefficient $2$ is different from $4$ in (\ref{normH}).
\subsection{Cabling}
The standard way to obtain the colored HOMFLY polynomials is to extract them from
those in the fundamental representation, but for different knots and links.
Namely, if one needs ${\cal H}^{\cal K}_R$, one considers instead
${\cal H}^{{\cal K}^{|R|}}_{[1]}$, where ${\cal K}^{|R|}$ is the {\it cabling}
of the knot ${\cal K}$, obtained by substituting the knot with a set of $|R|$
parallel ones (a "cable"),
i.e. actually a knot ${\cal K}$ is substituted by an $|R|$-component link.
However, to extract information about an arbitrary $R$ of a given size $|R|$
one should also allow additional intertwinings of the wires inside each cable,
which decreases the number of components in the link,
so that ${\cal K}^{|R|}$ is actually a linear combination of several links,
made in this way from the $|R|$-cabled ${\cal K}$.
If the knot ${\cal K}$ is represented by an $m$-strand braid,
the cabling involves $m|R|$ strands.
Since general formulas are known \cite{II}
for arbitrary $r$-strand knots in the fundamental representations,
one can actually demonstrate how the cabling procedure works for arbitrary
$2$-strand knots in symmetric and antisymmetric representations.
For the $3$-strand knots in these representations or for $2$-strand knots
in representations $[3],\ [21],\ [111]$ one needs the knowledge of the
$6$-strand knots in the fundamental representation, which is still not
available in full generality.
Thus, in the rest of this section we rederive ${\cal H}^{[2,n]}_{[2]}$
and ${\cal H}^{[2,n]}_{[11]}$ from ${\cal H}_{[1]}^{[2,n]^2}$.
Cabling is a tedious, but well known and widely used procedure,
we add this subsection for the sake of completeness.
\subsubsection*{Cabling the unknot}
Our first example is actually the $1$-strand knot: the unknot.
The $2$-cabling of a 1-strand braid implies that it is substituted with a $2$-strand one:
the two unlinked unknots and the answer is
\be
{H}_{[1]}^{[2,0]}\{p_k\} = S_{[1]}^2\{p_k\} = S_{[2]}\{p_k\} + S_{[11]}\{p_k\}
= {H}_{[2]}^{[1,0]}\{p_k\} + {H}_{[11]}^{[1,0]}\{p_k\}
\ee
a linear combination of unknot polynomials in two representations of the size $|R|=2$.
Similarly, the untwisted $p$-cabling gives a linear combination
\be
{H}_{[1]}^{[p,0]}\{p_k\} = S_{[1]}^{p}\{p_k\}
= \sum_{R: \ |R|=p} {\cal H}_R^{[1,0]}\{p_k\}
\ee
To extract the individual HOMFLY polynomials for two representations $[2]$ and $[11]$
one needs to consider not only the two non-intersecting strands,
but also to allow one intertwining.
One would naturally assume that one extra intersection just provides
$S_{[2]}-S_{[11]}$, but this is the case only for $q=1$.
If one associates an extra $R$-matrix with this additional intersection,
one gets $q$-dependent factors:
\be
{H}_{[1]}^{[2,1]}\{p_k\} \ \stackrel{(\ref{H2n})}{=}\
q S_{[2]}\{p_k\} - q^{-1} S_{[11]}\{p_k\} =
q{H}_{[2]}^{[1,0]}\{p_k\} - q^{-1}{H}_{[11]}^{[1,0]}\{p_k\}
\ee
and finally the cabling of the unknot implies
\be
{H}_{[2]}^{[1,0]}\{p_k\} = \frac{1}{1+q^2}{H}_{[1]}^{[2,0]}\{p_k\}
+ \frac{q}{1+q^2}{H}_{[1]}^{[2,1]}\{p_k\}, \\
{H}_{[11]}^{[1,0]}\{p_k\} = \frac{q^2}{1+q^2}{H}_{[1]}^{[2,0]}\{p_k\}
- \frac{q}{1+q^2}{H}_{[1]}^{[2,1]}\{p_k\}
\label{projt}
\ee
If one restricts the answer to the topological locus {\it and restore the factors},
see s.\ref{ff}, to make contact with the standard calculations, one
would write the same relations as follows:
\be
{H}_{[2]}^{[1,0]}(A|q) =
\frac{1}{1+q^2} {{H}}_{[1]}^{[2,0]}(A|q)
+ \frac{qA}{1+q^2} \Big( A^{-1}{H}_{[1]}^{[2,1]}(A|q)\Big), \\
{H}_{[11]}^{[1,0]}(A|q) =
\frac{q^2}{1+q^2} {{H}}_{[1]}^{[2,0]}(A|q)
- \frac{qA}{1+q^2} \Big( A^{-1}{H}_{[1]}^{[2,1]}(A|q)\Big)
\label{projA}
\ee
Formulas (\ref{projt}) and (\ref{projA}) actually define {\it the projectors},
specifying the linear combinations of cabled knots with additional twistings \cite{chi},
which select particular representations $[2]$ and $[11]$.
Since they are actually independent of the knot,
the same projectors are used for the same purpose below,
when we switch to a little more interesting examples of $2$-cabling the
$2$-strand knots.
\subsubsection*{2-cabling the 2-strand knots}
A new thing as compared to the previous subsection is that one has
intersections in original 2-strand braid (there were none in the 1-strand case).
Each ${\cal R}$-matrix at the $2$-strand crossing is substituted by
four ${\cal R}$-matrices, after lifting to $4$ strands:
\be
{\cal R} \longrightarrow \Big({\cal R}\otimes I \otimes I\Big)
\Big({\cal R}\otimes{\cal R}\Big) \Big(I\otimes I\otimes {\cal R}\Big)
\ee
so that the $2$-strand braid $[2,n]$ is lifted to a
$4$-strand braid of the type
$(0,1,1|1,1,0)^n$.
Moreover, to separate representations $[2]$ and $[11]$ one also needs to
allow one twisting between the first two and the last two braids,
i.e. to consider the four slightly different links/knots
\be
(0,1,1|1,1,0)^n, \ \ \ (0,1,1|1,1,0)^n (1,0,0), \ \ \
(0,1,1|1,1,0)^n(0,0,1), \ \ \ (0,1,1|1,1,0)^n(1,0,1)
\ee
Making use of projectors (\ref{projt}) and (\ref{projA}), one gets,
in somewhat compressed notation:
\be
q^{-2n}{H}_{[2]}^{[2,n]}\{p_k\} = \frac{1}{(1+q^2)^2}
\Big({H}_{[1]}^{[4,(000),n]}\{p_k\}
+ q {H}_{[1]}^{[4,(001),n]}\{p_k\} + q {H}_{[1]}^{[4,(100),n]}\{p_k\}
+ q^2 {H}_{[1]}^{[4,(101),n]}\{p_k\}\Big), \\
q^{2n}{H}_{[11]}^{[2,n]}\{p_k\} = \frac{1}{(1+q^2)^2}
\Big(q^4{H}_{[1]}^{[4,(000),n]}\{p_k\}
- q^3 {H}_{[1]}^{[4,(001),n]}\{p_k\} - q^3 {H}_{[1]}^{[4,(100),n]}\{p_k\}
+ q^2 {H}_{[1]}^{[4,(101),n]}\{p_k\}\Big)
\label{combs4}
\ee
According to \cite{II}, substituting the peculiar braid $(a_1b_1c_1|a_2b_2c_2|a_3b_3c_3|\ldots)
= (\underbrace{011|110|\ldots|011|110}_{n \ {\rm times}})$
into the general formula \cite[eq.(65)]{II} for the 4-strand extended HOMFLY polynomials gives
\be
{H}_{[1]}^{[4,(000),n]} = {H}_{[1]}^{[4,(011|110)^n(000)]}
= q^{4n}S_{[4]} + \tr_{3\times 3} \Big(\hat{\cal R}_{[31]}
\hat V_{[31]}\hat U_{[31]} \hat{\cal R}_{[31]}
\hat U_{[31]}^\dagger \hat V_{[31]}^\dagger\hat U_{[31]}^\dagger
\hat{\cal R}_{[31]}\hat U_{[31]} \hat{\cal R}_{[31]} \Big)^n \cdot S_{[31]} + \\
+ \left(q \longleftrightarrow -\frac{1}{q}\right)
+ \tr_{2\times 2} \Big( \hat{\cal R}_{[22]} \hat U_{22}^\dagger \hat{\cal R}_{[22]}^2
\hat U_{[22]}\hat{\cal R}_{[22]} \Big)^n \cdot S_{[22]}
\label{000}
\ee
and for the other three twisted cablings:
{\footnotesize
\be
{H}_{[1]}^{[4,(100),n]}
= q^{4n+1}S_{[4]} + \tr_{3\times 3} \left\{\hat U_{[31]} \Big(\hat{\cal R}_{[31]}
\hat V_{[31]}\hat U_{[31]} \hat{\cal R}_{[31]}
\hat U_{[31]}^\dagger \hat V_{[31]}^\dagger\hat U_{[31]}^\dagger
\hat{\cal R}_{[31]}\hat U_{[31]} \hat{\cal R}_{[31]} \Big)^n
\hat U_{[31]}^\dagger\hat{\cal R}_{[31]}
\right\} \cdot S_{[31]} + \\
+ \left(q \longleftrightarrow -\frac{1}{q}\right)
+ \tr_{2\times 2}\left\{ \Big( \hat{\cal R}_{[22]} \hat U_{22}^\dagger \hat{\cal R}_{[22]}^2
\hat U_{[22]}\hat{\cal R}_{[22]} \Big)^n
\hat U_{[22]}^\dagger\hat{\cal R}_{[22]}\hat U_{[22]}\right\}\cdot S_{[22]}, \\
{H}_{[1]}^{[4,(001),n]}
= q^{4n+1}S_{[4]} + \tr_{3\times 3}\left\{ \Big(\hat{\cal R}_{[31]}
\hat V_{[31]}\hat U_{[31]} \hat{\cal R}_{[31]}
\hat U_{[31]}^\dagger \hat V_{[31]}^\dagger\hat U_{[31]}^\dagger
\hat{\cal R}_{[31]}\hat U_{[31]} \hat{\cal R}_{[31]} \Big)^n
\hat V_{[31]}\hat U_{[31]}\hat{\cal R}_{[31]}\hat U_{[31]}^\dagger\hat V_{[31]}^\dagger
\right\} \cdot S_{[31]} + \\
+ \left(q \longleftrightarrow -\frac{1}{q}\right)
+ \tr_{2\times 2}\left\{ \Big( \hat{\cal R}_{[22]} \hat U_{22}^\dagger \hat{\cal R}_{[22]}^2
\hat U_{[22]}\hat{\cal R}_{[22]} \Big)^n \hat U_{[22]}^\dagger
\hat{\cal R}_{[22]}\hat U_{[22]}
\right\} \cdot S_{[22]}, \\
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
{H}_{[1]}^{[4,(101),n]}
= q^{4n+2}S_{[4]} + \tr_{3\times 3} \left\{\Big(\hat{\cal R}_{[31]}
\hat V_{[31]}\hat U_{[31]} \hat{\cal R}_{[31]}
\hat U_{[31]}^\dagger \hat V_{[31]}^\dagger\hat U_{[31]}^\dagger
\hat{\cal R}_{[31]}\hat U_{[31]} \hat{\cal R}_{[31]} \Big)^n
\hat U_{[31]}^\dagger\hat{\cal R}_{[31]}\hat U_{[31]}
\hat V_{[31]}\hat U_{[31]}\hat{\cal R}_{[31]}\hat U_{[31]}^\dagger\hat V_{[31]}^\dagger
\right\}\cdot S_{[31]} + \\
+ \left(q \longleftrightarrow -\frac{1}{q}\right)
+ \tr_{2\times 2}\left\{ \Big( \hat{\cal R}_{[22]} \hat U_{22}^\dagger \hat{\cal R}_{[22]}^2
\hat U_{[22]}\hat{\cal R}_{[22]} \Big)^n
\hat U_{[22]}^\dagger\hat{\cal R}_{[22]}^2\hat U_{[22]}
\right\}
\cdot S_{[22]}
\label{111}
\ee
}
\!\!Now let us look at the coefficient in front of $S_{[4]}$.
The two linear combinations, corresponding to (\ref{combs4})
for this coefficient give just
\be
\frac{q^{4n} + 2q\cdot q^{4n+1} + q^2\cdot q^{4n+2}}{(1+q^2)^2} = q^{4n}, \\
\frac{q^4\cdot q^{4n} - 2q^3\cdot q^{4n+1} + q^2\cdot q^{4n+2}}{(1+q^2)^2} = 0
\ee
Similarly, for two linear combinations in front of $S_{[22]}$ one has
\be
\frac{ (q^{2n}+q^{-2n}) + 2q\cdot(-q^{2n-1}+q^{1-2n})
+ q^2\cdot(q^{2n-2}+q^{2-2n})}{(1+q^2)^2} = q^{-2n}, \\
\frac{ q^4\cdot(q^{2n}+q^{-2n}) - 2q^3\cdot(-q^{2n-1}+q^{1-2n})
+ q^2\cdot(q^{2n-2}+q^{2-2n})}{(1+q^2)^2} = q^{2n}
\ee
and for those in front of $S_{[31]}$ the intermediate expressions are
different for knots and links:
for $n$ odd
\be
\frac{ -1 + 2q\cdot(-q )
+ q^2\cdot(-q^2)}{(1+q^2)^2} = -1 \\
\frac{ q^4\cdot(-1) - 2q^3\cdot(-q )
+ q^2\cdot(-q^2)}{(1+q^2)^2} = 0
\ee
while for $n$ even
\be
\frac{ (2q^{2n}+1) + 2q\cdot(q^{2n+1}-q^{2n-1}+q)
+ q^2\cdot(q^2-2q^{2n})}{(1+q^2)^2} = 1 \\
\frac{ q^4\cdot(2q^{2n}+1) - 2q^3\cdot(q^{2n+1}-q^{2n-1}+q)
+ q^2\cdot(q^2-2q^{2n})}{(1+q^2)^2} = 0
\ee
Thus, one finally obtains
\be
q^{-2n} H^{2,n}_{[2]}
= q^{4n} S_{[4]} \mp S_{[31]} + q^{-2n}S_{[22]} \\
q^{2n}H^{2,n}_{[11]}
= q^{-4n} S_{[1111]} \mp S_{[211]} + q^{2n}S_{[22]}
\label{H2H11}
\ee
which coincides with (\ref{H2n}).
\section{Summary}
In this paper we continued our program of constructing simple matrix expressions for the colored HOMFLY
polynomials of arbitrary knots/links started in \cite{I,II}. In practice, we always deal with braid representations
of knots. Here we considered the symmetric and antisymmetric representations $[2]$ and $[1,1]$ for 3-strand braids.
One can construct the result inductively, using the representation group theory, however, in this paper we used an
indirect way of using the known answers for the torus knot/link polynomials in order to restore all the necessary
ingredients (in particular, the mixing matrices) for the generic answer. We return to using the group theory
approach elsewhere \cite{IV}.
Using the formulas, obtained in this paper (we listed various knot polynomials for the knots that can be described
by 3-strand braids with no more than 8 crossings in the Appendix) we tested various conjectures, from the Ooguri-Vafa
conjecture \cite{OV} and its generalization \cite{LMV} to the conjecture of the representation dependence of the
"special" polynomials. The HOMFLY polynomials calculated in the paper were partly obtained earlier in a series of papers
by the Indian group \cite{Rama00,inds} within a different though close approach. In these cases our results confirm
these earlier calculations.
The results presented here are substantially extended in \cite{IV} to include higher symmetric representations
but this requires a deeper insight into the structure of the mixing matrices and, hence, is beyond the scope of the
present paper.
\section*{Note added}
After this paper was published there appeared a paper \cite{Rama}
with calculations of the HOMFLY polynomials in the first symmetric representation and of the corresponding
Ooguri-Vafa polynomials for various knots and links. Their results for the 3-strand knots coincide with formulas
of this paper for the only exception of the HOMFLY polynomial for knot $7_5$ where we made a misprint (the Ooguri-Vafa
polynomial was written in our paper correctly). We are grateful to the authors of \cite{Rama} for the correction.
\section*{Acknowledgements}
Our work is partly supported by Ministry of Education and Science of
the Russian Federation under contract 14.740.11.0081, by NSh-3349.2012.2,
by RFBR grants 10-02-00509 (A.Mir.), 10-02-00499 (A.Mor.), 11-02-01220 (And.Mor.) and
by joint grants 11-02-90453-Ukr, 12-02-91000-ANF,
11-01-92612-Royal Society.
The research of H.~I.~ and A.Mir.
is supported in part by the Grant-in-Aid for Scientific Research (23540316)
from the Ministry of Education, Science and Culture, Japan, and that of A.Mor. by
by JSPS Invitation Fellowship Program for Research in Japan (S-11137).
Support from JSPS/RFBR bilateral collaboration "Synthesis of integrabilities
arising from gauge-string duality" (FY2010-2011: 12-02-92108-Yaf-a) is gratefully appreciated.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,378 |
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} | 9,793 |
\section{Introduction}
The \gls{vc} task consists of modifying a speech signal uttered by some source speaker as another target speaker uttered it. Cross-lingual \gls{vc} allows using different source and target languages. In such a task, the linguistic information is preserved, but speaker-dependent features are changed, which requires semantic reasoning about the input signal.
\Gls{vc} is an inherently ill-posed problem; there are multiple correct outputs. Even if it is easy for humans to identify the concepts of content and style (assessed as naturalness and speaker similarity), it is difficult to quantify the conversion's overall speech quality. On the one hand, the lack of objective measures hinders choosing a training strategy and an objective function. On the other hand, most of the works in \gls{vc} only report subjective scores. Some works on parallel \gls{vc} report objective metrics such as the Root Mean Square Error (RMSE) \cite{rmse}, but those are not always correlated with human perception \cite{sisman2020overview}. This is especially the case for non-parallel scenarios, where even \gls{pesq}, which predicts the human-perceived speech quality with respect to a target signal, does not correlate with subjective scores (see \cref{sec:objective}).
The subjective score indicates how a system works and may be biased depending on the test setup. Moreover, it is impossible to compare available systems evaluated only by different evaluators' subjective tests and under different conditions. The \gls{vc} Challenge\footnote{\url{http://www.vc-challenge.org/}} circumvents this problem by providing a common evaluation dataset and performing large-scale crowd-sourced perceptual evaluations.
\gls{vc} requires the factorization of speech into linguistic and non-linguistic information. Therefore, one natural approach is to use representations that lack speaker information, which are well studied in the speech recognition domain. Such representations are then concatenated with the information of the target speaker (studied in the speaker identification and verification domains) and mapped to the waveform domain yielding the \gls{vc} output. One example of such an approach achieving high-quality conversions consists of first transcribing audio to text using an \gls{asr} system and then using a \gls{tts} model conditioned on the target speaker and the obtained transcription \cite{hayashi2019espnettts, inaguma2020espnet, watanabe2018espnet}.
Another common approach when dealing with non-parallel data is to train an \gls{ae} model in speech reconstruction and enforce speaker independence in the latent representations. Such latent features are then used in the same fashion as the speaker-independent representations of the former approach. This method can suppress the need for annotated data and find representations that potentially preserve para-linguistical information.
In line with the \gls{vc} Challenge 2020, this work uses non-parallel data for the \gls{vc} model training. In particular, the proposed method is based on conditional \glspl{ae} as in AutoVC~\cite{autovc}, but with focus on fast conversions.
\section{Related Works}
The approaches using speaker-independent representations leverage pre-trained models for its computation. In general, these systems require large amounts of transcribed data, whose collection is very costly and time-consuming \cite{Pascual2019}. Moreover, such speaker-independent features usually lack para-linguistical information such as the intonation, which can potentially lead to conversions having different meanings in some cases. In the case of using text, the timing information is also lost, and an additional mapping to phonetic transcriptions is needed in cross-lingual settings.
Speaker independence can be explicitly enforced with an adversarial setting, where a classifier is trained to predict the speaker from the latent representation. The loss from such classifiers can then be used to learn the mapping from the input signal to the latent space \cite{chou2018multitarget,starganvc}.
One can also implicitly enforce speaker independence by reconstructing the speech from the low-dimensional latent representation and uncompressed speaker information. Speaker disentanglement, in this case, follows from the redundancy principle \cite{barlow-redundancy}. Since the waveform generator is explicitly conditioned on the speaker identity, the feature extractor does not have to capture speaker-dependent information in the latent features. The desired speaker independence is achieved if the dimension of the latent space satisfies the following trade-off. On the one hand, it has to be sufficiently small to factor out the speaker's information. On the other hand, it has to be large enough to allow for perfect reconstruction and capture as much of the input data as possible.
AutoVC achieves a speaker-independent latent space by implementing the previous approach with a simple conditional \gls{ae}. The encoder network of the \gls{ae} applies an information bottleneck that both reduces the number of features and downsamples the signal in the temporal dimension.
CycleVAE \cite{cyclevae} models the information bottleneck with a \gls{vae}, which means that the latent features are enforced to follow a known distribution. \gls{vq}-\gls{vae} \cite{vq-vae} uses \gls{vq} as an additional information bottleneck on the latent features obtained with a \gls{vae}. Time-jitter regularization, consisting of replacing each latent vector with either one or both of its neighbors, was also shown to be useful as an additional information bottleneck in top of the former approach~\cite{unsupervised-representation}. This regularization helps to model the slowly-changing phonetic content by avoiding the use of latent vectors as individual units. The authors claimed that the latent representations found by \glspl{ae} do not factor the speaker out. Instead, their findings showed that a \gls{vae} or its \gls{vq} version was required to achieve speaker disentanglement. However, \cite{spk-encoder} showed that conditional \glspl{ae} with speaker-dependent encoders effectively achieve the desired speaker-independent latent representations.
\section{FastVC}
FastVC is an end-to-end model that performs fast many-to-many \gls{vc} and is trained using non-parallel data. This system performs \gls{vc} by learning a mapping between the source and converted waveforms. This latter has the same linguistic information as the source speech but different speaker information. In particular, FastVC learns this mapping with a conditional \gls{ae} framework that is trained on the reconstruction of Mel-spectrograms similarly to \cite{autovc}. FastVC performs \gls{vc} with the three-stage model depicted in \cref{fig:fastvc}. Its \gls{ae} module is depicted in \cref{fig:fastvc_autoencoder}.
\begin{figure*}[ht]
\centering
\includegraphics[width=.9\textwidth]{img/fastvc.pdf}
\caption[FastVC model architecture.]{FastVC model architecture during conversion mode. During training, both the the source and target speaker identities are the same.}
\label{fig:fastvc}
\end{figure*}
\begin{figure*}[ht]
\centering
\includegraphics[width=.8\textwidth]{img/fastvc_autoencoder.pdf}
\caption[AutoEncoder module of FastVC.]{Diagram of the \gls{ae} module for FastVC during conversion mode. The \texttt{AutoEncoder} comprises the \texttt{Encoder} and the \texttt{Decoder}, but also the \texttt{PostNet}. The \texttt{PostNet}, which is proposed in \cite{autovc}, builds the finer details of the spectrogram, which is excessively smooth before this module.}
\label{fig:fastvc_autoencoder}
\end{figure*}
Equally to AutoVC~\cite{autovc}, FastVC uses log-scale Mel-spectrograms with 80 Mel channels as inputs. However, the Mel-spectrogram module is a~\gls{cnn}-based learnable module and not a fixed transformation as in AutoVC. This module can be trained if desired and is initialized to provide exact Mel-spectrograms. This allows using raw speech waveforms as input instead of Mel-spectrograms.
The theoretical guarantees justifying the \gls{vc} capabilities of \cite{autovc} hold under the assumption that the speaker embeddings of different utterances of the same speaker are the same, and those from different speakers are distinct. In order to satisfy this assumption, FastVC uses one-hot encoded speaker embeddings similarly to \cite{vq-vae}.
In FastVC, both the encoder $E(\cdot,\cdot)$ and the decoder $D(\cdot,\cdot)$ are conditioned on the speaker identity of the source and target speaker as in~\cite{spk-encoder}. This speaker identity is concatenated with the other input signal at every time step. One of the most key design choices in implementing a conversion function for \gls{vc} learned on speech reconstruction is choosing adequate information bottlenecks.
FastVC uses dimensionality reduction in the frequency dimension and temporal downsampling as in \cite{autovc}. The latent features are then upsampled to match the original time rate using a causal variant of the nearest neighbor interpolation technique. This can be seen as a causal version of the time-jitter regularization proposed in \cite{unsupervised-representation}, with the time jitter as a hyper-parameter that corresponds to the downsampling factor.
The information bottleneck introduces two pivotal hyperparameters for the speaker disentanglement; the latent features' dimension and the downsampling factor. In particular, FastVC doubles the temporal downsampling factor with respect to \cite{autovc}. This design choice achieves speaker-independent phoneme-like latent features that solve the pitch inconsistency problems of AutoVC reported in \cite{autovc2}. For more details on this, refer to \cite{barbany_msc}.
FastVC generates speech with a sampling rate of 22050 Hz, which makes \gls{ar} models unsuitable, especially if fast conversions are desired. \cite{autovc} used WaveNet~\cite{wavenet} conditioned on the log-scaled Mel-spectrogram as a generative model for raw speech. To achieve fast inference, FastVC resorts to using a non-\gls{ar} generative model. This design choice is the main reason for the fast conversions obtained with this approach. In particular, the Mel-spectrogram inverter is chosen to be MelGAN, introduced by \cite{melgan}.
\section{Experimental setup}
\subsection{Datasets}
FastVC only trains on raw speech waveforms and speaker identities. This means that it does not requires any additional annotation. The main dataset used for this project is the \gls{vctk} described in \cite{vctk}. The \gls{vctk} dataset is chosen as the main dataset for its widespread use for the \gls{vc} task \cite{vq-vae,chou2018multitarget,autovc,autovc2}.
The model comparison of the \gls{vc} Challenge is performed with samples generated using the dataset of the same Challenge. The use of the Challenge training dataset is essential but not enough to train a model such as FastVC. In this project, the \gls{vctk} and \gls{vc} Challenge datasets were simply merged, which is allowed in the Challenge.
The TIMIT dataset \cite{timit} is chosen for the latent features' analysis (see \cref{sec:latents}). This dataset is public and contains speech data and hand-verified time-aligned phoneme transcriptions. In particular, only the test partition of this dataset is used to avoid incorporating many new speakers.
\subsection{Training}
\glspl{gan} are notoriously difficult to train, with mode collapse and oscillations being a common problem \cite{liang2018generative}. For this reason, the basic FastVC model (denoted FastVC in \cref{tab:pesq}) uses the pre-trained weights for MelGAN provided by \cite{melgan}. The \gls{ae} module in FastVC is trained from scratch to match the conditioning signal required by the Mel-spectrogram inverter. The basic version of FastVC is obtained by only training the \gls{ae} module using the ADAM optimizer \cite{adam} with a learning rate of 0.001, $\beta_1=0.9$, and $\beta_2=0.99$ for 200 epochs. The training objective for this setting is presented in \eqref{eq:basic_loss}, where $\mX$ is the input Mel-spectrogram, $\ss$ its correspondent speaker and $\hat{\mX}$ the \gls{ae} output.
\begin{align}
\cL_{content} = \norm{E(\mX,\ss) - E(\widehat{\mX},\ss)}^2
\label{eq:content}
\end{align}
\begin{align}
\min \E_{\mX,\ss}\big[&\norm{\mX-\widehat{\mX}}^2 + \norm{\mX-D(E(\mX,\ss),\ss)}^2 +\cL_{content} \big]
\label{eq:basic_loss}
\end{align}
\emph{FastVC with end-to-end training}: to allow the model to use information that may be not included in the Mel-spectrogram and generate more efficient representations for the task of \gls{vc}, FastVC also allows end-to-end training. The weights obtained in the only-\gls{ae} training are used as a starting point. In this setup, FastVC behaves as the generator of a \gls{gan}, which takes raw speech as input.
FastVC uses multiple discriminators that run at different rates, as proposed in \cite{melgan}. To ensure that the linguistic information is captured, the content loss term \eqref{eq:content} is added to the generator objective in \cite{melgan}. The content loss term enforces the codes of the original and converted speech to be the same, i.e., it enforces \gls{vc} to be idempotent. We speculate that this is enough to achieve quality speech that preserves the lexical content.
The content loss is weighted by a factor of $20$ and added to the generator's total objective. The regularization amount is chosen to ensure that the losses have the same order of magnitude, and they both decrease individually. For this setting, ADAM is also used as the optimization algorithm. In this case, however, with a learning rate of $10^{-4}$, $\beta_1=0.5$, and $\beta_2=0.9$ for 200 epochs. These specific values are suggested in \cite{gan_training} to train \glspl{gan} with ADAM, and also used in \cite{melgan}.
All the experiments use a batch size of 16, and the merged dataset is randomly split into 90\% for training and 10\% for testing purposes. During training, FastVC is fed chunks of 8192 samples of speech sampled at 22050 Hz, while in inference, the model's input is the whole waveform.
\section{Results}
FastVC converts voices $4\times$ faster than real-time, and $500\times$ faster than AutoVC, measured on Intel(R) Core(TM) i7-8700K at 3.70GHz.
\subsection{Objective assessment}
\label{sec:objective}
One of the main difficulties in building \gls{vc} models is that there are no standardized objective measures. The lack of such metrics hinders the system comparison and the performance of ablation studies. The variant of FastVC submitted to the \gls{vc} Challenge was chosen based on the value given by \gls{pesq}, an objective method that rates the speech quality by predicting the \gls{mos}.
The fact that \gls{pesq} is not used as a standardized measure to substitute \gls{mos} is that the former requires both the desired waveform and the one generated with the evaluated system. The approach that FastVC takes to deal with non-parallel data is to learn the conversion function on the task of speech reconstruction. In this case, the \gls{pesq} measure is more suited since self-reconstruction was learned during training, and mapping to the same speaker is a valid \gls{vc} instance.
\cref{tab:pesq} shows the obtained results. Note that the reconstruction performance alone is not a useful metric to evaluate a \gls{vc} system because it does not measure the speaker's disentanglement. In particular, perfect reconstruction can be achieved if there is no information bottleneck and thus the latent features are not speaker-independent. Therefore, this metric should be used for systems with speaker-independent inputs.
\begin{table}[t]
\centering
\begin{tabular}{p{5.4cm}p{1.6cm}}
\toprule
Experiment & PESQ\\
\midrule
AutoVC -- baseline \cite{autovc} & $\mathbf{2.56 \pm 0.23}$ \\
\midrule
FastVC with information bottleneck proposed in~\cite{autovc} & $2.57 \pm 0.25$\\
FastVC with 10 Hz latent features & $2.61 \pm 0.24$\\
FastVC with adversarial speaker classifier & $2.62 \pm 0.24$\\
FastVC (VCC20 submission) & $\mathbf{2.68 \pm 0.22}$\\
\midrule
FastVC with end-to-end training & $1.56 \pm 0.29$ \\
FastVC with learnable Mel-spectrogram & $1.50 \pm 0.40$ \\
PhonetVC (\cref{sec:phonetvc}) & $\mathbf{1.67 \pm 0.30}$\\
\bottomrule
\end{tabular}
\caption{Objective results performed over 100 utterances of less than 5 seconds from the test partition. The reported values are the mean and the standard deviation of the sample. First three FastVC variants are described later in \cref{sec:latents}.}
\label{tab:pesq}
\vspace{-2em}
\end{table}
FastVC with end-to-end training performs worse in terms of \gls{pesq} than FastVC. This can be justified because, in the end-to-end training, the aim is not to match the input Mel-spectrogram but to maximize the \gls{gan} objective. A future subjective evaluation would be needed to confirm if the \gls{pesq} also correlates with the perceived quality in such cases.
The use of this metric with parallel utterances aligned using \gls{dtw} was also explored. However, in this case, the results were inconclusive and not related at all with perceptual scores. The ill-posedness of the problem can justify this; a sound output other than the time-aligned parallel utterance (or the input utterance in the evaluation of reconstruction, especially on the end-to-end case) may be obtained.
\subsection{Subjective assessment}
The subjective scores for the cross-lingual \gls{vc} task are presented in the \gls{vc} Challenge 2020 paper. FastVC is represented with the label \textbf{T15}. You can also compare the \gls{vc} Challenge baselines, AutoVC, and the proposed FastVC at \url{https://barbany.github.io/fast-vc/}.
\subsection{Latent space analysis}
\label{sec:latents}
\subsubsection{Speaker independence}
Prosodic information leaks through the bottleneck of AutoVC, causing the target pitch to fluctuate unnaturally~\cite{autovc2}. To tackle this issue, the authors proposed in \cite{autovc} to remove the speaker identity from the latent representations and the prosodic information. The temporal downsampling factor proposed in \cite{autovc2} matches the design choice of FastVC. With this value, FastVC outputs do not have the unnatural pitch jumps of AutoVC without the need of disentangling the prosodic information from the latent features and introducing the synthetic target prosody. Refer to \cite{barbany_msc} for more details.
FastVC requires that the latent features are speaker-independent, but this is not explicitly enforced. The fact that the encoder disentangles the speaker in an unsupervised fashion can be explained with the redundancy principle \cite{barlow-redundancy}. However, adversarial training of the latent representations as in \cite{chou2018multitarget} could further disentangle the speaker's information and downplay the information bottleneck's design choices.
To confirm if the redundancy principle suffices, a variant of FastVC with an adversarial speaker classifier was implemented. In particular, an adaptation of the discriminator used to achieve class-independent latent representations in \cite{musictranslation} is implemented. The minimax game here is for the encoder to seek class-independent latent features and the classifier to classify them correctly. The classifier is trained with the cross-entropy loss on the speaker labels using the ADAM optimizer with a learning rate of 0.001, $\beta_1=0.9$, and $\beta_2=0.99$. The negative loss, termed as domain confusion loss in \cite{musictranslation}, is added as a regularizer to \eqref{eq:basic_loss} with a weighting of $0.1$ so that each individual objective had the same order of magnitude.
Even if the classifier network was trained simultaneously as FastVC, the prediction accuracy was 0\% when the speaker-independence signal was used with the latent features and when it was not. These results were obtained with a model trained using the 278 speakers resulting from the mix of the \gls{vctk} corpus and the test partition of the TIMIT dataset. The speaker-independence results suggest that the redundancy principle is enough to achieve speaker-independence, which is in line with the results reported in \cite{autovc,autovc2}.
\subsubsection{Phonetic similarity}
Similarly to \cite{vq-vae,unsupervised-representation}, the latent features of FastVC lack speaker information and are potentially similar to phonemes. A perceptron is used to find a hypothetically simple correspondence between phonemes and the latent features. This model is trained using the latent representations extracted from the TIMIT test data with a trained FastVC network. The obtained latent features are randomly split into the train (70\%), validation (10\%), and test (20\%) sets
The information bottleneck on the temporal dimension of FastVC yields a latent representation with a 2.5 Hz rate. This rate is a factor of 10 lower than the rate of the latent features in \cite{vq-vae}. The average phoneme rate is around 10 Hz \cite{inforate_speech,phon_rate}, which means that each latent vector at a given time represents more than one phoneme. For the classification task, each latent vector was assumed to represent the phoneme with a larger intersection in the temporal domain.
The phoneme classifier was trained by minimizing the cross-entropy loss with \gls{sgd} and early stopping on the loss on the validation partition of the dataset containing the latent features from the TIMIT test data. This classifier correctly classified 42.45\% of the latent features. In contrast, a random classifier and a classifier always choosing the prior most likely phoneme on the train partition had an accuracy of 2.44\% and 9.43\%. These results suggest that there is indeed a correspondence between the latent features and phonemes. For comparison, \gls{vq}-\gls{vae} \cite{vq-vae} uses a 128-dimensional discrete space and obtains a classification accuracy of 49.3\%, while choosing the prior most likely phoneme gives a 7.2\%. A classification drop from the results in \cite{vq-vae} is expected due to the lower rate representation and the classifier's simplicity.
Even if the latent representations' low rate suggested that a latent vector represents a combination of sounds rather than a single phoneme, the number of distinct units with groups of phonemes exponentially grows with the group size. This growth implies that there are more classes to predict, and some may not even be seen during training
\subsection{PhonetVC}
\label{sec:phonetvc}
PhonetVC is a variant of the proposed model designed to confirm the benefits of using the latent features obtained by FastVC instead of speaker-independent speech features. PhonetVC uses an estimation of the \gls{pllr} features computed with Phonet~\cite{phonet} instead of the latent representations obtained by the encoder network in \cref{fig:fastvc_autoencoder}. The resulting model works with speech at 16 kHz, and the Decoder, Postnet, and Mel-spectrogram inverter are jointly trained from scratch using the MelGAN training objective \cite{melgan}.
\section{Conclusions}
This work proposed a fast and competitive \gls{vc} system. It is worse at capturing the speaker's style of speakers with little data in comparison to its quality performance (see subjective results in the \gls{vc} Challenge paper). This is justified by the fact that the training dataset is very imbalanced concerning the language, and the performance could be degraded for non-English speakers. One possible approach to tackle the language imbalance problem is to incorporate additional non-English speech datasets to balance the languages. However, the percentage of data per speaker on the \gls{vc} Challenge would be even smaller in this case. A different approach to tackle dataset imbalance is the multi-reader technique described in \cite{dvector}.
\section{Acknowledgements}
We thank Kaizhi Qian for providing the non open-sourced full code of AutoVC used as a starting point for this project.
\balance
\bibliographystyle{IEEEtran}
| {
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Q: Write a function that imports data and calculates summary statistics by variable conditions and writes output files So initially I had the following object:
> head(gs)
year disturbance lek_id complex tot_male
1 2006 N 3T Diamond 3
2 2007 N 3T Diamond 17
3 1981 N bare 3corners 4
4 1982 N bare 3corners 7
5 1983 N bare 3corners 2
6 1985 N bare 3corners 5
With that I computed general statistics: n, min, max, mean, and sd of tot_male for year within complex. I then merged these by year within complex into a single dataset using the following:
gsnew <- gs %>% group_by(year, complex) %>%
summarise(n = length(tot_male), male_min = min(tot_male), male_max = max(tot_male), male_mean = mean(tot_male), male_sd = sd(tot_male))
Resulting in:
> gsnew
Source: local data frame [119 x 7]
Groups: year [?]
year complex n male_min male_max male_mean male_sd
(int) (fctr) (int) (int) (int) (dbl) (dbl)
1 1967 Diamond 2 33 101 67.000000 48.083261
2 1969 Diamond 2 29 69 49.000000 28.284271
3 1970 3corners 1 26 26 26.000000 NA
4 1970 Diamond 4 3 51 26.250000 21.093048
5 1971 3corners 3 6 22 12.333333 8.504901
How would I instead write a general function in the following format
FunctionName=function(Argument1,...,ArgumentN) {Statement1,...,StatementN}
• Argument1-N are any variable from object(s) • Statement1-N are any valid R statements
That allows me to:
• Import the data
• Select from the data a specified year for which statistics are desired;
• Calculate mean, 2SD, n, and 90% Confidence Interval for the specified year within lek complex
• Write the yearly-based output as a separate *.csv file
year complex mean st.dev2 n lo90ci hi90ci
2007 3corners 26.28571 52.04760 7 -393.50827 446.07970
2007 Blue 18.87500 20.15476 8 -40.00856 77.75856
2007 book_cliffs 4.50000 13.19091 6 -24.62443 33.62443
2007 Diamond 13.25000 48.83431 20 -205.38461 231.88461
A: Um, I think you're pretty close. It might look something like this:
read_write = function(file_name, this_year) {
file_name %>%
read.csv %>%
filter(year == this_year) %>%
summarise(n = length(tot_male),
male_min = min(tot_male),
male_max = max(tot_male),
male_mean = mean(tot_male),
male_sd = sd(tot_male),
male_2sd = 2*male_sd,
male_upper_bound = male_mean + 1.645*male_sd,
male_lower_bound = male_mean - 1.645*male_sd) %>%
write.csv("out_" %>% paste0(filename), row.names = false)
}
A: Thanks to @bramtayl
Here is the final code:
> library(dplyr)
> annualleksummary = function(x1) {
+ x1 %>%
+ read.csv %>%
+ filter(tot_male, year == 2007) %>% group_by(year, complex) %>%
+ summarise(n = length(tot_male),
+ male_min = min(tot_male),
+ male_max = max(tot_male),
+ male_mean = mean(tot_male),
+ male_sd = sd(tot_male),
+ male_2sd = 2*male_sd,
+ male_upper_bound = male_mean + 1.645*male_sd,
+ male_lower_bound = male_mean - 1.645*male_sd) %>%
+ write.csv("2007_" %>% paste0(x1), row.names = F)
+ }
> annualleksummary("gsg_leks.csv")
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\section{\large{Introduction and main results}}
The Liouville type theorem is the nonexistence of solutions in the entire space or in half-space. The classical Liouville type theorem stated that a bounded harmonic (or holomorphic) function defined in entire space must be constant. This theorem, known as Liouville theorem, was first announced in 1844 by Liouville \cite{Liouville} for the special case of a doubly-periodic function. Later in the same year, Cauchy \cite{Cauchy} published the first proof of the above stated theorem. This classical result has been extended to nonnegative solutions of the semilinear elliptic equation
\begin{align}\label{equation0}
-\Delta u=|u|^{p-1}u\ \ \ \ \ \mbox{in}\,\, \R^n,\; p>1,
\end{align}
in the whole space $\R^n$ by Gidas and Spruck \cite{Gidas1, Gidas2} see also the paper of Chen and Li \cite{Chen}. They proved that if $1<p< \frac{n+2}{n-2}$, then the above equation only has the trivial solution $u \equiv 0$ and this result is optimal. In an elegant paper, Farina \cite{Farina} proved that nontrivial finite Morse index solutions (whether positive or
sign changing) to \eqref{equation0} exists if and only if $p\geq p_{c}(n)$ and $n\geq 11$, or $ p=\frac{n+2}{n-2}$ and $n\geq 3$, where $ p_{c}(n) $ is the so-called Joseph-Lundgren exponent. The study of stable solutions in the H\'{e}non type elliptic equation: $-\Delta u= |x|^a |u|^{p-1}u,\;\; \mbox{in}\,\, \R^n,\; p>1\; \mbox{and}\; a> -2$ has been studied recently,
Wang and Ye \cite{WY} gave a complete classification of stable weak solutions and those of finite Morse index solutions.
In the past years, the Liouville property has been refined considerably and emerged as one of the most powerful tools in the study of initial and boundary value problems for nonlinear PDEs. It turns out that one can obtain from Liouville-type theorems a variety of results on qualitative properties of solutions such as universal, pointwise, a priori estimates of local solutions; universal and singularity estimates; decay estimates; blow-up rate of solutions of nonstationary problems, etc., see \cite{Polacik, Quittner} and references therein.
Liouville-type theorems for degenerate elliptic equations have been attracted the interest of many mathematicians. The classical Liouville theorem was generalized to $p$-harmonic functions on the whole space $\R^n$ and on exterior domains by Serrin and Zou \cite{Serrin}, see also \cite{Cuccu} for related results. The Liouville theorems for some linear degenerate elliptic operators such as $X$-elliptic operators, Kohn-Laplacian (and more general sublaplacian on Carnot groups) and degenerate Ornstein-Uhlenbeck operators were proved in \cite{Kogoj1, Kogoj2}.
More recently, Yu \cite{Yu} studied the equation
$$- L_{\alpha} u = f(u)\;\;\mbox{ in }\;\; \R^{n_1} \times \R^{n_2}, $$
where $L_{\alpha}= \D_{x} + (1+\alpha)^2 \D_{y}$, $\alpha >0$ and $Q= n_1 + (1+\alpha ) n_2$ is the \textit{homogeneous dimension} of the space. Under some assumptions on the nonlinear term $f$ , he showed that the above equation possesses no positive solutions and the main technique used is the moving plane method in the integral form.
In this paper, we are concerned with the Liouville-type theorems for the following problem \begin{align}\label{equation}
-\Delta_{\lambda} u=|x|^{a}_{\lambda} |u|^{p-1}u,\quad \mbox{in}\,\, \R^n:= \R^{n_1}\times \R^{n_2}\times ...\times \R^{n_k},
\end{align}
where $n\geq 1$, $a \geq 0$, $p>1$, $$ \D_{\lambda} = \lambda^2_{1} \D_{x^{(1)}} +...+ \lambda^2_{k} \D_{x^{(k)}},\quad |x|_{\lambda} := \left(\sum_{j=1}^k \prod_{i\neq j} \lambda_i^2(x) \epsilon^2_j |x^{(j)}|^2 \right)^{\frac{1}{2\sigma }},$$ $\sigma= 1+\sum_{i=1}^k (\epsilon_i-1)$, $1\leq \epsilon_1 \leq ...\leq \epsilon_k$, $x =(x^{(1)}, ..., x^{(k)}) \in \R^n$.
Here the functions $\lambda_i: \R^n \rightarrow \R$ are continuous, strictly
positive and of class $C^1$ outside the coordinate hyperplanes, i.e. $\lambda_i >0$, $i=1,...,k$ in $\R^n \backslash \prod$, where $\prod = \{ x=(x_1,...,x_n)\in \R^n : \prod_{i=1}^n x_i=0\}$, and $\D_{x^{(i)}}$ denotes the classical
Laplacian in $\R^{n_i}$, $i=1,...,k$. As in \cite{Kogoj3} we assume that $\lambda_i$ satisfy the following properties:\\\\
\textbf{$(H_1)$} $\lambda_1(x)=1$, $\lambda_i(x)= \lambda_i(x^{(1)}, ..., x^{(i-1)})$, $i=2,..., k$.\\\\
\textbf{$(H_2)$} For every $x\in \R^n$, $\lambda_i(x)= \lambda_i(x^*)$, $i=1, ..., k$, where $x^*=( |x^{(1)}|,...,|x^{(k)}|)$ if $x=(x^{(1)},..., x^{(k)})$. \\\\
\textbf{$(H_3)$} There exists a group of dilations $\{\delta_t\}_{t>0}$ $$\delta_t : \R^n \rightarrow \R^n , \; \delta_t(x)= \delta_t(x^{(1)},..., x^{(k)})= (t^{\epsilon_1}x^{(1)},..., t^{\epsilon_k}x^{(k)}), $$
where $1 \leq \epsilon_1 \leq \epsilon_2 \leq ...\leq \epsilon_k$, such that $\lambda_i$ is $\delta_t$-homogeneous of degree $\epsilon_i-1$, i.e. $$\lambda_i (\delta_t(x))= t^{\epsilon_i -1} \lambda_i(x),\;\; \forall \; x\in \R^n , \; t>0, \; i=1,..., k. $$
This implies that the operator $\D_{\lambda}$ is $\delta_t$-homogeneous of degree two, i.e. $$\D_{\lambda}(u(\delta_t(x)))= t^2 (\D_{\lambda} u) (\delta_t(x)),\; \; \forall \; u \in C^{\infty}(\R^n) .$$
We denote by $Q$ the \textit{homogeneous dimension } of $\R^n$ with respect to the group of dilations $\{\delta_t\}_{t>0}$, i.e. $$Q:= \epsilon_1 n_1+ \epsilon_2 n_2 +...+\epsilon_k n_k.$$
The $\D_{\lambda}$-Laplace operator was first introduced by Franchi and Lanconelli \cite{Franchi}, and recently reconsidered in \cite{Kogoj3} under an additional assumption that the operator is homogeneous of degree two with respect to a group dilation in $\R^n$. It was proved in \cite{AnhMy}, that the autonomous case, i.e. $a=0$, \eqref{equation} has no positive classical solution if $1< p\leq \frac{Q}{Q-2}$, with $Q= \epsilon_1 + \epsilon_2 +...+\epsilon_n $, ($n_i=1$, $i=1, ..., n$).
The $\D_{\lambda}$-operator contains many degenerate elliptic operators. We now give some examples of $\D_{\lambda}$-Laplace operators (see also \cite{Kogoj3}). We use the following notation: we split $\R^n$ as follows $\R^n=\R^{n_1} \times ...\times \R^{n_k}$ and write $$x=(x^{(1)}, ..., x^{(k)}), \; x^{(i)}=(x^{(i)}_1, ..., x^{(i)}_{n_i}) \in \R^{n_i},$$
$$|x^{(i)}|^{2}= \sum_{j=1}^{n_i} |x^{(i)}_j|^2,\;\;\; i=1,2,...,k.$$
We denote the classical Laplace operator in $\in \R^{n_i}$ by $$\D_{x^{(i)}} = \sum_{j=1}^{n_i} \partial^2_{x^{(i)}_j}.$$
\textbf{Example 1.} Let $\alpha$ be a real positive constant and $k = 2$. We consider the Grushin-type
operator $$ \D_{\lambda}= \D_x + |x|^{2 \alpha } \D_y,$$
where $\lambda = (\lambda_1,\lambda_2)$ with $$\lambda_1(x)=1, \quad \lambda_2(x)= |x^{(1)}|^{\alpha },\quad x \in \R^{n_1} \times \R^{n_2}.$$
Our group of dilations is
$$\delta_t(x)= \delta_t(x^{(1)},x^{(2)})= (tx^{(1)}, t^{\alpha+1}x^{(2)}),$$
and the homogenous dimension with respect to $(\delta_t)_{t>0}$ is $Q = n_1 + (\alpha + 1)n_2$.\\\\
\textbf{Example 2.}
Given a multi-index $\alpha=(\alpha_1,..., \alpha_{k-1}) $, $\alpha_j \geq 1 $, $j=1,...,k-1$, define $$\D_{\alpha}:= \D_{x^{(1)}}+ |x^{(1)}|^{2 \alpha_1} \D_{x^{(2)}} + ...+ |x^{(k-1)}|^{2 \alpha_{k-1}} \D_{x^{(k)}}.$$
Then $\D_{\alpha}= \D_{\lambda}$ with $\lambda= (\lambda_1, ..., \lambda_k )$ and $\lambda_i= |x^{(i-1)}|^{\alpha_{i-1}} $, $i=1,...,k$. Here we agree to let $|x^{(0)}|^{\alpha_{0}}=1$. A group of dilations for which $\lambda$ satisfies $(H_3)$ is given by
$$\delta_t : \R^n \rightarrow \R^n , \; \delta_t(x)= \delta_t(x^{(1)},..., x^{(k)})= (t^{\epsilon_1}x^{(1)},..., t^{\epsilon_k}x^{(k)}), $$ with $\epsilon_1=1$ and $\epsilon_i= \alpha_{i-1} \epsilon_{i-1}+1$, $i=2,...,k$. In particular, if $\alpha_1=...=\alpha_{k-1}=1$, the operator $\D_{\alpha}$ and the dilation $\delta_t$ becomes, respectively
$$\D_{\alpha}= \D_{x^{(1)}}+ |x^{(1)}|^{2 } \D_{x^{(2)}} + ...+ |x^{(k-1)}|^{2 } \D_{x^{(k)}},$$
and
$$\delta_t(x)= (t x^{(1)},t^{2}x^{(2)},..., t^{k}x^{(k)}).$$
\textbf{Example 3.} Let $\alpha$, $\beta$ and $\gamma$ be positive real constants. For the operator
$$\D_{\lambda} = \D_{x^{(1)}} + |x^{(1)}|^{2\alpha } \D_{x^{(2)}}+ |x^{(1)}|^{2\beta } |x^{(2)}|^{2\gamma } \D_{x^{(3)}},$$
where $\lambda = (\lambda_1,\lambda_2,\lambda_3)$ with $$\lambda_1(x)=1, \quad \lambda_2(x)= |x^{(1)}|^{\alpha } , \quad \lambda_3(x)=|x^{(1)}|^{\beta } |x^{(2)}|^{\gamma },\quad x \in \R^{n_1} \times \R^{n_2} \times\R^{n_3},$$
we find the group of dilations
$$\delta_t(x)= \delta_t(x^{(1)},x^{(2)}, x^{(3)})= (tx^{(1)}, t^{\alpha+1}x^{(2)} , t^{\beta + (\alpha+1)\gamma +1}x^{(3)}).$$
The aim of the present paper was to establish the Liouville-type theorems with finite Morse index for the equation \eqref{equation}. In order to state our results we need the following:
\begin{definition}
We say that a solution $u$ of \eqref{equation} belonging to $C^2(\R^n)$\\
$\bullet$ is stable, if
$$Q_u(\psi):= \int_{\R^n} |\nabla_{\lambda} \psi|^2 - p\int_{\R^n} |x|^{a}_{\lambda} |u|^{p-1}\psi^2 \geq0\;,\;\;\forall\; \psi\in C_c^1( \R^n),$$
where $\nabla_{\lambda}= (\lambda_1 \nabla_{x^{(1)}},..., \lambda_k \nabla_{x^{(k)}})$.\\
$\bullet$ has Morse index equal to $K \geq 1$ if $K$ is the maximal dimension of a subspace $X_K$ of $C^1_c(\R^n)$ such that $ Q_u (\psi)< 0$ for any $\psi \in X_K\backslash \{0\}$.\\
$\bullet$ is stable outside a compact set $\mathcal{K}\subset \R^n$ if $ Q_u (\psi)\geq0$ for any $\psi \in C^1_c(\R^n\backslash \mathcal{K})$.
\end{definition}
\begin{rem}
\textbf{a)} Clearly, a solution stable if and only if its Morse index is equal to zero.\\
\textbf{b)} It is well know that any finite Morse index solution $u$ is stable outside a compact set $\mathcal{K}\subset \R^n$. Indeed, there exists $m_0\geq1$ and $X_{m_0}:= \mbox{Span}\{\phi_1,..., \phi_{m_0}\} \subset C^1_c(\R^n)$ such that $Q_u (\phi)<0$ for any $\phi \in X_{m_0} \backslash\{0\}$. Hence, $Q_u (\psi)\geq0$ for every $\psi \in C^1_c(\R^n \backslash \mathcal{K}) $, where $\mathcal{K}:= \displaystyle{ \cup_{j=1}^{m_0}} supp(\phi_{j})$.
\end{rem}
In the following, we state Liouville-type results for solutions $u \in C^2(\R^n)$ of \eqref{equation}. In what follows,
we divide our study to stable solutions and solutions which are stable outside a compact set.
\subsection{\textbf{\large{Stable solutions}}}
To state the following result we need to introduce some notation. We set $\Gamma_M(p)=2p-1+2\sqrt{p(p-1)}$ and denote by $\Omega_{R}= B_1(0, R^{\epsilon_1})\times B_2(0, R^{\epsilon_2}) \times ...\times B_k(0, R^{\epsilon_k})$, where $B_i(0, R^{\epsilon_i})\subset \R^{n_i}$, $i=1,...,k$, the balls of center $0$ and radius $R^{\epsilon_i}$.
\begin{prop}\label{proposition1}
Let $u \in C^2(\R^n)$ be a stable solution of \eqref{equation}. Then, for any $\gamma \in \left[1, \Gamma_M(p)\right)$, there exists a positive constant $C$ independent of $R$, such that
\begin{eqnarray} \label{stable} \int_{\Omega_R} \left(|x|^{a}_{\lambda} |u|^{p+\gamma}+ |\nabla_{\lambda} ( |u|^{\frac{\gamma-1}{2}}u)|^2 \right) dx \leq C R^{Q- \frac{2(p+\gamma)+ (\gamma+1)a }{p-1}},\quad \mbox{for all} \; R>0.
\end{eqnarray}
\end{prop}
Proposition \ref{proposition1} provides an important estimate on the integrability of $u$ and $\nabla_\lambda u$. As we will see, our nonexistence results will follow by showing that the right-hand side of \eqref{stable} vanishes under the right assumptions on $p$ when $R\rightarrow +\infty$. More precisely, as a corollary of Proposition \ref{proposition1}, we can state our first Liouville type theorem.
\begin{thm}\label{th1}
Let $u \in C^2(\R^n)$ be a stable solution of \eqref{equation} with, \begin{eqnarray*}
p_c(Q,a)=
\begin{cases}
+\infty\;\;&\text{if $Q\leq 10 + 4 a $},\\
\frac{(Q-2)^2-2(a+2)(a+Q)+2 \sqrt{(a+2)^3(a+2Q-2)}}{(Q-2)(Q-4a-10)}\;\; &\text{if $Q > 10 + 4 a $}.
\end{cases}
\end{eqnarray*} Then $u\equiv 0$.
\end{thm}
\subsection{\textbf{\large{Solutions which are stable outside a compact set}}}
In this subsection we prove some integral identities extending to the $\D_{\lambda}$ setting the classical Pohozaev identity for semilinear
Poisson equation \cite{Pohozaev}. Pohozaev identity has been extended by several authors to general elliptic equations and systems,
both in Riemannian and sub-Riemannian context, see, e.g., \cite{Bozhkov, Garofalo, Pucci} and the references therein. To prove our identities we
closely follow the original procedure of Pohozaev, just replacing the vector field $ P= \sum_{i=1}^n x_i \partial_{x_i} $ in \cite{Pohozaev}, page $1410$], by $$ T= \sum_{i=1}^k \epsilon_i x^{(i)} \nabla_{x^{(i)}},$$
the generator of the group of dilation $(\delta_t)_{t\geq 0}$ in \textbf{$(H_3)$}(we say that $T$ generates $(\delta_t)_{t\geq 0}$ since a function $u$ is $\delta_t$-homogeneous of degree $m$ if and only if $T u = m u$).
\begin{prop}\label{proposition2}
Let $u \in C^2(\R^n)$ be a solution of \eqref{equation} and $\phi \in C^1_c(\Omega_R)$. If $T\left( |x|_{\lambda}\right)= |x|_{\lambda}$, then
\begin{multline} \label{pohozaev}
\int_{\Omega_R} \left[ \frac{Q-2}{2} |\nabla_{\lambda} u|^2 -\frac{Q+a}{p+1} |x|^{a}_{\lambda}|u|^{p+1} \right]\phi = \int_{\Omega_R} \left[\nabla_{\lambda} u \nabla_{\lambda} \phi T(u) +\left[- \frac 12 |\nabla_{\lambda} u|^2 + \frac{|x|^{a}_{\lambda}}{p+1} |u|^{p+1}\right] T(\phi)\right].
\end{multline}
\end{prop}
Thanks to Proposition \ref{proposition2}, we derive
\begin{thm}\label{th2}
Let $u \in C^2(\R^n)$ be a solution of \eqref{equation} which is stable outside a compact set of $\R^n$, with \begin{eqnarray*}
p_s(Q,a)=\begin{cases}
+\infty \; \; &\text{if \; $Q\leq 2$,}\\
\frac{Q+2+2a}{Q-2} \; \; &\text{if \; $Q> 2$.}
\end{cases}
\end{eqnarray*} If $T\left( |x|_{\lambda}\right)= |x|_{\lambda}$, then $u\equiv 0$.
\end{thm}
\section{ \large{Example which satisfies $T\left( |x|_{\lambda}\right)= |x|_{\lambda}$} }
The degenerate elliptic operators we consider are of the form $$ \D_{\lambda} = \lambda^2_{1} \D_{x^{(1)}} +...+ \lambda^2_{k} \D_{x^{(k)}}. $$
We denote by $|x^{(j)}|$ the euclidean norm of $x^{(j)} \in \R^{n_j}$ and assume the functions $\lambda_{i}$ are
of the form
\begin{eqnarray} \label{equal}
\lambda_{i}(x)= \prod_{j=1}^k |x^{(j)}|^{\alpha_{ij}},\quad i=1,...,k,
\end{eqnarray}
such that\\
1) $\alpha_{ij} \geq 0$ for $i = 2, . . . , k$, $j = 1, . . . , i-1$.\\
2) $\alpha_{ij} = 0$ for $j\geq i$.\\
3) $\sum_{l=1}^k \epsilon_l \alpha_{jl} = \epsilon_j-1$, $j=1,...,k$ with $1=\epsilon_1\leq \epsilon_2 \leq ...\leq \epsilon_k$.\\
Clearly, $\lambda_i$ is $\delta_t$-homogeneous of degree $\epsilon_i-1$ with respect to a group of dilations $\{\delta_t\}_{t>0}$ $$\delta_t : \R^n \rightarrow \R^n , \; \delta_t(x)= \delta_t(x^{(1)},..., x^{(k)})= (t^{\epsilon_1}x^{(1)},..., t^{\epsilon_k}x^{(k)}). $$
Now, using the relation $\sum_{l=1}^k \epsilon_l \alpha_{jl} = \epsilon_j-1$, we get $T\left( |x|_{\lambda}\right)= |x|_{\lambda}$ is satisfied.\\
This paper is organized as follows. In section $3$, we give the proof of Proposition \ref{proposition1} and Theorem \ref{th1}. Section $4$ is devoted to the proof of Proposition \ref{proposition2} and Theorem \ref{th2}.
\section{\large{The Liouville theorem for stable solutions: proof of Theorem \ref{th1}}}
In this section we prove all the results concerning the classification of stable solutions, i.e., Proposition \ref{proposition1} and Theorem \ref{th1}. First, to prove Proposition \ref{proposition1}, we need the following technical Lemma.
Let $R>0$, $\Omega_{2R}= B_1(0, 2 R^{\epsilon_1})\times B_2(0, 2 R^{\epsilon_2}) \times ...\times B_k(0, 2 R^{\epsilon_k})$, where $B_i(0, 2 R^{\epsilon_i})\subset \R^{n_i}$, $i=1,...,k$, and consider $k$ functions $\psi_{1,R}$,..., $\psi_{k,R}$ such that $$\psi_{1,R}(r^{(1)})=\psi_{1}\left(\frac{r^{(1)}}{R^{\epsilon_1}}\right),\;...,\; \psi_{k,R}(r^{(k)})=\psi_{k}\left(\frac{r^{(k)}}{R^{\epsilon_k}}\right),$$
with $\psi_{1,R},\; ...\; \psi_{k,R} \in C^{\infty}_c([0, + \infty)),\; 0\leq \psi_{1,R},\;...\; \psi_{k,R} \leq 1,$ $$ \psi_i(t)=\begin{cases} 1\; & \text{in \;$[0,\;1]$},\\
0\; & \text{in \;$[2,\;+\infty )$},
\end{cases}$$
and for some constant $C>0$ and $\psi_{1,R},\; ...\; \psi_{k,R}$ satisfy
\begin{eqnarray*} \label{ineq1}
\left|\nabla_{x^{(1)}} \psi_{1,R}\right| \leq C R^{-\epsilon_1},\; ...,\; \left|\nabla_{x^{(k)}} \psi_{k,R}\right| \leq C R^{-\epsilon_k},
\end{eqnarray*}
\begin{eqnarray*} \label{ineq2}
\left|\D_{x^{(1)}} \psi_{1,R}\right| \leq C R^{-2\epsilon_1},\; ...,\; \left|\D_{x^{(k)}} \psi_{k,R}\right| \leq C R^{-2\epsilon_k},
\end{eqnarray*}
where $r^{(i)}= |x^{(i)}|$, $i=1,..., k$.
\begin{lem} \label{lem1}
\textbf{(1)} There exists a constant $C>0$ independent of $R$ such that
\textbf{a)} $|\lambda_{i}(x)| \leq C R^{\epsilon_i-1},\; \forall \; x \in \Omega_{2R},\; i=1,...,k.$
\textbf{b)} $|\nabla_{\lambda} \psi_R|^2 + |\D_{\lambda} \psi_R|\leq C R^{-2}$, where
$\psi_R= \prod_{i=1}^k \psi_{i,R}$.\\\\
\textbf{(2)} The homogeneous norm, $|.|_{\lambda}$, is $\delta_t$-homogeneous of degree one, i.e. $$|\delta_t(x)|_{\lambda}= t |x|_{\lambda},\;\; \forall \; x\in \R^n , \; t>0. $$
\textbf{(3)} There exists a constant $C>0$ independent of $R$ such that$$|x|_{\lambda}\leq C R,\;\forall \; x \in \Omega_{2R}.$$
\end{lem}
\textbf{Proof.}\\ \textit{Proof of (1)\;a).} For any $x=(x^{(1)},..., x^{(k)})\in \Omega_{2R}$, we have $x^{(i)} \in B_i(0, 2R^{\epsilon_i})$, $i=1,...,k$, this implies $\frac{|x^{(i)}|}{R^{\epsilon_i}}\leq 2$, $i=1,...,k$. Therefore, if we write $$x=(x^{(1)},..., x^{(k)})= \left( R^{\epsilon_1} \times \frac{x^{(1)}}{R^{\epsilon_1}}, ..., R^{\epsilon_k} \times \frac{x^{(k)}}{R^{\epsilon_k}}\right), $$
and let $y=(y^{(1)}, ..., y^{(k)})= \left( \frac{x^{(1)}}{R^{\epsilon_1}}, ..., \frac{x^{(k)}}{R^{\epsilon_k}}\right)$, then $y \in \overline{\Omega_{2}}$. Hence by assumption $(H_3)$ made on functions $\lambda_i$, we get
\begin{eqnarray} \label{p1}
\lambda_i(x)&=& \lambda_i(R^{\epsilon_1} y^{(1)}, ..., R^{\epsilon_k} y^{(k)} )\nonumber\\&=& R^{\epsilon_i-1} \lambda_i(y^{(1)}, ..., y^{(k)}) \nonumber\\&=& R^{\epsilon_i-1} \lambda_i(y).
\end{eqnarray}
Moreover, since $\lambda_i$, $i=1, ..., k$ are continuous, then
\begin{eqnarray} \label{p2}
|\lambda_i(y)|\leq C, \;\; \forall \; y \in \overline{\Omega_{2}}.
\end{eqnarray}
Therefore, from \eqref{p1} and \eqref{p2}, we obtain $$|\lambda_{i}(x)| \leq C R^{\epsilon_i-1},\; \forall \; x \in \Omega_{2R},\; i=1,...,k.$$
\textit{Proof of (1)\;b).} Using assumption $(H_2)$ made on functions $\lambda_i$, $i=1,...,k$, with $r=( r^{(1)}, ..., r^{(k)})=(|x^{(1)}|, ..., |x^{(k)}|)$, we have $$\lambda_1(r)=1,\; \lambda_i(r)=\lambda_i(r^{(1)}, ..., r^{(i-1)}),\; \forall\; i=2,...,k.$$
If we denote by $\psi_R= \prod_{i=1}^k \psi_{i,R}$, we get
\begin{eqnarray*}
\nabla_{\lambda} \psi_R &=&\left( \lambda_1(r) \nabla_{x^{(1)}} \psi_R,\; ...,\; \lambda_k(r) \nabla_{x^{(k)}}\psi_R\right)\nonumber\\&=&
\left(\lambda_1(r)\nabla_{x^{(1)}}\psi_{1,R}\prod_{i=2}^k \psi_{i,R}, ..., \lambda_k(r)\nabla_{x^{(k)}}\psi_{k,R}\prod_{i=1}^{k-1} \psi_{i,R} \right),
\end{eqnarray*}
and
\begin{eqnarray*}
\D_{\lambda} \psi_R &=& \lambda^2_1(r) \D_{x^{(1)}} \psi_R +...+ \lambda^2_k(r) \D_{x^{(k)}} \psi_R \nonumber\\ &=& \lambda^2_1(r) \D_{x^{(1)}}\psi_{1,R}\prod_{i=2}^k \psi_{i,R} +...+ \lambda^2_k(r) \D_{x^{(k)}}\psi_{k,R}\prod_{i=1}^{k-1} \psi_{i,R}.
\end{eqnarray*}
Since $|\lambda_{i}(r)|=|\lambda_{i}(x)| \leq C R^{\epsilon_i-1}$, $\forall \; x \in \Omega_{2R}$, $i=1,...,k$, then there exists a constant $C>0$ independent of $R$ such that
$$|\nabla_{\lambda} \psi_R|^2\leq C R^{-2}\; \; \mbox{and}\;\; |\D_{\lambda} \psi_R|\leq C R^{-2}.$$
\textit{Proof of (2).} Let $x \in \R^n$. The homogeneity of the functions $\lambda_i$ implies that
\begin{eqnarray}\label{nouveau}
|\delta_t(x)|_{\lambda} :&=& \left(\sum_{j=1}^k \prod_{i\neq j} (\lambda_i(\delta_t(x)))^2 \epsilon^2_j |t^{\epsilon_j} x^{(j)}|^2 \right)^{\frac{1}{2(1+\sum_{i=1}^k (\epsilon_i-1))}} \nonumber\\&=& \left(\sum_{j=1}^k \prod_{i\neq j} t^{2\epsilon_j} t^{2(\epsilon_i-1)} (\lambda_i(x))^2 \epsilon^2_j | x^{(j)}|^2 \right)^{\frac{1}{2(1+\sum_{i=1}^k (\epsilon_i-1))}} \nonumber\\&=& \left( t^{2(1+\sum_{i=1}^k(\epsilon_i-1))}\sum_{j=1}^k \prod_{i\neq j} (\lambda_i(x))^2 \epsilon^2_j | x^{(j)}|^2 \right)^{\frac{1}{2(1+\sum_{i=1}^k (\epsilon_i-1))}}\nonumber\\&=& t |x|_{\lambda}
\end{eqnarray}
\textit{Proof of (3).} For any $x=(x^{(1)},..., x^{(k)})\in \Omega_{2R}$, we have $x^{(i)} \in B_i(0, 2R^{\epsilon_i})$, $i=1,...,k$, this implies $\frac{|x^{(i)}|}{R^{\epsilon_i}}\leq 2$, $i=1,...,k$. Therefore, if we write $$x=(x^{(1)},..., x^{(k)})= \left( R^{\epsilon_1} \times \frac{x^{(1)}}{R^{\epsilon_1}}, ..., R^{\epsilon_k} \times \frac{x^{(k)}}{R^{\epsilon_k}}\right), $$
and let $y=(y^{(1)}, ..., y^{(k)})= \left( \frac{x^{(1)}}{R^{\epsilon_1}}, ..., \frac{x^{(k)}}{R^{\epsilon_k}}\right)$, then $y \in \overline{\Omega_{2}(0)}$.\\
Using \eqref{nouveau}, we get
\begin{eqnarray*}
|x|_{\lambda}&=& |(R^{\epsilon_1} y^{(1)}, ..., R^{\epsilon_k} y^{(k)} )|_{\lambda}\nonumber\\&=& R |(y^{(1)}, ..., y^{(k)})|_{\lambda} \nonumber\\&=& R|y|_{\lambda}.
\end{eqnarray*}
Since $|\lambda_{i}(y)| \leq C$, $\forall \; y \in \overline{\Omega_{2}}$, $i=1,...,k$, then there exists a constant $C>0$ independent of $R$ such that $$|x|_{\lambda} \leq C R,\;\; \forall \; x \in \Omega_{2R}.$$
This completes the proof of Lemma \ref{lem1}. \qed\\\\
\textbf{Proof of Proposition \ref{proposition1}.} The proof follows the main lines of the demonstration of proposition $4$ in \cite{Farina}, with more modifications. We split the proof into four steps: \\
\textit{\textbf{Step 1.}} For any $\phi \in C^2_c(\R^n)$ we have
\begin{eqnarray}\label{e1}
\int_{\R^n} |\nabla_{\lambda} (|u|^{\frac{\gamma-1}{2}}u)|^2 \phi^2 dx&=& \frac{(\gamma +1)^2}{4 \gamma}\int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} \phi^2dx + \frac{\gamma +1}{4 \gamma} \int_{\R^n} |u|^{\gamma+1} \Delta_{\lambda} (\phi^2)dx.
\end{eqnarray}
Multiply equation \eqref{equation} by $|u|^{\gamma-1} u \phi^2$
and integrate by parts to find
$$\gamma \int_{\R^n} |\nabla_{\lambda} u|^2 |u|^{\gamma-1} \phi^2 dx+\int_{\R^n}\nabla_{\lambda} u \nabla_{\lambda} (\phi^2)|u|^{\gamma-1}u \;dx = \int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} \phi^2 dx, $$
therefore
\begin{eqnarray*}
\int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} \phi^2 dx&=& \frac{4\gamma}{(\gamma+1)^2} \int_{\R^n} |\nabla_{\lambda} (|u|^{\frac{\gamma-1}{2}}u)|^2 \phi^2 dx+ \frac{1}{\gamma +1} \int_{\R^n} \nabla_{\lambda} (|u|^{\gamma+1}) \nabla_{\lambda} (\phi^2) dx\nonumber\\&=& \frac{4\gamma}{(\gamma+1)^2} \int_{\R^n} |\nabla_{\lambda} (|u|^{\frac{\gamma-1}{2}}u)|^2 \phi^2dx- \frac{1}{\gamma +1} \int_{\R^n} |u|^{\gamma+1} \Delta_{\lambda} (\phi^2) dx .
\end{eqnarray*}
Identity \eqref{e1} then follows by multiplying the latter identity by the factor $\frac{(\gamma +1)^2}{4 \gamma}$.\\\\
\textit{\textbf{Step 2.}} For any $\phi \in C^2_c(\R^n)$ we have
\begin{eqnarray} \label{e2}
\left( p-\frac{(\gamma +1)^2}{4 \gamma}\right)\int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} \phi^2dx & \leq & \int_{\R^n} |u|^{\gamma+1}\left[ |\nabla_{\lambda} \phi|^2 +\left(\frac{\gamma +1}{4 \gamma}- \frac{1}{2}\right) \Delta_{\lambda} (\phi^2) \right]dx .
\end{eqnarray}
The function $|u|^{\frac{\gamma-1}{2}}u \phi $ belongs to $C^1_c(\R^n)$, and thus it can be used as a test function in the quadratic form $Q_u$. Hence, the stability assumption on $u$ gives
\begin{eqnarray}\label{e3}
p\int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} \phi^2 dx \leq \int_{\R^n} |\nabla_{\lambda} (|u|^{\frac{\gamma-1}{2}}u \phi)|^2 dx.
\end{eqnarray}
A direct calculation shows that the right hand side of \eqref{e3} equals to
\begin{multline}\label{e4}
\int_{\R^n} \left[|u|^{\gamma+1} |\nabla_{\lambda} \phi|^2 + |\nabla_{\lambda}(|u|^{\frac{\gamma-1}{2}}u)|^2 \phi^2 + \frac{1}{2} \nabla_{\lambda} \phi^2 \nabla_{\lambda} (|u|^{\gamma+1}) \right] dx \\= \int_{\R^n} |u|^{\gamma+1} \left[ |\nabla_{\lambda} \phi|^2 - \frac{1}{2} \Delta_{\lambda} (\phi^2 ) \right] dx + \int_{\R^n} |\nabla_{\lambda}( |u|^{\frac{\gamma-1}{2}}u)|^2 \phi^2 dx.
\end{multline}
From \eqref{e3} and \eqref{e4}, we obtain that
\begin{eqnarray}\label{e5}
p\int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} \phi^2 dx &\leq& \int_{\R^n} |u|^{\gamma+1}\left[ |\nabla_{\lambda} \phi|^2 - \frac{1}{2} \Delta_{\lambda} (\phi^2) \right]dx + \int_{\R^n} |\nabla_{\lambda} (|u|^{\frac{\gamma-1}{2}}u)|^2 \phi^2 dx.
\end{eqnarray}
Putting this back into \eqref{e1} gives
\begin{eqnarray*}\left( p-\frac{(\gamma +1)^2}{4 \gamma}\right)\int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} \phi^2dx & \leq & \int_{\R^n} |u|^{\gamma+1}\left[ |\nabla_{\lambda} \phi|^2 +\left(\frac{\gamma +1}{4 \gamma}- \frac{1}{2}\right) \Delta_{\lambda} (\phi^2) \right]dx .
\end{eqnarray*}
\textit{\textbf{Step 3.}} For any $\gamma \in \left[1,\,\Gamma_M(p)\right)$ and any integer $m\geq \max\{\frac{p+\gamma}{p-1},\; 2\}$ there exists a constant $C(p, m, \gamma)>0$ depending only on $p$, $m$ and $\gamma$
\begin{eqnarray} \label{e6}
\int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} \psi_R^{2m}dx & \leq & C(p, m, \gamma) \int_{\R^n} |x|^{\frac{-(\gamma+1)a}{p-1}}_{\lambda} \left( |\nabla_{\lambda} \psi_R|^2 + |\psi_R| |\Delta_{\lambda} \psi_R| \right)^{\frac{p+\gamma}{p-1}}dx ,
\end{eqnarray}
\begin{eqnarray} \label{e07}
\int_{\R^n} |\nabla_{\lambda} (|u|^{\frac{\gamma-1}{2}}u)|^2 \psi_R^{2m} dx& \leq & C(p, m, \gamma) \int_{\R^n} |x|^{\frac{-(\gamma+1)a}{p-1}}_{\lambda} \left( |\nabla_{\lambda} \psi_R|^2 + |\psi_R| |\Delta_{\lambda} \psi_R| \right)^{\frac{p+\gamma}{p-1}}dx , \end{eqnarray}
where $\psi_R = \prod_{i=1}^k \psi_{i,R}$. Moreover, the constant $C(p,m,\gamma )$ can be explicitly computed.
From \eqref{e2}, we obtain that
\begin{eqnarray}\label{e7}
\alpha \int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} \phi^2dx & \leq & \int_{\R^n} |u|^{\gamma+1}|\nabla_{\lambda} \phi|^2 + \beta \int_{\R^n} |u|^{\gamma+1} \Delta_{\lambda} \phi dx .
\end{eqnarray}
where we have set $\alpha = \left( p-\frac{(\gamma +1)^2}{4 \gamma}\right)$ and $\beta = \frac{1-\gamma}{4 \gamma}$. Notice that $\alpha >0$ and $\beta<0$, since $p>1$ and $\gamma \in \left[1,\,\Gamma_M(p)\right)$.
Now, we set $\phi = \psi_R^m$. The function $\phi$ belongs to $C^2_c(\R^n)$, since $m\geq 2$ and $m$ is an integer, hence it can be used in \eqref{e7}. A direct computation gives
\begin{eqnarray}\label{e8}
\alpha \int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} \psi_R^{2m} dx & \leq & \int_{\R^n} |u|^{\gamma+1} \psi_R^{2m-2}\left(m^2 |\nabla_{\lambda} \psi_R|^2 + \beta m (m-1) |\nabla_{\lambda} \psi_R|^2+ \beta m \psi_R\Delta_{\lambda} \psi_R \right)dx, \nonumber\\
\end{eqnarray}
hence
\begin{eqnarray}\label{e9}
\int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} \psi_R^{2m} dx & \leq & C_1 \int_{\R^n} |u|^{\gamma+1} \psi_R^{2m-2}\left( |\nabla_{\lambda} \psi_R|^2 + |\psi_R ||\Delta_{\lambda} \psi_R| \right)dx ,
\end{eqnarray}
with $C_1= \frac{m^2+ \beta m (m-1)}{\alpha} > -\frac{\beta m}{\alpha} \geq 0$.
An application of Young's inequality yields
\begin{eqnarray}\label{e10}
\int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} \psi_R^{2m} &\leq & C_1 \int_{\R^n} |u|^{\gamma+1} \psi_R^{2m-2}\left( |\nabla_{\lambda} \psi_R|^2 + |\psi_R ||\Delta_{\lambda} \psi_R| \right)dx \nonumber\\ &=& C_1 \int_{\R^n} |x|^{\frac{(\gamma+1)a}{p+\gamma}}_{\lambda}|u|^{\gamma+1} \psi_R^{2m-2}|x|^{\frac{-(\gamma+1)a}{p+\gamma}}_{\lambda}\left( |\nabla_{\lambda} \psi_R|^2 + |\psi_R ||\Delta_{\lambda} \psi_R| \right)dx \nonumber\\ & \leq & \frac{\gamma+1}{p+\gamma} \int_{\R^n} |x|^{a}_{\lambda}|u|^{p+\gamma} \psi_R^{(2m-2)\frac{p+\gamma}{\gamma +1}} + \frac{(p-1) \;C_1}{ p+\gamma} \int_{\R^n} |x|^{\frac{-(\gamma+1)a}{p-1}}_{\lambda}\left( |\nabla_{\lambda} \psi_R|^2 + |\psi_R ||\Delta_{\lambda} \psi_R| \right)^{\frac{p+\gamma}{ p-1}}.\nonumber\\
\end{eqnarray}
At this point we notice that $m\geq \max\{\frac{p+\gamma}{p-1},\; 2\}$ implies $(2m-2)\frac{p+\gamma}{p-1}\geq 2m$ and thus
$\psi_R^{(2m-2)\frac{p+\gamma}{\gamma +1}} \leq \psi_R^{2m}$ in $\R^n$, since $0 \leq \psi_R \leq 1 $ everywhere in $\R^n$.
Therefore, we obtain
\begin{eqnarray*}
\int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} \psi_R^{2m} & \leq & \frac{\gamma+1}{p+\gamma} \int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} \psi_R^{ 2m} + \frac{(p-1) \;C_1}{ p+\gamma} \int_{\R^n} |x|^{\frac{-(\gamma+1)a}{p-1}}_{\lambda} \left( |\nabla_{\lambda} \psi_R|^2 + |\psi_R ||\Delta_{\lambda} \psi_R| \right)^{\frac{p+\gamma}{ p-1}}.\nonumber\\
\end{eqnarray*}
The latter immediately implies
\begin{eqnarray} \label{e12}
\int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} \psi_R^{2m} dx & \leq & C_1 \int_{\R^n} |x|^{\frac{-(\gamma+1)a}{p-1}}_{\lambda} \left( |\nabla_{\lambda} \psi_R|^2 + |\psi_R ||\Delta_{\lambda} \psi_R| \right)^{\frac{p+\gamma}{ p-1}}dx,
\end{eqnarray}
which proves inequality \eqref{e6} with $C(p,m,\gamma )= C_1$.
To prove \eqref{e07}, we combine \eqref{e1} and \eqref{e2}. This leads to
\begin{eqnarray*}
\int_{\R^n} |\nabla_{\lambda} (|u|^{\frac{\gamma-1}{2}}u)|^2 \phi^2 dx \leq A \int_{\R^n} |u|^{\gamma +1} |\nabla_{\lambda} \phi|^2 dx + B \int_{\R^n} |u|^{\gamma +1} \phi \D_{\lambda} \phi dx,
\end{eqnarray*}
where $A= \frac{(\gamma +1)^2}{4 \gamma \alpha }+ \frac{(\gamma +1)}{2 \gamma } >0$ and $B=\frac{\beta (\gamma +1)^2}{4 \gamma \alpha }+ \frac{(\gamma +1)}{2 \gamma } \in \R$.
Now, we insert the test function $\phi = \psi_R^m$ in the latter inequality to find,
\begin{eqnarray*}
\int_{\R^n} |\nabla_{\lambda} (|u|^{\frac{\gamma-1}{2}}u)|^2 \psi_R^{2m} dx \leq \int_{\R^n} |u|^{\gamma +1} \psi_R^{2m-2} \left(A m^2 |\nabla_{\lambda} \psi_R|^2 + B m (m-1)|\nabla_{\lambda} \psi_R|^2+ B m \psi_R \D_{\lambda} \psi_R \right) dx,
\end{eqnarray*}
and hence
\begin{eqnarray} \label{e11}
\int_{\R^n} |\nabla_{\lambda} (|u|^{\frac{\gamma-1}{2}}u)|^2 \psi_R^{2m} dx \leq C_2 \int_{\R^n} |u|^{\gamma +1} \psi_R^{2m-2} \left( |\nabla_{\lambda} \psi_R|^2 + |\psi_R| |\D_{\lambda} \psi_R| \right) dx,
\end{eqnarray}
with $C_2= \max\{A m^2 + B m (m-1), \; |B|m \}>0$. \\Using H\"{o}lder's inequality in \eqref{e11} yields
\begin{eqnarray*}
\int_{\R^n} |\nabla_{\lambda} (|u|^{\frac{\gamma-1}{2}}u)|^2 \psi_R^{2m} &\leq& C_2 \left(\int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} \psi_R^{(2m-2)\frac{p+\gamma}{\gamma +1}}\right)^{\frac{\gamma+1}{p+\gamma}} \left(\int_{\R^n} |x|^{\frac{-(\gamma+1)a}{p-1}}_{\lambda}\left( |\nabla_{\lambda} \psi_R|^2 + |\psi_R ||\Delta_{\lambda} \psi_R| \right)^{\frac{p+\gamma}{ p-1}}\right)^{\frac{p-1}{ p+\gamma}}\nonumber\\ &\leq & C_2 \left(\int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} \psi_R^{2m} \right)^{\frac{\gamma+1}{p+\gamma}} \left(\int_{\R^n} |x|^{\frac{-(\gamma+1)a}{p-1}}_{\lambda} \left( |\nabla_{\lambda} \psi_R|^2 + |\psi_R ||\Delta_{\lambda} \psi_R| \right)^{\frac{p+\gamma}{ p-1}}\right)^{\frac{p-1}{ p+\gamma}}.
\end{eqnarray*}
Finally, inserting \eqref{e12} into the latter we obtain
\begin{eqnarray*}
\int_{\R^n} |\nabla_{\lambda} (|u|^{\frac{\gamma-1}{2}}u)|^2 \psi_R^{2m} dx &\leq& C_2 C^{\frac{1+\gamma}{ p-1}}_1 \int_{\R^n} |x|^{\frac{-(\gamma+1)a}{p-1}}_{\lambda} \left( |\nabla_{\lambda} \psi_R|^2 + |\psi_R ||\Delta_{\lambda} \psi_R| \right)^{\frac{p+\gamma}{ p-1}}dx,
\end{eqnarray*}
which gives the desired inequality \eqref{e07}.\\\\
\textit{\textbf{Step 4.}} For any $\gamma \in \left[1,\,\Gamma_M(p)\right)$, there exists a constant $C>0$ independent of $R$ such that
\begin{eqnarray} \label{expr6}
\int_{\Omega_R} \left(|x|^{a}_{\lambda} |u|^{p+\gamma} + |\nabla_{\lambda} (|u|^{\frac{\gamma-1}{2}}u)|^2 \right) dx & \leq & C R^{Q- \frac{2(p+\gamma)+ (\gamma+1)a }{p-1}},\;\; \forall \; R>0.
\end{eqnarray}
The proof of \eqref{expr6} follows immediately by adding inequality \eqref{e6} to inequality
\eqref{e07} and using Lemma \ref{lem1}. \qed \\\\
\textbf{Proof of Theorem \ref{th1}.} By Proposition \ref{proposition1}, there exists a positive constant $C$ independent of $R$ such that
\begin{eqnarray} \label{e13}
\int_{\Omega_R} |x|^{a}_{\lambda} |u|^{p+\gamma} \leq C R^{Q- \frac{2(p+\gamma)+ a (\gamma+1) }{p-1}}.
\end{eqnarray}
Then it suffices to show that we can always choose a $\gamma \in \left[1,\,\Gamma_M(p)\right)$, such that $Q- \frac{2(p+\gamma)+ a(\gamma+1) }{p-1}<0$. Therefore, by letting $R \rightarrow + \infty$ in \eqref{e13}, we deduce that $$\int_{\R^n} |x|^{a}_{\lambda} |u|^{p+\gamma} =0, $$ which implies that $u \equiv 0$ in $\R^n$.\\
Next, we claim that, under the assumptions on the exponent $p$ assumed in Theorem \ref{th1}, we can always choose $\gamma \in [1,\,\Gamma_M(p))$ such that
\begin{eqnarray} \label{e09}
Q-\frac{2(p+\gamma) + a( \gamma +1)}{p-1}<0.
\end{eqnarray}
As in \cite{Farina}, we consider separately the case $Q\leq 10 + 4a $ and the case $Q> 10 + 4 a$.\\
\textbf{Case 1.} $Q\leq 10 +4 a$ and $p>1$. In this case we have $$2(p+\Gamma_M(p)) + a( \Gamma_M(p) +1)> 2(3p-1+2(p-1)) + a(2p + 2(p-1) >(10+ 4 a)(p-1)$$
and therefore
\begin{eqnarray} \label{e010}
Q-\frac{2(p+\Gamma_M(p)) + a( \Gamma_M(p) +1)}{p-1}<Q - (10 + 4 a)\leq0.
\end{eqnarray}
The latter inequality and the continuity of the function $x\mapsto Q-\frac{2(p+x)+a(x+1)}{p-1}$ immediately imply the existence of $\gamma \in [1,\,\Gamma_M(p))$ satisfying \eqref{e09}.\\\\
\textbf{Case 2.} $Q> 10 + 4 a$ and $1<p<p_c(Q,a)$. In this case we consider the real-valued function $x\mapsto g(x):= \frac{2(x+\Gamma_M(x)) + a(\Gamma(x)+1)}{x-1}$ on $(1,\,+\infty)$. Since $g$ is strictly decreasing function satisfying $\lim_{x\rightarrow 1^+}g(x)=+\infty$ and $\lim_{x\rightarrow +\infty}g(x)=10+4a$, there exists a unique $p_0>1$ such that $Q=g(p_0) $. We claim that $p_0=p_c(Q,a)$. Indeed,
$$Q=g(p)\;\Leftrightarrow\; (Q-2)(p-1)- (4+2a)p=(4+2a)\sqrt{p(p-1)}$$\;$$\Leftrightarrow\; (Q-10-4a)(Q-2)p^2+(-2(Q-2)^2+ 4(a+2)(Q+a))p+(Q-2)^2= 0, $$
which implies that
\begin{eqnarray} \label{e011}
(Q-10-4a)(Q-2)p_0^2+(-2(Q-2)^2+ 4(a+2)(Q+a))p_0+(Q-2)^2= 0,
\end{eqnarray}
and
\begin{eqnarray} \label{e012}
(Q-2)(p_0-1)- (4+2a)p_0 >(4+2a)(p_0-1).
\end{eqnarray}
The roots of \eqref{e011}
\begin{eqnarray} \label{e013}
p_1=\frac{(Q-2)^2-2(a+2)(a+Q)+2 \sqrt{(a+2)^3(a+2Q-2)}}{(Q-2)(Q-4a-10)}= p_c(Q,a),
\end{eqnarray}
\begin{eqnarray} \label{e014}
p_2=\frac{(Q-2)^2-2(a+2)(a+Q)-2 \sqrt{(a+2)^3(a+2Q-2)}}{(Q-2)(Q-4a-10)}< p_0,
\end{eqnarray}
while \eqref{e012} easily implies $p_0>\frac{Q-6-2a}{Q-4a-10}>p_2 $. This proves that $p_0= p_1$. Hence $$p_c(Q,a)=\frac{(Q-2)^2-2(a+2)(a+Q)+2 \sqrt{(a+2)^3(a+2Q-2)}}{(Q-2)(Q-4a-10)}$$ as claimed. Since we have just proven that $g(p_c(Q,a))=Q$ and $g$ is a strictly decreasing function, it follows that
\begin{eqnarray} \label{e015}
\forall \; 1<p<p_c(Q,a), \; Q<g(p).
\end{eqnarray}
Now we can conclude as in the first case, i.e, the continuity of $x\mapsto Q-\frac{2(p+x) + a(x+1)}{p-1}$ immediately implies the existence of $\gamma \in [1,\,\Gamma_M(p))$ satisfying \eqref{e09}. \qed
\section{\large{The Liouville theorem for solutions which are stable outside a compact set of $\R^n$: proof of Theorem \ref{th2}}}
In this section, we prove Proposition \ref{proposition2} and Theorem \ref{th2}. \\\\
\textbf{Proof of Proposition \ref{proposition2}.} Let $u \in C^2(\R^n)$ be a solution of \eqref{equation} and $\phi \in C^1_c(\Omega_R)$. Multiplying equation \eqref{equation} by $T(u) \phi $ and integrating by parts in $\Omega_R$, we obtain
\begin{eqnarray} \label{p3}
-\int_{\Omega_R} \D_{\lambda} u T(u) \phi dx &=&-\int_{\Omega_R} \D_{\lambda} u \;\epsilon_j\; x^{(j)}\; \nabla_{x^{(j)}} u \;\phi dx \nonumber\\&=& \int_{\Omega_R} \lambda^2_{i} \;\nabla_{x^{(i)}} u\; \nabla_{x^{(i)}} \left( \epsilon_j x^{(j)} \nabla_{x^{(j)}} u \phi \right) dx \nonumber\\&=& \int_{\Omega_R} \lambda^2_{i} \;\nabla_{x^{(i)}} u \;\epsilon_j \delta_{ij} \;\nabla_{x^{(j)}} u \; \phi dx + \int_{\Omega_R} \lambda^2_{i} \; \nabla_{x^{(i)}} u \; \epsilon_j\; x^{(j)} \; \nabla_{x^{(i)}}\left(\nabla_{x^{(j)}} u \right) \; \phi dx \nonumber\\&&+\;\; \int_{\Omega_R} \lambda^2_{i} \; \nabla_{x^{(i)}} u \; \epsilon_j \; x^{(j)} \; \nabla_{x^{(j)}} u \; \nabla_{x^{(i)}}\phi dx\nonumber\\&:=& I_1 + I_2 + I_3,
\end{eqnarray}
Here and in the sequel, we use the Einstein summation convention: an index occurring twice in a product is to be summed from $1$ up to the space dimension.
Obviously
\begin{eqnarray} \label{p4}
I_1 :&=& \int_{\Omega_R} \lambda^2_{i} \nabla_{x^{(i)}} u \; \epsilon_j \delta_{ij}\; \nabla_{x^{(j)}} u \; \phi dx \nonumber\\&=& \int_{\Omega_R} \lambda^2_{i} \left|\nabla_{x^{(i)}} u \right|^2 \epsilon_i \phi dx.
\end{eqnarray}
Moreover, an integration by parts in $I_2$ gives
\begin{eqnarray*}
I_2 :&=& \int_{\Omega_R} \lambda^2_{i} \nabla_{x^{(i)}} u \epsilon_j x^{(j)} \nabla_{x^{(i)}}\left(\nabla_{x^{(j)}} u \right) \phi dx \nonumber\\&=& - \int_{\Omega_R} \nabla_{x^{(j)}} (\lambda^2_{i}) \left|\nabla_{x^{(i)}} u \right|^2 \epsilon_j x^{(j)} \phi dx -I_2- \int_{\Omega_R} \lambda^2_{i} \left|\nabla_{x^{(i)}} u \right|^2 \epsilon_j n_j \phi dx -\int_{\Omega_R} \lambda^2_{i} \left|\nabla_{x^{(i)}} u \right|^2 \epsilon_j x^{(j)} \nabla_{x^{(j)}}\phi dx \nonumber\\&=& - 2\int_{\Omega_R} \lambda_{i} \left|\nabla_{x^{(i)}} u \right|^2 T(\lambda_{i}) \phi dx -I_2- Q \int_{\Omega_R} |\nabla_{\lambda}u |^2 \phi dx -\int_{\Omega_R} |\nabla_{\lambda}u |^2 T(\phi)dx.
\end{eqnarray*}
Since $\lambda_i$ is $\delta_t$-homogeneous of degree $\epsilon_i-1$, then $T(\lambda_{i}) = (\epsilon_i-1) \lambda_{i} $. Hence
\begin{eqnarray*}
I_2&=& - 2\int_{\Omega_R} (\epsilon_i-1)\lambda^2_{i} \left|\nabla_{x^{(i)}} u \right|^2 \phi dx -I_2- Q \int_{\Omega_R} |\nabla_{\lambda}u |^2 \phi dx-\int_{\Omega_R} |\nabla_{\lambda}u |^2 T(\phi)dx \nonumber\\&=&
(2-Q) \int_{\Omega_R} |\nabla_{\lambda}u |^2 \phi dx -2I_1-I_2- \int_{\Omega_R} |\nabla_{\lambda}u |^2 T(\phi)dx.
\end{eqnarray*}
Then
\begin{eqnarray} \label{p5}
I_2= \frac {2-Q}{2} \int_{\Omega_R} |\nabla_{\lambda}u |^2 \phi dx -I_1 - \frac 12 \int_{\Omega_R} |\nabla_{\lambda}u |^2 T(\phi)dx.
\end{eqnarray}
It is easily seen that
\begin{eqnarray} \label{p6}
I_3:&=&\int_{\Omega_R} \lambda^2_{i} \nabla_{x^{(i)}} u \epsilon_j x^{(j)} \nabla_{x^{(j)}} u \nabla_{x^{(i)}} \phi dx\nonumber\\&=& \int_{\Omega_R} \nabla_{\lambda}u \nabla_{\lambda} \phi T(u) dx.
\end{eqnarray}
Hence, by \eqref{p3},
\begin{eqnarray} \label{point1}
-\int_{\Omega_R} \D_{\lambda} u T(u) \phi dx = \frac{2- Q}{2} \int_{\Omega_R} |\nabla_{\lambda}u |^2 \phi dx - \frac 12 \int_{\Omega_R} |\nabla_{\lambda}u |^2 T(\phi) dx+ \int_{\Omega_R} \nabla_{\lambda}u \nabla_{\lambda} \phi T(u)dx.\nonumber\\
\end{eqnarray}
On the other hand, an integration by parts gives
\begin{multline*}
\int_{\Omega_R} |x|^{a}_{\lambda} |u|^{p-1} u T(u) \phi dx = \frac{1}{p+1} \int_{\Omega_{R}} |x|^{a}_{\lambda} \nabla_{x^{(j)}}(|u|^{p+1}) \epsilon_j x^{(j)} \phi dx \\= -\frac{Q}{p+1} \int_{\Omega_{R}} |x|^{a}_{\lambda} |u|^{p+1} \phi -\frac{a}{p+1} \int_{\Omega_R} |x|^{a-1}_{\lambda} |u|^{p+1}T(|x|_{\lambda})\phi-\frac{1}{p+1} \int_{\Omega_R} |x|^{a}_{\lambda} |u|^{p+1} T(\phi) dx.
\end{multline*}
If $T(|x|_{\lambda})= |x|_{\lambda}$, then
\begin{eqnarray} \label{point2}
\int_{\Omega_R} |x|^{a}_{\lambda} |u|^{p-1} u T(u) \phi dx &=& \frac{1}{p+1} \int_{\Omega_R} |x|^{a}_{\lambda} \nabla_{x^{(j)}}(|u|^{p+1}) \epsilon_j x^{(j)} \phi dx \nonumber\\&=& -\frac{Q+a}{p+1} \int_{\Omega_R} |x|^{a}_{\lambda} |u|^{p+1} \phi -\frac{1}{p+1} \int_{\Omega_R} |x|^{a}_{\lambda} |u|^{p+1} T(\phi) dx.
\end{eqnarray}
Clearly \eqref{pohozaev} follows directly from \eqref{point1} and \eqref{point2}. \qed \\\\
\textbf{Proof of Theorem \ref{th2}.} Let $u$ be a solution of \eqref{equation} which is stable outside a compact set. We begin defining some smooth compactly supported functions which will be used several times in the sequel. More precisely, for $R_*>0$, we choose a function $\zeta_{i,R}\in C^2_c(\R^{n_i})$, $i=1,...,k$, $0\leq \zeta_{i,R} \leq 1$, everywhere on $\R^{n_i}$ and
$$\begin{cases}
\zeta_{i,R}(x^{(i)})=0 \;\;&\text{if \;\; $|x^{(i)}|< R_*+1$ \; or \;$|x^{(i)}|> 2 R^{\epsilon_i} $}, \\ \zeta_{i,R}(x^{(i)})=1 \;\;\; &\text{if\;\;$R_* +2<|x^{(i)}|< R^{\epsilon_i} $}, \\
|\nabla_{x^{(i)}} \zeta_{i,R}|^2 + |\D_{x^{(i)}} \zeta_{i,R}|\leq CR^{-2 \epsilon_i}\;\; &\text{for \;\; $\{R^{\epsilon_i} < |x^{(i)}| < 2 R^{\epsilon_i}\}$}.
\end{cases}$$
The rest of the proof splits into several steps.\\
\textbf{\textit{Step 1.}} Let $p>1$. There exists $R_*>0$ such that for every $\gamma \in \left[1,\,\Gamma_M(p)\right)$ and every $R^{\epsilon_i}>R_* +2$, we have \begin{eqnarray} \label{e14}
\int_{\Sigma_0(R)} \left(|x|^{a}_{\lambda} |u|^{p+\gamma}+ |\nabla_{\lambda} ( |u|^{\frac{\gamma-1}{2}}u)|^2 \right) dx \leq C_{R_*}+C R^{Q- \frac{2(p+\gamma) + (\gamma +1) a}{p-1}},
\end{eqnarray}
where $\Sigma_0(R)= \Omega_R \backslash B_1(0, R_*+2)\times ...\times B_k(0, R_*+2)$, $C_{R_*}$ and $C$ are positive constants depending on $p$, $\gamma $, $R_*$ but not on $R$.
Since $u$ is stable outside a compact set of $\R^n$, there exists $R_*>0$ such that, similar to that of Proposition \ref{proposition1} we derive
\begin{eqnarray*}
\int_{\Sigma_0(R)} \left(|x|^{a}_{\lambda} |u|^{p+\gamma}+ |\nabla_{\lambda} ( |u|^{\frac{\gamma-1}{2}}u)|^2 \right) dx &\leq& C(p,m,\gamma)\int_{\R^n} |x|^{\frac{-a (\gamma +1)}{p-1}}_{\lambda}\left(|\nabla_{\lambda} \zeta_{R}|^2 + |\zeta_{R} | |\D_{\lambda} \zeta_{R} |\right)^{\frac{p+\gamma}{p-1}} dx \nonumber\\&\leq& C_{R_*} + C R^{Q- \frac{2(p+\gamma)+ (\gamma +1)a}{p-1}},
\end{eqnarray*}
where $\zeta_{R}=\prod_{i=1}^n \zeta_{i, R}$. Hence, the desired integral estimate \eqref{e14} follows.\\\\
\textbf{\textit{Step 2.}} If $Q=2$ and $1<p<+\infty$ or $Q\geq 3$ and $1<p<\frac{Q+2+ 2a }{Q-2}$, then $u\equiv 0$.\\
By choosing $\gamma=1$ and using \textit{Step $1$}, we get $|x|^{\frac{a}{p+1}}_{\lambda} u\in L^{p+1}(\R^n)$ and $|\nabla_{\lambda} u|\in L^{2}(\R^n)$ for $1<p<p_s(Q,a)$.\\
Take $\phi = \psi_{ R}= \prod_{i=1}^k \psi_{i,R}$ in \eqref{pohozaev} where $\psi_{i,R}$ defined as above. Since $|x|^{\frac{a}{p+1}}_{\lambda} u\in L^{p+1}(\R^n)$ and $|\nabla_{\lambda} u|\in L^{2}(\R^n)$, then
\begin{eqnarray} \label{homogene} \int_{\Sigma_R}|\nabla_{\lambda} u|^2 dx \rightarrow 0, \; \mbox{as}\; R \rightarrow +\infty \quad \mbox{ and } \quad \int_{\Sigma_R} |x|^{a}_{\lambda} |u|^{p+1}dx \rightarrow 0, \; \mbox{as}\; R \rightarrow +\infty,\end{eqnarray}
where $\Sigma_R= \Omega_{2R} \backslash \Omega_{R}$.\\
Recalling that $\lambda_i$ and $\lambda_i \nabla_{x^{(i)}} u$ are $\delta_t$-homogeneous of degree $\epsilon_i-1$ and one respectively. Then, since $T$ generates $(\delta_t)_{t\geq 0}$, we have
\begin{eqnarray} \label{Euler}
T(\lambda_i)= (\epsilon_i-1) \lambda_i \quad \mbox{and} \quad T(\lambda_i \nabla_{x^{(i)}} u)= \lambda_i \nabla_{x^{(i)}} u.
\end{eqnarray}
Integrating by parts and using \eqref{Euler}, we derive
\begin{eqnarray} \label{integration}
&&\int_{\Omega_{2R}} \nabla_{\lambda} u \nabla_{\lambda} \psi_{ R} T(u) = \int_{\Omega_{2R}} \lambda_i \nabla_{x^{(i)}} u \lambda_i \nabla_{x^{(i)}} \psi_{ R} \epsilon_j x^{(j)} \nabla_{x^{(j)}} u\nonumber\\ &=& - \int_{\Omega_{2R}} T(\lambda_i \nabla_{x^{(i)}} u) \lambda_i \nabla_{x^{(i)}} \psi_{ R}\; u - \int_{\Omega_{2R}} \lambda_i \nabla_{x^{(i)}} u T( \lambda_i) \nabla_{x^{(i)}} \psi_{ R} \;u -\int_{\Omega_{2R}} \lambda_i^2 \nabla_{x^{(i)}} u T(\nabla_{x^{(i)}} \psi_{ R}) \; u\nonumber\\&& -\;\; Q \int_{\Omega_{2R}} \nabla_{\lambda} u \nabla_{\lambda}\psi_{ R} \; u \nonumber\\&=& -(Q+1) \int_{\Omega_{2R}} \nabla_{\lambda} u \nabla_{\lambda}\psi_{ R} \; u - \int_{\Omega_{2R}} (\epsilon_i-1)\lambda_i^2 \nabla_{x^{(i)}} u \nabla_{x^{(i)}} \psi_{ R} \;u -\int_{\Omega_{2R}} \lambda_i^2 \nabla_{x^{(i)}} u T(\nabla_{x^{(i)}} \psi_{ R}) \; u \nonumber\\&=& \frac{Q+1}{2} \int_{\Omega_{2R}} u^2 \D_{\lambda}\psi_{ R} + \int_{\Omega_{2R}} \frac{\epsilon_i-1}{2} u^2 \lambda_i^2 \D_{x^{(i)}} \psi_{ R} + \frac 12 \int_{\Omega_{2R}} u^2 \lambda_i^2 \nabla_{x^{(i)}}[T(\nabla_{x^{(i)}} \psi_{ R})]
\end{eqnarray}
By Lemma \ref{lem1}, \eqref{integration} and using H\"{o}lder's inequality, we obtain
\begin{eqnarray}\label{label1}
\left| \int_{\Omega_{2R}} \nabla_{\lambda} u\nabla_{\lambda} \psi_{ R} T(u)\right| &\leq& \frac{C}{R^{-2}} \int_{\Sigma_R} u^2 \nonumber\\&=& \frac{C}{R^{-2}} \int_{\Sigma_R} |x|^{\frac{-2a}{p+1}}_{\lambda} |x|^{\frac{2a}{p+1}}_{\lambda} u^2 \nonumber\\&\leq& C R^{(Q-\frac{2a}{p-1}) \frac{p-1}{p+1}-2} \left(\int_{\Sigma_R}|x|^{a}_{\lambda} |u|^{p+1}\right)^{\frac{2}{p+1}}.
\end{eqnarray}
Similarly, we get
\begin{eqnarray} \label{label2}
\left| \int_{\Omega_{2R}}
\left[- \frac 12 |\nabla_{\lambda} u|^2 + \frac{|x|^{a}_{\lambda}}{p+1} |u|^{p+1}\right] T(\psi_{ R}) \right| \leq C \int_{\Sigma_R}\left(|\nabla_{\lambda} u|^2 + |x|^{a}_{\lambda} |u|^{p+1}\right).
\end{eqnarray}
From \eqref{homogene}, \eqref{label1} and \eqref{label2}, we obtain
\begin{eqnarray*} \label{e016}
\lim_{R \rightarrow +\infty}\left| \int_{\Omega_{2R}} \left( \nabla_{\lambda} u \nabla_{\lambda} \psi_{ R} T(u) +\left[- \frac 12 |\nabla_{\lambda} u|^2 + \frac{|x|^{a}_{\lambda}}{p+1} |u|^{p+1}\right] T(\psi_{ R}) \right) \right| = 0.
\end{eqnarray*}
As a consequence, \eqref{pohozaev} becomes
\begin{eqnarray} \label{e17}
\frac{Q-2}{2} \int_{\R^n} |\nabla_{\lambda} u|^2 dx -\frac{Q+a}{p+1} \int_{\R^n} |x|^{a}_{\lambda} |u|^{p+1} dx = 0.
\end{eqnarray}
On the other hand, multiplying equation \eqref{equation} by $u \psi_{ R} $ and integrating by parts yields
\begin{eqnarray*}
\int_{\R^n} |\nabla_{\lambda} u|^2 \psi_{ R} dx - \int_{\R^n} |x|^{a}_{\lambda} |u|^{p+1} \psi_{ R} dx = \frac{1}{2} \int_{\R^n} u^2 \D_{\lambda} \psi_{ R}dx.
\end{eqnarray*}
Since $1<p<p_s(Q,a)$, we get
\begin{eqnarray*}
\left|\int_{\R^n} u^2 \D_{\lambda} \psi_{ R}dx\right| &\leq& \left(\int_{\R^n} |x|^{a}_{\lambda} |u|^{p+1}dx \right)^{\frac{2}{p+1}} \left(\int_{\Sigma_R} |x|^{\frac{-2a}{p-1}}_{\lambda}|\D_{\lambda} \psi_{ R}|^{\frac{p+1}{p-1}}dx \right)^{\frac{p-1}{p+1}} \nonumber\\ &\leq& C R^{Q\frac{p-1}{p+1}- 2 -\frac{2 a}{p+1}} \rightarrow 0 \; \mbox{as}\; R \rightarrow +\infty.
\end{eqnarray*}
Then
\begin{eqnarray} \label{e18}
\int_{\R^n} |\nabla_{\lambda} u|^2 dx= \int_{\R^n} |x|^{a}_{\lambda} |u|^{p+1}dx .
\end{eqnarray}
To complete the proof we combine \eqref{e17} and \eqref{e18} to get $$\left(\frac{Q-2}{2}- \frac{Q+a}{p+1}\right)\int_{\R^n} |x|^{a}_{\lambda} |u|^{p+1}dx=0, $$
but $\frac{Q-2}{2} - \frac{Q+a}{p+1} \neq0$, since $p$ is subcritical, hence $u$ must be identically zero, as claimed. \qed
\section*{\large{References}}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,388 |
Chicogrande es una película filmada en 2009 y estrenada en mayo de 2010. Fue dirigida por Felipe Cazals y está basada en un texto de Ricardo Garibay.
Trama
Pancho Villa, después de la frustrada invasión a Columbus, emprende la retirada y en Ciudad Guerrero es herido en una pierna por tropas carrancistas. Los estadounidenses en territorio mexicano inician una persecución masiva para capturarlo vivo o muerto. Villa se refugia en la sierra, en lo más profundo de las montañas. Chicogrande, un soldado villista, tiene el encargo de conseguir asistencia médica y no duda en sacrificar su propia vida para lograrlo.
Comentarios
Inaugura el Festival Internacional de Cine de San Sebastián 2010
Reparto
Damián Alcázar como Chicogrande.
Alejandro Calva como Francisco Villa.
Daniel Martínez como Butch Fenton.
Referencias
Enlaces externos
Youtube: Tráiler película
Películas de Damián Alcázar
Películas de Bruno Bichir
Películas de Juan Manuel Bernal
Películas basadas en novelas | {
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} | 8,955 |
Sex Abuse News
Former Rhode Island Catholic Priest Faces Sexual Abuse Charges for Assaulting at Least Two Boys
By Jason SipeNovember 21, 2014June 20th, 2020No Comments
Father Meehan (image credit: Providence Journal)
The Catholic Diocese of Providence in Rhode Island may soon be on the defensive as one of its former priests, Rev. Barry Meehan, has been arraigned on five counts of allegedly sexually assaulting two boys, one while he was pastor of St. Mary Parish in Cranston, RI, in the 1980s and another during his tenure at St. Augustine Catholic Church in Providence in the 1990s.
According to an article by W. Zachary Malinowski of the Providence Journal, The 65 year-old former former pastor of St. Timothy's Church in Warwick was nearly refused bail, but his defense attorney argued that "bail should be set because the sexual assault allegations had been swirling around the retired priest for 2 ½ years."
According to WPRI News, Meehan resigned his position at St. Timothy's about one month after one of the victims reported the abuse they experienced nearly 25 years ago.
Often, survivors of abuse are either too traumatized or too scared to report their abuse, sometimes waiting for decades before reporting the crime. Our law firm often advocates for these survivors, and their courage often inspires other people who have been abused to come forward and to speak out about their abuse.
If you have any information about abuse by Father Barry Meehan or another member of the clergy, we would like to talk to you on a confidential basis. Please contact us toll-free at 888-407-0224, or email us.
Jason Sipe
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As the sun starts its descent behind the mountains, Sarah yawns a bit. Dax looks over. "Let's pull over." Sarah nods.
They find a cozy dirt path off the road that opens to a clearing in the woods, a perfect place to park for the night. Sarah rigs the mosquito net with the back door popped, while Dax goes to relieve himself outside. There is a cool breeze and it will be lovely to sleep with the door open for a change.
Dax and her spent the bulk of the summer working double shifts in part-time jobs, socking away all of their money. They sold both of their beat-up cars to buy this van, and kept the rest to finance the trip. It was important to both of them that they do this themselves and not rely on any loans from their parents. Besides, who knows what trouble they might run into along the way and they'd want to keep the call home for funds as a safety net.
Dax returns to help finish the ritual evening prep. The cooler and duffles go in the front seat. The curtains get hung on the windows. The blankets and pillows get fluffed and straightened. The battery-operated lantern gets turned on. While make-shift and too rough for some, it's home for them. They usually read for an hour before sleep, today has been too magical for that. Instead, they lie next to each other and listen to the night sounds outside the van.
Smiling in silence for quite awhile before they turn to look at each other, they're almost giddy. Dax pulls her closer and kisses her on the forehead and then the nose. Sarah giggles.
He kisses her eyelids and then her cheeks. His lips glaze over her earlobes and then her neck. Her giddiness quickly transforms into something else. His lips move to the other side of her neck and then gently brush her lips. They glide down her chin to her throat and then meet the barrier of her t-shirt's neckline.
It's only been a few short months since their first intimate encounter in the tool shed of the tulip field, but they've wasted no time since. They have explored every inch of the other and continue to do so everyday… almost. There is an ease that they feel with one another that allows them to be totally free.
And right now, Sarah wants more and she's not going to be patient about it. She abruptly jumps up, as much as the roof of the van will allow, and rips off her t-shirt and then her jeans. Dax only needs a second to see her down to her panties and wants to rip the rest off. He pulls her onto the blankets beneath him, then props himself up to rip off his t-shirt and button-flies. It's been a week since they hit a laundry-mat, so he has the benefit of wearing no briefs.
By the look on Sarah's face, she sees it solely as her own benefit. She doesn't wait for Dax to disrobe her remaining garments. She quickly rips them off. Dax growls under his breath as he's perched above her.
She reaches up to run her fingers along his cut abdomen. "You sound like a bear."
He smiles seductively at her. "I feel like a bear."
Sarah smiles as her eyes glance downward and she runs her fingers along his navel and down until she feels the soft hairs leading to the source of his aching. He growls again and quickly lowers himself on top of the naked length of her. This won't be too gentle as the bear takes over. He pushes her legs open and feels the warm folds between them with his fingers. He moves the tips, dancing over the soft flesh. Sarah's mouth falls open with a gentle moan. He covers her mouth with his and takes her moan inside of him until it builds within his belly and he can't take anymore wanting.
They both moan. She wraps her legs around him and her head tilts back. Dax is ignited. His pace quickens and Sarah can feel the vibration building from her toes. It's climbing her legs toward her thighs. His breath is erratic. Sarah pushes harder against him. Dax clutches the blankets below them to steady his need to explode. They both moan as they reach the precipice of release. She can feel the wave wash over her as both of their bodies stiffen and their ravenous desire escapes in a moment of rapture.
Dax collapses on top of her and they both breathe heavily from exhaustion. Dax buries his head in her neck and kisses it. Afraid that he's crushing her, he rolls to his back and pulls her on top of him. "Is that the first time we… together?" Sarah's head rests on his chest, as she nods against it without a word. He smiles with his eyes closed. "Let's do that more."
They quickly drift into a smiling sleep.
About a week later, the van travels the single highway that goes in and out of the keys. Dax drives south. Sarah is napping in the passenger seat. The turquoise blue water surrounds them and the sun beams through the sunroof. The windows are open and the warm breeze wafts through the van and blows Sarah's hair in front of her face. Dax looks at her and smiles. He carefully grabs her camera from the divider and snaps a few photos of her. He doesn't focus of course, he has to keep his eye on the road. But here is wishing for a few good ones. He puts the camera back on its perch and gently moves the hair behind her ear. She doesn't wake. He smiles to himself.
Hours later they are in Key West. Dax pulls into a parking lot of a convenience store just as Sarah stirs. She wipes the sleep from her eyes. "We're here?"
Dax nods.
After they restock the cooler, they find a beach-side watering hole that has hammocks for chairs. Dax and Sarah claim one for their own, along with a couple of Pina Colada's. Dax sucks the straw until he gets a brain freeze. Sarah laughs. "Slow down. I want to go swimming."
Dax holds his head and sucks down the rest of the drink. "Too late. What's the rush anyway."
Sarah laughs and follows his lead. The frozen drink slurps through the straw at the bottom of the empty cup. "Ugh." Sarah grabs her head.
Dax laughs and calls to the waitress. "Two more, please." They both laugh and swing in the hammock.
The waitress returns with the refills. Dax raises his glass toward Sarah's. "To only the beginning." They clink glasses and slurp until they get brain freeze then laugh.
It feels like they have all the time in the world, not a rush in sight. Maybe it's the feeling like their deep in the Caribbean or maybe it's just that they really don't have an agenda. It's whatever they feel like doing, whenever they feel like doing it. They swing in the hammock until the sun sets and they watch the stars twinkle overhead.
That night they pay for a camping spot on the beach and pitch a tent. They've been looking forward to this the whole ride down… sleeping by the surf, gentle ocean breeze, waking up to the waves.
Sarah and Dax lay on top of their sleeping bags with the tent flap open and the mosquito net on. The gentle surf of the Keys lies just beyond the tent opening. But it's not quite what they had imagined. The ocean breeze is non-existent. They've switched the direction of the tent opening several times thinking they were just out of position… but no, the air was as still as a jar. And each time they got out to turn the tent, the mosquitos had a feast on Sarah, even with the gallon of insect repellant that she applied.
Dax helps to rub the all-too-familiar pink lotion on the bites, but the itching still keeps Sarah awake. Dax tosses and turns from the heat and ridiculous lack of air circulation. "Where is the breeze? We're right be the water… but nothing." Sweat beads on his forehead.
Sarah stairs at the pitched ceiling of the tent, willing herself not to scratch, and trying to convince herself that it will all be worth it. "Wait until we wake up next to the water tomorrow."
Dax sits up. "I'm gonna pop in the van for a minute and run the air. Maybe I'll be able to sleep if I cool off for a bit. Dax unzips the mosquito net.
Sarah instantly panics. "Don't let the mosquitos in."
Dax exits quickly and a moment later Sarah hears the car engine hum. In the meantime, Sarah spreads all four limbs out to cool off better. Her skin is a patchwork of pink poke-a-dots. She's trying not to think of the constant itch, because once she starts scratching she won't be able to stop. Outside the car, engine stops and the door slams.
Dax comes back in the tent. "I think that should do it." He zips up the mosquito net again and lies down next to Sarah.
In the stillness, a buzzing sound fills the void. Sarah eyes dart around. Then she starts smacking her arms and legs. "Ew. Mosquitos got in." She continues to smack at her skin. It's enough to make her want to cry.
Dax turns on the flashlight to search for the critter, or critters. They're both swatting at the air. Sweat beads on Dax's forehead again. Sarah is scratching everywhere. The buzzing continues, so their quest to kill the critter is a bust. Dax is full-on sweating now. Sarah is up and dancing around from the itch, nearly bumping the roof and collapsing the tent.
Dax stops and looks at her then at his sweat drenched shirt. He busts out laughing. "Let's get out of here." Sarah nods emphatically. Dax rattles off a plan. "You grab the blankets and make a run for the car." He throws her the car keys. "I'll collapse the tent and be there in a few. Keep the air running." They quickly jump into disassembling action then head north.
Days later, Sarah and Dax exit Café Du Monde in New Orleans' French Quarter with a to-go bag and two coffees. A weathered raisin of a man drums on a set of white containers and a few pots, outside the patio restaurant. The street is full of music and aromas. Sarah has to admit that this city is truly intoxicating. It's like sweet sensory overload.
Dax pulls a beignet from the bag and powdered sugar goes everywhere. Sarah takes a finger to wipe some off his face and lick it. She smiles coyly. Dax grabs her and wipes the white dust all over her shirt. She laughs then grabs her own beignet and showers the powder all over him. They laugh like school children.
They sip coffee as they stroll the walk around Jackson's Square and the French Market. Street musicians, artists and horse drawn carriage drivers compete for the tourists' attention. It's autumn and all is decked out with pumpkins and ghostly themes. They walk and take it all in.
A charcoal artist calls Sarah and Dax over. "How about a sketch of the lovely couple?" They know that their strict budget doesn't allow for street art, no matter how tempting. They politely decline. As they turn to continue their walk, they bump into an older woman dressed like a gypsy. Sarah grabs her hand to steady them both. "I'm sorry."
The woman grasps her hand and looks into Sarah's eyes intently. Then she turns and smiles at Dax. "Your in love." Sarah and Dax smile but don't really respond. Of course, it's obvious they're in love. Sarah is in deep and Dax is right there with her. Yet neither one has uttered the words… not yet. It's been six months and although there were a thousand perfect moments for them to say those three words, they just haven't yet.
Sarah suddenly wants to somehow bind them together more permanently. She wants a symbol that what she's feeling is not going to evaporate into thin air. They continue their walk and pass a tattoo parlor. She stops. "Let's get tattoos."
Dax looks at her oddly. "We just passed on charcoal art, now we're okay with tattoos?"
Sarah points to the sign in the window. Two for One. "Budget be damned. Just something small."
Dax is surprised. "You're serious?"
Sarah nods.
Dax spies in the dusty storefront. "You realize there are needles involved?"
Sarah pulls him by the hand into the shop. The door jingles when the door opens. There is only one customer in the back. His shoes are visible under the privacy curtain and the sound of the machine hums on and off. A voice calls from behind the curtain. "Someone will be right with you. Take a look at the books while you wait. Sarah flips through one of the books.
Dax steps up to another then looks at Sarah. "You're serious serious?"
Sarah laughs.
Dax opens his book. "Ok. What are we getting?"
Sarah looks at him. "Let's not tell each other. It'll be a surprise."
Dax flips through more pages. "Ok. But a few guidelines. No names."
Sarah is giddy with anticipation. "It should be something that symbolizes us as a couple."
Dax flips to a page and holds his place with his hand. "Ankle or wrist only. I don't want any creepy tattoo guy checking out any other parts."
Sarah smiles, holds her place in her book and extends her other hand. "Deal." They shake.
Shortly after, Sarah sits in one artist's chair while he works on her ankle. Dax sits in another while he works on his wrist. Dax calls out to her. "No peeking."
Sarah laughs but bites her finger from the pain of the needles.
The artists finish their work. Both Dax and Sarah step out of the back and are ready to unveil their masterpieces.
Sarah smiles. "Ready?"
Dax nods. "One." "Two." They finish the count of three together and unveil their art. Both of their mouths drop open when they see what the other has done. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,033 |
\subsection{Datasets}
We use five image datasets, namely MNIST, Fashion-MNIST, CIFAR-10,
CIFAR-100, and Tiny-ImageNet. These datasets were often used to evaluate
the classification performance of KD methods \cite{hinton2015distilling,kimura2018fewshot,chen2019data,nayak2019zero,wang2020neural}.
\subsection{Baselines}
We compare our method \textbf{FS-BBT} with the following baselines:
\begin{itemize}
\item \textit{Student-Alone}: the student network is trained on the student's
training data ${\cal D}$ from scratch.
\item \textit{Standard-KD}: the student network is trained with the standard
KD loss in Equation (\ref{eq:KD-loss}). We choose the trade-off factor
$\omega=0.9$, which is a common value used in KD methods \cite{hinton2015distilling,tian2020contrastive,yuan2020revisiting,ma2021undistillable}.
\item \textit{FSKD} \cite{kimura2018fewshot}: this is a few-shot KD method,
which generates synthetic training images using adversarial technique.
It requires a \textit{white-box} teacher model to generate adversarial
samples to train the student network.
\item \textit{WaGe} \cite{kong2020learning}: this few-shot KD method integrates
a Wasserstein-based loss with the standard KD loss to improve the
student's generalization.
\item \textit{BBKD} \cite{wang2020neural}: this method uses few original
images and a \textit{black-box} teacher model to train the student
model. Its main idea is to use MixUp and active learning to generate
synthetic images. Since this is the closest work to ours, we consider
BBKD as our main competitor.
\end{itemize}
To have a fair comparison, we use the same teacher-student network
architecture, the same number of original and synthetic images $N$
and $M$ as in FSKD and BBKD. We also set the same hyper-parameters
(e.g. batch size and the number of epochs) for Student-Alone, Standard-KD,
and our \textbf{FS-BBT}. We use threshold $\alpha=0.05$ to select
qualified mixup images across all experiments. In an ablation study
in Section \ref{subsec:Hyper-parameter-analysis}, we will investigate
how different values for $\alpha$ affect our method's performance.
We repeat each experiment five times with random seeds, and report
the averaged accuracy. For the baselines FSKD, WaGe, and BBKD, we
obtain their accuracy from the papers \cite{kong2020learning,wang2020neural}\footnote{This is possible because we use benchmark datasets, and the training
and test splits are fixed.}. We also compare with several well-known zero-shot KD methods in
Section \ref{subsec:Comparison-with-zero-shot}.
\subsection{Results on MNIST and Fashion-MNIST}
\subsubsection{Experiment settings.}
Following \cite{kimura2018fewshot,nayak2019zero}, we use the LeNet5
architecture \cite{lecun2015lenet} for the teacher and LeNet5-Half
(a modified version with half number of channels per layer) for the
student. We train the teacher network with a batch size of 64 and
20 epochs. As shown in Table \ref{tab:Classification-results-MNIST},
our teacher model achieves comparable accuracy with that reported
by BBKD in \cite{wang2020neural} (99.18\% vs. 99.29\% for MNIST and
90.15\% vs. 90.80\% for Fashion-MNIST). We train the student network
with a batch size of 64 and 50 epochs. We train the CVAE with feed-forward
neural networks for both encoder and decoder, using a latent dimension
of 2, a batch size of 256, and 100 (200) epochs for MNIST (Fashion-MNIST).
Following FSKD \cite{kimura2018fewshot} and BBKD \cite{wang2020neural},
we set $N=2000$ and $M=24000$ for MNIST and $N=2000$ and $M=48000$
for Fashion-MNIST.
The MNIST and Fashion-MNIST datasets have 60K training images and
10K testing images from 10 classes ($[0,1,...,9]$).
\subsubsection{Quantitative results.}
From Table \ref{tab:Classification-results-MNIST}, we can see that
our method \textbf{FS-BBT} outperforms Student-Alone and Standard-KD
on both MNIST and Fashion-MNIST. \textbf{FS-BBT} achieves 98.42\%
(MNIST) and 84.73\% (Fashion-MNIST), which is much better than Student-Alone
achieving 95.97\% and 81.37\%. With a support from the teacher model,
Standard-KD is always better than Student-Alone, for example, 83.87\%
vs. 81.37\% on Fashion-MNIST.
\begin{table}
\caption{\label{tab:Classification-results-MNIST}Classification results on
MNIST and Fashion-MNIST. ``Teacher'' indicates the accuracy of the
teacher network on the test set. ``Model'' indicates whether the
teacher network is a \textit{black-box} model. ``$N$'' shows the
number of original images used by each method. ``Accuracy'' is the
accuracy of the student network on the test set. The results of\textbf{
}FSKD, WaGe, and BBKD\protect\textsuperscript{$\star$} are obtained
from \cite{kong2020learning,wang2020neural}. ``$\star$'' means
the BBKD\protect\textsuperscript{$\star$} and \textbf{FS-BBT}\protect\textsuperscript{$\star$}
methods use the same architecture (LeNet5) for both teacher and student
networks.}
\centering{}%
\begin{tabular}{|l|l|r|c|r|r|}
\hline
\textbf{Dataset} & \textbf{Method} & \textbf{Teacher} & \textbf{Model} & \textbf{$N$} & \textbf{Accuracy}\tabularnewline
\hline
\hline
\multirow{6}{*}{MNIST} & Student-Alone & - & - & 2,000 & 95.97\%\tabularnewline
\cline{2-6}
& Standard-KD & 99.18\% & Black & 2,000 & 95.99\%\tabularnewline
\cline{2-6}
& FSKD \cite{kimura2018fewshot} & 99.29\% & White & 2,000 & 80.43\%\tabularnewline
\cline{2-6}
& BBKD\textsuperscript{$\star$} \cite{wang2020neural} & 99.29\% & Black & 2,000 & 98.74\%\tabularnewline
\cline{2-6}
& \textbf{FS-BBT }(Ours) & 99.18\% & Black & 2,000 & 98.42\%\tabularnewline
\cline{2-6}
& \textbf{FS-BBT}\textsuperscript{$\star$} (Ours) & 99.18\% & Black & 2,000 & \textbf{98.91\%}\tabularnewline
\hline
\multicolumn{6}{c}{}\tabularnewline
\hline
\multirow{7}{*}{Fashion-MNIST} & Student-Alone & - & - & 2,000 & 81.37\%\tabularnewline
\cline{2-6}
& Standard-KD & 90.15\% & Black & 2,000 & 83.87\%\tabularnewline
\cline{2-6}
& FSKD \cite{kimura2018fewshot} & 90.80\% & White & 2,000 & 68.64\%\tabularnewline
\cline{2-6}
& WaGe \cite{kong2020learning} & 92.00\% & White & 1,000 & 85.18\%\tabularnewline
\cline{2-6}
& BBKD\textsuperscript{$\star$} \cite{wang2020neural} & 90.80\% & Black & 2,000 & 80.90\%\tabularnewline
\cline{2-6}
& \textbf{FS-BBT }(Ours) & 90.15\% & Black & 2,000 & 84.73\%\tabularnewline
\cline{2-6}
& \textbf{FS-BBT}\textsuperscript{$\star$} (Ours) & 90.15\% & Black & 2,000 & \textbf{86.53\%}\tabularnewline
\hline
\end{tabular}
\end{table}
Compared with FSKD and WaGe, \textbf{FS-BBT} significantly outperforms
FSKD on both MNIST and Fashion-MNIST while \textbf{FS-BBT} is similar
with WaGe on Fashion-MNIST.
Compared with BBKD, \textbf{FS-BBT} achieves a comparable accuracy
with BBKD on MNIST while \textbf{FS-BBT} outperforms BBKD by a large
margin on Fashion-MNIST, where our accuracy improvement is around
4\%. Since BBKD uses the same architecture LeNet5 for both teacher
and student networks, we also report the accuracy of our method with
this setting, indicated by \textbf{FS-BBT}\textsuperscript{$\star$}.
With LeNet5 for the student network, we further achieve 2\% gain (i.e.
an improvement of 6\% over BBKD) on Fashion-MNIST.
\subsection{Results on CIFAR-10 and CIFAR-100}
\subsubsection{Experiment settings.}
Following \cite{kimura2018fewshot,nayak2019zero}, we use AlexNet
\cite{krizhevsky2012imagenet} and AlexNet-Half (50\% filters are
removed) for teacher and student networks on CIFAR-10. We train the
teacher network with a batch size of 512 and 50 epochs. Our teacher
model achieves a comparable accuracy with that reported by BBKD in
\cite{wang2020neural} (84.07\% vs. 83.07\%). We train the student
network with a batch size of 128 and 100 epochs. We use ResNet-32
\cite{he2016deep} for the teacher and ResNet-20 for the student on
CIFAR-100. We train student and teacher networks with a batch size
of 16/32 and 200 epochs. For both CIFAR-10 and CIFAR-100, we train
the CVAE model with convolutional neural networks for both encoder
and decoder, using a latent dimension of 2, a batch size of 64, and
600 epochs. Like BBKD \cite{wang2020neural} and WaGe \cite{kong2020learning},
we set $N=2000$ for CIFAR-10, $N=5000$ for CIFAR-100, and $M=40000$
for both datasets.
CIFAR-10 is set of RGB images with 10 classes, 50K training images,
and 10K testing images while CIFAR-100 is with 100 classes, and each
class contains 500 training images and 100 testing images. Since neither
the accuracy reference nor the source code is available for BBKD on
CIFAR-100, we implement BBKD by ourselves, and use the same teacher
as in our method for a fair comparison.
\begin{table}
\caption{\label{tab:Classification-results-CIFAR10}Classification results
on CIFAR-10 and CIFAR-100. ``$N$'' shows the number of original
images used by each method. The results of\textbf{ }FSKD, WaGe, and
BBKD\protect\textsuperscript{$\star$} are obtained from \cite{kong2020learning,wang2020neural}.
``$\star$'' means the BBKD\protect\textsuperscript{$\star$} and
\textbf{FS-BBT}\protect\textsuperscript{$\star$} methods use the
same architecture (AlexNet) for both teacher and student networks.
``$\dagger$'' means the result is based on our own implementation.}
\centering{}%
\begin{tabular}{|l|l|r|c|r|r|}
\hline
\textbf{Dataset} & \textbf{Method} & \textbf{Teacher} & \textbf{Model} & \textbf{$N$} & \textbf{Accuracy}\tabularnewline
\hline
\hline
\multirow{7}{*}{CIFAR-10} & Student-Alone & - & - & 2,000 & 54.59\%\tabularnewline
\cline{2-6}
& Standard-KD & 84.07\% & Black & 2,000 & 58.96\%\tabularnewline
\cline{2-6}
& FSKD \cite{kimura2018fewshot} & 83.07\% & White & 2,000 & 40.58\%\tabularnewline
\cline{2-6}
& WaGe \cite{kong2020learning} & 89.00\% & White & 5,000 & 73.08\%\tabularnewline
\cline{2-6}
& BBKD\textsuperscript{$\star$} \cite{wang2020neural} & 83.07\% & Black & 2,000 & 74.60\%\tabularnewline
\cline{2-6}
& \textbf{FS-BBT} (Ours) & 84.07\% & Black & 2,000 & 74.10\%\tabularnewline
\cline{2-6}
& \textbf{FS-BBT}\textsuperscript{$\star$} (Ours) & 84.07\% & Black & 2,000 & \textbf{76.17\%}\tabularnewline
\hline
\multicolumn{6}{c}{}\tabularnewline
\hline
\multirow{5}{*}{CIFAR-100} & Student-Alone & - & - & 5,000 & 32.85\%\tabularnewline
\cline{2-6}
& Standard-KD & 69.08\% & Black & 5,000 & 36.79\%\tabularnewline
\cline{2-6}
& WaGe \cite{kong2020learning} & 47.00\% & White & 5,000 & 20.32\%\tabularnewline
\cline{2-6}
& BBKD\textsuperscript{$\dagger$} \cite{wang2020neural} & 69.08\% & Black & 5,000 & 53.41\%\tabularnewline
\cline{2-6}
& \textbf{FS-BBT} (Ours) & 69.08\% & Black & 5,000 & \textbf{56.28\%}\tabularnewline
\hline
\end{tabular}
\end{table}
\subsubsection{Quantitative results.}
From Table \ref{tab:Classification-results-CIFAR10} we observe the
similar results as in MNIST and Fashion-MNIST. Student-Alone does
not have a good accuracy. Standard-KD improves 4\% of accuracy over
Student-Alone with the knowledge transferred from the teacher.
On CIFAR-10, WaGe and BBKD greatly outperform FSKD, and our \textbf{FS-BBT}
is comparable with WaGe and BBKD. When we use the same architecture
AlexNet for both teacher and student as in BBKD, our variant \textbf{FS-BBT}\textsuperscript{$\star$}
is the best method, where it outperforms BBKD (the second best method)
by around 2\%. \textbf{FS-BBT}\textsuperscript{$\star$} outperforms
WaGe by around 3\% even though WaGe uses much more original training
samples than ours (5K vs. 2K), and more powerful teacher (89\% vs.
84\%).
On CIFAR-100, Student-Alone achieves low accuracy at around 32\%.
Standard-KD is better than Student-Alone around 4\% thanks to the
knowledge transferred from the teacher. Interestingly, WaGe works
very poorly (only 20.32\% of accuracy), becoming the worst method.
Its unsatisfactory performance can be a consequence of distilling
from a low-accuracy teacher. BBKD is significantly better than other
methods with an improvement around 20-30\%. Using the same number
of original and synthetic images, our method \textbf{FS-BBT} achieves
3\% gains over BBKD thanks to the CVAE images generated in Section
\ref{subsec:Proposed-method-FS-BBT}.
The above results suggest that replacing disqualified mixup images
by synthetic images generated from CVAE is an effective solution to
improve the robustness and generalization of the student network on
the unseen testing samples, as we discussed in Section \ref{subsec:Proposed-method-FS-BBT}.
\subsection{Results on Tiny-ImageNet}
\subsubsection{Experiment settings.}
We use ResNet-32 and ResNet-20 for the teacher and student. We train
teacher and student networks with a batch size of 32 and 100 epochs.
Our teacher model achieves a similar accuracy with literature \cite{bhat2021distill}
(52.02\% vs. 48.26\%). We train CVAE in the same way as in CIFAR-100.
We set $N=10000$ and $M=50000$. Tiny-ImageNet has 100K training
images, 10K testing images, and 200 classes.
\begin{table}
\caption{\label{tab:Classification-results-TinyImageNet}Classification results
on Tiny-ImageNet. ``$N$'' shows the number of original images used
by each method. ``$\dagger$'' means the result is based on our
own implementation.}
\centering{}%
\begin{tabular}{|l|l|r|c|r|r|}
\hline
\textbf{Dataset} & \textbf{Method} & \textbf{Teacher} & \textbf{Model} & \textbf{$N$} & \textbf{Accuracy}\tabularnewline
\hline
\hline
\multirow{5}{*}{Tiny-ImageNet} & Student-Alone (full) & - & - & 100,000 & 48.81\%\tabularnewline
\cline{2-6}
& Student-Alone & - & - & 10,000 & 23.19\%\tabularnewline
\cline{2-6}
& Standard-KD & 52.02\% & Black & 10,000 & 35.81\%\tabularnewline
\cline{2-6}
& BBKD\textsuperscript{$\dagger$} \cite{wang2020neural} & 52.02\% & Black & 10,000 & 40.01\%\tabularnewline
\cline{2-6}
& \textbf{FS-BBT} (Ours) & 52.02\% & Black & 10,000 & \textbf{43.29\%}\tabularnewline
\hline
\end{tabular}
\end{table}
\subsubsection{Quantitative results.}
Table \ref{tab:Classification-results-TinyImageNet} shows that Student-Alone
reaches a very low accuracy due to a large number of classes presented
in this dataset. Standard-KD is significantly better than Student-Alone
with an improvement more than 12\%. Our method \textbf{FS-BBT} achieves
3\% gains over BBKD (the second-best baseline).
We also train Student-Alone with full 100K original images and their
soft-labels provided by the teacher. This can be considered as an
upper bound of all few-shot KD methods as it uses the full set of
training images. \textbf{FS-BBT} drops only 5\% accuracy from Student-Alone
with full training data although it requires only 10\% of training
data. This proves the efficacy of our proposed framework.
\subsection{Comparison with zero-shot (or data-free) KD methods\label{subsec:Comparison-with-zero-shot}}
We also compare with several popular zero-shot KD methods, including
\textit{Meta-KD} \cite{lopes2017data}, \textit{ZSKD} \cite{nayak2019zero},
\textit{DAFL} \cite{chen2019data}, \textit{DFKD} \cite{wang2021data},
and \textit{ZSDB3KD} \cite{wang2021zero}.
Table \ref{tab:Classification-data-free} reports the classification
accuracy on MNIST, Fashion-MNIST, and CIFAR-10. Our method is much
better than other methods on Fashion-MNIST and CIFAR-10 while it is
comparable on MNIST.
\begin{table}
\caption{\label{tab:Classification-data-free}Classification comparison with
zero-shot KD methods. The results of baselines are obtained from \cite{wang2021zero}.}
\centering{}%
\begin{tabular}{|l|c|r|r|r|}
\hline
\textbf{Method} & \textbf{Model} & \textbf{MNIST} & \textbf{Fashion-MNIST} & \textbf{CIFAR-10}\tabularnewline
\hline
\hline
Meta-KD \cite{lopes2017data} & White & 92.47\% & - & -\tabularnewline
\hline
ZSKD \cite{nayak2019zero} & White & 98.77\% & 79.62\% & 69.56\%\tabularnewline
\hline
DAFL \cite{chen2019data} & White & 98.20\% & - & 66.38\%\tabularnewline
\hline
DFKD \cite{wang2021data} & White & \textbf{99.08\%} & - & 73.91\%\tabularnewline
\hline
ZSDB3KD \cite{wang2021zero} & Black & 96.54\% & 72.31\% & 59.46\%\tabularnewline
\hline
\textbf{FS-BBT }(Ours) & Black & 98.91\% & \textbf{86.53\%} & \textbf{76.17\%}\tabularnewline
\hline
\end{tabular}
\end{table}
\subsection{Ablation study}
As there are several components and a hyper-parameter $\alpha$ in
our method, we further conduct some ablation experiments to analyze
how each of them affects to our overall classification accuracy. We
select CIFAR-10 for this analysis.
\subsubsection{Different types of synthetic images.\label{subsec:Different-types-of}}
As described in Section \ref{subsec:Proposed-method-FS-BBT}, we generate
three types of synthetic images to train the student network. First,
we generate \textit{mixup images}. Second, we sample $z^{{\cal N}}\sim{\cal N}(0,I)$
to generate CVAE images within the distribution of the original images
(we call them \textit{CVAE-WD images}). Finally, we sample $z^{{\cal U}}\sim{\cal U}([-3,3]^{d})$
to generate CVAE images out-of the distribution of the original images
(we call them \textit{CVAE-OOD images}).
Figure \ref{fig:Original-synthetic-images} shows original images
and three types of synthetic images for four true classes ``car'',
``deer'', ``ship'', and ``dog''. Our synthetic images have good
quality, where the objects are clearly recognized and visualized.
These synthetic images provide a comprehensive coverage of real images
in the test set, resulting in the great improvement of the student
network trained on them.
\begin{figure}[H]
\begin{centering}
\includegraphics[scale=0.3]{figs/synthetic_images}
\par\end{centering}
\caption{\label{fig:Original-synthetic-images}Original images (1\protect\textsuperscript{st}
column) and three types of synthetic images: mixup images (2\protect\textsuperscript{nd}
column), CVAE-WD images (3\protect\textsuperscript{rd} column), and
CVAE-OOD images (4\protect\textsuperscript{th} column). The text
on the left indicates the true labels of original images.}
\end{figure}
Table \ref{tab:Effectiveness-of-different-synthetic} reports the
accuracy of various types of our synthetic images. The standard KD
method achieves only 58.96\% of accuracy. By utilizing mixup images,
our method achieves up to 71.67\% of accuracy. However, using solely
mixup images has disadvantages as we discussed in Section \ref{subsec:Proposed-method-FS-BBT}.
By combining mixup images with CVAE-WD images or CVAE-OOD images,
our method further improves its accuracy up to 72.60\% and 73.25\%
of accuracy respectively. Finally, when combining all three types
of synthetic images, our method achieves the best performance at 74.10\%
of accuracy.
\begin{table}
\caption{\label{tab:Effectiveness-of-different-synthetic}Effectiveness of
different types of synthetic images on our method \textbf{FS-BBT}.}
\centering{}%
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
& KD & \multicolumn{7}{c|}{\textbf{FS-BBT} (Ours)}\tabularnewline
\hline
\hline
mixup images & & $\checkmark$ & & & $\checkmark$ & $\checkmark$ & & $\checkmark$\tabularnewline
\hline
CVAE-WD images & & & $\checkmark$ & & $\checkmark$ & & $\checkmark$ & $\checkmark$\tabularnewline
\hline
CVAE-OOD images & & & & $\checkmark$ & & $\checkmark$ & $\checkmark$ & $\checkmark$\tabularnewline
\hline
\textbf{Accuracy} & 58.96\% & 71.67\% & 70.26\% & 69.42\% & 72.60\% & 73.25\% & 70.63\% & \textbf{74.10\%}\tabularnewline
\hline
\end{tabular}
\end{table}
The ablation experiments suggest that each type of synthetic images
in our method is meaningful, where it greatly improves the student's
classification performance compared to the standard KD method. By
leveraging all three types of synthetic images, our method improves
the generalization and diversity of the training set, which is very
effective for the training of the student network.
\subsubsection{Hyper-parameter analysis.\label{subsec:Hyper-parameter-analysis}}
Our method \textbf{FS-BBT} has one hyper-parameter, that is, the threshold
$\alpha$ to determine disqualified mixup images and replace them
by CVAE images (see Section \ref{subsec:Proposed-method-FS-BBT}).
We examine how the different choices of $\alpha$ affect our classification.
\begin{wrapfigure}{o}{0.5\columnwidth}%
\begin{centering}
\includegraphics[scale=0.3]{figs/accuracy_alpha}
\par\end{centering}
\caption{\label{fig:Hyper-parameter-sensitivity}\textbf{FS-BBT}'s accuracy
vs. threshold $\alpha$ on CIFAR-10.}
\end{wrapfigure}%
As shown in Figure \ref{fig:Hyper-parameter-sensitivity}, \textbf{FS-BBT}
is always better than the standard KD method regardless of $\alpha$
values. More importantly, it is stable with $\alpha\in[0.05,0.10]$,
where its accuracy just slightly changes. When $\alpha$ is too small
(i.e. $\alpha<0.05$), most of mixup images will be considered qualified
although many of them are very similar to the original images, leading
to few extra meaningful training samples added. The performance of
\textbf{FS-BBT} is decreased as expected. When $\alpha$ is too large
(i.e. $\alpha>0.10$), \textbf{FS-BBT} also slightly reduces its accuracy.
This is because many mixup images may become cluttered and semantically
meaningless due to a large proportion of two original images blended
together, making them difficult for the teacher network to label.
\subsection{Problem definition\label{subsec:Problem-definition}}
Given a small set of \textit{unlabeled} images ${\cal X}=\{x_{i}\}_{i=1}^{N}$
and a black-box teacher $T$, our goal is to train a student $S$
on ${\cal X}$ s.t. $S$'s performance is comparable to $T$'s.
A direct solution for the above problem is to apply the standard KD
method \cite{hinton2015distilling}. We first query the teacher to
obtain the \textit{hard-label} (i.e. one-hot encoding) $y_{i}$ for
each sample $x_{i}\in{\cal X}$, and then create a labeled training
set ${\cal D}=\{x_{i},y_{i}\}_{i=1}^{N}$. Finally, we train the student
network with the standard KD loss function:
\begin{equation}
{\cal L}=\sum_{(x_{i},y_{i})\in{\cal D}}(1-\omega){\cal L}_{CE}(y_{x_{i}}^{S},y_{i})+\omega{\cal L}_{KL}(y_{x_{i}}^{S},y_{x_{i}}^{T}),\label{eq:KD-loss}
\end{equation}
where $y_{x_{i}}^{S}$, $y_{x_{i}}^{T}$, $y_{i}$ are the student's
softmax output, the teacher's softmax output, and the hard-label of
a sample $x_{i}$, ${\cal L}_{CE}$ is the cross-entropy loss, ${\cal L}_{KL}$
is the Kullback--Leibler divergence loss, and $\omega$ is a trade-off
factor to balance the two loss terms. Equation (\ref{eq:KD-loss})
does not use the \textit{temperature} factor as in Hinton\textquoteright s
KD method \cite{hinton2015distilling} since this requires access
to the pre-softmax activations (logits) of teacher, which violates
our assumption of \textquotedblleft black-box\textquotedblright{}
teacher.
Although training the student network via Equation (\ref{eq:KD-loss})
is a possible way, it is not a good solution as ${\cal X}$ only contains
very few samples while standard KD methods typically require lots
of training samples \cite{hinton2015distilling,kim2018paraphrasing,ahn2019variational,tian2020contrastive}.
\subsection{Proposed method FS-BBT\label{subsec:Proposed-method-FS-BBT}}
We propose a novel method to solve the above problem, which has three
main steps: (1) we generate mixup images from original images contained
in ${\cal X}$, (2) we replace disqualified mixup images by images
generated from CVAE, and (3) we train the student with a combination
of original, mixup, and CVAE images.
\textbf{Generating mixup images.} Our idea is to use MixUp \cite{zhang2018mixup}
-- one of recently proposed data augmentation techniques to expand
the training set ${\cal X}$.
Inspired by BBKD \cite{wang2020neural}, we generate $M$ mixup images
from $N$ original images (typically, $N\ll M$). Given two original
images $x_{i},x_{j}\in{\cal X}$, we use MixUp to generate a synthetic
image by a weighted combination between $x_{i}$ and $x_{j}$:
\begin{equation}
x_{mu}(\lambda)=\lambda x_{i}+(1-\lambda)x_{j},\label{eq:mixup-image}
\end{equation}
where the coefficient $\lambda\in[0,1]$ is sampled from a Beta distribution.
Let $X=[x_{1},x_{2},...,x_{N}]$ be the vector of original images.
We first sample two $M$-length vectors $X^{1}=[x_{1}^{1},x_{2}^{1},...,x_{M}^{1}]$
and $X^{2}=[x_{1}^{2},x_{2}^{2},...,x_{M}^{2}]$, where $x_{i}^{1},x_{i}^{2}\sim X$.
We then sample a vector $\lambda=[\lambda_{1},\lambda_{2},...,\lambda_{M}]$
from a Beta distribution, and mixup each pair of two images in $X^{1}$
and $X^{2}$ using Equation (\ref{eq:mixup-image}):
\begin{equation}
X_{mu}=\left[\begin{array}{c}
\lambda_{1}x_{1}^{1}+(1-\lambda_{1})x_{1}^{2}\\
\lambda_{2}x_{2}^{1}+(1-\lambda_{2})x_{2}^{2}\\
...\\
\lambda_{M}x_{M}^{1}+(1-\lambda_{M})x_{M}^{2}
\end{array}\right]\label{eq:X-mixup}
\end{equation}
The goal of mixing up original images is to expand the initial set
of training images ${\cal X}$ as much as possible to cover the manifold
of natural images.
\begin{wrapfigure}{o}{0.5\columnwidth}%
\begin{centering}
\includegraphics[scale=0.5]{figs/mixup_images}
\par\end{centering}
\caption{\label{fig:Examples-of-mixup_images}Desirable vs. disqualified mixup
images. At $\lambda=0.58$, the mixup image shows a good combination
between two original images ``horse'' and ``ship'' but at $\lambda=0.98$,
it looks almost the same as ``horse''.}
\end{wrapfigure}%
However, when mixing up two original images, there is a case that
the mixup image is very similar to one of two original images, making
it useless. This problem happens when $\lambda_{i}\approx0$ or $\lambda_{i}\approx1$.
Figure \ref{fig:Examples-of-mixup_images} shows two examples of desirable
vs. disqualified mixup images.
To remove disqualified mixup images, we set a threshold $\alpha\in[0,0.5]$,
and discard mixup images generated with coefficient $\lambda_{i}\leq\alpha$
or $\lambda_{i}\geq(1-\alpha)$.
Let $M_{1}$ be the number of remaining mixup images after we filter
out the disqualified ones. Our next step is to generate $M_{2}=M-M_{1}$
synthetic images from CVAE (we call them CVAE images).
\textbf{Generating CVAE images.} We first query the teacher model
to obtain the hard-label $y_{i}$ for each sample $x_{i}\in{\cal X}$
to create a labeled training set ${\cal D}=\{x_{i},y_{i}\}_{i=1}^{N}$.
We then train a Conditional Variational Autoencoder (CVAE) model \cite{sohn2015learning}
using ${\cal D}$ to learn the distribution of the latent variable
$z\in\mathbb{R}^{d}$, where $d$ is the dimension of $z$. CVAE is
a generative model consisting of an encoder and a decoder. We use
the encoder network to map an image along with its label $(x,y)\in{\cal D}$
to a latent vector $z$ that follows $P(z\mid y)$. From the latent
vector $z$ conditioned on the label $y$, we use the decoder network
to reconstruct the input image $x$. Following \cite{sohn2015learning},
we train CVAE by maximizing the variational lower bound objective:
\begin{align}
\log P(x & \mid y)\geq\mathbb{E}(\log P(x\mid z,y))-\text{KL}(Q(z\mid x,y),P(z\mid y)),\label{eq:cvae-loss}
\end{align}
where $Q(z\mid x,y)$ is parameterized by the encoder network that
maps input image $x$ and its label $y$ to the latent vector $z$,
$P(x\mid z,y)$ is parameterized by the decoder network that reconstructs
input image $x$ from the latent vector $z$ and label $y$, $\mathbb{E}(\log P(x\mid z,y))$
is the expected likelihood, which is implemented by a cross-entropy
loss between the input image and the reconstructed image, and $P(z\mid y)\equiv{\cal N}(0,I)$
is the prior distribution of $z$ conditioned on $y$.
After the CVAE model is trained, we can generate images via $G(z,y)$,
where $z\sim{\cal N}(0,I)$, $y$ is a label, and $G$ is the trained
decoder network.
\textbf{Covering both in-distribution and out-of-distribution samples.}
To generate $M_{2}$ CVAE images, we sample ($\frac{M_{2}}{2}$)-length
vector $z^{{\cal N}}$ from the normal distribution ${\cal N}(0,I)$
and ($\frac{M_{2}}{2}$)-length vector $z^{{\cal U}}$ from the uniform
distribution ${\cal U}([-3,3]^{d})$ (we choose the range $[-3,3]$
following \cite{higgins2017beta,gyawali2019semi}). We create vector
$z=z^{{\cal N}}\oplus z^{{\cal U}}$, where $\oplus$ is the concatenation
operator. We manually define a $M_{2}$-length vector $y_{cvae}$,
which contains the classes of generated images such that the number
of generated images for each class is equivalent. Finally, we generate
CVAE images $x_{cvae}=G(z,y_{cave})$.
The intuition behind our generation process is that: (1) Generating
images from $z^{{\cal N}}\sim{\cal N}(0,I)$ will provide \textit{synthetic
images within the distribution} of ${\cal X}$. These images are interpolated
versions of original images. (2) Generating images from $z^{{\cal U}}\sim{\cal U}([-3,3]^{d})$
will provide \textit{synthetic images out-of the distribution} of
${\cal X}$. These images are far way from the original ones, but
they are expected to better cover \textit{unseen images,} which improves
the student's generalization.
\textbf{Discussion.} One can sample $\lambda_{i}\in[\alpha,1-\alpha]$
to generate $M_{1}$ qualified mixup images, then generate $M_{2}$
CVAE images. This way requires two hyper-parameters $M_{1}$ and $M_{2}$.
While this is definitely possible, for simplicity we choose to aggregate
these two hyper-parameters into a single hyper-parameter $M$ that
controls the total number of synthetic images. In experiments, we
set the same values for $M$ as those in other few-shot KD methods
\cite{kimura2018fewshot,wang2020neural} while $M_{1}$ and $M_{2}$
are automatically computed based on $M$ and $\alpha$.
\textbf{Training the student network.} After the above steps, we obtain
two types of synthetic images -- mixup and CVAE images. We send them
to the teacher model to obtain their softmax outputs (i.e. their class
probabilities) as the \textit{soft-labels} for the images. We train
the student network with the original and synthetic images along with
their soft-labels using the following loss:
\begin{equation}
{\cal L}=\sum_{x_{i}\in{\cal X}\cup{\cal X}_{mu}\cup{\cal X}_{cvae}}{\cal L}_{CE}(y_{x_{i}}^{S},y_{x_{i}}^{T}),\label{eq:our-loss}
\end{equation}
where $y_{x_{i}}^{S}$, $y_{x_{i}}^{T}$ are the student's and the
teacher's softmax output, ${\cal X}$, ${\cal X}_{mu}$, ${\cal X}_{cvae}$
are the set of original, mixup, and CVAE images, and ${\cal L}_{CE}$
is the cross-entropy loss. Although we train the student by matching
the teacher's softmax outputs, our loss function is still applicable
in case the teacher only returns top-1 labels \cite{wang2021zero}.
Algorithm \ref{alg:The-proposed-KDFS} summarizes our proposed method
\textbf{FS-BBT}.
\begin{algorithm}
\caption{\label{alg:The-proposed-KDFS}The proposed \textbf{FS-BBT} algorithm.}
\LinesNumbered
\KwIn{$T$: pre-trained \textit{black-box} teacher network}
\KwIn{$\mathcal{X}=\{x_{i}\}_{i=1}^{N}$: \textit{unlabeled} training
set}
\KwIn{$M$: number of synthetic images}
\KwIn{$\alpha$: threshold to select mixup images}
\KwOut{$S$: student network}
\Begin{
query teacher $T$ to obtain hard-label $y_{i}$ for each $x_{i}\in{\cal X}$\;
train CVAE model using ${\cal D}=\{x_{i},y_{i}\}_{i=1}^{N}$\;
sample $\lambda=[\lambda_{1},...,\lambda_{M}]$ from a Beta distribution\;
select $M_{1}$ instances of $\lambda_{i}$ s.t. $\alpha<\lambda_{i}<1-\alpha$\;
generate $M_{1}$ mixup images ${\cal X}_{mu}$ using Eq. (\ref{eq:X-mixup})\;
compute $M_{2}=M-M_{1}$\;
sample ($\frac{M_{2}}{2}$)-length vector $z^{{\cal N}}\sim{\cal N}(0,I)$\;
sample ($\frac{M_{2}}{2}$)-length vector $z^{{\cal U}}\sim{\cal U}([-3,3]^{d})$\;
create vector $z=z^{{\cal N}}\oplus z^{{\cal U}}$\;
design $M_{2}$-length vector $y_{cvae}$ with class balance\;
generate $M_{2}$ CVAE images ${\cal X}_{cvae}=G(z,y_{cvae})$\;
query teacher $T$ to obtain soft-labels for ${\cal X}$, ${\cal X}_{mu}$,
${\cal X}_{cvae}$\;
train student $S$ with ${\cal X},{\cal X}_{mu},{\cal X}_{cvae}$
and their soft-labels using Eq. (\ref{eq:our-loss})\;
}
\end{algorithm}
\section{Introduction\label{sec:Introduction}}
\input{introduction.tex}
\section{Related Works\label{sec:Related-Works}}
\input{relatedwork.tex}
\section{Framework\label{sec:Framework}}
\input{framework.tex}
\section{Experiments and Discussions\label{sec:Experiments}}
\input{experiment.tex}
\section{Conclusion}
\input{conclusion.tex}
\textbf{Acknowledgment:} This research was fully supported by the
Australian Government through the Australian Research Council\textquoteright s
Discovery Projects funding scheme (project DP210102798). The views
expressed herein are those of the authors and are not necessarily
those of the Australian Government or Australian Research Council.
\bibliographystyle{splncs04}
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Socha kan syfta på följande platser:
Colombia
Socha (kommun), Boyacá,
Socha Viejo, ort, Boyacá, | {
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{"url":"http:\/\/surgery.matrix.jp\/math\/kamiya\/program.html","text":"# Complex Hyperbolic Geometry and Related Topics\n\n### Meeting in honor of Shigeyasu Kamiya's Retirement\n\n(Okayama University of Science, 1-1 Ridaicho, Kita-ku, Okayama, Okayama)\n\n## Program\n\npdf(English, 2 sheets), pdf(English, 1 sheet), pdf(Japanese, 2 sheets), pdf(Japanese, 1 sheet) : for A4 paper\n\nAbstracts: abstracts.pdf\n\n January 9th, Friday (Rm 10241) 2:00 -- 3:00 Seade, Jos\u00e9 On the limit set of complex Kleinian groups [slides] 3:20 -- 4:20 Kim, Joonhyung A characterization of complex hyperbolic Kleinian groups with trace fields contained in $${\\mathbb R}$$ 4:40 -- 5:40 Hasegawa, Keizo Non-K\u00e4hler complex geometric structures on homogeneous spaces [slides] January 10th, Saturday (Rm 22115) 10:00 -- 11:00 Parker, John Parabolic isometries of hyperbolic spaces and discreteness [slides] 11:15 -- 12:15 Kim, Youngju Parabolic quasiconformal conjugacy classes in the Heisenberg group 1:30 -- 2:30 Tan, Ser Peow Polynomial automorphisms of $${\\mathbb C}^n$$ preserving the Markoff-Hurwitz polynomial 2:50 -- 3:50 Komori, Yohei Arithmetic aspects of growth rates for hyperbolic Coxeter groups [slides] 4:10 -- 5:10 Kamiya, Shigeyasu Complex hyperbolic triangle groups of type $$(n,n,\\infty)$$ 5:45 -- 7:45 Party (Bldg 11, 8th floor) January 11th, Sunday (Rm 22115) 10:00 -- 11:00 Morimoto, Masaharu An additively closed subset of the Smith set in the real representation ring 11:15 -- 12:15 Apanasov, Boris Deformations of hyperbolic structures, nontrivial 4-cobordisms and homomorphisms of their fundamental groups","date":"2019-05-25 15:25:28","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\": 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\": 0.7011140584945679, \"perplexity\": 14133.590131839608}, \"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-2019-22\/segments\/1558232258120.87\/warc\/CC-MAIN-20190525144906-20190525170906-00329.warc.gz\"}"} | null | null |
Q: What is the word of Spanish or Portuguese origin starting with "a" and meaning enthusiast? There is a word starting with "a" (along the lines of "afinados") meaning enthusiast, connoisseur or fan. What is it?
A: Aficionado?
: a person who likes, knows about, and appreciates a usually fervently pursued interest or activity : devotee
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Q: Positive operators in $B_2(H)$ (Hilbert Schmidt operators) I was reading the paper Sum of small numbers of idempotents where they have proved that every operator on a Hilbert space can be written as a complex linear combination of 16 projections. Which also means that positive operators are dense in $B(H)$.
I was wondering if this is true in $B_2(H)$ as well. Does the set $$S=\{T\in B_2(H) : \left<Tx,x\right>\geq 0 ~\forall x\in H\}$$
span whole of $B_2(H)$ in Hilbert Schmidt norm?
| {
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var socket = io();
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Q: Vue js 2 - Accessing data in App.vue from child components I'm pretty new to Vue and I'm trying to create my first app with it (I have worked through a whole tutorial though so it's not like I'm completely new to it). I come from a Java/Grails background so this new "frontend oriented webapps" is still pretty confusing to me. I'm using Vue 2 with webpack.
The problem I'm having is running methods when the app initializes which creates data in the App.vue component (which I'm assuming is the root component, is that correct?) and then accessing this data in child components. So specifically what I'm trying to do is on in the 'created' life cycle hook I want to check if the user is logged in or not and then I want to update my navbar accordingly (show a login button if not, else show the user's name, for example).
I haven't even quite figured out how exactly I'm gonna determine if the user is logged in yet coz so far I've only been trying to create dummy data in the App.vue component and then accessing it in child components. Everywhere that I've researched says that I should use an event bus but (I think) that's not gonna work for me (correct me if I'm wrong) coz the only examples I can find is all $emit functions being called on user events (like button clicks) and I want it to have the data globally accessible (and mutate-able) everywhere all the time.
So I'll try to show you some code of what I had in mind:
App.vue:
...stuff...
data() {
return {
authToken = '',
userdetails = {},
loggedIn = false
}
},
created: function() {
// check browser storage for authToken
if(authToken) {
// call rest service (through Vue Resource) to Grails backend that I set up
// beforehand and set this.userdetails to the data that gets returned
if(this.userdetails) {
this.loggedIn = true;
}
}
}
...stuff...
Home.vue:
<template>
<div class="home">
<nav-bar></nav-bar>
</div>
</template>
...stuff
NavBar.vue:
<template>
<div class="navBar">
<div v-if="loggedIn">Hi {{ userdetails.name }}</div>
<div v-else>Please log in before continuing.</div>
</div>
</template>
Please excuse if any of that code has any mistakes in it, it's just to show more or less what I'm trying to do and I made most of it up right now. So the main question: How do I go about getting the v-if="loggedIn" and {{ userdetails.name }} part to work (coz obviously the way it's set up now that won't work, right?). And then besides that, any general advice on "global variables" and data flow in Vue js will be appreciated coz I believe that my server-side app mentality might not work in front-end javascript apps.
A: To get Data from parent component you can use this.$parent.userdetails in the child component.
But a much better way is to use props like this
https://v2.vuejs.org/v2/guide/components.html#Passing-Data-with-Props
A: Lite pattern, not for complex sharing tasks
You can access global app component everywhere (routes, components, nested components) with the predisposed field:
this.$root
Notes for overengineers: It is officially suggested as "lite" approach in vue guide: https://v2.vuejs.org/v2/guide/state-management.html.
Obviously, it is not a architectural patternized solution, but ok for just sharing a variable or two.
In these case this lean approach avoids developer to import a full sharing framework like vuex, heaving development, know-how requirement and page download, just for such a trivial sharing task.
If you dislike lite approaches, even when appropriated and officially suggested, maybe is not how to share the problem, but which framework to use.
For instance, Angular is probably best fitted for heavy engineered approach, and I not mean one is better than the other, I mean different instruments suite best for different task and approaches, I too use it when need to realize a heavier and more complex solution, in these case even with vuex I could not handle all the components interaction and realtime data sharing.
A: Following methods are not recommended by other developers or vue itself.
this.$root;
or
this.$parent.userdetails
In my opinion the best way to share any data across components is using vuex state management.
Here is the official documentation link: https://vuex.vuejs.org/#what-is-a-state-management-pattern
| {
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Paine, Thomas (1737-1809), American writer and philosopher.
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Perry, Oliver Hazard (1785-1819), American naval officer.
Peter, Laurence J. (1919-1990), Canadian writer.
Phelps, William Lyon (1865-1943), American educator and literary critic.
Picasso, Pablo Ruiz y (1881-1973), Spanish artist.
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Prochnow, Herbert Victor (1897-1998), American banker.
Proust, Marcel Valentin-Louis-George-Eugene (1871-1922), French author.
Putnam, Israel (1718-1790), American general.
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LA CROSSE, Wis. (WXOW) – La Crosse Police continue their investigation after a body is found in a backyard in a south side neighborhood Thursday afternoon.
Around 1:40 p.m., officers were called to a home in the 1600 block of Johnson Street.
La Crosse Police Sgt. Tom Walsh said a person shoveling snow discovered the body in a fenced backyard and called police.
At this point, investigators are attempting to determine the identity of the person.
Sgt. Walsh wasn't able to say what the connection is between the home and the person.
He added that it appears at this time there is no threat to the community. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,090 |
<?php
/* guestbook_add_action-starter.php Revision 4 4-25-13 5:30 pm
Requirements:
a) Please do not remove any of my comments in this code. I need them for grading.
b) Type code for only one step at a time, then run it in your browser to test it before moving to the next step.
Some of the code will not display any output, but you still need to test it to be sure there are no error messages.
Trouble-shooting is very difficult if many steps are entered at once. Do not ask me for help with your code if you are
not testing one step at a time!
==========================================================================================================
PLEASE DO NOT ATTEMPT TO WORK THROUGH THIS FILE WITHOUT FOLLOWING MY INSTRUCTIONS IN ASSIGNMENT 4, PART 2.
I HAVE A SPECIFIC SERIES OF STEPS I WOULD LIKE YOU TO FOLLOW.
==========================================================================================================
*/
// HEADER
// 1. Update the Header information below (all 3 lines).
/*
File Name: guestbook_add_action.php
Date: 10/17/2015
Programmer: Keith Murphy
*/
// ==========================================================
// VARIABLES
// 2. Add a variable named $missing_count after the $heading variable, and assign a value of 0 (zero) to it.
// Remember that quote marks are *not* used around numeric values. This variable will be used to track the
// number of fields that are missing data.
$styles_path = "styles";
$styles_page = "home.css";
$styles_2_page = "contact.css";
$link_1_page = "home.php";
$link_1_text = "Home";
$link_2_page = "guestbook_add.php";
$link_2_text = "Guestbook: Add";
$heading = "Contact Us";
$missing_count = 0;
// ==========================================================
// FUNCTIONS
// 3. Enter your 4 functions below this sentence. Put a blank line between each of the functions.
// Notice that we load the functions *before* we run any HTML, so they are available to the rest of our code below this.
function check_submitted($field_name, $field_type, &$missing_count) {
// Check for undefined variable (not submitted) on all but checkbox
if(!isset($_POST[$field_name])) {
$_POST[$field_name]=""; // set a default value if no value was submitted, to prevent errors in later code
if($field_type <> "checkbox") { // checkboxes usually don't have to be checked -- they are usually optional
echo "Missing data for <strong>" . $field_name . "</strong>.<br />";
$missing_count++;
}
}
// For text, text area, and select check for present but empty
// Note use of elseif instead of if, which means only one of the if blocks will run but not both
elseif($field_type == "text" || $field_type == "textarea" || $field_type == "select") {
if(trim($_POST[$field_name]) == "") {
echo "Missing data for <strong>" . $field_name . "</strong>.<br />";
$missing_count++;
} // end if($_POST...)
} // end if($field_type...)
} // end function
function count_errors($missing_count) {
if($missing_count > 0) {
echo "<br />Please <a href='guestbook_add.php'>Go Back</a> and fill in the missing data.<br /><br /></div></body></html>";
exit;
}
}
function sanitize($field_name, $field_type, &$field_value) {
if($field_type == "text" || $field_type == "textarea") {
$_POST[$field_name] = trim($field_value);
$_POST[$field_name] = stripslashes(strip_tags($field_value)); // strip html tags and back slashes
$_POST[$field_name] = str_replace("/","",$field_value); // removes forward slashes
$_POST[$field_name] = addslashes($field_value); // escapes quote marks so they will work in SQL statements
echo "The field <b>" . $field_name . "</b> has been sanitized.<br>";
}
}
function display_data($field_name, $field_type, &$field_value) {
if($field_type == "checkbox") {
if($field_value != "") {
echo $field_name . ": <strong>" . $field_value . "</strong>";
}
else {
echo $field_name . ": <strong>no</strong>";
}
} // end field type checkbox
else {
echo $field_name . ": <strong>" . $field_value . "</strong><br /><br />";
} // end else for field type checkbox
} // end function
// =======================================
// HTML HEADER ?>
<!DOCTYPE html>
<html>
<head>
<meta charset = "UTF-8">
<title><?php echo $heading; ?></title>
<link rel = "stylesheet" href = "<?php echo $styles_path . '/' . $styles_page; ?>" />
<link rel = "stylesheet" href = "<?php echo $styles_path . '/' . $styles_2_page; ?>" />
</head>
<body>
<div class = "shade">
<br />
<!-- We will code the line below later in the class, when we learn how to do logins. -->
Logged in as: <b>(no login code yet)</b>
<p>
<!-- NAVIGATION BAR -->
<a href="<?php echo $link_1_page; ?>"><?php echo $link_1_text; ?></a> |
<a href="<?php echo $link_2_page; ?>"><?php echo $link_2_text; ?></a>
</p>
</div>
<!-- ====================================== -->
<!-- The line below is a class that I created to mimic the action of the 'blockquote' tag in HTML, which is now deprecated. -->
<div class="blockquote">
<?php
// FUNCTION CALLS
// -- CHECK EACH FIELD FOR MISSING DATA
/* 4. Enter your check_submitted() function calls below this comment. There should be a separate call for each of
the four fields in your form.
Put a blank line between each function call. */
check_submitted("name", "text", $missing_count);
check_submitted("email", "text", $missing_count);
check_submitted("comment", "textarea", $missing_count);
check_submitted("mail", "checkbox", $missing_count);
// 4a. Enter your function call for count_errors below this comment. There should be only one line of code.
count_errors($missing_count);
// Below this point is your our old code for checking for missing data.
// Notice that you had more code, and it did less -- it didn't track how many fields were missing.
// Once you create the functions and call them, please delete the $counter line and the 'if' blocks in this section.
// -- SANITIZE FIELDS (REMOVE DANGEROUS CHARACTERS) -- text boxes and textarea only
// 5. Enter your sanitize() function calls below this comment. There should be a call for every text box or text area
// box on your form. Put a blank line between each function call.
sanitize("name", "text", $_POST["name"]);
sanitize("email", "text", $_POST["email"]);
sanitize("comment", "text", $_POST["comment"]);
// Below this point is your our old code for checking for sanitizing the data.
// Notice that you had a lot more code, and it did less -- we didn't escape quote marks in the previous version.
// Once you create the functions and call them, please delete the old code in this section.
// -- DISPLAY OUTPUT
// 6. Enter your display_data() function calls below this comment. There should be a call for each of the fields in your
// form. Put a blank line between each function call.
echo "<h3><i>You submitted the following information:</i></h3>";
echo "<div id='formData'>";
display_data("name", "text", $_POST["name"]);
display_data("email", "text", $_POST["email"]);
display_data("comment", "text", $_POST["comment"]);
display_data("mail", "checkbox", $_POST["checkbox"]);
echo "</div>";
// Below this point is your our old code for displaying the data.
// Once you create the functions and call them, please delete the old code in this section.
?>
<br><br><a href="guestbook_add.php">Return to Form</a>
</div>
<hr />
<!-- ===================================================== -->
<!-- FOOTER -->
</body>
</html> | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,272 |
{"url":"https:\/\/iitutor.com\/factorising-non-monic-quadratic-trinomials-perfect-squares\/","text":"# Factorising Non-Monic Quadratic Trinomials: Perfect Squares\n\n## Transcript\n\nWhat I\u2019m going to do? I\u2019m going to start with 4x and 4x. Remember I said I usually like to start with the ones that are closer together. I don\u2019t know I always have a tendency to start with the ones that are closer together because usually, they\u2019re the right answer. So I\u2019ll try 4x and 4x and if that doesn\u2019t work I\u2019ll switch around and try a different one like two and eight.\n\nBut I\u2019ll start with this first of all. This one\u2019s easy because it\u2019s just one, so it\u2019s just going to be one and one. Now, look! 4x times 1 is 4x, 4x times 1 is 4x, and together they make 8x which is this one. If you did have 2 and 8, you really can\u2019t make 8 with those numbers, can you? That\u2019s why trying to pick the right number at the very beginning if you can.\n\nSo, it\u2019s simply going to be 4x plus 1, 4x plus 1 which is, how do we simplify that guys? 4x plus 1 squared! the perfect square! Because we\u2019ve got two of a kind, okay? Now it\u2019s not x\u2019s we\u2019ve got a is and b is involved but that\u2019s okay we\u2019ll stick to that, that\u2019s all right. 4a squared, I\u2019m going to use 2a and 2a. You could use 4a and 1a but again I just like to use the ones that are closer together.\n\nAnd then, 9b squared that would be 3b and 3b. I\u2019m assuming this might turn into a perfect square, don\u2019t you think? Let\u2019s try. 2a or sorry 2a and 3b is 6ab, 2a and 3b is also 6ab, so together they make 12ab which is exactly what we\u2019ve got. So, 2a plus 3b, 2a plus 3b. How do we simplify that? So so simple. 2a plus 3b squared, okay? So perfect square again.","date":"2022-08-09 04:59:36","metadata":"{\"extraction_info\": {\"found_math\": false, \"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\": 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\": 0.8279300332069397, \"perplexity\": 757.5527189971172}, \"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-2022-33\/segments\/1659882570901.18\/warc\/CC-MAIN-20220809033952-20220809063952-00400.warc.gz\"}"} | null | null |
Q: Wording on SOFU About page The About page on SOFU currently says:
You can register if you want to
collect karma and win valuable flair
that will appear next to your name,
but otherwise, it's just free. And
fast. Very, very fast.
Shouldn't karma be replaced with reputation?
A: No, because we are using the terms generically here "flair" and "karma" in the abstract sense to refer to the concepts.
Not literally.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,264 |
Board index » Other than Jersey City, NJ » NOT Jersey City » IF YOU PHONE AND DRIVE, YOU'RE AN IDIOT - IT'S OFFICIAL!
notcrazy4u
Re: IF YOU PHONE AND DRIVE, YOU'RE AN IDIOT - IT'S OFFICIAL!
AMEN--------I just laugh to myself when I see Jersey City's finest chatting away and DRIVING. But I forgot, they are above the law . Better yet, I saw the best stunt yet, by our jcpd. The cop was chatting on the phone and eating a powder donut WHILE DRIVING. Talk about multitasking. They want us to abide by the law, but what type of example are they setting? The majority of them are OVERWEIGHT,SENIOR,BOLD AND HAVE A HUGH BUTT . They should lead by example------NOT BE A POOR EXAMPLE.
Posted on: 2008/1/7 5:51
jennymayla
Vigilante wrote:
I see JCPD yapping away while on "patrol" all the time. Always setting a good example!!
I see that all the time. Mind boggling.
I know a lady who keeps a plastic banana in her car and pretends to talk on it when she sees people driving and talking on their cell phones. Pretty hilarious and makes the point in a clever way.
Drive safely, folks.
From Leashless Glory.
fat-ass-bike
From 280 Grove Street
You're also an idiot if you don't cross at the crosswalk while on the phone or more interested in social talking on the phone when you have a customer infront of you.
My humor is for the silent blue collar majority - If my posts offend, slander or you deem inappropriate and seek deletion, contact the webmaster for jurisdiction.
My favorite is seeing the light-rail conductors chatting away as they're whizzing past me...you've got to wonder!
Eleanor_A
From Hamilton Park
I'm used to seeing some idiot driving his car while on the phone but the other day I witnessed one of those mini bull dozers in traffic changing lanes a little erratically. I pull up to the guy at the next light and sure enough he's on his flippin' cell phone gabbing away.
IF YOU PHONE AND DRIVE, YOU'RE AN IDIOT - IT'S OFFICIAL!
An article I thought could be of interest.
If you phone and drive, you're also a bloody idiot
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Eamonn Duff
USING a hands-free mobile phone behind the wheel could be more dangerous than drink driving, new research has shown.
Motorists under the influence of alcohol performed better than those driving while talking on a hand-held or hands-free phone, the study by Britain's Transport Research Laboratory found.
It also found that the risk of a crash was four times higher when the driver was on the phone.
The research has now prompted one of the world's biggest transport firms, FirstGroup, to last week ban all its 135,000 workers on bus and train services in Britain and North America from using mobile phones, including hands-free kits, while driving on company business.
The laboratory's Nick Reed said the latest research Conversations in cars: the relative hazards of mobile phones , showed that drivers revealed a significant impairment when making mobile phone calls while driving. "In some aspects of driving behaviour, speaking on a mobile phone is worse than being at the legal alcohol limit," he said. (The limit is 0.8 in Britain).
"The observed impairment was similar regardless of whether the call was made using a hand-held phone or using a hands-free kit."
Dr Reed said the results matched other studies showing that drivers had less control of their vehicle, and showed reduced awareness of their surroundings when making mobile phone calls. "This results in an accident risk that is four times greater and that persists for up to 10 minutes after the call has been completed," he said.
Part of the danger was that the person at the other end was not aware of how distracting the conversation could be, and was not aware of surrounding road conditions, Dr Reed said.
"Chatting to a passenger can be distracting, but its less so than having a mobile call. The passenger can see the traffic around you and can maybe pick up on your body language cues, then modify conversation accordingly."
In a 2002 study, the laboratory found that drivers on mobiles had slower reaction times and stopping times than those under the influence of drink. Drivers who were just over the British legal drink-driving limit stopped in an average distance of 35 metres. But drivers using hand-free mobile phones took 39 metres to stop.
The chairman of the Pedestrian Council of Australia, Harold Scruby, said there was no point calling for a blanket ban on mobile phones in cars until the "basic laws" surrounding hand-held phones were tightened.
"The offence for using a hand-held should become one of 'dangerous driving' instead of the current slap on the wrist," he said. "That includes the loss of 11 demerit points attached to the penalty, and confiscation of the phone ? it would stop the trend overnight."
Mr Scruby said while the issue remained a state jurisdiction, it was time the Federal Government took the lead.
Prime Minister Kevin Rudd was the first prime minister he had ever heard talking about driving safely at Christmas.
"His father was killed in a crash," he said.
"I would love to now see him and the Federal Government become more active in road safety, instead of handing it on to the states who are doing next to nothing." | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 274 |
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"redpajama_set_name": "RedPajamaC4"
} | 7,846 |
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Copyright © 2017 The Shack Coffee Shop & Beer and Wine Garden, Outer Banks NC. All Rights Reserved. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,065 |
\section{#1}}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\renewcommand\Im{\mathrm{Im}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{hypothesis}[theorem]{Hypothesis}
\newcommand{\arxivlink}[1]{\href{http://arxiv.org/abs/#1}{arXiv:#1}}
\begin{document}
\title[Weak Interactions in an uniform magnetic field]{ spectral theory for the weak decay of muons in a uniform magnetic field}
\author[J.-C. Guillot]{Jean-Claude Guillot}
\address[J.-C. Guillot]{Centre de Math\'ematiques Appliqu\'ees, UMR 7641, \'Ecole
Polytechnique - C.N.R.S, 91128 Palaiseau Cedex, France}
\email{Jean-Claude.Guillot@polytechnique.edu}
\subjclass[2010]{81V15, 81V10, 81Q10}
\keywords{Weak Decay of muons,Fermi's theory, Uniform Magnetic Field, Spectral Theory,}
\maketitle
\begin{center}
\textit{In memory of Raymond Stora}
\end{center}
\begin{abstract}
In this article we consider a mathematical model for the weak decay of muons in a uniform magnetic field according to the Fermi theory of weak interactions with V-A coupling. With this model we associate a Hamiltonian with cutoffs in an appropriate Fock space. No infrared regularization is assumed. The Hamiltonian is self-adjoint and has a unique ground state. We specify the essential spectrum and prove the existence of asymptotic fields from which we determine the absolutely continuous spectrum. The coupling constant is supposed sufficiently small.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction.}
\setcounter{equation}{0}
In this paper we consider a mathematical model for the weak decay of muons into electrons, neutrinos and antineutrinos in a uniform magnetic field according to the Fermi theory with V-A (Vector-Axial Vector)coupling,
\begin{equation}\label{1}
\mu_{-}\rightarrow e_{-} + \overline{\nu_{e}} + \nu_{\mu}
\end{equation}
\begin{equation}\label{2}
\mu_{+}\rightarrow e_{+} + \nu_{e} + \overline{\nu}_{\mu}
\end{equation}
\eqref{2} is the charge conjugation of \eqref{1}.
This is a part of a program devoted to the study of mathematical models for the weak interactions as patterned according to the Fermi theory and the Standard model in Quantum Field Theory. See \cite{GM}.
In this paper we restrict ourselves to the study of the decay of the muon $\mu_{-}$ whose electric charge is the charge of the electron \eqref{1}. The study of the decay of the antiparticle $\mu_{+}$, whose charge is positive, \eqref{2} is quite similar and we omit it.
In \cite{G17} we have studied the spectral theory of the Hamiltonian associated to the inverse $\beta$ decay in a uniform magnetic field. We proved the existence and uniqueness of a ground state and we specify the essential spectrum and the spectrum for a small coupling constant and without any low-energy regularization.
In this paper we consider the weak decay of muons into electrons, neutrinos associated with muons and antineutrinos associated with electrons in a uniform magnetic field according to the Fermi theory with V-A coupling. Hence we neglect the small mass of neutrinos and antineutrinos and we define a total Hamiltonian H acting in an appropriate Fock space involving three fermionic massive particles-the electrons, the muons and the antimuons - and two fermionic massless particles- the neutrinos and the antineutrinos associated to the muons and the electrons respectively.In order to obtain a well-defined operator, we approximate the physical kernels of the interaction Hamiltonian by square integrable functions and we introduce high-energy cutoffs. We do not need to impose any low-energy regularization in this work but the coupling constant is supposed sufficiently small.
We give a precise definition of the Hamiltonian as a self-adjoint operator in the appropriate Fock space and by adapting the methods used in \cite{G17}we first state that H has a unique ground state and we specify the essential spectrum for sufficiently small values of the coupling constant.
In this paper, our main result is the location of the absolutely continuous spectrum of H. For that we follow the first step of the approach to scattering theory in establishing, for each involved particle,the existence and basic properties of the asymptotic creation and annihilation operators for time $t$ going to $\pm\infty$. We then have a natural definition of unitary wave operators with the right intertwining property from which we deduce the absolutely continuous spectrum of H. Scattering theory for models in Quantum Field Theory without any external field has been considered by many authors. See, among others, \cite{AF,Ammari2004,BFS, DG, FGS, H, HK1, HK2, HK3,HK4, HK5, HS, KM1, KM2, KM3,T1, T2} and references therein. A part of the techniques used in this paper are adapted from the ones developed in these references. Note that the asymptotic completeness of the wave operators is an open problem in the case of the weak interactions in the background of a uniform magnetic field. See \eqref{1} for a study of scattering theory for a mathematical model of the weak interactions without any external field.
In some parts of our presentation we will only give the statement of theorems referring otherwise to some references.
The paper is organized as follows. In the second section we define the regularized self-adjoint Hamiltonian associated to \eqref{1}. In the third section we consider the existence of a unique ground state and we specify the essential spectrum of H. In the fourth section we carefully prove the existence of asymptotic limits, when time t goes to $\pm\infty$, of the creation and annihilation operators of each involved particle, we define a unitary wave operator and we prove that it satisfies the right intertwining property with the hamiltonian and we deduce the absolutely continuous spectrum of H. In Appendices A and B we recall the Dirac quantized fields associated to the muon and the electron in a uniform magnetic external field together with the Dirac quantized free fields associated to the neutrino and the antineutrino.
\section{The Hamiltonian.}\mbox{}
\setcounter{equation}{0}
In the Fermi theory the decay of the muon $\mu$ is described by the following four fermions effective Hamiltonian for the interaction in the Schr\"{o}dinger representation ( see \cite{GM}, \cite{GS} and \cite{WII}):
\begin{equation}\label{2.1}
\begin{split}
&H_{int} =\\&\frac{G_F}{\sqrt{2}} \int \d^3 \!x\,\big( \overline{\Psi}_{\nu_\mu}(x) \gamma^\alpha
(1-\gamma_5)\Psi_{\mu}(x)\big)\big( \overline{\Psi}_{e}(x) \gamma_{\alpha}(1-\gamma_5)\Psi_{\nu_e}(x)\big)\\
& + \frac{G_F}{\sqrt{2}} \int \d^3 \!x\,\big( \overline{\Psi}_{\nu_e}(x) \gamma_{\alpha}(1-\gamma_5)\Psi_{e}(x)\big)
\big( \overline{\Psi}_{\mu}(x) \gamma^\alpha(1-\gamma_5)\Psi_{\nu_\mu}(x)\big)
\end{split}
\end{equation}
Here $\gamma^{\alpha}$, $\alpha=0,1,2,3$ and $\gamma_5$ are the Dirac matrices in the standard representation. $\Psi_{(.)}(x)$ and $\overline{\Psi}_{(.)}(x)$ are the quantized Dirac
fields for $e$, $\mu$, $\nu_\mu$ and $\nu_e$. $\overline{\Psi}_{(.)}(x)= \Psi_{(.)}(x)^\dag \gamma^{0}$. $G_{F}$ is the Fermi coupling constant with $G_{F}\simeq 1.16639(2)\times10^{-5}GeV^{-2}$. See \cite{B}.
We recall that $m_e<m_\mu$. $\nu_\mu$ and $\nu_e$ are massless particles.
\subsection{The free Hamiltonian.}\mbox{}
Throughout this work notations are introduced in appendices A and B.
Let
\begin{equation}\label{2.2}
\begin{split}
\mathfrak{F}&=\mathfrak{F}_e\otimes\mathfrak{F}_\mu\otimes\mathfrak{F}_\mu\otimes\mathfrak{F}_{\nu_\mu}\otimes\mathfrak{F}_{\overline{\nu}_e}
\end{split}
\end{equation}
Let
\begin{equation}\label{2.3}
\begin{split}
&\omega(\xi_1)=E_n^{(e)}(p^3)\quad for\ \xi_1= (s,n,p^1,p^3) \\
&\omega(\xi_2)=E_n^{(\mu)}(p^3)\quad for\ \xi_2= (s,n,p^1,p^3) \\
&\omega(\xi_3)=|\textbf{p}|\quad for\ \xi_3= (\textbf{p} , \frac{1}{2})\\
&\omega(\xi_4)= |\textbf{p}|\quad for\ \xi_4= (\textbf{p} , -\frac{1}{2}).
\end{split}
\end{equation}
Let $H_D^{(e)}$ \big( resp.$H_D^{(\mu_-)}$,$H_D^{(\mu_+)}$, and $H_D^{(\nu)}$ \big) be the Dirac Hamiltonian for the electron \big(resp.the muon, the antimuon and the neutrino \big).
The quantization of $H_D^{(e)}$, denoted by $H_{0,D}^{(e)}$ and acting on $\mathfrak{F}_{e}$, is given by
\begin{equation}\label{2.4}
H_{0,D}^{(e)}==\int \omega(\xi_1) b^{*}_{+}(\xi_{1}) b_{+}(\xi_{1})\d \xi_{1}
\end{equation}
Likewise the quantization of $H_D^{(\mu_-)}$,$H_D^{(\mu_+)}$,$H_D^{(\overline{\nu}_e)}$ and $H_D^{(\nu_\mu)}$, denoted by $H_{0,D}^{(\mu)}$, $H_{0,D}^{(\overline{\nu}_e)}$ and $H_{0,D}^{(\nu_\mu)}$ respectively, acting on $\mathfrak{F}_\mu$, $\mathfrak{F}_{\overline{\nu}_e}$ and $\mathfrak{F}_{\nu_\mu}$ respectively, is given by
\begin{equation}\label{2.5}
\begin{split}
&H_{0,D}^{(\mu_-)}=\int \omega(\xi_2) b^{*}_{+}(\xi_{2}) b_{+}(\xi_{2})\d \xi_{2}\, \\
&H_{0,D}^{(\mu_+)}=\int \omega(\xi_2) b^{*}_{-}(\xi_{2}) b_{-}(\xi_{2})\d \xi_{2}\, \\
&H_{0,D}^{(\overline{\nu}_e)}=\int \omega(\xi_3) b^{*}_{-}(\xi_{3}) b_{-}(\xi_{3})\d \xi_{3} \\
&H_{0,D}^{(\nu_\mu)}= \int \omega(\xi_4) b^{*}_{+}(\xi_{4}) b_{+}(\xi_{4})\d \xi_{4}.
\end{split}
\end{equation}
We set $H_{0,D}^{(\mu)}$ = $H_{0,D}^{(\mu_-)}\otimes \1 $ +$ \1 \otimes H_{0,D}^{(\mu_+)}$. $H_{0,D}^{(\mu)}$ is defined on $\mathfrak{F}_\mu \otimes\mathfrak{F}_\mu$.
For each Fock space $\mathfrak{F}_{.}$ let $\mathfrak{D}^{(.)}$ denote the set of vectors $\Phi\in\mathfrak{F}^{(.)}$ for which each component $\Phi^{(r)}$ is smooth and has a compact support and $\Phi^{(r)} = 0$ for all but finitely many $r$. Then $H_{0,D}^{(.)}$ is
well-defined on the dense subset $\mathfrak{D}^{(.)}$ and it is essentially self-adjoint on $\mathfrak{D}^{(.)}$ . The
self-adjoint extension will be denoted by the same symbol $H_{0,D}^{(.)}$ with domain $D(H_{0,D}^{(.)})$).
The spectrum of $H_{0,D}^{(e)}$ in $\mathfrak{F}_{(e)}$ is given by
\begin{equation}\label{2.6}
\mathrm{spec}(H_{0,D}^{(e)})=\{0\}\cup [m_e,\infty)
\end{equation}
$\{0\}$ is a simple eigenvalue whose the associated eigenvector is the vacuum in $\mathfrak{F}^{(e)}$ denoted by $\Omega^{(e)}$. $[m_e,\infty)$ is the absolutely continuous spectrum of $H_{0,D}^{(e)}$.
Likewise the spectra of $H_{0,D}^{(\mu)}$, $H_{0,D}^{(\overline{\nu}_e)}$ and $H_{0,D}^{(\nu_\mu)}$ in $\mathfrak{F}_{(\mu)}\otimes \mathfrak{F}_{(\mu)}$, $\mathfrak{F}_{(\overline{\nu_e})}$ and $\mathfrak{F}_{(\nu_\mu)}$ respectively are given by
\begin{equation}\label{2.7}
\begin{split}
&\mathrm{spec}(H_{0,D}^{(\mu)})=\{0\}\cup [m_{\mu},\infty) \\
&\mathrm{spec}(H_{0,D}^{(\overline{\nu_e})})=[0,\infty)\\
&\mathrm{spec}(H_{0,D}^{(\nu_\mu)})=[0,\infty).
\end{split}
\end{equation}
$\Omega^{(\mu)}$, $\Omega^{(\overline{\nu}_e)}$ and $\Omega^{(\nu_\mu)}$ are the associated vacua in $\mathfrak{F}_{(\mu)}\otimes \mathfrak{F}_{(\mu)}$, $\mathfrak{F}_{(\overline{\nu_e})}$ and $\mathfrak{F}_{(\nu_\mu)}$ respectively and are the associated eigenvectors of $H_{0,D}^{(\mu)}$,$H_{0,D}^{(\overline{\nu}_e)}$ and $H_{0,D}^{(\nu_\mu)}$ respectively for the eigenvalue $\{0\}$.
The vacuum in $\mathfrak{F}$, denoted by $\Omega$, is then given by
\begin{equation}\label{2.8}
\Omega=\Omega^{(e)} \otimes \Omega^{(\mu)} \otimes \Omega^{(\overline{\nu_e})} \otimes \Omega^{(\nu_\mu)}
\end{equation}
The free Hamiltonian for the model, denoted by $H_0$ and acting in $\mathfrak{F}$, is now given by
\begin{equation}\label{2.9}
\begin{split}
&H_0=H_{0,D}^{(e)} \otimes \1 \otimes \1 \otimes \1 \otimes \1 + \1 \otimes H_{0,D}^{(\mu)}\otimes \1 \otimes\1 \otimes 1 \\
& + \1 \otimes \1 \otimes \1 \otimes H_{0,D}^{(\overline{\nu_e})}\otimes \1 + \1 \otimes \1 \otimes \1\otimes \1 \otimes H_{0,D}^{(\nu_\mu)}.
\end{split}
\end{equation}
$H_0$ is essentially self-adjoint on $\mathfrak{D}=\mathfrak{D}^{(e)}\widehat\otimes\mathfrak{D}^{(\mu)}\widehat\otimes\mathfrak{D}^{(\overline{\nu_e})}\widehat\otimes\mathfrak{D}^{(\nu_\mu)}$.
Here $\widehat\otimes$ is the algebraic tensor product.
$\mathrm{spec}(H_0)=[0,\infty)$ and $\Omega$ is the eigenvector associated with the simple eigenvalue $\{0\}$ of $H_0$.
Let $S^{(e)}$ be the set of the thresholds of $H_{0,D}^{(e)}$:
$$S^{(e)}=\big(s^{(e)}_n; n\in \N \big)$$
with $s^{(e)}_n=\sqrt{m_{e}^2 + 2neB}$.
Likewise let $S^{(\mu)}$ be the set of the thresholds of $H_{0,D}^{(\mu)}$:
$$S^{(\mu)}=\big(s^{(\mu)}_n; n\in \N \big)$$
with $s^{(\mu)}_n=\sqrt{m_{\mu}^2 + 2neB}$.
Then
\begin{equation}\label{2.10}
\mathfrak{S}= S^{(e)}\cup S^{(\mu)}
\underline{}\end{equation}
is the set of the thresholds of $H_0$.
Throughout this work any finite tensor product of annihilation or creation operators associated with the involved particles will be denoted for shortness by the usual product of the operators (see e.g \eqref{2.13} and \eqref{2.14}).
\subsection{The Interaction.}\mbox{}
Similarly to \cite{BG} \cite{ABFG}, \cite{BFG1}, \cite{BFG2} \cite{BFG3},\cite{G15}and \cite{G17}, in order to get well-defined operators on $\mathfrak{F}$, we have to substitute
smoother kernels $F(\xi_2,\xi_4)$ and $G(\xi_1,\xi_3)$ for the $\delta$-distribution associated with \eqref{2.1}( conservation of momenta) and for introducing ultraviolet cutoffs.
Let
\begin{equation}\label{2.11}
\textbf{r}= \textbf{p}_3 + \textbf{p}_4
\end{equation}
We get a new operator denoted by $H_I$ and defined as follows
\begin{equation}\label{2.12}
H_I= H_I^1 + (H_I^1)^{*} + H_I^{2} + (H_I^{2})^{*}
\end{equation}
Here
\begin{equation}\label{2.13}
\begin{split}
&H^{(1)}_{I}= \int \d \xi_1 \d \xi_2 \d \xi_3 \d \xi_4\,\Big(\int \d x^2 \mathrm{e}^{-ix^2r^2} \\
&\big( \overline{U}^{(\nu_\mu)}(\xi_4) \gamma^\alpha(1-\gamma_5)U^{(\mu)}(x^2,\xi_2)\big)\big( \overline{U}^{(e)}(x^2,\xi_1) \gamma_{\alpha}(1-\gamma_5)U^{(\nu)}(\xi_4)\big)\Big)\\
&F(\xi_{2}, \xi_4)\,G(\xi_1,\xi_3)b_{+}^{*}(\xi_4)b_{+}^{*}(\xi_1)b_{-}^{*}(\xi_3)b_{+}(\xi_2).
\end{split}
\end{equation}
and
\begin{equation}\label{2.14}
\begin{split}
&H^{(2)}_{I}= \int \d \xi_1 \d \xi_2 \d \xi_3 \d \xi_4\,\Big(\int \d x^2 \mathrm{e}^{-ix^2r^2} \\
&\big( \overline{U}^{(\nu_\mu)}(\xi_4) \gamma_{\alpha}(1-\gamma_5)W^{(\mu)}(x^2,\xi_2)\big)\big( \overline{U}^{(e)}(x^2,\xi_1) \gamma^\alpha(1-\gamma_5)W^{(\overline{\nu_e})}(\xi_3)\big)\Big)\\
&F(\xi_{2}, \xi_{4})\,G(\xi_1,\xi_3)b_{+}^{*}(\xi_4)b_{-}^{*}(\xi_2)b^{*}_{+}(\xi_1)b^{*}_{-}(\xi_3).
\end{split}
\end{equation}
$H^{(1)}_{I}$ describes the decay of the muon and $H^{(2)}_{I}$ is responsible for the fact that the bare vacuum will not be an eigenvector of the total Hamiltonian as expected from physics.
We now introduce the following assumptions on the kernels $F(\xi_{2}, \xi_{4})$ and $G(\xi_1,\xi_3)$ in order to get well-defined Hamiltonians in $\mathfrak{F}$.
\begin{hypothesis}\label{hypothesis:2.1}
\begin{equation}\label{2.15}
\begin{split}
&F(\xi_2, \xi_4)\in
L^2(\Gamma_1\times \R^3) \\
&G^(\xi_1, \xi_3)\in
L^2(\Gamma_1\times \R^3)
\end{split}
\end{equation}
\end{hypothesis}
These assumptions will be needed throughout the paper.
By \eqref{2.12}-\eqref{2.15} $H_I$ is well defined as a sesquilinear form on $\mathfrak{D}$ and one can construct a closed operator associated with this form.
The total Hamiltonian is thus
\begin{equation}\label{2.16}
H= H_0 + gH_I, \quad g>0.
\end{equation}
$g$ is the coupling constant that we suppose non-negative for simplicity. The conclusions below are not affected if $g\in\R$.
The self-adjointness of $H$ is established by the next theorem.
Let
\begin{equation}\label{2.17}
\begin{split}
C= &\|\gamma^0\gamma_\alpha (1-\gamma_5)\|_{\C^4}\|\gamma^0\gamma^\alpha (1-\gamma_5)\|_{\C^4}. \\
\frac{1}{M}=&\frac{1}{m_e} + \frac{1}{m_\mu}
\end{split}
\end{equation}
For $\phi \in D(H_0)$ we have
\begin{equation}\label{2.18}
\begin{split}
\|H_I\phi\| & \\
\leq & 2C\|F(.,.)\|_{L^2(\Gamma_1\times\R^3)}\|G(.,.)\|_{L^2(\Gamma_1\times\R^3)}\big(\frac{2}{M}\|H_0\phi\| + \|\phi\| \big).
\end{split}
\end{equation}
\eqref{2.18} follows from standard estimates of creation and annihilation operators in Fock space (the $N_\tau$ estimates, see \cite{GJ}). Details can be found in \cite[proposition 3.7]{BDG}.
\begin{theorem}\label{2.2}(Self-adjointness).
Let $g_0>0$ be such that
\begin{equation}\label{2.19}
4g_0\frac{C}{M}\|F(.,.)\|_{L^2(\Gamma_1\times\R^3)}\|G(.,.)\|_{L^2(\Gamma_1\times\R^3)}<1.
\end{equation}
Then for any g such that $g\leq g_0$ H is self-adjoint in $\mathfrak{F}$ with domain $\mathcal{D}(H)= \mathcal{D}(H_0)$. Moreover any core for $H_0$ is a core for $H$.
\end{theorem}
By \eqref{2.18} and \eqref{2.19} the proof of the self-adjointness of $H$ follows from the Kato-Rellich theorem.
$\sigma(H)$ stands for the spectrum and $\sigma_{ess}(H)$ denotes the essential spectrum. We have
\begin{theorem}\label{2.3}(The essential spectrum and the spectrum)
Setting
\begin{equation*}
E=\inf\sigma(H)
\end{equation*}
we have for every $g\leq g_0$
\begin{equation*}
\sigma(H)=\sigma_{\mathrm{ess}}(H)= [E,\infty)
\end{equation*}
with $E\leq 0$ .
\end{theorem}
In order to prove the theorem \ref{2.3} we easily adapt to our case the proof given in \cite{BFG3} (see also \cite{A}, \cite{T3} and \cite{G17}) . The mathematical model considered in \cite{BFG3} involves also one neutrino and one antineutrino. We omit the details.
\section{Existence of a unique ground state.}
In the sequel we shall make some of the following additional assumptions on the kernels $F(\xi_2,\xi_4)$ and $G(\xi_1,\xi_3)$.
\begin{hypothesis}\label{hypothesis:3.1} There exists a constant $K(F,G)>0$
such that for $\sigma>0$
\begin{equation*}
\begin{split}
\mbox{(i)}\quad &
\int_{\Gamma_1\times\R^3}
\frac{|
F(\xi_2,\xi_4)|^2}{|\textbf{p}_{4}|^2}
\d\xi_1 \d\xi_4 <\infty .
\end{split}
\begin{split}
\mbox{(ii)}\quad &
\int_{\Gamma_1\times\R^3}
\frac{|
G(\xi_1,\xi_3)|^2}{|\textbf{p}_{3}|^2}
\d\xi_1 \d\xi_3 <\infty .
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\mbox{(iii)}\quad &
\left(\int_{\Gamma_1\times\{|\textbf{p}_{4}|\leq \sigma\}}
|F(\xi_2,\xi_4)|^2
\d\xi_2 \d\xi_4 \right)^{\frac{1}{2}} \leq K(F,G) \sigma.
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\mbox{(iv)}\quad &
\left(\int_{\Gamma_1\times\{|\textbf{p}_{3}|\leq \sigma\}}
|G(\xi_1,\xi_3)|^2
\d\xi_1 \d\xi_3 \right)^{\frac{1}{2}} \leq K(F,G) \sigma.
\end{split}
\end{equation*}
\end{hypothesis}
We then have
\begin{theorem}\label{3.1}
Assume that the kernels $F(.,.)$ and $G(.,.)$ satisfy Hypothesis 2.1 and 3.1. Then there exists $g_1\in (0,g_0]$ such that $H$ has a unique ground state for $g \leq g_1$.
\end{theorem}
In order to prove theorem 3.1 it suffices to mimick the proofs given in \cite{ABFG}, \cite{BFG3} and \cite{G17}. We omit the details.
In \cite{AFG} fermionic hamiltonian models are considered without any external field. Without any restriction on the strengh of the interaction a self-adjoint hamiltonian is defined for which the existence of a ground state is proved. Such a result is an open problem in the case of magnetic fermionic models.
\section{The absolutely continuous spectrum.} \mbox{}
\setcounter{equation}{0}
As stated in the introduction , in order to specify the absolutely continuous spectrum of H, we follow the first step of the approach to scattering theory in establishing, for each involved particle, the existence and basic properties of the asymptotic creation and annihilation operators for time $t$ going to $\pm\infty$. The existence of a ground state is quite fundamental in order to get a Fock subrepresentation of the asymptotic canonical anticommutation relations from which we localize the absolutely continuous spectrum of H.
\subsection{Asymptotic fields.} \mbox{}
Let
\begin{equation}\label{4.1}
\begin{split}
b_{1,+,t}^{\sharp}(f_1) &= e^{itH}e^{-itH_0} b_{1,+}^{\sharp}(f_1)e^{itH_0}e^{-itH} \\
b_{2,\pm,t}^{\sharp}(f_2) &= e^{itH}e^{-itH_0} b_{2,\pm}^{\sharp}(f_2)e^{itH_0}e^{-itH} \\
b_{3,-,t}^{\sharp}(f_3)&= e^{itH}e^{-itH_0} b_{3,-}^{\sharp}(f_3)e^{itH_0}e^{-itH} \\
b_{4,+,t}^{\sharp}(f_4)&= e^{itH}e^{-itH_0} b_{4,+}^{\sharp}(f_4)e^{itH_0}e^{-itH}.
\end{split}
\end{equation}
where, for $i=1,2$, $f_i \in L^2(\Gamma_1)$ and, for $j=3,4$, $f_j \in L^2(\R^3)$.
The strong limits of $b_{.,t}^{\sharp}(.)$ when the time t goes to $\pm\infty$ for models in Quantum Field Theory have been considered for fermions and bosons by \cite{KM1}-\cite{KM3} and \cite{HK1}-\cite{HK5} and , more recently, by \cite{DG}, \cite{Ammari2004}, \cite{H}, \cite{T1}, \cite{T2} and \cite{BDH} and references therein.
In the sequel we shall make some of the following additional assumptions on the kernels $F(\xi_2,\xi_4)$ and $G(\xi_1,\xi_3)$.
\begin{hypothesis}\label{hypothesis: 4.1}
\begin{equation*}
\begin{split}
\mbox{(i)}\quad &
\int\left|
\frac{\partial F}
{\partial p_{\mu}^3}(\xi_2,\xi_4)\right|^2
\d\xi_2 \d\xi_4 < \infty\ ,
\int\left|
\frac{\partial G}
{\partial p_{e}^3}(\xi_1,\xi_3)\right|^2
\d\xi_1 \d\xi_3 < \infty\ .\\
\mbox{(ii)}\quad &
\int\left|
\left(\frac{\partial}
{\partial p_{\mu}^3}\right)^2F(\xi_2,\xi_4)\right|^2
\d\xi_2 \d\xi_4 < \infty\ ,
\int\left|
\left(\frac{\partial}
{\partial p_{e}^3}\right)^2G(\xi_1,\xi_3)\right|^2
\d\xi_1 \d\xi_3 < \infty\ .
\end{split}
\end{equation*}
\end{hypothesis}
\begin{hypothesis}\label{hypothesis: 4.2}
\begin{equation*}
\begin{split}
\mbox{(i)}\quad &
\int\left|
\nabla_{p_\mu}F
(\xi_2,\xi_4)\right|^2
\d\xi_2 \d\xi_4 < \infty\ ,
\int\left|
\nabla_{p_e}G
(\xi_1,\xi_3)\right|^2
\d\xi_1 \d\xi_3 < \infty\ . \\
\mbox{(ii)}\quad &
\int\left|\frac{\partial^2 F}
{\partial p_{\nu_\mu}^1 \partial p_{\nu_\mu}^3}(\xi_2,\xi_4,)\right|^2
\d\xi_1 \d\xi_2 < \infty\ ,
\int\left|\frac{\partial^2 G}
{\partial p_{\overline{\nu}_e}^1 \partial p_{\overline{\nu}_e}^3}(\xi_1,\xi_3,)\right|^2
\d\xi_1 \d\xi_3 < \infty\ .
\end{split}
\end{equation*}
\end{hypothesis}
We then have
\begin{theorem}\label{4.1}
Suppose Hypothesis 2.1-Hypothesis 4.2 and $g \leq g_0$. Let $f_1,f_2 \in L^2(\Gamma_1)$ and $f_3,f_4 \in L^2(\R^3)$ . Then the following asymptotic fields
\begin{equation}\label{4.2}
\begin{split}
b_{1,+,\pm\infty}^{\sharp}(f_1) &:=\slim_{t\to\pm\infty} b_{1,+,t}^{\sharp}(f_1) \\
b_{2, \pm,\pm\infty}^{\sharp}(f_2)&:=\slim_{t\to\pm\infty} b_{2,\pm,t}^{\sharp}(f_2) \\
b_{3,-,\pm\infty}^{\sharp}(f_3)&:=\slim_{t\to\pm\infty}b_{3,-,t}^{\sharp}(f_3) \\
b_{4,+,\pm\infty}^{\sharp}(f_+)&:=\slim_{t\to\pm\infty} b_{4,+,t}^{\sharp}(f_4) .
\end{split}
\end{equation}
exist.
\end{theorem}
\begin{proof}
The norms of the $b_{.,.,t}(f_.)'s$ are uniformly bounded with respect to t. Hence, in order to prove theorem 4.1 it suffices to prove the existence of the strong limits on $D(H)=D(H_0)$ with smooth $f_.$.
\noindent \textbf{Strong limits of $b_{1,+,t}^{\sharp}(f_1)$ and $b_{2,\pm,t}^{\sharp}(f_2)$ }.
Let
\begin{multline}\label{4.3}
\mathfrak{D}= \{f \in l^2(\Gamma_1)| f(s,n,.,.) \in C_0^\infty(\R^2 \setminus \{0\}) \mbox{ for all s and n} \ , \mbox{and} \\
f(.,n,.,.)=0 \mbox{ for all but finitely many n} \}.
\end{multline}
Let $f_1,f_2\in \mathfrak{D}$ . According to \cite[lemma1]{HK1} we have
\begin{equation}\label{4.4}
b_{1,+}^{\sharp}(f_1)D(H) \subset D(H)\quad \mbox{and}\quad b_{2,\pm}^{\sharp}(f_2)D(H)\subset D(H).
\end{equation}
Moreover we have
\begin{equation}\label{4.5}
\begin{split}
e^{itH_0}b_{1,+}^{\sharp}(f_1)e^{-itH_0}\Psi=&b_{1,+}^{\sharp}(e^{itE^e}f_1)\Psi, \\
e^{itH_0}b_{2,\pm}^{\sharp}(f_2)e^{-itH_0}\Psi=&b_{2,\pm}^{\sharp}(e^{itE^\mu}f_2)\Psi.
\end{split}
\end{equation}
where $\Psi \in D(H)$.
Let us first prove the existence of $ b_{1,+,\pm\infty}^{\sharp}(f_1)$.
Let $\Psi,\Phi \in D(H)$ and $f_{1,t}(\xi_1)=(e^{-itE^e}f_1)(\xi_1)$. By \eqref{4.4},\eqref{4.5} and the strong differentiability of $e^{itH}$ we get
\begin{equation}\label{4.6}
\begin{split}
& \left(\Phi, b_{1,+,T}(f_1)\Psi\right)-\left(\Phi, b_{1,+,T_{0}}(f_1)\Psi\right)\\
& = g \int_{T_{0}}^{T} \frac{d}{dt} \left(\Phi, b_{1,+,t}(f_1)\Psi\right)=ig \int_{T_{0}}^{T} \left(\Phi,e^{itH}[ H_I,b_{1,+}(f_{1,t})] e^{-itH}\Psi\right)dt
\end{split}
\end{equation}
By using the usual canpnicol anticommutation relations (CAR) (see\eqref{A.4}) we easily get for all $\Psi \in D(H)$
\begin{equation}\label{4.7}
\begin{split}
&\left[H_I^{(1)},b_{1,+}(f_{1,t})\right]\Psi=
\int \d \xi_1 \d \xi_2 \d \xi_3 \d \xi_4 \,\Big(\int \d x^2 \mathrm{e}^{-ix^2r^2} \\
&\big( \overline{U}^{(\nu_\mu)}(\xi_4) \gamma_{\alpha}(1-\gamma_5)U^{(\mu)}(x^2,\xi_2)\big)
\big(\overline{U}^{(e)}(x^2,\xi_1) \gamma^\alpha(1-\gamma_5)W^{(\overline{\nu}_{e})}(\xi_3)\big)\Big)\\ &\overline{f_{1,t}(\xi_1)}F(\xi_{2}, \xi_{4})G(\xi_1,\xi_3)b_{+}^{*}(\xi_4)b_{-}^{*}(\xi_3)b_{+}(\xi_2)\Psi.
\end{split}
\end{equation}
\begin{equation}\label{4.8}
\begin{split}
&\left[H_I^{(2)},b_{1,+}(f_{1,t})\right]\Psi=
- \int \d \xi_1 \d \xi_2 \d \xi_3 \d \xi_4 \Big(\int \d x^2 \mathrm{e}^{-ix^2r^2} \\
&\big( \overline{U}^{(\nu_\mu)}(\xi_4) \gamma^\alpha(1-\gamma_5)W^{(\mu)}(x^2,\xi_2)\big)
\big(\overline{U}^{(e)}(x^2,\xi_1)\gamma_{\alpha}(1-\gamma_5)W^{(\overline{\nu}_{e})}(\xi_3)\big)\\ &\overline{f_{1,t}(\xi_1)}F(\xi_{2},\xi_{4})G(\xi_1,\xi_3)\Big)
b_{+}^{*}(\xi_4)b^{*}_{-}(\xi_2)b^{*}_{-}(\xi_3)\Psi.
\end{split}
\end{equation}
\begin{equation}\label{4.9}
\left[(H_I^{(1)})^*,b_{1,+}(f_{1,t})\right]\Psi=\left[(H_I^{(2)})^*,b_{1,+}(f_{1,t})\right]\Psi=0
\end{equation}
where $\overline{U}= U^\dagger\gamma^0$.
Similarly we get
\begin{equation}\label{4.10}
\begin{split}
& \left(\Phi, b^*_{1,+,T}(f_1)\Psi\right)-\left(\Phi, b^*_{1,+,T_{0}}(f_1)\Psi\right)\\
& = g\int_{T_{0}}^{T} \frac{d}{dt} \left(\Phi, b^*_{1,+,t}(f_1)\Psi\right)=ig \int_{T_{0}}^{T} \left(\Phi,e^{itH} [ H_I,b^*_{1,+}(f_{1,t})] e^{-itH}\Psi\right)dt
\end{split}
\end{equation}
with
\begin{equation}\label{4.11}
\left[H_I^{(1)},b^*_{1,+}(f_{1,t})\right]\Psi=\left[H_I^{(2)},b^*_{1,+}(f_{1,t})\right]\Psi=0
\end{equation}
and
\begin{equation}\label{4.12}
\begin{split}
&\left[(H_I^{(1)})^*,b^*_{1,+}(f_{1,t})\right]\Psi=
- \int \d \xi_1 \d \xi_2 \d \xi_3 \d\xi_4 \,\Big(\int \d x^2 \mathrm{e}^{ix^2r^2} \\
&\big( \overline{W}^{(\overline{\nu}_e)}(\xi_3) \gamma^\alpha(1-\gamma_5)U^{(e)}(x^2,\xi_1)\big)
\big(\overline{U}^{(\mu)}(x^2,\xi_2) \gamma_\alpha(1-\gamma_5)U^{(\nu_\mu)}(\xi_4)\big)\Big)\\ &\overline{F(\xi_{2}, \xi_{4})}\, \overline{G(\xi_{1}, \xi_{3})} f_{1,t}(\xi_1)
b_{+}^{*}(\xi_2)b_{-}(\xi_3)b_{+}(\xi_4)\Psi.
\end{split}
\end{equation}
\begin{equation}\label{4.13}
\begin{split}
&\left[(H_I^{(2)})^{*},b^*_{1,+}(f_{1,t})\right]\Psi=
\int \d \xi_1 \d \xi_2 \d \xi_3 \d \xi_4\,\Big(\int \d x^2 \mathrm{e}^{ix^2r^2} \\
&\big(\overline{W}^{(\overline{\nu}_e)}(\xi_3) \gamma^\alpha(1-\gamma_5)U^{(e)}(x^2,\xi_1)\big)
\big( W^{(\mu)}(x^2,\xi_2) \gamma_\alpha(1-\gamma_5) U^{(\nu_\mu)}(\xi_4)\big)\Big)\\ &\overline{F(\xi_{2}, \xi_{4})}\, \overline{G(\xi_1,\xi_3)} f_{1,t}(\xi_1)
b_{-}(\xi_3)b_{-}(\xi_2)b_{+}(\xi_4)\Psi.
\end{split}
\end{equation}
By \eqref{4.6} and \eqref{4.10}, in order to prove the existence of $ b_{1,+,\pm\infty}^{\sharp}(f_1)$, we have to estimate $$e^{itH}[ H_I,b_{1,+}(f_{1,t})] e^{-itH}\Psi$$
and $$e^{itH}[ H_I,b^*_{1,+}(f_{1,t})] e^{-itH}\Psi$$
for large $|t|$.
By \eqref{B.5}, the $N_\tau$ estimates (see \cite{GJ} and \cite[Proposition 3.7]{BDG}), \eqref{A.8}, \eqref{A.11} and \eqref{A.13} we get
\begin{equation}\label{4.14}
\begin{split}
&\left\|e^{itH}[ H_I^{(1)},b_{1,+}(f_{1,t})] e^{-itH}\Psi\right\|\leq \\
&C\bigg( \int \d x^2 \bigg( \int \d \xi_3 \left\| \int \d \xi_1 U^{(e)}( x^2,\xi_1) f_{1,t}(\xi_1)\overline{G(\xi_1,\xi_3)}\right\|_{\C^4}^{2} \bigg) \bigg)^{\frac{1}{2}}\times\\
&\left\|F(.,.)\right\|_{L^2(\Gamma_1\times\R^3)} \|(N_{\mu_-} + 1)^{\frac{1}{2}} e^{-itH}\Psi\|.
\end{split}
\end{equation}
and
\begin{equation}\label{4.15}
\begin{split}
&\left\|e^{itH}[ H_I^{(2)},b_{1,+}(f_{1,t})] e^{-itH}\Psi\right\|\leq \\
&C\bigg( \int \d x^2 \bigg( \int \d \xi_3 \left\| \int \d \xi_1 U^{(e)}( x^2,\xi_1) f_{1,t}(\xi_1)\overline{G(\xi_1,\xi_3)}\right\|_{\C^4}^{2} \bigg) \bigg)^{\frac{1}{2}}\times\\
&\left\|F(.,.)\right\|_{L^2(\Gamma_1\times\R^3)} \|(N_{\mu_+} + 1)^{\frac{1}{2}} e^{-itH}\Psi\|.
\end{split}
\end{equation}
By \eqref{2.18}and \eqref{2.19} we have
\begin{equation}\label{4.16}
\|H_I\Psi\|\leq a\|H_0\phi\| +b\|\Psi\|
\end{equation}
with $$a=4\frac{C}{M}\|F(.,.)\|_{L^2(\Gamma_1\times\R^3)}\|G(.,.)\|_{L^2(\Gamma_1\times\R^3)}$$ and $$b=2C\|F(.,.)\|_{L^2(\Gamma_1\times\R^3)}\|G(.,.)\|_{L^2(\Gamma_1\times\R^3)}$$.
Hence we obtain
\begin{equation}\label{4.17}
\|H_0\Psi\|\leq \tilde{a}\|H\Psi\| +\tilde{b}\|\Psi\|
\end{equation}
with
\begin{equation*}
\tilde{a}=\frac{1}{1-g_0a}\, \mbox{and} \; \tilde{b}=\frac{g_0b}{1-g_0a}
\end{equation*}
Therefore we have
\begin{equation}\label{4.18}
\begin{split}
\|(N_{e} + 1)^{\frac{1}{2}} e^{-itH}\Psi\| &\leq \frac{1}{m_e}( \tilde{a}\|H\Psi\| +(\tilde{b}+ m_e)\|\Psi\|) \\
\|(N_{\mu_{\pm}} + 1)^{\frac{1}{2}} e^{-itH}\Psi\| &\leq \frac{1}{m_\mu}( \tilde{a}\|H\Psi\| +(\tilde{b}+ m_\mu)\|\Psi\|).
\end{split}
\end{equation}
where $m_{\mu}$ is the mass of the muon.
Hence we get
\begin{equation}\label{4.19}
\begin{split}
&\left\|e^{itH}[ H_I,b_{1,+}(f_{1,t})] e^{-itH}\Psi\right\|\leq \\
&2C\bigg( \int \d x^2 \bigg( \int \d \xi_3 \left\| \int \d \xi_1 U^{(e)}( x^2,\xi_1) f_{1,t}(\xi_1)\overline{G(\xi_1,\xi_3)}\right\|_{\C^4}^{2} \bigg) \bigg)^{\frac{1}{2}}\times\\
&\left\|F(.,.)\right\|_{L^2(\Gamma_1\times\R^3)}\frac{1}{m_\mu}( \tilde{a}\|H\Psi\| +(\tilde{b}+ m_\mu)\|\Psi\|.
\end{split}
\end{equation}
Moreover we have
\begin{equation}\label{4.20}
\begin{split}
& \int \d x^2 \bigg( \int \d \xi_3 \left\| \int \d \xi_1 U^{(e)}( x^2,\xi_1) f_{1,t}(\xi_1)\overline{G(\xi_1,\xi_3)}\right\|_{\C^4}^{2} \\
&=\sum_{j=1}^4 \int \d x^2 \bigg( \int \d \xi_3 \left| \int \d \xi_1 U_{j}^{(e)}( x^2,\xi_1)e^{-itE_n^(e)(p_e^3)} f_{1}(\xi_1)\overline{G(\xi_1,\xi_3)}\bigg)\right|^{2}
\end{split}
\end{equation}
where $\big(\bigcup_{j=1}^4 U_{j}^{(e)}( x^2,\xi_1)\big)$ are the four components of the vectors \eqref{A.8} and \eqref{A.11} $\in \C^4$.
Note that
\begin{equation}\label{4.21}
e^{-itE_n^{(e)}(p_e^3)}=\frac{1}{it}\frac{E_n^{(e)}(p_e^3)}{p_e^3}\frac{d}{dp_e^3}e^{-itE_n^{(e)}(p_e^3)} .
\end{equation}
By \eqref{4.20} and\eqref{4.21}, by a two-fold partial integration with respect to $p_e^3$ and by Hypothesis 4.1 one can show that there exits for every $j$ a function, denoted by $H_j^{(e)}(\xi_1,\xi_3)$, such that
\begin{equation}\label{4.22}
\begin{split}
&\sum_{j=1}^4 \int \d x^2 \bigg( \int \d \xi_3 \left| \int \d \xi_1 U_{j}^{(e)}( x^2,\xi_1)e^{-itE_n^{(e)}(p_e^3)} f_{1}(\xi_1)\overline{G(\xi_1,\xi_3)}\bigg)\right|^{2} \\
&= \sum_{j=1}^4\frac{1}{t^4} \int \d x^2 \bigg( \int \d \xi_3 \left| \int \d \xi_1 U_{j}^{(e)}( x^2,\xi_1)H_j^{(e)}(\xi_1,\xi_3)e^{-itE_n^{(e)}(p_e^3} \bigg)\right|^{2}\\
&\leq C_{f_1} \frac{1}{t^4}\sum_{j=1}^4\big(\int \d \xi_1 \d \xi_3 \chi_{f_1}(\xi_1)|H_j^{(e)}(\xi_1,\xi_3)|^2 \big)<\infty.
\end{split}
\end{equation}
Here $\chi_{f_1}(.)$ is the characteristic function of the support of $f_1(.)$ and \eqref{A.13} is used.
By \eqref{4.6} and \eqref{4.19}-
\eqref{4.22} the strong limits of $ b_{1,+,t}(f_1)$ on $\mathfrak{F}$ when t goes to $\pm\infty$ and for all $f_1 \in L^2(\Gamma_1)$ exist for every $g \leq g_0$.
By \eqref{4.11}-\eqref{4.13} and by mimicking the proof of \eqref{4.14} and \eqref{4.15} we get
\begin{equation}\label{4.23}
\begin{split}
&\mbox{Sup}\bigg(\left\|e^{itH}[ (H_I^{(1)})^*,b_{1,+}^*(f_{1,t})] e^{-itH}\Psi\right\|, \left\|e^{itH}[ (H_I^{(2)})^*,b_{1,+}^*(f_{1,t})] e^{-itH}\Psi\right\|\bigg) \\
&\leq C\bigg( \int \d x^2 \bigg( \int \d \xi_3 \left\| \int \d \xi_1 U^{(e)}( x^2,\xi_1) f_{1,t}(\xi_1)\overline{G(\xi_1,\xi_3)}\right\|_{\C^4}^{2} \bigg) \bigg)^{\frac{1}{2}}\\
&\times \left\|F(.,.)\right\|_{L^2(\Gamma_1\times\R^3)}\frac{1}{m_\mu}( \tilde{a}\|H\Psi\| +(\tilde{b}+ m_\mu)\|\Psi\|).
\end{split}
\end{equation}
It follows from \eqref{4.10} and \eqref{4.20}-\eqref{4.23} that the strong limits of $ b_{1,+,t}^*(f_1)$ exist when t goes to $\pm\infty$, for all $f_1 \in L^2(\Gamma_1)$ and for every $g \leq g_0$.
We now consider the existence of $ b_{2,\epsilon,\pm\infty}^{\sharp}(f_2)$
Let $\Psi,\Phi \in D(H)$ and $f_{2,t}(\xi_2)=(e^{-itE^\mu}f_2)(\xi_2)$ with $f_2\in \mathfrak{D}$. By \eqref{4.4}, \eqref{4.5} and the strong differentiability of $e^{itH}$ we get
\begin{equation}\label{4.24}
\begin{split}
& \left(\Phi, b_{2,+,T}(f_)\Psi\right)-\left(\Phi, b_{2,+,T_{0}}(f_2)\Psi\right)\\
& = g \int_{T_{0}}^{T} \frac{d}{dt} \left(\Phi, b_{2,+,t}(f_2)\Psi\right)=ig \int_{T_{0}}^{T} \left(\Phi,e^{itH}[ H_I,b_{2,+}(f_{2,t})] e^{-itH}\Psi\right)dt
\end{split}
\end{equation}
with
\begin{equation}\label{4.25}
\left[H_I^{(1)},b_{2,+}(f_{2,t})\right]\Psi=\left[(H_I^{(2)}),b_{2,+}(f_{2,t})\right]\Psi=\left[(H_I^{(2)})^*,b_{2,+}(f_{2,t})\right]\Psi=0
\end{equation}
and
\begin{equation}\label{4.26}
\begin{split}
&\left[(H_I^{(1)})^*,b_{2,+}(f_{2,t})\right]\Psi=
- \int \d \xi_1 \d \xi_2 \d \xi_3 \d\xi_4 \,\Big(\int \d x^2 \mathrm{e}^{ix^2r^2} \\
&\big( \overline{W}^{(\overline{\nu}_e)}(\xi_3) \gamma^\alpha(1-\gamma_5)U^{(e)}(x^2,\xi_1)\big)
\big(\overline{U}^{(\mu)}(x^2,\xi_2) \gamma_\alpha(1-\gamma_5)U^{(\nu_\mu)}(\xi_4)\big)\Big)\\ &\overline{F(\xi_{2}, \xi_{4})}\,\overline{G(\xi_{1}, \xi_{3})}\, \overline{f_{2,t}(\xi_2)}
b_{-}(\xi_3)b_{+}(\xi_1)b_{+}(\xi_4)\Psi.
\end{split}
\end{equation}
Similarly we obtain
\begin{equation}\label{4.27}
\begin{split}
&\left\|e^{itH}[ H_I,b_{2,+}(f_{2,t})] e^{-itH}\Psi\right\|\leq \\
&C\bigg( \int \d x^2 \bigg( \int \d \xi_4 \left\| \int \d \xi_2 U^{(\mu)}( x^2,\xi_2)F(\xi_2,\xi_4) f_{2,t}(\xi_1)\right\|_{\C^4}^{2} \bigg) \bigg)^{\frac{1}{2}}\times\\
&\left\|G(.,.)\right\|_{L^2(\Gamma_1\times\R^3)} \|(N_{e} + 1)^{\frac{1}{2}} e^{-itH}\Psi\|.
\end{split}
\end{equation}
It follows from \eqref{4.16}-\eqref{4.18} that
\begin{equation}\label{4.28}
\|(N_{e} + 1)^{\frac{1}{2}} e^{-itH}\Psi\|\leq \frac{1}{m_e}( \tilde{a}\|H\Psi\| +(\tilde{b}+ m_e)\|\Psi\|.
\end{equation}
Hence
\begin{equation}\label{4.29}
\begin{split}
&\left\|e^{itH}[ H_I,b_{2,+}(f_{2,t})] e^{-itH}\Psi\right\|\leq \\
&C\bigg( \int \d x^2 \bigg( \int \d \xi_4 \left\| \int \d \xi_2 U^{(\mu)}( x^2,\xi_2)F(\xi_2,\xi_4) f_{2,t}(\xi_2)\right\|_{\C^4}^{2} \bigg) \bigg)^{\frac{1}{2}}\times\\
&\left\|G(.,.)\right\|_{L^2(\Gamma_1\times\R^3)}\frac{1}{m_e}( \tilde{a}\|H\Psi\| +(\tilde{b}+ m_e)\|\Psi\| .
\end{split}
\end{equation}
Moreover we have
\begin{equation}\label{4.30}
\begin{split}
& \int \d x^2 \bigg( \int \d \xi_4 \left\| \int \d \xi_2 U^{(\mu)}( x^2,\xi_2) f_{2,t}(\xi_2)F(\xi_2,\xi_4)\right\|_{\C^4}^{2} \\
&=\sum_{j=1}^4 \int \d x^2 \bigg( \int \d \xi_4 \left| \int \d \xi_2 U_{j}^{(\mu)}( x^2,\xi_2)e^{-itE_n^{(\mu)}(p_{\mu}^3)} f_{2}(\xi_2)F(\xi_2,\xi_4) \bigg)\right|^{2}.
\end{split}
\end{equation}
where $\big(\bigcup_{j=1}^4 U_{j}^{(\mu)}( x^2,\xi_2)\big)$ are the four components of the vectors \eqref{A.8} and \eqref{A.11} $\in \C^4$ for $\alpha=\mu$.
By\eqref{4.30}, by a two-fold partial integration with respect to $p_{\mu}^3$ and by Hypothesis 4.1 one can show that there exits for every $j$ a function, denoted by $H_j^{(\mu)}(\xi_2,\xi_4)$ , such that
\begin{equation}\label{4.31}
\begin{split}
&\sum_{j=1}^4 \int \d x^2 \bigg( \int \d \xi_4 \left| \int \d \xi_2 U_{j}^{(\mu)}( x^2,\xi_2)e^{-itE_{n}^{(\mu)}(p_{\mu}^3)} f_{2}(\xi_2)F(\xi_2,\xi_4)\bigg)\right|^{2} \\
&= \sum_{j=1}^4\frac{1}{t^4} \int \d x^2 \bigg( \int \d \xi_4 \left| \int \d \xi_1 U_{j}^{(\mu)}( x^2,\xi_2)H_j^{(\mu)}(\xi_2,\xi_4)e^{-itE_n^{(\mu)}(p_{\mu}^3)} \bigg)\right|^{2}\\
&\leq C_{f_2} \frac{1}{t^4}\sum_{j=1}^4\big(\int \d \xi_2 \d \xi_4 \chi_{f_2}(\xi_2)|H_j^{(\mu)}(\xi_2,\xi_4)|^2 \big)<\infty.
\end{split}
\end{equation}
Here $\chi_{f_2}(.)$ is the characteristic function of the support of $f_2(.)$ and \eqref{A.13} is used.
Similarly we have
\begin{equation}\label{4.32}
\begin{split}
& \left(\Phi, b_{2,+,T}^*(f_2)\Psi\right)-\left(\Phi, b_{2,+,T_{0}}^*(f_2)\Psi\right)\\
& = g \int_{T_{0}}^{T} \frac{d}{dt} \left(\Phi, b_{2,+,t}^*(f_2)\Psi\right)=ig \int_{T_{0}}^{T} \left(\Phi,e^{itH}[ H_I,b_{2,+}^*(f_{2,t})] e^{-itH}\Psi\right)dt
\end{split}
\end{equation}
with
\begin{equation}\label{4.33}
\left[(H_I^{(1)})^*,b_{2,+}^*(f_{2,t})\right]\Psi=\left[H_I^{(2)},b_{2,+}^*(f_{2,t})\right]\Psi=\left[(H_I^{(2)})^*,b_{2,+}^*(f_{2,t})\right]\Psi=0
\end{equation}
and
\begin{equation}\label{4.34}
\begin{split}
&\left[(H_I^{(1)}),b_{2,+}^*(f_{2,t})\right]\Psi=
- \int \d \xi_1 \d \xi_2 \d \xi_3 \d\xi_4 \,\Big(\int \d x^2 \mathrm{e}^{-ix^2r^2} \\
&\big(\overline{U}^{(e)}(x^2,\xi_1)\gamma^\alpha(1-\gamma_5) W^{(\overline{\nu}_e)}(\xi_3)\big)G(\xi_{1}, \xi_{3})
\big(\overline{U}^{(\nu_\mu)}(\xi_4)\gamma_\alpha(1-\gamma_5)U^{(\mu)}(x^2,\xi_2)\big)\Big)\\ &F(\xi_{2}, \xi_{4})G(\xi_{1}, \xi_{3})f_{2,t}(\xi_2)
b_{+}^*(\xi_4)b_{+}^*(\xi_1)b_{-}^*(\xi_3)\Psi.
\end{split}
\end{equation}
Similarly we obtain
\begin{equation}\label{4.35}
\begin{split}
&\left\|e^{itH}[ H_I,b_{2,+}^*(f_{2,t})] e^{-itH}\Psi\right\|\leq \\
&C\bigg( \int \d x^2 \bigg( \int \d \xi_4 \left\| \int \d \xi_2 U^{(\mu)}( x^2,\xi_2)F(\xi_2,\xi_4) f_{2,t}(\xi_1)\right\|_{\C^4}^{2} \bigg) \bigg)^{\frac{1}{2}}\times\\
&\left\|G(.,.)\right\|_{L^2(\Gamma_1\times\R^3)}\frac{1}{m_e}( \tilde{a}\|H\Psi\| +(\tilde{b}+ m_e)\|\Psi\| .
\end{split}
\end{equation}
It follows from \eqref{4.29},\eqref{4.31} and \eqref{4.35} that the strong limits of $b^{\sharp}(2,+,t)(f_2)$ exist when t goes to $\pm\infty$, for all $f_2\in L^2(\Gamma_1\times\R^3)$ and for every $g \leq g_0$.
Let us now consider the strong limits of $b^{\sharp}(2,-,t)(f_2)$.
We have for all $f_2\in \mathfrak{D}$
\begin{equation}\label{4.36}
\begin{split}
& \left(\Phi, b_{2,-,T}(f_)\Psi\right)-\left(\Phi, b_{2,-,T_{0}}(f_2)\Psi\right)\\
& = g \int_{T_{0}}^{T} \frac{d}{dt} \left(\Phi, b_{2,-,t}(f_2)\Psi\right)=ig \int_{T_{0}}^{T} \left(\Phi,e^{itH}[ H_I,b_{2,-}(f_{2,t})] e^{-itH}\Psi\right)dt
\end{split}
\end{equation}
with
\begin{equation}\label{4.37}
\left[H_I^{(1)},b_{2,-}(f_{2,t})\right]\Psi=\left[(H_I^{(1)})^*,b_{2,-}(f_{2,t})\right]\Psi=\left[(H_I^{(2)})^*,b_{2,-}(f_{2,t})\right]\Psi=0
\end{equation}
\begin{equation}\label{4.38}
\begin{split}
&\left[(H_I^{(2)}),b_{2,-}(f_{2,t})\right]\Psi=
- \int \d \xi_1 \d \xi_2 \d \xi_3 \d\xi_4 \,\Big(\int \d x^2 \mathrm{e}^{-ix^2r^2} \\
&\big(\overline{U}^{(e)}(x^2,\xi_1)\gamma^\alpha(1-\gamma_5) W^{(\overline{\nu}_e)}(\xi_3)\big)G(\xi_{1}, \xi_{3})
\big(\overline{U}^{(\nu_\mu)}(\xi_4)\gamma_\alpha(1-\gamma_5)W^{(\mu)}(x^2,\xi_2)\big)\Big)\\ &F(\xi_{2}, \xi_{4})G(\xi_{1}, \xi_{3})f_{2,t}(\xi_2)
b_{+}^*(\xi_4)b_{+}^*(\xi_1)b_{-}^*(\xi_3)\Psi.
\end{split}
\end{equation}
By mimicking the proofs given above we get
\begin{equation}\label{4.39}
\begin{split}
&\left\|e^{itH}[ H_I,b_{2,-}(f_{2,t})] e^{-itH}\Psi\right\|\leq \\
&C\bigg( \int \d x^2 \bigg( \int \d \xi_4 \left\| \int \d \xi_2 W^{(\mu)}( x^2,\xi_2)F(\xi_2,\xi_4)\overline{f_{2,t}(\xi_1)}\right\|_{\C^4}^{2} \bigg) \bigg)^{\frac{1}{2}}\times\\
&\left\|G(.,.)\right\|_{L^2(\Gamma_1\times\R^3)}\frac{1}{m_e}( \tilde{a}\|H\Psi\| +(\tilde{b}+ m_e)\|\Psi\| .
\end{split}
\end{equation}
and
\begin{equation}\label{4.40}
\begin{split}
& \int \d x^2 \bigg( \int \d \xi_4 \left\| \int \d \xi_2 W^{(\mu)}( x^2,\xi_2) \overline{f_{2,t}(\xi_2)}F(\xi_2,\xi_4)\right\|_{\C^4}^{2} \\
&=\sum_{j=1}^4 \int \d x^2 \bigg( \int \d \xi_4 \left| \int \d \xi_2 W_{j}^{(\mu)}( x^2,\xi_2)e^{-itE_n^{(\mu)}(p_{\mu}^3)} \overline{f_{2}(\xi_2)}F(\xi_2,\xi_4) \bigg)\right|^{2}.
\end{split}
\end{equation}
where $\big(\bigcup_{j=1}^4 W_{j}^{(\mu)}( x^2,\xi_2)\big)$ are the four components of the vectors \eqref{A.14} - \eqref{A.16} $\in \C^4$ for $\alpha=\mu$.
By\eqref{4.40}, by a two-fold partial integration with respect to $p_{\mu}^3$ and by Hypothesis 4.1 one can show that there exits for every $j$ a function, denoted by $\widetilde{H}_j^{(\mu)}(\xi_2,\xi_4)$ , such that
\begin{equation}\label{4.41}
\begin{split}
&\sum_{j=1}^4 \int \d x^2 \bigg( \int \d \xi_4 \left| \int \d \xi_2 W_{j}^{(\mu)}( x^2,\xi_2)e^{-itE_{n}^{(\mu)}(p_{\mu}^3)} \overline{f_{2}(\xi_2)}F(\xi_2,\xi_4)\bigg)\right|^{2} \\
&= \sum_{j=1}^4\frac{1}{t^4} \int \d x^2 \bigg( \int \d \xi_4 \left| \int \d \xi_2 W_{j}^{(\mu)}( x^2,\xi_2)\widetilde{H}_j^{(\mu)}(\xi_2,\xi_4)e^{-itE_n^{(\mu)}(p_{\mu}^3)} \bigg)\right|^{2}\\
&\leq C_{f_2} \frac{1}{t^4}\sum_{j=1}^4\big(\int \d \xi_2 \d \xi_4 \chi_{f_2}(\xi_2)|\widetilde{H}_j^{(\mu)}(\xi_2,\xi_4)|^2 \big)<\infty.
\end{split}
\end{equation}
Here $\chi_{f_2}(.)$ is the characteristic function of the support of $f_2(.)$ and \eqref{A.17} is used.
It follows from \eqref{4.36},\eqref{4.39}-\eqref{4.41} that the strong limits of $b(2,-,t)(f_2)$ exist when t goes to $\pm\infty$, for all $f_2\in L^2(\Gamma_1\times\R^3)$ and for every $g \leq g_0$.
We now have for all $f_2\in \mathfrak{D}$
\begin{equation}\label{4.42}
\begin{split}
& \left(\Phi, b_{2,-,T}^{*}(f_2)\Psi\right)-\left(\Phi, b_{2,-,T_{0}}^*(f_2)\Psi\right)\\
& = g \int_{T_{0}}^{T} \frac{d}{dt} \left(\Phi, b_{2,-,t}^*(f_2)\Psi\right)=ig \int_{T_{0}}^{T} \left(\Phi,e^{itH}[ H_I,b_{2,-}^*(f_{2,t})] e^{-itH}\Psi\right)dt
\end{split}
\end{equation}
with
\begin{equation}\label{4.43}
\left[H_I^{(1)},b_{2,-}^*(f_{2,t})\right]\Psi=\left[(H_I^{(1)})^*,b_{2,-}^*(f_{2,t})\right]\Psi=\left[H_I^{(2)},b_{2,-}^*(f_{2,t})\right]\Psi=0
\end{equation}
\begin{equation}\label{4.44}
\begin{split}
&\left[(H_I^{(2)})^*,b_{2,-}^*(f_{2,t})\right]\Psi=
- \int \d \xi_1 \d \xi_2 \d \xi_3 \d\xi_4 \,\Big(\int \d x^2 \mathrm{e}^{ix^2r^2} \\
&\big( \overline{W}^{(\overline{\nu}_e)}(\xi_3)\gamma^\alpha(1-\gamma_5)U^{(e)}(x^2,\xi_1)\big)
\big(\overline{W}^{(\mu)}(x^2,\xi_2)\gamma_\alpha(1-\gamma_5)U^{(\nu_\mu)}(\xi_4)\big)\Big)\\ &\overline{F(\xi_{2}, \xi_{4})}\,\overline{G(\xi_{1}, \xi_{3})}f_{2,t}(\xi_2)
b_{-}(\xi_3)b_{+}(\xi_1)b_{+}(\xi_4)\Psi.
\end{split}
\end{equation}
Similarly to \eqref{4.39} we get
\begin{equation}\label{4.45}
\begin{split}
&\left\|e^{itH}[ H_I,b_{2,-}^*(f_{2,t})] e^{-itH}\Psi\right\|\leq \\
&C\bigg( \int \d x^2 \bigg( \int \d \xi_4 \left\| \int \d \xi_2 W^{(\mu)}( x^2,\xi_2)F(\xi_2,\xi_4)\overline{f_{2,t}(\xi_1)}\right\|_{\C^4}^{2} \bigg) \bigg)^{\frac{1}{2}}\times\\
&\left\|G(.,.)\right\|_{L^2(\Gamma_1\times\R^3)}\frac{1}{m_e}( \tilde{a}\|H\Psi\| +(\tilde{b}+ m_e)\|\Psi\| .
\end{split}
\end{equation}
It follows from \eqref{4.43},\eqref{4.45},\eqref{4.40} and \eqref{4.41} that the strong limits of $b(2,-,t)^*(f_2)$ exist when t goes to $\pm\infty$, for all $f_2\in L^2(\Gamma_1\times\R^3)$ and for every $g \leq g_0$.
\noindent \textbf{Strong limits of $b_{3,-,t}^{\sharp}(f_3)$ and $b_{4,+,t}^{\sharp}(f_4)$ }.
Let
\begin{equation}\label{4.46}
\mathfrak{D'}= \{f(.)\in C_0^\infty(\R^3 \setminus \{(0,0,p^3)\}); p^3\in\R \}.
\end{equation}
Let $f_1,f_2\in \mathfrak{D'}$ . According to \cite[lemma1]{HK1} we have
\begin{equation}\label{4.47}
b_{4,+}^{\sharp}(f_4)D(H) \subset D(H)\quad \mbox{and}\quad b_{3,-}^{\sharp}(f_3)D(H)\subset D(H).
\end{equation}
Moreover we have
\begin{equation}\label{4.48}
\begin{split}
e^{itH_0}b_{3,-}^{\sharp}(f_3)e^{-itH_0}\Psi=&b_{3,-}^{\sharp}(e^{it|\textbf{p}_3|}f_3)\Psi, \\
e^{itH_0}b_{4,+}^{\sharp}(f_4)e^{-itH_0}\Psi=&b_{4,+}^{\sharp}(e^{it|\textbf{p}_4|}f_4)\Psi.
\end{split}
\end{equation}
where $\Psi \in D(H)$.
Let $\Psi,\Phi \in D(H)$ and $f_{j,t}(\xi_j)=(e^{-it|\textbf{p}_j|}f_j)(\xi_j)$ where $j=3,4$.. By \eqref{4.4},\eqref{4.5} and the strong differentiability of $e^{itH}$ we get
\begin{equation}\label{4.49}
\begin{split}
& \left(\Phi, b_{3,-,T}(f_3)\Psi\right)-\left(\Phi, b_{3,-,T_{0}}(f_1)\Psi\right)\\
& = g \int_{T_{0}}^{T} \frac{d}{dt} \left(\Phi, b_{3,-,t}(f_3)\Psi\right)=ig \int_{T_{0}}^{T} \left(\Phi,e^{itH}[ H_I,b_{3,-}(f_{3,t})] e^{-itH}\Psi\right)dt
\end{split}
\end{equation}
By using the usual anticommutation relations (CAR)(see \eqref{A.4} and \eqref{B.4}) we easily get for all $\Psi \in D(H)$
\begin{equation}\label{4.50}
\begin{split}
&\left[H_I^{(1)},b_{3,-}(f_{3,t})\right]\Psi=
-\int \d \xi_1 \d \xi_2 \d \xi_3 \d \xi_4 \,\Big(\int \d x^2 \mathrm{e}^{-ix^2r^2} \\
&\big( \overline{U}^{(\nu_\mu)}(\xi_4) \gamma_{\alpha}(1-\gamma_5)U^{(\mu)}(x^2,\xi_2)\big)
\big(\overline{U}^{(e)}(x^2,\xi_1) \gamma^\alpha(1-\gamma_5)W^{(\overline{\nu}_{e})}(\xi_3)\big)\Big)\\ &\overline{f_{3,t}(\xi_3)}F(\xi_{2}, \xi_{4})G(\xi_1,\xi_3)
b_{+}^{*}(\xi_4)b_{+}^{*}(\xi_1)b_{+}(\xi_2)\Psi.
\end{split}
\end{equation}
\begin{equation}\label{4.51}
\begin{split}
&\left[H_I^{(2)},b_{3,-}(f_{3,t})\right]\Psi=
\int \d \xi_1 \d \xi_2 \d \xi_3 \d \xi_4 \Bigg(\int \d x^2 \mathrm{e}^{-ix^2r^2} \\
&\big( \overline{U}^{(\nu_\mu)}(\xi_4) \gamma^\alpha(1-\gamma_5)W^{(\mu)}(x^2,\xi_2)\big)
\big(\overline{U}^{(e)}(x^2,\xi_1)\gamma_{\alpha}(1-\gamma_5)W^{(\overline{\nu}_{e})}(\xi_3)\big)\Big)\\ &\overline{f_{3,t}(\xi_3)}F(\xi_{2},\xi_{4})G(\xi_1,\xi_3)
b_{+}^{*}(\xi_4)b^{*}_{-}(\xi_2)b^*_{+}(\xi_1)\Psi.
\end{split}
\end{equation}
and
\begin{equation}\label{4.52}
\left[(H_I^{(1)})^*,b_{3,-}(f_{3,t})\right]\Psi=\left[(H_I^{(2)})^*,b_{3,-}(f_{3,t})\right]\Psi=0
\end{equation}
By \eqref{B.5} we get
\begin{equation}\label{4.53}
\begin{split}
&\left\|\left[H_I^{(1)},b_{3,-}(f_{3,t})\right]\Psi\right\|\leq \\
&\int \d x^2 \bigg(\int \d \xi_1\left|\big(\overline{U}^{(e)}(x^2,\xi_1) \gamma^\alpha(1-\gamma_5) \int \d\xi_3 \mathrm{e}^{-ip^2_{3}x^2} W^{(\overline{\nu}_{e})}(\xi_3)G(\xi_1,\xi_3)\overline{f_{3,t}(\xi_3)}\big)\right|^2\bigg)^\frac{1}{2}\\ &\left \|\int \d \xi_2 \d \xi_4 \mathrm{e}^{-ip^2_{4}x^2}\big( \overline{U}^{(\nu_\mu)}(\xi_4) \gamma_{\alpha}(1-\gamma_5)U^{(\mu)}(x^2,\xi_2)\big)b_{+}^{*}(\xi_4)b_{+}(\xi_2)\Psi\right\|.
\end{split}
\end{equation}
and
\begin{equation}\label{4.54}
\begin{split}
&\left\|\left[H_I^{(2)},b_{3,-}(f_{3,t})\right]\Psi\right\|\leq \\
&\int \d x^2 \bigg(\int \d \xi_1\left|\big(\overline{U}^{(e)}(x^2,\xi_1) \gamma^\alpha(1-\gamma_5) \int \d\xi_3 \mathrm{e}^{-ip^2_{3}x^2} W^{(\overline{\nu}_{e})}(\xi_3)G(\xi_1,\xi_3)\overline{f_{3,t}(\xi_3)}\big)\right|^2\bigg)^\frac{1}{2}\\ &\left \|\int \d \xi_2 \d \xi_4 \mathrm{e}^{-ip^2_{4}x^2}\big( \overline{W}^{(\nu_\mu)}(\xi_4) \gamma_{\alpha}(1-\gamma_5)U^{(\mu)}(x^2,\xi_2)\big)b_{+}^{*}(\xi_4)b_{-}^*(\xi_2)\Psi\right\|.
\end{split}
\end{equation}
Moreover we have
\begin{equation}\label{4.55}
\begin{split}
& \int \d \xi_1 \left\| \int \d \xi_3 \mathrm{e}^{-ip^2_{3}x^2} W^{(\overline{\nu_e})}(\xi_3) \overline{f_{1,t}(\xi_1)}G(\xi_1,\xi_3)\right\|_{\C^4}^{2} \\
&=\sum_{j=1}^4 \int \d \xi_1 \left| \int \d \xi_3 \mathrm{e}^{-ip^2_{3}x^2}W_j^{(\overline{\nu_e})}(\xi_3)(e^{it|\textbf{p}_3|}\overline{f_{3}(\xi_3)}G(\xi_1,\xi_3)\right|^{2}.
\end{split}
\end{equation}
where $\big(\bigcup_{j=1}^4 W_{j}^{(\overline{\nu_e})}(\xi_3)\big)$ are the four components of the vector \eqref{B.12} $\in \C^4$.
By a two-fold partial integration with respect to $p^3$ and $p^1$ and by Hypothesis 4.2 one can show that there exits for every $j$ a function , denoted by $H_j^{(\overline{\nu_e})}(\xi_1,\xi_3)$, such that
\begin{equation}\label{4.56}
\begin{split}
&\sum_{j=1}^4 \int \d \xi_1 \left| \int \d \xi_3\mathrm{e}^{-ip^2_{3}x^2} W_{j}^{(\overline{\nu_e})}(\xi_3)e^{it|\textbf{p}_3|} \overline{f_{3}(\xi_3)}G(\xi_1,\xi_3)\right|^{2}\\
&= \sum_{j=1}^4\frac{1}{t^4} \int \d \xi_1 \left| \int \d \xi_3 \mathrm{e}^{-ip^2_{3}x^2} W_{j}^{(\overline{\nu_e})}(\xi_3)H_j^{(\overline{\nu_e})}(\xi_1,\xi_3)e^{it|\textbf{p}_3|}\right|^{2}\\
&\leq C_{f_3}^2 \frac{1}{t^4}\sum_{j=1}^4\int \d \xi_1 \d \xi_3 \chi_{f_3}(\xi_3)e^{it|\textbf{p}_3|}|H_j^{(\overline{\nu_e})}(\xi_1,\xi_3)|^2<\infty.
\end{split}
\end{equation}
Here $\chi_{f_3}(.)$ is the characteristic function of the support of $f_3(.)$.
By the $N_\tau$ estimates and by \eqref{4.18},\eqref{A.13}, \eqref{A.17} and \eqref{B.14} it follows from \eqref{4.52}-\eqref{4.56} that, for every $\Psi \in D(H)$,
\begin{equation}\label{4.57}
\begin{split}
&\left\|e^{itH}\left [H_I,b_{3,-}(f_{3,t})\right]e^{-itH}\Psi\right\|\leq \\
&CC_{f_3}\frac{1}{t^2}\bigg(\sum_{j=1}^4\int \d \xi_1 \d \xi_3 \chi_{f_3}(\xi_3)|H_j^{(\overline{\nu_e})}(\xi_1,\xi_3)|^2\bigg)^{\frac{1}{2}}\times \\
&\left\|F(.,.)\right\|_{L^2(\Gamma_1\times\R^3)}\frac{1}{m_\mu}( \tilde{a}\|H\Psi\| +(\tilde{b}+ m_\mu)\|\Psi\|)
\end{split}
\end{equation}
Furthermore we have
\begin{equation}\label{4.58}
\begin{split}
& \left(\Phi, b^*_{3,-,T}(f_3)\Psi\right)-\left(\Phi, b^*_{3,-,T_{0}}(f_3)\Psi\right)\\
& = g\int_{T_{0}}^{T} \frac{d}{dt} \left(\Phi, b^*_{3,-,t}(f_3)\Psi\right)=ig \int_{T_{0}}^{T} \left(\Phi,e^{itH} [ H_I,b^*_{3,-}(f_{3,t})] e^{-itH}\Psi\right)dt
\end{split}
\end{equation}
with
\begin{equation}\label{4.59}
\begin{split}
&\left[(H_I^{(1)})^*,b_{3,-}^*(f_{3,t})\right]\Psi=
- \int \d \xi_1 \d \xi_2 \d \xi_3 \d\xi_4 \,\Big(\int \d x^2 \mathrm{e}^{ix^2r^2} \\
&\big( \overline{W}^{(\overline{\nu}_e)}(\xi_3)\gamma^\alpha(1-\gamma_5)U^{(e)}(x^2,\xi_1)\big)
\big(\overline{U}^{(\mu)}(x^2,\xi_2)\gamma_\alpha(1-\gamma_5)U^{(\nu_\mu)}(\xi_4)\big)\Big)\\ &\overline{F(\xi_{2}, \xi_{4})}\,\overline{G(\xi_{1}, \xi_{3})}f_{3,t}(\xi_2)
b_{+}^*(\xi_2)b_{+}(\xi_1)b_{+}(\xi_4)\Psi.
\end{split}
\end{equation}
\begin{equation}\label{4.60}
\begin{split}
&\left[(H_I^{(2)})^{*},b^*_{3,-}(f_{3,t})\right]\Psi=
\int \d \xi_1 \d \xi_2 \d \xi_3 \d \xi_4\,\Big(\int \d x^2 \mathrm{e}^{ix^2r^2} \\
&\big(\overline{W}^{(\overline{\nu}_e)}(\xi_3) \gamma^\alpha(1-\gamma_5)U^{(e)}(x^2,\xi_1)\big)
\big( W^{(\mu)}(x^2,\xi_2) \gamma_\alpha(1-\gamma_5) U^{(\nu_\mu)}(\xi_4)\big)\Big)\\ &\overline{F((\xi_{2}, \xi_{4})}\,\overline{G(\xi_1,\xi_3)}f_{3,t}
b_{+}(\xi_1)b_{-}(\xi_2)b_{+}(\xi_4)\Psi.
\end{split}
\end{equation}
and
\begin{equation}\label{4.61}
\left[(H_I^{(1)}),b_{3,-}^*(f_{3,t})\right]\Psi=\left[(H_I^{(2)}),b_{3,-}^*(f_{3,t})\right]\Psi=0
\end{equation}
By adapting the proof of\eqref{4.53}-\eqref{4.57} to \eqref{4.58}-\eqref{4.61} we obtain
\begin{equation}\label{4.62}
\begin{split}
&\left\|e^{itH}[ H_I,b_{3,-}^*(f_{3,t})] e^{-itH}\Psi\right\|\\
&\leq CC_{f_3}\frac{1}{t^2}\bigg(\sum_{j=1}^4\int \d \xi_1 \d \xi_3 \chi_{f_3}(\xi_3)|H_j^{(\overline{\nu_e})}(\xi_1,\xi_3)|^2\bigg)^{\frac{1}{2}}\times \\
&\times \left\|F(.,.)\right\|_{L^2(\Gamma_1\times\R^3)}\frac{1}{m_\mu}( \tilde{a}\|H\Psi\| +(\tilde{b}+ m_\mu)\|\Psi\|).
\end{split}
\end{equation}
Here $\chi_{f_3}(.)$ is the characteristic function of the support of $f_3(.)$.
It follows from \eqref{4.49},\eqref{4.47},\eqref{4.58} and \eqref{4.62} that the strong limits of $b^\sharp_{3,-,t}(f 3 )$ exist when t goes to $\pm\infty$, for all $f_3\in L^2(\R^3)$ and for every $g \leq g_0$.
\begin{equation}\label{4.63}
\begin{split}
& \left(\Phi, b_{4,+,T}(f_1)\Psi\right)-\left(\Phi, b_{4,+,T_{0}}(f_1)\Psi\right)\\
& = g \int_{T_{0}}^{T} \frac{d}{dt} \left(\Phi, b_{4,+,t}(f_4)\Psi\right)=ig \int_{T_{0}}^{T} \left(\Phi,e^{itH}[ H_I,b_{4,+}(f_{4,t})] e^{-itH}\Psi\right)dt
\end{split}
\end{equation}
By using the usual canonical anticommutation relations (CAR)(see \eqref{A.4} and \eqref{B.4}) we easily get for all $\Psi \in D(H)$
\begin{equation}\label{4.64}
\begin{split}
&\left[H_I^{(1)},b_{4,+}(f_{4,t})\right]\Psi=
-\int \d \xi_1 \d \xi_2 \d \xi_3 \d \xi_4 \Big(\int \d x^2 \mathrm{e}^{-ix^2r^2} \\
&\big( \overline{U}^{(\nu_\mu)}(\xi_4) \gamma_{\alpha}(1-\gamma_5)U^{(\mu)}(x^2,\xi_2)\big)
\big(\overline{U}^{(e)}(x^2,\xi_1) \gamma^\alpha(1-\gamma_5)W^{(\overline{\nu}_{e})}(\xi_3)\big)\Big)\\ &\overline{f_{4,t}(\xi_4)}F(\xi_{2}, \xi_{4})G(\xi_1,\xi_3)
b_{+}^{*}(\xi_1)b_{-}^{*}(\xi_3)b_{+}(\xi_2)\Psi.
\end{split}
\end{equation}
\begin{equation}\label{4.65}
\begin{split}
&\left[H_I^{(2)},b_{4,+}(f_{4,t})\right]\Psi=
-\int \d \xi_1 \d \xi_2 \d \xi_3 \d \xi_4 \Bigg(\int \d x^2 \mathrm{e}^{-ix^2r^2} \\
&\big( \overline{U}^{(\nu_\mu)}(\xi_4) \gamma^\alpha(1-\gamma_5)W^{(\mu)}(x^2,\xi_2)\big)
\big(\overline{U}^{(e)}(x^2,\xi_1)\gamma_{\alpha}(1-\gamma_5)W^{(\overline{\nu}_{e})}(\xi_3)\big)\Big)\\ &\overline{f_{4,t}(\xi_4)}F(\xi_{2},\xi_{4})G(\xi_1,\xi_3)
b_{-}^{*}(\xi_2)b^{*}_{+}(\xi_1)b^*_{-}(\xi_3)\Psi.
\end{split}
\end{equation}
and
\begin{equation}\label{4.66}
\left[(H_I^{(1)})^*,b_{4,+}(f_{4,t})\right]\Psi=\left[(H_I^{(2)})^*,b_{4,+}(f_{4,t})\right]\Psi=0
\end{equation}
By \eqref{B.5} we get
\begin{equation}\label{4.67}
\begin{split}
&\left\|\left[H_I^{(1)},b_{4,+}(f_{4,t})\right]\Psi\right\|\leq \\
&\int \d x^2 \bigg(\int \d \xi_2 \left| \left\langle \int \d \xi_4 U^{(\nu_\mu)}(\xi_4)f_{4,t}(\xi_4)\mathrm{e}^{ip_{4}^2 x^2} \overline{F(\xi_{2},\xi_{4})}, \gamma^{0}\gamma^\alpha(1-\gamma_5) U^{(\mu)}(x^2,\xi_2) \right\rangle \right|^2 \bigg)^{\frac{1}{2}}\\ &\left \|\int \d \xi_1 \d \xi_3 \mathrm{e}^{-ip^2_{3}x^2}\left\langle U^{(e)}(x^2,\xi_1),\gamma^0\gamma_{\alpha}(1-\gamma_5)W^{(\overline{\nu_e})}(\xi_3)G(\xi_1,\xi_3)\right\rangle b_{+}^{*}(\xi_1)b_{-}^*(\xi_3)\Psi\right\|.
\end{split}
\end{equation}
where $\left\langle .,.\right\rangle$ is the scalar product in $\C^4$.
and
\begin{equation}\label{4.68}
\begin{split}
&\left\|\left[H_I^{(2)},b_{4,+}(f_{4,t})\right]\Psi\right\|\leq \\
&\int \d x^2 \bigg(\int \d \xi_2 \left| \left\langle \int \d \xi_4 U^{(\nu_\mu)}(\xi_4)f_{4,t}(\xi_4)\mathrm{e}^{ip_{4}^2 x^2} \overline{F(\xi_{2},\xi_{4})}, \gamma^0 \gamma^\alpha(1-\gamma_5) W^{(\mu)}(x^2,\xi_2) \right\rangle \right|^2 \bigg)^{\frac{1}{2}}\\ &\left \|\int \d \xi_1 \d \xi_3 \mathrm{e}^{-ip^2_{3}x^2}\left\langle U^{(e)}(x^2,\xi_1),\gamma^0
\gamma_{\alpha}(1-\gamma_5)W^{(\overline{\nu_e})}(\xi_3)G(\xi_1,\xi_3)\right\rangle b_{+}^{*}(\xi_1)b_{-}^*(\xi_3)\Psi\right\|.
\end{split}
\end{equation}
By adapting the proof of \eqref{4.57} to \eqref{4.67} and \eqref{4.68} one can show that there exists for every $j$ a function, denoted by $H^{\nu_\mu}(\xi_2,\xi_4)$, such that
\begin{equation}\label{4.69}
\begin{split}
&\left\|e^{itH}\left[H_I,b_{4,+}(f_{4,t})\right]e^{-itH}\Psi\right\|\leq \\
&CC_{f_4}\frac{1}{t^2}\bigg(\sum_{j=1}^4\int \d \xi_2 \d \xi_4 \chi_{f_4}(\xi_4)|H_j^{(\nu_\mu)}(\xi_2,\xi_4)|^2\bigg)^{\frac{1}{2}}\times \\
&\left\|G(.,.)\right\|_{L^2(\Gamma_1\times\R^3)}\frac{1}{m_e}( \tilde{a}\|H\Psi\| +(\tilde{b}+ m_e)\|\Psi\|)
\end{split}
\end{equation}
with
\begin{equation*}
\sum_{j=1}^4\int \d \xi_2 \d \xi_4 \chi_{f_4}(\xi_4)|H_j^{(\nu_\mu)}(\xi_2,\xi_4)|^2<\infty
\end{equation*}
Here $\chi_{f_4}(.)$ is the characteristic function of the support of $f_4(.)$.
Similarly we have
\begin{equation}\label{4.70}
\begin{split}
& \left(\Phi, b^*_{4,+,T}(f_4)\Psi\right)-\left(\Phi, b^*_{4,+,T_{0}}(f_4)\Psi\right)\\
& = g\int_{T_{0}}^{T} \frac{d}{dt} \left(\Phi, b^*_{4,+,t}(f_4)\Psi\right)=ig \int_{T_{0}}^{T} \left(\Phi,e^{itH} [ H_I,b^*_{4,+}(f_{4,t})] e^{-itH}\Psi\right)dt
\end{split}
\end{equation}
with
\begin{equation}\label{4.71}
\begin{split}
&\left[(H_I^{(1)})^*,b_{4,+}^*(f_{4,t})\right]\Psi=
\int \d \xi_1 \d \xi_2 \d \xi_3 \d\xi_4 \,\Bigg(\int \d x^2 \mathrm{e}^{ix^2r^2} \\
&\big( \overline{W}^{(\overline{\nu}_e)}(\xi_3)\gamma^\alpha(1-\gamma_5)U^{(e)}(x^2,\xi_1)\big)
\big(\overline{U}^{(\mu)}(x^2,\xi_2)\gamma_\alpha(1-\gamma_5)U^{(\nu_\mu)}(\xi_4)\big)\Big)\\ &\overline{F(\xi_{2}, \xi_{4})}\,\overline{G(\xi_{1}, \xi_{3})}f_{4,t}(\xi_4)
b_{+}^*(\xi_2)b_{-}(\xi_3)b_{+}(\xi_1)\Psi.
\end{split}
\end{equation}
\begin{equation}\label{4.72}
\begin{split}
&\left[(H_I^{(2)})^{*},b^*_{4,+}(f_{4,t})\right]\Psi=
\int \d \xi_1 \d \xi_2 \d \xi_3 \d \xi_4\,\Bigg(\int \d x^2 \mathrm{e}^{ix^2r^2} \\
&\big(\overline{W}^{(\overline{\nu}_e)}(\xi_3) \gamma^\alpha(1-\gamma_5)U^{(e)}(x^2,\xi_1)\big)
\big(\overline{W}^{(\mu)}(x^2,\xi_2) \gamma_\alpha(1-\gamma_5) U^{(\nu_\mu)}(\xi_4)\big)\Big)\\ &\overline{F((\xi_{2}, \xi_{4})}\,\overline{G(\xi_1,\xi_3)}f_{4,t}
b_{-}(\xi_3)b_{+}(\xi_1)b_{-}(\xi_2)\Psi.
\end{split}
\end{equation}
and
\begin{equation}\label{4.73}
\left[H_I^{(1)},b_{4,+}^*(f_{4,t})\right]=\left[(H_I^{(2)},b^*_{4,+}(f_{4,t})\right]=0
\end{equation}
By \eqref{B.5} we get
\begin{equation}\label{4.74}
\begin{split}
&\left\|\left[(H_I^{(1)})^*,b^*_{4,+}(f_{4,t})\right]\Psi\right\|\leq \\
&\int \d x^2 \bigg(\int \d \xi_2 \left| \left\langle U^{(\mu)}(x^2,\xi_2), \gamma^{0}\gamma^\alpha(1-\gamma_5)\int \d \xi_4 U^{(\nu_\mu)}(\xi_4)f_{4,t}(\xi_4)\mathrm{e}^{ip_{4}^2x^2} \overline{F(\xi_{2},\xi_{4})}\right\rangle \right|^2 \bigg)^{\frac{1}{2}}\\ &\left \|\int \d \xi_1 \d \xi_3 \mathrm{e}^{-ip^2_{3}x^2}\left\langle W^{(\overline{\nu_e})}(\xi_3),\gamma^0\gamma_{\alpha}(1-\gamma_5)U^{(e)}(x^2,\xi_1)\overline{G(\xi_1,\xi_3)}\right\rangle b_{+}(\xi_1)b_{-}(\xi_3)\Psi\right\|.
\end{split}
\end{equation}
where $\left\langle .,.\right\rangle$ is the scalar product in $\C^4$.
and
\begin{equation}\label{4.75}
\begin{split}
&\left\|\left[(H_I^{(2 )})^*,b^*_{4,+}(f_{4,t})\right]\Psi\right\|\leq \\
&\int \d x^2 \bigg(\int \d \xi_2 \left| \left\langle U^{(\mu)}(x^2,\xi_2), \gamma^{0}\gamma^\alpha(1-\gamma_5)\int \d \xi_4 U^{(\nu_\mu)}(\xi_4)f_{4,t}(\xi_4)\mathrm{e}^{ip_{4}^2x^2} \overline{F(\xi_{2},\xi_{4})}\right\rangle \right|^2 \bigg)^{\frac{1}{2}}\\ &\left \|\int \d \xi_1 \d \xi_3 \mathrm{e}^{-ip^2_{3}x^2}\left\langle W^{(\overline{\nu_e})}(\xi_3),\gamma^0\gamma_{\alpha}(1-\gamma_5)U^{(e)}(x^2,\xi_1)\overline{G(\xi_1,\xi_3)}\right\rangle b_{+}(\xi_1)b_{-}(\xi_3)\Psi\right\|.
\end{split}
\end{equation}
By adapting the proof of \eqref{4.57} and \eqref{4.67} to \eqref{4.74} and \eqref{4.75} one gets
\begin{equation}\label{4.76}
\begin{split}
&\left\|e^{itH}\left[H_I,b^*_{4,+}(f_{4,t})\right]e^{-itH}\Psi\right\|\leq \\
&CC_{f_4}\frac{1}{t^2}\bigg(\sum_{j=1}^4\int \d \xi_2 \d \xi_4 \chi_{f_4}(\xi_4)|H_j^{(\nu_\mu)}(\xi_2,\xi_4)|^2\bigg)^{\frac{1}{2}}\times \\
&\left\|G(.,.)\right\|_{L^2(\Gamma_1\times\R^3)}\frac{1}{m_e}( \tilde{a}\|H\Psi\| +(\tilde{b}+ m_e)\|\Psi\|)
\end{split}
\end{equation}
It follows from \eqref{4.63},\eqref{4.69},\eqref{4.70} and \eqref{4.76} that the strong limits of $b^\sharp_{4,+,t}(f_4)$ exist when t goes to $\pm\infty$, for all $f_4\in L^2(\R^3)$ and for every $g \leq g_0$.
This concludes the proof of theorem 4.3.
\end{proof}
\subsection{Existence of a Fock space subrepresentation of the asymptotic CAR}\mbox{}
From now on we only consider the case where the time t goes to $+\infty$. The following proposition is an easy consequence of theorem 4.1.
\begin{proposition}\label{4.5}\mbox{}
Suppose Hypothesis 2.1-Hypothesis 4.2 and $g \leq g_1$. We have
i)Let $f_1,g_1,f_2,g_2 \in L^2(\Gamma_1)$ and $f_3,g_3,f_4,g_4 \in L^2(\R^3)$. The following anticommutation relations hold in the sense of quadratic form.
\begin{equation*}
\begin{split}
\{b_{1,+,\infty}(f_1),b_{1,+,\infty}^{*}(g_1)\}=& \langle f_1,g_1\rangle_{L^2(\Gamma_1)}\mathbf{1} \\
\{b_{2, \epsilon,\infty}(f_2),\,b_{2, \epsilon',\infty}^*(g_2)\} =&\langle f_2,g_2\rangle_{L^2(\Gamma_1)}\delta_{\epsilon\epsilon'}\mathbf{1} \\
\{b_{3,-,\infty}(f_3),b_{3,-,\infty}^{*}(g_3)\}=& \langle f_3,g_3\rangle_{L^2(\R^3)}\mathbf{1} \\
\{b_{4,+,\infty}(f_4),b_{4,+,\infty}^{*}(g_4)\}=& \langle f_4,g_4\rangle_{L^2(\R^3)}\mathbf{1}\\
\{b_{1,+,\infty}(f_1),b_{1,+,\infty}(g_1)\}=&\{b_{1,+,\infty}^*(f_1),b_{1,+,\infty}^{*}(g_1)\}= 0\\
\{b_{1,+,\infty}(f_1),\,b_{2,\epsilon,\infty}^{\sharp}(f_2)\}=&\{b_{1,+,\infty}(f_1),b_{3,-,\infty}^{\sharp}(f_3)\}=0\\
\{b_{1,+,\infty}(f_1),b_{4,+,\infty}^{\sharp}(f_4)\}=&0.\\
\{b_{2, \epsilon,\infty}(f_2),\,b_{2, \epsilon',\infty}(g_2)\} =&\{b_{2, \epsilon,\infty}^*(f_2),\,b_{2, \epsilon',\infty}^*(g_2)\} =0\\
\{b_{2, \epsilon,\infty}(f_2),\,b_{3,-,\infty}^\sharp(f_2)\} =&\{b_{2, \epsilon,\infty}(f_2),\,b_{4,+,\infty}^\sharp(f_4)\} =0\\
\{b_{3,-,\infty}(f_3),b_{3,-,\infty}(g_3)\}=&\{b_{3,-,\infty}^*(f_3),b_{3,-,\infty}^{*}(g_3)\}=0\\
\{b_{3,-,\infty}(f_3),b_{4,+,\infty}^\sharp(f_4)\}=&0\\
\{b_{4,+,\infty}(f_4),b_{4,+,\infty}(g_4)\}=&\{b_{4,+,\infty}^*(f_4),b_{4,+,\infty}^{*}(g_4)\}=0.
\end{split}
\end{equation*}
Here $\epsilon=\pm$.
ii)
\begin{equation*}
\begin{split}
e^{itH}b_{1,+,\infty}^{\sharp}(f_1)&=b_{1,+,\infty}^\sharp(e^{i\omega(\xi_1)t}f_1)e^{itH}. \\
e^{itH}b_{2,\pm,\infty}^{\sharp}(f_2)&=b_{2,\pm,\infty}^\sharp(e^{i\omega(\xi_2)t}f_2)e^{itH}.\\
e^{itH}b_{3,-,\infty}^{\sharp}(f_3)&=b_{3,-,\infty}^\sharp(e^{i\omega(\xi_3)t}f_3)e^{itH}.\\
e^{itH}b_{4,+,\infty}^{\sharp}(f_4)&=b_{4,+,\infty}^\sharp(e^{i\omega(\xi_4)t}f_4)e^{itH}.
\end{split}
\end{equation*}
and the following pulltrough formulae are satisfied:
\begin{align*}
[H,b_{1,+,\infty}^{*}(f_1)]= b_{1,+,\infty}^{*}(\omega(\xi_1)f_1), & \; [H,b_{1,+,\infty}(f_1)]=- b_{1,+,\infty}(\omega(\xi_1)f_1) \\
[H,b_{2,\pm ,\infty}^{*}(f_2)]= b_{2,\pm ,\infty}^{*}(\omega(\xi_2)f_2), & \; [H,b_{2,\pm ,\infty}(f_2)]=- b_{2,\pm ,\infty}(\omega(\xi_2)f_2) \\
[H,b_{3,-,\infty}^{*}(f_3)]= b_{3,-,\infty}^{*}(\omega(\xi_3)f_3), & \; [H,b_{3,-,\infty}(f_3)]=- b_{3,-,\infty}(\omega(\xi_3)f_3) \\
[H,b_{4,+,\infty}^{*}(f_4)]= b_{4,+,\infty}^{*}(\omega(\xi_4)f_4), & \; [H,b_{4,+,\infty}(f_4)]=- b_{4,-,\infty}(\omega(\xi_4)f_4).
\end{align*}
iii)
\begin{equation*}
b_{1,+,\infty}(f_1)\Omega_g=b_{2,\pm ,\infty}(f_2)\Omega_g=b_{3,-,\infty}(f_3)\Omega_g=b_{4,+,\infty}(f_4)\Omega_g=0
\end{equation*}
Here $\Omega_g$ is the ground state of H.
\end{proposition}
Our main result is the following theorem
\begin{theorem}\label{4.5}
Suppose Hypothesis 2.1-Hypothesis 4.2 and $g \leq g_1$. Then we have
\begin{equation*}
\sigma_{ac}=[E,\infty).
\end{equation*} \end{theorem}
\begin{proof}
By \eqref{2.2} we have, for all sets of integers $(p,q,\bar q,r,s)$ in $\N^5$,
\begin{equation}\label{4.77}
\mathfrak{F} = \bigoplus_{(p,q,\bar q,r,s)} \mathfrak{F}^{(p,q,\bar q, r,s)}.
\end{equation}
with
\begin{equation}\label{4.78}
\mathfrak{F}^{(p,q,\bar q,r,s)}
=(\otimes _a^{p} L^2(\Gamma_1))\otimes (\otimes _a^{q} L^2(\Gamma_1))\otimes
(\otimes _a^{\bar q} L^2(\Gamma_1))\otimes
(\otimes _a^{r} L^2(\R^3))\otimes
(\otimes _a^{s} L^2(\R^3))\ .
\end{equation}
Here $p$ is the number of electrons, $q$ (resp. $\bar q$) is the number of muons
(resp. antimuons), $r$
is the number of antineutrinos $\overline{\nu_e}$ and $s$ is the number of neutrinos $\nu_\mu$.
Let $\{e^1_i|i=1,2,..\}$,$\{e^2_j|j=1,2,..\}$ and $\{f^2_k|k=1,2,..\}$ be tree orthonormal basis of $L^2(\Gamma_1)$. Let $\{e^3_l|l=1,2,..\}$ and $\{e^4_m|m=1,2,..\}$ be two orthonormal basis of $L^2(\R^3)$.
Consider the following vectors of $\mathfrak{F}$
\begin{multline}\label{4.79}
\produ_{1\leq\alpha\leq p}b_{1,+}^*(e^1_{i_\alpha})\prod_{1\leq\alpha\leq q}b_{2,+}^*(e^2_{j_\alpha})\prod_{1\leq\alpha\leq \bar q}b_{2,-}^*(f^2_{k_\alpha})
\produ_{1\leq\alpha\leq r}b_{1,+}^*(e^3_{l_\alpha}) \prod_{1\leq\alpha\leq s}b_{4,+}^*(e^4_{m_\alpha})\Omega
\end{multline}
The indices are assumed ordered, $i_1<...<i_p$, $j_1<...<j_q$, $k_1<...<k_{\bar q}$, $l_1<...<l_r$ and $m_1<...<m_s$.
The set, for $(p,q,\bar q,r,s)$ given in $\N^5$,
\begin{multline*}
\mathbb{D}^{(p,q,\bar q,r,s)}= \{\Phi\in \mathfrak{F}^{(p,q,\bar q,r,s)}\;|\;\Phi \; \mbox{is a finite linear combination of basis vectors}\\ \mbox{of the form \eqref{4.79}} \}
\end{multline*}
is a dense domain in $ \mathfrak{F}^{(p,q,\bar q,r,s)}$.
The set of vectors of the form \eqref{4.79} is an orthonormal basis of $\mathfrak{F}^{(p,q,\bar q,r,s)}$ (see \cite[Chapter 10]{Thaller1992}). Hence the vectors obtained in this way for $ p,q,\bar q,r,s,= 0,1,2,...$ form an orthonormal basis of $\mathfrak{F}$ and the set
\begin{multline*}
\mathbb{D}= \{\Psi\in \mathfrak{F}\;|\;\Psi \; \mbox{is a finite linear combination of basis vectors}\\ \mbox{of the form \eqref{4.79} for $p,q,\bar q,r,s=0,1,2,\dotsb$} \}
\end{multline*}
is a dense domain in $ \mathfrak{F}$.
On the other hand we now introduce the following vectors of $\mathfrak{F}$
\begin{multline}\label{4.80}
\produ_{1\leq\alpha\leq p}b_{1,+,\infty}^*(e^1_{i_\alpha})\prod_{1\leq\alpha\leq q}b_{2,+,\infty}^*(e^2_{j_\alpha})\prod_{1\leq\alpha\leq \bar q}b_{2,-,\infty}^*(f^2_{k_\alpha})\\
\produ_{1\leq\alpha\leq r}b_{3,-,\infty}^*(e^3_{l_\alpha})\prod_{1\leq\alpha\leq s}b_{4,+,\infty}^*(e^4_{m_\alpha})
\Omega_g
\end{multline}
Let $\mathfrak{F}_\infty^{(p,q,\bar q,r,s)}$ denote the closed linear hull of vectors of the form \eqref{4.80}. It follows from proposition 4.4 that the set of vectors of the form \eqref{4.80} is an orthonormal basis of $\mathfrak{F}_\infty ^{(p,q,\bar q,r,s)}$.
The set, for $(p,q,\bar q,r,s)$ given in $\N^5$,
\begin{multline*}
\mathbb{D}_\infty^{(p,q,\bar q,r,s)}= \{\Phi\in \mathfrak{F}_\infty\;|\;\Phi \; \mbox{is a finite linear combination of basis vectors}\\ \mbox{of the form \eqref{4.80}}\}.
\end{multline*}
is a dense domain in $ \mathfrak{F}_\infty^{(p,q,\bar q,r,s)}$.
The asymptotic outgoing Fock pace denoted by $\mathfrak{F}_\infty$ is then defined by
\begin{equation}\label{4.81}
\mathfrak{F}_\infty = \bigoplus_{p,q , \bar q, r, s} \mathfrak{F}_\infty^{(p,q,\bar q,r,s)}.
\end{equation}
The vectors of the form \eqref{4.80} obtained for $ p,q,\bar q,r,s,= 0,1,2,...$ form an orthonormal basis of $\mathfrak{F}$ and the set
\begin{multline*}
\mathbb{D}_\infty= \{\Phi\in \mathfrak{F}_\infty\;|\;\Phi \; \mbox{is a finite linear combination of basis vectors}\\ \mbox{of the form \eqref{4.80}\,for $p,q,\bar q,r,s= 0,1,2,\dotsb$}\}
\end{multline*}
is a dense domain in $ \mathfrak{F}_\infty$.
We now introduce the following linear operators, denoted by $W^{(p,q,\bar q,r,s)}_\infty$, and defined on $\mathbb{D}^{(p,q,\bar q,r,s)}$ by
\begin{equation}\label{4.82}
\begin{split}
&W^{(p,q,\bar q,r,s)}_\infty \produ_{1\leq\alpha\leq p}b_{1,+}^*(e^1_{i_\alpha})\prod_{1\leq\alpha\leq q}b_{2,+}^*(e^2_{j_\alpha})\prod_{1\leq\alpha\leq \bar q}b_{2,-}^*(f^2_{k_\alpha})\produ_{1\leq\alpha\leq r}b_{1,+}^*(e^3_{l_\alpha})\\ &\quad\qquad\qquad\prod_{1\leq\alpha\leq s}b_{4,+}^*(e^4_{m_\alpha})\Omega \\
&=\produ_{1\leq\alpha\leq p}b_{1,+,\infty}^*(e^1_{i_\alpha})\prod_{1\leq\alpha\leq q}b_{2,+,\infty}^*(e^2_{j_\alpha})\prod_{1\leq\alpha\leq \bar q}b_{2,-,\infty}^*(f^2_{k_\alpha})\produ_{1\leq\alpha\leq r}b_{3,-,\infty}^*(e^3_{l_\alpha})\\ &\quad\prod_{1\leq\alpha\leq s}b_{4,+,\infty}^*(e^4_{m_\alpha})\Omega_g.
\end{split}
\end{equation}
$W^{(p,q,\bar q,r,s)}_\infty$ can be uniquely extended to linear operators from $\mathbb{D}^{(p,q,\bar q,r,s)}$ to $\mathbb{D}_\infty^{(p,q,\bar q,r,s)}$. It then follows from prposition 4.4. that the operators $W^{(p,q,\bar q,r,s)}_\infty$ can be uniquely extended to unitary operators from $\mathbb{D}^{(p,q,\bar q,r,s)}$ to $\mathbb{D}_\infty^{(p,q,\bar q,r,s)}$
Let
\begin{equation}\label{4.83}
W_\infty = \bigoplus_{p,q , \bar q, r, s}W^{(p,q,\bar q,r,s)}_\infty .
\end{equation}
Hence $W_\infty$ is a unitary operator from $ \mathfrak{F}$ to $ \mathfrak{F}_\infty$.
The operators $b_{1,+,\infty}(f_1)$,$b_{1,+,\infty}^*(g_1)$,$b_{2,+,\infty}(f_2)$, $b_{2,+,\infty}^*(g_2)$,$b_{2,-,\infty}(f_2)$, $b_{2,-,\infty}^*(g_2)$,$b_{3,-,\infty}(f_3)$, $b_{3,-,\infty}^*(g_3)$, $b_{4,+,\infty}(f_4)$ and $b_{4,+,\infty}^*(g_4)$ defined on $\mathfrak{F}_\infty $ generate a Fock representation of the ACR (see Proposition 4.4 i)).
By proposition 4.4 ii) we have
\begin{equation}\label{4.84}
\begin{split}
&e^{itH}\produ_{1\leq\alpha\leq p}b_{1,+,\infty}^*(e^1_{i_\alpha})\prod_{1\leq\alpha\leq q}b_{2,+,\infty}^*(e^2_{j_\alpha})\prod_{1\leq\alpha\leq \bar q}b_{2,-,\infty}^*(f^2_{k_\alpha})\produ_{1\leq\alpha\leq r}b_{3,-,\infty}^*(e^3_{l_\alpha})\\ &\qquad \produ_{1\leq\alpha\leq s}b_{4,+,\infty}^*(e^4_{m_\alpha})\Omega_g \\
&=e^{iEt}\produ_{1\leq\alpha\leq p}b_{1,+,\infty}^*(e^{i\omega(\xi_1)t}e^1_{i_\alpha})\produ_{1\leq\alpha\leq q}b_{2,+,\infty}^*(e^{i\omega(\xi_2)t} e^2_{j_\alpha})\produ_{1\leq\alpha\leq \bar q}b_{2,-,\infty}^*(e^{i\omega(\xi_2)t}f^2_{k_\alpha})\\ &\quad\qquad\produ_{1\leq\alpha\leq r}b_{3,-,\infty}^*(e^{i\omega(\xi_3)t}e^3_{l_\alpha})\produ_{1\leq\alpha\leq s}b_{4,+,\infty}^*(e^{i\omega(\xi_4)t}e^4_{m_\alpha})\Omega_g.
\end{split}
\end{equation}
Hence $e^{iHt}$ leaves $\mathfrak{F}_\infty$ invariant and $H$ is both reduced by $\mathfrak{F}_\infty$ and $\mathfrak{F}_\infty^{\bot}$. Thus
\begin{equation*}
\mathfrak{F}\simeq \mathfrak{F}_\infty\oplus\mathfrak{F}_\infty^{\bot}
\end{equation*}
In view of \eqref{4.5}, \eqref{4.48} and \eqref{4.84} we get
\begin{equation}\label{4.85}
\begin{split}
&W_\infty e^{itH_0}\produ_{1\leq\alpha\leq p}b_{1,+}^*(e^1_{i_\alpha})\prod_{1\leq\alpha\leq q}b_{2,+}^*(e^2_{j_\alpha})\prod_{1\leq\alpha\leq \bar q}b_{2,-}^*(f^2_{k_\alpha})\produ_{1\leq\alpha\leq r}b_{3,-}^*(e^3_{l_\alpha})\\ &\qquad \produ_{1\leq\alpha\leq s}b_{4,+}^*(e^4_{m_\alpha})\Omega \\
&=e^{i(H-E)t}W_\infty \produ_{1\leq\alpha\leq p}b_{1,+}^*(e^1_{i_\alpha})\produ_{1\leq\alpha\leq q}b_{2,+,}^*( e^2_{j_\alpha})\produ_{1\leq\alpha\leq \bar q}b_{2,-}^*(f^2_{k_\alpha})\\ &\quad\qquad\produ_{1\leq\alpha\leq r}b_{3,-}^*(e^3_{l_\alpha})\produ_{1\leq\alpha\leq s}b_{4,+}^*(e^4_{m_\alpha})\Omega.
\end{split}
\end{equation}
This yields
\begin{equation}\label{4.86}
W_\infty e^{it(H_0+ E)}= =e^{iHt}W_\infty
\end{equation}
Hence the reduction of $H$ to $\mathfrak{F}_\infty$ is unitarily equivalent to $H_0 + E$. Thus $ \sigma_{ac}(H)= [E,\infty)$ . This concludes the proof of theorem 4.5.
\end{proof}
\section*{Acknowledgements.}
J.-C.G. acknowledges J.-M~ Barbaroux, J.~Faupin and G.~Hachem for helpful discussions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,715 |
Q: where does the people picker control get its data from? I have a sharepoint hosted add-in which includes some people picker controls. In these people picker there are all of the users available which are also available through the Active directory but if I check the users on my site collection I can only see a few of them (not all of the AD user where created here).
So where does the people picker control get its data from? Directly from the AD or from a "boss"-DB on the sharepoint server?
A: The People Picker has three sources:
*
*Active Directory (or the configured identity provider)
*The User Information List. The UIL is stored in the SharePoint Content Database UserInfo table. These are users you've already added to the Site Collection.
*The User Profile Service Application for Audiences only, as well as augmenting information to People pulled from one of the two sources above.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,657 |
Q: Server open index in wrong direction I have a React app which I start from express server. I have public directiory with /public/index.html file and copy of that file in '/' directory.
Code:
const express = require('express')
const path = require('path')
const port = process.env.PORT || 8080
const app = express()
app.use(express.static(__dirname))
app.get('*', (req,res) => {
res.sendFile(path.resolve(__dirname, './public/index.html'))
})
app.listen(port)
console.log('Server started: '+ port)
My server.js file is in '/' directory.
Express opens index.html in '/' directory instead of one this one in public directory, any suggestions why ?
A: I think you need app.use(express.static(path.join(__dirname, '/public/')))
Your use statement handles get requests first and makes app.get(* obsolete.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,226 |
Ranmau is a village in Dalmau block of Rae Bareli district, Uttar Pradesh, India. The nearest large town is Lalganj, which is 8 km away. As of 2011, it has a population of 710 people, in 119 households. It has one primary school and no healthcare facilities.
The 1961 census recorded Ranmau as comprising 1 hamlet, with a total population of 313 people (152 male and 161 female), in 56 households and 47 physical houses. The area of the village was given as 195 acres. There were 5 small shoe manufacturers in the village at that point.
The 1981 census recorded Ranmau as having a population of 463 people, in 77 households, and having an area of 77.70 hectares. The main staple foods were listed as wheat and rice.
References
Villages in Raebareli district | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,779 |
Kingsley Coman (París, 13 de juny de 1996) és un futbolista professional francès que juga com a davanter o mitjapunta, actualment al Bayern de Munic.
Ha jugat a Paris Saint-Germain FC, Juventus FC i Bayern de Munic.
Palmarès
Paris Saint-Germain
2 Ligue 1: 2012-13, 2013-14.
1 Copa de la lliga francesa: 2013-14.
1 Supercopa francesa: 2013.
Juventus FC
1 Serie A: 2014-15.
1 Copa italiana: 2014-15.
1 Supercopa italiana: 2015.
FC Bayern Munic
1 Campionat del Món: 2020.
1 Lliga de Campions de la UEFA: 2019-20.
1 Supercopa d'Europa: 2020.
5 Lliga alemanya: 2015-16, 2016-17, 2017-18, 2018-19, 2019-20.
3 Copa alemanya: 2015-16, 2018-19, 2019-20.
4 Supercopa alemanya: 2016, 2017, 2018, 2020.
Referències
Enllaços externs
Fitxa a Eurosport
Kingsley Coman a TBPlayers
Futbolistes parisencs
Futbolistes internacionals amb França de la dècada de 2010
Futbolistes al Campionat d'Europa de futbol 2016
Futbolistes internacionals amb França de la dècada de 2020
Futbolistes al Campionat d'Europa de futbol 2020
Futbolistes de la Copa del Món de Futbol de 2022
Futbolistes de la Juventus
Futbolistes del Paris Saint-Germain
Futbolistes del Bayern de Munic | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,712 |
\section{Introduction}
Color coherence effects in vacuum are found by TASSO and OPAL experiments \cite{Braunschweig:1990yd, Abbiendi:2002mj}, by the depletion of particle spectrum for small energies. Medium-induced soft gluon radiation off a single quark in the final state was also considered long ago \cite{Baier:1996kr, Zakharov:1997uu, Wiedemann:2000za, Gyulassy:2000er}. Our goal is to check the color coherence effects between initial and final states in the presence of a medium which could be of applicability in DIS on nucleus. It is a setup complementary to the antenna in the $s$-channel \cite{MehtarTani:2010ma, CasalderreySolana:2011rz, Armesto:2011ir, MehtarTani:2011tz}. In high energy DIS the virtual photon scatters on one quark inside the nucleus to change its transverse momentum, and then such quark can rescatter on the fields generated by the other components of the nucleus. The process we consider is just a first step in studying eA collisions \footnote{It still needs to be further improved. If the exchanging particle is a highly virtual gluon, then one can consider the process as pA collisions.}, which will be studied in the future LHeC and EIC colliders. When the scattering angle between the incoming and outgoing quarks becomes 0, our calculation matches the one for gluon production in the totally coherent limit in the CGC framework \cite{Kovchegov:1998bi}.
\section{The antenna spectrum in $t$-channel in vacuum}
\begin{figure*}
\begin{center}
\includegraphics[width=0.8\textwidth]{vacuum.pdf}
\end{center}
\caption{Antenna radiation in $t$-channel in vacuum}
\label{fig:vacuum}
\end{figure*}
Antenna radiation in $t$-channel in vacuum is shown in Fig.\ref{fig:vacuum}. We work in the infinite momentum frame. Small scattering angle between incoming and outgoing quarks ($p$ and $\bar p$, respectively) is assumed, i.e. $\theta_{p {\bar p}} \ll 1$. The eikonal approximation is employed, i.e. $p^+ \sim {\bar p}^+ \gg k^+ \gg |\bf k|$, where the relation between the forward light-cone momentum and the energy of the emitted soft gluon reads $k^+ = \sqrt{2} \, \omega$. The light-cone gauge $n \cdot A = A^+ = 0$ is chosen, with the axial vector $n = (0, 1, {\bf 0})$. A virtual photon $\gamma^*$ with transverse size $|\Delta {\bf x}| \sim 1 / |{\bar {\bf p}} - {\bf p}|$ is absorbed by an incoming quark. The soft gluon spectrum off the incoming quark in vacuum in which the azimuthal angle is integrated with respect to the direction of quark (analogously for the outgoing quark) reads
\begin{equation}
d N^{\rm vac}_{\rm in} = \frac{\alpha_s \, C_F}{\pi} \, \frac{d \omega}{\omega} \, \frac{\sin\theta \, d \theta}{1 - \cos\theta} \, \Theta (\cos\theta - \cos \theta_{p \bar{p}}),
\end{equation}
where $\alpha_s$ is the strong coupling constant, $C_F$ is the Casimir factor of the fundamental representation, $\theta \approx |{\boldsymbol k}| / \omega$ is the gluon emission angle, and the heaviside step function gives the angular constraint $\theta < \theta_{p {\bar p}}$, i.e. the soft gluon emission is constrained to be inside the cone set by the scattering angle between the incoming and outgoing quarks \cite{Basics of Perturbative QCD}. As long as $|{\boldsymbol \lambda}| = 1 / |{\boldsymbol k}|$, the transverse wave length of the emitted gluon, is shorter than $|{\boldsymbol r}| \sim t_{\rm form} \, \theta_{p {\bar p}}$ (the change of position of the quark in the transverse plane due to the photon scattering when the gluon is formed \footnote{The formation time is defined as $t_{\rm form} \sim \omega / {\boldsymbol k}^2$.}), the gluon resolves the color structure and the bremsstrahlung is therefore off either the incoming quark or the outgoing quark. In terms of angles, one gets immediately $\theta < \theta_{p {\bar p}}$ from the simple analysis above.
\section{The medium-induced antenna spectrum in $t$-channel}
\begin{figure*}
\begin{center}
\includegraphics[width=0.8\textwidth]{medium.pdf}
\end{center}
\caption{Medium-induced antenna radiation in $t$-channel}
\label{fig:medium}
\end{figure*}
Medium-induced antenna radiation in $t$-channel is shown in Fig.\ref{fig:medium}. We work in the same setup as the one in vacuum. Medium is assumed to act right after the photon scattering. Note that there is no need to go into the details of how the medium is modeled in our study. We consider dilute medium scenario, i.e. one gluon exchange. The opacity is defined as $L / \lambda$, and when we say ``opacity expansion'', we mean the expansion in number of scatterings. Hadronization is considered to occur outside the medium, therefore the process we consider is a pure perturbative one. The medium-induced antenna spectrum in $t$-channel at 1st order in opacity expansion reads:
\begin{equation}\label{eq:Total_spectrum}
\begin{split}
\omega\frac{dN^\text{med}}{d^3\vec{k}} = & \, \frac{\alpha_s C_F \,\hat q}{\pi} \int \frac{d^2 {\boldsymbol q}}{(2\pi)^2}{\cal V}^2({\boldsymbol q})~\int_0^{+ \infty} dx^+\\
& \, \Biggl[\frac{{\boldsymbol \nu}^2}{x^2\,(p\cdot v)^2}-\frac{{\boldsymbol \kappa}^2}{x^2\,(p\cdot k)^2}\\
& \, + \frac{2}{\bar x^2}\Biggl(\frac{\bar{{\boldsymbol \nu}}^2}{(\bar {p}\cdot v)^2}-\frac{\bar{{\boldsymbol \kappa}}\cdot\bar{{\boldsymbol \nu}}}{(\bar {p}\cdot v)(\bar{p} \cdot k)}\Biggr)(1-\cos\bigl[\Omega_{\bar{p}}{\boldsymbol x}+\bigr])\\
& \, + \frac{2}{x\,\bar{x}}\,\Biggl(\frac{{\boldsymbol \kappa}\cdot\bar{{\boldsymbol \kappa}}}{(\bar{p} \cdot k)\,(p\cdot k)}\,-\frac{{\boldsymbol \nu}\cdot\bar{{\boldsymbol \kappa}}}{(\bar{p} \cdot k)\,(p\cdot v)}\\
& \, + \Bigl(\frac{{\boldsymbol \nu}\cdot\bar{{\boldsymbol \kappa}}}{(\bar{p} \cdot k)\,(p\cdot v)}-\,\frac{{\boldsymbol \nu}\cdot\bar{{\boldsymbol \nu}}}{(\bar {p}\cdot v)\,(p\cdot v)}\Bigr)\,\bigl(1-\cos\bigl[\Omega_{\bar{p}}{\boldsymbol x}+\bigr])\Biggr)\Biggr],
\end{split}
\end{equation}
where ${\hat q}=\alpha_s C_A n_0 m_D^2$ denotes the medium transport coefficient and $\Omega_{\bar p} = ({\bf k} - {\bf q})^2 / (2 \, k^+)$ is the inverse of the gluon formation length. The second line in Eq.(\ref{eq:Total_spectrum}) is the contribution from the gluon radiation off the initial quark. The third line in Eq.(\ref{eq:Total_spectrum}) comes from the medium-induced gluon radiation off the outgoing quark, i.e. the independent spectrum \cite{Armesto:2011ir}. The last two lines of Eq.(\ref{eq:Total_spectrum}) is the interference between the incoming and outgoing quarks. Note that when the scattering angle $\theta_{p {\bar p}} = 0$, the spectrum becomes \footnote{The case of $\theta_{p {\bar p}} = 0$ was studied in the multiple soft scattering approach in \cite{Kovchegov:1998bi}.}
\begin{equation}\label{eq:0_scattering_angle}
\omega\frac{dN^{\rm med}}{d^3\vec{k}} = \frac{4 \, \alpha_s C_F \, {\hat q} \, L^+}{\pi} \int \frac{d^2 {\bf q}}{(2\pi)^2}{\cal V}^2({\bf q}) \, {\boldsymbol L}^2,
\end{equation}
where ${\boldsymbol L}= ({\boldsymbol \kappa} - {\boldsymbol q}) / ({\boldsymbol \kappa} - {\boldsymbol q})^2 - {\boldsymbol \kappa} / {\boldsymbol \kappa}^2$ is the transverse component of the Lipatov vertex in the light-cone gauge. The notations are ${\boldsymbol \kappa} = {\boldsymbol k} - x \, {\boldsymbol p}$, $x = k^+ / p^+$ and $\bar{{\boldsymbol \kappa}} = {\boldsymbol k} - {\bar x} \, {\bar {\boldsymbol p}}$, ${\bar x} = k^+ / {\bar p}^+$. The structure of the transverse component of the gauge invariant Lipatov vertex indicates that Eq.(\ref{eq:0_scattering_angle}) is the genuine medium-induced gluon radiation off an on-shell quark which comes from the $- \infty$ and goes to the $+\infty$.
\section{Soft limit}
In the soft gluon emission limit ($\omega \rightarrow 0$), the antenna spectrum in $t$-channel in medium, adding the medium-induced and the vacuum contributions, reads
\begin{equation}\label{eq:Soft_limit}
\omega \frac{d N^{\rm vac}}{d^3 \vec{k}} + \omega \frac{d N^{\rm med}}{d^3 \vec{k}} = \frac{4 \, \alpha_s \, C_F}{( 2 \, \pi )^2} \left[ (1 - \Delta) \left( \frac{1}{{\boldsymbol \kappa}^2} - \frac{{\boldsymbol \kappa} \cdot \bar{{\boldsymbol \kappa}}}{{\boldsymbol \kappa}^2 \, \bar{{\boldsymbol \kappa}}^2} \right) + \frac{1}{\bar{{\boldsymbol \kappa}}^2} - (1 - \Delta) \, \frac{{\boldsymbol \kappa} \cdot \bar{{\boldsymbol \kappa}}}{{\boldsymbol \kappa}^2 \, \bar{{\boldsymbol \kappa}}^2} \right],
\end{equation}
where $\Delta = {\hat q} \, L^+ / m_D^2$ at 1st order in opacity. The first term in the brackets shows the angular constraint for a reduced number of soft gluon emission off the incoming quark if one performs the azimuthal angle integration for the emitted soft gluon. The reason for the reduction of the soft gluon multiplicity inside the cone is that, the emitted soft gluon off the incoming quark will suffer rescattering when it goes through the medium. It therefore slows down the evolution of the gluon density. The rest in the brackets is the soft gluon emission off the outgoing quark in medium. When the medium is switched off, i.e. $\Delta \rightarrow 0$, one naturally gets the vacuum contribution. In the opaque medium limit, i.e. $\Delta \rightarrow 1$, one has
\begin{equation}\label{eq:opaque_medium}
\omega \frac{d N^{\rm vac}}{d^3 \vec{k}} + \omega \frac{d N^{\rm med}}{d^3 \vec{k}} = \frac{4 \, \alpha_s \, C_F}{( 2 \, \pi )^2} \, \frac{1}{\bar{{\boldsymbol \kappa}}^2}.
\end{equation}
After comparing Eq.(\ref{eq:opaque_medium}) with Eq.(\ref{eq:Soft_limit}), one can see that the soft part of the medium-induced gluon energy spectrum off the incoming quark is suppressed, i.e. the gluon density is saturated, and the bremsstralung contribution between vacuum and medium-induced parts get canceled with each other. In Eq.(\ref{eq:opaque_medium}), one gets complete color decoherence for the outgoing quark, i.e. in the opaque medium the outgoing quark loses the color coherence with the incoming quark, and then the soft gluon radiation off the outgoing quark is like the soft gluon radiation off a single quark in vacuum. A similar property was discovered in the antenna spectrum in $s$-channel \cite{Armesto:2011ir, MehtarTani:2011tz}.
Generalizing the results to the multiple soft scattering limit is in progress.
{\raggedright
\begin{footnotesize}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,031 |
This is the first time since I moved to California that I've seen leafhoppers actually injuring rice. Leaf tips turned yellow, and from the road it looks like salt injury, but once you get in the field you can see leafhopper nymphs and adults jumping and flying around.
What we have is aster leafhopper. They are reported to feed on broadleaf weeds and rice. I have some research plots in this field, with some areas where rice was not planted between plots. Weeds grew on the open areas, and after a propanil application that killed the weeds the leafhoppers might have moved to the rice.
From the UC IPM: Rice website: "Although leafhoppers can be present in fields during most of the growing season, the heaviest populations usually occur from early July through mid-August. Leafhoppers feed on rice plants by sucking up plant fluids through their long, piercing mouthparts. Although they are not known to vector of any rice pathogens in California, leafhoppers may occasionally occur in sufficient numbers to cause damage by their feeding. Injury associated with leafhoppers include stippling, yellowing, and drying leaves. Leafhoppers prefer senescing leaves, and symptoms usually occur on older leaves first. Leafhoppers are very mobile; adults fly and nymphs jump. Thus, infestations are rarely localized but appear generally throughout the field."
We haven't treated this field yet. There are still plenty of leafhoppers, but the plants are growing fast and putting out new leaves. If the leafhoppers remain and start injuring the flag leaf, a treatment might be appropriate. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,020 |
Болеслав Берут (, псевдоніми Яно́вський, Іваню́к, То́маш, Бенько́вський, Рутко́вський, * 18 квітня 1892, с. Рури Єзуїтське біля Любліна, Російська імперія — 12 березня 1956, Москва, РРФСР, СРСР) — польський політичний і державний діяч прорадянського спрямування. Перший і єдиний президент ПНР. Агент НКВД. Берут, людина-самоучка, з повним знанням і залізною рішучістю, прагнув впровадити в Польщі гнітючу сталіністську систему. Проте, разом з Владиславом Гомулкою, його головним суперником, Берут несе головну відповідальність за історичні зміни, які відбулися в Польщі після Другої світової війни. На відміну від будь-яких своїх комуністичних спадкоємців, Берут правив Польщею до своєї смерті.
Біографія
Отримав середню освіту.
З 1912 брав участь у лівацькому революційному русі, будучи членом ПСП — лівиці.
З грудня 1918 — член комуністичної партії Польщі (КПП).
У 1915—1923 — активіст профспілок кооперативів Польщі, потім — у КПП.
У 1923-1924 — член Виконавчого комітету окружної парторганізації в Заглембу-Домбровському.
У 1928 емігрував до СРСР, де з 1930 — слухач Міжнародної ленінської школи в Москві. Берута завербувало НКВС, після цього його переправлено для агентурної роботи в Австрію, потім Чехословаччину й Болгарію (1930—1932). За шпигунство та диверсійну діяльність засуджено до 7-ми років ув'язнення, але 1938 звільнено за амністією; втік до СРСР.
У 1939-1941 роках працював на окупованих територіях Польщі у структурах НКВС.
У роки Другої світової війни — один із послідовних сталіністів, ідеолог окупації Польщі та нав'язування країні радянського державного ладу. За рознарядкою НКВС СРСР, його включено в керівні органи Польської робітничої партії (ПРП, 1942). Вів підривну роботу на території Білорусі.
У 1943 закинуто на територію Польщі, що перебувала під контролем Німеччини. Працює над створенням мережі шпигунів-сталіністів.
Партійна кар'єра
У 1943—1944 — член Секретаріату, 1944—1956 — член Політбюро ЦК Польської робітничої партії. 1 січня 1944 р. обрано на голову Крайової Ради Народової, перебував на цьому пості до 4 лютого 1947 року.
21 липня 1944 створено Польський комітет національного визволення, що 31 грудня проголосив себе тимчасовим керівництвом.
У 1947—1952 р. — президент Польської Республіки (Польщі) й голова Державної Ради.
З вересня 1948 голова ЦК і Секретаріату ЦК.
У 1952—1954 р. — голова Ради Міністрів ПНР. З березня 1954 р. і до своєї смерті — перший секретар ПОРП.
З вересня 1948 — генеральний секретар ЦК ППР, з грудня 1948 — член Політбюро ЦК і голова ЦК Польської об'єднаної робітничої партії (ПОРП).
З березня 1954 р. — перший секретар ЦК ПОРП.
Був присутній як почесний гість на XX з'їзді КПРС. Після скандальної доповіді М. С. Хрущова «Про культ особистості та його наслідки» Берута розбив параліч, і він незабаром помер.
Пам'ять
Іменем Болеслава Берута названо вулицю в Мінську.
Примітки
Джерела та література
Andrzej Garlicki, Bolesław Bierut, Warszawa 1994, ISBN 83-02-05434-8
Література
Р. Кривонос. Берут (Bierut) Болеслав //
Посилання
Президенти Польщі
Польські кооператори
Члени ПОРП
Кавалери ордена «Хрест Грюнвальда»
Особи, увічнення яких підпадає під закон про декомунізацію
Польські атеїсти | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,484 |
package org.hivesoft.confluence.rest.representations;
import org.junit.Test;
import static org.hamcrest.CoreMatchers.is;
import static org.hamcrest.CoreMatchers.not;
import static org.hamcrest.MatcherAssert.assertThat;
import static org.junit.Assert.assertFalse;
import static org.junit.Assert.assertTrue;
public class VoteRepresentationTest {
@Test
public void test_gettersSetters_success() {
VoteRepresentation classUnderTest = new VoteRepresentation("someBallotTitle", "someChoice", "vote");
assertThat(classUnderTest.getBallotTitle(), is("someBallotTitle"));
assertThat(classUnderTest.getVoteChoice(), is("someChoice"));
assertThat(classUnderTest.getVoteAction(), is("vote"));
VoteRepresentation anotherInstance1 = new VoteRepresentation("someBallotTitle", "someChoice", "vote");
assertTrue(classUnderTest.equals(anotherInstance1));
assertFalse(classUnderTest.equals(null));
assertFalse(classUnderTest.equals("someString"));
assertThat(classUnderTest.hashCode(), is(anotherInstance1.hashCode()));
VoteRepresentation notThisBallotTitle = new VoteRepresentation("notThisBallotTitle", "someChoice", "vote");
assertFalse(classUnderTest.equals(notThisBallotTitle));
assertThat(classUnderTest.hashCode(), is(not(notThisBallotTitle.hashCode())));
VoteRepresentation notThisChoice = new VoteRepresentation("someBallotTitle", "notThisChoice", "vote");
assertFalse(classUnderTest.equals(notThisChoice));
assertThat(classUnderTest.hashCode(), is(not(notThisChoice.hashCode())));
VoteRepresentation notThisVote = new VoteRepresentation("someBallotTitle", "someChoice", "notThisVote");
assertFalse(classUnderTest.equals(notThisVote));
assertThat(classUnderTest.hashCode(), is(not(notThisVote.hashCode())));
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,794 |
<?php
class Google_Service_ShoppingContent_Row extends Google_Collection
{
protected $collection_key = 'cells';
protected $cellsType = 'Google_Service_ShoppingContent_Value';
protected $cellsDataType = 'array';
public function setCells($cells)
{
$this->cells = $cells;
}
public function getCells()
{
return $this->cells;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,226 |
Сезон НБА 1959/1960 — стал 14-м сезоном Национальной баскетбольной ассоциации (НБА). Чемпионом стала команда «Бостон Селтикс». В финале «Селтикс» победили в семи играх команду «Сент-Луис Хокс».
На драфте НБА 1959 года под первым номером клубом «Цинциннати Роялз» был выбран тяжёлый форвард Боб Бузер из университета штата Канзас. Под 2-м номером на том драфте был выбран Бейли Хауэлл, под 3-м — Том Хокинс, под 4-м — Дик Барнетт, а под 10-м — Руди Ларуссо. На территориальном драфте был выбран Уилт Чемберлен.
С этого года в регулярном чемпионате каждая из команд стала проводить по 75 матчей, вместо 72-х в прошлом.
Регулярный сезон
В = Выигрышей, П = Поражений, П% = Процент выигранных матчей
Плей-офф
Лидеры сезона по средним показателям за игру
Награды по итогом сезона
Самый ценный игрок НБА: Уилт Чемберлен, Филадельфия Уорриорз
Новичок года НБА: Уилт Чемберлен, Филадельфия Уорриорз
Первая сборная всех звёзд:
Ф Боб Петтит
Ф Элджин Бэйлор
Ц Уилт Чемберлен
З Боб Коузи
З Джин Шу
Вторая сборная всех звёзд:
Ф Джек Тваймен
Ф Дольф Шейес
Ц Билл Расселл
З Ричи Герин
З Билл Шерман
Ссылки
1959-60 NBA Season Summary
НБА по сезонам
НБА в сезоне 1959/1960 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 346 |
2019 was Earth's second-warmest year on record
NASA tests its water-hunting lunar rover VIPER
SpaceX plans to test Crew Dragon's launch escape system on January 18th
Latest in Science
Most habitable planets may be completely covered in water
Nearby 'super-Earth' may be our best shot yet at finding alien life
Alphabet starts collecting health info to better predict disease
OSIRIS-REx is complete and will collect asteroid samples in 2016
Mariella Moon, @mariella_moon
It seems like only yesterday that OSIRIS-REx got its first scientific instrument: a thermal emission spectrometer that can take the temp of an asteroid called Bennu every two seconds. Now, Lockheed Martin is done building the spacecraft, and NASA's slated to send it off to space in 2016 to collect samples from the near-Earth asteroid. Before it blasts off, though, the probe will undergo more rigorous testing as a whole spacecraft within the next five months. That includes subjecting it to extreme temps, vibrations, electromagnetic interference and vacuum that simulate space travel.
OSIRIS-REx principal investigator Dante Lauretta said:
We now move on to test the entire flight system over the range of environmental conditions that will be experienced on the journey to Bennu and back. This phase is critical to mission success, and I am confident that we have built the right system for the job.
If the spacecraft passes all the tests Lauretta and his team have in store for it, then it'll be shipped off to NASA in May to prepare for its launch in September 2016. NASA's hoping that the data and samples it collects can give them clues on the beginning of life on Earth. They're also depending on it to help them conjure up a way to deal with any asteroid that could hit our planet and to keep us from going the way of the dinosaur.
[Image credit: Lockheed Martin]
Source: Lockheed Martin
In this article: asteroid, nasa, osiris-rex, space | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,578 |
<!doctype html>
<html>
<head>
<meta charset="UTF-8">
<title>List Group</title>
<link href="http://fonts.googleapis.com/css?family=Patua+One" rel="stylesheet">
<link href="../css/typography.css" rel="stylesheet">
<link href="../css/list-group.css" rel="stylesheet">
</head>
<body>
<h1>List Group</h1>
<ul="list-group">
<li class="list-group-item">Dimetrodon</li>
<li class="list-group-item">Stegasaurus</li>
<li class="list-group-item">Triceratops</li>
</ul>
<h2>List Group Horizontal</h2>
<ul="list-group">
<li class="list-group-item list-group-item-inline">Dimetrodon</li>
<li class="list-group-item list-group-item-inline">Stegasaurus</li>
<li class="list-group-item list-group-item-inline">>Triceratops</li>
</ul>
</body>
</html>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,084 |
Cantu Construction Co started out as a small business 30 years ago, and was founded in Beeville, Texas. Our founder developed a strong passion in creating, which led him to start a project that was going to help other people transform their favorite place in the world: their homes. Our enthusiasm and commitment in this business has made many clients happy and satisfied.
Our clients have opened the door of success for our company, which is why we keep on growing in the field. Our professionals would like to help you through the process of the projects you plan, and we promise to be there from start to finish; our company is always up to dated and informed about the new machinery and tools, to make sure we meet with the quality standards required.
We are certified to do your construction projects properly, and our professionals count with many years of practice to paint, remodel and install everything correctly. Allow yourself to be part of this wonderful experience, you will thank us after we improve the aesthetic look of your property, and the final results will generate a smile in your face.
Our mission at Cantu Construction Co is to develop efficient and detailed plans in each construction project, in order to reach our client's satisfaction, and accomplish the expected results in a timely manner.
Our vision at Cantu Construction Co is to be the masters in construction in the area of Beeville, Texas by performing a quality work and a remarkable customer service, in order to expand our business and offer quality solutions to more people.
Our specialists are committed in delivering a detailed and well constructed job to our clients. The skills and years of experience in construction can only be found with our professionals, and the superior equipment and tools we use, will ensure quality at good prices. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,001 |
"ANTIQUE PAUL STORR GEORGE III GEORGIAN SILVER SERVING SPOON / BASTING SPOON 1810"
A superb quality Paul Storr Antique Solid Silver Serving Spoon or Basting Spoon in a traditional Fiddle & Thread Pattern. The Pattern is double struck from and back and there is a small crest engraved on the back of the handle. The Spoon is of notably heavy weight.
Made in London in 1810 by Paul Storr.
This Spoon is in very good condition with no damage. A wonderful patina commensurate with age and use. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,529 |
Jouvençon ist eine französische Gemeinde im Département Saône-et-Loire in der Region Bourgogne-Franche-Comté. Sie gehört zum Arrondissement Louhans und zum Kanton Cuiseaux. Die Gemeinde hat Einwohner (Stand ), sie werden Jouvençonnais, resp. Jouvençonnaises genannt.
Geografie
Die Gemeinde liegt in der Landschaft Bresse, im Südwesten des Arrondissement Louhans. Die nördliche und westliche Gemeindegrenze bildet die Seille, ein kurzes Stück der östlichen Gemeindegrenze die Sâne, die auf dem Gemeindegebiet von La Genête und Brienne weiter nach Westen verläuft. Durch die Gemeinde verläuft die Departementsstraße D971 (Cuisery–Rancy) in Südwest-Nordost-Richtung. Die Gemeinde weist nördlich der Siedlungsgebiete einige Waldgebiete auf, von den bewaldeten Stellen bis zur Seille finden sich keine Siedlungen oder Gebäude, da es sich um Überschwemmungsgebiete handelt. Zur Gemeinde gehören die folgenden Weiler und Fluren: Bief-de-Four, Bon de Caravattes, Écluse, Genetières, Grande-Rippe, Layer, Platières, Portail, Volatiers.
Klima
Das Klima in Jouvençon ist warm und gemäßigt. Es gibt das ganze Jahr über deutliche Niederschläge, selbst der trockenste Monat weist noch hohe Niederschlagsmengen auf. Die effektive Klimaklassifikation nach Köppen und Geiger ist Cfb. Die Temperatur liegt im Jahresdurchschnitt bei 11,1 °C. Innerhalb eines Jahres fallen 797 mm Niederschläge.
Toponymie
Die erste Erwähnung geht zurück auf das Jahr 981, in dem Capella Sancti-Mauricii in villa Gevencion erwähnt wurde. Man geht davon aus, dass das Gebiet bereits in gallo-römischer Zeit besiedelt war und einem Gevencius gehörte. Diese frühe Besiedlung wird durch archäologische Funde bestätigt, 1978 wurden Reste eines Hauses aus dem 3. Jahrhundert gefunden, dessen Grundriss zwei auf fünf Meter betrug.
Geschichte
Reste einer römischen Siedlung wurden bei les Pérouses gefunden, die Sankt-Mauritius-Kapelle wird beschrieben als bei einer Motte liegend, in einer Schleife der Seille. 981 vermachte Henri, Herzog von Burgund die Kirche von Huilly dem Kloster Tournus, behielt sich aber die Nutzniessung vor. Er erhielt dafür die Kapelle von Jouvençon, die sich bei seinem Schloss, aber auf der gegenüberliegenden Seite der Seille befand. Im 15. Jahrhundert wurde das Schloss zerstört und die Bewohner von Jouvençon bemächtigten sich wieder der Kapelle, die zur Dorfkirche wurde, jedoch lediglich als Filialkirche von Brienne. 1755 fand eine Auseinandersetzung statt, die Kirche war eingestürzt und die Kosten für den Wiederaufbau beliefen sich auf 2222 Livres. Die Bewohner von Layer sollten 613 Livres bezahlen, wollten jedoch nur 520 Livres beitragen. 1838 befürchtete der Conseil municipal, dass die Kirche wegen Bodenbewegungen erneut einstürze. Zudem war sie zu klein geworden und man erwarb ein Grundstück im Dorfzentrum und errichtete eine neue Kirche mit Pfarrhaus. Für die neue Kirche verwendete man Material von der alten Kapelle, von der lediglich noch der Chor bestehen blieb, und die zu einem kleinen Wallfahrtsort wurde. Der Lehrer berichtete Ende des 19. Jahrhunderts, es bestünden immer noch alte Bräuche, wie das Rattenfest zwischen Weihnachten und Neujahr, oder das Feuerfest am 5. Februar. An diesem Tag werde ein gesegneter Buchsbaumzweig im Waschtrog verbrannt, um das Haus gegen Feuer zu schützen. 1988 bestanden noch 11 Landwirtschaftsbetriebe.
Bevölkerung
Wirtschaft und Infrastruktur
In der Gemeinde befinden sich fünf Landwirtschaftsbetriebe, drei Betriebe der Baubranche. Als AOC-Produkte sind in Jouvençon Crème et beurre de Bresse zugelassen, ferner Volaille de Bresse und Dinde de Bresse.
Bildungseinrichtungen
In der Gemeinde besteht eine École maternelle, die der Académie de Dijon untersteht und von 45 Kindern besucht wird. Für die Schule gilt der Ferienplan der Zone A.
Literatur
Lucien Guillemaut (1842–1917): Histoire de la Bresse Louhannaise. Bd. 1, Louhans 1897.
Weblinks
Einzelnachweise
Ort in Bourgogne-Franche-Comté | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,712 |
module.exports.run = async (client, message, args) => {
let pingedPerson = message.mentions.users.first;
if (pingedPerson == null || pingedPerson.length == 0) {
message.reply(`Here ya go! ${message.author.avatarURL}`);
}
else {
message.reply(`Here ya go! ${pingedPerson.avatarURL}`)
}
}
module.exports.help = {
name: 'getavatar',
args: '[pingedPerson]',
notes: 'Gets an avatar of a user.'
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,264 |
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Praise for _Raising Wild_
"This book is not exactly about wild landscapes but the life of a house-holding family placed out there with two verge-of-puberty daughters. It is about our daily reality, not our fantasy possibilities, and who knows today what these girls will have to say later? So it is remarkably interesting, lively, non-theoretical, and hopeful. The wild might be wildfire or bushy-tailed woodrats under the floor—not just to live with but to know them. Michael Branch's book points forward, not back."
—GARY SNYDER
"I have long considered Michael Branch one of the true visionaries of western American literature—and here is further proof. This beautiful, often raucous account of fatherhood and (wild) faith takes us even deeper into his remarkable kinship with northwestern Nevada. A place where, through the 'daily practices of love, humility, and humor,' we can all learn to be at home in this world."
—JOHN T. PRICE, author of _Daddy Long Legs: The Natural Education of a Father_
Raising Wild
DISPATCHES FROM A HOME IN THE WILDERNESS
Michael P. Branch
ROOST BOOKS
BOULDER
2016
Roost Books
An imprint of Shambhala Publications, Inc.
4720 Walnut Street
Boulder, Colorado 80301
roostbooks.com
© 2016 by Michael P. Branch
Cover art by Westend61/Getty Images
Cover design by Jess Morphew and Daniel Urban-Brown
All rights reserved. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher.
"Witness" from _The Rain in the Trees_ by W. S. Merwin, copyright © 1988 by W. S. Merwin. Used by permission of Alfred A. Knopf, an imprint of the Knopf Doubleday Publishing Group, a division of Penguin Random House LLC. All rights reserved.
The Credits section constitutes a continuation of the copyright page.
Title page illustration © 2016 by grop/Shutterstock
LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA
Names: Branch, Michael P.
Title: Raising wild: dispatches from a home in the wilderness / Michael P. Branch.
Description: First edition. | Boulder: Roost Books, 2016.
Identifiers: LCCN 2016000871 | eISBN 9780834840553 | ISBN 9781611803457 (hardcover: acid-free paper)
Subjects: LCSH: Branch, Michael P. | Branch, Michael P.—Homes and haunts—Great Basin. | Wilderness areas—Great Basin. | Branch, Michael P.—Family. | Parenting—Great Basin. | Sustainability—Great Basin. | Great Basin—Biography. | Great Basin—Description and travel. | Great Basin—Environmental conditions. | BISAC: NATURE / Essays. | BIOGRAPHY & AUTOBIOGRAPHY / Personal Memoirs. | FAMILY & RELATIONSHIPS / Parenting / Fatherhood.
_For Hannah Virginia and Caroline Emerson_
contents
PRE-AMBLE
Learning to Walk
PART ONE
Birthing
1. Endlessly Rocking
2. The Nature within Us
3. Tracking Stories
4. Ladder to the Pleiades
PART TWO
Wilding
5. The Adventures of Peavine and Charlie
6. The Wild within Our Walls
7. Playing with the Stick
8. Freebirds
PART THREE
Humbling
9. Finding the Future Forest
10. My Children's First Garden
11. The Hills Are Alive
12. Fire on the Mountain
CODA
The V.E.C.T.O.R.L.O.S.S. Project
Acknowledgments
Credits
About the Author
E-mail Sign-Up
PRE-AMBLE
Learning to Walk
_"Off the trail" is another name for the Way, and sauntering off the trail is the practice of the wild. That is also where—paradoxically—we do our best work._
—GARY SNYDER, _The Practice of the Wild_
I earned my whiskers as a desert rat out in the remote, hilly, high-elevation western Great Basin Desert, scurrying across the land, scrabbling up every dome, running each ridge, scouring all the canyon draws, poking my snout into any rocky crevice I could find. My life's ambition had been to inhabit a place so remote as to provide immediate access to solitude and big wilderness. After many years of making incremental decisions that inched me closer to this dream, at last I took the leap. I put my life savings down on a parcel of raw land in the hinterlands of northwestern Nevada. My new property was not only suitably isolated and rendered nearly inaccessible by mud, snow, and unmaintained access roads, it was also at 6,000 feet in elevation and adjacent to public lands stretching west all the way to the foot of the Sierra Nevada mountains in neighboring California. Never mind walking the dog—I'd be able to launch solo backpacking trips from my own front door. Of course I had no front door and no house to hang it on—just an exposed, windy patch of sand and sagebrush out on the threshold of a vast, beautiful high desert wilderness.
My first hike from the secluded hilltop where I would eventually build a passive solar home began at sunrise. I set off alone, tromping west up a nearby hill that I would later name Moonrise. From the crest of Moonrise I looked out across a sage-filled draw that swept gracefully up to the rocky cliffs ornamenting a higher summit ridge I would later call Palisades. Another hour of stepping out brought me to the top of Palisades, from which an even higher ridge came into view. Harder scrambling brought me to the boulder-strewn crest of that third ridge, which I would name Prospect. From Prospect I gazed west over my dream landscape. Below me steep slopes of sage and scree, dotted yellow-green with mounds of ephedra, fell away to a deep, broad, sweeping valley. Several miles away, on the far side of that wild valley, rose my home mountain, an impressive, 8,000-foot-tall, fifteen-mile-long sleeping giant whose rocky north-facing flanks held streaks of late-season snow. From Prospect Ridge I enjoyed an expansive view of a high desert montane landscape on the monumental, inhuman scale that is its signature. A small cluster of pronghorn antelope could be seen gliding across the bitterbrush-stippled flank of the distant mountain. Two glossy black ravens wheeled silently beneath me. I felt the exhilaration of solitude.
Descending the ridge in a long, semicontrolled slide down the scree slope, I soon reached the broad canyon and made my way across its sunny, sage-strewn expanse to the foot of the big mountain. From the wildfire-scorched bitterbrush flats at the mountain's base I began an 1,800-foot ascent into the cloudless azure sky. Halfway up the mountain I discovered a small seep, where I paused to refill my water bottles. Looking back from there across the Great Basin I saw broken hills and alkali-white playas separated by juniper-dotted ridges rolling east to the horizon. Above me to the west twisted a faint game trail, rising through copses of bitter cherry and coyote willow and dodging between granite cliffs tattooed with chartreuse lichen.
Finally cresting the mountain's 8,000-foot ridge, I found myself in a sweeping summit meadow that reclined between a brace of rocky peaks and was graced with groves of gnarled aspens and the occasional green dome of a snowberry bush. Spreading out before me in all directions was an undulating yellow blanket of flowering tower butterweed. As I stood in the swell and ripple of that wild meadow, I gazed even farther west, first to the valley 3,000 feet below and then out over green California, its thick conifer forests and towering granite turrets and gables ignited by shafts of high-elevation sunlight. I was a man alone in the wilderness, and in that moment of summit bliss I imagined there was nothing that I could not see. Only later would I discover that, even from that high peak, what mattered most remained invisible to me.
Ever since they were toddlers, our daughters, Hannah and Caroline, have had as their life's ambition to achieve the summit of Moonrise, to finally stand with their fists in the air on the crest of what is in fact a modest bump in this immense landscape. An officially nameless little knoll of the sort that are numberless here in the Great Basin, Moonrise is less than a mile from our front door and perhaps only four or five hundred feet above us. It is the kind of "summit" a desert rat scurries over on his way to bigger, wilder quarry. But to a kid Moonrise looks imposing, and the idea of gaining a ridge carved so high into the sapphire desert sky has proven irresistible to the girls. The first ridge westward from our house, Moonrise is in sight on every walk we take together. It is what you notice when you walk Beauregard the dog, what you look up at when Dad pulls you overland in the sky-blue toboggan. The girls see it clearly from their play structure and from their tree house, too. If the window shades are up, Hannah and Caroline can see the etched crest of the ridge as they lie reading stories with my wife, Eryn, in their bunk beds. To a little kid, Moonrise fills the western sky.
On her fourth birthday, Hannah blew out the candles on her birthday cake and then turned immediately to me.
"Dad, am I old enough to climb Moonrise now?" she asked, as if the process of extinguishing the candles had suddenly brought her to a new level of readiness for outdoor adventure.
"Not yet, Bug. How about if we go up on your seventh birthday?" I replied.
"How about birthday six instead?" she persisted. The look on her face was both plaintive and determined.
"OK, kid, six it is. Two years from today. Put it on your calendar."
"Can CC come?" Hannah asked, looking down at her six-month-old sister, Caroline, who was at that moment pulling herself upright by grasping the seat of the kitchen stool upon which Hannah sat. "It wouldn't be fair not to take her with us for something this important."
"Good point," Eryn replied. "Sure, you guys can take CC. It will be the first ascent of Moonrise for both Branch girls!"
During the two intervening years, Moonrise was never out of sight, either literally or figuratively. For Hannah it was as if the days were being counted down, the anticipation rising as her sixth birthday approached. She often told Caroline stories about the upcoming "Moonrise Expedition," as she called it, an expedition that also appeared in Hannah's writings and drawings and even in her dreams. In an attempt to document insights gleaned from one dream of the great expedition, Hannah drew a detailed map, which she reasoned we could use to navigate safely to summit and thus avoid the explorer's fate of becoming forever lost in this vast desert wilderness. I explained that because our house would never be out of sight during the hike, we'd probably be fine, but I thanked her for the map just the same, and we agreed that it would be a good idea for her to add it to the day pack whose contents she had begun assembling more than a month before the epic climb was to take place.
When the day of her sixth birthday arrived, Hannah shouldered her little red day pack, and Eryn lifted Caroline into the baby backpack I had buckled around my waist and chest. After we made sure we had everything needed for the Moonrise Expedition, Eryn took pictures of the momentous occasion, and I set off with the girls toward the western horizon.
By the time we finished bushwhacking up the brushy draw behind the house, Hannah was tired and thirsty, and she had been caught a few times on the sharp thorns of the desert peach bushes that choked the ravine. But she was tough, continuing to slog uphill, her sinewy little legs driving downward as she leaned into the grade, pumping her elbows side to side like an old Scotsman striding into the wind. Hannah kept this up for two more hours, when at last she plopped down on the steep southern exposure of Moonrise. At this elevation the brushy canyon slopes had given way to open, sandy desert dotted with wild rye and ephedra. Patches of snow hid in the shade of a few boulders across the canyon, while the delicate yellow flowers of sagebrush buttercup were visible here and there in damp spots in the granitic sand.
On this steep desert slope we all drank water, ate snacks from Hannah's pack, and marveled at how tiny our house appeared, so far below us down in the sage. "You've come a long way today, Bug," I affirmed, looking homeward. "This is your biggest hike ever. How do you feel?"
"I'm pretty pooped, Dad. Do you think I can make it?" She looked tired, flushed, and also a little worried about the fate of the expedition.
"I do. It is so steep from here to summit that you'll have to use your hands, though, and scramble up like a desert monkey. We can go home anytime you want. Moonrise will always be here for you. But if you want to go all the way up, I think you can make it." She sat quietly, gazing out over the Great Basin. I couldn't tell what she was thinking or if she even knew. There were countless occasions on which I had taken the long view across our home desert with that same exhausted, exhilarated, open-hearted gaze.
"What do you think we'll find at the top?" she asked, as if weighing the prize of the summit against the cost of her fatigue.
"The thing about climbing a mountain is you never know what you'll find at the top. You'll just have to go up and see. But you don't have to do it today, honey."
Hannah paused for a moment and then stood up, though I still wasn't sure what decision she had made. "Let's go see what's at the top of Moonrise," she said resolutely, as she began to clamber up the final pitch with renewed energy. I trudged along behind, using my hands now and then for balance, watching Hannah rising into the sky above me as Caroline bounced in her pack on my shoulders.
Thirty minutes later our expedition's leader shouted back, "Dad, I think I'm almost there!"
"Just a little farther, Bug," I replied. "You go ahead and be the first up there, and we'll be along soon."
Five minutes later, panting with the weight of Caroline on my back, I crested the ridge and lifted my head. There, on the rocky summit of Moonrise, stood my six-year-old daughter, wearing the widest grin her little face could contain and holding up an immense, gracefully curved, many-fingered, bone-white mule deer antler—the largest and most perfect imaginable.
"Look what was on top of Moonrise!" Hannah exclaimed in delight. "Did you know this was up here?"
I stood in silence, genuinely astonished at the remarkable antler. "No, honey, I didn't," I finally managed to reply, breaking into a wide smile to match hers. "I think your expedition was a success. Happy birthday, Bug."
I looked down at our little home, a small island in a vast sagebrush ocean, a sanctuary huddled among the scattered green dots of wild juniper trees below. It was a more meaningful view than I'd ever achieved from a higher ridge or even from the summit of my majestic home mountain. In that moment it occurred to me that maybe Moonrise actually _was_ my home mountain. Our home mountain.
For more than a decade now we have lived together—my wife, Eryn, and our two girls, Hannah and Caroline—in these high, dry wilds, on this windy knoll, amid the remote ridges and canyons of the western Great Basin. A vast 200,000-square-mile expanse of sagebrush steppe high desert, the Great Basin extends from the Columbia Plateau up north to the Mojave and Sonoran Deserts down south, from the Rocky Mountains all the way out here to northwestern Nevada, where we live among the arid hills that ripple out beneath the rain shadow of the towering granitic escarpment of the Sierra Nevada.
The Great Basin is the largest and least familiar of American deserts, and it is among the most extreme landscapes in North America. To call conditions here "extreme" hardly does justice to the apocalyptic nature of life in this isolated high desert place. Since making our home here we've seen temperatures close to 110 degrees on the top end and 20 below at the bottom, while day-night swings of 40 degrees occur often. We've had high winds whip up into impenetrable sandstorms of swirling alkali dust, while other periods have been so breathless and stultifying that it seemed the earth had ceased to turn on its axis. Some spells have been so dry that we went six months without feeling a drop of rain, while other years have brought thunderstorms so intense as to trigger flash floods, which blasted through this open desert in broad sheets led by roiling, muddy, foot-tall snouts, sweeping our road away and leaving us stranded. Because we're in one of the most active seismic zones in the country, earthquakes have on occasion rattled books off our shelves, and smaller tremors are common. And, of course, this is fire country. There have been wildfires on the nearby public lands almost every summer or fall, and we have twice been subject to emergency evacuation as a curtain of wind-driven flames raced toward us across the desert plains beneath billowing black clouds of acrid smoke. Although we've occasionally been forced out by fire, we've more often been snowbound on our isolated hill, sometimes for days and often without electricity, huddled by the wood stove, gazing out at the beauty of snow that has sometimes been wind driven into drifts five feet deep.
The critters here are as wild as the weather. Mountain lions hunt our high valley each winter, and on several occasions a bobcat has tried to make a meal of the few laying hens we keep. This is winter range for mule deer, which join us in autumn to avoid becoming the snowbound prey of lions up in the Sierra, and it is also the year-round home of pronghorn antelope, which we see gliding effortlessly through the open desert at speeds of up to sixty miles per hour. Coyote are everywhere in this landscape, loping through the sage and bitterbrush by day and yipping by night in bands whose shrill chorus is carried downcanyon on moonlit wind. Desert cottontails are dwarfed by seven-pound black-tailed jackrabbits, which are struck from above and eviscerated by big redtails and by the fierce, silent great horned owls that have repurposed our home's peaked roof as a hunting perch. The golden eagles are much larger—so immense that they prey successfully on pronghorn fawns—while the white pelicans that glide high above us in splintered Vs on their way from one remote desert lake to another are larger still, each one a cross of alabaster drifting through the azure sky on a ten-foot wingspan that is second in length only to that of the California condor.
There are plenty of smaller brute neighbors, too. Our home ground is also residence to California and antelope ground squirrels, kangaroo rats, pocket gophers, broad-footed moles, grasshopper mice, and that most amazing of desert rodents, the bushy-tailed pack rat. Snakes are common, and while most are Great Basin gopher snakes, I once discovered an eleven-button rattler shading himself in our garage. It seems that a scorpion hides beneath every rock, with plenty left over to inhabit the ductwork of our house. One year hordes of shield-backed katydids (a large sagebrush-country insect often called a "Mormon cricket") invaded, blanketing these hills so thickly that their mushed guts rendered the paved roads slick as ice, even prompting the closure of a major highway. Whenever we get a big rain after an especially hot summer day, the western harvester ants whose colonies dot this desert sprout wings and rendezvous to mate at the highest point in the local landscape, which is the top of our chimney, on top of our house, on top of our home hill. From there they descend the chimney in untold thousands and writhe in a foot-deep mass behind the glass doors of the wood stove's firebox. Our shared life on this isolated patch of high desert has provided an unbroken string of such challenges.
If this place doesn't sound like paradise, then perhaps I've taken the wrong tack in describing it. Maybe I should say that this is an open, wild country of extreme beauty, that its undulating, muscular hills hide life-giving springs and seeps, that these canyons and arroyos wind sinuously between knobs crowned with palisades of granite slung gracefully between ridges of sand and sage. Perhaps I should mention that the western horizon is etched by the serrated ridgeline of our spectacular 8,000-foot home mountain, which conceals a soaring split summit brimming with tangled stands of mountain mahogany and rolling meadows flowing in lavender waves of wild iris. Maybe I should explain that the elevation and aridity here create a sky bluer than any imagining of it could ever be. This landscape has a clarity and presence unmatched by any other I've experienced, a quality of light that keeps the hard splendor of the world before us in high relief. This is the place where the moon draws your long shadow across snow and granite, the place where the forking path of the Milky Way seems always within reach.
I could offer a great deal more in this lyrical vein, but I don't want to do that. I'd prefer to return to a celebration of our home landscape as vast, alien, and fierce. This place is not remarkable in spite of its blizzards and droughts, its fires and floods, its rattlers and scorpions. It is astonishing because of them. This is a landscape so full of energy, surprise, and struggle that it constantly challenges our ideas about nature and about ourselves. To dwell in this vast desert requires that we relinquish any pretense of control over the circumstances of our wild existence. Living here offers the salutary reminder that control is nothing more than a human idea, an abstract concept that this marvelous landscape is under no obligation to recognize. It is the fantasy of control that is itself most vulnerable, because in this land a humbling corrective is only one fire, blizzard, or rattler strike away.
The passionate desire to inhabit this extreme landscape is not easily explained, even by those few of us who have been so rash as to act upon it. Yet harder to explain is why anybody would choose to raise their children out in this open wilderness. This is a question I am still trying to answer, one that I find intriguing and useful even after a decade of meditating on it. I could, of course, point to the exquisite beauty of the high desert landscape, to the way a single glint of alpenglow on a distant snowcapped desert peak sweeps away all doubt. Or I might claim that there is a certain resourcefulness or strength of character that develops in those who struggle to make a life in this unforgiving place. I could say that our geographical isolation has strengthened us as a family, providing opportunities to bond deeply through shared experiences that other modes of living do not provide. I could suggest that for kids, outdoor adventure and play are essential forms of engaging with the world, and I might offer a corollary lament that for many of us grown-ups the thrill of adventure and the joy of free play have by now receded into the ghostly world of memory. Speaking as a man who is still learning to be a father, I could share my deep belief that this wild desert is teaching my daughters things that I cannot.
These sorts of claims point to a deeper issue concerning the relationship between domesticity and wildness. Wilderness, and the wildness it both embodies and expresses, has received eloquent praise within American literary culture. In many ways the flight to nature in order to transcend the perceived limitations of the domestic world describes the arc of the quintessential American journey. Consider Lewis and Clark hunting their way across the broad continent; Herman Melville's Ishmael bolting to a life on the high seas; Henry Thoreau retreating to his writer's shack at Walden Pond; John Wesley Powell shooting the rapids of the uncharted Colorado; John Muir riding out a windstorm in the tossing crown of a towering Doug fir up in the nearby Sierra; Mark Twain's Huck Finn rolling on the big river and then "lighting out for the territory"; the heroes of Jack London's tales answering the call of the wild western wilderness; my fellow desert rat "Cactus Ed" Abbey soloing across the glowing red-rock mesas of the Colorado Plateau. Even those who never returned tell a similar story: Everett Ruess vanishing into a labyrinth of canyons in southern Utah's Escalante or Chris McCandless perishing in the sublimity of the Alaskan wilderness.
While plenty of counterexamples might be cited, the dominant narrative of engagement with wildness in American culture has been one that features men, often operating in solitude, removing themselves from the sphere of home and children in order to enter a distant green world where opportunities for heroism and adventure abound. To judge from the mainstream of American environmental literature, a reader might be forgiven for assuming that the concepts of family and wildness are in fact mutually exclusive. But when we assume that the wild does not exist within the family—or that the family cannot exist within the wild—we radically limit our conception of what wildness means and so also limit what it can teach us.
In perpetuating an understanding of wildness that depends upon the far-flung exploits of men heroically working alone (or with other men), we overlook the wildness that is inherent both to parenting and to children. The wildness of conception and birth only initiate young lives in which a spontaneous wildness is strikingly apparent. I am keen to avoid romanticizing the relationship of children to nature, because to do so often leads to a self-indulgent, Wordsworthian sentimentality that is rife with problems. That said, anybody who has attended carefully to how a young kid's universe operates can't help but notice the many ways in which children appear to be more like wild animals than human beings. Watch a kid climb a tree, build a shelter, dig a tunnel, imagine and then inhabit a magical forest or underwater hideout or secret desert cave, and you'll witness a visceral form of animal engagement. Children's stories are so often framed as animal parables, not only because we grown-ups aim to use all those rabbits and coyotes to impart moral lessons but also because children are already in such close communication with the animal world. Of course we grown-ups are animals too, but we've had the time and training necessary to forget that, while our children have not. In this sense our kids are the keepers of a wild flame that may be nearly extinguished in ourselves; they are emissaries between the adult world and the wild world from which we emerged and upon which we have so often turned our backs.
This book's title, _Raising Wild,_ is intended to suggest a very different approach to how we conceive the relationship between wildness and domesticity. We tend to think that something that is "raised" cannot also be "wild" and that something that is "wild" must not have been "raised." (Think salmon here.) But rather than figure the wild as other than and apart from the family, this book explores the ways in which living as a family in a wild landscape reveals the wildness at the heart of both childhood and parenthood. Raising daughters in this amazing place has tested many of my own assumptions, leading me to question our culture's association of wildness with adulthood, masculinity, and solitude. Some of what I've gleaned from my experience appears in the following dispatches from our remote home in this high desert wilderness.
The complete text of W. S. Merwin's elegant little poem "Witness" reads as follows:
_I want to tell what the forests_
_were like_
_I will have to speak_
_in a forgotten language_
I am moved by Merwin's suggestion that the language we use to express our understanding of nature is itself the fruit of nature—and also by his warning that an ancient connection between words and world is weakened when the world upon which language depends is imperiled.
Among the innumerable joys and challenges of parenthood is the recognition that kids often speak in this forgotten language. We see in them some glimmer of how the natural world once appeared to us: immediate, new, strange, funny, waiting to be touched and played with. While it is we who teach our children the names of things, it is they who engage the things themselves, often spontaneously employing modes of perception, imagination, and intimacy that are no longer immediately available to us. If it is a truism that children can teach as well as be taught, it is equally certain that in order to fully communicate with our kids we'll need to remember some of what we've forgotten about how we once saw this world. While some might argue that a language, once forgotten, is destined for extinction, my experience suggests that substantial relearning is not only possible but necessary to parenting, especially in this wild place.
I do not subscribe to the notion that wildness is reserved for adult male adventurers who are defined by their escape from the constraints of home and family. Instead, I have come to view childhood as a valuable repository of wildness from which we grown-ups might derive insight and draw inspiration. As the father of daughters, I am also convinced that our association of wilderness and wildness with masculinity is not only archaic but profoundly misinformed from the start. And I am troubled that the so-called retreat narrative—a stylized way of telling the wild that valorizes solitude as the correct mode of encounter with the natural world—authorizes a view of wildness that necessarily excludes children. It is certainly not that I fail to appreciate the value of solitary wilderness experiences. Since the day of that first epic hike over Moonrise, Palisades, and Prospect—those three ridges still unnamed on every map—and up to the forked summit of our high home mountain, I have walked more than 13,000 solitary miles in the desert hereabouts (during which I encountered a grand total of two recreational hikers). Having logged so many miles in all seasons and all weathers, I've witnessed miracles and wonders too numerous to tell. Nevertheless, I have found the experiences I share in nature with my daughters to be instructive and liberating in ways that my solo hikes are not.
I too have retreated to the wild, but I have retreated _with_ my family, rather than _from_ them. In so doing I have discovered wildness in my children and in myself, as well as in the remote hills and canyons of the high desert. This experience has been more fascinating and valuable than any heroic male wilderness adventure could possibly be. Here, in the wildness of home, Moonrise is our Everest and our Denali.
One of the most surprising and useful things I've learned is that inhabiting the desert and raising children have many things in common. Both enterprises begin with genuine passion, but it is a passion born of an uninformed idealism that blinds us to the challenges and blessings of the actual condition we're about to enter. I look back on my pilgrimage to this desert place in much the same way I view the fact that I once read how-to books to prepare myself for fatherhood. Even as a confirmed bibliophile, I now freely acknowledge as laughable the idea that anyone could become prepared for fatherhood by any means other than being a father. So too with this hard, bright landscape: language can only obliquely approximate the experience of it. One must be immersed in desertness to have an inkling of what ennobling challenges full engagement with it might offer.
Eking out a living in this unforgiving landscape requires a high tolerance for frustration and failure, for this is a place that, if it is not exactly hostile, is impressively indifferent to human ambition. Parenting can be similarly exasperating, because, like the desert, it constantly forces upon us an awareness of our limitations. In this sense, both raising children and dwelling in arid lands help us to acquire a necessary humility, an acknowledgment of our weaknesses, without which either enterprise would be perilous. Both experiences strip us of superfluity, deprive us of naive assumptions about ourselves and the world, render absurd any coveted delusion that we are superior to our kids or to the nonhuman natural world. Self-congratulation is an affectation that neither parents nor desert dwellers can long sustain.
Making a home in the high desert and making a home for children also have in common that they entail a constant process of self-examination and growth—one facilitated by the enforced humility I've described. _Raising Wild_ shares the story of how my girls' intuitive understanding of self and nature has provided a profound challenge to my own. In their disarmingly honest approach to me and to their home landscape, Hannah and Caroline have innocently exposed many of my unquestioned assumptions about myself as a man, a father, an environmentalist, and a lover of wild places. My shared experiences with the girls have often forced me to admit the absurdity or hypocrisy of my own values or actions, ultimately helping me to revisit core questions about why we engage both children and nature as we do. Even my most cherished self-image—that of an independent man whose journey to this remote place was motivated by a passion for galvanic, unmediated contact with wildness—has been sorely tested, as I have confronted both the daunting realities of desert living and also my own weaknesses as a man who is still learning how to be a good father.
"As soon as you have made a thought, laugh at it." So proclaimed Lao-tzu, who understood that humor is an inevitable and redeeming by-product of humility. Laughter is the sound we make in the moment we acknowledge, perhaps even begin to accept, our own mistakes or inadequacies. A laugh is a marker of recognition, a sign that we have momentarily seen ourselves in some new light. As the humor theorist Gina Barreca puts it, "Humor, like shame or wisdom, is a product of understanding." My efforts to make a home in the high desert and to become a good dad have been so riddled with missteps, absurd inconsistencies, and failed grand plans that I couldn't survive a day without humor. For this reason, _Raising Wild_ often approaches the braided challenge of parenting and environmental engagement in comical ways that differ considerably from the solemn attitude often adopted by environmental writers. Unfortunately, a great deal of writing about nature (and, for that matter, about children) is humorless and predictable, relying on threadbare tropes that give readers the false impression that the writer's relationship to nature is a fait accompli—a static achievement that can be used to manufacture calculated epiphanies celebrating the sacredness of nature. But those of us who care deeply about the natural world have good reasons to laugh, because our relationship to nature is often inherently comical, and also because humor provides a sustainable form of resilience that we desperately need in this trying time of environmental crisis. James Thurber was right in identifying humor as "one of our greatest earliest natural resources, which must be preserved at all cost." Humor is also an essential element of parenting, because laughter generates the flexibility and acceptance that are necessary for one to develop patience and express love. I have found that, even in its most earnest moments, parenting is very much a laughing matter.
Much like living in the desert, becoming a mindful father requires an ongoing and often chastening process of attempt, failure, insight, and growth—and then, ever and always, _attempt_. Practice does not make perfect but rather makes more practice, and it is this practice of reflective parenting and attentive dwelling in place that has led me to discover enriching forms of intimacy within my family and within the wild landscape we inhabit. There is no place I love more than this high desert, no people I love more than my daughters. And yet there is a wonderful sense in which I do not know them yet. Even in the open desert much remains hidden. Hannah and Caroline too are deserts—each an exquisite mystery I have not yet solved. I need one more day, or year, or another decade; just one more walk, another mile, then 13,000 more. I am still learning to pay attention.
Parenting, like walking in the desert, is a meandering art of improvisation. The adventure never turns out quite as we planned it, the inner and outer weather remain unpredictable. We are in sympathy but never in control. We take conditions as we find them, putting one foot in front of the other as we navigate a topography of uncertainty, trying to make the most of each day's pilgrimage back to the heart and to the land. We carry water toward our lovely garden, but end up using it to put out a fire. We set out for the spring but follow pronghorn tracks instead, intend to witness the rise of the crescent moon but are drawn to the huddled glow of the Pleiades. We search for ways to reshape our experiences into narrative, to transform a shared life into a story that can be given to our children, mapped onto the high, dry landscape of their only home.
In this wide open desert there are no trails, which is to say that any route you choose becomes a trail only as it is walked, becomes a story only as it is told. Before our children can fully engage the world they must first learn to walk. As a father I am still exploring the hard beauty of this wild landscape, learning slowly, with the help of my family, how to walk with and within it.
PART ONE
Birthing
_To know the spirit of a place is to realize that you are a part of a part and that the whole is made of parts, each of which is whole. You start with the part you are whole in._
—GARY SNYDER, _The Practice of the Wild_
_Chapter 1_
1. Endlessly Rocking
It's true that on the day Eryn and I decided to have a kid we had been drinking quite a lot of gin. Gin, the product of fermented juniper berries; juniper, the wild trees that surround our home in the high-elevation, western Great Basin Desert— _Juniperus osteosperma,_ the seminal one. It is best to achieve a state of extreme lucidity before making a sober determination about something as weighty as the eternal fate of one's sperm or eggs.
After many years of wandering in the glaring sun and desiccating wind of the Great Basin, I had come, as we all eventually must, back to the sea, to the cradle Walt Whitman rightly described as "endlessly rocking." The sea here was the late-winter Pacific, gray-green and breaking gently along the rocky shores of Monterey Bay, on California's central coast. In addition to the harbor seals, sea otters, and sea lions that hang around the wharves and rocky islands, you see here a variety of shorebirds and occasionally notice the rolling of dolphins or the spouting of whales—mostly gray whales this time of year, though the big humpbacks and hundred-foot-long blues will return come summer. What remains invisible is even more remarkable, for not far offshore is a submarine canyon of incredible proportions. The top of the walls of Monterey Canyon are a mile beneath the ocean's surface, and from there the canyon descends another mile—the approximate depth of the Grand Canyon—to the frigid darkness of the seafloor far below. This remarkable canyon was cut by a giant river, and though the river hasn't run for eight million years, its massive canyon remains, a precipitous chasm snaking from the bay out to the broad Pacific beyond. What swims in the nearly 12,000-foot-deep ocean in and above this grand submarine canyon? Better to ask what doesn't swim there, so wild and vast is that invisible labyrinthine world beneath the waves.
My main objective in leaving my home desert to visit this place was to sit on the chilly beach and stare at the horizon. Maybe study the tip of a surf rod stuck in a sand spike by the cooler. Maybe unwrap a C harp from a green bandana and bend a few blues lines around the booming G-ish bass of surf on sand. Maybe decide, once and for all, who would take the National League pennant in the upcoming season. Maybe resolve to have a child. It was a modest agenda, but I have always believed that with enough gin and time all problems are solvable. Or at least soluble: capable of being diluted with equal parts distilled juniper berries and seawater.
Hiking on an exposed expanse of bare beach in February, squinting into the wind, pelted by flying sand, buried beneath the sound of roaring waves—these things are surprisingly comforting to a desert dweller. If you can excuse there being water present, the rest is keenly familiar: leaning into the gust-driven gyre that lifts surging blasts of sand, you tilt toward a deep gray horizon of dusty green swells that rise like shiny billows of mountain mahogany and creosote bush and bitterbrush—breakers undulating like shimmering waves of _Artemisia tridentata,_ big sage, each desiccated three-lobed leaf reminiscent of Neptune's trident. Even the distant battleship clouds rise in broken, serrated ridgelines like desert mountains, low ranges lipping an overflowing world-round cup that contains both gray whales and pronghorn antelope.
It is best to visit visited places when they are unvisited, both to avoid the throng of folks who shatter the solitude necessary for problem solving—and questions of pennant races and procreation promise to be close calls this year—and also because we sometimes enjoy people's presence most when we register their absence. The best kind of solitude is created when people not only aren't around but _might_ have been around and aren't. Even in praising the beauty of a "deserted" beach we reveal the awareness that it was once inhabited—betray the recognition that its charm is created not by its beauty alone but also by the people who once were there but have now moved on, blown away, _deserted_.
Melville observed that all paths lead to water—that an irresistible force constantly and silently pulls us benighted terrestrials back to our watery home. Even in the desert it is true that all paths terminate at either a glistening spring or a pile of powdered bones. Like everything else in life, it's simply a matter of choosing the correct fork in the canyon's sandy wash-bottom game trail. But there is something compelling about this limitless mass of life-filled water, roiling around the globe, pulled by moon and pushed by wind. It is a truism that we carry the ocean in our veins and tears, but that seems a thinly clinical way to measure the affiliation. My body is a gin-powered carbon-based flesh satchel that is essentially saltwater—so far so good. But think of the wildness of the sea, with its innumerable underwater canyons and mountain peaks, its turreted and gabled reefs, its fissures and crypts, vents and vaults. Think of a myriad of minute life-forms spiraling around towering spires of swaying kelp, of the high-pressure, frigid, eternal darkness above which bright, fish-filled rivers of animated current run. Think of the battle between whale and giant squid that is raging in the depths at this moment somewhere, the giant cephalopod frenetically twining its eighty-foot tentacles around the snapping jaws of a hundred-foot cetacean that is glaring, coldly, out of its tiny eye.
But think, too, of ourselves. Of how we crawled, frame by time-lapse frame, out of the pond, rose to our feet, grabbed an ash or maple stick just as our flippers became hands with digits, and smacked a soaring dinger into the left-field bleachers—or invented the quadrant, or wrote _Hamlet,_ or created the smartphone, or whatever you think of as the pinnacle of hominid evolution. And just when a giant squid seems the ultimate nasty neighbor, try living on land for a while, always worried about finding shade and fresh water and paying rent and taxes, flinching constantly at all the looming things that can spear you through the back of the neck while you're only trying to grub a few roots. Maybe the whale and its air-breathing marine cousins got it right when they crawled back into the drink: any sensible terrestrial mammal will tell you that leaving the pond wasn't exactly a cakewalk.
I slice another lime with my bait knife, thinking to myself that what would be wildest—and what would connect us with the wildness of our watery home even more powerfully than knowing that we cry salt tears—would be to crawl back in. Not in an underwater robot, like Jacques Cousteau, or with an oxygen tank on our back, like Sean Connery's Bond, James Bond (impossibly cool even in those British secret agent diaper-white swim trunks), but silently and unassisted, simply breathing water as we once did, returning quietly to the calm of our coral caves, leaving the windy beach without regret. _Deserting_.
Since this doesn't seem possible—although the seals and dolphins have managed it rather gracefully—I've been contemplating the terrestrial mammal's best alternative: my wife's suggestion that perhaps we should give birth to a tiny human. This proposition seems at once perfectly natural and extremely reckless. For us nonmarine mammals, being a fluid-breathing fetus floating in the amniotic ocean of our mother's uterus is as close as we'll ever get to turning our backs on this troubled land and sliding back into the silent sea. Still, I can't help but think of the more mundane implications: How long does it take before a thing like that can run a Weedwacker, cut stove wood, or slice limes, even? My father always said that the perfect age for a kid is when they're old enough to run the lawn mower but not old enough to drive the car. Fair enough, but think of the magnitude of the investment, given the extremely narrow preautomotive mowing phase of child development. And these infants—what, exactly, do they do all day? And doesn't much of what they do smell? I've heard poet Galway Kinnell's scatophilic assertion that those who don't poop don't live, while those who do do doo doo do. But still.
As I look out over Uncle Walt's endlessly rocking cradle and consider this question further, I no longer picture the epic battle of cetacean and cephalopod or the spiraling, undulating towers of kelp, but instead imagine a pudgy little human baby, rosy cheeked, bulging eyed, wide smiled, wearing bunchy diapers attached with those big pins (for some reason) and doing the breaststroke underwater as a curtain of bubbles periodically covers its fat face like a belch. An amphibious cherub, more monstrous than cute and not at all as advertised. As the thing swims slowly toward me with its sweet, trusting grin, I think how unlikely it is to survive very long down there, with all the hungry fish folk, so red in tooth and fin—and it so corpulent and awkward and slow-moving, and probably not too chewy. Do I really want to take responsibility for this defenseless monster, neither fish nor ape, that can't hide in a coral nook, or out-swim a shark, or even cut a lime? I'll mow my own damned lawn.
As the tiny beast paddles yet closer, a huge mushroom cloud of brown bubbles suddenly bursts from beneath its diaper, blowing the cloth to shreds in underwater slo-mo. I suspect that sharks can smell this. I wince in disgust. Lifting the fruit jar from the sand, I take a healthy belt of sandy gin and tonic, then slowly raise my eyes and look out across the sea again. Somewhere beneath its rocking green cradle is a hypothetical baby—an amphibious infant that, like me, has saltwater in its veins and tears. I look beneath the surf again: against a trailing curtain of brown butt chum the child is still swimming at me placidly, still approaching land, ready to crawl out and stand up and swing a bat. And it is still smiling.
The woman who calls me her husband is from California. But Eryn isn't blond, and she doesn't surf. (As it turns out, California is loaded with brunettes, several of whom don't even know how to surf. Who knew?) She's one of the Crackers of the West, that sturdy Okie stock whose kin came across the Great Basin and Sierra like Ma and Pa Joad, piloting a ramshackle jalopy and looking for the endless orchards of what everybody from Moses to Chuck Berry called the Promised Land. Eryn is the kind of woman who makes you want to do a rash thing like get married, even if you've had a good, long run of knowing better than to enter what I once disparagingly referred to as "the condition."
Make no mistake, marriage is one of the few institutions I respect. Ralph Waldo Emerson was right that most institutions are dead forms: ossified, impersonal, ineffective, inertial, disingenuous, self-promoting, tautological, hermetic, superficial, and fucked up (Emerson didn't actually say "fucked up," but that's what he was thinking). In front of a bus stop at a remote rural crossroads in central Nevada I once saw an old drunk preaching, most righteously, into the vastness of the glaring desert: "Beware the _institution,_ for there's two things, _two things_ that it never can do, _never can do,_ and that is anything, _anything,_ for the first or the last time . . . _first or last,_ brother, first or last, _beware_!" There are prophets everywhere. But I don't think the crazy wise man intended his divinely inspired admonition to apply to the institution of marriage, which is, as none less than lascivious old Ben Franklin recognized, a fine condition into which even freedom-loving men should rightly enter. But it somehow never seemed a good idea for _me_ to enter it. Marriage wasn't like baseball—a game meant to be both played and watched—but rather like horse racing, something you watched, wagered on, and drank at, but didn't actually participate in.
Even with the weight of the evidence regarding "the condition" on the other side of the question, I married Eryn—or, more accurately, she was generous enough to marry me. Eryn is smart, patient, and generous. She's also witty, stubborn, and optimistic. And though she is Californian, her peach-picking lineage in the Central Valley is substantially redeeming. She's a good friend, and she's resourceful and interesting, which is saying something. It would have been good if I had thought of some of this stuff to put in my crappy, bootlegged, eleventh-hour wedding vows—if you're hung over and you end-run the Bible you're left with crap for vows, as it turns out. I love being married to Eryn.
But just as I'm feeling at peace with my new and improved life, the specter of the swimming diaper-blasting insanely grinning non-lime-slicing not-yet-lawn-mowing amphibious proto-dinger-smacking belching cherub has come upon me from right field—a place things should go to rather than come from. Somehow this strikes me as unfair. I've just taken a deep breath and said, yes, I'm quite pleased with this whole marriage condition, when this fat-faced hypothetical baby comes along, with all its expulsing bodily fluids, to sour my gin and trouble the wide oceans and attract poop-sniffing sharks. It's like sliding safely across home plate and then being tagged out—and by the umpire. But safety, so hard to come by in this world, is especially elusive when freakish babies are paddling around the juniper juice in your noggin.
It happened this way. Eryn and I had just come back from a nearly ideal lovers' evening walk along the chilly, deserted strand of beach with my dog, a thick-headed mystery mutt Eryn generously characterized as "good-natured." I should explain that I had vexed my family by foolishly naming the dog Cat, which I thought sounded cool (as in "cool cat") but which I bestowed primarily, and smugly, to illustrate the power of behavior modification and operant conditioning. "This dog doesn't think about the fact that it's a dog," I philosophized over an IPA one summer afternoon after returning from the SPCA with the new addition to the family. "I could call him Cat, and he'd still come when I called him, so long as he was trained, like Pavlov's dog, through use of a clearly structured series of rewards and punishments." By the time the words tumbled out of my mouth I was already in trouble. First of all, it should have occurred to me that this sort of conditioning had failed when my parents tried it on me. In characteristic form, that morning at the campsite Cat had licked the coagulated bacon grease out of the bottom of the frying pan while I was peeing on the other side of the dune, and when we later walked him down to the ocean he immediately rushed into the surf and attacked the first wave he could reach as it broke on shore, shotgunning a bucketful of ocean as a chaser for his slimy breakfast, after which he dragged along all morning, hacking up sand and saltwater as he went.
So we had just come back from a wonderful evening walk along the beach, and we were sitting comfortably in the tent as a light sea breeze rippled the sloping nylon walls and the glow of the rising moon poured through the mesh windows. We were half-tucked into our sleeping bags when evening damp began to fall, and we were on the sandy shoulder of the infinite sea, and we were laughing, and we were simultaneously playing and drinking gin. Cat, who was curled up in the corner of the tent, snoring happily and occasionally farting, had even stopped yacking. I was fully inhabiting the role of the proverbial happily married man. The situation was as close to ideal as it is likely to get on this side of the vale of tears.
"Do you ever think about having a baby?" Eryn asked, absolutely unprovoked. I could hear her voice winging in from right field as I stood, incredulous, once again tagged out after safely crossing the plate. At that exact moment I was slicing a lime, and I damn near cut my finger off, though it did flash through my mind that if nine and a half fingers was good enough for the Grateful Dead guitarist Jerry Garcia, it ought to be good enough for me.
"Huh?" I replied. Before she could rephrase the question I rebounded, wittily: "I'd like to, but I don't think it's anatomically possible. Perhaps you've mistaken me for a sea horse?"
"Michael, be serious," she said. My long first name plus a command, encapsulated incisively in a three-word sentence. This was clearly inauspicious. Happily married man meets buzz-crushing topic of adult conversation.
"Hey, feel free to call me Mike. Besides, what do you want with one of those?" I asked, desperately invoking levity where it had so little chance of success. "I hear they're expensive and noisy and they smell bad. Really, things are so perfect right now."
"But maybe they would be _more_ perfect if we were a real family," Eryn said with disturbing sincerity.
I objected, grasping at semantic straws. "You can't have ' _more_ perfect'—'perfect' is as good as it gets. Besides, we _are_ a real family. What are you talking about? Look at us: happy family!" At this moment I spontaneously spread my arms wide apart, gesticulating grandly to suggest the impressive expansiveness of said happy family, when the gin-soaked gyroscope in my inner ear caused me to lose my balance and, as I fell over, snag my hand on the taut laundry cord above me, spilling my icy drink in my crotch while catapulting a pair of boxer shorts, formerly on the line, onto the extended snout of the sleeping Cat, who snuffled loudly.
I looked up at my sweet wife, who looked back in silence at my undoubtedly plaintive expression, and my soaked crotch, and my strewn underwear, and my flatulent dog, and then dropped her pretty forehead into her open palm to hide a smile. She had to be fantasizing about what her life was like before she married me.
"Let me think on it some," I said.
"OK," she replied, looking up with a labored straight face. "Now cut the lime—the round green one, not the long brown one with the fingernail on it. It's your deal, Bubba." I freshened our drinks; removed the boxers from Cat's snout and placed them, officiously, upside-down on my head, waistband-as-headband style; and began to deal a hand of gin and to sing an old Sleepy John Estes song:
_When that wind, that chilly breeze,_
_Come blowin' through your BVDs,_
_You gotta move, you gotta move child,_
_You gotta move._
Inspired by my flapping lid, I was belting out the blues with what I took to be the accent of a French chef.
I was pretty sure that adults didn't sit in tents, half-drunk, wearing underwear on their heads, losing repeatedly at gin, and singing the blues in Franco-phony. Adults, I had been led to believe, made mature, considered decisions about things like whether or not to have children. Long after Eryn fell asleep I was still singing mournful old Sleepy John:
_There's a change in the ocean, a change in the sea,_
_I declare now, mama, there'll be a change in me,_
_Everybody, they ought to change sometime,_
_Sooner or later, you got to go down in that lonesome ground._
Unlike John, I wasn't sleepy at all. As I lay in my bag listening to the breakers walk up the beach under the pull of the big moon, I thought about change, and love, and water. These are expansive topics, but I tend to sleep about three hours less each night than Eryn, which gives me plenty of extra time for these contemplations—roughly one thousand hours extra each year, the equivalent of around forty days per annum of dangerously abstract and self-absorbed metaphysical musings. In fact, my unique habit of combining strong java, hard liquor, and excessive contemplation makes me an excellent candidate for spontaneous human combustion. I once pointed out to Eryn that thanks to my superhuman frenetic insomnia, I'd get in years more of sentient contemplations than she would before they threw the dirt on the box. Her reply, without hesitation: "What if you're only given a certain number of waking hours in your mortality allowance? You're squandering time thinking about the shape of the universe, while I'm having wonderful dreams about being a kid again." As usual, she had the more gracious and intelligent side of the argument. But I couldn't sleep anyway, so I lay there singing Sleepy John, listening to my snoring dog, and picturing that sleepless ocean rocking on the far side of the dune.
Water. We're made of it, it surrounds us, and we buy the farm if we go more than a few days without taking some of it in or even without squirting some of it out. Mark Twain said that in the West, "whiskey is for drinking and water is for fighting over," and W. C. Fields, pontificating about water while drinking rye, claimed he "never touched the stuff " because, as he put it so memorably, "fish fuck in it." But there's a good reason why a guy with a name like McKinley Morganfield would call himself Muddy Waters. You wouldn't see a guy named Muddy Waters changing his name to McKinley Morganfield, because he'd end up being an accountant rather than wailing the blues. We wouldn't get far—physically or imaginatively—without water. Newborns must revisit the hydrant of their mother's breast ten times a day, and old men must make pilgrimages to the places where they swam and fished and paddled in their youth before they can die well. _Well_ . . . another place from which life-giving water flows.
I fell asleep thinking of old friends who were pulled to water. Not just Herman in the South Pacific and Henry at the pond and Walt crossing Brooklyn Ferry, but also Mister Jefferson admiring the dramatic confluence of the Potomac and Shenandoah, Lewis and Clark portaging around the sublime falls of the Missouri, Twain dodging snags in deep fog on the moonlit Mississippi. One-armed John Wesley Powell lashed in his straight-backed chair to the deck of a wooden boat, shooting the gorges of the unknown Colorado. Obsessed Hemingway cruising the Caribbean in a fishing boat he had outfitted to attack German U-boats. John Muir trying in vain to explain the sacredness of Hetch Hetchy water to folks who believed that water comes from a tap. Cactus Ed Abbey floating in the unspeakable beauty of Glen Canyon before it was buried beneath sunburned water skiers. Ellen Meloy, one of the very finest of our desert writers, running the graceful bows of the Green River year after year. Norman Maclean reaching with the tip of a fly rod to touch the heart of his lost brother on the Big Blackfoot River. I think especially of young Nathaniel Hawthorne, sitting in the upper window of the Old Manse in Concord, looking up from his manuscript and out to the sweeping bend of the Concord River as it glides through swollen spring meadows and under the wooden arch of the historic Old North Bridge. He had just become a father, and he was ecstatic with joy and creative energy.
"Rise and whine, Bubba," said my wife, through the sound of the ocean beyond her. I wriggled from my mummy bag, crawled out of the tent, and stood up, bleary but proud in my American-flag boxer shorts, out of which fell a desiccated slice of lime. "Eeeew," Eryn said, exaggerating for effect as she handed me a cup of cowboy coffee. Cat sniffed the leathery lime, licked it up off the sand, and then spat it out again, shaking his head a little. I pulled on some clothes and scaled the dune to have a look out over the sea. The breeze was up, and the surf was booming. Three gulls circled above me on the spokes of an invisible wheel, and a few light clouds on the horizon said good weather. It was chilly, but not nearly as cold as I had expected it to be.
After breakfast Eryn instructed me to go fishing, which is another of her many fine qualities. "Take your gear and your limes and whatever nastiness you use for bait and go stare at the sea. That's why you came, right? I'm going to read and discuss current events with Cat, the boy genius. Come back whenever." So off I went, shuffling along the shore with my sand spike and rod in one hand, my little cooler in the other, and my libational day pack on my back. Because I was raised as a fisherman and also as a slave to a puritanical work ethic, I must fish in order to think—or to think about something other than how lazy I'm being by just thinking instead of doing something productive, like fishing. So I was following a comfortable routine, except that now I had the burdensome assignment of contemplating progeny, which seemed intimidating. Setting up by the deserted seashore, I sat in the sand and stared at the world, as planned, and I fished and thought, as usual. Although I had pleasant solitary meditations on the important subjects of water and baseball and evolution, I ultimately began to envision the vulnerable, belching, defecating, grinning underwater ape baby I've already described. Unable to shake this vision, I decided to forego the pleasures of angling—old Izaak Walton would have been scandalized—and instead trudged back to camp by early afternoon. I had decided that procreation is a subject more fit for lively discussion than solipsistic meditation—and so I reckoned that, given her central role in the would-be plan, it might be best if Eryn had an opportunity to weigh in on the subject.
Despite my disturbing vision and unproductive musings, I had struck upon one brilliant idea: if I could imagine a name for the child that we might, perhaps, possibly someday have at some unspecified time in the future, I could humanize, personalize, the thing, thus making it easier to imagine without having weird visions. _Baby, infant, toddler_ : these words cause mild discomfort and sound like they would trigger the need for substantial responsibility. But somehow it didn't sound so bad to imagine hanging around and listening to a ball game with _Joe_ or _Jane_. No big deal. Then Joe or Jane mows the lawn a couple times and goes off to college, right? So I entered camp with what I took to be a superb conversation starter: baby names.
"Honey," I asked, setting my rod and spike aside by the tent, "if we did have a kid, what would you want to name it?" She stared at me blankly.
"That's what you came up with after contemplating the great sea?"
"No, really. You know how when you give a name to something that's nameless, you know, anonymous, like a disease that you have, or somebody faceless like a criminal, it really humanizes the whole thing?" I urged.
"A disease or a criminal?"
"OK, bad examples. But just for fun, come on, let's talk about what names you like. Let's sit in the sand and work on this a little. I'll mix the G and Ts, you start tossing out some names. Let's say it's a girl. Whatcha got?" She paused, looking at me suspiciously, but she couldn't resist, which was when I first realized she had already been thinking about the dangerous subject of kid names. So we sat facing each other, a light breeze easing down the dune and the rocking ocean stretching out beyond us to the western horizon.
"Well, I had a wonderful great-aunt on my mother's side who I really loved, Aunt Mabel," she said, with inexplicable seriousness.
"Mabel," I choked. "Thou shittest me, yes?"
"OK, how about Phyllis—it means 'leafy bow' in Greek. I think that's pretty."
"Don't you think somebody would call her _Sy_ -phyllis? We don't need that."
"Well, what about something fun, like Jasmine?" she suggested in frustration.
"Perfect! That is, if you want your daughter to major in pole dancing. Hey, we could just name her Jasmine Syphyllis. Of course, we'd never get affordable health insurance on somebody with a name like that." Eryn was enjoying this exchange, though she also enjoyed pretending that she wasn't.
"Bubba, you're appalling. OK, what girl names do you like?" she asked.
This was a predictable turn in the discussion, but, as usual, I was unprepared for it anyway. I tried to buy some time: "Did I ever tell you about the time I went to Sleepy Hollow Cemetery in Concord to visit the graves of Emerson and Thoreau? No? Well, I made a pilgrimage to the holy burying place, called Author's Ridge, a beautiful, breezy knoll, covered with big white pines and, well, dead writers. It was a beautiful day, midsummer, and I had just come from skinny-dipping in Walden Pond. Emerson's headstone is this big-ass rock—just a giant, unhewn granite boulder. Thoreau, on the other hand, is napping under this dinky little stone that just says Henry. Tasteful, you know. Modest, restrained, not too showy. Still, you'd think they could have taken up a collection or something. Even Hawthorne and Alcott had better stones than Henry."
"So you want to name our daughter Henrietta?"
" _As I was saying,_ once I checked out the big boys up on the ridge, I decided to wander around the graveyard looking for the oddest name I could find. Believe me, there were some pretty funkified old-fashioned names there. But when I found what I was looking for, I knew I couldn't possibly do better, and to this day I've never forgotten her name." I paused. "Are you ready for this?" Again, dramatic pause. "Fucius Barzilla Holdenbum."
She rolled her eyes. "First, I don't believe you. Second, that's got to be a boy's name. And, third, it wouldn't be pronounced 'Fewshus'—it would be 'Fewkaius.'"
"First, I do solemnly swear on the grave of Fucius Barzilla Holdenbum," I said, moving my drink to my left hand so I could raise my right in a solemn pledge, "that I have spoken the gospel truth. Second, I don't know how you could think Barzilla is a guy's name. And, third, only an Okie with strong Old Testament leanings could get 'Fewkaius' out of 'Fewshus.'"
"Well, which of these delightful names are you proposing for your poor daughter? Not Holdenbum, certainly?"
"Witty," I said with fake condescension. "I had to marry the witty one." She smiled, a little proudly. "How about Melissa," I offered spontaneously. I had been humming Allman Brothers tunes while fishing, and it was all I could think of besides Fucius Barzilla Holdenbum, which had clearly played out.
"No good. People will call her Mel, and then everybody will think she's a boy—and a truck driver or a short-order cook."
I hesitated, thinking about the fact that two of my best friends were engaged in these noble occupations. But I had already thought of a good follow-up name. Then Eryn continued: "Besides, Melissa is like Jessica—it just sounds kind of trashy." She had an uncanny way of anticipating and blocking my next move, though I suppose I shouldn't have stuck inflexibly with Allman Brothers song titles. But I loved those smooth, sonorous, sibilant southern names.
"Well, how about Althea?" I blurted.
"Althea? That's what you named your guitar!"
"Well, yeah, but it's a great name. From the Greek. Means 'healing herb.' What's not to like?" She stared me down with that great fake-serious look of hers.
"I could rename my ax. No, never mind." I really didn't want to rename that old Martin D-18. It would be like renaming my dog—which I obviously would have done long ago if I could have. I'd just have to figure out the girl's name without borrowing from any of my instruments, pets, or nicknames for body parts.
The longer this went on, the more I realized that if I were to become the father of a daughter I would be compelled, about seventeen years from now, to kick the ass of somebody who would probably look and act a lot like me. I quickly retrieved and refiled for future use the first line my father-in-law had used in welcoming me into his home: "Son, let me show you my gun collection." But beneath the laughter of the name game I felt a real fear, some impossible-to-describe sense that I wasn't ready, that I somehow just wouldn't know what to do—that I'd be a bumbling father to a baby girl, a flawed, obsessive father to a girl kid, an alien species to a teenaged woman. Now I could feel the limey G and T, which I had ingested with a fair amount of sand, begin to roil in my gut. Maybe the escape hatch was to imagine being a father to a son instead. I knew this wasn't something you could count on, but I figured the odds weren't any worse than the ones people wager on at the roulette table—though I also realized that the simple choice of red or black didn't involve changing diapers or saving for college and would, at least, come with free drinks. Still, it seemed worth a try.
"This ain't workin,'" I said. "How about boys' names? Whatcha got?"
"Well, what about Jeremiah? You've always liked strong names," she said.
"Strong, yes; apocalyptic, no. I'd feel like it was the Last Supper every time I called the kid for dinner: 'Oh, Jeremiah, boy, on the way home from school, could you ask the Lord to have mercy on Daddy's soul? Now put down your flaming cross and go mow the lawn, then get washed up for your loaves and fishes.' On the other hand," and now I leaned over toward her, half-turned for dramatic effect, and sang loudly: "' _Jeremiah was a bullfrog!_ ' A lot of people wouldn't want to have a son who is a bullfrog," I said, "but I'm very accepting. To _Rana catesbeiana_. Long may he jump!" We raised our plastic glasses in a silent toast.
"So, Bubba McBluffer, you don't have a decent boy name, do you?" she asked, starting the next round.
"Nope. I've got three: Diogenes Asclepiades Themistocles. Real name. Means 'fat man with healthy testicles.' Believe me, you could do worse—with those fine testes you'd have good prospects for grandchildren."
"I had to marry the witty one," she said, smiling. "Let's eat. Maybe we're too famished to think clearly."
"Impossible! We're drinking the juice of junipers and limes here—vegetables and fruits, very nutritious. Come on, one last shot at naming the poor boy. Let's use the trusty blues naming formula—works especially well for boys."
Eryn knew this was a trap, but she didn't care. We were by the great ocean, our ancestral home, and we were with our dog, such as he was, and we were drinking G and Ts with fresh lime and sitting in the soft sand. And in our own laughing, indirect way, we were discussing the idea of starting a family.
"Whatcha got?" she asked, knowing how much I would enjoy this. It was like hitting a homer off a tee, but it still felt great.
"Here's the formula: disability plus fruit or vegetable plus last name of US president. Works every time—you know, as in haunted Texas bluesman Blind Lemon Jefferson." This was an example Eryn could appreciate, since my own blues nickname, given to me by bandmates who wanted to encourage my blues harp playing while also forcing me to maintain absolute humility, was a riff on this one: Blind Lemon Pledge.
"Give it a try," I said, with sincere encouragement in my voice.
"Deaf . . . Watermelon . . . Washington." She grinned.
"Perfect! Deef Melon, get in this house and eat your macaroni and cheese! Deef Melon, you damned rounder, you be in by ten!" We toasted again, our plastic cups coming together silently. Cat, disturbed by our laughter, opened one eye briefly before he resumed snoozing.
"How about Bald Pineapple Wilson?" she offered. "Bald Pineapple, you get out there and mow the lawn this minute! Bald Pineapple, put a little elbow grease into those dirty dishes!"
"Bald Pineapple," I continued, sternly, "if I catch you hangin' round the crossroads 7-Eleven I'll slice your noggin and stick toothpicks in the pieces! Why, now, Bald Pineapple, I can't believe you swung a D in math. Way to go, son!" I could hardly speak for laughing. I truly think if we had given birth to a child that moment I would have insisted on naming it Bald Pineapple. It somehow seemed perfect. Then again, everything seemed perfect.
"Bowlegged Broccoli Adams!" Eryn said.
"Specify Adams," I insisted, sounding serious.
"John, of course. The other one is Crippled Quince Adams," she replied instantly.
"I remember now, he ended up in a jam."
"Pigeon-toed Asparagus Taft," she continued.
"Is that the one they used to call Stinky Pee?" I asked.
"The very same. And they said he'd never amount to anything. He was in office just after Rheumatoid Brussels Sprout Roosevelt."
"Teddy, then?" I said, again calling for clarification.
"Of course. FDR was much later—you know, Flatulent Dewberry Roosevelt." We both looked at Cat and laughed.
"Of course. He filled the power vacuum created by Hoover," I said. She smiled.
"Right. Psychotic Carrot Hoover. He was quite unstable, but he had great vision in a dark time, may he rest in peace," she said, momentarily grieving his loss until she broke out laughing again. Now neither of us could stop laughing at the fact that the other was laughing so hard at something so ridiculous. We knew that none of this was really funny, but we didn't care, which made us laugh even more. That is the irrational, liberating nature of joy.
"Wife," I heard myself say unexpectedly. "Do you think I'd be a good father?"
"Yes, Bubba, I do." She smiled. I paused, laughed quietly, and then lowered my head and nodded it left to right—but I meant yes, the way you shake your head and raise your eyebrows and laugh before you start skiing down or climbing up the biggest, most beautiful mountain you've ever seen in your life.
I sat silently now, washed over by a feeling of quiet certainty. Eryn's face was glowing as it must have when she was a curly-headed little baby girl, and the woman who calls me her husband had never looked so beautiful before. I could see in her face a child, and I could also see a mother and an old woman. I heard the swaying ocean and felt the evening breeze and witnessed the bone moon lifting slowly out of the dunes. I was immersed in the moment and yet also somehow already looking back at it with deep satisfaction, as if I was seeing this place and time from an old wicker rocker, rocking with my old wife, endlessly rocking, on some crooked porch—ninety years old, maybe toothless and incontinent, but somehow happy anyway, and happier still to have the great gift of one clear memory of the moment I was now living. It was like sitting backward on a bale of hay in the bed of a speeding pickup: the first moment you see what's around you it's already racing away toward the receding horizon. Only in such a moment can we wrinkle up our lives to make the best parts touch—fold the cascading narrative of days to see ourselves being told by a larger story that, however haltingly, is still being written.
_Chapter 2_
2. The Nature within Us
During the first trimester of Eryn's pregnancy strange things began to happen to me. As Eryn started to experience nausea in the mornings, my own appetite, usually reliable as a plow horse, began to falter. My back started to ache, and I was so fitful at night and so bleary in the morning that I despaired of ever feeling rested again. At the time I supplied a litany of possible explanations: fatigue, old sports injuries, hassles at work. But as Eryn's pregnancy progressed, these things became harder to explain. I began to have headaches and to feel bloated, and I developed disconcerting cravings for foods I had always disliked. I felt as if my body were being taken over by an alien force that I didn't understand and couldn't name. As I treated my ailments with midnight doses of dill pickles, hot and sour soup, and tequila, I wondered if I could have contracted Lyme disease or be suffering from chronic fatigue syndrome. In a desperate moment I even enumerated the karmic missteps by which my health could possibly have been compromised in such a systemic way. I complained little to Eryn, though, and there was a good reason for my reticence: I felt awkward bemoaning my nausea, bloating, appetite loss, cravings, insomnia, fatigue, and headaches when she was so uncomfortable because of her nausea, bloating, appetite loss, cravings, insomnia, fatigue, and headaches.
Strange as it now seems, at the time I simply didn't realize how completely my own physical discomfort mirrored Eryn's. But then one morning I experienced something I had never felt before: my teeth began to hurt—not one or two, but every one of them—with a dull, throbbing pain. Along with this perplexing face ache came a more humiliating and inexplicable symptom: I was salivating excessively, which was easily the grossest of the bizarre tricks my insubordinate body had lately pulled on me. I recall staring at my own incredulous, drooling face in the bathroom mirror, the way the Wolf Man does in that helpless moment when the hair begins to sprout from his once-human forehead. Just as my disgust and confusion reached their heights, Eryn called out from the bedroom: "Ugh, my teeth hurt, and my mouth is so watery. The books say it's pretty common for this stage of pregnancy, though." That was the breakthrough moment. I went to Eryn and, drooling like a rabid coyote, expressed my deep sympathy with her discomfort. Then I confessed my own strange symptoms and swore that I'd figure out what was going on. Eryn was pregnant, but what was I?
Couvade syndrome, sometimes called sympathetic pregnancy, refers to the experiencing by men of some of the physical symptoms of pregnancy during their partner's forty-week journey to delivery. The term comes from the French _couver_ , "to hatch." The phenomenon is poorly understood but widespread and has been observed since antiquity in various cultures in Africa, China, Japan, and India, as well as among Native peoples in North and South America and the Basques of France and Spain. The ancient Greek geographer Strabo recorded sympathetic pregnancies, as did the thirteenth-century Venetian traveler Marco Polo, who observed the strange phenomenon around the globe.
Among the first Westerners to offer detailed descriptions of male responses to pregnancy were the twentieth-century anthropologists Margaret Mead, working in the South Pacific, and George Gorer, up in the Himalayas. It turns out that many non-Western native cultures practice rituals that mimic couvade in fascinating ways. In many of these cultures, the husband of a woman in labor will occupy a kind of maternity bed of his own during the time the woman is in childbirth, as if he were the one bearing the child. Such cultural practices, which involve imitation by the man of the experiences of the pregnant or laboring woman, have been explained in a number of ways. Some believe that the practices are purely superstitious and are intended to ward off evil that might otherwise be visited upon the new baby through supernatural means. A related theory is that the man, by occupying his own birthing bed, attracts the evil spirits toward himself and away from the newborn. It appears that all of these practices involve some ritualized assertion of paternity—a symbolic participation by the man in the birth of his son or daughter.
Perhaps the most dramatic example of this odd ritual occurs among descendants of the mighty Aztecs: the native Huichol people, who still inhabit the mountainous wilderness of the Sierra Madre Occidental in western Mexico. During childbirth, the Huichol father goes not to his own bed but rather into the rafters of the family hut, where he positions himself directly above his laboring wife. Once perched there, he ties one end of a rope around his testicles and lowers the other end to the point where it is within reach of his wife. During labor the wife grasps her end of the rope, yanking it each time she experiences a painful contraction. In this amazing way the husband is made to experience his wife's pain during labor, thus sharing the physical discomfort associated with his child's entry into the world. It is a participatory gesture I suspect few would-be fathers would consent to. If you are unlucky enough to have a female partner who insists that you wear one of those ridiculous artificial pregnancy simulators called Empathy Bellies, you might want to pour yourself a glass of whiskey, recall what is required of the Huichol man, and then silently strap it on.
While most of these practices involve the man's ritual _imitation_ of pregnancy and childbirth, couvade syndrome is instead a _physical_ response by the man to the pregnancy and childbirth experiences of his partner. Modern scientific studies have found that couvade is surprisingly common among expectant fathers in the United States, Europe, Thailand, and elsewhere. Somewhere between 10 and 60 percent of expectant fathers experience at least some physical symptoms of pregnancy, which run the gamut from changes in appetite, food cravings, nausea, insomnia, weight gain, indigestion, diarrhea, and constipation to headaches, toothaches, nosebleeds, itchy skin, backaches, mood swings, and, in acute cases, even abdominal cramps during the woman's delivery. Symptoms tend to be more intense during the first and third trimesters, and they are more common among men who have a strong emotional involvement with the pregnancy.
Couvade may be related to false pregnancy among women, which has been documented since the time of Hippocrates in 300 B.C. Called pseudocyesis—the word comes from the Greek _pseudes_ (false) and _kyesis_ (pregnancy)—false pregnancy occurs when a woman manifests objective signs of pregnancy but is not in fact pregnant. Perhaps the most famous victim of pseudocyesis was Mary Tudor, queen of England, but this odd response is surprisingly common, especially among the female relatives and close friends of the pregnant woman. But if the physical effects of false pregnancy in women are strange, couvade syndrome—the experiencing of those same physical symptoms of pregnancy in men—is stranger still.
While couvade syndrome is well documented, it is not easily explained. Some researchers believe couvade is the man's physical reaction to a feeling of marginalization in the processes of pregnancy and parturition. Some argue that the symptoms are instead triggered by the man's sympathy for the woman's discomfort. Others consider couvade an expression of generalized anxiety about the substantial life change parenthood represents. Yet others, drawing upon psychoanalytic principles, propose that the response is a form of "womb envy" triggered by the man's jealousy that he is unable to be the bearer of his own child.
The etiology of couvade is likewise mysterious. Most scientists believe it is a psychosomatic syndrome, but others claim that it cannot be purely psychosomatic since it includes symptoms like nosebleeds, which have mechanical triggers. Others consider couvade a form of Munchausen syndrome, in which individuals feign or imagine symptoms in order to draw attention to themselves. Still others argue that couvade is hormonal, as studies have shown substantial hormonal fluctuations in expectant fathers as well as in expectant mothers. For example, the male partners of pregnant women experience changes in their levels of estradiol (a form of estrogen), prolactin (a female hormone associated with milk production), and cortisol (a hormone related to stress responsiveness). New fathers also have higher levels of estrogen and lower levels of testosterone than do other men. The scientific evidence makes clear that pregnancy has biological and physical—not to mention psychological and emotional—effects on men as well as on women.
As it turned out, the unnatural symptoms I was experiencing were perfectly natural after all. That said, nobody wants to have a syndrome, and being able to name my bizarre symptoms didn't make me feel much better about them—even if confused men have been drooling prodigiously since old Strabo's day. In fact, the choice of explanations for my condition seemed vaguely threatening. I imagined trying to enlighten my male friends: "Look, man, it's not as weird as it sounds," I'd say. "It's just a psychosomatic syndrome with a French name that's triggered by my selfishness, angst, womb envy, and skyrocketing estrogen levels." While the caveman in me felt that my impending fatherhood should make me more of a man, my body was instead being hijacked by an inscrutable second-order pregnancy characterized by a disturbing influx of lady hormones.
My couvade symptoms also highlighted the awkward hitch that only half of us are anatomically equipped to experience pregnancy. For the rest of us, pregnancy is mysterious and also strangely alienating, and if men were entirely candid about this, many of us would confess that the pregnancy of our other half seems by turns terrifying, comical, and bizarre. However much we may want to be a full partner in the experience of pregnancy and birth—a longing revealed by our odd use of the locution " _We_ are going to have a baby"—we simply can't be. It is a fact of nature—though one we resist admitting—that it isn't _our_ body that is miraculously transformed, and so there are limits to how informed our sympathies can ultimately be. Despite a good deal of hype to the contrary, the male role in the proverbial miracle of birth is marginal. We're the sidekick, supporter, witness, cheerleader, coach, observer, assistant, and though we have box seats we'll never have a chance to step up to the plate. This marginalized status of men is affirmed by every book ever written for prospective parents—books in which countless pages of fascinating descriptions of the development of the fetus are only occasionally broken by small, shaded boxes containing "Dad Tips," which helpfully instruct men to do things like save the toxic outgassing fumes of the newly painted nursery for ourselves.
While men aren't sea horses, many of us secretly wish to be. We want desperately to experience this celebrated miracle of gestation and birth, to feel it rather than simply have it reported to us, to be able to tell our children that we held them before they were born as well as after—to assert that we were more than sperm donors, foot masseurs, nursery painters, and breathing coaches. It is precisely this longing that sympathetic pregnancy expresses and addresses. Mysterious as it may be, couvade syndrome helps to narrow the inevitable gap that exists between the sexes when it comes to the experience of pregnancy and childbirth. In the odd misery of my couvade symptoms, I registered the depth of my connection to Eryn and to our unborn child. And my responses to this wonderful impending change in my life were physical, which made all the difference. What I was experiencing was the product of biology, even if the hormonal wheels inside me were turning magically on the axle of emotion.
If no egg was implanted in me, something was. As it grew, I caught my first glimpse of what all parents must eventually come to see: that their connection to their children is not only deeper than they know but deeper than they _can_ know. It is an ancient connection that has been in preparation throughout the long journey of hominid evolution, and not only through the sprint of a flailing spermatozoon. My symptoms provided the first hint that parenthood is a form of love magnified by the deep forces of biology, a double mystery that involves both the wonderful machine of our body and the most beautiful of the ghosts that inhabit it.
My weird symptoms continued apace until the night Eryn's water broke. We rushed to the hospital at 4:00 A.M., where we would begin a marathon that exhausted three shifts of doctors and nurses. Nearly spent after twenty hours of labor, Eryn decided to give pushing one last try before resorting to a C-section—an eventuality I had already prepared for by signing paperwork and snapping on one of those poofy green plastic hair restraints used by old ladies in the shower. When the baby finally crowned, the doctor donned a costume that looked very like police riot gear, complete with a giant Plexiglas face shield, which made me wonder if he was as scared of what was about to happen as I was. Next he attached to our yet unborn daughter's bald scalp a suction device that looked a little too much like a common bathroom plunger, and he pulled as Eryn pushed. Under the pressure of the suction the baby's skull distended so as to resemble the attenuated, aerodynamic teardrop shape of a bicycle racer's helmet, and I confess that in that moment I was more frightened than ecstatic. I'm not squeamish, but a close inspection of the scene from my necessarily limited male point of view had momentarily convinced me that the much vaunted miracle of birth was in fact a terrifying mess. The sheer animal physicality of human birth was unexpected and startling, thrilling but also deeply disconcerting.
When our daughter, who in that moment of nativity and grace had not yet found her name, was finally born, she was whisked away to the neonatal intensive care unit. It would turn out that even after her ordeal the baby was perfectly healthy. As I stood exhausted, with my forehead pressed against the window of the infant ICU, looking down at her weird, bald, beautiful head—which now, unaccountably, had a pink plastic bow glued to it—I wondered where her energy, and Eryn's, had come from during those twenty-two hours.
In some way that I have yet to fully understand, my couvade experience helped prepare me for the challenge and blessing of my daughter's birth. Couvade was a narrow bridge that connected the nature of Eryn's pregnant body and of our baby's tiny, hidden body to the nature of my own body. If I was a bloated, drooling insomniac I was also a father in training, and couvade was a kind of physically enforced education in sensitivity and sympathy. I now know that this capacity for sensitivity is precisely what fathers need most. My symptoms may have been psychosomatic, but they were also an adaptive response to the growth and development of my unborn child. Parenting is physically and emotionally demanding, and while my symptoms disappeared with the birth of our first daughter, other challenges requiring sensitivity, compassion, and empathy emerged with her.
Pregnancy and childbirth are among the most natural events humans experience, and in them we recognize the naturalness of our own bodies—a naturalness that everything from clothes to makeup to plastic surgery is designed to help us forget. While I tend to think of owls' nests and desert springs as "nearby nature," there is no nature so close to us as the nature of our own bodies—and there is nothing that connects us so directly to the nature outside ourselves as the gestation of a child within us. To be pregnant is to hold and grow a world inside yourself, to be joined to elemental natural forces that turn the invisible wheels within wheels throughout nature. We are always in nature, but when we carry a child we are reminded in a profound way that nature is also always in us.
How then are we to distinguish between nature and the self, between physical and emotional experience, between the machine and the ghosts that inhabit it? When our body is gripped by disease we know that our anguish is real, but how should we understand the physical suffering caused by the emotional trauma of grief or the visceral feelings of pleasure sometimes produced by aesthetic experiences? For example, we wouldn't tell an injured war veteran that he doesn't _really_ feel the pain in the limb that has been amputated, because this "ghost" pain is ghostly only to us; to the sufferer, it is excruciatingly real. When you feel a pleasant tingle or chill while listening to a song you love, you're actually experiencing increased mood-enhancing endogenous dopamine transmission. When you laugh at a stand-up comic whose work you enjoy, her humor is also raising your heart rate and pulmonary ventilation, increasing your brain activity and alertness, stimulating the production of endorphins in your ventromedial prefrontal cortex, and even reducing your perception of pain.
Pregnancy's effects on the body—even, as in my own case, its effects on the male body—remind us of the deep reciprocity between our bodies and the body of nature. In her 1903 classic _The Land of Little Rain_ , Mary Austin wrote of the desert that "there are hints to be had here of the way in which a land forces new habits on its dwellers." Consider something as simple as the effects of this landscape on Great Basin flora. The high winds showed these desert flowers how to keep a low profile, while the sun taught their seeds the patience necessary to lay dormant for years, even decades, waiting for a rare wet spring. The drought-sculpted steppe shrubs learned to give one another space, while the gnarled junipers mastered the enviable trick of allowing parts of themselves to die off in order to keep their core alive during stressful times. Balsamroot and mule's ears, those wild relatives of the sunflower, perfected the art of lifting their hairy, palm-shaped leaves and slowly rotating them on edge, tracking the desert sun throughout the day so as to minimize exposure to its desiccating rays. Many desert plants have allowed fire to prove the value of keeping their true hearts underground, the lovely part we humans see being little more than the replaceable efflorescence of a life whose endurance remains secure beneath the sand. We too are like these desert plants: the landscape we inhabit shapes who we are and who we may become. As Austin wrote, "not the law, but the land sets the limit."
Because the intense emotional feelings we often experience with one another or while we are immersed in a remarkable landscape—like this high desert—are accompanied by very real chemical changes in our bodies, it is a tricky business to say where the nature of our body ends and the nature outside it begins. Derived from the Greek _soma_ , which means "body," the _somatic_ in _psychosomatic_ reminds us that even imagined pleasure and pain register in our bodies. The word _compassion_ , after all, means "to suffer together with another," and _sympathy_ has at its etymological root the experience by which our own state of being mirrors that of another with whom we feel a deep connection. And isn't this sympathetic mirroring at the heart of our deepest connections to one another and to the natural world? Perhaps I should not have been surprised to discover that, much like my journeys through this wild landscape, the growth of my unborn daughter was altering my body as well as my mind. In the inner and outer landscapes of nature, something invisible is always being born.
We later named our little daughter Hannah Virginia. Hannah, an ancient name meaning "grace," and a name that makes sense both forward and backward, as she races into the unknown future and then returns in her own growing memories of the past. Virginia, the name of the homeland I carried with me so long ago when I came west, as I hope she too will have a homeland within her always. The body that nurtured and shaped Hannah was Eryn's, but that home body was embodied within the larger, visceral body of this open desert. Who can know whether some unique quality of Hannah's body or spirit will be the product of high-elevation sunlight or intense aridity, of this desert that shapes us suddenly, like a flash flood cutting an arroyo, and also incrementally, like wind sculpting sandstone?
Both before and after her birth, Hannah helped connect me to my own body, to Eryn, to this desert place, and to the future. When Hannah hurts, I hurt with her. When she laughs, so do I. When we watch the big moon rise together, we feel something rising within us also, calibrating our small, vulnerable bodies to the body of this vast landscape. For me, couvade was not a disease but rather a cure: a preparation for the remarkable changes that becoming a father would bring. There is no wall but only a permeable membrane between physical and emotional experience—between nature and self, body and mind, ourselves and those we love. The strange mystery of my couvade experience taught me that nature is not only the machine but also its resident ghosts. Look hard at the strange beauty of a child, the lines and curves of her little body repeated in the ridgelines and hilltops of the cold, bright desert that surrounds us. It's a fool's game to try to name the place where nature isn't.
_Chapter 3_
3. Tracking Stories
Little Hannah Virginia witnessed her first pronghorn antelope the week before her second birthday. It was early April, and the wind swept light snow around the sage and rabbitbrush as I shoved her three-wheeled stroller up a sandy wash a few miles from our house. Although I was going overland through the high desert, I had customized the stroller for off-road travel, complete with knobby tires, slimed to protect against puncture by desert peach thorns. The sky was cloudless, that shimmering azure that distinguishes winter days in the high desert, and we were surrounded by the dry rattling of last season's balsamroot finning stiffly in the wind. As I dug the toes of my boots into the snow to get traction enough to push Hannah uphill through a thick stand of desert peach, she suddenly took the pacifier out of her mouth with her left hand, pointed to the southern horizon with her right, and said, "Moon." At that age she loved to find the moon, and she often spotted the thinnest crescent even in the brightest sky.
"Good job, honey," I said routinely, keeping my eyes on the terrain ahead and pushing hard.
"Daddy, _moon_!" Her tone was so urgent that I stopped pushing, knelt next to the stroller, and looked with her into the southern sky, where I saw nothing but blue depth and distant snowcapped mountains. Then I noticed that beneath the sky, where no moon hung, two pronghorn does stood on a ridgeline several hundred yards away, staring directly at us. Hannah was not saying "moon" but rather "moo," a sound whose association with cows had made it her shorthand for "big, nonhuman mammal."
What happened next was as memorable as an encounter with wildlife could be. Hannah again said, " _Moo!_ " and one of the does responded with a breathy snort, "Cha-oo." Hannah's extended arm shot down into her lap, and she whipped her head toward me with eyes as big as a fawn's. This was the look of alarmed joy that she shot me every time we saw a strange miracle in the desert, which was often. When I smiled, Hannah turned back toward the pronghorn, leaned forward in her stroller, and yelled, " _Moo!_ " again into the wind. The second doe replied: " _Cha-oo!_ " Now Hannah was thrilled, and she began flapping her arms like a magpie, shouting " _Moo! Moo! Moo!_ " at the pronghorn as they stood frozen against the moonless sky. Then she stopped and held absolutely still, listening intently. The sound of the wind seemed more textured than before, surging and lilting like invisible surf. The desiccated balsamroot leaves scraped as if we were hearing them through a stethoscope. Again: "Cha-oo. Cha. _Cha-oo_." Hannah grinned, wide-eyed. And then the two does eased a few steps back and were out of sight on the other side of the ridge. In another five weeks they would each give birth to twin fawns, somewhere out there.
We live on a wind-ripped hilltop on the eastern slope of the Sierra Nevada mountains, in the broken sagebrush steppe hills and high valleys on the northern Nevada-California borderlands. It is a land of lizards and eagles, pack rats and ravens, jackrabbits and mountain bluebirds. Our home mountain, which is three miles to our west and looms 2,000 feet above us, has California on its summit crest. Much of the country around us is managed by the Bureau of Land Management (BLM). And while the BLM is so politicized and underfunded that its ability to manage much of anything remains ever in question, these public lands have made it possible for big mammals to live here too. We're fortunate that our valley and mountain remain wild enough to support not only Old Man Coyote but also mule deer, bobcat, mountain lion, and pronghorn.
Although I am admittedly lococentric and thus strongly biased in favor of my own neighbors, to my mind the pronghorn is the most charismatic of megafauna. It is very unlike any other species on earth, and that is because it is—among the many antelope-like ungulates that once inhabited the prairies of prehistoric North America—the sole survivor of the Pleistocene extinctions that erased three-quarters of this continent's large mammals around eleven thousand years ago. _Antilocapra americana_ is alone in its genus because it is a living relic of the late Cenozoic savannah and in fact has no kinship with present-day antelope—nor is it related to goats, despite the _capra_ in its name. The pronghorn is a true North American native, having evolved here over the past twenty million years. Sometimes called the "prairie ghost" for its elusiveness and speed, the pronghorn is also the ghost of evolution itself. It is all that remains of at least twelve distinct pronghorn-like genera of animals that once inhabited the ancient North American prairies—some species with two horns, some with four, some with six, some with branching horns, others with horns that spiraled fantastically to a point. Of what were probably dozens of species of antilocaprids, only the pronghorn has survived the crucible of evolution. _Survived_ may be too grim a word for so beautiful a product of so beautiful a process. Say instead that pronghorn have been turned on the lathe of evolution for twenty million years, sculpted by predator and place, fired in evolution's prairie and desert furnace. Seen in the light of its evolutionary history, pronghorn is not a thing but rather an outcome—one as inevitable as it was unlikely.
The evolutionary adaptations of these living Paleolithic ghosts to prairie and desert environments are many and remarkable. Their coloration is tawny as dust, and their tan necks are garlanded at the throat with a white shield and higher up with a crescent ring, creating a broken tan-white pattern of camouflaging so effective that when a pronghorn turns to look at you it often seems to magically disappear. Bucks have elegant black patches from the ears downward to beneath the chin, and older, stronger males have intimidating black masks that sometimes cover their eyes. Pronghorn hair, whatever color it may be, is adapted to allow the animal to endure almost inconceivable extremes of heat and cold, from 50 below zero in winter on the northern part of its range, on the windswept Canadian prairies, to summer temperatures of 120 degrees or more on its southern range, in the scorching deserts of the Southwest and Mexico. Each hair is perfectly insulated, spongy and air filled, and can be flattened to create a waterproof and windproof shield in winter or lifted up to allow heat dissipation in summer. The bright white hairs on the rump are controlled by an extreme version of the same process of piloerection that makes the hair on the back of your neck stand up when you're hiking in grizzly country. These three-inch-long hairs, when flared, make the rump appear larger and brighter than usual and so function as a warning signal to other pronghorn, especially in flight. Illuminated in the brittle glare of the high desert sun, these distinctive white patches are often visible from miles away, and I have seen them glide across a distant hillside even when the rest of the animal had vanished into the sage-dotted land.
While pronghorn females sometimes grow short horns, males carry the distinctive pronged horns, which may grow to twenty inches in length and are used as weapons against competing bucks during the autumn rut. These cranial meat hooks are so powerful and sharp that the mortality rate in buck fights sometimes runs to 10 percent or even higher. The horn of the pronghorn is not an antler; while antlers are shed each year and are made of bone, horns are kept for life and consist of keratin—the same material used to build hair and hooves. Yet even here the pronghorn is anomalous. Not only is it the only animal in the world with horns that branch, but it is also the only animal that sheds its horns annually—or, to be more precise, sheds the keratinous sheath that covers the horn's bony core. Although this odd combination of antler and horn qualities has resulted in the pronghorn's notoriously imprecise genus name, _Antilocapra_ , which means "antelope goat," the pronghorn is neither antelope nor goat. Most of the several hundred recorded Native American names for pronghorn are more elegant and accurate. Many, including those of the Indian peoples who have long inhabited this region, mimic the pronghorn's breathy snorts. The Northern Paiute of nearby Honey Lake and Surprise Valley call the pronghorn _dü'ná_ ; to the Western Shoshone of central Nevada, _Antilocapra americana_ is _wahn'-ze_ ; it is called _á-yĭs_ by the Washoe people of the eastern Sierra.
Considering that pronghorn are fairly small bodied, with adults typically weighing not much more than a hundred pounds, their eyes are unusually large, almost as large as an elephant's—an adaptation that allows them to see great distances across open landscapes. While they have good hearing and a decent sense of smell, it is their superb vision that they depend upon most, a fact that became clear to me the first time I spooked pronghorn from more than a mile away. The animal's large black eyes are set unusually high in the head and far apart, socketed in bony turrets that allow for nearly panoramic vision. Because pronghorn are native to shortgrass prairie and desert, where cover is low, the high placement of their eyes allows them to scan for cruising predators even as they forage.
Pronghorn are such choosy and itinerant browsers that you can approach an area where they have just been grazing and yet search in vain for any sign of cropped plants. Far from being the rototillers or Weedwackers so common among their fellow ungulates, pronghorn snip individual leaves and stems so selectively that their presence on the range is barely discernible. This discriminating browsing for vascular forbs is also related to their ability to survive with so little water, even in conditions that immediately threaten dehydration for other desert mammals. While pronghorn prefer to drink some free water during most times of the year, they can manage to derive most—or, under extreme conditions, all—of their water from the plants they so carefully select. Pronghorn have a four-chambered stomach, the second chamber of which is loaded with bacteria, protozoa, and fungi that digest cellulose and other plant material the standard-issue mammalian gastric system can't handle. They have coevolved with the microorganisms in their gut, creating an ecosystem-within-an-ecosystem without which they would starve. All of these formidable adaptations to the often fatal extremes of their home environments have been crafted by twenty million years of trial and error in which error soundly dominated. Nevertheless, only a few successes are necessary if you have twenty million years to perfect them. The pronghorn we see today are what remain after twenty million years of errors have been culled from the gene pool, each weakness chipped away until the sculpted form of this evolutionary ghost emerged from the unhewn rock of ages.
Over the past thirty million years, hoofed mammals have evolved to become faster, an acceleration driven by an evolutionary increase in the speed of their predators. That said, pronghorn speed remains in a class of its own. Pronghorn can run thirty miles per hour almost indefinitely, in an effortless trot that barely resembles running. They can run for many miles at forty-five miles per hour, and even their incredible top speed of around sixty miles per hour is sustainable for surprisingly long distances. While their bodies appear much like sausages, an improbable form for a speedster, their stilt-like legs are unusually strong and fast, and their stride, almost inconceivably long. It is this unique combination of stride turnover rate and stride length that propels pronghorn forward at such unbelievable speeds. They complete a full stride in a third of a second, a motion so fast that John James Audubon aptly described running pronghorn as appearing to float above the spokes of a wheel turning so quickly that he could see only a gauzy blur where the legs should be. The stride of a pronghorn running at full speed is about twenty-nine feet, which is almost identical to the world record long jump—only a pronghorn does it stride after stride for mile after mile, and never foot faults, wears goofy shorts, or falls on its ass in the sand. I've watched pronghorn glide effortlessly across our nearby high desert basin in what looked like a relaxed canter, only to find their track sets separated by eighteen or twenty feet. When tracking them I can't help doubling back on the assumption that I've missed a set of hoof prints, as the tracks seem impossibly far apart. No wonder that Meriwether Lewis, when he first saw pronghorn on the western plains, wrote that their running "appeared rather the rapid flight of birds than the motion of quadrupeds."
Flight is an apt metaphor for pronghorn speed. It is difficult to describe how strangely beautiful these animals look simply drifting across the hills at forty or forty-five miles per hour. They don't appear to be straining in the least, and their effortless gait combined with the ridiculous amount of ground they cover makes them seem, by the laws of physics, uncalibrated—magically unmoored from any sense of motion our eyes are equipped to register. Pronghorn are more than twice as fast as white-tailed deer, and even a five-week-old fawn can outrun any Kentucky Derby winner. And pronghorn do their speeding through sagebrush and bitterbrush, over sandy and rocky terrain, upslope and downslope, in conditions of extreme heat and cold. Not only can a pronghorn run the length of a football field in about three seconds, but if you could somehow lean that field up on the shoulder of a desert mountain, fill it with sage and boulders and ground squirrel holes and rattlesnakes, and then crank the temperature up to 100 degrees, a pronghorn would still go end zone to end zone in about three seconds—and would take fewer than a dozen strides to do it.
So how do pronghorn run so fast and so far under such inhospitable environmental circumstances? Practice. Twenty million years of practice. There is no evolutionary paradigm shift at work here, no genetic equivalent of a hydrogen fuel cell, no revolution but evolution. The pronghorn has simply refined traits that are common to most mammals, including ourselves. Where we have five fingers on our hands, pronghorn have fused the third and fourth digits into their perfectly engineered padded cloven hooves. Where we have a wrist with squarish bones a few inches long, the pronghorn has elongated its hand bones until they are almost as long as your forearm. This marvelous cannon bone, nine inches in length but only as thick as your finger, is what allows pronghorn to achieve the stride length necessary for their remarkable speed. I once found one of these impossibly attenuated cannon bones beneath a bitter cherry bush out in the desert scrub, and I now keep it on my writing desk to remind me that even in this broken world, refinement and delicacy, rather than brute strength, can still translate into power.
The downside of this skeletal refinement is that pronghorn find jumping dangerous, as an unstable landing on these delicate, elongated wrist bones can shatter them and thus prove fatal. Pronghorn consider jumping so disconcerting that they rarely attempt to clear anything higher than a few feet, and as a result they often find their movement to winter range or calving grounds blocked by fencing, which proves no significant obstacle to woodland-evolved leapers like deer. Unlike deer, pronghorn have spent millions of years matching themselves to the demands of shortgrass prairie and sagebrush steppe, where running fast is much more important than jumping high. Taking advantage of this reluctance to jump, at one time some Native peoples drove pronghorn into improvised corrals, building immense V-shaped drift fences that were often several miles apart at the mouth but funneled to a chute where pronghorn were killed or captured. The long walls that kept the animals within the V were constructed of rocks, juniper snags, or, more often, sagebrush. If the barrier was even a few feet high, pronghorn would continue to run alongside it rather than jump over it to freedom. About fifteen miles north of where Hannah spotted the two pregnant does is a pronghorn funnel called the Fort Sage Drift Fence, a thousand-year-old Paleo-Indian rock structure, thirty inches high and more than a mile long on each side. In more recent times, the Paiute people of Honey Lake Valley—whose country is visible from the top of our home mountain—wove thick sagebrush-bark ropes more than a mile long, which they shook in order to keep spooked pronghorn running forward rather than attempting to escape by jumping laterally.
Several other pronghorn adaptations for speed are specific to does and fawns. The first has to do with the pronghorn's distinctive reproductive strategy of birthing twins. While a doe breeding for the first time usually calves a single fawn, nearly all her subsequent births—and every healthy doe will be pregnant every year for the rest of her life—will result in twin fawns weighing about eight pounds each. This twinning is ensured not only by a complex process of intrauterine competition among embryos, but also by the fact that the pronghorn uterus, unlike that of a human, is twin horned, thus virtually ensuring that an embryo will develop in each uterine horn. The pronghorn's primary defense against predators is speed, so the development of twins ensures that the doe's body will remain balanced and fast, even as she becomes heavier. Because Eryn is an identical twin, as were my uncles (who were named Robert and Bruce after Robert the Bruce, the warrior who led Scotland to independence during the early fourteenth century), I'm enamored of the idea that for pronghorn twinning is not an anomaly but rather an essential means of bringing a new generation of children into this world.
A related adaptation is the pronghorn's exceptionally long gestation period, which, at about 252 days, is much longer than that of larger ungulates, such as bighorn sheep (about 180 days) or mule deer (about 200 days). This protracted gestation is an evolutionary adaptation that reduces the window of vulnerability for fawns. Because fawns stay in the oven so long, they remain protected from predation longer and are more fully developed at birth. Pronghorn fawns can stand up just thirty minutes after they are born, and four days later they can outrun you or me. Although many fawns fall prey to coyotes, the fact that pronghorn gain size and speed so quickly allows many more to survive than would otherwise be possible. Because I am now a father, it is impossible for me not to see compelling correspondences between pronghorn fawns and human kids. The pronghorn's extended gestation of about thirty-six weeks is remarkably close to the human gestation period of forty weeks, and I too have a fawn who weighed not much less than eight pounds when she entered the world. Whenever I find fawn bones in the desert hills I struggle to negotiate my own allegiances, for both the pronghorn and coyote, like me, have children to feed.
The primary fuel of a pronghorn's speed is oxygen, and it is here that the animal's evolutionary adaptations are most impressive. Where you and I do a respectable job metabolizing oxygen so we can have muscular strength sufficient to thumb text messages and hoist pints, the pronghorn consumes three times the oxygen we do. Its heart is enormous, and its lungs are triple the size of those found in comparably sized mammals. In fact, all the organs and processes associated with burning oxygen are radically developed in pronghorn. Compared with other mammals their size they have a third more blood cells, half again as much blood, and two-thirds more hemoglobin; their muscle cells are loaded with the mitochondria necessary to process all this oxygen. Unlike the three-quarter-inch tube through which we pathetically slow bipeds suck wind when we're out for a jog, the pronghorn has a two-inch windpipe that allows it to shotgun air while running. In fact, the minute it takes off sprinting it lolls its tongue out and opens its mouth to clear the tracheal opening for a full blast. If you don't think this form of air gathering works, try sticking your open-mouthed noggin out the window of your car while going sixty miles per hour. One unfortunate by-product of all this oxygen-stoked muscular performance is tremendous heat, which is especially threatening to the animal's brain. But pronghorn have developed a number of novel thermoregulation strategies, among which is a fascinating circulatory system that shunts overheated blood through a network of superficial arteries and veins that cool the blood before it is redirected to the brain.
Although these and many other adaptations for speed have been sculpted by the pressures of natural selection, the creation of the amazing feat of form and function that we call the pronghorn has required eons. When I despair at how shortsighted we humans often are, I find it comforting to remember that _Homo not-so-sapiens_ has had only a quarter of a million years or so of practice. Even the aches in our backs and knees remind us that our ancestors haven't been standing upright for very long—and, in fact, that the evolutionary jury is still out on whether this bold uprising to verticality really pays off for anybody other than NBA players and chiropractors. Pronghorn, by contrast, have been trying to get it right for at least eighty times longer than we have, which provides some hope that in another nineteen and a half million years we might gain wisdom enough to be worthy of our name, _sapiens_. Then again, since we were the ones foolish enough to name ourselves in celebration of our wisdom, it is a good bet that we may lack the humility necessary to survive that long.
To understand something of _how_ pronghorn run so fast remains a far cry from understanding _why_ they do so, especially in the context of such core evolutionary concepts as selective pressure and speciation. The problem with pronghorn speed is that it is anomalous. No predator in North America can catch an adult pronghorn, and none even comes close. While coyotes exact a heavy toll on each year's crop of pronghorn fawns—and golden eagles and bobcats pick off a few as well—Old Man Coyote can't begin to catch an adult pronghorn. The sophisticated pack-hunting strategies used by wolves gives them a better shot at relay racing a pronghorn to death, but even here predation is minimal. In fact, some studies suggest that the presence of wolves on the range increases pronghorn populations because wolves regularly haze the coyotes that are the major predator of pronghorn fawns. Like every other major predator in North America, coyotes, wolves, and mountain lions are so incredibly slow compared with pronghorn that predation by these animals cannot have provided the selection pressure necessary to create the pronghorn's speed. Where, then, did this amazing speed come from? Evolutionary theory allows only one explanation: pronghorn run so fast because they are being chased by ghosts.
Imagine a lovely, sunny spring day out on the North American prairie, a million years ago. You are one of tens of millions of pronghorn that live on the savannah, which you share with a great many other mammals, most of which will be extinct a million years from now. As you stand eyeing the prairie horizon—or perhaps you are reclined, chewing your cud—you are worried, and it isn't because you have bills to pay or your car is making a funny sound. You have real, urgent, and incessant reasons to be terrified. Of course Old Man Coyote wants to snap your fawns' necks today—just as he will on this day a million years from now—but at least he isn't fast enough to catch you. Wolves are a more serious concern, as they'll separate you from your herd and pack hunt you if they can. Thank goodness the gluttonous plundering dog _Borophagus_ finally went extinct after seven million years of crunching your kids' spines in its powerful jaws. Although evolution is busy replacing _Borophagus_ with the huge, bone-crushing dire wolf ( _Canis dirus_ ), it will be a few hundred thousand years before you really have to worry about that. In the meantime, you can handle most of these _Canis_ types. After all, how scared can you be of a genus that will eventually produce animals that prance along on a leash and wear little sweaters? The hyenas are a bigger problem. Ever since they crossed the Bering land bridge back in the Pliocene, they've been getting better and better at chasing you. In fact, when he's hot on your white rump, the running hyena _Chasmaporthetes_ looks more like a jaguar than it does like the slouchy little hyena that will be around a million years from now.
That's just for starters. The short ambush predators can really ruin your day. Of these, the gnarliest is _Arctodus simus_ , the giant short-faced bear. Don't try any of your excellent short-face jokes on him, because he weighs eighteen hundred pounds, is seven feet tall at the shoulder and ten feet tall on hind legs, and has a reach of about fourteen feet. He needs to eat thirty-five pounds of meat each day, which is about what you weigh once he picks his teeth with your ribs and gets to the juicy part. What's worse, unlike bears a million years from now—the ones that lumber over to the park dumpster to scavenge from discarded buckets of KFC—his feet are oriented with toes straight forward, which means he can really run. He can't outrun you in a prairie race, of course, but if you spend one second too long daydreaming about the upcoming rut while drinking from the spring, you're a goner.
The stalk-and-ambush crew also includes the scimitar cats ( _Homotherium_ ) and the saber-toothed cats—which aren't named _Smilodon fatalis_ for nothing. There are a bunch of species of each, all of them weighing five or six hundred pounds, with six- or seven-inch-long teeth hanging out of their mouths, quiet as a mouse when stalking, and not too shabby running short distances. The "smaller" cats also cause problems. The American lion ( _Panthera leo atrox_ ) is five hundred pounds of bad-tempered nuisance, and the cougar ( _Felis concolor_ ) and jaguar ( _Panthera onca_ ) are always staking out your migration routes. A good rule of thumb with all these cats is that you should just stick with your herd, eat a nutritious diet of succulent forbs, get lots of exercise, and stay out of situations where you can get bushwhacked. Of course they're going to eat most of your children, but as long as you stay out in the open and remain vigilant, you can outrun any of these carnivores in a pinch. Still, I'll bet you can hardly wait for the Pleistocene extinctions to get some of these giant monsters off your back.
But here's the real problem, the one you'd really prefer not to discuss. There's one thing out there that is faster than you are, at least in a short sprint—and if you get taken down in the first hundred yards, then your ability to run fast mile after mile won't help you, will it? What's worse, _you_ made that thing fast. You made it fast by being so fast yourself. It is thanks to you that the American cheetah can sprint seventy miles per hour and is thus among the swiftest of terrestrial mammals. His adaptations for speed are numerous, fabulously inventive, and all your fault. Many's the day you pranced around racking your brain for a new way to pick up just a few MPHs. Put your ears back to reduce wind resistance? Stick out your tongue to shotgun air? Find some way to stretch your stride length from twenty-eight to twenty-nine feet? Seems like every time you came up with something, though, he did too; no matter how fast you became, the American cheetah was a little faster, at least in the short run. But if it bothers you that he's faster than you are, remember that you also have him to thank for making you so fast. If you weren't running from him all the time, you'd never have become fast enough to outrace all those other voracious predators, and then where would you be? Your speed and his have coevolved, each causing the other. I know you hate it when he eats you, but he really has made you the best pronghorn you can be.
Nobody knows for certain why most large mammals went extinct around eleven thousand years ago. In North America, roughly thirty-three of forty-five genera of mammals larger than about ninety pounds disappeared in a wave that erased the mastodons, mammoths, dire wolves, giant short-faced bears, saber-toothed cats, and many other mammals—including the American cheetah and all species of antilocaprids except the pronghorn. Some believe that overhunting by the humans who were then on the scene and wielding the new Clovis spearpoint technology was a major factor. Others find it more likely that a changing climate disturbed the ecosystems upon which large carnivores depended for survival. Yet others subscribe to the theory that the late Pleistocene extinctions are attributable to a wave of highly infectious diseases against which few large mammals had adequate defenses.
We do know that the lightning-fast American cheetah ( _Miracinonyx trumani_ ) went extinct about ten thousand years ago, while its prey, the pronghorn, did not. As a result, _Antilocapra americana_ has been alone out there for the past ten thousand years, running much faster than anything in North America, though the pronghorn doesn't seem to know that yet. And this is the most fascinating part of the pronghorn's story. Evolutionary theory provides no way to account for _Antilocapra_ 's excessive speed except to say that the pronghorn still sees the sleek shadow of the American cheetah over its shoulder. The pronghorn has experienced twenty million years of directional selection for speed: it just kept getting faster and faster in an attempt to outrun its ever speedier predators. For the last ten thousand years, though, that selection pressure has been relaxed. Now you might think that if you had ten thousand years to change your ways you could do so (though my experience with humans suggests this isn't a sure thing), but the pronghorn's speed—and a number of its other adaptations to avoid Pleistocene predators that no longer exist—is inscribed in its body. It can't simply decide that it really doesn't need so large a windpipe or even that it can abandon its vigilance while browsing. If we assume a twenty-million-year evolutionary history for the pronghorn, then the ten-thousand-year period during which directional selection for speed has been relaxed amounts to just 0.0005 percent of that history. The pronghorn is a product of millions of years, not thousands. Pronghorn run fast because evolution hasn't yet had time to slow them down.
Unlike many people, I have a neighbor who has cheetahs. My friend Aaron, who lives just four miles from our place as the kestrel flies, has for more than twenty-five years managed an animal rehabilitation center that cares for injured wild animals and for exotic animals confiscated from traders trafficking illegally in endangered species. Many years ago a pair of orphaned African cheetah cubs found their way to Aaron, who immediately began working to figure out how best to keep them healthy. Veterinarians at zoos had long observed that captive cheetahs developed a wide variety of health problems and that gastrointestinal diseases often claimed their lives. Using the kind of common sense that is as valuable to a human as speed is to a cheetah, Aaron reckoned that animals evolved to run seventy miles per hour would be healthiest if they could run seventy miles per hour. In his early work with the cheetahs, Aaron took them out to a desert playa not far from here, attached a flagged lure to a rope behind his pickup truck, and literally ran the cats across the desert as fast as he could zigzag without rolling his F-150. Aaron tells a wonderful story of once seeing a group of pronghorn lining a ridge above him, watching intently as the cheetahs sprinted across the playa at top speed. I love to imagine what kind of evolutionary ticking must have been going on behind those huge antilocaprid eyes. Were they curious, surprised, terrified? Or had evolution hardwired them so effectively that they simply thought the pronghorn version of "You again, huh?" Indeed, if advocates of the radical idea of "Pleistocene rewilding" ever get their way, African cheetahs ( _Acinonyx jubatus_ ) will be introduced to the pronghorn range as a substitute for the extinct American cheetah. And if that wild day ever comes, pronghorn will be mighty glad they didn't become slackers just because the Pleistocene extinctions wiped out their fastest predator.
It isn't only pronghorn physiology and speed that are vestigial but many pronghorn behaviors as well. These "relict behaviors" are another kind of evolutionary shadow—another indication that it takes tens of thousands of years to unlearn the lessons that millions of years have taught. Among relict behaviors in pronghorn, the most dramatic is the animals' congregation into large herds during winter. Typically, herding is a behavior that increases energy-draining competition for food, but it is worthwhile because it is an effective way to guard against predation. After all, your chances of seeing spring will be much better if a few dozen or a few hundred of your buddies are scanning the ridges while you're pawing away snow to get at forage. And the pronghorn winter herd displays the additional skill of being synchronized to a degree of precision that is incomprehensible to a bumbling biped like me. When the herd moves across the land, it exhibits a 97 percent synchrony of gait, which means that whether there are twenty pronghorn or two hundred, 97 percent of them will put their four hooves on the ground in exactly the same sequence and at exactly the same time. Pronghorn have at least four distinct gaits, or running patterns, but even when they shift from one to another at high speed (which is analogous to your shifting the gears of your bike or car as you increase speed), there remains better than 90 percent synchrony in the herd. Even in the exact instant they change their hoof sequence and timing while running at forty or fifty miles per hour, 93 percent of them will do so at precisely the same time—and the rest will correct within a fraction of a second.
Although I have witnessed the full winter herd only once, there is no possibility that I will ever forget it. The flawless synchronization of the animals' gait makes the herd's movement indescribably graceful. These animals don't appear to run at all but instead seem to flow across the land like a single organism. At first glance I assumed that I was seeing the shadow of a cloud rippling across the desert hills; when I looked again, my home desert had been fantastically transformed into the Serengeti, with graceful herds pouring out beneath a towering sky. As a result of the pronghorn's incredible synchronization, predators find it difficult to single out an individual to attack, and some scientists have even suggested that the weird effect of the perfectly coordinated herd—which looks more like a school of fish than it does a bunch of long-legged mammals—may actually have a hypnotic effect on predators.
Of course the fleet predators this herding instinct evolved to protect against no longer exist. So how long will it take before pronghorn will slow down, before evolutionary pressures will select for something different, like the ability to jump a fence or to recognize the glint of sunlight off gunmetal? Or will they remain incredibly fast, while their slower predators evolve to become as fast as cheetahs? Or will the ongoing biodiversity and climate change crises, which threaten the extinction of numberless species, wipe pronghorn out before evolution can correct for their anomalous speed? Nobody knows. In the meantime, the pronghorn is still out there, calving twins, hanging on, running as fast as it can. It is still out there, a ghost being chased by the ghosts of predators past.
We don't know how many pronghorn inhabited North America before their slaughter began in earnest during the mid-nineteenth century. An educated guess is that a hundred and fifty years ago there were around forty million pronghorn in North America, ranging from the oxbows of the Mississippi west almost to the breakers of the Pacific, and they flourished in numbers comparable to those of the iconic American bison. Then came White Guy with Fire Stick, and pronghorn were shot for meat, for pelts, and often just for fun. To cite a particularly egregious example, in 1859 a single pelt hunter killed five thousand pronghorn at one watering hole without using a pound of the resulting hundred tons of meat. After a venerable twenty-million-year evolutionary history, it took only a century for pronghorn numbers to be reduced from forty million or so to fewer than fifteen thousand by the early twentieth century. Not only was the species in imminent danger of extinction; many conservation biologists felt that the numbers had already fallen too low—that the prairie ghost was doomed, regardless of any efforts that might be made to prevent the curtain of time from being drawn on it forever.
Fortunately, this grim forecast didn't stop some people from trying anyway, and their ultimately successful efforts are the reason pronghorn are still being born on our prairies and in our deserts today. The odd combination of realistic pessimism and idealistic optimism necessary to take up the gauntlet of pronghorn conservation was expressed by Charles Sheldon, after whom the Sheldon National Wildlife Refuge in northern Nevada is named: "I think that the antelope are doomed, yet every attempt should be made to save them." Thanks to Sheldon and many others, most western states imposed moratoriums on hunting pronghorn—prohibitions that here in Nevada gave complete protection to the species from 1917 through 1943. Refuges were created, management guidelines instituted, and trap-and-relocate programs initiated to return pronghorn to parts of the range from which they had been extirpated. In the half century between 1925 and 1975, pronghorn increased from what may have been as few as thirteen thousand animals to a half million, and the population now stands at around one million.
Despite this inspiring conservation success story, the future of the species remains uncertain. It is now development rather than wanton slaughter that impinges on ancient pronghorn migration routes, deprives the animals of winter range, and covers their calving grounds with asphalt or drilling pads. Most of the valley just south of where Hannah saw her first pronghorn was recently proposed for transformation into "Evans Ranch." If you live in the West, you already know that "ranch" is the going euphemism for "massive subdevelopment," and so Evans Ranch will be no ranch at all but rather a dense concentration of more than five thousand houses and apartments in a valley where pronghorn now forage, water, and calve. This valley has remained mostly wild because of our federally protected aquifer: we have so little water here—about seven inches of precipitation each year (most of it snow) and almost no surface water—that anything more than a rural scattering of homes is plainly unsustainable. But that was before trans-basin water importation, the latest in a long line of get-rich-quick schemes to force the intermountain West to bloom and fructify. The plan for Evans Ranch is to drain a desert aquifer someplace else—most likely in the Honey Lake basin, where Paiutes have honored the pronghorn for millennia—and pipe the liquid gold to this valley.
When Evans Ranch was proposed to city government, it was described in a planning document more than a thousand pages in length. That document never mentioned pronghorn. The property to be developed borders public land on two-thirds of its huge perimeter; despite this, the BLM, which represents the public interest, was never even consulted about the negative effects the development would have on public lands. When the project came before the city's planning commission, one commissioner was forced to recuse herself because she is the wife of the developer's lead PR man; another insisted that people who care about wildlife shouldn't try to stop progress but should instead pick up trash in the desert. Then the commissioners voted unanimously to approve the development. When Evans Ranch subsequently went before the Reno City Council, letters opposing the development arrived in a landslide, while statements of support came in a feeble trickle; testimony ran about five to one against the development, with nearly all of those who spoke in favor of the project being the developer's own employees. In advance of the hearing, the developer told the media that in twenty years he had never seen pronghorn in the valley; in response, many citizens who testified showed photographs of pronghorn grazing on or near their properties.
I was the citizen appellant to the Evans Ranch decision, which meant that while Eryn was hosting Hannah's second birthday party I was in city council chambers speaking as best I could for public lands, for sustainable growth, for sparing the pronghorn. On one side was a row of experts on traffic flow, water importation, effluent disposal, and a number of other bleak arts. On the other side were a bunch of my heartbroken neighbors and I trying to explain why an ineffable mystery like the pronghorn might matter—and trying to explain it to people whose minds were already made up. When I rose to offer my rebuttal to the developer's PR man, the mayor of Reno instructed me to sit down and be quiet—until his legal counsel reminded him of his statutory obligation to allow me to speak, at which point he acquiesced and permitted me two minutes to make the case for protecting a species that has lived here for twenty million years. In the end, the arguments made by those of us who value wildlife, open space, and public lands were compelling. One staunchly prodevelopment member of the growth-at-all-costs city council even complimented us on what he characterized as the most passionate and well-reasoned citizen appeal ever heard before council. Then he voted, along with every other council member, to approve the development. It was painfully apparent to everyone in those packed chambers that the vital process of citizen appeal and testimony had been reduced to a form of window dressing intended to superficially legitimate a deal long since struck between local power brokers.
I've seen pronghorn yearlings alone and in pairs, does alone and in groupings, males alone and in bachelor herds, huge bucks herding and hiding harems of as many as fourteen does; and I've witnessed the winter herd of scores of animals together flowing across the land in synchrony. That's here, in our home valley, where the developer apparently hasn't managed to spot a pronghorn in twenty years. Some may be mercenary enough to argue that we humans, developers and environmentalists alike, are just maximizing our opportunity to prosper, to make homes for our children, to ensure that our way of life—or, considered sociobiologically, our genetic heritage—will be borne into future generations. If we erase the astonishing evolutionary phenomenon that is the pronghorn in order to do so, that's simply collateral damage.
But the counterargument comes from the depth of ages: it cannot be evolutionarily adaptive for us to proceed so recto-cranially, with our selfish heads so far up our asses that we can no longer see the beauty of this world. We don't have nineteen and a half million years to come up with intelligent, sustainable practices of inhabitation that will allow the pronghorn miracle to live with us. If we do not evolve ethically, and do so quickly, we will suffer what the ecologist and writer Robert Michael Pyle calls "the extinction of experience." We will deprive ourselves of rich contact with the physical world, which is something evolution has taught us to need. We will replace the textured earth that has literally made us human with an impoverished remnant earth in which our children will no longer have the chance to converse with pronghorn.
In an ancient cave not far from here there is an infant interred with a rattle made from the horn sheaths of the pronghorn. At another Paleo-Indian burial site, the body of an eight-year-old girl is found at repose; she has been laid to rest wearing a necklace made of the hooves of unborn pronghorn fawns, a sacred ornament possessing the power to help her travel swiftly across worlds. As a father, I cannot conceive the species of grief caused by the loss of a daughter—what terrifying ghosts this unspeakable loss might summon. I do not know what kind of necklace I could make. I do know that when unchecked development has erased the unique evolutionary phenomenon that is the pronghorn from our home valley and mountain, something that animates this place will be extinguished. Its loss will ripple through the life of this fragile montane desert ecosystem and through the lives of anyone capable of appreciating what twenty million years of striving for perfection has wrought. There will be a new ghost here, and it will haunt us for a very long time to come.
_Chapter 4_
4. Ladder to the Pleiades
Hannah, who recently turned three years old, is teaching me about the stars. Far from being a liability to her, my own profound astronomical ignorance has turned out to be her boon and, through her, a boon to me as well. The most important thing the kid has taught me about the stars is the brilliant open secret that if you don't go outside and look up, you won't see anything. Every night before bedtime she takes my hand and insists that I get my bedraggled ass up and take her outside to look at the stars. If this sounds easy, ask yourself if you can match her record of going out every single night to observe the sky—something she has done without fail for more than a year now. It seems to me that Hannah has accomplished something impressive: she has perfect attendance at the one-room schoolhouse of night. That she has somehow brought her celestially illiterate father along is more amazing still.
Following the inexorable logic that makes a kid's universe so astonishing, Hannah insists on looking for stars no matter the weather. At first I attempted the rational, grown-up answer: "It just isn't clear enough to see anything tonight, honey." But her response, which is always the same, is so emphatic that it is irresistible: "Dad, we can always _check_." And so we check. And it is when we check that the rewards of lifting my head up and out of another long day come into focus. One cold and windy night we stepped out and discovered, through a momentary break in an impossibly thick mat of clouds, a stunning view of Sirius blazing low in the southeast. Another evening we stood in an unusual late winter fog and saw nothing—but then we heard the courtship hooting of a nearby great horned owl, followed immediately by the distant yelping of coyotes up in the hills. At Hannah's insistence we even stand out in snowstorms to stargaze, and while we've never seen any stars on those wild, white nights, we've seen and felt and smelled the crisp shimmering that arrives only on the wings of a big January storm. Snow or no snow, she knows those stars are up there, and so she does easily what is somehow difficult for many of us grown-ups: she looks for them. Whether Hannah actually sees stars or not, in seeking them every evening she has forged an unbreakable relation with the world-within-a-world that is night.
Questions are the waypoints along which Hannah's orbit around things can be plotted, and she has asked so many questions about stars for so many nights in a row that at last I've been compelled to learn enough to answer some of them. In doing so I've stumbled into placing myself, my family, my home on the cosmic map whose points of reference wheel across the sky. We've learned a surprising number of stars and constellations together, and we each have our favorites. Now that we're in our second year of performing this unfailing nightly ritual, we're also having the gratifying experience of seeing our favorite summer stars, long gone in the high desert winter, come round again on the dark face of the year's towering clock of night.
The other evening after supper, Eryn asked Hannah to make a wish. Without hesitating she replied, "I wish I could have a ladder tall enough to reach the stars." As usual, I didn't know what to say. It is impossible to dismiss a three-year-old kid—who, among other things, discovered the cosmos without much help from me—when she articulates a hope that is at once so perfectly reasonable and so beautifully impossible. Before she goes to sleep, Hannah and I look at the six-dollar cardboard star wheel I bought to help us identify constellations. Too tired to make much of it, I toss the disk down on her bed in mild frustration. But she picks it up, holds it upright in front of her with both hands, stares earnestly out beyond the walls of her room, and begins to turn it left and right as if it were a steering wheel.
"Where you going?" I ask.
"Pleiades," she replies. "Want to come?"
Despite its faintness relative to many other celestial objects, the Pleiades—which Hannah still pronounces Tweety Bird style, "Pwee-a-deez"—are Hannah's favorite thing in the sky. If you're in the northern hemisphere, this lovely cluster is relatively easy to locate: when looking south in summer, it appears above and to the right of Orion the hunter, the three bright stars of whose belt align to form an unmistakable field mark. To the naked eye, the Pleiades resemble the dipper shape that is better known in the northern sky, where the Big and Little Dippers of Ursa Major and Ursa Minor, the two bears, revolve endlessly around the axle of Polaris, the North Star. Known to myth as the Seven Sisters and to science as Messier object 45 (M45), the Pleiades are an open cluster of stars—not a random constellation but a close grouping of intimately related stars that were born together from a single nebula cradled in the arms of a spiral galaxy. Dominated by hot blue stars, the Pleiades cluster consists of at least five hundred members, which formed together as recently as 100 million years ago. It is a tightly woven nest of baby stars, a galactic brood perhaps only one-fiftieth the age of our own sun. The cluster is expected to survive only another 250 million years or so, after which its individual stars will flee the spiral arms of our home galaxy and its giant molecular clouds and will break their sisterly gravitational bonds, causing them to forget their common origin. They will light out for the territory, wayward, each on its own new path.
Among the nearest to the earth of all open clusters, the Pleiades cluster is 12 light-years in diameter and a mere skip of 440 light-years away from Hannah's upturned face. Despite the tranquillity of their huddled glow, some Pleiades stars are actually spinning at up to three hundred kilometers per second at their surfaces. The Pleiades are also celebrated for their remarkable nebulosity, which was revealed by the first astrophotographs, taken during the 1880s. Otherworldly pictures capture spectacular blue reflection nebulae, illuminated clouds of interstellar dust attenuated into ghostly, cirrus-shaped wings by the intense stellar radiation emanating from Pleiades stars. The lyricism of Tennyson's 1835 poem "Locksley Hall" reflects this wonderful nebulosity: "Many a night I saw the Pleiads, rising thro' the mellow shade, / Glitter like a swarm of fire-flies tangled in a silver braid."
I still don't know why Hannah loves the Pleiades so much, and I wonder if I'll ever fully understand. She gives different answers on different nights, and though she's patient with me, mostly she seems to think I'm asking the wrong question. What fascinates me most about the Pleiades is not any arcana of astrogeekery but rather the simple fact that they are beautiful yet barely visible. The nine brightest stars in the cluster all have magnitudes hovering just around the limit of sharp human vision under excellent viewing conditions. But how many you actually see depends not only upon your eyesight and the weather and the position and phase of the moon, but also upon altitude, humidity, dust, pollution, and the most critical factor: light levels in the night sky.
"Let there be light" may have sounded good at first, but for there to be Pleiadian light there must first be earthly darkness. Although even the dimmest of the major Pleiades stars is forty times brighter than our own sun would appear at a similar distance (the brightest, Alcyone, is a thousand times brighter than our sun), only six stars are usually visible to the unaided eye. Although some people can see the seventh sister as well, I myself have witnessed it only three nights in thirty years of admiring this cluster—a once-in-a-decade pace that I find perfectly satisfying. Under ideal viewing conditions it is occasionally possible for extremely sharp-sighted people to see nine stars. In 1579, before the invention of the telescope, the astronomer Michael Maestlin accurately drew the positions of eleven of the cluster's stars. In Egyptian tombs archaeologists have found, buried along with mummies, ancient calendars ornamented with a dozen Pleiades stars, and aboriginal cultures in remote desert regions of Australia produced art depicting thirteen Pleiades. There is even some evidence that in the rarefied air of the high Andes some Incan people may once have been able to see fourteen member stars. It was common among Native American peoples in North America—just as it was among stargazers in ancient Greece—to measure the acuity of one's vision by the number of Pleiades stars they could see. Simply by being there, the Pleiades test the limits of our vision. These stars are easy to find but also easy to lose. Like most things that are precious, they are there but barely, and how well we see them—or if we see them at all—matters enormously.
In Greek mythology, the Pleiades are the celebrated Seven Sisters, celestial daughters of the sea nymph Pleione and the Titan Atlas, whose punishment from Zeus was to bear the weight of the heavens upon his shoulders. The Seven Sisters—Alcyone, Celaeno, Electra, Maia, Merope, Sterope, and Taygete—were nymphs in the train of Artemis, and I owe them a deep debt of gratitude because they served as nursemaids to baby Bacchus, the feisty little god of booze. The Pleiades sisters must have been easy on the eyes, as many well-heeled Olympian gods, including Poseidon, Ares, and even Zeus himself, had affairs with ladies from this fine retinue of astral maidens. Only Merope, whose name means "mortal," resisted seduction by the gods, a mistake for which she took a lot of grief. Or perhaps the problem was that the mortal she chose to marry, that old coyote Sisyphus, was a fast-talking con man whose infamous boulder-rolling punishment made Atlas's bad day at the office seem like a moonlight stroll.
So while Daddy Atlas was too busy holding up the cosmos to stand on his crooked porch with the twelve gauge, Orion the great hunter set eyes on the Pleiades gals and decided to follow a new quarry. This turned out to be a bigger challenge than Orion expected, and after seven frustrating years of pursuit his lust remained unsatisfied. In some mythological genealogies Atlas and Orion are related, which suggests that Orion wasn't above cruising for chicks at the family reunion (in most old European languages, the Pleiades were indeed called "the chicks"). Tired of fielding complaints from the harried Pleiades maidens and their kin, Zeus finally agreed to perform an Olympian feat of astromorphosis: he turned the seven sisters into doves and placed them in the sky, where they had good reason to suppose they would be free from sexual harassment. So much for that; later, when Orion was killed, Zeus placed him in the heavens just east of the sisters, where he might pursue them across the wheeling night for all eternity.
The word _Pleiades_ , a Greek name lovely to both eye and ear, has a mysterious etymology. A likely origin is _plein,_ which means "to sail." Scientifically speaking, because Pleiades is a cluster, its member stars do sail across the wine-dark ocean of space together, as the unrelated stars of most constellations and asterisms do not. The more direct connotation of sailing, though, is hinted at by the ancient Greek nickname for the Pleiades: "Sailors' Stars." The cluster's conjunction with the sun in spring and opposition with the sun in fall marked the opening and closing of the season for safe sailing in the ancient Mediterranean world. Around 700 B.C. the Greek poet Hesiod included this lyrical admonition in his "Works and Days":
_And if longing seizes you for sailing the stormy seas,_
_when the Pleiades flee mighty Orion_
_and plunge into the misty deep_
_and all the gusty winds are raging,_
_then do not keep your ship on the wine-dark sea_
_but, as I bid you, remember to work the land._
Hesiod well knew the winds of trade and adventure that drove Greek ships and Greek ambition, but he also knew that to sail after the heliacal setting of the Pleiades was to risk a permanent visit to Poseidon's coral caves. Our limits are written in the stars, he seems to say: know the time to plow the earth and not the waves, planting spring's hope in soil and not upon the furrows of the deep.
But the word _Pleiades_ may instead be derived from _pleos_ , meaning "full," which perhaps alludes to the wind-filled sails of Aegean summer but suggests other meanings as well. The plural of _pleos_ means "many," an apt description of the cluster's stars. Or perhaps the name is derived matriarchally, from Pleione, the mythological sea nymph who is mother to the Seven Sisters and whose own name means "to increase in numbers." But the etymology most consistent with the mythic astromorphosis of the sisters comes from _peleiades,_ which means "flock of doves." Although some believe this derivation to be merely poetic, having perhaps originated with the ancient Greek poet Pindar, many etymologists maintain that the literal meaning of _Pleiades_ is "constellation of the doves." The only certainty is that the origin of this name has vanished into the dark ocean of time. _Pleiades_ stands as a lovely marker of a harbor now lost, a lyrical name that remains unmoored and sailing.
I spin for Hannah the yarn of the seven dove sisters, and she performs morphoses of her own, changing them into the ravens and meadowlarks who are her neighbors, and then into girls, and back into stars once more. She also changes the Pleiadian doves into her own sisters, whom she plans to visit up in the sky whenever she pleases. Narratives of sisterly bonds have special appeal for Hannah these days, because she knows that her mommy now carries within her a five-month-old fetus—a little sister who, on the ultrasound's photograph of galactic inner space, appears gently lit and curved, a crescent moon floating in a dark sky of amniotic fluid. The magic of transformation is closer to Hannah's reality than mine, a fact made plain by her fluidity in imagining and describing transformations of all kinds: in herself and her unborn sister, in her nonhuman neighbors, in the night sky. On the mountain behind our house, she tells me, pronghorn drink together at the same springs with a local dinosaur she has imagined and named Braucus. She's certain that pronghorn and plesiosaurs sleep in the same caves, have breakfast together, even go to the same birthday parties wearing homemade garlands of lupine, sidalcea, and balsamroot flowers. The walls of the world—the ones that delineate the boxes within boxes where we grown-ups live and work—are invisible to her. For Hannah, the net of relations that makes the universe cohere is as interstitial as it is connective: strong and flexible, but full of inviting passages between worlds and permeable in ways that have long since been lost to me.
I tell Hannah about the science of the Pleiades as well as their mythology, but to her these two modes of perception offer equally compelling explanations for something she experiences in a relatively unmediated way. Orbital gravity, escape velocity, open cluster, reflection nebulae—that story is fine with her. Seven sisters who, while their daddy tries to hold up the world, become doves in the night sky—that makes perfect sense, too. Hannah loves stories, but behind the stories she sees only the world, just as it is, entirely full of possibility, every night when the light of the sun goes out.
Many ancient Greek temples, including the celebrated Parthenon, were constructed precisely so as to align with the heliacal rising or setting of the Pleiades. By orienting their temples in this way the Greeks were also orienting themselves within a universe in which this tiny, dipper-shaped net of stars formed the hub. The Seven Sisters are vitally important to ancient Greek culture—they figure in _The Iliad, The Odyssey,_ and other foundational Greek texts—and their beauty is also sung by the Romans, including Cicero, Ovid, Varro, Pliny, and Virgil. But the delicate beauty of the cluster, its visibility from both hemispheres, and its importance in marking the seasons of navigation and agriculture have given it prominence in cosmologies and agricultural calendars from around the planet.
In many ancient cultures the Pleiades cluster was considered the _axis mundi_ —the hub of the universe, around which all else revolves. In ancient Arabic cultures the cluster was known as Al Na'ir ("the bright one"), and in ancient Egypt the ritual of the dead included speaking the names of the seven Hathors—the Pleiades—to assist the dead on their journey to the distant stars. In Hindu mythology the Pleiades stars were called the Krittika (Sanskrit for "the cutters"), the seven daughters of Brahma who married the Seven Rishis, or Seven Sages. The skilled sailors of ancient Polynesia celebrated these "highborn" stars as crucial navigational guides whose arrival initiated their new year. The Chaldeans called the cluster Chimah, meaning "hinge," because it was thought to be the point upon which the universe pivots. In China this cluster of "Blossom Stars" was recorded in a 2357 B.C. reference, which may be the first in astronomical literature. The indigenous Ainu people of Japan saw not a blossom but instead a great tortoise that they called Subaru, a designation we still see on the six-star logo of the automotive conglomerate that borrowed its name from the Pleiades. To the Maori of New Zealand the Pleiades are Matariki, the beloved "little eyes of heaven." Among the aboriginal peoples of Australia, who believe the Seven Sisters came to earth from the celestial sphere during the ancient Dreamtime, each tribe calls the cluster by its own name: to the Kulin people the Pleiades were Karagurk; to the Adnyamathanha, Magara; to the Bundjalung, Meamai; and to the Walmadjeri, Gungaguranggara.
I can also tell Hannah stories that are closer to home, since most of the indigenous peoples of North America have rich cultural connections to the Pleiades. The Navajo, who called the cluster Dilyehe, claimed that these special stars were placed in the heavens by the Fire God. The Blackfeet have a myth of lost children who eventually find safety in the sky, and the Lakota believed that after death one's soul returns to the Pleiades. The Wyandot tell a story of seven girls who fly up to the star cluster in a big basket, in part to escape an amorous hunter. The Hopi called the Pleiades the Chuhukon ("those who cling together") and considered themselves descended from these stars, while Cree myths claim that protohumans came to earth from the Pleiades in spirit form before becoming corporeal bodies here. The Iroquois performed a prayer to the Seven Sisters, while Dakota myths relate that the home of the ancestors was among the Pleiades.
Hannah's favorite of these many wonderful stories, which is shared by the Arapaho, Crow, Cheyenne, Kiowa, Lakota, and our own local Shoshone peoples, closely links the Pleiades to Devils Tower (in Lakota, Mathó Thípila, which means "Bear Lodge"), a sacred site in present-day northeastern Wyoming that functions as the axis mundi within several Native American cosmographies. Seven Indian girls who were fleeing a great bear climbed to the top of the volcanic tower and prayed to the Great Spirit for help, after which the tower magically soared upward until the girls were delivered into the heavens for their protection. Once her little sister is born, Hannah tells me, she hopes the two of them can visit the top of that tower, just to see how close they can get to the stars.
The Pleiades were also used to calibrate many ancient agricultural and sacred calendars, and festivals devised to honor the Seven Sisters have even survived into our own day. The Aztecs based their famous calendar on the Pleiades, whose heliacal rising, ritually marked by priests, began their calendrical year. At Cusco, the capital of the ancient Incan empire, the 328-night year also began with the celebrated vernal rising of the Sisters. The Mayan sacred calendar, the Tzolk'in, describes a 26,000-year period and is likewise based upon the cycles of the Pleiades. And throughout much of pre-Columbian Mesoamerica there was, every fifty-two years, a monumental celebration when the Pleiades culminated (reached their highest point in the sky) just at midnight. In both spatial and temporal terms, these ancient cultures located themselves in the universe by calculating their relationship to the sacred star cluster.
Because the heliacal rising of the Pleiades in spring coincides with the life-giving season of new growth, many ancient traditions associate the cluster with fertility and plenitude. But the cycles of the Pleiades have also marked the end of seasons and the end of life itself. In old Europe the cluster was powerfully associated with mourning because its acronychal rising (rising in the east just as the sun sets in the west) occurred on the cross-quarter day between the autumnal equinox and the winter solstice and thus marked All Hallows' Eve, the feast of the dead that was later Christianized as All Saints' Day and eventually secularized as Halloween. Traced back far enough, Halloween leads to the ancient Celtic feast of Samhain, which was also calibrated to the midnight culmination of the Pleiades. In fact, the two prominent cross-quarter times in the old Dorian calendar were both marked by the Pleiades, and they evolved into May Day and Halloween, those vital pagan celebrations of life and death. The Pleiades are thus an illuminated bridge between ourselves and our ancestors. When little Hannah dons her Halloween butterfly costume and completes her metamorphosis from girl into butterfly, she is actually participating in an ancient ritual of death and rebirth that has for millennia been marked by the movement of her favorite stars.
Given that astral worship was one of the forms of pagan idolatry against which the ancient Hebrews defined their faith, it is fascinating that the Old Testament—a book not rich in stars—contains explicit references to the Pleiades. Both are in the Book of Job, and both are assertions of God's authority as supreme creator of heaven and earth. Job's problem is that, like Hannah, he asks a lot of questions. In Job 38:31–33, the Old Testament God—who is master of, among other things, the high-stakes rhetorical question—asks Job the sort of question one had better answer correctly: "Canst thou bind the sweet influences of Pleiades, or loose the bands of Orion?" The correct answer, of course, is "No way, boss." Job's inability to govern the stars defines the limit of his mortal gifts, for only omniscient God can penetrate the nebular veil shrouding the Seven Sisters. Like Job, Hannah and I must get along as best we can, asking much but understanding little.
Among the innumerable myths about the Pleiades are many that attempt to account for the "Lost Pleiad"—that missing seventh sister we so seldom see when gazing skyward at night. Some say that Merope, having married the mortal Sisyphus, went dim with shame. Others claim that it is Electra, who in mourning hid her bright face in grief upon the death of her son, Dardanus. Scientists tend to identify the missing Pleiad as Celaeno because it is the dimmest of the seven stars and just at the limit of human vision. But even here there is uncertainty, for the fluctuations in apparent magnitude that occur over time in these stars may mean that the ancients saw a brighter seventh sister than we see today. I do not know which Pleiad is the lost sister. In one story I often tell Hannah, she herself is the seventh sister, now invisible in the sky because she descended to earth to become part of our family. She likes that story, and whenever I tell it she smiles and sings an incisive line of verse: "Twinkle, twinkle, little star, how I wonder what you are."
The myth of the lost Pleiad has more serious implications in our own day, for now all seven sisters are in danger of becoming lost. Because the Pleiades stars are barely visible, even moderate levels of human-generated light in the night sky wash them away as if they had never existed. Insofar as our visceral experience of them is concerned, these stars are critically endangered. If the light of the Pleiadian doves is extinguished, it will not be the same kind of extinction suffered by the passenger pigeon and Carolina parakeet—but it will be extinction nonetheless. When I crest the last dark desert ridge before entering the city, I see in the southern sky the shining stars of Orion's belt. But above and to the right of the hunter I see nothing—the sisters are lost in the hazy glow of city lights, and where there should be beauty there is instead a hole in the sky. I know that beyond the cloud of light generated by the Walmart and its satellites floats a delicate net of exquisite stars that has played a vital role in human culture for millennia, but this abstract knowledge cannot substitute for the experience of seeing the sisters for myself.
When the pagans celebrated feasts to honor the Pleiades, they first extinguished every fire in the land so as to better view the Seven Sisters ignited in the heavens. Our age, too, must find some ritual to honor the stars, for darkness is the only mother from which starlight can be born. Modern connotations of the word _benighted_ are strictly pejorative: "to be unenlightened; involved in intellectual or moral darkness." But the word's archaic meaning sings a different tune: "to be overtaken or affected by the darkness of the night." We cannot be enlightened without first becoming benighted. In _Nature_ Emerson tried to rekindle in his readers the miraculous wonder of stargazing: "If the stars should appear one night in a thousand years, how would men believe and adore; and preserve for many generations the remembrance." What new myth can inspire us to restore the darkness that now lies hidden beneath the bright map of America?
How long will my little daughter continue to check nightly for the Pleiades if the light spewing from the sprawling city sends the other six stars into exile with their seventh sister? How can we measure what is lost when something that connects us viscerally to the universe simply ceases to be part of our sensory experience? I wonder what will have vanished when the nebulous glow of this celebrated cluster recedes, leaving only a blank spot on Hannah's treasure map of night. I have no way of knowing how the woman she grows to become will have been enriched by the presence of her sisterly stars or, conversely, impoverished by their silent vanishing into the artificially illuminated night. I do know that Hannah cannot bear the thought of her own little sister being born into a world from which the light of the lovely Pleiadian sister doves has been exiled.
PART TWO
Wilding
_Life in the wild is not just eating berries in the sunlight. I like to imagine a "depth ecology" that would go to the dark side of nature—the ball of crunched bones in a scat, the feathers in the snow, the tales of insatiable appetite._
—GARY SNYDER, _The Practice of the Wild_
_Chapter 5_
5. The Adventures of Peavine and Charlie
Late one summer afternoon, while out for a walk, Hannah and I decide to follow some pronghorn tracks, just for practice. Tracks, as Hannah knows, are story. To follow a track is to pursue a single story line through a palimpsest landscape, a richly imbricated world of interlacing narrative possibilities, the ultimate hypertext. How fast is the animal moving, in what direction, and for what purpose? Where does it pause to rest or forage? Why is it here at this time of year and this time of day? Where is it coming from, and where is it going? The pronghorn's trail intersects with those of jackrabbit and coyote, kangaroo rat and quail, father and little girl; its story unfolds in a particular place at a specific time and for certain reasons, only some of which are discernible by us. We engage with this text because we relish the language of hoofprint and fur tuft and scat in which it is told, but also because, like all readers, we crave narrative resolution.
After we have followed the antelope trail for a few hundred yards, Hannah asks, "Dad, what's at the end of these tracks?"
"A pronghorn," I reply.
"How do you know?" she insists.
"I can't be sure, but that's what I believe."
"OK," she says, "let's keep going until we find it." We follow the dual crescent moons of the socketed hoofprints another hundred yards until they crest a rise, from which we look down and see a large pronghorn buck looking up at us. " _Dad, just like you said!_ " Hannah whispers excitedly. It is as if, in turning the pages of an autobiography, we have somehow read our way up to the author as he sits at the desk, writing.
I never figured out how to explain to Hannah that in more than thirty years of following tracks I have never—either before that day or since—walked a track right up to the animal that was making it. It was as if Hannah's question about the outcome of the antelope's trail had somehow called the animal into being—as if by reading the trail's story carefully, she was able to write for herself a satisfying conclusion to it. If a trail is certain evidence of our faith in things unseen, the pronghorn was for Hannah the substance of the thing hoped for.
The exposed hilltop on which we live is desiccated by extreme aridity, raked by howling winds, inundated with deep snowdrifts, rocked by frequent earthquakes, incinerated by raging wildfires, and inhabited by rattlesnakes, scorpions, and vultures. Hannah Virginia and her little sister, Caroline Emerson, have never known another home. To them, living in this place and under these conditions is simply what little girls do. This is the place our stories come from, by which I mean this stark high desert landscape holds and inspires fantastic tales, but also that it profoundly shapes the larger story of our shared lives—our human attempt to structure, articulate, revise, and interpret the narrative of our experience in the world.
When I was Caroline's age, my father told me bedtime stories about Harry the Duck, a character of his own invention. Like my father himself, Harry was a great friend and also a comic figure. He was a resilient creature, always in a little trouble but perpetually able, through resourcefulness, generosity, and good humor, to escape his various predicaments—though I recall worrying that Harry might not survive some of his more harrowing adventures. When, at about two years old, Hannah seemed ready for bedtime stories, I considered making Harry an intergenerational narrative figure in our family. But as I reflected on how attached Hannah was already becoming to her desert home, a duck seemed the narrative equivalent of an invasive species, an alien figure ill suited to the dry hills among which we live. Instead, the very first story I offered to tell Hannah was about a pair of black-tailed jackrabbits that frequented the weedy sand flat that passes for our yard.
"What are their names?" she asked.
"Well, the older one is named . . . Peavine," I said, quickly appropriating the name of the mountain to our south to cover for my lack of preparation.
"And his little brother is Charlie!" she declared, out of nowhere.
"Sure," I said. "That's right. Peavine and Charlie."
From that moment on, Peavine and Charlie became a part of our family, and each evening I'd detail their latest adventure for Hannah before she slipped into slumber. In the early going I wasn't a very resourceful storyteller, and I noticed to my disappointment that Peavine and Charlie, endearing as they were, tended to become dull, especially when I was tired, or when the frustrations of work impinged on the joy the jackrabbits would otherwise have sought and found. Worse still, many of my narratives seemed heavy-handed and didactic, as if the brother rabbits existed only to serve my adult need to foist lessons upon my daughter. I often felt that I had imprisoned poor Peavine and Charlie—who were, after all, wild animals—in a narrative cage fabricated of my own fatigue or anxiety. There were some evenings when I almost sensed the jackrabbits' frustration with me, and they might well have wished they could step out of my bland tales and wrest control of their telling, if only to liberate themselves from the constraints of my tepid imagination.
Over time, though, Peavine and Charlie began to push back, to defy my urge to control them, occasionally rupturing the boundaries within which my stories tried to enclose them. They became less predictable, wilder, at times even irreverent. They would suddenly do or say things I hadn't anticipated, taking my tales in new and sometimes dangerous directions. When the rabbit boys were challenged to a soccer match by Raven and Magpie, they secretly persuaded Pack Rat to piss on the ball, since they suspected—correctly, as it turned out—that their corvid opponents would find the stench intolerable. When Peavine and Charlie were warned by their parents not to venture beyond the rim of their desert basin, they instead persuaded Golden Eagle to fly them all the way to the summit of the highest mountain in our valley. Whereas in earlier stories the boys might have learned a valuable lesson from their disobedience, they now had the time of their lives, returned home before supper, and were never found out. In some stories the jackrabbits even shared adventures with Old Man Coyote, an unsavory companion borrowed from Native trickster tales: a witty, energetic, ingenious, libidinous, transgressive, impertinent outlaw who drinks heavily, has sex frequently, farts lustily, and boasts loudly about it all.
The most important developments in the Peavine and Charlie tales, however, came from Hannah, whose increasing involvement in their telling helped the brother rabbits to grow into three-dimensional characters. When I first began this bedtime ritual I would announce, with unassailable adult authorial intention, the topic of the evening's story: "Tonight I'm going to tell you about the time Peavine and Charlie went to a birthday party with the white pelicans." And that was that. Later, Hannah began asking me to repeat stories, which I soon discovered I could not do to her satisfaction, since her memory of a story's details was so much keener than my own. As a result of this incapacity on my part, Hannah inadvertently became a more active participant, a co-narrator who corrected and inserted details as necessary: "No, Dad, the king of the white pelicans baked a _chocolate_ birthday cake, remember?"
As Hannah's role in the evening storytelling grew, a new dimension emerged in our narrative collaboration. She began to ask me if I knew a particular story—but in asking she would refer to an original story rather than one previously told. For example, she might ask, without provocation or precedent, whether I knew the story about the time Peavine and Charlie flew from Nevada to Australia in a hot air balloon—a story I had never dreamed of. Hannah's presumption was that the story existed independently of my ability to create it, and so the question was not whether Peavine and Charlie had actually flown to Australia in a balloon (of course they had!), but instead whether my own limited life experience had made me familiar with that particular story. As our bedtime stories evolved, we no longer told tales of my own imagining, and we ceased repeating tales. Instead, each evening Hannah would ask if I knew the story of the time Peavine and Charlie took a nap on top of a thundercloud, or the time the jackrabbits ate a bunch of fence lizards just to see if they would get a tummy ache, or the time they saw Old Man Coyote drink up the whole ocean, or the time they helped Kestrel fly so fast that it made the sun dizzy to watch.
What came of this new phase in our evening storytelling was as remarkable as it was simple, as illuminating as it was obvious: it turned out that I _did_ know the stories. All of them. While Hannah expressed amazement each night that I knew every tale she suggested (asking once, "Dad, when did you have time to learn all these stories?"), I too was amazed. I was astonished at the elasticity and power of narrative to spontaneously express any set of experiences that two jackrabbits—or, by extension, a father and his daughter—might have, and in that amazement I was liberated, along with Peavine and Charlie, from the repetitive plots and moralistic conclusions that had previously constrained me. Instead of forcing my furry protagonists forward at narrative gunpoint toward a preconceived and didactic conclusion, the jackrabbits now led us into fields of story we had never imagined. While the new stories were fragmented, often implausible, sometimes even unfathomable, they were utterly spontaneous and fresh. They were also funnier and more engaging, though less conclusive. Telling Peavine and Charlie stories was now like following tracks: the more attention we paid to each meandering narrative hoofprint, the more certain we were that a big buck was out there, even if it remained invisible from where we stood or lay while telling the tale.
Soon enough Hannah was no longer simply suggesting the stories but actively helping me to tell them. I'd ramble up to a particular plot point and then ask her if she "remembered" what happened next—even though the story was being told for the first time. So if Peavine and Charlie were swimming across Pyramid Lake on the back of Lahontan Cutthroat Trout, I'd ask if she remembered where they swam to.
"Oh, sure!" Hannah would always say at first, perhaps to give herself a moment to formulate the next passage of the story. "They swam all the way to Anaho Island so they could eat rattlesnake eggs and visit the magic well near where the white pelicans live!"
"Exactly," I'd reply, affirming her acts of imagination. "And do you remember how they got the water out of the magic well?"
"Oh, sure! They drank it up with a really, really long straw!"
On we'd go like this, taking turns telling, stitching together stories off the cuff and on the fly, without the slightest idea where we were headed, enjoying the experience immensely. We never corrected each other or felt that one person's story had been stifled by some outrageous turn the other person had interjected into the narrative. On the contrary, the delight for me came from the way the collaborative dynamic prevented the narrative from conforming to my own narrow, linear adult expectations for how a story should proceed. Because I'm a grown-up—which I can't help and for which I might well be pitied—I would have thought Peavine and Charlie would retrieve Anaho Island's well water using a bucket. But, of course, I was wrong. Since they had instead used a really, really long straw, that magical water was now inside them, and it would be up to me to discern what wonderful effects it might produce. The godlike omniscient narrator was at last dead and gone; what remained was a world of uncertainty and flux, which is to say a world where it is easy to become lost and where everything you experience feels inimitable and enchanted.
During this phase of our collaborative bedtime storytelling, Hannah never once implied that she viewed our stories as the product of invention or imagination. Instead, she continued to speak of the tales as if they preceded both the tellers and the telling. As far as she was concerned, all possible stories existed, and all existed independently of our speaking them. Even in their most unexpected and peculiar details, the tales were to Hannah an infinite number of tracks that we had not made but could always follow. To her, the stories of the nonhuman beings whose lives we narrated already possessed an uneditable fullness and integrity of their own—a fantastic, preexistent logic we could never contain or control but were free to discover and express. As our collaborative storytelling progressed, I began to wonder if Hannah wasn't somehow correct that even our wildest and most improbable narrative imaginings simply describe the world as it is. Because the world was there before us, maybe she's right that the stories were there before us too.
Hannah, who was three years old at the time, began telling stories to baby Caroline on the day Caroline was born. This introduced yet another phase in the lives of Peavine and Charlie, one that positioned Hannah as the primary teller of tales in our family. Many of the stories Hannah told her sister in that first year seemed designed to convey practical information about our home landscape. There were stories about how Peavine and Charlie learned to recognize and avoid the buzz of the rattler, and how they used snowshoes to keep from becoming trapped in the drifts, and what precautions they took when wildfire scoured the hills behind our house. In one tale, Peavine and Charlie strayed too far from home one evening but navigated back by walking toward Polaris, the North Star, which Hannah explained could be found by drawing an imaginary line through the Big Dipper's bowl. In another, the jackrabbits wisely avoided going out after dark, since Mountain Lion had been hunting the area. In yet another, the rabbit brothers failed to carry enough water on a hike up our home mountain and so would have become dehydrated had not Raven flown to their rescue from Summit Spring carrying water in a bag of balsamroot leaves woven together with strips of sage bark.
In addition to being dramatic admonitions about scorpions and rattlers, Hannah's stories for Caroline also functioned as an introduction to the natural history of this high desert environment. In the single story of "Peavine and Charlie's Bird World Series," for example, Caroline was introduced to most of the avian species that live here in summer. In the tale of the rabbit boys' long walk to Lone Tree, Caroline was offered a clear overview of kid's-eye geomorphology: the best climbing rock, the canyon where the snow stays deep, the ridge where the stars look _so_ bright. The story of the jackrabbits' "Great Water Race" identified the general locations of the two natural springs within hiking distance of our home, while the account of how Peavine and Charlie enjoyed a midday nap in the shade of some boulders seemed calculated to instill a potentially life-saving respect for the intensity of the desert sun at high elevation.
I listened to these stories but never interfered with their telling, except when I was asked to help—as when Hannah couldn't quite fill out the rosters of Peavine and Charlie's bird baseball teams without me suggesting Juniper Titmouse, Say's Phoebe, and Western Tanager. As I listened, I was amazed not only at how much Hannah had learned about her local environment but also at how clearly she could convey information through the vehicle of her Peavine and Charlie narratives. It was telling, too, that while Hannah's stories were replete with joy and humor, they also seemed crafted to convey lessons that one's little sister, if she intended to become a permanent inhabitant of the sagebrush steppe, would need to learn. In this sense, Hannah's stories adopted a touch of Dad's earlier didacticism, the key difference being that her jackrabbit tales were not contrived morality plays, as my stories had so often been, but instead were engaging, detailed, informed local narratives that used nonhuman lives to convey valuable information about how to live well in this place. And this, it seemed to me, must be very close to the root of all stories, which from time immemorial must have sprung from the desire of one person to teach someone they love how to feel at home in the world.
If Hannah had in some senses taken on the adult role of storyteller, it was interesting how quickly Caroline consequently grew into the role that Hannah had once played. Not content with only listening to and learning from the tales, Caroline by age three wanted to help shape and guide the narratives, often blurting out suggestions that Hannah was challenged to accommodate. So, for example, Hannah would announce that she was about to tell the story of the time Peavine and Charlie discovered the buried treasure, when Caroline would stubbornly insist that the treasure was not buried but instead hidden in the crown of a cottonwood tree. Or Hannah would prepare to tell of how the jackrabbits helped California Ground Squirrel and Antelope Ground Squirrel get married, when Caroline would instead demand to hear how the rabbit boys taught the ground squirrels to play blues harmonica. I confess that it was amusing to see Hannah's exasperation in these moments; each time Caroline shouted out a suggestion, Hannah was compelled to relinquish her own narrative intentions and expectations and instead follow the tracks that Caroline's imagination was laying down. Hannah had to learn to share the stories: to view them as the fruit of collaboration with her sister rather than as the product of her own unmediated authorial intent.
Then another wonderful thing happened. Caroline began to ask Hannah, just as Hannah had once asked me, if she "remembered the story about the time" the jackrabbits had this or that adventure—referring, as Hannah once had, to entirely new stories of her own invention. Hannah, for her part, fell naturally into the role I had formerly played, which was to reply joyfully that yes, in fact, she did remember that particular story. Although little sister was duly amazed at big sister's wonderful capacity to "remember" all the stories she requested, Caroline nevertheless participated actively in their telling. Rather than seeing her involvement as a disruption of a stable or proprietary text, Caroline instead assumed, precisely as Hannah once had, that all the stories were simply out there in the world, preexistent and waiting to be recalled by their tellers. To listen to the girls tell Peavine and Charlie stories was not only to marvel at their imaginative capacity to empathize with the lives of jackrabbits and their other nonhuman neighbors; it was also to realize that in the girls' world no story's path was ever blocked, no detail unchangeable, no conclusion assumed, no possibility foreclosed. For Hannah and Caroline, the tracks of a story could lead anywhere, and did.
We are each the hero of our own life story, which we write daily with our actions and ambitions, failings and fantasies. If we're fortunate, we may be able to construct ourselves as comic rather than tragic heroes, but our attempt to write our lives ultimately remains embattled. We struggle to control a recalcitrant protagonist, to impose narrative structure on a disorderly reality, to extract meaning from a personal story that is by turns fragmented, unfathomable, or mundane. And we often adhere obsessively to a preconceived story line, even when our lived experience is painfully incongruous with it. I'm certain, for example, that those of us who are parents remain bound by a story line within which we teach and inspire our children, helping them to mature into truly good people while also demonstrating our wisdom and affirming our values. But even as we wish our lives to proceed according to the stories we tell about them, we also struggle with the obdurate tensions those lives invariably present: their fractured narrative arcs and rough transitions, glaring stylistic flaws and troubling ambiguities, undesirable settings and underdeveloped characters, tediously predictable plot points and frustrating lack of closure. Rather than embrace the inevitability of this uncertainty, difficulty, and flux, we instead stubbornly pretend that we possess infallible authorial control. As a consequence, the writing of our lives reflects an impoverishment of narrative possibility—an overreliance upon a limited set of plot lines designed to force our story toward the denouement we most desire.
I try to remember how the world looks to a kid, but I find this imaginative leap increasingly difficult to make. There is something about adult perception, however finely honed it might be, that struggles to attain the sense of possibility that is instinctive to children. While I reject sentimental rhapsodies about the angelic nature of children—romantic propositions that the odor of a single diaper plainly refutes—it does seem to me that children possess the enviable capacity to imagine and thus inhabit a world in which all stories remain possible and in which any story may be told by any person at any time. The fluidity with which Hannah and Caroline's spontaneous tales narrated imaginative voyages across the permeable boundary between themselves and their nonhuman neighbors showed me how wild are the stories that link our domestic sphere to the natural world that is our wildest home. Even here, by the wood stove's hearth, the girls discovered a wilderness of possibility in the expansive relations those words helped them to imagine.
What if adults lived in a world of comparable imaginative richness? What if, instead of choosing desperately from among the half-dozen threadbare plots our popular culture sells, we asked a broader range of questions about our stories? "Do you know the one about the man who learned to love his wife?" "Will you tell about how the lady in the cubicle discovered that her work really mattered in the world?" "Do you remember the tale about the old man who played guitar for the very first time?" "Please spin the yarn of that father who, while splitting a bucked juniper stump just at dusk, suddenly remembered his daughters' coyote story and so looked beyond himself and witnessed alpenglow igniting the snowy flanks of his home mountain." Who knows what new questions we might ask, what new language we might ask them in, what new answers our stories might inspire? After all, no fine story unfolds without surprising plot twists, and no real story can know its own conclusion. Perhaps our lives may only be fully written once we relinquish narrative control, allowing the tale to tell the teller—once we renew our belief in a world in which a little girl reading the trail of a pronghorn can imagine the animal and, in that imagining, can summon a breathing ungulate on a dusty desert mountainside.
_Chapter 6_
6. The Wild within Our Walls
"Honey, get the camera!" I've just peeled back the tarp covering a half-used pallet of quartzite flagstone to reveal the cutest thing I've ever seen. From between the slats of the empty part of the pallet pokes a furry little head with large, dark, shining eyes, a tiny, sniffing nose, and twitching whiskers. Its big, rounded ears are backlit in the early morning summer sun, and its bushy tail is partially visible through the slats just behind its body. Then a second curious little head appears, and a third, and a fourth! By now Eryn has arrived with Hannah and Caroline, and with each furry noggin that pops up the girls' eyes open wider with surprise and joy.
"They're so cute!" Eryn exclaims, snapping away with her camera.
"What are their names?" Hannah asks, as she holds the hand of little Caroline, who is squealing with delight.
"Pack rats." It is the gravelly voice of my mason and surly neighbor, Charlie, who is coming around the corner of the house, finishing his third beer and getting ready to lay stone. In building this passive solar house of our own design here at 6,000 feet in the remote high desert, we've employed mostly rural neighbors, and there have been some hardpan characters among them. Charlie the mason—whom Eryn refers to as Charlie Manson—is an irredeemable desert rat, and though he's wild-eyed and sullen, I like him very much. He knows stone and he knows the desert, which is good enough for me.
"Pack rats?" Eryn asks. "Is that really a kind of rat? It looks more like a baby squirrel."
"Pack rat," Charlie repeats, scowling a little from behind his beer. "Wood rat. Trade rat. Pack rat. Rat. Deal with him." He's looking straight at me. My daughters are still smiling, but my wife is now frowning. I reach over and slowly slide the glass door closed in front of them. I can see the girls' lips moving but can no longer hear their voices.
I turn to Charlie. "What do you mean, 'Deal with him'?"
Charlie takes a big step closer to me, crushing and dropping his beer can as he does. "You don't know him like I know him, Nature Boy." I hated when he called me that, but I tried never to show it. "One day you'll step out of your house to take a piss behind that woodpile, and when you come back inside you'll find your bottle of single-barrel empty and that rat bastard sitting up in bed with your wife, smoking a Marlboro." Charlie is as earnest as a shark hunter, and he's in my face, reeking and squinting as he growls out his apocalyptic warning.
"Come on, Charlie, it's just a little squirrely thing." I gesture toward the pallet, where one of the wee, timorous beasties is still peeping up at us, cute as can be.
"Deal with him," he says over his shoulder as he walks back to his pickup to fetch another beer.
That night I heard no hooting from the resident great horned owls, no yelping from the upcanyon band of coyotes. Instead, strange scurrying sounds filled the darkness, unmistakably the acoustic trail of rodents, though the sounds were as loud as the jackrabbits out here are big. Scurrying on the deck, the walls, even the sill of the window just behind the headboard of our bed—the scratching and scrabbling sounds of claws, so loud that they kept me awake most of the night. In the morning I found small, sausage-shaped lumps of poop on the stoops of the house, with the greatest concentration on the doormats, as if the calling card had been left intentionally. Brownish-yellow streaks ran down our stucco walls, as if pints of porter had been spilled from the window ledges. The flowers in the pots on our porch had been clear-cut, leaving sheared stems where their colorful stalks had recently stood.
By breakfast I was already hunched over my field guides, reciting preliminary findings to Eryn. "Got to be _Neotoma cinerea,_ the bushy-tailed wood rat," I concluded. "There are twenty-one species of wood rats, spread out across the United States and from Arctic Canada south to the jungles of Nicaragua, but this is the only one that fits: fairly boreal, widely distributed in the Great Basin, and has that big, fluffy tail. Nocturnal, good climber, likes to hole up in crevices in rock faces but also builds stick houses in juniper country like this. Will carry stuff off and pack it into its house. Generalist herbivore, but relishes succulent plants like, say, your snapdragons. _Neotoma cinerea_. That's our boy." Now I can answer Hannah's question. "He's named 'bushy tail,'" I tell her. Caroline likes that.
In the days that followed, the signs of a rodent invasion became increasingly evident. Each night brought the Mardi Gras of scurrying, and every morning, a new harvest of the signature butt pellets and the proliferation of nasty yellow streaks down the walls of the house. The potted plants were soon completely gone, and their stems invariably showed the distinctive angular cut of rodent teeth, which got the maligned desert cottontails off the hook. Bugs Bunny just chomps and gnaws, while rodents slash stems on a perfect angle, like a carrot julienned with the precision of a sous chef's blade.
One morning I went to fetch our hidden house key to loan it to Charlie, and it was unaccountably missing from its secret hiding place. Then in the afternoon I opened the hood of my truck to add windshield wiper fluid and discovered that over a single night the bushy tails had built a respectable nest atop my battery, one that was not only well wrought but also colorful. As I removed the nest I called out to Eryn: "Honey, I found your snapdragons." Then the inevitable happened. Evolutionary programming told the pack rats that our stucco house was the sheer face of a rock cliff and the cement roof tiles were the innumerable doorways to a honeycomb of crevices and cavities. Once under the tiles, they had the run of the joint and could enter the soffits and at their leisure chew their way into the attic space, where they proceeded to gnaw insulation off pipes and coating off wiring. Now we had a critter issue that no amount of cute was going to fix. Despite the girls' enthusiasm for these furry visitors, if something wasn't done soon we'd be driven from our home by an army of furry little _Neotoma_. I had to "deal with him" and without delay.
But in a matter of twenty-four hours, before I could even formulate a strategy, something happened that should happen only in tales born in the dark imagination of Edgar Allan Poe: scratching, scrabbling, clawing sounds came from _within_ the interior walls of the house. Even Nature Boy was forced to admit that things were now out of hand. I winced, steeling myself for what I knew was necessary: a belly wriggle through the dark, claustrophobic crawl space beneath the house. If the rats were down there (at some point I had stopped calling them "bushy tails" and started calling them "rats"), then the levee was breached, and they would not only be under the roof and in the walls but would also have access to a labyrinth of ducts.
I drained a glass of single-barrel sour mash before strapping on my headlamp and dropping through the trap hole in the floor as the girls held Eryn's hands and watched me vanish into the darkness. It was immediately clear from the acrid smell that something was down there, and the stench wasn't coming from the glossy, bulbous black widow spider whose orange hourglass tattoo was the first thing illuminated in the beam of my small headlamp. With less than two feet of clearance I commando crawled across the vapor barrier on the ground, where I soon discovered that the plastic sheeting was littered with turds and sticky with the same Grey Poupon that streaked the walls of the house. They were down here, all right, but I didn't know where, or what I'd do if I found them. How many would there be? What diseases would they carry? Would they julienne my fingers with their razor-sharp choppers, leaving me with ten precisely angled stumps?
As I pulled myself forward with my forearms, I began to identify trash that had been collected from around our property: bottle cap, tinfoil, coyote scat, wood screw, tuft of dog hair, flagging tape, masonry nail, owl pellet, snapdragon stalk, tabs from Charlie's beer cans, even my spare house key. There were clipped juniper twigs lying all around, most of them still fresh and green. It was immediately clear that here, within the very foundation of our home, something quite wild had been going on. This was my house but their territory, and since they were accustomed to crawling around the dark in the choking ammonia stench of urine, they had an immediate advantage over me. Continuing my commando crawl, in the dusty beam of my headlamp, I noticed a pile of sticks a few yards ahead, tucked against a foundation wall and behind the elbow of a large duct. Nudging forward in the dark, I expected at any moment to feel a rat run across my back or tangle itself in my hair.
At last I eased up close to the duct behind which the stick nest was concealed and then slowly craned my neck down and around the duct, turning my head to allow the beam of my headlamp to fall onto the bundle of sticks. There he was, a foot from my face and exactly at eye level, and he was all attitude, standing straight up on his hind legs and glaring at me as if ready to rumble. I had somehow assumed that if I found the little beast he would simply scurry away, but clearly this bruiser had no intention of stepping off. Which of us would blink first? In the next moment I reached the turning point in my increasingly troubled relationship with _Neotoma cinerea_. As I tilted my head a little, the lamp revealed the treasures he was defending in his nest. Nestled among juniper berries and beer can tabs rested Caroline's pink pacifier. Now I could feel my jaws clench. "Listen here, you wife-stealing rat bastard," I said aloud, "you are _not_ going to drink my good bourbon." He didn't move a whisker, and he didn't look at all scared.
Since moving out to this remote hilltop in the western Great Basin I've dealt with plenty of critters. A great horned owl once swooped in front of us in an attempt to airlift our cat in its talons. On another occasion I intercepted a scorpion crawling along the baseboard beneath Caroline's crib. Then there was the time the big bobcat walked coolly beneath the girls' swing set on his way to raid our chicken coop. I'd even had my truck towed to the shop only to have the mechanic call to say he couldn't work on it because there was a four-foot gopher snake snoozing on the engine block. But it was Caroline's pilfered pacifier that finally pushed me over the edge. In a way that perhaps only a father can understand, that was going too far. Nature Boy was on the warpath.
I started by doing a lot of research quickly. I reckoned that to defeat an enemy you must first understand his strengths and weaknesses and compare them to your own. What would make my adversary hungry, thirsty, tired, cold, worried, existentially despondent? What might a bad day for a pack rat look like? I read what I could find on wood rat social biology and then made a two-column list documenting a point-by-point comparison of the relative strengths of Rat Bastard and Nature Boy. I expected, given my status as a member of the species widely celebrated as the pinnacle of evolution (albeit only among ourselves), that I'd have impressive advantages over my foe. As the two-columned list shaped up, I soon discovered that the opposite was true. Rat Bastard was such a dietary generalist that I could never stop him by cutting off his food supply—he'd eat almost anything, from plants, berries, seeds, twigs, and bark to small invertebrates, fungi, and even his own feces. I couldn't chase him down, as he could climb like lightning up vertical and even inverted faces, and I couldn't shoot him because he was nocturnal and, besides, he was always crawling across my damned house. I couldn't track him, as he could navigate by kinesthetic memory and scent markings—detailed maps I couldn't even see to take away. Because he was hydrated by the plant matter he ate, he could go his whole life without taking a drink, while I, by contrast, couldn't go a day without whiskey, let alone water. When I was roasting in summer he'd be chillin' in a cool rock crevice, and when I was freezing in winter the master of thermoregulation would be toasty in the heart of his insulated den, curled into a ball and wrapped snugly in a long, furry tail designed especially for the purpose.
As for breeding, it's no wonder he's a threat to people's wives. Bushy tails produce one to three litters of two to six young each season, so they can feed plenty of owls and still fill your crawl space with the next generation. Their social structure is polygynous, which means that each male shacks up with a harem of several females. The females, for their part, are capable of running up vertical walls even with young dangling from their mammary glands, and they will often begin copulating within twelve hours of giving birth (two facts that, as it turns out, one should not bring to the attention of human mothers). Finally, pack rats are meaner than they are cute. We aren't talking field mice here. Because of sexual dimorphism in the species, a male _Neotoma cinerea_ can weigh in at a whopping six hundred grams, which is about the size of a big burrito—only a ferocious burrito with razor-sharp claws and teeth. In fighting with other pack rats, which he does viciously and from a young age, he'll stand upright on his hind legs and bite and scratch for all he's worth, even leaping into the air to use his powerful hind claws to slash at and lacerate his opponents.
So what advantages are enumerated in the Nature Boy column of my list? Not many. I have an overdeveloped cerebral cortex, which, though helpful for executing such abstract tasks as making lists, is proving virtually useless in the practical conflict at hand. I do find that I have some things in common with Rat Bastard, who is described as "unsocial, solitary, and strongly territorial," but identifying his weaknesses is another matter. I did locate a few general sources devoted to how to get rid of rodents, but all the prescribed methods had attendant perils. "Exclusion" sounded obvious, but it came with the assumption that I knew exactly where they were getting into the house, which I didn't. "Toxicants" wasn't a great option, as pack rats will carry poison bait away and cache it in their dens, sometimes not eating it for weeks or even months; besides, my collateral damage assessment suggested that, considering both the domestic and wild animals around, there were plenty of "non-target species" (including our daughters) to be concerned about. "Biological control" for pack rats is more or less limited to cats, and our cat, Lucy, who has an impressively fat ass and who in any case spends most of her time trying not to become dinner for a coyote, is far too slow to dream of going to the mat with a fierce, burrito-sized street fighter like a pack rat; indeed, Lucy is the kind of feline who would have her paws full in a smackdown with an actual burrito. "Ultrasound repellers" sounded encouragingly high-tech but had been proven ineffective after a very short period. This left only "Trapping," which had plenty of complications. Rat Bastard was too big for a glue board, which seemed cruel anyhow, and a conventional snap trap would certainly catch me or my kid or my obese cat before it ever caught him. And anyway, think of the splatter. I knew my desperation had peaked when I found myself actually considering a plan enthusiastically proposed by my father-in-law, who is an impressively resourceful guy: fabricate a steel mat and wire it to a battery system by which we could electrocute the pack rats.
It was then that the owls showed me the light. Eryn and the girls and I went away for a single night, mostly to get a short break from my ratty nemesis, and when we returned it looked as if a fifty-five-gallon drum of half-and-half had been poured on the peaks of our roof and onto the ground below. "That's bird doo-doo," I explained in answer to Hannah's question, though it seemed inconceivable that anything smaller than a pterodactyl could produce crap this voluminous. "Yay, the birdie went poopie all by his self!" Caroline screeched, hopping up and down. Soon afterward I noticed that for once there were no turds on the porches—not one pellet. Aha! Rat Bastard, I have found your weakness! My presence here had disrupted a predatory cycle and had given _Neotoma_ prime digs—not only a good place to hang out but also protection from predation by Ye Olde Nocturnal Raptor. Combining what I learned from reading about wood rats with my feces-induced epiphany about the vulnerabilities of my furry antagonist, I rededicated myself to getting something other than my frontal lobes on the Nature Boy side of the power tally.
How does the genius detective catch the genius criminal? By thinking like the criminal. If I were Rat Bastard, what would I want, and what would I fear? He's got plenty of chow and women, and he doesn't even need to drink, but he has the willies about being out in the open. He needs deep cover to nest, cache food, and thermoregulate—and to avoid being suddenly eviscerated by that silent-winged death from above. And pack rats are fiercely territorial and agonistic. As do other agonistic species, like alligators, they'll live close together but do not like each other much, and they'll fight fiercely for cover when it is scarce. As one study put it, "Possession of a house is so important to wood rat survival that a high level of aggression and solitary house occupancy are basic to the genus." A glance around our property suggested that I'd inadvertently created pack rat nirvana. There were several tarped pallets of rock, old PVC pipe lying around the foundation of the house, and heaps of scrap lumber here and there. Because we heat primarily with wood, there was a pile of stove wood the size of a train car—at least eight cords that I'd hauled, bucked, and split. How many rats might be living inside there? At the top of Rat Bastard's column of advantages over me was the superb cover that my own trash had provided him.
Heartened by these new observations, I decided to run my insights by an expert, if I could find one. After a number of phone calls I tracked down a guy nicknamed "the Ratman" at the state wildlife office. I didn't catch his real name, but everybody I talked to swore he was the pack rat guru in Nevada. When I finally reached him, he turned out to be another of the quirky, knowledgeable desert characters I enjoy so much.
"You up in the juniper? Yeah, _cinerea_ for sure. Cute, aren't they? Ever seen them climb? They're amazing!" The Ratman is not only knowledgeable and friendly but also perfectly evangelical about the wonders of wood rats. He talks like a guy with nowhere to be, which is oddly pleasant, and he clearly wants to educate me as well as solve my problem. I go over my observations and options with him, and he concludes that he can help, but first he extorts from me a promise that I'll bring him a frozen pack rat. "Not one that's poisoned!" he insists. I agree reluctantly, though I haven't the slightest idea how I'll keep my promise.
"First, you've got to stop them from getting in," he says. "Then you need to deprive them of protective habitat around your place—that's their real vulnerability. They've got a small home range, so four or five hundred meters is plenty. Then you need to trap like crazy. Listen to the Ratman." Passing over his creepy use of the third person, I proceed to resist the apparent complexity of his plan.
"I can't figure out where they're getting in, Ratman, and I have about six or eight cords of wood I'd have to move. Besides, I'm really not the hunter-trapper type."
"If they're getting in under the roof tiles, stick steel wool or blow foam in there. If they're coming in around the foundation, mortar it up. You have to move all the stove wood—that's ideal habitat. The best trap for _Neotoma_ is the two-foot, two-door Havahart." I'm comforted that he's been so humane as to suggest a live trap, when he follows with: "And do _not_ go soft and release them! Best to drown them in a trash can." This from a guy who likes pack rats enough to call himself the Ratman.
"I know it's hard to tell over the phone," I reply, "but I'm really not the killing type. I'm one of those tender-hearted environmentalists. You know, a peace-to-all-creatures, live-and-let-live sort of guy."
"Listen, nothing you do will dent the general population—we're just talking about making an NFZ, a _Neotoma_ Free Zone, in a five-hundred-meter arc around your place. Besides, you're only icing some of this year's model, the young who are being kicked out of the nests by the big adults. The home nests will still be out there in the scrub. You've got to tear up that woodpile like a coyote and hunt them at night like an owl. Listen to the Ratman. Peanut butter in the trap, then swimming lessons in the trash can. You have to get busy." I thank the Ratman for his help and, at his insistence, renew my pledge to bring him a bushy-tailed Popsicle.
Then, over a solid week of summer, I set out to do everything the Ratman recommended. First I spent several days up on our roof, broiling in the sun as I lay on my belly, spraying foam into every rat-sized gap. The remainder of the backbreaking week was devoted to moving stove wood and debris to another part of the property—fourteen pickup loads in all. As I did so, I found no less than a dozen individual pack rat nests in the woodpile alone—downy soft, palm-sized cups of shredded juniper bark, more delicately woven than a bird's nest. Every night I set a trap, and most mornings I had a hopping mad incarcerated pack rat to deal with. At first I allowed my tender green mercies to trump the Ratman's directive, and I would drive my catch out into the desert each morning for its ritual release. But soon I admitted that the scarcity of quality habitat, the fiercely agonistic nature of _Neotoma_ , the extreme climatic conditions, and the summer's bumper crop of dispersing, aggressive young pack rats made this approach a death sentence for the animal, as well as being a very stinky and time-consuming prebreakfast activity for me each day.
And that is how I became a cold-blooded killer of my fellow creatures. Every morning I would wake up, drink a big mug of strong java, trudge off to my improvised trash-can water-tank death chamber, and use the unfortunately named Havahart trap to capture and give terminal swimming lessons to animals that, even if they are shameless stealers of baby pacifiers, are handsome, intelligent creatures that really just want to share our peanut butter.
Perhaps it was my guilt about the fact that my furry pupils invariably failed their swimming lessons that fueled my fascination with them, for as I became a murderer of pack rats I also became their admirer. Once I had evicted the bushy tails from our house and had, after repeated failures, finally managed to keep them out, I read more about _Neotoma_ —both in books and in the landscape. I perused scientific studies of wood rats, gleaned what I could, and then hiked out into the desert to make my own observations. I discovered pack rat bones in owl pellets, and I climbed cliffs to examine _Neotoma_ nests in granite notches. I identified many pack rat stick houses, which I visited regularly and watched closely, noticing if they had been built up, lined with fresh juniper, or disturbed by predators. Based on exposure, cover, and vegetation, I learned to anticipate where pack rats were likely to live, and eventually I even developed the ability to locate pack rat houses by following their distinctive acrid odor across the sagebrush steppe. I made regular visits to inspect the urination "posts" that pack rats visit repeatedly as a way of sharing olfactory information, and I watched the artistic patterns of their urine streaks emerge on rock faces after rain washed organic materials away, leaving expressionistic striations of dissolved calcium carbonate behind.
As I explained to Hannah and Caroline, the pack rat is no Johnny-come-lately, like the Old World rats whose pedigree dates only to the _Mayflower_ and which now infest the sewers beneath every city in America. _Neotoma_ is instead a true native, and there are fossil pack rat nesting sites in the Great Basin that have seen more or less continuous use for fifty thousand years. As a result of this long residency—and because pack rats have not changed their basic social behavior since the last ice age—we have learned amazing things from them. Of special value is the pack rat's powerful but indiscriminate penchant for collecting and storing things, often caching them deep within cliff faces and protecting them within stick houses, where the worst weather never reaches. Better still for posterity, it is often the case that generations of _Neotoma_ will urinate on the collected materials, causing them to be encased in a virtually impervious, amber-like gem of crystallized pee—a substance the earliest western explorers couldn't identify but thought looked a lot like candy (some who took the confectionary appearance literally reported that the stuff actually tastes sweet but causes a terrific bellyache). Because it is an assemblage not only of food but also trash, a pack rat's midden (the debris pile it gathers in or around its nest) is a wild miscellany of objects that functions as a time capsule of the local environment at the time of collection. What some of us call trash an anthropologist calls an artifact and a paleoecologist calls a macrofossil.
Within the pack rat's trash we have discovered answers to some of the most important questions we ask of the past. How do we know what level ancient lakes in the West reached, and how have we managed to chart their rising and recession over tens of thousands of years? By documenting the elevations of fossil pack rat middens, the ages of which are determined by studying the easily dated materials they contain. How do we know whether martens or jackrabbits or pikas or rattlers lived in a certain place twenty thousand years ago? Because pack rats collect and store small bones, as well as coyote scat and owl pellets, in which bone fragments of larger animals may be found. How do we know that here in the Great Basin subalpine conifers once grew 1,000 meters below their lowest distribution today? Because the twigs of bristlecone and limber pine are found in ancient pack rat middens that are far lower in elevation than the areas where those trees now exist.
This sort of information isn't important only to paleogeeks. Knowing the relative distribution of various plant species over a period of thirty or forty thousand years allows us to extrapolate changes in climate during that period, which in turn provides a paleoclimatologic baseline against which we can measure current rates of climate change. The pack rat's appetite for gathering trash provides a crucial means to gauge the degree to which the human appetite for consuming fossil fuels is causing a planetary environmental crisis. And we know all this for a very simple reason: pack rats gather trash near their houses and store it safely. The pack rat is a preservationist of the first order, one without whose collections of scat and berries and bone we would be hard pressed to understand the environment of the late Pleistocene and early Holocene periods.
It is no doubt true that one person's trash is another's treasure, but to the pack rat all trash is treasure—and I have discovered that it is in this treasuring of trash, as well as in the trash itself, that treasures are often found. Although we made sure that the pink pacifier I found beneath the house never made it back into Caroline's mouth, we still keep it in a shoebox along with scores of other objects the girls and I have collected from pack rat nests. And while our box of pack rat treasures includes tufts of fur and shards of bone, most of the objects we've discovered in local pack rat houses are rediscovered items—those ponytail holders and bobby pins, flat washers and beer bottle caps that are the unmistakable artifacts of our own unusual twenty-first-century life lived out on this high desert hill. Although the pack rat is a truly wild presence here, he also traffics in domestic goods and in that sense might be considered part of our extended household.
In fact, _Neotoma_ and I are still cohabitants of sorts. Although he no longer lives within the walls of our house, he does have his own houses nearby, and we visit each other from time to time. In this truce we both prosper, even if my way of "dealing with him" doesn't involve the apocalyptic extermination Charlie advocated. I realize, of course, that there is blood on my hands. Indeed, I've remembered my debt to the Ratman and have in the back of our freezer, where Eryn isn't likely to notice it, a wrapped and taped butcher-paper package clearly labeled NOT A BURRITO. Despite this frozen corpse, my relationship with my bushy-tailed neighbor is mostly peaceful, and détente has its advantages. I still have my wife and my single-barrel hooch, and he still has his wives and his beer can tabs. If my spare key vanishes occasionally, I trust that my furry neighbor won't use it to let himself in. Perhaps in another fifty thousand years, some earnest researcher may discover that key enshrined in crystalline pee in the heart of an ancient midden and wonder what kind of life was lived behind the long-vanished door it once would have opened.
_Chapter 7_
7. Playing with the Stick
The photograph of Curator Man that hit all the wire services and accompanied most of the online stories is of a tall, thin, well-groomed, friendly looking fellow (the kind of guy you'd actually call a "fellow"), with short hair, prominent ears, wire-rimmed glasses, and what looks like an expensive tie. In his hands he displays an elegantly framed item that in a few moments will become the most prized and celebrated treasure in his museum's collections. Curator Man's proud smile tells us that this is a big day for him. And what is the treasure behind the glass in the mahogany case? The stick.
This stick is at once just any old stick and not at all just any old stick. It is the stick that on November 6, 2008, was inducted into the National Toy Hall of Fame at the Strong National Museum of Play in Rochester, New York. As yet another anniversary of the stick's induction rolls around, I'm reminded of this photo of proud Curator Man, who could not have anticipated the media circus his museum's stick would provoke. When news of the stick's induction was announced in a ceremony and accompanying press release, the story was picked up by hundreds of online news sites and blogs and was even featured prominently in the last sixty seconds of many local TV news programs, right in the slot where the quintuplets usually go—which proves that even quintuplets can have a bad media cycle. Reporters invariably skipped the obvious question, "Is there really a National Museum of Play?" and went straight to the kind of penetrating journalism that helps a benighted public understand the complexities of so important an issue. "What can you do with a stick?" they wondered in print. "Who plays with sticks, and just how do they do it?" Since the stick doesn't come with directions and doesn't cost anything, they worried, how will Americans figure out how to use or value it? And, the tabloid sites asked, what do we _really_ know about the panel of nineteen so-called experts whose deliberations resulted in its selection? In short, everyone demanded to know what's so great about a stick.
I'm intrigued by this famous stick for a number of reasons, not the least of which is that I still can't figure out if it is profound or absurd, or profoundly absurd, or absurdly profound. There's a little of the emperor's clothes phenomenon going on here, I think. When I tell people about the celebrated stick, the response is nearly always the same. "You're kidding, right? A _stick_? You mean a _real_ stick? Like one you'd pick up off the ground?" There follows a long, uncertain pause. Then, almost invariably, comes the grinning reply: " _Hell, yeah,_ the stick. Greatest toy ever. Totally brilliant!" And after proclaiming something "Totally brilliant!" it is difficult for people to turn back. But I do want to turn back, to ask whether the museum's stick was nature masquerading as culture or culture masquerading as nature. I want to return to the moment in which we had to decide for ourselves what to make of the idea that a stick, rather than being viewed as a natural object, needed to be displayed in a museum.
If Curator Man thought any of this was funny, he certainly didn't let on. First, he pointed out that the selection panel of esteemed judges—intellectuals, artists, curators, pooh-bahs of various stripes—had a very difficult decision to make. They also had to adhere strictly to a formally articulated set of explicit criteria when choosing a toy to join the vaunted ranks of already inducted classics like crayons, marbles, the teddy bear, and Mr. Potato Head. To be selected, the toy must: (1) possess icon status, (2) have longevity, (3) encourage discovery, and (4) promote innovation. Curator Man went on to extol the many virtues and uses of the stick: "It can be a Wild West horse, a medieval knight's sword, a boat on a stream, or a slingshot," he pointed out. "No snowman is complete without a couple of stick arms, and every campfire needs a stick for toasting marshmallows." The media's immoderate love of Curator Man and his stick even spawned a widely syndicated "news" story actually called "Notable Suggestions for How to Play with a Stick," which made it evident that Hannah and Caroline were already as intelligent as at least some journalists.
It is at this point that the strange stick story jumps journalism's slick tracks and begins tearing through the weedy field of American popular culture, no longer under anyone's spin control. In Rochester there was still a stick in a case on a wall, but the story of that stick had gone viral. The first wave of responses to the stick was uniformly positive. What we might call the Good Old Stick! crowd rushed to expand Curator Man's already long list of noble uses for the stick, and they were mighty hard to argue with. I wasn't so impressed that a javelin and a golf club may be considered sticks—finding one so dangerous and the other so dangerously boring as to have no use for either—but a fishing rod and a baseball bat were sticks of an entirely different sort, and it was painful to imagine life without them. What about a conductor's baton or a pair of drumsticks? The fretted neck of my guitar is a kind of stick, and even the wooden combs of my harmonicas are little ten-notched sticks. The more I thought about it, the more impossible life without stick play seemed, and for a while I teetered on the brink of conversion.
But then the intellectuals got involved, and before I could make up my mind about the stick, all hell broke loose. First the developmental psychologists more or less said that kids would all be retarded without sticks, and some careless readers concluded by extrapolation that ADD, ADHD, OCD, LH, SLD, SLI, HDTV, THC, PCP, and LSD could all be blamed on the condition of tragic sticklessness to which "kids these days" had been so brutally subjected. Evolutionary biologists then asserted that it was the use of sticks that caused humans to develop immense cerebral cortexes, which apparently we needed in order to ensure that the really sharp sticks would poke the saber-toothed tiger and not our brother-in-law—that being the kind of "accident" that might halt activities leading to procreation and would surely have been selected against by evolutionary pressures. The sociobiologists went even further, asserting that the human affinity for sticks was evident in our fort-building behaviors and in our innate desire to have pickets in front of our houses when somebody came over to kill and eat us.
Then, predictably, the closet Luddites—who might best be described as "really old white guys who somehow learned to use social media"—got involved in the debate, and they were so elated to see the triumph of the good old stick that they felt their lives fully vindicated. The excruciatingly detailed "When I was a boy . . ." stories about sticks proliferated so quickly as to crash several servers, even as young IT people scrambled to figure out how a lowly stick could have brought down their networks. These old-guy stick lovers were soon joined by the TV haters, who didn't care about sticks one way or the other but judged them better than what they called the "mind-numbing cancer" of television, never mind that they were sitting in front of glowing computer screens and posting their views on blogs with disturbing names like Turned Off Moms.
At last, the very worst occurred. We eco-geeks got hold of the story, and that was when the shit that was already hitting the fan began to stick. According to its green defenders, the stick is important not because it is iconic or because it promotes discovery or innovation—indeed, even the detail that sticks might actually be played with by children drops out of the story at this point—but rather because it is "ecofriendly," "the ultimate disposable, biodegradable, versatile, multipurpose plaything." These ecobloggers celebrated the stick as "sustainable, recyclable, and upcyclable." One exclaimed euphorically that "you can even turn it into mulch when you're done playing with it!" which made me imagine tearing a stick from Caroline's little hands and jamming it into my tractor's wood chipper.
I don't want to rain on any parade that puts a humble stick in the lead float—after all, if Silly Putty and the Easy-Bake Oven can make the National Toy Hall of Fame, who am I to whine about the stick having its day in the sun?—but there's something creepy about this whole business. As the viral contagion of the stick story spread, I found myself possessed by a desire to shake Curator Man and his army of zombie bloggers and yell, "Hey! Wake up! Y'all are talking about a fucking stick!" But once the stick's coronation was hijacked, what had been a plaything was transformed into Captain Ahab's doubloon, Hester Prynne's scarlet letter, Citizen Kane's Rosebud: not a window onto childhood play but, rather, a mirror in which obsessed grown-ups saw nothing more than the reflections of their own faces. The stick's induction had been distorted from a celebration of how kids play into an ideological skirmish into which adults brought their own values and obsessions. At this point something in the stick story was lost forever. After all, isn't the beauty of a kid playing with a stick precisely that it is never our stick but always already theirs, that their imaginative powers define its shape, name, and use? It seemed to me that we pathetic grown-ups wanted to usurp the magic of the wand: to name and claim it, to wield it as a shield against time and tide.
That's the first thing that's suspicious about this stick story. Who could be so pretentious as to assume that a bunch of grown-ups—even worse, "expert" grown-ups—could possibly be capable of selecting toys for a National Museum of Play? The real experts, who are obviously the kids, hadn't been asked about any of this—including whether the idea of a National Toy Hall of Fame makes any damned sense in the first place. And what about the fact that all the negative connotations of sticks were being glossed over by these blithe stick enthusiasts? The sordid etymology and usage of the word _stick_ offers a powerful reminder that the stick we might imagine as a medieval knight's sword in fact has a double edge. What about "stick-in-the-mud," "stick it to them," or "beat him with a stick"? What about the wonderfully imaginative denigration of a pompous person as having "a stick up their ass" or the fact that soft speaking is enabled only by the carrying of a "big stick"? How about the derogatory slang terms "dip stick," "dumb stick," "dick stick," and "weak stick," or "to give stick," which means to disparage or criticize, or the suggestion that one "stick it" (in their ear or elsewhere)? Or the unfortunate transformation of perfectly decent food like bread and cheese _into_ sticks; or, conversely, the use of the stick to skewer and roast things, like squirrels? And what about chopsticks, which Americans would starve if forced to eat with, or the stick shift, which we can't drive, or the 1970s hair band Styx—which isn't quite the same, I know, but still makes my point that for every two sticks lashed together to make a mast or rubbed together to make fire, two others are used to make nunchucks or a crossbow. For every bouncing pogo stick or stirring swizzle stick, every forked dowsing stick or sacred rain stick, some poor stick figure ends up swinging from the hangman's gallows. For every burnished walking stick there is a cancerous fire stick, for every joy stick, a night stick, for every prayer stick, at least one stick of dynamite.
Of course the stick lovers don't tell you any of this. They'd also like you to forget the main thing sticks do, which is to poke your eye out. Even if a lot of things in life are "better than a poke in the eye with a sharp stick," one thing that is not in fact better is actually being poked in the eye with a sharp stick. Indeed, the same people who are now swooning nostalgically over their own stick-blessed childhoods are also yelling at their grandchildren to put down the goddamned stick before they put somebody's eye out. Let's face it, sticks are _dangerous_. And if you look at what kind of imaginative play the old-guy stick lovers valorize, it is invariably martial. One man unironically opined that what he most missed about his lost youth was the nurturing imaginative play by which he "could pretend that a stick was a big bazooka." Bazooka Lover had plenty of company. The most treasured memories of childhood play reported by these respondents featured the stick as rifle, shotgun, machine gun, sword, knife, spear, bow, arrow, harpoon, spear gun, blowgun, and even pipe bomb (good old pipe bomb!). One guy enthusiastically described the good fun he had while attacking his siblings with a stick that he pretended was a "Borg prosthetic arm/gun." Another waxed sentimental over the character-building effects of a spirited round of "Dodge the Stick," a game that, from what I could make out, basically amounts to throwing sticks as hard as you can at another person's head. But in addition to the Good Old Stick! crowd valorizing the violent imaginative and literal uses of the stick, they were also smug. Here is a representative posting: "The toys we in the older generation grew up with, like the stick, fostered the imagination. Nowadays, children sit in front of a computer screen playing video games that teach them violence and disrespect. It's no wonder kids these days are obese and ignorant."
The targets of this abuse didn't waste any time in putting down the Game Boy and chicken nuggets to give Gramps a piece of their mind. To their credit, the folks in this second wave of responses to the stick's ascension were more playful than those in the Good Old Stick! faction. Some mocked the stickophilic sentimentalists with sarcastic remarks like this one: "The sticks we had when I was growing up were way better than the ones they have now." Others used humor to fight back against the characterization of American youth as depraved because they play with computers instead of sticks. My favorite of these technophile back-talkers was the kid who wrote wryly, "I have an old Atari 2600 that I use as a makeshift stick." Yet others used hyperbole to ridicule the violent tendencies of the Good Old Stick! folks. "In a related story," wrote one mockumentarian, "the National Child Toy Safety Commission has issued a recall on the stick, identifying it as the nation's most dangerous toy. The Commission is now in negotiations with leading environmentalists, who make access to sticks easier every year."
One especially witty blogger imagined comments that might have been posted to Amazon.com by consumers who had heard of the stick's new fame and rushed out to buy one. One of these fake postings, from a mom and stick purchaser, describes the trauma suffered by her son after he discovered the troubling indeterminacy of the stick's meaning. She advises that parents "speak to the neighborhood kids in advance to reach a consensus as to what The Stick represents." Another, posted by the wonderful "Grandpa Dan" (who, of course, writes from Florida), reads as follows: "The Stick will never be beat. And it's a great bargain, too! The wife and I bought a single Stick, sawed it into five pieces, and now all our young grandchildren are having a grand time talking on their 'cell phones.'"
But the best was yet to come. The debate about the stick soon spawned a number of playful mock campaigns to have various other items inducted into the National Toy Hall of Fame. Among these nominees were the leaf, bubble wrap, the Popsicle stick, the log, the egg carton, shadows, the pillow, the vibrating dildo, the shoebox, dirt, the snowball, and Pete Rose (after all, Rochester is only 174 miles from Cooperstown). But the mock campaign that gathered the most momentum was the one agitating for inclusion of the rock in the National Toy Hall of Fame. As a first move, the rock advocates appropriated the discourse of racial justice to argue that the elevation of the stick over the rock was a clear case of bias, pointing out that sticks had received preferential treatment for far too long. They also observed that "sticks and stones" had long been associated with one another—in various cultural contexts, including the breaking of people's bones—and it was thus unfair that the stick alone should receive recognition. The rock folks gave hundreds of examples of the wonderful ways in which rocks foster imaginative play. Taking a page from the battle plan of Bazooka Lover and his ilk, for example, they pointed out that a stick's ability to be a gun is in no way superior to a rock's ability to be a grenade. I found this hard to argue with. Finally, the rock people emphasized the precedent of the toy Pet Rock, which in the 1970s swept the nation and made so much money for its creator that the guy became a millionaire overnight and at last achieved his lifelong dream: to own a bar in Los Gatos, California.
The persuasiveness of the rock campaign caused me to wonder not only about sticks and stones but about all the toys that have been inducted into and rejected from the National Toy Hall of Fame. As it turns out, debate has surrounded these selections from the very beginning. When the inaugural class of 1998 included Barbie but not Ken, a group of college students complained of sexual discrimination, arguing that Ken deserves equal billing with his female counterpart—who, they pointed out, is insipid, emaciated, and nippleless and, furthermore, has poor taste in purses and terrible gaydar. Some Marxist critics declared that the induction of the Radio Flyer wagon, the Duncan yo-yo, and the Crayola crayon constituted the baldest form of product placement advertising. Wouldn't the generic wagon or crayon have been good enough, or was the Hall of Fame taking kickbacks from these companies? When Monopoly was the only board game included in that first class, the aficionados of everything from Candy Land to Parcheesi to backgammon went wild—not to mention the evangelical Scrabble-ites, who had plenty of choice words for the Hall of Fame after their snubbery (if that's even a word).
This debate surrounding the choice of inductees became an annual ritual; most interesting is how regularly adult obsessions were reflected in these bizarre skirmishes over toys. While the Ken doll faction was clearly in it for the laughs, the Raggedy Ann fans—who actually call themselves "Raggedy Fans" and who in many ways disturbingly resemble a cult—were in genuine fits from the beginning. It wasn't so much that Raggedy Ann, whose oft-recited pedigree dates to 1915, was rejected—it was instead the fact that Barbie, that mindless slut, had been inducted with the very first class. The Raggedy Fans took to the warpath and for four long years endured repeated defeats until, at last, in 2002, came the "magical moment" (their words) when Raggedy Ann became the twenty-sixth toy to join the ranks. During those four years the Ann cultists collected more than eight thousand petitions but still had to endure the humiliation of having been outgunned by the Mr. Potato Head lobby, which, after suffering a similar defeat in the inaugural year, had their man in office straightaway in year two.
My study of inducted and rejected toys also revealed the precedent that indirectly enabled the stick's ultimate success: the surprising choice, in November 2005, of the cardboard box. The box was an influential inductee, because it was the first plaything not produced by a toy manufacturer to have made the National Toy Hall of Fame. Once the humble box had cracked the dam of the hall's logic, other toys not made to be toys couldn't be far behind. The affinity between the cardboard box and the stick was in fact remarked upon by many folks who responded to the stick's induction. One would-be parodist offered the _Onion-_ esque headline "Stick Enters Toy Hall of Fame, Cardboard Box Snubbed," only to be informed that, in fact, the box was already in. Many parents liked the choice of the box because it confirmed their observation that no matter how much dough they shelled out for toys, their kids preferred to play with the boxes in which the toys came. As a parent who has spent too much time repairing overengineered toys, I too approved of the box and stick, both of which I added to my personal list of Things That Actually Work, which until that time had included only WD-40, bourbon, and _Moby-Dick_.
I also found it instructive to consider some of the National Toy Hall of Fame's selections in light of its explicit criteria for inclusion. For example, while I'll fight the man who claims that the Slinky doesn't "possess icon status," it is harder to see how the Atari can be said to "have longevity." The Atari was inducted in 2007, by which time it had been obsolete for decades, and to make matters worse the Atari shared the class of 2007 with the kite, which is a three-thousand-year-old toy. It is also difficult to see how some of the inaugurated toys "encourage discovery," unless, as in the case of Play-Doh and Silly Putty, the discovery is simply that it is better if you don't swallow it. And can we legitimately claim that the jack-in-the-box works to "promote innovation," given that playing with this toy amounts to mindlessly cranking it up, scaring the shit out of yourself, and cranking it up again, over and over? It even took the play "experts" until 2009 to admit the ball and until 2013 to open their doors to the rubber duck, thus finally ending their ill-advised stonewalling of two of the most universally beloved toys of all time.
Then there's the problem of the still-rejected toys. I note that after the embattled first year of the National Toy Hall of Fame's existence, when every nut who could click a mouse raised hell that their favorite toy had been left out in the cold, the panel of wise toy experts responded in year two by rejecting both the soccer ball and baseball glove, thus ensuring that they would piss off every person on earth. As with the Raggedy Ann standoff, adult obsessions were at the heart of these debates. For example, after being judged unfit for service in the Hall of Fame for several years running, GI Joe went commando and was carried into the hall in 2004 on a testosterone-driven groundswell of support from advocates whose appeals sounded as if they had been excerpted from speeches by General Patton. Gender politics were equally transparent in the induction the following year of the Easy-Bake Oven, which, though reviled by feminists as a symbol of the subjugation of women within a hegemonic patriarchal system of exploitative domestic servitude, was celebrated by other women as " _really_ cute." The museum capped off its long run of poor choices in 2014, when it rejected more than a hundred worthy nominees—including such truly pleasurable toys as the Slip'n Slide—in order to induct the Rubik's cube, which is not a toy but rather a heinous torture device intentionally designed to fatally deprive innocent children of their self-esteem.
I ultimately decided that to settle the troubling matter of the famous stick I would have to consult a real play expert. Hannah seemed to me the right choice. She's thoughtful, asks good questions, and doesn't jump to conclusions about anything other than the need to take care of her little sister and eat ice cream immediately. She has informed opinions about things she has experience with, and clearly she has experience playing. One morning, while driving the girls to school, I told Hannah all about the Strong National Museum of Play, and the National Toy Hall of Fame contained within it, and about the stick. She listened carefully, raising her eyebrows a few times.
"Who are the kids who get to decide which toys are allowed to be in the Hall of Fame?" she asked.
"They aren't kids," I explained. "They're all grown-ups." "That's weird," she said. Little Caroline nodded in agreement. "Kids have a lot more practice playing. Why don't they ask kids?" I told her I didn't know. Hannah said she could understand why somebody might think of a stick as a toy, since kids could use sticks to . . . and then she breathlessly listed about fifty uses of the stick that had never occurred to Curator Man: a bridge for an ant to walk across, a hole poker for making secret caves, a fencing foil for sister Caroline, a key to a magic ice castle, a cloud scratcher. "Yeah, a cloud scratcher!" Caroline repeated enthusiastically.
Next, Hannah wanted to know how the grown-ups decide what's a toy and what isn't. "If a stick is in there, how about a whole tree, which is better, because most kids love to climb trees. Can that be in there?" I told her I didn't know. Hannah has always loved learning the names of flowers and trees, and so she also wanted to know what kind of stick it was. Was it a stick from a Utah juniper, or a Jeffrey pine, or maybe a Fremont cottonwood?
"Nobody ever said what kind of stick it was," I replied. Now Hannah frowned in earnest. Caroline followed suit, shaking her head side to side as if perfectly disgusted.
"They put it in a museum without even learning its name?" Caroline asked, incredulous.
I was nonplussed by how quickly the girls' simple questions were exploding the pretensions of the National Toy Hall of Fame, and I was quietly embarrassed that their best questions had never occurred to me. But Hannah's next question was especially provocative.
"When kids visit this Hall of Fame, can they _play_ with the stick?"
I paused before replying.
"No. They can't. The stick is in a display case on a wall in the museum."
" _Really?_ " she asked, her voice alive with genuine surprise. "Why do they call it a Museum of Play if you can't play with the stuff there? Maybe they could make the case with a lid, so you could just get the stick out. Or maybe they could have lots of sticks, so if me and Caroline and a bunch of other kids showed up we could all have a stick to play with. Why don't they do something like that?" Caroline nodded her assent as I once again told Hannah that I just didn't know.
Hannah's critique reminded me of the debate we've been having about modern art since the early twentieth century. Does the display of an object—an African mask, a bicycle wheel, an antique milk jug—deprive that object of its life? When we put a vernacular object in a museum and declare it "art," are we celebrating the meanings of that object, or are we impoverishing our understanding and enjoyment of it? And what becomes of the stick's status as a natural object once we define and limit its use? Is a stick in a case just another elephant in a zoo, another butterfly with a pin through it, yet one more grown-up way of attempting to domesticate the wildness that is inherent to natural play and to the children who benefit from it? Is a stick on display in a museum even a stick at all?
Hannah was still thinking hard, and she sat quietly for a while before reaching her conclusion. "Dad, since the stick isn't made by people, it really is different than a Hula-Hoop and stuff like that. And I think natural things that belong together should stay together, so if the stick is in there, then it isn't fair not to put in the whole tree, plus leaves, and rocks, and everything else around it."
"And bugs too," added Caroline, "but it wouldn't be nice to keep bugs inside like that."
"Right," Hannah agreed. "I think they ought to just leave the stick outside. That way it can be in the wind and rain, which it's used to, and bugs can use it to crawl on, and also kids can play with it."
I'm aware that we've been waxing rhapsodic about the wisdom of children since the romantic poets tromped euphorically around the Lake District (without children, I might add), but this struck me as a sensible verdict, rendered by a thoughtful judge and based on a sound interrogation of the facts. Caroline's energetic nodding in support of her sister's argument suggested that even a little kid could grasp the problem Hannah's logic had exposed. We grown-ups had insisted upon turning the stick into everything from a three wood to a bazooka, but the girls' imaginations had effortlessly, magically transformed it back into a stick.
I suppose we could say that adults crave play, too, and that playing with the famous stick's meanings is the grown-up way of trying to think up something as cool as using a stick as a cloud scratcher. By eliciting the two most powerful forms of nostalgia—the loss of nature and the loss of childhood—the celebrated stick had captured our adult imaginations. But while we were arguing over its meaning, turning its induction into the National Toy Hall of Fame into a cause for celebration or complaint, we also forgot to go outside and play. And I suspect that it is this failure to play—this atrophying of the ability to imaginatively engage nature and then also leave it as we found it—that somehow separates us from our childhoods and perhaps also from our children. Maybe we've been grasping at the stick because we need to recover something that we dropped on the ground a long time ago.
_Chapter 8_
8. Freebirds
One afternoon a few days before Thanksgiving, Eryn came home from town with Hannah and Caroline, both of whom had been administered the traditional Thanksgiving myth at school that day. Three-year-old Caroline was as proud as could be of her paper turkey, made from a cutout drawing of her own little hand. And Hannah, the loquacious six-year-old, began blurting out her holiday lecture the moment she came through the door: "Dad, I bet you didn't know that Thanksgiving comes from the Pilgrims and the Indians helping each other a bunch and then having a peace party and eating a really big supper with crazy colored corn and turkeys and those turkeys were _wild_!" With this she donned her construction-paper Pilgrim hat with its big fake buckle and gave me a huge smile.
I took a long sip of my whiskey and tried to formulate a response. The Thanksgiving feast the girls had learned about did in fact occur—at Plymouth Plantation in 1621—but by the following year violent conflict between colonists and Native Americans had erupted, and devastating Indian wars soon swept New England. There weren't many turkeys shared at Mystic River in 1637, for example, when the Pilgrims burned and hacked to death at least four hundred Pequot, mostly women and children, as they slept. The Pilgrim leader William Bradford—who had actually been present at that much-celebrated first Thanksgiving—had this to say about the slaughter: "It was a fearful sight to see [the Indians] thus frying in the fire and the streams of blood quenching the same, and horrible was the stink and scent thereof; but the victory seemed a sweet sacrifice, and [we] gave the praise thereof to God." Just as I was pondering how best to explain this act of violence in a way that might somehow be compatible with the ennobling concept of Thanksgiving the girls had learned at school, Hannah pointed excitedly at the muted TV behind me and shouted, "It's turkey time!" I turned to see that the news had given way to the image of a large white turkey. A turkey at the White House, in fact. A turkey that was about to receive a formal pardon from the president of the United States.
For many people, Thanksgiving is about bringing together family and friends; for some, it is centered around the ancient autumnal harvest festival; for others, it is an opportunity to count and express our most precious blessings; for yet others, it is a holiday devoted to copious amounts of football and alcohol. I believe deeply in all these versions, but for me Thanksgiving is very much about the pardoning of turkeys.
The tradition of the presidential turkey pardon is wonderfully rife with distortion, ambiguity, and error—as all good stories should be—but what is most perplexing about this bizarre ritual is our uncertainty about its origins. Some claim that the turkey pardon began with President Lincoln, who, hoping to promote national unity amid the social fragmentation of the Civil War, did in fact declare our first official day of national thanksgiving in 1863. That same year Lincoln's ten-year-old son, Tad, so the story goes, became so attached to a Christmas turkey that the president relented and agreed to spare "Jack" from the family table. More common is the claim that Harry Truman was the first president to save a turkey, but while Truman was indeed the first commander in chief to receive a holiday gift bird from the National Turkey Federation—a custom begun in 1947 and continued to this day—the evidence suggests that Truman, like most presidents who followed him, hadn't the slightest compunction about eating his gift. It was President Kennedy who first broke with his predecessors by declaring, just four days before he was assassinated, that—despite the sign reading GOOD EATING that the National Turkey Federation had hung around the bird's neck—he would let his fifty-five-pound gobbler live.
Even though President Reagan delivered a few respectable one-liners about sparing his turkey (he was every bit as charismatic with his bird as he was with that cute chimp in the movie _Bedtime for Bonzo_ ), the Gipper promptly gobbled up all of his gobblers. And it is here that bird-pardoning lore moves from speculation to historical fact, for in 1989 George Herbert Walker Bush had the honor of becoming the first president to formally issue a pardon to a turkey—an innovative leadership move that no doubt helped to secure his legacy. Since Bush Senior, every president has participated annually in this strange ritual—which held special pleasure for Presidents Bill Clinton and George W. Bush, each of whom embraced the event as an occasion for the kind of political theater that offered welcome distraction from the kind of political theater that occupied them at all other times.
It seemed to me that President Clinton always shot his birds an amorous look while pardoning them, and his uncharacteristic restraint in looking but not touching may have been indirectly attributable to our old friend William Bradford, who, in his seventeenth-century page turner _Of Plymouth Plantation,_ carefully documented the execution of one of his fellow Pilgrims for the unpardonable sin of sodomizing a turkey. Bradford's troubling account leaves me with three questions: Can there be anything more disgusting than having sex with poultry? How, exactly, would you go about doing it anyway? Is this really something you ought to kill a guy for? I think it would have been more humane, more punishing, and also more entertaining to simply make fun of him for the rest of his life. It wouldn't take much—you could just gobble a little under your breath as he passed your pew in church. Of course, none of the Pilgrims' distasteful Indian killing or turkey raping stopped President George W. Bush from executing a 2007 pardon for "May" and "Flower," birds whose names offered a clear allusion to Bradford's intrepid congregation.
The tradition of the presidential turkey pardon has continued to evolve in surprising ways. In the early years the exonerated gobblers were sent to Kidwell Farm, a petting zoo in northern Virginia where, as turkey rock stars, they lived a life featuring excessive drug use and constant media attention, but only the brief fame their overbred and steroid-addled condition would allow. Since 2005, however, the ritual has become more surreal: the pardoned bird is now immediately flown to Disneyland or Disney World, where it serves as grand marshal of the Thanksgiving Day parade at the self-proclaimed "Happiest Place on Earth." And if the idea of Americans spending their Thanksgiving holiday at a theme park watching a fat bird lead a Mickey Mouse parade seems depressing, it is encouraging to note that the birds are flown to their new posts first class, so while in transit they enjoy comfortably wide seats and a lot of free gin and tonics. It beats the hell out of that cramped poultry yard with its hormone-dusted cracked corn, and since the birds are so overbred as to find it barely possible to waddle (pardon the pun), much less fly, their trip to the Happiest Place on Earth is in fact the only flight they will ever know.
Soon after he took office, President Obama recognized the surreal quality of the ceremony when he remarked, "There are certain days that remind me of why I ran for this office. And then there are moments like this, where I pardon a turkey and send it to Disneyland." Now that the pardoned bird is a national celebrity, it has become necessary to pardon an alternate bird each year, in case the National Turkey is unable to fulfill its duties—as occurred in 2008, when "Pecan" fell suddenly ill and required his understudy, "Pumpkin," to receive the honors.
In our family it is a hallowed tradition—one as sacred and as ceremoniously performed as cheering on the opening day of baseball season—to witness and celebrate the annual presidential pardoning of the turkeys. Although Hannah and Caroline are necessarily recent participants in this annual custom, I consider myself the Cal Ripken of turkey pardoning, having never missed one since the initiation of the ritual more than twenty years ago. As is the case with other Thanksgiving traditions, I find it helpful to drink while participating in this one, so I annually toast the birds' reprieve with stout tumblers of what I call _Meleagris gallopavo_ cocktail, which is Wild Turkey straight up, the "cocktail" mixed in only as an avian pun to sweeten the experience of watching the president talk turkey. After all, nothing is more threatening to one's mental health than to be caught uncomfortably sober when it comes time for the leader of the free world to issue a televised and legally binding pardon to a bird.
Although I have long found the pardoning of the turkeys to be among the more entertaining things to transpire in our nation's capital each year, the levity of this ritual has become compromised by politics. In particular, the vegetarian lobby has complained that the annual pardoning amounts to free advertising for the poultry industry, and it has suggested that the president would set a better example by accepting a cruelty-free Tofurky, whose life before being pressed into a shimmering loaf of gelatinous curd presumably consisted of cavorting innocently through fragrant bean fields while in absolutely no danger of being sodomized. The Humane Society has also objected, making the hard-to-dispute point that turkeys produced by industrial poultry farming have about as unpleasant a life as one can imagine, and while two birds do get to fly first class to Anaheim or Orlando each year, 250 million others aren't so lucky. Each year following the pardoning, PETA is served a generous slice of free media pie when it describes in gory detail the miserable lives of these factory-farmed birds.
My objection to such complaints is certainly not that they are groundless—they must be at least as compelling as the idea that the leader of 300 million people should not waste his time, not to mention his political capital, pardoning a turkey—but rather that they are unpardonably lacking in humor. It is not, after all, a Supreme Court deliberation we are talking about, but rather a turkey pardoning. So here, perhaps, is a useful rule of thumb for animal rights activists: if the president and a turkey are more entertaining than you are, it can hardly be surprising that your client is headed for decapitation. With a little creativity, such activists might dramatize their objections in ways that would be more in the spirit of the event. How about staging a parody of the turkey pardoning in which a PETA activist, costumed as a giant turkey, pardons the sitting president for _his_ misdeeds?
The potential for humor here is also suggested by the comic irony of an actual event involving none other than the media superstar and former governor of Alaska, Sarah Palin. Back in the days before her Fox News stint and failed reality shows and her rewriting of the history of the American Revolution, the governor, having just pardoned a turkey (yes, many state governors also participate in this tomfoolery), waxed rhapsodic on camera about the virtues of compassion and forgiveness, while unbeknownst to her a worker in the background was busy decapitating and bleeding out turkeys. The YouTube video of this interview, which is far funnier than any _Saturday Night Live_ send-up of it could possibly be, has been watched more than a million times.
If I were to object to the turkey pardoning—which, of course, I haven't the slightest intention of doing—I would do so on the grounds that to render a turkey a fit subject for pardon, we must presume the bird's guilt. To be pardoned, one must first be in violation of some communal law or code. While enjoying Thanksgiving dinner, for example, we don't "beg your pardon" unless we belch, fart, or otherwise violate the community ethic by which the ritual meal is conducted. A pardon is both an expression of mercy and a certificate of absolution; it is both amnesty and exoneration. To pardon, after all, is to forgive. And if we're talking about a turkey, it becomes difficult to discern what criminal or immoral behavior on the bird's part may be said to establish the necessary preconditions for its forgiveness.
Now if Benjamin Franklin had won the argument and the turkey had become our national symbol, the case might be different. You may recall that wise old Ben claimed that the bald eagle made a poor national symbol because "he is a bird of bad moral character." This, incidentally, from a man who advocated choosing for a mistress an older woman because "there is no hazard of children, which irregularly produced may be attended with much inconvenience"; who invented bifocals so he might focus on prostitutes both up close and from slightly farther away; and who is credited with proffering the timeless verity that "beer is proof that God loves us and wants us to be happy." So much for moral character. Unfortunately, Ben's failed lobbying prevented the fat gobbler from making it onto the presidential seal, though I do still like to imagine a plump tom turkey with an olive branch in one scrabbly claw and a sheaf of gleaming arrows in the other, as if to say: "I'm a jolly, peaceful old bird, but _don't fuck with me_." I would argue that, given its failure to achieve the status of national icon, the turkey—being innocent of everything save its cowardly squandering of the rare opportunity to peck viciously at the president of the United States—cannot in fact be legally pardoned. And if the annual pardon is both presidentially sanctioned and demonstrably illegal, then it is also necessarily unconstitutional and therefore constitutes legitimate grounds for impeachment.
My point here is not that a US president should be impeached for pardoning a turkey—though I won't stand in the way if that's how it ultimately goes down—but rather that we might benefit from asking what human vanity or lust for power inspired the presumption that we _could_ pardon a bird. On his final day in office, Bill Clinton pardoned 140 people, including a few whose deeds might lead you to conclude, by comparison, that even the turkey sodomizer wasn't such a bad guy. Although most industrialized democracies on the planet have long since abolished capital punishment, more than two-thirds of US states continue to respond to violent crime with the awkwardly violent response of sending people to the death chamber—not to mention the excruciating frequency with which these killings are not even executed properly. And we still aren't done quibbling about what constitutes torture and whether our nation should sanction its use in extreme circumstances. But, as Thomas Jefferson well knew, it is the political expediency of those in power that defines the extremity of the circumstances. As Jefferson's buddy James Madison wisely observed, men are not angels.
I realize this is pretty heavy stuff to include in an essay about pardoning turkeys—and for that I hope I too may be pardoned—but the plain truth that we _are_ so flawed, so very far from being angels, is directly relevant to this story. It is we who burn the village, justify the torture, execute the criminal. How is it that we are so sure of ourselves, so certain about the infallibility of our judgment and authority? I wonder if there is some relationship between our presumption of power and this desire to pardon—even the desire to pardon an innocent, feathered, nonhuman being. I wonder if perhaps we have a vague sense that it is some guilt of our own that must be assuaged: that we, whose power is so often used to judge, might be redeemed by some corollary power to forgive, that exoneration might at the eleventh hour become the bright shadow of a looming condemnation.
Of course the National Turkey—which, for all we know, might wisely prefer death to Disney World in any case—doesn't require our mercy in the slightest. It is we who need the bird, desperately so, for through its salvation we are permitted to express our deep human desire to grant amnesty to those who would otherwise suffer. From where I sit it is difficult to determine whether the granting of a pardon constitutes an assertion of power or a relinquishment of it. But for allowing us a momentary, if symbolic, reprieve from our role as judge and executioner, we have ample reason to give thanks to these turkeys—so many thanks, in fact, that it probably is a good idea to be on the safe side and pardon one every now and then.
I pour another tumbler of bourbon and look again at Caroline's sweet little handprint turkey. Then I look at Hannah's beaming face, which so clearly registers her innocent excitement that President Obama—with his own two daughters by his side—has made it possible for these otherwise doomed gobblers to go free. I think about that mythic first Thanksgiving that we describe to our children, even as a long shadow of violence threatens constantly to reduce it to historical insignificance. I think of the presidential turkey pardoning being performed in a world so replete with greed and conflict, suffering and injustice. I think of the fact that the ratio of turkeys annually pardoned and given free gin and tonics to those raised under horrendous conditions and unceremoniously decapitated is approximately 1:125,000,000.
"Girls!" I suddenly hear myself exclaim. "This is the best day _ever_! The president has made sure that the turkeys will be set free, and now they get to fly in a plane to Disney World, and they even get to be the stars in the big parade! And today you girls have learned all about how Thanksgiving is a holiday of peace and forgiveness, and soon we'll have a wonderful Thanksgiving dinner of our own. This is truly a day to count our blessings!"
Eryn instantly furrows her brow, as if contemplating whether to take my Wild Turkey away. Then Caroline starts counting aloud, "One, two, free, four!" and Hannah claps her hands and chants, "The birds are free! The birds are free!" I glance at Eryn, who is looking at me as if I've once again started something she will have to finish. It is a difficult moment, I admit. And so I do the only thing I can. I do what I think any father would do under the circumstances. I set my whiskey down slowly. Then I begin jumping up and down, clapping and shouting along with Hannah, and then Caroline follows suit, and, at last, even Eryn joins in: "The birds are free! The birds are free! The birds are free!" Before our celebration reaches its breathless finish, we have segued from our avian freedom chant into "Turkey in the Straw," "Five Fat Turkeys Are We," and, for our big closer, "Free Bird." There is much jamming on air guitar, and when we finish singing we all stand panting, heads bowed, holding our imaginary lighters ceremoniously above our heads.
All too soon my daughters' veneration of the first Thanksgiving will give way to a painful awareness of the Mystic River massacre. In the meantime we will celebrate, not history, which is so often a monument to human failure, but rather myth, which is the necessary dream that a better future might redeem the errors of our past. Perhaps we each deserve a pardon. Maybe, whether we are doomed prisoner or executioner, we each need to receive that last-minute phone call in what would otherwise be our death chamber. We forgive the birds, and in so doing, we hope desperately that they might forgive us.
PART THREE
Humbling
_Two conditions—gravity and a livable temperature range between freezing and boiling—have given us fluids and flesh. The trees we climb and the ground we walk on have given us five fingers and toes. The "place" (from the root_ plat _, broad, spreading, flat) gave us far-seeing eyes, the streams and breezes gave us versatile tongues and whorly ears. The land gave us a stride, and the lake a dive. The amazement gave us our kind of mind. We should be thankful for that, and take nature's stricter lessons with some grace._
—GARY SNYDER, _The Practice of the Wild_
_Chapter 9_
9. Finding the Future Forest
I find it easy to praise the high desert, because it is a landscape so open, wild, and resistant to human inhabitation that it speaks simultaneously to my love of nature, my longing for challenge, and my desire to get in touch with my Inner Curmudgeon. But there are two weeks each July when my idealization of this arid land is tested by the scorching high-elevation sun, and I begin to wonder how anything besides leopard lizards and Great Basin rattlers could make a home in this withering heat. During this period I avoid reading anything I've ever written about the desert, because it invariably strikes me as fluffy pastoral crap. If Wordsworth were here now, he'd lay down his glowing pen and take up a shotgun and a fifth of tequila and head for the shade of a juniper. During these two weeks I have no choice but to relinquish my lyricism and admit the obvious: this place is a baking hell, and I was insane for building a house without air conditioning. This time of year it is easy to understand why the old miners who passed through this area named local landscape features things like Hell's Furnace Hill, Devil Canyon, and Inferno Ridge.
There is, however, summer consolation nearby. From my sweltering perch I can see the distant Sierra Nevada crest, and I don't need John Muir's ghost to tell me how cool and lovely it is up there right now. Fifty miles south of our desert home is one of the most beautiful alpine lakes on earth. Lake Tahoe is often described as a gem, a translucent blue jewel set among the snowcapped peaks that ring its spectacular basin. But this gemlike image of Tahoe as static and isolated disconnects the celebrated lake from the watershed that not only encircles it but also extends downstream from it into the desert below. It might be better to say of Tahoe, as Thoreau said of Walden Pond, that it is not a jewel but rather an eye: something deep, clear, reflective, but also intimately related to a larger living body through a complex arterial system. The Tahoe watershed is spectacular not only for the gem lake at its fountainhead but also for the life it brings to the western Great Basin, these beautiful, blazing landscapes where the snowmelt of the mountains becomes the lifeblood of the desert.
Five rivers drain eastward from the northern Sierra into the Great Basin. From south to north they are the Owens, Walker, Carson, Truckee, and Susan. Each terminates in a lake, marsh, or sink from which it never escapes. This is the origin of the term _Great Basin_ : we live on the extreme western edge of a vast desert from which no water ever reaches the sea. Of these five rivers, our home river is the Truckee, which winds 121 miles from Tahoe to its terminus in Pyramid Lake, the eye of the desert and the final home of that shimmering alpine water. Among the most magnificent desert terminus lakes in the world, Pyramid is a twenty-six-mile-long azure expanse cradled among the bare, glowing hills of the desert and surrounded by formations of tufa—towers of calcium carbonate that stand in surreal poses along the shores of the lake. Pyramid is as ancient as it is beautiful; it is one of the few surviving remnants of Pleistocene-era Lake Lahontan, which at its height thirteen thousand years ago covered more than 8,000 square miles of what is now Nevada. Although Tahoe and Pyramid are in starkly different ecosystems and are separated by 2,400 vertical feet, they are linked by the flowing bridge of the Truckee River. They are sister lakes that share the same origin in Sierra snow and whose fates remain intimately intertwined.
Of the river systems along the eastern flank of the Sierra, none has escaped severe damage, and several are critically endangered. Beginning with the federal Reclamation Act in 1902, enterprising westerners tried to make the desert bloom by replumbing Sierra watersheds in the vain hope that the vast, arid expanses of the Great Basin might be converted through irrigation into a prosperous agricultural empire. The Newlands Reclamation Project resulted in the construction of Derby Dam, which diverted much of the Truckee's flow into the Carson River watershed, where it could be put to use in commercial agriculture instead of "going to waste" out in the desert. This is why Fallon, Nevada, which receives an average of five inches of rain each year, is now known for its trademark Hearts of Gold cantaloupes.
But while the Truckee was being diverted to grow cantaloupes in the desert, the magnificent terminal lakes it fed began to die. Starting in the early twentieth century, water levels at Pyramid and Winnemucca went into freefall, and by 1939 Winnemucca Lake had simply vanished. Once a twenty-eight-mile-long, tule-filled, wildlife-rich lake, today it is an immense stretch of bare alkali flat, a white ghost lake rippled only by wind and sand. Soon after the diversion of Truckee water the native Pyramid Lake Lahontan cutthroat trout—the largest cutthroat in the world, which weighed up to forty pounds and which at one time was shipped to mining camps at the rate of one million pounds per year—could no longer reach its spawning grounds and began to go extinct. So too the now-endangered Pyramid Lake cui-ui fish, whose critical importance to local Native American culture is reflected in the ancestral name of the Pyramid Lake Paiute people: _Kuyuidokado_ , or "cui-ui eaters." As the marshes and lakes of the region dried up and their fisheries collapsed, the great eastern Sierra flyway also began to collapse, with the result that the millions of migrating eagles, ibis, loons, dowitchers, plovers, phalaropes, and other migratory birds that depended on their waters were reduced to a trickle.
The Lahontan cutthroat, Nevada's state fish, provides an inspiring example of the fall and rise of some desert species over the past century. At Pyramid Lake, the triple threat of introduced non-native fish, overfishing, and plummeting water levels posed a severe threat to these fish. During the 1920s, anglers were landing monster cutthroat here, but by the 1930s this native fish had begun to disappear. Their last river spawn was in 1938, and less than a decade later the Lahontan cutthroat had vanished from Pyramid. The species was federally listed as endangered in 1970. The mistakes we made in managing the liquid gold that is the lifeline of this watershed had resulted in the loss of one of the most remarkable fish species in the West, and now the desert lake was teeming with swimming ghosts.
Thirty years after the cutthroat's disappearance we tried to correct our mistake by attempting an ambitious restoration effort—an example of the sort of creative rewilding upon which the environmental health of the Intermountain West will ultimately depend. During the 1970s, a transplanted population of Lahontan cutthroat was discovered in a remote stream up in the Pilot Mountains on the Nevada-Utah border. Using genetics from late-nineteenth-and early-twentieth-century museum mounts of the extirpated native trout, it was established that the Pilot Mountains population was indeed descended from the original Truckee Basin stock. A hatchery program was established here in northwestern Nevada, and genetically native Lahontan cutthroat trout were raised and, beginning in 2006, reintroduced to Pyramid Lake. In 2014 a long-awaited miracle occurred: the fish spawned in the Truckee River for the first time in seventy-six years. Fishermen are already catching twenty-pounders in the lake. Once vanished, the Lahontan cutthroat trout has become native here again.
"Keep Tahoe Blue" is the rallying cry of folks who care about the integrity of this alpine lake (though "Drink Tahoe Red," which celebrates an excellent local beer, also has a place in our sloganeering lexicon). "Keep Tahoe Blue," which we sometimes render "Mantenga Tahoe Azul," refers not so much to the color of the lake's water as to its clarity. Tahoe's clarity is measured using a Secchi disk, a black-and-white platter, ten inches in diameter, which is lowered down into the water until it finally becomes invisible. Back in 1968, when clarity measurements began, the disk was visible more than a hundred feet below the surface of the water—a remarkable depth, but one that has been decreasing ever since. By 1997 average visibility was down to a mere sixty-four feet, and since then a lot of folks have been working to bring that number back up. It isn't just that we want to be able to see kokanee salmon seventy-five feet down in the blue depths, though of course we do; we also realize that Tahoe's clarity is a prime indicator of the health of our entire watershed.
Despite setbacks, the Tahoe-Pyramid watershed has many advocates who've worked, with some notable successes, to restore these great alpine and desert lakes and the spectacular river corridor connecting them. Activists in the Tahoe Basin are educating the public about the value of minimizing runoff, erosion, and pollution, and ambitious restoration efforts in the Truckee's riparian zone have stabilized stream banks and improved fish spawning areas. One dramatic success story is Anaho Island in Pyramid Lake, which is now protected as a National Wildlife Refuge. Here breeding colonies of many species of gulls, terns, cormorants, herons, and egrets thrive. Anaho is also home to one of the West's largest colonies of American white pelicans, majestic birds that soar above the lake on snow-white wings that span an incredible ten feet. The astonishing sight of a gyre of immense pelicans wheeling in the desert sky is a graceful reminder that the creatures of the Great Basin are utterly dependent upon the Tahoe Sierra waters that nourish them.
Although I'm a confirmed desert rat and would rather camp out in the bright dust of a remote alkali flat than spend a weekend in a chalet overlooking Lake Tahoe, I do know which side my bread is buttered on. The health of the Tahoe Basin and its watershed is our lifeline; if the chalet dwellers don't send us enough water, or don't send it at the right time, or don't send it in decent condition, the ecosystem we inhabit is stressed—and the desert is a place where the margin of survival is razor thin to begin with. Because the closing of an upstream spigot can cause a magnificent desert lake to simply vanish, the stakes could hardly be higher. While I like cantaloupes as much as the next guy, lakes for 'lopes is not a trade-off I am prepared to make. This is why I ultimately decided that in order to help protect the desert I would have to start in the mountains. If forestry practices in the Tahoe Basin could be improved, so would the volume and quality of water we receive down here in the Great Basin. If we manage to "Keep Tahoe Blue," we'll also achieve something less visible but equally important: we'll keep the lifeblood of the desert flowing. If I wanted to see white pelicans in desert skies, I would have to start by working high above me, up in the alpine forests.
I'd been at home in other forests: first in the mixed deciduous woods of my southern Appalachian boyhood and then in the tangled cypress strands and shaded mangrove hammocks of the Everglades detour my life took before I swooned for the high desert. But the stately coniferous forest of the Tahoe Basin looked to me like someone else's woods, its picturesque slopes, open stands, and sculpted boulders too like an advertisement for property you couldn't afford, a ski lodge waiting to be built, a place that only existed for two weeks in August. These are the limitations of vision we carry with us on our scattered voyages through new landscapes. If it bothered me that people so often failed to appreciate the skeletal beauty of the desert, it was equally true that as a desert rat I had failed not only to appreciate the beauty of Tahoe but also to acknowledge my dependence upon it.
I had also been a forest activist in other places, and I had always been especially devoted to watchdogging logging roads—those arteries of pain invading the healthy body of wild country—and helping to ensure road closures and obliteration in the interest of forest health. As an admirer of poetry as well as of forests, my motto had been, in the spirit of Robert Frost, "Two roads diverged in a yellow wood, and I . . . got them both closed." Now it seemed time to get behind the postcard image and try to discover what might be missing from my own way of inhabiting the Truckee River watershed.
As a first step, I contacted a guy named Rich Kentz at the League to Save Lake Tahoe and volunteered to serve as a forest monitor up in the Tahoe Basin. Rich was a good guide and teacher as we took long hikes through forests that were slated to be logged. Like rafting a river just before it's dammed, we walked in the shadow of the tractor and the saw—a long, chilling shadow you could feel falling across something that had reached evening before its time. On some days monitoring felt like little more than measuring the waning life of something inexorably doomed; on others, though, the woods still felt as restorative and quiet and endless as they had seemed when I was Hannah's age.
The objective of the forest monitor is to compare the US Forest Service's plan for logging (the "prescription") to the ecological reality of the forest ("on the ground"). For example, a forest monitor might discover that a riparian zone (or SEZ: "stream environment zone") has not been adequately protected by a logging prescription or might note that the removal of large trees would be inconsistent with the objectives of the forest management plan. A monitor may express concern that tractor logging in certain areas will create excessive soil compaction or erosion, threatening critical wildlife habitat or endangering watershed health. In other cases, a monitor might note that the stated reason for logging an area—heavy fuel loading posing fire danger or die-off from insect kill, for example—isn't justified by observable conditions in the forest. The monitor then communicates concerns to the Forest Service (whose employees are often nicknamed "Freddies" by forest activists). If all goes well, the prescription for logging that tract will be adjusted toward a more sustainable approach to harvesting timber.
One summer afternoon I had just hiked out of a timber sale unit on Tahoe's North Shore with Rich. As we sat on the tailgate sweating, drinking Tahoe red, and looking out over the windy expanse of the sapphire lake, Rich offered a brief explanation of forest history in the basin. During the Comstock era of the mid-to late-nineteenth century, the basin's forests had been severely cut. Ancient ponderosa and sugar pine were the first to go, providing the beams and braces that supported many miles of mineshafts beneath then-booming Virginia City. Almost overnight, a towering ancient forest in the mountains became the skeleton of a labyrinth of subterranean catacombs in the desert. A century later, the clear-cut basin forests have grown up into same-age stands of trees, rather than the mixed-age stands that would have prevailed under an undisturbed forest regime. Within these unnatural same-age forests there is fierce competition for the vital resources of sunlight, water, and soil nutrients—competition that often triggers die-off, especially in thick stands of fir. At the same time, increased human inhabitation of the basin has prompted ambitious fire-suppression efforts, and while a lot of oversized vacation homes have been spared, the forest has become loaded with an unnaturally high concentration of dead and downed trees, which intensifies the risk of catastrophic wildfire—a risk that is exacerbated by drought and by the long-term effects of climate change.
In response to the fire danger posed by this heavy fuel loading, the Forest Service strategy has usually been to log these areas in order to thin dense forests—which are dense precisely because the natural fire cycle has been interrupted—and eliminate the ladder fuels by which a low-intensity ground fire might climb into the canopy and escalate to become a holocaust. Unfortunately, such logging has often proven to be environmentally destructive. The economic reality is that the trees most in need of removal to reduce the risk of stand-replacing fire—dead and downed trees, ladder fuels, jackstrawed ground fuels, and saplings and young trees, especially white fir—are those with the least economic value. In order to make logging pay, it is necessary to cut larger trees, green trees, and pines as well as firs, an approach that often results in a disturbed and less biodiverse forest of mostly younger trees. Add to this the fact that logging commonly causes soil compaction and erosion, destruction of wildlife habitat, and threats to water quality and that it sometimes leaves behind so much residual slash that fire danger is actually increased, and you can understand why concerned citizens worry that the solution is often worse than the problem.
When Rich finished explaining this environmentally destructive cycle of fuel loading, salvage logging, and habitat destruction, we began to wonder aloud what the forest would look like if the Forest Service could afford to manage these public lands according to their own stated objectives of watershed health, wildlife habitat, and fire control.
"To begin with," said Rich, "you wouldn't bring tractors in here. This unit was logged too recently. It's already taken a beating."
"And you wouldn't cut so many big trees," I added.
"Right, and you'd select for dominant species and thin for diversity," he continued. "Basically, you'd be removing ladder fuels and reducing ground fuels to mimic the effects of a low-intensity ground fire."
"I like to close roads," I added, handing Rich another ice-cold Tahoe red. "Could we obliterate a few skid trails while we're at it?"
"Why not?" he laughed. "Let's go ahead and restore those blown-out landing zones too."
Although the impossibility of the plan was painfully obvious to us both, we continued drinking and talking until we had fantasized a massive volunteer effort to hand-thin this part of the forest, sparing it from logging while also managing for key forest values, including wildlife habitat and fire control. It would be a model forest project combining activism and education—a collaboration between citizens and land management agencies that might result in a healthy forest, one that could also provide a baseline against which industrially logged parts of the basin could be compared.
"No problem," said Rich as he gazed out over the lake. "All we need is a workable plan, a team of field coordinators, a shitload of free equipment, an army of volunteers, the blessing of the Freddies, and some juicy media coverage."
That night I had a strange dream. I was standing in a primeval forest in which every tree was marked to be taken—a looming clear-cut with telltale orange blazes on every last trunk—when the markings slowly faded away, leaving a healthy forest of straight, clean, furrowed bark, standing tall and ready to catch the flakes of the first snow.
A tailgate is as good a place as any from which to begin the reformation of the world. Rich wrote an astute proposal, with which he and the League to Save Lake Tahoe succeeded in persuading the Forest Service to remove 240 acres from the planned North Shore timber sale. Instead, we would go in and care for the forest ourselves. From this urge to take care, the Tahoe Forest Stewardship project was born. Collaborating with the league and other groups, we worked out a treatment plan with the Forest Service, begged and borrowed the necessary equipment, recruited a group of team leaders, and had a series of preparatory meetings, usually held out on the forest unit we would be working. All this required a huge effort, but we were buoyed by the fact that so many people were willing to help. My favorite part of the project was working throughout the year with the other field coordinators, a diverse and talented group of foresters, activists, whistle-blowers, fly-fishers, smartasses, ski bums, and rednecks, many of whom combined these various identities to impressive effect.
One year from the tailgate, more than two hundred volunteers participated in the first Tahoe Forest Stewardship Day. Bird watchers, mountain bikers, Boy and Girl Scouts, ROTC teams, teachers, students, and scientists worked side by side to thin dense stands of saplings, eliminate ladder fuels, chip ground fuels, regrade landing zones, obliterate skid trails, and protect wildlife habitat. We had hydrologists, wildlife biologists, and fire ecologists on hand to help volunteers understand forest ecology and to add a crucial educational component to the event. Native Washoe elders also participated, telling their old stories and teaching our kids how to roast pine nuts and how to build a wickiup—an elegant, dome-shaped shelter constructed of arched willow saplings. Even the media coverage was uncharacteristically well informed. As we gathered at the end of the day for hot barbecue and cold beer, there was little left to wish for.
In the intervening thirteen years we have continued our cooperative forest protection work. Indeed, I achieved dubious fame among my fellow field coordinators the year I opted to run volunteer crews at the event rather than go on a honeymoon with my newlywed wife. My rationale, which Eryn agreed with, was simple: What the hell good is a great marriage if you don't have a healthy watershed to enjoy it in? The success of our initial effort resulted in an expansion of the project, which is now conducted annually at various sites around Lake Tahoe. The Forest Service, at first hesitant to work with us, is now enthusiastic about this project as a model of collaborative, community-based forest management. Sensing which way the wind is blowing, the Freddies often send a film crew to the event, not to mention a guy in a Smokey the Bear costume. Having been forced by my wife to occupy half of a horse costume at Caroline's last birthday party—the blazing hot ass half, needless to say—I'm always vigilant about slipping poor Smokey a few chilly brewskies. This act of generosity backfired last year when I discovered, too late, that the bear costume was being worn by a very sweet (and also by that time somewhat tipsy) thirteen-year-old Cub Scout. But even the incensed scoutmaster had to admit that at least my heart was in the right place.
For thirteen years I have continued to return to the patch of forest we worked in that first year of the project. Although, as today, I often bring Hannah and Caroline with me, the presence of those volunteers—by now around two thousand of them—is with me whenever I visit. Even at age seven, Hannah can appreciate what has been accomplished here; and while three-and-a-half-year-old Caroline may not yet get the big picture, she has a sense that this place is special to Dad, and she loves to come along whenever I visit. Thanks to the hard work of a lot of determined folks, this site has now become a dot on the map of what western American writer Wallace Stegner once called "the geography of hope." It is a place that has been given new layers of narrative, and for a change it isn't the same old story of cut-and-run exploitation. The chill of that long shadow is gone from this one place at least, and the forest here is healing itself and learning to hold water again. I'm especially glad that the girls have come along with me today. Their enjoyment of this place reminds me that our desire to care for the world is intuitive and spontaneous.
For Caroline it is a fun ride into the forest in her three-wheeled off-road stroller, though she climbs out pretty often to investigate pinecones and play in the forest duff. At lower elevations I'm struck by the glowing incense of cedars, their thick, aromatic bark reddish, twisted, and dotted with acorn-sized holes. Chartreuse moss adorns the north side of the trees in symmetrical rings. Hannah notices a pair of Steller's jays as they flash indigo through the lower branches of the surrounding trees. The trail steepens, and we pause to rest. I point out a noiseless arc of black and gray gliding into the crown of a big cedar below us: a Clark's nutcracker. Even the girls have a sense that this is a good place to be quiet. It occurs to me that silence, so commonly driven from our daily lives, is another balm that the forest produces, that it stores and holds, as it does water.
In following the trail's grade, we rise above the cedars and into the thick Jeffrey pine forest and then climb higher still, into mixed stands of large white fir and even larger red fir, a tree whose scientific name, _Abies magnifica,_ gives a fair idea of its grandeur. On the way to the former site of our volunteer work we pass the telltale signs of previous logging operations: skid trails that disappear like jagged scars into the forest; landing zones perhaps a decade old that remain lunar and nearly unvegetated; piles of slash that weren't worth loading; a scrapped skid cable, splayed out like intestines. But even the areas stripped to bare mineral soil have begun to show signs of life, with small sprigs of stunted lupine struggling to colonize this miniature wasteland in the heart of the forest.
As we summit a ridgeline, Hannah asks the meaning of what for me is a too-familiar yellow sign, nailed high into the bole of a big red fir.
"That marks the border of a timber sale," I reply.
"They're going to cut them down?" she asks. When I nod my reply, Hannah frowns but doesn't comment. Much of what we've walked through today will be auctioned and logged, even as the forest struggles to recover from having been cut over so recently. As we pause to rest, Caroline gathers pine needles into a pile, then places four cones inside it, like eggs in a nest.
When Caroline has finished playing with her nest, she climbs back into the stroller, and we hike ahead, past the yellow sign and into the unit that is not for sale—into the woods that are the home of our by now multigenerational forest project. A magical line is crossed as we step from a world that is up for sale into a world of trees that will be allowed to stand their ground. The forest looks, sounds, even smells different on this side of the invisible line, and you don't have to be an enviro-geek—or even a grown-up—to tell the difference between land that's been abused and land that's been left to do its natural work. Successive winters have piled snow over the old skid trails we worked so hard to repair, pressing organic material into contact with the ground, building soil, rewilding this place. Each spring this forest gets a little better at holding and filtering water rather than shooting it off the mountain in erosion gullies that carry living soils off to cloud the lake and silt the fish-filled river that runs, invisible, in the valley below. Goshawk, California spotted owl, and other native birds have begun to return to their familiar perches in the canopy. I show the girls the berry-spangled scat of a black bear, which now has a recovering home to return to. We all have a place worth returning to—a place that is healing and in healing can somehow make the world feel right again.
It is a slow process, this work of finding home, and slower still for a stubborn desert rat like me. I suppose we need time to see the wheels within wheels of our home ecosystems turn, to understand these landscapes through the slowly deposited layers of sensibility and perception. Bringing Hannah and Caroline to this place—and knowing that I've had the humbling privilege of helping to make it a place worth bringing them to—reminds me that the story of this forest is now braided with the desert strand of our family's life narrative. The rewilding project has helped me attach to this green place and to attach this place to my desert home below. Now when I gaze up to the Sierra crest during a lizard-blistering summer in the Great Basin I visualize not Lake Tahoe chalets but instead this refreshingly cool patch of forest, this fountainhead of my home watershed.
I hope to return to this place during some hot summer in the autumn of my life, when my girls are grown, and walk them through a forest that feels whole again. By then it may even appear as if our volunteers were never here at all, which would be a wonderful legacy of their hard work. By then the skid trails I helped to repair will also be invisible, and the telltale orange markings on the trunks will have slowly faded away, leaving a healthy forest of straight, clean, furrowed bark, standing tall and ready to catch the flakes of the first snow.
_Chapter 10_
10. My Children's First Garden
For several years I had been promising Hannah that we would plant a special garden together just as soon as little Caroline was old enough to be able to share the experience with us. Now that Hannah was eight years old and her little sister was four and a half, it seemed that the time was finally right. I had long wanted the girls to have a garden of their own at home. I wanted to share with them a sustainable practice of labor, meditation, and production that has been essential to my life and that I hoped would become important in their lives as well. I wanted them to experience the miracle of a seed germinating, have the satisfaction of seeing things they planted with their own little hands grow to flower and fruit. I wanted not so much to watch my children's garden grow as to watch them watch it grow and to see them grow with it. To quote Ken Kesey, I wanted to help Hannah and Caroline "plant a garden in which strange plants grow and mysteries bloom." And even if some of the strange plants in Kesey's garden were illegal, his admonition to cultivate mysteries speaks directly to the higher law that governs true gardening.
The garden is not only the place where we grow food but also the place where we plant hopes and nurture ideas, where many plans grow from each thought that is sown. The garden is also a metaphor for what is peaceful, harmonious, and productive in our lives, and even our language reveals how deeply our imaginations are rooted in the garden. We celebrate ideas that are _seminal_ , because so much can grow from the _seed_ of a single thought. When we deeply question our culture's values we are considered _radical_ , because we attempt to address problems at their _root_. We work hard to bring our projects to _fruition,_ for nothing is sweeter than the _fruits_ of our own labor. The garden is the imaginatively fertile ground where we harvest metaphors along with squash and beans.
Without realizing it at the time, in planning my children's first garden I had conflated parenting and gardening in my mind. After all, both are practices of love, attention, and creativity that result in healthy growth. Both require vigilance, nurturing, and care, and both yield sustaining harvests. Gardening and parenting are disciplines of sustainability, acts of faith in the future that must be renewed through daily practice. Of course this is the same sort of thinking employed by people who believe they are ready to have children because they managed to own a dog. Be that as it may, a gardener's universe is all sweetness and light, and the gardener's mind, like his or her garden, must remain an inviolable space that is impervious to the world's heartless logic. I had been a good gardener, and now I intended to use a garden to be a good father.
Even in my glowing optimism, I knew that my children's first garden might present a challenge. In this high-elevation desert we live amid conditions no gardener would relish: short growing season, extreme aridity, severe temperature fluctuations, desiccating winds, ravenous critters of every stripe. This is a place where even our human neighbors sometimes dry up, burn down, or blow away, which hardly bodes well for a pepper. Still, to a true gardener the unplanted garden is ever a canvas before paint, an idea not yet gone wrong, a sweet dream from which he or she has yet to awaken. As with any good idea, the trouble with a garden begins only when we attempt to realize what we have imagined.
As I spent the last nights of winter by the wood stove, sipping bourbon and strategizing to ensure the success of my children's first garden, I realized that the pressure was on. What if the garden I promised these sweet little girls didn't prosper? Given the metaphorical and philosophical significance of the garden, what larger, darker conclusions might they draw about their small place in the hard world if their own garden failed to come to fruition? I had to get this one right, and so I planned carefully. First, I would site the garden on the lee side of our big hill, where the hot wind would be more forgiving, and I'd stake and cage everything that grew upright to provide reinforcement against the scorching afternoon wind known locally as the Washoe Zephyr. I would also go to the trouble of constructing a large garden box out of railroad ties; this way I could backfill the raised bed with the primo double-mix soil I call "black gold" while also discouraging the desert cottontails and, especially, the big jackrabbits that can graze a bed to its roots in the time it takes to fetch an ale. To be safe, I'd wrap the whole garden in an impervious wire fence. In the interest of ensuring my daughters' success as neophyte gardeners, I would also set aside my environmentalist scruples and declare the plot a xeric-free zone where the girls could water to their heart's content. Finally, I would swallow my gardener's pride and grow only hardy, reliable plants, those that are fast growing, difficult to kill, and, if possible, spicy enough to be unpalatable to rodents. As winter waned, I came to think of my children's first garden as "the radish plot," as radishes have long been the mascot vegetable for amateur gardeners: fast, easy, cheap, unfailing, and plenty spicy. In a world of waiting and suffering, the lowly radish signifies short-term gratification and certain success.
When spring came, which it does mighty late at 6,000 feet, I built the garden just as I had designed it. Reminding myself of Rudyard Kipling's admonition that "gardens are not made by sitting in the shade," I worked all day in the glaring desert sun, transforming a bare plot of weedy caliche hardpan into my children's first garden. After the mattock, shovel, rake, spading fork, and field hammer were put up, I grabbed a trowel and hand cultivator and called Hannah and Caroline out to plant their garden. It was a momentous occasion, and I insisted that Eryn photograph this moment for posterity. I still treasure those pictures. In one, Hannah is deep-setting her first tomato plant with all her might; in another, Caroline is broadcasting radish seeds from her little green-gloved hand; in a third the sisters stand together, smiling broadly from behind the frame I built to trellis their beans. Throughout this photo op, though, Eryn's face wore a look of bemusement that suggested pleasure tinged with concern.
"Hey, what's the matter?" I asked as I stepped away from the garden, leaving Hannah and Caroline to water in their new plants. "Isn't this great?"
"I don't know, Bubba, they're pretty into this. Are you sure this is going to work?"
I shot a glance over my shoulder at the girls in their garden, then turned back to Eryn with a look of wounded pride.
"Come on, honey, I've been gardening my whole life. Besides, I've got this nailed. Look at what we're growing," I said, lowering my voice to a whisper. "I've stooped to radishes, for crying out loud. This is foolproof."
Only a fool believes that anything in this broken world is foolproof, and so the proliferation of things called foolproof is proof only that the world has fools and plenty. But the troubles in Caroline and Hannah's garden didn't start immediately. There was a golden two-week honeymoon during which seeds germinated, plants grew, and we gathered around the heavily armored plot to witness our little garden prevailing, even against daunting desert odds. We had planted tomatoes, squash, beans, sweet and hot peppers, collards, basil, and, of course, plenty of radishes. It was an unambitious, low-risk starter plot—a prosperous, shining little garden upon a hill that would make the desert bloom according to promises of old. We showered our plants with well water and attention and screened them with burlap to protect them from the scorching wind. My fence kept out the cottontails and big jackrabbits, just as I had hoped. As our plants grew we also took more pictures, proudly documenting for posterity the triumph of two little girls' first efforts at gardening.
Despite Bobby Burns's admonition that "the best laid schemes o' mice an' men gang aft agley," around our place the mice have aft done extremely well, while it is only my own schemes that have gang very, very damned agley. One aspect of my garden fortress scheme that was not well laid was my failure to foresee that slack spots along the base of my hardware cloth barricade would be virtual highways to the field mouse, which has no difficulty squeezing through a gap less than a half inch in diameter. So the "wee, sleekit, cow'rin, tim'rous beastie" got some seeds, as did the long-tailed, chisel-toothed, hopping kangaroo rats ( _Dipodomys microps_ ) that abound here, foraging nocturnally for just such simple treasures as radish seeds. But this was sustainable damage, and I explained to Caroline that we were just being nice by sharing with our nonhuman neighbors. As an added form of insurance, however, I secretly overseeded the plot, and I tightened the fence carefully in hopes of mending the breaches. Within hours of fixing the fence, though, I discovered a mixed flock of bushtits, chickadees, and sparrows tearing up the seedbeds, and so I rushed to string in a half-dozen aluminum pie tins, which whipped in the Washoe Zephyr and made enough racket to keep out the hungry birds.
The next morning we were happy to find the seedbeds undisturbed, but just as I began to relax into a feeling of satisfaction I noticed that the squash plants had been shorn to the ground. After the time-consuming process of unwrapping various layers of fence, I was at last able to get close enough to the plants to examine them closely. Judging from the size of the lacerated stems and the clean, sharp angle of their cuts, I was certain that bushy-tailed pack rats had been the culprits.
Hope is the coin of the gardener's realm, but it is a devalued currency, worth increasingly less as it is traded against the incessantly rising tender of the gardener's despair. It was several days before I made it to town to get plastic bird netting—which I now needed to keep out not birds but pack rats—and by then I was forced to buy replacement seed and bedding plants for the garden, which had already been substantially ravaged by the birds, mice, kangaroo rats, and pack rats. Indeed, the all-you-can-eat salad bar that was once my children's first garden was now so heavily frequented by rodents that the local great horned owl took up a hunting perch on the peak of our roof. While the final stages of the garden's destruction proceeded apace despite the owl's nightly vigil, I now had the additional chore of cleaning from the sidewalk many overlapping bursts of snow-white owl crap, occasionally accompanied by owl "pellets"—a delicate euphemism for a compact, puked-up wad of undigested fur, bones, teeth, and feathers.
During this time the girls continued to water what was left of their garden, and they had a great time playing together in the mud—even as I was constantly humiliated by having to explain to their mother why I, the seasoned gardener, was finding it so difficult to grow a few radishes. The ripe fruit of my gardener's pride began to wither on the blasted vine of my children's first garden, and while I was relieved that Hannah and her little sister seemed unfazed by what Eryn had begun to call "the Vegpocalypse," I had no way of knowing how long this grace period might continue. Grace, like good gardening weather, is welcome when it comes, but you're in deep, uncomposted manure if your success depends upon it. At some point, I reckoned, this garden must produce.
Returning from town one day, despondent but with plenty of bird netting and extra-hoppy IPA, I once again unwrapped the various layers of artillery separating the little garden from the cruel world, and we started over. I dug out the root balls of the clear-cut squash and removed the hard stems of the limbless tomatoes while Hannah and Caroline gathered the scattered leaves of the savaged sweet peppers. We retilled and replanted my children's first garden from end to end, and while I was pretty surly about it, the girls remained in great spirits the entire time. Caroline even told me innocently that she wished we could start the garden over "every single day!" When our replanting was done, the girls went in to wash up while I spent another hour reinforcing my defense systems, concluding with the double layer of plastic netting that I draped over the top of the fortress and snapped down with bright yellow bungee cords. Then I called the girls and sat down in my old aluminum lawn chair, with Hannah at my feet and Caroline on one knee, and stared at my children's second first garden.
I stared, and yet I could see nothing that we had planted. With all the railroad ties, posts, trellises, stakes, tomato cages, hardware cloth fences, burlap windbreaks, shimmering pie tins, panels of bird netting, and neon bungee cords, it was now virtually impossible to see any living thing through the overlapping walls of the garden. I could in fact see no garden at all, but only a hideous monument to my own determination to establish a barrier where my brute neighbors would have none. My children's second first garden looked like a miniature factory, or prison, or maybe the tangled bones of a steel barn after it meets with the tornado's funnel. What the hell kind of garden was that, especially for a kid? "Something there is that doesn't love a wall, that wants it down," wrote Robert Frost in his poem "Mending Wall." I had instead abided by the maxim Frost's neighbor endorses in the poem: "Good fences make good neighbors." If the furry and feathered would just stay on their side of the fence, all would be well. It was a simple concept, and I was exasperated by the lengths I was driven to in order to make sure my kids would have the chance to pull a few radishes.
Discouraged as I was, as a gardener I had been toughened by disappointment as well as by wind and sun. Even the Vegpocalypse, I reasoned, wasn't worse than some other gardening catastrophes I had endured, and as I sat meditating on my failure I was reminded of the heartening words of Thomas Jefferson, one of our most philosophical gardeners. In the garden, wrote Mr. Jefferson, "the failure of one thing is repaired by the success of another." Perhaps, as T. J. suggested, there would be some yet hidden consolation to compensate for these losses. Just at that moment, however, something tragic and unexpected happened—something that fatally radicalized my approach to gardening in the high desert. As I sat in my bent lawn chair, with Hannah leaning against my shoulder, Caroline sitting in my lap, and one half of my ass poking down through the ripped canvas seat, two tiny white-tailed antelope ground squirrels skittered across the rocky patch of dirt that is our "yard," raced beneath us, and ran over my right foot in their gleeful sprint to get to our newly replanted garden. There they disappeared momentarily behind the railroad ties forming the garden box before popping up in the garden itself, wasting no time in starting to crop our squash.
All of this happened in a matter of seconds, but in that moment I felt the swell of failure curl over me in a slow-breaking wave of despair. My gardener's hubris—that profound, unsubstantiated delusion of superiority without which no gardener could long endure his or her trials—had bitten me on the half of my ass that was exposed to the cruel world. In allowing myself to believe that the replanted and rearmored garden was impenetrable, I had foolishly ignored one of the basic principles of fatherhood, a principle so fundamental that it belongs with such core insights as "The only people who like change are babies in diapers" or "When your kid tells you to cover your eyes, do it with one hand and cover your nuts with the other." Humility is the alpha and omega of both parenting and gardening. I had failed to be humble, and in failing to be humble I was now humbled by failure.
In mute frustration I now reckoned that my daughters' ultimate happiness—their futures as good gardeners, great women, excellent human beings—was staked on the uncertain success of the little garden these white-tailed beasts were devouring. And, damn it, they had stepped on my foot. Both the gardener and the father in me now felt called to battle. As I sat with my jaw locked, watching my children's second first garden being destroyed, I could feel that I was about to cross an invisible line—that extraordinary and perhaps violent measures had become necessary. For their part, however, the girls greeted this new problem with screeches of joy. "Daddy! Look at the cute little chipmunks in our garden!" exclaimed Hannah, pointing excitedly toward the ravaged plot. "Hello, friends!" added Caroline, as she held her hands to her face and wriggled her fingers to mimic whiskers.
As the cracked stubs of what were once our plants baked in the heat of a sun whose azimuth had grown troublingly higher in recent weeks, I began a calculated assault on the "chipmunks" that had now become more numerous than flies around our place. The white-tailed antelope squirrel ( _Ammospermophilus leucurus_ ) is a remarkable creature, one whose adaptations to the grueling excesses of the desert environment can only have been shaped by a zillion evolutionary near misses, each one resulting in a crunchy snack for a harrier or coyote. Until you've been in the desert awhile, the antelope squirrel really does look like a chipmunk, though the differences are many and extraordinary. For starters, optimal habitat for the antelope squirrel is sparse juniper, sagebrush steppe, and desert scrub; unlike the chipmunk, it is never seen in the alpine forest. The white-tailed antelope—the species of _Ammospermophilus_ most widely distributed throughout the arid West and the one that was chowing down on my kids' second first garden—sports a narrow white stripe from shoulder to rump, has white cheeks, and carries none of those unsightly chipmunk stripes on its face. Antelopes have thin, coarse fur that is light brown with a gray or, as in our local population, reddish tint. The infallible means to distinguish them from chipmunks, though, is by their amazing ability to run with their tail pulled up over their bodies, a feat your average chipmunk could only dream of attempting. The underside of that raised tail flashes as white as the rump of a pronghorn in flight, an unmistakable visual correspondence that is the source of the antelope squirrel's odd name.
The antelope lives up to the name "squirrel" perfectly. _Squirrel_ comes from the Vulgar Latin _scuriolus,_ which is a variant of the Latin _scurius,_ which comes from the Greek _skiouros,_ which gives this family its name, Sciuridae, and which is almost certainly the source of the word _scurry_. Antelope squirrels are so small—four ounces or so—that you could mail one across the country for about a buck, but they scurry with such inconceivable swiftness as to make sprinting chipmunks look like they are shuffling from couch to cupboard to get a bag of Cheetos during halftime. Once running, the antelope squirrel—unlike its namesake, the pronghorn—never stops or looks back, instead flying across the earth like a white-flagged shot that vanishes into the ground at hyperspeed.
Unlike chipmunks and even other kinds of ground squirrels, the antelope is active throughout the year, which frees it from the need to hoard food in preparation for hibernation. It is not particularly territorial and in fact will huddle with others of its kind to conserve heat during the winter, so you can't count on them to off one another the way an aggressive, territorial rodent like a pack rat might. They excavate tunnels with multiple entrances, have several burrows in their large home range, construct separate tunnels for food caches and quick escapes, and often appropriate the burrow systems of kangaroo rats, all of which makes them virtually impossible to locate underground. Somewhere in one of those many tunnels the antelope will make a nest six inches in diameter, which it has likely lined with the golden strands set adrift when Eryn brushes the girls' hair outside on sunny mornings. You can't easily starve an antelope squirrel, as they are generalist omnivores that will eat almost anything, from seeds, plants, roots, grasses, and fruits to insects, lizards, and carrion, including the flesh of fellow rodents. Virtually invulnerable to dehydration, they need no free water to survive, instead deriving all their hydration from the plants and bugs they eat. Their kidneys are so highly efficient in saving water by reducing nitrogenous waste that if we were as well designed we could go several years without needing a replacement roll of toilet paper. Around here the little antelopes usually mate in late February and are born around Hannah's birthday in early April—a reproductive cycle that is precisely calibrated to the emergence of leafy annual plants here in the high desert.
What is most amazing about the antelope ground squirrel is its astonishing ability to keep itself cool in the brutal desert heat. Unlike the nocturnal foragers whose lifestyle seems more compatible with this scorching environment, the antelope is diurnal, cruising around above ground even on the hottest days. It has a number of tricks that make this marvelous feat possible. Its water conservation strategies help, of course, as does its habit of keeping its back to the sun and shading its body with its tail and its ability to climb into bushes to catch a breeze. Most important, the antelope doesn't cool itself evaporatively, as humans and most other mammals do, so it isn't water stressed by cooling. For this little squirrel, keeping cool is literally no sweat. Instead, it is so well adapted to the desert environment that it can allow its body temperature to rise to an incredible 110 degrees Fahrenheit—a thermal load that would cause brain damage and likely death in an adult human. When it begins to overheat, it returns to its burrow, splays its legs, drops its sparsely furred belly against the earth, and lets the ground pull the heat right out of its body—after which it pops up again and resumes razing our garden. If they happen to be well hydrated when they heat up, they will also use the neat trick of rubbing saliva on their faces to cool themselves, a behavior that makes me suspect Caroline may in fact be a ground squirrel.
The antelope squirrel's adaptive strategies are so many and so effective as to give it a daunting home field advantage, and as a beer-guzzling mammalian biped with few effective adaptations to desert life I do not feel myself a worthy opponent. After driving to town to buy several live traps smaller than those used to nab pack rats, a backup case of IPA, and a large jar of crunchy peanut butter (which I labeled NOT FOR KIDS), I began my attempt to capture the cute little antelope squirrels. Following some misadventures in which I trapped and released a piñon jay, a kangaroo rat, a bunny-sized desert cottontail, and my own fingers (twice), I did at last discover that the wee squirrels like Uncle Crunkle's Old-Fashioned Peanut Butter at least as well as they like squash stems. During the next two weeks I captured sixteen antelopes—many of which I released on public lands far from my children's garden (some studies claim that this six-inch-long creature will find its way home from several miles away) but some of which received swimming lessons in my improvised pack rat dunk tank.
Having more or less succeeded in my efforts to remove the antelope tribe from the neighborhood of our garden (there was one smug little urchin I never did catch), I raked together what was left of my gardener's self-respect and helped the girls put on their gardening boots and gloves. We completely replanted our plot yet again, which was perfectly fine with Hannah and Caroline, even as I feared that the seasonal window would slam shut on our plants before they could come to fruition. I was now sufficiently desperate that I hatched a plan to photograph the girls next to the plants as they grew, later interpolating these shots of their third first garden with the earlier ones of them planting the actual first garden so as to create the appearance of success where in fact there had been two complete failures. The fact that I had sprung for larger, more developed bedding plants this time would help support the illusion. When I suggested to Eryn that some photo editing might also help out where evidence of the Double Vegpocalypse was inadvertently revealed (for example, I had carelessly planted a pepper where a tomato had been), she smilingly dismissed my idea as only the most extreme of "the many clear signs" that my obsession with my children's first garden had become desperate. Undeterred, I insisted that the plan would work and so took a raft of pictures of my children's third first garden, which really did look quite nice. We then put up our tools, got out the old lawn chair, and sat together, admiring our neat little garden as Venus arced toward the serrated summit crest of our home mountain.
The next morning I awoke before daybreak with a deeply unsettled feeling. I had experienced a disturbingly vivid nightmare in which I stood in my children's garden, leaning at a slight angle but with arms straight out, like a scarecrow, immobilized by a strange paralysis as rodents of every kind crawled up and down my body, even clinging to my beard with their claws as they grinned directly into my bloodshot eyes. In the dream I was in a state of heightened sensory awareness, and I could not only smell the dank fur but feel the tiny whiskers and even the quivering breath of the mice, voles, moles, pocket gophers, kangaroo rats, pack rats, and antelope ground squirrels as they clambered over me. When I awoke I was just beginning to lose sight as one especially enterprising antelope squirrel nibbled on my exposed, unblinking eyeball.
Trying to chase away my bad night with java that I brewed a good bit stronger than usual, I stood at the slider door sipping from my mug and awaiting sunrise over my children's third first garden. At last the ascending Venus dimmed, the sky brightened, and the little plot was bathed in a golden, effulgent light. There it was, suddenly, in all its shining glory, the little garden for my little girls, and I felt that somehow my struggles had been rewarded. Here was the garden I had envisioned, the sweet little plot that would grow with my kids, teaching them to nurture the flower and fruit that bind us to the nonhuman world. Here they would learn the ethic of care that is the highest mark of a moral person; here they would practice techniques of sustainability that would give them healthy food to eat and a harvest basket overflowing with metaphors to live by.
As I was admiring the garden and contemplating my inspiring success with earth-centered parenting, one of the tomato plants suddenly vanished. I quickly scurried back and forth in front of the slider to make sure of what I was seeing, and then I yanked open the door and sprinted to the garden, leaning over the various fences and pressing my face against the nylon bird netting to get a closer view. In the soil directly beneath the center of one tomato cage, which was still rocking slightly, there was a hole where the plant had stood only a moment before.
In that moment I did not cry out, like Job, to the unjust heavens to demand explanation for why I was being punished for a crime I did not commit. I did not observe the natural historical evidence before me in search of a scientific understanding of the depredation nature had here wrought. I did not engage in the inimitable brand of breathtaking, blue-streaking profanity for which I have been reviled by some and celebrated by others. Instead, I felt something deep inside me begin to uncoil, some mainspring in the engine of my tolerance for my fellow creatures irreversibly unwinding, the psychic rivets of my identity as a father and an environmentalist popping off as that spring unwound.
I have only a hazy recollection of what happened next, but Eryn reports that she awoke to see me walking slowly through the house "like a zombie," wearing only lime-green boxer shorts decorated with orange ladybugs, blaze-orange sound-protection muffs over my ears, and carrying my shotgun. I vaguely recall the muffled sound of Caroline shouting, "Daddy has the fire stick!" as I passed the open door of the girls' bedroom, but this too remains foggy. I do remember vividly how different the garden looked when sighted down the barrel to the bead, and I recall the feeling of the trigger moving beneath my finger when it popped its furry little head out of the hole where the tomato plant had recently stood.
It is true that the shot blasted apart the fence on both sides of the garden—creating gaping holes through which other ravenous critters would enter and finish off my children's third first garden in the days ahead. And it is true that the buckshot perforated the leaves of many plants, which hardly mattered since they were soon eaten by the animals that came in through the fence. It is also true that trying to stop rodents is like trying to dig a hole in the ocean: a bucketful of water closes in where every bucketful is lifted out, forever. And it is very true that no knuckle-dragging human is any match for animals that are so brilliantly adapted to this desert environment. It is further true that the proliferation of these rodents was my own fault—not only because the caloric easy money of the garden drew them in, but also because our presence here created a charmed circle that coyotes hesitated to enter. And it is of course true that gunplay in the garden is not especially consistent with the parental and environmental ethic of care I had hoped the garden would teach my little girls. And it is disconcertingly true that once a pacifist nature lover blows something's head off with a shotgun, it generates in him a certain amount of cognitive dissonance, which in turn is deeply threatening to the identity said nature lover may have spent a lifetime cultivating. But it is also indisputably true that, when driven far enough, even a father with a firm grip on the ethical steering wheel can rattle off the washboarded road of his own morality and slide into the humbling ditch of hypocrisy.
Wise Cicero, who wrote that "if you have a garden and a library, you have everything you need," clearly didn't know about the California ground squirrel, or he would have had a shotgun too. _Spermophilus beecheyi_ is even more remarkable than his little cousin the antelope squirrel, though I did not know that at the time I decapitated him with a fire stick. Almost a foot long, with a tail that can extend another nine inches, often more than two pounds in weight, and a prodigious excavator and vegetarian with a huge appetite, this beautiful monster is a real threat to agricultural enterprises on both the commercial and domestic scale. Females often mate with more than one male, sometimes doing so more than once each season, and may give birth to a dozen or more pups in each litter, a reproductive strategy that keeps them well ahead of hawks and guys with fire sticks. Their digging capabilities are truly impressive. One study found that a single California ground squirrel complex containing only eleven animals consisted of a tunnel system extending more than seven hundred feet, including thirty-three openings, and descending nearly thirty feet below ground.
The ground squirrel's tunnels usually protect it from predators other than rattlesnakes, but even here its defenses are daunting. Adult squirrels are actually immune to rattler venom, so when a buzz worm slithers into a tunnel system it is not unusual for a squirrel to harass it, even kicking sand in the snake's face. And while squirrel pups do not share their parents' immunity to rattler venom, female squirrels collect sloughed rattlesnake skins, masticate them, and lick the snake-scented saliva onto their pups, thus using smell to trick the rattler into thinking the baby ground squirrels are actually fellow snakes and encouraging the predator to seek elsewhere for its supper. The California ground squirrel is fast, agile, intelligent, and resourceful and has vision as sharp as yours. It protects others of its kind with an unmistakable high-pitched alarm call by which it communicates danger across miles of desert—a piercing, surging, metallic cry that now rings in my ears as the soundtrack to my own defeat as a gardener. Although diurnal like its little cousin the antelope, the California ground squirrel goes into estivation (a period of strategic inactivity) when the weather becomes too hot and goes into true hibernation in the winter—a physiological shutdown so amazingly like suspended animation that the animal's heartbeat is reduced to a tenth of its normal rate, and it draws a breath only once every few minutes.
After I learned all this about my neighbor _Spermophilus,_ it seemed clear that, as usual, I was overmatched. He didn't need to chew through or climb the wire protection around our garden because he could tunnel under from anywhere he pleased and pop up beneath a tomato plant—which, as the literature on ground squirrel crop damage shows, is among his favorite foods. But everything I read suggested that if I didn't stem this invasion the ground squirrels would overrun the place, undermining the foundation of our house with their tunnels, eating ornamental plants as well as vegetables, spreading fleas that can carry bubonic plague, and perhaps ultimately driving my old pickup to town to buy expensive sour mash with my credit card. They are in fact so destructive to croplands and irrigation systems that it is illegal to release a squirrel that has been livetrapped, and a sense of how far folks will go to try to kill them is suggested by this discouraging remark, which I discovered somewhere in the voluminous antisquirrel literature: "truck-mounted vacuum devices that suck ground squirrels out of their burrows have not demonstrated sufficient efficacy to justify their use." Still, I made up my mind that if my children were ever to have a first garden I would have to try everything this side of burrow vacuuming.
Having already become a gun-wielding killer, at first I decided, like Huck Finn, that "I would take up wickedness again" and so resolved to keep blasting away at my scurrilous neighbors. After all, once you've crossed the line and become a heartless murderer, what are a few dozen perforated corpses, more or less? But Eryn talked me out of the gunplay, not so much by pointing out its incompatibility with the environmental values I aspired to inculcate in my daughters, but instead by reminding me that if I was going to walk around heavily armed I wouldn't be able to drink at the same time—something I prognosticated could become imperative in the battle ahead.
I began with attempts to livetrap the big ground squirrels, as I had their smaller cousins, but they proved too wily to be snared, and as my traps sat empty, my children's third first garden was wiped out completely. At this point I could have acknowledged that after three strikes you're out, but instead I did what my species does best: I chose to believe, foolishly and against all evidence, that nature doesn't bat last, that I could still somehow win one for the humanoids by knocking it into the bleachers in the bottom of the ninth. _Spermophilus_ had become my white whale. In a weak moment I went online and ordered a case of Wild Bill's Shure Kill Varmint Hole Fumigating Bombs. I soon prepared a new strategy to defend my children's garden—which, granted, was now an entirely hypothetical construct—by smoking out my subterranean opponent. On Saturday morning I dressed for the occasion, in boots, long pants, long-sleeved shirt, gloves, hat, safety goggles over my eyes, and bandana over my mouth. Eryn observed that I closely resembled the drunken-looking hillbilly Wild Bill, whose scowling visage appeared on the cylinder of each bomb. Carefully following Bill's directions, I first located what I felt certain were all the holes to the squirrels' tunnel system and then began to execute my plan.
For a moment there was a wonderful rush of excitement, as I sprinted from hole to hole amid swirls of fuchsia smoke, dropping lit canisters into the four burrow entrances I had found. The girls stood at a safe distance with Mom. Hannah gave me a two-thumbs-up sign, while Caroline windmilled her arms enthusiastically, like a swimmer warming up for her next event. There followed an ominous hiatus during which nothing at all seemed to happen. As I stood perched over the squirrel hole nearest the house, I finally looked up to see Hannah and Caroline smiling widely and Eryn pointing at something that was apparently behind me. Lifting my goggles from my eyes to my grimy forehead, I turned slowly around. One, two, three, four . . . eleven columns of fuchsia smoke curled gracefully off into the cobalt-blue desert sky. It was a lovely sight, in an _Apocalypse Now_ sort of way. Thus was it colorfully brought to my attention that the tunnel system had far more escape hatches than I knew of and that this superb ventilation system had prevented my aerial gas attack from being anything more harmful than a fireworks show for the girls, who in fact liked it so much that they made me a lovely thank-you card out of fuchsia-colored construction paper.
Being averse to using poisons, and having now given up on the trap, gun, and bomb, I had but one weapon remaining in my armory: piss. Relinquishing the treasured idea that a mere human could defeat these squirrels, in my desperation I resorted to an unlikely, indirect form of biological control. The ground squirrel used smell to trick the rattlesnake into believing that _Spermophilus_ pups were not what they seemed. What if I could rip a page from the ground squirrels' own stinking playbook and make the scurrilous beast think I was his lethal archpredator, the coyote? Having failed in my roles as Mr. McGregor, Elmer Fudd, and Wild Bill, I now prepared to transform myself into Old Man Coyote. And how is this trickster, _Canis latrans,_ known unmistakably to his neighbors in the wild world? By his wicked grin and by the reek of his piss. Having become the first person I know of to mail order a jug of coyote urine, I now possessed both the grin and the pee. And while I tried not to think about just how one would go about collecting a gallon of coyote whiz, I did feel a late surge of hope that my final plan had a chance of working because I was now going with the flow of nature, so to speak, rather than against it. If the crucible of evolution hadn't taught these squirrels to fear gardeners or smoke bombers, it had certainly taught them to dread the loping death that is Old Man Coyote.
The problem with my "desert doggie wee-wee plan," as Eryn unsympathetically taught Caroline to call it, was that in order to test its results, we would have to plant my children's first garden a fourth time, even as I was now virtually certain that after the time consumed by the Triple Vegpocalypse and its attendant skirmishes, frost would kill the garden if _Spermophilus_ did not. And so the girls and I spent the next Saturday replanting the plot from scratch, of course using the most humiliatingly large bedding plants that a raided college savings account can buy. While I experienced the replanting as a Sisyphean labor, big sister Hannah effortlessly preserved a Jeffersonian equanimity that demonstrated a healthy resilience of which I was incapable. Little Caroline expressed her happiness that the squirrels had eaten such a healthy dinner. "Daddy, vegetables make them strong too!" she said, holding up her tiny arms in a futile attempt at a bicep flex. Once replanted, the garden had yet again to be rearmored and in fact now looked even more unsightly, especially with the addition of the hardware cloth patches I had wired in to cover the large holes blown out by the buckshot. None of this bothered the girls, who "made rainbows" while watering their new plants, after which we put up our tools and went inside to wait for late afternoon, when I would administer the final deterrent in my dwindling arsenal.
I should admit that as the day wore on I became increasingly nervous about the outcome of my looming experiment and that I drank a fair amount of whiskey in an attempt to knock back my growing uncertainty. Eryn, who is both more sensible and more intelligent than her husband, thought the "desert doggie wee-wee plan" absurd, which seemed inauspicious. I had of course failed in every other attempt, and my poor track record suggested that I was the only creature in my local environment completely ill suited to inhabit it. What could it signify that in my hour of greatest need I had resorted not to the dual consolations of acceptance and prayer but rather to the twin elixirs of bourbon and coyote urine? If I failed in this last and most desperate attempt, I would be forced to admit that I was not only a humiliated gardener, an environmentalist pariah, and an ineffectual father, but also a half-drunk, first-order, second-grade, third-string, fourth-first-garden-planting, gas-, gun-, and pee-toting five-star vigilante.
By late afternoon I was sufficiently lubricated that I should perhaps have reconsidered my plan to resume work in my children's fourth first garden, but I knew that the garden would soon be gone without some form of protection, and I still had a spendy Jug O' Whiz with _Spermophilus'_ s name on it. I drained one last whiskey, fetched my secret weapon, and approached the garden with the jug clenched tightly in my right hand. I climbed up onto the garden's railroad-tie frame and balanced myself there, slowly wrapping the fingers of my left hand around the jug's screw-top cap. I paused, taking one last deep breath before cracking the seventh seal.
I still remember how lovely the girls' newly planted garden looked in the glow of the low-angle afternoon sun, how the light breeze rippled the leaves of the tomatoes and squash, how moist and fertile the seedbeds seemed, how neat and well tended the plot appeared. I vividly recall feeling that I was witnessing a perfect garden in a perfect moment, though this transcendental epiphany was no doubt intensified by the blush and tingle of the hooch that had by then loosened all my muscles. I remember how lovely that moment felt, how hesitant I was to twist that cap and lose the wonderful feeling that little garden had raised in me. I remember, with a dreamlike sense of distance, an overflowing feeling that, despite the many trials it had presented, this tender plot was a noble monument to my love for my daughters and for the earth. Maybe, I recall thinking, there is some cosmic plan within which this struggle has been an indispensable part of my own journey both as a father and as a gardener. Then I twisted off the lid.
There can be no word in any human language that even begins to suggest the overpowering, unspeakable stench that exploded from the jug the instant that lid came off. Nothing in human evolutionary biology could have prepared me for this reeking bomb, the first whiff of which instantly flooded my eyes with tears, filled my mouth with a choking metallic tang, and set the whiskey roiling violently in my gut. I felt as if my body had suddenly become a permeable membrane through which the worst stink in the universe was blowing at gale force, carrying off my flesh as it howled through and reducing me to a shattered pile of smoldering bones. This was not just the urine of who knows how many very angry (and presumably catheterized) coyotes, it was a highly concentrated _gallon_ of the stuff, and it had been stored for who knows how long in this vacuum-sealed container. I now held, at the end of my quivering, hyperextended arm, a vessel of the kind of stench that could make a gagging human hope to be sprayed by a polecat just to cover it up.
Of course this all happened in a flash, but in the instant that I recoiled from the revolting stink, which blasted out of the jug and attacked my face like a swarm of yellow jackets, I heard that signature metallic chirp of victory ring out from the sagebrush behind me. It was _Spermophilus,_ either laughing at me or warning his kin that a thousand coyotes had just simultaneously taken a huge leak. Spinning my contorted face away from the jug and toward that piercing cry, I suddenly felt my boot sole begin to slip on the edge of the railroad tie atop which I was perched. And it is at this moment that time seemed to slow almost to a stop, and I experienced the next few seconds in that frame-by-frame fashion that the human brain reserves for only the most unimaginable of accidents.
I fell for what seemed quite a long time, and I even remember seeing the pee that splashed out of the jug floating in midair, as if in the zero gravity of a space capsule. Eventually the handle of the jug was released from the grip of my fingers, and it too turned slowly in midair, as if it would remain spinning there forever. And then, at last, came the splintering crash of my body as it landed on my children's beautiful fourth first garden, taking down fences and netting and stakes and cages as it did and crushing the plants that by now had assumed a symbolic significance very different from what I had originally intended. As I looked up through my bleary stink tears and through the fragments of the garden in which I now lay, I could just make out a matched pair of sisters in the distance, both of them pinching their noses with one hand and waving at me with the other. And they were smiling. As always, my little girls were smiling.
I scrubbed until I had about peeled my skin off, but Eryn still made me sleep out on the deck that first night. In the weeks that followed I couldn't get near enough to the Superfund site that was my first children's fourth first garden to initiate remediation, though we diluted the terrible pee stench by hosing the garden down from about thirty feet upwind, an ablution that I performed twice each day in order to make it tolerable for the girls to play outside. Five weeks later an early frost hit, and the cold snap knocked the stench down enough that I could approach the wrecked garden to clean it up before the first snow. When I pulled away the broken fences and cages, I found that a few plants had actually survived, unmolested because the not-quite-empty Jug O' Doom still rested openmouthed near their stems. One of these was a tomato plant, and while its tiny yellow flowers were now burned by frost, it had set some fruit, and a single tomato looked pink enough that it might ripen off the vine. I harvested the little tomato, washed it well, and gave it to the girls to put on the sill of their bedroom window. Over the next few days that tomato ripened, and so our family huddled around the kitchen table to celebrate the ritual of the first fruits—well, fruit—of the season, even as a snow sky gathered outside. Caroline took a bite, wrinkled her nose, and said "Thanks for the tomato, Daddy. I don't like it. Can we have a garden again next year? Let's plant pineapples!"
In parenting and in gardening we risk failure every moment of every day, and how could it be otherwise? Through these daily practices of love, humility, and humor we just keep trying, not because our success is certain, but because it certainly is not. We hope, and yet we fail; we fail, and yet we hope. I have promised the girls that we will plant their garden in the spring. And it will again be their first garden, just as every garden is a first garden, just as every day with those we love is a chance to start over, to plant something again.
_Chapter 11_
11. The Hills Are Alive
My grandmother's highest compliment for a natural landscape was to say that it was "as pretty as a picture." Even as a kid I remember thinking that this aesthetic was upside-down, that the loveliness of art should be judged according to the inimitable standard of natural beauty rather than the other way around. During the late eighteenth and early nineteenth centuries, well-heeled European travelers scoured the countryside looking for views that would be as pretty as a picture—or, to be more precise, as pretty as a painting. Because they had a certain kind of painting in mind as embodying their standard of natural beauty, these early eco-tourists often carried with them a "Claude glass," a small, convex, tinted mirror that was nicknamed for the seventeenth-century landscape painter Claude Lorrain. When a picturesque landscape was encountered—say, for example, the snowcapped Alps—the tourist would first turn his or her back to the mountains, then whip out the Claude glass, holding it up so as to frame the mountains. The peaks appeared reflected and also color shifted to a tonal range that made them look more painterly. Voilà! Actual Alps not only become pretty as a picture but _become_ a picture, as the pleased eco-tourist stands admiring, not the mountains, but rather an image of them that he or she has created. But why must we turn our backs on the land in order to see it in a way that we find aesthetically pleasing? Have we so lost a sense of humility before nature that we've come to love our own representations of the world more dearly than we love the world itself?
It might be fair to say that the Claude glass of the nineteenth century was photography and that the twentieth-century Claude glass was film, since these technologies of representation have profoundly conditioned our landscape aesthetics. They have allowed us to frame the world. As with the Claude glass, there is a sense in which cinema's stylized, controlled, and color-corrected depiction of nature has thoroughly mediated our relationship to the physical world, reshaping our environmental aesthetics and implying that a representation of nature is an improvement upon nature itself. Cinema has the power to show us the environment in remarkably dramatic fashion, but to see the land in film we must temporarily turn our backs on the land itself. To climb into the bright mountains of the screen we must first descend into the dark cave of the theater.
From a very early age I've held the deep and unwavering conviction that musicals—especially movie musicals—represent the most intolerable and misguided aesthetic form in the checkered history of human civilization. In addition to being uniformly hokey and boring, musicals are also cloying and saccharine, which is even more offensive. I make it a policy never to trust a person who would spontaneously break into song for no reason, especially when he's about to begin a knife fight ( _West Side Story_ ), he's adopting an orphan as a publicity stunt ( _Annie_ ), or she's confessing her unwanted pregnancy ( _Grease_ ). It is not simply that the suspension of disbelief required in such cases would daunt Hercules, it is also that it is so obviously inappropriate to croon about things like gang violence, homeless waifs, and bastard children. The world would be a better place if the urge to sing of such things could be soundly repressed—if this upswelling, confessional, tuneful emoting could instead become a stoical moment of shutting the piehole good and tight.
If I sound testy about this issue of movie musicals, I have good reason. As the father of young daughters, I have been subjected—wholly against my will—to musicals too numerous and nauseating to be enumerated. The most frequently repeated of these abominations is the much-beloved 1965 "timeless classic" _The Sound of Music,_ whose perennial popularity confirms every curmudgeonly thing I've ever said or written about my fellow human beings. Indeed, the National Association of Misanthropes might consider screening this gem at its annual convention, if only to reassure members that they really are on the right track. Despite my personal aversion to the picture, _The Sound of Music_ not only bailed out a sinking 20th Century Fox in the mid-1960s but, adjusted for inflation, has gone on to net more than a billion dollars. That's "billion" with a _b,_ just like the _b_ in "blockbuster," or "banal," or "bullshit."
So much beloved is this appalling movie—which, by the way, won five Academy Awards and was nominated for five more—that the first-ever reunion of its nine principal actors had to be held as part of the final season of _Oprah_. The film was even ranked number 55 on the American Film Institute's centennial "100 Years . . . 100 Movies" list of the most important American films, where it was judged superior to _actual_ "timeless classics" including _The Third Man_ and _Vertigo, Stagecoach_ and _The Searchers, The Gold Rush, City Lights,_ and _Modern Times_. Can there be any doubt that Carol Reed, Alfred Hitchcock, John Ford, and Charlie Chaplin—the directors of these amazing pictures—are spinning in their graves? Among the few people ever to tell the truth about _The Sound of Music_ was the film critic Pauline Kael, who called it "the sugar-coated lie that people seem to want to eat." "We have been turned into emotional and aesthetic imbeciles," wrote Kael, "when we hear ourselves humming [this film's] sickly, goody-goody songs." In a simultaneous blow to free speech and good taste, Kael was fired from _McCall's Magazine_ for the heresy of making this astute opinion public.
I've meditated at length on what disturbs me so much about this awful film. It isn't simply the gratuitous singing, which is endemic to the form, or the appalling sentimentality of the characters, which I might have predicted, or even that I'm asked to believe that a guy with seven children could be happy instead of insane, even were he not on the run from the Nazis—which, as you'll recall, he is. No, the problem runs much deeper, and it is this: _The Sound of Music_ is an expression of my own values. First, there is an emphasis upon the centrality, resilience, and importance of family, which is a principle I hold dear. Then there is, in the romance plot, an assertion of the power of love to pull down interpersonal barriers, including those related to class. This too I believe. And the good guys in this movie seem to feel that the Nazis are bad guys, which I have no difficulty going along with.
But what is the core value at the true heart of this film? It is the protagonist's deep love of nature. You'll recall that the Julie Andrews character, Maria (soon to become Mrs. Maria von Trapp), is from the beginning an irresponsible and negligent nun in training who fails miserably at her religious duties. Why? Because she is so busy spinning around flowery mountaintops in implausibly orgasmic nature reveries. Here we recognize the oldest of the tricks in the book written by Wordsworth and Coleridge, Beethoven and Schubert, Bierstadt and Cole, Emerson and Thoreau: indulge orthodox rejoicing and piety, but while your parents aren't looking just swap out the divinity of God for the divinity of nature. Maria isn't a bad nun. She is a good transcendentalist. She believes deeply in grace and in the divine, but for her the locus of divinity is the Alps rather than the abbey. So moved is she by nature that, well, damn it, she just has to "climb every mountain." And she's none too quiet about it.
If this film reflects so many of my own values, why then do I find it intolerable? You know that feeling that comes over you when you discover that some bloviating asshat is a huge fan of your favorite baseball team or an ardent admirer of your favorite band—when the purity of your ineffable love for something is sullied because it must be shared with an obnoxious knothead? _The Sound of Music_ is so incredibly trite that I can't help but resent its superficial dramatization of my own beliefs—particularly my core faith in the spiritual value of nature. Is this how I appear to others, a self-indulgent, dirt-worshipping, gushy, feeble-witted tree hugger who twirls around in fields bursting forth in earth-loving song?
Inspired by their immoderate affection for Maria, Hannah and Caroline propose that we climb our local hill and reenact the opening scene of _The Sound of Music_. As a man who despises musicals and is deeply suspicious of Chautauquans, Civil War reenactors, and department store Santas—all of whom I consider not only counterfeits but also drunkards and pedophiles—I am a poor choice for this mission. But here's the thing: I'm their dad. Among the many blessings of being the father of daughters is the constant opportunity to operate entirely outside my comfort zone. What choice do I have, especially after Eryn, with a wry smile, tells the girls how certain she is that Dad would _love_ to be a part of this project? "Daddy even teaches film classes," she says enthusiastically. "I'm _sure_ he can help you understand why this movie is so great!" This is my punishment for having married someone with a sense of humor, which now seems less charming than it did during our courtship.
"OK," I finally assent, "but if I help you reenact the 'hills are alive' scene, then I get to choose another scene that someday y'all will help _me_ to reenact." When the girls agree, I reveal my choice: the scene in which the dad, a grumpy sea captain, imposes martial discipline upon his children, controlling their every behavior through a series of coded orders tooted out shrilly on a dog whistle. This promises to be a refreshing change from my usual domestic life, in which my agency has been reduced to running the chainsaw and hoisting IPAs.
As we screen the opening sequence of the film in order to observe every excruciating nuance of the "hills are alive" scene, I'm reminded that the movie begins with a montage of lovely establishing shots of the snowy Alps and verdant Salzkammergut foothills—helicopter shots that are plenty respectable for sixties cinema. Just as I begin to enjoy these rich images, however, the aerial camera makes the unhappy discovery of Julie Andrews doing those orgasmic hilltop pirouettes, after which she promptly destroys the moment by bursting into song. It is the kind of cinematic moment that, rescreened a few times, could make spies talk. I find it difficult not to fantasize about some way— _any_ way—to make Julie stop. I imagine that the studio helicopter is in fact a helicopter gunship, its sweeping descent toward warbling Maria accompanied by the satisfying rat-a-tat-tat of machine gun strafing. Or perhaps that she might be skewered by the chopper skid, a chirruping Maria-kabob rising joyfully into the clouds. Maria's song, "The Sound of Music," turns out to be a kind of environmentalist anthem, replete with natural images including hills, birds, lakes, trees, breezes, brooks, and stones. The degree to which Oscar Hammerstein's gift as a lyricist has been exaggerated is made evident by the line in which Maria's heart wants to sing "like a lark who is learning to pray." This is a moment so insufferable that we ourselves might pray, along with the hapless lark, that Maria would just shut her von Trapp. But there it is again: my personal belief in the divinity of nature, being expressed in the most saccharine and clichéd manner possible. Of course Hannah and Caroline absolutely love it.
The girls and I make our plans for the reenactment, and Eryn costumes them to look suitably Maria-ish. I fill a day pack with snacks, water, and sunscreen, and we begin our afternoon ascent of Moonrise, the nearby hill that we've so named because it's an especially fine spot from which to watch the rising moon on summer nights. These Great Basin foothills could not be more different from the lush hills of the film's Austrian Alps. Here we push through high desert scrub, including thorny desert peach and scratchy bitterbrush, an unbroken carpet of big sage and rabbitbrush rolling out before us to the distant horizon. It is a desiccated and brown landscape in which we must guard against sunstroke, dehydration, and Great Basin rattlesnakes, which are common on the rocky slopes of Moonrise. Here are no babbling brooks to meditate beside, no azure lakes into which to dip our oars, no trees to stroll romantically beneath, no emerald grass to loll upon. Nothing here is green, save for the yellowish green of an ephedra bush here and there. The glare of the high-elevation sun is intense as we push up the dusty slope of Moonrise and into the hot blast of the Washoe Zephyr. This is not the land of the Claude glass but rather the land of the emergency signal mirror—not a place for twirling but rather for hunkering down in order to survive.
Wallace Stegner wisely observed that we need to "get over the color green." "You have to quit associating beauty with gardens and lawns," he admonished. Stegner realized that our fantasy landscape remains closer to that of _The Sound of Music_ than to the austere geophysical reality of the arid West and that this aesthetic preference has environmental consequences that are all too real. My girls have been raised in this open, windy desert, and they know instinctively that its power lies precisely in its gorgeous starkness, in its effortless resistance to our intentions. This land is sublimely inhospitable, and its grandeur inspires a humility that is the greatest gift it has to offer. Stegner was right. Until we get over the color green we'll remain doomed to view the West through a Claude glass of our own imaginative construction. We'll continue to see the world indirectly, artificially framed, color shifted to conform to an environmental aesthetic that is disconnected from the visceral reality of this astonishing place. Here in the western Great Basin, green is the color of the lawns that don't belong and the money that buys the vanishing water that keeps them that way. The high desert is not the green world of the Austrian Alps, but neither was it meant to be. This is our home landscape, and to us it is far more beautiful than the Alps could ever be.
On the way up Moonrise I ask little Caroline what her favorite part of _The Sound of Music_ is. Without hesitating she replies, "I like the part with those bad guys, Daddy. What are they called again?"
"Nazis," Hannah replies.
I cringe. This is the same kid who, during our earlier reenactment of scenes from _The Wizard of Oz,_ insisted upon playing the role of one of the terrifying flying monkeys, even in scenes where they had no credible reason to appear. Hoping to shift the conversation, I ask Hannah what her favorite part is.
"I like Liesl the best, especially when she's singing in the rain."
I wince again. The scene Hannah has in mind depicts the courting of Liesl, the eldest von Trapp daughter, by an Aryan messenger boy named Rolfe—a scene in which Liesl croons the insipid teen anthem "Sixteen Going on Seventeen." This awful ditty includes the girl singing sweetly to her Hitler wannabe beau: "I need someone older and wiser, / Telling me what to do." As the father of daughters, this is not the sort of thing I want to hear. I note that this antifeminist narrative isn't much of an improvement over Hannah's favorite kid flick, Disney's _The Little Mermaid_ , in which a mermaid girl—basically an aquatic Liesl—disobeys her father, leaves her home place, and relinquishes her own voice, all in order to be with a guy just because he's human.
"Hannah, do me a favor," I implore. "When you're sixteen going on seventeen, remember that _I_ am the one who is older and wiser. Not some boy, _me_! And don't forget that charming Rolfe ends up joining the Nazis."
"Right," interjects little Caroline. "Nazis!"
At last we reach the summit of Moonrise, where we pause in the shade of a granite palisade to hydrate and snack. We are well above 6,000 feet now, and the cloudless cobalt sky shimmers as it can only here in the high desert. The scat of pronghorn and coyote are nearby, and the faint tracks of black-tailed jackrabbits, and some orange lichen that has eked out a living in a fissure in the rock. Once rested, we choose the site for our reenactment, and I clamber up into the cliffs above in order to approximate the film's memorable high-angle opening shot. The girls are down below, practicing their lyrics and poised to pirouette. They look adorable in their corny dresses and makeshift aprons. At last I yell, " _Action!_ " and they begin to twirl like crazy, stumbling a little over the rocks, bumping into each other and also into the sage and rabbitbrush. I catch a word here and there as the hot wind sweeps their song away toward Utah. The sere, brown land is treeless and flowerless. In the viewfinder of my camera I frame the little stars of my own life story, spinning in their mountain-top reverie. They are laughing, and dancing, and singing, right here, in this place, among the rattlers and scorpions. It is, I admit to myself, a strange and wonderful kind of musical. In the glare of the high desert sun and the sweep of the scorching wind, the irony of the reenactment suddenly dissipates, and I feel a rush of genuine sentiment. My little girls are dancing in their home hills, and the hills are alive.
_Chapter 12_
12. Fire on the Mountain
Here in the remote western Great Basin Desert we dwell as guests in the house of fire. Of course wildfire has helped to shape most natural landscapes, but that fact is easy to forget in urban and suburban settings, while out on this wildlands interface the presence of fire is tangible. From the windows of our home we look out over open desert that is dotted green with Utah juniper trees but also dotted gray and black with the charred snags of junipers that have been consumed by flames. We see fire-protected, rocky spots with immense, seven-foot-tall sagebrush, while other areas have burned over so recently as to harbor only the sprigs of bunchgrasses, rabbitbrush, desert peach, gooseberry, bitter cherry, ephedra, and balsamroot. Fire visits our area almost every summer, and the vibrant mosaic of trees, plants, grasses, and flowers visible on the distant hillsides has in fact been produced by flames. Because its presence on the land is so conspicuous, we tend to think of fire as a neighbor—albeit a neighbor who, like rattlesnake and mountain lion, is deserving of immense respect.
As a parent I've worked deliberately to avoid demonizing fire, just as I've resisted misinformed violence toward maligned endemic species like rattlers and scorpions. Hannah and Caroline understand that this desert place was the residence of coyote, buzzworm, and wildfire long before we chose to make our home here. The girls have come to appreciate these neighbors—to offer respect by giving them a wide berth, but also to learn about them, and never to encounter them with the irrational fear that too often characterizes human relationships with the desert environment and its creatures. I hope this desert will teach my daughters to accept that we are subject to natural forces larger than ourselves and often beyond our control, since it is the humility inspired by that awareness that helps lead us from a blind assumption of dominance toward an enriching imagination of reciprocity with the natural world.
We are fortunate that the desert we inhabit has been sculpted by fire. In a normal fire regime, low-intensity ground fires have substantial ecological benefits. These fires often wipe out invasive plants that compete with fire-evolved natives, and they help to balance the populations of many insects. Their passage opens the way for the growth of grasses and forbs that provide excellent forage for mule deer, pronghorn, and many other animals. The ashes they produce add nutrients to the soil, and there are even species of trees and plants that are unable to reproduce without fire. Unless the land has been excessively degraded by destructive forces such as road building, overgrazing, or previous colonization by invasive exotic plants, the presence of fire tends to catalyze biodiversity and improve ecosystemic health. The fire history revealed by dendrochronology suggests that in a natural fire regime the wildfire cycle here might be as short as twenty or twenty-five years. The northern Great Basin sagebrush steppe is a biome that needs its fire.
Knowing that we are fire's neighbors, we've made extensive preparations for cohabitation. We've designed and built a wildfire-resistant home with a stucco exterior and concrete tile roof, and we've sited our propane tank far from the house and away from trees. We've done extensive fuels-reduction work on our property, engaging in the kind of thinning performed by fire while also decreasing the fuel load available for combustion when the flames do come. We maintain firebreaks and defensible space, and we've landscaped with native plants chosen for their resistance to fire. We've installed freestanding hose bibs on all sides of our home site, and we have a propane generator that can be used to run the well pump and thus provide water to fight a ground fire even if the electricity has been lost (or cut, which often occurs during fires). We've laid out our long driveway with wide turns and a loop up near the house to facilitate access by emergency vehicles. We've even made sure that down at the gravel road our address numbers, which are reflective, are mounted on a noncombustible post. We've done all this because out here we are so far away from help that we can't afford to count on good luck. In our years on this remote hill we've witnessed dozens of fires, and twice we've been evacuated while emergency crews occupied the firebreaks that we maintain with such vigilance. Someday wildfire will return to our home hill, and it is our responsibility to prepare for that day while also respecting fire's ecological value and resisting the urge to view it through the distorted lens of fear.
Ecologists employ the paired concepts of resistance and resilience to understand and describe the effects of environmental disturbance. _Resistance_ refers to the capacity of an ecosystem to _retain_ its essential structure and functioning despite stressors, which might come in the form of extreme weather, invasive species, human activity, or wildfire; _resilience_ describes the capacity of the ecosystem to _regain_ its essential structure and functioning once altered by a disturbance, like wildfire. In other words, we should take care not to compromise the health of this desert ecosystem, because its well-being is essential to its ability to withstand and recover from environmental stress. The concepts of resistance and resilience are helpful to us in imagining our own lives as well. If we remain healthy, we're better able to resist illness and disease; in the unfortunate event that by some accident we are overwhelmed by a major environmental stress, our chances of being sufficiently resilient to restore ourselves to health are vastly improved.
When the smoke alarm went off upstairs, Hannah and Caroline had already been asleep for an hour or so, snuggled in their bunk beds in their shared bedroom on the ground floor of the house. Eryn and I were sitting on the stone hearth, enjoying one drink before the warmth of the wood stove, talking over our day and sharing that sweet, brief calm that comes to parents after their children are asleep and before they have themselves succumbed to the day's fatigues. It was Valentine's eve (which happened to be Ash Wednesday), and we had just finished cleaning up from helping the girls make their Valentine's cards, which they would give to their friends at school the following day.
"Here we go again," I said in exasperation, taking one more sip of rye before setting the tumbler down to go deal with the problem.
"I'll check on the girls in case the noise wakes them," Eryn said.
"Thanks," I replied. "I'll get the ladder and see which one it is this time." I had yet to make it through a full year without an alarm going rogue, and while the problem was usually solved with a fresh nine-volt battery, I had also replaced several of the devices that simply wouldn't keep quiet. As I headed down the hall to fetch the stepladder from the garage, I felt certain this was another false alarm—until a second alarm also began to sound from upstairs.
I paused for a single moment before turning on my heel and sprinting down the hall to the stairs, which I took three at a time as I raced up to my scribble den. Although I smelled no smoke, the two alarms continued to blare convincingly in an echoing, alternating rhythm, as if synchronized. As I grabbed the doorjamb and swung myself around the corner and into the room, my eyes went first to my writing table, where my laptop sat open, surrounded by stacks of papers important to the book project I had been working on that afternoon. Beneath the table were piled boxes full of books, notes, papers, and manuscripts associated with the various other projects I had finished in recent years or hoped to write in the years to come. My eyes then shifted a few feet to the left of my writing table. There I saw bright orange and yellow flames several feet high blasting up through a sizable hole in the floor.
The physiology of alarm is not easily translated into words. For a single instant I felt as if the air had been sucked from my collapsed lungs, while my legs had suddenly turned to molten rubber and my mouth was filled with a nauseating metallic taste. And then the adrenaline rush hit with such intensity that it felt as if my body had been struck by lightning from the inside. My breathing suddenly quickened, and I could feel my pulse pounding in my neck and my heart hammering in my ribs. It was then that I smelled the smoke.
" _Eryn, this is for real! Call 911 and get the girls out, NOW!_ " I yelled down the stairs. Over the blaring of what were now at least three smoke alarms, I heard her acknowledge me, and I caught a glimpse of her running down the hall toward the girls' bedroom. Turning back into the study, I grabbed the fire extinguisher from the top of a nearby bookshelf, yanked the pin as I ran across the room, and unloaded the extinguisher's contents into the flaming hole in the floor. As I sprayed the foamy fire retardant the flames disappeared, and for a brief, hopeful moment I imagined that the situation was under my control. The instant the extinguisher ran empty, however, the flames shot back up out of the hole, as if I had done nothing at all to discourage the fire. I then scrambled over to the upstairs bathroom, where I removed a second extinguisher from beneath the sink, ran back to the study, and repeated my efforts. Once again the flames vanished only while the retardant was flying, immediately surging back out of the hole as the extinguisher ran dry. It was now clear that the fire was intense and that it was burning within the floor beneath me.
I sprinted down the stairs and then down the hall and into the garage, hitting the garage door button as I dashed out. I grabbed an orange five-gallon bucket from the garage floor, ran to my workbench, and lifted the large fire extinguisher I keep there into the bucket, which I left sitting on the bench. I then jumped into my pickup, started it, and backed out onto the gravel driveway. I jammed it into park, threw open the door, and jumped out.
" _Eryn?_ " I called loudly.
"We're out front!" she replied, from around the corner of the house.
"The truck is running here—come get the girls off the hill!"
"Where's the puppy?" she hollered back.
"No time! Get off the hill _now_!" I shouted, dashing back into the garage before I could catch a glimpse of Eryn, Hannah, or Caroline.
Barely breaking stride, I grabbed the plastic grip on the wire handle of the bucket, swung it off the workbench, and ran into the kitchen, leaving the door open behind me. I retrieved a small fire extinguisher from beneath the kitchen sink and tossed it into the bucket. I then sprinted with my bucket to the hearth, where I kept another extinguisher near the kindling rack. As I crossed the living room in a rush to the hearth, I had a strange experience. The fight-or-flight dump of dopamine from my exploding adrenal glands had produced the odd effect of time slowing down, and so even as I worked frenetically, I had bizarre moments in which it seemed that I had paused to engage in leisurely contemplation of a particular image or sensation. Now, as I sprinted across the room, I noticed Eryn's half-full wineglass and her cell phone and my rye tumbler sitting placidly on the stone, as if nothing were amiss. Behind the glass doors of the wood stove lapped our lovely little fire, just as before. The scene looked cozy, and although the thought must have taken only a millisecond to register in my racing mind, I reflected on how strange it was that the fire behind the glass appeared so controlled while just upstairs burned a sister fire that was fierce. This protracted moment of slow-motion perceptual intensity was so dreamlike that it caused me to wonder if I might awaken to discover that this strange fire had been ignited only within the wildness of a dream.
I snatched the extinguisher from the hearth, added it to the bucket, and ran back upstairs with my load. Although I had been gone only a short time, the situation upstairs had grown a good deal worse. The hole in the floor had at least doubled in size, and the flames too had increased, now leaping three or four feet out of the burning floor. Worse still, the room had begun to fill with smoke, and while visibility was decent at eye level, the ceiling was now a wall-to-wall mat of thick smoke. I set the bucket down and lowered myself to one knee next to my writing table, hoping to achieve a low angle that might allow me to blast retardant further up under the floor, toward the unseen source of the flames. I emptied one extinguisher into the flaming hole, and then the second, and finally the third. Each time the flames retreated as the foam splattered into the hole; each time the fire flared up the instant the extinguisher was emptied.
It was clear from the five empty extinguisher canisters and the increasingly intense flames rising toward the ceiling that the fire was raging hot and fast within the floor. I now began to realize, for the first time, that it was entirely possible our home would burn to the ground. The fate of the house would be determined by the outcome of a race between the fire and the people who, unlike me, have the skill and equipment to fight it—people who, because of the remoteness of our home, are a very long way away when you need them most. All I could do was try to slow the spread of the fire, buying precious time until help arrived.
Having exhausted all my extinguishers, I now ran with my empty bucket to the upstairs bathroom, where I slid open the glass shower door and turned the tub faucet on full blast. I jammed the empty bucket beneath it and watched as time once again slowed to a crawl. How long had I been fighting this fire? How long would it take for help to arrive? How many years before this bucket would fill with water—water that fell as if it were a viscous plasma, with a slowness that made me question whether the law of gravity had been repealed? In the eternity it took for that bucket to fill I kept hearing, over and over, the percussion-driven rhythm of the Grateful Dead surrounding Robert Hunter's provocative lyrics for "Fire on the Mountain":
_Long distance runner, what you holdin' out for?_
_Caught in slow motion in a dash for the door._
_The flame from your stage has now spread to the floor._
_You gave all you had, why you want to give more?_
_The more that you give, the more it will take_
_To the thin line beyond which you really can't fake._
_There's a fire . . . fire on the mountain._
I finally lifted the bucket out half-full, left the water running, and ran back into the scribble den. Fearing the collapse of the burning floor, I didn't dare put my weight too close to the blazing, ever-expanding hole. Instead, I stood back and flung the contents of the bucket all over the floor in the general area of the flames. I then ran back and refilled the bucket, again dousing the room, but this time slamming water onto the walls as well as the floor, trying desperately to slow the spread of the fire.
It suddenly occurred to me that although the fire in the wood stove appeared small and controlled, it might somehow be contributing to the fire in the floor above. Filling the bucket a third time, I hauled it downstairs, threw open the wood stove door, and tossed the full bucket of water directly onto the coals and logs that were burning within the stove box. The dousing produced a loud hissing sound, audible even over the piercing blasts of every smoke alarm in the house, as an immense cloud of acrid, sulfurous, ashy smoke billowed from the wood stove and rose up the rock hearth to the ceiling.
I ran back upstairs and continued my bathroom-to-study shuttle run, splashing buckets of water onto the floor, carpets, cabinets, walls, and, eventually, even the stacks of boxes, books, and papers that comprised all of my writing projects. The full wisdom of my adrenaline-driven logic might be summarized this way: "Wet stuff doesn't burn, right?" As the room continued to fill with choking smoke, I finally noticed the flashing lights of emergency vehicles out in the distance, though their movement through the blackness of the desert seemed preternaturally slow. Encouraged by the prospect of help, I resumed my sprint-and-douse dash.
Finally, two firefighters arrived in my study. They were dressed in full battle regalia, complete with helmets, face shields, head-to-toe yellow fire suits, giant fireproof boots and gloves, and air masks with hoses that led to the oxygen tanks they wore on their backs. I was wearing old jeans, a Buddy Guy concert T-shirt, and broken-down trail running shoes. Worse still, I was standing in a flaming, smoke-filled room in a burning house holding a half-full bucket of water. Ignorant of the social protocol surrounding house fires, I wasn't sure if I should toss those last few gallons onto the fire or instead address the men. "Thanks for coming," I offered, sheepishly setting my little bucket down. "Can I do anything to help?"
"Sir, you need to evacuate this structure _immediately,_ " said one of the large yellow men, who seemed surprised to see me. He then spoke a string of commands into his walkie-talkie. It was clear that my shift was over. In a matter of moments our "home," which had already become a "structure," would be what emergency responders call a "fireground." Reaching the bottom of the stairs, I had to jump aside as three guys hauling fire hoses over their shoulders came rushing in the front door and charged upstairs into the smoke. As I stepped slowly out into the cold, carrying nothing at all, I was surprised at how many emergency vehicles had already arrived: a command car, two fire trucks, a water truck, and a sheriff's deputy car—and I could see more flashing red lights off in the distance.
I stood out in our firebreak, between patches of old snow, shivering in my T-shirt. I first looked west, in the direction I always imagined the fire would come from—the wildfire I had worked so hard to prepare for. Out there in the black depths of the open desert there was no fire save the stars ignited in the moonless sky. I then turned and looked back at our house, where the wild force of fire was traveling through the floors and perhaps also through the walls and ceilings. The power had been cut, but through the unshaded windows I could see the firefighters' headlamps, laser-like, slicing through the thick smoke as they worked in the murky darkness. I heard chainsaws revving and then droning while I caught glimpses of large yellow men cutting holes into the walls and ceilings as they chased fire through the bones of our home.
After what seemed a very long time, the fire captain came over to talk with me. When I asked if they could save the house, he answered only that they would do their best. I waited for a more hopeful prognostication, but none came. The visible flames were extinguished, he said, but the fire was still smoldering within the floors or walls, and they had not yet located it. This sort of fire could blow up quickly, he explained. I asked him about our pets—two dogs and a cat—and he reported that Darcy, our old dog, had been found cowering beneath the water truck, but that the puppy, Beauregard, and Lucy the cat had not been seen either inside or outside the house.
At some point my father arrived, and the two of us stood together in the cold, staring at the house, watching firefighters rush in and out, listening to the sound of chainsaws, receiving no assurances. My dad was the primary designer of our passive solar house, and he worked tirelessly as our general contractor when it was built. It was as efficient, creative, and beautiful a home as any desert rat could ever hope for—the result of a multigenerational collaboration in which every member of our family was deeply and personally invested. Hannah was an infant when we brought her to this place, and it has been Caroline's only home. This was the place where we had built our shared life together—the place our stories came from. And now, as we stood silently out in the cold, thick smoke billowed from beneath all sides of the roof.
While I was battling the fire on my own, Eryn was making sure the girls were safe. Hearing me shout from upstairs, Eryn called 911 and reported the emergency. She then dropped her phone on the hearth and ran to the kids' bedroom, where both girls were sound asleep despite the blaring smoke alarms. She first woke Caroline and instructed her to "give Mommy your best monkey hug," a neat trick by which Caroline can hold on to you with the vice grip of her muscular little legs. With Caroline firmly attached, she then woke Hannah, took her by the hand, and led the girls quickly out of the house, where she sat them on a railroad-tie wall out in the cold February night. Now having a moment to gather herself, Eryn realized that the girls were barefoot and wearing nothing but their pajamas. She told Hannah to put her arm around Caroline and under no circumstances to go back into the house.
"I am coming back, but you two stay together no matter what. And don't come back into the house. That's your job," she told them. Then Eryn dashed back into the girls' bedroom, pulling their robes off the dragonfly-shaped hook on the back of their door and snatching the first shoes she could grab from the closet, which turned out to be the two little pairs of bright green cowgirl boots we'd bought for Hannah and Caroline on our last family expedition across the Great Basin.
Although Eryn had been gone only moments, by the time she ran back outside Caroline was scared and upset, and Hannah was hugging her little sister and comforting her. Caroline is fortunate to have such a caring, loving big sister. After all, the girls had gone from a deep sleep to being awakened to the sound of alarms and hauled out into the dark night without so much as putting their shoes on. They were disoriented and cold. They didn't know where their dad was.
When I appeared with the truck, Eryn brought the girls around the corner of the house in their green cowgirl boots and red bathrobes and piled them into the idling pickup. She then drove down our dark, muddy, half-mile-long driveway, fielding a battery of fearful questions from the girls. What was going on in the house? Why wasn't Daddy coming with us? Was he going to be OK? Eryn's answer to all these questions was to say, over and over and with great confidence, "Dad always knows what to do."
When Eryn reached the bottom of the driveway, she jumped out and propped open our green farm gate with a rebar stake. Then she drove out onto the gravel road, spun the truck around, and backed it up so as to face the road and be just beyond our driveway—a position from which she intended to direct emergency vehicles. During the long wait for help to arrive, she realized that she didn't have her phone, or her glasses, or any warm clothes, or anything at all for the girls to read or play with. And there they sat, with the truck running and the heater blowing, headlights tunneling into the darkness, waiting in the breathless silence of the desert night. The remote location of our house and the hilly topography in which it is nestled prevented her from seeing our place or hearing anything from it, and because she had no way of knowing what might be happening up on the hill—except to know that I was there alone and that there was a fire—it was an agonizing wait.
It was during that lonely, frightening eternity in the truck, waiting for help and knowing nothing, that Eryn gave the girls a gift she has also given me: a magnificent strength born of love. Her courage and poise calmed Hannah and Caroline, helping them navigate this difficult and chaotic experience. As Eryn hugged the girls on the old truck's bench seat, she explained to them that some people would be driving out to help up at the house and that the first one to spot the flashing lights would be allowed to honk the truck horn to greet the helpers. Then she turned on the radio and suggested a family sing-along, which she used to buy time between repeating, in response to Hannah's worried questions, that "Dad always knows what to do."
At last the girls spotted the distant flashing red lights of an emergency vehicle barreling up the gravel road. Eryn declared the competition a tie and told the girls that they could both honk the truck's horn as soon as she started waving a friendly hello to the helpers. Then Eryn climbed out of the truck and stood in the middle of the road, in the middle of the desert, in the middle of the night, and flagged the fire truck with a "friendly wave" that was actually a desperate gesture toward the mouth of our driveway, into which the truck sped without slowing down. Climbing back into the pickup, she then told the girls that it would be important to greet every vehicle in just that way and also to observe the number and types of vehicles so they could report to me later exactly what they had seen. Caroline was especially enthusiastic about the game, and so Eryn and the girls repeated their honking and waving another half-dozen times over the next half hour.
Soon, however, the eerie silence fell once again, and while many emergency vehicles had raced up to the house, none had come back down. The last of the vehicles to arrive looked to Eryn like an ambulance, which redoubled her concern. Because she had no phone, could not leave the girls, and did not want to expose them to whatever emergency operations might be going on at the house, she was once again stranded in the dark without support or information.
At only six years of age and possessed of a naturally buoyant personality, Caroline had no trouble playing along with Eryn's games. Ten-year-old Hannah, however, was a great deal more worried, her concern fueled both by her greater understanding of the seriousness of what was happening and also by her naturally caring and fretful disposition. When Hannah began to cry again, Caroline jumped right in to help out.
"Hannahbug, don't worry. An awesome fireman came to my class, and he was a daddy too! It's all good. You can really trust these guys!" Then Caroline did something that was as remarkable as it was mundane, something that was quintessentially Caroline. First she retrieved my broken sunglasses from the side panel of the truck door—cheap shades I had shown her the previous day when one of the lenses popped out and couldn't be refitted—and put them on. Then she took a bright orange ice scraper from the same door panel. And then she threw her head back, curled her lip, transformed herself into a six-year-old Elvis, and began singing passionately into the ice scraper microphone: "You ain't nothin' but a hound dog / Cryin' all the time!" She rocked out until her big sister, still crying, also began to laugh. And once Hannah had laughed good and hard through her tears, Caroline concluded with a drawly "Thankyah, thankyah vury mush." Hannah had comforted Caroline out in the cold, and now her little sister was returning the favor in her own inimitable way. I still consider Caroline's cover of "Hound Dog," performed in my truck that night, to be one of the greatest acts of resistance and resilience imaginable. Although I did not even witness it, this is my most poignant memory from the night of the fire. It plays over and over in my mind, especially in times of stress. I have kept the broken sunglasses and orange ice scraper microphone to remind me of that moment, of its spontaneous, albeit transitory, triumph over circumstances beyond our control.
After another hour or hour and a half of waiting, a second sheriff's deputy rolled up. Eryn was able to flag the officer down and ask him to radio up to the site for information. He did so and reported that I was safe. In that moment Eryn cried for the first time that night. Soon afterward my mom and dad arrived. When Hannah asked why her grandparents were driving out here in the middle of the night, Eryn replied simply that "G and G always come when we need them." We later learned that a neighbor who picked up news of the incident on her police scanner had recognized our address and called my parents, who rushed out from town as quickly as they could.
"I see you girls will do anything to get a slumber party out of me," said my mother, coolly normalizing the situation for the girls. "Let's the four of us go back to town and have hot chocolate," she suggested.
"Great idea," added my father, with his signature calm. "You girls can sleep at our place while Grandpa goes up to check on things. I'll give your dad a lift to town."
It would be another three hours before the firefighters were satisfied that the fire they had chased through the bones of our house was fully extinguished. Thanks to their expertise and efforts, our home was saved. I used my father's phone to call Eryn, update her briefly, and tell her that we hoped to be back in town by daybreak. No, I hadn't been able to locate Lucy the cat, but I had found the puppy, Beau, far out in the desert and had put both dogs safely in the garage for what little remained of the night. I asked if there was anything she needed from the house.
"My phone and my glasses, if you can find them," Eryn replied. "That's not important," she added, "but please be sure to get the girls' valentines off the kitchen counter. They really want to give them out at school tomorrow."
The next morning, after dropping Hannah and Caroline off at school with their valentines and having discreet conversations with their teachers about what had happened, Eryn and I drove with my folks back out to the house to meet the insurance agent and fire inspector. By the time we arrived, the part of the scribble den floor that had not burned away had already been removed. There were large chainsawed holes in walls and ceilings throughout the upstairs and plenty of visible water damage, but the place didn't look as bad as I had feared it might. We even found Lucy the cat hiding behind the headboard of our bed, where she had weathered the fire's storm. Looking down through the missing floor of the scribble den to the room below felt odd, but all things considered, I found myself thinking that we might come out of this OK.
"This doesn't look so terrible," Eryn said to the insurance agent, hopefully. "What do you think?"
The woman hesitated, choosing her words carefully. "Look, I can see that you guys are coping. I think that's great, really. But at some point this whole thing is going to hit you. I just don't want you to be surprised."
As Eryn continued the conversation with the agent, I gained a better appreciation for what "this whole thing" would mean. Because of extensive smoke damage, every piece of electronics in the house—including the wristwatch I had worn while fighting the fire—would have to be thrown out. Any furniture not made of solid wood or tile would also have to go. Every inch of carpet would have to be replaced throughout the house, and the wood floors, which were water damaged, would have to be sanded and refinished. Every wall and ceiling would need to be scrubbed and repainted. Every piece of clothing and bedding would have to be professionally cleaned. Every single book would need to be hand wiped to remove fine ash and then treated in some kind of ozone chamber to remove the smell of smoke. Water damage had destroyed many of my papers and manuscripts. The items that didn't have to be either discarded or professionally cleaned, which were few, would be hauled away to storage during the many months it would take to complete reconstruction. As for reconstruction itself, most of the stone hearth wall would have to come down, as would the walls in the study and the room beneath it, neither of which would be safe to enter during reconstruction. The entire wood stove system would have to be replaced and substantial rewiring done in several rooms. The only room in the house that had escaped significant water and smoke damage was Hannah and Caroline's bedroom.
Before I could process this information, it was time for a debriefing with the fire inspector. Eryn and my mom continued talking with the agent while my dad and I went upstairs so the inspector could walk us through his findings.
"Your fire started here, under the floor," he said, pointing to a hypothetical spot in the immense hole that now gaped where floor struts had once run. "Then it spread laterally, along the joists. This was a true sill-to-sill burn," he added, showing us the charred ends of timbers adjacent to both the interior and exterior walls of the room. "The ignition source was associated with the wood stove system."
"A chimney fire?" my dad asked.
"Absolutely not. This is one of the safest stack and chase installations I've seen. And look at the condition of that pipe. No overheating there. The fire was caused by the wood stove system and started under the floor. Somewhere in the firebox or joint or pipe, there must have been a leak that released a small amount of heat that rose and gathered under the floor."
"Yeah, but I wasn't burning the stove hot last night. I've had it hotter hundreds of times. Besides, how could a little hot air start a fire?" I asked.
"It's more a matter of time than heat," he answered. "You burn this stove every night for four or five months a year for eight or ten years. Even if we're talking a pinhole, over time the timbers get so dried out that their combustion point is lowered. This fire wasn't caused by overburn in the stove. Just a straw-that-broke-the-camel's-back situation."
"What about the leak, this pinhole? Can you tell where it was or what caused it?" I asked.
"Nope, we'll never know. That's what I'm putting in my report. That we'll never know."
" _Never know?_ " I asked in mild frustration.
"Listen," he said, with a hint of sternness in his voice, "I'm a forensic fire specialist, and I've been doing this for a lot of years. If there was a way to know I'd know. It was just bad luck."
"So if nothing could have been done to keep this from happening, it could happen again?" I asked.
"That's right, but this sort of fire is very rare. It's a million-to-one chance," replied the inspector.
"It was a million-to-one chance this time too, right?" I asked.
"That's right," the inspector replied. Then he hesitated for a moment before looking directly at me. "I hear you stayed on the fireground."
"Yeah. Me and a couple of dinky extinguishers and a bucket. Doubt it did any good."
"You see that," he said, pointing to a massive post that had charring within a foot of it. "What does that hold up?"
"Everything," said my father.
"Right, _everything,_ " repeated the man. "The entire structure of both roofs. That post burns, the whole place comes down. You got more water on this than you might think. Clear signs that you cooled and slowed it. It didn't hurt that you've got so much blown-in insulation in this floor—that helped slow it too. But no question that what you did here saved the house."
A few minutes later I walked outside with the inspector, shook his hand, and thanked him for his time. Before he climbed into his truck I wanted to say one more thing.
"It's bothering me that this was just some kind of bad luck that I can't do anything to control. I mean, if it happened before, it could happen again, or something like it." I paused, searching for words. "I've got kids."
"Yeah," he said, in a tone of genuine sympathy. "I know." Then he climbed into his truck and pulled its door shut. He looked at me through the partially opened window, as if it was he who now had something to say but wasn't sure he wanted to say it.
"Look," he continued. "I said that you saved your place. You did. But you shouldn't have. You know how often a civilian is killed in a house fire in this country? I do, because I've seen it. Every two hours and forty-two minutes. That's 24-7, 365, no time off for holidays." Then he started his truck and drove away.
The experience of a traumatic event can result in a variety of stress-induced symptoms, among which are memory loss, flashbacks, and increased worry about the safety of loved ones. But the most fascinating in a long list of stress effects from trauma is fragmentation, which is produced by the inability to tell the story of a distressing event. This incapacity to narrate the experience may be caused by any number of factors, including a desire to avoid painful recollections, an inability to discern meaning in the event, a concern about judgment from others, or an uncertainty about how to represent oneself in the retelling. But in our inability to tell a story—in our fear that we may get the story wrong, or that we might misinterpret its significance, or that its telling will cause us to reexperience a painful event—we risk allowing the story to tell us.
Of course my retelling of what Eryn and the girls went through while I was fighting the fire is a narrative reconstruction. After all, I wasn't with them. But I must explain that the story of what I experienced while fighting the fire is also a reconstruction, one that has been halting and uncertain and that remains unfinished.
In the wake of the fire I remembered little of what had happened during it, and so I've had to resort to physical evidence and to other people's accounts to reconstruct what actually happened that night. I have indelibly sharp memories of certain things: my first glimpse of the flames leaping out of a hole in the floor, the cozy look of our hearth as I passed it in full sprint, the bizarre sluggishness with which water fell from the tub faucet into the bucket. I remember with odd lucidity the sensation of the smooth plastic grip of the bucket handle in my fist, but I do not remember how it came to be there. Because I must have retrieved the bucket from the garage, that is what I have told you I did. The truth is that I do not know how it came to be in my hand. Perhaps it was given to me by an angel or by a ghost. I know what happened that night only in the way we know black holes—by the meticulous extrapolation of data that remain unverifiable by direct observation. I do not know about the fire in the same sense that I know about things I remember.
During the three years that have passed since the fire, pieces of that night have come back to me, a flash here and a flash there, vanishingly thin, crescent-moon slivers of memory often accompanied by powerful emotion. These memories do not arrive in either priority or temporal order. Rather, they are loose pieces of a jumble, fragments that emerge randomly from the bottomless black box of my brain. And many things remain shrouded in the mystery of blackout. To this day I have no memory of pulling my truck out of the garage or telling Eryn to get the girls into it and off the hill. I learned this part of the story from her later, when she also confessed her fear in that moment. "What if that had been the last time I ever saw you, Bubba?" she asks. It is a question I cannot answer and do not try to. When I later discovered five spent fire extinguisher canisters on the floor of my burned-over scribble den—several of which lay beneath the writing table that held my laptop, which was utterly destroyed by heat, smoke, and ash—I wondered how they came to rest there. I had a memory of using two extinguishers, not five, yet there they were. I recall hurling buckets of water many times but only remember filling the bucket once, and I couldn't say whether I ran my alarm shuttle ten times or a hundred. Nor do I recollect the number or type of emergency vehicles that had arrived at my burning house by the time I walked out of it. That is information I recovered from the girls' close observations. The story I have told you is in fact a composite narrative and a reconstruction. If it remains conjectural and elliptical, it is nevertheless as accurate as I am able to make it.
This partial memory loss has been disquieting, but also disorienting has been the fact that my memories of this important event are not only fragmentary but also disjointed, wrenched from the usual calibration of temporal flow. Some things that occurred within seconds in real time still feel to me as if they took much longer, while other events that took hours feel as if they were almost instantaneous. For example, I had guessed the emergency vehicle response time at well over an hour until I was informed that in real time it was only twenty-six minutes. In my memory my father was with me from the time I walked out of the house, while in real time he did not arrive until much later. This experience, and my difficult attempt to recover it, has introduced me to a concept of time that is disturbingly elastic, subjective, and unreliable. For me, "real time" is considerably less real than it once was.
Even more disturbing is the way the fire has challenged my sense of myself as a father. I love my children more than life, and I hope always to be their protector. I want to remove risk of harm from my daughters' home environment. And I want to believe that when a bad thing does happen, I can arrive at an explanation that doesn't resort to terms like "bad luck" or "one in a million." The fire statistics I later read suggest that, on average, one child in the United States dies in a house fire every single day. Behind that cold statistic is someone's son or daughter, somebody's Hannah or Caroline. Well, sure, we might think, but how likely is it that something like that would happen to _me_? I now understand that this question of probability is the wrong one to ask. I can recall a time when one-in-a-million odds of danger sounded reassuring. That's no longer the case.
If, through no fault of my own, my children's lives can be put at risk by some unidentifiable, inexplicable pinhole, how then am I to keep them safe in this troubled and dangerous world? I understand rattlers and scorpions, blizzards and flash floods, and even wildfires, but I have no battle plan for pinholes—no certainty that I can protect my girls from harm in a universe whose fabric is shot through with invisible, long-odds hazards against which no amount of preparation will ever be sufficient. I feel the prick of those pinholes all around me. But what threat should I throw my little bucket of water on next?
I've called parenting an art of improvisation. Much of the ad-libbing we do involves telling our children that everything is going to be fine when we haven't the slightest idea whether it will be fine or not. This "fake it 'til you make it" aspect of being a parent is imperative, because nothing is more important to a child than reassurance. But the poignancy of our sanguine assertions of security weighs more heavily on me than it once did, because my sense of our perpetual vulnerability is now so visceral. How are we to assure our children that we can keep them safe in a world of perils when there is so much evidence to the contrary? Eryn told the girls that "Dad always knows what to do," and it was exactly the right thing for her to have told them in that moment of crisis. But Dad did not have everything under control. Far from it. I was being humbled by the unbridled wildness of fire.
It may be because our memories of traumatic events are so unreliable that writing is widely acknowledged to be an effective technique in the treatment of stress-induced trauma. Beginning with groundbreaking work in the late 1980s, hundreds of studies have investigated the positive health effects of writing about emotionally difficult experiences. Widely replicated and validated, the results of these studies show clearly that the health benefits of this kind of writing are not only psychological but also physical. Blood tests employed in this research show that those who write about their traumatic experiences develop stronger immune systems, and other tracking data show that these writers feel less pain, use fewer medications, require fewer trips to the doctor, and function better in the daily tasks associated with their roles as workers, spouses, and parents. People who write about traumatic events are a great deal more resilient than are trauma victims who do not create a narrative of their experience.
Although the health benefits of writing about trauma are well established, the underlying mechanisms that make this writing so effective remain uncertain. Some researchers posit that writing allows people a safe distance from which to consider and process their most stressful experiences. Others believe that making an event into a story transforms experience into something that can then be let go of. It is also likely that writing triggers a creative form of imaginative reexperiencing, one that, unlike the traumatic event itself, remains under the writer's control. Whatever the reason, we do know that therapeutic writing is effective in relieving trauma-induced stress and that stress correlates strongly with a wide range of health problems.
The novelist John Berger's incisive observation that the work of a writer is "to struggle to give meaning to experience" has special importance when the experience is a traumatic one. And yet the struggle to make sense—perhaps even to make art—of a difficult experience is foundational to the work of any storyteller. Aren't our lives always elliptical, our memories partial, the ultimate significance of our days uncertain? That is the adventure of experience, its beauty and its mystery. I can testify that fire, which has shaped our home landscape, has now also shaped me. And while I still buck, split, and haul the fuelwood that warms our home, my complicated relationship with fire reminds me that we abide with a wildness that can never be fully comprehended. The story of this fire, like the story of a life, must ultimately remain a narrative reconstruction, however fastidiously it may be told.
As a writer myself, I believe the nearly magical positive health effects of narrating trauma must be related to the way telling a story naturally encourages us to wrest cohesion from the disparate, recalcitrant fragments of our life experience. To write is necessarily to fill in gaps where memory fails or information is missing, to reconcile inconsistent facts and impressions, to fabricate a discernible narrative arc, to weave a tale that expresses significance, or at least to craft a narrative artifact from which meaning might later be made. Writing must also be performed in a specific voice, from a particular point of view, and thus represents an important choice on the part of the writer—a choice that the writer lacked while living through the event he or she now struggles to understand and represent. For even the teller of a true tale is a narrator, and every narrator must also be a character. Essential to the writer's work, then, are meaningful decisions about who we are and who we will become. I write not to report what has occurred but to transform what has occurred into a story that makes more sense to me than does reality, which I often find ambiguous and uninstructive. What scientists call "trauma writing" I simply call "writing."
So who, finally, is the protagonist of this fire story? He could be a man who was prepared for any eventuality, who evacuated his family safely, who owned five fire extinguishers and knew where to find them and how to use them. That is the resourceful narrator. Or he could be a courageous man, one who refused to flee in the face of danger, whose bravery saved his family's home. That is the heroic narrator. Then again he may be a man who views the outcome of the fire as a miracle, a blessing, an act of salvation perhaps associated with divine intervention. That is the faithful narrator. Alternatively, he could be a man overwhelmed by circumstances beyond his control, who was overcome by an unforeseen hazard, and who is now immobilized by fear. That is the powerless narrator. Or perhaps he is a hostile man who can neither forgive nor forget, who lashes out blindly at the injustice visited by fire upon his family. That is the angry narrator. Conversely, he may be a philosophical man, one who understands the universe to be comprised of a series of events that remain beyond human ken. That is the accepting narrator.
My narrator is only moderately resourceful, and if he is not quite weak, he is far from heroic. He is neither angry nor fully accepting. He places his faith in his family and in this hard desert and in other wild blessings unrecognizable to orthodoxy. But as the teller of my own tale I am empowered to choose who my narrator will be, and after a good deal of consideration of this important question I've finally made my decision. My narrator is . . . a narrator. The man in my story is one who somehow finds a way to tell it. He is a man who must turn an experience over slowly in his hands until he at last discovers the angle from which it shines. My narrator can braid a rope from sentences and cast it down to himself in a dark well of fire; he can climb to safety by grasping at the knots of his own words. I am crafting these sentences in the same scribble den where I encountered the wildness of fire within the sanctuary of home. I am remembering and transforming, braiding and knotting, climbing hand over hand, word by word, from the fire toward the light.
Hannah and Caroline, you are too young to read this unfinished fire story, but someday you will. And when you do, I want you to remember this: my narrator is above all and forever your father. The truth is that he is not a man who always knows what to do. But he will walk with you, beneath these gorgeous, flaming pinholes of light burning in the unknowable darkness of our high desert night.
_Coda_
The V.E.C.T.O.R.L.O.S.S. Project
During the winter of the big blizzard, Hannah was not yet six years old, while Caroline was just two. It was mid-January when the first storm hit, dumping nearly three feet of snow. Another foot fell a day later and almost as much again a few days after that. By the time the weather finally cleared, we were buried. Because we live so remotely, we enjoy no public maintenance of the terrible dirt road whose dead end leads to our home. Worse still is our own half-mile-long driveway, which is slick with perilous caliche mud whenever it isn't packed with snow and ice. Back in those days we had no tractor, and so a big winter storm meant staying put and waiting it out until a few sunny desert winter afternoons could render the driveway passable.
Anticipating the arrival of the first storm, we had parked our old pickup at the foot of the driveway down at the dirt road. If a sympathetic neighbor with heavy equipment cleared the road, allowing us to chain up and make it to town, we would simply hike the half mile up to our house for a day or two until our hilly, curvy driveway melted out enough to offer safe passage. To our surprise, however, a rare cold front settled in, and that day or two turned into a solid three weeks, during which every trip up and down the half-mile driveway had to be made on foot.
We became quite resourceful during those three unusual weeks. I snowshoed groceries up the hill in a backpack that I learned to load as quickly and safely as a store clerk packs a sack. I switched from beer to whiskey to reduce the haul weight of our supplies. When I had to come or go from the house before daybreak or after nightfall, I snowshoed by the beam of my headlamp, following the route of my own deepening tracks. If the snow became too icy, I resorted to strapping crampons onto my boots. When shuttling the kids, Eryn would often carry little Caroline in a backpack, while I pulled Hannah along in the girls' blue two-kid toboggan. Because I sometimes had to haul both girls up and down the hill myself, I devised a harness system by which their toboggan was connected by ropes to straps that I wore crossed over my chest and shoulders. Using this arrangement, I learned to walk in an entirely new way, leaning into the hill, driving hard with the toes of my boots and the tips of my ski poles as I pulled the girls the half mile up to the house each afternoon. In the mornings I reversed my route, using my harness and ropes to ease their sled down in front of me so as to keep it from schussing away. I soon became as adept as a plow horse, maintaining my rhythm and balance as I slid Hannah and Caroline down to the truck each morning and towed them up to the house every afternoon. For their part, the girls devised a system of their own: Caroline sat between Hannah's knees, with big sister's arms wrapped around her for a "seat belt."
We felt quite isolated during that time, when our home existed as a desert island floating high in a shoreless sea of snow. Huddled by the wood stove, with the electricity sometimes on and sometimes lost, Eryn and I wondered aloud when the sun would return. It was hard work hauling stove wood through the deep, wind-driven drifts, and each night my aching lower back reminded me that every loaf of bread and pint of rye, each library book and school project that came or went from the house, required an arduous trek. But what I remember most vividly from that time is how much fun we had with Hannah and Caroline, who were convinced that coming and going from home by toboggan was as good as life could get. The girls soon referred to me not as Dad but rather as Snow Donkey, and as the days turned into weeks they began to despair that warmer weather would deprive them of an experience that had become the highlight of their winter.
One afternoon, while I was grunting my way uphill with my giggling progeny in tow, Hannah offered a comment that captured the novelty of the entire three-week adventure. "I asked around at school today," she told me. "Do you know how many other kids get to start every day by sledding down a mountain? _None_!"
"You're right, Bug. This is pretty cool, isn't it?" I admitted. "Maybe we should do something special to make sure we never forget how fun this was. It's hard to imagine right now, but someday all this snow is going to be gone."
"How about if I show CC how to make a snow angel?" Hannah suggested.
"Bug, show me the angel!" Caroline replied, enthusiastically. With that Hannah took her little sister's hands, helped her up from the toboggan, and then plopped her down in a spot that, if not for several feet of snow, would have been in the middle of our driveway. There she splayed Caroline's snow-booted feet and mittened hands until the kid looked like a four-armed, bright purple starfish. Then she helped Caroline to flap her arms and legs up and down in the snow, after which Caroline took over and did plenty of energetic flapping of her own. There was laughter all around, and when Hannah and I each grabbed a mitten and lifted Caroline back to her feet, we discovered that she had created a striking little work of snow art. Where her arms had flown, the snow did in fact resemble wings in motion, while the pattern left by her flailing legs impressed the snow with twin marks that looked like a pair of ringing Christmas bells.
Within a week the snow was gone and with it any visible sign of Caroline's angel. But if her snow art was short-lived, my recollection of it has not been. In the many years since that blizzard I have walked over the angel spot many hundreds of times, and I never pass it without reflecting on that beautiful winter's day and the small ritual of appreciation we devised to celebrate it. Caroline's snow art was ephemeral, but my memory of it is indelible. Every trace of her angel is gone, but the angel is not.
A storied landscape is one in which memorable associations have been forged between specific places and the meaningful experiences we've had in those places. Looking out across our home desert, I see layers upon layers of these associations—strata of rich experiences now so textured, so interwoven with the land, as to have created the fabric of a shared life. Here is where Hannah saw her first pronghorn. There is where fire crested the ridge. Here is where I discovered the paw prints of that wayward Sierra bear. There is where the redtail dropped a live rattlesnake from the cloudless sky. Here is the juniper where the great horned owls nested last year. There is the spot where Caroline became, for one cherished, unforgettable moment, an angel.
Take a moment to imagine the landscape you now inhabit or, alternatively, a treasured landscape from your past. What memories have you attached to that place? How has that place helped to shape the person you are today? How have your experiences there informed your way of seeing yourself, your family, the place itself? Now try this thought experiment. Imagine a landscape—every landscape—in which emotionally, intellectually, aesthetically, or spiritually significant vernacular experiences are geographically located, recorded, and commemorated. What might a place like that look like? How would our imagination of the landscape change if the small but vitally important events that happen to regular people every day were neither privatized nor erased but instead communally celebrated—memorialized in our conception of our local geography? "I was engaged to be married here, standing waist-deep with her in this snaky slough in the Fakahatchee swamp." "My sister had a life-changing realization there, under that broken-topped cottonwood tree across the arroyo." _You_ had an intimation of mortality or immortality _there_ , you fell in love, or perhaps you breathed your final breath right _here_ —on _this_ spot. Now go further: imagine a map of the world containing an individualized, place-based record of every important vernacular experience that has ever occurred. We, our people, our sons and daughters, were born here, somewhere, in a place. Someplace is where we experienced ineffable joy or grief; _some place,_ a poignant, hidden place unmarked on every map. In the cartography of memory—the cartography of the spirit—our local landscapes are rich with such invisible associations.
Imagine from your childhood a place that is rich with layers of memory, experience, and imagination. This is not simply a place _where_ things happened, it is a place _because_ things happened—the events of perception, memory, and experience cause this place to glow forth from an otherwise undifferentiated quotidian geography. Even if they build a light-spewing Walmart on your place, would the place beneath it be any less hallowed? If we pave paradise and put up a parking lot, wouldn't you still care about the exact place where you were born, or fell in love, or where you will die, even if it were entombed in asphalt? Shouldn't we somehow mark the spots where we gave or received compassion—where we witnessed the triumph of the better angels of our own or a fellow traveler's nature? Some of us believe we should.
Allow me to briefly explain the inspiration, goals, and benefits of a new environmental and emotional global mapping project called V.E.C.T.O.R.L.O.S.S.—an acronym that stands for Vernacular Experiential Cartographic Traces on Regions, Landscapes, or Specific Sites. The project seeks to map—in both space and time, and both in narrative forms and through accurate Global Positioning System (GPS) coordinates—the precise locations of transformative vernacular experiences. Our project aims to significantly change how landscapes are mapped, described, and remembered, and it has the potential to influence not only the ways places are conceived, named, and marked but also how they are treated. At present, we are using seed grants to conduct a pilot version of the project.
Phase one of V.E.C.T.O.R.L.O.S.S. involves gathering significant place-based vernacular narratives, however brief or informal, accompanied by a set of GPS coordinates identifying the exact spot upon which the described event took place. In the project's second phase we construct precise maps of various scales depicting landscapes as they have been animated by potentially innumerable vernacular experiences. The third phase involves GPS-matrixed cartographic draping (to use the technical term), through which we generate experientially specific layers of mapped data that can be overlaid, like multiple transparencies, to allow for more sophisticated terrain visualizations. For example, here is a map of all the spots in the township of Washington Park (in Washington County, Washington) where people reported falling suddenly, hopelessly in love. Now we can "drape" over this love map any other data set—say, a map of all the places in the township where someone picked up a beautiful rock, or saw a ghost, or witnessed the sun break through the clouds in absolute majesty. The fourth phase requires that we project these draped data sets along an infinitely extended temporal axis. So here is a map of all the places in your town where someone now long gone experienced a transformative moment of inspiration or, conversely, a painful moment of religious doubt. As our data sets proliferate over time, we can successively drape onto this other maps depicting different sites and events in the same area over time. For example, we can overlay on the inspirational moment or religious doubt map an infinite number of other maps—say, maps of Native American geoglyph or of hidden veins of iron ore, of copper bootlegger kettles, or of the powdered bones of buried dogs.
We do not presently have adequate technology to achieve the fifth and final phase of the project, but we're confident that we soon will. This fifth phase is built upon the platform of all the accrued and draped vernacular event cartographies I've described—it is a kind of single meta-hyper-draped map—but also extends to include the record of all important events in the nonhuman natural world. Of course we can already map a fossil bed or earthquake fault, but we do not know the precise spot where, about ten thousand years ago, the last American cheetah crushed the skull of a pronghorn antelope in her jaws and filled her gut before lying down to die beneath the glowing light of the Pleiades. We know that this event did occur; we simply don't know precisely where and when. In its fullest articulation, this final phase of V.E.C.T.O.R.L.O.S.S. is so ambitious as to be difficult for some people to imagine. But who wouldn't want to know the location of the exact branch upon which the last passenger pigeon to survive in the wild built its nest?
If phase five sounds unrealistic, consider this: we've already mapped the imbricated labyrinth of the human genome. And we mapped the human genome because that's what we _wanted_ to map. We made a choice about what to observe, record, and remember—just as every map is a mirror that reflects, in its scale, frame, and minutiae, the cartographer's values and sensibility, fears and dreams. Or consider that we have an excellent record—through written accounts, photographs, and video—of the precise spot on the particular tree where the ivory-billed woodpecker, long believed extinct, was once again sighted, thus bringing a momentary infusion of hope to the world. _That_ place in the forest, at _that_ moment on _that_ day, can be located in space and time—it can and has been recorded—so as never to be forgotten. To map such a place in time is vitally important for spiritual as well as scientific reasons, for what would it say about us as a people if we simply forgot which tree it was or which day? What if we felled that tree and made ammunition crates from it because nobody bothered to remember?
Now simply extrapolate from this example. All that is required for the success of V.E.C.T.O.R.L.O.S.S. phase five is that we make a collective commitment to continue this sort of attentiveness for the next, say, ten thousand years, after which, if we have cared enough to notice, we should know the exact location of each sparrow's fall—of every flash of black and red and ivory that restores hope to the world—at least within the sphere of human observation. To extend our project's scope beyond the radically circumscribed sphere of human observation will be our children's work.
The V.E.C.T.O.R.L.O.S.S. pilot program involves the collection of what we call "vernacular witnessings," which are GPS-located narratives of meaningful personal experiences in place.
For example, here is vernacular witnessing number 780, received by text on October 14, 2014—and, of course, accompanied by accurate GPS coordinates: "I had just come off the flight after visiting my mother in LA. As I was going through the security gate a man in front of me began to stagger, and a woman—his wife, I'm sure—tried to hold him up, but he fell forward onto the floor, then rolled, clutching his chest and moaning. The poor woman was so scared, and she started screaming for help. I'll never forget the look of terror on her face. The security guards ran over and cleared people back, and the EMTs arrived quickly, lifted the man onto a stretcher, rushed through a security door to an ambulance that almost in a single motion pulled up, loaded the gurney in, and sped away. There was this odd silence for a few seconds. Then one of the guards bent down and picked up a pen that had fallen out of the man's breast pocket and slid it into his own breast pocket. After that they just started running everybody through again, and in less than a minute there was absolutely no trace of the man. I sat on a bench and tried to calm myself down. Anybody who got in that line even a minute later had no idea what had happened there, and by the end of the day thousands of people had stepped on the exact spot where his cheek had hit the floor. Maybe millions have stepped there by now. And I don't even know if he lived or died. Shouldn't there be a trace of him—some trace?"
And here is vernacular witnessing number 1270, received by e-mail with GPS coordinates on April 9, 2016. "My wife was pregnant with our first kid, and we were right around the due date when her water broke in the middle of the night. So I jump up and I'm tired and mixed up, like when you wake up from a weird dream and then realize you're still dreaming it. So I run around grabbing our stuff up, and then we head out the front door. It's real frosty outside. I turn on the porch light, and we start walking down the walk to the truck. In the eaves of our house, right over the walk, there's this swallow's nest—just mud and sticks and horsehair. Barn swallows. They've been there five years running and raised up a batch of young every year and two batches last year. Even though they crap on the walk I never wanted to chase them out—especially with those babies in there. So I'm walking to the truck to drive into town for my kid to get born, and as I'm walking a feather falls out of this nest, somehow, and it drifts down. Well, without changing my stride I just catch it in my hand and keep going. That next morning our daughter was born, and healthy. I know it sounds like a steaming pile of horseshit, but that's what happened. And that's _where_ it happened—right there, between the front door and the truck, under that nest."
V.E.C.T.O.R.L.O.S.S. has already generated a number of provocative research questions that will guide the future of the project. Here are several of them:
**QUESTION 1**
_What is the nature of vernacular experiential hot spots and clusters?_
Our initial research suggests that an exhaustive analysis of any single geographical spot over time is extremely revealing. For example, as we gather more data and begin draping data set upon data set, map upon map, we find that certain places have been especially active loci of significant vernacular events. Even preliminary layered cartography reveals, for example, that in the past half century a variety of important events have occurred at one spot on a high bank above the north bend in the Washington Run River (near Washingtonville, Washington): thirty-seven people saw the full moon rise, twenty-two people had good ideas, twelve performed acts of kindness, eight people had sex (two of them for the first time), four wrote all or part of a song or poem, three children saw their first shooting stars, two couples were engaged to be married (though years later one of those couples decided, on the same spot, to divorce), and one pregnant woman went into labor. Yet comparable draped data sets for a similar bank on the next bend of the Washington Run reveal a sort of experiential wasteland. Here two day hikers contracted giardiasis, and one old fisherman caught a catfish that already had a hook in its mouth. Nothing more—at least so far.
Obviously, extending our analysis into the future might produce very different results. We hope that enriched data and increasingly sophisticated analysis will help us explain the nature of these mysterious vernacular hot spots within local landscapes. Once we have a complete map of these unusual places, we feel it would be appropriate to construct holographic memorials that convey the remarkable richness of these places over time. Obviously it will be necessary to develop and integrate mobile apps for use by the many who will make pilgrimages to these special places.
**QUESTION 2**
_Is there also a vernacular landscape of missed opportunities?_
On April 6, 1327, during "the first hour of the day," in one particular spot in the Church of St. Claire in Avignon, France, Francesco Petrarch cast his eyes upon Laura, a young woman whom he would never know but a mere glimpse of whose beauty would inspire in him a lifetime of radiant poems. On the one hand, this is obviously a transformative vernacular experience in a local landscape; on the other, it is also a poignant missed opportunity. While we have had good initial success gathering vernacular witnessings of events that did happen, we also seek a way to map those events that _didn't_ happen—or, put otherwise, to chart events that might have occurred but did not.
Imagine a windy, dust-colored crossroads beneath a single power line, under an immense sky, in the middle of a vast ocean of sagebrush. Here a young man waits in the dry wind for the bus that will take him to town for the weekend. On that bus is the woman with whom, unbeknownst to him, he is about to fall deeply in love. Perhaps one of their children will cure cancer or negotiate peace in the Middle East—who can know? In any case, these two people are destined for a lifetime of shared affection, perhaps fifty fine Christmases together. As it turns out, however, this crossroads is not the locus of the significant vernacular experience. That spot is a mile and a half up the road, where the bus sits motionless at the roadside while its driver sweats in the desert heat as he changes the left rear tire, which has just blown out. The man, becoming impatient, at last walks back to his little house. Twenty-two minutes later the bus speeds by. Will the man eventually die after a long and empty life? Perhaps not, but who can know? The crossroads at which the man stands must now be placed on a different kind of vernacular map—a map of missed opportunities. That is, the spot where the man is not standing when the bus passes is a vitally important place in a global matrix of places where life-changing events almost happened. We hope to develop more sensitive mechanisms to record such incidents, since we cannot depend upon vernacular witnesses to record events that did not occur. Do you have such missed appointments in your own life? In time, V.E.C.T.O.R.L.O.S.S. might help you to find out.
**QUESTION 3**
_Is there an unexperienced inch of ground on earth?_
While the answer will depend upon the temporal and experiential parameters we establish for our inquiry, this question remains extremely important to our project's staff. We ask one another this: When V.E.C.T.O.R.L.O.S.S. is complete, will it be a map of everything? Most of our researchers now agree that, assuming sufficient temporal depth in the model, results will show that every point on earth, however remote or minute, has at some time been the site of some significant human experience. But there are still a few holdouts, mostly among the senior researchers, who cling to the perhaps romantic belief that somewhere on earth (maybe only in a few spots, they acknowledge) there exists some bit of ground—a few centimeters, perhaps, of tundra, ice, or rock—that yet remains unmarked by human experience. The search for these unexperienced vernacular landscapes has become a specialty for certain researchers, whose quest is not for sites of vernacular experience but rather for the mythic purity of a place that remains free from them. On one thing, however, we all agree: if such places do exist and can be located, they will need to be placed on an entirely new kind of map.
**QUESTION 4**
_Can the project's findings be used to protect places?_
If you are Native American and can prove that a developer's planned golf course is about to be constructed atop your people's ancestral burial ground, you have at least a chance of saving your sacred site. But some of the V.E.C.T.O.R.L.O.S.S. scientists have taken this kind of resistance to a radical extreme. These researchers—closet activists, really—are motivated by the belief that the project, in proving the richness of vernacular experience in places, can provide ammunition to the enemies of development. I might as well confess that I am one of these people.
Consider this. Even in our current judicial system certain relationships between people and nature are constitutionally protected as expressions of religious freedom. I once read of a case in which some folks prevented construction of a Walmart in their local forest by persuading a judge that they used that forest as a place to pray. The day after reading the article I joined the Universal Pantheist Society (I've been a dues-paying member ever since) and began documenting my devotional use of my favorite place—a hanging canyon hidden at 8,000 feet in the forked embrace of the split summit of my home mountain. I now have hundreds of pages of dated journal entries describing the spiritually illuminating experiences I've had in this high, wet canyon here in the heart of sage country, and I've taken more than seven hundred photographs, now spanning almost fourteen years, of myself "praying" in front of juniper and mountain mahogany, pack rat nests and granite boulders, the perennial spring at the head of the canyon, the carcass of a mule deer freshly killed by a mountain lion.
When they try to build the ski lodge or drill the fracking well there, I'll be ready. I have demonstrable evidence that I am a long-term member of a recognized religious sect (we pantheists even have a newsletter) and that my religious freedom would be threatened if I am deprived of this specific site. And because I worship all the natural elements of this montane desert ecosystem in their complex relationship to one another, my conscience will not allow me to sacrifice even a single serrated leaf of a snow-bent cottonwood, or a button from the vibrating tail of a Great Basin rattler, or a tuft of reddish fur from the ear of a coyote pup crossing this high meadow full of snowberry on its way to seek shelter among the aspen. I won't burn your church, you don't pave my mountain. I have incontrovertible evidence, and I'll take my case to the Supreme Court if I have to.
Now extrapolate from this to your own important place. If V.E.C.T.O.R.L.O.S.S. can scientifically prove that your place is comparably rich with associations that are crucial to your faith—whatever that faith might be—then it might help you make the case that they should not build the Walmart on the spot where your grandfather set the cornerstone of his farmhouse, or where your son caught his first glistening brook trout, or where your little brother saw a tanager flash scarlet in the emerald canopy of a shagbark hickory . . . or maybe that spot on the summit of Moonrise where your daughter, on her sixth birthday, discovered the curving, bone-white arch of a perfect mule deer antler.
In mathematics, a _vector_ is "a quantity that has direction as well as magnitude," and it is represented symbolically by "a line drawn from its original to its final position." Each of our lives is also a vector whose representative story line—the narrative of its trajectory through time and space—can be plotted only along the points of our significant vernacular experiences in place. The tragedy of our age is how discontinuous and fractured that line has become, how in desperation we are forced to splice its missing fragments together with misinformation from distorted, alien maps that omit or obscure the lost story of our own being in the world.
I have a dream that someday the V.E.C.T.O.R.L.O.S.S. witnessing archives will occupy hundreds of acres of buildings full of vaults in which all of our most important vernacular experiences are expressed and located, never to be lost. Numberless letters, e-mails, postcards, text messages, tweets—even scribbled cocktail napkins or photographs of drawings etched with sticks into damp sand—each with a small story and an accompanying set of numbers that describes a specific location in space. Countless moments of illumination and disappointment and transformation, placed carefully onto a fully textured map of the living world. Even in its earliest phases, the project has demonstrated that every map, however fastidiously drawn, obscures a rich, invisible landscape of important vernacular associations with the land. And these local experiences do not simply accrue in places over time, piling up like stones in a rising cairn that marks a trail along the ridgeline leading to the future. Instead, the vernacular landscape is a palimpsest, a page of geo-experiential manuscript upon which holograph records are written, overwritten, erased, elided, interpolated, inserted, canceled, and then written again, sometimes in languages not yet spoken, sometimes in a forgotten language.
I believe that the infinitely draped cartography of significantly experienced places, if it is ever completed, will simply be the map of all that is holy. Every spot it depicts will be sacred ground—will be the site of some otherwise invisible deer antler or snow angel. And who among us would not kneel, if in kneeling we could touch a furrow of bark from that tree where the shining ivory bill momentarily reappeared, ghostlike, from the world where all that is lost is said to remain forever? Only then will we finally know what to tell our children when they ask. We'll tell them that all of the earth is a reliquary—that every chip of bark and stone and bone turned out to be a chip of the true cross. If they don't believe us they can check the map, where the first thing they will discover is the place and moment of their own birth, so precious to us, already recorded there.
Acknowledgments
Writers are very much in need of friends, and I have been fortunate to have so many in my life and in my corner. Here I offer my sincere thanks, along with equally sincere apologies to anyone I may have neglected to include.
Among fellow writers of environmental creative nonfiction, my thanks go to Rick Bass, Paul Bogard, John Calderazzo, SueEllen Campbell, Laird Christensen, Casey Clabough, Jennifer Cognard-Black, Chris Cokinos, John Elder, Andy Furman, Dimitri Keriotis, Ian Marshall, Kate Miles, Kathy Moore, John Murray, Nick Neely, Sean O'Grady, Tim Palmer, Bob Pyle, David Quammen, Eve Quesnel, Janisse Ray, Suzanne Roberts, Chris Robertson, Leslie Ryan, Terre Ryan, Gary Snyder, John Tallmadge, David Taylor, and Rick Van Noy. Very special thanks to David Gessner, John Lane, and John Price, whose support has been decisive.
Thanks also for the encouragement I've received from other friends in the environmental literature community, including Tom Bailey, Patrick Barron, Jim Bishop, Kate Chandler, Ben Click, Nancy Cook, Jerry Dollar, Ann Fisher-Wirth, Tom Hillard, Heather Houser, Richard Hunt, Dave Johnson, Rochelle Johnson, Mark Long, Tom Lynch, Kyhl Lyndgaard, Annie Merrill, Clint Mohs, David Morris, Dan Philippon, Steve Railton, Heidi Scott, Robert Sickels, Dave Stentiford, Jim Warren, and Alan Weltzien.
I've been fortunate to benefit from productive collaborations with many talented and industrious editors. Following are a few of these folks, along with the magazine or press at which they worked at the time I received their help: Chip Blake, Jennifer Sahn, Hannah Fries, and Kristen Hewitt ( _Orion_ ); David Gessner, Ben George, and Anna Lena Phillips ( _Ecotone_ ); Stephanie Paige Ogburn, Jodi Peterson, Paul Larmer, Tay Wiles, Michelle Nijhuis, Diane Sylvain, Cally Carswell, Emily Guerin, and Kate Schimel ( _High Country News_ ); Kate Miles ( _Hawk & Handsaw_); Chris Cokinos ( _Isotope_ ); Nick Neely ( _Watershed_ ); Rowland Russell ( _Whole Terrain_ ); Nancy Levinson ( _Places Journal_ ); Jamie Iredell ( _New South_ ); Mike Colpo (Patagonia's _The Cleanest Line_ ); Tara Zades ( _Reader's Digest_ ); Justin Raymond ( _Shavings_ ); Jeanie French ( _Red Rock Review_ ); Bruce Anderson ( _Sunset_ ); Caleb Cage and Joe McCoy ( _The Nevada Review_ ); Fil Corbitt ( _Van Sounds_ ); Jason Leppig ( _Island Press Field Notes_ ); Brad Rassler ( _Sustainable Play_ ); Barry Tharaud ( _Nineteenth-Century Prose_ ); Greg Garrard (Oxford University Press); George Thompson (GFT Publishing); Jonathan Cobb (Island Press); and Boyd Zenner (University of Virginia Press).
I want to express my sincere gratitude to the terrific team at Roost Books, whose work on _Raising Wild_ has been exemplary from the start. Thanks to assistant editor Julia Gaviria, copy editor Diana Rico, and proofreader Emily White for seeing the manuscript down the final stretch, and to art director Daniel Urban-Brown and designer Jess Morphew for making it a thing of beauty. My thanks to sales and marketing manager KJ Grow, publicity director Steven Pomije, and publicity and marketing coordinator Stephany Daniel, whose excellent work has helped this book to find its readers. Most important, I offer my deepest and most heartfelt thanks to my editor Jennifer Urban-Brown. My collaboration with Jenn has been among the most productive and enjoyable of my career, and I can only hope that folks who believe that a writer's relationship with their editor must be adversarial might someday be as fortunate as I have been in having such a supportive, patient, and insightful partner in their work.
It is fitting that _Raising Wild_ should have found a home at Shambhala, given the strange and wonderful way in which my own path and that of the press came to cross. Shambhala Publications emerged from Shambhala Booksellers, which began in 1968 in the back of the storied Moe's Books, on Telegraph Avenue in Berkeley. Back in early August 2002, my wife, Eryn, and I were on our way from the high desert over to the San Francisco Bay to root for the Giants as they took on the Pirates. As usual, we left time to visit the Berkeley Hills and peruse the shelves at Moe's. It was there, as I admired the plates in a beautiful edition of Audubon's _The Birds of America_ , that Eryn emerged from the bathroom waving the small wand of a pregnancy test above her head. Through the wand's tiny window we caught a glimpse of our future: two parallel magenta lines that gave the first indication we would become parents. The kid presaged by that magic wand at Moe's Books turned out to be Hannah, who is now twelve years old. She and her little sister, nine-year-old Caroline, are the figures at the heart of _Raising Wild,_ serendipitously published by Shambhala, which also traces its origin story to Moe's.
Closer to home, I'd like to offer thanks to fellow Great Basin writers Bill Fox, Shaun Griffin, Ann Ronald, Rebecca Solnit, Steve Trimble, Claire Watkins, and Terry Tempest Williams, with a nod to the desert writers who led my way: Mary Austin, Ed Abbey, Ellen Meloy, and Chuck Bowden. Thanks to my colleagues in the MFA program at the University of Nevada, Reno: Steve Gehrke, Ann Keniston, Gailmarie Pahmeier, Susan Palwick, and, especially, Chris Coake. And thanks to my students in the courses on American humor writing, place-based creative nonfiction, and western American literary nonfiction that I taught during 2014, 2015, and 2016. I have also been encouraged by the stalwart readers of my "Rants from the Hill" essay series at _High Country News_ online, where more than a hundred thousand folks have been kind enough to spend five minutes with my unusual way of seeing the world.
Among Reno friends, I've received valuable support from Pete Barbieri, Mike Colpo, Fil Corbitt, Dondo Darue, David Fenimore, Daniel Fergus, Mark Gandolfo, Betty Glass, Torben Hansen, Aaron and Diana Hiibel, Kent Irwin, Rich Kentz, Tony Marek, Ashley Marshall, Katie O'Connor, Eric Rasmussen, and Meri Shadley. Special thanks to my closest friends, Colin and Monica Robertson and Cheryll and Steve Glotfelty. The most significant support I have received outside my family came from Cheryll, whose encouragement has been essential to my growth as a writer.
I am blessed with a family that is exceptionally tolerant of my eccentricities and ambitions, my fierce sense of place and idiosyncratic sense of humor. On the other side of the Sierra, thanks to our Central Valley people: O. B. and Deb Hoagland, Sister Kate and Uncle Adam Myers, Troy and Scott Allen, and all the cousin critters. Here on the Great Basin side of the big hill, more thanks than I will ever manage to express go to my folks, Stu and Sharon Branch, who have directly or indirectly enabled everything I've accomplished in life. My wife, Eryn, is all I ever dreamed of in a partner—loving, patient, smart, creative, funny, generous, and encouraging; this book could never have been written without her constant support.
I often tell our daughters that "it takes a family to make a book." The dedication of a book is the most sincere gesture of gratitude available to a writer, and I have dedicated _Raising Wild_ to Hannah and Caroline, whose wild desert upbringing inspired it. I hope my record of our shared experiences up on this high desert hilltop will seem to them even sweeter as the years go by.
Credits
Many of the chapters in this book had their first life as essays published in magazines, though the versions that appear here are very much expanded and revised (and in several cases retitled). Information on first publication (and, as necessary, original titles) appears below. I am deeply grateful to these magazines and their editors for their support of my work.
"Endlessly Rocking." _Ecotone: Reimagining Place_ 2 _,_ no. 1 (Fall/Winter 2006): 20–37.
"The Nature within Us." Originally published as "Couvade Days." _Whole Terrain: Reflective Environmental Practice_ 15 (2008): 44–47.
"Tracking Stories." Originally published as "Ghosts Chasing Ghosts: Pronghorn and the Long Shadow of Evolution." _Ecotone: Reimagining Place_ 4, nos. 1 and 2 (January 2009): 1–19.
"Ladder to the Pleiades." In _Let There Be Night: Testimony on Behalf of the Dark,_ edited by Paul Bogard, 74–84. Reno: University of Nevada Press, 2008.
"The Adventures of Peavine and Charlie." Originally published as "The Adventures of Peavine and Charlie: A Journey through the Imaginative Landscape of Childhood." _Orion_ 30 _,_ no. 1 (January/ February 2011): 58–63.
"The Wild within Our Walls." Originally published as "Nothing Says Trash Like Packrats: Nature Boy Meets Bushy Tail." In _Trash Animals: The Cultural Perceptions, Biology, and Ecology of Animals in Conflict with Humans,_ edited by Phillip David Johnson and Kelsi Nagy, 139–49. Minneapolis: University of Minnesota Press, 2013.
"Playing with the Stick." Originally published as "Sticking with the Stick." _Hawk & Handsaw: The Journal of Creative Sustainability_ 5 (2012): 68–73.
"Freebirds." Originally published as "Freebirds: A Thanksgiving Lesson in Forgiveness." _Orion_ 30 _,_ no. 6 (November/December 2011): 44–49.
"Finding the Future Forest." Some passages in this chapter are derived from the following two sources: "Lifeblood of the Desert," _Tahoe Quarterly_ (Fall 2007): 55–57; "Finding the Forest," _Orion Afield_ 3 _,_ no. 4 (Autumn 1999): 10–14.
"My Children's First Garden." Originally published as "My Child's First Garden." _Hawk & Handsaw: The Journal of Creative Sustainability_ 1 (2008): 56–65.
"The Hills Are Alive." _Places Journal,_ January 2012. https://places-journal.org/article/the-hills-are-alive.
"The V.E.C.T.O.R.L.O.S.S. Project." _Isotope: A Journal of Literary Nature and Science Writing_ 5 _,_ no. 2 (Fall/Winter 2007): 2–9.
About the Author
Michael P. Branch is professor of literature and enviornment at the University of Nevada, Reno, where he teaches creative nonfiction, American literature, environmental studies, and film studies. He has published five books and more than two hundred essays, articles, and reviews, and his creative nonfiction includes pieces that have received Honorable Mention for the Pushcart Prize and been recognized as Notable Essays in _The Best American Essays_ (three times), _The Best American Science and Nature Writing_ , and _The Best American Nonrequired Reading_. His work has appeared in many magazines, including _Orion_ , _Ecotone_ , _Utne Reader_ , _Slate_ , _Places Journal_ , _Whole Terrain_ , and _Red Rock Review_. His widely read monthly essay series, "Rants from the Hill," has received more than one hundred thousand page views at _High Country News_ online (hcn.org); a book-length collection of those essays is forthcoming from Roost Books.
Mike lives with his wife, Eryn, and daughters, Hannah Virginia and Caroline Emerson, in a passive solar home of their own design at 6,000 feet in the remote high desert of northwestern Nevada, in the ecotone where the Great Basin Desert and Sierra Nevada mountains meet. There he writes, plays blues harmonica, drinks sour mash, curses at baseball on the radio, cuts stove wood, and walks at least 1,200 miles each year in the surrounding hills, canyons, ridges, arroyos, and playas.
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Lane 1 : HeLa Cell Lysate;\r\nLane 2 : A431 Cell Lysate;\r\nLane 3 : PC12 Cell Lysate;\r\nLane 4 : NIH3T3 Cell Lysate; probed with MEK1 (32G3) Monoclonal Antibody, unconjugated (bsm-50303M) at 1:5000 overnight at 4°C followed by a conjugated secondary antibody at 1:5000 for 60 minutes at Room Temperature.
Storage HEPES with 0.15M NaCl, 0.01% BSA, 0.03% sodium azide, and 50% glycerol. Store for 1 year at -20°C from date of shipment.
Background Dual specificity protein kinase which acts as an essential component of the MAP kinase signal transduction pathway. Binding of extracellular ligands such as growth factors, cytokines and hormones to their cell-surface receptors activates RAS and this initiates RAF1 activation. RAF1 then further activates the dual-specificity protein kinases MAP2K1/MEK1 and MAP2K2/MEK2. Both MAP2K1/MEK1 and MAP2K2/MEK2 function specifically in the MAPK/ERK cascade, and catalyze the concomitant phosphorylation of a threonine and a tyrosine residue in a Thr-Glu-Tyr sequence located in the extracellular signal-regulated kinases MAPK3/ERK1 and MAPK1/ERK2, leading to their activation and further transduction of the signal within the MAPK/ERK cascade. Depending on the cellular context, this pathway mediates diverse biological functions such as cell growth, adhesion, survival and differentiation, predominantly through the regulation of transcription, metabolism and cytoskeletal rearrangements. One target of the MAPK/ERK cascade is peroxisome proliferator-activated receptor gamma (PPARG), a nuclear receptor that promotes differentiation and apoptosis. MAP2K1/MEK1 has been shown to export PPARG from the nucleus. The MAPK/ERK cascade is also involved in the regulation of endosomal dynamics, including lysosome processing and endosome cycling through the perinuclear recycling compartment (PNRC), as well as in the fragmentation of the Golgi apparatus during mitosis. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,065 |
"What is the World Mountain?" Slava asked Sensei.
"Wow, it's the mountain for the select ones!" Kostya made his conclusion with admiration.
Stas could hardly wait until Sensei replies Kostya's questions and uttered with impatience, "Does it mean the World Mountain doesn't look like this mountain?" he motioned to the disappearing heavenly image of the mountain with melting-down snowy hat on its sharp-edged top.
"Does the World Mountain really exist?!" Andrew asked with distrust. Like all of us, he held his breath before that in order to hear Sensei's answer to Stas better.
"Is it associated even with Shambala?" Volodya got surprised.
"Sure. There is plenty of such interweaving of legends about the World Mountain with legends about Shambala. But it is understandable. For people who know nothing about Shambala or the cosmic Mount Meru, both located 'somewhere there', nobody knows where, all these concepts will certainly merge together in one and the same idea. Though in fact there is a big difference between them. Shambala is located between the real eternal world of God and the temporarily existing material universe. It's the abode of Bodhisattvas. Whereas the World Mountain is located amidst the worlds of the material universe, and it is a place of visit of wise beings, like our enlightened people or demigods as they were called by ancient people, that is those who reached a high spiritual level. Therefore in ancient legends this mountain was associated with achievement of the genuine human happiness and immortality.
… Buddha became able to visit the World Mountain. Owing to Buddha and some of his followers who possessed this knowledge, the information about Mount Meru became known across the Orient. Buddhists, by the way, like Hindu, described this mountain as a pistil of the lotus flower – the sacred flower for both religions, and this image was more associated not with the mountain itself, but with main elements of a dynamic meditation for preparation to visit Meru.
Or let us take our Slavonic 'heathenism'. What is noteworthy about it is that knowledge and practices about the World Mountain were accessible to ordinary people, unlike 'usurpation' of this knowledge by priests of other peoples. For a long time on Slavonic territories people practiced ancient rituals of magic 'flights' to the cosmic mountain, which were held on a sacred hill similar by its shape to outlines of the World Mountain. By the way, in old times people celebrated big holidays not because they wanted to enjoy themselves. From time immemorial holidays were arranged on the days that coincided with certain cosmic cycles, and people started celebrations not in the morning as it is habitual today, but in the evening, at the rise of the first star.
In the course of time, when 'heathenism' was extirpated by fire and sword among Slavonic peoples and the Christian faith was imposed, they began to persecute servants of 'heathen cults who were labelled as 'devil's offspring', 'servants of the devil'. At that, 'pagan' ancient rituals were labelled as sabbaths and festivals of 'satanic forces' hostile to people. Generally speaking, after physical annihilation of most of the magi, the knowledge about the World Mountain was lost, and what remained was distorted to a negative side beyond recognition. Holidays were renamed, although due to such renaming the occurrence of certain cosmic cycles didn't change of course, and neither did the holiness of relevant places. Thus, Christian churches were built on the sites of former heathen temples of chief pagan gods. But, as they say, this is already history.
A similar image of the World Mountain in the centre of the universe was known to Altaians, though they called it Altyn-Tu. According to their beliefs, the foundation of this golden cosmic mountain was fastened to the heaven (i.e. its widest part was above and its narrowest part was beneath), and its top hung above the earth at the distance 'equal to the length of a human shin'. Moreover, they had other widespread names of the World Mountain, e.g. Mount Sumeru, with stars revolving around it. It was also known by Kalmyks and many other peoples in Central Asia. According to myths of the Altai people, there are 33 tengri, or 33 gods on it.
The Chinese call the World Mountain Kunlun. According to their beliefs, it is possible to go through it to the highest spheres of the universe. It was considered to be something like 'paradise'. There is a following record in one of ancient texts, "The one who would go up from Kunlun twice as high will reach the Mountain of Cool Wind and gain immortality; the one who would go up twice as high will reach the Hanging Ground and gain miraculous abilities, having learnt to manage the wind and the rain; the one who would go up twice as high will reach the heaven, the abode of Tai Di – the supreme lords, and will become a spirit." For unconversant people it's a nice fairy tale, while for knowledgeable people it is only a hint.
By the way, such association of the World Mountain with paradise is mentioned in the Bible as well. There you can find separate evidences pointing out to echoes of knowledge about the World Mountain, e.g. that it is located in the centre of the universe, that God descends to it, that there is kind of a paradise on it, that the mountain is surrounded with rivers that symbolize the primeval ocean. The text also says that only a righteous person with 'innocent hands and pure heart' is able to ascend it. It was associated with Mount Zion, and even with Mount Ararat to which Noah's Ark moored according to the legend.
Ideas of the World Mountain, just like of the World Tree, date back to the remote antiquity. These cosmological images are recorded in rock paintings of the Upper Palaeolithic, i.e. have a history of several tens of thousands of years. In any case, we can speak of people's knowledge of the axis mundi during the Stone Age with certainty. This is exactly how we can interpret numerous images found by archaeologists on Neolithic ceramics and widespread from Western Asia through China. Having appeared on ceramics in the period preceding the emergence of the first great civilizations, the motif of the universe pillar preserved its popularity in visual arts in subsequent periods as well.
Thus, for instance, Scythians believed Mount Meru was in the north, in the region of darkness and snow, "where stars, the Moon and the Sun revolve". A common plot of many myths and legends was description of the magic abode behind the sacred mountains – the so-called "land of blissful", situated on the northern slope of Mount Meru, on the shore of the Milky Ocean (Sea) – the Arctic Ocean.
Altai Tatars imagine Bay Ulgen sitting amidst the Heaven on a golden mountain. Abakan Tatars call it the Iron Mountain; Mongols, Buryats and Kalmyks know it as Sumbur, Sumur or Sumeru. In Central Asian traditions and for many Altai peoples, the World Mountain, based on the image of Mount Meru (Sumeru, Sumur, Sumbur, etc.), is often represented as an iron pillar (the Iron Mountain) located in the middle of the earthly disc and joining heaven and earth, with its top touching the Pole Star. Sometimes the Mountain (Sumbur) stands on a navel of a turtle turned to its side, on each leg of which a particular continent is located. According to other versions, the Pole Star itself is the edge of God's palace built on top of the Mountain. Kalmyks believe that stars revolve around Mount Sumeru. According to myths of some Altai peoples, 33 tengri live on top of the Mountain. There is a myth saying that Sumeru is three times surrounded by the enormous serpent Losun.
Mongols and Kalmyks believe the World Mountain has three or four tiers; for Siberian Tatars the Mountain has seven tiers. In their mystic journeys Yakut shamans also ascend a seven-tier mountain. Its top rests against the Pole Star – "the navel of the heaven". Buryats say the Pole Star is attached to the Mountain top.
The most classical World Mountain is the great Mount Meru in Hindu mythology and cosmography. It is situated in the centre of the earth under the Pole Star and surrounded by the world ocean. On its three tops – golden, silver and iron – there reside Brahma, Vishnu and Shiva, or (according to other versions) 33 deities making up the pantheon; underneath the mountain there is the Asur Kingdom. A huge tree stands on each of the four mountains that surround Meru (including Ashvattha and Pippala which represent the World Tree), indicating a corresponding cardinal direction. In Buddhist texts, in addition to Mount Meru there is Himavat ("the King of Mountains") serving as a pillow for Tathāgata.
On the northern side, the mighty Meru stands, shining in its great glory; Brahma's abode is on it, where the soul of all beings resides, Prajapati that created everything movable and immovable… The great Meru, the chaste, good abode… Here (over the mountain) seven divine Rishis with Vasishtha (constellation of the Bid Dipper) at their head set and rise again.
All heavenly bodies revolve around Meru. The polar star hangs on it motionlessly, while Cassiopeia and Bootes circle around it together with the Bid Dipper. Here, the day lasts for six months and the night lasts for six months, one night and one day being equal to a year. To the north of the Milky Ocean there is a big island known as Shveta Dvila ("the Radiant White Island").
This land is described as "the land of everlasting happiness", "the tribe knows neither diseases nor age weakness", "flocks of antelopes and birds are everywhere", "having gone there, one does not return to his world again". It's "the Land of the Select", "the Land of Saints", "the Land of Blissful".
… and started beating the water to obtain Amrita. When devas and suras churned the ocean with Mandara, the great noise began, similar to rumble of monstrous clouds. Various water inhabitants, crushed by the great mountain, found their death in the salt water. Diverse creatures of the world of Varuna, as well as inhabitants of lower regions of the world, were destroyed by the mountain, the pillar of the earth. While it was rotating, mighty trees populated by birds collided and fell from the mountain top. The fire that emerged from their friction, blazing with lightning every minute as if a blue cloud, wrapped Mount Mandara round. It burned elephants and lions that turned out to be there. All other diverse beings lost their life, too. Then Indra, the best of the immortal, extinguished that burning fire with water begotten from the clouds. Thereafter variegated secretions of mighty trees and numerous herb juices flew into the ocean waters. Devas achieved immortality exactly from drinking of those juices endowed with an immortal power, as well as from the flow of gold.
A way to Amaravati – the royal city of Indgra – runs through its top. Serpent Vasuki encircles Meru.
Mount Mandara, used by devas and suras for churning of the Milky Ocean, is an obvious "synonym" of the World Mountain – Mount Meru. The resemblance between the mountains is clearly seen in description given in the Mahabharata.
all over with intertwining herbs.
and beasts of prey roam about.
The gods, the Apsaras and the Kinnaras visit the place.
and descends downwards as much.
and looking like a heap of effulgence.
that mountain is the haunt of the gods and the Gandharvas.
by men of manifold sins.
and it is illuminated by many divine life-giving herbs.
and is the first of mountains.
Ordinary people cannot even think of ascending it.
According to one of myths, Shiva used Mandara as an axle for his chariot and an arc for his bow. As for Mount Meru, it is considered to be the centre of the earth and the universe; its top rises 84,000 leagues above the ground. The sun, the moon, planets and stars revolve round Meru. The heavenly river Ganga flows from the heaven first to this mountain, and only thereafter it flows to the world of people. On top of Mount Meru the city of Brahma is located, stretching for 14,000 leagues. Cities of Indra and other gods are situated nearby.
The Bhagavata Purana describes one of the versions of emergence of river Ganga and explains how Ganga gets to various planets from the highest point of the universe. Once, when Maharaja Bali performed yajna, Vishnu came to him as Vamana and asked three steps of the earth from him. His request was satisfied: in two steps Vamana crossed all three planetary systems (lokas) and pierced a hole in the universe shell with the big toe of his left foot. Several drops of water from the Primordial Ocean leaked through the hole in the shell, fell on Shiva's head and remained there for thousand yugas. These drops of water were the sacred river Ganga. It is described that it first runs to Dhruvaloka (the Pole Star) and purifies it, then it washes planets of the seven great Rishis (Marichi, Vasishtha, Atri and others) who reside on the planets located under Dhruvaloka, and then billions of heavenly spaceships carry its waters along the ways of devas – first to the Moon (Chandraloka) and finally to the abode of Brahma, which is situated on top of Mount Meru. Here is divides into four branches – Sita, Alakananda, Chakshu and Bhadra – which flow down from Meru slopes and reach middle-level planets, one of which is the Earth. From the Himalayan peaks the river branches run down, flow through Haridvar across India plains, cleaning everything on their way.
The foundation of Mount Meru rests upon the klobuk of the world serpent Shesha lying on the back of a giant turtle that swims in primordial waters. Under another version of the myth, Meru (and the earth as a whole) is supported by four elephants.
In the same way gods in a Kalmyk myth used Sumer as a stick to "beat up" the Ocean and thus create the Sun, the Moon, and the stars. Another Central Asian myth reflects penetration of Hindu elements into the myth: having assumed the appearance of eagle Garida (Garuda), god Ochirvani (Indra) attacked serpent Losun in the primordial Ocean; he wound the serpent round Mount Sumer three times and broke his head. An idea was spread across Eastern and Central Asia that the main and most important pillar of the world is the mythical Mount Meru located in an inaccessible place (it was usually placed in the Himalayas). The source of such ideas was myths of Ancient India, later on adopted by Buddhism that made them very popular. Mount Meru was not only considered the centre of cosmos, but was regarded as the sacred abode of gods. In numerous legends and tales Meru was described via the brightest epithets and definitions: it was called golden, shining, brilliant, and was associated with happiness, abundance and immortality. Myths draw this mountain in different ways. Buddhists often depicted it as a colossal cylinder directed to the superior height. Sometimes it was described as the pistil of lotus – the sacred flower. Likening of the cosmic mountain to a plant, i.e. basically to the World Tree, is quite interesting. An unusual feature is that the foundation and the top as if swap places, thus an impression arises that the mountain grows from the heaven.
It is of gold and shines like fire with no tinge of darkening smoke. Its four sides are of four different colours. The colour of the eastern side is white like the colour of brahmans; the colour of the northern side is red like the colour of kshatriyas, the colour of the southern side is yellow like the colour of vaishyas; the colour of the western side is black like the colour of shudras. Its height is 86,000 yojanas, of which 16,000 are inside the earth. Every edge of the four sides makes up 34,000 yojanas. On the mountain, there are rivers with fresh water and marvellous golden dwellings where diverse spiritual beings reside: devas together with gandharva singers and their mistresses apsaras, as well as asurs, daytias and rakshasas. Manasa reservoir is around the mountain, and on four sides of the reservoir lokapalas live – the keepers of the world and its inhabitants. Mount Meru has seven nodes, i.e. big mountains called Mahendra, Malaya, Sahya, Shuktibam, Pikshabam, Vindhya, Pariyatra, and there are so many small mountains that it's almost impossible to count them; these are the mountains on which people live. As for the big mountains around Meru, they include: Himavat covered with eternal snows and populated by rakshasas, pishachis and yakshi; Hemakuta made of gold, on which gandharvas reside.
The internal shell of the world egg endowed with great Atman was Mount Meru, while mountains were its external shell; the amniotic fluid is formed of the oceans. In this egg, oh brahman, there were mountains, continents, oceans, planets, worlds, devas, asurs, and people. On its outer side the egg is covered with water, fire, air, space, and the source of primary elements, the primary elements endowed with ten qualities and the great principle of creation.
Such scheme of the world egg structure is common for Purana texts, epos and the Upanishads. However, the quantity and names of different worlds vary.
In Buddhist cosmology the earth is deemed to be plane with Mount Meru (or Sumeru) in its centre. On Buddhist mandalas it is also depicted in the centre, surrounded by four big dvipas (islands) with eight small dvipas behind them. Mount Sumeru, according to Buddhist cosmology, consists of four jewels, namely: its entire eastern side consists of silver, the southern side of lazurite, the western side of ruby, and the northern side of gold. For this reason, on the four sides of the eminence made on the mandala and intended to depict Mount Sumeru, lamas insert separate pieces of silver, lunar caustic, lazurite, ruby and gold.
A similar image was known to Altaians who believed that the golden cosmic mountain Altyn-Tu was attached to the heaven and hung over the earth, not reaching the earth by a minimal distance of the length of a human shin. However, most traditionally Meru was depicted as a round or tetrahedral mountain gradually narrowing towards its top. In such form is was depicted on various Buddhist works of art.
It is no mere chance that the four facets were associated with a mythical pillar of the universe. One world river nourishing the universe with its waters was flowing down on each side of the mountain. This detail indicates another aspect of the cosmic symbolism of Meru: the meaning of its four rivers is equivalent to the four oceans, which surround the world according to Hindu myths. Thus, Meru itself is a reduced model of cosmos, by which cosmos was exactly structured by gods.
In lamaist mythology Mount Meru (Sumeru) has a pyramidal shape and is surrounded with seven mountain ranges with seas between them. Each side of the pyramid has colour characteristics: the southern side is blue, the western is red, the northern is yellow, and the eastern is white. Same equivalents are known in India, Tibet, China, and even in traditions of some American Indian tribes. Thus, Navajo Indians believed that black (or northern) mountains covered the earth with darkness, blue (or southern) mountains brought the dawn, white (or eastern) mountains brought the day, whereas yellow (or western) mountains brought the shining sunlight.
The names of Babylonian temples and towers evidence identification of such structures with the World Mountain: "the Mountain of Home", "the House of the Mountain of All Lands", "the Mountain of Storms", "the Junction of Heaven and Earth", etc. Ziggurat – the cult tower in Babylon – was basically the World Mountain, a symbolic representation of Cosmos: its seven floors corresponded to the seven planetary heavens (as in Borsippa) or were painted in world colours (as in Ur). Borobudur Temple, the true imago mundi (image of the world), was built in a shape of a mountain.
Biblical texts give a ground to state that paradise is also located on a mountain. According to the Book of Genesis, the garden of Eden was "in the east", but Ezekiel (Chapter 28, Verse 13) specified it was on a mountain. To all appearances, paradise is identified with the sacred "Lord's mountain" mentioned in psalm (23, p. 3-5) with a note that only a righteous person with "innocent hands and pure heart" may ascend it. The biblical reference to the heavenly river washing the entire earth with its four streams is also quite interesting (Genesis, Chapter 2, Verse 10).
Having collected separate evidence from the Bible, we will get a complete set of features of the mythic World Mountain: it is situated in the centre of the universe, God descends onto it, the heavenly garden is located there with a marvellous tree in the middle, whereas the mountain itself is surrounded by rivers which symbolize the primordial ocean. A well-known biblical myth says that during the Flood only Mount Ararat to which Noah's ark moored was not submerged. At that, in other biblical legends Mount Zion is mentioned, which once again emphasizes its particular cosmic role as of the centre of the world, "the navel of the earth", the highest point of the universe.
Even a more graphic demonstration of this is the Muslim world outlook. According to Islamic mythology, the earth was initially very unstable, permanently shuddered and addressed complaints to Allah. Taking pity on it, Allah created the enormous Mount Qaf that encircled the populated world and firmly supported the universe. Behind this stone ring, the Creator created another earth, seven times larger in size. That earth is populated by angels so densely that even a needle cannot fall down between them. Angels incessantly glorify Allah and pray over human sins. If we omit details and single out the gist, it will turn out that the world populated by people is situated inside the cosmic mountain, i.e. the mountain itself is exactly the universe.
According to the Indian mythology, the Universe is a giant global snake biting its tail and wrapping the universe in a ring. Inside the ring it was carrying a giant turtle, on whose back there were four elephants supporting the world. In the centre of the world was the inhabited land Jambudvipa, the shape of which reminded of a blossoming lotus flower with Mount Meru in the middle.
6) the turtle resting on the serpent Ananta is an embodiment of the ancient Indian guardian god Vishnu (the universal reviving principle).
In Altai mythology influenced by Buddhism, the World Mountain name was somewhat altered and sounds as Sumeru. By this very name the mountain is known to other peoples of Central Asia, too. In Altai legends, Buddhist ideas have fancifully mixed with ancient local beliefs. It is exactly the World Mountain which a future shaman ascends during initiation and visits later on in his astral journeys. Ascent of a mountain always signifies a journey to the World Centre. As we have seen, such "centre" is one way or another present even in the structure of a human dwelling, but only shamans and heroes can ascend the World Mountain. Moreover, only a shaman, climbing up a ritual tree, actually climbs up the World Tree, thus reaching the peak of the universe, the highest Heaven.
Besides Meru (Sumeru) and Altyn-Tu, the Altaians placed a mythic mountain Ak-Toshon Altai-syny in the centre of the universe. On its top there is a milky lake in which shaman souls bathe on their way to the heavenly world.
The same milky lake in the heaven is described by the Hungarian narrator of fairy tales Lajos Ami (with angels, not shamans, bathing in it). This is no mere coincidence. Once, before migrating to Europe, forefathers of Hungarians inhabited the territory of the contemporary Trans-Urals and maintained close contacts with South Siberian tribes. Yet, let us go back to Altai legends. A wonderful poplar rises from the milky lake, and shamans use it to get into the kingdom of heavenly spirits, while the mountain itself represents the first station on the way to heaven. Here a strong shaman takes a respite, whereas a weak one does not even dare to move further and returns. On the flat top of the sacred mountain the main spirit of the earth resides, as well as many other spirits being in charge of the souls of livestock and wild animals. The spirits are light-minded and enjoy gambling. It often happens so that one of them loses all animal souls that belong to him to another spirit at cards or dice. In such case the livestock on earth dies, and wild animals move to another locality. Concerned people send a shaman who ascends the cosmic mountain and ascertains which spirit is currently the winner, in order to gain his favour with sacrifices.
The ancient Chinese also worshipped mountains, real and mythical. They regarded any elevation as a holy site, because they believed the light power Yang was concentrated there, whereas bottom lands and depressions were the region of the dark principle Yin. In ancient times in China there was a cult of the five sacred mountains located in the south, west, north, east, and in the centre. Mount Tai Shan (literally "the Great Mountain"), which really existed and was located in the east of the country, was particularly revered. It was believed to patronize the imperial dynasty, and the Sons of Heaven offered sacrifices to it. Just like in the case of Mount Meru of India, the numerical symbolism of Chinese World Mountains has its deep meaning: the five points of space by which they were placed were the most sacred and determined the structure of mythological cosmos.
At that, the Chinese revered Mount Kunlun – the centre of the earth – the most. They believed one could get into the higher realms of the world by ascending it. One of ancient texts say: "The one who would go up from Kunlun twice as high will reach the Mountain of Cool Wind and gain immortality; the one who would go up twice as high will reach the Hanging Ground and gain miraculous abilities, having learnt to manage the wind and the rain; the one who would go up twice as high will reach the heaven, the abode of Tai Di – the supreme lords, and will become a spirit." This is a true description of a shaman journey to the other world! By the way, shamanism existed in Ancient China, too. It is evidenced by frequent mentions of sacred mountains and trees via which priests and gods of Ancient China ascended to the heaven and descended to the earth. Mount Kunlun itself was regarded as something like an earthly paradise: rivers of five different colours flew down from it (including the largest Chinese river Huang He), and diverse cereals drew there in abundance. The Chinese mountain worship tradition and an attempt to unite mountains into a classification system are reflected in the Classic of Mountains and Seas (Shan Hai Ching). The appearance and location of mountains was associated with activities of a mythical subjugator of the flood and manager of the earth – the great Yul who was not only cleaving and moving mountains to eliminate consequences of the flood, but also gave names to three hundred mountains.
In ancient Greek myths gods lived on Mount Olympus where Zeus the thunderer resided, Mount Etna (Zeus put this mountain on Typhon; Hephaestus's smithy was on it, and Demeter was looking for Persephone with the fire obtained on Etna), Mount Ida where Zeus was hiding from Cronus, and the Caucasus where Prometheus was chained.
One of the founders of the "Arctic theory of the origin of Aryans" was the famous Indian politician Bal Gangadhar Tilak (1856-1920). He is the author of a hypothesis that the ancestral home of Indo-Europeans (or Aryans) was northern regions of Eurasia (Kola Peninsula, Karelia, the White Sea coast, and Taimyr). The meaning of the word Arctic: Greek ἄρκτος – "female bear", ἀρκτικός – "being under the constellation of the Great Bear", "northern".
In Tayttiria Brahman (III, 9, 22.1) and Avesta (Vendidad, Fargard II) a year is compared to one day, whereas the sun sets and rises only once a year.
Numerous hymns of Rigveda are dedicated to the goddess of the morning sun Ushas. Moreover, it is mentioned that dawn lasts very long and there are very many stars moving along the horizon, which may be indicative of circumpolar regions. A weakness of such hypothesis is the impossibility to link it with any archaeological culture.
In the Divine Avesta there is a description of not a separate peak, but a whole mountain range Hara Berezaiti (perhaps, these are the Urals), although Avesta provides us with another supporting element mentioned neither in the Vedas nor in the Puranas: this mountain range is stretched in latitudinal direction "from the sunset to the sunrise", i.e. from the west to the east.
Which flow on the earth.
Slavs regard a mountain (а hill, a mound, etc.) as a vertical line binding the top and the bottom, which determines the duality of ideas of the mountain, on the one hand, as of a pure locus and, on the other hand, as of a demonic locus. The link between the mountain and the heaven is reflected in Slavic vocabulary (in Slavic gora mans "a top", and gorniy means "heavenly") and in the ritual practice. Russian chronicles tell that Slavs worshipped heathen deities on mountains. Spring rituals were performed on mountains (the Russan gorka – "gathering of youth in spring and in summer for the round dance"). In Russian charms a mountain, on the one hand, is a place where God, Christ and Our Lady reside, and on the other hand it is a place associated with evil spirits.
A mountain is associated with the other world. In Russian "to leave for a mountain" means "to die"; the kingdom of the dead is a land with golden moutains; paradise is located on an iron mountain or behind mountains.
Collections of Russian bylinas (heroic epics) usually start with Sviatogor, which is a solely archaic character. The image is based on Slavic heathen beliefs, therefore it is very difficult to interpret it for traditional researchers who thoughtlessly date the time described in bylinas to the 10th century only because Prince Vladimir is mentioned there. Sviatogor is particularly notable, since in none of extant bylinas he fights with any other epical hero – he simply has no adequate rivals among mortals. For instance, Sviatogor can easily put Ilya of Murom together with his horse into his pocket, and he measures his strength with Mother Earth itself or starts building a stone pole up to the heaven.
In Greek mythology there is Sviatogor's analogue – giant Atlant. He also performs a function of maintaining the world order – holds the firmament on his shoulders. The earth cannot endure its mass, so he stands on the Atlas Mountains. Those mountains are located at the end of the earth, on the coast of the outer ocean. In front of the mountains, boundless sunburnt deserts stretch for many days of journey (just like those in front of Meru and the Riphean Mountains!). The only treasure of the heaven-holder is a marvellous garden where a tree with golden apples grows – Heracles had to commit his eleventh feat in order to get the apples. Furthermore, both giants ended their lives in the same way – they petrified.
The Pigeon Book refers to the Alatyr Stone, "the father of all stones". It is a sacred rock crowning the top of the Alatyr Mountain and covering the entrance to the underworld, while Sviatogor guards this entrance.
According to Slavic genealogy, Maya of the Golden Mountain was the daughter of Sviatogor and the first wife of Dajdbog whose son she gave birth to – the god of the calendar cycle Kolyada (the word calendar in Slavic means "the gift of Kolyada", while kolyadki means "festive carols" that nowadays are usually performed on the Christmas Eve). Maya of the Golden Mountain herself was the goddess of summer and was depicted with golden braids that symbolized ripe ears.
Many books can be written about the World Mountain, and the fact that it is known in all cultures and on all the continents represents another evidence that the spiritual knowledge is, first and foremost, single, and secondly it has been given to humanity from the earliest times. However, today for someone it's a nice fairy tale, while for knowledgeable people it is a hint! | {
"redpajama_set_name": "RedPajamaC4"
} | 8,147 |
\section{Introduction}
Studies have focused on seizure prediction for decades, but reliable prediction of seizure activity many minutes before a seizure has been elusive. Constructed features like wavelet, energy of spike and spectral power \cite{bandarabadi2015epileptic,li2013seizure,eftekhar2014ngram,gadhoumi2012discriminating, netoff2009seizure, chua2009automatic,sorensen2010automatic,temko2011eeg,acharya2011automatic} are applied on electroencephalogram (EEG) or electrocorticographic (ECoG) data with coarse resolution for most of current neurological analysis works. Prior work has focused on predicting seizures minutes or hours in advance of the seizure, using supervised datasets with labeled examples of seizures. However, with the rich spatial and temporal patterns unveiled by high resolution micro-electrocorticographic ($\mu$ECoG) \cite{viventi2011flexible} which is very similar to a high-frame rate video signal, accurate prediction of neural activities at the sub-second level could become a very interesting and tractable problem. Neural signal prediction on this time frame would allow responsive stimulation to suppress seizures. This kind of neural signal prediction could also find a compact representation for neural activity which could lead to understanding non-pathologic neural activity. To capture the highly non-linear dynamics in neural activities, deep learning neural networks appear to be a promising solution. But, learning a compact representation for neural video prediction gets more challenging when trying to predict in a longer future and the deep neural network model develops a more severe vanishing gradient problem.
To model long term dependencies, Long Short Term Memory (LSTM) units \cite{hochreiter1997long} were proposed as an improvement over vanilla RNN to solve vanishing gradients problem by introducing gate functions. Gated Recurrent Unit (GRU) \cite{cho2014learning} as simplified version of LSTM units has achieved better performance in a number of applications \cite{kaiser2015neural, trischler2016natural,bahdanau2014neural}. Even though LSTM and GRU tries to solve vanishing gradient problem by preserve long term dependency in their cells, modeling long term dependencies is still difficult. For neural language translation, instead of decoding a sequence from a compact feature learnt through an encoding network, \cite{bahdanau2014neural,rocktaschel2015reasoning} use word-by-word attention mechanism which allow direct connection between premise and hypothesis sentences. By using such direct connections, it alleviates the vanishing gradient problem for long sentence translation. Another different approach is to use memory augmented neural networks as Neural Turing Machines \cite{graves2014neural}. By using external memory to store information, the explicit storage of hidden states creates a shortcut through time. \cite{santoro2016one, gulcehre2017memory} both achieve good performance by using external memory network.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.98\textwidth}
\includegraphics[width=\textwidth]{Framework.pdf}
\end{subfigure}
\caption{Video prediction framework. The generative model is built using convolutional LSTM \cite{xingjian2015convolutional}. The network flow is represented with solid arrow, whereas the losses for the generative model are represented with dashed arrows. }
\label{framework_fig}
\end{figure*}
Because convolution account for short-range dependencies, to capture long term correlation, CNN based models have to either increase the depth of the model or use a larger receptive field and larger stride. Either way is generally considered not optimal in time series prediction. Work in \cite{srivastava2015highway, he2016deep} creates efficient information flow from lower layer to high layer by using residual modules and skip connections. \cite{oord2016pixel} used residual module between depth layer in both CNN and RNN based image pixel prediction. Because time series such as audio and video have high temporal correlation, to increase the receptive field with same number of parameters \cite{van2016wavenet} used diluted convolution kernel and achieve good performance.
Inspired by diluted convolution in \cite{van2016wavenet}, we propose a LSTM network that uses multi-resolution layers. The higher layer skips each temporal connections to
create a shortcut, while lower layer is temporally connected and preserves the fine grained information. We also experiment with an explicitly multi-resolution LSTM structure that resembles a temporal pyramid. We demonstrate both multi-resolution representations improve long term prediction compare to a benchmark LSTM.
Learning long term dependencies not only needs an appropriate network structure but also needs a suitable loss function. For video prediction, to overcome blurry predictions caused by using pixel-wise mean square error (MSE), \cite{mathieu2015deep} added total variation loss. Video prediction could be consider as a special case of domain transfer, where the past observed frames lies on one data manifold and future frame lies on another one. Adversarial training finds the relationship between these two manifolds. \cite{lotter2015unsupervised, mathieu2015deep} add adversarial loss on top of MSE. But how video prediction benefits from adversarial training are not fully understood. To further understand how adversarial training benefits video prediction, we use a encoder-decoder 3D CNN structure for the discriminative model. The discriminative model uses reconstruction error as its loss rather than KL-divergence measure. This resembles energy-based GAN in \cite{zhao2016energy} versus GAN \cite{goodfellow2014generative}.
\section{Framework}
In this section, we describe the general structure of our neural video prediction model. The structure consists of two different models, a generative model and a discriminative model. The generative model first takes past observations of video sequences as input and learns a compact feature representation, from which the generative model then reconstructs the past frames and predicts future frames. We explore different model structures during the experiments, and use convolutional LSTM \cite{xingjian2015convolutional} as basic building block for generative model. The discriminative model structure is nearly the same in all experiments. Its main goal is to determine whether the future frames are generated conditioned on the true past frames. Together with the generative model, these two models are considered as adversarial training \cite{goodfellow2014generative}. The general structure is shown in Fig.~\ref{framework_fig}.
\subsection{Generative Model}
Let $X = \{x_1 \cdots x_{t+n} \}$ denotes a video sequence, where $x_t$ denotes current observation, $x_{t+n}$ denotes the $n$th frame in the future to predict. The generative network takes $\{x_1 \cdots x_{t}\}$ as input and outputs a sequence $Y =\{y_1 \cdots y_{t+n} \}$.
In our approach the generative model has an encoder network, a decoder network and a predictor network similar as \cite{srivastava2015unsupervised}. These networks all use convolutional LSTM \cite{xingjian2015convolutional} as basic computation module. The encoder network takes $\{x_1 \cdots x_t\}$ as input and generates a representation $l$. The decoder network reconstructs $\{y_1 \cdots y_t\}$ from $R_{x_1,\cdots,x_t}$. The decoder LSTM is set to be a conditional model namely the decoder reconstructs $y_{t-m}$ from $y_{t-m+1}$. The predictor network generates $\{y_{t+1} \cdots y_{t+n}\}$ from $R_{x_1,\cdots,x_t}$. The predictor model is also a conditional model and it conditions on $y_{t+m}$ to predict $y_{t+m+1}$. The loss for the generative model consists four parts:
\begin{equation}
\begin{aligned}
\mathcal{L}_G &= \lambda_{rec} \mathcal{L}_{rec} + \lambda_{pred}\mathcal{L}_{pred} +\lambda_{adv} \mathcal{L}_{adv} \\
\mathcal{L}_{rec} &= \sum_{i=1}^t || x_i - y_i||^2_2 \\
\mathcal{L}_{pred} &= \sum_{i={t+1}}^{t+n}|| x_i - y_i||^2_2 \\
\mathcal{L}_{adv} &= ||Dec(Enc(Z)) - Z||^2_2 \\
\end{aligned}
\label{total_loss}
\end{equation}
Z is a four dimensional tensor of size $c \times (t+n) \times h \times w$, with $c$, $h$ and $w$ represents channel, height and width of the frame respectively. Z is constructed by stacking $\{x_1 \cdots x_{t}\}$ and $\{y_{t+1} \cdots y_{t+n}\}$ in time order. $\mathcal{L}_{rec}$, $\mathcal{L}_{pred}$ are the pixel domain loss for the reconstructed frames and predicted frames respectively. Whereas $\mathcal{L}_{adv}$ is the adversarial loss from the discriminative model.
\begin{figure*}
\centering
\includegraphics[width=0.95\textwidth]{D_model_understanding.pdf}
\caption{Understanding the benefit from adversarial training: The input to the discriminative model is either true history with true future or true history with predicted future. Third and fifth row of each example shows the activation of second to last layer output across all channel. The activation by true data is distributed almost evenly in both space and time domain to reconstruct the entire sequence. The activation by the sequence with predicted future however concentrates on spatial and temporal inconsistencies. For example, in the first sequence, the discriminative model finds the inconsistency in the last few frames.}
\label{model_understanding}
\end{figure*}
\subsection{Discriminative Model}
\begin{figure}
\centering
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{regular_generative.pdf}
\caption{benchmark network}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{temporal_pyramid.pdf}
\caption{multi-resolution LSTM}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=\textwidth]{highway_generative.pdf}
\caption{LSTM with multi-resolution layers}
\end{subfigure}
\caption{Benchmark network, multi-resolution LSTM and LSTM with multi-resolution layers. The multi-resolution LSTM has two scales, and in each scale it has two layer structure. Only one layer is drawn per scale for simplicity. The dotted box represents the predicted frames and $\hat{y}$ represents linear interpolated frames.}
\label{framework}
\end{figure}
Generative adversarial network(GAN) were introduced by \cite{goodfellow2014generative}, where image is generated from random noise by using two networks trained in a competing manner. The discriminative model in \cite{goodfellow2014generative} minimize the KL-divergence between true image distribution and generated image distribution. The original GAN structure suffers from convergence problem and collapsing mode \cite{salimans2016improved,arjovsky2017wasserstein}. To solve those problems, \cite{salimans2016improved} introduce several techniques including feature matching, minibatch discrimination and historical averaging. \cite{zhao2016energy} use image reconstruction loss instead of KL-divergence loss. \cite{radford2015unsupervised} derive a stable deep convolutional GAN structure by modifying modules in both generator and discriminator. \cite{arjovsky2017wasserstein} show the true data distribution and generative data distribution manifolds in high dimensional space hardly have any overlap, and Wasserstein distance is better compare to other distance measures for non-overlapping distribution. \cite{arjovsky2017wasserstein} achieved the state of art performance for image generation.
For adversarial training for domain transfer problem, where one generates a sample in target domain condition on the data in source domain. In domain transfer unlike GAN, whose generative model could easily suffer from mode collapsing \cite{salimans2016improved, radford2015unsupervised}, overlaps between source domain and target domain manifold is easier to find. \cite{ledig2016photo, taigman2016unsupervised,lotter2015unsupervised,mathieu2015deep} all used modest model structure to perform domain transfer task and achieved good performance. Video prediction could be considered as a domain transfer problem, where the past frames embedding lies on one manifold and future embedding lies on another manifold. \cite{lotter2015unsupervised} concatenates the LSTM features of past frames and CNN feature of generated frame to train a separate multilayer perceptron. \cite{mathieu2015deep} uses a multi-scale 2d convolutional network, the discriminative model stacks all input frames in the channel dimension and output a single scalar indicating whether the video frames are generated or from ground truth future. But both networks fail to model the temporal correlation between frames explicitly.
More importantly, it is not fully understood how adversarial training benefits video prediction. To exploit the temporal dependancies, we use an auto encoder and decoder 3D CNN structure as our discriminative model. The discriminative model uses the energy as the loss function. Energy-based model finds compact representation for the sequence which lives on a low dimension manifold. \cite{zhao2016energy} demonstrate the energy-based GAN training has advantage over GAN for image generation. Another benefit of using encoder-decoder structure is by mapping the activation into pixel space, it helps understanding how adversarial training benefits video prediction. Figure~\ref{model_understanding} shows the activation in the second to last layer of the discriminative model when provided different input.
The loss for discriminative model is:
\begin{equation}
\begin{aligned}
\mathcal{L}_{D} =&||Dec(Enc(X)) - X||^2_2 - \\
&||Dec(Enc(Z)) - Z||^2_2
\end{aligned}
\label{D_loss}
\end{equation}
The Dec and Enc in Eq.~\ref{D_loss} refers to the encoder and decoder in the discriminative model.
\section{Benchmark and Multi-resolution Network}
For neural video prediction, to capture the long term dependencies, we propose two different network structures: multi-resolution LSTM and LSTM with multi-resolution layers. For all generative models, the basic building block is ConvLSTM \cite{xingjian2015convolutional}. Each ConvLSTM layer at each time takes $X_t$ as input, and has memory cell state $C_t$, hidden state $H_t$ and gates $i_t, f_t, o_t$. The equation we use for ConvLSTM are shown in Eq.~\ref{ConvLSTM}, where $\ast$ denotes convolution operator and $\circ$ denotes Hadamard product. For all generative models, they all have an encoder, a decoder and a predictor.
\begin{equation}
\begin{aligned}
i_t &= \sigma ( W_{xi} \ast {X}_t + W_{hi} \ast H_{t-1} + W_{ci} \circ C_{t-1} + b_i) \\
f_t &= \sigma ( W_{xf} \ast {X}_t + W_{fi} \ast H_{t-1} +W_{cf} \circ C_{t-1}+b_f) \\
C_t &= f_t\circ C_{t-1}+i_t \circ tanh(W_{xc}\ast X_t+ W_{hc}\ast H_{t-1}+b_c) \\
o_t &= \sigma(W_{xo} \ast {X}_t + W_{ho} \ast H_{t-1} +W_{co }\circ C_{t}+b_o) \\
H_t &= o_t \circ tanh(C_t) \\
\end{aligned}
\label{ConvLSTM}
\end{equation}
\subsection{Benchmark Network}
First we introduce the benchmark ConvLSTM model, which is a two layer ConvLSTM structure shown in Fig.~\ref{framework}(a). In the benchmark model, each convolution LSTM layer uses a convolution kernel of size $5 \times 5$. For the predictor and decoder convLSTM in the model, the output of both layers go through a deconvolution layer. The deconvolution layer uses kernel size of $1\times 1$ and outputs a frame, which is essentially a weighted average of all input feature maps followed by a $tanh$ function.
\subsection{Multi-resolution LSTM}
In this approach, in general, we generate $k$ temporal scales of the training sequences. The original sequence constitutes scale 0, and the upper scales are recursively down-sampled from the lower scale by a factor of 2. The top scale (coarsest resolution) works in the same way as the benchmark network over its samples only. The lower scale considers both the samples in that scale as well as the interpolated samples from the upper scale. We only present the 2-scale case here for simplicity. In order to avoid delay, we use the simple averaging of the current and the previous sample in the lower scale as the anti-aliasing filter for downsampling. Specifically let $x_i^0$ represents the true video frame at time $i$. Scale 1 signal at even time samples is produced by:
\begin{equation}
x^1_{i} = \frac{ x^0_{i-1} + x^0_{i}}{2},i=\text{even}
\end{equation}
To interpolate the odd samples at scale 1 from even samples, we use simple averaging interpoation filter. The interpolated signal from scale 1 is
\[
u(x^1)(i)=
\begin{cases}
x^1_i \quad i= \text{even} \\
\frac{ x^1_{i-1} + x^1_{i+1}}{2}, i =\text{odd} \\
\end{cases} \\
\]
As shown in Fig.~\ref{framework}(b), we first predict even samples at scale 1. We then interpolate the odd future samples from the predicted even samples, to generate all predicted samples, $u(y^1)(t+i)$, from scale 1. We then predict samples at scale 0 using both past samples at scale 0 and current predicted sample at scale 1. Specifically, we predict samples at $t+i$ using the features learned up to time $t+i-1$ (from both scales), the actual or predicted sample at $t+i-1$ at scale 0, as well as the predicted sample at $t+i$ at scale 1, i.e., $y^0_{i+i} = G(f_{t+i-1}, y^0_{t+i-1}, u(y^1)(t+i))$. The two inputs $y^0_{t+i-1}$ and $u(y^1)(t+i)$ to the ConvLSTM predictor are simply stacked as two channels at the same time. In each scale, the generative model is trained by minimizing the loss function at that scale $(k=0, 1)$:
\begin{equation}
\begin{aligned}
\mathcal{L}_G^k &= \lambda_{rec}^k \mathcal{L}_{rec}^k + \lambda_{pred}^k\mathcal{L}_{pred}^k +\lambda_{adv}^k \mathcal{L}_{adv}^k \\
\mathcal{L}_{rec}^k &= \sum_{i \in \text{scale} \, k} ||x^k_i - y_i^k||_2^2 \\
\mathcal{L}_{pred}^k &= \sum_{i \in \text{scale} \, k} ||x^k_{t+i} - y_{t+i}^k||_2^2 \\
\mathcal{L}_{adv}^k &= ||Dec(Enc(Z^k)) - Z^k||^2_2 \\
\end{aligned}
\label{scale_loss}
\end{equation}
where $Z^k$ is a four dimensional tensor by stacking the true past frames and predicted frames for scale $k$ in time order. The illustration of multi-scale structure is shown in Fig.~\ref{framework}(b). In each scale, the LSTM network has exactly the same two-layer ConvLSTM structure as the benchmark model, except the input frame of scale 0 have twice the number of channels compared to scale 1: half from the current scale and another half from the upper scale. The comparison of 2-scale multi-resolution LSTM and single scale prediction are shown in Fig.~\ref{temporal_pyramid_seq}.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.9\textwidth}
\includegraphics[width=\textwidth]{temporal_pyramid_seq.pdf}
\end{subfigure}
\caption{Comparison between 2-scale and single scale model for video prediction. For single scale video prediction, the encoder, decoder and predictor in the generative model each uses two layer convolutional LSTM. In 2-scale prediction, for each scale the model have the same network structure as the single scale benchmark. The single scale model and multi-resolution LSTM each corresponds to model 6 and 7 in Tab.~\ref{PSNR_compare}.}
\label{temporal_pyramid_seq}
\end{figure*}
\subsection{LSTM with Multi-resolution layer}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{temporal_highway_display.pdf}
\caption{Demonstration of the high correlation of the first layer output of the two-layer LSTM model at the original resolution. Each layer used 2 layer convolutional LSTM each with 128 convolutional kernels with receptive field of $5 \times 5$. Row 1 and 3 show the encoder observation for two different sequences. Row 2 and 4 show the
average of the 128 feature maps produced by layer 1. }
\label{temporal_highway_display}
\end{figure}
For time series prediction because the temporal correlation is high, in order to achieve a larger receptive field with the same amount of parameters, \cite{van2016wavenet} uses diluted convolution where the convolutional filter in higher layers of the CNN network are structured with zero coefficients every other connections.
Inspired by \cite{van2016wavenet}, we propose a LSTM network that has multi-resolution layers. The network have a higher layer and a lower layer. The fine-grained temporal resolution is preserved by the lower layer shown in in Fig.~\ref{temporal_highway_display}. The higher layer of the convolutional LSTM model use a skip temporal connection shown in Fig.~\ref{framework}(c). Compare to the lower layer, the higher layer creates a temporal highway, which alleviates the vanishing gradient problem. Different from the benchmark network shown in Fig.~\ref{framework}(a), the deconvolution layer in multi-resolution layer network (Fig.~\ref{framework}(c)) use different parameters to predict. In our implementation, the deconvolution layer is performing $1\times1$ spacial convolution on the feature map outputs, and the increase number of parameter compare to Fig.~\ref{framework}(a) is almost negelectable. The number of parameters used in different models are shown in Tab.~\ref{PSNR_compare}.
\section{Experiment}
\subsection{Dataset}
We analyzed $\mu$ECoG data from an acute in vivo feline model of seizures. The 18 by 20 array of high-density active electrodes has 500 $\mu$m spacing between nearby channels. The in vivo recording has a temporal sampling rate of 277.78 Hz and lasts 53 minutes. We obtained a total of 894K frames. In total, there are 788 K consecutive training frames and 106K consecutive testing frames. During training, we use 16 frames as observation to predict the next 16 frames.
\subsection{Results}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{error_comparision}
\caption{PSNR of predicted frames against prediction time. The benchmark model, benchmark model with adversarial, multi-resolution LSTM, LSTM with multi-resolution layer correspond to models 5,6,7,8 respectively in Tab.~\ref{PSNR_compare}. LSTM with multi-resolution layer has a better long term prediction accuracy compared to other models. The PSNR is obtained by first computing MSE by averaging squared errors over all pixels over all frames and all sequences, and then converting the resulting MSE to PSNR.}
\label{error_comparison}
\end{figure}
For the discriminative model, we use a 3D convolutional neural network with an encoder-decoder structure. The encoder uses three 3D strided convolution layers with all layers using $7 \times 7 \times 7 $ convolutional kernel. The decoder uses three 3D strideded deconvolution layers with all layers with the same receptive field. We use batch normalization \cite{ioffe2015batch} and leaky ReLU \cite{he2015delving} except for the last layer. For multi-resolution network, the discriminative for the higher scale uses only two 3D convolution layer and deconvolution layer as the sequence length is reduced by 2.
We present results for several models. The weights for reconstruction error and prediction error are set to be $\lambda_{rec}=1, \lambda_{pred} =1$ in all models. For models use adversarial training, we set the weight $\lambda_{adv} =0.1$. For multi-resolution LSTM network, the weights are set the same in different scale. The discriminative model and generative model are trained both use Adam algorithm\cite{kingma2014adam} both with learning rate of 0.001 decreasing by a factor of 10 halfway through training. To avoid exploding gradient for generative model, we perform gradient clipping by setting the $l_2$ norm maximum at 0.001. In all adversarial training case, the discriminative model is updated once every two iterations.
During testing stage, the observation sequences lasts 16 frames and the predictor generates 16 future frames based on the observed frames. To evaluate the performance of different approaches we compute the Peak Signal to Noise Ratio (PSNR) between the true future frames $X$ and predicted future frames $Y$.
\begin{equation*}
PSNR = 10*log_{10}\frac{(max_X^2)}{\frac{1}{N}\sum_{i=1}^N(x_i-y_i)^2}
\label{PSNR_def}
\end{equation*}
Sample prediction comparison are shown in Tab.~\ref{PSNR_compare}. The adversarial training brings improvement compared to using $l_2$ loss alone. It is interesting to note that even the PSNR is based on $l_2$ metric, adding adversarial training into the loss function gets better prediction accuracy. Discriminative model helps generative model to learn the long term dependencies. The further the prediction the more significant is the accuracy gain from adversarial training, which is shown in Fig.~\ref{error_comparison}. Comparing among all structures using adversarial learning, LSTM with multi-resolution layer and multi-resolution LSTM both have a significant gain compared to the benchmark model. Even though the multi-resolution LSTM achieved more gains for prediction up to 10 frames ahead, the LSTM with multi-resolution layers take over after 10 frames. PSNR increase by using LSTM with multi-resolution layer at $16th$ frames gets as high as 1.92 dB compared to the benchmark model. This is remarkable as the LSTM with multi-resolution layers have about the same number of parameters as the benchmark model. In Fig.~\ref{all_sequences}, we show sample results of different models.
\begin{table*}
\scalebox{0.8}{
\begin{tabular}{ p{2cm} | p{2cm} p{2cm} p{2cm} p{2.5 cm} | p{2 cm} p{2 cm} p{2 cm} p{2cm}}
\toprule
Generative model & ConvLSTM 64-64 & ConvLSTM 64-64 & Multi-resolution LSTM 64-64, 64-64 & LSTM with multi-resolution layer 64-64 & ConvLSTM 128-128 & ConvLSTM 128-128 & Multi-resolution LSTM 128-128, 128-128 & LSTM with multi-resolution layer 128-128 \\
\hline
Number of parameters in the generative model
& 4123266 & 4123266& 4123266 and 8265732 & 4123524 &15619330 & 15619330 & 15619330 and 31277060 & 15619844 \\
\toprule
Discriminative model number of feature maps per layer & None & 32,32,4,32,32 & 32,4,32 and 32,32,4,32,32 & 32,32,4,32,32 & None & 32,32,4,32,32 & 32,4,32 and 32,32,4,32,32 & 32,32,4,32,32 \\
\toprule
PSNR of all frames & 27.8737 & 28.3426 &28.8903 & \textbf{29.0372} & 27.9942 & 28.5317 & 29.0931 & \textbf{29.1741} \\
\bottomrule
\end{tabular}}
\caption{Comparison of the accuracy and number of parameters of all models. 64-64 represents the number of convolution LSTM cells in layer 1 and 2 are both 64. All the convolution LSTM cells uses $5 \times 5$ kernel. The multi-resolution LSTM structure has two scales, each scale has two layer convolution LSTM cells. 64-64, 64-64 means each scale uses a 64-64 two layer LSTM. Model 1 and 5 are trained with $l_2$ loss alone. }
\label{PSNR_compare}
\end{table*}
\section{ACKNOWLEDGEMENT}
This work was funded by National Science Foundation award CCF-1422914.
\vspace{-0.05in}
\section{Conclusion}
In this work, we have proposed two ways to do video prediction using multi-resolution presentations. The first approach uses a novel LSTM structure with multi-resolution layers for long term video prediction. The network creates a temporal highway in the upper-layer to capture the long-term dependencies between video frames.
The second approach uses two scale multi-resolution LSTM. We compare the performance of these two approaches against single resolution benchmark model and demonstrate the advantage of using multi-resolution representation of LSTM. Both multi-resolution LSTM and LSTM with multi-resolution layers have better performance than single resolution representation when they all use adversarial training. The long term prediction accuracy using LSTM with multi-resolution layers are much higher than the benchmark models with similar number of parameters. We also demonstrate that all models benefit from energy-based adversarial training which is accomplished by using a 3D CNN based encoder-decoder structure.
\begin{figure*}
\centering
\includegraphics[width=0.95\textwidth,height = 1.2 \textwidth]{sequence_1.pdf}
\caption{Prediction result comparison between different methods: generative model, adversarial training, multi-resolution LSTM and LSTM with multi-resolution layers correspond to model 5,6,7,8 respectively in Tab.~\ref{PSNR_compare}. }
\label{all_sequences}
\end{figure*}
{\small
\bibliographystyle{ieee}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,023 |
package com.sequenceiq.cloudbreak.service.decorator.responseprovider;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import javax.annotation.PostConstruct;
import javax.inject.Inject;
import org.springframework.stereotype.Component;
@Component
public class ResponseProviders {
@Inject
private List<ResponseProvider> responseProviders;
private final Map<String, ResponseProvider> map = new HashMap<>();
@PostConstruct
public void init() {
responseProviders.forEach(rp -> map.put(rp.type(), rp));
}
public ResponseProvider get(String type) {
return map.get(type);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,258 |
{"url":"https:\/\/blog.marco.ninja\/tags\/windows\/","text":"## The WSL Onion\n\n### Calling powershell.exe from PowerShell inside WSL\n\nThe other day I was playing around in WSL with a colleague of mine and we did this: # We start out in WSL Debian and enter PowerShell Core marco@box:~$pwsh PowerShell 7.2.2 Copyright (c) Microsoft Corporation. https:\/\/aka.ms\/powershell Type 'help' to get help. # Then we get the major version of the active PowerShell session PS \/home\/marco> ($PSVersionTable.PSVersion).Major 7 # Then we call powershell.exe and get a completely different version! PS \/home\/marco> (powershell. [Read More]\n\n## Hidden WSL Fileshare\n\nWSL file systems get exposed as a hidden share network share: \\\\wsl$\\<WSL Name>\\<path\\to\\file> For example, my Debian home folder is at: \\\\wsl$\\Debian\\home\\kamner\n\n## Windows Terminal: Open New WSL Tab In Linux Home Folder\n\nThe path you are in when opening a new WSL tab is determined by startingDirectory. This parameter needs to be a valid Windows path, which isn\u2019t great if we want to end up in \/home\/kamner inside WSL. The nice thing about WSL is that it will resolve windows paths into their equivalent WSL\/linux path if possible. For example, C:\\Scripts would resolve to \/mnt\/c\/Scripts. Using this and the neat trick that the WSL filesystem is exposed as a a hidden fileshare ([[technology\/windows\/wsl-hidden-fileshare]]) we can get to where we want. [Read More]\n\n## Why You Want To Install PowerShell On Windows\n\n### Why installing PowerShell on Windows may actually be a good idea\n\nQ: Wait a minute! Don\u2019t I already have PowerShell? A: Yes. However you may want to continue reading because things are never as simple as they appear. The PowerShell version you have installed on Windows 10 is 5.X, Microsoft even says so in their own documentation. On the initial release of Windows 10, with automatic updates enabled, PowerShell gets updated from version 5.0 to 5.1. If the original version of Windows 10 is not updated through Windows Updates, the version of PowerShell is 5. [Read More]","date":"2022-08-16 13:56:04","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\": 0.4155760109424591, \"perplexity\": 4334.640190835367}, \"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-2022-33\/segments\/1659882572304.13\/warc\/CC-MAIN-20220816120802-20220816150802-00733.warc.gz\"}"} | null | null |
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*<A NAME="usr_29.txt"></A><B>usr_29.txt</B>* For Vim version 7.2. Last change: 2008 Jun 28
VIM USER MANUAL - by <A HREF="intro.html#Bram">Bram</A> <A HREF="intro.html#Moolenaar">Moolenaar</A>
Moving through programs
The creator of Vim is a computer programmer. It's no surprise that Vim
contains many features to aid in <A HREF="editing.html#writing">writing</A> programs. Jump around to find where
identifiers are defined and used. Preview declarations in a separate <A HREF="windows.html#window">window</A>.
There is more in the next chapter.
|<A HREF="#29.1">29.1</A>| Using <A HREF="tagsrch.html#tags">tags</A>
|<A HREF="#29.2">29.2</A>| The preview <A HREF="windows.html#window">window</A>
|<A HREF="#29.3">29.3</A>| Moving through a program
|<A HREF="#29.4">29.4</A>| Finding global identifiers
|<A HREF="#29.5">29.5</A>| Finding local identifiers
Next chapter: |<A HREF="usr_30.html">usr_30.txt</A>| Editing programs
Previous chapter: |<A HREF="usr_28.html">usr_28.txt</A>| <A HREF="fold.html#Folding">Folding</A>
Table of contents: |<A HREF="usr_toc.html">usr_toc.txt</A>|
<hr class="doubleline">
*<A NAME="29.1"></A><B>29.1</B>* Using <A HREF="tagsrch.html#tags">tags</A>
What is a <A HREF="tagsrch.html#tag">tag</A>? It is a location where an identifier is defined. An example
is a function definition in a C or C++ program. A list of <A HREF="tagsrch.html#tags">tags</A> is kept in a
<A HREF="tagsrch.html#tags">tags</A> file. This can be used by Vim to directly jump from any place to the
<A HREF="tagsrch.html#tag">tag</A>, the place where an identifier is defined.
To generate the <A HREF="tagsrch.html#tags">tags</A> file for all C files in the current directory, use the
following command:
<B> ctags *.c</B>
"<A HREF="tagsrch.html#ctags">ctags</A>" is a separate program. Most <A HREF="os_unix.html#Unix">Unix</A> systems already have <A HREF="motion.html#it">it</A> installed.
If you <A HREF="diff.html#do">do</A> not have <A HREF="motion.html#it">it</A> yet, you can find Exuberant <A HREF="tagsrch.html#ctags">ctags</A> here:
<B><FONT COLOR="PURPLE"> http://ctags.sf.net </FONT></B>
Now when you are in Vim and you want to go to a function definition, you can
jump to <A HREF="motion.html#it">it</A> by using the following command:
<B> :tag startlist</B>
This command will find the function "startlist" even if <A HREF="motion.html#it">it</A> is in another file.
The <A HREF="tagsrch.html#CTRL-]">CTRL-]</A> command jumps to the <A HREF="tagsrch.html#tag">tag</A> of the <A HREF="motion.html#word">word</A> that is under the cursor.
This makes <A HREF="motion.html#it">it</A> <A HREF="starting.html#easy">easy</A> to explore a tangle of C code. Suppose, for example, that
you are in the function "write_block". You can see that <A HREF="motion.html#it">it</A> calls
"write_line". But what does "write_line" <A HREF="diff.html#do">do</A>? By placing the cursor on the
call to "write_line" and pressing <A HREF="tagsrch.html#CTRL-]">CTRL-]</A>, you jump to the definition of this
function.
The "write_line" function calls "write_char". You need to figure out what
<A HREF="motion.html#it">it</A> does. So you position the cursor over the call to "write_char" and press
<A HREF="tagsrch.html#CTRL-]">CTRL-]</A>. Now you are at the definition of "write_char".
+-------------------------------------+
|void write_block(char **s; int cnt) |
|{ |
| int <A HREF="insert.html#i">i</A>; |
| for (i = 0; <A HREF="insert.html#i">i</A> <A HREF="change.html#<"><</A> cnt; ++i) |
| write_line(s[i]); |
|} | |
+-----------|-------------------------+
|
<A HREF="tagsrch.html#CTRL-]">CTRL-]</A> |
| +----------------------------+
+--> |void write_line(char *s) |
|{ |
| while (*s != 0) |
| write_char(*s++); |
|} | |
+--------|-------------------+
|
<A HREF="tagsrch.html#CTRL-]">CTRL-]</A> |
| +------------------------------------+
+--> |void write_char(char <A HREF="change.html#c">c</A>) |
|{ |
| putchar((int)(unsigned char)c); |
|} |
+------------------------------------+
The "<A HREF="tagsrch.html#:tags">:tags</A>" command shows the list of <A HREF="tagsrch.html#tags">tags</A> that you traversed through:
<A HREF="tagsrch.html#:tags">:tags</A>
<B><FONT COLOR="PURPLE"> # TO tag FROM line in file/text </FONT></B>
<B><FONT COLOR="PURPLE"> 1 1 write_line 8 write_block.c </FONT></B>
<B><FONT COLOR="PURPLE"> 2 1 write_char 7 write_line.c </FONT></B>
<B><FONT COLOR="PURPLE"> > </FONT></B>
Now to go back. The <A HREF="tagsrch.html#CTRL-T">CTRL-T</A> command goes to the preceding <A HREF="tagsrch.html#tag">tag</A>. In the example
above you get back to the "write_line" function, in the call to "write_char".
This command takes a <A HREF="intro.html#count">count</A> argument that indicates how many <A HREF="tagsrch.html#tags">tags</A> to jump
back. You have gone forward, and now back. Let's go forward again. The
following command goes to the <A HREF="tagsrch.html#tag">tag</A> on top of the list:
<B> :tag</B>
You can prefix <A HREF="motion.html#it">it</A> with a <A HREF="intro.html#count">count</A> and jump forward that many <A HREF="tagsrch.html#tags">tags</A>. For example:
":3tag". <A HREF="tagsrch.html#CTRL-T">CTRL-T</A> also can be preceded with a <A HREF="intro.html#count">count</A>.
These commands thus allow you to go down a call tree with <A HREF="tagsrch.html#CTRL-]">CTRL-]</A> and back
up again with <A HREF="tagsrch.html#CTRL-T">CTRL-T</A>. Use "<A HREF="tagsrch.html#:tags">:tags</A>" to find out where you are.
SPLIT WINDOWS
The "<A HREF="tagsrch.html#:tag">:tag</A>" command replaces the file in the current <A HREF="windows.html#window">window</A> with the one
containing the new function. But suppose you want to see not only the old
function but also the new one? You can split the <A HREF="windows.html#window">window</A> using the "<A HREF="windows.html#:split">:split</A>"
command followed by the "<A HREF="tagsrch.html#:tag">:tag</A>" command. Vim has a shorthand command that does
both:
<B> :stag tagname</B>
To split the current <A HREF="windows.html#window">window</A> and jump to the <A HREF="tagsrch.html#tag">tag</A> under the cursor use this
command:
<B> CTRL-W ]</B>
If a <A HREF="intro.html#count">count</A> is specified, the new <A HREF="windows.html#window">window</A> will be that many lines high.
MORE TAGS FILES
When you have files in many directories, you can create a <A HREF="tagsrch.html#tags">tags</A> file in each of
them. Vim will then only be able to jump to <A HREF="tagsrch.html#tags">tags</A> within that directory.
To find more <A HREF="tagsrch.html#tags">tags</A> files, set the <A HREF="options.html#'tags'">'tags'</A> option to include all the relevant
<A HREF="tagsrch.html#tags">tags</A> files. Example:
<B> :set tags=./tags,./../tags,./*/tags</B>
This finds a <A HREF="tagsrch.html#tags">tags</A> file in the same directory <A HREF="motion.html#as">as</A> the current file, one
directory level higher and in all subdirectories.
This is quite a number of <A HREF="tagsrch.html#tags">tags</A> files, but <A HREF="motion.html#it">it</A> may still not be enough. For
example, when editing a file in "~/proj/src", you will not find the <A HREF="tagsrch.html#tags">tags</A> file
"~/proj/sub/tags". For this situation Vim offers to search a whole directory
tree for <A HREF="tagsrch.html#tags">tags</A> files. Example:
<B> :set tags=~/proj/**/tags</B>
ONE TAGS FILE
When Vim has to search many places for <A HREF="tagsrch.html#tags">tags</A> files, you can hear the disk
rattling. It may get a bit slow. In that <A HREF="change.html#case">case</A> it's better to spend this
time while generating one big <A HREF="tagsrch.html#tags">tags</A> file. You might <A HREF="diff.html#do">do</A> this overnight.
This requires the Exuberant <A HREF="tagsrch.html#ctags">ctags</A> program, mentioned above. It offers an
argument to search a whole directory tree:
<B> cd ~/proj</B>
<B> ctags -R .</B>
The <A HREF="todo.html#nice">nice</A> thing about this is that Exuberant <A HREF="tagsrch.html#ctags">ctags</A> recognizes various file
types. Thus this doesn't work just for C and C++ programs, also for Eiffel
and even Vim scripts. See the <A HREF="tagsrch.html#ctags">ctags</A> documentation to tune this.
Now you only need to tell Vim where your big <A HREF="tagsrch.html#tags">tags</A> file is:
<B> :set tags=~/proj/tags</B>
MULTIPLE MATCHES
When a function is defined multiple times (or a method in several classes),
the "<A HREF="tagsrch.html#:tag">:tag</A>" command will jump to the first one. If there is a match in the
current file, that one is used first.
You can now jump to other matches for the same <A HREF="tagsrch.html#tag">tag</A> with:
<B> :tnext</B>
Repeat this to find further matches. If there are many, you can select which
one to jump to:
<B> :tselect tagname</B>
Vim will present you with a list of choices:
<B><FONT COLOR="PURPLE"> # pri kind tag file </FONT></B>
<B><FONT COLOR="PURPLE"> 1 F f mch_init os_amiga.c </FONT></B>
<B><FONT COLOR="PURPLE"> mch_init() </FONT></B>
<B><FONT COLOR="PURPLE"> 2 F f mch_init os_mac.c </FONT></B>
<B><FONT COLOR="PURPLE"> mch_init() </FONT></B>
<B><FONT COLOR="PURPLE"> 3 F f mch_init os_msdos.c </FONT></B>
<B><FONT COLOR="PURPLE"> mch_init(void) </FONT></B>
<B><FONT COLOR="PURPLE"> 4 F f mch_init os_riscos.c </FONT></B>
<B><FONT COLOR="PURPLE"> mch_init() </FONT></B>
<B><FONT COLOR="PURPLE"> Enter nr of choice (<CR> to abort): </FONT></B>
You can now enter the number (in the first column) of the match that you would
like to jump to. The information in the other columns give you a good idea of
where the match is defined.
To move between the matching <A HREF="tagsrch.html#tags">tags</A>, these commands can be used:
<A HREF="tagsrch.html#:tfirst">:tfirst</A> go to first match
:[count]tprevious go to <A HREF="intro.html#[count]">[count]</A> previous match
:[count]tnext go to <A HREF="intro.html#[count]">[count]</A> next match
<A HREF="tagsrch.html#:tlast">:tlast</A> go to last match
If <A HREF="intro.html#[count]">[count]</A> is omitted then one is used.
GUESSING TAG NAMES
Command line completion is a good way to avoid typing a long <A HREF="tagsrch.html#tag">tag</A> name. Just
type the first bit and press <A HREF="motion.html#<Tab>"><Tab></A>:
<B> :tag write_<Tab></B>
You will get the first match. If it's not the one you want, press <A HREF="motion.html#<Tab>"><Tab></A> until
you find the right one.
Sometimes you only know part of the name of a function. Or you have many
<A HREF="tagsrch.html#tags">tags</A> that start with the same string, but end differently. Then you can tell
Vim to use a <A HREF="pattern.html#pattern">pattern</A> to find the <A HREF="tagsrch.html#tag">tag</A>.
Suppose you want to jump to a <A HREF="tagsrch.html#tag">tag</A> that contains "block". First type
this:
<B> :tag /block</B>
Now use command line completion: press <A HREF="motion.html#<Tab>"><Tab></A>. Vim will find all <A HREF="tagsrch.html#tags">tags</A> that
contain "block" and use the first match.
The "<A HREF="pattern.html#/">/</A>" before a <A HREF="tagsrch.html#tag">tag</A> name tells Vim that what follows is not a literal <A HREF="tagsrch.html#tag">tag</A>
name, but a <A HREF="pattern.html#pattern">pattern</A>. You can use all the items for search patterns here. For
example, suppose you want to select a <A HREF="tagsrch.html#tag">tag</A> that starts with "write_":
<B> :tselect /^write_</B>
The "<A HREF="motion.html#^">^</A>" specifies that the <A HREF="tagsrch.html#tag">tag</A> starts with "write_". Otherwise <A HREF="motion.html#it">it</A> would also
be found halfway a <A HREF="tagsrch.html#tag">tag</A> name. Similarly "<A HREF="motion.html#$">$</A>" at the end makes sure the <A HREF="pattern.html#pattern">pattern</A>
matches until the end of a <A HREF="tagsrch.html#tag">tag</A>.
A TAGS BROWSER
Since <A HREF="tagsrch.html#CTRL-]">CTRL-]</A> takes you to the definition of the identifier under the cursor,
you can use a list of identifier names <A HREF="motion.html#as">as</A> a table of contents. Here is an
example.
First create a list of identifiers (this requires Exuberant ctags):
<B> ctags --c-types=f -f functions *.c</B>
Now start Vim without a file, and edit this file in Vim, in a vertically split
<A HREF="windows.html#window">window</A>:
<B> vim</B>
<B> :vsplit functions</B>
The <A HREF="windows.html#window">window</A> contains a list of all the <A HREF="eval.html#functions">functions</A>. There is some more stuff,
but you can ignore that. Do "<A HREF="options.html#:setlocal">:setlocal</A> ts=99" to clean <A HREF="motion.html#it">it</A> up a bit.
In this <A HREF="windows.html#window">window</A>, define a <A HREF="map.html#mapping">mapping</A>:
<B> :nnoremap <buffer> <CR> 0ye<C-W>w:tag <C-R>"<CR></B>
Move the cursor to the line that contains the function you want to go to.
Now press <A HREF="intro.html#<Enter>"><Enter></A>. Vim will go to the other <A HREF="windows.html#window">window</A> and jump to the selected
function.
RELATED ITEMS
You can set <A HREF="options.html#'ignorecase'">'ignorecase'</A> to make <A HREF="change.html#case">case</A> in <A HREF="tagsrch.html#tag">tag</A> names be ignored.
The <A HREF="options.html#'tagbsearch'">'tagbsearch'</A> option tells if the <A HREF="tagsrch.html#tags">tags</A> file is sorted or not. The default
is to assume a sorted <A HREF="tagsrch.html#tags">tags</A> file, which makes a <A HREF="tagsrch.html#tags">tags</A> search a lot faster, but
doesn't work if the <A HREF="tagsrch.html#tags">tags</A> file isn't sorted.
The <A HREF="options.html#'taglength'">'taglength'</A> option can be used to tell Vim the number of significant
characters in a <A HREF="tagsrch.html#tag">tag</A>.
When you use the SNiFF+ program, you can use the Vim interface to <A HREF="motion.html#it">it</A> |<A HREF="if_sniff.html#sniff">sniff</A>|.
SNiFF+ is a commercial program.
<A HREF="if_cscop.html#Cscope">Cscope</A> is a free program. It does not only find places where an identifier is
declared, but also where <A HREF="motion.html#it">it</A> is used. See |<A HREF="if_cscop.html#cscope">cscope</A>|.
<hr class="doubleline">
*<A NAME="29.2"></A><B>29.2</B>* The preview <A HREF="windows.html#window">window</A>
When you edit code that contains a function call, you need to use the correct
arguments. To know what values to pass you can look at how the function is
defined. The <A HREF="tagsrch.html#tags">tags</A> mechanism works very well for this. Preferably the
definition is displayed in another <A HREF="windows.html#window">window</A>. For this the preview <A HREF="windows.html#window">window</A> can be
used.
To open a preview <A HREF="windows.html#window">window</A> to display the function "write_char":
<B> :ptag write_char</B>
Vim will open a <A HREF="windows.html#window">window</A>, and jumps to the <A HREF="tagsrch.html#tag">tag</A> "write_char". Then <A HREF="motion.html#it">it</A> takes you
back to the original position. Thus you can continue typing without the need
to use a <A HREF="index.html#CTRL-W">CTRL-W</A> command.
If the name of a function appears in the text, you can get its definition
in the preview <A HREF="windows.html#window">window</A> with:
<B> CTRL-W }</B>
There is a <A HREF="usr_41.html#script">script</A> that automatically displays the text where the <A HREF="motion.html#word">word</A> under
the cursor was defined. See |<A HREF="windows.html#CursorHold-example">CursorHold-example</A>|.
To close the preview <A HREF="windows.html#window">window</A> use this command:
<B> :pclose</B>
To edit a specific file in the preview <A HREF="windows.html#window">window</A>, use "<A HREF="windows.html#:pedit">:pedit</A>". This can be
useful to edit a header file, for example:
<B> :pedit defs.h</B>
Finally, "<A HREF="windows.html#:psearch">:psearch</A>" can be used to find a <A HREF="motion.html#word">word</A> in the current file and any
included files and display the match in the preview <A HREF="windows.html#window">window</A>. This is
especially useful when using library <A HREF="eval.html#functions">functions</A>, for which you <A HREF="diff.html#do">do</A> not have a
<A HREF="tagsrch.html#tags">tags</A> file. Example:
<B> :psearch popen</B>
This will show the "stdio.h" file in the preview <A HREF="windows.html#window">window</A>, with the function
prototype for popen():
<B><FONT COLOR="PURPLE"> FILE *popen __P((const char *, const char *)); </FONT></B>
You can specify the height of the preview <A HREF="windows.html#window">window</A>, when <A HREF="motion.html#it">it</A> is opened, with the
<A HREF="options.html#'previewheight'">'previewheight'</A> option.
<hr class="doubleline">
*<A NAME="29.3"></A><B>29.3</B>* Moving through a program
Since a program is structured, Vim can recognize items in <A HREF="motion.html#it">it</A>. Specific
commands can be used to move around.
C programs often contain constructs like this:
<B><FONT COLOR="PURPLE"> #ifdef USE_POPEN </FONT></B>
<B><FONT COLOR="PURPLE"> fd = popen("ls", "r") </FONT></B>
<B><FONT COLOR="PURPLE"> #else </FONT></B>
<B><FONT COLOR="PURPLE"> fd = fopen("tmp", "w") </FONT></B>
<B><FONT COLOR="PURPLE"> #endif </FONT></B>
But then much longer, and possibly nested. Position the cursor on the
"#ifdef" and press <A HREF="motion.html#%">%</A>. Vim will jump to the "#else". Pressing <A HREF="motion.html#%">%</A> again takes
you to the "#endif". Another <A HREF="motion.html#%">%</A> takes you to the "#ifdef" again.
When the construct is nested, Vim will find the matching items. This is a
good way to check if you didn't forget an "#endif".
When you are somewhere inside a "#if" - "#endif", you can jump to the start
of <A HREF="motion.html#it">it</A> with:
<B> [#</B>
If you are not after a "#if" or "#ifdef" Vim will beep. To jump forward to
the next "#else" or "#endif" use:
<B> ]#</B>
These two commands skip any "#if" - "#endif" blocks that they encounter.
Example:
<B><FONT COLOR="PURPLE"> #if defined(HAS_INC_H) </FONT></B>
<B><FONT COLOR="PURPLE"> a = a + inc(); </FONT></B>
<B><FONT COLOR="PURPLE"> # ifdef USE_THEME </FONT></B>
<B><FONT COLOR="PURPLE"> a += 3; </FONT></B>
<B><FONT COLOR="PURPLE"> # endif </FONT></B>
<B><FONT COLOR="PURPLE"> set_width(a); </FONT></B>
With the cursor in the last line, "<A HREF="motion.html#[#">[#</A>" moves to the first line. The "#ifdef"
- "#endif" block in the middle is skipped.
MOVING IN CODE BLOCKS
In C code blocks are enclosed in <A HREF="intro.html#{}">{}</A>. These can get pretty long. To move to
the start of the outer block use the "<A HREF="motion.html#[[">[[</A>" command. Use "<A HREF="motion.html#][">][</A>" to find the end.
This assumes that the "<A HREF="motion.html#{">{</A>" and "<A HREF="motion.html#}">}</A>" are in the first column.
The "<A HREF="motion.html#[{">[{</A>" command moves to the start of the current block. It skips over
pairs of <A HREF="intro.html#{}">{}</A> at the same level. "<A HREF="motion.html#]}">]}</A>" jumps to the end.
An overview:
function(int <A HREF="motion.html#a)">a)</A>
+-> {
| if (a)
| +-> {
<A HREF="motion.html#[[">[[</A> | | for (;;) --+
| | +-> { |
| <A HREF="motion.html#[{">[{</A> | | foo(32); | --+
| | <A HREF="motion.html#[{">[{</A> | if (bar(a)) --+ | <A HREF="motion.html#]}">]}</A> |
+-- | +-- break; | <A HREF="motion.html#]}">]}</A> | |
| } <-+ | | <A HREF="motion.html#][">][</A>
+-- foobar(a) | |
} <-+ |
} <-+
When <A HREF="editing.html#writing">writing</A> C++ or Java, the outer <A HREF="intro.html#{}">{}</A> block is for the class. The next level
of <A HREF="intro.html#{}">{}</A> is for a method. When somewhere inside a class use "<A HREF="motion.html#[m">[m</A>" to find the
previous start of a method. "<A HREF="motion.html#]m">]m</A>" finds the next start of a method.
Additionally, "<A HREF="motion.html#[]">[]</A>" moves backward to the end of a function and "<A HREF="motion.html#]]">]]</A>" moves
forward to the start of the next function. The end of a function is defined
by a "<A HREF="motion.html#}">}</A>" in the first column.
int func1(void)
{
return 1;
+----------> }
|
<A HREF="motion.html#[]">[]</A> | int func2(void)
| +-> {
| <A HREF="motion.html#[[">[[</A> | if (flag)
start +-- +-- return flag;
| <A HREF="motion.html#][">][</A> | return 2;
| +-> }
<A HREF="motion.html#]]">]]</A> |
| int func3(void)
+----------> {
return 3;
}
Don't forget you can also use "<A HREF="motion.html#%">%</A>" to move between matching (), <A HREF="intro.html#{}">{}</A> and <A HREF="motion.html#[]">[]</A>.
That also works when they are many lines apart.
MOVING IN BRACES
The "<A HREF="motion.html#[(">[(</A>" and "<A HREF="motion.html#])">])</A>" commands work similar to "<A HREF="motion.html#[{">[{</A>" and "<A HREF="motion.html#]}">]}</A>", except that they
work on () pairs instead of <A HREF="intro.html#{}">{}</A> pairs.
<B> [(</B>
<--------------------------------
<-------
<B><FONT COLOR="PURPLE"> if (a == b && (c == d || (e > f)) && x > y) </FONT></B>
-------------->
-------------------------------->
<B> ])</B>
MOVING IN COMMENTS
To move back to the start of a comment use "<A HREF="motion.html#[/">[/</A>". Move forward to the end of a
comment with "<A HREF="motion.html#]/">]/</A>". This only works for /* - */ comments.
+-> +-> /*
| <A HREF="motion.html#[/">[/</A> | * A comment about --+
<A HREF="motion.html#[/">[/</A> | +-- * wonderful life. | <A HREF="motion.html#]/">]/</A>
| */ <-+
|
+-- foo = <A HREF="motion.html#bar">bar</A> * 3; --+
| <A HREF="motion.html#]/">]/</A>
/* a short comment */ <-+
<hr class="doubleline">
*<A NAME="29.4"></A><B>29.4</B>* Finding global identifiers
You are editing a C program and wonder if a variable is declared <A HREF="motion.html#as">as</A> "int" or
"unsigned". A quick way to find this is with the "<A HREF="tagsrch.html#[I">[I</A>" command.
Suppose the cursor is on the <A HREF="motion.html#word">word</A> "column". Type:
<B> [I</B>
Vim will list the matching lines <A HREF="motion.html#it">it</A> can find. Not only in the current file,
but also in all included files (and files included in them, etc.). The result
looks like this:
<B><FONT COLOR="PURPLE"> structs.h </FONT></B>
<B><FONT COLOR="PURPLE"> 1: 29 unsigned column; /* column number */ </FONT></B>
The advantage over using <A HREF="tagsrch.html#tags">tags</A> or the preview <A HREF="windows.html#window">window</A> is that included files are
searched. In most cases this results in the right declaration to be found.
Also when the <A HREF="tagsrch.html#tags">tags</A> file is out of date. Also when you don't have <A HREF="tagsrch.html#tags">tags</A> for the
included files.
However, a few things must be right for "<A HREF="tagsrch.html#[I">[I</A>" to <A HREF="diff.html#do">do</A> its work. First of all,
the <A HREF="options.html#'include'">'include'</A> option must specify how a file is included. The default value
works for C and C++. For other languages you will have to change <A HREF="motion.html#it">it</A>.
LOCATING INCLUDED FILES
Vim will find included files in the places specified with the <A HREF="options.html#'path'">'path'</A>
option. If a directory is missing, some include files will not be found. You
can discover this with this command:
<B> :checkpath</B>
It will list the include files that could not be found. Also files included
by the files that could be found. An example of the output:
<B><FONT COLOR="PURPLE"> --- Included files not found in path --- </FONT></B>
<B><FONT COLOR="PURPLE"> <io.h> </FONT></B>
<B><FONT COLOR="PURPLE"> vim.h --> </FONT></B>
<B><FONT COLOR="PURPLE"> <functions.h> </FONT></B>
<B><FONT COLOR="PURPLE"> <clib/exec_protos.h> </FONT></B>
The "io.h" file is included by the current file and can't be found. "vim.h"
can be found, thus "<A HREF="tagsrch.html#:checkpath">:checkpath</A>" goes into this file and checks what <A HREF="motion.html#it">it</A>
includes. The "functions.h" and "clib/exec_protos.h" files, included by
"vim.h" are not found.
Note:
Vim is not a compiler. It does not recognize "#ifdef" statements.
This means every "#include" statement is used, also when <A HREF="motion.html#it">it</A> comes
after "#if NEVER".
To fix the files that could not be found, add a directory to the <A HREF="options.html#'path'">'path'</A>
option. A good place to find out about this is the Makefile. Look out for
lines that contain "-I" items, like "-I/usr/local/X11". To add this directory
use:
<B> :set path+=/usr/local/X11</B>
When there are many subdirectories, you can use the "*" <A HREF="editing.html#wildcard">wildcard</A>. Example:
<B> :set path+=/usr/*/include</B>
This would find files in "/usr/local/include" <A HREF="motion.html#as">as</A> well <A HREF="motion.html#as">as</A> "/usr/X11/include".
When working on a project with a whole nested tree of included files, the "**"
items is useful. This will search down in all subdirectories. Example:
<B> :set path+=/projects/invent/**/include</B>
This will find files in the directories:
<B><FONT COLOR="PURPLE"> /projects/invent/include </FONT></B>
<B><FONT COLOR="PURPLE"> /projects/invent/main/include </FONT></B>
<B><FONT COLOR="PURPLE"> /projects/invent/main/os/include </FONT></B>
etc.
There are even more possibilities. Check out the <A HREF="options.html#'path'">'path'</A> option for info.
If you want to see which included files are actually found, use this
command:
<B> :checkpath!</B>
You will get a (very long) list of included files, the files they include, and
so on. To shorten the list a bit, Vim shows "(Already listed)" for files that
were found before and doesn't list the included files in there again.
JUMPING TO A MATCH
"<A HREF="tagsrch.html#[I">[I</A>" produces a list with only one line of text. When you want to have a
closer look at the first item, you can jump to that line with the command:
<B> [<Tab></B>
You can also use "[ CTRL-I", since <A HREF="motion.html#CTRL-I">CTRL-I</A> is the same <A HREF="motion.html#as">as</A> pressing <A HREF="motion.html#<Tab>"><Tab></A>.
The list that "<A HREF="tagsrch.html#[I">[I</A>" produces has a number at the start of each line. When you
want to jump to another item than the first one, type the number first:
<B> 3[<Tab></B>
Will jump to the third item in the list. Remember that you can use <A HREF="motion.html#CTRL-O">CTRL-O</A> to
jump back to where you started from.
RELATED COMMANDS
<A HREF="tagsrch.html#[i">[i</A> only lists the first match
<A HREF="tagsrch.html#]I">]I</A> only lists items below the cursor
<A HREF="tagsrch.html#]i">]i</A> only lists the first item below the cursor
FINDING DEFINED IDENTIFIERS
The "<A HREF="tagsrch.html#[I">[I</A>" command finds any identifier. To find only macros, defined with
"#define" use:
<B> [D</B>
Again, this searches in included files. The <A HREF="options.html#'define'">'define'</A> option specifies what a
line looks like that defines the items for "<A HREF="tagsrch.html#[D">[D</A>". You could change <A HREF="motion.html#it">it</A> to make
<A HREF="motion.html#it">it</A> work with other languages than C or C++.
The commands related to "<A HREF="tagsrch.html#[D">[D</A>" are:
<A HREF="tagsrch.html#[d">[d</A> only lists the first match
<A HREF="tagsrch.html#]D">]D</A> only lists items below the cursor
<A HREF="tagsrch.html#]d">]d</A> only lists the first item below the cursor
<hr class="doubleline">
*<A NAME="29.5"></A><B>29.5</B>* Finding local identifiers
The "<A HREF="tagsrch.html#[I">[I</A>" command searches included files. To search in the current file only,
and jump to the first place where the <A HREF="motion.html#word">word</A> under the cursor is used:
<B> gD</B>
Hint: Goto Definition. This command is very useful to find a variable or
function that was declared locally ("static", in C terms). Example (cursor on
"counter"):
+-> static int counter = 0;
|
| int get_counter(void)
<A HREF="pattern.html#gD">gD</A> | {
| ++counter;
+-- return counter;
}
To restrict the search even further, and look only in the current function,
use this command:
<B> gd</B>
This will go back to the start of the current function and find the first
occurrence of the <A HREF="motion.html#word">word</A> under the cursor. Actually, <A HREF="motion.html#it">it</A> searches backwards to
an empty line above a "<A HREF="motion.html#{">{</A>" in the first column. From there <A HREF="motion.html#it">it</A> searches forward
for the identifier. Example (cursor on "idx"):
int find_entry(char *name)
{
+-> int idx;
|
<A HREF="pattern.html#gd">gd</A> | for (idx = 0; idx <A HREF="change.html#<"><</A> table_len; ++idx)
| if (strcmp(table[idx].name, name) <A HREF="change.html#==">==</A> 0)
+-- return idx;
}
<hr class="doubleline">
Next chapter: |<A HREF="usr_30.html">usr_30.txt</A>| Editing programs
Copyright: see |<A HREF="usr_01.html#manual-copyright">manual-copyright</A>| vim:tw=78:ts=8:ft=help:norl:
<A HREF="#top">top</A> - <A HREF="index.html">main help file</A>
</PRE>
</div>
</div>
<footer class="site-footer">
<div class="wrap">
<div class="footer-content">
<i>vim document on </i>
<a href="/vim-en/usr_toc.html">www.4e00.com</a>
</div>
</div>
</footer>
</body>
</HTML>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,331 |
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using NeuroSystem.Workflow.UserData.UI.Html.Mvc.Extensions;
namespace NeuroSystem.Workflow.UserData.UI.Html.Mvc.UI
{
public class Transport : JsonObject
{
public CrudOperation Create
{
get;
private set;
}
public IDictionary<string, object> CustomCreate
{
get;
set;
}
public IDictionary<string, object> CustomDestroy
{
get;
set;
}
public IDictionary<string, object> CustomRead
{
get;
set;
}
public IDictionary<string, object> CustomUpdate
{
get;
set;
}
public CrudOperation Destroy
{
get;
private set;
}
public ClientHandlerDescriptor FunctionCreate
{
get;
set;
}
public ClientHandlerDescriptor FunctionDestroy
{
get;
set;
}
public ClientHandlerDescriptor FunctionRead
{
get;
set;
}
public ClientHandlerDescriptor FunctionSubmit
{
get;
set;
}
public ClientHandlerDescriptor FunctionUpdate
{
get;
set;
}
public string IdField
{
get;
set;
}
public ClientHandlerDescriptor ParameterMap
{
get;
set;
}
public string Prefix
{
get;
set;
}
public CrudOperation Read
{
get;
private set;
}
public bool SerializeEmptyPrefix
{
get;
set;
}
//public TransportSignalR SignalR
//{
// get;
// set;
//}
public bool StringifyDates
{
get;
set;
}
public CrudOperation Update
{
get;
private set;
}
public Transport()
{
this.Read = new CrudOperation();
this.Update = new CrudOperation();
this.Destroy = new CrudOperation();
this.Create = new CrudOperation();
this.FunctionRead = new ClientHandlerDescriptor();
this.FunctionUpdate = new ClientHandlerDescriptor();
this.FunctionDestroy = new ClientHandlerDescriptor();
this.FunctionCreate = new ClientHandlerDescriptor();
this.FunctionSubmit = new ClientHandlerDescriptor();
this.ParameterMap = new ClientHandlerDescriptor();
this.SerializeEmptyPrefix = true;
//this.SignalR = new TransportSignalR();
}
//protected override void Serialize(IDictionary<string, object> json)
// {
// if (this.CustomRead != null)
// {
// json["read"] = this.CustomRead;
// }
// else if (!this.FunctionRead.HasValue())
// {
// IDictionary<string, object> strs = this.Read.ToJson();
// if (strs.Keys.Any<string>())
// {
// json["read"] = strs;
// }
// }
// else
// {
// json["read"] = this.FunctionRead;
// }
// if (this.SerializeEmptyPrefix)
// {
// json["prefix"] = (this.Prefix.HasValue() ? this.Prefix : string.Empty);
// }
// else if (this.Prefix.HasValue())
// {
// json["prefix"] = this.Prefix;
// }
// if (this.CustomUpdate != null)
// {
// json["update"] = this.CustomUpdate;
// }
// else if (!this.FunctionUpdate.HasValue())
// {
// IDictionary<string, object> strs1 = this.Update.ToJson();
// if (strs1.Keys.Any<string>())
// {
// json["update"] = strs1;
// }
// }
// else
// {
// json["update"] = this.FunctionUpdate;
// }
// if (this.CustomCreate != null)
// {
// json["create"] = this.CustomCreate;
// }
// else if (!this.FunctionCreate.HasValue())
// {
// IDictionary<string, object> strs2 = this.Create.ToJson();
// if (strs2.Keys.Any<string>())
// {
// json["create"] = strs2;
// }
// }
// else
// {
// json["create"] = this.FunctionCreate;
// }
// if (this.CustomDestroy != null)
// {
// json["destroy"] = this.CustomDestroy;
// }
// else if (!this.FunctionDestroy.HasValue())
// {
// IDictionary<string, object> strs3 = this.Destroy.ToJson();
// if (strs3.Keys.Any<string>())
// {
// json["destroy"] = strs3;
// }
// }
// else
// {
// json["destroy"] = this.FunctionDestroy;
// }
// if (this.FunctionSubmit.HasValue())
// {
// json["submit"] = this.FunctionSubmit;
// }
// if (this.StringifyDates)
// {
// json["stringifyDates"] = this.StringifyDates;
// }
// if (!string.IsNullOrEmpty(this.IdField))
// {
// json["idField"] = this.IdField;
// }
// if (this.ParameterMap.HasValue())
// {
// json["parameterMap"] = this.ParameterMap;
// }
// IDictionary<string, object> strs4 = this.SignalR.ToJson();
// if (strs4.Keys.Any<string>())
// {
// json["signalr"] = strs4;
// }
// }
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 1,864 |
define(function () {
app.registerController('CreateTaskCtrl', ['$scope', '$http', 'Template', 'Project', 'Options', function ($scope, $http, Template, Project, Options) {
console.log(Template);
$scope.task = {};
$scope.tpl = Template;
$scope.options = Options || {};
if (Template.type === 'deploy' && Template.build_template_id) {
$http.get(Project.getURL() + '/templates/' + Template.build_template_id + '/tasks/last').then(function(Builds) {
$scope.builds = Builds ? Builds.data.filter(function(build) { return build.status === 'success'; }) : [];
if ($scope.options.build_task_id) {
$scope.task.build_task_id = $scope.options.build_task_id;
$scope.task.commit = $scope.options.commit;
} else if ($scope.builds[0]) {
$scope.task.build_task_id = $scope.builds[0].id;
}
});
}
$scope.getBuildTitle = function (build) {
var ret = '';
if (build.ver) {
ret += build.ver;
} else {
ret += "#" + build.id;
}
if (build.description) {
ret += ' - ' + build.description;
}
return ret;
};
$scope.run = function (task, dryRun) {
task.template_id = Template.id;
var params = angular.copy(task);
if (dryRun) {
params.dry_run = true;
}
$http.post(Project.getURL() + '/tasks', params).success(function (t) {
$scope.$close(t);
}).error(function (_, status) {
swal('Error', 'error launching task: HTTP ' + status, 'error');
});
};
}]);
app.registerController('TaskCtrl', ['$scope', '$http', function ($scope, $http) {
$scope.raw = false;
$scope.task = $scope.task;
var logData = [];
var onDestroy = [];
onDestroy.push($scope.$on('task.log', function (evt, data) {
var o = data.output + '\n';
var d = moment(data.time);
if (!$scope.raw) {
o = d.format('HH:mm:ss') + ': ' + o;
}
if ($scope.task.id !== data.task_id) {
return;
}
for (var i = 0; i < logData.length; i++) {
if (d.isAfter(logData[i].time)) {
// too far -- no point scanning rest of data as its in chronological order
break;
}
if (d.isSame(logData[i].time) && data.output == logData[i].output) {
return;
}
}
$scope.output_formatted += o;
if (!$scope.$$phase) $scope.$digest();
}));
onDestroy.push($scope.$on('task.update', function (evt, data) {
$scope.task.status = data.status;
$scope.task.start = data.start;
$scope.task.end = data.end;
if (!$scope.$$phase) $scope.$digest();
}));
$scope.reload = function () {
$http.get($scope.project.getURL() + '/tasks/' + $scope.task.id + '/output')
.success(function (output) {
logData = output;
var out = [];
output.forEach(function (o) {
var pre = '';
if (!$scope.raw) pre = moment(o.time).format('HH:mm:ss') + ': ';
out.push(pre + o.output);
});
$scope.output_formatted = out.join('\n') + '\n';
});
if ($scope.task.user_id) {
$http.get('/users/' + $scope.task.user_id)
.success(function (output) {
$scope.task.user_name = output.name;
});
}
}
$scope.remove = function () {
$http.delete($scope.project.getURL() + '/tasks/' + $scope.task.id)
.success(function () {
$scope.$close();
}).error(function () {
swal("Error", 'Could not delete task', 'error');
})
}
$scope.$watch('raw', function () {
$scope.reload();
});
$scope.$on('$destroy', function () {
logData = null;
onDestroy.forEach(function (f) {
f();
});
});
}]);
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,794 |
Das Craters of the Moon National Monument (seit dem Jahr 2000 auch Craters of the Moon National Monument and Preserve) ist ein Schutzgebiet vom Typ eines National Monuments im Zentrum der Ebene des Snake Rivers im US-Bundesstaat Idaho auf etwa 1750 m über dem Meer. Das Gebiet besteht aus großflächigen erkalteten Lavaströmen und einigen Schlackenkegeln. Die vulkanische Aktivität ruht, ist aber nicht erloschen. Der letzte Ausbruch im Craters-of-the-Moon-Gebiet liegt etwa 2000 Jahre zurück.
Die Craters of the Moon liegen in einer der abgelegensten Regionen der Vereinigten Staaten, wurden erst Anfang des 20. Jahrhunderts näher erkundet und 1924 unter Schutz gestellt. Zuvor mehrfach geringfügig erweitert, wurde das Gebiet im Jahr 2000 auf die 13-fache Fläche ausgedehnt. Seitdem stehen das ganze Vulkanfeld einschließlich des Wapi-Felds im Süden und die angrenzenden Präriegebiete unter Naturschutz. Die Erweiterungsgebiete unterliegen dem geringeren Schutzstatus einer National Preserve, weshalb die Jagd weiterhin zulässig ist und die bestehenden Verträge über die Nutzung der öffentlichen Flächen durch Herden privater Rinder-Rancher in Kraft bleiben. Das Schutzgebiet wird gemeinsam durch den National Park Service und das Bureau of Land Management verwaltet, zwei Behörden unter dem Dach des US-Innenministeriums.
Der Name des Schutzgebietes stammt vom lebensfeindlichen Eindruck, den frühe Besucher hatten. Im Rahmen des Apollo-Programms nutzten Astronauten das Gebiet kurzzeitig zur Ausbildung.
Geologie
Das Craters-of-the-Moon-Gebiet liegt im Zentrum der vulkanisch geprägten Ebene des Snake Rivers, die sich durch den Süden Idahos zieht. Wie die gesamte Ebene fällt es leicht und gleichmäßig nach Süden ab. Die heutige Landschaft und ihr geologischer Untergrund wurden in drei charakteristischen Phasen geprägt.
Yellowstone-Hotspot
Der Untergrund der Ebene sind rhyolithischer Tuff und Gesteine aus verdichteter vulkanischer Asche. Sie entstanden durch einen Hotspot, der von einem Plume mit Magma aus dem Erdmantel gespeist wurde. Über diesen Plume verschiebt sich die Nordamerikanische Platte, eine der tektonischen Platten der Erdkruste, so dass die vulkanische Aktivität scheinbar von Südwest nach Nordost wandert. Der Hotspot liegt heute unter dem Yellowstone-Nationalpark und ist für die vulkanische Aktivität des Yellowstone-Supervulkans, seine Caldera, die Geysire und die sonstigen vulkanischen Erscheinungsformen des Nationalparks verantwortlich.
Das heutige Schutzgebiet befand sich vor rund 11 Millionen Jahren über dem Hotspot. Das aus dem Erdmantel aufsteigende Magma schmolz Granitgestein der Erdkruste. Dabei wölbte sich die Erdoberfläche auf. Bei einer weiteren Steigerung der Energie kam es zur katastrophalen Eruption und der geschmolzene Granit wurde in Form von Rhyolith ausgeworfen. Dabei entstanden eine Caldera und großflächige Tuff- und Aschenschichten. Mit der scheinbaren Wanderung verschob sich der Ort der nächsten Eruption. Für die Snake River Plain sind 142 massive Eruptionen nachgewiesen, dazu kommen hunderte mittlere und kleinere. Die Calderen überlappen sich typischerweise. Auf diese Weise wurden rhyolitische Lava und die Aschen ausgestoßen, die heute den tiefen Untergrund der Snake River Plain prägen. In der Erdkruste, in einer Tiefe von mindestens 10 km hinterließen die Schmelzvorgänge des Hotspots eine rund 10–15 km dicke Schicht aus basaltischer Schlacke, die bis heute weitgehend in geschmolzenem Zustand ist. Sie liefert seither die Energie für die vulkanische Aktivität in der Region. Die Oberfläche sank nach dem Abkühlen ab, es kam zu einer Senkung, die den Zugweg der Nordamerikanischen Platte über den Hotspot hinweg markiert: die Tiefebene der Snake River Plain.
Basaltische Eruptionen
Vor rund sechs Millionen Jahren begann im westlichen Nordamerika ein tektonischer Prozess, der zu einer Dehnung der Erdkruste führte, die eine Vielzahl weitgehend paralleler, grob in Nord-Süd-Richtung verlaufender Grabenbrüche zur Folge hatte. Damals entstand die Basin-and-Range-Region mit ihren parallelen Graben-und-Horst-Strukturen, die von Mexiko im Süden bis im Norden gerade noch nach Idaho reicht. Im Süden des heutigen Idahos wirkten die Kräfte auf die abgesunkenen Schichten aus eruptivem Gestein auf dem Zugweg des Hotspots. Hier führte die Dehnung der Kruste zu einer Vielzahl von Dehnungsbrüchen in Nordwest-Südost-Orientierung. Der größte der parallelen Dehnungsbrüche liegt im Zentrum der Ebene, hat eine Länge von rund 80 km bei etwa 2,5 km Breite und wird als Great Rift (of Idaho) bezeichnet. Nach anderen Interpretationen entstand der Great Rift unter dem Druck aufsteigender Lava aus dem Reservoir, das durch Belastung von oben plastisch verformt wurde und dabei Risse erzeugte.
Im Great Rift, wie in den anderen Brüchen in der Snake River Plain, stieg basaltisches Magma in Dykes auf und trat in Form von Spaltenvulkanen, Schlackenkegeln und flachen Schildvulkanen an der Oberfläche aus. Im Zeitraum von vor rund 6 Millionen Jahren bis vor etwa 15.000 Jahren entstanden gewaltige Lavaströme, die mehrere bis zu 1200 m dicke Schichten im Zentrum der Snake River Plain bildeten, in die vereinzelt Schichten aus Sedimentgestein eingelagert sind. Sie bedecken etwa 95 % der Ebene.
Jüngste Aktivitäten
Die bisher letzte Phase vulkanischer Aktivität im Crater of the Moon Gebiet begann vor etwa 15.000 Jahren. Am Great Rift traten drei und weiteren Brüchen fünf weitere junge Lavafelder aus. Die drei Felder des Great Rift, Craters of the Moon, Kings Bowl und Wapi, liegen innerhalb des Schutzgebietes. Es ist nach dem Craters-of-the-Moon-Feld benannt, das das größte Lavafeld in der Snake River Plain und zugleich in den Continental United States ist, das weit überwiegend aus dem Holozän, also den letzten 10.000 Jahren stammt. Es ist aus über 60 einzelnen Lavaströmen zusammengesetzt, die sich gegenseitig überlappen. Rund 30 km³ basaltische Lava traten in acht eruptiven Perioden aus und bedecken eine zusammengesetzte Fläche von rund 1600 km².
Rund 80 % der Lavafelder bestehen aus der gering viskosen Pāhoehoe-Lava. In erstarrter Form ist sie durch glatte Oberflächen und runde Formen gekennzeichnet, die vereinzelt einen blauen oder grünen Glanz annehmen können. Sie wird durch lange Flüsse gespeist, die beim Auslaufen und Erkalten hohle Lavaröhren hinterlassen können, langgestreckte Höhlen in und unter den Lavaflüssen. Fünf Lava-Röhren können im Gebiet begangen werden. Den Rest der Lavafelder macht die unregelmäßige Aa-Lava aus. Sie hat eine scharfkantige Oberfläche, die aus einzelnen Brocken von unter einem bis zu wenigen Zentimetern Größe zusammengesetzt ist. Ihre Gestalt ist teilweise wellenförmig mit bis zu 3 m hohen Berg- und Talstrukturen. Ihre Grenzen sind in der Regel durch eine steile Front gekennzeichnet.
Geochemisch bestehen die basaltischen Lavaflüsse im Craters-of-the-Moon-Feld aus olivinischem Tholeiit. Sie sind besonders reich an Eisen, Phosphor, Titan und Alkalimetallen
Bei den Eruptionen wurden außer den Lavaströmen rund 25 Schlackenkegel aufgeworfen, die häufig durch mehrere naheliegende Schlote entstanden, so dass sich die Kegel überlappen oder vollkommen ineinander liegen. Außerdem wurden vulkanische Bomben verschiedener Typen und Größen ausgestoßen. Die ausströmende Lava bildete mehrere flache Schildvulkane, darunter das Wapi-Feld, das aus nur einer Eruption entstanden ist.
Die jüngsten Erscheinungen sind kleinere Lavaströme im Norden des Craters-of-the-Moon-Feld. Darunter ist der North Crater Flow, der auf ein Alter von etwa 2000 Jahren datiert wurde. Auch wenn die vulkanische Aktivität im Craters-of-the-Moon-Gebiet seither ruht, ist sie nicht erloschen.
Ökosysteme
Das Gebiet der Craters of the Moon liegt im Bereich der Östlichen Snake-River-Basalt-Ebene. Das Klima ist semi-arid, kontinental mit heißen, trockenen Sommern und kalten Wintern. Die Böden sind jung, soweit vorhanden ist die Humusauflage gering. Zu unterscheiden sind drei Bodentypen: die jungen Lavafelder, Flächen älterer Lavaströme, in denen bereits eine Bodenbildung stattgefunden hat, sowie Böden auf äolischen, durch Wind verfrachteten Sedimenten aus Sand und Löss. Spezifisch für das Schutzgebiet – und untypisch für die Region – ist das kleinräumige Mosaik der Lebensräume. Mehrere hundert Kipukas, mittelgroße und kleine Inseln von verwitterten Böden, sind in jüngere Lava-Flüsse eingelagert und bilden jeweils abgeschlossene Ökosysteme.
Die jungen Lavaflächen sind zumeist ohne jede Vegetation. Vereinzelt stehen Büschel von Federgräsern. In Spalten und Rissen wachsen diverse Arten, die extreme Trockenheit ertragen. Darunter sind die Bartfaden-Art Penstemon deustus, die Flammenblume Phlox hoodii, das Sperrkrautgewächs Aliciella leptomeria und Fingerkräuter. An den Vulkankegeln und einigen Hügelflanken gibt es lokal genug Wasser für montane Busch- und Waldgesellschaften. Hier wachsen Wacholder, die Amerikanische Zitterpappel und die Nevada-Zirbel-Kiefer (Pinus flexilis).
Auf verwitterten Lavafeldern ist die typische Pflanzengesellschaft eine Wüsten-Beifuß-Steppe. Der namensgebende Wüsten-Beifuß bildet großflächig niedriges Gebüsch, ansonsten ist der Boden locker mit Gräsern der Prärie bedeckt. Die häufigste Grasart ist das Blaubüschel-Weizengras (Agropyron spicatum – auch: Pseudoroegnaria spicata), das in Büscheln wächst. Daneben kommen verschiedene Federgräser und Indianisches Reisgras (Oryzopsis hymenoides) vor. Purshia tridentata ist ein häufiger Strauch. Die Dach-Trespe tritt als Neophyt auf.
Die gleichen Pflanzengesellschaften bedecken auch die Böden außerhalb der Lavafelder. Die Vegetation ist dichter, weshalb die Flächen zur extensiven Weidenutzung mit Rindern dienen. Die Prärien im Zentrum und Süden des Gebietes sind durch historische und anhaltende Beweidung stark gestört.
Die Tierwelt ist geprägt durch die große Trockenheit und die extremen Sommertemperaturen. Verschiedene Tiere passen sich durch eine nächtliche Lebensweise an. Darunter fallen Taschenratten, Stinktier, Rotfuchs, Rotluchs, Puma, mehrere Fledermausarten, Nachtschwalben, Eulen und die meisten Kleinnager. Den Tagesrand nutzen Maultierhirsch, Kojote, Baumstachler und Baumwollschwanzkaninchen sowie viele Singvögel. Tagesaktiv sind Erdhörnchen, Streifenhörnchen, Murmeltiere sowie Eidechsen, Schlangen, Adler und Bussarde. In den Lavaröhren leben einige Käferarten, die endemisch für das Gebiet sind.
Geschichte
Ursprünglich war die Snake-River-Ebene dünn von Shoshonen besiedelt. Die Lavafelder und die anderen vulkanischen Strukturen des Great Rifts waren jedoch lebensfeindlich und wurden nicht dauerhaft genutzt. An einer Lavaröhre, die heute als Indian Tunnel bezeichnet wird, wurden Steine künstlich um die Tunnelöffnung angeordnet.
Die Lewis-und-Clark-Expedition von 1805/6 brachte die ersten Weißen ins heutige Idaho, sie passierten das Gebiet aber nördlich vom heutigen Schutzgebiet. In den 1830er Jahren wurde das Snake-River-Gebiet von Trappern und Pelzhändlern der britischen Hudson's Bay Company besucht. Auch sie kamen wohl nie in das heutige Schutzgebiet. 1833 erkundete der US-Offizier Benjamin Bonneville den Snake River und Teile der Ebene. Der erste nachweisliche Aufenthalt eines Weißen im heutigen Schutzgebiet fand 1852 statt. John Jeffrey erkundete eine nördliche Abkürzung für die Siedler-Trecks auf dem Oregon Trail, um die weiten Schleifen des Snake Rivers zu umgehen. Seine Route führte durch den äußersten Nordwesten des heutigen National Monuments. Sie wurde in den ersten Jahren nur sporadisch genutzt, gewann aber 1862 an Bedeutung, als Schoschonen und Bannock-Indianer sich auf der Hauptroute am Snake River gegen die in ihr Land eindringenden Weißen wehrten. Um den Indianern zu entgehen, wurde der größte Siedlerzug auf dem Oregon-Trail aller Zeiten mit 338 Wagen und 1095 Menschen von Tim Goodale über die nördliche Route geführt, die ab da als Goodale's Cutoff bezeichnet und bevorzugt wurde.
1901 fand die erste Begehung des Gebietes durch einen Geologen der US Geological Survey statt. Harold Stearns, Geologe der USGS, besuchte das Gebiet 1921 und 1923 und verfasste eine erste Beschreibung, die 1924 in der Geographical Review veröffentlicht wurde. Der National Park Service forderte bei ihm ein Gutachten an, das die Gebietsstrukturen und geeignete Grenzen eines möglichen National Monuments enthalten sollte. 1920 waren Robert W. Limbert und sein Nachbar W. L. Cole, naturkundlich begeisterte Laien aus Boise, in 17 Tagen durch das Great-Rift-Gebiet gewandert. Sie erkundeten als erste die Region in voller Länge und beschrieben die vielfältigen vulkanischen Strukturen. Limbert veröffentlichte 1924 einen Bericht über die Expedition im populären Magazin National Geographic unter dem Titel "Among the 'Craters of the Moon'". Er schlug vor, das Gebiet als Nationalpark auszuweisen, und schickte sein Skizzenbuch mit Fotos und weiteren Notizen an Präsident Calvin Coolidge. Aufgrund des vorliegenden Gutachtens für den National Park Service und der öffentlichen Bekanntheit des Gebietes nach dem Erscheinen des National-Geographic-Artikels nutzte Coolidge die Ermächtigung des Antiquity Acts und widmete das Gebiet als National Monument.
In den ersten Jahrzehnten war das Schutzgebiet nur wenig ausgebaut. In Vorbereitung auf das 50. Jubiläum des National Park Services wurde Mitte der 1950er Jahre das Mission-66-Programm aufgelegt, bei dem rund eine Milliarde Dollar in die Nationalparks und sonstigen Schutzgebiete des Bundes investiert wurden. Auch Craters of the Moon bekam ab 1956 ein neues Besucherzentrum, einen Campingplatz, eine im Kreis durch das Schutzgebiet geführte Straße und weitere Einrichtungen zur Natur- und Kulturinterpretation und Besucherbetreuung.
1969 erkundeten Teams des Apollo-Programms geologisch besondere Regionen im Westen der Vereinigten Staaten. Die Astronauten Alan Shepard, Edgar Mitchell, Eugene Cernan und Joe Engle kamen unter anderem in das Crater-of-the-Moon-Gebiet und lernten die fachmännische Identifizierung und Beschreibung von vulkanischem Gestein, wie es auf dem Mond zu erwarten war.
1970 widmete der Kongress der Vereinigten Staaten den Großteil des National Monuments zusätzlich als Wilderness Area, die strengste Klasse von Naturschutzgebieten in den USA. Es war die erste offiziell geschützte Wildnis in einem Gebiet des National Park Service.
In den 1980er Jahren betrieben lokale Tourismus-Unternehmer und die Politik des Staates Idaho die Umwidmung des National Monuments in einen Nationalpark. Sie erhofften sich davon eine stärkere Anziehungskraft für den Tourismus der Region. Idaho hat als einziger Bundesstaat im Westen der Vereinigten Staaten keinen Nationalpark innerhalb seiner Grenzen, wenn man von einem schmalen Streifen am Rand des Yellowstone-Nationalparks absieht. Der National Park Service lehnte die Aufwertung ab: Das Gebiet habe mit dem Vulkanismus nur einen Typ von natürlichen Erscheinungsformen und sei daher nicht als Nationalpark geeignet. Der Status als National Monument sei angemessen. Anfang der 1990er Jahre wurde eine vermittelnde Position erarbeitet, nach der es bei der Ausweisung als National Monument bleiben solle, aber das Gebiet großräumig erweitert werden könne.
Im Jahr 2000 wurde das Schutzgebiet um die gesamten Lavafelder des Great-Rift-Gebietes erweitert und benachbarte Prärieflächen unter der Verwaltung des Bureau of Land Management hinzugenommen. Das Schutzgebiet unter den Namen Crater of the Moon National Monument and Preserve umfasst jetzt fast 2900 km² und alle vulkanischen Erscheinungen der Region.
Das Schutzgebiet heute
Der erschlossene Bereich des Craters of the Moon National Monument and Preserve liegt im Norden des Gesamtgebietes am US-Highway US-20, der hier gebündelt mit US-26 und US-93 verläuft. Vom Highway zweigt eine Stichstraße zum Besucherzentrum ab, sie führt weiter zum kleinen Campingplatz und einer als Einbahnstraße ausgewiesenen, etwa 11 km langen Schleife, die durch den Norden des wichtigsten Lavafeldes führt. Die Autoroute windet sich um mehrere Vulkankegel, von ihr zweigen wiederum Stichstraßen ab, die zu ausgewiesenen kurzen Wanderwegen führen.
Am meisten begangen sind der North Crater Flow Trail am Besucherzentrum, der in die Charakteristika einer Lavalandschaft einführt, der Devils Orchard Trail, auf dem Besucher die langsame Besiedlung der Lavalandschaft mit Pioniervegetation erkunden können, und der Caves Trail, der zu den Eingängen von vier für Besucher zugängliche und eine einsehbare Lavaröhren führt. Der Indian Tunnel kann ohne weitere Ausrüstung in der vollen Länge von knapp 250 m begangen werden, die anderen erfordern Höhlenausstattung mit mehreren unabhängigen Lichtquellen und die Bereitschaft zum Kriechen durch schmale Eingänge. Außerdem gibt es zwei ausgewiesene Wanderwege, die aus dem erschlossenen Teil des Gebietes in die Wildnisregion führen. Im Sommerhalbjahr werden von den Rangern täglich geführte Wanderungen und Vorträge angeboten. Im Winter ist die Straßenschleife für Autos gesperrt und als Skilanglauf-Loipe geöffnet. Führungen auf Schneeschuhen finden an den Wochenenden statt.
Craters of the Moon National Monument and Preserve ist mit deutlich über 200.000 Besuchern im Jahr das wichtigste touristische Ziel in der dünn besiedelten Region. Es ist jedoch typischerweise nur ein Abstecher für Besucher anderer Ziele im weiteren Umfeld und wird nur von rund 20 % der Besucher gezielt angesteuert. Die anderen kommen aus anderem Grund nach Idaho oder durchqueren die Region nur und nutzen das Schutzgebiet als Zwischenstopp.
Nahezu alle Besucher beschränken sich auf den erschlossenen Bereich im Norden, der aber nur den kleinsten Teil des Schutzgebietes ausmacht. Er liegt im ursprünglichen National Monument, das eine Fläche von etwa 216 km² hat. Seit der Erweiterung im Jahr 2000 erstreckt sich das Gebiet nach Süden bis auf fast die gesamte Breite der Snake River Plain. Damals kamen rund 1600 km² unter der Verwaltung des National Park Service als National Preserve hinzu. Hierbei handelt es sich nahezu ausschließlich um Lavafelder. Weitere rund 1000 km² stehen unter der Verwaltung des Bureau of Land Management und wurden ebenfalls 2000 dem Schutzgebiet zugeschlagen. Sie sind Steppe und die bestehende Weidenutzung durch Rinderherden privater Rancher wurde bei der Schutzgebietsausweisung beibehalten. Im Hinterland gibt es einige wenige unbefestigte Straßen und Pisten in den Steppenanteilen, auf den Lavafeldern kann man sich nur zu Fuß bewegen.
Literatur
Kathleen M. Haller; Spencer H. Wood: Geological field trips in southern Idaho, eastern Oregon, and northern Nevada. Boise, Idaho, Boise State University, 2004, ISBN 978-0-9753738-0-4
Weblinks
Bureau of Land Management: Craters of the Moon National Monument (offizielle Seite) (englisch)
Einzelnachweise
Geographie (Idaho)
National Monument (Vereinigte Staaten)
National Preserve (Vereinigte Staaten)
Schlacken- und Aschenkegel
Vulkangebiet
Blaine County (Idaho)
Power County
Minidoka County
Lincoln County (Idaho)
Butte County (Idaho)
Schutzgebiet der IUCN-Kategorie III
Vulkanismus in den Vereinigten Staaten
Geologie Amerikas
Schutzgebiet (Umwelt- und Naturschutz) in Nordamerika | {
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Home/Sports/Frittelli beats Atwal in playoff for Mauritius Open title
Frittelli beats Atwal in playoff for Mauritius Open title
BEL OMBRE, Mauritius (AP) Dylan Frittelli beat Arjun Atwal with a birdie on the first playoff hole to clinch the Mauritius Open title on Sunday.
The 27-year-old South African secured a second European Tour title in his first year on the tour. He was named the tour's best newcomer at the start of the week following a 19th place finish in the 2017 Race to Dubai.
He began the new season with victory in Mauritius after India's Atwal, who set the pace for much of the tournament at Heritage Golf Course, agonizingly missed out on an eagle on the last hole in regulation play that would have given him his first European title in nearly a decade.
Atwal saw his eagle putt lip out on No. 18, and settled for birdie to set up a playoff. He and Frittelli both finished on 16 under par after 18 holes.
Frittelli made five birdies and a bogey in his final round 4-under 67. Atwal, who set a course record with an opening-round 62, finished with six birdies but also two bogeys for his 68.
France's Romain Langasque was alone in third place, two shots behind Frittelli and Atwal after finishing with a 67. Louis de Jager was fourth.
Louis Oosthuizen, the highest-ranked player at the tournament, made a charge up the leaderboard on the final day with two eagles and a run of three straight birdies on Nos. 10-12. However, the former Open champion's challenge ended with a triple bogey seven on No. 16 and he finished seventh, five shots behind.
Cris Carter believes Carson Wentz will be a superstar in Week 1
Seema confident Gordinho will stay at Celtic
Swamp soccer tournament takes place in forest in Russia
Barcelona eyeing shock move for Manchester City captain Vincent Kompany
Football: Ivan Rakitic says goodbye to Croatia squad as he calls time on international career | {
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Martín Miguel Juan de la Mata de Güemes Montero Goyechea y la Corte (Salta, 8 de febrer de 1785 - Cañada de la Horqueta (província de Salta), 17 de juny de 1821) fou un militar argentí.
Martín era fill d'un alt funcionari espanyol i als catorze anys assentà plaça en el regiment d'infanteria de Buenos Aires, lluitant després contra la invasió anglesa i mereixent el grau d'oficial.
Fou dels primers a ajuntar-se a la Revolució i a causa de la seva cultura tècnica i els seus dots de valor, reuní al seu entorn un exèrcit compost de gautxos, però en el qual els oficials eren joves de les més distingides famílies. Especialment en la campanya empresa en l'Alt Perú (1815), a les ordres de José Casimiro Rondeau (1775-1844), en el setge conegut com per los Cerrillos, un conveni, pel qual es juraven <pau sòlida, amistat eterna, oblit del passat i amnistia general>, quedant Güemes amb els desertors de l'exèrcit vençut a Sipe-Sipe.
Des de llavors s'encarregà de la defensa del Nord, i complí amb tal força i brillantor la seva feina, que el seu nom es cobrí de glòria i es feu mereixedor de passar a la posteritat, i ocupa una bonica pàgina en la història militar i política de la República Argentina, en la qual se'l reconeix com un dels primers homes de la Revolució i dels millors generals de la Independència.
Fou un gran amic i admirador de la guerrillera Juana Azurduy de Padilla de la qual en feu grans elogis.
Bibliografia
Enciclopèdia Espasa tom. Núm. 27, pàg. 76,
Enllaços externs
http://www.iruya.com/content/view/258/41/
https://web.archive.org/web/20100102053916/http://www.camdipsalta.gov.ar/INFSALTA/biografia.htm
Militars argentins
Independentistes
Persones de Salta
Morts a l'Argentina
Activistes argentins | {
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De Mandjeswaard is een polder van circa 600 hectare ten oosten van het Kampereiland in de Nederlandse provincie Overijssel.
Beschrijving
In 1432 was er al sprake van verpachting van een erf in de Mandjeswaard. De polder, oorspronkelijk een eiland, behoort bij Kampen en ligt ten noorden van de Kamperzeedijk. Door aanslibbing vanuit de voormalige Zuiderzee vond landaanwinning van deze polder plaats. De Mandjeswaard wordt begrensd door het Ganzendiep, de Goot en het Zwarte Meer. De polder maakt deel uit van het, in 2005 tot Nationaal Landschap aangewezen, gebied de IJsseldelta. Vroeger werd de polder ook wel Mandemakersland genoemd, naar een van de pachters, ene Klaas Mandemaker. De polder is over land slechts bereikbaar via een brug over de Goot nabij Lutterzijl. Tussen het Kampereiland en de Mandjeswaard is een zelfbedienings veerpontje over het Ganzendiep, het H.C. Kleemanspontje, genoemd naar Henk Kleemans, die van 1978 tot 2000 burgemeester van Kampen was. Het voetveer over de Goot naar de polder de Pieper is opgeheven. Op 28 mei 2016 is op deze plek een zelfbedienings-fietspontje (de Pieperpont) in gebruik genomen. Een "belle" bij het Pieperstrand herinnert nog aan het verdwenen veer. Deze "belle" is erkend als provinciaal monument. Bij de polder stond het laatste in Nederland gebouwde schepradgemaal. Het scheprad is tentoongesteld bij het oude stoomgemaal Mastenbroek. In het noorden van de polder staat op een terp langs de Goot een obelisk ter ere van Johan Christiaan baron van Haersolte van Haerst, die als commissaris van de "Naamloze maatschappij ter bevordering van landaanwinning van het Zwolsche Diep", zich heeft ingespannen voor de ontwikkeling van het gebied. De obelisk is erkend als een rijksmonument.
Afbeeldingen
Polder in Overijssel
Geografie van Kampen | {
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{"url":"https:\/\/docs.mosaicml.com\/en\/latest\/api_reference\/composer.algorithms.seq_length_warmup.seq_length_warmup.html","text":"# composer.algorithms.seq_length_warmup.seq_length_warmup#\n\nCore code for sequence length warmup.\n\nFunctions\n\n set_batch_sequence_length Set the sequence length of a batch.\n\nClasses\n\n SeqLengthWarmup Progressively increases the sequence length during training.\nclass composer.algorithms.seq_length_warmup.seq_length_warmup.SeqLengthWarmup(duration=0.3, min_seq_length=8, max_seq_length=1024, step_size=8, truncate=True)[source]#\n\nProgressively increases the sequence length during training.\n\nChanges the sequence length of all tensors in the input batch. The sequence length increases from min_seq_length to max_seq_length in steps of step_size during the first duration fraction of training.\n\nThe sequence length is then kept at max_seq_length for the rest of training.\n\nTensors are either truncated (truncate=True) or reshaped to create new examples from the extra tokens (truncate=False).\n\nThis algorithm runs on AFTER_DATALOADER to modify the sequence length of a batch of data after the model and data have been moved to accelerators.\n\nNote\n\nstep_size should be a multiple of eight for optimal throughput on NVIDIA GPUs.\n\nNote\n\nVariable input lengths can create CUDA OOM errors. To avoid this, we follow the PyTorch notes and pre-allocate the memory with a blank forward and backward pass.\n\nSee the Method Card for more details.\n\nExample:\n\nfrom composer.algorithms import SeqLengthWarmup\nfrom composer import Trainer\n\nseq_length_warmup = SeqLengthWarmup(duration=0.5,\nmin_seq_length=8,\nmax_seq_length=1024,\nste_size=8,\ntruncate=False)\n\ntrainer = Trainer(model=model,\nmax_duration=\"1ep\",\nalgorithms=[seq_length_warmup])\n\nParameters\n\u2022 duration (float, optional) \u2013 Fraction of total training for sequential length learning. Default = 0.3.\n\n\u2022 min_seq_length (int, optional) \u2013 Minimum sequence length to start the warmup. Default = 8.\n\n\u2022 max_seq_length (int, optional) \u2013 Maximum sequence length to stop the warmup. Default = 1024.\n\n\u2022 step_size (int, optional) \u2013 Step size of sequence length. Default = 8.\n\n\u2022 truncate (bool, optional) \u2013 Truncate tensors or reshape extra tokens to new examples. Default = True.\n\ncomposer.algorithms.seq_length_warmup.seq_length_warmup.set_batch_sequence_length(batch, curr_seq_len, truncate=True)[source]#\n\nSet the sequence length of a batch.\n\nChanges the sequence length of all tensors in the provided dictionary to curr_seq_len by either truncating the tensors (truncate=True) or reshaping the tensors to create new examples from the extra tokens (truncate=False).\n\nNote\n\nThe schedule for curr_seq_len over training time should be managed outside of this function.\n\nNote\n\nVariable input lengths can create CUDA OOM errors. To avoid this, we follow the PyTorch notes and pre-allocate the memory with a blank forward and backward pass.\n\nParameters\n\u2022 batch (Dict[str, Tensor]) \u2013 The input batch to the model, must be a dictionary.\n\n\u2022 curr_seq_length (int) \u2013 The desired sequence length to apply.\n\n\u2022 truncate (bool, optional) \u2013 Truncate sequences early, or reshape tensors to create new examples out of the extra tokens. Default: True.\n\nReturns\n\nDict[str, Tensor] \u2013 a Mapping of input tensors to the model, where all tensors have curr_seq_len in the second dimension.\n\nExample:\n\nimport composer.functional as cf\n\nfor epoch in range(num_epochs):","date":"2022-06-29 19:51:25","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\": 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\": 0.4601442217826843, \"perplexity\": 7891.867703629468}, \"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-2022-27\/segments\/1656103642979.38\/warc\/CC-MAIN-20220629180939-20220629210939-00364.warc.gz\"}"} | null | null |
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\title[Controlling Several Atoms in a Cavity]{Controlling Several Atoms in a Cavity}
\author{Michael Keyl}
\address{Zentrum Mathematik, M5,
Technische Universit{\"a}t M{\"u}nchen,\\
Boltzmannstrasse 3, 85748 Garching, Germany}
\ead{michael.keyl@tum.de}
\author{Robert Zeier}
\address{Department Chemie,
Technische Universit{\"a}t M{\"u}nchen,\\
Lichtenbergstrasse 4, 85747 Garching, Germany}
\ead{robert.zeier@ch.tum.de}
\author{Thomas {Schulte-Herbr{\"u}ggen}}
\address{Department Chemie,
Technische Universit{\"a}t M{\"u}nchen,\\
Lichtenbergstrasse 4, 85747 Garching, Germany}
\ead{tosh@ch.tum.de}
\vspace{3mm}
\hspace{17mm}{\footnotesize (Dated: January 22, 2014)}\\
\vspace{-3mm}
\pacs{03.67.Ac, 02.30.Yy, 42.50.Ct, 03.65.Db}
\begin{abstract}
We treat control of several two-level atoms interacting
with one mode of the electromagnetic field in a cavity.
This provides a useful model to study pertinent
aspects of quantum control in infinite dimensions via
the emergence of infinite-dimensional system algebras.
Hence we address problems arising with infinite-dimensional Lie algebras and those of unbounded
operators. For the models considered, these problems can be
solved by splitting the set of control Hamiltonians into two subsets:
The first obeys an abelian symmetry and can be treated in terms
of infinite-dimensional Lie algebras and strongly closed subgroups of
the unitary group of the system Hilbert space. The second breaks this
symmetry, and its discussion introduces new arguments. Yet, full
controllability can be achieved in a strong sense: e.g., in a time dependent
Jaynes-Cummings model we
show that, by tuning coupling constants appropriately, every
unitary of the coupled system (atoms and cavity) can be approximated
with arbitrarily small error.
\end{abstract}
\maketitle
\BLUE{\scriptsize
\tableofcontents
}
\section{Introduction}
Exploiting controlled dynamics of quantum systems is becoming of increasing importance not only
for solving computational tasks or quantum-secured communication, but also for
simulating other physical systems \cite{Fey82, VC02, WRJB02, ZGB, JC03, Lewenstein12}. An interesting
direction in quantum simulation applies many-body correlations to create \/`quantum matter\/'.
E.g., ultra-cold atoms in optical lattices are versatile models for
studying large-scale correlations \cite{BDZ08, Lewenstein12}.
Tunability and control over the system parameters of optical lattices allows for switching between
several low-energy states of different quantum phases \cite{Sachdev99, QPT10} or in particular for following real-time
dynamics such as the quantum quench from the super-fluid to the Mott insulator regime \cite{GMEHB02}.
Thus manipulating several atoms in a cavity is a key step to this end \cite{Haroche06} at the same time
posing challenging infinite-dimensional control problems.
While in finite dimensions controllability can readily be assessed by the Lie-algebra
rank condition \cite{SJ72, JS72, Bro72, Bro73, Jurdjevic97}, infinite-dimensional systems are
more intricate \cite{LY95}. As exact controllability in infinite dimensions seemed daunting in earlier work \cite{HTC83,TR03,LTCC05,WTL06},
it took a while before approximate control paved the way to more realistic assessment \cite{AB05,CMS+09,BGR+13},
for a recent (partial) review see, e.g., also \cite{Borzi2011rev} and references therein.
Here we explore systems and control aspects for systems consisting of several two-level atoms coupled to a
cavity mode, i.e. the {\em Jaynes-Cummings model} \cite{JC63, TC68, TC69, BRX98}.
We build upon our previous symmetry arguments \cite{ZS11,ZZKS14} and moreover, we
apply appropriate operator topologies for addressing two controllability problems in particular:
(i) to which extent can pure states be interconverted and
(ii) can unitary gates be approximated with arbitrary precision.
In particular by treating the latter, we go beyond previous work,
which started out by a finite-dimensional truncation of a
two-level atom coupled to an oscillator \cite{RBM+04} followed by generalisations to infinite
dimensions \cite{BRB03,YL07,BBR10} both being confined to establishing
criteria of pure-state controllability. Note that \cite{BBR10} also treats one atom coupled to several
oscillators.
The general aim of this paper is twofold: On the one hand we study control
problems for atoms interacting with electromagnetic fields in cavities. On the
other hand, we address quantum control in infinite dimensions.
Therefore, the
purpose of Section~\ref{sec:controllability} is to provide enough material for a non-technical overview
on the second subject in order to understand the results on
the first (where the difficulties come from). Mathematical
details are postponed to Sections \ref{sec:lie-algebra-block}
and \ref{sec:strong-contr}, while results on cavity systems are presented in overview in
Section~ \ref{sec:atoms-cavity}.
\section{Controllability}
\label{sec:controllability}
The control of quantum systems poses considerable mathematical challenges when
applied to infinite dimensions. Basically, they arise from the fact that
anti-selfadjoint operators (recall that according to Stone's Theorem~\cite[VIII.4]{ReedSimonI},
they are generators of strongly continuous, unitary one-parameter groups)
do neither form a Lie algebra nor even a vector space. Or seen on the group level,
the group of unitaries equipped with the strong operator topology is a topological group yet not a Lie group.
So whenever strong topology has to be invoked, controllability cannot be assessed via a system Lie algebra.
Thus we address these challenges on the group level by employing the controlled time evolution of the quantum system
in order to approximate unitary operators, the action of which is measured with respect
to arbitrary, but finite sets of vectors. This is formalized in the notion of \emph{strong controllability}
(see Section~\ref{sec:strong-contr-1}) introduced here as a generalisation of pure-state controllability
already discussed in the literature.
Central to our discussion are abelian symmetries. Assuming that all but one of our Hamiltonians
observe such an ablian symmetry, we systematically analyze the infinite-dimensional control system
in its block-diagonalized basis. We obtain strong controllability (beyond pure-state controllability)
if one of the Hamiltonian breaks this abelian symmetry and some
further technical conditions are fullfilled.
\subsection{Time evolution}
\label{sec:time-evolution}
We treat control problems of the form
\begin{equation} \label{eq:1}
\dot{\psi}(t) = \sum_k u_k(t) H_k \psi(t) = H(t) \psi(t)
\end{equation}
where the $H_k$ with $k\in\{1,\dots,d\}$ are selfadjoint control Hamiltonians on an
infinite-dimensional, separable Hilbert space $\mathcal{H}$ and the controls $u_k: \ensuremath{\mathbb R} \to \ensuremath{\mathbb R}$ are piecewise-constant control
functions.
Since $\mathcal{H}$ is infinite-dimensional, the operators $H_k$ are usually
only defined on a dense subspace $D(H_k) \subset \mathcal{H}$ called the \emph{domain}
of $H_k$, the only exceptions being those $H_k$ which are bounded. However,
in this context,
control problems where all $H_k$ are bounded are not very interesting from a
physical point of view. In other words, there is no way around considering
those domains and many difficulties of control theory in infinite dimensions
arises from this fact\footnote{Note that domains of unbounded operators are
not just a mathematical pedantism. The domain is a crucial part of the
definition of an operator and contains physically relevant information. A
typical example is the Laplacian in a box which requires \emph{boundary
conditions} for a complete description. Up to a certain degree, domains can
be regarded as an abstract form of boundary conditions (possibly at
infinity).} .
We will also assume that Eq.~\eqref{eq:1} will have unique solutions for all
initial states $\psi_0 \in \mathcal{H}$ and all times $t$. So for
each pair of times $t_1 < t_2$ there is a unitary propagator
$U(t_1,t_2)\psi_0 = \mathcal{T} \int_{t_1}^{t_2} \exp(-i t H(t)) \psi_0$,
where $\mathcal{T}$ denotes time ordering. Observe that this condition is usually
\emph{not} satisfied, not even if the $H_k$ share a joint domain of essential
selfadjointness. Fortunately, the systems we are going to study do not
show such pathological behavior. Yet, a minimalistic way to avoid this problem
would be to restrict to control functions where only one $u_k$ is different from 0
at each time $t$. In this case the propagator $U(t_1,t_2)$ is just a
concatenation of unitaries $\exp(i t H_k)$ which are guaranteed to exist due
to selfadjointness of the $H_k$.
\subsection{Pure-state controllability}
A key-issue
in quantum control theory is \emph{reachability}: Given two pure states
$\psi_0$, $\psi \in \mathcal{H}$, we are looking for a time $T>0$ and control functions
$u_k$ such that $\psi = U(0,T)\psi_0$. In infinite dimensions, however, this
condition is too strong, since there might be states which can be reached only
in infinite time, or not at all. Yet, one may
find a reachable state
``close by'' with arbitrary small control error. Therefore we will call
$\psi$ reachable from $\psi_0$ if for all $\epsilon>0$ there is a
\emph{finite} time $T>0$
and control functions $u_k$ such that
$ \| \psi - U(0,T)\psi_0\| < \epsilon$
holds. Accordingly, we will call the system (\ref{eq:1}) \emph{pure-state
controllable}, if each pure state~$\psi$ can be reached from one~$\psi_0$
(and, by unitarity, also vice versa).
Since pure states are described by one-dimensional projections, two state
vectors describe the same state if they differ only by a global phase. Hence the
definition just given is actually a bit too strong. There are several ways
around this problem, like using the trace norm distance of $\kb{\psi}$ and
$\kb{\psi_0}$ rather then the norm distance of $\psi$ and $\psi_0$. For our
purposes, however, the most appropriate method is to assume that the unit
operator $\mathbbmss{1}$ on $\mathcal{H}$ is always among the control Hamiltonians. This
may appear somewhat arbitrary, but it helps to avoid problems with determinants and traces on
infinite-dimensional Hilbert spaces, which otherwise would arise.
\subsection{Strong controllability}
\label{sec:strong-contr-1}
Next, the analysis shall be lifted to the level of operators, i.e.\ to
unitaries~$U$ from the group $\mathcal{U}(\mathcal{H})$ of unitary operators
on the Hilbert space $\mathcal{H}$ such that a time $T>0$ and control functions $u_k$ exist with
$U= U(0,T)$. As in the last paragraph, this has to be generalized to an
approximative condition again. The best choice---mathematically as well as from a
practical point of view---is approximation in the \emph{strong sense}: We
look for unitaries $U$ such that for each set of (not necessarily
orthonormal or linearly independent) vectors $\psi_1, \dots, \psi_f \in
\mathcal{H}$ and each $\epsilon > 0$, there exists a time $T>0$ and control
functions $u_k$ such that
\begin{equation} \label{eq:26}
\| [U - U(0,T)]\, \psi_k\| < \epsilon\, \text{ for all }\, k\in\{1,\dots,f\}.
\end{equation}
In other words, we are comparing $U$ and $U(0,T)$ only on a finite set of
states, and the worst-case error one can get here is bounded by $\epsilon$. We
will call the control system (\ref{eq:1}) \emph{strongly controllable} if each
unitary $U$ can be approximated that way.
(NB, in strong controllability, one again has the choice of one single
joint global phase factor.)
Clearly, strong controllability implies pure-state controllability. To see this, choose an
arbitrary but fixed $\psi_0 \in \mathcal{H}$. For each $\psi \in
\mathcal{H}$, there is a unitary $U$ with $U\psi_0 = \psi$. Hence strong
controllability implies $\|\psi - U(0,T)\, \psi_{0}\| = \| [U - U(0,T)]\, \psi_0\| < \epsilon$.
\subsection{The dynamical group $\mathcal{G}$}
\label{sec:dynamical-group}
Strong controllability is concept-wise related to the
strong operator topology \cite[VI.1]{ReedSimonI} on the group
$\mathcal{U}(\mathcal{H})$ of unitary operators on $\mathcal{H}$.
To this end, consider the sets
\begin{equation} \label{eq:66}
\mathcal{N}(U;\psi_1,\dots,\psi_f;\epsilon) = \{ V \in
\mathcal{U}(\mathcal{H})\,|\,\|(V-U)\, \psi_k\| < \epsilon\; \text{ for all } \; k\in\{1,\dots,f\}
\}.
\end{equation}
They form a neighborhood base for the strong topology, and we will call them
\emph{(strong) $\epsilon$-neighborhoods}. The condition in Eq.~\eqref{eq:26} can now
be restated as: Any $\epsilon$-neighborhood of $U$ contains a time-evolution
operator $U(0,T)$ for appropriate time $T$ and control functions $u_k$. In turn, this
can be reformulated as: $U$ is an \emph{accumulation point} of the set
$\tilde{\mathcal{G}}$ of all unitaries $U(0,T)$. The set of all accumulation
points of $\tilde{\mathcal{G}}$ (which contains $\tilde{\mathcal{G}}$ itself)
is a strongly closed subgroup\footnote{There is a subtle point here: The
group $\mathcal{U}(\mathcal{H})$ is not
strongly closed as a subset of the bounded operators $\mathcal{B}(\mathcal{H})$. Actually its
strong closure is the set of all isometries;
cf.\ \cite[Prob. 225]{halmos82}. Hence whenever we talk about strongly closed groups
of unitaries, this has to be understood as the closure in the
restriction of the strong topology to $\mathcal{U}(\mathcal{H})$ (which
coincides with the restriction of the weak topology).}
of $\mathcal{U}(\mathcal{H})$, which we will
call the \emph{dynamical group} $\mathcal{G}$ generated by control Hamiltonians
$H_k$ with $k\in\{1,\dots,d\}$. If we choose the
controls as described in Subsection~\ref{sec:time-evolution} (i.e.\ piecewise
constant and only one $u_k$ different from zero at each time), $\mathcal{G}$
is just the smallest strongly closed subgroup of $\mathcal{U}(\mathcal{H})$
that contains all $\exp(itH_k)$ for all $k\in\{1,\dots,d\}$ and all $t \in
\Bbb{R}$. Note that it contains in particular all unitaries that can
be written as a strong limit s-$\lim_{T \rightarrow \infty} U(0,T) $. In
finite dimensions, $\mathcal{G}$ can be calculated via its system algebra,
i.e.\ the Lie algebra $\mathfrak{l}$ generated by the $i H_k$, since each $U
\in \mathcal{G}$ can be written as $U=\exp(H)$ for an $H \in \mathfrak{l}$.
In infinite dimensions, however, several difficulties can occur. First, unbounded
operators $H_k$ are only defined on a dense domain $D(H_k) \subset
\mathcal{H}$. The sum $H_k + H_j$ is therefore only defined on the
intersection $D(H_k) \cap D(H_j)$ and the commutator even only on a subspace
thereof. There is no guarantee that $D(H_k) \cap D(H_j)$ contains more than
just the zero vector. In this case, the Lie algebra cannot even be defined.
The minimal requirement to get around this difficulty is the existence of a
joint dense domain $D$, i.e.\ $D \subset D(H_j)$ and $H_j D \subset D$ for all
$j$. However, even then we do not know whether $\mathcal{G}$ can be generated from
$\mathfrak{l}$ in terms of exponentials. In general, it is impossible to
define some $\exp(H)$ for all $H \in \mathfrak{l}$.
There are several ways to deal with these problems. One is to consider cases
where the $H_k$ generate (i) a finite-dimensional Lie algebra and admit (ii) a
common, invariant, dense domain consisting of analytic vectors \cite{HTC83,LTCC05}. In
this case the exponential function is defined on all of $\mathfrak{l}$, and we
can proceed in analogy to the finite-dimensional case. The problem is that the
group $\mathcal{G}$ will become a finite-dimensional Lie group and its orbits through a
vector $\psi \in \mathcal{H}$ are finite-dimensional as well. Hence, we never
can achieve full controllability. This approach is well studied; cf.\ \cite{HTC83,LTCC05}
and references therein.
Another possibility which includes the possibility to study an
infinite-dimensional Lie algebra $\mathfrak{l}$ is to restrict to bounded generators $H_k$. In this
case, one can define $\mathfrak{l}$ as a norm-closed subalgebra of the Lie algebra
$\mathcal{B}(\mathcal{H})$ of bounded operators, and one ends up with a Banach-space theory which
works almost in the same way as the finite-dimensional analog; cf.~\cite{Lang96}
for details. Although this is a perfectly reasonable approach from the mathematical
point of view, it is not very useful for physical applications, since in most
cases at least some of the $H_k$ are unbounded.
In this paper, we will thus consider a different approach which splits the
generators into two classes. The first $d{-}1$ generators $H_1, \ldots, H_{d-1}$ admit an abelian
symmetry and can be treated---with Lie-algebra methods---along the lines
outlined in the next subsection. Secondly, the last generator $H_d$ breaks this symmetry and achieves full
controllability with a comparably simple argument.
The details will be explained
in Section~\ref{sec:lie-algebra-block} and \ref{sec:strong-contr}.
\subsection{Abelian symmetries}
\label{sec:abelian-symmetries}
One way to avoid the problem described in the last subsection, arises if the
control system admits symmetries. In this section, we will only sketch the
structure, while the details are postponed to Sect. \ref{sec:lie-algebra-block}.
Let us consider the case of a
$\mathrm{U}(1)$-symmetry\footnote{The generalization to multiple charges,
i.e.\ a $\mathrm{U}(1)^N$, is straightforward.}, i.e.\ a
(strongly continuous) unitary representation $z \mapsto \pi(z) \in \mathcal{U}(\mathcal{H})$
of the abelian group $\mathrm{U}(1)$ on
$\mathcal{H}$
where $\mathcal{U}(\mathcal{H})$ denotes the group of unitaries on
$\mathcal{H}$. It can be written in terms of a selfadjoint operator $X$ with
pure point spectrum consisting of (a subset of) $\ensuremath{\mathbb Z}$ as
$\mathrm{U}(1) \ni z = e^{i\alpha} \mapsto \pi(z) = \exp(i \alpha X) \in
\mathcal{U}(\mathcal{H})$.
If we denote the eigenprojection of $X$ belonging to the eigenvalue $\mu\in \ensuremath{\mathbb Z}$ as
$X^{(\mu)}$ (allowing the case $X^{(\mu)}=0$ if $\mu$ is not an
eigenvalue of $X$) we get a block-diagonal decomposition of $\mathcal{H}$
in the symmetry-adapted basis as
\begin{equation} \label{eq:37}
\mathcal{H} = \bigoplus_{\mu=-\infty}^\infty \mathcal{H}^{(\mu)} \, \text{ with }\,
\mathcal{H}^{(\mu)}=X^{(\mu)} \mathcal{H},
\end{equation}
and we can rewrite $\pi(z)$ again as
$\mathrm{U}(1) \ni z = e^{i\alpha} \mapsto \pi(z) =
\sum_{\mu=-\infty}^\infty e^{i \alpha \mu} X^{(\mu)} \in
\mathcal{U}(\mathcal{H})$.
Here we will make two assumptions representing substantial
restrictions of generality:
\begin{enumerate}
\item \label{item:4}
All eigenvalues of $X$ are of finite multiplicity, i.e.\ the
$\mathcal{H}^{(\mu)}$ are finite-dimensional. This is crucial for basically
everything we will discuss in this paper.
\item
All eigenvalues of $X$ are non-negative. This assumption can be relaxed at
certain points (e.g.\ all material in Sect. \ref{sec:commuting-operators} can
be easily generalized). However, it helps to simplify the discussion at a
technical level and all examples we are going to consider in the next
section are of this form.
\end{enumerate}
The first important consequence of
(\ref{item:4}) concerns the space
of \emph{finite particle vectors}
\begin{equation} \label{eq:28}
D_X = \{ \psi \in \mathcal{H} \, | \, X^{(\mu)}\psi = 0\ \text{for all but
finitely many $\mu$}\},
\end{equation}
since it becomes (due to finite-dimensionality of $\mathcal{H}^{(\mu)}$)
a ``good'' domain for basically all unbounded operators appearing in this
paper. Moreover one gets the following theorem:
\begin{thm} \label{thm:6}
Consider a strongly continuous representation $\pi$ of $\mathrm{U}(1)$ on
$\mathcal{H}$ and the corresponding charge-type operator $X$. Then the following
statements hold:
\begin{enumerate}
\item
A selfadjoint operator $H$ commuting with $X$ admits $D_X$ as an
invariant domain, i.e.\ $D_X \subset D(H)$ and $H D_X = D_X$. Hence the
space
$\mathfrak{u}(X) = \{ iH \, | \, H=H^*\, \text{ commuting with }\, X\}$
is a Lie algebra with the commutator as its Lie bracket.
\item
The exponential map is well defined on $\mathfrak{u}(X)$ and maps it onto
the strongly closed subgroup
$\mathcal{U}(X) = \{ U \in \mathcal{U}(\mathcal{H})\, | \, [U,\pi(z)]=0 \, \text{ for all }\, z
\in \mathrm{U}(1)\}$
of $\mathcal{U}(\mathcal{H})$.
\item
The subalgebra $\mathfrak{l} \subset \mathfrak{u}(X)$ generated by a family of
Hamiltonians $iH_1, \dots, iH_d \in \mathfrak{u}(X)$ is mapped by the
exponential map into the dynamical group $\mathcal{G}$ of the corresponding control
problem. The strong closure of $\exp(\mathfrak{l})$ coincides with $\mathcal{G}$.
\end{enumerate}
\end{thm}
The basic idea behind this theorem, is that one can cut off the decomposition
(\ref{eq:37}) at a sufficiently high $\mu$ without sacrificing strong
approximations as described in Subsection~\ref{sec:strong-contr-1}. One only has
to take into account that the cut-off on $\mu$ has to become higher when the
approximation error decreases. This strategy allows for tracing a lot of
calculations back to finite-dimensional Lie algebras. We will postpone a
detailed discussion of this topic---including the proof of Theorem~\ref{thm:6}---to
Section~\ref{sec:lie-algebra-block}.
The only additional material one needs at
this point, since it is of relevance for the next section, is a subgroup of $\mathcal{U}(X)$ and
its corresponding Lie algebra which relates unitaries
with determinant one and their traceless generators. Since the $iH \in
\mathfrak{u}(X)$ are unbounded and not necessarily positive, it is difficult
to give a reasonable definition of tracelessness, and the determinant of $U \in
\mathcal{U}(X)$ runs into similar problems. However, the elements of
$U \in \mathcal{U}(X)$ and $iH \in \mathfrak{u}(X)$ are block diagonal with
respect to the decomposition of $\mathcal{H}$ given in (\ref{eq:37}). In other
words $U=\sum_{\mu} U^{(\mu)}$ and $H=\sum_{\mu} H^{(\mu)}$ are infinite sums of operators\footnote{Two small remarks
are in order here: (i).~Infinite sums require a proper definition of
convergence in an appropriate topology. In Section
\ref{sec:lie-algebra-block}, this will be made precise. (ii).~Operator products of the form $X^{(\mu)} H
X^{(\mu)}$ are potentially problematic if $H$ is unbounded and therefore
only defined on a domain. In our case, however, $X^{(\mu)}$ projects onto
$\mathcal{H}^{(\mu)}$, which is a subspace of the domain $D_X$ on which $H$
is defined.}, where
$U^{(\mu)} = X^{(\mu)} U X^{(\mu)} \in \mathcal{U}(\mathcal{H}^{(\mu)})$,
$H^{(\mu)} = X^{(\mu)} H X^{(\mu)} \in \mathcal{B}(\mathcal{H}^{(\mu)})$, and
$X^{(\mu)}$ denotes the projection onto the $X$-eigenspace
$\mathcal{H}^{(\mu)}$. Since all the $U^{(\mu)}$ and $H^{(\mu)}$ are operators
on finite-dimensional vector spaces, one can define
\begin{align}
\mathcal{SU}(X) &:= \{ U \in \mathcal{U}(X)\, | \, \det U^{(\mu)} = 1\; \text{ for all } \;
\mu \in \Bbb{Z}\}, \label{eq:42}
\\ \mathfrak{su}(X) &:= \{ iH \in \mathfrak{u}(X)\, | \,
\tr(H^{(\mu)}) = 0\; \text{ for all } \; \mu \in \Bbb{Z}\}.
\end{align}
Obviously, $\mathcal{SU}(X)$ is a (strongly closed) subgroup of
$\mathcal{U}(X)$ and $\mathfrak{su}(X)$ is a Lie subalgebra of
$\mathfrak{u}(X)$. The image of $\mathfrak{su}(X)$ under the exponential map
therefore coincides with $\mathcal{SU}(X)$. Note that $\mathcal{SU}(X)$ is
effectively an infinite direct product of groups $\mathrm{SU}(d^{(\mu)})$, if
$d^{(\mu)} = \dim \mathcal{H}^{(\mu)}$ and not the ``special'' subgroup of
$\mathcal{U}(X)$.
\subsection{Breaking the symmetry}
To get a fully controllable system, one has to leave the group $\mathcal{U}(X)$,
which can be thought of as being represented as block diagonal, see Fig.~\ref{fig:1}~a.
To this end, we have to add control Hamiltonians that break the
symmetry. There are several ways of doing so, and a successful strategy depends
on the system in question (beyond
the treatment of the symmetric part of
the dynamics captured in Theorem~\ref{thm:6}). Here, we will present a special
result which covers the examples discussed in the next section. The first step
is another direct sum decomposition of
$\mathcal{H} = \mathcal{H}_- \oplus \mathcal{H}_0 \oplus \mathcal{H}_+$, where
$\mathcal{H}_\alpha = E_\alpha \mathcal{H}$, with $\alpha\in\{+,0,-\}$
are projections onto the subspaces
$\mathcal{H}_\alpha$ and should satisfy $[E_\alpha,X^{(\mu)}] = 0$.
Let in the following $\ensuremath{\mathbb N}:=\{1,2,3,\ldots\}$ denote the set of positive
integers and define $\ensuremath{\mathbb N}_0:=\ensuremath{\mathbb N}\cup\{0\}$.
Hence for $\mu \in \ensuremath{\mathbb N}_0$ we can introduce the
projections $X_\pm^{(\mu)} = X^{(\mu)} E_\pm$ which we require to be non-zero. For the exceptional case $\mu=0$
the relation $X_-^{(0)} = X^{(0)} E_- = X^{(0)}$ should hold.
Futhermore we write $X^{(\mu)}_0 = X^{(\mu)} E_0$ for the overlap of
$X^{(\mu)}$ and $E_0$ which can (in contrast to $X^{(\mu)}_\pm$) be equal to zero for
all $\mu$. The $X^{(\mu)}_\alpha$ are projections onto the subspaces
$\mathcal{H}^{(\mu)}_\alpha := X^{(\mu)}_\alpha \mathcal{H}$ satisfying
$X^{(\mu)} = X^{(\mu)}_- \oplus X^{(\mu)}_0 \oplus X^{(\mu)}_+$.
\begin{defi} \label{def:1}
A selfadjoint operator $H$ with domain $D(H)$ is called complementary to
$X$, if there exists a decompositon $\mathcal{H} = \mathcal{H}_- \oplus \mathcal{H}_0
\oplus \mathcal{H}_+$
as defined above such that:
\begin{enumerate}
\item
$\mathcal{H}_0 \subset D(X)$ and $H\psi = 0$ for all $\psi \in \mathcal{H}_0$.
\item \label{item:1}
$D_X \subset D(H)$ and for all $\mu > 1$ we have $H X_{+}^{(\mu+1)}\psi =
X_-^{(\mu)} H \psi$. The corresponding operator $X_-^{(\mu)} H
X_+^{(\mu+1)} \in \mathcal{B}(\mathcal{H})$ is a partial isometry with
$X_+^{(\mu+1)}$ as its source and $X_-^{(\mu)}$ as its target projection.
\item \label{item:2}
Given the projection $F_{[0]} = X^{(0)}_{\phantom{+}} \oplus X^{(1)}_-$ and the
corresponding subspace $\mathcal{H}_{[0]} = F_{[0]} \mathcal{H}$. The
group generated by $\exp(i t H)$ with $t \in \Bbb{R}$ and those $U \in
\mathcal{SU}(X)$ which commute with $F_{[0]}$ acts transitively on the
space of one-dimensional projections in $\mathcal{H}_{[0]}$.
\end{enumerate}
\end{defi}
\begin{figure}[tb]
(a)\hspace{70mm} (b)\\[-5mm]
\centerline{
\begin{tabular}{c@{\hspace{20mm}}c}
\hspace{12mm}\includegraphics[width=.37\textwidth]{ControlBlock-3} &
\raisebox{-2.3mm}{\includegraphics[width=.49\textwidth]{energy} }
\end{tabular}}
\caption{\small \label{fig:1} (a) Block structure of operators in $\mathfrak{u}(X)$ (red) and of
operators complementary to $X$ (blue) in the case where the projection
$E_0$ vanishes. (b) Energy diagram for the Jaynes-Cummings model (here two atoms in a cavity under individual controls
$\omega_I^{(1)}$ and $\omega_I^{(2)}$)
with combined atom-cavity transitions matching the block structure of (a) given in red (see Eqs.~(\ref{eq:15},\ref{eq:44})) since commuting with $X_1$ or $X_M$ of Eqs.~(\ref{eq:36}, \ref{eq:47}),
and complementary transitions solely within the atoms given in blue (see Eqs.~(\ref{eq:25},\ref{eq:45})). }
\end{figure}
At first sight, the definition may look somewhat clumsy, but it allows for
proving a controllability result which covers all examples we are going to
present in the next
section. We will state them here without a proof and postpone the latter to
Sect.~\ref{sec:strong-contr}.
\begin{thm} \label{thm:10}
Consider a strongly continuous representation $\pi: \mathrm{U}(1) \to
\mathcal{U}(\mathcal{H})$ with charge operator $X$ and a family of
selfadjoint operators $H_1, \dots, H_d$ on
$\mathcal{H}$. Assume that the following conditions hold:
\begin{enumerate}
\item
$H_1, \dots, H_{d-1}$ commute with $X$.
\item
The dynamical group generated by $H_1, \dots, H_{d-1}$ contains
$\mathcal{SU}(X)$.
\item
The operator $H_d$ is complementary to $X$.
\end{enumerate}
Then the control system \eqref{eq:1} with Hamiltonians
$H_0=\Bbb{1}, H_1, \dots, H_d$ is pure-state controllable.
\end{thm}
\begin{thm} \label{thm:9}
The
control system (\ref{eq:1}) is even strongly controllable
if in addition to the assumptions of Thm.~\ref{thm:10} the condition $\dim
\mathcal{H}^{(\mu)} > 2$ holds for at least one $\mu \in \ensuremath{\mathbb N}_0$.
\end{thm}
\section{Atoms in a cavity}
\label{sec:atoms-cavity}
An important class of examples that can be treated along the lines described
in the last section are atoms interacting with the light field in a cavity. We
will discuss the case of $M$ two-level atoms interacting with one mode in
detail and consider three particular scenarios: one atom in Sect.~\ref{sec:one-atom},
individually controlled atoms in Sect.~\ref{sec:many-atoms-indiv},
and atoms under collective control in Sect.~\ref{sec:many-atoms-coll}.
\subsection{One atom}
\label{sec:one-atom}
Let us start with the special case $M=1$, i.e.\ one atom and one mode as
discussed in a number of previous publications
mostly on pure-state controllability \cite{LE96,BBR10,YL07}. Our results go beyond this, in
particular because we are considering strong controllability not just pure-state controllability.
The Hilbert space of the system is given by
\begin{equation}
\mathcal{H} = \ensuremath{\mathbb C}^2 \otimes \mathrm{L}^2(\ensuremath{\mathbb R})
\end{equation}
and the dynamics is described by the well known Jaynes-Cummings Hamiltonian \cite{JC63}:
\begin{gather}
H_{\mathrm{JC}} := \omega_A H_{\mathrm{JC},1} + \omega_I H_{\mathrm{JC},2} +
\omega_C H_{\mathrm{JC},3} \; \text{ with } \label{eq:9}\\
H_{\mathrm{JC},1} := (\sigma_3 \otimes \mathbbmss{1})/2,\, H_{\mathrm{JC,2}}
:= (\sigma_+ \otimes a + \sigma_- \otimes a^*)/{2}, \,
H_{\mathrm{JC},3} := \mathbbmss{1} \otimes N \label{eq:15},
\end{gather}
where $\sigma_\alpha$ with $\alpha\in\{1,2,3\}$ are the Pauli matrices
($\sigma_\pm=\sigma_1 \pm i \sigma_2$), $a,a^*$ denote
the annihilation and creation operator, and $N = a^*a$ is the number operator. The
joint domain of all these Hamiltonians is the space
\begin{equation} \label{eq:40}
D = \operatorname{span}\{\ket{\nu}\otimes\ket{n}\, | \, \nu\in\{0,1\} \, \text{ and }\, n \in \ensuremath{\mathbb N}_0 \}
\end{equation}
with $\nu \in \ensuremath{\mathbb C}^2$ as canonical basis and $\ket{n} \in
\mathrm{L}^2(\ensuremath{\mathbb R})$ as number basis (Hermite functions).
We will assume that the frequencies $\omega_A$, $\omega_I$ and
$\omega_C$ can be controlled independently (or at least two of
them) such that we get a control system with control Hamiltonians
$H_{\mathrm{JC},j}$ where $j\in\{1,2,3\}$
corresponding to the lower half (1 atom) of the energy diagram in Fig.~\ref{fig:1}~b,
where we adopt the widely used convention of
forcing the atom (spin) state $\ket{\negthickspace\uparrow}$ to be of \/`higher\/' energy than $\ket{\negthickspace\downarrow}$
to compensate for negative Larmor frequencies, see, e.g., the note in \cite[p.~144]{Haroche06}.
The task is to determine the dynamical group $\mathcal{G}$. To this end, we use the
strategy described in Subsection~\ref{sec:abelian-symmetries}, which follows in
this particular case closely the exact solution of the Jaynes-Cummings model
\cite{JC63}. The charge-type operator $X_1$ (determining the block structure)
then takes the form
\begin{equation} \label{eq:36}
X_1 = \sigma_3 \otimes \mathbbmss{1} + \mathbbmss{1} \otimes N,
\end{equation}
again with $D$ from (\ref{eq:40}) as its domain, which in this case turns out
to be identical to the space $D_{X_1}$ of finite-particle vectors. The operator $X_1$ is
diagonalized by the basis $\ket{\nu}\otimes\ket{n}$. It is convenient to
relabel these vectors in order to get
\begin{equation} \label{eq:27}
\ket{\mu,\nu} = \ket{\nu} \otimes \ket{\mu - \nu} \in \mathcal{H}\,\text{ with }\, \mu =
n+\nu \geq 0.
\end{equation}
In this basis, we have $X_1 \ket{\mu,\nu} = \mu \ket{\mu,\nu}$ and the
subspaces $\mathcal{H}^{(\mu)}$ from (\ref{eq:37}) become
\begin{equation} \label{eq:38}
\mathcal{H}^{(\mu)} = \operatorname{span}\{ \ket{\mu,0},\ \ket{\mu,1} \}
\end{equation}
for $\mu>0$ and $\mathcal{H}^{(0)} = \ensuremath{\mathbb C} \ket{0,0}$ for $\mu=0$.
The
space $D_{X_1} \subset \mathcal{H}$ of finite-particle vectors turns out to be
identical with the domain $D$ from (\ref{eq:40}).
It is easy to see that the operators $H_{\mathrm{JC},j}$ from
Eq.~\eqref{eq:15} commute with $X_1$, and therefore we get $i H_{\mathrm{JC},j} \in
\mathfrak{u}(X_1)$. A more detailed analysis, as will be given in
Section~\ref{sec:lie-algebra-block}, shows that $i H_{\mathrm{JC},1}$ and $i H_{\mathrm{JC},2}$ generate
$\mathfrak{su}(X_1)$, and therefore we get according to Theorem~\ref{thm:6}:
\begin{thm} \label{thm:3}
The dynamical group $\mathcal{G}$ generated by $H_{\mathrm{JC},j}$ with $j\in\{1,2\}$ from
Eq.~(\ref{eq:15}) coincides with the group $\mathcal{SU}(X_1)$ defined in (\ref{eq:42}).
\end{thm}
To get a fully controllable system, apply Theorem~\ref{thm:9} to see that
one has to add a Hamiltonian which breaks the symmetry. A possible candidate is
\begin{equation} \label{eq:25}
H_{\mathrm{JC},4} = \sigma_1 \otimes \mathbbmss{1} \in \mathcal{B}(\mathcal{H}).
\end{equation}
If we define the spaces $\mathcal{H}_\alpha$ as
$\mathcal{H}_- = \operatorname{span} \{\ket{\mu,0}\, | \, \mu \in \ensuremath{\mathbb N}_0\}$, $\mathcal{H}_0 =
\{0\}$, and $\mathcal{H}_+ = \operatorname{span}\{\ket{\mu,1}\, | \, \mu \in \ensuremath{\mathbb N} \}$
the operator $H_{\mathrm{JC},4}$ becomes complementary to $X_1$, which can be
easily seen since $\mathcal{H}^{(\mu)}_+ = \Bbb{C} \ket{\mu,1}$,
$\mathcal{H}^{(\mu)}_- = \Bbb{C} \ket{\mu,0}$, and $\mathcal{H}^{(\mu)}_0 =
\{0\}$. Hence, according to Thm. \ref{thm:10}, the control system with Hamiltonians
of Eqs.~(\ref{eq:9},\ref{eq:15}).
\begin{equation} \label{eq:65}
H_0 = \Bbb{1},\; H_1 = H_{\mathrm{JC},1},\; H_2 =
H_{\mathrm{JC},2},\; H_3 = H_{\mathrm{JC},4}
\end{equation}
is pure-state controllable\footnote{We have
omitted the Hamiltonian $H_{\mathrm{JC},3}$ since it is not needed for the
result. However, it can be added as a drift term without changing the result.}, and we are
recovering a previous result from \cite{LE96,BBR10,YL07}. However, with our
methods we can go beyond this and prove even strong
controllability. Thm. \ref{thm:9} cannot be applied since $\dim
\mathcal{H}^{(\mu)} \leq 2$ for all $\mu$, but the analysis of
Sect. \ref{sec:strong-contr} will lead to an independent argument.
\begin{thm} \label{thm:1}
The control problem (\ref{eq:1}) with Hamiltonians $H_j$ and $j\in\{0,\dots,3\}$ from
Eq.~\eqref{eq:65} is strongly controllable.
\end{thm}
Hence any unitary $U$ on $\mathcal{H}$ can be approximated by varying the
control amplitudes $u_1=\omega_A$ and $u_{2}=\omega_{I}$ in the
Hamiltonian $H_{\mathrm{JC}}$ of (\ref{eq:9}) plus flipping ground and excited
state of the atom in terms of $H_{\mathrm{JC},4}$ (with strength $u_3$)---both in an
appropriate time-dependent manner. The approximation has to be understood in
the strong sense as described in Eq.~(\ref{eq:26}).
Finally, note that Theorem \ref{thm:1} implies that one can simulate (again in
the sense of strong approximations) any unitary $V \in
\mathcal{B}(\mathrm{L}^2(\ensuremath{\mathbb R}))$ operating on the cavity mode alone. One only
has to find controls $u_j$ such that $U(0,T)\, \phi \otimes \psi_k$ $\approx$
$\phi \otimes V \psi_k$ for a finite set of states $\psi_k$ of the cavity (and
an arbitrary auxiliary state $\phi$ of the atom).
\subsection{Many atoms with individual control}
\label{sec:many-atoms-indiv}
Next, consider the case of many atoms interacting with the same mode,
and under the assumption that each atom (including the coupling with the cavity)
can be controlled individually. Such a scenario is relevant for experiments
with ion traps, if the number of ions is not too big as have been studied
since \cite{CC00,GRL+03,LBM+03}.
The Hilbert space of the system is
\begin{equation}
\mathcal{H} = (\ensuremath{\mathbb C}^2)^{\otimes M} \otimes \mathrm{L}^2(\ensuremath{\mathbb R}),
\end{equation}
where $M$ denotes the number of atoms. We define the basis
$\ket{b}\otimes\ket{n} \in \mathcal{H}$
where
$n \in \ensuremath{\mathbb N}_0$,
$ \ket{b} = \ket{b_1} \otimes \dots \otimes \ket{b_M}$,
$ b = (b_1,\dots,b_M) \in \ensuremath{\mathbb Z}_2 \times \dots \times \ensuremath{\mathbb Z}_2 = \ensuremath{\mathbb Z}_2^M$,
and the canonical basis $\ket{b_j} \in \ensuremath{\mathbb C}^2$ with $b_j\in\{0,1\}$.
The control Hamiltonians become
\begin{equation}
\label{eq:44}
H_{\mathrm{IC},j} = \sigma_{3,j} \otimes \mathbbmss{1}\,\text{ and }\,
H_{\mathrm{IC},M{+}j} = \sigma_{+,j} \otimes a + \sigma_{-,j} \otimes a^*
\end{equation}
where
$j\in\{1,\dots,M\}$ and
$ \sigma_{\alpha,j} = \mathbbmss{1}^{\otimes (j-1)} \otimes \sigma_\alpha \otimes
\mathbbmss{1}^{\otimes (N-j)}$. As before,
$a$ and $a^*$ denote annihilation and creation operator.
The joint domain of all these operators is
\begin{equation} \label{eq:41}
D = \operatorname{span} \{ \ket{b}\otimes\ket{n}\, | \, b \in \ensuremath{\mathbb Z}_2^M\, \text{ and }\, n \in
\ensuremath{\mathbb N}_0 \},
\end{equation}
with the basis $\ket{b}\otimes\ket{n}$ as defined above.
As depicted by the red parts in Fig~\ref{fig:1},
all the $H_{\mathrm{IC},k}$ are invariant under the symmetry defined by the charge operator
\begin{equation}\label{eq:47}
X_M = S_3 \otimes \mathbbmss{1} + \mathbbmss{1} \otimes N\,\text{ with }\, S_3 = \sum_{j=1}^N \sigma_{3,j}
\end{equation}
where $N=a^*a$ denotes again the number operator and $D$ from (\ref{eq:41}) is the
domain of $X_M$. The eigenvalues of $X_M$ are $\mu \in
\ensuremath{\mathbb N}_0$ and the eigenbasis is given by
\begin{equation} \label{eq:48}
\ket{\mu,b} = \ket{b} \otimes \ket{\mu - |b|} \,\text{ for }\, |b|=\sum_{j=1}^M b_j \leq
\mu.
\end{equation}
In this basis, $X_M$ becomes
$X_M \ket{\mu,b} = \mu \ket{\mu,b}$
and the eigenspaces $\mathcal{H}^{(\mu)}$ are
$\mathcal{H}^{(\mu)} = \operatorname{span} \{\ket{\mu,b} \,|\, b \in \ensuremath{\mathbb Z}_2^M$ with $ |b| \leq
\mu\} $.
From now on, one may readily proceed as for one atom to arrive at the following analogy to
Theorem~\ref{thm:3}:
\begin{thm} \label{thm:8}
The dynamical group $\mathcal{G}$ generated by $H_{\mathrm{IC},k}$ with $k\in\{1,\dots,2M\}$ from
Eq.~\eqref{eq:44} coincides with group $\mathcal{SU}(X_M)$ of unitaries commuting with
$X_M$.
\end{thm}
To get strong controllability, one has to add again one Hamiltonian. As before
a $\sigma_1$-flip of one atom is sufficient (see the blue parts in Fig~\ref{fig:1}), and
\begin{equation} \label{eq:45}
H_{\mathrm{IC},2M+1} = \sigma_{1,1} \otimes \mathbbmss{1}.
\end{equation}
is complementary to $X_M$ with $\mathcal{H}_\alpha$
given by $\mathcal{H}_0 =\{0\}$,
$\mathcal{H}_- = \operatorname{span} \{ \ket{\mu;0,b_2,\dots,b_M}\,|\, \mu \in \ensuremath{\mathbb N}_0,\,
|(b_2,\dots,b_M)| \leq \mu \}$,
$\mathcal{H}_+ = \operatorname{span} \{ \ket{\mu;1,b_2,\dots,b_M}\,|\, \mu \in \ensuremath{\mathbb N},\,
|(b_2,\dots,b_M)| < \mu \}$.
Obviously, all the conditions of Thm. \ref{thm:9} are satisfied such that one
gets
\begin{thm} \label{thm:7}
The control problem (\ref{eq:1}) with $H_{\mathrm{IC},k}$ and $k\in\{1,\dots,2M{+}1\}$
from (\ref{eq:44}) and (\ref{eq:45}) is strongly controllable.
\end{thm}
As a special case of this theorem, one can approximate
any unitary $U$ acting on the atoms alone, i.e.\ $U \in
\mathcal{U}((\ensuremath{\mathbb C}^2)^{\otimes M})$, by applying Theorem~\ref{thm:7} to $U
\otimes \mathbbmss{1}$. That is, one can simulate~$U$ only by operations on one atom and
the interactions with the harmonic oscillator. This is used in
ion-trap experiments and is known as ``phonon bus''.
\subsection{Many atoms under collective control}
\label{sec:many-atoms-coll}
Now one may modify the setup from the last section by considering again $M$ atoms
interacting with one mode, but assuming that one can control the atoms only
collectively rather than individually. In other words instead of the
Hamiltonians $H_{\mathrm{IC},j}$ and $H_{\mathrm{IC},M{+}j}$ with $j\in\{1,\dots,M\}$
from Eq.~\eqref{eq:44} one only has their sums
\begin{equation} \label{eq:31}
H_{\mathrm{TC},1} = S_3 \otimes \mathbbmss{1}\,\text{ and }\, H_\mathrm{{TC},2} = S_+ \otimes
a + S_- \otimes a^*,
\end{equation}
where
$S_\alpha = \sum_{j=1}^M \sigma_{\alpha,j}$ and $\alpha\in\{1,2,3,\pm\}$,
combinded with the free evolution
\begin{equation} \label{eq:33}
H_{\mathrm{TC},3} = \mathbbmss{1} \otimes N
\end{equation}
of the cavity.
As before, all operators are defined on the domain $D$ from
(\ref{eq:41}). Note that one readily recovers the original setup from
Subsection~\ref{sec:one-atom} with Pauli operators $\sigma_\alpha$ replaced by
pseudo-spin operators $S_\alpha$. The multi-atom analogue of the Jaynes-Cummings
Hamiltonian, which can be formed from the $H_{\mathrm{TC},j}$ just defined, is called
Tavis-Cummings Hamiltonian \cite{TC68,TC69}.
All the Hamiltonians in Eqs.~(\ref{eq:31}) and (\ref{eq:33}) are invariant
under the $\mathrm{U}(1)$-action generated by $X_M$ of
Eq.~\eqref{eq:47}. However, this is not the only symmetry, since all these
$H_{\mathrm{TC},j}$ are also invariant under the permutation of the
atoms. Therefore, one may no longer exhaust the group $\mathcal{SU}(X_M)$ as in
Theorem~\ref{thm:8} (since the following operators
cannot be reached: those
commuting only with $X_M$ but not also with permutations
of the atoms). A minimal modification is to restrict the states of the
atoms to spaces on which permutation-invariant unitaries operate
transitively\footnote{An alternative strategy would be to treat
permutation symmetry in the same way as
$\mathrm{U}(1)$-symmetry. However, already the restriction to permutation-invariant
states will turn out to be difficult enough.}. The most natural
choice is the symmetric tensor product
$
(\ensuremath{\mathbb C}^2)^{\otimes M}_{\operatorname{sym}} \subset (\ensuremath{\mathbb C}^2)^{\otimes M},
$
i.e.\ the Bose subspace of $(\ensuremath{\mathbb C}^2)^{\otimes M}$. The preferred basis of
$(\ensuremath{\mathbb C}^2)^{\otimes M}_{\operatorname{sym}}$ is
$\ket{\nu} = \operatorname{Sym}_M \left( \ket{1}^{\otimes \nu} \otimes \ket{0}^{\otimes (M
- \nu)} \right)$
with $\nu \in \{0, \dots, M\}$ and the projection
$
\operatorname{Sym}_M $ from $(\ensuremath{\mathbb C}^2)^{\otimes M}$
onto the symmetric subspace $(\ensuremath{\mathbb C}^2)^{\otimes M}_{\operatorname{sym}}$. In other words $\ket{\nu}$ is the unique, pure,
permutation-invariant state with $\nu$ atoms in the excited state $\ket{1}$
and $M {-} \nu$ ones in the ground state $\ket{0}$. Therefore,
$(\ensuremath{\mathbb C}^2)^{\otimes M}_{\operatorname{sym}}$ can be identified with the Hilbert space
$\ensuremath{\mathbb C}^{M+1}$ of a (pseudo-)spin-$M/2$ system. Its
basis $\ket{\nu}$, with $\nu\in\{0,\dots,M\}$ becomes the canonical basis. Combining this
with $\mathrm{L}^2(\ensuremath{\mathbb R})$ for the cavity one gets
$\mathcal{H}_{\operatorname{sym}} = \ensuremath{\mathbb C}^{M+1} \otimes \mathrm{L}^2(\ensuremath{\mathbb R})$
as the new Hilbert space of the system.
All the operators defined above ($H_{\mathrm{TC},1}, H_{\mathrm{TC},2},
H_{\mathrm{TC},3}$ and $X_M$) can be restricted to $\mathcal{H}_{\operatorname{sym}}$ (and in
slight abuse of notation we will re-use the symbols after restriction) and
their domain becomes
\begin{equation} \label{eq:51}
D_{\operatorname{sym}} = \operatorname{span} \{ \ket{\nu} \otimes \ket{n}\, | \, \nu\in\{0,\dots,M\}\,\text{ and }\, n \in
\ensuremath{\mathbb N}_0 \},
\end{equation}
which is just the projection of $D$ from (\ref{eq:41}), i.e.\ $D_{\operatorname{sym}} =
\operatorname{Sym}_M D$. The eigenbasis of $X_M$ now takes the form
$\ket{\mu,\nu} = \ket{\nu}\otimes\ket{\mu-\nu}$ where $\mu \in \ensuremath{\mathbb N}_0$ and $\nu
< d_\mu = \min(\mu,M{+}1)$.
For the $X_M$-eigenspaces, we get again $X_M \ket{\mu,\nu} = \mu \ket{\mu,\nu}$ and
\begin{equation} \label{eq:52}
\mathcal{H}_{\operatorname{sym}}^{(\mu)} = \operatorname{span} \{ \ket{\mu,\nu} \, | \, \nu\in\{0,\dots,d_\mu \}\}.
\end{equation}
Now one can proceed as the in the previous cases: The operators
$H_{\mathrm{TC},1}, H_{\mathrm{TC},2}, H_{\mathrm{TC},3}$ are (as operators on
$\mathcal{H}_{\operatorname{sym}}$) invariant under the action generated by $X_M$ and
therefore elements of $\mathfrak{u}(X_M)$. However, one still
cannot exhaust all of $\mathcal{U}(X_M)$ (or $\mathcal{SU}(X_M)$). One only gets:
\begin{thm} \label{thm:4}
The dynamical group $\mathcal{G}$ generated by the operators $H_{\mathrm{TC},1},
H_{\mathrm{TC},2}, H_{\mathrm{TC},3}$ from Eqs.~\eqref{eq:31} and
(\ref{eq:33}) is a strongly closed subgroup of $\mathcal{U}(X_M)$. For each unitary $V
\in \mathcal{U}(X_M)$ and each $\mu \in \ensuremath{\mathbb N}_0$ we can find an element $U \in \mathcal{G}$ such
that
$U \psi^{(\mu)} = V \psi^{(\mu)}$
holds for all $\psi^{(\mu)} \in \mathcal{H}_{\operatorname{sym}}^{(\mu)}$.
\end{thm}
In other words: As long as the charge $\mu$ is fixed, one can still approximate
any $V \in \mathcal{U}(X_M)$, but if one considers superpositions of different charges
this is no longer the case, i.e.\ there are $\psi \in D_{X_M}$ and $V \in \mathcal{U}(X_M)$
such $U \psi \neq V \psi$ for all $U \in \mathcal{G}$. We have checked the latter
explicitly with the computer algebra system Magma \cite{magma} for the case
$M=2$.
To circumvent this problem, one has to add control Hamiltonians.
Unfortunately, it seems that one has to add quite a lot. The best result we
have got so far is to replace the operators from Eqs.~(\ref{eq:31}) and
(\ref{eq:33}) by
\begin{gather}
H_{\mathrm{CC},k} = \bigl(\kb{k} - \kb{k{-}1}\bigr) \otimes \Bbb{1} \, \text{ with }\,
k\in\{1,\dots,M\}, \nonumber \\
H_{\mathrm{CC},M+1} = H_{\mathrm{TC},2} = S_+ \otimes a + S_- \otimes
a^*\,\text{ and }\,
H_{\mathrm{CC},M+2} = \bigl(\KB{0}{1} + \KB{1}{0}\bigr) \otimes \Bbb{1}. \label{eq:30}
\end{gather}
The operators $H_{\mathrm{CC},k}$ with $k\in\{1,\dots,M+1\}$ commute with $X_M$ and
generate (as we will see in Sect.~\ref{sec:many-atoms-coll-1}) the Lie algebra
$\mathfrak{su}(X_M)$. In addition we have $H_{\mathrm{CC},M+2}$ which is
complementary to $X_M$ with Hilbert spaces
$\mathcal{H}_+ = \operatorname{span} \{ \ket{\mu;0}\, | \, \mu \in \ensuremath{\mathbb N}_0\}$,
$\mathcal{H}_- = \operatorname{span} \{ \ket{\mu;1}\, | \, \mu \in \ensuremath{\mathbb N} \}$, and
$\mathcal{H}_0 = \operatorname{span} \{ \ket{\mu,\nu},\ | \, \mu \in \ensuremath{\mathbb N},\, \mu > 2,\,
\nu\in\{3,\dots,\min(M,\mu)\} \}$.
Note that we get an example for Def. \ref{def:1} with a non-trivial
$\mathcal{H}_0$. Now one can apply Thms. \ref{thm:6} and \ref{thm:9} to get
the analogues of Theorems \ref{thm:3} and \ref{thm:1}:
\begin{thm} \label{thm:5}
The dynamical group $\mathcal{G}$ generated by $H_{\mathrm{CC},k}$ with $k\in\{1,\dots,M{+}1\}$ from
Eq.~(\ref{eq:30}) coincides with the group $\mathcal{SU}(X_M)$ of unitaries commuting with
$X_M$.
\end{thm}
\begin{thm} \label{thm:2}
The control problem (\ref{eq:1}) with $H_0=\Bbb{1}$ and
$H_{\mathrm{CC},k}$ for $k\in\{1,\dots, M{+}2\}$ from (\ref{eq:30})
is strongly controllable.
\end{thm}
To be able to control all diagonal traceless operators $H_{\mathrm{CC},k}$,
with $k\in\{1,\dots,M\}$ is a very strong assumption. Unfortunately, a detailed analysis
including computer algebra indicates that we cannot recover Theorem
\ref{thm:2} with fewer resources.
\section{A Lie algebra of block-diagonal operators}
\label{sec:lie-algebra-block}
The purpose of this section is to re-discuss abelian symmetries and to provide
technical details (in particular proofs) we omitted in
Sections~\ref{sec:controllability} and \ref{sec:atoms-cavity}. To this end,
re-use the notations already introduced in Section~\ref{sec:abelian-symmetries}.
In particular, the abelian symmetry induces a block-diagonal decomposition
which, in infinite dimensions, allows for defining a block-diagonal Lie algebra
and its exponential map onto a block-diagonal Lie group;
see Propositions~\ref{prop:4.1} and \ref{prop:4.2}. We identify
the set of all block-diagonal unitaries reachable by block-diagonal
time evolutions in Proposition~\ref{prop:5}
as the strong closure of exponentials of block-diagonal Lie algebra elements.
A central result is Corollary~\ref{kor:1}, in which the question of controllability
for the block-diagonal system of infinite dimensions
is reduced to analyzing controllability for all finite-dimensional blocks.
Using finite-dimensional commutator calculations one can now establish controllability
on the infinite-dimensional but block-diagonal space
for each of the three control systems analyzed.
\subsection{Commuting operators}
\label{sec:commuting-operators}
The first step is a closer look at the Lie algebra $\mathfrak{u}(X)$ and the
corresponding group $\mathcal{U}(X)$ introduced in Theorem~\ref{thm:6} (which we will prove
in this context). To this end, let us start with a unitary $U$ commuting with
the representatives $\pi(z)$, i.e.\ $[\pi(z),U]=0$ for all $z \in
\mathrm{U}(1)$. This is equivalent to
$U\psi = \sum_{\mu=0}^\infty U^{(\mu)}\psi^{(\mu)}$ for all $\psi \in
\mathcal{H}$ with $\psi^{(\mu)} := X^{(\mu)} \psi \in
\mathcal{H}^{(\mu)}$
given a sequence of unitaries $U^{(\mu)}$ on the $\mu$-eigenspaces
$\mathcal{H}^{(\mu)}$ of $X$. Similarly one can consider a selfadjoint $H$
with domain $D(H)$ commuting with $X$. By definition\footnote{Note
that the identity $[X,Y]\,\psi = 0$ for all $\psi$ on a common dense domain is--in contrast
to popular belief--{\em not a proper definition} for two commuting selfadjoint
operators; cf.\ the discussion in \cite[VIII.6]{ReedSimonI}. Fortunately, such
pathological cases do not occur in our set-up.}
this means the spectral projections
of $H$ commute with the $X^{(\mu)}$, which is equivalent to
\begin{equation}\label{eq:29}
D_X \subset D(H),\, H D_X \subset D_X \, \text{ and }\, H \psi =
\sum_{\mu=0}^\infty H^{(\mu)} \psi^{(\mu)} \,\text{ for }\, \psi \in D_X
\end{equation}
with a sequence of selfadjoint operators $H^{(\mu)}$ on the eigenspaces
$\mathcal{H}^{(\mu)}$ and the $\psi^{(\mu)}$ as defined above.
The
$\mathcal{H}^{(\mu)}$ are finite-dimensional, and therefore the $H^{(\mu)}$ are
bounded. Hence the unboundedness of $H$ is inherited only from the unboundedness of
the sequence of norms $\|H^{(\mu)}\|$. So it is easy to
see that all elements of $D_X$ are analytic for $H$ and therefore $D_X$
becomes a domain of essential selfadjointness for $H$ (i.e.\ $H$ is uniquely
determined by its restriction to $D_X$ as a consequence of Nelson's
analytic vector theorem \cite[Thm. X.39]{ReedSimonII}). Accordingly, we
will denote (in slight abuse of notation) the selfadjoint operator $H$ and
its restriction to $D_X$ by the same symbol. This proves very handy when
introducing, on the set $\mathfrak{u}(X)$ of anti-selfadjoint operators commuting
with $X$, the structure of a Lie algebra by
$(\lambda Q_1 + Q_2) \psi = \lambda Q_1 \psi + Q_2 \psi,\, [Q_1,Q_2] \psi
= Q_1 Q_2 \psi - Q_2 Q_1 \psi$ for
$Q_1, Q_2 \in \mathfrak{u}(X)$, $\lambda \in \Bbb{R}$, and
$\psi \in D_X$.
The linear combination $\lambda Q_1 + Q_2$ and the commutator $[Q_1,Q_2]$ are
defined only on the joint domain $D_X$ but since they are essentially
selfadjoint on it, their selfadjoint extensions exist and are uniquely
determined. This proves the first statement of Thm.~\ref{thm:6}, which we restate
as follows:
\begin{prop}\label{prop:4.1}
A selfadjoint operator $H$ commuting with $X$ admits $D_X$ as an invariant
domain of essential selfadjointness. The space
\begin{align}
\mathfrak{u}(X) &= \big\{ iH \, | \, H=H^*\ \text{commuting with}\ X\big\}\\
&= \big\{ iH \, | \, H \psi = \mbox{$\sum_\mu$} H^{(\mu)}
\psi^{(\mu)},\ \psi \in D_X,\ H^{(\mu)} = (H^{(\mu)})^* \in
\mathcal{B}(\mathcal{H}^{(\mu)})\big\}
\end{align}
becomes a Lie algebra with the commutator as its Lie bracket.
\end{prop}
Since all $iH \in \mathfrak{u}(X)$ are anti-selfadjoint, they admit a
well-defined exponential map $\exp(i H)$. Boundedness of the $H^{(\mu)}$
together with Eq.~\eqref{eq:29} allows to express $\exp(iH)$ very
explicitly. More precisely one has
\begin{equation} \label{eq:22}
\exp(iH)\, \psi = \sum_{\mu=-\infty}^\infty \exp(i H^{(\mu)})\,
\psi^{(\mu)}\, \text{ where }\, \psi^{(\mu)} = X^{(\mu)}\psi \in \mathcal{H}^{(\mu)}
\end{equation}
and $\exp(i H^{(\mu)}) = \sum_{n=0}^\infty (iH^{(\mu)})^n/(n!)$.
This shows that $\exp: \mathfrak{u}(X) \rightarrow \mathcal{U}(X)$ is well-defined and onto
as stated in Thm.~\ref{thm:6}, which we are now ready to prove:
\begin{prop}\label{prop:4.2}
The exponential map on $\mathfrak{u}(X)$ is well-defined and given in terms of
Equation~\eqref{eq:22}.
It maps $\mathfrak{u}(X)$ onto
the strongly closed subgroup
\begin{align}
\mathcal{U}(X) &= \big\{ U \in \mathcal{U}(\mathcal{H})\, | \, [U,\pi(z)]=0 \, \text{ for all }\, z
\in \mathrm{U}(1)\big\} \\
&= \big\{ U \, | \, U \psi = \mbox{$\sum_\mu$} U^{(\mu)} \psi^{(\mu)},\
\psi \in \mathcal{H},\
U^{(\mu)} \in \mathcal{U}(\mathcal{H}^{(\mu)})\big\} \,\text{ of }\, \mathcal{U}(\mathcal{H}).
\end{align}
\end{prop}
\begin{proof}
The only statement not yet proven is the closedness of $\mathcal{U}(X)$. To this end, we
have to show that for any net $(U_\lambda)_{\lambda \in \mathcal{I}}$
strongly converging to a bounded operator $U$ we have $U \in \mathcal{U}(X)$.
As $U_\lambda \in \mathcal{U}(X)$ we have $[\pi(z),U_\lambda] =0$
for all $\lambda$. Due to strong continuity of the map $A \mapsto [\pi(z),A]$ and
the convergence of the $U_\lambda$ to $U$ it follows that $[\pi(z),U]=0$.
Hence $U$ decomposes into a strongly converging series $U = \sum_\mu
U^{(\mu)}$ with $U^{(\mu)} \in \mathcal{B}(\mathcal{H}^{(\mu)})$, and for
each fixed $\mu$ we get $\lim_\lambda U_\lambda^{(\mu)} = U^{(\mu)}$. Since
$\mathcal{H}^{(\mu)}$ is finite-dimensional, the nets
$(U_\lambda^{(\mu)})_{\lambda \in \mathcal{I}}$ converge in norm and therefore $U^{(\mu)} \in
\mathcal{U}(\mathcal{H}^{(\mu)})$ which implies $U \in \mathcal{U}(X)$.
\end{proof}
Note that we actually proved more than what we stated. A strongly
convergent sequence (or net) of elements of $\mathcal{U}(X)$ cannot converge to an
isometry which is not unitary as well. Hence $\mathcal{U}(X)$ is strongly closed as a subset of
$\mathcal{B}(\mathcal{H})$---and not only as a subset of $\mathcal{U}(\mathcal{H})$
as generally is the case (cf.\ corresponding remarks in Sect.~\ref{sec:dynamical-group}).
The remaining statements in this subsection are devoted to the dynamical
group~$\mathcal{G}$ generated by a family of selfadjoint operators $H_1, \dots, H_d$.
Recall that we have introduced it as the smallest strongly closed subgroup of
$\mathcal{U}(\mathcal{H})$ containing all unitaries of the form $\exp(i t
H_k)$. If the $H_k$ are commuting with $X$, i.e.\ $iH_k \in \mathfrak{u}(X)$, then the
group $\mathcal{G}$ is a subgroup of $\mathcal{U}(X)$, and the simple structure of the latter
makes explicit calculations at least feasible. In the following, we show
how $\mathcal{U}(X)$ is related to the Lie algebra $\mathfrak{l}$ generated by the
$iH_k$. To this end, we need some additional notations. For each $K \in
\ensuremath{\mathbb N}$, $U \in \mathcal{U}(X)$, and $iH \in \mathfrak{u}(X)$, let us consider
\begin{equation} \label{eq:62}
U^{[K]} = \sum_{\mu=0}^K U^{(\mu)},\; H^{[K]} = \sum_{\mu=0}^K
H^{(\mu)},\; \mathcal{H}^{[K]} = \bigoplus_{\mu=0}^K \mathcal{H}^{(\mu)}.
\end{equation}
The operators $U^{[K]}$ and $H^{[K]}$ act on the finite-dimensional Hilbert space
$\mathcal{H}^{[K]}$. Therefore all operator topologies coincide and we can
apply the well-known finite-dimensional theory. The dynamical group $\mathcal{G}^{[K]}$
(generated by $H^{[K]}_k$ with $k\in\{1,\dots,d\}$) becomes a closed subgroup of the
unitary group $\mathcal{U}(\mathcal{H}^{[K]})$, which is a Lie group. Hence
$\mathcal{G}^{[K]}$ is a Lie group, too, and its Lie algebra $\mathfrak{l}^{[K]}$ is
generated by $iH^{[K]}_k$ with $k\in\{1,\dots,d\}$. Now, the crucial point is that one can
approximate the infinite-dimensional objects $\mathcal{G}$ and $\mathfrak{l}$
by the finite-dimensional $\mathcal{G}^{[K]}$ and $\mathfrak{l}^{[K]}$. To see this, the
first step is the following lemma.
\begin{lem} \label{lem:1}
Consider the Lie algebras $\mathfrak{l} \subset \mathfrak{u}(X)$ and
$\mathfrak{l}^{[K]} \subset \mathcal{B}(\mathcal{H}^{[K]})$ (with $K \in \ensuremath{\mathbb N}$) generated by
$iH_1, \dots, iH_d$ and $iH^{[K]}_1, \dots, iH^{[K]}_d$,
respectively. Each element $\tilde{Q} \in \mathfrak{l}^{[K]}$ can be written as
$\tilde{Q} = Q^{[K]}$ for some element $Q \in \mathfrak{l}$.
\end{lem}
\begin{proof}
Since $\tilde{Q} \in \mathfrak{l}^{[K]}$, it
is equal to a linear
combination $\sum_{\ell} c_{\ell} C_{\ell}(iH_{j_1}^{[K]},\dots,iH_{j_n}^{[K]})$ of repeated commutators
$C_{\ell}(iH_{j_1}^{[K]},\dots,iH_{j_n}^{[K]})$
containing the elements
$\{iH_{j_1}^{[K]},\dots,iH_{j_n}^{[K]}\}$ with $j_k \in\{1, \dots, d\}$. However,
$\mathfrak{l}$ is generated by $iH_1, \dots, iH_k$ and
it contains the same
commutators $C_{\ell}(iH_{j_1},\dots,iH_{j_n})$ yet with $H_j^{[K]}$ replaced by
$H_j$. Hence one can form a linear combination $Q$ such that $Q^{[K]} =
\tilde{Q}$ as stated.
\end{proof}
Moreover, we now have the tools to prove the relation between the Lie algebra
$\mathfrak{l}$ and the dynamical group $\mathcal{G}$ already stated in Thm. \ref{thm:6}.
\begin{prop} \label{prop:5}
Consider again $iH_1, \dots, iH_d \in \mathfrak{u}(X)$ and the Lie algebra
$\mathfrak{l}$ generated by them. Then the corresponding dynamical group $\mathcal{G}$
coincides with the strong closure of $\exp(\mathfrak{l}) \subset \mathcal{U}(X)$.
\end{prop}
\begin{proof}
Each $U \in \mathcal{G}$ can be written as the limit of a net $(U_\lambda)_{\lambda
\in \mathcal{I}}$ of operators $U_\lambda$, which are monomials in
$\exp(it_kH_k)$ with $k\in\{1,\dots,d\}$ with appropriate times $t_k$. This implies in
particular that the $U_\lambda$ commute with $\pi(z)$ for all $z$, and, by
continuity, the same is true for $U$. Hence $U\in \mathcal{U}(X)$, and for each $K \in
\ensuremath{\mathbb N}$ we can define $U^{[K]}$ which is the limit of the net
$(U^{[K]}_\lambda)_{\lambda \in \mathcal{I}}$. The latter converges in norm
(since $\mathcal{H}^{[K]}$ is finite-dimensional), and therefore $U^{[K]}
\in \mathcal{G}^{[K]}$. This implies $U^{[K]} = \exp(Q_K)$ with $Q_K \in
\mathfrak{l}^{[K]}$ as $\mathcal{G}^{[K]}$ is a Lie group and $\mathfrak{l}^{[K]}$
its Lie algebra.
For $U$ to be in the strong closure of $\exp(\mathfrak{l})$, each strong
$\epsilon$-neighborhood of $U$, i.e.\ the sets
$\mathcal{N}(U;\psi_1,\dots,\psi_f;\epsilon)$ introduced in
Eq.~\eqref{eq:66}, should contain an element of $\exp(\mathfrak{l})$
for all $\psi_1, \dots, \psi_f$ and all $\epsilon > 0$.
However, the unitary group is contained in the unit
ball of $\mathcal{B}(\mathcal{H})$, and thus it is sufficient to consider
only those $\mathcal{N}(U;\psi_1,\dots;\psi_f,\epsilon)$ with vectors
$\psi_1, \dots, \psi_f$ from a dense subspace of $\mathcal{H}$;
cf.\ \cite[I.3.1.2]{blackadar06}. Hence, in turn, it is sufficient to consider only
neighborhoods with $\psi_j \in D_X$. But then there is a $K \in \ensuremath{\mathbb N}$
such that $\psi_j \in \mathcal{H}^{[K]}$ for all $j\in\{1,\dots,f\}$. Now take
the operator $Q_K$ from the last paragraph and $\tilde{Q}_K \in \mathfrak{l}$
with $\tilde{Q}_K^{[K]} = Q_K$, which exists due to Lemma \ref{lem:1}. By
construction we have
$\| [U - \exp(\tilde{Q}_K)]\psi_j\| = \| [U^{[K]}
- \exp(\tilde{Q}_K)^{[K]}]\psi_j\| = \| [(U^{[K]} -
\exp(\tilde{Q}_K^{[K]})]\psi_j\|
= \| [(U^{[K]} - \exp(Q_K)]\psi_j\| = 0 $
since $U^{[K]} = \exp(Q_K)$, as was also seen in the previous paragraph. Hence
$\exp(\tilde{Q}_K) \in \mathcal{N}(U;\psi_1,\dots,\psi_f;\epsilon)$ which
shows that $U$ is in the strong closure of $\exp(\mathfrak{l})$. This shows that
the dynamical group $\mathcal{G}$ is contained in the strong closure of $\exp(\mathfrak{l})$.
Conversely, consider $\exp(Q)$ for $Q \in \mathfrak{l}$. We have to show
that $\exp(Q)$ is in the dynamical group $\mathcal{G}$. To this end we observe, for each $K
\in \ensuremath{\mathbb N}$, that $\exp(Q^{[K]}) = \exp(Q)^{[K]}$, which is obviously in
$\mathcal{G}^{[K]}$. Hence there is a $U_K = \exp(iH_{j_1}^{[K]}) \cdots
\exp(iH_{j_n}^{[K]})$ with $j_k \in \{1,\dots,d\}$ which is $\epsilon$-close (in
norm) to $\exp(Q^{[K]})$. As in the last paragraph, this implies that
$\tilde{U} = \exp(iH_{j_1})\cdots\exp(iH_{j_n})$ is in
$\mathcal{N}(\exp(Q);\psi_1, \dots, \psi_f;\epsilon)$ provided $\psi_j \in
\mathcal{H}^{[K]}$ for all $j\in\{1,\dots,f\}$. Hence $\exp(Q)$ is in the strong
closure of the group of monomials in the $\exp(iH_j)$, but this is just the
dynamical group $\mathcal{G}$. Since $\mathcal{G}$ is strongly closed, the strong closure of
$\exp(\mathfrak{l})$ is contained in $\mathcal{G}$, too. Since we have shown the other
inclusion before, the entire proposition is proven.
\end{proof}
Moreover, with this proposition the proof of Thm.~\ref{thm:6} is complete. -- The rest of
this subsection is devoted to analyzing a related question: If, in finite
dimension, two Lie algebras $\mathfrak{l}_1, \mathfrak{l}_2$ generate the same
group, then they are actually identical. However, in infinite dimensions this no longer true.
Therefore, the next proposition is meant to decide if dynamical groups generated by two
different sets of Hamiltonians do in fact coincide.
\begin{prop} \label{prop:6}
Consider two Lie algebras $\mathfrak{l}_1, \mathfrak{l}_2 \subset
\mathfrak{u}(X)$. Assume that for each $Q \in \mathfrak{l}_1$ and each $K \in
\ensuremath{\mathbb N}$, there is a $\tilde{Q} \in \mathfrak{l}_2$ such that $Q^{[K]} =
\tilde{Q}^{[K]}$ holds (note that we can have different
$\tilde{Q}$ for the same $Q$ but different $K$). Then
$\exp(\mathfrak{l}_1)$ is contained in the strong closure of
$\exp(\mathfrak{l}_2)$.
\end{prop}
\begin{proof}
One may readily use the same strategy as in the proof of Prop. \ref{prop:5}: If the given
condition holds, one can find in each neighborhood
$\mathcal{N}(\exp(Q);\psi_1,\dots,\psi_f;\epsilon)$ of $\exp(Q)$ with
$\psi_1,\dots,\psi_f \in D_X$ an $\exp(\tilde{Q})$ with $\tilde{Q} \in
\mathfrak{l}_2$. Hence $\exp(Q)$ is in the strong closure of
$\exp(\mathfrak{l}_2)$.
\end{proof}
Inserting $\mathfrak{su}(X)$ for $\mathfrak{l}_2$ provides a useful criterion to
check whether the dynamical group $\mathcal{G}$ generated by $H_1, \dots, H_d
\in \mathfrak{su}(X)$ is as large as possible in the sense that $\mathcal{G} =
\mathcal{SU}(X)$. To this end, let us introduce the truncated versions
\begin{align}
\mathfrak{su}^{[K]}(X) &= \{ Q^{[K]}\,|\,Q \in \mathfrak{su}(X)\} = \oplus_{\mu=0}^K\,
\mathfrak{su}(\mathcal{H}^{(\mu)}),\nonumber \\
\mathcal{SU}^{[K]}(X) &= \{U^{[K]}\,|\, U \in \mathcal{SU}(X)\}=
\oplus_{\mu=0}^K\, \mathcal{SU}(\mathcal{H}^{(\mu)}),
\end{align}
where we have used for any finite-dimensional subspace $\mathcal{K}$ of
$\mathcal{H}$ the notations $\mathfrak{su}(\mathcal{K})$ for the Lie algebra of
traceless operators on $\mathcal{K}$ and similarly $\mathcal{SU}(\mathcal{K})$
for the Lie group of unitaries on $\mathcal{K}$ with determinant $1$. Note
that elements of $\mathfrak{su}(\mathcal{K})$
and $\mathcal{SU}(\mathcal{K})$ have---as operators on $\mathcal{H}$---a finite rank and their support and range
are both contained in $\mathcal{K}$.
\begin{kor} \label{kor:1}
Consider Hamiltonians $iH_1, \dots, iH_d \in \mathfrak{su}(X)$, the corresponding
dynamical group $\mathcal{G}$ and the generated Lie algebra
$\mathfrak{l}$. If $\mathfrak{su}^{[K]}(X) = \mathfrak{l}^{[K]}$ holds for all
$K \in \ensuremath{\mathbb N}$, then one finds $\mathcal{G} = \mathcal{SU}(X)$.
\end{kor}
\begin{proof}
Simple application of Props.~\ref{prop:5} and \ref{prop:6}.
\end{proof}
\subsection{One atom}
\label{sec:one-atom-1}
The material just introduced readily applies to the systems studied in
Sect.~\ref{sec:atoms-cavity}. This includes in particular the proofs of
Thms.~\ref{thm:3}, \ref{thm:8}, \ref{thm:4} and \ref{thm:5}. The first step is
again one atom interacting with a cavity (Sect.~\ref{sec:one-atom}). Hence the
Hilbert space is $\mathcal{H} = \Bbb{C}^2 \otimes \mathrm{L}^2(\Bbb{R})$ and
the $\mathrm{U}(1)$-symmetry under consideration is generated by the operator
$X_1 = \sigma_3 \otimes \Bbb{1} + \Bbb{1} \otimes N$ already defined in
(\ref{eq:36}). The domain of $X_1$ is $D$ from Eq.~\eqref{eq:40}, which is
identical to $D_{X_1}$ introduced in (\ref{eq:28}).
The next step is to characterize the Lie algebra $\mathfrak{l}$ generated by
the control Hamiltonians $H_{\mathrm{JC},1}$ and $H_{\mathrm{JC},2}$ as defined in
(\ref{eq:15}). They admit $D=D_{X_1}$ as a joint common domain, and it is easy to
see that they commute with $X_1$ (in the sense introduced in the previous
subsection). Hence $\mathfrak{l} \subset \mathfrak{u}(X_1)$, and all the machinery
from Subsection~\ref{sec:commuting-operators} applies. This includes in particular
the block-diagonal decomposition of operators $A \in \mathfrak{u}(X_1)$ given in
Eq.~\eqref{eq:29}. In our case the subspaces $\mathcal{H}^{(\mu)}$ with $\mu \in
\ensuremath{\mathbb N}$ are given by (cf.\ Eq.~\eqref{eq:38}) $\mathcal{H}^{(\mu)} = \operatorname{span}\{
\ket{\mu,0},\ \ket{\mu,1} \}$ using the basis $\ket{\mu,\nu} \in \mathcal{H}$
introduced in (\ref{eq:27}). For $\mu=0$, we get the one-dimensional space
$\mathcal{H}^{(0)} = \Bbb{C} \ket{0,0}$.
The restrictions
$H_{\mathrm{JC},j}^{(\mu)}$ of the operators
$H_{\mathrm{JC},j}$ to the subspaces $\mathcal{H}^{(\mu)}$ are given
by (for $\mu \geq 1$):
\begin{equation} \label{eq:67}
H_{\mathrm{JC},1}^{(\mu)} = - \varsigma_3^{(\mu)}/2 ,\, H_{\mathrm{JC},2}^{(\mu)} = \sqrt{\mu}
\varsigma_1^{(\mu)},\, H_{\mathrm{JC},3} = (\mu + 1/2) \varsigma_0^{(\mu)} -
\varsigma_3^{(\mu)}/2,
\end{equation}
where we have introduced the operators $\varsigma_{\alpha}=\sum_{\mu} \varsigma_{\alpha}^{(\mu)}$
with $\alpha \in \{0,\ldots,3\}$
via their projections
$\varsigma_0^{(\mu)} = \Bbb{1}^{(\mu)} = X^{(\mu)} = \kb{\mu,0} + \kb{\mu,1}$,
$\varsigma_1^{(\mu)} = \KB{\mu,0}{\mu,1} + \KB{\mu,1}{\mu,0}$,
$\varsigma_2^{(\mu)} = i \bigl(\KB{\mu,1}{\mu,0} - \KB{\mu,0}{\mu,1}\bigr)$, and
$\varsigma_3^{(\mu)} = \kb{\mu,0} - \kb{\mu,1}$.
Hence, for each fixed $\mu$, the operator $\varsigma_\alpha^{(\mu)}$ is just the
corresponding Pauli operator on $\mathcal{H}^{(\mu)}$ given in the basis
$\ket{\mu,0},\ket{\mu,1}$. We have used the core symbol $\varsigma$ rather than
$\sigma$ in order to avoid confusion with the operators $\sigma_\alpha \otimes
\Bbb{1}$ acting only on the atom. In addition we introduce the operators
$A_{\alpha,k} \in \mathfrak{u}(X_1)$ with $\alpha\in\{0,\dots,3\}$ and $k \in \ensuremath{\mathbb N}_0$ by
\begin{equation} \label{eq:68}
A_{\alpha,k} = \sqrt{X_1} X_1^k \varsigma_\alpha\,\text{ for }\, \alpha \in \{1,2\},\, A_{3,k} =
X_1^k \varsigma_3,\, A_{0,k} = X_1^{k}.
\end{equation}
In terms of the $A_{\alpha,k}$, now the $H_{\mathrm{JC},j}$ can readily be re-expressed as
\begin{equation} \label{eq:39}
H_{\mathrm{JC},1} = - A_{3,0}/2,\, H_{\mathrm{JC},3} = A_{1,0}/2,\, H_{\mathrm{JC},2} = A_{0,1} +
(A_{0,0} - A_{3,0})/2.
\end{equation}
The next lemma shows that the Lie algebra $\mathfrak{l}$ generated by the
$H_{\mathrm{JC},j}$ is spanned as a vector space by a subset of the
$A_{\alpha,k}$.
\begin{lem} \label{lem:2}
The Lie algebra $\mathfrak{l}$ generated by $iH_{\mathrm{JC},j}$ with $j\in\{1,2\}$
is spanned as a vector space by the operators $i A_{\alpha,k}$ with
$\alpha\in\{1,2,3\}$ and
$k \in \ensuremath{\mathbb N}_0$.
\end{lem}
\begin{proof}
Obviously the operators $iA_{\alpha,k}$ are in $\mathfrak{su}(X_1)$. Hence, they
span a subspace $\tilde{\mathfrak{l}} \subset \mathfrak{su}(X_1)$.
To prove that $\tilde{\mathfrak{l}}$ is a Lie subalgebra of
$\mathfrak{su}(X_1)$ one only has to check that $[A_{\alpha,k},A_{\beta,j}] \in
\tilde{\mathfrak{l}}$ for all $\alpha,\beta \in\{1,2,3\}$ and $j,k \in
\mathbb{N}_0$. This follows easily, because the $A_{\alpha,k}$ are just
products of powers of $X_1$ and the $\varsigma_\alpha$. But the latter are
representatives of the Pauli operators. Hence
\begin{equation} \label{eq:18}
[A_{1,k},A_{2,\ell}] = 2 i A_{3,k+\ell+1},\, [A_{3,k},A_{1,\ell}] = 2 i
A_{2,k+\ell},\, [A_{2,k}, A_{3,\ell}] = 2 i A_{1,k+\ell},
\end{equation}
All operators vanish in the case of $\mu=0$. Hence $\tilde{\mathfrak{l}}$ is a
Lie algebra and Eq.~\eqref{eq:39} proves that $\mathfrak{l} \subset
\tilde{\mathfrak{l}}$.
For proving $\tilde{\mathfrak{l}} = \mathfrak{l}$, one has to express the
$A_{\alpha,k}$ for $\alpha\in\{1,2,3\}$ and $k \in \ensuremath{\mathbb N}_0$
in terms of repeated commutators of the $H_{\mathrm{JC},2}$ and $H_{\mathrm{JC},3}$.
By the commutation relations in Equation \eqref{eq:18} it is obvious that
$\tilde{\mathfrak{l}}$ is generated (as a Lie algebra) by $A_{\alpha,0}$ with
$\alpha \in\{1,2,3\}$. Therefore, the statement follows from Eq.~\eqref{eq:39},
which in turn shows that $A_{1,0}$ and $A_{3,0}$ are just $H_{\mathrm{JC},3}$ and
$H_{\mathrm{JC},1}$, while $A_{2,0}$ can be derived from the commutator
$[H_{\mathrm{JC},1},H_{\mathrm{JC},3}]$.
\end{proof}
With this Lemma and the material developed in the last subsection, one can proceed to
determine the structure of the dynamical group generated by
$H_{\mathrm{JC},1}$ and $H_{\mathrm{JC},2}$. This is the content of
Thm.~\ref{thm:3}, which is restated (and proven) here as a proposition.
\begin{prop} \label{prop:7}
The dynamical group generated by $H_{\mathrm{JC},1}$ and $H_{\mathrm{JC},2}$
is equal
to $\mathcal{SU}(X)$.
\end{prop}
\begin{proof}
According to Prop.~\ref{prop:5} the dynamical group $\mathcal{G}$ is the strong
closure of $\exp(H)$ with $H \in \mathfrak{l}$, i.e.\ the Lie algebra generated
by $H_{\mathrm{JC},1}$ and $H_{\mathrm{JC},2}$, while $\mathcal{SU}(X)$ is the
strong closure of $\exp(\mathfrak{su}(X))$. Hence, by Cor.~\ref{kor:1}
we have to show that the truncated algebras $\mathfrak{l}^{[K]}$ and
$\mathfrak{su}^{[K]}(X)$ are identical. The inclusion $\mathfrak{l}^{[K]}
\subset \mathfrak{su}^{[K]}(X)$ is trivial, since all the blocks
$H_{\mathrm{JC},j}^{(\mu)}$ with $j\in\{1,2\}$ are traceless. To show the other
inclusion, first note that
$\mathfrak{l}^{[0]} = \mathfrak{su}(X)^{[0]} = \{0\}$. Hence it is sufficient to
check that for each fixed $0< \mu_0 \leq K$ and each $iH \in
\mathfrak{su}^{[K]}(X)$ with $H^{(\mu)} = 0$ for $\mu \neq \mu_0$ there is
an $i A \in \mathfrak{l}$ such that $i A^{(\mu_0)} = i H^{(\mu_0)}$ and
$A^{(\mu)} = 0$ for all $0 < \mu \leq K$ with $\mu \neq \mu_0$. The rest
follows by linearity.
For constructing such an $A$, recall from Lemma \ref{lem:2} that $\mathfrak{l}$
is spanned (as a vector space) by the $A_{\alpha,k}$ with $\alpha\in\{1,2,3\}$ and $k \in
\ensuremath{\mathbb N}_0$. Now consider a polynomial $f$ in one real variable satisfying
$f(\mu) = 0$ for all $0 < \mu \leq K$ with $\mu \neq \mu_0$ and $f(\mu_0) =
1$. The operators
$B_{\alpha,f} = f(X) \sqrt{X} \varsigma_\alpha$ with $\alpha \in \{1,2\}$ and
$B_{3,f} = f(X) \varsigma_3$
are linear combinations of the $A_{\alpha,k}$, and they satisfy the
condition
$B_{\alpha,f}^{(\mu)} = 0$ for all $0 < \mu \leq K$ such that $\mu\neq\mu_0$ and $B_{\alpha,f}^{(\mu_0)} =
c_\alpha \varsigma_\alpha^{(\mu_0)}$
for a constant $c_\alpha$ given by $c_{1}=c_{2}=\sqrt{\mu_0}$ and $c_3 = 1$. But
all traceless operators $H^{(\mu_0)} \in \mathcal{B}(\mathcal{H}^{(\mu_0)})$
can be written as a linear combinations of the $\varsigma_\alpha^{(\mu_0)}$,
which concludes the proof.
\end{proof}
Before proceeding to the next subsection, consider the free
Hamiltonian of the cavity $H_{\mathrm{JC},3}$. We have omitted it from the
discussion of the dynamical group, and the reason can be seen easily from
(\ref{eq:18}): $H_{\mathrm{JC},2}$ differs from $\mathrm{H}_{\mathrm{JC},1}$
only by $X_1 + \Bbb{1}/2$ which commutes with all elements of
$\mathfrak{su}(X)$. Hence adding $H_{\mathrm{JC},3}$ as a control Hamiltonian
would just add a one-dimensional center to the dynamical group
$\mathcal{G}=\mathcal{SU}(X)$. For the same reason, $H_{\mathrm{JC},3}$ could
be easily added as a drift term. Any effect it may have can be undone by evolving
the system with $H_{\mathrm{JC},1}$, and the remaining relative phase between
sectors of different charge $\mu$ does not affect the
discussion of strong controllability in Sect.~\ref{sec:strong-contr}.
Finally, let us remark that---due to the same reasons just discussed---we
could exchange $H_{\mathrm{JC},1}$ and $H_{\mathrm{JC},3}$ almost without
changes to the results of this subsection.
\subsection{Many atoms with individual control}
First, recall some notations from Sect.~\ref{sec:many-atoms-indiv}. The
Hilbert space is $\mathcal{H}_M = (\Bbb{C}^2)^{\otimes M} \otimes
\mathrm{L}^2(\Bbb{R})$ using the distinguished basis $\ket{\mu;\vec{b}}$ with $\vec{b} \in
\Bbb{Z}_2^M$ from Eq.~\eqref{eq:48}. The charge operator is $X_M = S_3
\otimes \mathbbmss{1} + \mathbbmss{1} \otimes N$, cf.\ Eq.~\eqref{eq:47}, with domain $D_M$
from Eq.~\eqref{eq:41}. In addition, let us introduce the re-ordered tensor
product (where $\ket{\mu,b_1,\dots,b_M} \in \mathcal{H}_M$ and $b\in\Bbb{Z}_2$)
\begin{equation}
\ket{\mu,\vec{b}} \hat{\otimes}_k \ket{b} =
\ket{\mu+b;b_1,\dots,b_{k-1},b,b_k,\dots,b_M} \in \mathcal{H}_{M+1}.
\end{equation}
The key result of this section is split into the following three lemmas,
which eventually will lead to a proof of Thm.~\ref{thm:8}.
\begin{lem} \label{lem:5}
The complexification $\mathfrak{su}_\Bbb{C}(\mathcal{H}_M^{(\mu)})$ of the
real Lie algebra $\mathfrak{su}(\mathcal{H}_M^{(\mu)})$ is generated by
elements $\KB{\mu;\vec{b}}{\mu;\vec{c}}$ with $\vec{b},\vec{c} \in
\Bbb{Z}_2^M$ satisfying $\vec{b} \neq \vec{c}$.
\end{lem}
\begin{proof}
$\mathfrak{su}_\Bbb{C}(\mathcal{H}_M^{(\mu)})$ is isomorphic to the Lie algebra
$\mathfrak{sl}(\mathcal{H}_M^{(\mu)})$ of traceless operators on
$\mathcal{H}_M^{(\mu)}$. The $\KB{\mu;\vec{b}}{\mu;\vec{c}}$ with $\vec{b}
\neq \vec{c}$ span the vector space of all $A \in
\mathcal{B}(\mathcal{H}_M^{(\mu)})$ satisfying $\langle \mu; \vec{b}\,|\, A
\,|\, \mu; \vec{b}\rangle = 0$ for all $\vec{b} \in \Bbb{Z}_2^M$ i.e.\ all
operators which are off-diagonal in the basis $\ket{\mu;\vec{b}}$. The
smallest Lie algebra containing this space is
$\mathfrak{sl}(\mathcal{H}_M^{(\mu)})$.
\end{proof}
\begin{lem}\label{lem_prev}
The Lie algebra $\mathfrak{su}_\Bbb{C}(\mathcal{H}_{M+1}^{(\mu)})$ is generated by the union of
the subalgebras $\mathfrak{su}_\Bbb{C}(\mathcal{H}_M^{(\mu-b)} \hat{\otimes}_k
\ket{b})$ with $b \in \Bbb{Z}_2$ and $k\in\{1,\dots,M\}$.
\end{lem}
\begin{proof}
First of all, note that (by definition) $\ket{\mu{-}b;\vec{b}} \in
\mathcal{H}_M^{(\mu-b)}$. Hence $\ket{\mu{-}b;\vec{b}} \hat{\otimes}_k \ket{b} \in
\mathcal{H}_{M+1}^{(\mu)}$ which shows that all the Hilbert spaces
$\mathcal{H}^{(\mu-b)} \hat{\otimes}_k \ket{b}$ are subspaces of
$\mathcal{H}_M^{(\mu)}$. According to the previous lemma, we have to show
that operators $A = \KB{\mu;\vec{b}}{\mu;\vec{c}}$ with $\vec{b}, \vec{c} \in
\Bbb{Z}_2^{M+1}$ and $\vec{b} \neq \vec{c}$ can be written as commutators from
operators in the $\mathfrak{su}_\Bbb{C}(\mathcal{H}_M^{(\mu+b)} \hat{\otimes}_k
\ket{b})$. We have to distinguish two cases:
In the first case, there is at least one $k\in\{1,\dots,M\}$ with $b_k = c_k = b$. If this holds,
$A$ can be written as
$
\KB{\mu{-}b;b_1,\dots,b_{k-1},b_{k+1},\dots,b_{M+1}}{\mu{-}b;c_1,\dots,c_{k-1},c_{k+1},\dots,c_{M+1}}
\otimes \kb{b}
\in\mathfrak{su}_\Bbb{C}(\mathcal{H}_M^{(\mu-b)} \hat{\otimes}_k \ket{b})$. The second
case arises if $b_k \neq c_k$ for all $k$. Now consider the commutator
of the operators $B =
\KB{\mu;\vec{b}}{\mu;b_1,c_2,\dots,c_{M+1}}$ and $C =
\KB{\mu;b_1,c_2,\dots,c_{M+1}}{\mu;\vec{c}}$ obviously $A = [B,C]$, $B
\in \mathfrak{su}_\Bbb{C}(\mathcal{H}_M^{(\mu-b_1)} \hat{\otimes}_1 \ket{b_1})$, and $C
\in \mathfrak{su}_\Bbb{C}(\mathcal{H}_M^{(\mu-c_k)} \hat{\otimes}_k \ket{c_k})$ for
$k>1$. This concludes the proof.
\end{proof}
\begin{lem} \label{lem:6}
The Lie algebra $\mathfrak{su}(\mathcal{H}_{M+1}^{(\mu)})$ is contained in
the Lie algebra $\mathfrak{g}$ generated by
$\mathfrak{su}(\mathcal{H}_M^{(\mu)}) \hat{\otimes}_k \Bbb{1}$ and
$\mathfrak{su}(\mathcal{H}_M^{(\mu-1)}) \hat{\otimes}_k \Bbb{1}$.
\end{lem}
\begin{proof}
First of all note that it is sufficient to prove the statement for the
corresponding complexified Lie algebras
$\mathfrak{su}_\Bbb{C}(\mathcal{H}_{M+1}^{(\mu)}) =
\mathfrak{su}(\mathcal{H}_{M+1}^{(\mu)}) \oplus i (\mathcal{H}_{M+1}^{(\mu)})$
and $\mathfrak{g}_\Bbb{C} = \mathfrak{g} \oplus i \mathfrak{g}$, since we
get the original statement back by restricting the inclusion
$\mathfrak{su}_\Bbb{C}(\mathcal{H}_{M+1}^{(\mu)}) \subset
\mathfrak{g}_\Bbb{C}$ to anti-selfadjoint elements on both sides.
The elements of $\mathfrak{su}_\Bbb{C}(\mathcal{H}^{(\mu)}) \hat{\otimes}_k \Bbb{1}$
are of the form $A = a \hat{\otimes}_k \kb{0} + a \hat{\otimes}_k \kb{1}$ with
$a \in \mathfrak{su}_\Bbb{C}(\mathcal{H}^{(\mu)})$. We will show that both summands
are elements of $\mathfrak{g}_\Bbb{C}$, i.e.\ $a \hat{\otimes}_k \kb{b} \in
\mathfrak{g}_\Bbb{C}$ for $b\in\{0,1\}$. The same holds for $\mu{-}1$. The statement then
follows from Lemma~\ref{lem_prev}.
Use again Lemma~\ref{lem:5} and choose $a = \KB{\mu;\vec{b}}{\mu;\vec{c}}$
with $\vec{b}, \vec{c} \in \Bbb{Z}_2^M$ and $\vec{b} \neq \vec{c}$.
We rewrite $A = a \hat{\otimes}_k \kb{0} + a \hat{\otimes}_k
\kb{1}$ as
\begin{align}
&\KB{\mu;(b_1,\dots,b_k,0,b_{k+1},\dots,b_M)}{\mu;(c_1,\dots,c_k,0,c_{k+1},\dots,c_M)}\nonumber \\ + &
\KB{\mu+1;(b_1,\dots,b_k,1,b_{k+1},\dots,b_M)}{\mu+1;(c_1,\dots,c_k,1,c_{k+1},\dots,c_M)}.
\end{align}
Moreover,
$\vec{b}_0:= (b_2,\dots,b_k,0,b_{k+1},\dots,b_M)$, $\vec{b}_1:=
(b_2,\dots,b_k,1,b_{k+1},\dots,b_M)$,
$\vec{c}_0:= (c_2,\dots,c_k,0,c_{k+1},\dots,c_M)$, and $\vec{c}_1:=
(c_2,\dots,c_k,1,c_{k+1},\dots,c_M)$
allows us to simplify
\begin{equation} \label{eq:46}
A=(\KB{\mu{-}b_1;\vec{b}_0}{\mu{-}c_1;\vec{c}_0}
+ \KB{\mu{-}b_1{+}1;\vec{b}_1}{\mu{-}c_1{+}1;\vec{c}_1}) \hat{\otimes}_1
\KB{b_1}{c_1}.
\end{equation}
Next, consider a second operator $B = (\kb{\mu{-}c_1;\vec{c}_0}
- \kb{\mu{-}c_1;\vec{c}_1}) \hat{\otimes}_1 \Bbb{1}$ and assume that $M > 1$
holds. Then there is a $\ell\in\{1,\dots,M\}$ with $b_\ell \neq c_\ell$. Without loss of
generality one can assume that $\ell\neq 1$ (otherwise rewrite $A$ in
(\ref{eq:46}) as $\tilde{A} \hat{\otimes}_j \KB{b_j}{c_j}$ with another
index $j$). The commutator now equals
$[A,B] = \KB{\mu-b_1;\vec{b}_0}{\mu-c_1;\vec{c}_0} \hat{\otimes}_1
\KB{b_1}{c_1} = a \hat{\otimes}_k \kb{0}$.
If $M=1$ one has two possible cases: either $b=0$ and $c=1$ or $b=1$ and $c=1$. In the first
case choose $B=(\kb{\mu{-}c;0} - \kb{\mu{-}c;1}) \otimes \Bbb{1}$, and in the
second case pick $B = (\kb{\mu{-}b;0} - \kb{\mu{-}b;1}) \otimes \Bbb{1}$. Then the commutator
$[A,B]$ leads again to $\pm \KB{\mu{-}b;0}{\mu{-}c;0} \otimes
\KB{b}{c}$.
Therefore, one can conclude that $\mathfrak{su}_\Bbb{C}(\mathcal{H}_M^{(\mu)}
\hat{\otimes}_k \ket{0}) \subset \mathfrak{g}_\Bbb{C}$ for all $k$. The same
reasoning holds for $\mathfrak{su}_\Bbb{C}(\mathcal{H}_M^{(\mu-1)}
\hat{\otimes}_k \ket{1})$. Hence the statement follows from the previous
lemma.
\end{proof}
Now let us consider the control Hamiltonians $H_{\mathrm{IC},j},
H_{\mathrm{IC},M+j}$ from Equation (\ref{eq:44}). We will use Lemma
\ref{lem:6} and an induction in $M$ to prove Thm. \ref{thm:8}, which we restate
here as a proposition.
\begin{prop} \label{prop:8}
The dynamical group generated by the control Hamiltonians
$H_{\mathrm{IC},j}$ with $j\in\{1,\dots,2M\}$ is identical to $\mathcal{SU}(X_M)$.
\end{prop}
\begin{proof}
According to Corollary \ref{kor:1} we have to show that for each $K$, we find that
$\mathfrak{l}_M^{[K]} = \mathfrak{su}^{[K]}(X_M)$, where
$\mathfrak{l}_M$ denotes the Lie algebra generated by the
$H_{\mathrm{IC},j}$ with $j\in\{1,\dots,2M\}$. Since $\mathfrak{l}_M \subset
\mathfrak{su}(X_M)$ is trivial, only the other inclusion has to be
shown. This will be done by induction. By Prop.~\ref{prop:7} the statement
is true for $M=1$. Now we assume it is true for $M$ to show that it is
true for $M{+}1$, too. To this end, consider for
each $k\in\{1,\dots,M{+}1\}$ the Hamiltonians $H_{\mathrm{IC},j}$,
$H_{\mathrm{IC},M+1+j}$ with $j\in\{1,\dots,M{+}1\}$ and $j\neq k$. They can be
regarded as operators on the Hilbert space $\mathcal{H}_M$ and they generate
a Lie algebra $\mathfrak{l}_M$ which satisfies by assumption
\begin{equation} \label{eq:49}
\mathfrak{l}_M^{[K]} = \mathfrak{su}^{[K]}(X_M) = \bigoplus_{\mu=1}^K
\mathfrak{su}(\mathcal{H}_M^{(\mu)})
\end{equation}
for all $K$. As operators on $\mathcal{H}_{M+1}$, they generate the
Lie algebra $\mathfrak{l}_M \hat{\otimes}_k \Bbb{1} \subset \mathfrak{l}_{M+1}$
and according to (\ref{eq:49}) one finds that
$\mathfrak{su}(\mathcal{H}_M^{(\mu)}) \hat{\otimes}_k \Bbb{1} \subset
\mathfrak{l}_{M+1}^{[K+1]}$ holds for all $\mu \leq K$
and
$k\in\{1,\dots,M{+}1\}$. Thus, we can apply Lemma~\ref{lem:6}
and $\mathfrak{su}(\mathcal{H}_{M+1}^{(\mu)})$ is contained
in the Lie algebra $\mathfrak{l}_{M+1}^{[K+1]}$ for all $\mu \leq K$. But since
$\mathfrak{l}_{M+1}^{(K)} \subset \mathfrak{su}^{[K]}(X_{M+1}) =
\mathfrak{su}(\mathcal{H}_{M+1}^{(K)})$, one even gets
$\mathfrak{su}^{[K]}(X_{M+1}) \subset \mathfrak{l}_{M+1}^{[K]}$, just as was
to be shown.
\end{proof}
\subsection{Many atoms under collective control}
\label{sec:many-atoms-coll-1}
As a last topic in this section, we provide proofs for Thms.~\ref{thm:4} and
\ref{thm:5}. To this end, recall the notation from
Sect.~\ref{sec:many-atoms-coll}. The Hilbert space is $\mathcal{H}_{\operatorname{sym}} =
\Bbb{C}^{M+1} \otimes \mathrm{L}^2(\Bbb{R})$ with basis
\begin{equation} \label{eq:23}
\ket{\mu;\nu} = \ket{\nu} \otimes \ket{\mu {-} \nu}\,\text{ where }\, \nu \in\{0, \dots,
d_\mu\}\, \text{ and }\, d_\mu= \min(\mu,M).
\end{equation}
The charge operator is again $X_M = S_3 \otimes \mathbbmss{1} + \mathbbmss{1} \otimes N$
from Eq.~\eqref{eq:47} but now as an operator on $\mathcal{H}_{\operatorname{sym}}$ with
domain $D_{\operatorname{sym}}$ defined in (\ref{eq:51}) and the $\mu$-eigenspaces
$\mathcal{H}_{\operatorname{sym}}^{(\mu)}$ become $\mathcal{H}_{\operatorname{sym}}^{(\mu)} = \operatorname{span}
\{\ket{\mu;\nu}\, | \, \nu \in \{0,\dots,d_\mu\}\}$; cf.\ Eq.~\eqref{eq:52}. The
control Hamiltonians are $H_{\mathrm{TC},j}$ with $j\in\{1,\dots,3\}$ defined in
\eqref{eq:31} and \eqref{eq:33}. In addition let us introduce the operators $Y_3, Y_\pm \in
\mathfrak{su}_\Bbb{C}(X_M)$ (which denotes again the complexification of
$\mathfrak{su}(X_M)$) given by
\begin{equation} \label{eq:43}
Y_3 \ket{\mu;\nu} = \nu \ket{\mu;\nu}, \, Y_+^{(\mu)} = \sum_{\nu=0}^{d_\mu-1}
\KB{\mu;\nu{+}1}{\mu;\nu}, \, Y_-^{(\mu)} = \sum_{\nu=1}^{d_\mu}
\KB{\mu;\nu{-}1}{\mu;\nu}.
\end{equation}
They are related to the $H_{\mathrm{TC},j}$ by
\begin{align}
H_{\mathrm{TC},1} &= Y_3 - (M/2)\, \Bbb{1},\, H_{\mathrm{TC},3} = X_M
- Y_3,\nonumber \\
H_{\mathrm{TC},+} &= S_+ \otimes a = f(X_M,Y_3) Y_+,\, H_{\mathrm{TC},-}=S_- \otimes a^* = Y_-
f(X_M,Y_3) \label{eq:53}
\end{align}
where $f$ is a function in two variables $x,y$ given by
\begin{equation} \label{eq:59}
f(x,y) = h_1(x,y) h_2(y) \sqrt{y},\, h_1(x,y) = \sqrt{x+1-y},\, h_2(y)
= \sqrt{M+1-y},
\end{equation}
and $f(X_M,Y_3)$ has to be understood in the sense of functional caculus (both
operators commute). As operators on $\mathcal{H}_{\operatorname{sym}}^{(\mu)}$ for fixed
$\mu$, the $Y_\pm$ satisfy
\begin{equation} \label{eq:54}
Y_+Y_- = \Bbb{1} - \kb{\mu,0},\, Y_-Y_+ = \Bbb{1} - \kb{\mu,d_\mu}
\end{equation}
and for any function $g(y)$ which is continuous on the spectrum of $Y_3$, one finds
\begin{equation} \label{eq:55}
Y_+ g(Y_3) = g(Y_3-\Bbb{1}) Y_+,\, Y_- g(Y_3) = g(Y_3+\Bbb{1}) Y_-.
\end{equation}
We are now prepared for the first lemma.
\begin{lem}\label{lem:4.13}
The operators $H_{\mathrm{TC},1}$, $H_{\mathrm{TC},+} = S_+ \otimes a$,
and $H_{\mathrm{TC},-} = S_- \otimes a^*$ satisfy the following commutation
relations (as operators on $\mathcal{H}^{(\mu)}$)
(i) $[Y_3^{n-1} H_{\mathrm{TC},+},H_{\mathrm{TC},-}] = (X_M - Y_3) Y_3^n + (N
\Bbb{1} - Y_3)Y_3^n - (X_M - Y_3)(N \Bbb{1} - Y_3) \sum_{k=0}^{n-1} \binom{n}{k} Y_3^k$
and
(ii) $[Y_3^{n+1},H_{\mathrm{TC},+}] = \sum_{k=0}^n \binom{n}{k} (-1)^{n-k} Y_3^k H_{\mathrm{TC},+}$.
\end{lem}
\begin{proof}
Using Eq.~\eqref{eq:53} to re-express $H_{\mathrm{TC},\pm}$ in terms of
$Y_\pm$, $Y_3$ and $X_N$, we get for the first commutator
\begin{equation} \label{eq:56}
[Y_3^{n-1} H_{\mathrm{TC},+},H_{\mathrm{TC},-}] = Y_3^{n-1}
f(X_M,Y_3)Y_+Y_f(X_M,Y_3) - Y_- f^2(X_M,Y_3)Y_3^{n-1}Y_+.
\end{equation}
It is easy to check that $f(X_M,Y_3) \ket{\mu;0} = 0$ holds. Together with
(\ref{eq:54}) this leads to
\begin{equation} \label{eq:57}
Y_3^{n-1}f(X_M,Y_3) Y_+Y_- = Y_3^{n-1}f(X_M,Y_3).
\end{equation}
With (\ref{eq:55}) we get on the other hand $Y_-f^2(X_M,Y_3)Y_+ =
f^2(X_M,Y_3+\Bbb{1})(Y_3+\Bbb{1})^{n-1} Y_-Y_+$. Now observe that
$h_1^2(X_M,Y_3+\Bbb{1}) h_2^2(Y_3+\Bbb{1}) \ket{\mu;d_\mu} = 0$ and use again
(\ref{eq:54}) to get
\begin{equation} \label{eq:58}
Y_-f^2(X_M,Y_3)Y_+ = f^2(X_M,Y_3+\Bbb{1})(Y_3+\Bbb{1})^{(n-1)}.
\end{equation}
Inserting (\ref{eq:57}) and (\ref{eq:58}) into (\ref{eq:56}) leads to
$[Y_3^{n-1} H_{\mathrm{TC},+},H_{\mathrm{TC},-}] = Y_3^{n-1}
f^2(X_M,Y_3)-f^2(X_M,Y_3+\Bbb{1}) (Y_3+\Bbb{1})^{n-1}$,
where we have used the fact that $f(X_M,Y_3)$ and $Y_3$ commute. Inserting
the definition of $f$ in (\ref{eq:59}) and expanding $(Y_3+\Bbb{1})^{n-1}$
leads to the first commutator.
The second commutator follows similarly from
$[Y_3^{n+1},H_{\mathrm{TC},+}] = Y_3^{n+1} f(X_M,Y_3) Y_+ -
f(X_M,Y_3)Y_+Y_3^{n+1}$
and applying (\ref{eq:55}) to commute $Y_+$ to the right.
\end{proof}
We are now ready to prove Thm.~\ref{thm:4}. The statement about the dynamical
group $\mathcal{G}$ as a subgroup of $\mathcal{U}(X_M)$ is an easy consequence
of the discussion in Sect.~\ref{sec:commuting-operators}. The second
statement in Thm.~\ref{thm:4} is rephrased in the following Proposition.
\begin{prop}
Consider the Lie algebra $\mathfrak{l}_{\mathrm{TC}} \subset
\mathfrak{u}(X_M)$ generated by the $H_{\mathrm{TC},j}$ with $j\in\{1,\dots,3\}$ and
$\mu \in \ensuremath{\mathbb N}$. The restriction
$\mathfrak{l}_{\mathrm{TC}}^{(\mu)}$ of $\mathfrak{l}_{\mathrm{TC}}$ to
$\mathcal{H}_{\operatorname{sym}}^{(\mu)}$ coincides with the Lie algebra
$\mathfrak{u}(\mathcal{H}_{\operatorname{sym}}^{(\mu)})$ of anti-hermitian operators on
$\mathcal{H}_{\operatorname{sym}}^{(\mu)}$.
\end{prop}
\begin{proof}
We will prove the corresponding statements for the complexifications:
$\mathfrak{l}_{\mathrm{TC},\Bbb{C}}= \mathfrak{l}_{\mathrm{TC}} \oplus i
\mathfrak{l}_{\mathrm{TC}} = \mathcal{B}(\mathcal{H}_{\operatorname{sym}}^{(\mu)})$. The
proposition then follows from taking only anti-hermitian operators on both
sides. Now note that $H_{\mathrm{TC},\pm} \in
\mathfrak{l}_{\mathrm{TC},\Bbb{C}}$ since we can express them as linear
combinations of $H_{\mathrm{TC},2}$ with the commutator of
$H_{\mathrm{TC},1}$ and $H_{\mathrm{TC},2}$. Furthermore, $X_M$ act as $\mu
\Bbb{1}$ on $\mathcal{H}_{\operatorname{sym}}^{(\mu)}$. Hence, Eq.~\eqref{eq:53} shows
that the restriction $\mathfrak{l}_{\mathrm{TC},\Bbb{C}}^{(\mu)}$ is
generated by $\Bbb{1}$, $Y_3$ and $H_{\mathrm{TC},\pm}$ considered as
operators on $\mathcal{H}_{\operatorname{sym}}^{(\mu)}$. Note that all operators in this
proof are operators on $\mathcal{H}_{\operatorname{sym}}^{(\mu)}$, and therefore we
simplify the notation by dropping \emph{temporarily} the superscript $\mu$,
when operators are concerned.
The first step is to show that $Y_3^k, Y_3^jH_{\mathrm{TC},\pm} \in
\mathfrak{l}_{\mathrm{TC},\Bbb{C}}^{(\mu)}$ holds for all $k, j \in \ensuremath{\mathbb N}_0$. This
is done by induction. The statement is true for $k\in\{0,1\}$ and $j=0$. Now
assume it holds for all $k\in\{0,\dots,n\}$ and $j\in\{1,\dots,n{-}1\}$. Lemma~\ref{lem:4.13}(i)
shows that the commutator $[Y_3^{n-1} H_{\mathrm{TC},+},H_{\mathrm{TC},-}]$
is a polynomial in $Y_3$ with $-(n+2) Y_3^{n+1}$ as leading term. Since
$Y_3^j \in \mathfrak{l}_{\mathrm{TC},\Bbb{C}}^{(\mu)}$ for $j\in\{0,\dots,n\}$ we can
subtract all lower order terms and get $Y_3^{n+1} \in
\mathfrak{l}_{\mathrm{TC}}^{(\mu)}$. To handle $Y_3^nH_{\mathrm{TC},\pm}$ we
use Lemma~\ref{lem:4.13}(ii). The commutator $[Y_3^{n+1},H_{\mathrm{TC},+}]$ is of the
form $P(Y_3) H_{\mathrm{TC},+}$ with an $n^{\mathrm{th}}$-order polynomial $P$. Since
$Y_3^k H_{\mathrm{TC},+} \in \mathfrak{l}_{\mathrm{TC},\Bbb{C}}^{(\mu)}$, we can
subtract all terms of order $k<n$ and conclude that $Y_3^n H_{\mathrm{TC},+}
\in \mathfrak{l}_{\mathrm{TC},\Bbb{C}}^{(\mu)}$.
Now consider a polynomial $P$ with $P(\nu) = 0$ for $\nu \neq \kappa$ and
$P(\kappa)=1$ with $\nu,\kappa \in\{ 0, \dots, d_\mu\}$. Since all $Y_3^n$ are
in $\mathfrak{l}_{\mathrm{TC},\Bbb{C}}^{(\mu)}$, we get $\kb{\mu;\kappa}
= P(Y_3) \in \mathfrak{l}_{\mathrm{TC},\Bbb{C}}^{(\mu)}$. Applying the same argument
to $Y_3^n H_{\mathrm{TC},\pm}$, we also get
$\KB{\mu;\kappa}{\mu;\kappa\pm 1} \in \mathfrak{l}_{\mathrm{TC},\Bbb{C}}^{(\mu)}$
and the general case $\KB{\mu;\nu}{\mu;\lambda}$ with $\mu\neq\lambda$ can be
treated with repeated commutators of $\KB{\mu;\kappa}{\mu;\kappa\pm 1}$ for
different values of $\kappa$.
\end{proof}
This proposition says that the control system with Hamiltonians
$H_{\mathrm{TC},j}$ with $j\in\{1,2,3\}$ can generate any special unitary
$U^{(\mu_0)}$ on $\mathcal{H}_{\operatorname{sym}}^{(\mu_0)}$ for any $\mu_0$. However,
some calculations using computer algebra, we have done for the case $M=2$
indicate that we cannot exhaust all of $\mathcal{SU}(X_M)$. In other words:
After $U^{(\mu_0)}$ is fixed, we loose the possibility to choose an
\emph{arbitrary} $U^{(\mu)} \in \mathcal{SU}(\mathcal{H}_{\operatorname{sym}}^{(\mu)})$ for
another $\mu$. Our analysis for two atoms suggests that the Lie algebra
generated by the $H_{\mathrm{TC},j}$ is almost as big as
$\mathfrak{su}(X_2)$, but does not contain operators of the form $A \otimes
\Bbb{1}$ with a diagonal traceless operator $A$ (except
$H_{\mathrm{TC},1}$). This observation suggests the choice of the Hamiltonians
$H_{\mathrm{CC},k}$ with $k\in\{1,\dots,M{+}1\}$ in Eq.~\eqref{eq:30}, which lead to a
dynamical group exhausting $\mathcal{SU}(X_M)$. This is shown in the next
proposition, which completes the proof of Thm.~\ref{thm:5}.
\begin{prop}
The dynamical group generated by $H_{\mathrm{CC},k}$ with $k\in\{1,\dots,M{+}1\}$
coincides with $\mathcal{SU}(X_M)$.
\end{prop}
\begin{proof}
Let us introduce the operators $\kappa(k,j) \in \mathfrak{u}_\Bbb{C}(X_M)$
(the complexification of $\mathfrak{u}(X_M)$) given by $\kappa(k,j)^{(\mu)}
=\KB{\mu;k}{\mu;j}$ with $k,j\in\{0,\dots,M\}$ and $\kappa(k,j) = 0$ if $k \geq
d_\mu$ and $j \leq d_\mu$, where $d_\mu = \min(\mu,M{+}1)$;
cf.\ Eq.~\eqref{eq:23}. We can re-express $Y_\pm$ in terms of $\kappa(k,j)$ as
$Y_+ = \sum_{k=0}^{M{-}1} \kappa(k{+}1,k)$, $Y_-=\sum_{k=1}^M \kappa(k{-}1,k)$.
Compare this to the definition of $Y_\pm$ in (\ref{eq:43}). The truncation
of the sums occuring for $\mu < M$ is now built into the definition of the
$\kappa(k,j)$. Similarly we can write the $H_{\mathrm{CC},j}$ for $j\in\{1,\dots,M\}$
as $H_{\mathrm{CC},j} = \kappa(k,k) - \kappa(k{-}1,k{-}1)$. The $\kappa(k,j)$
are particularly useful because their commutator has the following simple
form: $[\kappa(k,j),\kappa(p,q)] = \delta_{jp}\kappa(k,q) - \delta_{kq}
\kappa(p,j)$. Note that all truncations for small $\mu$ are automatically
respected. This can be used to calculate the commutator of
$H_{\mathrm{CC},k}$ and $Y_\pm$. To this end we introduce the $M \times M$
matrix $(A_{jk})$ with $A_{jj}=2$, $A_{j,k}=-1$ if $|j-k|=1$ and $A_{jk}=0$
otherwise. Using $(A_{jk})$ we can write $[H_{\mathrm{CC},j},Y_+] = \sum_k A_{jk}
\kappa(k,k{-}1)$. The matrix $(A_{jk})$ is tridiagonal, and therefore its
determinant can be easily calculated and it equals $M{+}1$. Hence $(A_{jk})$ is
invertible, and we can express $\kappa(j,j{-}1)$ for $j\in\{1,\dots,M\}$ as linear
combination of the commutators $[H_{\mathrm{CC},k},Y_+]$.
Now consider the Lie algebra $\mathfrak{l}_{\mathrm{CC}}$ generated by
$H_{\mathrm{CC},k}$ with $k\in\{1,\dots,M{+}1\}$ and its complexification
$\mathfrak{l}_{\mathrm{CC},\Bbb{C}}$. We have $H_{\mathrm{TC},1} \in
\mathfrak{l}_{\mathrm{CC}}$ since it can be written as a linear combination
of the $H_{\mathrm{CC},j}$. In addition $H_{\mathrm{TC},3} =
H_{\mathrm{CC},M+1} \in \mathfrak{l}_{\mathrm{CC}}$ and since $S_+ \otimes
a$, $S_-\otimes a^*$ can be written as (complex) linear combinations of
$H_{\mathrm{TC},3}$ and its commutator with $H_{\mathrm{TC},1}$ we get $S_+
\otimes a, S_-\otimes a^* \in \mathfrak{l}_{\mathrm{CC},\Bbb{C}}$. To
calculate the commutators $[H_{\mathrm{CC},j},S_+\otimes a]$ note that
according to (\ref{eq:43}) we have $S_+ \otimes a = f(X_M,Y_3) Y_+$ and
$f(X_M,Y_+)$ commutes with $H_{\mathrm{CC},k}$. Hence
$[H_{\mathrm{CC},j},S_+\otimes a] = [H_{\mathrm{CC},j}, f(X_M, Y_3) Y_+]
= f(X_M, Y_3) [H_{\mathrm{CC},j},Y_+] = \sum_k A_{jk} f(X_M, Y_3) \kappa(k,k{-}1)$.
Using the reasoning from the last paragraph, we see that $f(X_M,Y_3)
\kappa(k,k{-}1) \in \mathfrak{l}_{\mathrm{CC}}$. Similarly we can show by
using commutators with $S_- \otimes a^*$ that all $\kappa(k,k{+}1) f(X_M,Y_3)$
are in $\mathfrak{l}_{\mathrm{CC}}$, too. By expanding the function $f$ we
see in this way that for $k\in\{1,...,M\}$ the operators
\begin{equation}\label{eq:64}
A_+ = P(k) \, \kappa(k,k{-}1),\, A_- =
P(k) \, \kappa(k{-}1,k),\,
A_3 = \kappa(k,k) - \kappa(k{-}1,k{-}1)
\end{equation}
with $P(k):= \sqrt{X_M +(1{-}k) \Bbb{1}} $ are elements of $\mathfrak{l}_{\mathrm{CC},\Bbb{C}}$.
To conclude the proof, we apply again Corollary \ref{kor:1}. Hence we have to
consider the truncated algebra $\mathfrak{l}_{\mathrm{CC}}^{[K]}$. To this
end, look at the subalgebra $\mathfrak{l}_{\mathrm{CC},k}$ of
$\mathfrak{l}_{\mathrm{CC}}$ generated by the operators in
(\ref{eq:64}). They are acting on the subspace generated by basis vectors
$\ket{\mu;k}$, $\ket{\mu;k{-}1}$ and if we write $A_1 = A_+ + A_-$, $A_2 =
i(A_+ - A_-)$ we get (up to an additive shift in the operator $X_M$) the
same structure already analyzed in Lemma \ref{lem:2} (cf.\ also the operators
$A_{\alpha,k}$ in Eq.~\eqref{eq:68}). Hence we can apply the method from
Sect.~\ref{sec:one-atom-1} to see that for all $\mu \in\{ 0, \dots, K\}$ the
operators $\kb{\mu;k} - \kb{\mu,k{-}1}$, $\KB{\mu;k}{\mu;k{-}1}$ and
$\KB{\mu;k{-}1}{\mu;k}$ are elements of
$\mathfrak{l}_{\mathrm{CC},\Bbb{C}}^{[K]}$ (provided $k \leq d_\mu$). Now we
can generate all operators $\KB{\mu;p}{\mu,j}$ with $p,j \leq d_\mu$ by
repeated commutators of $\KB{k}{k{-}1}$ and $\KB{k{-}1}{k}$ for different values
of $k$. This shows that $\mathfrak{su}_{\Bbb{C}}(\mathcal{H}_{\operatorname{sym}}^{(\mu)})
\subset \mathfrak{l}_{\mathrm{CC},\Bbb{C}}^{[K]}$ for all $\mu \leq K$. By
passing to anti-selfadjoint elements we conclude that
$\mathfrak{l}_{\mathrm{CC}}^{[K]} = \mathfrak{su}(X_M)^{[K]}$ holds for all
$K$. Hence the statement follows from Corollary \ref{kor:1}.
\end{proof}
\section{Strong controllability}
\label{sec:strong-contr}
The purpose of this section is to show how one can complement the block-diagonal
dynamical groups from the last section to get strong controllability.
We add one generator which
breaks the abelian symmetry of the block-diagonal decomposition.
The proofs for pure-state controllability and strong controllability are given in Proposition~\ref{prop:9}
and Proposition~\ref{prop:10}, respectively.
This completes the proof of Theorems \ref{thm:1}, \ref{thm:7} and \ref{thm:2}.
\subsection{Pure-state controllability}
Consider a family $H_1, \dots, H_n$ of control Hamiltonians on the Hilbert
space $\mathcal{H}$ with joint domain $D \subset \mathcal{H}$ admitting a
$\mathrm{U}(1)$-symmetry defined by a charge operator $X$ with the same
domain. Since all the subspaces $\mathcal{H}^{(\mu)}$ are invariant under all
time evolutions, which can be constructed from the $H_k$, pure-state controllability
cannot be achieved. For rectifying this problem, we have to
add a Hamiltonian that breaks this symmetry in a specific way. We will do
so by using complementary operators as in Definition
\ref{def:1}. Hence in addition to the projections $X^{(\mu)}$, $\mu \in
\ensuremath{\mathbb N}_0$ we have the mutually orthogonal projections $E_\alpha$,
$\alpha\in\{+,0,-\}$ introduced in Sect.~\ref{sec:atoms-cavity} and the
corresponding derived structures. This includes in particular the
subprojections $X^{(\mu)}_\alpha \leq X^{(\mu)}$, $\mu \in \ensuremath{\mathbb N}_0$ and the
Hilbert spaces $\mathcal{H}^{(\mu)}_\alpha$ onto which they project. Recall,
that they satisfy $X^{(\mu)}_\alpha = E_\alpha X^{(\mu)}$ and $X^{(\mu)} =
X^{(\mu)}_- \oplus X^{(\mu)}_0 \oplus X^{(\mu)}_+$, and that for $\mu > 0$ the
$X^{(\mu)}_\pm$ are required to be non-zero. For the following discussion we
need in addition the Hilbert spaces $\mathcal{H}_{[K]} = \mathcal{H}^{[K]}
\oplus \mathcal{H}^{(K+1)}_-$, the projections $F_{[K]}$ onto them and the
group $\mathcal{SU}(X,F_{[K]})$ of $U \in \mathcal{SU}(X)$ commuting with
$F_{[K]}$. Furthermore we will indicate restrictions to the subspaces
$\mathcal{H}_{[K]}$ by a subscript $[K]$, e.g. $\mathcal{SU}_{[K]}(X,F_{[K]})$
denotes the corresponding restriction of $\mathcal{SU}(X,F_{[K]})$ which has the
form $\mathcal{SU}_{[K]}(X,F_{[K]}) = \mathcal{SU}^{[K]}(X) \oplus
\mathcal{SU}(X^{(K+1)}_-)$. Now one can prove the following lemma, which will be of
importance in the subsequent subsections.
\begin{lem} \label{lem:7}
Consider a strongly continuous representation $\pi: \mathrm{U}(1) \to
\mathcal{U}(\mathcal{H})$ with charge operator $X$, an operator $H$
complementary to $X$, and the objects just introduced. For
all $K \in \ensuremath{\mathbb N}$, introduce the Lie group $\mathcal{G}_{X,F,K}$ generated by
$\mathcal{SU}_{[K]}(X,F_{[K]})$, $\exp(i t H)$, $t\in \Bbb{R}$ and global
phases $\exp(i \alpha) \Bbb{1}$, $\alpha \in [0,2\pi)$. Then the group
$\mathcal{G}_{X,F,K}$ acts transitively on the unit sphere of
$\mathcal{H}_{[K]}$.
\end{lem}
\begin{proof}
Consider $\phi \in \mathcal{H}_{[K]}$ and choose $\tilde{U}_1 \in
\mathcal{SU}_{[K]}(X,F_{[K]})$ such that $X_+^{(\mu)} \tilde{U}_1 \phi =
0$ for all $\mu > 0$. This is possible, since
$\mathcal{SU}(\mathcal{H}^{(\mu)})$ acts transitively (up to a phase) on the
unit vectors of $\mathcal{H}^{(\mu)} = \mathcal{H}_-^{(\mu)} \oplus
\mathcal{H}_0^{(\mu)} \oplus \mathcal{H}^{(\mu)}_+$.
According to item (\ref{item:1}) of Def. \ref{def:1} we can find $t \in
\Bbb{R}$ (e.g. $t=\pi/2$ will do) such that $\exp(i t H)
\mathcal{H}_+^{(K+1)} = \mathcal{H}_-^{(K)}$ holds. Hence $\exp(i t H) \phi
\in \mathcal{H}^{[K]}$ and we can find a $\tilde{U}_2 \in
\mathcal{SU}_{[K]}(X,F_{[K]})$ with $\phi_1 = \tilde{U}_2 \exp(i t H)
\tilde{U}_1 \phi \in \mathcal{H}_{[K-1]}$. Applying this procedure $K$
times we get $\phi_K = U_K \cdots U_1 \phi \in \mathcal{H}_{[0]}$ with $U_j
\in \mathcal{G}_{X,F,k}$. Similarly we can find $V_1, \dots, V_K \in
\mathcal{G}_{X,F,k}$ with $\psi_K = V_k \cdots V_1 \psi \in
\mathcal{H}_{[0]}$.
Now note that the group $\mathcal{G}_{X,F,0}$ can be regarded as a subgroup
of $\mathcal{G}_{X,F,k}$ (which acts trivially on the orthocomplement of
$\mathcal{H}_{[0]}$ in $\mathcal{H}_{[K]}$). Hence, the statement of the
lemma follows from the fact that, due to condition (\ref{item:2}) of
Def. \ref{def:1}, the group $G_{X,F,0}$ acts transitively on the unit
vectors in $\mathcal{K}_{[0]} = F_{[0]} \mathcal{H}$.
\end{proof}
The first easy consequence of this lemma is the following result which is a
proof of Thm. \ref{thm:10} which we restate here as a proposition.
\begin{prop} \label{prop:9}
Consider a strongly continuous representation $\pi: \mathrm{U}(1) \to
\mathcal{U}(\mathcal{H})$ with charge operator $X$ and a family of
selfadjoint operators $H_1, \dots, H_d$ on $\mathcal{H}$. Assume that the
following conditions hold:
\begin{enumerate}
\item
All eigenvalues $\mu$ of $X$ are greater than or equal to $0$.
\item
$H_1, \dots, H_{d-1}$ commute with $X$.
\item
The dynamical group generated by $H_1, \dots, H_{d-1}$ contains
$\mathcal{SU}(X)$.
\item
The operator $H_d$ is complementary to $X$.
\end{enumerate}
Then the system (\ref{eq:1}) with Hamiltonians $H_0 =
\Bbb{1}, H_1, \dots, H_d$ is pure-state controllable.
\end{prop}
\begin{proof}
We have to show that for each pair of pure
states $\psi, \phi \in \mathcal{H}$ and each $\epsilon>0$ there is a finite
sequence $U_k \in \mathcal{U}(\mathcal{H})$ with $k \in\{1, \dots, N\}$ and either $U_k
\in \mathcal{SU}(X)$, $U_k=\exp(i t H_d)$, or $U_k = \exp(i \alpha)
\Bbb{1}$ such that $\| \psi - U_N \cdots U_1 \phi \| < \epsilon$. To this
end, first note that we can find $K \in \ensuremath{\mathbb N}$ such that $\| \psi - F_{[K]}
\psi\| < \epsilon/3$ and $\|\phi - F_{[K]} \psi\| < \epsilon/3$, where
$F_{[K]}$ is the projection defined in the first paragraph of this
subsection. Therefore
$
\| \psi - U_N \cdots U_1 \phi \| \leq \| \psi - F_{[K]} \psi\| + \|
F_{[K]} \psi - U_N \cdots U_1 F_{[K]} \phi \| + \| U_N \cdots U_1 F_{[K]}
\phi - U_N \cdots U_1 \phi \| < \epsilon
$
provided $\| F_{[K]} \psi - U_N \cdots U_1 F_{[K]} \phi \| <
\epsilon/3$. Hence we can assume that $\psi, \phi \in \mathcal{H}_{[K]}$ and
apply Lemma \ref{lem:7}. This leads to a sequence $V_1 ,\dots, V_N \in
\mathcal{G}_{X,F,K}$ with $V_N \cdots V_1 \phi = \psi$. Now note that the
dynamical group $\mathcal{G}$ generated by $H_0, \dots, H_{d}$ contains by
assumption the group $\mathcal{SU}(X)$, the unitaries $\exp(i t H_{d})$
and the global phases $\exp(i \alpha) \Bbb{1}$. Hence with the definition of
$\mathcal{G}_{X,F,K}$, we get for $j\in\{1,\dots,N\}$ a $W_j \in \mathcal{G}$ with
$[W_j,F_{[K]}] = 0$ and $F_{[K]} W_j = V_j$, and therefore $\psi = W_N \cdots
W_1 \phi$. But by definition the dynamical group is the strong closure of
monomials $U_N \cdots U_1$ with $U_j = \exp(i t_j H_{k_j})$ for some $t_j \in
\Bbb{R}$ and $k_j \in\{ 0, \dots, d+1\}$. In other words for all $U \in
\mathcal{G}$, $\xi \in \mathcal{H}$ and $\epsilon >0$ we can find such a
monomial satisfying $\| U_N \cdots U_1 \xi -U \xi\| < \epsilon$. Applying
this statement to the operators $W_j$ and the vectors $W_{j-1} \cdots W_1
\phi$ concludes the proof.
\end{proof}
This proposition can be applied to all systems studied in
Sect. \ref{sec:atoms-cavity}. Therefore, {\em they are all pure-state
controllable}. However, as already stated, one can even prove strong
controllability, which is the next goal.
\subsection{Approximating unitaries}
\label{sec:appr-unit}
Lemma \ref{lem:7} shows that the group $\mathcal{G}_{X,F,K}$ acts transitively
on the pure states in the Hilbert space $\mathcal{H}_{[K]}$. This implies
that there are only two possibilities for this group: either
$\mathcal{G}_{X,F,K}$ coincides with group of symplectic unitaries on
$\mathcal{H}_{[K]}$ (which is only possible if the dimension of
$\mathcal{H}_{[K]}$ is even), or it is the whole unitary group
\cite{SchiSoLea02,SchiSoLea02b,AA03}. At the same time we have seen in
Prop. \ref{prop:9} that (under appropriate
conditions on the control Hamiltonians) each $U \in \mathcal{G}_{X,F,K}$
admits an element $W$ in the dynamical group satisfying $W \xi = U \xi$ for
all $\xi \in \mathcal{H}_{[K]}$. Proving full controllability can therefore be
reduced to two steps:
\begin{enumerate}
\item
Find arguments that for an infinite number of $K \in \ensuremath{\mathbb N}$, the group
$\mathcal{G}_{X,F,K}$ cannot be unitary symplectic, such that it has to
coincide with the full unitary group on $\mathcal{H}_{[K]}$.
\item \label{item:3}
Show that each unitary $U \in \mathcal{U}(\mathcal{H})$ can be approximated
by a sequence $W_K$, $K \in \ensuremath{\mathbb N}$ of unitaries of the form $W_K = U_k
\oplus V_k$, where $U_k \in \mathcal{U}(\mathcal{H}_{[K]})$ can be chosen
arbitrarily, while $V_K$ is a unitary on $(\Bbb{1} - F_{[K]}) \mathcal{H}$
which is (at least partly) fixed by the choice of $U_k$.
\end{enumerate}
The purpose of this subsection is to prove the second statement, while the
first one is postponed to Section~\ref{sec:strong-contr-2}. We start with the
following lemma:
\begin{lem} \label{lem:8}
Consider a sequence $F_{[K]}$, $K \in \ensuremath{\mathbb N}$ of finite-rank projections
converging strongly to $\Bbb{1}$ and satisfying $F_{[K]} \lneq
F_{[K+1]}$. For each unitary $U \in \mathcal{U}(\mathcal{H})$ there is a
sequence $U_{[K]}$, $K \in \ensuremath{\mathbb N}$ of partial isometries, which converges
strongly to $U$ and satisfies $U_{[K]}^*U_{[K]} = U_{[U]} U_{[K]}^* =
F_{[K]}$; i.e.\ $F_{[K]}$ is the source and the target projection of $U_{[K]}$.
\end{lem}
\begin{proof}
Let us start by introducing the space $D \subset \mathcal{H}$ of vectors
$\xi \in \mathcal{H}$ satisfying $F_{[K]} \xi = \xi$ for a $K \in
\ensuremath{\mathbb N}$. It is a dense subset of $\mathcal{H}$ and we can define the map
$m: D \rightarrow \ensuremath{\mathbb N}$, $m(\xi) = \min \{ K \in \ensuremath{\mathbb N}\,|\, F_{[K]} \xi
= \xi \}$. All operators in this proof are elements of the unit ball
$\mathcal{B}_1(\mathcal{H}) = \{ A \in \mathcal{B}(\mathcal{H})\, | \, \|A\|
\leq 1\}$ in $\mathcal{B}(\mathcal{H})$. A sequence $A_K$ of elements of
$\mathcal{B}_1(\mathcal{H})$ converges to $A \in
\mathcal{B}_1(\mathcal{H})$ iff $\lim_{K\rightarrow \infty} A_K \xi = A \xi$
holds for all $\xi \in D$; \cite[I.3.1.2]{blackadar06}.
Now define $A_{[K]} = F_{[K]} U F_{[K]}$. For $\xi \in D$, we have $U F_{[K]}
\xi = U \xi$ if $K > m(\xi)$ and $\lim_{K \rightarrow \infty} F_{[K]} U \xi
= U \xi$ since $F_{[K]}$ converges strongly to $\Bbb{1}$. Hence the strong
limit of the $A_{[K]}$ is $U$, similarly one can show that the strong
limit of $A_{[K]}^*$ is $U^*$.
The $A_{[K]}$ are not partial isometries. We will rectify this problem by
looking at the polar decomposition. To this end, first consider
$|A_{[K]}|^2 = A_{[K]}^* A_{[K]}$ and
$
\| A_{[K]}^* A_{[K]} \xi - \xi \| = \| A_{[K]}^* A_{[K]} \xi - U^* U \xi
\| \leq \| A_{[K]}^* (A_{[K]} -U) \xi\| + \| (A_{[K]}^* - U^*) U\xi\|
\leq \|(A_{[K]} - U) \xi\| + \|(A_{[K]}^*-U^*) U\xi\|
$
where we have used that $\|A_{[K]}^*\|\leq 1$ holds. Strong convergence of
$A_{[K]}$ and $A_{[K]}^*$ implies $\lim_{K\rightarrow\infty} \| A_{[K]}^*
A_{[K]} \xi - \xi \| =0$. Hence $|A_{[K]}|^2$ converges strongly to
$\Bbb{1}$.
The operators $A_{[K]}$ are of finite rank with support and range contained
in $\mathcal{H}_{[K]} = F_{[K]} \mathcal{H}$. Hence the $|A_{[K]}|$ have pure point
spectrum and their spectral decomposition is $\sum_{\lambda \in
\sigma(|A_{[K]}|)} \lambda P_\lambda$ with eigenvalues $0 \leq \lambda
\leq 1$ and spectral projections $P_\lambda$ satisfying $P_\lambda \leq
F_{[K]}$ for $\lambda > 0$. Using the fact that the $P_\lambda$ are
mutually orthogonal, we get for $|A_{[K]}|^2$:
$
\| |A_{[K]}|^2 \xi - \xi\| = \| \sum_{\lambda \in \sigma(|A_{[K]}|)}
(\lambda^2 -1) P_\lambda \phi\| = \sum_{\lambda \in
\sigma(|A_{[K]}|)} |\lambda^2-1| \|P_\lambda \xi\|
= \sum_{\lambda \in \sigma(|A_{[K]}|)} |\lambda-1|(\lambda+1)
\|P_\lambda \xi\| \geq \sum_{\lambda \in \sigma(|A_{[K]}|)} |\lambda-1|
\|P_\lambda \xi\|
$.
Hence strong convergence of $|A_{[K]}|^2$ implies strong convergence of
$|A_{[K]}|$.
Now we can look at the polar decomposition $A_{[K]} = W_{[K]}
|A_{[K]}|$. The $W_{[K]}$ are partial isometries, and moreover, since support and range
of the $A_{[K]}$ are contained in $\mathcal{K}_{[K]}$, they satisfy
$W_{[K]}^* W_{[K]} \leq F_{[K]}$ and $W_{[K]} W_{[K]}^* \leq F_{[K]}$. In
other words, we can look upon the $W_{[K]}$ as partial isometries on the finite
dimensional Hilbert space $\mathcal{H}_{[K]}$. As such we can extend them
to untaries $U_{[K]} \in \mathcal{U}(\mathcal{H}_{[K]})$ without sacrificing
the relation to $A_{[K]}$, i.e.\ $A_{[K]} = U_{[K]} |A_{[K]}|$. As operators
on $\mathcal{H}$, the $U_{[K]}$ are still partial isometries, but now with
source and target projection equal to $F_{[K]}$ as stated in the lemma.
The only remaining point is to show that the $U_{[K]}$ converges strongly to
$U$. This follows from
$
\| U_{[K]} \xi - U \xi\| \leq \| U_{[K]}\xi - A_{[K]}\xi\| + \|A_{[K]}\xi
- U\xi \|$
and
$ \|U_{[K]}\xi - A_{[K]}\xi\| = \| U_{[K]} (\Bbb{1} - |A_{[K]}|)\xi\|
$
since $A_{[K]}$ converges strongly to $U$ and $|A_{[K]}|$ to $\Bbb{1}$.
\end{proof}
Now we come back to the case discussed in the beginning of this subsection
under item (\ref{item:3}):
\begin{lem} \label{lem:10}
Consider $U$, $F_{[K]}$ and $U_{[K]}$ as in Lemma \ref{lem:8}, and an
additional sequence of partial isometries $V_{[K]}$, $K \in \ensuremath{\mathbb N}$ with
$V_{[K]}^*V_{[K]} = V_{[K]} V_{[K]}^* = \Bbb{1} - F_{[K]}$. The operators
$W_{[K]} = U_{[K]} + V_{[K]}$ are unitary, and if $U$ is the strong limit of
the $U_{[K]}$, the same is true for the $W_{[K]}$.
\end{lem}
\begin{proof}
The kernels of $U_{[K]}$ and $V_{[K]}$ are $(\Bbb{1} - F_{[K]}) \mathcal{H}$
and $\mathcal{H}_{[K]} = F_{[K]} \mathcal{H}$, respectively. These spaces are
complementary, and therefore $W_{[K]} = U_{[K]} + V_{[K]}$ is unitary for all
$K$. To show strong convergence, recall the space $D$ and the function
$D \ni \xi \mapsto m(\xi) \in \ensuremath{\mathbb N}$ introduced in the last proof. For
$\xi \in F$ we have $W_{[K]} \xi = U_{[K]} \xi$ if $K > m(\xi)$. Hence by
assumption $\lim_{K \rightarrow \infty} W_{[K]} \xi = \lim_{K \rightarrow
\infty} U_{[K]} \xi = U\xi$, which implies strong convergence of
$W_{[K]}$ to $U$.
\end{proof}
\subsection{Strong controllability}
\label{sec:strong-contr-2}
We are now prepared to prove Theorem~\ref{thm:7}. The first
step is the following lemma announced already at the beginning
of Subsection~\ref{sec:appr-unit}.
\begin{lem} \label{lem:9}
Consider the group $\mathcal{G}_{X,F,K}$ introduced in Lemma \ref{lem:7} and
assume that there is a $\mu \leq K$ with $d^{(\mu)} =
\dim(\mathcal{H}^{(\mu)}) > 2$. Then $\mathcal{G}_{X,F,K} =
\mathcal{U}(\mathcal{H}_{[K]})$.
\end{lem}
\begin{proof}
Consider the group $\mathcal{SG}_{X,F,K}$ consisting of elements of
$\mathcal{G}_{X,F,K}$ with determinant $1$. By Lemma \ref{lem:7} this group
acts transitively on the set of pure states of the Hilbert space
$\mathcal{H}_{[K]}$. Hence, there are only two possibilities
left\footnote{\label{fn:1} Note that $\mathcal{H}_{[K]}$ is a finite-dimensional Hilbert
space. Hence after fixing a basis $e_1, \dots, e_d$ it can be identified with
$\Bbb{C}^d$.}: $\mathcal{SG}_{X,F,K}$ coincides
either with the unitary symplectic group $\mathrm{USp}(\mathcal{H}_{[K]})$
or with the full unitary group $\mathcal{U}(\mathcal{H}_{[K]})$; cf.\
\cite{SchiSoLea02,SchiSoLea02b,AA03}. Assume $\mathcal{SG}_{X,F,K} =
\mathrm{USp}(\mathcal{K}_{[K]})$ holds. This would imply that
$\mathcal{SG}_{X,F,K}$ is self-conjugate (or more precisely the
representation given by the identity map on $\mathcal{SG}_{X,F,K} \subset
\mathcal{B}(\mathcal{H}_{[K]})$ is self-conjugate). In other words, there
would be a unitary $V \in \mathcal{U}(\mathcal{H}_{[K]})$ with $V U V^* =
\bar{U}$ for all $U \in \mathcal{SG}_{X,F,K}$. Here $\bar{U}$ denotes
complex conjugation in an arbitrary but fixed basis (cf.\ footnote~\ref{fn:1}).
Now consider $\mathcal{SU}(\mathcal{H}^{(\mu)})$ with $d^{(\mu)} > 2$. It
can be identified with $\mathrm{SU}(d)$ in its first fundamental
representation $\lambda_1$ (i.e.\ the ``defining'' representation). At the
same time it is a subgroup of $\mathcal{SG}_{X,F,K}$ (one which acts
nontrivially only on $\mathcal{H}^{(\mu)} \subset
\mathcal{H}_{[K]}$). Existence of a $V$ as in the last paragraph would imply
that $\lambda_1$ is unitarily equivalent to its conjugate representation,
which is the $d-1^{\mathrm{st}}$ fundamental representation. This is
impossible if $d^{(\mu)} > 2$ holds. Hence $V$ with the described properties
does not exist and $\mathcal{SG}_{X,F,k}$ has to coincide with
$\mathcal{SU}(\mathcal{H}_{[K]})$ and therefore $\mathcal{G}_{X,F,K} =
\mathcal{U}(\mathcal{H}_{[K]})$ as stated.
\end{proof}
Finally we can conclude the proof of Thm. \ref{thm:7} which we restate here as
the following proposition:
\begin{prop} \label{prop:10}
A control system (\ref{eq:1}) with control Hamiltonians
$H_0=\Bbb{1}, \dots, H_d$ satisfying the conditions from
Prop.~\ref{prop:9} is strongly controllable, if $d^{(\mu)} = \dim
\mathcal{H}^{(\mu)} > 2$ for at least one $\mu \in \ensuremath{\mathbb N}$.
\end{prop}
\begin{proof}
Consider an arbitrary unitary $U \in \mathcal{U}(\mathcal{H})$. By Lemma
\ref{lem:8}, there is a sequence of partial isometries $U_{[K]}$ converging
strongly to $U$, and by Lemma \ref{lem:9} we can assume that $U_{[K]} \in
\mathcal{G}_{X,F,K}$. Now considering the dynamical group $\mathcal{G}$ generated by
the $H_j$, define the subgroup $\mathcal{G}(F_{[K]})$ of $U \in
\mathcal{G}$ commuting with $F_{[K]}$, and the restriction
$\mathcal{G}_{[K]}$ of $\mathcal{G}(F_{[K]})$ to $\mathcal{H}_ {[K]}$. The assumptions
on the $H_j$ imply that $\mathcal{G}_{[K]} = \mathcal{G}_{X,F,K} =
\mathcal{U}(\mathcal{H}_{[K]})$. Hence there is a sequence $W_K$, $K \in
\ensuremath{\mathbb N}$ of unitaries with $W_{[K]} \in \mathcal{G}(F_{[K]}) \subset
\mathcal{G}$ and $F_{[K]} W_{[K]} = U_{[K]}$. Since $U_{[K]}$ converges to $U$
strongly, Lemma \ref{lem:10} implies that the strong limit of the $W_{[K]}$
is $U$, which was to show.
\end{proof}
This proposition shows strong controllability for all the systems studied in
Sect.~\ref{sec:atoms-cavity}. The only exception is one atom interacting with
one harmonic oscillator (Sect. \ref{sec:one-atom}). Here we have $d^{(\mu)} =
\dim \mathcal{H}^{(\mu)} \leq 2$ and we \emph{can} actually find a unitary $V$
with $V U V = \bar{U}$ for all $U \in
\mathcal{SU}_{[K]}(X_1,F_{[K]})$. However, the elements
$U$ of $\mathcal{SU}(X_1)$ are block diagonal where the blocks $U^{(\mu)} \in
\mathcal{SU}(\mathcal{H}^{(\mu)})$ can be chosen independently. This implies $V
\in \mathcal{SU}_{K]}(X_1,F_{[K]})$, which is incompatible with $V
H_{\mathrm{JC},4} V^* = - H_{\mathrm{JC},4}$ (cf.\ Eq.~\eqref{eq:25} for
the definition of $\mathcal{H}_{\mathrm{JC},4}$) which would be necessary for the
group $\mathcal{G}_{X_1,F,K}$ to be self-conjugate. Hence we can proceed as in
the proof of Prop.~\ref{prop:10} to prove Thm.~\ref{thm:1}.
\section{Conclusions and Outlook}
Many of the difficulties of quantum control theory in infinite dimensions
arise from the fact that, due to unbounded operators, the
group $\mathcal{U}(\mathcal{H})$ of all unitaries on an infinite-dimensional
separable Hilbert space $\mathcal{H}$ is in fact no Lie group as long as it is
equipped with the strong topology, which inevitably is the correct choice when studying
questions of quantum dynamics. Yet $\mathcal{U}(\mathcal{H})$ contains
\begin{table}[Ht!]
\centering
\caption{Controllability results for several 2-level atoms in a cavity as derived here.}
\begin{tabular}{l@{\hspace{2mm}}l@{\hspace{2mm}}ll}
\hline \\[-4.8mm]\hline\\[-4.8mm]
System & Control Hamiltonians & \multicolumn{2}{c}{------------ Controllability ------------}\\[-1mm]
& & \multicolumn{2}{l}{system algebra\ $\mathfrak{g}$, dynamic\ group $\mathcal{G}$}\\
\hline\\[-4.8mm]\hline\\[-4mm]
one atom & $H_{\mathrm{JC},j}$, $j=1,2$, Eq.~\eqref{eq:9} & $\mathfrak{g}=\mathfrak{su}(X_1)$,
$\mathcal{G}=\mathcal{SU}(X_1)$ &[Thm.~\ref{thm:3}] \\[-1mm]
& \multicolumn{3}{c}{\xhrulefill{black}{.5pt}}\\[-0mm]
& $H_{\mathrm{JC},j}$, $j=1,2$, Eq.~\eqref{eq:9} & strongly controllable$^{a}$ & \\
& $H_{\mathrm{JC},4}$, Eq.~\ref{eq:25} & \quad with $\mathcal{G}=\mathcal{U}(\mathcal{H})$ & [Thm.~\ref{thm:1}]\\
\hline\\[-5.5mm]\hline\\[-4mm]
$M$ atoms & $H_{\mathrm{IC},j}$, $j=1,\dots 2M$ &$\mathfrak{g}=\mathfrak{su}(X_M)$ and &\\
\multicolumn{2}{l}{\quad with individual controls of Eq.~\eqref{eq:44} } &
$\mathcal{G} = \mathcal{SU}(X_M)$ & [Thm.~\ref{thm:8}]\\[-1mm]
& \multicolumn{3}{c}{\xhrulefill{black}{.5pt}}\\[-0mm]
& $H_{\mathrm{IC},j}$, $j=1,\dots 2M+1$ & strongly controllable$^{a}$ &\\
\multicolumn{2}{l}{\quad with individual controls of Eqs.~(\ref{eq:44},\ref{eq:45})} & \quad with $\mathcal{G}=\mathcal{U}(\mathcal{H})$ & [Thm.~\ref{thm:7}]\\
\hline\\[-5.5mm]\hline\\[-3mm]
$M$ atoms & $H_{\mathrm{TC},j}$, $j=1,2,3$ & $\mathfrak{g} \subset \mathfrak{u}(X_M)$ and &\\
\multicolumn{2}{l}{\quad under collective control of Eq.~\eqref{eq:31}} & $\mathcal{G} \subset \mathcal{U}(X_M)$ & [Thm.~\ref{thm:4}]\\[-1mm]
& \multicolumn{3}{c}{\xhrulefill{black}{.5pt}}\\[-0mm]
& $H_{\mathrm{CC},j}$, $j=1,\dots,M+1$ & $\mathfrak{g}=\mathfrak{su}(X_M)$ and &\\
\multicolumn{2}{l}{\quad under collective control of Eq.~\eqref{eq:30}} & $\mathcal{G} = \mathcal{SU}(X_M)$ & [Thm.~\ref{thm:5}]\\[-1mm]
& \multicolumn{3}{c}{\xhrulefill{black}{.5pt}}\\[-0mm]
& $H_{\mathrm{CC},j}$, $j=1,\dots,M+2$ & strongly controllable$^{a}$ &\\
\multicolumn{2}{l}{\quad under collective control of Eq.~\eqref{eq:30}} & \quad with $\mathcal{G}=\mathcal{U}(\mathcal{H})$ & [Thm.~\ref{thm:2}]
\\
\hline\\[-4.8mm]\hline
\multicolumn{4}{l}{$^{a}$\footnotesize{Here in the strong topology, no system algebra or exponential map exists.}}
\end{tabular}
\label{tab:1}
\end{table}
a plethora of subgroups which are still infinite-dimensional while admitting a
proper Lie structure -- including in particular a Lie algebra~$\mathfrak{l}$ consisting of
unbounded operators and a well-defined exponential map. An important
example are those unitaries with an abelian $\mathrm{U}(1)$-symmetry,
which in the Jaynes-Cummings model relates to a kind of particle-number operator.
As shown here, this infinite-dimensional
system Lie algebra $\mathfrak{l}$ can be exploited for control theory
in infinite dimensions in close analogy to the finite-dimensional case.
Due to the in-born symmetry of $\mathfrak{l}$ and
the corresponding Lie group $\mathcal{G}$, full controllability cannot be achieved
that way. Yet we have also shown that this problem can readily be overcome by
complementary methods directly on the group level.
For several
2-level atoms interacting with one harmonic oscillator (e.g.,\ a cavity mode or a phonon mode),
these methods allowed us to extend previous results substantially, in particular
in two aspects also summarized in Table \ref{tab:1}:
(A) We have answered approximate control and convergence questions for
asymptotically vanishing control error. (B) Our results include not only
reachability of states, but also its operator lift, i.e.\ simulability of unitary gates.
To this end, we have introduced the notion of \emph{strong controllability}, and we
have shown that all systems under consideration require only a fairly small set of
control Hamiltonians for guaranteeing strong controllability, i.e.\ simulability.
--- Thus we anticipate the methods introduced here will
find wide application to systematically characterize experimental set-ups of
cavity QED and ion-traps in terms of pure-state controllability and simulability.
\section*{Acknowledgements}
\footnotesize{
This work was supported in part by the {\sc eu}
through the integrated programmes \mbox{{\sc q-essence}} and {\sc siqs},
and the {\sc eu-strep} {\sc coquit}, and moreover
by the Bavarian Excellence Network {\sc enb}
via the international doctorate programme of excellence
{\em Quantum Computing, Control, and Communication} ({\sc qccc}),
by {\em Deutsche Forschungsgemeinschaft} ({\sc dfg}) in the
collaborative research centre {\sc sfb}~631 as well as the international
research group {\sc for} 1482 through the grant {\sc schu}~1374/2-1.
}
\section*{References}
\providecommand{\newblock}{}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,275 |
{"url":"https:\/\/yeokhengmeng.com\/2020\/06\/radar-theory-and-teardown-of-aircraft-transponder-rt-459a\/","text":"# Yeo Kheng Meng\n\nMaker, Coder, Private Pilot, Retrocomputing Enthusiast\n\n## Radar Theory and Teardown of Aircraft Transponder RT-459A\n\n11 minutes read\n\nI recently got my hands on an old unserviceable aircraft transponder from my flying club. The tinkerer me was obviously very curious on what\u2019s inside this piece of equipment. What better way to find out then to do a teardown!\n\nAircraft Radio Corporation (ARC) RT-459A\n\nApproximate size: 16.5cm x 29cm x 4.5cm\n\n# Background info\n\nBefore going to the teardown, it is helpful to provide some information as I know not every reader has a background in aviation.\n\nAir traffic control (ATC) has to use to know the position of every aircraft in the area of its responsibility in order to manage or advice the air traffic. Without any special equipment, ATC has to rely on pilots to report their position. In fact, in some areas of the world, relying on pilot position reports is still the case!\n\nThere are many problems with this approach as pilots can sometimes report their position incorrectly or not even be in communication at the right frequency with the ATC facility in charge of that area.\n\n## Here is where radar come in\n\nRadar actually stands for Radio Detection and Ranging. I didn\u2019t know Radar is an acronym until a few years after I first encountered that term as a kid.\n\nChain Home radar deployed in Britain in 1940 to detect incoming German planes. Source\n\nRadar came into more widespread practical use during WW2 as a means to detect enemy aircraft in advance so the defenders have sufficient time to prepare and know the position to intercept the threat.\n\nSource\n\nThe basic concept is to basically emit radio waves and analyse the properties of the returned echo.\n\nBased on the direction of the emitted wave, speed of light and the time delay between the pulse and echo, radar can determine the azimuth and distance of the remote object with respect to the radar station. Extra information like size and speed can also be determined.\n\nThis approach is called Primary Surveillance Radar (PSR).\n\n## Limitations of PSR\n\n### 1. Range\n\nThe PSR has to emit very powerful radio waves in order for the wave to have sufficient energy to reach the distant object and bounce back to be source.\n\nThis is the equation I researched into when it comes into calculating power density of a transmitted wave from a directional antenna in a cone of solid angle.\n\n$P(R) = \\frac{G_tP_t}{4\\pi R^2}$\n\n\u2022 $P(R)$: Power Density at a distance R away from transmitter\n\u2022 $P_t$: Transmit Power\n\u2022 $G_t$: Antenna Gain\n\nSimplifying the above equation, we can therefore see the waves\u2019 power decays with respect to the inverse square law. $\\rho \\propto \\frac{1}{R^2}$\n\nHowever, that is just a one way transmission. The PSR requires that the radar waves be reflected off the target object back the source.\n\nThe amount of power that will be reflected by the distance object is given with this value\n\n$(RCS)\\frac{G_tP_t}{4\\pi R^2}$\n\n\u2022 RCS is short for Radar Cross Section in $m^2$\n\nThe returned wave will decay at $\\frac{1}{4\\pi R^2}$ due to isotropic spreading. The following equation is thus the power that will be received back at the source.\n\n$P(Reflected) = (RCS) \\frac{G_tP_t}{4\\pi R^2} * \\frac{1}{4\\pi R^2}$\n\n->\n\n$P(Reflected) = (RCS) \\frac{G_tP_t}{(4\\pi)^2 R^4}$\n\nThis shows that the signal decreases in power with the distance R to the target in the 4th power.\n\nSource: Principles of Avionics (4th Edition) and https:\/\/en.wikipedia.org\/wiki\/Radar_cross-section\n\nEmission of powerful radio waves makes the radar more expensive to build and it could be a health hazard to people nearby.\n\n### 2. Identity\n\nFor a controller manning a PSR station, every aircraft in range of the station will appear as just anonymous dots on the screen. There is no way to easily tell the dots apart. What planes are those? How do I know the plane I\u2019m talking to is associated to which dot?\n\nBBC Source\n\nIn this case, every dot has some form of identification, direction, callsign, groundspeed and altitude.\n\n### 3. Altitude\n\nMost civilian PSR do not have elevation information so they can only produce a 2D picture of radar returns in the area. Reading this Stackexchange post, the limitation is due to cost. Military radars have the elevation function as the defence budget has \u201cless of an issue\u201d with money.\n\nUnlike ground-based vehicles, aircraft operate in a 3D environment.\n\nIt is entirely possible for 2 dots to overlap safely if only because the planes are at different altitudes. Yet it is also possible that they ARE at the same altitude which pose a collision risk.\n\nCertain airspaces also have altitude restrictions, say at certain altitudes you cannot fly at certain speed or need special permission to enter or just plain disallowed altogether.\n\nATC therefore needs the altitude information.\n\n## Transponder to the rescue\n\nThe transponder (TRANSmitter-resPONDER) was invented to tackle the above 3 problems. Aviation authorities worldwide mandate planes have to be installed with a transponder in order to fly into certain airspaces. A radar system that uses transponders is called Secondary Surveillance Radar (SSR).\n\n### 1. Solving the range problem\n\nWhen the transponder receives a signal from a radar station at 1030Mhz, it will reply back at 1090Mhz after a 3us delay. The delay allows time for the transponder to process the signal.\n\nSource\n\nThe returning wave will now be generated by the aircraft instead of having powerful radio waves reflected back to the source.\n\n### 2. Solving the identity problem (Mode A)\n\nSince the transponder is sending a return signal back, let\u2019s put it to good use. A special code can be embedded in the return signal to identify the aircraft to the radar station.\n\nSource\n\nWhen the transponder receives a special Pulse signal (P1\/P3) of 2us length and 8us pulse separation, it will reply back a 4-digit octal identification code. This is called a squawk code.\n\nThe weaker P2 pulse is sent as a form of sidelobe suppression to enable the transponder to determine if the P1 pulse is a genuine signal or signal leakage. More information can be found here\n\nThis is called interrogation Mode A.\n\n### 3. Solving the altitude problem (Mode C)\n\nWhen the transponder receives a special signal of 21us pulse separation, it will reply back the altitude information.\n\nSource\n\nThis is called interrogation Mode C.\n\nThe radar station alternates between Mode A and Mode C interrogation modes at regular intervals.\n\n### Transponder replies\n\nRegardless of Mode A or Mode C, the transponder reply is always in the same format.\n\nSource\n\n\u2022 F1: Frame start bit to indicate transmission start\n\u2022 F2: Frame end bit to indicate transmission end\n\u2022 X: Unused bit\n\u2022 SPI (Special Purpose Identification): To indicate pressing of ident button\n\u2022 A\/B\/C\/D (1,2,4): Squawk code or altitude data\n\nLook up this table to see the what altitude\/squawk code corresponds to what values of A\/B\/C\/D (1,2,4).\n\nBasically the 12 transmission slots between F1 and F2 is used to transmit the 12 data bits in Gillham code which is similar to Gray Code.\n\n### Transponder failure\n\nHere is a video snippet of my solo flight when my transponder failed. It seems my aircraft is now invisible to Paya Lebar Approach which is the ATC facility in charge of that area. Paya Lebar Approach is manned by military controllers so I\u2019m unsure why they did not have PSR to identify me.\n\n## External inspection\n\nSo now with all the dry theoretical information let\u2019s come down to the hardware.\n\n### Front view and explanation\n\n\u2022 OFF - Turns transponder off\n\u2022 SBY - Transponder on standby. Usually set on the ground after engine start to give time for device to warmup.\n\u2022 ON - Only replies to Mode A interrogation. Should not be used in the air but I usually set to this mode on the ground.\n\u2022 ALT - Replies to both Mode A and C interrogation. Set here just before takeoff.\n\u2022 Reply Lamp: Indicate transponder is replying\n\u2022 Dim - Adjust brightness of reply lamp\n\u2022 ID - Push button to send an SPI signal which will cause a flash on the ATC screen. Only done when instructed by ATC if they have problems locating the plane.\n\nThis transponder is currently set to squawk 1200 which is the standard squawk code used by aircraft flying by Visual Flight Rules in Singapore, USA, etc.\n\n### Rear view and explanation\n\nAt the back of the unit, we have a 23-pin connector and a coaxial connector for connecting to an antenna outside the plane.\n\nGiven the age of this unit, usage and maintenance manuals are difficult to locate online. However I did manage to locate this image based on a similar unit.\n\nSource\n\nThe 23-pin connector is meant for connecting to an external altitude encoder. An external altimeter will measure the current pressure altitude, encodes the data into Gillham Code format and supply it to the transponder. The transponder will then relay this data during a Mode C interrogation.\n\n## Teardown\n\nSo now let\u2019s finally crack open the device and look inside! Given the lack of schematic documentation of such an old design and with my personal lack of experience on RF electronics, I can only make guesses of the components used and their functions.\n\n### First impressions\n\nEverything is based on through-hole components! Not a single surface-mount component which is indicative of a device designed at least many decades ago.\n\nI also noticed that there is no single processing core, everything is done by individual chips in stages.\n\nTo drill home further the age of the device, we can see the PCB traces are all designed by hand before the age of Electronic Design Automation (EDA) software in the early 1980s.\n\n### Antenna portion\n\nI suspect the components in the rectangular top section are low-pass filters. The red spring-looking inductors as well as the resistors and capacitors beside them. They help to filter out any unwanted higher frequencies above the 1030 Mhz from the radar station.\n\nI can\u2019t identify what the large cyclindrical component is although it\u2019s likely a likely a tube amplifier. It is connected directly to the coaxial output. The logo is that of General Electric. Googling the values \u201cC-2080B\u201d and \u201cARC 42255-0250\u201d don\u2019t turn up anything meaningful.\n\nThe 79-13 code suggests a 1979 Week 13 manufacturing date.\n\nOpening the square cover on the side produces this. It\u2019s likely a duplexer (T\/R switch) to isolate the transmitter and receiver portion.\n\nThere is a crystal of frequency 161.6670 Mhz. It seems like a very odd frequency so I did some multiplier calculations. Multiplying 161.667 by 6 gives an almost round number of 970.002\u202c Mhz. Googling 970 Mhz gave me this diagram.\n\nSource\n\nThe frequency in this diagram is 107.77 Mhz with 2 x3 multipliers. I\u2019m guessing the multipliers in this RT-459A is a x2 and x3 multiplier.\n\nWith the 970Mhz, the input 1030Mhz is shifted to 60 Mhz which is easier for the subsequent electronics to process. I\u2019m no RF person so this is as much as I understand it.\n\n### High voltage power supply\n\nThe first indication that high-voltage is being used is the 1600V written on those gigantic brown capacitors. They are at 0.033uF and 0.015uF. The yellow coil beside them should be the step-up transformer although googling \u201c41454f\u201d produces no meaningful results.\n\nAccording to the diagram, high voltage is used to power the transmission tube amplifier.\n\n### Trimpots\n\nI noticed many trimpots used. I measured the resistance while turning them and confirmed they are trimpots. Probably for post-manufacturing calibration or maintenance tuning.\n\n## Plenty of NAND chips\n\nI found many similar ICs on the board.\n\nNames like the following:\n\n\u2022 Motorola MC846P\n\u2022 Motorola MC830L\n\u2022 Fairchild F7400PC\n\u2022 TI SN15830N\n\nGoogling says they are long obsolete NAND gate chips.\n\nFollowing the PCB traces from these chips indicates that the pilot-facing squawk code switches and altitude encoder pins are connected to them. No wonder there are so many of them.\n\nBut as to why the NAND logic, I really have no idea. According to the comments I received, it\u2019s because NAND gates are the most universal of gates and can be used to built any number of gates and decide whether to use a particular input bit.\n\n# Conclusion\n\nWith my almost non-existent RF knowledge, I did my best effort to try to understand how this transponder works. Given the age of this equipment, there was hardly any user documentation, maintenance manual much less a schematic I can use to work on. The components are so old that even component markings are useless.\n\nNevertheless, writing this blog post especially the theory portion has increased my knowledge on how the aviation radar system works. Amazing to think people have thought of and implemented this so many decades ago.\n\nIt also shows the snail\u2019s pace in the improvement of aviation technology where equipment built so long ago still can function usefully today unlike our rapidly obsolete modern technology like computers and smartphones.\n\nMany thanks to Seletar Flying Club for giving me this old transponder to play with! Now if only I can do a teardown of the equipment behind those ultra-modern glass cockpits. It\u2019ll probably be more interesting but those are probably too expensive to just dissect and analyse.\n\nEdit (24 June 2020): I made some changes to this post due to new information suggested by others.\n\nOnline presentation I gave of this topic at Hackware.\n\n\u2022 None\ncomments powered by Disqus","date":"2020-10-25 16:42:30","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\": 0.41450369358062744, \"perplexity\": 2656.3266481800215}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"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-2020-45\/segments\/1603107889574.66\/warc\/CC-MAIN-20201025154704-20201025184704-00068.warc.gz\"}"} | null | null |
from annoying.functions import get_object_or_None
from django.contrib.auth.models import User
from django.http import HttpResponseRedirect
from django.shortcuts import get_object_or_404
class BlogMiddleware:
def process_request(self, request):
request.blog_user = None
host = request.META.get('HTTP_HOST', '')
host_s = host.replace('www.', '').split('.')
if host != 'snipt.net' and \
host != 'snipt.localhost' and \
host != 'local.snipt.net':
if len(host_s) > 2:
if host_s[1] == 'snipt':
blog_user = ''.join(host_s[:-2])
if '-' in blog_user:
request.blog_user = \
get_object_or_None(User,
username__iexact=blog_user)
if request.blog_user is None:
request.blog_user = \
get_object_or_404(User,
username__iexact=blog_user
.replace('-', '_'))
else:
request.blog_user = \
get_object_or_404(User, username__iexact=blog_user)
if request.blog_user is None:
pro_users = User.objects.filter(userprofile__is_pro=True)
for pro_user in pro_users:
if pro_user.profile.blog_domain:
if host in pro_user.profile.blog_domain.split(' '):
request.blog_user = pro_user
if host != \
pro_user.profile.get_primary_blog_domain():
return HttpResponseRedirect(
'http://' +
pro_user
.profile
.get_primary_blog_domain())
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,923 |
\section{Introduction}
\label{s_intro}
This paper is the second in the series
initiated with Ref.~\cite{Isacker15}, henceforth referred to as I.
The overall purpose of this series
is the study of nuclear shapes with a higher-rank discrete symmetry,
in particular of the tetrahedral or octahedral type,
in the framework of a variety of interacting boson models.
For the general context of this study we refer the reader to the introduction in I,
of which only the essential points and references are mentioned here.
The paper by Li and Dudek~\cite{Li94}
pointed out the possibility of intrinsic nuclear shapes with a higher-rank discrete symmetry.
Subsequent publications by the Strasbourg group and their collaborators~\cite{Dudek02,Dudek03,Dudek06,Rouvel14}
showed the possible occurrence of tetrahedral, octahedral and icosahedral symmetries
through combinations of deformations of specific multipolarity.
In parallel with these theoretical developments,
a search was initiated for experimental manifestations
of such symmetries in nuclei~\cite{Curien10,Curien11,Jentschel10}.
In addition to these experimental searches cited in I,
a more recent study of this type appeared in 2018~\cite{Dudek18},
formulating criteria for the identification of tetrahedral and/or octahedral symmetries in nuclei.
From the theory side most work related to higher-rank discrete symmetries
has been carried out in the context of mean-field models.
The aim of this series of papers is to address the question
of the possible occurrence of higher-rank discrete symmetries
from a different theoretical perspective.
Our study is carried out in the context of algebraic collective models
inspired by the interacting boson model (IBM)
of Arima and Iachello~\cite{Arima76,Arima78,Arima79},
which proposes a description of quadrupole collective nuclear states
in terms of $s$ and $d$ bosons with angular momentum $\ell=0$ and $\ell=2$.
The application of this idea to collective states of different nature,
notably of octupole and hexadecapole character,
requires the introduction of other bosons~\cite{Iachello87},
in particular $f$ and $g$ bosons with angular momentum $\ell=3$ and $\ell=4$.
Since shapes with tetrahedral symmetry
arise in lowest order through a particular octupole deformation
and those with octahedral symmetry
emanate from a combination of hexadecapole deformations,
the study of such shapes in an algebraic context
requires the introduction of $f$ and $g$ bosons, respectively.
We initiated this program in I with the study of octahedral shapes in the \mbox{$sdg$-IBM}.
We considered the most general rotationally invariant Hamiltonian
with up to two-body interactions between the bosons
and derived the conditions for this Hamiltonian
to have in its classical limit a minimum with an intrinsic shape with octahedral symmetry.
Owing to the general nature of the analysis in I,
only qualitative conclusions could be drawn
with regard to the (non-)existence of such minima in the \mbox{$sdg$-IBM}.
In the present paper we arrive at more concrete conclusions
by considering a subset of all possible Hamiltonians of the \mbox{$sdg$-IBM},
namely those that have a dynamical symmetry and are analytically solvable.
Although none of the symmetry Hamiltonians leads to a stable shape with octahedral symmetry,
a subset can be used to propose a generalization
with additional interactions between the $g$ bosons
that drive the system toward such a shape.
The paper is structured as follows.
To avoid repeatedly referring to equations in I,
we list in Section~\ref{s_sdgibm} formulas from I that are needed in this paper.
In Section~\ref{s_limits} we focus on two dynamical symmetries of the \mbox{$sdg$-IBM}
of particular relevance in our quest for octahedral shapes
and in Section~\ref{s_clas} the classical limit is derived for the Hamiltonian
that is transitional between these two limits.
The central results of this paper are presented in Section~\ref{s_octa},
with a catastrophe analysis of the energy surface
associated with the transitional symmetry Hamiltonian,
and suitable generalizations thereof,
to uncover the existence of minima at stable shapes with octahedral symmetry.
Finally, in Section~\ref{s_conc} the conclusions
of the second paper in this series are summarized.
\section{The \mbox{$sdg$-IBM} and its classical limit}
\label{s_sdgibm}
A boson-number-conserving, rotationally invariant Hamiltonian of the \mbox{$sdg$-IBM}
with up to two-body interactions is of the form
\begin{eqnarray}
\hat H&=&\epsilon_s\hat n_s+
\epsilon_d\hat n_d+
\epsilon_g\hat n_g
\nonumber\\&&
+\sum_{\ell_1\leq\ell_2,\ell'_1\leq\ell'_2,L}
\frac{(-)^Lv^L_{\ell_1\ell_2\ell'_1\ell'_2}}{\sqrt{(1+\delta_{\ell_1\ell_2})(1+\delta_{\ell'_1\ell'_2})}}
[b^\dag_{\ell_1}\times b^\dag_{\ell_2}]^{(L)}\cdot
[\tilde b_{\ell'_2}\times\tilde b_{\ell'_1}]^{(L)},
\label{e_ham}
\end{eqnarray}
where $\hat n_\ell$ is the number operator for the $\ell$ boson,
$\epsilon_\ell$ is the energy of the $\ell$ boson
and $v^L_{\ell_1\ell_2\ell'_1\ell'_2}$ is a boson--boson interaction matrix element.
A geometric understanding of this quantum-mechanical Hamiltonian
is obtained by considering its expectation value in a coherent state.
As discussed in I,
a study of shapes with octahedral symmetry in \mbox{$sdg$-IBM}
requires a coherent state of the form
\begin{equation}
|N;\beta_2,\beta_4,\gamma_2,\gamma_4,\delta_4\rangle=
\sqrt{\frac{1}{N!(1+\beta_2^2+\beta_4^2)^N}}
\Gamma(\beta_2,\beta_4,\gamma_2,\gamma_4,\delta_4)^N|{\rm o}\rangle,
\label{e_coh}
\end{equation}
with
\begin{eqnarray}
\Gamma(\beta_2,\beta_4,\gamma_2,\gamma_4,\delta_4)&=&
s^\dag+
\beta_2\Bigl[\cos\gamma_2d^\dag_0+
\sqrt{\textstyle{\frac 1 2}}\sin\gamma_2(d^\dag_{-2}+d^\dag_{+2})\Bigr]
\label{e_coh1}\\&&+
\beta_4\Bigl[\Bigl(\sqrt{\textstyle{\frac{7}{12}}}\cos\delta_4+
\sqrt{\textstyle{\frac{5}{12}}}\sin\delta_4\cos\gamma_4\Bigr)g^\dag_0
\nonumber\\&&\phantom{+\beta_4\Bigl[}-
\sqrt{\textstyle{\frac 1 2}}\sin\delta_4\sin\gamma_4(g^\dag_{-2}+g^\dag_{+2})
\nonumber\\&&\phantom{+\beta_4\Bigl[}+
\Bigl(\sqrt{\textstyle{\frac{5}{24}}}\cos\delta_4-
\sqrt{\textstyle{\frac{7}{24}}}\sin\delta_4\cos\gamma_4\Bigr)
(g^\dag_{-4}+g^\dag_{+4})\Bigr],
\nonumber
\end{eqnarray}
in terms of the deformation parameters
$\beta_2$, $\beta_4$, $\gamma_2$, $\gamma_4$ and $\delta_4$,
following the convention of Rohozi\'nski and Sobiczewski~\cite{Rohozinski81}.
A shape with octahedral symmetry is obtained for $\beta_2=0$, $\beta_4\neq0$
and $\delta_4=0$ (octahedron) or $\delta_4=\pi$ (cube) with arbitrary $\gamma_4$.
Another solution with octahedral symmetry
exists for $\gamma_4=0$ and $\delta_4=\arccos(1/6)$,
corresponding to a rotated octahedron,
i.e. with the same intrinsic shape as for $\delta_4=0$.
The parameterization introduced in Ref.~\cite{Rohozinski81} therefore
does {\em not} define a unique intrinsic state.
The expectation value of the Hamiltonian~(\ref{e_ham})
in the coherent state~(\ref{e_coh}) (or the classical limit of $\hat H$)
leads to the energy surface
\begin{eqnarray}
\langle\hat H\rangle&\equiv&
E(\beta_2,\beta_4,\gamma_2,\gamma_4,\delta_4)
\nonumber\\&=&
\frac{N(N-1)}{(1+\beta_2^2+\beta_4^2)^2}
\sum_{kl}\beta_2^k\beta_4^l
\left[c'_{kl}+\sum_{ij}c^{ij}_{kl}\cos(i\gamma_2+j\gamma_4)\phi_{kl}^{ij}(\delta_4)\right],
\label{e_climit}
\end{eqnarray}
where $\phi_{kl}^{ij}(\delta_4)$ are trigonometric functions defined in I.
The coefficients $c'_{kl}$ are known
in terms of the scaled single-boson energies $\epsilon'_\ell\equiv\epsilon_\ell/(N-1)$
and the interaction matrix elements $v^L_{\ell_1\ell_2\ell'_1\ell'_2}$,
\begin{eqnarray}&&
\textstyle
c'_{00}={\frac 1 2}v_{ssss}^0+\epsilon'_s,
\nonumber\\&&
\textstyle
c'_{20}=\sqrt{\frac 1 5}v_{ssdd}^0+v_{sdsd}^2+\epsilon'_s+\epsilon'_d,
\nonumber\\&&
\textstyle
c'_{02}={\frac 1 3}v_{ssgg}^0+v_{sgsg}^4+\epsilon'_s+\epsilon'_g,
\nonumber\\&&
\textstyle
c'_{40}=\frac{1}{10}v_{dddd}^0+{\frac 1 7}v_{dddd}^2+\frac{9}{35}v_{dddd}^4+\epsilon'_d,
\nonumber\\&&
\textstyle
c'_{22}=
\frac{1}{\sqrt{45}}v_{ddgg}^0
+\frac{7}{\sqrt{715}}v_{ddgg}^4
+\frac{1}{6}v_{dgdg}^2
+\frac{4}{11}v_{dgdg}^4
+\frac{1}{6}v_{dgdg}^5
+\frac{10}{33}v_{dgdg}^6+\epsilon'_d+\epsilon'_g,
\nonumber\\&&
\textstyle
c'_{04}=
\frac{1}{18}v_{gggg}^0
+\frac{38}{693}v_{gggg}^2
+\frac{89}{1001}v_{gggg}^4
+\frac{62}{495}v_{gggg}^6
+\frac{1129}{6435}v_{gggg}^8+\epsilon'_g.
\label{e_coef1}
\end{eqnarray}
Only a single coefficient $c^{ij}_{kl}$ is needed in the subsequent analysis, {\it viz.}
\begin{equation}
\textstyle
c^{00}_{04}=
-\frac{2}{693}v_{gggg}^2
+\frac{4}{3003}v_{gggg}^4
+\frac{2}{495}v_{gggg}^6
-\frac{16}{6435}v_{gggg}^8,
\label{e_coef2}
\end{equation}
introducing a $\delta_4$ dependence
in the energy surface~(\ref{e_climit}) since
\begin{equation}
\phi^{00}_{04}(\delta_4)=2\cos2\delta_4+17\cos4\delta_4.
\label{e_phi}
\end{equation}
It is assumed in the following that the boson Hamiltonian is Hermitian
and therefore that $v^L_{\ell_1\ell_2\ell'_1\ell'_2}=v^L_{\ell'_1\ell'_2\ell_1\ell_2}$.
With this assumption the expressions for the coefficients $c_{kl}$ and $c^{ij}_{kl}$,
given in Eqs.~(20) and~(21) of I,
are valid with $v^L_{\ell_1\ell_2\cdot\ell'_1\ell'_2}=v^L_{\ell_1\ell_2\ell'_1\ell'_2}$.
Note that for a general Hamiltonian one has
$v^L_{\ell_1\ell_2\cdot\ell'_1\ell'_2}=(v^L_{\ell_1\ell_2\ell'_1\ell'_2}+v^L_{\ell'_1\ell'_2\ell_1\ell_2})/2$,
which corrects by a factor 2 the expression given in I.
\section{The U$_g$(9) and SO$_{sg}$(10) limits}
\label{s_limits}
Since the Hamiltonian of the \mbox{$sdg$-IBM} conserves the total number of bosons,
it can be written in terms of the $(1+5+9)^2=225$ operators $b_{\ell m}^\dag b_{\ell' m'}$.
The 225 operators generate the Lie algebra U(15) with a substructure
that determines the dynamical symmetries of the \mbox{$sdg$-IBM}.
A comprehensive list of the dynamical symmetries of the \mbox{$sdg$-IBM}
is given by De Meyer {\it et al.}~\cite{Meyer86},
and their group-theoretical properties
are extensively discussed by Kota {\it et al.}~\cite{Kota87}.
It is found that the model has seven major dynamical symmetries,
four of strong coupling, SU(3), SU(6), SU(5) and SO(15),
and three of weak coupling,
${\rm U}_s(1)\otimes{\rm U}_{dg}(14)$,
${\rm U}_{sd}(6)\otimes{\rm U}_g(9)$
and ${\rm U}_{sg}(10)\otimes{\rm U}_d(5)$.
The question treated in this paper is
whether any of the dynamical symmetries of the \mbox{$sdg$-IBM}
corresponds to a shape with octahedral symmetry.
In this section a choice of the limits that possibly have such property
is made on the basis of intuitive arguments.
Subsequently, in Sections~\ref{s_clas} and~\ref{s_octa},
the conditions for the existence of a shape with octahedral symmetry
are derived rigorously on the basis of the results obtained in I.
A minimum with octahedral shape
requires mixing of $s$ and $g$ bosons,
so as to induce hexadecapole deformation,
and no or weak mixing of these with the $d$ boson
to ensure zero quadrupole deformation.
These conditions rule out all limits
where $s$, $d$ and $g$ bosons are strongly mixed on an equal footing,
that is, they discard the SU(3), SU(6), SU(5) and SO(15) limits~\cite{Bouldjedri05}.
A strict decoupling of the $s$ and $g$ from the $d$ bosons
is obtained by the reduction
\begin{equation}
\begin{array}{ccccc}
{\rm U}(15)&\supset&{\rm U}_d(5)&\otimes&{\rm U}_{sg}(10)\\
\downarrow&&\downarrow&&\downarrow\\[0mm]
[N]&&n_d&&n_{sg}
\end{array},
\label{e_ds}
\end{equation}
and, furthermore, zero quadrupole deformation
follows from a U(5) classification for the $d$ bosons,
\begin{equation}
\begin{array}{ccccc}
{\rm U}_d(5)&\supset&{\rm SO}_d(5)&\supset&{\rm SO}_d(3)\\
\downarrow&&\downarrow&&\downarrow\\[0mm]
n_d&&\upsilon_d&\nu_d&L_d
\end{array},
\label{e_u5}
\end{equation}
where underneath each algebra the associated quantum number is given.
Specifically, $N$ is the total number of bosons
while $n_\ell$ is the number of $\ell$ bosons
and $n_{\ell\ell'}$ is the number of $\ell$ plus $\ell'$ bosons.
The seniority label associated with an $\ell$ boson is denoted as $\upsilon_\ell$
and corresponds to the number of $\ell$ bosons
not in pairs coupled to angular momentum zero.
Additional (or missing) labels, not associated with any algebra, are indicated with $\nu_\ell$.
Finally, the angular momentum generated by the $\ell$ bosons is denoted as $L_\ell$.
The ${\rm U}_{sg}(10)$ algebra in Eq.~(\ref{e_ds}) allows two classifications of interest.
The first is obtained by eliminating from the generators of ${\rm U}_{sg}(10)$
those that involve the $s$ boson, leading to
\begin{equation}
({\rm I})\qquad
\begin{array}{ccccccc}
{\rm U}_{sg}(10)&\supset&{\rm U}_g(9)&\supset&{\rm SO}_g(9)&\supset&{\rm SO}_g(3)\\
\downarrow&&\downarrow&&\downarrow&&\downarrow\\
n_{sg}&&n_g&&\upsilon_g&\nu_g&L_g
\end{array}.
\label{e_u9}
\end{equation}
In this limit, which for brevity shall be referred to as ${\rm U}_g(9)$ or limit I,
the separate boson numbers $n_s$ and $n_g$ are conserved.
The resulting spectrum is vibrational-like with a spherical ground state
and excited states that correspond to oscillations in the hexadecapole degree of freedom.
The second classification of ${\rm U}_{sg}(10)$ is specified by the following chain of nested algebras:
\begin{equation}
({\rm II})\qquad
\begin{array}{ccccccc}
{\rm U}_{sg}(10)&\supset&{\rm SO}_{sg}(10)&\supset&{\rm SO}_g(9)&\supset&{\rm SO}_g(3)\\
\downarrow&&\downarrow&&\downarrow&&\downarrow\\
n_{sg}&&\upsilon_{sg}&&\upsilon_g&\nu_g&L_g
\end{array},
\label{e_so10}
\end{equation}
which for brevity shall be referred to as ${\rm SO}_{sg}(10)$ or limit II.
The defining feature of the reduction~(\ref{e_so10})
is the appearance of the algebra ${\rm SO}_{sg}(10)$ and its label $\upsilon_{sg}$,
associated with the pairing of $s$ and $g$ bosons.
As is shown in Section~\ref{s_octa},
the ground state in this limit acquires a permanent hexadecapole deformation
and the limit is therefore of interest in our quest for octahedral shapes.
On its own, however, limit II implies degenerate energies of the $s$ and $g$ boson,
and as such it is not realistic.
It is therefore necessary to study a combination of the two limits I and II.
In Sections~\ref{s_clas} and~\ref{s_octa}
we investigate to what extent non-degenerate energies
can be taken for the $s$ and $g$ boson
that still lead to a hexadecapole-deformed minimum
and whether that minimum can have octahedral symmetry.
In the remainder of this section we list some of the properties of limits I and II
that are necessary to carry out this analysis.
The classification of limits I and II can be summarized with the algebraic lattice
\begin{equation}
\begin{array}{ccccc}
{\rm U}(15)&\supset&{\rm U}_d(5)&\otimes&{\rm U}_{sg}(10)\\
&&|&&\swarrow\quad\searrow\\
&&\downarrow&&\quad {\rm U}_g(9)\quad {\rm SO}_{sg}(10)\\
&&{\rm SO}_d(5)&&\searrow\quad\swarrow\\
&&|&&{\rm SO}_g(9)\\
&&\downarrow&&\downarrow\\
&&{\rm SO}_d(3)&&{\rm SO}_g(3)\\
&&\qquad\searrow&&\swarrow\qquad\\
&&&{\rm SO}(3)&
\end{array},
\label{e_lat}
\end{equation}
where SO(3) is associated with the total angular momentum $L$,
which results from the coupling of $L_d$ and $L_g$.
The generators of the different algebras in the lattice~(\ref{e_lat})
are as follows:
\begin{eqnarray}
{\rm U}_d(5)&:\quad&
\{[d^\dag\times\tilde d]^{(\lambda)}_\mu,\lambda=0,\dots,4\},
\nonumber\\
{\rm SO}_d(5)&:\quad&
\{[d^\dag\times\tilde d]^{(\lambda)}_\mu,\lambda=1,3\},
\nonumber\\
{\rm SO}_d(3)&:\quad&
\{\hat L_{d,\mu}\equiv\sqrt{10}[d^\dag\times\tilde d]^{(1)}_\mu\},
\nonumber\\
{\rm U}_{sg}(10)&:\quad&
\{[s^\dag\times\tilde s]^{(0)}_0,[s^\dag\times\tilde g]^{(4)}_\mu,[g^\dag\times\tilde s]^{(4)}_\mu,
[g^\dag\times\tilde g]^{(\lambda)}_\mu,\lambda=0,\dots,8\},
\nonumber\\
{\rm U}_g(9)&:\quad&
\{[g^\dag\times\tilde g]^{(\lambda)}_\mu,\lambda=0,\dots,8\},
\nonumber\\
{\rm SO}_{sg}(10)&:\quad&
\{[s^\dag\times\tilde g+g^\dag\times\tilde s]^{(4)}_\mu,
[g^\dag\times\tilde g]^{(\lambda)}_\mu,\lambda=1,3,5,7\},
\nonumber\\
{\rm SO}_g(9)&:\quad&
\{[g^\dag\times\tilde g]^{(\lambda)}_\mu,\lambda=1,3,5,7\},
\nonumber\\
{\rm SO}_g(3)&:\quad&
\{\hat L_{g,\mu}\equiv\sqrt{60}[g^\dag\times\tilde g]^{(1)}_\mu\},
\nonumber\\
{\rm SO}(3)&:\quad&
\{\hat L_\mu\equiv\hat L_{d,\mu}+\hat L_{g,\mu}\}.
\label{e_gens}
\end{eqnarray}
The linear and quadratic Casimir operators
of the algebras appearing in the lattice~(\ref{e_lat})
can be expressed as follows
in terms of the generators~(\ref{e_gens}):
\begin{eqnarray}
\hat C_1[{\rm U}(15)]&=&\hat N=\hat n_s+\hat n_d+\hat n_g,
\nonumber\\
\hat C_2[{\rm U}(15)]&=&\hat N(\hat N+14),
\nonumber\\
\hat C_1[{\rm U}_d(5)]&=&\hat n_d,
\nonumber\\
\hat C_2[{\rm U}_d(5)]&=&\hat n_d(\hat n_d+4),
\nonumber\\
\hat C_2[{\rm SO}_d(5)]&=&
2\sum_{\lambda\;{\rm odd}}[d^\dag\times\tilde d]^{(\lambda)}\cdot[d^\dag\times\tilde d]^{(\lambda)},
\nonumber\\
\hat C_2[{\rm SO}_d(3)]&=&\hat L_d\cdot\hat L_d,
\nonumber\\
\hat C_1[{\rm U}_{sg}(10)]&=&\hat n_s+\hat n_g,
\nonumber\\
\hat C_2[{\rm U}_{sg}(10)]&=&(\hat n_s+\hat n_g)(\hat n_s+\hat n_g+9),
\nonumber\\
\hat C_1[{\rm U}_g(9)]&=&\hat n_g,
\nonumber\\
\hat C_2[{\rm U}_g(9)]&=&\hat n_g(\hat n_g+8),
\nonumber\\
\hat C_2[{\rm SO}_{sg}(10)]&=&
[s^\dag\times\tilde g+g^\dag\times\tilde s]^{(4)}\cdot
[s^\dag\times\tilde g+g^\dag\times\tilde s]^{(4)}+
\hat C_2[{\rm SO}_g(9)],
\nonumber\\
\hat C_2[{\rm SO}_g(9)]&=&
2\sum_{\lambda\;{\rm odd}}[g^\dag\times\tilde g]^{(\lambda)}\cdot[g^\dag\times\tilde g]^{(\lambda)},
\nonumber\\
\hat C_2[{\rm SO}_g(3)]&=&\hat L_g\cdot\hat L_g,
\nonumber\\
\hat C_2[{\rm SO}(3)]&=&\hat L\cdot\hat L.
\label{e_cas}
\end{eqnarray}
The expressions for the quadratic Casimir operators of unitary algebras
are not general but are valid in a symmetric irreducible representation.
A rotationally invariant Hamiltonian with up to two-body interactions
can be written in terms of the Casimir operators~(\ref{e_cas}):
\begin{eqnarray}
\hat H_{\rm sym}&=&
\epsilon_d\,\hat n_d+
a_d\,\hat C_2[{\rm U}_d(5)]+
b_d\,\hat C_2[{\rm SO}_d(5)]+
c_d\,\hat C_2[{\rm SO}_d(3)]
\nonumber\\&&+
\epsilon_s\,\hat n_s+\epsilon_g\,\hat n_g+
a_{sg}\,\hat C_2[{\rm U}_{sg}(10)]+
a_g\,\hat C_2[{\rm U}_g(9)]+
b_{sg}\,\hat P_{sg}^\dag\hat P_{sg}
\nonumber\\&&+
b_g\,\hat C_2[{\rm SO}_g(9)]+
c_g\,\hat C_2[{\rm SO}_g(3)]+
c\,\hat C_2[{\rm SO}(3)],
\label{e_hamlat}
\end{eqnarray}
where $\epsilon_\ell$, $a_\ell$, $a_{\ell\ell'}$,
$b_\ell$, $b_{\ell\ell'}$ and $c_\ell$ are parameters.
The quadratic Casimir operator of U(15) is omitted for simplicity
since it gives a constant contribution for a fixed boson number $N=n_s+n_d+n_g$.
Furthermore, it is convenient to define,
instead of the quadratic Casimir operator $\hat C_2[{\rm SO}_{sg}(10)]$,
the combination
\begin{equation}
\hat P^\dag_{sg}\hat P_{sg}=
\hat C_2[{\rm U}_{sg}(10)]-\hat C_1[{\rm U}_{sg}(10)]-\hat C_2[{\rm SO}_{sg}(10)],
\label{e_sfpairing}
\end{equation}
where $\hat P^\dag_{sg}\equiv s^\dag s^\dag-g^\dag\cdot g^\dag$
is the pairing operator for $s$ and $g$ bosons.
The symmetry Hamiltonian~(\ref{e_hamlat}) is less general than Eq.~(\ref{e_ham})
but it is the most general one that can be written
in terms of invariant operators of the lattice~(\ref{e_lat})
and as such it is intermediate between the limits I and II.
The ${\rm U}_g(9)$ limit occurs for $b_{sg}=0$,
leading to the eigenvalues
\begin{eqnarray}
E_{\rm I}&=&
\epsilon_d\,n_d+
a_d\,n_d(n_d+4)+
b_d\,\upsilon_d(\upsilon_d+3)+
c_d\,L_d(L_d+1)
\nonumber\\&&+
\epsilon_s\,n_s+\epsilon_g\,n_g+
a_{sg}\,n_{sg}(n_{sg}+9)+
a_g\,n_g(n_g+8)+
\nonumber\\&&+
b_g\,\upsilon_g(\upsilon_g+7)+
c_g\,L_g(L_g+1)+
c\,L(L+1).
\label{e_eigu9}
\end{eqnarray}
The ${\rm SO}_{sg}(10)$ limit is attained
for $\epsilon_s=\epsilon_g\equiv\epsilon_{sg}$ and $a_g=0$,
in which case the Hamiltonian's eigenstates have the eigenvalues
\begin{eqnarray}
E_{\rm II}&=&
\epsilon_d\,n_d+
a_d\,n_d(n_d+4)+
b_d\,\upsilon_d(\upsilon_d+3)+
c_d\,L_d(L_d+1)
\nonumber\\&&+
\epsilon_{sg}\,n_{sg}+
a_{sg}\,n_{sg}(n_{sg}+9)+
b_{sg}[n_{sg}(n_{sg}+8)-\upsilon_{sg}(\upsilon_{sg}+8)]
\nonumber\\&&+
b_g\,\upsilon_g(\upsilon_g+7)+
c_g\,L_g(L_g+1)+
c\,L(L+1).
\label{e_eigso10}
\end{eqnarray}
The eigenspectra are then determined
with the help of the necessary branching rules.
The reduction ${\rm U}(15)\supset{\rm U}_d(5)\otimes{\rm U}_{sg}(10)$
implies the relation $N=n_d+n_{sg}$ or the branching rule
\begin{equation}
[N]\mapsto(n_d,n_{sg})=(0,N),(1,N-1),\dots,(N,0).
\label{e_branch}
\end{equation}
The branching rules for the classification~(\ref{e_u5})
are known from the U(5) limit of the \mbox{$sd$-IBM}~\cite{Arima76}
and those for the classifications~(\ref{e_u9}) and~(\ref{e_so10})
can be found in Ref.~\cite{Kota87}.
\begin{figure}
\centering
\includegraphics[width=6.5cm]{fig1a.pdf}
\includegraphics[width=6.5cm]{fig1b.pdf}
\caption{Energy spectra in the ${\rm U}_g(9)$ and ${\rm SO}_{sg}(10)$ limits
of the \mbox{$sdg$-IBM} for $N=5$ bosons.
For the ${\rm U}_g(9)$ spectrum the non-zero parameters in the Hamiltonian~(\ref{e_hamlat})
are $\epsilon_d-\epsilon_s=800$, $\epsilon_g-\epsilon_s=1000$,
$b_d=40$, $c_d=10$, $b_g=25$ and $c_g=c=5$~keV.
For the ${\rm SO}_{sg}(10)$ spectrum the non-zero parameters
are $\epsilon_d-\epsilon_s=800$, $\epsilon_g-\epsilon_s=0$,
$b_d=40$, $c_d=10$, $b_{sg}=60$, $b_g=50$ and $c_g=c=5$~keV.}
\label{f_limits}
\end{figure}
Typical energy spectra in the ${\rm U}_g(9)$ and ${\rm SO}_{sg}(10)$ limits
are shown in Fig.~\ref{f_limits}.
The ${\rm U}_g(9)$ spectrum displays quadrupole- and hexadecapole-phonon multiplets
characterized by a fixed number of $d$ and $g$ bosons.
The multiplets are further structured by a seniority quantum number:
for example, the $n_d=2$ multiplet has $\upsilon_d=2$ except for the $0^+$ level,
which has $\upsilon_d=0$,
and similarly for the $n_g=2$ multiplet and the $\upsilon_g$ seniority.
Also combined quadrupole--hexadecapole multiplets occur in the spectrum.
The ${\rm SO}_{sg}(10)$ spectrum
contains sets of levels with $\upsilon_{sg}=N,N-2,\dots$ (for $n_d=0$)
with $\upsilon_{sg}=N-1,N-3,\dots$ (for $n_d=1$) etc.,
and $\upsilon_{sg}=N$ levels are lowest in energy due the repulsive $sg$-pairing.
Multiplets characterized by the seniority quantum number $\upsilon_g=0,1,\dots$
occur within each ${\rm SO}_{sg}(10)$ multiplet.
Note that in this limit, unless $b_{sg}$ is small, the first-excited state has $J^\pi=4^+$.
This is a consequence of the unrealistic condition
that the energies of the $s$ and $g$ bosons are degenerate, $\epsilon_s=\epsilon_g$.
\section{Classical limit of the symmetry Hamiltonian}
\label{s_clas}
The classical limit of the symmetry Hamiltonian~(\ref{e_hamlat})
can be obtained with the general procedure outlined in I.
First, a conversion to the standard representation~(\ref{e_ham}) is carried out.
A quadratic Casimir operator $\hat C_2(G)$
is given generically as an expansion over the generators,
\begin{equation}
\hat C_2(G)=
\sum_{\lambda r}a_{\lambda r}
\left(\sum_{\ell_1\ell_2}\alpha^{\lambda r}_{\ell_1\ell_2}
\left(b_{\ell_1}^\dag\times\tilde b_{\ell_2}\right)^{(\lambda)}\right)
\cdot
\left(\sum_{\ell'_1\ell'_2}\alpha^{\lambda r}_{\ell'_1\ell'_2}
\left(b_{\ell'_1}^\dag\times\tilde b_{\ell'_2}\right)^{(\lambda)}\right),
\label{e_casexp}
\end{equation}
with coefficients $\alpha^{\lambda r}_{\ell_1\ell_2}$ and $a_{\lambda r}$
that are specific to each algebra $G$, as given in Eq.~(\ref{e_cas}).
The overall sum in Eq.~(\ref{e_casexp})
is over the multipolarity $\lambda$ of the generators
and over an additional index $r$
to distinguish different generators with the same $\lambda$.
With use of the expansion~(\ref{e_casexp})
one finds the following expression
for the matrix element of $\hat C_2(G)$ between two-boson states~\cite{note1}:
\begin{eqnarray}
\lefteqn{\langle \ell_1\ell_2;L|\hat C_2(G)|\ell'_1\ell'_2;L\rangle}
\nonumber\\&=&
\left[f_{\ell_1}(G)+f_{\ell_2}(G)\right]
\delta_{\ell_1\ell_3}\delta_{\ell_2\ell_4}+
\frac{2(-)^{\ell_2+\ell_3}}{\sqrt{(1+\delta_{\ell_1\ell_2})(1+\delta_{\ell_3\ell_4})}}
\sum_{\lambda r}a_{\lambda r}(2\lambda+1)
\nonumber\\&&\times
\Biggl[\alpha^{\lambda r}_{\ell_1\ell_4}\alpha^{\lambda r}_{\ell_2\ell_3}
\Biggl\{\begin{array}{ccc}
\ell_1&\ell_4&\lambda\\
\ell_3&\ell_2&L
\end{array}\Biggr\}+
(-)^L\alpha^{\lambda r}_{\ell_1\ell_3}\alpha^{\lambda r}_{\ell_2\ell_4}
\Biggl\{\begin{array}{ccc}
\ell_1&\ell_3&\lambda\\
\ell_4&\ell_2&L
\end{array}
\Biggr\}\Biggr],
\label{e_convert}
\end{eqnarray}
where the symbol between curly brackets is a Racah coefficient~\cite{Talmi93}.
The quantity $f_\ell(G)$ is the expectation value
of the operator $\hat C_2(G)$ between single-boson states,
\begin{equation}
f_\ell(G)\equiv\langle \ell|\hat C_2(G)|\ell\rangle=
\sum_{\lambda r}\frac{2\lambda+1}{2\ell+1}a_{\lambda r}
\sum_{\ell'}(-)^{\ell+\ell'}
\alpha^{\lambda r}_{\ell\ell'}
\alpha^{\lambda r}_{\ell'\ell}.
\label{e_conone}
\end{equation}
The Hamiltonian~(\ref{e_hamlat}) can,
with use of Eqs.~(\ref{e_convert}) and~(\ref{e_conone}),
be converted into its standard representation
on which the classical-limit expression~(\ref{e_climit}) can be applied.
The procedure results in the energy surface
\begin{equation}
E(\beta_2,\beta_4)=
\frac{N(N-1)}{(1+\beta_2^2+\beta_4^2)^2}
\sum_{kl}c'_{kl}\beta_2^k\beta_4^l,
\label{e_climit1}
\end{equation}
where the non-zero coefficients $c'_{kl}$ are
\begin{eqnarray}
c'_{00}&=&
a_{sg}+b_{sg}+\Gamma_s,
\quad
c'_{20}=
\Gamma_s+\Gamma_d,
\quad
c'_{02}=
2a_{sg}-2b_{sg}+\Gamma_s+\Gamma_g,
\nonumber\\
c'_{40}&=&
a_d+\Gamma_d,
\quad
c'_{22}=
\Gamma_d+\Gamma_g,
\quad
c'_{04}=
a_{sg}+b_{sg}+a_g+\Gamma_g,
\label{e_coeflat}
\end{eqnarray}
in terms of the combinations
\begin{eqnarray}
\Gamma_s&\equiv&\frac{1}{N-1}(\epsilon_s+10a_{sg}),
\nonumber\\
\Gamma_d&\equiv&\frac{1}{N-1}(\epsilon_d+5a_d+4b_d+6c_d+6c),
\nonumber\\
\Gamma_g&\equiv&\frac{1}{N-1}(\epsilon_g+10a_{sg}+9a_g+8b_g+20c_g+20c).
\label{e_comb}
\end{eqnarray}
All coefficients $c^{ij}_{kl}$ of Eq.~(\ref{e_climit}) vanish identically
in the classical limit of the symmetry Hamiltonian~(\ref{e_hamlat}).
\section{Octahedral shapes}
\label{s_octa}
From the outset it should be clear that energy surface~(\ref{e_climit1})
cannot have an isolated minimum with octahedral shape
since it is independent of $\gamma_2$, $\gamma_4$ and $\delta_4$.
What can still happen, however, is the occurrence of a minimum
with zero quadrupole and non-zero hexadecapole deformation
($\beta^*_2=0$ and $\beta^*_4\neq0$),
which, given the instability in $\gamma_4$ and $\delta_4$,
{\em includes} a shape with octahedral symmetry.
Therefore, the goal of this section is
to establish the conditions on the parameters in the symmetry Hamiltonian~(\ref{e_hamlat})
such that its classical limit displays a minimum with $\beta^*_2=0$ and $\beta^*_4\neq0$,
and, subsequently, to identify the interactions in the general Hamiltonian~(\ref{e_ham})
that generate a dependence on $\delta_4$,
enabling the formation of an isolated minimum with octahedral shape.
This problem can be investigated with the procedure outlined in I.
According to the analysis of the previous section,
the classical energy~(\ref{e_climit1}) is a two-variable function
$E(\beta_2,\beta_4)$.
The conditions for this energy surface
to have an extremum are
\begin{equation}
\left.\frac{\partial E}{\partial\beta_2}\right|_{p^*}=
\left.\frac{\partial E}{\partial\beta_4}\right|_{p^*}=0,
\label{e_extr}
\end{equation}
where $p^*\equiv(\beta^*_2,\beta^*_4)$
is a short-hand notation for an arbitrary critical point.
Furthermore, a critical point at an extremum
with $\beta^*_2=0$ and $\beta^*_4\neq0$ shall be denoted as $h^*$.
The condition~(\ref{e_extr}) in $\beta_2$ is identically satisfied for $p^*=h^*$
and does not lead to any constraints on the coefficients $c'_{kl}$.
The condition in $\beta_4$ leads to a cubic equation with the solutions
\begin{equation}
\beta_4^*=0,
\qquad
\beta_4^*=\pm\sqrt{\frac{2c'_{00}-c'_{02}}{2c'_{04}-c'_{02}}}.
\label{e_extrb4}
\end{equation}
Only the last solution corresponds to an extremum $h^*$
and implies the following condition on the ratio of coefficients:
\begin{equation}
\frac{2c'_{00}-c'_{02}}{2c'_{04}-c'_{02}}>0.
\label{e_cond1}
\end{equation}
While the condition~(\ref{e_cond1}) is necessary and sufficient
to have an {\em extremum} at $\beta_2^*=0$ and $\beta_4^*\neq0$,
a {\it minimum} at these values implies further constraints.
They are obtained by requiring that the eigenvalues of the stability matrix
[{\it i.e.}, the partial derivatives of $E(\beta_2,\beta_4)$
of second order] are all positive.
Since the off-diagonal element of the stability matrix vanishes for the energy surface~(\ref{e_climit1}),
the existence of a minimum follows from the uncoupled conditions
\begin{equation}
\left.\frac{\partial^2 E}{\partial\beta_2^2}\right|_{h^*}>0,
\qquad
\left.\frac{\partial^2 E}{\partial\beta_4^2}\right|_{h^*}>0,
\label{e_stab1}
\end{equation}
or, in terms of the coefficients $c'_{kl}$ in Eq.~(\ref{e_climit1}),
\begin{eqnarray}
\frac{(2c'_{04}-c'_{02})[2c'_{00}(c'_{22}-2c'_{04})+c'_{02}(c'_{02}-c'_{20}-c'_{22})+2c'_{04}c'_{20}]}
{(c'_{00}-c'_{02}+c'_{04})^2}&>&0,
\nonumber\\
\frac{(2c'_{00}-c'_{02})(2c'_{04}-c'_{02})^3}{(c'_{00}-c'_{02}+c'_{04})^3}&>&0.
\label{e_stab2}
\end{eqnarray}
If we write Eq.~(\ref{e_cond1}) as $A/B>0$,
the second inequality in Eq.~(\ref{e_stab2}) becomes $AB^3/(A+B)^3>0$
and therefore both $A$ and $B$ should be positive,
$A\equiv2c'_{00}-c'_{02}>0$ and $B\equiv2c'_{04}-c'_{02}>0$,
leading to the constraint
\begin{equation}
-4b_{sg}-2a_g<\Gamma_g-\Gamma_s<4b_{sg}.
\label{e_cond2}
\end{equation}
The first inequality in Eq.~(\ref{e_stab2}) can be reduced to
\begin{equation}
2b_{sg}(2\Gamma_d-\Gamma_s-\Gamma_g-4a_{sg}-a_g)
+a_g(\Gamma_d-\Gamma_s-2a_{sg})>0.
\label{e_cond3}
\end{equation}
The conditions~(\ref{e_cond2}) and~(\ref{e_cond3}) are necessary and sufficient
for the energy surface $E(\beta_2,\beta_4)$
to have a minimum at zero quadrupole and non-zero hexadecapole deformation.
To obtain an intuitive understanding of them,
we note that $\Gamma_g-\Gamma_s$, for a reasonable choice of parameters, is positive.
The upper part of the inequality~(\ref{e_cond2}) therefore expresses
the need for $b_{sg}$ to be positive and sufficiently large,
corresponding to a repulsive $sg$-pairing interaction
that puts the configuration with {\em maximal} $sg$ seniority $\upsilon_{sg}=n_{sg}$ at lowest energy.
For $b_{sg}>0$ and $a_g>0$,
the lower part of the inequality~(\ref{e_cond2}) is automatically satisfied.
The condition~(\ref{e_cond3}) is easier to appreciate
if it is assumed that the coefficients in front of the quadratic Casimir operators
of the unitary algebras ${\rm U}_d(5)$, ${\rm U}_g(9)$ and ${\rm U}_{sg}(10)$ vanish,
$a_d=a_{sg}=a_g=0$.
This assumption is justified if anharmonicities are neglected in the various limits.
Given that $b_{sg}>0$, it then follows that
\begin{equation}
2\Gamma_d-\Gamma_s-\Gamma_g>0.
\label{e_cond4}
\end{equation}
In terms of the original parameters in the symmetry Hamiltonian~(\ref{e_hamlat})
(assuming $a_d=a_{sg}=a_g=0$)
the conditions to have a minimum at $\beta^*_2=0$ and $\beta^*_4\neq0$
can be summarized as
\begin{eqnarray}
-4(N-1)b_{sg}<\epsilon_g-\epsilon_s+8b_g+20(c_g+c)&<&4(N-1)b_{sg},
\nonumber\\
2\epsilon_d-\epsilon_s-\epsilon_g+8(b_d-b_g)+4(3c_d-5c_g-2c)&>&0.
\label{e_cond}
\end{eqnarray}
We now ask the question whether two-body interactions
can be added to the symmetry Hamiltonian~(\ref{e_hamlat}),
which lift the $(\gamma_4,\delta_4)$ instability
and create a minimum at $\delta_4=0$, $\delta_4=\arccos(1/6)$ or $\delta_4=\pi$.
To achieve this goal, we recall the result from I
that for a general Hamiltonian of the \mbox{$sdg$-IBM}
an extremum with $\beta_2^*=0$ and $\beta_4^*\neq0$ occurs for
\begin{equation}
\beta_4^*=\pm\sqrt{\frac{2c'_{00}-c'_{02}}{2c'_{04}-c'_{02}+38c^{00}_{04}}}.
\label{e_extrb4g}
\end{equation}
The term in $c^{00}_{04}$ introduces a dependence in $\delta_4$
that may lead to an isolated minimum with octahedral symmetry.
Given the expression~(\ref{e_coef2}) for $c^{00}_{04}$,
this argument suggests adding $g$-boson interactions $v_{gggg}^L$ to the Hamiltonian~(\ref{e_hamlat}).
The classical energy~(\ref{e_climit1}) then becomes a three-variable function
$E(\beta_2,\beta_4,\delta_4)$,
for which the above catastrophe analysis can be repeated.
With these additional two-body interactions
the extremum and stability conditions~(\ref{e_extr}) and~(\ref{e_stab1}) become
\begin{eqnarray}
&&2c'_{00}-c'_{02}>0,
\qquad
2c'_{04}-c'_{02}+38c^{00}_{04}>0,
\qquad
c^{00}_{04}<0,
\label{e_cong1}\\
&&2c'_{00}(c'_{22}-2c'_{04})+c'_{02}(c'_{02}-c'_{20}-c'_{22})+2c'_{04}c'_{20}-38c^{00}_{04}(2c'_{00}-c'_{20})>0.
\nonumber
\end{eqnarray}
For $a_d=a_{sg}=a_g=0$,
these conditions imply the inequalities
\begin{eqnarray}
&&-2b_{sg}-\tilde v<\Gamma_g-\Gamma_s<4b_{sg},
\nonumber\\
&&\bar v\equiv 65v_{gggg}^2-30v_{gggg}^4-91v_{gggg}^6+56v_{gggg}^8>0,
\nonumber\\
&&4b_{sg}(2\Gamma_d-\Gamma_s-\Gamma_g)
+(\Gamma_d-\Gamma_s-2b_{sg})(\tilde v-2b_{sg})>0,
\label{e_cong2}
\end{eqnarray}
where $\tilde v$ is the following linear combination of $g$-boson interaction matrix elements:
\begin{equation}
\textstyle
\tilde v=
\frac{1}{9}v_{gggg}^0
+\frac{98}{429}v_{gggg}^4
+\frac{40}{99}v_{gggg}^6
+\frac{10}{39}v_{gggg}^8.
\label{e_vtilde}
\end{equation}
For the Hamiltonian~(\ref{e_hamlat}) the linear combination $\bar v$ vanishes identically
and the second inequality in Eq.~(\ref{e_cong2}) is not fulfilled.
This expresses the $\delta_4$ independence of the symmetry Hamiltonian
and the fact that its classical limit does not acquire an isolated minimum with octahedral shape.
Furthermore, for the symmetry Hamiltonian one has $\tilde v=2b_{sg}$
and the conditions~(\ref{e_cong2}) reduce to Eq.~(\ref{e_cond}).
There are clearly many ways to find matrix elements $v_{gggg}^L$
that satisfy all conditions~(\ref{e_cong2}) but one way is particularly simple.
Note that the quadrupole matrix element $v_{gggg}^2$
does not appear in the combination $\tilde v$.
By making this matrix element more repulsive,
the second inequality in Eq.~(\ref{e_cong2}) is satisfied
while the other two conditions are not modified
with respect those in Eq.~(\ref{e_cond}) valid for the symmetry Hamiltonian~(\ref{e_hamlat}).
A possible procedure to construct an \mbox{$sdg$-IBM} Hamiltonian
whose classical energy displays a minimum with octahedral shape
is therefore to add to a hexadecapole-deformed symmetry Hamiltonian~(\ref{e_hamlat})
a repulsive $v_{gggg}^2$ interaction.
\begin{figure}
\centering
\includegraphics[width=7cm]{fig2.pdf}
\caption{Energy spectrum of a ${\rm U}_g(9)$--${\rm SO}_{sg}(10)$ transitional Hamiltonian
of the \mbox{$sdg$-IBM} for $N=5$ bosons.
The non-zero parameters of the Hamiltonian~(\ref{e_hamlat})
are $\epsilon_d-\epsilon_s=1200$, $\epsilon_g-\epsilon_s=1500$,
$b_d=40$, $c_d=10$, $b_{sg}=150$, $b_g=25$ and $c_g=c=5$~keV.}
\label{f_u9so10}
\end{figure}
Let us illustrate this procedure with an example.
The starting point is a ${\rm U}_g(9)$--${\rm SO}_{sg}(10)$ transitional Hamiltonian
associated with the lattice~(\ref{e_lat}),
giving rise to the spectrum shown in Fig.~\ref{f_u9so10}.
Note that the choice of the single-boson energies $\epsilon_\ell$ for this figure
is realistic in the sense that the $g$-boson energy
is higher than that of the $d$ boson.
The $sg$-pairing strength $b_{sg}$,
of which little is known either empirically or microscopically,
is chosen such that a hexadecapole-deformed minimum occurs in the classical limit.
Other parameters in the Hamiltonian~(\ref{e_hamlat}) are of lesser importance
and are chosen as to lift degeneracies in the spectrum.
Note that with this choice of parameters
the resulting spectrum, as shown in Fig.~\ref{f_u9so10},
is rather closer to the ${\rm U}_g(9)$ and than to the ${\rm SO}_{sg}(10)$ limit.
\begin{figure}
\centering
\includegraphics[width=6cm]{fig3.pdf}
\caption{The energy surface $E(\beta_2,\beta_4)$
obtained in the classical limit
of a ${\rm U}_g(9)$--${\rm SO}_{sg}(10)$ transitional Hamiltonian of the \mbox{$sdg$-IBM}.
Parameters of the Hamiltonian~(\ref{e_hamlat})
are given in the caption of Fig.~\ref{f_u9so10}.
Black corresponds to low energies
and the lines indicate changes by 10~keV.}
\label{f_b2b4}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=6cm]{fig4a.pdf}
\includegraphics[width=6cm]{fig4b.pdf}
\caption{Energy surfaces $E(\beta_4,\delta_4)$
obtained in the classical limit
of two different Hamiltonians of the \mbox{$sdg$-IBM} for $N=5$ bosons.
The dependence on $\beta_4>0$ and $0\leq\delta_4\leq\pi$ is shown for $\beta_2^*=0$.
Black corresponds to low energies
and the lines indicate changes by 10~keV.
(a) The ${\rm U}_g(9)$--${\rm SO}_{sg}(10)$ transitional Hamiltonian is taken
with the parameters given in the caption of Fig.~\ref{f_u9so10}.
(b) The Hamiltonian of (a) is modified by taking a repulsive interaction $v_{gggg}^2=500$~keV.}
\label{f_b4d4}
\end{figure}
The parameters quoted in the caption of Fig.~\ref{f_u9so10}
satisfy both conditions~(\ref{e_cond}).
As a result, the energy surface in the classical limit of the corresponding Hamiltonian
has a minimum for $\beta_2^*=0$ and $\beta_4^*\approx0.34$,
as shown in Fig.~\ref{f_b2b4}.
According to the preceding discussion,
the surface is independent of $\delta_4$,
which is indeed confirmed by Fig.~\ref{f_b4d4}(a).
If the $v_{gggg}^2$ matrix element is modified,
a dependence in $\delta_4$ is introduced,
as illustrated in Fig.~\ref{f_b4d4}(b) for the value $v_{gggg}^2=500$~keV.
It is seen that the energy surface displays {\em three} isolated minima
that are exactly degenerate.
The three minima all have an octahedral symmetry,
corresponding to either an octahedron [$\delta_4^*=0$ and $\delta_4^*=\arccos(1/6)\approx84.4^{\rm o}$]
or a cube ($\delta_4^*=\pi$).
\begin{figure}
\centering
\includegraphics[width=3cm]{fig5a.pdf}
\includegraphics[width=7cm]{fig5b.pdf}
\includegraphics[width=3cm]{fig5c.pdf}
\caption{Energy spectrum of a general Hamiltonian of the \mbox{$sdg$-IBM} for $N=5$ bosons.
The same Hamiltonian is taken as in Fig.~\ref{f_u9so10}
but one $g$-boson two-body matrix element is modified to $v_{gggg}^2=500$~keV.
On the left- and right-hand sides are shown
the shapes at the minima in the energy surface
obtained in the classical limit of this Hamiltonian.
They have octahedral symmetry and correspond to either an octahedron or a cube.}
\label{f_u15}
\end{figure}
Although this analysis shows that
isolated minima with octahedral symmetry can be obtained
in the classical limit of an \mbox{$sdg$-IBM} Hamiltonian with reasonable parameters,
it can be expected that such minima are rather shallow.
Even for the fairly large value of the interaction matrix element
in the above example, $v_{gggg}^2=500$~keV,
the three minima are separated by a barrier of $\sim20$~keV,
inducing only very weak observable effects.
This point is illustrated with Fig.~\ref{f_u15},
which shows the spectrum of the ${\rm U}_g(9)$--${\rm SO}_{sg}(10)$ transitional Hamiltonian
with the modified $v_{gggg}^2$ matrix element.
Except for some minute changes the spectrum is essentially the same
as that shown in Fig.~\ref{f_u9so10}.
One subtle point made clear by the current study is that
it is not sufficient to carry out a catastrophe analysis
of the generic surface~(\ref{e_climit})
obtained in the classical limit of the most general \mbox{$sdg$-IBM} Hamiltonian~(\ref{e_ham})
with up to two-body interactions between the bosons.
The coefficients $c'_{kl}$ and $c^{ij}_{kl}$ cannot be treated as free parameters
but their expressions in terms of the single-bosons energies
and boson--boson interactions are an essential part of the analysis.
To illustrate this point consider the energy surface shown in Fig.~\ref{f_b4d4}(b).
The minima at $\delta_4^*=0$ and $\delta_4^*\approx84.4^{\rm o}$
correspond to the same intrinsic shape
(an octahedron, shown on the left-hand side of Fig.~\ref{f_u15})
and, as a consequence, the minima {\em must} be exactly degenerate.
This behavior is generally valid.
Therefore, whatever single-boson energies and boson--boson interactions
one adopts in the Hamiltonian~(\ref{e_ham}),
the energy surface in its classical limit must satisfy the constraint
that points on the surface with same intrinsic shape
(e.g., $\delta_4^*=0$ and $\delta_4^*\approx84.4^{\rm o}$)
are at the same energy.
\section{Conclusions}
\label{s_conc}
The main conclusion of this paper is
that no \mbox{$sdg$-IBM} Hamiltonian with a dynamical symmetry
that includes Casimir operators of up to second order
displays in its classical limit an isolated minimum with octahedral shape.
Nevertheless, a degenerate minimum that {\em includes} a shape with octahedral symmetry
can be obtained from a Hamiltonian transitional between two limits.
In the limits in question the $d$ boson is decoupled from $s$ and $g$ bosons.
Furthermore, limit I, ${\rm U}_g(9)$, has hexadecapole vibrational characteristics
while in limit II, ${\rm SO}_{sg}(10)$, $s$- and $g$-boson states
are mixed through an $sg$-pairing interaction.
A catastrophe analysis of the energy surface
obtained in the classical limit of this transitional symmetry Hamiltonian
indicates that a minimum with zero quadrupole and non-zero hexadecapole deformation
can be obtained with reasonable parameters.
However, this minimum is always $\delta_4$ independent,
meaning that it ranges from an octahedron to a cube
and includes intermediate shapes without octahedral symmetry.
Isolated minima with octahedral symmetry can be obtained
by adding two-body interactions between the $g$ bosons
to the transitional symmetry Hamiltonian.
The resulting energy surface displays in this case minima with octahedral symmetry,
with the shape of either an octahedron or a cube,
separated by a barrier with low energy
even for fairly strong interactions between the $g$ bosons.
The conclusion of this analysis in the context of the \mbox{$sdg$-IBM}
is therefore that it will be difficult to find
experimental manifestations of octahedral symmetry in nuclei.
\section*{Acknowledgements}
This work has been carried out
in the framework of a CNRS/DEF agreement,
project N 13760.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,921 |
In software engineering, software system safety optimizes system safety in the design, development, use, and maintenance of software systems and their integration with safety-critical hardware systems in an operational environment.
Overview
Software system safety is a subset of system safety and system engineering and is synonymous with the software engineering aspects of Functional Safety. As part of the total safety and software development program, software cannot be allowed to function independently of the total effort. Both simple and highly integrated multiple systems are experiencing an extraordinary growth in the use of computers and software to monitor and/or control safety-critical subsystems or functions. A software specification error, design flaw, or the lack of generic safety-critical requirements can contribute to or cause a system failure or erroneous human decision. To achieve an acceptable level of safety for software used in critical applications, software system safety engineering must be given primary emphasis early in the requirements definition and system conceptual design process. Safety-critical software must then receive continuous management emphasis and engineering analysis throughout the development and operational lifecycles of the system. Software with safety-critical functionality must be thoroughly verified with
objective analysis.
Functional Hazard Analyses (FHA) are often conducted early on - in parallel with or as part of system engineering Functional Analyses - to determine the safety-critical functions (SCF) of the systems for further analyses and verification. Software system safety is directly related to the more critical design aspects and safety attributes in software and system functionality, whereas software quality attributes are inherently different and require standard scrutiny and development rigor. Development Assurance levels (DAL) and associated Level of Rigor (LOR) is a graded approach to software quality and software design assurance as a pre-requisite that a suitable software process is followed for confidence. LOR concepts and standards such as DO-178C are NOT a substitute for software safety. Software safety per IEEE STD-1228 and MIL-STD-882E focuses on ensuring explicit safety requirements are met and verified using functional approaches from a safety requirements analysis and test perspective. Software safety hazard analysis required for more complex systems where software is controlling critical functions generally are in the following sequential categories and are conducted in phases as part of the system safety or safety engineering process: software safety requirements analysis; software safety design analyses (top level, detailed design and code level); software safety test analysis, and software safety change analysis. Once these "functional" software safety analyses are completed the software engineering team will know where to place safety emphasis and what functional threads, functional paths, domains and boundaries to focus on when designing in software safety attributes to ensure correct functionality and to detect malfunctions, failures, faults and to implement a host of mitigation strategies to control hazards. Software security and various software protection technologies are similar to software safety attributes in the design to mitigate various types of threats vulnerability and risks. Deterministic software is sought in the design by verifying correct and predictable behavior at the system level.
Goals
Functional safety is achieved through engineering development to ensure correct execution and behavior of software functions as intended
Safety consistent with mission requirements, is designed into the software in a timely, cost effective manner.
On complex systems involving many interactions safety-critical functionality should be identified and thoroughly analyzed before deriving hazards and design safeguards for mitigations.
Safety-critical functions lists and preliminary hazards lists should be determined proactively and influence the requirements that will be implemented in software.
Contributing factors and root causes of faults and resultant hazards associated with the system and its software are identified, evaluated and eliminated or the risk reduced to an acceptable level, throughout the lifecycle.
Reliance on administrative procedures for hazard control is minimized.
The number and complexity of safety critical interfaces is minimized.
The number and complexity of safety critical computer software components is minimized.
Sound human engineering principles are applied to the design of the software-user interface to minimize the probability of human error.
Failure modes, including hardware, software, human and system are addressed in the design of the software.
Sound software engineering practices and documentation are used in the development of the software.
Safety issues and safety attributes are addressed as part of the software testing effort at all levels.
Software is designed for human machine interface, ease of maintenance and modification or enhancement
Software with safety-critical functionality must be thoroughly verified with objective analysis and preferably test evidence that all safety requirements have been met per established criteria.
See also
Software assurance
IEC 61508 - Functional Safety of Electrical/Electronic/Programmable Electronic Safety-related Systems
ISO 26262 - Road vehicles – Functional safety
Functional Safety
Software quality
System accident
References
Software quality | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 303 |
package com.artemis.io;
import com.artemis.Component;
import com.artemis.EntitySubscription;
import com.artemis.annotations.Wire;
import com.artemis.utils.IntBag;
import java.util.HashMap;
import java.util.IdentityHashMap;
import java.util.Map;
/**
* <p>The default save file format. This class can be extended if additional
* data requires persisting. All instance fields in this class - or its children -
* are persisted.</p>
*
* <p>Two default de/serializer back-ends are provided: {@link KryoArtemisSerializer}
* and {@link JsonArtemisSerializer}. These know how to serialize entities
* and metadata, but little else. Custom serializers can be written for both
* backends, and are required when extending the save file format.</p>
*
* <p>The typical custom serializer works on type, e.g. a <code>GameStateManager</code>
* contains additional data not available to components directly. A serializer would
* be registered to only interact with that class; during loading and saving, the
* serializer interacts directly with the manager and reads/writes the data as needed.</p>
*
* <p><b>Nota Bene:</b> PackedComponent types are not yet supported.</p>
*
* @See {@link JsonArtemisSerializer}
* @See {@link KryoArtemisSerializer}
* @See {@link EntityReference}
*/
@Wire
public class SaveFileFormat {
// all fields are automatically serialized
public Metadata metadata;
public final IdentityHashMap<Class<? extends Component>, String> componentIdentifiers;
public final IntBag entities;
public SaveFileFormat(IntBag entities) {
this.entities = (entities != null) ? entities : new IntBag();
componentIdentifiers = new IdentityHashMap<Class<? extends Component>, String>();
metadata = new Metadata();
metadata.version = Metadata.LATEST;
}
public SaveFileFormat(EntitySubscription es) {
this(es.getEntities());
}
private SaveFileFormat() {
this((IntBag)null);
}
protected Map<String, Class<? extends Component>> readLookupMap() {
Map<String, Class<? extends Component>> lookup
= new HashMap<String, Class<? extends Component>>();
for (Map.Entry<Class<? extends Component>, String> entry : componentIdentifiers.entrySet()) {
lookup.put(entry.getValue(), entry.getKey());
}
return lookup;
}
public static class Metadata {
public static final int VERSION_1 = 1;
public static final int LATEST = VERSION_1;
public int version;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,741 |
Help your neighbors who require special assistance--infants, elderly, people with disabilities.
After an earthquake it takes about 3x as long to get from point A to B.
Be prepared for aftershocks in the first hours, days and weeks after the quake.
If (and ONLY if) you smell gas or hear blowing or hissing noise, open a window and quickly leave the building. Turn off the gas at the outside main valve if you can.
Call the gas company from a neighbor's home. If you turn off the gas for any reason, it should be turned back on by a professional.
If you see sparks or broken or frayed wires, or if you smell hot insulation, turn off the electricity at the main fuse box or circuit breaker.
If you have to step in water to get to the fuse box or circuit breaker, call an electrician first for advice.
If you suspect sewage lines are damaged, avoid using toilets and call a plumber.
If water pipes are damaged, contact the water company and avoid using tap water. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,443 |
Ethel Cuff Black (October 17, 1890 – September 17, 1977) was one of the founders of Delta Sigma Theta Sorority, Incorporated. She was elected the sorority's first vice president and attended the Deltas' first public event, the Woman Suffrage Procession in Washington, D.C., in March 1913. Prominent suffragist Mary Church Terrell lobbied on behalf of the Deltas to win them a place in the parade, where they were the only African-American organization represented.
Biography
Ethel L. Cuff was born in Wilmington, Delaware. Her father, Richard Cuff, was a tanner in an African-American owned business. Her maternal grandfather was a Civil War veteran. In Bordentown, New Jersey, she attended the Industrial School for Colored Youth and graduated with the highest grade point average. At Howard University, she was chairwoman of the collegiate chapter of the YWCA. During college, she was also the vice-president of Alpha Kappa Alpha, but later voted to reorganize the sorority and formed Delta Sigma Theta with twenty-one other women Due to illness, she graduated Howard in 1915. She was also the first African-American teacher in Rochester, New York. She was married in 1939 to real estate agent David Horton Black.
References
External links
Delta Sigma Theta Founder History at the University of Texas
"Black Greek-letter organizations in the twenty-first century", Parks, Gregory, 2008
1890 births
1977 deaths
African-American schoolteachers
Bordentown School alumni
Schoolteachers from Delaware
American women educators
Delta Sigma Theta founders
Howard University alumni
People from Wilmington, Delaware
20th-century African-American women
20th-century African-American people
20th-century American people | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,148 |
{"url":"http:\/\/mathhelpforum.com\/advanced-algebra\/115390-normal-subgroups.html","text":"# Thread: Normal Subgroups\n\n1. ## Normal Subgroups\n\nLet H and K be normal subgroups of G such that H\u2229K = {e} and HK = G.\n\nProve that G \u223c= H \u00d7 K\n\n2. Originally Posted by elninio\nLet H and K be normal subgroups of G such that H\u2229K = {e} and HK = G.\n\nProve that G \u223c= H \u00d7 K\nHint - Can you define an isomorphism from G->HxK. (Use the fact G=HK and H\u2229K={e})\n\n3. Originally Posted by elninio\nLet H and K be normal subgroups of G such that H\u2229K = {e} and HK = G.\n\nProve that G \u223c= H \u00d7 K\n\nFirst show that if $h_1\\,,\\,h_2\\in H\\,,\\,\\,k_1\\,,\\,k_2\\in K\\,,\\,\\,then\\,\\,h_1k_1=h_2k_2\\Longleftrightarrow h_1=h_2\\,,\\,k_1=k_2$ , next define $f: G\\rightarrow H\\times K \\,\\,\\,as\\,\\,\\,f(g=hk)=(h,k)$ and show f is a isomorphism of groups.\n\nTonio\n\n4. Tonio, I'm not sure if this is the same thing as what you've posted but i've worked on this problem for over an hour and I know now that I can simply show that f is in isomorphism in a map defined f: H x K -> G by f((h,k)) = hk (Where hk is in G).\n\nDoes this seem correct so far? This is where I am having my trouble. I worked on one isomorphism problem similar to this before but I cant seem to apply any of the same methods or logic to this one.\n\nHow do I show f is an isomorphism in this problem?\n\n5. Originally Posted by elninio\nTonio, I'm not sure if this is the same thing as what you've posted but i've worked on this problem for over an hour and I know now that I can simply show that f is in isomorphism in a map defined f: H x K -> G by f((h,k)) = hk (Where hk is in G).\n\nDoes this seem correct so far? This is where I am having my trouble. I worked on one isomorphism problem similar to this before but I cant seem to apply any of the same methods or logic to this one.\n\nHow do I show f is an isomorphism in this problem?\n\nAfter you show what I proposed in my prior post, you have:\n$f\\left((h_1,k_1)+(h_2,k_2)\\right)=f(h_1h_2,k_1k_2) =h_1h_2k_1k_2=h_1k_1h_2k_2=f(h_1,k_1)f(h_2,k_2)$ since by now it must be clear H, K commute POINTWISE.\nAlso, $f(h,k)=hk=1=1\\cdot 1 \\Longleftrightarrow h=1=k\\Longrightarrow\\,Ker(f)=1$ , and surjectivity is immediate.\n\nTonio","date":"2017-01-19 11:10:26","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\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 4, \"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\": 0.9783340096473694, \"perplexity\": 431.99313668184016}, \"config\": {\"markdown_headings\": true, \"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-2017-04\/segments\/1484560280587.1\/warc\/CC-MAIN-20170116095120-00138-ip-10-171-10-70.ec2.internal.warc.gz\"}"} | null | null |
Sezemice kan verwijzen naar de volgende Tsjechische gemeenten:
Sezemice (okres Pardubice)
Sezemice (okres Mladá Boleslav) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,902 |
Q: groovy or java: how to retrieve a block of comments using regex from /** ***/? This might be a piece of cake for java experts. Please help me out:
I have a block of comments in my program like this:
/*********
block of comments - line 1
line 2
.....
***/
How could I retrieve "block of comments" using regex?
thanks.
A: Something like this should do:
String str =
"some text\n"+
"/*********\n" +
"block of comments - line 1\n" +
"line 2\n"+
"....\n" +
"***/\n" +
"some more text";
Pattern p = Pattern.compile("/\\*+(.*?)\\*+/", Pattern.DOTALL);
Matcher m = p.matcher(str);
if (m.find())
System.out.println(m.group(1));
(DOTALL says that the . in the pattern should also match new-line characters)
Prints:
block of comments - line 1
line 2
....
A: Pattern regex = Pattern.compile("/\\*[^\\r\\n]*[\\r\\n]+(.*?)[\\r\\n]+[^\\r\\n]*\\*+/", Pattern.DOTALL);
This works because comments can't be nested in Java.
It is important to use a reluctant quantifier (.*?) or we will match everything from the first comment to the last comment in a file, regardless of whether there is actual code in-between.
/\* matches /*
[^\r\n]* matches whatever else is on the rest of this line.
[\r\n]+ matches one or more linefeeds.
.*? matches as few characters as possible.
[\r\n]+ matches one or more linefeeds.
[^\r\n]* matches any characters on the line of the closing */.
\*/ matches */.
A: Not sure about the multi-line issues, but it were all on one line, you could do this:
^\/\*.*\*\/$
That breaks down to:
^ start of a line
\/\*+ start of a comment, one or more *'s (both characters escaped)
.* any number of characters
\*+\/ end of a comment, one or more *'s (both characters escaped)
$ end of a line
By the way, it's "regex" not "regrex" :)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,388 |
London-based Scottish chocolatier William Curley has created a lavish Easter egg exclusively for Blakes Hotel. Housed in a sophisticated black box, the design and flavours have been inspired by the decadent Anouska Hempel-designed hotel and its fusion of Western and Eastern influences. Crack open the milk chocolate egg to find exotic orangettes and ginger coated in dark chocolate.
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Claridge's has unveiled a limited-edition easter egg, and it's just as sophisticated as one might expect. Inspired by its iconic black and white lobby of the five-star hotel, the elegant egg has been handcrafted in milk, white and dark versions. Each one is generously filled with a selection of hand-made salted-caramel gull's eggs.
A chocolate teapot may not be the most practical, but it certainly proves the most delicious option this Easter. Inspired by the Mad Hatter's Tea Party, this whimsical creation is crafted from creamy milk chocolate and finished with a lustrous golden shimmer. Heston's design is accompanied by cocoa nib 'tea leaves' and crunchy 'sugar cubes' for the perfect, chocolate-fuelled tea party.
Certain to ruffle some feathers, Hotel Chocolat's Ostrich Easter egg is one for sharing (or not, we won't tell). The giant Easter egg comes in two flavours - dark and original - both of which are paired with decadent chocolate boxes. This unique product comprises a total of over a kilo's worth of chocolate – maybe it should be shared after all.
Eggs with legs? Yup, you read that right. This Harvey Nichols Easter egg comes in a hand-painted cup, so you can have a colourful keepsake long after the chocolate is consumed.
Master Belgian chocolatier Godiva has launched a sweet selection of Easter treats, including a milk chocolate Pixie Easter egg. Packaged in a spring-inspired box, the textured milk chocolate egg comes with a selection of brightly wrapped chocolate pops in white, blanc, and lait praliné.
This egg, created by Harrods' head pastry chef Alistair Birt and his team, features nineteen circular layers of decadent chocolate. The delicious work of art is topped with an edible gold leaf and the Harrods stamp. With only 50 eggs made, and time running out, make sure to get your hands on one before you miss out.
The luxury French brand has produced a range of gorgeous Easter products, including cute bunnies and classic eggs. The delicious eggs are filled with chocolate figurines so you'll be gorging for weeks.
The ultimate indulgent Easter treat comes from Cutter and Squidge, known for using only natural ingredients. The British confectioner's Billionaire Easter egg packs a milk chocolate half eggshell with crunchy chocolate crumb, salted caramel and creamy dark chocolate ganache, topped with mini eggs and crispy pearls. There is also a vegan-friendly option available, which uses dried fruit. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,577 |
\section*{Abstract}
Given a weighted graph $G$ with $n$ vertices and $m$ edges, and a positive integer $p$, the Hamiltonian $p$-median problem consists in finding $p$ cycles of minimum total weight such that each vertex of $G$ is in exactly one cycle. We introduce an $O(n^6)$ 3-approximation algorithm for the particular case in which $p \leq \lceil \frac{n-2\lceil \frac{n}{5} \rceil}{3} \rceil$. An approximation ratio of 2 might be obtained depending on the number of components in the optimal 2-factor of $G$. We present computational experiments comparing the approximation algorithm to an exact algorithm from the literature. In practice much better ratios are obtained. For large values of $p$, the exact algorithm is outperformed by our approximation algorithm.
\noindent {\bf Keywords:} Hamiltonian $p$-median problem, approximation algorithms, 2-factor, matching
\section{Introduction}
Given a graph $G=(V,E)$ with $n$ vertices and $m$ edges, a positive integer $p$, and a set of edge weights $\{c_e \in \mathbb{R}_+: e \in E\}$, the Hamiltonian $p$-median problem (H$p$MP) asks for $p$ cycles of minimum total weight, such that each vertex of $G$ is in exactly one cycle. A cycle is defined as a sequence of vertices $(v_1, v_2, \ldots, v_k, v_1)$ such that $k \geq 3$, $\{v_k, v_1\} \in E$, $\{v_i, v_{i+1}\} \in E$ for $1 \leq i < k$, and $v_i \neq v_j$, for $1 \leq i < j \leq k$. Note that the problem has a feasible solution only if $p \leq \left\lfloor \frac{n}{3} \right\rfloor$. In what follows, we assume that $E$ is complete and the edge weights satisfy the triangle inequality.
A particular case of the H$p$MP, when $p=1$, is the traveling salesman problem (TSP) \cite{DanFulJoh54}, one of the best known NP-Hard combinatorial optimization problems. Consequently, the H$p$MP is also NP-Hard. In fact, the TSP can be reduced to the H$p$MP for any value of $p$.
The H$p$MP was introduced by Branco and Coelho \cite{BraCoe90}.
Formulations and exact approaches were studied in \cite{Glaab00, Zohrehbandian07, HpMP-INOCold, GolGouLap14, MarCenUst16, ErdLapChi18}. A polyhedral study was conducted in \cite{HupLie13}. An iterated local search heuristic was introduced in \cite{ErdLapChi18}. Not many heuristic approaches have been proposed for the problem.
Since polynomial time approximation algorithms exist for the TSP \cite{Vaz01}, a natural research question is whether such algorithms exist for the H$p$MP. In this paper, we introduce a 3-approximation algorithm that runs in $O(n^6)$ for the particular case of the H$p$MP in which $p \leq \lceil \frac{n-2\lceil \frac{n}{5} \rceil}{3} \rceil$. To the best of our knowledge, this is the first approximation algorithm proposed for the problem.
The rest of the paper is organized as follows. The approximation algorithm is introduced in Section \ref{sec:alg}, along with complexity and approximation results. In Section \ref{sec:exp}, we present some experimental results, comparing the approximation to an exact algorithm. Concluding remarks are given in Section \ref{sec:conc}.
\section{The Approximation Algorithm}\label{sec:alg}
Before presenting the algorithm, we introduce some notation. The notation $S \subseteq G$ is used to indicate that $S$ is a subgraph of $G$. Given $S \subseteq G$, we let $V(S)$ denote its set of vertices and $E(S)$ denote its set of edges. We define the weight of $S$ as $c(S) = \sum_{e \in E(S)} c_e$. We refer to cycles containing exactly $k$ vertices as $k$-cycles. The optimal H$p$MP solution is denoted $H^*$.
The approximation algorithm we propose is based on the concept of 2-factors. A 2-factor of a graph is a set of cycles such that each vertex is in exactly one cycle. A related concept is that of 2-matchings, the difference being that 2-matchings might have cycles containing only 2 vertices. Given a weighted graph, a 2-factor having minimum total weight can be found in polynomial time, using matching techniques. One possible approach, presented in \cite{CooCunPul97}, is reproduced in Section \ref{sec:2f}.
As any feasible H$p$MP solution is a 2-factor, the minimum weight 2-factor yields a lower bound on the optimal H$p$MP solution.
The approximation algorithm is presented in Algorithm \ref{alg}.
\begin{algorithm}[htpb]
\begin{algorithmic}[1]
\State Find a minimum weight 2-factor $F$ of $G$. Note that $c(F) \leq c(H^*)$. Let $q$ be the number of cycles in $F$.
\If {$q = p$}
\State $H \leftarrow F$.
\Else
\If {$q > p$}
\State Find a spanning tree $T$ of $G$, remove its $p-1$ edges with largest weight. Note that after the removal $c(T) \leq c(H^*)$ and the number of components in $T$ is $p$, but some of its components might be singletons.
\State Since $T$ has $p$ components, its union with $F$ will create a graph with $p$ or less components. Add to $F$ edges of $T$ connecting different components of $F$ until it has $p$ components. Note that after this $c(F) \leq 2c(H^*)$.
\State Transform $F$ into a multigraph by duplicating the edges that were added from $T$, $c(F) \leq 3c(H^*)$.
\ElsIf {$q < p$}
\While {$F$ does not have $p$ components}
\State Select a component of $F$ with at least six vertices. For the special case of $p \leq \lceil \frac{n-2\lceil \frac{n}{5} \rceil}{3} \rceil$ such a component always exists. This component is either a cycle $(v_1, v_2, \ldots, v_k, v_1)$ or a path $(v_1, v_2, \ldots, v_k)$. Irrespective of the case, replace it by two paths $(v_1, v_2, v_3)$ and $(v_4, \ldots, v_k)$. Note that this does not increase $c(F)$, i.e., $c(F) \leq c(H^*)$.
\EndWhile
\State Transform $F$ into a multigraph by duplicating all of its edges, $c(F) \leq 2c(H^*)$.
\EndIf
\State Note that $F$ is a multigraph with $p$ connected components in which the degree of each vertex is even. The approximate solution $H$ is obtained by following an Eulerian tour for each component of $F$, skipping vertices already visited.
\EndIf
\State {\bf Return} $H$. If case $q=p$ was executed $H$ is an optimal solution. If case $q>p$ was executed, $H$ is a 3-approximation. If case $q<p$ was executed, $H$ is a 2-approximation.
\end{algorithmic}
\caption{H$p$MP approximation algorithm.}
\label{alg}
\end{algorithm}
For general values of $p$, it might happen that in step 11 the number of components in $F$ is less than $p$ and there are no components with six or more vertices. As an example, if $n=10$, $p=3$, and the optimal 2-factor has 2 connected components with 5 vertices each, the algorithm will fail. This type of problem is caused by components of $F$ whose number of vertices is not a multiple of 3. When splitting components in step 11, a component with $k$ vertices will have $k \mod 3$ vertices that cannot be left alone, as they are not enough to form a cycle. If we try to connect them to vertices coming from other components we might violate the approximation ratio guarantee. For the special case we consider, $p \leq \lceil \frac{n-2\lceil \frac{n}{5} \rceil}{3} \rceil$, a component with at least six vertices will always exist in step 11. This is proven below, we will need the following result first:
\begin{lemma}
Given a 2-factor $F$ of $G$ composed of $q$ cycles $\{C_1, \ldots, C_q\}$ let $l(F) = \sum_{i = 1}^{q} (|V(C_i)| \mod 3)$ be the number of vertices that cannot form cycles by themselves in step 11 of Algorithm \ref{alg}. Then, the maximum possible value of $l(F)$ over all 2-factors of $G$ is not greater than $2\left\lceil \frac{n}{5} \right\rceil$.
\end{lemma}
\begin{proof}
We first show that the maximum of $l(F)$ is attained when $F$ is a 2-factor having the maximum possible number of 5-cycles. Let $F^*$ be a 2-factor attaining the maximum value of $l$. Let $r$ be the size of the largest cycle of $F^*$. If $F^*$ is not a 2-factor with the maximum possible number of 5-cycles, we can always construct another two factor from $F^*$ having one more 5-cycle than $F^*$ without decreasing $l(F^*)$.
If $r \geq 8$, replace the corresponding cycle by two cycles, one containing 5 and the other containing the remaining vertices. If $r \leq 7$, there must be another cycle with number of vertices $r'$ such that $r'\leq 7$ and $r' \neq 5$, otherwise $F^*$ has the maximum possible number of 5-cycles. Now, it is a matter of examining the possibilities and showing that for each of them we can construct another 2-factor having more 5-cycles than $F^*$ yielding the same value (or a larger value) of $l(F^*)$. The possibilities are:
\begin{itemize}
\item $r = 7$ and $r' = 7$, we can generate cycles with 5, 5, and 4 vertices, respectively.
\item $r = 7$ and $r' = 6$, we can generate cycles with 5, 5, and 3 vertices, respectively.
\item $r = 7$ and $r' = 4$, we can generate cycles with 5 and 6 vertices, respectively.
\item $r = 7$ and $r' = 3$, we can generate cycles with 5 and 5 vertices, respectively.
\item $r = 6$ and $r' = 6$, we can generate cycles with 5 and 7 vertices, respectively.
\item $r = 6$ and $r' = 4$, we can generate cycles with 5 and 5 vertices, respectively.
\item $r = 6$ and $r' = 3$, we can generate cycles with 5 and 4 vertices, respectively.
\item $r = 4$ and $r' = 4$, we can generate cycles with 5 and 3 vertices, respectively.
\item $r = 4$ and $r' = 3$, $F^*$ already has the maximum number of 5-cycles.
\end{itemize}
Now, if $F$ has the maximum number of 5-cycles, than it has at least $\left\lfloor \frac{n}{5} \right\rfloor-1$ 5-cycles. The possible values for the remaining $n-(\left\lfloor \frac{n}{5} \right\rfloor-1)$ vertices are $5, \ldots, 9$. If there are 5 remaining vertices, the maximum of $l(F)$ is attained by organizing them into a 5 cycle, in which case $l(F) = 2\frac{n}{5}$. If there are 6 remaining vertices, the maximum of $l(F)$ is attained by organizing them into a 6-cycle, in which case $l(F) = 2(\left\lfloor \frac{n}{5} \right\rfloor - 1)$. If there are 7, the maximum is attained by organizing them into a 7-cycle, $l(F) = 2(\left\lfloor \frac{n}{5} \right\rfloor -1) + 1$. If there are 8, by organizing them into a 5-cycle and a 3-cycle, $l(F) = 2\left\lfloor \frac{n}{5} \right\rfloor$. And if there are 9, by organizing them into a 5-cycle and a 4-cycle, $l(F) = 2\left\lfloor \frac{n}{5} \right\rfloor + 1$. In all these cases, $l(F) \leq 2\left\lceil \frac{n}{5} \right\rceil $.
\end{proof}
\begin{proposition}
\label{pro}
It is possible to split any 2-factor of a graph $G$ into at least $\left\lceil \frac{n - 2\left\lceil \frac{n}{5} \right\rceil }{3} \right\rceil$ connected components containing at least 3 vertices each.
\end{proposition}
\begin{proof}
Given a 2-factor $F$ of $G$ composed of $q$ cycles $\{C_1, \ldots, C_q\}$, the maximum number of connected components with at least 3 vertices it can be split into is $\frac{n-l(F)}{3} \geq \left\lceil \frac{n - 2\left\lceil \frac{n}{5} \right\rceil }{3} \right\rceil$, since $l(F) \leq 2\left\lceil \frac{n}{5} \right\rceil$ and the number of resulting components is integer.
\end{proof}
Thus, for $p \leq \left\lceil \frac{n - 2\left\lceil \frac{n}{5} \right\rceil }{3} \right\rceil$, Algorithm \ref{alg} is guaranteed to find a feasible solution.
\begin{theorem}
For the special case of the H$p$MP with $p \leq \lceil \frac{n-2\lceil \frac{n}{5} \rceil}{3} \rceil$, Algorithm \ref{alg} finds a solution $H$ with $c(H) \leq 3c(H^*)$ in $O(n^6)$ time.
\end{theorem}
\begin{proof}
The dominating term in the computational complexity of Algorithm \ref{alg} is given by step 1. The minimum weight 2-factor of a graph $G$ can be found by executing a minimum weight perfect matching (MWPM) algorithm on an auxiliary graph $G'$ with $O(n^2)$ vertices and $O(m)$ edges, obtained from $G$. The MWPM of a graph $G$ can be found in $O(n^2m)$ \cite{Edm65}, thus, finding the MWPM of $G'$ is $O(n^6)$ as we assume $E$ to be complete. The approximation factor was proven along the algorithm description.
\end{proof}
For the sake of completeness, in the next section we present a scheme for finding a minimum weight 2-factor of $G$.
\subsection{Finding minimum weight 2-factors of $G$}\label{sec:2f}
Minimum weight 2-factors of $G$ can be found on a appropriately defined auxiliary graph, as described in \cite{CooCunPul97}, pp. 185.
First, define a graph $G' = \{V', E'\}$: Replace each edge $e =\{u,v\}$ of $E$ by two new vertices $a_e^u$ and $a_e^v$ and three new edges $\{u, a_e^u\}$, $\{a_e^u, a_e^v\}$, and $\{a_e^v, v\}$.
Now, define a graph $G'' = \{V'', E''\}$ from $G'$: For each original vertex $v \in V \cap V'$, replace it by two new vertices $b'_v$ and $b_v''$. Replace every edge $\{v, a_e^v\}$ by two new edges $\{b'_v, a_e^v\}$ and $\{b''_v, a_e^v\}$, both with weight $c''_{\{b'_v, a_e^v\}} = c''_{\{b''_v, a_e^v\}} = \frac{c_e}{2}$. Let $c''_e = 0$ for any other edge in $E''$.
An edge $e = \{u,v\} \in E$ is in the minimum weight 2-factor of $G$ if and only if $\{b'_u, a_e^u\}$ or $\{b''_u, a_e^u\}$ is in the minimum weight perfect matching of $G''$, in which case either $\{b'_v, a_e^v\}$ or $\{b''_v, a_e^v\}$ will also be in the matching.
\section{Computational Experiments}\label{sec:exp}
In this section, we present results from computational experiments that were conducted to evaluate the practical approximation ratio of the algorithm introduced here, as well as its performance in comparison to an exact algorithm.
We consider the instances with 100 vertices among those proposed in Gollowitzer et al. \cite{HpMP-INOCold}, they comprise 5 complete Euclidean graphs whose vertices were randomly generated by sampling 100 points in the square $[0,100]^2$. These instances are numbered 1-5. Besides these 5 instances, 5 more instances were generated following the same scheme. These instances are numbered 6-10. Since $n=100$ in all instances, the special H$p$MP case for which our algorithm was devised consists of the instances with $p \leq 20$. Besides values of $p$ in this range we also considered larger values, in order to evaluate the algorithm's behavior for more general values of $p$. The following values of $p$ were considered: $\{2, 10, 18, 26, 33\}$.
Gollowitzer et al. \cite{GolGouLap14} propose several models from the H$p$MP. The exact algorithm we use in the experiments is the one based on ``model 1'', the best performing algorithm in that reference. We had access to its source code and ported it to the version 12.63 of the IBM CPLEX solver.
The computational experiments were carried in an Intel XEON machine with 8GB of RAM and 8 cores running at 3.5GHz, running under Linux Ubuntu 14.04. Both the exact and the approximation algorithm were implemented in C++ and evaluated on this same computational environment. A time limit of 1 hour was imposed for the exact algorithm. The source code for the approximation algorithm can be found in \cite{GITHUB}.
The computational results are presented in Tables \ref{tab1} and \ref{tab2}. Table \ref{tab1} presents results for $p \in \{2, 10, 18\}$ while Table \ref{tab2} presents results for $p \in \{26, 33\}$. Under the heading ``Instance'' we present instance information: the value of $p$ and a number identifying the instance. Under the heading ``Exact Algorithm'' we present results for the exact algorithm: the lower ($lb$) and upper bound ($ub$) upon the algorithm termination, and the total computational time in seconds ($t(s)$). Under the heading ``Approximation Algorithm``, we present results for the approximation algorithm: the obtained upper bound ($ub$), the total computational time ($t(s)$), the approximation factor (apx) in relation to the lower bound provided the exact algorithm and the number of cycles ($q$) in the 2-factor computed in step 1 of Algorithm \ref{alg}.
\begin{table}[htbp]
\centering
\footnotesize
\begin{tabular}{@{}lrrrrrrrrrr@{}}
\toprule
\multicolumn{2}{c}{Instance} & & \multicolumn{3}{c}{Exact Algorithm} & & \multicolumn{4}{c}{Approximation Algorithm}\\
\cmidrule{1-2}\cmidrule{4-6}\cmidrule{8-11}
$p$ & ID & & $lb$ & $ub$ & $t(s)$ & & $ub$ & $t(s)$ & apx & $q$\\
\midrule
2 & 1 & & 782.29 & 782.29 & 665.7 & & 1018.72 & 2.57 & 1.3 & 16\\
& 2 & & 781.52 & 781.52 & 38.7 & & 920.02 & 2.99 & 1.17 & 11\\
& 3 & & 732.24 & 732.24 & 132.95 & & 850.01 & 2.49 & 1.16 & 9\\
& 4 & & 789.62 & 789.62 & 559.4 & & 873.28 & 2.87 & 1.1 & 9\\
& 5 & & 748.19 & 748.19 & 134.2 & & 962.85 & 2.62 & 1.28 & 16\\
& 6 & & 761.12 & 761.12 & 73.12 & & 815.12 & 2.63 & 1.07 & 7\\
& 7 & & 754.79 & 754.79 & 76.56 & & 922.95 & 2.57 & 1.22 & 14\\
& 8 & & 752.55 & 783.93 & 3600 & & 1067.03 & 2.62 & 1.41 & 22\\
& 9 & & 778.12 & 778.12 & 235.31 & & 1059.02 & 2.61 & 1.36 & 19\\
& 10 & & 787.75 & 787.75 & 19.31 & & 965.22 & 2.45 & 1.22 & 14\\
\cmidrule{2-11}
& avg. & & 766.81 & 769.95 & 553.52 & & 945.42 & 2.64 & 1.22 & 13.7\\
\midrule
10 & 1 & & 758.52 & 758.52 & 5.9 & & 838.53 & 2.54 & 1.1 & 16\\
& 2 & & 770.25 & 770.25 & 3.9 & & 780.97 & 2.93 & 1.01 & 11\\
& 3 & & 714.49 & 714.49 & 2.39 & & 742.5 & 2.63 & 1.03 & 9\\
& 4 & & 764.74 & 764.74 & 2.44 & & 796.01 & 2.77 & 1.04 & 9\\
& 5 & & 724.61 & 724.61 & 15.03 & & 798.97 & 2.58 & 1.1 & 16\\
& 6 & & 748.96 & 748.96 & 50.68 & & 776.61 & 2.61 & 1.03 & 7\\
& 7 & & 737.49 & 737.49 & 7.17 & & 771.79 & 2.44 & 1.04 & 14\\
& 8 & & 721.92 & 721.92 & 95.25 & & 879.43 & 2.53 & 1.21 & 22\\
& 9 & & 751.18 & 751.18 & 18.49 & & 885.37 & 2.6 & 1.17 & 19\\
& 10 & & 767.77 & 767.77 & 1.09 & & 811.65 & 2.45 & 1.05 & 14\\
\cmidrule{2-11}
& avg. & & 745.99 & 745.99 & 20.23 & & 808.18 & 2.60 & 1.07 & 13.7\\
\midrule
18 & 1 & & 754.15 & 754.15 & 11.04 & & 779.62 & 2.55 & 1.03 &16 \\
& 2 & & 776.21 & 782.54 & 3600 & & 843.03 & 2.98 & 1.08 & 11 \\
& 3 & & 720.43 & 735.43 & 3600 & & 764.55 & 2.64 & 1.06 & 9 \\
& 4 & & 769.5 & 770.23 & 3600 & & 883.84 & 2.69 & 1.14 & 9 \\
& 5 & & 723.34 & 723.34 & 12.19 & & 773.38 & 2.56 & 1.06 & 16 \\
& 6 & & 750.09 & 750.09 & 32.62 & & 871.85 & 2.63 & 1.16 & 7 \\
& 7 & & 734.91 & 734.91 & 18.61 & & 817.68 & 2.51 & 1.11 & 14 \\
& 8 & & 709.13 & 709.13 & 1.63 & & 740.2 & 2.53 & 1.04 & 22 \\
& 9 & & 744.04 & 744.04 & 1.83 & & 754.58 & 2.6 & 1.01 & 19 \\
& 10 & & 772.45 & 772.72 & 3600 & & 820.37 & 2.45 & 1.06 & 14 \\
\cmidrule{2-11}
& avg. & & 745.42 & 747.65 & 1447.79 & & 804.91 & 2.614 & 1.07 & 13.7\\
\bottomrule
\end{tabular}
\caption{Results for instances with $n=100$ and $p \in \{2, 10, 18\}$.}
\label{tab1}
\end{table}
\begin{table}[htbp]
\centering
\footnotesize
\begin{tabular}{@{}lrrrrrrrrrr@{}}
\toprule
\multicolumn{2}{c}{Instance} & & \multicolumn{3}{c}{Exact Algorithm} & & \multicolumn{4}{c}{Approximation Algorithm}\\
\cmidrule{1-2}\cmidrule{4-6}\cmidrule{8-11}
$p$ & ID & & $lb$ & $ub$ & $t(s)$ & & $ub$ & $t(s)$ & apx & $q$\\
\midrule
26 & 1 & & 763.22 & 779.39 & 3600 & & 821.87 & 2.56 & 1.07 &16 \\
& 2 & & 780.01 & 838.97 & 3600 & & 882.28 & 3.1 & 1.13 & 11 \\
& 3 & & 724.36 & 776.42 & 3600 & & 852.75 & 2.48 & 1.17 & 9 \\
& 4 & & 773.17 & 856.18 & 3600 & & 885.26 & 2.75 & 1.14 & 9 \\
& 5 & & 730.45 & 752.31 & 3600 & & 825.69 & 2.56 & 1.13 & 16 \\
& 6 & & 757.35 & 814.33 & 3600 & & 917.27 & 2.63 & 1.21 & 7 \\
& 7 & & 743.99 & 768.43 & 3600 & & 853.86 & 2.51 & 1.14 & 14 \\
& 8 & & 713.33 & 714.26 & 3600 & & 750.86 & 2.52 & 1.05 & 22 \\
& 9 & & 750.11 & 757.43 & 3600 & & 784.2 & 2.59 & 1.04 & 19 \\
& 10 & & 777.39 & 795.51 & 3600 & & 881.95 & 2.44 & 1.13 & 14 \\
\cmidrule{2-11}
& avg. & & 751.33 & 785.32 & 3600 & & 845.59 & 2.61 & 1.12 & 13.7\\
\midrule
33 & 1 & & 769.34 & 1006.93 & 3600 & & 826.08 & 2.57 & 1.07 &16 \\
& 2 & & 783.96 & 1086.46 & 3600 & & 877.14 & 3.1 & 1.11 & 11 \\
& 3 & & 729.49 & 962.52 & 3600 & & 838.51 & 2.59 & 1.14 & 9 \\
& 4 & & 778.92 & 1092.88 & 3600 & & 869.3 & 2.74 & 1.11 & 9 \\
& 5 & & 736.38 & 893.25 & 3600 & & 825.47 & 2.56 & 1.12 & 16 \\
& 6 & & 762.55 & 1195.72 & 3600 & & 930.87 & 2.75 & 1.22 & 7 \\
& 7 & & 749.7 & 1044.82 & 3600 & & 858.56 & 2.66 & 1.14 & 14 \\
& 8 & & 717.85 & 1045.19 & 3600 & & 766.35 & 2.5 & 1.06 & 22 \\
& 9 & & 754.22 & 891.91 & 3600 & & 833.5 & 2.62 & 1.1 & 19 \\
& 10 & & 783.52 & 1149.15 & 3600 & & 908.93 & 2.47 & 1.16 & 14 \\
\cmidrule{2-11}
&avg. & & 756.59 & 1036.88 & 3600 & & 853.47 & 2.65 & 1.12 & 13.7\\
\bottomrule
\end{tabular}
\caption{Results for instances with $n=100$ and $p \in \{26, 33\}$.}
\label{tab2}
\end{table}
In terms of performance, the approximation algorithm took between 2.4 and 3.1 seconds to solve the instances, with no significant difference in time across the different values of $p$. The exact algorithm, on the other hand, was not able to find the optimal solution for large values of $p$ under the time limit of one hour, and for the smallest values the time taken varied significantly from instance to instance.
In terms of solution quality, the approximation algorithm achieved practical approximation ratios between 1.01 and 1.41. The exact algorithm was able to find the optimal solution for low values of $p$, and obtained relatively small optimality gaps for instances with mid-range values of $p$. However, under the time limit of one hour, for the $p$ value of 33, the approximation algorithm was able to find better solutions for all instances.
On average, the value of $q$ was 13.7, which explains the fact that a better practical approximation ratio was obtained for instances with $p=10$ and $p=18$. The quality of the approximation also seems to be better among the $p > q$ cases, in which the average gap was 1.10 as opposed to 1.16 for the other case.
These results highlight the efficiency of the the approximation algorithm: small computational times and a good practical approximation factor, which in practice was much better than the theoretical factor of 3.
\section{Conclusion}\label{sec:conc}
In this paper, we introduced a 3-approximation algorithm for the special $p \leq \lceil \frac{n-2\lceil \frac{n}{5} \rceil}{3} \rceil$ case of the H$p$MP. The first H$p$MP approximation algorithm in the literature to the best of our knowledge.
We conducted computation experiments to assess the practical approximation ratio of the proposed algorithm, comparing it to an exact algorithm from the literature. The practical approximation ratio was 1.1, on average, with computational times orders of magnitude smaller than the exact algorithm. In fact, the approximation algorithm proves to be specially helpful for higher values of $p$, for which the exact algorithm has a hard time finding the optimal solution.
As future research, it might be interesting to look for new approximation algorithms, with better approximation ratios and time complexities, and capable of handling general values of $p$. Another possible line of research is the combination of the approximation algorithm with local search procedures.
\bibliographystyle{plainnat}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,580 |
The USDA says this year's U.S. corn and soybean harvests are both about a third of the way complete, but the soybean pace has fallen behind average because of rain in some key U.S. growing areas.
As of Sunday, 34% of corn is harvested, compared to the five-year average of 26%, and 93% of the crop has reached maturity. 68% of U.S. corn is in good to excellent condition, 1% less than last week, but 4% more than this time last year.
32% of U.S. soybeans are harvested, compared to 36% on average, with several states slower than normal, including Iowa, Michigan, Minnesota, Ohio, South Dakota, and Wisconsin. 91% of beans are dropping leaves, compared to the usual pace of 85%. 68% of this year's U.S. crop is rated good to excellent, unchanged on the week and up 7% on the year.
57% of U.S. winter wheat is planted and 30% has emerged, both slightly ahead of average.
48% of U.S. pastures and rangelands are in good to excellent condition, steady with a week ago. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,172 |
Q: Retrieving the latest result in sql I have written sql query that is pulling agreements data. I need to pull the latest agreement. The latest version is determined based on the most recent version. As you can see currently my query is displaying two records. Please note that version is varchar field
Query
SELECT ua.ID AS UserAgreementID ,
A.ID AS AgreementID ,
A.Code ,
A.ComplianceCode ,
A.Name ,
A.Description ,
A.Version ,
ua.UserAgreementStateID ,
uas.Name AS UserAgreementStateName ,
ua.AcceptanceWindowExpiry ,
declaration.GetDifferenceInDaysOrHours(ua.AcceptanceWindowExpiry) AS TimeLeft ,
A.Data ,
pa.ID AS AuthoredByID ,
pa.FirstName + ' ' + pa.LastName AS AuthoredByName ,
A.Authored ,
ia.ID AS IssuedByID ,
ia.FirstName + ' ' + pa.LastName AS IssuedByName ,
A.Issued
FROM declaration.Agreement AS A
INNER JOIN declaration.UserAgreement AS ua ON A.ID = ua.AgreementID
INNER JOIN declaration.UserAgreementState AS uas ON ua.UserAgreementStateID = uas.ID
LEFT JOIN common.Person AS pa ON A.AuthoredBy = pa.ID
LEFT JOIN common.Person AS ia ON A.IssuedBy = ia.ID WHERE ua.UserID = 607
AND uas.Code IN ('ISS',
'DEF','EXP')-- Issued, Deferred
AND A.Draft = CONVERT(BIT, 0) -- Not a draft.
AND A.Deleted = CONVERT(BIT, 0) -- Not deleted.
AND (A.Issued <= GETUTCDATE()
OR A.Issued IS NULL)
AND (A.Expires > GETUTCDATE()
OR A.Expires IS NULL)
ORDER BY UA.AcceptanceWindowExpiry ASC
As you can see there are two versions. I need my query should pull only 2.1.0.000 version. Do I need to do max of the version or any other way ? If I do Max() , it shows the latest record on the top but i need only the latest and not the second record
A: You can use top (1) :
SELECT TOP (1) ua.ID AS UserAgreementID, A.ID AS AgreementID,
. . . .
FROM declaration.Agreement AS A INNER JOIN
declaration.UserAgreement AS ua
ON A.ID = ua.AgreementID INNER JOIN
declaration.UserAgreementState AS uas
ON ua.UserAgreementStateID = uas.ID LEFT JOIN
common.Person AS pa
ON A.AuthoredBy = pa.ID LEFT JOIN
common.Person AS ia
ON A.IssuedBy = ia.ID
WHERE ua.UserID = 607 AND
uas.Code IN ('ISS','DEF','EXP') AND -- Issued, Deferred
A.Draft = CONVERT(BIT, 0) AND -- Not a draft.
A.Deleted = CONVERT(BIT, 0) AND -- Not deleted.
(A.Issued <= GETUTCDATE() OR A.Issued IS NULL) AND
(A.Expires > GETUTCDATE() OR A.Expires IS NULL)
ORDER BY A.Version DESC;
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,426 |
\section{Introduction}
\
Myelination plays a major role in the neurological development of infants.
In the central nervous system, oligodendrocytes, which are myelinogenic cells, wrap their cytoplasm around neuronal axons, forming a multilayered structure. \cite{Dobbing1973Quant}
The multiple myelin layers around an axon functions as an insulator, enabling extremely high-speed transmission of electric signals by saltatory conduction.
This high-speed electrical signal transmission enables rapid neurological development of infants. Myelination is already initiated before birth; at birth, myelination is seen in the brain stem, cerebellar white matter, posterior limb of the internal capsule, and subcortical white matter of primary motor and sensory cortices. After birth, myelination progresses in the brain caudally to rostrally, posteriorly to anteriorly, and centrally to peripherally. Myelin maturation progresses on a monthly basis. (Figure 1)
Magnetic resonance imaging (MRI) visualizes myelin maturation in white matter as a high signal intensity on T1-weighted images and a low signal intensity on T2-weighted images.
\cite{Holland BA1986MRI}
T1 / T2 shortening reflects an increase of lipid and protein and a decrease of water content associated with myelin maturation. \cite{McArdle CB1987Developmental} MRI evaluation of myelin maturation based on an alteration of MR signal is one of the objective evaluation methods of neurological development of infants. In evaluation of the neurological development of infants and children, T1-weighted images are useful in the early stage of myelination and T2-weighted images in the late stage. Clinically, radiologists and pediatricians often spend their precious time and effort to accurately predict neurological growth in children using brain MRI.
\
A convolutional neural network (CNN) is a machine learning technique that excels in image recognition and classification. CNN architecture is composed of a multilayered convolutional layer and a classifier that can extract features of objects such as size, shape and density patterns from various levels of pixel units in each region. The superior CNN image recognition / classification ability may be able to contribute to the evaluation of myelination using brain MRI. The purpose of this study was to construct a machine learning model to estimate the age of infants from MR signal alteration of myelination using CNN architecture.
\section{Materials and methods}
This retrospective non-invasive and non-intervention study was approved by the institutional review board of Juntendo University Hospital and opt-out alternatives was approved to obtaining informed consent for subjects.
\subsection{Participants}
The potentially eligible participants were 908 infants and children from birth to 24 months after birth, whom head MR imaging was performed in our institution from January 2014 to December 2015.
Participants with obvious neurological symptoms or received some treatment intervention in the course of the subsequent two years were excluded. 133 participants were extracted under this exclusion condition. In addition, 14 participants with insufficient MR image quality with a motion artifact were excluded. As a result, 119 infants and children up to 2 years old, 40 boys and 79 girls were recruited for this study.(Figure 2)
Each participants age was corrected by converting gestational age to a full 40 weeks, the corrected ages were from - 1.71 to 23.54 months.
\subsection{MR imaging}
Whole brain 2D T1- and T2-weighted images were obtained using a 3T MR unit (Achieva, Philips) in 95 subjects and a 1.5T MR unit (Magnetom Avanto, Siemens) in 24 subjects. The sequence parameters of MRI were as follows. Achieva, Philips 3T MR unit: T1WI; Turbo-Inversion recovery, TR/TE/TI: 2155/10/1000 msec, echo train length: 4, flip angle: 90 degrees; T2WI; turbo spin echo, TR/TE: 4000/80 msec, echo train length: 11, NEX: 2, FOV 180–230 mm, matrix 512x512, slice thickness: 5 mm. Magnetom Avanto, Siemens 1.5T MR unit: T1WI; spin echo; TR/TE: 530/11msec, echo train length: 1, flip angle: 80 degrees; T2WI; turbo spin echo; TR/TE: 4100/94, echo train length: 9, NEX: 2, FOV: 200–260 mm, matrix: 256x256, slice thickness: 5.5 mm.
\subsection{Data sets}
Five T1- and T2-weighted images at the levels of (a) the middle cerebellar peduncle of the pons, (b) midbrain, (c) internal capsule and splenium of corpus callosum, (d) central semiovale, and (e) subcortical white matter, were extracted by a neuroradiologist with 25 years of clinical experience (A.W) and adapted for input of machine learning. The corrected age from a subject's gestational age was adopted as the label of the supervised learning. We prepared two types of data sets to evaluate the effect of preprocessing of data sets on the accuracy of machine learning. One data set consisted of all 119 subjects and the other was composed of two components: from birth to 16 months and from 8 to 24 months.
\subsection{Machine learning framework}
Eight layers CNN architecture was consisting of 2 convolutional layers and 6 fully connected layers were developed using Neural network console ver. 1.2 (https://dl.sony.com/) on a Windows PC Intel Core i7 2.2 GH, with 32 GB memory and an NVIDIA GeForce GTX 1070 graphical processing unit. (Figure 2) The input image of 128x128 gray scale mediated by 2 convolution layers (5x5 convolutional kernel with valid padding, strides 1, 1 and out map was 16 and 8) and 2 Max Pooling layers (2x2 kernel and stride 2, 2) and 29 x 29 x8 feature maps was transmitted to Dense / fully connected layer (Dense) with 100 nodes. In each layers the rectified linear unit (ReLU) was applied as activation function. The output from 5 level T1 weighted images are concatenated and transmitted to 2 Dense layers with 100 nodes and concatenated with output from T2 weighted images and transmitted to Dense layers with 200, 50, 25 nodes with activation function ReLU and the learning model was trained to minimize the loss function Squared Error. Dense layer, which works as a classifier, was set in multiple levels at each MR image levels, T1 and T2weighted contrast levels and in all concatenated data. Two algorithms were constructed for infant age estimation using different two types of data sets. (Figure 3) One algorithm was a simple machine learning model that learned using the all-age data set. The other was an ensemble algorithm combining two learning models; one model learned with the data set from birth to 16 months and another model learned with the data set from 8 to 24 months. The latter split data set adoption is a configuration that takes into account the weight of myelin-related T1 / T2 signals up to 12 months and after that to 24 month.
The following hyperparameters for training were common in both models: 1000 epochs; base learning rate for the untrained model, Adam (learning rate = 0.001, beta1 = 0.9, beta2 = 0.999, epsilon = 0.00000001).
\subsection{Performance evaluation of the machine learning model}
We compared the two machine learning models by Spearman's rank-order correlation test, the root mean square error (RMSE) and the mean absolute error (MAE) to investigate correlations between the estimated age and a subject's corrected age.
\section{Result}
There was a strong correlation between the estimated age by the two machine learning models and the corrected age of subjects. In model A, which learned with the all-age data set, the correlation coefficient, RMSE and MAE were 0.81 (p < 0.001), 3.40 and 2.28. In model B, which consisted of an ensemble of two data sets, learning showed a superior correlation coefficient, RMSE and MAE, of 0.93 (p < 0.001), 2.12 and 1.34, respectively. (Figure 4, Table 1)
\section{Discussion}
Myelination of axons enables high-speed transmission of electrical signals by fast saltatory conduction. Fast neural networks constructed with myelinated nerve fibers support rapid neurological growth and maturation of higher order cognitive functions in neonates and infants. In the central nervous system, the processes of oligodendrocytes, which are myelin-forming cells, become flattened and wind around axons like a roll of paper. The cytoplasm within the process gradually disappears and adjacent membranes adhere to each other, forming a sheath around axons. The myelin sheath is constructed of a multilayered spiral containing lipid layers and proteins. The lipid layers are composed of cholesterol, phospholipids and glycolipids and the ratio is approximately 4:3:2.
\cite{Barkovich AJ2000Concepts} Lipid layers support the saltatory conduction of myelinated fibers, allowing high-speed electrical transmission by acting as an insulator. Proteins maintain stability between the multiple membrane layers surrounding an axon. The protein component of myelin sheathes is composed of myelin basic protein and proteolipid protein, which are myelin-specific proteins. Associated with myelination of an infant's brain, the water content of white matter decreases and glycolipids and proteins of cell membranes constituting the myelin sheath increase.
By MRI, myelination of white matter leads to shortening of the T1 / T2 relaxation time; mature myelin exhibits a high T1 signal and low T2 signal compared to gray matter.
\cite{Holland BA1986MRI}
The high T1 signal originates from an increase of cholesterol and galactocerebroside in the glycolipids that are the main components of the myelin sheath lipid layers.
\cite{Kucharczyk W1994Relaxivity, Koenig W1990Relaxometry}
These molecules act on the magnetization transfer effect in adjacent free water to shorten the T1 relaxation time. Myelin maturation begins before birth; the full-term infant brain presents a high signal on the dorsal side of the brainstem, cerebellar peduncle and thalamus on T1-weighted images at birth. These MR signal alterations proceed through the brain from caudal to rostral, posteriorly to anteriorly and centrally to peripherally, along with myelin maturation. Three months after birth, the high T1 signal reaches the cerebellar white matter, the posterior limb of internal capsule, the periventricular white matter, and the subcortical white matter near the central sulcus. By 6 months, it reaches the optic radiation and the splenium of the corpus callosum and by 12 months, it reaches the entire white matter of the cerebral hemispheres.
T1-weighted imaging is sensitive at an early stage of myelination, and its signal pattern visually exhibits an adult type approximately 8 months after birth.
\cite{Barkovich AJ1988Normal, Bird CR1989MR, Christophe C1900Mapping, van der Knaap1990MR, Martin E1988Developmental, Martin E1991MR}
Conversely, alteration of the T2 signal is slower than the T1 signal and becomes significant in the late stage of myelin maturation. The T2 signal alteration is derived from the reduction of free water content that accompanies maturity of white matter beyond the biochemical changes. The water molecules inside myelin sheath and outside (at axon and interstitial tissue) directly affect the T2 signal intensity.
\cite{Barkovich AJ1988Normal, Bird CR1989MR, Christophe C1900Mapping, van der Knaap1990MR, Martin E1988Developmental, Martin E1991MR}
As the density of the spiral structure of the myelin sheath increases, interstitial water molecules decrease and the hydrocarbon chains present in the lipid layers in the membrane increase the hydrocarbon bundle bands. The completion of alteration of the T2 signal of whole brain white matter is at around 2 years old.
\cite{Barkovich AJ2000Concepts}
CNN is a machine learning method suitable for image recognition that can capture the spatial features of an object. CNN was devised from experimental results of human visual cortex.
\cite{Fukushima K1982new,LeCun Y2015Deep}
In the internal structure of CNN, the multilayered convolutional layers extract features of the object at various levels. In the convolutional layer, the matrix format input data is connected to the local receptive field and the convolutional process extracts the features of an object and creates a feature map. Multilayered convolutional layers extract the spatial features of subjects at different levels. The features of the object can be caught step by step from a simple element to a complex structure, from simple units such as pixels, lines and angles to structures such as a person's eyes and nose, as well as more complex parts such as face and body. The feature map created by the convolutional layers is transformed into a vector and transmitted to the following fully connected layer. Multilayered fully connected layers perform a classification or regression analysis of the object as a classifier.
However, a proper pre-processing that makes it easy to recognize features of an object leads to improvement in the accuracy of machine learning. As effective preprocessing for image recognition and classification, selection of input data from a large amount of data, cutting of an object from an input image, and adjustment of contrast are worthy of mention. In this study, two pre-processing steps were applied to improve the accuracy of the learning model. As input data, 5 slice images that had characteristics of myelin maturation were selected from 20 or more whole-brain MR images. By this pre-processing, it was expected that the feature map created by CNN would be more related to myelination. In addition, considering that the mechanisms associated with MR signal changes during myelin maturation are different between T1 and T2 signals, two data sets with different target ages were created, and ensemble architecture of two machine learning models was constructed. Improvement of the accuracy of the learning model by these methods was shown by an improvement of the correlation coefficient and error loss.
As a next step, we are preparing for a pipeline that extracts the input data from whole brain MR images, estimates the myelination age by the ensemble machine learning model, and evaluates the delay of myelination in a subject's brain automatically.
There are some limitations in our study. The first limitation is that the number of samples was not large. Generally, in machine learning of image classification, sample sizes from several thousands to tens of thousands are applied. \cite{Krizhevsky A2012ImageNet}
The number of samples in our study was less than this. However, it is equivalent to the median number of samples in a survey study of machine learning of medical images in brain regions within the last 5 years (mean 231, median 120).
\cite{Sakai K2018Machine}
The second is that our estimation was performed with two static magnetic field strength MR units of 1.5T and 3T.
\cite{Bottomley PA1998review, Schmitz BL2005Advantages, Stanisz GJ2005T1}
With 3T MRI, the T1 relaxation time is longer than with 1.5T and may affect myelination evaluation in the early stage of myelination. Creation of a data set and construction of a learning model corresponding to each static magnetic field intensity may improve the accuracy of myelination evaluation.
\section{Conclusion}
The CNN machine learning model can estimate the age of infants and children with high accuracy from the MR signal intensity and is adaptable to the assessment of infant brain myelination. Our machine learning model will support the work of radiologists and pediatricians in evaluating the neurological development of infants.
\begin{figure}
\centering
\includegraphics{Figure_01.png}
\label{fig.myelin}
\caption{Normal myelin maturation on T1-weighted images}
\flushleft
The top row is full-term infant, the 2nd row is a child 3 months old, the 3rd row is 6 months old and the bottom row is 12 months old. The full-term infant brain presents a high signal on the dorsal side of the brain stem, cerebellar peduncle and thalamus on T1-weighted images at birth (top row). These MR signal alterations proceed through the brain from caudal to rostral, posteriorly to anteriorly and centrally to peripherally.
\end{figure}
\begin{figure}
\centering
\includegraphics{Figure_02.png}
\label{fig.architecture}
\caption{The architecture of convolution neural network.}
\flushleft
Eight layers convolution neural network with 5 levels T1 and T2 weighted images as input (a), and detail of the 2 layers convolutional part (b). In convolution part, 128x128 gray scale images mediated by 2 convolution layers; Conv1(5x5 convolutional kernel with valid padding, strides 1, 1 and out map was 16, Conv2 (5x5 convolutional kernel with valid padding, strides 1, 1 and out map was 8) and 2 Max Pooling layers (2x2 kernel and stride 2, 2) and 29 x 29 x8 feature maps was transmitted to following Dense / fully connected layer. In each layers the rectified linear unit (ReLU) was applied as activation function. The output from 5 convolution parts and following Dense / fully connected layer are concatenated and transmitted to 2 Dense layers with 100 nodes and concatenated with output from T2 weighted images and transmitted to Dense layers with 200, 50, 25 nodes with activation function ReLU and the learning model was trained to minimize the loss function Squared Error.
\end{figure}
\begin{figure}
\centering
\includegraphics{Figure_03.png}
\label{fig.augumentation}
\caption{Augmentation of dataset}
\flushleft
A simple learning model using all ages for a data set (a) and an ensemble learning model using two data sets from birth to 16 months and from 8 months to 24 months (b).
\end{figure}
\begin{figure}
\centering
\includegraphics{Figure_04.png}
\label{fig.scatter}
\caption{Scatter diagram of corrected age of subject and estimated age by machine learning in simple model}
\flushleft
Scatter diagram of corrected age of subject and estimated age by machine learning in simple model (a) and ensemble model (b).
\end{figure}
\begin{table}
\caption{Estimation of two regression machine learning models}
\label{tab:table}
\centering
\begin{tabular}{c c c} \hline
\ & simple model & ensemble model \\ \hline \hline
r & 0.901 & 0.964 \\
RMSE & 3.40 & 2,.12 \\
MAE & 2.26 & 1.34 \\ \hline
\end{tabular}
\centering
\end{table}
\bibliographystyle{unsrt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,264 |
{"url":"https:\/\/www.hackmath.net\/en\/math-problem\/109?tag_id=79,4","text":"# Density - simple example\n\nMaterial of density of 532 kg\/m3 occupies a container volume of 79 cm3. What is its mass?\n\nResult\n\nm = \u00a042 g\n\n#### Solution:\n\n$h = 532 \\cdot \\ 1000 = 532000 \\ g\/m^3 \\ \\\\ \\ \\\\ V = 79 \\ cm^3 = 79 \/ 1000000 \\ m^3 = 7.9 \\cdot 10^{ -5 } \\ m^3 \\ \\\\ \\ \\\\ m = h \\cdot \\ V = 532000 \\cdot \\ 0.0001 = 42.028 = 42 \\ \\text { g }$\n\nLeave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):\n\nBe the first to comment!\n\n#### Following knowledge from mathematics are needed to solve this word math problem:\n\nDo you know the volume and unit volume, and want to convert volume units? Tip: Our Density units converter will help you with the conversion of density units. Do you want to convert mass units?\n\n## Next similar math problems:\n\n1. Sea on the Moon\nAssume that the Moon has sea, the same composition as on the Earth (has same density of salt water). Calculate dive of boat floating in the sea on the Moon, when on Earth has dive 3.6 m. Consider that the Moon has 6.5-times smaller gravitational accelerati\n2. Orl\u00edk hydroelectric plant\nThe Orl\u00edk hydroelectric power plant, built in 1954-1961, consists of four Kaplan turbines. For each of them, the water with a flow rate of Q = 150 m3\/s is supplied with a flow rate of h = 70.5 m at full power. a) What is the total installed power of the p\n3. Heptagonal pyramid\nA hardwood for a column is in the form of a frustum of a regular heptagonal pyramid. The lower base edge is 18 cm and the upper base of 14 cm. The altitude is 30 cm. Determine the weight in kg if the density of the wood is 0.10 grams\/cm3.\n4. Floating of wood - Archimedes law\nWhat will be the volume of the floating part of a wooden (balsa) block with a density of 200 kg\/m3 and a volume of 0.02 m3 that floats in alcohol? (alcohol density is 789 kg\/m3)\n5. Cheops pyramid\nThe Pyramid of Cheops is a pyramid with a square base with a side of 233 m and a height of 146.6 m. It made from limestone with a density of 2.7 g\/cm3. Calculate the amount of stone in tons. How many trains with 30 twenty tons wagons carry the stone?\n6. Hydrostatic force\nWhat hydrostatic force is applied to an area of 30 cm2 in water at a depth of 20 m? (Water density is 1000 kg\/m3)\n7. Butter fat\nA quarter of kg butter contains 82% fat. How many grams of fat are in four cubes of butter?\n8. Car\nAt what horizontal distance reaches the car weight m = 753 kg speed v = 74 km\/h when the car engine develops a tensile force F = 3061 N. (Neglect resistance of the environment.)\n9. Flywheel\nFlywheel turns 450 rev\/min (RPM). Determine the magnitude of the normal acceleration of the flywheel point which are at a distance of 10 cm from the rotation axis.\n10. Shot\nShot with a mass 43 g flying at 256 m\/s penetrates into the wood to a depth 25 cm. What is the average force of resistance of wood?\n11. Forces on earth directions\nA force of 60 N [North] and 80 N [East] is exerted on an object wigth 10 kg. What is the acceleration of the object?\n12. A car\nA car weighing 1.05 tonnes driving at the maximum allowed speed in the village (50 km\/h) hit a solid concrete bulkhead. Calculate height it would have to fall on the concrete surface to make the impact intensity the same as in the first case!\n13. Accelerated motion - mechanics\nThe delivery truck with a total weight of 3.6 t accelerates from 76km\/h to 130km\/h in the 0.286 km long way. How much was the force needed to achieve this acceleration?\n14. Positional energy\nWhat velocity in km\/h must a body weighing 60 kg have for its kinetic energy to be the same as its positional energy at the height 50 m?","date":"2019-12-13 00:18:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 1, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"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\": 0.4653666317462921, \"perplexity\": 1364.6826513427459}, \"config\": {\"markdown_headings\": true, \"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-2019-51\/segments\/1575540547536.49\/warc\/CC-MAIN-20191212232450-20191213020450-00147.warc.gz\"}"} | null | null |
{"url":"http:\/\/fivethirtyeight.com\/live-blog\/2016-election-second-republican-presidential-debate\/","text":"12:16 AM\nWhere Does Carly Fiorina Go From Here?\n\nIt can be dangerous to predict how the polls will move in response to debates. On the one hand, journalists and political pundits don\u2019t have so much in common with the Republican voters who are watching the debates at home. On the other hand, the post-debate narratives and \u201cspin\u201d sometimes matter more than what happened on the debate stage itself.\n\nBut for what it\u2019s worth, a lot of media types in my Twitter feed seem to think Carly Fiorina did really well, and my colleagues and I at FiveThirtyEight agree. I asked our team to grade each candidate from A to F on the basis of how much they helped their chances of winning the nomination or otherwise running a \u201csuccessful\u201d campaign. The grades were sent to me individually so we didn\u2019t have the opportunity to crib off one another (not that we\u2019re immune from other types of groupthink). Twelve people responded, including me.\n\nFiorina A A+ A-\nRubio B A C\nBush B A- C\nChristie B- B+ D\nCarson B- B+ D\nCruz C+ B D\nTrump C+ B+ D\nKasich C+ B+ D\nPaul C+ B C-\nWalker C+ B- D\nHuckabee C B D\n\nEveryone gave Fiorina an A-, A or A+. No one else was really that close, with Marco Rubio and Jeb Bush averaging a B. Donald Trump averaged a C+ in our straw poll, although with a wide range of responses including one \u201cWTF.\u201d\n\nIf Fiorina gets a further boost in the polls, it will be interesting to see where her support comes from. Fiorina doesn\u2019t have traditional credentials for a nominee, having never held elected office before (she lost to the Democrat Barbara Boxer in the California senate race in 2010). But Fiorina\u2019s policy positions are fairly conventional, mirroring those of the Republican establishment, and her ability to claim \u201coutsider\u201d status could be a boon given the mood of the GOP electorate. So check to see how Fiorina does in the polls \u2014 but also whether she picks up any endorsements or other signs of support from the Republican establishment.\n\n11:38 PM\nWho Spoke The Most?\n\nDonald Trump dominated the debate, and he had the help of the moderators to do it. They asked him more questions than any other candidate (and only one fewer than they asked Huckabee, Kasich, Rubio and Walker combined). Bush was the second-most-frequent speaker, relying on the favor of the moderators and the other candidates\u2019 propensity to attack him (and thus give him the chance to reply). Fiorina pushed her way into the top three with her frequent interruptions.\n\nThis analysis excludes the opening statements and the final, lighthearted questions posed to everyone.\n\n11:28 PM\n\n11:22 PM\n\nHere\u2019s what the moderators thought was most important for the candidates to cover. We tallied all the topics that got at least two questions (excluding the silly session\u00a0at the end).\n\n11:13 PM\n\nTrump and Carson both said we give too many vaccines, too close together. The Institute of Medicine took a look at\u00a0whether the immunization schedule is safe\u00a0and found that there was no evidence of safety concerns. But it went further, saying \u201crather than exposing children to harm, following the complete childhood immunization schedule is strongly associated with reducing vaccine-preventable diseases.\u201d\n\n11:12 PM\nReagan Led Carter Through Much Of The 1980 Campaign\n\nScott Walker just said that Ronald Reagan trailed just before the 1980 election. That\u2019s not true. As this Monkey Cage blog post details, Reagan led consistently over Jimmy Carter in an aggregate of polls from June 1980 through the election.\n\n11:10 PM\nFinally, Someone Talks About The Economy\n\nWe\u2019re down to the final minutes of the debate, and finally someone is talking about the economy. Jeb Bush correctly notes that poverty is up under Obama and incomes and workforce participation are down. Whether his policies will reverse those trends is another question. Bush has pledged to deliver 4 percent economic growth, something most experts believe will be difficult at a time when baby boomers are retiring. Still, it\u2019s good to hear someone talking about serious economic problems in this debate \u2014 even if it\u2019s likely come long after most people have tuned out.\n\n11:05 PM\n\nHuckabee asked: \u201cWhy doesn\u2019t this country focus on cures rather than treatment?\u201d Nixon declared a war on cancer in 1971. This year the National Institutes of Health will spend $5.4 billion on cancer research. We still have no cure. 10:57 PM 10:50 PM Trump\u2019s brain, like all of ours, is primed to reach false conclusions. 10:48 PM Yes, Trump should stop saying vaccines cause autism. There is no link between vaccines and autism. 10:47 PM Yeah, if I\u2019m Clinton, I\u2019m happy I don\u2019t have to stand on a stage for three hours. 10:46 PM If I\u2019m Hillary Clinton, Micah, then I want Donald Trump to be a factor in the Republican race for as long as possible, and I don\u2019t think Trump has had a particularly good night. But we\u2019ll see \u2014 it\u2019s not very easy to predict how the polls move in immediate response to a debate. The one thing I feel fairly confident about is that Fiorina is going to get some kind of bounce. 10:45 PM Nate and Harry, If you\u2019re Hillary Clinton watching this debate (though apparently Clinton isn\u2019t watching), are you happy with how it\u2019s going? 10:45 PM We had a brief respite from Trump, but he\u2019s still gotten at least twice as many questions from the moderator (12) as anyone but Bush (nine). Walker and Huckabee bring up the rear with three each. 10:44 PM Chris Christie just spoke about his Social Security plan, which includes eliminating benefits for those making more than$200,000 a year. As I noted when Christie first proposed overhauling\u00a0Social Security,\u00a0polling indicates that Christie will not benefit electorally from his stance.\n\n10:42 PM\n\nThe data on Christie\u2019s claim that marijuana is a gateway drug:\u00a0111 million Americans have tried marijuana\u00a0in their lifetime; 4.6 million people have tried heroin. Marijuana \u201cdoes not appear to be a gateway drug to the extent that it is the cause or even that it is the most significant predictor of serious drug abuse,\u201d according to the federal Institute of Medicine.\n\n10:42 PM\n\n10:36 PM\nRand Paul Sounds Social Justice Message On Pot\n\nEnding mass incarceration was once a predominantly liberal issue, but it\u2019s increasingly drawing bipartisan support. Conservatives worry about the high cost of locking up millions; libertarians worry about state overreach; and progressives worry about the impact on minority communities.\n\nOne thing that was interesting about Rand Paul\u2019s answer, though, was that he justified his position on explicitly social justice grounds. The\u00a0data backs him up: The people locked up for drug crimes are disproportionately poor men of color. As my colleague Oliver Roeder\u00a0has written, however, ending mass incarceration will require far more than releasing nonviolent drug offenders.\n\n10:34 PM\n\n10:33 PM\n\nIf you believe the polls, most Republicans haven\u2019t lit up a joint. According to a recent Gallup survey, only 31 percent of Republicans say they\u2019ve done the doobie. Moreover, only 36 percent of Republicans in an April 2015 Fox News poll were in favor of legalizing marijuana.\n\n10:26 PM\n\n10:22 PM\n\nTed Cruz and Jeb Bush are going at each\u00a0other over the Supreme Court nomination of John Roberts. As I pointed out during the first debate, no Republican senator voted against the Roberts nomination, and the vast majority of Republicans nationwide were for it.\n\n10:18 PM\n\nThere\u2019s an advantage in being attacked. A quarter of candidates\u2019 chances to speak have come when they\u2019re mentioned by opponents and get a right of reply. Trump and Bush have gotten to do this the most. Bush\u2019s opponents have given him half of his chances to speak.\n\n10:16 PM\nI'm The Only \u2026\n\n## Cruz\n\nIf you\u2019re fed up with Washington and looking for somebody to stand up to career politicians, I\u2019m the only one on this stage that\u2019s done that over and over again.\n\nI am the only candidate on this stage who has never supported amnesty.\n\n## Christie\n\nI\u2019m the only person on the stage who spent seven years as the United States attorney after Sept. 11.\n\n## Kasich\n\nI am the only person on this stage and one of the few people in this country that led the effort as the chief architect of the last time we balanced the federal budget.\n\n## Walker\n\nI\u2019m the only one on the stage that has a plan introduced to repeal Obamacare on day one.\n\n## Trump\n\nI\u2019m the only person up here that fought against going into Iraq.\n\n10:05 PM\n\nTrump is going after George W. Bush; that\u2019s dangerous if not executed correctly. According to a May 2015 CNN\/ORC survey, 89 percent of Republicans have a favorable view of him.\n\n10:01 PM\n\nNearly half of the times Fiorina has spoken, it\u2019s been because she has interrupted others.\n\n10:00 PM\n\nI think Walker has been better than in the previous debate. The problem is that Rubio and Bush have each been sharper and had more memorable lines than Walker, and they\u2019re the ones who Walker\u2019s competing against most directly for the establishment\u2019s support.\n\n9:59 PM\n\nIt\u2019s September 2015. The Iowa caucus is scheduled for Feb. 1, 2016. Unless Walker is running out of money, it\u2019s not that dire.","date":"2017-01-21 17:29:53","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\": 0.2163814753293991, \"perplexity\": 5424.480794410209}, \"config\": {\"markdown_headings\": true, \"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-2017-04\/segments\/1484560281162.88\/warc\/CC-MAIN-20170116095121-00246-ip-10-171-10-70.ec2.internal.warc.gz\"}"} | null | null |
Q: How to use Select Case in SQL with datetime field I have a table utblAdvertise and one of the field is PublishedDate and which Contains Datetime data like this
PublishedDate
2014-03-21 15:07:22.173
2014-02-11 15:05:22.223
2014-03-21 15:15:22.673
2014-01-01 15:15:22.272
2014-02-11 15:15:22.173
2014-03-20 15:15:22.372
2014-03-26 15:15:22.393
2014-02-25 15:15:22.273
I want the time only in case published date is between 24 hours eg. 5 Hours ago, 15 Hours ago etc. else I want to show date eg. 11-Feb-2014 I have tried for time my query is as follows
Select PublishedDate,RIGHT(CONVERT(VARCHAR, PublishedDate, 100),7) as Time From CLF.utblAdvertise
which give the result time like 3:09PM, 11:27AM.
I want the output as
3 hours ago -- Incase in between 24 hours
12-Feb-2014 -- incase in not between 24 hours
any help are surely appretiated.
A: SELECT
CASE WHEN DATEDIFF(HOUR, PublishedDate, GETDATE()) < 24 THEN
CASE DATEDIFF(HOUR, PublishedDate, GETDATE())
WHEN 1 THEN CONVERT(VARCHAR, DATEDIFF(HOUR, PublishedDate, GETDATE())) + ' hour ago'
ELSE CONVERT(VARCHAR, DATEDIFF(HOUR, PublishedDate, GETDATE())) + ' hours ago'
END
ELSE REPLACE(CONVERT(VARCHAR, PublishedDate, 6), ' ', '-')
END
FROM Something
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,564 |
{"url":"https:\/\/socratic.org\/questions\/how-do-you-find-all-real-solutions-to-the-following-equation-x-2-7x-11-x-2-2x-35","text":"# How do you find all real solutions to the following equation: (x^2-7x+11)^(x^2-2x-35)=1?\n\nFeb 12, 2017\n\n$x = \\left\\{- 5 , 7\\right\\}$\n\n#### Explanation:\n\nWe know that ${a}^{0} = 1$ so making ${x}^{2} - 2 x - 35 = 0$ the solutions are $x = \\left\\{- 5 , 7\\right\\}$ but we know also that $a$ must be non null. Solving now\n\n${x}^{2} - 7 x + 11 = 0$ we have $x = \\frac{1}{2} \\left(7 \\pm \\sqrt{5}\\right) \\ne \\left\\{- 5 , 7\\right\\}$\n\nFinally the equation is meaningful for\n\n$x = \\left\\{- 5 , 7\\right\\}$","date":"2019-11-13 14:14:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 8, \"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\": 0.9216501116752625, \"perplexity\": 276.91595726240666}, \"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\/1573496667262.54\/warc\/CC-MAIN-20191113140725-20191113164725-00383.warc.gz\"}"} | null | null |
Most of my readers are from the Northern Hemisphere, and I usually do make mention that I'm referring to "Up Over." LOL! Having never experienced the seasons from the Southern perspective, I'd love to learn more about how natives of Australia, New Zealand, etc. use the imagery of the signs when the seasons are flipped and right-side-up from your perspective.
I began contemplating this from watching Miss Fisher's Murder Mysteries. Lady detective Phryne Fisher was born on the Summer Solstice in Australia, and I've been wondering what that's like ever since. BTW, I've been able to unearth the birth data of Phryne's character and will be writing a post soon about her and her chart.
PS - I was born Sept. 22, cusp of autumn up here. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,623 |
\section{A barrier}
In this final section, we prove a barrier result, which shows that previous techniques cannot be used to extract from local sources with min-entropy below \(\sqrt{n}\).
In more detail, previous techniques \cite{DW12,Vio14} show that local sources are close to a convex combination of affine sources (with almost the same min-entropy), which means that a black-box affine extractor can also extract from local sources. However, the arguments in \cite{DW12,Vio14} only work if the local sources start out with min-entropy \(\gg\sqrt{n}\). On the other hand, it is not completely clear if this is an artifact of the exact arguments made in those works, or if local sources with min-entropy below \(\sqrt{n}\) simply cannot be shown to be close to a convex combination of affine sources, using \emph{any} argument.
In our barrier result, stated below, we show that the latter is the case.
\begin{theorem}[\cref{claim:barrierintro}, formal version]\label{thm:barrier:technical-section}
There exists a \(2\)-local source \(\mathbf{X}\sim\{0,1\}^n\) with min-entropy \(k\geq\sqrt{n}\) that is \((1/2)\)-far from a convex combination of affine sources with min-entropy \(5\).
\end{theorem}
This barrier result confirms that it is impossible to use affine extractors in a standard black-box manner (i.e., using convex combinations) to extract from local sources with min-entropy below \(\sqrt{n}\), for two reasons:
\begin{enumerate}
\item The statistical distance to a convex combination of affine sources is too large: since the statistical distance in \cref{thm:barrier:technical-section} is \(\geq1/2\), a black-box application of an affine extractor result in error \(\geq1/2\), which is no better than the error of the trivial extractor (which just outputs a constant).
\item The entropy of the affine sources in the convex combination is too low: the entropy of the affine sources in \cref{thm:barrier:technical-section} is \(5\), which is much less than the \(\Omega(\log n)\) entropy required for an affine extractor to even exist \cite{AggarwalBGS21}.
\end{enumerate}
In order to prove \cref{thm:barrier:technical-section}, we use a \(2\)-local source \(\mathbf{X}\sim\{0,1\}^n\) called the \emph{clique source}. We define the clique source over \(n=k+\binom{k}{2}\) bits to be the random variable \(\mathbf{X}\sim\{0,1\}^n\) generated by sampling \(k\) independent and uniform bits \(\mathbf{Y}_1,\dots,\mathbf{Y}_k\) and outputting
\[
\mathbf{X}=(\mathbf{Y}_1,\dots,\mathbf{Y}_k, (\mathbf{Y}_i\cdot\mathbf{Y}_j)_{1\leq i < j \leq k}).
\]
Then, we prove the following, from which \cref{thm:barrier:technical-section} is immediate.
\begin{theorem}\label{thm:far-convex-combo}
Let \(\mathbf{X}\sim\{0,1\}^n\) be the clique source over \(n={k+1\choose2}\) bits. Then for any \(t\), it holds that \(\mathbf{X}\) is \((1-\varepsilon)\)-far from a convex combination of affine sources with min-entropy \(t\), where
\[
\varepsilon=2^{-(t-3)/2}.
\]
\end{theorem}
The main lemma we use to prove \cref{thm:far-convex-combo} roughly shows that the collection of all cliques in the space of \(n\)-vertex graphs is a \emph{subspace-evasive set} (a well-studied object in pseudorandomness: see, e.g., \cite{DL12}). More formally, fix any \(n\in\mathbb{N}\) and let \(m:=n+{n\choose 2}\). We identify \(\mathbb{F}_2^m\) with the set of all \(n\)-vertex (undirected) graphs as follows. First, fix an arbitrary identification of \([m]\) with \([n] \cup {[n]\choose2}\), and let \(x\in\mathbb{F}_2^m\) represent the graph \(G_x=(V_x,E_x)\) where \(i \in V_x\) if and only if \(x_i = 1\) and \(e\in E_x\) if and only if \(x_e=1\). Then, we say that \(x\in\mathbb{F}_2^m\) is a clique if (and only if) \(E_x={V_x\choose2}\), and prove the following.
\begin{lemma}\label{lem:main-subspace-evasive}
Let \(\mathcal{Q}\subseteq\mathbb{F}_2^m\) denote the collection of all cliques. Then for any affine subspace \(S\subseteq\mathbb{F}_2^m\) of dimension \(k\),
\[
\frac{|\mathcal{Q}\cap S|}{|S|}\leq2^{-(k-3)/2}.
\]
That is, an exponentially small fraction of graphs in \(S\) are cliques.
\end{lemma}
Before we prove \cref{lem:main-subspace-evasive} (in the following subsection), we show how it implies \cref{thm:far-convex-combo}.
\begin{proof}[Proof of \cref{thm:far-convex-combo}]
Let \(\mathbf{Y}\sim\{0,1\}^n\) be a convex combination of affine sources \(\{\mathbf{Y}^{(i)}\}_i\), each with min-entropy \(t\). Let \(S=\text{support}(\mathbf{X})\) and \(\overline{S}:=\{0,1\}^n-S\). We have
\begin{align*}
|\mathbf{X}-\mathbf{Y}|&\geq\Pr[\mathbf{Y}\in\overline{S}]-\Pr[\mathbf{X}\in\overline{S}]=\Pr[\mathbf{Y}\in\overline{S}]\\
&\geq\min_i\Pr[\mathbf{Y}^{(i)}\in\overline{S}]=1-\max_i\Pr[\mathbf{Y}^{(i)}\in S]\\
&=1-\max_i\frac{|\text{support}(\mathbf{X})\cap\text{support}(\mathbf{Y}^{(i)})|}{|\text{support}(\mathbf{Y}^{(i)})|}.
\end{align*}
It is easy to verify that each element \(x\in\text{support}(\mathbf{X})\) is a clique, and thus by \cref{lem:main-subspace-evasive} the above equality is at least \(1-2^{-(t-3)/2}\), as desired.
\end{proof}
\subsection{Cliques are subspace-evasive}
Now, all that remains in the proof of our barrier result is to show that cliques are subspace-evasive (\cref{lem:main-subspace-evasive}). We do so below.
\begin{proof}[Proof of \cref{lem:main-subspace-evasive}]
We start with the standard observation that it suffices to prove a slightly stronger version of the result just for vector subspaces, since every affine subspace \(T\) of dimension \(\kappa\) is contained in a vector subspace \(S\) of dimension \(k\leq \kappa+1\) \cite{PR04}. Thus \begin{equation*}
\frac{|\mathcal{Q} \cap T|}{|T|} > 2^{-(\kappa-3)/2} \implies \frac{|\mathcal{Q} \cap S|}{|S|} > 2^{-k/2+1}
\end{equation*}
so it suffices to show \(|\mathcal{Q}\cap S|/|S|\leq2^{-k/2+1}\) for every vector subspace \(S\) of dimension \(k\). Indeed, we proceed by proving this result.
Define $\mathcal{Q}_S:=\mathcal{Q}\cap S$. Notice that for any $x, y \in \mathcal{Q}_S$, we know that $x+y \in S$ as they each belong in $S$. By this observation, we see that $\mathcal{Q}_S + \mathcal{Q}_S \subseteq S$. Furthermore, we claim for any nonzero $z \in \mathcal{Q}_S + \mathcal{Q}_S$, there are at most $2$ pairs $(x,y) \in \mathcal{Q}_S \times \mathcal{Q}_S$ such that $z = x+y$. In other words, $\mathcal{Q}_S$ is \emph{a Sidon set}. Indeed, once we prove this, we find that $|\mathcal{Q}_S|(|\mathcal{Q}_S| - 1)/2 \le |\mathcal{Q}_S + \mathcal{Q}_S| - 1 \le |S|-1$. Thus $|\mathcal{Q}_S|^2/4 < |S|$, and so $|\mathcal{Q}_S|/|S|<2/\sqrt{|S|} = 2^{-k/2+1}$, which is what we wanted to show.
Consider the linear map $\varphi : \mathbb{F}_2^m \to \mathbb{F}_2^{k \times k}$ where the matrix $M = \varphi(Y)$ is defined by $M_{ij} = M_{ji} = Y_{ij}$. The key observation with this map is noticing that the set $\varphi(Q)$ corresponds exactly to the rank-1 matrices in $\text{im}(\varphi)$ as $\varphi(Y) = xx^\top$ for any $Y \in Q$. Now, for any nonzero vector $v \in Q_S + Q_S$, let us count the number of $y,y' \in \mathcal{Q}_S$ such that $v = y+y'$. By linearity and injectivity of $\varphi$, this is equivalent to counting the number of $x,x' \in \mathbb{F}_2^l$ satisfying $M_v = xx^\top + x'x'^\top$ where $M_v = \varphi(v)$.
Consider one such solution $M_v = zz^\top + z'z'^\top$. From the equation $M_v = zz^\top + z'z'^\top$, we see that the row space of $M_v$, which we denote by $\mathcal{R}(M_v)$, is contained in the span of $z$ and $z'$. Furthermore, since $M_v$ is nonzero, then $z \neq z'$. Thus at least two of the elements in $\{z,z', z+z'\}$ must be rows in $M_v$. This implies that the span of $z$ and $z'$ is contained in $\mathcal{R}(M_v)$. Hence $\mathcal{R}(M_v) = \{0, z, z', z+z'\}$.
For any other such solution $M_v = xx^\top + x'x'^\top$, we also find by repeating the same steps that $x \neq x'$ and $\mathcal{R}(M_v) = \{0, x, x', x+x'\} = \{0, z, z', z+z'\}$. Let us now consider 2 possible cases:
\begin{enumerate}
\item If either $z$ or $z'$ is zero, then without loss of generality, say $z'=0$. Then either $(x,x') = (z,0)$ or $(x,x') = (0,z)$. Thus there are at most $2$ possible pairs $(x,x')$ for this case.
\item If both $z$ and $z'$ are nonzero, then both $x$ and $x'$ must be nonzero. Furthermore, $x, x'$ must be one of the elements in $\{z,z',z+z'\}$. From all $6$ possible pairs, only the pairs $(z,z')$ and $(z',z)$ satisfy the equation $M_v = xx^\top + x'x'^\top$. Thus there are at most $2$ possible pairs $(x,x')$ for this case.
\end{enumerate}
Since these are the only two possible cases, this therefore shows that $\mathcal{Q}_S$ is a Sidon set, which is what we wanted to show.
\end{proof}
\dobib
\end{document}
\section{Introduction}\label{sec:intro}
Randomness is a fundamental resource in many areas of computer science, such as algorithm design and cryptography. However, such tasks often assume access to a source of independent and uniform bits, while real-world physical processes (e.g., electromagnetic noise, timings of user keystrokes) generate randomness that is far from perfect.
This state of affairs motivates the problem of \emph{randomness extraction}. The goal is to design a deterministic function, called an \emph{extractor}, that can distill a (nearly) uniform bit from any source belonging to a certain family.\footnote{In general, one would actually like to distill (from the source) as many uniform bits as possible. However, we focus here on the simpler goal of obtaining just a single uniform bit, as there remain significant barriers even in this most basic setting.}
\begin{definition}[Extractor]\label{def:intro:extractor}
A function $\mathsf{Ext}:\{0,1\}^n\to\{0,1\}$ is an \emph{extractor} for a family of distributions $\mathcal{X}$ over \(\{0,1\}^n\) with error \(\varepsilon\) if, for every \(\mathbf{X}\in\mathcal{X}\),
\begin{equation*}
\left|\Pr[\mathsf{Ext}(\mathbf{X})=1]-\frac{1}{2}\right|\leq \varepsilon.
\end{equation*}
We also call \(\mathsf{Ext}\) an \emph{\(\varepsilon\)-extractor for \(\mathcal{X}\).}
\end{definition}
Besides their practical motivation, randomness extractors (and other related pseudorandom objects such as dispersers, condensers, and expander graphs) have deep connections to coding theory, combinatorics, and complexity theory.
In order to construct an extractor for a family \(\mathcal{X}\) of sources, the most general assumption one can make about \(\mathcal{X}\) is that each \(\mathbf{X}\in\mathcal{X}\) has some ``randomness.'' Here, it is typical to measure the randomness of a source \(\mathbf{X}\) by its min-entropy \(H_\infty(\mathbf{X}):=-\log\max_x\Pr[\mathbf{X}=x]\). However, even if we assume each \(\mathbf{X}\in\mathcal{X}\) has a very high amount of this very strong notion of entropy, extraction is still impossible: indeed, one cannot hope to extract from \(\mathcal{X}\) even if each source \(\mathbf{X}\in\mathcal{X}\) is guaranteed to have min-entropy \(k\geq n-1\)~\cite{CG88}. To enable extraction, one must make additional assumptions on the structure of each \(\mathbf{X}\in\mathcal{X}\).
\paragraph*{Extractors for local sources, \(\mathsf{AC^0}\) sources, and small-space sources.}
In a seminal work, Trevisan and Vadhan~\cite{TV00} initiated the study of randomness extraction from sources that can be sampled by ``simple'' processes. In addition to the generality of such sources, it can be argued that they serve a reasonable model of randomness that might actually be found in nature.
More formally, Trevisan and Vadhan studied sources that can be sampled by polynomial size circuits that are given uniform bits as input. However, extracting randomness from this class of sources requires strong computational hardness assumptions.
This motivated De and Watson~\cite{DW12} and Viola~\cite{Vio14} to consider \emph{unconditional} extraction from sources sampled by more restricted, but still natural, circuit families.
To this end, they introduced the notion of \emph{local sources}.
Intuitively, a local source \(\mathbf{X}\) is one that can be sampled by a low-depth circuit with bounded fan-in (a low-complexity process).
\begin{definition}[Local source~\cite{DW12,Vio14}]\label{def:localsources}
A distribution $\mathbf{X}\sim\{0,1\}^n$ is a \emph{$d$-local source} if \(\mathbf{X}=g(\mathbf{U}_m)\), where \(\mathbf{U}_m\) is the uniform distribution over \(m\) bits (for some \(m\)), and \(g:\{0,1\}^m\to\{0,1\}^n\) is a function where each output bit depends on at most \(d\) input bits.
\end{definition}
Local sources are closely connected to other models of sources sampled by simple processes. Viola~\cite{Vio14} proved that every source generated by $\mathsf{AC}^0$ circuits is (close to) a convex combination of local sources with small locality and slightly lower min-entropy. More recently, Chattopadhyay and Goodman~\cite{CG21} showed a similar result for sources generated by bounded-width branching programs~\cite{KRVZ11}. Thus, extractors for local sources also work for sources generated by these classical computational models. In fact, the current state-of-the-art extractors for sources generated by \(\mathsf{AC}^0\) circuits and bounded-width branching programs are extractors for \(1\)-local sources.
\paragraph*{A barrier at \(\sqrt{n}\) min-entropy.}
Despite the applications above and being introduced over a decade ago, little progress has been made on constructing extractors for local sources \cite{DW12,Vio14,Li16}. In particular, all known constructions require min-entropy at least \(\sqrt{n}\), and follow via a reduction to extractors for affine sources (i.e., sources that are uniform over affine subspaces of \(\mathbb{F}_2^n\)). Thus, there appears to be a ``barrier'' at \(\sqrt{n}\) min-entropy, at least when using affine extractors \cite{Vio20}. It is natural to ask how we might break this \(\sqrt{n}\) barrier, which raises the question:
\begin{center}
\emph{Can affine extractors be used to extract from local sources with min-entropy \(k\ll \sqrt{n}\)?}
\end{center}
As motivation, we start by providing strong evidence that the answer to the above question is negative. In particular, we prove the following.
\begin{customthm}{0}[Barrier result]\label{claim:barrierintro}
It is not possible to extract randomness from $2$-local sources with min-entropy $k\geq\sqrt{n}$ by applying an affine extractor in a black-box manner.
\end{customthm}
Thus, if we would like to construct extractors for local sources with min-entropy significantly below \(\sqrt{n}\), new techniques are needed.
\paragraph*{Towards breaking the \(\sqrt{n}\) barrier using low-degree polynomials.}
While explicit extractors that break the \(\sqrt{n}\) min-entropy barrier for local sources are the end goal, these still seem beyond reach.
We believe that the next best thing are non-explicit extractors that are of "low complexity."
Our hope is that such extractors may help us eventually construct truly explicit extractors, as non-explicit extractors are more likely to be easier to derandomize if they belong to a low complexity class. At the same time, such non-explicit extractors may have applications in complexity theory (i.e., since the current state-of-the-art circuit lower bounds are against extractors \cite{li20213}). There is a long line of work~\cite{Vio05,GVW15,CL18,Lu04,Vad04,BogG13,CL18,DY21,HIV21,CT15} on the power of low-complexity computational models for extracting from various families of sources.
We choose to study the class of \emph{low-degree \(\mathbb{F}_2\)-polynomials} as our low-complexity computational model. In particular, we ask whether (non-explicit) low-degree polynomials can help break the \(\sqrt{n}\) barrier for extracting from local sources, and more generally we seek to answer the following question:
\begin{question}\label{q:mainq}
How powerful are low-degree $\mathbb{F}_2$-polynomials as extractors for local sources?
\end{question}
Beyond being a natural algebraic class, low-degree $\mathbb{F}_2$-polynomials have a natural combinatorial interpretation.
We can represent a degree-$2$ $\mathbb{F}_2$-polynomial $f$ as a graph $G_f$ on $n$ vertices, with edges representing monomials included in $f$.
Then, $f$ being a good extractor for local sources translates into a parity constraint on the number of edges in certain induced subgraphs of $G_f$.
Likewise, a degree-$3$ $\mathbb{F}_2$-polynomial can be represented as a 3-hypergraph, and so on. Given the correspondence between low-degree polynomials and hypergraphs with small edge sizes, we hope that tools from combinatorics can be leveraged to make our constructions explicit and break the $\sqrt{n}$ min-entropy barrier for extracting from local sources (which would also give improved extractors for small-space sources).
Our motivation to study low-degree polynomials as our ``low complexity'' model also comes from the work of Cohen and Tal \cite{CT15}, which studied the same question in the context of \emph{affine} sources. In their work, they showed that there exist degree-$r$ $\mathbb{F}_2$-polynomials that extract from affine sources with min-entropy $O(r n^{\frac{1}{r-1}})$, and that this is tight. In order to answer \cref{q:mainq}, we aim to provide a local source analogue of this result.
\subsection{Summary of our results}\label{sec:summary}
In this paper, we fully characterize the power of low-degree polynomials as extractors for local sources, answering \cref{q:mainq} and proving a local-source analogue of Cohen-Tal. Along the way, we rely on several new ingredients which may be of independent interest. We present these results in \cref{subsub:first-part-summary} and \cref{subsub:second-part-summary}, respectively.
\subsubsection{Main result}\label{subsub:first-part-summary}
Our main result gives a tight characterization of the power of low-degree polynomials as extractors for local sources. We state it formally, below.
\begin{customthm}{1}[Main result]\label{thm:absolute-main}
For every \(d\in\mathbb{N}\) and \(r\geq2\), there are constants \(C,c>0\) such that the following holds. For every \(n\in\mathbb{N}\) there exists a (not necessarily explicit) degree \(\leq r\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\) that is a \((2^{-ck})\)-extractor for \(d\)-local sources with min-entropy
\[
k\geq C (n\log n)^{1/r},
\]
but for every degree \(\leq r\) polynomial \(g\in\mathbb{F}_2[x_1,\dots,x_n]\) there exists a \(d\)-local source with min-entropy
\[
k\geq c (n\log n)^{1/r}
\]
on which it is constant.
\end{customthm}
\cref{thm:absolute-main} implies that degree-\(3\) polynomials are already enough to extract from min-entropy \(k=O((n\log n)^{1/3})\), which (non-explicitly) breaks the \(\sqrt{n}\) min-entropy barrier of previous techniques (\cref{claim:barrierintro}). Furthermore, given known reductions from \(\mathsf{AC}^0\) sources and small-space sources to local sources~\cite{Vio14,CG21}, it also follows that low-degree polynomials can be used to break the current min-entropy barriers for extracting from these other models of weak sources. Finally, \cref{thm:absolute-main} can be viewed as a generalization of a result by Viola \cite{Vio16}, who showed the existence of quadratic functions \(f\) such that \(\mathsf{AC}^0\) cannot sample \((\mathbf{U}_n,f(\mathbf{U}_n))\): our result shows that there exist degree \(\leq r\) polynomials such that \(\mathsf{AC}^0\) cannot sample \((\mathbf{X},f(\mathbf{X}))\), where \(\mathbf{X}\) can be \emph{any} distribution with min-entropy \(k\geq Cn^{1/r+\delta}\) for a large enough \(C\) and arbitrarily small \(\delta>0\).
While \cref{thm:absolute-main} is stated for constant locality \(d\) and constant degree \(r\), we actually prove stronger results that hold for superconstant \(d,r\). In particular, \cref{thm:absolute-main} follows immediately from the following two results, which provide upper and lower bounds on the entropy required to extract from \(d\)-local sources using a degree \(\leq r\) polynomials (where \(d,r\) need not be constant).
\begin{customthm}{1.1}[Technical version of \cref{thm:absolute-main}, Upper Bound]\label{thm:localextintro}
There are universal constants \(C,c>0\) such that for all \(n,d,r\in\mathbb{N}\), the following holds. With probability at least \(0.99\) over the choice of a random degree \(\leq r\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\), it holds that \(f\) is an \(\varepsilon\)-extractor for \(d\)-local sources with min-entropy \(k\), for any
\[
k\geq C2^dd^2r\cdot(2^dn\log n)^{1/r},
\]
where \(\varepsilon=2^{-\frac{ck}{r^32^dd^2}}\).
\end{customthm}
\begin{remark}[Informal version of \cref{thm:formal-version-of-main-remark}]\label{rem:main-remark-here}
If instead of extractors we aim to construct \emph{dispersers},\footnote{A function $\mathsf{Disp}:\{0,1\}^n\to\{0,1\}$ is a \emph{disperser for a class of sources $\mathcal{C}$} if the support of $\mathsf{Disp}(\mathbf{X})$ is $\{0,1\}$ for all sources $\mathbf{X}\in\mathcal{C}$.} then we are able to improve \cref{thm:localextintro} to hold for min-entropy $k\geq Cd^3r\cdot(n\log n)^{1/r}$.
\end{remark}
\begin{customthm}{1.2}[Technical version of \cref{thm:absolute-main}, Lower Bound]\label{thm:localtightintro}
There exists a universal constant \(c>0\) such that for all \(n,d,r\in\mathbb{N}\) with \(2\leq r\leq c\log(n)\) and \(d\leq n^{\frac{1}{r-1}-2^{-10r}}/\log(n)\), the following holds. For any degree \(\leq r\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\), there is a \(d\)-local source \(\mathbf{X}\sim\{0,1\}^n\) with min-entropy at least
\[
k\geq cr (dn\log n)^{1/r}
\]
such that $f(\mathbf{X})$ is constant.
\end{customthm}
\subsubsection{Key new ingredients}\label{subsub:second-part-summary}
Our main result follows from a collection of new ingredients, which may be of independent interest. In order to prove our upper bound on min-entropy (\cref{thm:localextintro}), we prove a new reduction from \(d\)-local sources to \emph{$d$-local non-oblivious bit fixing (NOBF) sources}. Informally, a $d$-local NOBF source $\mathbf{X} \sim \{0,1\}^n$ of min-entropy $k'$ is a source that has $k'$ uniform independent bits, with all other bits depending on at most $d$ of the $k'$ bits.
\begin{customthm}{2}[Reduction from \(d\)-local sources to \(d\)-local NOBF sources]
\label{thm:red-to-nobf}
There exists a universal constant \(c>0\) such that for any \(n,k,d\in\mathbb{N}\), the following holds. Let \(\mathbf{X}\sim\{0,1\}^n\) be a \(d\)-local source with min-entropy \(\geq k\). Then \(\mathbf{X}\) is \(\varepsilon\)-close to a convex combination of \(d\)-local NOBF sources with min-entropy \(\geq k^{\prime}\), where \(\varepsilon=2^{-ck^{\prime}}\) and
\[
k^{\prime}=\frac{ck}{2^d d^2}.
\]
\end{customthm}
The family of \(d\)-local NOBF sources, introduced in~\cite{CGGL20}, is a significant specialization of \(d\)-local sources. The above reduction shows that, at least for constant locality \(d\), we can just focus on extracting from this simpler class - even in future explicit constructions.
To prove our lower bound on min-entropy (\cref{thm:localtightintro}), we actually prove this lower bound for the special class of \(d\)-local sources known as \emph{$d$-local affine sources}: such a source \(\mathbf{X}\sim\mathbb{F}_2^n\) is uniform over a \(d\)-local affine subspace \(X\subseteq\mathbb{F}_2^n\), which is a special type of affine subspace that admits a basis \(v_1,\dots,v_k\in\mathbb{F}_2^n\) where each coordinate \(i\in[n]\) holds the value \(1\) in at most \(d\) of these vectors. For this special class of sources, our lower bound is actually tight, and can be viewed as a ``local'' version of a result by Cohen and Tal~\cite{CT15}.
\begin{customthm}{3}[Local version of Cohen-Tal]\label{thm:local-cohen-tal}
There exist universal constants \(C,c>0\) such that for every \(n,r,d\in\mathbb{N}\) such that \(2\leq r\leq c\log(n)\) and \(d\leq n^{\frac{1}{r-1}-2^{-10r}}/\log(n)\), the following holds. For any degree \(\leq r\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\), there exists a \(d\)-local affine subspace \(X\subseteq\mathbb{F}_2^n\) of dimension
\[
k\geq cr (d n\log n)^{1/r}
\]
on which \(f\) is constant.
This is tight: there exists a degree \(\leq r\) polynomial \(g\in\mathbb{F}_2[x_1,\dots,x_n]\) which is an extractor for \(d\)-local affine sources of dimension \(k\geq Cr(dn\log n)^{1/r}\), which has error \(\varepsilon=2^{-ck/r}\).
\end{customthm}
We prove \cref{thm:local-cohen-tal} by extending the techniques of Cohen and Tal \cite{CT15}, while leveraging a key new ingredient: a ``low-weight'' Chevalley-Warning theorem. This result, which may be of independent interest, shows that any small system of low-degree polynomials admits a (nontrivial) solution of low Hamming weight.
\begin{customthm}{4}[Low-weight Chevalley-Warning]
\label{thm:low-wt-cw}
Let \(\{f_i\}\subseteq\mathbb{F}_2[x_1,\dots,x_n]\) be a set of polynomials with linear degree\footnote{The \emph{linear degree} $D$ is the sum of the degrees of the $f_i$'s which have degree \(1\).} at most \(D\) and nonlinear degree\footnote{The \emph{nonlinear degree} is the sum of the degrees of the $f_i$'s which have degree at least $2$.} at most \(\Delta\), such that \(0\) is a common solution and \(D+\Delta<n\). Then there is a common solution \(x\neq 0\) with Hamming weight
\[
w\leq 8\Delta + 8D/\log(n/D)+8.
\]
\end{customthm}
\subsection{Open problems}
Our work leaves open several interesting avenues for future work. We highlight three of them here:
\begin{itemize}
\item For any constant $r \ge 2$, construct an explicit $\mathbb{F}_2$-polynomial of degree $\leq r$ that extracts from 2-local sources of min-entropy $o(n)$.
\item In \cref{thm:red-to-nobf}, we showed a reduction from \(d\)-local sources of min-entropy $k$ to \(d\)-local non-oblivious bit fixing (NOBF) sources of min-entropy \(\Omega(k/(2^dd^2))\). It would be interesting to show a reduction from a \(d\)-local source of min-entropy \(k\) to a \(d\)-local NOBF source of min-entropy \(\Omega(k/\text{poly}(d))\) (or show that such a reduction is impossible).
\item \cref{thm:low-wt-cw} shows that if a collection of low-degree \(n\)-variate polynomials of linear degree $D$ and nonlinear degree $\Delta$ has the zero vector as a solution (and \(D+\Delta<n\)), then there exists a nonzero solution of weight at most $O(\Delta + D/\log(n/D))$. When $\Delta = 0$, this becomes asymptotically tight by the Hamming bound. Moreover, when $D = 0$, this will also be tight by picking the polynomial $f(x) = \sum_{1 \le |S| \le \Delta}{\prod_{i \in S}{x_i}}$. However, if we had $\Delta/2$ quadratic polynomials, will the upper bound of $O(\Delta)$ still be tight?
\end{itemize}
\end{document}
\section{Entropy lower bounds}\label{sec:lower-bounds-new}
In this section, we obtain lower bounds on the entropy required to extract from \(d\)-local sources using degree \(\leq r\) polynomials, proving the following theorem.
\begin{theorem}[\cref{thm:localtightintro}, restated]\label{thm:localtightintro:restated}
There is a constant \(c>0\) such that for all \(n,d,r\in\mathbb{N}\) with \(2\leq r\leq c\log(n)\) and \(d\leq n^{\frac{1}{r-1}-2^{-10r}}/\log(n)\), the following holds. For any degree \(\leq r\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\), there is a \(d\)-local source \(\mathbf{X}\sim\{0,1\}^n\) with min-entropy at least
\[
k\geq cr (dn\log n)^{1/r}
\]
such that \(f(\mathbf{X})\) is constant.
\end{theorem}
In order to prove \cref{thm:localtightintro:restated}, we actually prove a stronger theorem, which can be viewed as a \emph{local} version of a result by Cohen and Tal~\cite{CT15}. In particular, Cohen and Tal show that every \(n\)-variate polynomial of degree \(\leq r\) admits a subspace of dimension \(\Omega(rn^{\frac{1}{r-1}})\) on which it is constant, and that this is tight. We show that a similar result holds even for the special subclass of \emph{local} subspaces.
\begin{theorem}[\cref{thm:local-cohen-tal}, restated]\label{thm:local-cohen-tal:restated}
There exist universal constants \(C,c>0\) such that for every \(n,r,d\in\mathbb{N}\) such that \(2\leq r\leq c\log(n)\) and \(d\leq n^{\frac{1}{r-1}-2^{-10r}}/\log(n)\), the following holds. For any degree \(\leq r\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\), there exists a \(d\)-local subspace \(X\subseteq\mathbb{F}_2^n\) of dimension
\[
k\geq cr (d n\log n)^{1/r}
\]
on which \(f\) is constant.
This is tight: there exists a degree \(\leq r\) polynomial \(g\in\mathbb{F}_2[x_1,\dots,x_n]\) which is an extractor for \(d\)-local affine sources of dimension \(k\geq Cr(dn\log n)^{1/r}\), which has error \(\varepsilon=2^{-ck/r}\).
\end{theorem}
Notice that \cref{thm:local-cohen-tal:restated} immediately implies \cref{thm:localtightintro:restated}, since (the uniform distribution over) a \(d\)-local subspace is not only an affine source, but it is also a \(d\)-local source. Thus, in the remainder of the section, we focus on proving \cref{thm:local-cohen-tal:restated}. The key new ingredient we rely on is a so-called ``low-weight Chevalley-Warning theorem,'' which may be of independent interest. We prove this result in \cref{subsec:low-wt-cw}, and we show how it can be used to obtain \cref{thm:local-cohen-tal:restated} in \cref{subsec:cohen-tal-local-sec}. This will conclude our discussion of entropy lower bounds (and \cref{sec:lower-bounds-new}).
\subsection{A low-weight Chevalley-Warning theorem}\label{subsec:low-wt-cw}
The \emph{Chevalley-Warning theorem} is a classical result from number theory, which guarantees that a small set of low-degree polynomials admits a nontrivial common solution. More formally, it states the following.
\begin{theorem}[Chevalley-Warning theorem~\cite{War36}]\label{thm:absolutely-standard-Chevalley-Warning}
Let \(\{f_i\}\subseteq\mathbb{F}_2[x_1,\dots,x_n]\) be a set of polynomials with degree at most \(D\) such that \(0\) is a common solution. Then there are at least \(2^{n-D}\) common solutions to \(\{f_i\}\). In particular, if \(D<n\), then there must be a nontrivial common solution.
\end{theorem}
In this subsection, we prove a ``low-weight'' version of this theorem (which will be instrumental in our proof of \cref{thm:local-cohen-tal:restated}). In more detail, we believe it is natural to ask not only if \(\{f_i\}\) contains a nontrivial common solution, but if \(\{f_i\}\) contains a nontrivial common solution of \emph{low Hamming weight}.
In the case where all the \(f_i\) are linear, this is a question about the distance-dimension tradeoff of linear codes. Here, this question is answered by the \emph{Hamming bound}, which says that there must be a nontrivial common solution of weight \(w\leq O(D/\log(n/D))\).\footnote{More formally, it says that if all nontrivial solutions have weight \(>w\), then it must hold that \(\binom{n}{\leq\lfloor w/2\rfloor}\leq 2^D\).}
On the other hand, for general collections \(\{f_i\}\) that may have nonlinear polynomials, it is straightforward to use classical Chevalley-Warning (\cref{thm:absolutely-standard-Chevalley-Warning}) to show that there is always a nontrivial common solution of weight \(w\leq D+1\),\footnote{Given a collection \(\{f_i\}\subseteq\mathbb{F}_2[x_1,\dots,x_n]\) of degree \(D\), apply \cref{thm:absolutely-standard-Chevalley-Warning} to the collection \(\{f_i\}\cup\{x_1,x_2,\dots,x_{n-(D+1)}\}\).} and one can show that this is tight in general.\footnote{Consider the singleton set \(\{f\}\subseteq\mathbb{F}_2[x_1,\dots,x_n]\), where \(f(x)=\sum_{\emptyset\subsetneq S\subseteq[n]:|S|\leq D}x^S\).}
Thus, the upper bound on Hamming weight is much better when we know that all the polynomials in \(\{f_i\}\) are linear. This begs the question: if we know that \emph{most} of the polynomials in \(\{f_i\}\) are linear, can we get an upper bound on Hamming weight that is almost as strong as in the purely linear case? And, more generally, can we get a granular bound that takes into account exactly how many \(f_i\) are linear and nonlinear?
Our \emph{low-weight Chevalley-Warning theorem}, which we state next, provides a result of exactly this form.
\begin{theorem}\label{thm:low-wt-cw:technical}
Let \(\{f_i\}\subseteq\mathbb{F}_2[x_1,\dots,x_n]\) be a set of polynomials with linear degree at most \(D\) and nonlinear degree at most \(\Delta\), such that \(0\) is a common solution and \(D+\Delta<n\). If all nontrivial common solutions to \(\{f_i\}\) have Hamming weight \(>w\), then
\begin{align*}
\binom{n}{\leq\lfloor w/2\rfloor}\leq2^{D+\Delta+1}\cdot\binom{n}{\leq\lfloor\Delta/2\rfloor}.
\end{align*}
\end{theorem}
As we will see, a relatively straightforward (but tedious) calculation yields the following corollary.
\begin{corollary}[\cref{thm:low-wt-cw}, restated]\label{cor:low-wt-cw:main-technical-cor}
Let \(\{f_i\}\subseteq\mathbb{F}_2[x_1,\dots,x_n]\) be a set of polynomials with linear degree at most \(D\) and nonlinear degree at most \(\Delta\) such that \(0\) is a common solution and \(D+\Delta<n\). Then there is a nontrivial common solution with Hamming weight
\[
w\leq 8\Delta + 8D/\log(n/D)+8.\footnote{We define the expression \(8D/\log(n/D)\) to be \(0\) if \(D=0\).}
\]
\end{corollary}
Notice that the weight upper bound in \cref{cor:low-wt-cw:main-technical-cor} tightly interpolates (up to constant factors) between the linear and nonlinear cases discussed at the beginning of this section. We will also see that it is not difficult to extend \cref{cor:low-wt-cw:main-technical-cor} to obtain the following, which is the result that we will actually end up using in our proof of \cref{thm:local-cohen-tal:restated}.
\begin{corollary}\label{cor:projection-cw}
Let \(\{f_i\}\subseteq\mathbb{F}_2[x_1,\dots,x_n]\) be a set of polynomials with linear degree at most \(D\) and nonlinear degree at most \(\Delta\) such that \(0\) is a common solution. Then for any \(S\subseteq[n]\) such that \(D+\Delta<|S|\), there is a nontrivial common solution supported on \(S\) with Hamming weight
\[
w\leq 8\Delta + 8D/\log(|S|/D) + 8.
\]
\end{corollary}
In the remainder of this section, we prove \cref{thm:low-wt-cw:technical} and \cref{cor:low-wt-cw:main-technical-cor,cor:projection-cw}. The proof of \cref{thm:low-wt-cw:technical} is most interesting, and can be found in \cref{subsubsec:proof-low-wt-cw:main}. The proofs of \cref{cor:low-wt-cw:main-technical-cor,cor:projection-cw} are less interesting and somewhat tedious, and can be found in \cref{subsubsec:proof-low-wt-cw:corollaries}.
\subsubsection{A proof of our low-weight Chevalley-Warning theorem}\label{subsubsec:proof-low-wt-cw:main}
We now prove our low-weight Chevalley-Warning theorem (\cref{thm:low-wt-cw:technical}). The key ingredient we need is the following lemma, which says that for any collection of polynomials \(\{f_i\}\) and any big enough set of common solutions \(A\), it holds that \(A+A\) contains a nontrivial common solution.
\begin{lemma}\label{lem:key-ingredient:low-wt-cw:technical}
Let \(\{f_i\}\subseteq\mathbb{F}_2[x_1,\dots,x_n]\) be a set of polynomials with nonlinear degree at most \(\Delta\) such that \(0\) is a common solution. Then for any set \(A\subseteq\mathbb{F}_2^n\) of common solutions of size
\[
|A|>2\binom{n}{\leq\lfloor\Delta/2\rfloor},
\]
it holds that \(A+A\) contains a nontrivial common solution.
\end{lemma}
Before proving this result, we show how it can be combined with the Hamming bound and the classical Chevalley-Warning theorem to get \cref{thm:low-wt-cw:technical}, our low-weight Chevalley-Warning theorem.
\begin{proof}[Proof of \cref{thm:low-wt-cw:technical}]
Let \(Q\subseteq\mathbb{F}_2^n\) be the set of common solutions to \(\{f_i\}\), and suppose that all \(v\in Q-\{0\}\) have Hamming weight \(>w\). Then for any ball \(\mathcal{B}\) of radius \(\lfloor w/2\rfloor\), it must hold that
\[
|Q\cap\mathcal{B}|\leq 2\binom{n}{\leq\lfloor\Delta/2\rfloor},
\]
since otherwise \cref{lem:key-ingredient:low-wt-cw:technical} (combined with the triangle inequality) implies that that there is a nontrivial solution of weight at most \(2\lfloor w/2\rfloor\leq w\). Thus, we see that \(Q\) is a \((\rho,L)\)-list-decodable code, where
\begin{align*}
\rho &= \lfloor w/2\rfloor / n,\\
L &= 2\binom{n}{\leq\lfloor\Delta/2\rfloor}.
\end{align*}
Now, the Hamming bound (\cref{thm:hamming-bound}) implies that \(|Q|\leq 2^nL/\binom{n}{\leq\rho n}\), whereas the Chevalley-Warning theorem (\cref{thm:absolutely-standard-Chevalley-Warning}) implies that \(2^{n-(D+\Delta)}\leq |Q|\). Combining these inequalities yields
\[
2^{n-(D+\Delta)}\leq \frac{2^n L}{\binom{n}{\leq\rho n}}\leq \frac{2^n\cdot 2\binom{n}{\leq\lfloor\Delta/2\rfloor}}{\binom{n}{\leq\lfloor w/2\rfloor}},
\]
which immediately implies the result.
\end{proof}
All that remains is to prove our key ingredient, \cref{lem:key-ingredient:low-wt-cw:technical}. As it turns out, it follows quite readily from the following \emph{CLP lemma}, which was instrumental in the recent resolution of the cap set conjecture.
\begin{lemma}[\hspace{1sp}\cite{CLP17}]\label{lem:clp}
Let \(f\in\mathbb{F}_2[x_1,\dots,x_n]\) be a polynomial of degree at most \(r\), and let \(M\) denote the \(2^n\times 2^n\) matrix with entries \(M_{x,y}=f(x+y)\) for \(x,y\in\mathbb{F}_2^n\). Then
\[
\mathsf{rank}(M)\leq2\binom{n}{\leq\lfloor r/2\rfloor}.
\]
\end{lemma}
Given the CLP lemma, we are ready to prove \cref{lem:key-ingredient:low-wt-cw:technical}, which will conclude our proof of \cref{thm:low-wt-cw:technical}.
\begin{proof}[Proof of \cref{lem:key-ingredient:low-wt-cw:technical}]
First, let \(\{g_i\}\subseteq\{f_i\}\) be the set of polynomials in \(\{f_i\}\) that have degree \(>1\). Notice that if \(A+A\) contains a nontrivial common solution to the system \(\{g_i\}\), then it also contains a nontrivial common solution to \(\{f_i\}\): this follows from the linearity of the polynomials of \(\{f_i\}-\{g_i\}\) and the fact that every \(a\in A\) is a common solution (by definition of \(A\)). Thus, it suffices to show the result for the set \(\{g_i\}\).
Next, consider the polynomial \(g\in\mathbb{F}_2[x_1,\dots,x_n]\) defined as
\[
g(x):=\prod_i(1+g_i(x)).
\]
It is straightforward to verify that \(g\) has degree at most \(\Delta\), and that \(g(x)=1\) if and only if \(x\) is a common solution to \(\{g_i\}\). Now, suppose for contradiction that \(A+A\) contains no nontrivial common solution to \(\{g_i\}\): that is, for every distinct \(x,y\in A\) it holds that \(g(x+y)=0\). Then, consider the \(2^n\times2^n\) matrix \(M\) with entries \(M_{x,y}=g(x+y)\) for every \(x,y\in\mathbb{F}_2^n\). Define \(k:=|A|\), and let \(M[A,A]\) denote the \(k\times k\) submatrix of \(M\) obtained by taking the rows and columns of \(M\) indexed by \(A\). Since \(0\) is a common solution to \(\{g_i\}\), we get that \(M[A,A]=I_k\) and thus
\[
\mathsf{rank}(M)\geq\mathsf{rank}(M[A,A])=\mathsf{rank}(I_k)=k>2\binom{n}{\lfloor \Delta/2\rfloor},
\]
which directly contradicts \cref{lem:clp}.
\end{proof}
\subsubsection{Proofs of its corollaries}\label{subsubsec:proof-low-wt-cw:corollaries}
Now that we have proven our general low-weight Chevalley-Warning theorem (\cref{thm:low-wt-cw:technical}), we are ready to prove its corollaries. These proofs are relatively straightforward, but somewhat tedious. We start with the proof of \cref{cor:low-wt-cw:main-technical-cor}, which says that we can always find a nontrivial common solution of Hamming weight \(w\leq 8\Delta + 8D/\log(n/D)+8\).
\begin{proof}[Proof of \cref{cor:low-wt-cw:main-technical-cor}]
First, note that the result is true if \(8\Delta+8D/\log(n/D)+8\geq D+\Delta+1\), as the existence of a nontrivial common solution with Hamming weight \(\leq D+\Delta+1\) is immediate from Chevalley-Warning (see the discussion following \cref{thm:absolutely-standard-Chevalley-Warning}). Thus we henceforth assume, without loss of generality, that
\begin{align}\label{eq:excellent-inequality}
8\Delta+8D/\log(n/D)+8<\Delta+D+1\leq n,
\end{align}
where the last inequality comes from the given hypothesis \(D+\Delta<n\). This string of inequalities will come in handy later.
Now, by \cref{thm:low-wt-cw:technical}, we know that for any \(W\) satisfying both
\begin{align}
\binom{n}{\leq\lfloor W/2\rfloor}&>2^{(D+\Delta+1)\cdot2},\label{eq:first-cw}\\
\binom{n}{\leq\lfloor W/2\rfloor}&>\binom{n}{\leq\lfloor\Delta/2\rfloor}^2,\label{eq:second-cw}
\end{align}
it holds that there is a nontrivial common solution of Hamming weight \(\leq W\). We seek to find the smallest \(W\) satisfying both \cref{eq:first-cw,eq:second-cw}.
We start with \cref{eq:second-cw}. First, by relatively straightforward binomial inequalities, observe
\[
\binom{n}{\leq\lfloor\Delta/2\rfloor}^2\leq\binom{2n}{\leq 2\lfloor\Delta/2\rfloor}\leq\binom{n}{\leq4\lfloor\Delta/2\rfloor},
\]
where the last inequality uses the fact that \(8\Delta<n\) (as implied by \cref{eq:excellent-inequality}). Thus, any \(W\) satisfying
\begin{align*}
\binom{n}{\leq\lfloor W/2\rfloor}>\binom{n}{\leq 4\lfloor\Delta/2\rfloor}
\end{align*}
also satisfies \cref{eq:second-cw}. And the above inequality is satisfied by any \(W\) satisfying
\[
\lfloor W/2\rfloor>4\lfloor\Delta/2\rfloor,
\]
where we have again used the assumption that \(8\Delta<n\) to ensure \(4\lfloor\Delta/2\rfloor<n\). Thus, any \(W\) satisfying
\[
W>4\Delta+1.
\]
also satisfies \cref{eq:second-cw}.
We now consider \cref{eq:first-cw}. By \cref{eq:excellent-inequality}, we may assume \(D>7\Delta+7\), which means that \(\Delta<D/7-1\) and thus \((D+\Delta+1)\cdot2<16D/7\). As a result, any \(W\) satisfying
\begin{align}\label{eq:tough-inequality}
\binom{n}{\leq\lfloor W/2\rfloor}>2^{16D/7}
\end{align}
also satisfies \cref{eq:first-cw}. We consider two cases.
\emph{Case (i): \(D\leq \log n\):} By \cref{eq:excellent-inequality}, we may assume \(n>8\). It is now easy to verify that for any \(W\geq 8\),
\[
\binom{n}{\leq\lfloor W/2 \rfloor}\geq\binom{n}{\leq 4} > n^{16/7}\geq 2^{16D/7},
\]
thereby satisfying \cref{eq:tough-inequality}.
\emph{Case (ii): \(D>\log(n)\):} Let \(1\leq k\leq n\) be a parameter we will pick later, and notice that for any \(W\geq 2k\) we have
\[
\binom{n}{\leq\lfloor W/2\rfloor}\geq\binom{n}{\leq\lfloor k\rfloor}\geq\binom{n}{\lfloor k\rfloor}\geq\left(\frac{n}{\lfloor k\rfloor}\right)^{\lfloor k\rfloor}.
\]
Thus if \(1\leq k\leq n\) is a parameter satisfying \(\lfloor k\rfloor\log(n/\lfloor k\rfloor)>16D/7\)
then any \(W\geq 2k\) satisfies \cref{eq:tough-inequality}. Now, define \(k:=4D/\log(n/D)\). By the case condition and \cref{eq:excellent-inequality}, we have \(4\leq k<n\), and thus
\[
3D/\log(n/D)\leq\lfloor k\rfloor\leq 4D/\log(n/D).
\]
By \cref{eq:excellent-inequality}, we may assume \(\log(n/D)\geq 8\), and thus
\[
\lfloor k\rfloor \log(n/\lfloor k\rfloor)\geq\frac{3D}{\log(n/D)}\log\left(\frac{n\log(n/D)}{4D}\right)\geq\frac{3D}{\log(n/D)}\log\left(\frac{2n}{D}\right)\geq3D > 16D/7.
\]
Thus, every \(W\geq2k=8D/\log(n/D)\) satisfies \cref{eq:tough-inequality}.
To conclude, we get that every
\[
W\geq\max\{4\Delta+2,8,8D/\log(n/D)\}
\]
satisfies both \cref{eq:first-cw,eq:second-cw}, which completes the proof.
\end{proof}
We now turn towards proving \cref{cor:projection-cw}, which says that, not only is it possible to find a low weight common solution, but it is even possible to force this solution to be supported on a target set of coordinates \(S\). The proof follows quite directly by combining \cref{cor:low-wt-cw:main-technical-cor} with the idea of function restrictions.
\begin{proof}[Proof of \cref{cor:projection-cw}]
For any \(y\in\mathbb{F}_2^{|S|}\), let \(y^+\in\mathbb{F}_2^n\) denote the unique string where \(y^+_S=y\) and \(y^+_{\overline{S}}=0^{n-|S|}\). Now, for each \(f_i\) let \(g_i\in\mathbb{F}_2[x_1,\dots,x_{|S|}]\) denote the polynomial \(g_i(x):=f_i(x^+)\). Notice that each \(\deg(g_i)\leq\deg(f_i)\). Now, for each \(i\) create a polynomial \(h_i\in\mathbb{F}_2[x_1,\dots,x_{|S|}]\) as follows. If \(\deg(f_i)>1\) but \(\deg(g_i)=1\): assume without loss of generality that \(g_i(x)\neq x_1\), and set \(h_i(x):=1+(1+x_1)(1+g_i(x))\); otherwise, just define \(h_i(x):=g_i(x)\).
Consider the set of polynomials \(\{h_i\}\). Notice that the nonlinear degree of \(\{h_i\}\) is at most \(\Delta\), and the linear degree of \(\{h_i\}\) is at most \(D\). Thus, by \cref{cor:low-wt-cw:main-technical-cor}, there is a nontrivial common solution \(y\) to \(\{h_i\}\) with weight \(w\leq8\Delta+8D/\log(|S|/D)+8\). It is then straightforward to verify that \(y\) must be a common solution to \(\{g_i\}\), and that \(y^+\) must be a common solution to \(\{f_i\}\). Furthermore, \(y^+\) is clearly supported on \(S\) and has the same Hamming weight as \(y\).
\end{proof}
\subsection{A local version of Cohen-Tal}\label{subsec:cohen-tal-local-sec}
Equipped with our low-weight Chevalley-Warning theorem, we are now ready to prove our local version of Cohen-Tal (\cref{thm:local-cohen-tal:restated}). Recall that this result says every degree \(\leq r\) polynomial admits a large local subspace on which it is constant, and that the lower bounds that we get on its size are tight. For convenience, we split this theorem into two lemmas: one which claims the lower bounds (\cref{lem:lower-bound-cohen-tal-us}), and one which claims their tightness (\cref{lem:upper-bound-cohen-tal-us}). We prove these lemmas in \cref{subsubsec:the-lower-bounds,subsubsec:tightness-of-the-lower-bounds}, respectively. \cref{thm:local-cohen-tal:restated} will then follow immediately.
\subsubsection{The lower bounds}\label{subsubsec:the-lower-bounds}
We start with a proof of the lower bounds in \cref{thm:local-cohen-tal:restated}, which is much more challenging to prove than the tightness result.
\begin{lemma}[Lower bound of \cref{thm:local-cohen-tal:restated}]\label{lem:lower-bound-cohen-tal-us}
There exists a universal constant \(c>0\) such that for every \(n,r,d\in\mathbb{N}\) such that \(2\leq r\leq c\log(n)\) and \(d\leq n^{\frac{1}{r-1}-2^{-10r}}/\log n\), the following holds. For any degree \(\leq r\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\), there exists a \(d\)-local affine subspace \(X\subseteq\mathbb{F}_2^n\) of dimension
\[
k\geq cr (d n\log n)^{1/r}
\]
on which \(f\) is constant.
\end{lemma}
Our proof of \cref{lem:lower-bound-cohen-tal-us} will combine our low-weight Chevalley-Warning theorem (\cref{cor:projection-cw}) with a (known) ingredient from Cohen and Tal \cite{CT15} that involves \emph{directional derivatives}. Given a polynomial $f\in\mathbb{F}_2[x_1,\dots,x_n]$ and a set of vectors \(S\subseteq\mathbb{F}_2^n\), we let \(f_S\) denote the \emph{derivative of \(f\) in the directions of \(S\)}, where
\begin{equation*}
f_S(x):=\sum_{T\subseteq S} f\left(x+\sum_{v\in T}v\right).
\end{equation*}
One useful observation is that \(\deg(f_S)\leq\max\{0,\deg(f)-|S|\}\). In addition to this, the key ingredient about directional derivatives that we import from Cohen and Tal is the following.
\begin{lemma}[\hspace{1sp}\cite{CT15}]\label{lem:derivatives-yahoo}
For any degree \(\leq r\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\) and \(B\subseteq\mathbb{F}_2^n\),
\[
f\left(x + \sum_{v\in S}v \right)=0\text{ for all }S\subseteq B\iff f_S(x)=0\text{ for all }S\subseteq B\text{ of size }|S|\leq r.
\]
\end{lemma}
This shows that \(f\) is \(0\) on the entire affine space \(x+\mathsf{span}(B)\) if and only if \(0\) is a common solution to all \((\leq r)\)-wise directional derivatives of \(f\) across B. Now, given \cref{lem:derivatives-yahoo} and our low-weight Chevalley-Warning theorem (\cref{cor:projection-cw}), we are ready to prove our lower bound for extracting from \(d\)-local affine sources (\cref{lem:lower-bound-cohen-tal-us}).
\begin{proof}[Proof of \cref{lem:lower-bound-cohen-tal-us}]
Assume without loss of generality that \(f(0)=0\), for otherwise we can work with the polynomial \(1+f\). We will iteratively build a basis \(B\subseteq\mathbb{F}_2^n\) of a \(d\)-local subspace \(X\subseteq\mathbb{F}_2^n\) on which \(f\) is constantly \(0\).
Towards this end, we start by initializing several sets:
\begin{itemize}
\item \(B\gets\emptyset\).
\item \(\mathsf{UNIQUE}\gets\emptyset\).
\item \(\mathsf{SATURATED}\gets\emptyset\).
\item \(P\gets\{f\}\).
\end{itemize}
Now, \textbf{while} there exists a common nontrivial solution \(b\in\mathbb{F}_2^n\) to the set \(P\subseteq\mathbb{F}_2[x_1,\dots,x_n]\) such that \(b\) is supported on \([n]\setminus(\mathsf{UNIQUE}\cup\mathsf{SATURATED})\), \textbf{do} the following:
\begin{itemize}
\item Let \(b^\ast\) be the lowest (Hamming) weight vector of this type (breaking ties arbitrarily).
\item Set \(B\gets B\cup\{b^\ast\}\).
\item Define \(\alpha\in[n]\) as the smallest index such that \(b^\ast_\alpha=1\).
\item Set \(\mathsf{UNIQUE}\gets\mathsf{UNIQUE}\cup\{\alpha\}\).
\item Set \(\mathsf{SATURATED}\gets\{i\in[n] : \text{ there are \(d\) vectors }v\in B\text{ with }v_i=1\}\).
\item For each \(S\subseteq B\) of size \(|S|\leq r\), set \(P\gets P\cup\{f_S\}\).
\end{itemize}
This completes the construction of \(B\).
We now proceed with the analysis. First, we argue that at the end of the above construction, \(B\) will hold a basis for a \(d\)-local subspace \(X:=\mathsf{span}(B)\). To see why, first define \(T\) to be the number of times the above loop executes, and for each \(i\in[T]\) let \(b^i\in\mathbb{F}_2^n\) denote the vector added to \(B\) in iteration \(i\). Then, let \(\alpha(i)\in[n]\) denote the smallest index such that \(b^i_{\alpha(i)}=1\), and observe that \(b^j_{\alpha(i)}=0\) for all \(j>i\), because of bullets 3-4 above (combined with the loop condition). Thus \(B=\{b^1,\dots,b^T\}\) is a collection of linearly independent vectors: in other words, a basis. Furthermore, bullet 5 above (combined with the loop condition) ensures that \(X:=\mathsf{span}(B)\) is \(d\)-local.
Thus we have argued that \(X\) is a \(d\)-local subspace of dimension \(T\). Now, by combining bullet 6 above with the loop condition and \cref{lem:derivatives-yahoo}, we immediately get that \(f(x)=0\) for all \(x\in X\). Thus, all that remains is to get a lower bound on \(T\), the number of times that the loop executes.
We call the first iteration of the loop \emph{iteration \(1\)}, and we call the pseudocode that precedes the while loop \emph{iteration \(0\)}. Now, for all \(t=0,\dots,T\), we define the following variables, for convenience:
\begin{itemize}
\item Let \(D_t\) denote the degree of the system \(P\) upon completing iteration \(t\).
\item Let \(\Delta_t\) denote the nonlinear degree of the system \(P\) upon completing iteration \(t\).
\item Let \(u_t\) denote the size of \(|\mathsf{UNIQUE}|\) upon completing iteration \(t\).
\item Let \(s_t\) denote the size of \(|\mathsf{SATURATED}|\) upon completing iteration \(t\).
\item Let \(w_t\) denote the Hamming weight of the vector \(b^\ast\) selected in iteration \(t\).
\end{itemize}
Note that \(w_0\) is undefined above, and for convenience we define \(w_0:=0\). Furthermore, for any \(t>T\), it will be convenient to define \(D_t := D_T, \Delta_t := \Delta_T\), and so on. Now, before we turns towards getting a lower bound on \(T\), we \emph{upper bound} the above quantities.
\begin{itemize}
\item \(D_t\leq\sum_{i=0}^{r-1}\binom{t}{i}(r-i)\leq r\binom{t}{\leq r-1}\). This is because, at the end of iteration \(t\) we have: \(|B|=t\); and \(P\) holds the set of polynomials \(f_S\) for all \(S\subseteq B\) of size \(|S|\leq r\); and \(\deg(f_S)\leq\max\{0,\deg(f)-|S|\}\).
\item \(\Delta_t\leq\sum_{i=0}^{r-2}\binom{t}{i}(r-i)\leq r\binom{t}{\leq r-2}\). This holds for the same reasons as above, except that we are now ignoring the polynomials \(f_S\) for all \(S\subseteq B\) of size \(|S|\geq r-1\) (since these have degree at most \(1\)).
\item \(u_t=t\), since we grow \(\mathsf{UNIQUE}\) by a (unique) coordinate in every iteration.
\item \(s_t\leq\frac{1}{d}\sum_{i=1}^tw_i\), since otherwise the sum of the Hamming weights of the vectors \(b^\ast\) selected in iterations \(1,\dots,t\) is at least \(ds_t>\sum_{i=1}^tw_i\), a contradiction.
\item \(w_t\leq 8\Delta_{t-1}+8D_{t-1}/\log(\frac{n-{u_{t-1}-s_{t-1}}}{D_{t-1}})+8\), by \cref{cor:projection-cw}.
\end{itemize}
We now turn towards getting a lower bound on \(T\). Here, the key observation is that it \emph{must} hold that
\[
D_T \geq n - (u_T + s_T),
\]
because otherwise the degree of the set of polynomials is strictly less than the size of the set on which it must be supported, so \cref{cor:projection-cw} trivially implies that there is \emph{some} nontrivial common solution supported on \(S\) (thereby forcing the next execution of the while condition to pass). A straightforward consequence of this is \(T\geq1\) since \(D_0 = r < n=n-(u_0+s_0)\).
Instead of getting a lower bound on \(T\), it will facilitate the analysis to instead lower bound some \(\tau\leq T\). In particular, let \(\tau\in\{0,1,\dots,T-1\}\) be the smallest integer such that
\begin{align}\label{eq:geqnover2}
D_{\tau+1}+u_{\tau+1}+s_{\tau+1}\geq n/2.
\end{align}
Such a \(\tau\) must exist, since we saw above that \(D_T+u_T+s_T\geq n\). We now seek to lower bound \(\tau\), which will in turn lower bound \(T\).
We start with a basic (but useful) lower bound on \(\tau\). In particular, we claim that \(\tau\geq2r\). To see why, suppose for contradiction \(\tau<2r\). Then by definition of \(\tau\) it holds that \(D_{2r}+u_{2r}+s_{2r}\geq n/2\). But by our upper bounds on these quantities (and the fact that \(r\geq 1\)) we have
\begin{align*}
D_{2r}+u_{2r}+s_{2r}&\leq D_{2r} + u_{2r} + 8\sum_{i=0}^{2r-1}(\Delta_i + D_i + 1)\\
&\leq D_{2r}+u_{2r} + 16r\cdot(\Delta_{2r} + D_{2r}+1)\\
&\leq r\cdot\binom{2r}{\leq r-1} + 2r + 16r\cdot(r\cdot\binom{2r}{\leq r-2} + r\cdot\binom{2r}{\leq r-1}+r)\\
&\leq r\cdot 2^{2r} + 2r + 16r\cdot(r\cdot 2^{2r} + r\cdot 2^{2r}+r)\\
&\leq 2^{12r}.
\end{align*}
Thus we get that \(n/2\leq D_{2r}+u_{2r} + s_{2r}\leq 2^{12r}\), which implies that \(r\geq \log(n)/13\). But the theorem hypothesis claims that \(r\leq c\log(n)\) for some universal constant \(c\); thus, we can just pick \(c\) later to ensure that \(c<\frac{1}{13}\), which would yield a contradiction and conclude the proof of the claim that \(\tau\geq2r\). And since \(r\geq1\) by the theorem hypothesis, we also know that \(\tau\geq2\).
Given the above, we now proceed to get a more general lower bound on \(\tau\). We will do so by sandwiching the quantity \(D_{\tau+1}+u_{\tau+1}+s_{\tau+1}\) between two inequalities. Towards this end, the key observation is that for any \(i\in[\tau]\), it holds that \(u_i+s_i\leq u_i+s_i+D_i < n/2\).\footnote{This useful inequality is the reason we work with \(\tau\) instead of \(T\), as this is not necessarily true for any \(i\in[T]\).} Combining this with \cref{eq:geqnover2} and our earlier upper bounds on \(s_t,w_t\), we have
\begin{align*}
n/2\leq D_{\tau+1}+u_{\tau+1}+s_{\tau+1}&\leq D_{\tau+1}+u_{\tau+1}+\frac{1}{d}\sum_{i=1}^{\tau+1}w_i\\
&\leq D_{\tau+1}+u_{\tau+1}+\frac{8}{d}\sum_{i=0}^{\tau}\left(\Delta_i+\frac{D_i}{\log\left(\frac{n-u_i-s_i}{D_i}\right)}+1\right)\\
&\leq D_{\tau+1}+u_{\tau+1} + \frac{8}{d}\sum_{i=0}^\tau\left(\Delta_\tau + \frac{D_\tau}{\log\left(\frac{n-u_\tau-s_\tau}{D_\tau}\right)}+1\right)\\
&\leq D_{\tau+1}+(\tau+1) + \frac{8}{d}(\tau+1)\left(\Delta_\tau + \frac{D_\tau}{\log\left(\frac{n-u_\tau-s_\tau}{D_\tau}\right)}+1\right)\\
&\leq D_{\tau+1}+(\tau+1) + \frac{8}{d}(\tau+1)\left(\Delta_\tau + \frac{D_\tau}{\log\left(\frac{n}{2D_\tau}\right)}+1\right)\\
&\leq D_{\tau+1} + 2\tau + \frac{16\tau}{d}\left(\Delta_\tau+\frac{D_\tau}{\log\left(\frac{n}{2D_\tau}\right)}+1\right)\\
&\leq D_{\tau+1} + 18\tau + \frac{16\tau}{d}\left(\Delta_\tau + \frac{D_\tau}{\log\left(\frac{n}{2D_\tau}\right)}\right).
\end{align*}
Thus we know that if we define
\begin{align*}
K_1 &:= D_{\tau+1}+18\tau + \frac{16\tau}{d}\Delta_\tau,\\
K_2 &:= \frac{16\tau D_\tau}{d\log(\frac{n}{2D_\tau})},
\end{align*}
it must hold that \(K_1 + K_2\geq n/2\), or rather we know that either \(K_1\geq n/4\) or \(K_2\geq n/4\) must hold. We analyze these cases separately, and get a lower bound on \(\tau\) in each case. But before we do so, it will be useful to recall that for any integers \(b>a\geq0\) it holds that \(\binom{b}{a}\leq\binom{b}{a+1}\) if \(a+1\leq b/2\). Since we saw earlier that \(\tau\geq2r\), we have \(\binom{2\tau}{\leq r-1}\leq r\cdot\binom{2\tau}{r-1}\) and \(\binom{\tau}{\leq r-2}\leq r\cdot\binom{\tau}{r-2}\) and \(\binom{\tau}{\leq r-1}\leq r\cdot \binom{\tau}{r-1}\). We now proceed with the case analysis.
\emph{Case (1): \(K_1\geq n/4\)}: By applying the upper bounds on \(D_t,\Delta_t\) obtained earlier in the proof, the standard binomial inequality \(\binom{n}{k}\leq(en/k)^k\), and the fact that \(r\geq2\), we have:
\begin{align*}
n/4 \leq K_1 = D_{\tau+1}+18\tau + \frac{16\tau}{d}\Delta_\tau &\leq r\cdot\binom{\tau+1}{\leq r-1}+18\tau + \frac{16\tau}{d}r\cdot\binom{\tau}{\leq r-2}\\
&\leq r\cdot\binom{2\tau}{\leq r-1}+18\tau+\frac{16\tau}{d}r\cdot\binom{\tau}{\leq r-2}\\
&\leq r^2\cdot\binom{2\tau}{r-1}+18\tau+\frac{16\tau r^2}{d}\cdot\binom{\tau}{r-2}\\
&\leq r^2\cdot(4e)^{r-1}\cdot\left(\frac{\tau}{r}\right)^{r-1} + 18r\left(\frac{\tau}{r}\right)^{r-1}+\frac{16}{d}r^3\cdot (3e)^r\cdot\left(\frac{\tau}{r}\right)^{r-1}\\
&\leq 3\cdot18\cdot(4e)^r\cdot r^3\cdot\left(\frac{\tau}{r}\right)^{r-1}\\
&\leq 2^6\cdot2^{7r}\cdot\left(\frac{\tau}{r}\right)^{r-1},
\end{align*}
which implies that
\[
n\leq 2^8\cdot2^{7r}\cdot\left(\frac{\tau}{r}\right)^{r-1}\leq 2^{15r}\cdot\left(\frac{\tau}{r}\right)^{r-1},
\]
which gives
\[
\tau\geq r\cdot\left(n\cdot2^{-15r}\right)^{\frac{1}{r-1}}\geq2^{-30}\cdot rn^{\frac{1}{r-1}}.
\]
\emph{Case (2): \(K_2\geq n/4\)}: In this case we have
\[
n/4\leq K_2 = \frac{16\tau D_\tau}{d\log\left(\frac{n}{2D_\tau}\right)},
\]
which of course implies
\[
64\tau D_\tau\geq dn\log\left(\frac{n}{2D_\tau}\right).
\]
Now, notice that our earlier upper bounds on \(D_\tau\) yield
\[
D_\tau\leq r\cdot\binom{\tau}{\leq r-1}\leq r^2\cdot\binom{\tau}{r-1}\leq r^2\cdot(2e)^{r-1}\cdot\left(\frac{\tau}{r}\right)^{r-1}\leq 2^{5r}\cdot\left(\frac{\tau}{r}\right)^{r-1},
\]
and combining this with the previous inequality yields
\[
64\tau\cdot 2^{5r}\cdot\left(\frac{\tau}{r}\right)^{r-1}\geq 64\tau D_\tau\geq dn\log\left(\frac{n}{2D_\tau}\right)\geq dn\log\left(\frac{n}{2^{5r+1}\cdot\left(\frac{\tau}{r}\right)^{r-1}}\right).
\]
Then, applying straightforward bounds on both sides of the above chain of inequalities yields
\begin{align}\label{eq:almost-done-with-hard-direction}
2^{12r}\cdot\left(\frac{\tau}{r}\right)^r\geq dn\log\left(\frac{n}{2^{6r}\cdot\left(\frac{\tau}{r}\right)^{r-1}}\right).
\end{align}
We now seek to get a lower bound on all \(\tau\) that satisfy the above inequality. Towards this end, notice that for any \(\alpha>0\) such that
\begin{align}\label{eq:actually-almost-done-with-hard-direction}
2^{12r}\cdot\left(\frac{\alpha}{r}\right)^r< dn\log\left(\frac{n}{2^{6r}\cdot\left(\frac{\alpha}{r}\right)^{r-1}}\right),
\end{align}
it holds that all \(\tau\) that satisfy \cref{eq:almost-done-with-hard-direction} must also satisfy \(\tau\geq\alpha\): this is because as \(\alpha\) decreases, the left hand side of the above inequality decreases, while its right hand side increases.
It is now straightforward to verify that
\[
\alpha=2^{-30}\cdot r\cdot \left(dn\log n\right)^{1/r}
\]
satisfies \cref{eq:actually-almost-done-with-hard-direction}, as long as \(d\log n\leq n^{\frac{1}{r-1} - 2^{-10r}}\). Thus in this case we have
\[
\tau\geq\alpha = 2^{-30}\cdot r\cdot (dn\log n)^{1/r},
\]
provided \(d\log n\leq n^{\frac{1}{r-1}-2^{-10r}}\).
To conclude, since one of the cases must hold, we get that
\[
\tau\geq 2^{-30}\cdot\min\{rn^{\frac{1}{r-1}},r(dn\log n)^{1/r}\}
\]
if \(d\log n\leq n^{\frac{1}{r-1}-2^{-10r}}\). But given this condition on \(d\log n\), it always holds that \(r(dn\log n)^{1/r}\leq rn^{\frac{1}{r-1}}\). Thus, as long as \(d\log n\leq n^{\frac{1}{r-1} - 2^{-10r}}\), we get that
\[
T\geq \tau\geq 2^{-30}r(dn\log n)^{1/r},
\]
as desired.
\end{proof}
\subsubsection{On the tightness of our lower bounds}\label{subsubsec:tightness-of-the-lower-bounds}
At last, all that remains is to show our lower bounds (in \cref{lem:lower-bound-cohen-tal-us}) are tight, in the following sense.
\begin{lemma}[Upper bound of \cref{thm:local-cohen-tal:restated}]\label{lem:upper-bound-cohen-tal-us}
There exist universal constants \(C,c>0\) such that a random degree \(\leq r\) polynomial \(f\sim\mathbb{F}_2[x_1,\dots,x_n]\) is an extractor for \(d\)-local affine sources with min-entropy
\[
k\geq Cr\cdot(dn\log n)^{1/r}
\]
and error \(\varepsilon=2^{-c k/r}\), except with probability at most \(2^{-c{k\choose\leq r}}\) over the selection of \(f\).
\end{lemma}
Previously, we showed (in \cref{sec:new-upper-bounds}) that low-degree polynomials extract from \(d\)-local sources (\cref{thm:localextintro:restated}), and our main ingredients were: (i) a reduction from \(d\)-local sources to \(d\)-local NOBF sources; and (ii) a result showing that low-degree polynomials extract from \(d\)-local NOBF sources (which relied heavily on a certain correlation bound we proved in \cref{subsubsec:correlation-bounds-sec}).
To prove \cref{lem:upper-bound-cohen-tal-us}, we wish to show that low-degree polynomials extract from \(d\)-local \emph{affine} sources. These are much more structured than general \(d\)-local sources, and we are able to optimize the above framework to skip step (i) and rely on a stronger known correlation bound in step (ii). In particular, our key ingredient will be the following, which shows that a random low-degree polynomial has small bias (i.e., small correlation with the constant \(1\) function).
\begin{lemma}[\hspace{1sp}\cite{low-bias-polys}]\label{lem:low-bias-poly}
For any fixed \(\varepsilon>0\) there exist \(0<c_1,c_2<1\) such that the following holds. Let \(f\sim\mathbb{F}_2[x_1,\dots,x_n]\) be a random polynomial of degree \(\leq r\) where \(r\leq (1-\varepsilon)n\). Then
\[
\Pr[\mathsf{bias}(f)>2^{-c_1n/r}]\leq 2^{-c_2{n\choose\leq r}}.
\]
\end{lemma}
Given this result, we are now able to prove \cref{lem:upper-bound-cohen-tal-us} without too much trouble.
\begin{proof}[Proof of \cref{lem:upper-bound-cohen-tal-us}]
Let \(\mathbf{X}\sim\mathbb{F}_2^n\) be a \(d\)-local affine source of dimension \(k\). It is straightforward to show, via Gaussian elimination, that the following holds: there exists a subset \(S\subseteq[n]\) of size \(k\) such that \(\mathbf{X}_S\) is uniform over \(\mathbb{F}_2^k\), and for every other \(j\in[n]\) there exists an affine function \(\ell_j:\mathbb{F}_2^k\to\mathbb{F}_2\) such that \(\mathbf{X}_j=\ell_j(\mathbf{X}_S)\). Without loss of generality, assume that \(S=[k]\). Thus if we consider the function \(\ell:\mathbb{F}_2^k\to\mathbb{F}_2^n\) defined as
\[
\ell(y_1,\dots,y_k):=(y_1,\dots,y_k,\ell_{k+1}(y_1,\dots,y_k),\dots,\ell_n(y_1,\dots,y_k))
\]
we of course have \(\mathbf{X}=\ell(\mathbf{U}_k)\), where \(\mathbf{U}_k\sim\mathbb{F}_2^k\) denotes the uniform random variable.
Now, let \(f\sim\mathbb{F}_2[x_1,\dots,x_n]\) be a random polynomial of degree \(\leq r\). Notice that
\[
|f(\mathbf{X})-\mathbf{U}_1|=|f(\ell(\mathbf{U}_k))-\mathbf{U}_1|=\mathsf{bias}(f(\ell))/2
\]
So we want to upper bound the probability that \(\mathsf{bias}(f(\ell))\) is large, over our random selection of \(f\). Well, consider fixing whether each monomial \(x^S, S\not\subseteq[k]\) exists. Notice that under any such fixing, \(f(\ell)\) actually turns into a polynomial of the form \(f^{\prime} + g\), where \(f^{\prime}\sim\mathbb{F}_2[x_1,\dots,x_n]\) is a uniformly random degree \(\leq r\) polynomial, and \(g\) is a sum of monomials, each of size at most \(r\), where each term in each monomial is linear (i.e., degree \(1\)). In other words, \(g\) is a fixed degree \(\leq r\) polynomial, and thus \(f^{\prime}+g\) is a uniformly random degree \(\leq r\) polynomial over \(k\) variables.
Thus, by \cref{lem:low-bias-poly}, we get that under \emph{every} such fixing, \(\bias(f(\ell))\leq 2^{-c_1k/r}\) except with probability \(\leq 2^{-c_2{k\choose\leq r}}\). Thus for a given \(d\)-local affine source \(\mathbf{X}\sim\mathbb{F}_2^n\) of min-entropy \(k\), it holds that a random degree \(\leq r\) polynomial extracts from \(\mathbf{X}\) with error \(\leq 2^{-c_1k/r}\), except with probability \(\leq 2^{-c_2{k\choose\leq r}}\).
Now, it is not hard to show that there are at most \({k\choose\leq d}^n\cdot2^n\) \(d\)-local affine sources of dimension \(k\). Thus, by a union bound, we know that a random degree \(\leq r\) polynomial extracts from \emph{all} \(d\)-local affine sources \(\mathbf{X}\sim\mathbb{F}_2^n\) of min-entropy \(k\), except with probability at most
\[
2^{-c_2{k\choose \leq r}}\cdot{k\choose\leq d}^n\cdot 2^n.
\]
Thus as long as \({k\choose\leq d}^n\cdot 2^n\leq 2^{c_2{k\choose\leq r}/2}\), it holds that a random degree \(\leq r\) polynomial extracts from \(d\)-local affine sources with min-entropy \(k\) with error \(\leq 2^{-c_1 k/r}\), with probability \(\geq 1-2^{-c_2{k\choose\leq r}/2}\). It is now easy to verify that the inequality \({k\choose\leq d}^n\cdot2^n\leq 2^{c_2{k\choose\leq r}/2}\) holds for the stated lower bound on \(k\).
\end{proof}
\dobib
\end{document}
\section{Overview of our techniques}\label{sec:overview}
In this section, we provide an overview of the techniques that go into our three main results:
\begin{itemize}
\item An entropy \emph{upper bound} for low-degree extraction from local sources (\cref{thm:localextintro}).
\item An entropy \emph{lower bound} for low-degree extraction from local sources (\cref{thm:localtightintro}).
\item A \emph{barrier} for extracting from local sources using black-box affine extractors (\cref{claim:barrierintro}).
\end{itemize}
Along the way, we will overview the several new key ingredients (Theorems~\ref{thm:red-to-nobf},~\ref{thm:local-cohen-tal}, and~\ref{thm:low-wt-cw}) that go into these main results.
\subsection{Upper bounds}\label{sec:techoverUB}
We begin by discussing our entropy upper bounds for low-degree extraction from local sources. By this, we mean that we upper bound the entropy \emph{requirement} for extracting from \(d\)-local sources using degree \(\leq r\) polynomials. In other words, we show that low-degree polynomials extract from local sources.
\subsubsection{Low-degree extractors for local sources}
We start by sketching the techniques behind our main upper bound (\cref{thm:localextintro}), which shows that most degree \(\leq r\) polynomials are low-error extractors for \(d\)-local sources with min-entropy at least \[k=O(2^dd^2r\cdot(2^dn\log n)^{1/r}).\]
\paragraph*{A strawman application of the probabilistic method.}
A natural first attempt at proving our result would use a standard application of the probabilistic method, which looks something like the following. First, let \(f\in\mathbb{F}_2[x_1,\dots,x_n]\) be a uniformly random polynomial of degree \(\leq r\), meaning that each monomial of size \(\leq r\) is included in \(f\) with probability \(1/2\). Then, we let \(\mathcal{X}\) be the family of \(d\)-local sources over \(\{0,1\}^n\), each with min-entropy at least \(k\). To prove that most of these polynomials are low-error extractors for this family, a standard application of the probabilistic method would suggest that we:
\begin{enumerate}
\item Prove that \(f\) is an extractor for a single \(\mathbf{X}\in\mathcal{X}\) with extremely high probability.
\item Show that the family \(\mathcal{X}\) does not contain too many sources.
\item Conclude, via the union bound, that \(f\) is an extractor for \emph{every} \(\mathbf{X}\in\mathcal{X}\) with high probability.
\end{enumerate}
It is not too hard to complete Steps \(2\) and \(3\) in the above framework, but Step \(1\) turns out to be much more challenging. To see why, let us consider an arbitrary \(d\)-local source \(\mathbf{X}\sim\{0,1\}^n\) with min-entropy at least \(k\). By definition of \(d\)-local source, there exists some \(m\in\mathbb{N}\) and functions \(g_1,\dots,g_n:\{0,1\}^m\to\{0,1\}\) such that each \(g_i\) depends on just \(d\) of its inputs, and such that given a uniform \(\mathbf{Y}\sim\{0,1\}^m\), we have
\[
\mathbf{X}=(g_1(\mathbf{Y}),g_2(\mathbf{Y}),\dots,g_n(\mathbf{Y})).
\]
Now, we want to argue that a random degree \(\leq r\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\) is a low-error extractor for \(\mathbf{X}\). To do so, consider the function \(F:\{0,1\}^m\to\{0,1\}\) defined as
\[
F(y_1,\dots,y_m):=(f\circ g)(y_1,\dots,y_m)=f\left(g_1(y_1,\dots,y_m),\dots,g_n(y_1,\dots,y_m)\right).
\]
Notice that by the definition of \(\mathbf{X}\) and by \cref{def:intro:extractor} of extractor, we know that \(f\) is an extractor for \(\mathbf{X}\) with error \(\varepsilon\) if
\[
|\bias(F)|:=\left|\Pr_{y\sim\mathbf{U}_m}[F(y)=1]-\Pr_{y\sim\mathbf{U}_m}[F(y)=0]\right|\leq2\varepsilon.
\]
Thus, to argue that a random degree \(\leq r\) polynomial \(f\in \mathbb{F}_2[x_1,\dots,x_n]\) is a low-error extractor for \(\mathbf{X}\), it suffices to argue that the function \(F=f\circ g\) has low bias (with high probability over the selection of \(f\)).
Of course, the question now becomes: how can we ensure that \(F\) has low bias? We can start by noticing some properties of \(F\). First, we know \(F=f\circ g\), where \(f\) is a random degree \(\leq r\) polynomial and \(g\) is a fixed function where each output bit depends on \(\leq d\) input bits. Thus, it is not hard to argue that \(F\) will have degree \(\leq rd\). Furthermore, since \(f\) is random and \(g\) is fixed, one may hope to argue that \(F\) is a \emph{uniformly random} polynomial of degree \(\leq rd\): in this case we would be done, since it is well-known that uniformly random low-degree polynomials have extremely low bias (with extremely high probability) \cite{low-bias-polys}.
Unfortunately, it is too much to hope that \(F\) is a uniformly random low-degree polynomial. Indeed, it is not hard to see that the distribution of \(F\) over degree \(\leq rd\) polynomials depends heavily on the exact selection of \(g\). Furthermore, for most selections of \(g\), the random function \(F\) is \emph{not} uniformly distributed over degree \(\leq t\) polynomials for \emph{any} \(t\).
Thus, there is no obvious way to apply \cite{low-bias-polys} in order to argue that \(F\) will have low bias. To proceed, it seems like we will somehow need to argue that the distribution of \(F\) over low-degree polynomials is guaranteed to have some specific \emph{structure}, and then somehow argue that a random polynomial from any such structured distribution is guaranteed to have low-bias. Each of these steps seems quite challenging.
\paragraph*{Reductions to the rescue.}
As it turns out, there is a simple trick we can use to greatly simplify the above approach. The key idea is to \emph{reduce} local sources to a simpler class of sources. Given two familes \(\mathcal{X},\mathcal{Y}\) of distributions over \(\{0,1\}^n\), we say that \(\mathcal{X}\) \emph{reduces} to \(\mathcal{Y}\) if each \(\mathbf{X}\in\mathcal{X}\) is (close to) a convex combination of \(\mathbf{Y}\in\mathcal{Y}\).\footnote{By this we mean that each \(\mathbf{X}\in\mathcal{X}\) can be written in the form \(\mathbf{X}=\sum_ip_i\mathbf{Y}_i\), where each \(\mathbf{Y}_i\in\mathcal{Y}\), \(\sum_ip_i=1\), and \(\mathbf{X}\) samples from \(\mathbf{Y}_i\) with probability \(p_i\).} Reductions are extremely useful, because of the following well-known fact: if \(\mathcal{X}\) reduces to \(\mathcal{Y}\), and \(f:\{0,1\}^n\to\{0,1\}\) is an extractor for \(\mathcal{Y}\), then \emph{\(f\) is also an extractor for \(\mathcal{X}\)}.
Thus, in order to show that low-degree polynomials extract from \(d\)-local sources, a key new ingredient we use is a reduction from \(d\)-local sources to a simpler class of sources called \emph{\(d\)-local non-oblivious bit-fixing (NOBF) sources} \cite{CGGL20}. Using the above discussion, it then suffices to show that low-degree polynomials extract from \(d\)-local NOBF sources. Thus, we proceed by:
\begin{enumerate}
\item Defining local NOBF sources, and showing how we can appropriately tailor our previous attempt at the probabilistic method so that it works for local NOBF sources.
\item Providing a new reduction from local sources to local NOBF sources.
\end{enumerate}
\paragraph*{Low-degree extractors for local NOBF sources.}
A \emph{\(d\)-local NOBF source} \(\mathbf{X}\sim\{0,1\}^n\) is a natural specialization of a \(d\)-local source where the entropic bits of the source must show up ``in plain sight'' somewhere in the source.\footnote{The relationship between local sources and local \emph{NOBF} sources is not dissimilar to the relationship between error-correcting codes and \emph{systematic} error-correcting codes.} More formally, a \(d\)-local NOBF source with min-entropy \(k\) is a random variable \(\mathbf{X}\sim\{0,1\}^n\) for which there exist functions \(g_1,\dots,g_n:\{0,1\}^k\to\{0,1\}\) such that the following holds: each \(g_i,i\in[n]\) depends on \(\leq d\) input bits; for every \(i\in[k]\) there is some \(i^{\prime}\in[n]\) such that \(g_{i^{\prime}}(y)=y_i\); and for uniform \(\mathbf{Y}\sim\{0,1\}^k\) we have
\[
\mathbf{X}=(g_1(\mathbf{Y}),g_2(\mathbf{Y}),\dots,g_n(\mathbf{Y})).
\]
In other words, some \(k\) ``good'' bits in \(\mathbf{X}\) are uniform, and the remaining \(n-k\) ``bad'' bits are \(d\)-local functions of the good bits.
We must now show that a random degree \(\leq r\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\) extracts from \(d\)-local NOBF sources with entropy \(k\). As in our strawman application of the probabilistic method, consider an arbitary \(d\)-local NOBF source \(\mathbf{X}=(g_1(\mathbf{Y}),\dots,g_n(\mathbf{Y}))\) and let \(f\in\mathbb{F}_2[x_1,\dots,x_n]\) be a uniformly random degree \(\leq r\) polynomial. To show that \(f\) extracts from all \(d\)-local NOBF sources, recall that we just need to show that the function \(F:\{0,1\}^k\to\{0,1\}\) defined as
\[
F(y):=(f\circ g)(y)=f\left(g_1(y),\dots,g_n(y)\right)
\]
has extremely low bias with extremely high probability. Furthermore, recall that if we can show that \(F\) itself is a uniformly random low-degree polynomial, then we know via \cite{low-bias-polys} that this is true.
It is still too much to hope that \(F\) is a uniform low-degree polynomial, but \(F\) is now ``close enough in structure'' to one so that we can make this work. To see why, we can first assume without loss of generality (by definition of local NOBF source) that \(g_1(y)=y_1,\dots,g_k(y)=y_k\). Thus, we can define \(c_S\sim\{0,1\}\) as an independent uniform bit (for each \(S\subseteq[n]\) of size \(\leq r\)) and write
\begin{align*}
F(y) = \sum_{S\subseteq[k]:|S|\leq r}c_S\prod_{i\in S}g_i(y) + \sum_{S\subseteq[n]:|S|\leq r, S\not\subseteq[k]}c_S\prod_{i\in S}g_i(y) = A(y) + B(y),
\end{align*}
where \(A\in\mathbb{F}_2[y_1,\dots,y_k]\) is a uniformly random polynomial of degree \(\leq r\), and \(B\in\mathbb{F}_2[y_1,\dots,y_k]\) is a random polynomial whose selection of monomials is \emph{not uniformly random}, but is nevertheless \emph{independent} of the selections made by \(A\).
Thus, to show that \(F\) has extremely low bias, it suffices to show that \(A+B\) has extremely low bias. And to show that \(A+B\) has extremely low bias, it suffices to show that \(A+B^{\prime}\) has low bias for any fixed polynomial \(B^{\prime}\) induced by fixing the random monomials selected by \(B\).
To conclude, we actually show something stronger: recalling that \(A\in\mathbb{F}_2[y_1,\dots,y_k]\) is a uniformly random polynomial of degree \(\leq r\), we show that: for \emph{any} fixed function \(B^\ast:\{0,1\}^k\to\{0,1\}\), it holds that \(A+B^\ast\) has low bias. This does not follow immediately from the result \cite{low-bias-polys} that a random low-degree polynomial has low bias: indeed, it is more general, since \cite{low-bias-polys} is the special case where \(B^\ast=0\). However, it \emph{does} follow immediately from known upper bounds on the list size of Reed-Muller codes \cite{KLP12}.
In the language we are using here, an upper bound on the list size of a Reed-Muller code is equivalent to saying that for any fixed function \(D\in\mathbb{F}_2[x_1,\dots,x_k]\), \(A\) will differ from \(D\) on many inputs, with very high probability. Thus, such bounds tell us that \(A\) differs from \(B^\ast\) on many inputs with very high probability, and \(A\) differs from \(1+B^\ast\) (or rather, \emph{equals} \(B^\ast\)) on many inputs with very high probability. In other words, \(A\) is completely uncorrelated with \(B^\ast\), meaning that \(\bias(A+B^\ast)\) is extremely small, as desired.
Thus a random low-degree polynomial \(f\) extracts from the \(d\)-local NOBF source \(\mathbf{X}\) with min-entropy \(k\) with very high probability. In other words, it fails to do so with some very small probability \(\delta=\delta(k)\) which decreases rapidly as \(k\) grows. By applying the union bound, we get that \(f\) extracts from the entire family \(\mathcal{X}\) of \(d\)-local NOBF sources, provided \(\delta(k)\cdot|\mathcal{X}|\ll 1\). All that remains is to upper bound the size of \(\mathcal{X}\), which can easily be done using the \(d\)-locality of the sources.
\paragraph*{A reduction to local NOBF sources.}
Above, we saw that random low-degree polynomials extract from \emph{local NOBF sources}. To complete the proof that they also extract from more general \emph{local sources}, recall that we need to provide a reduction from local sources to local NOBF sources. In other words, we need to show that every \(d\)-local source with min-entropy \(k\) is (close to) a convex combination of \(d\)-local NOBF sources with min-entropy \(k^{\prime}\approx k\). This is the main key ingredient in our result that low-degree polynomials extract from local sources (\cref{thm:localextintro}).
Our reduction works as follows. First, pick an arbitrary \(d\)-local source \(\mathbf{X}\sim\{0,1\}^n\) with min-entropy \(k\). Let \(k^{\prime}\) be a parameter which is slightly smaller than \(k\), which will be picked later. We start by arguing that \(\mathbf{X}\) is (close to) a convex combination of \(d\)-local NOBF sources where there are \(k^{\prime}\) good bits, but the good bits may be biased (but not constant).
Towards this end, recall that \(\mathbf{X}=(g_1(\mathbf{Y}),\dots,g_n(\mathbf{Y}))\) for some \(d\)-local functions \(g_1,\dots,g_n:\{0,1\}^m\to\{0,1\}\) and uniform \(\mathbf{Y}\sim\{0,1\}^m\). The key idea is to consider the \emph{largest possible set} \(T\subseteq[n]\) of ``good bits,'' i.e., such that \(\{\mathbf{X}_i\}_{i\in T}\) are independent (and none are constants). Then, we let \(T^{\prime}\subseteq[m]\) be the bits of \(\mathbf{Y}\) on which \(\{\mathbf{X}_i\}_{i\in T}\) depend. The key observation is that \emph{every} bit in \(\mathbf{X}\) depends on \emph{some} bit in \(\{\mathbf{Y}_i\}_{i\in T^{\prime}}\), by the maximality of \(T\). Using this observation, there are \emph{two possible cases}, over which we perform a \emph{win-win analysis}.
First, it is possible that \(T\) contains \(\geq k^{\prime}\) bits. In this case, we consider fixing all bits \(\{\mathbf{Y}_i\}_{i\notin T^{\prime}}\). It is then not too hard to show that \(\mathbf{X}\) becomes a source which contains \(\geq k^{\prime}\) good bits (which are mutually independent and not constants), and the remaining bad bits in \(\mathbf{X}\) are deterministic \(d\)-local functions of these good bits.\footnote{Technically, we need to fix a little more randomness to make this happen, but this can be done without much trouble by invoking some standard tricks from the extractor literature.} Thus in this case, we get that \(\mathbf{X}\) is a convex combination of NOBF sources of the desired type.
Second, it is possible that \(T\) contains \(<k^{\prime}\) bits. In this case, we consider fixing all bits \(\{\mathbf{Y}_i\}_{i\in T^{\prime}}\). But since all bits in \(\mathbf{X}\) depend on \emph{some} bit in this set, this fixing \emph{decrements} the locality \(d\to d-1\). And furthermore, since this fixes \(|T^{\prime}|\leq d|T|<dk^{\prime}\) bits, the entropy only decreases from \(k\to k-dk^{\prime}\) by the entropy chain rule. We then recurse until we hit the first case, or until we hit \(d=1\). If we eventually hit the first case, we already know that \(\mathbf{X}\) is a convex combination of NOBF sources of the desired type. On the other hand, it is easy to show that a \(1\)-local source is actually a \(1\)-local NOBF source! Thus we will always arrive at a (biased) \(d^{\prime}\)-local NOBF source with \(d^{\prime}\leq d\), proving that \(\mathbf{X}\) is always convex combination of NOBF sources of the desired type. Depending on when this recursion stops, we will arrive at an NOBF source with the number of good bits equal to at least
\[
\min\{k^{\prime}, k-dk^{\prime},k-d(d-1)k^{\prime},\dots,k-k^{\prime}\prod_{i\in[d]}i\}\geq\min\{k^{\prime},k-d^2k^{\prime}\},
\]
which is always at least \(k^{\prime}\) provided \(k^{\prime}\leq \frac{k}{2d^2}\).
Thus we see that any \(d\)-local source with min-entropy \(k\) can be written as a convex combination of \(d\)-local NOBF source with \(\Omega(k/d^2)\) good bits, where the good bits are mutually independent (and nonconstant), but they may be heavily biased. So all that remains is to show that such biased \(d\)-local NOBF sources can be written as a convex combination of \emph{unbiased} \(d\)-local NOBF sources (as they were originally defined). This step is not difficult, by applying a standard Chernoff bound. However, since each good bit depends on up to \(d\) bits, each such good bit \(\mathbf{X}_i\) may have \(|\bias(\mathbf{X}_i)|=1-2\cdot2^{-d}\). As a result, we end up with \(\Omega(\frac{k}{d^22^d})\) unbiased good bits.
This completes the reduction from local to local NOBF sources. Given our earlier proof sketch that low-degree polynomials extract from local NOBF sources, we finally get that low-degree polynomials also extract from local sources, as desired.
\subsubsection{Low-degree dispersers for local sources}
We now proceed to sketch the proof of \cref{rem:main-remark-here}, which shows that most degree \(\leq r\) polynomials are dispersers for \(d\)-local sources with min-entropy at least
\[
k=O(d^3r\cdot(n\log n)^{1/r}).
\]
This improves our min-entropy requirement for extractors (which was \(k=O(2^dd^2r\cdot(2^d n\log n)^{1/r})\)) by removing two terms of the form \(2^d\). We use a different key idea to remove each \(2^d\) term. While the outer exponential term \(2^d\) is the more dramatic one to remove, it turns out that it is also the easier one. To do this, we simply note that for dispersers, we can forego the last step in our local to local NOBF reduction, which incurs a factor of \(2^d\) by making the biased local NOBF source into an unbiased one. This improves the entropy requirement for dispersers from
\[
k=O(2^dd^2r\cdot(2^d n\log n)^{1/r})\to k=O(d^2r\cdot(2^d n\log n)^{1/r}).
\]
\paragraph{Removing the inner exponential term.} Next, we focus on improving the entropy requirement for dispersers from
\[
k=O(d^2r\cdot(2^d n\log n)^{1/r})\to k=O(d^3r\cdot(n\log n)^{1/r}),
\]
turning the inner exponential term \(2^d\) into an outer linear term \(d\). This improvement is more challenging: while our first improvement relied on improving the local to local-NOBF reduction, this improvement relies on improving the entropy requirement for dispersing from local NOBF sources.
In order to show that low-degree polynomials extract from local NOBF sources, recall that we: (1) showed that a random low-degree polynomial extracts from an arbitrary local NOBF source with extremely high probability; and (2) used a union bound over the family \(\mathcal{X}\) of local NOBF sources to conclude that it extracts from \emph{all} local NOBF sources with high probability. To get our second improvement on the min-entropy requirement for dispersers, we get improved upper bounds on the size of \(\mathcal{X}\), so that our union bound is over fewer terms.
Towards this end, the key new idea is to show that in order to disperse from the family of \(d\)-local NOBF sources \(\mathcal{X}\), it actually suffices to disperse from the much smaller family \(\mathcal{X}^{\prime}\) of so-called \emph{\(d\)-local, degree \(\leq r\) NOBF sources}.\footnote{Technically, we require something slightly stronger than dispersion from such sources, but this does not make a huge difference. We will go into more details below.} This source family is the exact same as \(d\)-local NOBF sources, except it has the added restriction that the bad bits (which still depend on \(\leq d\) good bits each) can each be written as a degree \(\leq r\) polynomials. However, this family is significantly smaller: natural estimates on the sizes of \(\mathcal{X},\mathcal{X}^{\prime}\) give
\begin{align*}
|\mathcal{X}|&\leq{n\choose k}\cdot\left({k\choose d}\cdot 2^{2^d}\right)^{n-k},\\
|\mathcal{X}^{\prime}|&\leq{n\choose k}\cdot\left({k\choose d}\cdot2^{{d\choose \leq r}}\right)^{n-k}.
\end{align*}
After plugging in these improved size bounds, it is straightforward calculation to see that the inner \(2^d\) term from the entropy requirement drops out. So all that remains is to show the above claim that a disperser for \(d\)-local, degree \(\leq r\) NOBF sources automatically works for the more general family of \(d\)-local NOBF sources.
The key ingredient that goes into this claim is a simple lemma on polynomial decomposition. We show the following: for any function \(f:\{0,1\}^n\to\{0,1\}\), any degree \(\leq r\) polynomials \(a_1,\dots,a_n:\{0,1\}^k\to\{0,1\}\), and any polynomials \(b_1,\dots,b_n:\{0,1\}^k\to\{0,1\}\) that have no monomials of size \(\leq r\), the following holds. There exists a polynomial \(h:\{0,1\}^k\to\{0,1\}\) with no monomials of size \(\leq r\) such that
\[
f\left(a_1(y)+b_1(y), \dots, a_n(y)+b_n(y)\right)=f\left(a_1(y),\dots,a_n(y)\right)+h(y).
\]
Then, given an arbitrary \(d\)-local NOBF source \(\mathbf{X}\sim\{0,1\}^n\), the idea is to write it in the form
\[
\mathbf{X}=(a_1(\mathbf{Y})+b_1(\mathbf{Y}),\dots,a_n(\mathbf{Y})+b_n(\mathbf{Y})),
\]
where \(\mathbf{Y}\sim\{0,1\}^k\) is uniform and \(a_i,b_i\) are as before. Using our polynomial decomposition lemma, it then (roughly) holds that \(f\) is a disperser for \(\mathbf{X}\) if \(f\) is a disperser for the simpler class of \(d\)-local, degree \(\leq r\) NOBF sources. More precisely, we actually require from \(f\) a property that is ever-so-slightly stronger than being a disperser: we require that for any \(d\)-local, degree \(\leq r\) NOBF source \(\mathbf{X}^{\prime}=(a_1(\mathbf{Y}),\dots,a_n(\mathbf{Y}))\), it holds that the polynomial \(f(a_1,\dots,a_n)\) has a monomial of degree \(\leq r\). Our polynomial decomposition lemma then guarantees that \(f\) will also have this property for the more general \(d\)-local source, since the polynomial \(h\) in our decomposition lemma does not have any monomials of degree \(\leq r\). Intuitively, \(h\) is not able to ``destroy'' the monomial of degree \(\leq r\) guaranteed to pop out of \(f(a_1(y),\dots,a_n(y))\).
Thus, it suffices to ``disperse'' from \(d\)-local, degree \(\leq r\) NOBF sources in order to disperse from more general \(d\)-local NOBF sources, meaning that we can leverage our improved bound on the size of \(\mathcal{X}^{\prime}\) to get our claimed improvement on the disperser's entropy requirement.
\subsection{Lower bounds}
We now discuss our entropy lower bounds for low-degree extraction from local sources. By this, we mean that for every degree \(\leq r\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\) we can find a \(d\)-local source \(\mathbf{X}\sim\{0,1\}^n\) with relatively high min-entropy \(k\) on which \(f\) is constant. In other words, we show that in order to disperse (and thus extract) from \(d\)-local sources, they must have min-entropy exceeding this value \(k\).
We show that every degree \(r\leq\Omega(\log n)\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\) must admit a \(d\)-local source \(\mathbf{X}\sim\{0,1\}^n\) of min-entropy at least
\[k=\Omega(r(dn\log{n})^{1/r})\]
on which it is constant. That is, we sketch the proof of our lower bound (\cref{thm:localtightintro}).
In order to prove this result, we actually prove a slightly stronger result: we show that we can find a \(d\)-local source \(\mathbf{X}\sim\{0,1\}^n\) with the above parameters such that it is also \emph{affine}.
Our starting point is a tight result of Cohen and Tal \cite{CT15}, which shows that any degree \(\leq r\) polynomial \(f:\mathbb{F}_2^n\to\mathbb{F}_2\) admits a subspace \(V\subseteq\mathbb{F}_2^n\) of dimension \(\Omega(rn^{1/(r-1)})\) on which it is constant. Here, we obtain a (tight) ``local'' version of their result, and show that any degree \(\leq r\) polynomial \(f\) admits a \(d\)-local subspace \(X\subseteq\mathbb{F}_2^n\) of dimension \(k=\Omega(r(dn\log n)^{1/r})\) on which it is constant. Here, we say that \(V\) is \emph{\(d\)-local} if \(V\) has a basis \(v_1,\dots,v_k\in\mathbb{F}_2^n\) such that for any index \(i\in[n]\), at most \(d\) of these basis vectors equal \(1\) at this index. It is straightforward to verify that the uniform distribution \(\mathbf{X}\) over \(V\) is a \(d\)-local source with min-entropy \(k\), so we focus now on proving the existence of such a \(V\).
At a high level, the proof of Cohen and Tal proceeds by iteratively growing a subspace \(V\) on which \(f\) is constant. At each phase, they define a set \(A\subseteq\mathbb{F}_2^n\) such that \(f\) is constant over \(\mathsf{span}(V,x)\) for every \(x\in A\). They note that if \(|A|\) has size \(>2^{\mathsf{dim}(V)}\), then of course there is some \(x\in A\setminus V\) and furthermore we already know that \(f\) is constant on \(\mathsf{span}(V,x)\). Thus, they can grow their monochromatic subspace by one dimension.
In order to get a lower bound on \(|A|\), they note that this set can be defined as the common solutions to a small collection of low-degree polynomials. A classical result known as the \emph{Chevalley-Warning theorem} (\cref{thm:absolutely-standard-Chevalley-Warning}) then shows that \(|A|\geq2^{n-t}\), where \(t\) is the sum of degrees across the collection of polynomials. To complete their proof, they grow their subspace \(V\) until they are no longer able to show \(|A|>2^{\mathsf{dim}(V)}\).
In our lower bound, we show that \(f\) is monochromatic on a \emph{\(d\)-local subspace}. To prove this, we start with the same approach as Cohen and Tal. However, at each phase, we add extra constraints to \(A\) which guarantee the following: if we take any \(x\in A\) and add it to our current subspace \(V\) (with basis, say, \(v_1,\dots,v_{|\dim(V)|}\)), then the location of the \(1\)s appearing in vectors \(x,v_1,\dots,v_{|\dim(V)|}\) satisfy the \(d\)-locality constraint defined above. Again, as long as \(A\) is large enough, we can find some \(x\in A\) that grows the dimension of our \(d\)-local subspace.
In order to ensure that \(A\) remains large for as many iterations as possible, we would like to minimize the impact of the new ``locality'' constraints that we have added to \(A\). Given the description of these constraints above, we observe that these constraints are minimized if we grow \(V\) by carefully selecting vectors that have the lowest possible Hamming weight. However, we now need an upper bound on the Hamming weight of the lightest (nontrivial) common solution to a system of polynomial equations. Thus, our key new ingredient will be a result of this type, which we call a ``low-weight Chevalley-Warning theorem.''
\paragraph*{A low-weight Chevalley-Warning theorem.}
Above, we saw how the classical Chevalley-Warning theorem is critical in lower bounding the size of \(A\), thereby showing that there is some (nontrivial) vector \(v\in A\) by which we can grow our monochromatic subspace. Now, we need an additional guarantee that there is such a \(v\in A\) that also has low Hamming weight. We prove such a result, and call it a \emph{low-weight Chevalley-Warning theorem}. Our theorem roughly says the following. Given a collection \(\{f_i\}\) of polynomials that have cumulative degree \(D\) (and a common solution \(0\)), if most of these polynomials have degree \(\leq 1\) then they admit a nontrivial solution of Hamming weight at most
\[
w=O(D/\log(n/D)).
\]
In order to prove our result, our key observation is that for any large enough subset \(A\) of common solutions to \(\{f_i\}\), it holds that \(A+A\) also contains a (nontrivial) common solution to \(\{f_i\}\). As it turns out, this observation follows quite readily from the CLP lemma \cite{CLP17} - a result which was instrumental in the recent resolution of the cap set conjecture. Furthermore, given the above observation, we obtain an elementary proof of our low-weight Chevalley-Warning theorem, as follows. First, we let \(Q\) denote the set of common solutions to our collection of polynomials, and then:
\begin{enumerate}
\item We assume for contradiction that there is no nontrivial solution \(q\in Q\) of weight \(\leq w\).
\item We use our key observation to conclude that every Hamming ball of radius \(w/2\) cannot have too many elements of \(Q\) in it, which implies that \(Q\) is a list-decodable code with small list size at radius \(w/2\).
\item Using the list-decoding properties of \(Q\), we use the Hamming bound (for list-decodable codes) to get an \emph{upper bound} on its size.
\item Using the fact that \(Q\) holds the common solutions to a small set of low-degree polynomials, we use the classical Chevalley-Warning theorem to get a \emph{lower bound} on its size.
\item We observe that the lower bound is greater than the upper bound, which yields a contradiction.
\end{enumerate}
Equipped with our low-weight Chevalley-Warning theorem, our entropy lower bound for low-degree extraction from \(d\)-local affine spaces follows immediately via the proof sketch described above.
\subsection{A barrier}
To conclude our overview, we provide a proof sketch of our barrier result (\cref{claim:barrierintro}), which shows that affine extractors (applied in a black-box manner) cannot extract from local sources with min-entropy \(k=\Omega(\sqrt{n})\), even if the locality is \(2\). More formally, to show that an affine extractor also extracts from a different family \(\mathcal{Q}\) of distributions with min-entropy \(k\), the standard technique is to show that each source \(\mathbf{Q}\in\mathcal{Q}\) with min-entropy \(k\) is (close to) a convex combination of affine sources with min-entropy slightly less than \(k\). Here, we show that this is simply not possible for local sources with min-entropy \(\sqrt{n}\). In particular, we show that there is a very simple \(2\)-local source \(\mathbf{Q}\sim\{0,1\}^n\) with min-entropy \(\Omega(\sqrt{n})\) that has statistical distance \emph{exponentially close to \(1\)} from any convex combination of affine sources with min-entropy \(k^{\prime}\).
In more detail, we consider the \(2\)-local ``clique'' source \(\mathbf{Q}\sim\{0,1\}^n\) defined as follows: first, pick any \(\ell\in\mathbb{N}\) and set \(n:=\ell+{\ell\choose 2}\). Then, pick uniform and independent bits \(\mathbf{q}_1,\dots,\mathbf{q}_\ell\sim\{0,1\}\) and set \(\mathbf{Q}\) to be the concatenation of all \(\mathbf{q}_i\) over \(1 \le i \le n\) and \(\mathbf{q}_i\cdot\mathbf{q}_j\) over \(1\leq i<j\leq \ell\). Now, let \(\mathbf{X}\sim\{0,1\}^n\) be a convex combination of affine sources, each with min-entropy \(\ell^{\prime}\). We argue that \(|\mathbf{Q}-\mathbf{X}|\geq1-2^{-\Omega(\ell^{\prime})}\) by showing that for any \(\ell^{\prime}\)-dimensional affine \(\mathbb{F}_2\)-subspace \(S\subseteq\{0,1\}^n\), it holds that \(|\text{support}(\mathbf{Q})\cap S|/|S|\leq 2^{-\Omega(\ell^{\prime})}\). That is, we wish to show that \(Q=\text{support}(\mathbf{Q})\) is \emph{subspace-evasive}.
To show that cliques are subspace-evasive, we use the following key observation:
For any nonempty set \(Q^{\prime}\subseteq Q\) of cliques, the set \(Q^{\prime} + Q^{\prime} := \{u+v : u\in Q^{\prime},v\in Q^{\prime}, u\neq v\}\) (where the sum is over \(\mathbb{F}_2^n\)) has a ``Sidon property:'' each element \(x\) in \(Q^{\prime} + Q^{\prime}\) has a unique pair \(u,v \in Q^{\prime}\) such that \(x = u+v\). This observation is proven by noticing that by making another copy of each coordinate of the form \(\mathbf{q}_i\cdot\mathbf{q}_j\), the set \(Q\) will correspond precisely to the symmetric rank-1 matrices of \(\mathbb{F}_2^{\ell \times \ell}\). Thus all the elements in \(Q^{\prime} + Q^{\prime}\) would correspond to symmetric \(\mathbb{F}_2^{\ell \times \ell}\) matrices of rank at most $2$. Hence by looking at the row space of \(x \in Q^{\prime} + Q^{\prime}\), we can precisely find its symmetric rank-1 decomposition. That is, we can find \(u,v \in Q^{\prime}\) such that $x=u+v$. Now, pick \(Q^{\prime} = Q \cap S\). Since addition is closed in \(S\), we see that \(Q^{\prime} + Q^{\prime} \subseteq S\). Thus \(|S| \geq |Q^{\prime} + Q^{\prime}| \geq {|Q^{\prime}|\choose 2} \geq \Omega(|Q\cap S|^2)\). Hence we find that \(|Q\cap S|/|S|\leq O(\sqrt{|S|}/|S|)=O(1/\sqrt{|S|})\), which is \(2^{-\Omega(\ell^{\prime})}\) as \(|S| = 2^{\ell^{\prime}}\).
\end{document}
\section{Preliminaries}\label{sec:new-preliminaries}
We outline some basic notation, definitions, and facts that will be used throughout the paper.
\subsection{Notation}
We let \(\mathbb{N}:=\{1,2,\dots,\}\) denote the natural numbers, and for any \(n\in\mathbb{N}\), we define \([n]:=\{1,2,\dots,n\}\). Throughout, we use \(\log\) to denote the base-\(2\) logarithm, and we write \(\binom{n}{\leq r}:=\sum_{i=0}^r\binom{n}{i}\). For a bitstring \(x\in\{0,1\}^n\), we let \(x_i\in\{0,1\}\) denote the bit it holds at coordinate \(i\), and for a set \(S\subseteq[n]\) we let \(x_S\in\{0,1\}^{|S|}\) denote the the concatenation of bits \(x_i,i\in S\) in increasing order of \(i\). On the other hand, we define \(x^S:=\prod_{i\in S}x_i\), and let \(x^\emptyset:=1\).
We define the \emph{support} of \(x\in\{0,1\}^n\) as the set of coordinates \(i\) where \(x_i=1\).
For any \(n\in\mathbb{N}\), we let \(0^n\) denote the string of \(n\) \(0\)s in a row, and we let \(1^n\) denote the string of \(n\) \(1\)s in a row. Given a set \(S\subseteq[n]\), we write \(\overline{S}\) to denote its complement. We let \(\mathbb{F}_2\) denote the finite field of size \(2\), and we let \(\mathbb{F}_2^n\) denote the \(n\)-dimensional vector space over \(\mathbb{F}_2\). Given sets \(A,B\subseteq\mathbb{F}_2^n\), we let \(A+B\) denote the \emph{sumset} of \(A\) and \(B\):
\[
A+B:=\{a+b : a\in A, b\in B\}.
\]
\subsection{Probability theory}
We now overview some basic notions from (discrete) probability theory. First, random variables are denoted by boldface letters such as \(\mathbf{X}\), and we let \(\text{support}(\mathbf{X})\) denote its support. For a set \(S\), we write \(\mathbf{X}\sim S\) to denote that \(\mathbf{X}\) is supported on a subset of \(S\). Next, the uniform distribution over \(\{0,1\}^m\) is denoted by \(\mathbf{U}_m\), and whenever \(\mathbf{U}_m\) appears multiple times in the same expression or formula, this represents multiple copies of the \emph{same} random variable. For example, the random variable \((\mathbf{U}_m,\mathbf{U}_m)\) hits each string \((x,x)\in\{0,1\}^m\times\{0,1\}^m\) with probability \(2^{-m}\).
Throughout, we will measure ``randomness'' of a random variable by its min-entropy:
\begin{definition}[Min-entropy]\label{def:minent}
The \emph{min-entropy} of a random variable $\mathbf{X}$ supported on a set $S$, denoted by $H_\infty(\mathbf{X})$, is defined as
\begin{equation*}
H_\infty(\mathbf{X})=-\log \max_{x\in S}\Pr[\mathbf{X}=x].
\end{equation*}
\end{definition}
The following lemma about min-entropy is not difficult to show, but is extremely useful.
\begin{lemma}\label{lem:clean-entropy-drop}
Suppose $\mathbf{X}$ and $\mathbf{Y}$ are arbitrary random variables such that $\mathbf{Y}$ is uniformly distributed over its support.
Then, for every \(y\in\text{support}(\mathbf{Y})\), it holds that
\begin{equation*}
H_\infty(\mathbf{X}|\mathbf{Y}=y)\geqH_\infty(\mathbf{X})-\log|\text{support}(\mathbf{Y})|.
\end{equation*}
\end{lemma}
We measure the similarity of two random variables (or, rather, their distributions) via statistical distance.
\begin{definition}[Statistical distance]
The \emph{statistical distance} between two random variables $\mathbf{X}$ and $\mathbf{Y}$ supported on a set $S$, denoted by $\Delta(\mathbf{X},\mathbf{Y})$, is defined as
\begin{equation*}
\Delta(\mathbf{X},\mathbf{Y})=\max_{T\subseteq S}|\Pr[\mathbf{X}\in T]-\Pr[\mathbf{Y}\in T]|=\frac{1}{2}\sum_{s\in S}|\Pr[\mathbf{X}=s]-\Pr[\mathbf{Y}=s]|.
\end{equation*}
Moreover, we say $\mathbf{X},\mathbf{Y}$ are \emph{$\varepsilon$-close}, denoted $\mathbf{X}\approx_\varepsilon \mathbf{Y}$, if $\Delta(\mathbf{X},\mathbf{Y})\leq \varepsilon$.
\noindent On the other hand, if \(\Delta(\mathbf{X},\mathbf{Y})\geq\varepsilon\), we say that \(\mathbf{X},\mathbf{Y}\) are \emph{\(\varepsilon\)-far}.
\end{definition}
Next, we say that \(\mathbf{X}\) is a \emph{convex combination} of distributions \(\{\mathbf{Y}_i\}\) if there exist probabilities \(\{p_i\}\) summing to \(1\) such that \(\mathbf{X}=\sum_i p_i\mathbf{Y}_i\), meaning that \(\mathbf{X}\) samples from \(\mathbf{Y}_i\) with probability \(p_i\). The following standard fact about convex combinations is extremely useful in extractor research, and is straightforward to show.
\begin{fact}\label{fact:convex-combo-lifted-extractor}
Let \(\mathsf{Ext}:\{0,1\}^n\to\{0,1\}\) be an extractor for a family of distributions \(\mathcal{X}\) over \(\{0,1\}^n\) with error \(\varepsilon\). Then \(\mathsf{Ext}\) also extracts with error \(\varepsilon\) from any \(\mathbf{X}\) which is a \emph{convex combination} of distributions from \(\mathcal{X}\).
\end{fact}
For the easier setting of dispersers, it is easy to show something even stronger than the above fact.
\begin{fact}\label{fact:convex-combo-lifted-disperser-optimized}
Let \(\mathsf{Disp}:\{0,1\}^n\to\{0,1\}\) be a disperser for a family of distributions \(\mathcal{X}\) over \(\{0,1\}^n\). Then \(\mathsf{Disp}\) is also a disperser for any \(\mathbf{X}\) whose support contains the support of some \(\mathbf{X}^{\prime}\in\mathcal{X}\).
\end{fact}
Finally, we will make use of the following standard concentration bound.
\begin{lemma}[Chernoff bound, lower tail]\label{lem:chernoff}
Let \(\mathbf{X}_1,\dots,\mathbf{X}_n\) be independent random variables over \(\{0,1\}\), and define \(\mathbf{Z}:=\sum_{i\in[n]}\mathbf{X}_i\). Then for every \(0<\delta<1\), it holds that
\[
\Pr[\mathbf{Z}\leq(1-\delta)\cdot\mathbb{E}[\mathbf{Z}]]\leq e^{-\mathbb{E}[\mathbf{Z}]\cdot\delta^2/2}.
\]
\end{lemma}
\subsection{Coding theory}
We will now review some basics of coding theory, which will be used in both our entropy upper bounds and lower bounds. First, given a string \(v\in\{0,1\}^n\), we let \(\Delta(v):=\{i\in[n] : v_i\}\) denote its \emph{Hamming weight}, and for any strings \(u,v\in\{0,1\}^n\), we let \(\Delta(u,v):=\{i\in[n] : u_i\neq v_i\}\) denote their \emph{Hamming distance}. Then, given any string \(v\in\{0,1\}^n\) and integer \(r\in\{0,1,\dots,n\}\), we let
\[
\mathcal{B}(v,r):=\{v^{\prime}\in\{0,1\}^n : \Delta(v,v^{\prime})\leq r\}
\]
denote the (closed) Hamming ball of radius \(r\) around (i.e., centered at) \(v\). If we replace the inequality ``\(\leq\)'' above with a strict inequality ``\(<\)'', we call this object the \emph{open} Hamming ball of radius \(r\) around \(v\), and denote it by \(\mathcal{B}^-(v,r)\). Notice that the closed Hamming ball has size \(\binom{n}{\leq r}\), whereas the open Hamming ball has size \(\binom{n}{\leq r-1}\).
An \((n,k,d)\) code is a subset \(Q\subseteq\{0,1\}^n\) of size \(2^k\) such that the pairwise Hamming distance between any distinct \(x,y\in Q\) is at least \(d\). We call an \((n,k,d)\) code \(Q\subseteq\{0,1\}^n\) \emph{linear} if it is an \(\mathbb{F}_2\)-linear subspace, and subsequently call it an \([n,k,d]\) code.
A standard relaxation of \((n,k,d)\) codes are \emph{list-decodable codes}. We say that a subset \(Q\subseteq\{0,1\}^n\) is a \((\rho,L)\) list-decodable code if every Hamming ball of radius at most \(\rho n\) contains at most \(L\) codewords (i.e., elements of \(Q\)). The following classical bound places a limit on the tradeoff between the size and list-decodability of arbitrary codes.
\begin{theorem}[Hamming bound for list-decodable codes]\label{thm:hamming-bound}
For any \((\rho,L)\)-list-decodable code \(Q\subseteq\{0,1\}^n\),
\[
|Q|\leq\frac{2^n L}{\binom{n}{\leq \rho n}}.
\]
\end{theorem}
\begin{proof}
Consider the quantity \(\sum_{q\in Q}|\mathcal{B}(q,\rho n)|\). Notice that for any fixed \(v\in\{0,1\}^n\), there are at most \(L\) codewords \(q\in Q\) such that \(v\in\mathcal{B}(q,\rho n)\), because otherwise \(|\mathcal{B}(v,\rho n)\cap Q|>L\), contradicting the list-decodability of \(Q\). Thus
\[
|Q|\cdot\binom{n}{\leq\rho n}=\sum_{q\in Q}|\mathcal{B}(q,\rho n)|\leq\sum_{v\in\{0,1\}^n}L\leq2^nL,
\]
and the inequality follows.
\end{proof}
Finally, a classic linear code that will be used in this paper is the \emph{Reed-Muller code}, defined below.
\begin{definition}
The \emph{Reed-Muller code} \(\mathsf{RM}(m,r)\) is the subset \(Q\subseteq\mathbb{F}_2^{2^m}\) defined as follows:
\[
Q:=\{ (p(\alpha))_{\alpha\in\mathbb{F}_2^m} : p\text{ is a multilinear polynomial of degree \(\leq r\).}\}
\]
It is a \([2^m,\binom{m}{\leq r},2^{m-r}]\) linear code.
\end{definition}
\subsection{Boolean functions and \(\mathbb{F}_2\)-polynomials}\label{subsec:prelims:polynomials}
We now review some standard definitions and facts about boolean functions and \(\mathbb{F}_2\)-polynomials, which will be used throughout the paper. First, every function \(f:\mathbb{F}_2^n\to\mathbb{F}_2\) has a unique representation as a multilinear \(\mathbb{F}_2\)-polynomial. This means that there exist (unique) coefficients \(\{c_S\}_{S\subseteq[n]}\), each from \(\mathbb{F}_2\), so that
\[
f(x) = \sum_{S\subseteq[n]}c_Sx^S.
\]
The \emph{degree} of \(f\), denoted \(\deg(f)\), is the size of the largest \(S\) such that \(c_S=1\), and we say that \(\deg(f)=-\infty\) if all \(c_S=0\). A \emph{random degree \(\leq r\) polynomial} (independently) sets each \(c_S\) to \(1\) with probability \(1/2\) if \(|S|\leq r\), and otherwise sets \(c_S=0\). We often do not distinguish between a function \(f:\{0,1\}^n\to\{0,1\}\) and its natural interpretation as a function \(f:\mathbb{F}_2^n\to\mathbb{F}_2\), nor do we distinguish between a function \(f:\mathbb{F}_2^n\to\mathbb{F}_2\) and its unique representation as a multilinear \(\mathbb{F}_2\)-polynomial. This allows us to slightly abuse language, and gives meaning to phrases like ``the degree of \(f:\{0,1\}^n\to\{0,1\}\).''
Next, we let \(\mathbb{F}_2[x_1,\dots,x_n]\) denote the set of (multilinear) \(\mathbb{F}_2\)-polynomials, and we let \(\{f_i\}\subseteq\mathbb{F}_2[x_1,\dots,x_n]\) refer to a set (of unspecified size) of such polynomials. The \emph{linear degree} of \(\{f_i\}\) is the sum of the degrees of the \(f_i\)'s which have degree \(1\), the \emph{nonlinear degree} of \(\{f_i\}\) is the sum of the degrees of the \(f_i\)'s which have degree \(>1\), and the \emph{degree} of \(\{f_i\}\) is the sum of the degrees of all the \(f_i\)'s. We say that \(x\) is a \emph{common solution} to \(\{f_i\}\) if \(f_i(x)=0\) for all \(i\), and we say that \(x\) is \emph{nontrivial} if \(x\neq 0\).
The notion of \emph{bias} and \emph{correlation} of boolean functions will be important for our entropy upper bounds. Given arbitrary functions \(f,g:\{0,1\}^n\to\{0,1\}\), we define their \emph{correlation} as
\[
\corr(f,g):=|\mathbb{E}_{x\sim\{0,1\}^n}[(-1)^{f(x)+g(x)}]|=|\Pr[f(x)=g(x)]-\Pr[f(x)\neq g(x)]|,
\]
and we define the \emph{bias} of \(f\) as
\[
\bias(f):=|\mathbb{E}_{x\sim\{0,1\}^n}[(-1)^{f(x)}]|=|\Pr[f(x)=0]-\Pr[f(x)=1]|.
\]
Observe that \(\bias(f)=\corr(f,0)\).
To conclude this section, we record some notation about function combination and composition. For any two functions \(a,b:\mathbb{F}_2^k\to\mathbb{F}_2\), we let \(a+b:\mathbb{F}_2^k\to\mathbb{F}_2\) denote the function \((a+b)(x):=a(x)+b(x)\), and we let \(ab:\mathbb{F}_2^k\to\mathbb{F}_2\) denote the function \(ab(x):=a(x)b(x)\). For any functions \(f:\mathbb{F}_2^n\to\mathbb{F}_2\) and \(a_1,\dots,a_n:\mathbb{F}_2^k\to\mathbb{F}_2\), we let \(f(a_1,\dots,a_n):\mathbb{F}_2^k\to\mathbb{F}_2\) denote the function
\(
f(a_1,\dots,a_n)(x):=f(a_1(x),\dots,a_n(x))
\).
Finally, given functions \(a_1,\dots,a_n:\mathbb{F}_2^k\to\mathbb{F}_2\) and \(S\subseteq[n]\), we defined \(a^S:\mathbb{F}_2^k\to\mathbb{F}_2\) as \(a^S:=\prod_{i\in S}a_i\), and we let \(a^\emptyset:=1\).
\end{document}
\section{Entropy upper bounds}\label{sec:new-upper-bounds}
In this section, we obtain upper bounds on the entropy required to extract from \(d\)-local sources using degree \(\leq r\) polynomials, proving the following theorem.
\begin{theorem}[\cref{thm:localextintro}, restated]\label{thm:localextintro:restated}
There are universal constants \(C,c>0\) such that for all \(n,d,r\in\mathbb{N}\), the following holds. With probability at least \(0.99\) over the choice of a random degree \(\leq r\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\), it holds that \(f\) is an \(\varepsilon\)-extractor for \(d\)-local sources with min-entropy \(\geq k\) and error \(\varepsilon=2^{-\frac{ck}{r^32^dd^2}}\), as long as
\[
k\geq C2^dd^2r\cdot(2^dn\log n)^{1/r}.
\]
\end{theorem}
In order to prove \cref{thm:localextintro:restated}, we combine two key ingredients. Our first key ingredient reduces \(d\)-local sources to the more specialized family of \(d\)-local NOBF sources:
\begin{theorem}[\cref{thm:red-to-nobf}, restated]\label{thm:upper-bounds:first-key-ingredient}
There exists a universal constant \(c>0\) such that for any \(n,k,d\in\mathbb{N}\), the following holds. Let \(\mathbf{X}\sim\{0,1\}^n\) be a \(d\)-local source with min-entropy \(\geq k\). Then \(\mathbf{X}\) is \(\varepsilon\)-close to a convex combination of \(d\)-local NOBF sources with min-entropy \(\geq k^{\prime}\), where \(\varepsilon=2^{-ck^{\prime}}\) and
\[
k^{\prime}=\frac{ck}{2^d d^2}.
\]
\end{theorem}
Our second key ingredient gives upper bounds on the entropy required to extract from \(d\)-local NOBF sources using degree \(\leq r\) polynomials:
\begin{theorem}\label{thm:upper-bounds:second-key-ingredient}
There are universal constants \(C,c>0\) such that for all \(n,d,r\in\mathbb{N}\), the following holds.
With probability at least \(0.99\) over the choice of a random degree \(\leq r\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\), it holds that \(f\) is an \(\varepsilon\)-extractor for \(d\)-local NOBF sources of min-entropy \(\geq k\) with error \(\varepsilon=2^{-ck/r^3}\), as long as
\[
k\geq C r\cdot(2^d\cdot n\log n)^{1/r}.
\]
\end{theorem}
Recall that if an extractor works for a family \(\mathcal{X}\) of sources over \(\{0,1\}^n\), then it also works for any \(\mathbf{X}\sim\{0,1\}^n\) which is a convex combination of sources from \(\mathcal{X}\) (\cref{fact:convex-combo-lifted-extractor}). As a result, by combining \cref{thm:upper-bounds:first-key-ingredient,thm:upper-bounds:second-key-ingredient}, we immediately get \cref{thm:localextintro:restated}. Thus, in the remainder of this section, we just focus on proving \cref{thm:upper-bounds:first-key-ingredient,thm:upper-bounds:second-key-ingredient}. We prove \cref{thm:upper-bounds:first-key-ingredient} in \cref{subsec:upper-bounds:first-key-ingredient}, and we prove \cref{thm:upper-bounds:second-key-ingredient} in \cref{subsec:upper-bounds:second-key-ingredient}. We conclude the section in \cref{subsec:optimized-dispersers}, where we show that \cref{thm:localextintro:restated} can be optimized for the easier setting of \emph{dispersers}, allowing us to obtain a formal version of \cref{rem:main-remark-here}.
\subsection{A reduction from \(d\)-local sources to \(d\)-local NOBF sources}\label{subsec:upper-bounds:first-key-ingredient}
We start by proving our reduction, \cref{thm:upper-bounds:first-key-ingredient}. In order to reduce \(d\)-local sources to \(d\)-local NOBF sources, we use an intermediate model called \emph{biased \(d\)-local NOBF sources}.
\begin{definition}
A random variable \(\mathbf{X}\sim\{0,1\}^n\) is a \emph{\((\delta,k)\)-biased \(d\)-local NOBF source} if there exists a set \(S\subseteq[n]\) of size \(k\) such that both of the following hold:
\begin{itemize}
\item The bits in \(\mathbf{X}_S\) are mutually independent (but need not be identically distributed), and each \(\mathbf{X}_i,i\in S\) has bias \(|\Pr[\mathbf{X}_i=1]-\Pr[\mathbf{X}_i=0]|\leq\delta\).
\item Every other bit \(\mathbf{X}_j,j\notin S\) is a deterministic function of at most \(d\) bits in \(\mathbf{X}_S\).
\end{itemize}
\end{definition}
Notice that this intermediate model generalizes \(d\)-local NOBF sources of min-entropy \(k\), which are just \((0,k)\)-biased \(d\)-local NOBF sources. Now, given this intermediate model, we prove \cref{thm:upper-bounds:first-key-ingredient} by combining two lemmas.
The first lemma reduces \(d\)-local sources to biased \(d\)-local NOBF sources.
\begin{lemma}[Reduction, Part 1]\label{lem:hard-part-of-reduction}
Let \(\mathbf{X}\sim\{0,1\}^n\) be a \(d\)-local source with min-entropy \(\geq k\). Then \(\mathbf{X}\) is a convex combination of \((\delta,k^{\prime})\)-biased \(d\)-local NOBF sources, where \(\delta\leq1-2^{-d}\) and \(k^{\prime}\geq k/(2d^2)\).
\end{lemma}
The second lemma reduces biased \(d\)-local NOBF sources to (unbiased) \(d\)-local NOBF sources.
\begin{lemma}[Reduction, Part 2]\label{lem:simulate-coins}
Let \(\mathbf{X}\sim\{0,1\}^n\) be a \((\delta,k)\)-biased \(d\)-local NOBF source. Then \(\mathbf{X}\) is \(\varepsilon\)-close to a convex combination of \((0,k^{\prime})\)-biased \(d\)-local NOBF sources, where \(k^{\prime}\geq(1-\delta)k/4\) and \(\varepsilon=2^{-(1-\delta)k/4}\).
\end{lemma}
By combining these two lemmas, \cref{thm:upper-bounds:first-key-ingredient} follows immediately. Thus, all that remains in the proof of our reduction (\cref{thm:upper-bounds:first-key-ingredient}) is to prove \cref{lem:hard-part-of-reduction,lem:simulate-coins}, which we do in \cref{subsubsec:first-part-reduction,subsubsec:second-part-reduction}, respectively.
\subsubsection{A reduction from \(d\)-local sources to biased \(d\)-local NOBF sources}\label{subsubsec:first-part-reduction}
We now prove the first part of our reduction (\cref{lem:hard-part-of-reduction}), which shows that every \(d\)-local source is a convex combination of biased \(d\)-local NOBF sources.
\begin{proof}[Proof of \cref{lem:hard-part-of-reduction}]
Let \(\mathbf{X}\sim\{0,1\}^n\) be a \(d\)-local source with min-entropy \(\geq k\). We wish to show that \(\mathbf{X}\) is a convex combination of \((\delta,k^{\prime})\)-biased \(d\)-local NOBF sources, where \(\delta\leq 1-2^{-d}\) and \(k^{\prime}\geq k/(2d^2)\).
The key observation that we will prove is that for any \(t\), one of the following \emph{must} hold: either
\begin{itemize}
\item \(\mathbf{X}\) is a convex combination of \((\delta,t)\)-biased \(d\)-local NOBF sources, for \(\delta\leq1-2^{-d}\); or
\item \(\mathbf{X}\) is a convex combination of \((d-1)\)-local sources with min-entropy \(>k-td\).
\end{itemize}
Before we prove this key observation, let us see how we can use it to prove the desired result. First, recall that convex combinations ``stack'' in the following sense: if a source \(\mathbf{X}\) is a convex combination of convex combinations of sources from a family \(\mathcal{X}\), then \(\mathbf{X}\) is just a convex combination of sources from \(\mathcal{X}\). Thus, by repeatedly applying the key observation until either the first item becomes true or we arrive at a \(1\)-local source (the ``base case''), we see that \(\mathbf{X}\) is a convex combination of sources \(\{\mathbf{Z}_i\}\), where each \(\mathbf{Z}_i\) is either:
\begin{itemize}
\item A \((\delta,t)\)-biased \(d\)-local NOBF source, for \(\delta\leq1-2^{-d}\); or
\item A \(1\)-local source with min-entropy \(>k - t\cdot(d + (d-1) + \dots + 2)=k-t\cdot(d^2+d-2)\).
\end{itemize}
However, it is clear from the definitions that a \(1\)-local source with min-entropy \(k^{\prime}\) is a \(1\)-local NOBF source with min-entropy \(k^{\prime}\). Furthermore, it is easy to see that a \(1\)-local NOBF source with min-entropy \(\geq k^{\prime}\) is a convex combination of \(1\)-local NOBF sources with min-entropy exactly \(k^{\prime}\), by fixing any additional random ``good'' bits. Thus, for any \(t\leq k^{\prime}\), we know that a \(1\)-local source with min-entropy \(\geq k^{\prime}\) is a convex combination of \((\delta,t)\)-biased \(d\)-local NOBF sources, for \(\delta\leq1-2^{-d}\).
By the above discussion, we see that for any \(t\leq k-t\cdot(d^2+d-2)\), \(\mathbf{X}\) is a convex combination of \((\delta,t)\)-biased \(d\)-local NOBF sources, where \(\delta\leq1-2^{-d}\). Setting \(t=\frac{k}{2d^2}\) yields the result.
Thus, all that remains is to prove the key observation stated at the beginning of the proof. Towards this end, let \(\mathbf{X}\sim\{0,1\}^n\) be a \(d\)-local source with min-entropy \(\geq k\). By definition of \(d\)-local source, there exists some \(\ell\) and \(f:\{0,1\}^\ell\to\{0,1\}^n\) such that \(\mathbf{X}=f(\mathbf{Y})\) for uniform \(\mathbf{Y}\sim\mathbf{U}_\ell\), such that each bit \(\mathbf{X}_i\) is a deterministic function of at most \(d\) bits in \(\mathbf{Y}\). In other words, there exist sets \(S_1,\dots,S_n\subseteq[\ell]\) of size \(d\) and functions \(f_1,\dots,f_n:\{0,1\}^d\to\{0,1\}^n\) such that
\[
\mathbf{X}=(\mathbf{X}_1,\mathbf{X}_2,\dots,\mathbf{X}_n)=(f_1(\mathbf{Y}_{S_1}),f_2(\mathbf{Y}_{S_2}),\dots,f_n(\mathbf{Y}_{S_n})).
\]
Now, let \(T\subseteq[n]\) be any set of coordinates of \emph{maximal size} such that:
\begin{itemize}
\item \(H_\infty(\mathbf{X}_i)>0\) for all \(i\in T\); and
\item \(S_i\cap S_j=\emptyset\) for any distinct \(i,j\in T\).
\end{itemize}
Suppose \(T\) has size \(\tau\). Without loss of generality, assume \(T=[\tau]\). We conclude with two cases.
\textbf{Case (i)}: \(\tau<t\). In this case, we fix the random variable \(\mathbf{Y}_{S_1},\dots,\mathbf{Y}_{S_\tau}\). We know that with probability \(1\) over this fixing, all bits \(\mathbf{X}_i,i\in[n]\) become deterministic functions of at most \(d-1\) unfixed variables in \(\mathbf{Y}\), by the maximality of \(T\) and its intersection property. In other words, \(\mathbf{X}\) becomes a \((d-1)\)-local source. Furthermore, by \cref{lem:clean-entropy-drop}, we know that with probability \(1\) over this fixing, \(\mathbf{X}\) loses \(\sum_{i\in[\tau]}|S_i|=d\tau<dt\) bits of min-entropy. Thus in this case, \(\mathbf{X}\) is a convex combination of \((d-1)\)-local sources of min-entropy \(>k-dt\).
\textbf{Case (ii)}: \(\tau\geq t\). In this case, define \(\overline{S}:=[n]-(\bigcup_{i\in[\tau]}S_i)\) and notice that \(S_1,S_2,\dots,S_\tau,\overline{S}\) partition the coordinates of \(\mathbf{Y}\). Next, define the random variables \(\mathbf{Z}_i:=\mathbf{Y}_{S_i}\) for each \(i\in[\tau]\), and define \(\overline{\mathbf{Z}}:=\mathbf{Y}_{\overline{S}}\). Notice that \(\mathbf{X}_i=f_i(\mathbf{Z}_i)\) for each \(i\in[\tau]\). Furthermore, it is straightforward to verify that for all \(j>\tau\), there exists a set \(Q_j\subseteq[\tau]\) of size at most \(d\) and a deterministic function \(f^{\prime}_j\) such that \(\mathbf{X}_j=f^{\prime}_j(\mathbf{Z}_{Q_j},\overline{\mathbf{Z}})\). In other words, we can rewrite \(\mathbf{X}\) as
\begin{align*}
\mathbf{X}&=(\mathbf{X}_1,\dots,\mathbf{X}_{\tau},\mathbf{X}_{\tau+1},\dots,\mathbf{X}_n)\\
&=(f_1(\mathbf{Z}_1),\dots,f_\tau(\mathbf{Z}_\tau),f_{\tau+1}^{\prime}(\mathbf{Z}_{Q_{\tau+1}},\overline{\mathbf{Z}}),\dots,f_n^{\prime}(\mathbf{Z}_{Q_n},\overline{\mathbf{Z}})).
\end{align*}
Now, for each \(i\in[\tau]\), define \(\mathbf{A}_i:=f_i(\mathbf{Z}_i)\). Furthermore, it is straightforward to show that we can define a new random variable \(\mathbf{B}\) independent of \(\mathbf{Y}\), and for each \(i\in[\tau]\) a deterministic function \(g_i\) such that \(g_i(\mathbf{A}_i,\mathbf{B})=\mathbf{Z}_i\) for all \(i\in[\tau]\). Thus, for any subset \(Q\subseteq[\tau]\) we have \(\mathbf{Z}_Q=g^{\prime}_Q(\mathbf{A}_Q,\mathbf{B})\) for some deterministic function \(g^{\prime}_Q\). And finally, for each \(j>\tau\) there must be some deterministic function \(\psi_j\) such that
\[
f^{\prime}_{j}(\mathbf{Z}_{Q_j},\overline{\mathbf{Z}})=\psi_j(\mathbf{A}_{Q_j},\mathbf{B},\overline{\mathbf{Z}}).
\]
Thus we can rewrite \(\mathbf{X}\) as:
\[
\mathbf{X}=(\mathbf{A}_1,\dots,\mathbf{A}_\tau,\psi_{\tau+1}(\mathbf{A}_{Q_{\tau+1}},\mathbf{B},\overline{\mathbf{Z}}),\dots,\psi_n(\mathbf{A}_{Q_n},\mathbf{B},\overline{\mathbf{Z}})).
\]
Notice that the collection \(\{\mathbf{A}_i\}_{i\in[\tau]}\) are mutually independent, and each has bias at most \(1-2^{-d}\) since it is a non-constant deterministic function of \(d\) uniform bits. Thus no matter how \(\mathbf{B},\overline{\mathbf{Z}}\) are fixed, \(\mathbf{X}\) becomes a \((\delta,t)\)-biased \(d\)-local NOBF source, for \(\delta\leq1-2^{-d}\).
\end{proof}
\subsubsection{A reduction from biased \(d\)-local NOBF sources to (unbiased) \(d\)-local NOBF sources}\label{subsubsec:second-part-reduction}
We now prove the second part of our reduction (\cref{lem:simulate-coins}), which shows that every biased \(d\)-local NOBF source is (close to) a convex combination of (unbiased) \(d\)-local NOBF sources.
\begin{proof}[Proof of \cref{lem:simulate-coins}]
Let \(\mathbf{X}\sim\{0,1\}^n\) be a \((\delta,k)\)-biased \(d\)-local NOBF source. We wish to show that \(\mathbf{X}\) is \(\varepsilon\)-close to a convex combination of \((0,k^{\prime})\)-biased \(d\)-local NOBF sources, where \(k^{\prime}\geq(1-\delta)k/4\) and \(\varepsilon=2^{-(1-\delta)k/4}\).
Without loss of generality, assume that the first \(k\) bits in \(\mathbf{X}\) are the ``good bits'': that is, there exist \(d\)-local functions \(g_{k+1},\dots,g_n : \{0,1\}^k\to\{0,1\}\) such that
\[
\mathbf{X}=(\mathbf{X}_1,\dots,\mathbf{X}_k,g_{k+1}(\mathbf{X}_1,\dots,\mathbf{X}_k),\dots,g_n(\mathbf{X}_1,\dots,\mathbf{X}_k),
\]
where each \(\mathbf{X}_i\) is independent and has bias at most \(\delta\). To remove the bias from this source, the key idea will be to simulate each \(\mathbf{X}_i\) by two independent coins: one which is biased, and one which is not.
In more detail, for every \(i\in[k]\) we construct a pair of independent random variables \(\mathbf{B}_i,\mathbf{A}_i\sim\{0,1\}\) as follows. First, let \(\gamma_i\in\{0,1\}\) be the value favored by \(\mathbf{X}_i\), breaking ties arbitrarily. Then, define \(p_i:=\Pr[\mathbf{X}_i=\gamma_i]\), and notice that \(\frac{1}{2}\leq p_i\leq\frac{1+\delta}{2}\), where the lower bound holds because \(\mathbf{X}_i\) favors \(\gamma_i\) over \(1-\gamma_i\), and the upper bound holds because the good bits have bias at most \(\delta\). Next, define \(\mathbf{B}_i\) such that
\begin{align*}
\Pr[\mathbf{B}_i=0]&=2p_i-1,\\
\Pr[\mathbf{B}_i=1]&=2-2p_i.
\end{align*}
Finally, let \(\mathbf{A}_i\sim\{0,1\}\) simply be a uniform bit, and define the function \(h_i:\{0,1\}\times\{0,1\}\to\{0,1\}\) as
\[
h_i(b,a):=
\begin{cases}
\gamma_i &\textbf{ if }b=0,\\
a &\textbf{ if }b=1.
\end{cases}
\]
Given these definitions, it is straightforward to verify that \(\mathbf{X}_i\) has the same distribution as \(h_i(\mathbf{B}_i,\mathbf{A}_i)\), and thus we may rewrite \(\mathbf{X}\) as
\[
(h_1(\mathbf{B}_1,\mathbf{A}_1), \dots, h_k(\mathbf{B}_k,\mathbf{A}_k), g_{k+1}(h_1(\mathbf{B}_1,\mathbf{A}_1), \dots, h_k(\mathbf{B}_k,\mathbf{A}_k)), \dots, g_n(h_1(\mathbf{B}_1,\mathbf{A}_1), \dots, h_k(\mathbf{B}_k,\mathbf{A}_k))).
\]
Now, define the random variable \(\mathbf{B}=(\mathbf{B}_1,\dots,\mathbf{B}_k)\). Given the above description of \(\mathbf{X}\), it is not too difficult to see that for any \(b\in\{0,1\}^k\), the conditional distribution \((\mathbf{X}\mid\mathbf{B}=b)\) is a \(d\)-local (unbiased) NOBF source, which has min-entropy equal to the Hamming weight of \(b\). Thus, we may write \(\mathbf{X}\) as the convex combination
\[
\mathbf{X}=\sum_{b\in\{0,1\}^k}\Pr[\mathbf{B}=b]\cdot(\mathbf{X}\mid\mathbf{B}=b).
\]
This means that if \(\mathbf{B}\) has Hamming weight \(\geq k^{\prime}\) with probability \(\geq1-\varepsilon\), then \(\mathbf{X}\) is \(\varepsilon\)-close to a convex combination of \(d\)-local NOBF sources with min-entropy \(\geq k^{\prime}\). Such a claim will follow almost immediately from a Chernoff bound.
In more detail, define a random variable \(\mathbf{Z}:=\sum_{i\in[k]}\mathbf{B}_i\) and notice that the value of \(\mathbf{Z}\) is exactly the Hamming weight of \(\mathbf{B}\). Furthermore, recall that \(\mathbf{B}_i=1\) with probability \(2-2p_i\) for some \(p_i\in[\frac{1}{2},\frac{1+\delta}{2}]\). Thus
\[
\mu:=\mathbb{E}[\mathbf{Z}]=\sum_{i\in[k]}\mathbb{E}[\mathbf{B}_i]\geq k\cdot(2-2\cdot((1+\delta)/2))=(1-\delta)k.
\]
Thus, by a standard Chernoff bound on the lower tail (\cref{lem:chernoff}), we get that
\[
\Pr[\mathbf{Z}\leq \mu/4]\leq e^{-\mu\cdot(3/4)^2/2}\leq 2^{-\mu/4},
\]
which means that \(\mathbf{B}\) has Hamming weight \(\geq\mu/4\) with probability \(\geq1-2^{-\mu/4}\). Thus, \(\mathbf{X}\) is \(2^{-\mu/4}\)-close to a convex combination of \(d\)-local NOBF sources with min-entropy \(\geq\mu/4\), as desired.
\end{proof}
\subsection{Low-degree polynomials extract from \(d\)-local NOBF sources}\label{subsec:upper-bounds:second-key-ingredient}
Now that we have proven our reduction from \(d\)-local sources to \(d\)-local NOBF sources (\cref{thm:upper-bounds:first-key-ingredient}), all that remains is to prove an upper bound on the entropy required to extract from \(d\)-local NOBF sources using degree \(\leq r\) polynomials (\cref{thm:upper-bounds:second-key-ingredient}). In this section, we prove \cref{thm:upper-bounds:second-key-ingredient}.
In order to prove \cref{thm:upper-bounds:second-key-ingredient}, our main tool will be the following lemma, which shows that for \emph{any} fixed NOBF source, a random low-degree polynomial can extract from it with very high probability.
\begin{lemma}\label{lem:random-extractor}
There is a universal constant \(c>0\) such that for any \(r\leq ck^{1/4}\), the following holds. Let \(\mathbf{X}\sim\{0,1\}^n\) be an NOBF source with min-entropy \(k\), and let \(f:\{0,1\}^n\to\{0,1\}\) be a random \(\mathbb{F}_2\)-polynomial of degree \(\leq r\). Then
\[
\Pr_f[f\text{ is an extractor for \(\mathbf{X}\) with error \(\varepsilon=2^{-ck/r^3}\)}]\geq1-2^{-c{k\choose\leq r}}
\]
\end{lemma}
Before we prove this lemma, we show how it can be used to prove \cref{thm:upper-bounds:second-key-ingredient}. The proof is a straightforward application of the probabilistic method, which combines \cref{lem:random-extractor} with a basic upper bound on the number of \(d\)-local NOBF sources over \(\{0,1\}^n\) with min-entropy \(k\).
\begin{proof}[Proof of \cref{thm:upper-bounds:second-key-ingredient}]
Let \(\mathcal{X}\) be the family of \(d\)-local NOBF sources over \(\{0,1\}^n\) with min-entropy \(k\). Notice
\[
|\mathcal{X}|\leq{n\choose k}\cdot\left({k\choose d}\cdot2^{2^d}\right)^{n-k}.
\]
By \cref{lem:random-extractor}, a random degree \(\leq r\) polynomial \(f : \{0,1\}^n\to\{0,1\}\) fails to extracts from any fixed NOBF source of min-entropy \(k\) with error \(\varepsilon=2^{-ck/r^3}\) with probability at most \(2^{-c{k\choose\leq r}}\). Thus, a union bound shows that a random degree \(\leq r\) polynomial \(f\) extracts from all sources in \(\mathcal{X}\) with error \(\varepsilon=2^{-ck/r^3}\) with probability at least \(0.99\) (over the selection of \(f\)) as long as
\[
|\mathcal{X}|\cdot2^{-c{k\choose\leq r}}\leq{n\choose k}\cdot\left({k\choose d}\cdot2^{2^d}\right)^{n-k}\cdot2^{-c{k\choose\leq r}}\leq0.01,
\]
which is true if
\[
k\geq Cr \cdot (2^d\cdot n\log n)^{1/r}
\]
for a large enough constant \(C\).
\end{proof}
Now, all that remains is to prove \cref{lem:random-extractor}, which says that for any fixed NOBF source, a random low-degree polynomial can extract from it. In order to prove this result, our main tool will be following, which shows that for any fixed function \(g\), a random low-degree polynomial \(f\) has very low correlation with it.
\begin{lemma}\label{lem:main-correlation-bound}
There is a universal constant \(c>0\) such that for any fixed \(g:\mathbb{F}_2^n\to\mathbb{F}_2\), the following holds. For any \(1\leq r\leq cn^{1/4}\), let \(f:\mathbb{F}_2^n\to\mathbb{F}_2\) be a random degree \(\leq r\) polynomial. Then
\[
\Pr[\mathsf{corr}(f,g)>2^{-cn/r^3}]\leq2^{-c{n\choose\leq r}}.
\]
\end{lemma}
Before we prove these correlation bounds (\cref{lem:main-correlation-bound}), we show how they can be used to show that for any fixed NOBF source, a random low-degree polynomial can extract from it (\cref{lem:random-extractor}). Then, we conclude the section in \cref{subsubsec:correlation-bounds-sec} by proving \cref{lem:main-correlation-bound}.
\begin{proof}[Proof of \cref{lem:random-extractor}]
The proof is via the probabilistic method over the selection of \(f\). In more detail, let \(\mathbf{X}\sim\{0,1\}^n\) be an arbitrary NOBF source with min-entropy \(k\), and let \(f:\{0,1\}^n\to\{0,1\}\) be a random \(\mathbb{F}_2\)-polynomial of degree \(\leq r\). Assume (without loss of generality) that \(\mathbf{X}_1,\dots,\mathbf{X}_k\) are the ``good bits'' in the NOBF source.
Now, let \(f^{\prime}\) be the (sum of the) monomials in \(f\) that do not use any variables outside \(\mathbf{X}_1,\dots,\mathbf{X}_k\), and let \(g\) be the (sum of the) monomials in \(f\) that use at least one variable outside \(\mathbf{X}_1,\dots,\mathbf{X}_k\). This gives us \(f(\mathbf{X})=f^{\prime}(\mathbf{X})+g(\mathbf{X})\), where \(f^{\prime}\) is a random degree \(\leq r\) polynomial over \(\mathbf{X}_1,\dots,\mathbf{X}_k\), and \(g\) is an (independent) random polynomial over \(\mathbf{X}_1,\dots,\mathbf{X}_n\). By definition of NOBF source, each of the bits \(\mathbf{X}_{k+1},\dots,\mathbf{X}_n\) is a deterministic function of the good bits \(\mathbf{X}_1,\dots,\mathbf{X}_k\), and thus we have
\[
f(\mathbf{X})=f^{\prime}(\mathbf{X}_1,\dots,\mathbf{X}_k)+g^{\prime}(\mathbf{X}_1,\dots,\mathbf{X}_k),
\]
where \(f^{\prime}\) is a random degree \(\leq r\) polynomial over \(\mathbf{X}_1,\dots,\mathbf{X}_k\) and \(g^{\prime}\) is an independent (but not necessarily uniform) random polynomial over the same set of variables.
Since \(\mathbf{X}_1,\dots,\mathbf{X}_k\) are independent uniform bits, we get that \(f\) is an extractor with error
\begin{align*}
\varepsilon=\left|\Pr_{x\sim\mathbf{U}_k}[f(x)=1]-\frac{1}{2}\right|=\left|\Pr_{x\sim\mathbf{U}_k}[f^{\prime}(x)\neq g^{\prime}(x)]-\frac{1}{2}\right|=\frac{1}{2}\corr(f^{\prime},g^{\prime}).
\end{align*}
Thus
\begin{align*}
\Pr_f[f\text{ is \emph{not} an extractor for \(\mathbf{X}\) with error \(\varepsilon\)}]=\Pr_{f^{\prime},g^{\prime}}[\corr(f^{\prime},g^{\prime})>2\varepsilon]\leq\Pr_{f^{\prime}}[\corr(f^{\prime},g^\ast)>2\varepsilon],
\end{align*}
where \(g^\ast=\argmax_{g^{\prime}}\Pr_{f^{\prime}}[\corr(f^{\prime},g^{\prime})>2\varepsilon]\). Thus \(g^\ast:\mathbb{F}_2^k\to\mathbb{F}_2\) is an arbitrary fixed function and \(f^{\prime}:\mathbb{F}_2^k\to\mathbb{F}_2\) is a random degree \(\leq r\) polynomial. Thus, by combining the above inequality with \cref{lem:main-correlation-bound},
\begin{align*}
\Pr_f[f\text{ is \emph{not} an extractor for \(\mathbf{X}\) with error \(2^{-ck/r^3}\)}]&\leq \Pr_{f^{\prime}}[\corr(f^{\prime},g^\ast)>2\cdot2^{-ck/r^3}]\\
&\leq\Pr_{f^{\prime}}[\corr(f^{\prime},g^\ast)>2^{-ck/r^3}]\\
&\leq2^{-c\binom{k}{\leq r}},
\end{align*}
which completes the proof.
\end{proof}
\subsubsection{Correlation bounds against a single arbitrary function}\label{subsubsec:correlation-bounds-sec}
At last, all that remains in our proof of \cref{thm:upper-bounds:second-key-ingredient} is a proof of \cref{lem:main-correlation-bound}, which says that for any fixed function \(g\), a random low-degree polynomial \(f\) has very low correlation with it. As it turns out, \cref{lem:main-correlation-bound} follows quite readily from the following bounds on the list-size of Reed-Muller codes.
\begin{theorem}[\hspace{1sp}\cite{KLP12}]\label{thm:list-decodability-Reed-Muller}
There is a universal constant \(C>0\) such that for any \(n,r\in\mathbb{N}\) and \(\varepsilon>0\), the Reed-Muller code \(\mathsf{RM}(n,r)\) is \((\frac{1-\varepsilon}{2},L)\)-list-decodable, where
\[
\log L\leq C\cdot r\cdot(r+\log(1/\varepsilon))\cdot{n\choose\leq r-1}.
\]
\end{theorem}
Equipped with these bounds, we proceed to prove \cref{lem:main-correlation-bound}.
\begin{proof}[Proof of \cref{lem:main-correlation-bound}]
Let \(N:=2^n\), and consider any arbitrary functions \(f,g:\{0,1\}^n\to\{0,1\}\). If we let \(\widehat{f},\widehat{g}\in\{0,1\}^N\) denote the truth tables of \(f,g\), notice that
\[
N\cdot\corr(f,g)=|2\Delta(\widehat{f},\widehat{g})-N|,
\]
which implies that \(\corr(f,g)\leq\varepsilon\) if and only if \(\Delta(\widehat{f},\widehat{g})\in [\frac{1-\varepsilon}{2}\cdot N,\frac{1+\varepsilon}{2}\cdot N]\). Thus, if we let \(Q\subseteq\{0,1\}^N\) denote the Reed-Muller code \(\mathsf{RM}(n,r)\), the following holds: for any fixed function \(g\) and a random degree \(\leq r\) polynomial \(f\),
\begin{align*}
\Pr_f[\corr(f,g)>\varepsilon]&=\Pr_{q\sim Q}\left[\Delta(q,\widehat{g})\notin\left[\frac{1-\varepsilon}{2}N, \frac{1+\varepsilon}{2}N\right]\right]\\
&\leq\max_{v\in\{0,1\}^N}\Pr_{q\sim Q}\left[\Delta(q,v)\notin\left[\frac{1-\varepsilon}{2}N, \frac{1+\varepsilon}{2}N\right]\right]
\end{align*}
Thus, we would like to upper bound the probability that a random Reed-Muller codeword \(q\in\{0,1\}^N\) is \emph{not} within relative Hamming distance \([(1-\varepsilon)/2,(1+\varepsilon)/2]\) from an arbitrary point \(v\in\{0,1\}^N\). Towards this end, suppose that the Reed-Muller code \(Q\) is \((\frac{1-\varepsilon}{2},L)\)-list decodable. We claim that for any \(v\in\{0,1\}^N\),
\[
\Pr_{q\sim Q}\left[\Delta(q,v)\notin\left[\frac{1-\varepsilon}{2}N, \frac{1+\varepsilon}{2}N\right]\right]\leq\frac{2L}{|Q|}.
\]
Indeed, this holds for \emph{any} \((\frac{1-\varepsilon}{2},L)\)-list decodable code: to see why, simply note that \(\Delta(q,v)\notin\left[\frac{1-\varepsilon}{2}N,\frac{1+\varepsilon}{2}N\right]\) if and only if \(q\) is in one of the (open) balls \(\mathcal{B}(v,\frac{1-\varepsilon}{2}N)\) or \(\mathcal{B}(\overline{v},\frac{1-\varepsilon}{2}N)\), where \(\overline{v}\in\{0,1\}^N\) denotes the vector \(v\) with all of its bits flipped. By definition of list-decodability, each of these balls contains \(\leq L\) codewords, and the claim follows.
To conclude, recall that the Reed-Muller code \(\mathsf{RM}(n,r)\) has size \(|Q|=2^{\binom{n}{\leq r}}\), and \cref{thm:list-decodability-Reed-Muller} tells us that, for any \(\varepsilon>0\), \(Q\) is \((\frac{1-\varepsilon}{2},L)\)-list decodable, where \(
\log L\leq C\cdot r\cdot(r+\log(1/\varepsilon))\cdot\binom{n}{\leq r-1}
\) and \(C\) is a universal constant. It is a straightforward calculation to verify that there is a universal constant \(c>0\) such that for \(\varepsilon=2^{-cn/r^3}\), it holds that
\[
\log L\leq C\cdot r\cdot (r+\log(1/\varepsilon))\cdot\binom{n}{\leq r-1}\leq\binom{n}{\leq r}/2,
\]
as long as \(r\leq cn^{1/4}\). Combining all of our inequalities, we get that as long as \(r\leq cn^{1/4}\), it holds that for any fixed function \(g\) and a random degree \(\leq r\) polynomial \(f\),
\[
\Pr_f[\corr(f,g)>\varepsilon=2^{-cn/r^3}]\leq\frac{2L}{|Q|}\leq\frac{2\cdot 2^{\binom{n}{\leq r}/2}}{2^{\binom{n}{\leq r}}}=2^{-\binom{n}{\leq r}/2+1}\leq 2^{-\binom{n}{\leq r}/4}\leq 2^{-c\binom{n}{\leq r}},
\]
as desired.
\end{proof}
\subsection{Optimized upper bounds for dispersers}\label{subsec:optimized-dispersers}
In this subsection, we show that we can improve our entropy upper bounds if we only wish to \emph{disperse} from \(d\)-local sources using degree \(\leq r\) polynomials (instead of \emph{extract}). We prove the following theorem, which is an optimized version of \cref{thm:localextintro:restated} for the setting of dispersers.
\begin{theorem}[Formal version of \cref{rem:main-remark-here}]\label{thm:formal-version-of-main-remark}
There is a universal constant \(C>0\) such that for all \(n,d,r\in\mathbb{N}\), the following holds. With probability at least \(0.99\) over the choice of a random degree \(\leq r\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\), it holds that \(f\) is a disperser for \(d\)-local sources with min-entropy \(\geq k\), as long as
\[
k\geq Cd^2r\cdot(dn\log n)^{1/r} + Cd^3\cdot n^{1/r}.
\]
\end{theorem}
As \cref{thm:formal-version-of-main-remark} is an optimized version of \cref{thm:localextintro:restated} for the setting of dispersers, we set out to prove \cref{thm:formal-version-of-main-remark} by re-examining our proof of \cref{thm:localextintro:restated}. Recall that \cref{thm:localextintro:restated} shows that low-degree polynomials can extract from \(d\)-local sources, and in order to prove it, we combined two main ingredients:
\begin{enumerate}
\item A reduction from \(d\)-local sources to \(d\)-local NOBF sources (\cref{thm:upper-bounds:first-key-ingredient}).
\item A result showing that low-degree polynomials can extract from \(d\)-local NOBF sources (\cref{thm:upper-bounds:second-key-ingredient}).
\end{enumerate}
In order to optimize \cref{thm:localextintro:restated} for the setting of dispersers (and thereby prove \cref{thm:formal-version-of-main-remark}), we show that both of these ingredients can be optimized for dispersers.
In more detail, in order to prove \cref{thm:formal-version-of-main-remark}, we combine two key ingredients. Just like the first key ingredient in \cref{thm:localextintro:restated} (\cref{thm:upper-bounds:first-key-ingredient}), our first key ingredient for \cref{thm:formal-version-of-main-remark} reduces \(d\)-local sources to the more specialized family of \(d\)-local NOBF sources. However, as we are dealing with the easier setting of \emph{dispersing} (instead of \emph{extracting}), the definition of \emph{reduction} here is not as strict.
\begin{theorem}[Optimized version of \cref{thm:upper-bounds:first-key-ingredient} for dispersers]\label{thm:upper-bounds:first-key-ingredient:optimized-for-dispersers}
There exists a universal constant \(c>0\) such that for any \(n,k,d\in\mathbb{N}\), the following holds. Let \(\mathbf{X}\sim\{0,1\}^n\) be a \(d\)-local source with min-entropy \(\geq k\). Then there is a \(d\)-local NOBF source \(\mathbf{X}^{\prime}\sim\{0,1\}^n\) with min-entropy \(\geq k^{\prime}\) such that \(\text{support}(\mathbf{X}^{\prime})\subseteq\text{support}(\mathbf{X})\), where
\[
k^{\prime}=\frac{ck}{d^2}.
\]
\end{theorem}
Next, while the second key ingredient in \cref{thm:localextintro:restated} (\cref{thm:upper-bounds:second-key-ingredient}) shows that a random low-degree polynomial \emph{extracts} from \(d\)-local NOBF sources, the second key ingredient for \cref{thm:formal-version-of-main-remark} just needs to show that a random low-degree polynomial \emph{disperses} from \(d\)-local NOBF sources. Given this weaker requirement, we are able to improve the min-entropy requirement and prove the following, which is our second key ingredient.
\begin{theorem}[Optimized version of \cref{thm:upper-bounds:second-key-ingredient} for dispersers]\label{thm:upper-bounds:second-key-ingredient:optimized-for-dispersers}
There is a universal constants \(C>0\) such that for all \(n,d,r\in\mathbb{N}\), the following holds.
With probability at least \(0.99\) over the choice of a random degree \(\leq r\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\), it holds that \(f\) is a disperser for \(d\)-local NOBF sources of min-entropy \(\geq k\), as long as
\[
k\geq Cr\cdot(d n\log n)^{1/r}+Cd\cdot n^{1/r}
\]
\end{theorem}
Now, recall that if a disperser works for a family \(\mathcal{X}\) of sources over \(\{0,1\}^n\), then it also works for any \(\mathbf{X}\sim\{0,1\}^n\) whose support contains the support of some \(\mathbf{X}^{\prime}\in\mathcal{X}\) (\cref{fact:convex-combo-lifted-disperser-optimized}). As a result, by combining \cref{thm:upper-bounds:first-key-ingredient:optimized-for-dispersers,thm:upper-bounds:second-key-ingredient:optimized-for-dispersers}, we immediately get \cref{thm:formal-version-of-main-remark}. Thus, in the remainder of this section, we just focus on proving \cref{thm:upper-bounds:first-key-ingredient:optimized-for-dispersers,thm:upper-bounds:second-key-ingredient:optimized-for-dispersers}. We prove \cref{thm:upper-bounds:first-key-ingredient:optimized-for-dispersers} in \cref{subsubsec:first-disperser-optimization}, and we prove \cref{thm:upper-bounds:second-key-ingredient:optimized-for-dispersers} in \cref{subsubsec:second-disperser-optimization}. This will conclude our discussion of entropy upper bounds (and \cref{sec:new-upper-bounds}).
\subsubsection{A reduction from \(d\)-local sources to \(d\)-local NOBF sources (optimized for dispersers)}\label{subsubsec:first-disperser-optimization}
We start by proving \cref{thm:upper-bounds:first-key-ingredient:optimized-for-dispersers}, a reduction from \(d\)-local sources to \(d\)-local NOBF sources, which is optimized for dispersers. As it turns out, \cref{thm:upper-bounds:first-key-ingredient:optimized-for-dispersers} follows quite readily from our reduction from \(d\)-local sources to biased \(d\)-local NOBF sources (\cref{lem:hard-part-of-reduction}).
\begin{proof}[Proof of \cref{thm:upper-bounds:first-key-ingredient:optimized-for-dispersers}]
Let \(\mathbf{X}\sim\{0,1\}^n\) be a \(d\)-local source with min-entropy \(k\). By \cref{lem:hard-part-of-reduction}, \(\mathbf{X}\) is a convex combination of \((\delta,k^{\prime})\)-biased \(d\)-local NOBF souces, where \(\delta<1\) and \(k^{\prime}\geq k/(2d^2)\). Let \(\mathbf{Y}\sim\{0,1\}^n\) be an element of this convex combination (that is assigned probability \(>0\)). Clearly \(\mathbf{Y}\) is a \((\delta,k^{\prime})\)-biased \(d\)-local NOBF source, and furthermore \(\text{support}(\mathbf{Y})\subseteq\text{support}(\mathbf{X})\).
Now, let \(\mathbf{X}^{\prime}\sim\{0,1\}^n\) be the ``unbiased'' version of \(\mathbf{Y}\): that is, let \(\mathbf{X}^{\prime}\) have the same set of \(k^{\prime}\) ``good'' bits as \(\mathbf{Y}\), and let the bad bits in \(\mathbf{X}^{\prime}\) depend on the good bits via the same deterministic functions as in \(\mathbf{Y}\). However, make the good bits in \(\mathbf{X}^{\prime}\) have \(0\) bias. Then \(\mathbf{X}^{\prime}\) is an (unbiased) \(d\)-local NOBF source with min-entropy \(k^{\prime}\), and \(\text{support}(\mathbf{X}^{\prime})=\text{support}(\mathbf{Y})\subseteq\text{support}(\mathbf{X})\).
\end{proof}
\subsubsection{Low-degree polynomials disperse from \(d\)-local NOBF sources (optimized for dispersers)}\label{subsubsec:second-disperser-optimization}
Now that we have proven our reduction from \(d\)-local sources to \(d\)-local NOBF sources (\cref{thm:upper-bounds:first-key-ingredient:optimized-for-dispersers}), all that remains is to prove an upper bound on the entropy required to disperse from \(d\)-local NOBF sources using degree \(\leq r\) polynomials (\cref{thm:upper-bounds:second-key-ingredient:optimized-for-dispersers}). In this section, we prove \cref{thm:upper-bounds:second-key-ingredient:optimized-for-dispersers}.
In order to prove \cref{thm:upper-bounds:second-key-ingredient:optimized-for-dispersers}, our key ingredient is a lemma about \emph{function composition} and how it affects \emph{which degrees are ``hit.''} More formally, for any nonnegative integer \(r\), we say that a function \(f:\mathbb{F}_2^n\to\mathbb{F}_2\) \emph{hits degree \(r\)} if the unique multilinear \(\mathbb{F}_2\)-polynomial computing \(f\) includes a monomial of size \(r\),\footnote{The monomial of size \(0\) is the constant \(1\).} and we say that \(f\) \emph{hits degree \(\leq r\)} if there is some \(0\leq r^{\prime}\leq r\) such that \(f\) hits degree \(r^{\prime}\). Then, using standard notation for function composition, we prove the following.
\begin{lemma}\label{lem:hitting-lemma}
Let \(f:\mathbb{F}_2^n\to\mathbb{F}_2\) and \(a_1,\dots,a_n:\mathbb{F}_2^k\to\mathbb{F}_2\) be functions such that \(f(a_1,\dots,a_n)\) hits degree \(r\). Furthermore, let \(b_1,\dots,b_n:\mathbb{F}_2^k\to\mathbb{F}_2\) be functions that do not hit degree \(\leq r\). Then the function \(f(a_1+b_1,\dots,a_n+b_n):\mathbb{F}_2^k\to\mathbb{F}_2\) hits degree \(r\).
\end{lemma}
Before we prove this lemma, we show how it can be used to to show that degree \(\leq r\) polynomials disperse from \(d\)-local NOBF sources, proving \cref{thm:upper-bounds:second-key-ingredient:optimized-for-dispersers}. Then, we conclude this section by proving \cref{lem:hitting-lemma}.
\begin{proof}[Proof of \cref{thm:upper-bounds:second-key-ingredient:optimized-for-dispersers}]
We start with some basic definitions. First, recall that a \(d\)-local NOBF source of min-entropy \(k\) is a random variable \(\mathbf{X}\sim\{0,1\}^n\) that is generated as follows: there exist \(a_1,\dots,a_n:\mathbb{F}_2^k\to\mathbb{F}_2\) such that \(\mathbf{X}=(a_1(\mathbf{U}_k),\dots,a_n(\mathbf{U}_k))\), where each \(\mathbf{U}_k\) is a copy of the same uniform random variable. Furthermore, the following must hold: there exists a ``good'' set \(S=\{\alpha_1,\dots,a_k\}\subseteq[n]\) such that each \(a_{\alpha_i}\) simply outputs its \(i^\text{th}\) input, and for every other (\emph{bad}) coordinate \(j\notin S\) we have that \(a_j\) depends on just \(d\) inputs. For convenience, we refer to the set of functions \((a_1,\dots,a_n)\) as the \emph{generating functions} of the \(d\)-local NOBF source. If each \(a_i\) is a degree \(\leq r\) polynomial, we say that \(\mathbf{X}\) is not just a \(d\)-local NOBF source, but a \emph{(\(d\)-local, degree \(\leq r\)) NOBF source}.
Now, the goal is to use \cref{lem:hitting-lemma} to show that, in order for a function \(f:\mathbb{F}_2^n\to\mathbb{F}_2\) to disperse from all \(d\)-local NOBF sources of min-entropy \(k\), it suffices to show that \(f\) ``hits'' some degree in \([1,r]\) on all (\(d\)-local, degree \(\leq r\)) NOBF sources of the same min-entropy. Then, a straightforward application of the probabilistic method over the latter (smaller) family will yield the result.
\\
More formally, fix an arbitrary function \(f:\mathbb{F}_2^n\to\mathbb{F}_2\). Suppose that for any (\(d\)-local, degree \(\leq r\)) NOBF source \(\mathbf{X}\sim\{0,1\}^n\) with min-entropy \(k\) and generating functions \(a_1,\dots,a_n:\mathbb{F}_2^k\to\mathbb{F}_2\), there is some \(r^{\prime}\in[1,r]\) such that \(f(a_1,\dots,a_n)\) hits degree \(r^{\prime}\). Then, we \textbf{claim} that \(f\) is not only a disperser for all (\(d\)-local, degree \(\leq r\)) NOBF sources of min-entropy \(k\), but it is also a disperser for all \(d\)-local NOBF sources of min-entropy \(k\).
To see why, let \(\mathbf{X}^{\prime}\sim\{0,1\}^n\) be a \(d\)-local NOBF source with generating functions \(h_1,\dots,h_n:\mathbb{F}_2^k\to\mathbb{F}_2\), and for each \(i\in[n]\) define the functions \(a_i^{\prime},b_i^{\prime}:\mathbb{F}_2^k\to\mathbb{F}_2\) such that \(a_i^{\prime}\) is the sum of all monomials in \(h_i\) of degree \(\leq r\), and \(b_i^{\prime}\) is the sum of all monomials in \(h_i\) of degree \(>r\). Consider now the source
\[
\mathbf{X}:=(a_1^{\prime}(\mathbf{U}_k),\dots,a_n^{\prime}(\mathbf{U}_k)),
\]
and notice that \(\mathbf{X}\) is a (\(d\)-local, degree \(\leq r\)) NOBF source with min-entropy \(k\) and generating functions \(a_1^{\prime},\dots,a_n^{\prime}:\mathbb{F}_2^k\to\mathbb{F}_2\). By the hypothesis, \(f(a_1^{\prime},\dots,a_n^{\prime})\) hits degree \(r^{\prime}\), where \(r^{\prime}\in[1,r]\). And since \(b_1^{\prime},\dots,b_n^{\prime}:\mathbb{F}_2^k\to\mathbb{F}_2\) do \emph{not} hit degree \(\leq r\), we know by \cref{lem:hitting-lemma} that \(f(a_1^{\prime}+b_1^{\prime},\dots,a_n^{\prime}+b_n^{\prime})=f(h_1,\dots,h_n)\) hits degree \(r^{\prime}\in[1,r]\). This means that \(f(h_1,\dots,h_n)\) is not constant, and since \(f(\mathbf{X}^{\prime})=f(h_1,\dots,h_n)(\mathbf{U}_k)\), it holds that \(f(\mathbf{X}^{\prime})\) is not constant: in other words, \(f\) is a disperser for \(\mathbf{X}^{\prime}\). Thus, \(f\) is a disperser for all \(d\)-local NOBF sources of min-entropy \(k\).
\\
Thus, all that remains is to show that with probability \(\geq0.99\) over the choice of a random degree \(\leq r\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\), the following holds: for any (\(d\)-local, degree \(\leq r\)) NOBF source \(\mathbf{X}\sim\{0,1\}^n\) with min-entropy \(k\) and generating functions \(a_1,\dots,a_n:\mathbb{F}_2^k\to\mathbb{F}_2\), there is some \(r^{\prime}\in[1,r]\) such that \(f(a_1,\dots,a_n)\) hits degree \(r^{\prime}\).
\\
Towards this end, let \(f\in\mathbb{F}_2[x_1,\dots,x_n]\) be a (uniformly) random degree \(\leq r\) polynomial, and let \(\mathbf{X}\sim\{0,1\}^n\) be an arbitrary NOBF source with min-entropy \(k\) and generating functions \(a_1,\dots,a_n:\mathbb{F}_2^k\to\mathbb{F}_2\). Without loss of generality, assume that \(a_i(y)=y_i\) for all \(i\in[k]\). Now, we want to upper bound the probability that for all \(r^{\prime}\in[1,r]\), it holds that \(f(a_1,\dots,a_n)\) does \emph{not} hit degree \(r^{\prime}\). We \textbf{claim} that this probability is at most \(2^{-\binom{k}{\leq r}+1}\).
To see why, note that since \(f:\mathbb{F}_2^n\to\mathbb{F}_2\) is a uniformly random degree \(\leq r\) polynomial, we have
\[
f(x):=\sum_{S\subseteq[n] : |S|\leq r}c_Sx^S,
\]
where each \(c_S\sim\{0,1\}\) is independent and uniform. Now, note that for \(y\in\mathbb{F}_2^k\),
\begin{align*}
f(a_1,\dots,a_n)(y)&=\sum_{S\subseteq[n]:|S|\leq r}c_S a^S(y)\\
&=\sum_{S\subseteq[k]:|S|\leq r}c_Sa^S(y) + \sum_{S\subseteq[n]:|S|\leq r,S\not\subseteq[k]}c_Sa^S(y)\\
&=\sum_{S\subseteq[k]:|S|\leq r}c_Sy^S+\sum_{S\subseteq[n]:|S|\leq r, S\not\subseteq[k]}c_Sa^S(y).
\end{align*}
Now, notice that for each fixing of \(\{c_S\}_{S\subseteq[n]:|S|\leq r,S\not\subseteq[k]}\), we get that
\[
f(a_1,\dots,a_n)(y)=f^{\prime}(y)+g(y),
\]
where \(f^{\prime}:\mathbb{F}_2^k\to\mathbb{F}_2\) is a uniformly random polynomial of degree at most \(\leq r\), and \(g:\mathbb{F}_2^k\to\mathbb{F}_2\) is a fixed polynomial. Now, for any \(S\subseteq[k]\) of size at most \(r\), the probability that \(f^{\prime}+g\) \emph{does not} include the monomial \(y^S\) is the probability that \(f^{\prime}\) copies the decision of \(g\) on whether or not to include \(y^S\): this probability is \(1/2\). More generally, the probability that \(f^{\prime}+g\) \emph{does not} hit degree \(r\) will be \(2^{-{k\choose r}}\), and the probability that it does not hit any degree in \([1,r]\) will be \(2^{-{k\choose\leq r}+1}\). Thus
\begin{align}\label{eq:single-hit-failure}
\Pr_f[f(a_1,\dots,a_n)(x)\text{ does not hit any degree in }[1,r]]\leq2^{-{k\choose\leq r}+1},
\end{align}
as claimed.
\\
Now, we say that \(f\) \emph{fails} if there exists a (\(d\)-local, degree \(\leq r\)) NOBF source \(\mathbf{X}\sim\{0,1\}^n\) of min-entropy \(k\) with generating functions \(a_1,\dots,a_n\) such that \(f(a_1,\dots,a_n)\) does not hit any degree in \([1,r]\). By combining \cref{eq:single-hit-failure} with a simple union bound, we get that
\[
\Pr_f[f\text{ fails}]\leq|\mathcal{X}|\cdot2^{-\binom{k}{\leq r}+1},
\]
where \(\mathcal{X}\) is the family of (\(d\)-local, degree \(\leq r\)) NOBF sources \(\mathbf{X}\sim\{0,1\}^n\) of min-entropy \(k\). To upper bound the size of \(\mathcal{X}\), simply note that each \(\mathbf{X}\in\mathcal{X}\) can: (i) choose the locations of its \(k\) good bits; (ii) choose the \(d\) good bits that each bad bit depends on; and (iii) choose the degree \(\leq r\) polynomial that computes each bad bit. Thus
\[
|\mathcal{X}|\leq \binom{n}{k}\cdot\left(\binom{k}{d}\cdot2^{\binom{d}{\leq r}}\right)^{n-k}.
\]
It is now a straightforward calculation to verify that there is a sufficiently large constant \(C>0\) such that for all \(k\geq Cr(d n\log n)^{1/r}+Cdn^{1/r}\), it holds that
\[
\Pr_f[f\text{ fails}]\leq|\mathcal{X}|\cdot2^{-\binom{k}{\leq r}+1}\leq \binom{n}{k}\cdot\left(\binom{k}{d}\cdot2^{\binom{d}{\leq r}}\right)^{n-k}\cdot2^{-\binom{k}{\leq r}+1}\leq0.01.
\]
To conclude, we get that with probability \(\geq0.99\) over the choice of a random degree \(\leq r\) polynomial \(f\in\mathbb{F}_2[x_1,\dots,x_n]\), it holds that \(f\) hits some degree in \([1,r]\) on every (\(d\)-local, degree \(\leq r\)) NOBF source with min-entropy \(k\), provided that \(k\geq Cr\cdot(dn\log n)^{1/r}\). By our claim from the beginning of the proof, it follows that \(f\) is also a disperser for all \(d\)-local NOBF sources of min-entropy \(k\), as desired.
\end{proof}
At last, all that remains is to prove \cref{lem:hitting-lemma}, which says that if \(f:\mathbb{F}_2^n\to\mathbb{F}_2\) and \(a_1,\dots,a_n:\mathbb{F}_2^k\to\mathbb{F}_2\) are functions such that \(f(a_1,\dots,a_n)\) hits degree \(r\), and \(b_1,\dots,b_n:\mathbb{F}_2^k\to\mathbb{F}_2\) are functions that do not hit degree \(\leq r\), then \(f(a_1+b_1,\dots,a_n+b_n)\) hits degree \(r\).
\begin{proof}[Proof of \cref{lem:hitting-lemma}]
Since we can write any function \(f:\mathbb{F}_2^n\to\mathbb{F}_2\) as a unique multilinear \(\mathbb{F}_2\)-polynomial, we know that
\[
f(x)=\sum_{S\subseteq[n]}c_S x^S
\]
for some constants \(c_S\in\{0,1\}\), \(S\subseteq[n]\). Using this, we have
\begin{align*}
f(a_1+b_1,\dots,a_n+b_n)(x)&=\sum_{S\subseteq[n]}c_S\prod_{i\in S}(a_i(x)+b_i(x))\\
&=\sum_{S\subseteq[n]}c_S\sum_{T\subseteq S}a^T(x)b^{S-T}(x)\\
&=\sum_{S\subseteq[n]}c_S\left(a^S(x)+\sum_{T\subsetneq S}a^T(x)b^{S-T}(x)\right)\\
&=\sum_{S\subseteq[n]}c_Sa^S(x) +\sum_{S\subseteq[n]}\sum_{T\subsetneq S}a^T(x)b^{S-T}(x)\\
&=f(a_1,\dots,a_n)(x)+\sum_{S\subseteq[n]}\sum_{T\subsetneq S}a^T(x)b^{S-T}(x)\\
&=f(a_1,\dots,a_n)(x)+\sum_{\substack{\emptyset\subsetneq S\subseteq[n]\\mathbf{T}\subsetneq S}}a^T(x)b^{S-T}(x).
\end{align*}
Now, for each \(S\subseteq[n],T\subsetneq S\) such that \(S\) is nonempty, define the function \(g_{S,T}(x):=a^T(x)b^{S-T}(x)\). Since \(f(a_1,\dots,a_n)\) hits degree \(r\), notice that in order to complete the proof, it suffices to show that each \(g_{S,T}\) does \emph{not} hit degree \(r\). Towards this end, pick an arbitrary element \(\gamma\in S-T\), and observe
\begin{align*}
g_{S,T}(x):&=a^T(x)b^{S-T}(x)\\
&=\prod_{i\in T}a_i(x)\prod_{j\in S-T}b_j(x)\\
&=\left(\prod_{i\in T}a_i(x)\prod_{j\in (S-T)-\{\gamma\}}b_j(x)\right)b_\gamma(x).
\end{align*}
Next, it is straightforward to verify that for any functions \(p,q:\mathbb{F}_2^k\to\mathbb{F}_2\) such that \(q\) does not hit degree \(\leq r\), it follows that the function \(pq:\mathbb{F}_2^k\to\mathbb{F}_2\) does not hit degree \(\leq r\).\footnote{This is because any monomial that ``pops out'' of \(pq\) must be at least the size of the smallest monomial in \(q\), which is \(>r\).} Thus, since \(b_\gamma\) does not hit degree \(\leq r\) by the hypothesis, it follows via the above observation and identity that \(g_{S,T}\) does not hit degree \(\leq r\), which is even stronger than what we needed to complete the proof.
\end{proof}
\dobib
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,291 |
Q: Vue - closing dialog from child component I'm using Vue and Vuetify and I'm having trouble closing a dialog from within a child component using $emit. In the main component I'm using v:on:close-dialog="closeDialog" and setting this.dialog = false. I'm trying to call that function from within the child. Trying three different ways:
*
*On the <v-icon>close</v-icon> in the child component, I'm calling a closeDialog method that calls this.$emit('close-dialog').
*On the <v-btn>Cancel</v-btn>, I have v-on:click="$emit('close-dialog')".
*On the <v-btn>Cancel 2</v-btn>, I have v-on:click="$emit('dialog',false)".
None of those close the dialog or fire off the closeDialog method in the main component. Code is below.
mainComponent:
<template>
<v-flex>
<v-flex xs12 class="text-xs-right">
<v-dialog v-model="dialog" fullscreen hide-overlay
transition="dialog-bottom-transition">
<v-btn fab slot="activator" small color="red" dark>
<v-icon dark >add</v-icon>
</v-btn>
<childComponent v:on:close-dialog="closeDialog" />
</v-dialog>
</v-flex>
</v-flex>
</template>
<script>
import childComponent from './childComponent'
export default {
data(){
return{
dialog: false
}
},
name: 'Test',
components: {
childComponent
},
methods:{
closeDialog: function(){
console.log('close dialog 2');
this.dialog = false;
}
}
}
</script>
childComponent:
<template>
<v-flex xs12>
<v-card>
<v-toolbar dark color="primary">
<v-btn icon dark v-on:click="closeDialog">
<v-icon>close</v-icon>
</v-btn>
<v-toolbar-title>Dialog Test</v-toolbar-title>
<v-spacer></v-spacer>
<v-toolbar-items>
<v-btn dark flat v-on:click="$emit('close-dialog')">Cancel</v-btn>
</v-toolbar-items>
<v-spacer></v-spacer>
<v-toolbar-items>
<v-btn dark flat v-on:click="$emit('dialog',false)">Cancel 2</v-btn>
</v-toolbar-items>
</v-toolbar>
<v-flex xs12 class="px-10">
<v-form ref="form">
<v-text-field
v-model="testField"
:counter="150"
label="Test field"
required
></v-text-field>
</v-form>
</v-flex>
</v-card>
</v-flex>
</template>
<script>
export default {
data: () => ({
testField: ''
}),
methods: {
closeDialog: function(){
console.log('close dialog 1');
this.$emit('close-dialog');
}
}
}
</script>
As you might have guessed, I'm new to Vue and still fumbling my way through it. Any help would be much appreciated.
A: In your parent you have:
<childComponent v:on:close-dialog="closeDialog" />
it should be (hyphen replaces colon in v-on):
<childComponent v-on:close-dialog="closeDialog" />
or @close-dialog altenatively.
This method, combined with this.$emit('close-dialog'); in your child should work.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,492 |
"What I need most right now is a wife."
But can their convenient marriage turn into forever?
Solitaire Saunders will do anything to save her family's café—even marry enigmatic billionaire Xavier McQueen! As his contracted bride, she's there to secure Xavier's inheritance—not to enjoy his delicious kisses... It's only meant to be temporary, until the pretense of being married starts to feel passionately real!
"Look, is there some sort of arrangement we could come to here?" Soli asked desperately, blinking back her tears and looking a little embarrassed about losing her cool.
Xavier looked hurriedly away, frowning down at his desk. "I've already held back on rolling out the new rent, and if I do it for you I'll have to—"
"Please. Have a heart," she broke in hoarsely, clearly aware she was losing the battle but seemingly not prepared to accept it. "I'll do anything. I'll do whatever it takes to keep our café running."
The ring of hope in her voice made something twist painfully inside him. He had to give her credit, she was certainly determined.
Or maybe just desperate.
His heart gave a hard thump in his chest. He knew what desperate felt like and he wouldn't wish it on anyone.
"Whatever it takes?" he asked slowly, meeting her eyes again now. He wasn't sure where he was going with this, but he had the strangest feeling there could be a solution here that he hadn't quite hit upon yet.
"Anything. Just name it," she said, her eyes wide with anticipation.
"What I need most right now is a wife."
Dear Reader,
I love a Cinderella rags-to-riches story, don't you? There's nothing quite as satisfying as seeing someone who deserves a break get the thing she most longs for in the world—something that's previously been well beyond her reach because of circumstances out of her control.
My heroine, Soli, has had it hard for quite some time when we first meet her, and it's at a point of crisis when my hero, Xavier, offers her a seemingly perfect way out of her troubles. The only problem? She has to put her life on hold and marry him—a practical stranger, albeit a handsome, charismatic and very rich one.
Soli was such a fabulous character to write. I found myself loving her for her determination to ensure a happy ending for the people she loves, even, potentially, at the expense of her own happiness. Though perhaps her positivity and compassion will turn out to be the very things emotionally reserved Xavier never knew he needed.
And maybe, ultimately, she'll end up saving him?
I hope you enjoy reading their story to find out.
With warmest wishes,
Christy
A CONTRACT, A WEDDING, A WIFE?
Christy McKellen
Formerly a video and radio producer, Christy McKellen now spends her time writing fun, impassioned and emotive romance with an undercurrent of sensual tension. When she's not writing, she can be found enjoying life with her husband and three children, walking for pleasure, and researching other people's deepest secrets and desires. Christy loves to hear from readers. You can get ahold of her at christymckellen.com.
Books by Christy McKellen
Harlequin Romance
Romantic Getaways
The Unforgettable Spanish Tycoon
Maids Under the Mistletoe
A Countess for Christmas
Unlocking Her Boss's Heart
One Week with the French Tycoon
His Mistletoe Proposal
Harlequin KISS
Holiday with a Stranger
Lessons in Rule-Breaking
Fired by Her Fling
Bridesmaid with Attitude
Visit the Author Profile page at Harlequin.com for more titles.
Join Harlequin My Rewards today and earn a FREE ebook!
Click here to Join Harlequin My Rewards
http://www.harlequin.com/myrewards.html?mt=loyalty&cmpid=EBOOBPBPA201602010002
This one is for you, lovely romance reader, for choosing to spend your precious time reading my words. I am truly grateful. May you always have romance in your heart.
Praise for
Christy McKellen
"Packed with compelling drama, moving romance and nail-biting emotional tension, Unlocking Her Boss's Heart is a first-rate romantic tale readers shouldn't dare miss!"
—Goodreads
Contents
CHAPTER ONE
CHAPTER TWO
CHAPTER THREE
CHAPTER FOUR
CHAPTER FIVE
CHAPTER SIX
CHAPTER SEVEN
CHAPTER EIGHT
CHAPTER NINE
CHAPTER TEN
CHAPTER ELEVEN
CHAPTER TWELVE
EPILOGUE
EXCERPT FROM _UNLOCKING THE MILLIONAIRE'S HEART_ BY BELLA BUCANNON
CHAPTER ONE
Risk—a game of strategy, conflict and diplomacy.
HE WAS NEVER going to find someone suitable to marry at this rate.
Xavier McQueen let out an exasperated sigh as the woman who had seemed like his best hope—on paper at least—gave a firm and very final no to his admittedly completely barmy-sounding proposal before putting the phone down on him.
Apparently only being married for a year before divorcing wouldn't look good on her dating CV. She was under the impression it could put off real prospects in the future because they'd be worried about her coming with baggage from such a short previous marriage.
Closing his eyes, he slumped back in his chair.
Three months he'd been wasting his time with this ridiculous endeavour and now he only had six weeks left before the Hampstead mansion where he'd lived for the last four years—the home that had been in his family for the last hundred and fifty years—would pass to his money-grubbing clown of a cousin.
Damn his great-aunt and her jeopardous eccentricity.
He thought she'd loved him—certainly more than his parents ever had—but this bizarre stunt she'd pulled with her will had made him wonder about that.
Shoving a hand through his hair and trying not to pull it out in his frustration, he stared out of the floor-to-ceiling window of his office, barely registering his view of the majestic Tower Bridge stretching out across the fast-moving River Thames.
He'd not wanted to widely advertise exactly what he was looking for in case it brought out the crooks and the crazies but that meant he'd quickly run out of people to ask to help him out. The problem was, the chosen candidate needed to be someone he could trust, as well as someone he'd be able to get along with, but all his good female friends were already married and he didn't fancy taking his chances with any of his exes. A year was a long time to live with someone who detested the very sight of you.
The other two women, who had also been put forward as possible candidates by his friend Russell—the only friend he'd trusted with his problem—hadn't worked out either. Not being able to have sex for a year hadn't appealed to either of them. They'd both been looking for the real deal. Soul mates. An ideal he had no faith in whatsoever any more, not after being left humiliated at the altar five years ago by the woman he'd thought he'd spend the rest of his life with. His disaster of a non-wedding, which he now liked to think of as a near miss, had put paid to that ridiculous notion.
Nope, it was short-term, uncomplicated relationships for him from here on in. Or a purely business one like this needed to be, thanks to the bizarre demands stipulated in Great-Aunt Faith's will.
Just as he was reaching for the glass of water on his desk to relieve his parched throat, there was a loud knock on the door and a petite woman with bright blue eyes and a riot of blonde curls walked purposefully into his office and placed a small basket of assorted cakes on his desk with a flourish.
He frowned down at them, then up at her. 'I didn't order any cakes.'
'I know. They're an excuse to get some face-to-face time with you,' she said, folding her arms and looking down at him with a determined expression that made his stomach sink.
'I've been trying to get a meeting with you for weeks but your PA keeps fobbing me off,' she went on before he had a chance to say anything. 'So I've been forced to take drastic action. On the other hand, I've brought you some really fantastic cakes. I made them myself. So it's actually a win for you.' She flashed him a half-smile that didn't entirely convince him she was as self-assured as her spirited speech had made her seem.
He leant back in his chair again and studied her in bemusement.
She looked young, maybe early-to-mid-twenties, with a sweetly pretty face. Her abundance of curly blonde hair, which she'd tried to tame with an Alice band, stuck out around her head, probably due to the windy day. She surveyed him back with intelligent eyes, her button nose, which was scattered with freckles, wrinkling a little under his gaze. She seemed to him to have the air of someone who could cause a great deal of mischief if she put her mind to it.
As he scrutinised her she shifted on the spot and visibly swallowed as if rapidly losing her nerve in the face of his silence. It seemed her blustery, confident entrance had all been an act to get past the temporary PA sitting outside his office. Soon to be his ex-temporary PA.
'And you are?' he said with a sigh. He really didn't need this extra hassle today; his nerves were already strung as tightly as they'd go and he had an important meeting in ten minutes which he needed to have his head in the game for.
'Solitaire Saunders. Soli for short. That's what everyone ends up calling me, anyway. It's a bit of a mouthful otherwise.'
His eyebrow twitched involuntarily upwards.
'Solitaire? Like the diamond?'
She gave a self-conscious grin. 'No, like the card game. My dad was a huge fan of games. He set up our board game café on Hampstead High Street—in the unit we rent from your company.'
Board game café?
He was surprised anyone could make a living from a business like that, though, judging by the increasingly irate letters he now remembered receiving from the woman running the place—presumably this woman—after they'd notified her of the upcoming rent raise, perhaps she didn't.
Despite his reluctance to get into this with her right now, he knew he ought to nip the issue in the bud while she was here in front of him. His executive assistant was fed up with having to field her constant phone calls asking to speak to him directly and he'd never been one to shy away from a legitimate business conflict when it reared its head. Its pretty, curly blonde head in this instance.
'The trouble is, Soli,' he said, splaying his hands on the desktop, 'the market's moved on a lot since you last signed the rental agreement a couple of years ago—'
'Four years ago,' she butted in. 'And it was my father who signed it. I've been running it without him for the last three of them.'
'Okay, I don't have the exact details to hand right now,' he said, trying to remain patient, 'but I do know that the market's moved even more since then.' He lifted his hands, palms towards her. 'We're not monsters here, we've actually held back on increasing the rent on a lot of our property because we know how hard it can be for small independent businesses to survive in London, but we have to move with the times.'
'You know how hard it is to run a struggling business, do you?' she shot back. 'How utterly heartbreaking it is when a once thriving business starts to fail? How demoralising that can be?' Her voice rose on each question. She glanced pointedly around his plush office with its high-end furniture and enviable London view then fixed him with a challenging look, her cheeks flushed a deep shade of pink but the expression in her eyes unwavering.
He experienced a shiver of guilt, but knew he couldn't let it get to him. Everyone he came across these days seemed to have a sob story to tell him so that he'd agree to charge them less money for the property they rented from his company. He couldn't let his personal feelings get in the way. This was business.
'We live above the café,' she said before he could form his careful reply. 'If we can't afford to keep the business going we'll lose our home as well, but then I don't expect you'd know how a threat like that feels either!'
If only that were the case.
He began to shake his head, but she took a step closer to his desk and put her hands over her heart, her cute little nose wrinkling again in a way that made something twist uncomfortably in his chest.
'Is there any way I can persuade you to hold off for a little while longer?' she asked in a voice wobbly with emotion. 'Please. Just give me a chance to get a bit more business in.'
'How do you intend to do that?' he asked, genuinely interested. 'Aren't there a lot of other café options on Hampstead High Street?'
Her bold stance deflated a little. 'Yes. Unfortunately there are. But they're all chains owned by big corporations.' She waved a dismissive hand. 'We offer a more local, family-run atmosphere. And board games! Who doesn't love playing board games?'
He shuffled a little in his chair. 'Can't say I'm a huge fan of them.'
'You just haven't played the right ones yet,' she persisted. 'If you come in you'll see how much fun they can be. We have four hundred games to choose from. Something for everyone. We'll even teach you how to play them.'
He shook his head, holding back the smile that was pushing at the corners of his mouth. Learning to play board games was the last thing he could imagine wanting to do with his precious time off. 'As appealing as that sounds,' he said, trying to keep the irony out of his voice, 'that doesn't tell me how you're going to start making enough profit to pay your rent.'
'I'm working on it,' she stated, but her gaze wasn't meeting his now; instead she was staring out towards the river, her hands clenched at her sides as if she was fighting to keep her composure. 'I just need to find some time to do a bit of local advertising, update the website and post to the social media sites we're on,' she said, almost to herself. 'Trouble is, I work long hours. I have a cleaning job at a gastro pub from seven thirty till ten, then I have to make the cakes and prepare the sandwiches we sell at the café, then we're open from eleven till three. When we close I have to go shopping for supplies for both the café and the family and take care of anything my mum needs and then the café's open again from five till ten pm. So there's not been a lot of time for developing a high-concept business strategy.'
More guilt tried to shoulder its way in as she looked back at him with tired eyes.
He shook it off. This wasn't his problem. He couldn't allow it to become his problem either. He had enough of his own troubles to deal with right now.
'Look, Soli, things are complicated for me at the moment and I'm afraid I don't have time to deal with this today. I have an important meeting in a few minutes, so if you leave your contact details with my PA—'
She flinched at the hard edge he'd given his voice now, but didn't move from where she stood.
'Complicated? You think your life's complicated? Beat this, buster.' She pointed her finger at him. 'I'm desperately trying to save the business my late father built from scratch, our family's legacy, so I can afford to get my mother, who's suffering with Parkinson's disease, the care she needs whilst also trying to scrape together enough money to support my younger sister, who's a brilliant mathematician with an offer from Oxford University, but who can't afford to take the place there. And you're making it even harder for me to do all that by raising our already extortionate rent. That's complicated!'
The ensuing silence rang out loudly in the still air of his office.
'Okay. Fine,' he said resignedly when he saw a glint of tears in her eyes. 'You win the "complicated" competition.' He made a placating gesture. 'But only just—believe me. My life isn't exactly easy right now either.'
'Look, is there some sort of arrangement we could come to here?' she asked desperately, blinking back her tears and looking a little embarrassed about losing her cool. 'Any sort of deal we could make which would give me a bit more time to try and turn the fortunes of the café around and make the money we need to afford the rent hike? I can't lose the place. Not after all the love and hard graft my father put into it. It's all we have left of him now.' Despite her efforts, a single tear ran down her cheek.
He looked hurriedly away, frowning down at his desk. 'I've already held back on rolling out the new rent and if I do it for you I'll have to—'
'Please. Have a heart,' she broke in hoarsely, clearly aware she was losing the battle but seemingly not prepared to accept it. 'I'll do anything. I'll come and work here for you when I'm not working at the café. I can type and make coffee, file things. Documents. Tidy up! I'll do whatever it takes to keep our café running.'
The ring of hope in her voice clawed at his chest. He had to give her credit, she was certainly determined.
Or maybe just desperate.
His heart gave a hard thump. He knew what desperate felt like and he wouldn't wish it on anyone.
'Whatever it takes?' he asked slowly, meeting her eyes again now. He wasn't sure where he was going with this, but he had the strangest feeling there could be a solution here that he hadn't quite hit upon yet.
'Anything. Just name it,' she said, her eyes wide with anticipation.
He sighed and shook his head. 'The thing is, I have a PA already. There wouldn't be anything for you to do here at the office.'
'At your house, then? I'm a great cleaner. Fast and totally reliable.'
'Got a whole team of those.'
'Then what do you need? There has to be something.'
And there it was.
The idea.
But he couldn't suggest that.
Could he?
No.
He shouldn't.
'Please,' she whispered in a broken voice, tears brimming in her eyes again.
'What I need most right now is a wife,' he said roughly, losing the grip on his restraint as the idea pushed harder at his brain and compassion loosened his tongue. 'At least, I need to find a woman that's prepared to get married in the next few weeks and stay married to me for a year.' Catching the expression of shock on her face, he silently cursed himself for letting that slip out.
She must think he was a total loony.
'Are you serious?' she asked in a faltering voice.
He sighed, feeling tiredness wash through him. 'Unfortunately, I am.'
'Why do you need a wife so fast?'
'Like I said, it's complicated.'
She surprised him by perching on the edge of his desk and fixing him with an intent stare. 'Well, you listened to my problems; let me hear yours.'
His pulse stuttered. 'I don't think it's appropriate—'
She held up her hands in a halting motion. 'Just tell me. Perhaps I can help.'
He frowned at her, taken aback by her unexpected forcefulness. 'I very much doubt it.'
'Look, I won't say anything to anyone if that's what you're worried about. I'm good with secrets. Maybe it would help to say it all out loud. That's what my dad used to do. He used me as a sounding board and often I didn't need to say a word: he already had the answer; he was just having trouble accessing it.'
He took a moment to study her, trying to judge whether he could trust her not to blab to all and sundry once she'd left his office. The last thing he needed right now was for this to be circulated around social media or the Press. He was already taking enough risks talking to the women he'd approached so far and it could only be a matter of time before his luck ran out.
'Go on. What harm can it do?' she murmured, giving him a reassuring smile. There was something about her that encouraged confidences, he realised, and for some reason he felt, deep down, that he could trust her.
He sighed, deciding that he may as well tell her the whole sorry tale since she knew most of it already anyway. Plus, he didn't really have anything more to lose at this point. And who knew, perhaps she could help in some way?
Stranger things had happened.
Getting up from his chair, he paced over to the window and stared out at the pleasure boats transporting tourists up and down the wide river. 'My late great-aunt owned the house I live in at the moment.' He swallowed past the dryness in his throat. 'It's the place I've considered to be my home for my entire life. It was meant to go to my father next, but he passed away a few years ago, so I'm next in line to inherit it,' he said, glancing back to check she was listening.
She was. She gazed back at him with an open, interested expression, her hands folded neatly in her lap.
'It's been in my family since 1875, ever since it was built for my great-great-grandfather,' he continued, turning back to look at the river again. 'It's the house where I spent all my holidays from boarding school and the home I intend to live in until I die.' He paused for a moment, feeling his throat tighten as he remembered how he used to say it was the place where he and Harriet would always live, before—well, before his whole life was turned upside down.
Shaking off the tension this memory produced, he moved away from the window and sat back down on his chair.
'In order to inherit the place, though, a covenant in the will states I have to be married within the next month.' He tried not to grimace as he said it.
She nodded slowly. 'Okay.' Frowning now as if a little puzzled, she said, 'Could I just ask—why the rush? Haven't you known about this for a while?'
'No. Apparently my great-aunt wrote it into her will a couple of years ago, but she was in a coma for eighteen months after suffering a massive stroke. I only found out about it three months ago when she passed away.'
He paused and swallowed, shaking his head as a wave of sadness at losing the woman he thought of as more of a mother figure than a great-aunt flooded through him. 'I only inherit it if I'm married by my thirtieth birthday and remain married for at least a year, otherwise it gets passed on to my cousin, who is already married,' he grimaced, 'and the most immoral, wasteful, tasteless man I've ever met. He'd sell the place to the highest bidder in the blink of an eye.'
There was a heavy pause where he watched her eyes widen and her mouth twitch at the corner.
'And before you ask, no, he wouldn't sell it to me. We don't exactly get on.'
'I kind of gathered that from your description of him,' she said with a smile.
He tried to smile back but he couldn't quite muster the energy needed. Mirth was a hard response to summon when you were about to lose the only place in the world that really meant something to you. The place that held all your childhood memories and felt like an integral part of your history.
Your home.
He'd feel baseless without it, adrift, disenfranchised.
'Well,' she said, her eyes alive with what looked suspiciously like amusement, 'that's quite a conundrum you have there. It's like something from a soap opera.' Her mouth twitched. 'And not a very good one.'
Rubbing his hand over his brow, he felt the tension this predicament had caused under his fingertips. 'I'd have to agree with you.'
'Your great-aunt sounds like a real character.' Her eyes still sparkled with amusement but her smile was warm.
'She was a little eccentric, yes.'
Crossing her arms, she peered down at him. 'And I'm guessing no one you've asked so far has said yes to this rather unusual proposal?'
'Correct. Not that there have been many suitable candidates.' He leant back in his chair and mirrored her by crossing his own arms. 'The fact we'd have to live together to make it look like we're a real couple—apparently a solicitor will be deployed at random times to check on this,' he added by way of explanation, 'but not have a real relationship hasn't exactly caught the attention of the women I've approached so far. I'm really only interested in getting married as a business arrangement; I'm not looking for true love.'
Her brow furrowed at this. 'You don't want to fall in love?'
'No.'
There was a small pause before she asked, 'Why not?'
He shrugged. 'It's just not for me, that's all. Despite my great-aunt's insistence that it was the best thing that ever happened to her, I don't believe falling in love with someone can really make you happy.' He sat up in his chair. 'In fact, I think it does the opposite. It didn't work out for my parents, or for a large population of the country, and I intend to learn from their mistakes.'
Not to mention his own near miss—though he wasn't about to tell her about that humiliating experience.
'Just out of interest, what does your temporary bride get out of this arrangement?' she asked in a faltering voice, jerking him out of his scrambled thoughts.
There was a tense pause where they looked at each other and he weighed up what he'd be prepared to offer her if she meant what he thought she meant by that.
'The candidate would be able to keep the rental cost on their property the same for the next five years,' he replied slowly.
'And would there be some sort of pay-out as soon as she'd signed the marriage register?' she asked, her gaze intent on his now.
'There could be, if it was a reasonable request.'
'But she'd have to live with you,' she appeared to swallow, 'in your house?'
Noting the renewed flush of her skin, he could guess what she actually meant by that.
'It would be a purely business arrangement,' he reassured her, 'which would mean she'd sleep in her own bedroom. There wouldn't be any conjugal expectations. In fact, it would be a totally platonic relationship, to avoid any complications.'
'I see,' she said, her shoulders seeming to relax a little.
Despite his wish to keep sex out of the deal, he couldn't help but feel a little miffed by her apparent horror at the idea of sleeping with him. Was it really that off-putting an idea? He shook off his irritation, telling himself not to be an idiot. The woman didn't know him from the next man, so of course she'd be nervous about the idea of any expected intimacy between them.
'We'd also both have to agree not to have any sexual relationships outside the marriage, again, to avoid complications.'
'Okay,' she said without expression, not giving him any clues about her feelings on that one. Would that be a deal-breaker for her? She was an attractive, sparky woman and he guessed she must get plenty of male attention. There was something really appealing about her, especially when she smiled.
'One of the other stipulations would be that she'd need to take my surname for the duration of the marriage,' he said, pulling his attention back to the matter at hand. 'It would just be for appearances and she could change it back again afterwards, of course.'
'Afterwards?'
'After the divorce. There'll be a pre-nuptial agreement to sign so she won't be able to petition for money or property during the legal severance of the marriage.'
There was a pause in which the air seemed to vibrate between them.
'Oka-a-ay,' she said slowly, her voice sounding a little breathy now.
He frowned, panicking for moment that she might be stringing him along for a laugh.
Before he could start to backpedal, though, she fixed him with a steady gaze, her lips quirking into a wide smile—triggering a warm, lifting sensation of hope in his chest—then took an audible breath and said, 'I'll do it. I'll be your wife.'
CHAPTER TWO
Monopoly—move around the board for the chance to collect money and new property.
SOLITAIRE SAUNDERS HEARD her father's voice in her head as she gazed anxiously back at the man who had the power to turn the course of her and her family's lives around with a mere nod.
'Your tendency to run headlong into things without thinking is going to get you in serious trouble some day, Soli,' her father's voice warned her.
He wasn't wrong.
She knew that.
But you're not here any more, Dad, and I'm doing the best I can.
There was a chance, of course, that she was actually dreaming all this and would wake up at any moment in bed with her heart racing and her palms as sweaty as they felt right now.
But she really hoped that wasn't the case.
In fact, she knew it wasn't possible because when she'd actually rolled out of bed this morning, and been unable to eat her breakfast because her stomach was jumping around so badly with nerves and worry, she'd never felt so awake—and afraid. The pressure of her mum and sister relying on her to stop both their home and livelihoods from being swept out from under them weighed heavily on her.
So she was hyper-aware, sitting there now in her smartest clothes with her wild hair as neat as she'd been able to get it, that how well she performed in this meeting could change all their lives for ever, one way or another. What she hadn't expected when she turned up here was to be confronted with such an unusual and nerve-racking way to do it.
This—this incredible stroke of luck—could be the answer to all her problems.
If she could handle it, that was.
As far as she could see, the most challenging thing about it would be having to see Xavier McQueen, property baron and high-society mover and shaker every day for the next year.
And be his wife.
The thought of living with this powerful, domineering stranger made her heart thump harder in her chest.
The guy was seriously attractive, with a lean but muscular physique which she imagined he kept looking that fit with regular trips to the gym. His face was angular, with high cheekbones and a strong jaw, and he had light green, almond-shaped eyes, framed with dark lashes, which gave him a nerve-jangling look of stark intensity. And he had really good hair. Thick and shiny and the colour of melted chocolate. It sat neatly against his scalp as if it had been styled deliberately to do that by a master hairdresser at a top salon. Which, she mused, it probably had. Her fingers twitched at her sides as she fought a powerful urge to reach out and touch the soft waves, to see if it was as soft and smooth as it looked.
'I have some non-negotiable demands if I'm going to do this,' she said, a little more loudly than she'd meant to out of nerves.
'I thought you might have,' Xavier replied, with an ironic tinge to his voice. He had to be the most sardonic person she'd ever met. Throughout all their exchanges it had seemed as though he'd been having trouble taking anything she'd said seriously.
Still, he wasn't exactly laughing now. In fact, despite his sarcasm, he was actually looking at her as if she might be the answer to all his problems.
'Okay. If I'm going to be your wife for a year I need to know that my mother is being taken care of properly, so I'll need to have a live-in carer provided for her while I'm away. She'll be mostly okay during the day, but she'll definitely need someone there overnight to help her get ready for bed and to get up when my sister's not there. Which leads me on to the next stipulation. I also want you to pay for my sister's tuition fees at university. She'll get a job to cover her living expenses, but it won't go any way towards the fees.' Her heart was racing as she laid all this out, wondering whether he'd just tell her to get up and get out because she was being too greedy.
But he needs you, a voice in the back of her head told her, so front it out.
There was a long pause while he looked at her with such an intense gaze she felt it right down to her toes.
'Okay, so let me get this straight,' he said eventually; 'you want a full-time carer for your mother, tuition fees paid for your sister, a stay on the rent on the café for the next five years and an as yet undisclosed sum of money as soon as we're married?'
She swallowed hard, but held her nerve. 'Yes.'
'And how much were you thinking of for your lump sum?'
Shakily, she said an amount that she thought would cover the wages at the café for the next year as well as giving her some spending money which she could use for marketing or renovations to the café once they were divorced.
He surveyed her for a moment, his right eyebrow twitching upwards by a couple of degrees.
Soli held her breath, aware of her pulse throbbing in her head.
Had she blown it by asking for too much?
'Okay. It's a deal,' he said finally. 'But, considering you'll be losing your wage from the cleaning job and you'll have to employ someone to cover your position in the café, I'm prepared to give you an additional twenty per cent on top of that.'
Soli swallowed hard, his unexpected generosity bringing tears to her eyes.
'As long as you agree to marry to me within the next month and spend the majority of your time in my home,' he added quickly. 'I don't mind you visiting your mother and working part-time at the café, perhaps one or two days a week so you can keep an eye on it, but it needs to look as though the majority of your time is spent living there with me. Particularly in the evenings.'
'So I can only work during the day?'
'Yes. I'd like it if you were able to attend any work or social events at the drop of a hat. For that, I need you focused on your life with me as much as possible.'
She suspected that what he wasn't saying out loud was that he wasn't the sort of man to have the owner of a board game café for a wife and he didn't want to have to explain himself to anyone.
'So what will I do for the rest of the time?' she asked as indignation rippled through her. What was wrong with working in a board game café? She really enjoyed it. It was sociable and kept her fit because she was on her feet all day.
He frowned, momentarily stumped by her question. 'Perhaps you could work on that "high-concept business strategy" you haven't had time for?' He waved a hand. 'I'm sure you'll find plenty of things to do with your day.'
'And what do you want me to tell people when they ask what I do for a living?' she asked, still riled by her suspicion that he didn't value her choice of livelihood. 'What do the kind of women you normally date do for a job?' she added, perhaps a little tetchily.
He rubbed a hand over his forehead, looking taken aback by the directness of her question. 'Most of the women I've dated have either had a media job or been a doctor or solicitor.'
'Well, I don't think I'm going to convince anyone I'm a doctor or lawyer,' Soli said, raising a wry eyebrow. 'My sister got all the brains in the family.'
He frowned, apparently a little bemused by her now. 'Okay, well, if you want to choose a different career for yourself, go right ahead. What would you have done if you hadn't taken over the café? Do you have any burning ambitions?'
His question stumped her for a second. It had been a long time since she'd thought about doing anything but running the café. 'I don't know. I wasn't exactly focused at school so I never expected to have a high-flying career. I liked designing clothes, but I did that in my spare time. My dad pressganged me into taking academic subjects to "give me a better chance in life".' She put this in air quotes, remembering with a sting of shame how she'd rallied against this notion, thinking it would bore her to tears to have a professional job in the future. All she'd wanted when she was in her mid-teens was to have a family of her own and perhaps make a living in some sort of arty career.
How naïve she'd been.
'Well, why don't you have a think about what you'd feel comfortable telling people you do? You're a business owner; why don't you go with that?'
She nodded slowly, her earlier irritation at his imagined snobbery subsiding. 'Okay. Business owner it is.'
He nodded. 'And what do you intend to tell your family about our arrangement?' he asked in a careful tone.
'I'm going to say I've taken a job as your live-in housekeeper, for which you're going to pay me an exorbitant wage.'
He nodded, then pulled out his phone and began to type onto the touch screen, presumably making a note of her demands, and his, so they'd have something to refer back to should there be any issues in the future.
'They'd buy that much more readily than the truth—that I'm marrying a total stranger,' she added with a strange tingling feeling in her throat.
It felt so odd to say those words. Whenever she'd imagined getting married, which hadn't been very often recently, owing to her life being too complicated for her to think that far into the future, she'd imagined herself meeting a guy, their mutual love of board games bringing them together, and dating him for a couple of years before moving in together, then him proposing to her out of the blue in some far-flung romantic destination, like Hawaii or Morocco, or maybe on a Mediterranean island whilst sailing through the clear blue water in a yacht.
They'd get married in a quaint little church with all their friends and family watching and throw a huge party afterwards, where they'd dance the night away together. Then, a year or two later, after they'd had some time together as a couple, they'd have kids, maybe three or four of them.
She'd always wanted a big family.
When she was younger, sitting bored and frustrated at school during subjects she couldn't get a handle on no matter how hard she tried, she'd fantasised about what it would be like to be a mother. How she'd make her kids big bowls of hearty food, which they'd gobble down gratefully before going off to play happily with their toys, or do finger-painting with her at the kitchen table, laughing about the mess they were making together. Or she'd imagine ruffling their hair at the school gates and receiving rib-crushing hugs in return before they ran in, with her shouting that she loved them, which they'd pretend to find embarrassing but would secretly adore. Then later in the evening she'd tuck her sleepy, happy kids up into bed before spending the rest of the evening with her gorgeous husband, chatting about the day they'd had before retiring to bed together hand in hand.
That all seemed a million miles away now though.
It had been ages since she'd been on a proper date with anyone and even then they'd barely got to the kissing stage before her lifestyle and responsibilities had got in the way of things developing any further. She'd made it clear that her family came first and that had destroyed the chances of a relationship.
Not that she blamed her mother and sister. Not a bit. In fact, despite their difficult circumstances, she quite liked being the head of the family. The one that everyone relied on. It gave her a sense of purpose that had previously been lacking in her life.
Yes, anyway, it was a good thing that Xavier had insisted on a purely platonic relationship. It wasn't like she had any time for romance.
'How old are you, Soli?' Xavier asked brusquely, jolting her back to the present.
A shiver of disquiet tickled down her spine. Was he worried she wasn't mature enough to deal with this?
'I'm twenty-one,' she said, setting back her shoulders and fixing him with a determined stare. 'Old enough to know my own mind,' she added firmly.
His eyes assessed her for a couple of beats more before he nodded. 'Okay, then. I guess that's everything we need to discuss today.' He put his phone down on his desk, arranging it so it sat parallel with his keyboard, before looking up and giving her his full attention again. 'Look, I appreciate this is a lot to take in right now, so why don't you go away and have a think about it, to make sure you're comfortable with everything we've discussed? It's a big decision to make and I don't expect you to sign up for it until you've had a chance to check me out first.'
She nodded jerkily. Despite her bravado, she was actually glad of the chance to go and think about this away from his discombobulating presence, just to make sure she hadn't overlooked something important. 'Okay. I'll do that. It really wouldn't do to marry an axe murderer by mistake,' she said, flashing him a jokey grin.
Ignoring her attempt at levity, he opened a drawer in his desk and took out a business card which he handed to her. 'This has my personal mobile number and address on it. Give me a call when you're ready to talk again.' He paused and frowned. 'But don't leave it too long or I might find someone else to marry in the meantime.'
For a second she wasn't sure whether he was joking or not. He didn't seem to do smiling, at least not the kind that made him look as though he was genuinely happy. Cynical. That was what he came across as. And reserved.
She wondered fleetingly what had happened to him to make him like that, but pushed the thought away. It wasn't important right now and she really shouldn't allow herself to get emotionally attached to him anyway, not if this was going to work as a purely business arrangement.
'Okay, thanks. I'll get in touch very soon,' she replied, taking the card from his fingers.
She shot him a tense smile, then got up from the desk on shaky legs and turned to go.
'And Solitaire.'
She turned back.
'If I find out the details of this proposition have been leaked to the Press I'll know where to find you.' There was a heavy pause before he added, 'And you'll find your business and your family swiftly evicted from my property.'
'Understood,' she said, then left the office of her potential future husband, wondering what in the heck she'd just got herself into.
* * *
Back at the café, she relieved Callie, who waitressed for them a lot and had kindly agreed to work an extra shift that morning so Soli could go to the McQueen Property office. Once she'd caught up with the daily tasks and served a sudden rush of customers, she sat behind the serving counter with her laptop and typed Xavier's name into the search engine with trembling fingers.
She'd already looked him up before the meeting, of course, scouring the web pages for something she could use in her defence against him, but to her frustration had found him to be squeaky clean. At least at first glance. She needed to put in more thorough due diligence here though if she was going to commit to live with the man for a year. The last thing she needed was to find herself sucked into something she'd not anticipated and then couldn't escape from without causing more harm to her situation.
But as hard as she looked, she couldn't find anything that threw even the meanest of shadows over his reputation.
The only things that came up about him were on gossip sites, where they mentioned him in relation to the high-society women he'd had flings with over the last few years. The man appeared to be some kind of international playboy, always showing up at high-profile fundraisers and gallery openings with a different, instantly recognisable woman on his arm. He was like a character from one of the romantic novels she liked to gobble up like sweets for escapism from her busy, stressful existence. She'd never really believed such a person could exist in real life, but here he was, a living, breathing, alpha male business tycoon.
So he checked out okay online.
Picking up her phone, she called a friend who was a police officer in the Met and asked him if there was any way he could have a check around about Xavier, pretending she was doing it for business reasons concerning the café. Mercifully, her friend seemed to buy that and asked her to leave it with him.
She spent the rest of the day in a jumpy, nerve-filled state and was mightily relieved when her friend called her back in the early evening to let her know that nothing negative at all had come back to him with regard to Xavier, either personally or with his business. It seemed he was an upstanding citizen of the realm.
The only thing left to do now was to check out exactly where his house was using an online map app—just to make sure he wasn't expecting her to live in some kind of broken-down hovel. Not that she expected to encounter that. Judging by the high-end furniture and breathtaking elegance of his office, she couldn't imagine his house being a place she wouldn't like to spend time in. She could have happily lived right there in his office if he'd asked her to, with that wonderful view over the water. It certainly beat the one she had from their living room window over the busy, vehicle-choked high street, or the one of the bins in their small back yard from the bedroom she shared with her sister.
Not that she was complaining about her lot. Home was where her family was and she'd been happy living here above the café with them. Staying in this flat had made her feel closer to her father somehow. She could still picture him sitting in the battered old leather armchair by the window after long shifts in the café, with a paperback resting on his knee and his requisite triple-shot black coffee on the small table beside him. He'd hated working at the bank and after twenty years he'd finally given up corporate life and they'd all downsized so he could run the board game café, a dream he'd had for years.
Sadly, he'd only worked there for five years before he died. Still, Soli was glad he'd had the opportunity to realise his dream. Ever since she'd lost him the café had become a symbol of hope for her, as well as a reminder that hard work and dedication paid off—something she'd been slow to learn in her younger years, to her everlasting shame.
Shaking off the guilt that always gave her a painful jab when she remembered how selfishly she'd acted in her teens, she got up from behind the counter to close up after the last stragglers made their way out onto the street, waving cheerily to her and calling their thanks. If only they had more regulars like them, the type that bought food and drink every hour as they played, the café would have some hope of survival.
She just needed to find a way to entice those types of people to walk through the door.
After locking up behind them and giving the floor a sweep and the tables one last wipe, Soli walked into the middle of the room and tried to survey it with objective eyes. Why weren't people coming in as much as they'd used to? Sure, it was a bit shabby-looking now after years of wear and tear and it could probably do with a bit of sprucing up, but it had a friendly, comfortable aura to it, and didn't people love shabby chic these days?
She hated the idea of messing with what her father had done to the café. He'd sanded and varnished the wooden tables himself, painted the walls, chosen the now slightly chipped crockery, and she couldn't imagine any of it changing. It would be like wiping her father's soul from the place.
She shuddered, hating the very thought of that.
No, she'd try advertising first, then think about any alterations they might have to make once the money was flowing in again.
Assuming they didn't lose the tenancy in the meantime.
Taking a breath, she focused on calming her suddenly raging pulse. All she needed to do was marry Xavier McQueen and everything would be okay.
The utter bizarreness of that thought made her laugh out loud.
Shaking her head at the surreal turn her life had taken, she went to the till to make sure it had been cashed up properly, grimacing at the sight of the meagre takings for the day. Yes, something definitely needed to change.
Picking up her phone, she tapped in the number he'd given her. He picked up after two rings.
'Xavier McQueen.'
'It's Soli.'
'Hi,' was all he said in reply.
There was a pause in which the weight of expectation hung heavily in the air.
'So I checked up on you and it turns out you're not an axe murderer,' she quipped nervously.
There was a uncomfortable pause when he didn't respond.
Okay, then. Jokes weren't deemed appropriate right now. Wow, this guy was so businesslike.
Probably best just to get down to business, then.
'So I've thought about it and I still want to go ahead with our deal.'
'Great, that's great.' She could hear the relief in his voice. 'I'll arrange for a solicitor to draw up a pre-nuptial contract and another one that states the terms of our deal, which we'll both need to sign.' His tone was professional again now.
'I'll give notice at the register office that we want to get married but we'll have to wait twenty-eight days before we can legally perform the ceremony. The closest one is near St Pancras Station, but I'm assuming you won't have an issue with where the formality of it takes place.' It wasn't a question, she realised. 'It's not like we'll be having a big celebration with friends and family,' he added when she didn't reply right away.
'Er, no, that's fine.' The words came out sounding confident, but something deep in her chest did a strange, sickening sort of flip. This really wasn't the way she'd imagined it happening. Getting married. But, as he'd rightly pointed out, this wasn't meant to be a romantic event, it was a business transaction and should be treated as such. There was no room for any kind of emotional attachment. She'd make sure her real wedding, to the guy who loved and cherished her, was a big, exciting affair, with all her friends and family present. That one would be a cause for a true celebration. She just needed to keep that in mind when she signed the register. True love would come later in her life, when she finally had the time and energy to consider it a possibility.
'Okay, good. I'll let you know the details as soon as I've set it up. I'll need some personal documents from you which I'll swing by and pick up tomorrow, if that works for you?'
'N-no problem,' she stuttered, feeling suddenly as though her life was running away from her a little.
It's not surprising; you're getting married in a month.
A shiver of nerves tickled down her spine.
There was a lot to sort out before then, not least accepting the university place for Domino and finding a full-time carer for her mum, as well as giving notice at the gastro pub and hiring someone to cover her shifts at the café.
The mere thought of all the work and organisation ahead of her was exhausting.
This is for the family, she reminded herself as panic threatened to engulf her. And it's only temporary.
In a year's time her life would have taken on a whole new shape. She was doing this for all the right reasons and once she and Xavier were divorced she'd be free to fall in love and get married for real.
With that thought in mind, she told Xavier goodbye and hung up.
Trying to ignore the now almost overwhelming wave of nerves, she turned off all the lights in the café, hid a yawn behind her hand and trudged up the narrow staircase to the flat, first to check that her mother didn't need anything, then to spend the next hour or so planning how best to kick-start the beginning of her brand-new life.
CHAPTER THREE
Scrabble—choose your words carefully.
THEIR WEDDING DAY was glorious. At least the weather was, with the sun pouring in on them through the large picture widows of the register office as they stood at the desk reciting the lines they were asked to say.
The huge room, with its rows of chairs facing the desk, was eerily empty except for Xavier and Soli, the registrar, Xavier's friend Russell—the only friend he'd confided in and who had drawn up the contracts in his other role as a solicitor—and one other witness, who was a complete stranger to them all. Xavier had approached him outside on the street, pretending that their second witness had been delayed in traffic, and offered him a wad of cash for half an hour of his time.
Glancing around the room, he remembered all too well the last time he'd been in a place like this as echoes of a clawing sense of shame and dread pricked at his skin. He'd promised himself he'd never set foot in a register office again and hadn't attended a wedding since his own disastrous debacle. He'd actually intended to avoid them for the rest of his life, if at all possible.
But he hadn't counted on his Aunt Faith's iron-like will.
So here he was again.
At least this time the bride had turned up and actually married him.
Well, you got what you wanted, Aunty. I hope you're happy now.
Soli, to her credit, didn't say a thing about the lack of guests or the stranger signing the marriage register beneath her name. In fact, she'd seemed more than happy to let him deal with all the arrangements and go along with whatever he'd asked her to do. She'd told him it had meant she'd been able to focus fully on making the necessary arrangements for her family and the café before she came to live with him. Apparently her sister was off to live in Oxford over the summer to earn rent money at a job she'd found there before her first year began and her mother now had a full-time carer living in the flat with her. All thanks to his money.
Not that he resented it. It meant he was able to achieve exactly what he wanted after all.
In his experience, money always smoothed the way. It was the only thing he could ever really rely on.
'Congratulations,' the registrar said to the two of them once the ceremony had come to a close. She didn't seem at all fazed by the lack of guests or the sombreness of the occasion, but Xavier guessed she must have seen it all in the course of her duties.
'Thank you,' he said, giving her a nod of gratitude.
'Yes, it was a lovely service,' Soli added with a barely discernible quaver in her voice.
He glanced at her, wondering whether she was having a moment of regret, but she just smiled back at him as if nothing in the world was wrong. He appreciated her professionalism.
He'd not really looked at what she was wearing when they'd met in the lobby only minutes before their slot because the registrar had come straight over to introduce herself then whisked them straight in, but as he surveyed Soli now he realised she'd made a real effort with her appearance today.
Her wild curls had been tamed into an elegant up do and she'd put on more make-up than he'd previously seen her wear, which accentuated her big bright eyes and full, rosebud mouth.
The simple cream-coloured sheath dress she wore exposed her slim, toned arms and flowed over her curves, drawing his gaze to the tantalising swell of her breasts under the thin fabric.
Hoping she'd assume he was looking at the small posy of flowers she clutched in front of her, he cleared his throat and raised his eyes to give her a tight smile.
Yes, she definitely looked the part. She was a very attractive woman and no one would find it strange that he'd chosen to marry her. At least on the surface. As long as she kept her mouth shut about the terms of the deal they'd worked out, his secret would be safe.
Hopefully there wouldn't be many opportunities for their charade to be discovered anyway. He'd asked her to be ready to attend functions with him but he wasn't actually intending to take her along to many. Just one or two, so it didn't look odd if anyone checked up on them.
He'd already alerted his great-aunt's solicitor to the fact he was getting married and had been told to expect spot checks in the next few months, just to satisfy her conditions. After a year the title deeds to the house would pass into his name.
Then he'd be free to live his life as he chose again.
One year wasn't too long a time to maintain this farce. He could manage it.
'Well, Mrs McQueen, now that's over, shall we get out of here?' he suggested once the registrar had departed, more than ready to leave the place now.
To his surprise Soli pressed her lips together and pulled a mock horrified face. 'You know, I thought Solitaire Saunders was bad, but Solitaire McQueen?' She raised both eyebrows. 'My father will be dancing with glee in his grave.'
The sad edge to her voice gave him pause. 'How did your father die? If you don't mind me asking?'
She shrugged. 'I don't mind. I guess you should know now that we're husband and wife.' Taking a breath, she pushed her shoulders back a little, as if using the action to give her courage. 'He was knocked off his bike by a guy who was texting whilst driving. He died instantly.'
A prickle of horror rushed up Xavier's spine. 'Ah, hell, that's awful. I'm so sorry.'
There was an awkward pause while she blinked back the tears that had pooled in her eyes.
'Thank you,' she whispered, smiling bravely. 'I still miss him every day, but he'd want to know we were all getting on with our lives without him.' She glanced down at the slim white gold ring he'd placed on her finger only minutes ago with an expression of incredulity on her face, then flashed him a wry smile. 'I'm not sure what he'd think about me marrying a stranger though.'
'I'm sure he'd approve if he knew you were doing it for the right reasons,' Xavier pointed out.
Nodding, she let out a small chuckle. 'Yeah, I'd like to think so. He always said I'd get myself in a knotty situation one day with my impulsiveness, but I don't think this was quite the scenario he had in mind.'
Her cheeks had flushed an attractive shade of pink and he had the strangest urge to stroke his fingers across her skin and feel the heat he knew must be there.
Don't be a fool, McQueen.
Instead he nodded jerkily in response to her joke, then gestured towards the exit. 'Well, anyway, we should leave the room before the next wedding party arrives,' he said stiffly, wishing he didn't sound like such an uptight prig.
Giving her body a small jiggle, as if shaking off her melancholy, Soli nodded in agreement.
He marched ahead of her, trying to blank his mind of the way her voluptuous body had shimmied in his vision as he held the door open for her.
A hubbub of noise surrounded them as they entered the lobby and walked through a large group of people that had gathered there, presumably to attend the next marriage that was taking place in the room they'd just vacated.
Russell and the other witness appeared beside them as they made their way towards the exit. Xavier hadn't noticed them slipping out while they were talking to the registrar, but he suspected Russell had suggested they made themselves scarce so he wouldn't find himself having to answer any awkward questions.
'Let's go out to the front of the building and I'll take a couple of photos of you both in your wedding gear, then we'll see if we can grab a passer-by to take one with the four of us in it,' Russell murmured into his ear.
'Good idea,' Xavier agreed, heading towards the large doors at the other side of the vestibule.
Once outside, they posed next to the register office sign while Russell fiddled with his XLR camera, which he stood on a tripod. Once it was set up, he directed them to stand closer together, with Xavier's arm around Soli's waist and her body pressed close to his. They shuffled awkwardly into the pose and Russell had just taken the first photo when a loud and uncomfortably familiar voice boomed out behind them.
'McQueen? Is that you, old boy?'
Turning reluctantly, with his heart in his mouth, Xavier came face to face with the one person he really could have done without bumping into today.
'Hugo. Good to see you. What brings you here today?' he said, letting go of Soli and taking a deliberate step in front of her so she was obscured from Hugo's line of sight as he shook the man's hand.
'A colleague's getting married and I promised to attend.' He leaned towards Xavier conspiratorially and cocked an eyebrow. 'He's on track to become my boss one day soon so I thought I'd do the smart thing and turn up today. Show willing, you know?'
'Sure. I hear you,' Xavier said. He knew exactly how these old boy networks worked. It wasn't wise to snub someone who had the potential to either help your career or ruin it for you in the future.
'Is Veronica with you?' Xavier asked, a little panicked at the thought of having to save face in front of Hugo's scarily perceptive wife as well.
'No, she's off on some girls' retreat, lucky mare!' he said loudly, adding in a jovial guffaw for good measure. It seemed Xavier's attempt to hide Soli hadn't been successful, though, because Hugo leaned to one side to peer past him. 'And who is this, may I ask?'
Xavier swallowed down his exasperation. 'This is Soli.'
She took a step forwards and held out her hand, giving Hugo a warm smile. 'Solitaire McQueen,' she said, as if happy to have the opportunity to test out her new name for the first time.
Bad timing, Soli. Very bad.
Not that she could have known that.
'McQueen, you say?' Hugo boomed, giving Xavier a confused glance, then looking towards where Russell stood with the camera. 'Have the two of you—' he waggled a finger between them '—just got married?'
'Yes, just a few minutes ago,' Soli confirmed, to Xavier's chagrin.
'Well. You are a dark horse, McQueen. We had no idea marriage was on the cards for you.' Hugo's confused frown deepened as he looked between the two of them.
'No, well, it all happened very quickly,' Xavier said, his heart sinking through his chest. 'We've dated on and off for years but only recently decided we should make a proper go of it,' he lied, silently begging his friend to take him at his word.
'Really?' Hugo said with a tinge of disbelief in his voice. 'It happened so quickly you couldn't even wait to invite your friends to the wedding?'
Damn. He was well and truly busted. He'd never hear the end of it from his old friends now.
'Neither of us wanted a big do,' Xavier said gruffly, feeling heat rise up his throat. The last thing he needed was Hugo and his old social group to find out he'd had to pay Soli to marry him in order to keep his family home. He hated the idea of that getting back to Harriet. His humiliation really would be complete then.
'We thought we'd have a party for close friends and family some time in the near future,' he said, deciding the only thing to do was to bluster his way through this.
Hugo flashed him a knowing smile. 'Fair enough, old chap. I suppose I can understand why you'd choose not to shout about it to all and sundry.' Turning away before Xavier could comment on that, he asked, 'And what do you do, Soli?'
He noticed her shoulders stiffen at the question and silently prayed she'd be able to handle this unexpected confrontation without making Hugo suspicious about the real state of their relationship. 'I'm a small-business owner in the catering industry and one of Xavier's clients,' she said, somewhat mechanically.
Xavier cringed at how that must have sounded to Hugo, but mercifully he didn't seem to find anything odd about it.
'Well, I must congratulate you, Soli; I never thought I'd see a woman manage to make an honest man of Xavier McQueen,' he said, aiming a cheerful grin in her direction. 'Any particular reason for getting married right now though? Do we have the patter of tiny McQueen feet to look forward to?' Hugo asked with a sly wink.
'No,' Xavier stated coldly, feeling the atmosphere thicken between them. 'It just felt like the right time for us both,' he added, trying to smooth over the extremity of his reaction.
Hugo didn't take offence though and slapped him hard on the arm. 'Sorry, old chap, didn't mean to put my foot in it. The wife's always telling me off for that! Very happy for you both, obviously. I'll have to tell Veronica I saw you; she'll be delighted.'
He felt Soli look round at him but didn't turn his head.
'In fact,' his friend went on, completely oblivious to the discomfort he was inflicting, 'if you're not going on honeymoon right away—' He paused and looked at them expectantly.
The only thing they could do was shake their heads dumbly, caught out by the question.
'Well, in that case, why don't the two of you come over to our place next weekend? We're having a bit of a do to celebrate our fifth wedding anniversary—all Veronica's idea, you know,' he added with a pseudo grimace towards Soli. 'I know she'd be delighted to see you, McQueen, and to meet you too, Soli.'
'I'm not sure—' Xavier began to argue.
'Don't be a bore, McQueen!' Hugo broke in before Xavier had chance to air an excuse. 'You can't hide from us for ever. And Veronica will never forgive you if you don't accept at least one of our social invitations. We've not seen hide nor hair of you for years! Now you're married you've no excuse not to come along to see the old crowd. You don't want folks thinking you're shunning them, now, do you?' he said this with a laugh in his voice, but Xavier knew it covered a real sense of hurt. Clearly Hugo was nursing a sense of resentment about being ignored and avoided for so long.
He was trapped. Damned if he did and damned if he didn't.
They'd have a week to prepare for it though. That ought to be enough time for him and Soli to get to know each other well enough to convince Hugo and Veronica, and anyone else they'd invited, that they were a real, loving couple.
'I'm pretty sure we're free then,' he conceded. 'We'll check our diaries and let you know.'
'Great! I'll get Veronica to send you an official invite,' Hugo said with affable gumption. 'You still in your aunt's Hampstead pad?'
'Yes. I'm still there,' Xavier said, feeling a desperate urge to get away from his friend now so he could regain his shaky composure. 'I look forward to receiving the invite, Hugo. Anyway, we'd better get on. We need to take a couple more photos, then we have some celebrating to do,' he added in an upbeat voice that didn't sound like his own, forcing himself to give Hugo a happy-looking smile.
'You've picked a real charmer there, Soli,' Hugo said with another wink in her direction. 'I hope he's intending to treat you like a princess today.'
'Oh, I fully expect him to,' Soli replied, smiling back. 'He's the most generous man I've ever met.'
Xavier experienced a rush of gratitude towards her for that.
'I tell you what, Hugo,' Xavier said, as a flash of inspiration struck him. 'Since you're here, let's have you in the photo too. It'll only take two seconds. We just need to grab someone to press the button for us—it's all set up.'
'Sure! Be happy to!' Hugo boomed, clearly pleased to be included. 'This lovely lady will do it for us, won't you?' he said, making a large beckoning motion to a woman in a big red hat who was just about to enter the doors of the register office.
'Er...yes, of course,' she said, looking over at first Xavier, then Soli, and giving them a warm, indulgent smile.
So they all bunched together and the lady took a couple of pictures of the five of them.
'Thanks so much,' Xavier said, pleased with his quick thinking. It would look much better to have at least two of his friends in the wedding photos.
'We should do one more of just you and Soli, for safety,' Russell said to Xavier, before they all dispersed.
'Ooh, yes. You make such a gorgeous couple,' the woman said, beaming at the two of them. 'You should do one with the two of you kissing. I always think they're the nicest ones to have displayed.'
There was a small pause where neither of them reacted.
'Er...yes, good idea,' Soli said a little too loudly beside him and he felt her slip her arm around his back and lean in towards him, giving him a slightly awkward cuddle.
For a moment he stiffened under her touch before realising how odd that would seem to the small group that were looking at them intently now.
Without allowing himself to think about it, he slid his hand against her jaw, tipped her head towards him and kissed her fully on the mouth.
She drew in a small, breathy gasp, but didn't pull away, instead sinking into the kiss and wrapping her arms tightly around his waist as he instinctively opened his mouth against hers. The soft, flowery scent of her enveloped his senses and he breathed her in deeply, struck by the distinctiveness of her taste and smell.
Delicious.
Her lips were so soft and perfectly pliant he experienced the strangest sensation that they were somehow made to perfectly fit with his. His stomach swooped at the thought and he became aware of something deep inside him—long buried—beginning to stir.
Sensation fizzed along his veins, causing his breath to shorten and his heart to pound against his chest.
Oh, Lord, that's not good.
Pulling abruptly away, he didn't dare look her in the eye again in case she saw even a hint of the heavy need that now pulsed through him. 'Was that okay? Did you get it?' he asked Russell, who was standing behind the camera looking at the two of them with an odd expression on his face.
'Yes, perfect. I got it,' his friend replied, quickly rearranging his features into a smile.
Xavier's stomach twisted as he realised that the spark he'd felt between them had outwardly showed. Not that that was necessarily a bad thing. At least to Hugo it would have looked as though they were genuinely attracted to each other.
When he glanced back at Soli he noticed she looked as bewildered as he felt and his stomach knotted even tighter.
'Well, I'd better get inside before I miss the beginning of this thing,' Hugo said loudly, strolling over to give Xavier a slap on the back and jolting him out of his tangled thoughts. 'Looking forward to seeing the two of you next weekend.' And with that he gave them both a farewell salute and strode quickly away.
Xavier swallowed, his mouth suddenly dry and his head tight around the temples.
'I'd better go too!' the lady in the red hat said, glancing at her watch and moving towards the door of the register office. 'Congratulations, you two, and good luck for the future!'
Luck? Yes, they might need a bit of that if this fake marriage was going to go without a hitch.
'I guess we'd better get on, then,' he said, pulling himself together and turning to look at Soli.
'Okay. Sure. Great,' she replied, her expression still a little shell-shocked. Something tugged hard inside him as a strangely protective instinct appeared from nowhere.
He shook it off.
He needed to get on top of this weird, edgy feeling that was messing with his head. It was this place, it had to be; it was bringing back too many long-suppressed emotions. He couldn't allow himself to develop any kind of feelings for Soli, or encourage her to have any for him. It could cause all sorts of problems.
And extra problems were something neither of them needed right now.
CHAPTER FOUR
Snakes & Ladders—a frustrating game of ups and downs.
SOLI STOOD IN front of the register office, having just kissed her new husband for the very first time, trying to deal with a strange sort of unease at finding she'd enjoyed it much more than was probably healthy for a woman who was now expected to live a life of celibacy for the next twelve months.
How had it been possible to feel so much in so few seconds?
She'd thought it would be fine, kissing Xavier for the camera with everyone looking on, but she'd been shocked by how her body had responded to him. Her skin had flushed all over as if the sun had concentrated all its power on her in those moments and her heart had done a triple flip before sinking to somewhere in the region of her stomach.
It had been wonderful and terrible all at the same time.
She wished she hadn't liked it quite so much because she now had a year to think about how wonderful Xavier's mouth felt on hers without being allowed to experience it again. He'd made it very clear this relationship wasn't ever going to be anything more than as friends.
Friends. But were they really? Could they be?
'I'll go and get your bags from the cloakroom,' Xavier said brusquely.
All she could do was nod in agreement, then watch him stride back into the building where she'd stashed the two cases she'd packed so carefully the day before. They represented the sum total of her worldly goods, apart from a few items of clothing she'd given to Domino and a small box of mementoes from her childhood which she'd left back at the flat because she'd not wanted to lug them over to his house.
Taking a moment to compose herself while Xavier wasn't around, she took a breath and pushed back her shoulders, uncomfortably aware that her legs were still wobbling like mad after she'd put herself through the most nerve-racking half-hour of her life—first the actual marriage ceremony, during which she'd felt as though she were looking down on herself from above, then the whole surreal exchange when Xavier's friend Hugo had appeared out of nowhere and she'd had to scrabble for the correct way to act in front of him.
In the heat of the moment she'd just gone for it and introduced herself as Solitaire McQueen without considering that this might not be something Xavier would approve of, but she'd realised from the way he'd stiffened, then scowled at her, that it had been a mistake. But then, how was she supposed to have known how he wanted her to act in front of his acquaintances? They'd been so busy sorting out the legal side of things they hadn't got round to discussing the day-to-day business of being married yet. It had all been such a whirl.
She'd tried hard not to take offence at his obvious reluctance to introduce her to his friend, but it still rankled. Obviously Xavier wouldn't want the marriage of convenience part to be public knowledge, she understood that—he was clearly a private and proud man and if people found out the amounts of money he'd promised to pay her to go through with it neither of them would come out looking particularly good—but surely he wasn't planning on not telling his friends that he was married to her.
Judging by Hugo's reaction it sounded as though Xavier had kept the marriage a secret from all of his friends too, apart from Russell, of course, who had written up the legal documents for them to sign and so was clearly a necessary confidant.
She turned to look at Russell now, who was standing quietly beside her, and wondered what he thought about the whole strange undertaking.
He must have felt her gaze on him because he turned to look at her and asked, 'How well do you know Xavier?' as if he'd been wondering the same things that she had. From his expression she suspected he was actually a bit concerned about what his friend had just done.
'Uh, hardly at all,' she said with a pained grimace. 'We've not spent a lot of time together because we've been too busy sorting out our personal situations before the ceremony. It's all been a bit of a whirlwind to be honest.'
Russell nodded thoughtfully, then gave her an encouraging smile. 'Listen, I know he comes across as a bit distant sometimes, but he's a good guy. He's just been through a lot during his life, that's all, and it's made him a bit hard to reach. Emotionally, I mean.'
'Really? What happened?' Soli asked, intrigued by what Russell might have to tell her. If Xavier was going to keep her at a distance, talking to his friends would probably be the only way to really get to know more about him.
Russell looked uncomfortable. 'I should probably let him tell you all that himself. It's not really my place.' He rocked back onto his heels and crossed his arms. 'You've got a year to get him to lower his barriers, after all,' he said, his smile a little strained now. 'I'll just say this—' he paused, as if searching for the right words to use '—you should keep in mind that he's unlikely to want to commit to the marriage fully. What I mean is, don't get your hopes up that he'll let it become a real relationship. I don't think he's really cut out for that sort of commitment. Not any more.'
Soli nodded, but couldn't help but frown, feeling a bit overwhelmed by it all now. 'Don't worry, I'm not looking for it to turn into a real marriage either,' she said assertively, though something in the back of her mind let out a small squeak of protest.
No. She'd be a fool to even consider that happening. She was only doing this for the year, then she'd be in a much better position to commit her heart for real. With someone who truly cared about her.
The momentousness of what she'd just done suddenly hit her full force, sucking her breath away.
She was married and about to move in with a man she barely knew.
Adrenaline surged through her body, making her hands shake.
'A year suddenly feels like an awfully long time to live with a stranger,' she blurted.
Russell gave her a reassuring smile. 'Don't worry, he'll look after you. He's nothing if not a gentleman.'
Xavier reappeared through the doors with her bags then and came over to thank Russell for his help, giving him a friendly slap on the back, before turning to face her.
'Are you ready to go home?'
Home.
His home though, not hers.
'Yes, I'm ready when you are,' she said, summoning a smile, which she hoped wouldn't give away how nervous she was.
Xavier nodded, not seeming to notice her jitters, and set off at a brisk pace with her bags to where he'd left his car. He loaded her cases into the boot while she slipped into the passenger seat.
The car smelt wonderful: of new leather and Xavier's distinctive scent, a mixture of the aftershave he wore and a musky, masculine fragrance all of his own. She'd been hyper-aware of it in his office when she'd first met him and it had haunted her ever since, the olfactory memory appearing in the air at random moments, though she'd known she was only imagining it. Her mind was good at playing tricks on her like that. It had done the same thing after her father died, conjuring his scent at odd moments, bringing with it a surge of such painful grief she'd often been immobilised by it.
But this definitely wasn't a time for immobility. She needed to make good on this opportunity and she was determined to do everything in her power to make this deal work out well—for both of them.
Sensing Xavier needed a few minutes to process what had just happened too, Soli stayed quiet as they pulled away from the kerb and stared out of the window, watching the busy London streets slip by.
When he still hadn't said a word to her as they began to drive through Hampstead Village towards the road where he lived—where they lived—she found she couldn't stand the silence any longer. Turning to look at him, she experienced a wave of concern when she caught sight of his rigid profile.
'Is everything okay?' she asked quietly. 'I'm sorry if I did something wrong back there. I thought Hugo was a friend of yours so it'd be okay to introduce myself. Surely you weren't hoping to keep me a secret for the next year.' Pausing, she took a shaky breath. 'Were you?' She laughed nervously, concern creeping over her skin as she considered the possibility that she'd hit upon the correct answer.
'No, no, of course not, it's fine,' he said, but his tone wasn't exactly convincing. 'I hadn't really thought about how I'd handle telling people about us, so it caught me off guard, that's all. But I think he bought the whirlwind marriage thing.' He turned to look at her now and his set expression softened a little. 'No point dwelling on it though. It's done.'
'No. Okay.' His answer hadn't done much to calm her nerves but she decided to push her concern to the back of her mind. He was right—there was no point worrying about it. She'd figure him out eventually.
They'd need to spend some quality time together this week if they were going to look like a convincing couple at Hugo's party next weekend though. The last thing she wanted was to put her foot in it again with his friends. She hated the idea of making a fool of herself in front of them. If she and Xavier weren't careful something like that could potentially cause resentment and tension between them, which would make for a really uncomfortable home life. She really didn't want that. Not if they were going to have to live with each other for the next year.
After driving along the long, wide road locally referred to as 'Millionaire's Row' she'd expected Xavier's house to be impressive, but as they swung in through the automatic gates at the end of the driveway, which magically opened for his car, the true magnificence of the place struck her like a blow to the stomach. Built in the arts and crafts style, it loomed above her like an enormous geometric citadel, its two wings standing like sentries either side of the grand entrance.
'Home, sweet home,' he said, turning momentarily to raise both eyebrows at her as he pulled the car up to the front of the house, then turned off the engine. 'I'll grab your cases, then I'll show you around.'
Taking a moment to get another swell of nerves under control, she watched him get out of the car and take her bags out of the boot, then dragged in a deep, steadying breath and got out too, following him to the front door, which he was opening with a swipe card.
It was like walking into another world as Soli took her first step into the house and she let out a gasp of wonder.
'You weren't expecting me to carry you over the threshold, were you?' Xavier asked gruffly, possibly mistaking her stunned awe for upset as she stood there, gazing around the cavernous, marble-floored entrance hall with wide eyes.
'No, of course not,' she said, giving him a reassuring smile before returning her gaze to the dark wooden-banister staircase, which drew the eye upwards towards an ornate mullioned window, its many panes of glass winking in the late-afternoon sunshine. Looking at it, she wouldn't be surprised to find this one room had the same square footage as her entire café.
'Wow. I can see why you wouldn't want to lose this place. It's spectacular!' she said, turning to flash him an impressed expression.
He glanced around him as if checking out what she meant, then gave her a taut smile back.
'It's been in my family for nearly a hundred and fifty years, but I've only had the privilege of living here for the last four—since my great-aunt was taken into hospital after her first stroke.'
'That must have been hard. Coming to live here on your own when she was so ill.'
He shrugged but didn't say anything. There was a glimmer of sadness in his eyes though, she was sure of it.
'Have you done much to it?' she asked, sensing his intention to keep the subject on a non-emotional level.
'Hardly a thing, which was great for me because I could just move straight in.' He gesticulated around the large, elegant entrance hall with its neat marble-topped table and large vase of fresh flowers sitting invitingly in the middle of the space. The subtly coloured walls were hung with striking pieces of modern art, and there was a huge gilt-framed mirror on the far one which reflected their images back to them.
'She had really good taste and a love of interior design, so kept up with all the trends. She was always poring over those house and garden magazines,' he said with a faraway look in his eyes, as if remembering her fondly. 'I'm sure she would have been an interior designer if she'd had the chance but my great-uncle didn't want her to work. He was pretty traditional like that.'
'Right. Wow.' She couldn't imagine a world in which she wouldn't be allowed to work. She'd be bored to tears.
'Let me give you the tour,' he said, already moving towards one of the large mahogany doors that stood open to the right of them.
He guided her around the frankly massive ground floor: through the sitting room with its classy antique furniture, the library with its shelves stuffed with old books, the snug with a huge widescreen TV on the wall and a squashy-looking sofa facing it, and then on to what he called the morning room, which looked as though no one ever used it. She guessed the William Morris wallpaper in there was original, due to its slightly faded look.
She couldn't help but watch Xavier closely as he walked through the rooms ahead of her, his broad back straight and his long-legged gait a little tense. He intrigued her. Why was such an attractive, successful man living here alone? Maybe it had something to do with the not-believing-in-love thing.
Best not to think about that, though. She didn't want to get herself in any kind of emotional tangle. She had enough to deal with right now.
He then led her towards the back of the house, where there was a fully equipped gym, and on through another frosted glass door leading to an indoor swimming pool, which was surrounded by green-leafed plants in pots standing against the beautiful mosaic-tiled walls.
'You can use this any time you like,' he said, waving his hand at it as if everyone had one and it wasn't anything special.
A little bubble of nervous excitement, that had begun to form in the pit of her stomach as soon as she'd entered the house, rose up to her throat and tickled her tongue.
This incredible place was going to be her home for the next year.
They ended the tour of the main house in a huge kitchen diner, which was the most well-worn looking place in the house. Even so, she guessed the oak kitchen cabinets and marble-topped work surfaces would have cost a pretty penny.
This room was clearly the heart of the home and Soli immediately felt much more relaxed in here. The rest of the house was beautiful, but it had been a bit like being shown around a stately home where you weren't allowed to touch anything.
She could imagine spending lots of time in this room though, making meals for them both and perhaps baking her locally famous cakes and biscuits for Xavier to sample. She'd welcome the chance to impress him with her cooking skills. It would make her feel less insignificant in the face of his overwhelming prowess.
He'd leant back against a scrubbed oak table in the middle of the tiled floor as she looked around, and she glanced over at him, wondering how many times he'd sat there to eat in his lifetime. She could imagine him as a bright-eyed, but serious, little boy with a wicked grin, when he chose to deploy it. Not that she'd seen any evidence of it so far. Any smiles he'd given her had seemed perfunctory and lacking in any real emotion.
What must he have gone through to not have any warmth in his smile? The thought of it made her inordinately sad, especially when it occurred to her that he might well have lost his spirit when he was a little boy.
But perhaps that wasn't the case. He seemed to have genuine love and affection for his great-aunt and clearly adored living here judging by the reverent tone he'd used when showing the rooms to her.
'Did your great-aunt have any children?' she asked, thinking what a wonderful house this would be for games of hide and seek. You could probably go for hours without being discovered with all the nooks and crannies available.
'No. I think she wanted them but it never happened. My great-uncle died before I was born so I never met him, but I used to spend a lot of time with Aunt Faith and I think she considered me the child she never had. She always invited me here during my holidays from boarding school.'
'And your parents were okay with that? Didn't they want you at home with them?'
He let out a low snort. 'They didn't mind at all. They're not exactly "kid people".'
'Oh.' The sharp edge of tension in his voice disturbed her. Was he telling her that his parents didn't want anything to do with him? How heartbreaking.
'Anyway,' Xavier said loudly, making her jump, 'let me show you the room you'll be staying in. Part of the ground floor was converted into a bedroom for my great-aunt to live in, but wasn't used because she had the second stroke before she could move into it.'
She followed him out of the kitchen and back to the entrance hall, suspecting there would probably be a lot of Xavier suddenly changing the subject when things started to get too personal for his liking—which would be frustrating, considering she needed to get to know this enigmatic man a lot better in a very short space of time if they were going to come across as a convincing couple.
'It's down here,' he said, guiding her along a hallway towards the back of the house, then through a door with its own mortice lock and into a large, airy bedroom.
So her bedroom was to be downstairs? As far away from Xavier's as possible, perhaps. Not that she had any right to question this. It was his home after all and she was, to all intents and purposes, his guest.
Like the kitchen in the main house, the bedroom was decorated in a warm, homey style, which immediately made her feel comfortable. There was a queen-sized bed against the wall on the far side and the rest of the room was kitted out in tasteful modern furniture, which, she suspected from its pristine gleam, had never been used before. Her heart fluttered as she realised there was a walk-in wardrobe. She'd only ever had half a small wardrobe at home, where she'd shared a room with Domino.
'You'll need to put some of your things, like toiletries and clothes, in my room too, just in case one of the solicitor's people drops round without giving us any notice and goes snooping. We don't want to give ourselves away by overlooking details like that,' Xavier said, crossing his arms, making him seem even more intimidating than usual.
The thought of being caught out like that only increased Soli's anxiety about them not knowing each other well enough yet. What if she had to answer questions about him that she didn't know the answers to? He could potentially lose his inheritance if the solicitor didn't believe they were a real couple, which would mean their deal would fall through and that could signal the end of the café.
'I can't lose this place, Soli, and I'm definitely not going to let it go to my money-grubbing cousin because we messed up the small stuff,' Xavier said, echoing her thoughts.
'Okay. No problem,' she said, trying to sound reassuring. She'd do everything in her power not to let that happen.
'Good. Well, now you've had a look around I'll bring your cases in here and you can get settled in.'
Following him out of the bedroom and back to the grand entrance hall, she tried not to let a feeling of being on the very edge of control overwhelm her.
She needed to focus on the positives, such as this amazing place being her home for the next year.
It would be so exciting to be here.
Or terrifying—depending on which way you chose to look at it.
Xavier seemed like a good guy though, if a little cold and reserved. Everything she'd found out about him had been positive, she reminded herself. Especially the things she'd read about his business practices. And her father wouldn't have rented a property from a shyster after all; he'd always been a very cautious and thorough businessman himself.
'Why did your great-aunt want you to be married in order to inherit this place?' she blurted as they reached the front door. It had been playing on her mind as he'd shown her around. 'It seems a little extreme in this day and age.'
He turned back to face her with a grimace. 'Yes, well, my great-aunt had very traditional values. Her marriage was arranged by her family and she stayed married to my great-uncle for forty-three years, until he died of a heart attack. I think she had some romantic notion that if she forced me into getting married I'd end up the same way she did. Blissfully happy.' He pulled a face.
'Not convinced, huh?'
'Not one bit.'
'Shame.'
He frowned. 'What do you mean?' From his tone she suspected he was unnerved to hear her talking about marriage in such a positive way. Perhaps because he was worried she might become more attached to the idea of being his wife than he was comfortable with.
'Don't worry,' she reassured him quickly. 'You're really not my type. When I get married for real it'll be because I'm in love with my partner and I want to spend the rest of my life with him. That won't happen with us.'
He continued to look at her in that unnervingly intense way he had, as if he was trying to read her innermost thoughts and catch sight of any lies she might be telling him. She stared boldly back, trying not to think about how devastatingly attractive he was.
'I promise you, I will not want to stay married to you after the year is up,' she reiterated firmly. 'I've got too much going on in my life to be a wife and mother right now.'
She was sure she saw him flinch when she said that, but before she could say anything more he nodded curtly and said, 'Anyway, you'll need these for the doors.' He pulled a plastic key card and a key for her apartment out of his pocket and handed them to her.
'So you're not planning on keeping me locked up inside all day, then?' she asked in as jovial a tone as she could muster.
'Of course not,' he said, waving away her words as if they were completely ridiculous.
Her skin prickled as she remembered how reluctant he'd been to introduce her to Hugo as his wife earlier, but she bit her tongue. She really didn't want to have a row with him on their wedding day. Not that it seemed they were actually going to celebrate it in any way.
This was confirmed when Xavier said, 'Well, I've a lot of work to do today so I'll let you get settled in. There's food in the fridge or takeaway menus by the phone in the kitchen if you want to eat in tonight.'
Disappointment trickled through her. 'We won't be eating together?'
'Not tonight. I need to deal with something that's just cropped up at work right away.' He gestured towards his phone. 'I'll put your bags in your apartment and catch up with you tomorrow.'
'No, don't worry, I can take them.'
He paused, then nodded distractedly, and she watched him walk away and mount the stairs, heading up the wide staircase to the top landing.
She suddenly felt very small and alone in the huge, dark house.
Looking down at the sheath dress she'd made especially for the ceremony out of one of her mother's old dresses, she felt a heavy sense of trepidation sink through her.
No, Soli, don't let it get to you.
Why she'd thought she needed to look good for this farce of a wedding today, she had no idea now. But it had been important to her to make an effort, even if Xavier hadn't appreciated it. He'd not said a word about how she looked.
She hadn't sewed her own clothes for a very long time, but being strapped for cash and not wanting to waste money on buying a proper wedding dress, even a second-hand one, she'd decided to make her own. She'd worked for three nights straight on it and was really pleased with the results. Whilst working on it she'd remembered how much she'd enjoyed designing and making her own clothes before her mother had become too ill to look after herself and her father had died, requiring her to step into his role as carer, parent and breadwinner. Experiencing that had made her appreciate just how hard he'd worked to keep them all in the lifestyle she'd taken for granted. She wished fervently now that she'd had the opportunity to tell him how grateful she was to him for providing that for her.
She hoped he would have been proud of her for what she was doing here—making sure that her sister and mother were well looked after.
The way she needed to look at it was that Xavier was providing her with a unique opportunity to set them all up for the rest of their lives.
All she had to do was make sure she didn't do anything to jeopardise it.
CHAPTER FIVE
Frustration (UK)/Trouble (US)—roll a six before you can make a move.
THE NEXT MORNING Xavier came down to the kitchen at seven a.m. to find to his relief that it was empty of his new wife. He'd been hoping that Soli wasn't an early riser and wouldn't expect to have breakfast with him so he could continue with his usual morning routine of sitting at the kitchen table and reading the news on his tablet whilst sipping his first cup of coffee of the day in peace.
It looked as if he was in luck.
As he set up the coffee machine, he noticed some cake tins and spatulas and a few bags of ingredients on the worktop. The sight of them sitting in what he thought of as his personal space sent a tingle of annoyance through him.
Telling himself to relax, he tamped down on his irritation, knowing he was going to have to get used to sharing his house with Soli for the next year and getting uptight about a few pieces of kitchen equipment lying around wasn't a good way to start. Anyway, it needed to look as if she lived in this house, he reminded himself, so having a few of her things scattered around would actually be a good thing.
He was just pouring the coffee into a mug when there was the sound of footsteps behind him and he spun around to see Soli standing in the doorway wearing a slouchy pair of pyjamas with a cartoon character on the front and her hair wild and sticking up around her head.
'Morning,' she said, hiding a yawn behind her hand. 'You're up bright and early.'
'I always leave at this time,' he said, averting his gaze as the idea of seeing her as she'd just rolled out of bed suddenly felt way too intimate.
'So you don't have time to have breakfast with me?' she asked, moving towards the kettle, which she flicked on to boil.
'No, sorry. I need to leave in a minute.'
'Oh, okay.'
She looked disappointed, but he pushed aside the sting of guilt this brought about. He couldn't just change his routine to fit in with her. She'd need to work around him.
As if she'd sensed this, she leant back against the work surface, smiled at him and said, 'Perhaps we could spend some time together this evening instead? It'd be nice to get to know you a bit better since we'll be sleeping under the same roof for the next year. We could spend a bit of time drawing up a list of dos and don'ts for the relationship.'
He frowned as the uncomfortable reality of having someone at home waiting for him every night struck him. 'Yes, of course, but perhaps not tonight. I have a really heavy day at work and I'll want to relax this evening. Once my workload's calmed down a bit we'll have plenty of time to do that.'
'Perhaps we could just play a board game or two, then. It'll be a fun, unpressurised way to learn more about each other,' she suggested. 'They're great icebreakers and it'll give us something else to concentrate on so we can chat freely. Our café is a popular destination for first dates precisely because of that. At least it used to be.' A frown flickered across her face. 'I'll cook us a light meal and we can play afterwards?' she suggested brightly.
His phone had beeped for the third time in as many minutes and he plucked it out of his pocket distractedly and glanced at the screen, seeing with a wave of concern that it was a message from his financial director. There must be an urgent issue for him to get in contact before the working day had begun. He glanced up from the screen to see Soli was looking at him with a questioning expression. 'Er, yes, okay. I should be back by about eight o'clock,' he said, wondering what could have happened for Rob to try him three times already.
'It'll be useful for us to know what makes each other tick,' she added.
'Yes. Quite right. We'll need to put up a good front at Hugo and Veronica's party and it'll be good to be prepared for that,' he said with a sigh, running his hand over his jaw as the idea of it sent a twinge of frustration up his back.
Would they really believe he'd married someone like Soli of his own free will? She really didn't dress, or act, like the type of woman he usually dated.
'I've been thinking about your question about what you're going to do during the day when you're not working at the café.'
'Oh, yes?'
'I'm going to arrange for a monthly stipend to be put into your bank account which you can use to go shopping for new clothes and book hair and beauty appointments and the like. Those sorts of things ought to keep your free days busy enough and they'll be practically expected of a woman who's married to me and not working full-time, so it'll fit our story.'
She lifted a hand to smooth down her wild curls, then adjusted her pyjama top, her brow creasing into a bewildered sort of frown. 'Okay. Well, thanks.'
Feeling satisfied with this act of generosity, he cleared away his empty mug and gave her one last nod before striding out of the house and setting off for the office, feeling a strange sense of relief at having something to focus on outside of Soli and his brand-new marriage.
* * *
Soli let out the sigh of frustration she'd been holding in when she finally heard the front door slam closed behind Xavier.
Well, that had been an incredibly frustrating meet-up.
Awkward didn't even cover it.
It was clear he wanted them to look like a convincing couple at the party, but he seemed reluctant to actually spend any time with her.
Work, apparently, was going to take precedence.
Her heart sank at the anticipation of the fight she might have on her hands to get his full attention.
She shook off her worry. She'd find a way to make it work. Her family was relying on her and there was no way she was going to let them down now.
Pouring herself a reviving cup of coffee, she mulled over what he'd said to her just before he'd left. She really hadn't expected him to give her even more money—not that it wasn't welcome. She'd been acutely aware as she'd hung up her clothes in the wardrobe in her room that the sort of things she wore—mostly high street store outfits or things she'd picked up from a great little vintage clothes stall in Camden Market—probably wouldn't look quite right for someone married to Xavier McQueen, but his insinuation that she wasn't the sleek, sophisticated-looking woman he'd hoped for in a partner had still stung a little.
Not that she couldn't fix that if she splashed the cash around a bit.
It was going to feel pretty strange spending his money on frivolous things like that though, and she was going to have to get over that. If he was happy to give it to her, she should just be grateful for it.
She took a breath and straightened her spine, imagining herself into the role of the lady of the house.
The first thing she needed to do today was plan what she was going to make for their 'getting to know each other' meal. She wanted it to be something that looked as if she'd made a bit of effort, which of course actually meant making a lot of effort. Despite the fact they were only pretending, for the sake of her pride she wanted to be as good a partner as she could be.
Perhaps she could do steak with a peppercorn sauce and some lovely fresh seasonal vegetables. And Dauphinoise potatoes. Her sister loved it when she made that dish—which wasn't often because it was pretty labour intensive.
Yes, something like that perhaps. And she'd make a dessert from scratch too. Something with lots of fresh fruit, like a summer pudding.
Her spirits rose again.
While she was out shopping for all the ingredients she could pick up a couple of board games for them to play this evening too. Games that might lead them to interesting discussions and help them to get to know each other a bit better.
With a sense of positivity and purpose surging through her now, she sat down at the table and began to make a list, planning a wonderful evening of food and entertainment for her and her new husband.
It would be great to finally feel as if she was on top of things and acting like the kind of daughter her father always wanted to have.
Yup, she was a grown-up now and determined to prove to Xavier that he'd made a good choice in her and that it would be money well-spent.
He was going to be so pleased he'd married her.
* * *
It was ten thirty before Xavier made it home that evening, after having to deal with the crisis at work that had kept him, his PA and his financial director in the office, scrambling to close a property deal that they'd been working on for the last three months.
It had been a taxing day, but Xavier was pleased with the way it had gone in the end. He felt buzzed with success as he let himself in through his front door and made his way across the entrance hall towards the back of the house.
Striding into his kitchen, he experienced a shiver of disquiet as something niggled at the back of his brain.
He'd not even had a chance to let Soli know he'd be back late; in fact, he'd been so engrossed in what he was doing he'd not noticed how late it was until his PA had jokily pointed out they should eat before all the takeaway outlets shut for the night, but he'd figured it wouldn't matter. Soli had plenty of things to entertain her here in the house and she seemed like the resourceful type.
After flicking on the kettle, he leant back against the kitchen counter and took a moment to look around the kitchen. There was something different in here, he was sure of it. It smelt different. A bit like the French restaurant he loved to go to on the bank of the Thames in Southwark. Garlicky and delicious.
The kettle boiled and he made himself a cup of tea, lifting the teabag out with the spoon after swishing it around in the boiling water for a few seconds. He never had the patience to let tea brew properly. When he lifted up the lid of the food-waste bin to dispose of the teabag, the garlicky smell grew even stronger and he paused, staring down inside the bin's depths. It looked as though there was a whole meal in there—what looked like Dauphinoise potatoes and cooked vegetables. What was Soli doing throwing so much food away? How wasteful.
And then it struck him and his stomach turned over with unease. She'd made dinner for him and he'd not turned up for it. He hadn't even phoned her to let her know he'd be too late to eat with her.
He snapped the bin lid shut and stepped away from it, feeling a strange mixture of self-righteousness and guilt. It wasn't as though he'd deliberately not turned up for the 'getting to know each other' dinner, he'd just forgotten about it. Work had had to take precedence today; it had been imperative to get on top of the problem before it had snowballed.
He'd explain all that to her tomorrow and apologise for missing dinner. There was no point in feeling guilty about it though. They had plenty of time to get to know each other and she'd have to get used to him having to work late without giving her any notice. That was how his life worked and he wasn't about to change it for someone who was fundamentally in his employ.
She'd understand that.
Frankly, he was paying her a hefty chunk of money to understand and accept that.
With that assuring thought in mind he added a splash of milk to his tea then took it through to the sitting room to drink it, determined to enjoy a few minutes of his evening before he had to retire to bed.
* * *
When Soli turned up in the kitchen at seven o'clock the next morning, hoping to catch Xavier before he went to work, she was frustrated to find he'd already been and gone.
Had he done it on purpose so he didn't have to see her?
She'd been disappointed and a bit hurt when he hadn't come home in time to eat the food she'd spent so much time and energy on, but she'd tried not to take it to heart. She'd decided to give him the benefit of the doubt and assume he'd just forgotten about it.
And her.
The uncomfortable twisting sensation that she'd experienced when she'd finally accepted he wasn't coming home last night reappeared.
She needed to get a grip. There were bound to be a few misunderstandings until they got to know each other better. He was a very busy man who ran his own company, so of course he was going to be working long hours and would be prone to forgetting she was at home, waiting for him.
But the dissenting voice in her head whispering that he was deliberately avoiding her wouldn't shut up.
She felt wired and restless now as if there was something portentous in the air.
Perhaps it was the ghost of Xavier's great-aunt who had come to check up on the state of the marriage and was most displeased with what she saw.
Not that I blame you, Aunt Faith—I'd be pretty annoyed too if I found out he was playing the system to get round my wily attempt to force him to emotionally connect with life.
By eight o'clock that evening she'd just about given up hope of him appearing for dinner again and was about to start making enough food for one when she heard the front door open, then slam closed.
Heart thumping hard, she waited with bated breath to see whether he'd come to try and find her in the kitchen.
When he appeared in the doorway he seemed almost shocked to see her, as if he'd completely forgotten he had a wife.
'Soli, hi,' he said, frowning at her.
Steeling herself against a wave of disquiet, she said, 'Hi. How was your day?'
Oh, man, why did she feel so awkward talking to him? Perhaps because he was still frowning at her as if wondering what she was still doing here.
'It was fine,' he said distractedly, glancing around the kitchen.
'You're back late again.' She forced herself to smile graciously then waited to see whether this would trigger an apology for missing the dinner she'd made him the night before.
He ran a hand over his eyes and let out a sigh. 'Yes. It's not an uncommon occurrence.'
'I see.' So she wasn't getting an apology, then.
'How was your day?' he asked instead.
He still wasn't looking at her though; instead his gaze ran over the kitchen surfaces where she'd left some of her cake-making equipment.
'Pretty good, thanks,' she replied, pleased that he'd at least asked about her even if he wasn't entirely engaged in the conversation.
'Are you planning on making stock for the café here?' he asked abruptly, the terseness in his tone shooting a shiver of discomfort down her spine.
'No, this is just some of my baking stuff from home. You don't mind if I keep it in here, do you?'
He seemed to seriously consider this request for a second or two as if deciding whether he'd be prepared to share the space with her. 'Sure. Why not?' he said eventually.
'Thanks,' she said, slightly discomforted, hoping he wasn't going to be this possessive about the rest of the house. Clearly he wasn't used to having someone invading his territory.
Perhaps a goodwill gesture would make him more tolerant of her presence here.
'You know, I make a mean chocolate fudge cake. It's a particular favourite in the café. I can make one for you, if you like. It's yummy.'
She looked at him expectantly, hoping for some spark of interest.
'No. Thanks. I'm not a big fan of desserts.'
The pleasure she'd initially felt at the thought of spending time with him tonight was rapidly draining out of her.
'Oh. Okay.' She forced an undaunted smile, despite the sting of rejection she felt. 'No problem.' She swallowed. 'Have you eaten supper? I can make us some chilli. I've got all the ingredients right here.'
'No. Thanks,' he said again. 'I had a late lunch and I've got a few calls to make to the US so it'll be a while before I'm done. You go ahead and eat without me.' He gave her a curt nod, then turned to leave the room.
The last dregs of her optimism drained away, leaving her totally deflated.
His businesslike attitude towards her was seriously denting her excitement at living here with him. It was becoming starkly clear that he didn't want to spend any time with her and that he was deadly serious about keeping their relationship emotion-free.
Turning to stare down at the chopping board and the pile of ingredients for the food she didn't really feel like eating any more, she was just about to pick up the knife to start chopping enough onions for one portion, determined not to let him totally disturb her equilibrium, when she realised he was still in the room.
When she turned to look at him she saw he was leaning against the doorjamb, watching her with a thoughtful expression on his face.
'I'm sorry I forgot about our "getting to know each other" dinner last night,' he said when he noticed her questioning eyebrow.
He didn't exactly sound sorry though. In fact, from the tone of his voice she got the impression he was actually quite irritated about having to explain himself.
'There's a good chance I won't be around at regular times in the evenings so don't worry about making food for me. I'll eat when I get in,' he went on when she didn't react right away.
So she was going to be eating on her own every evening? How horrible. She hated the thought of sitting in this huge empty house all on her own night after night, not having anyone to talk to. She was so used to being around people all day in the café and then chatting to her mum and sister over their family meals it made her spirits sink to think she'd miss out on all that life while she was here.
For a whole year.
Her stomach knotted at the thought of it.
'You know, this house is such a big, lonely place for one person. Perhaps your great-aunt wanted you to get married and raise a family here so you're not on your own all the time,' she muttered, unable to keep her agitated thoughts to herself any longer.
She saw his shoulders stiffen and the air felt suddenly leaden with tension.
'Yes, well, she'll have to be for ever disappointed in me for not having children, I'm afraid,' he said tersely.
'You don't want kids?' She was surprised to hear that.
'No.'
'Why not?'
'Because they're an inconvenience. They mess up your life.'
The bluntness of his tone bothered her.
'You really believe that?'
'Yes.'
But she could have sworn she saw a hint of uncertainty in his eyes. Just for a second.
'That's sad.'
'Sad? Why?' He was scowling at her now as if she was talking utter nonsense.
She shrugged. 'I don't know. I guess I can imagine you being a great dad.'
He looked at her steadily for a couple more beats and she got the feeling he was trying to decide how to handle this without it turning into a big deal.
'Well, thanks. But I don't think I'm the sort of person who could give a kid the kind of love they need.'
'Because your parents didn't give it to you?' she blurted without thinking, her frustration at his aloofness getting the better of her.
Anger flashed across his face, but he covered it quickly. 'I'd rather not talk about my parents.'
'Okay. Sorry.' But she didn't feel sorry, she felt annoyed with him for being so obstructive. Was he going to treat her like this for the entire year? Perhaps he'd hoped she'd squirrel away in her downstairs bedroom like an animal in a cage, never asking any awkward questions or getting in his way and only showing herself when he summoned her.
He must have seen the irritation on her face because he blinked in surprise. 'Anyway, I have a lot to do this evening,' he said, to her mounting ire. 'I'll leave you to it.'
Before she could utter another word, he strode out of the room, leaving her alone.
Again.
CHAPTER SIX
Articulate—use words cleverly to figure out what your partner is trying to tell you.
THE NEXT FEW days followed a similar pattern, with Soli barely catching Xavier for two minutes over his morning coffee before he left for work and then him coming home at random times in the evening, regularly texting her to let her know he wouldn't be back in time for supper and to eat without him.
As her anxiety grew about how they would fare at the impending party, her concentration levels waned, making it increasingly difficult to concentrate on the marketing she had planned for the café.
So when Friday night came around she crossed her fingers that Xavier would be back at a reasonable hour and finally willing to give her a bit of his precious time so she could finally put her mind at rest.
But when he walked in after nine o'clock he only poked his head into the sitting room, where she'd been trying to distract herself by half watching TV and half checking social media, and said a curt 'hello' before excusing himself to go up to his office.
Having sat on her own, fretting, as she drained a large glass of wine, Soli suddenly found she'd had enough of being ignored.
Springing up off the sofa, she ran into the hallway, where Xavier was already mounting the stairs, his long legs making short work of the winding staircase.
'Xavier!' she shouted, determined to get his attention before he disappeared on her again.
He stopped climbing and turned to look back down on her with a frown of surprise.
'Look,' she said, throwing up her hands in exasperation, 'I appreciate you're busy, but how am I supposed to convincingly pretend I'm your wife—someone that loves you and knows you intimately—if you won't even talk to me?' She held up both hands, palms forward. 'Can you please just give me half an hour of your time? Is that really too much to ask?'
He was looking at her now as if completely stunned by her outburst.
'I'm sorry to snap,' she said quickly, worried that she'd gone too far in her agitation, 'but I've reached my limit of pretending not to mind you treating me like a piece of furniture.' She tried to smile but her mouth refused to play ball. Instead, to her horror, her muscles began to tug downwards at each side and her throat constricted painfully as she fought back tears.
But she wasn't going to cry. No way. She was going to be an adult about this.
'If you really want us to appear like a proper couple you're going to have to let me in a bit,' she pointed out in a measured tone.
A muscle in his jaw flickered, but after a few seconds of seeming to seriously contemplate what she was saying he gave her a curt nod of agreement.
'Okay, then. And how do you propose I do that?'
She moved to the bottom of the stairs and leant on the newel post, looking up at him, a mixture of excitement and relief surging through her. 'There's this game I've played at a party which helps you get to know the other guests better. First of all you have to look into each other's eyes for three minutes—to begin to feel more comfortable with that person in a physical sense.' She paused, gauging his reaction.
'Go on,' was all he said, walking down one step towards her.
'Then we ask each other a set of questions which are meant to give us some insight into each other's lives—how we see the world, what makes us feel good and bad. Personal self-disclosure, I think it's called.'
'Right.' He seemed less certain about this, but he hadn't said no, so she decided to forge on.
'The shared vulnerability is supposed to make us feel closer and help us trust each other more. I appreciate this is a bit of an ask at this early stage, but I think it'll be a great way to get comfortable with each other pretty quickly, especially since we don't have a lot of time to do that, what with you being so busy at work.'
She tried to keep her scepticism out of her voice about how busy she thought he really was, as opposed to how he'd probably been using it as an excuse to avoid her, but from the twitch in his eyebrow she could tell she'd failed.
'Okay, Soli, fine.' He rubbed his hand over his jaw. 'I guess we should do this now since Hugo and Veronica's party is tomorrow.' He walked down the rest of the stairs to where she stood. 'Where do you want to do it?'
'The sitting room would be good. Somewhere we can sit comfortably.'
'Okay. Lead the way,' he said.
She ignored the weariness in his voice, determined not to give him an excuse not to go through with this. Hopefully once they started communicating properly he'd start to relax around her.
In the sitting room she chose the three-seater sofa and sat down on it, patting the cushion next to her to encourage him to sit close by.
He followed her suggestion and turned to face her, laying his arm along the top of the sofa and looking anything but excited about the prospect of doing this.
'Okay, I'm going to set the timer going, then we have to sit looking at each other's faces, particularly the eyes, until the beeper goes.' She shifted in her seat, trying to ignore the wave of heat rising up her neck at the thought of actually doing this now with Xavier.
'Are you ready?' she asked him.
'As I'll ever be,' he drawled.
'Okay, then, I'm starting the timer—now,' she said, tapping on the screen of her phone.
Turning to face him and settling her body into a comfortable position, she fought back a ridiculous urge to giggle, knowing it would spoil the exercise before they'd even started, and took a breath, locking her eyes with his.
He nodded as if resigned to letting this happen and looked back at her with that shrewd, intelligent gaze of his.
Soli swallowed, suddenly acutely aware of every breath she took, every facial muscle she moved. It was intensely intimate, having him looking at her so thoroughly without a break, but she was determined not to look away.
All the hairs on her body stood up and a hot tingle rushed over her skin as the seconds ticked by slowly, but she still didn't look away.
He really was an immensely attractive man, she mused as she gazed at his olive skin with its five o'clock shadow and his long, dark lashes that almost brushed his cheeks every time he blinked.
She became aware of some strange feeling building inside her, something that made her pulse jump in her throat, but before she had a chance to figure out exactly what it meant the timer went off, making them both jump.
'Sorry, I didn't realise I'd set the volume that high,' she said, flustered and grabbing for the phone with fumbling fingers. Finally managing to turn it off, she turned back to him and shot him an apologetic grin.
'Well, that was fun,' she joked.
His mouth actually twitched up at the corner at that, which she considered a personal victory. Of sorts.
'At least we'll be able to describe each other's faces to an outsider in detail now too,' she said, acutely aware of a tell-tale wobble in her voice. 'You know, I hadn't realised your eyes had yellow flecks in them; I thought it was just a lighter shade of green. Oh, and how did you get the small scar by your lip? It's so tiny I hadn't noticed it till now.'
'I fell off my bike here in the garden when I was eight,' he said, lifting his hand to touch the scar she'd mentioned, almost absently. 'Aunt Faith bought it for my birthday and the front brakes were really fierce. I went right over the handlebars.'
'Ouch!' she said, with a grin.
'Precisely,' he agreed, his mouth twitching upwards again.
Yes, progress! It seemed the forced intimacy had opened something up a crack between them. She just needed to press her advantage now and get him talking to her some more.
'Okay, then, now we've done that let's move on to the questions. I'll ask you some first, okay?'
He began to frown, but seemed to change his mind. 'Sure. Fire away.' Shuffling back against the sofa cushions, he crossed his arms in front of him and gave her his full attention.
Soli felt herself flush again under his gaze, but tried not to let it distract her.
'What would be a perfect day for you?' she asked as a starter question.
'Hmm.' He rubbed his hand over his jaw again, but in a thoughtful manner this time.
'Well, I rarely get the chance for a lie-in, so I'd have one of those.'
Soli tried hard not to picture him lying naked and rumpled in bed. And failed.
'Then I'd have a long, lazy breakfast and perhaps a walk across the heath. Maybe have a swim in the lake and a picnic lunch.'
From the faraway look in his eyes now, she got the impression he was actually enjoying thinking this up. The thought of it warmed her.
'In the afternoon I'd go and play tennis then head out for a slap-up meal in town.'
'Wow, that all sounds great,' she said with a grin. 'And when was the last time you spent a day doing things like that?'
This seemed to stump him. 'I don't think I've ever had a day like that. I've always been too busy with work or had other social engagements.'
'Oh. What a shame.'
'Yes, I guess it is,' he said, shrugging his shoulder.
There was a small pause where neither of them said anything.
'When was the last time you sang in front of someone?' Soli asked, to fill the silence.
'Never,' he said with a shake of his head.
'Really? Never?'
'I'm not really into performing,' he said with finality to his tone.
'Oh. Okay, then,' she said, recognising his need to move on.
'How about this: if you could choose one ability that you don't already have, what would it be?'
'To predict the future,' he said with confidence.
Soli thought this was interesting. It clearly pointed to a need for complete control.
'What are you most grateful for?'
He paused infinitesimally before replying, 'My health, wealth and happiness.'
Glib, but okay.
'What would you never joke about?'
'Money.' There was no pause before that answer.
'Is there something you've always dreamt of doing but have never got round to? Tell me about it, then tell me why you haven't done it yet.'
'Hmm.' This gave him pause. 'I think I'm doing what I dreamt of. I wanted to run my own company and live in this house.'
'Okay. Well...well done,' she said with a smile. 'What's your biggest accomplishment?'
'Same answer. My company and finding a way to live in this house.' He looked particularly pleased with himself for that answer.
'Tell me about a happy memory from your school days.'
Suddenly the buoyant atmosphere seemed to drop like a stone.
'I can't think of one right now,' he said tersely, his gaze skimming away from hers now.
There was something heartbreakingly raw about the way he said this, but she didn't press it. From the way his shoulders had stiffened she got the impression he'd happily call an end to the session if she did and that was the last thing she wanted when he was finally starting to open up to her a little.
'What has been your most embarrassing moment?' she asked with a smile, hoping to flip the mood, but was a little taken aback to see his eyes harden at this. 'It can be something really silly,' she added quickly, desperately trying to rescue the lightness they'd had previously.
'Pass. I can't think of anything right now,' he said again, his tone warning her not to push it. Clearly she was treading on dodgy ground.
Okay. She could come back to that another time. She didn't want to ruin the progress they'd made. But something still pushed her to ask the next question anyway.
'What's your relationship with your mother like now you're grown up?'
The light went out of his eyes. She realised with a shiver of disappointment that she'd blown it and that he'd probably clam up completely now, but to her surprise he didn't. Instead, he hooked an arm across the back of the sofa again and looked directly into her eyes as if actively deciding not to dodge her interest in the question any more. Perhaps he was hoping she'd leave him alone if he finally gave her an answer to it.
'I don't really know her, to be honest. We have very little contact these days. She's not exactly the maternal type. I think she fell pregnant with me by accident—at least, that's what I overheard one day when my great-aunt and a friend of hers were chatting. Apparently my father convinced her to keep me, but she and I never really bonded. Not that my relationship with my father was much better. He was always being sent away overseas with work. He was a foreign diplomat. My mother often went with him but they kept me here in England at boarding school. It was for my own good, apparently, so I wouldn't feel unsettled.'
From the expression on his face she gleaned that it had actually had the opposite effect. No wonder he was so attached to this house. It seemed to be the only place he'd ever felt secure. She couldn't imagine how horrible it must have been not to be allowed to live with your family. There were so many good memories from her own childhood, she'd be devastated not to have had the opportunity to experience them. Some of them were only snatched, random moments in her memory, but they still held so much meaning for her. They'd help her grow and form as a person and the knowledge that she'd be able to come home to her family and a safe, loving environment every day after school had kept her going through her most taxing years.
'I suspect it was really because they thought I'd cramp their style if I was living with them,' Xavier went on, his eyes taking on a far-away, troubled look now. 'They were always big socialisers, according to my great-aunt...' He paused, as if weighing up whether he wanted to say the next thing out loud, obviously deciding that he did when he added, 'And not exactly faithful to each other.'
'Oh. I'm sorry to hear that. It must have been really unsettling for you,' she said quietly.
He looked at her again, his expression softer now with what she thought might be appreciation for her understanding.
'It wasn't great, but then no one's life is perfect, right?'
'True,' she said, giving him a supportive smile.
There was another heavy pause as they just looked at each other again and Soli felt a strange sort of pulse beat between them.
'Any more?' Xavier said, breaking the tension.
'Any more what?' she asked, a little shaken by the atmosphere that had formed.
'Questions,' he said pointedly.
'Oh! Yes. Okay.' Pulling herself together, she asked, 'What would you regret not having said to someone if you were to unexpectedly die this evening?'
He raised a wry eyebrow, his eyes twinkling with mirth.
'Don't worry, I'm not thinking about doing you in,' she added with a grin, then muttered, 'yet,' waggling her eyebrows in jest.
He came really close to properly smiling at that and her heart did a little dance of joy in her chest.
'Hmm, I don't know,' he said thoughtfully. 'I guess I wish I'd had a chance to tell my great-aunt how much I appreciated her taking me under her wing like she did. I don't know what would have happened to me if she hadn't.'
Soli became aware of tears pooling in her eyes. 'I'm glad you had her. She sounds like an amazing woman.'
'She was.'
Blinking away her tears and pointing at her eyes, then wafting her hands at either side of them with a strained smile of embarrassment, she asked, 'When did you last cry in front of someone?'
He frowned, but didn't meet her eyes. 'I'm not a crier.'
'Really? You never cry?'
'Not in front of other people, no.' He shifted a little in his seat and crossed his arms. 'And it's been years since I cried on my own.'
'Oh. Okay, then.' There was something so heart-wrenching about this it actually caused her physical pain deep in her chest. How awful that he didn't feel he could express his sorrow in front of someone else.
'I cry in front of people all the time,' she said with a self-deprecating grin. 'I find it cathartic. I always feel better afterwards, though sometimes I'm embarrassed by how easily I do it. I cry at anything even remotely sad,' she said, feeling tears pushing at the backs of her eyes again just from thinking about it.
His shoulders had stiffened as if he was really uncomfortable now and he glanced down at his watch, as if wanting to escape from the conversation.
This was confirmed when he said, 'Anyway, Soli, it's been an interesting exercise, but I really do have some work to do this evening, so I'm going to say goodnight.'
'Okay,' she said, watching him get up from the sofa, feeling a swell of satisfaction from getting as far as she had.
It was early days yet, but at least she knew a bit more about him now.
His emotional unavailability made more sense now she knew more about how he'd been ignored by his parents during his childhood. She felt truly sorry for the poor, lonely little boy he must have been growing up, not having a family who loved him or a steady base to come home to during his breaks from school. What must that do to a child? To not feel wanted by the people who were supposed to love you unconditionally?
It was a horrible thought.
Well, she'd make sure she did her very best to support him in the months to come. She'd need to be careful not to get dragged into an emotional quicksand where Xavier was concerned—it wouldn't be sensible to allow herself to actually fall for the guy, she reminded herself with a strange pulse of panic—but she could be a friend to him, as he'd suggested.
Yes, that was exactly what they both really needed at this juncture in their lives.
A good friend.
CHAPTER SEVEN
Would I Lie to You?—keep your cool and think quickly to win this game.
XAVIER SLEPT BADLY that night and woke up later than his usual six a.m. start.
After having that conversation with Soli his dreams had been tangled with memories of his time at boarding school—feeling isolated and humiliated when he'd been the only boy whose parents hadn't turned up to watch an end-of-term performance before they all left for the Christmas break. It had then changed to him standing in the mostly empty register office with Soli. In the dream they were reciting the lines they'd been asked to say, except it wasn't the registrar conducting the service, it was Great-Aunt Faith, who was barking instructions at them as if they were disobedient children.
As he'd turned to apologise to Soli, he'd realised it wasn't her standing next to him after all—it was Harriet.
'Really, Xavier? You had to pay her to marry you? How pathetic,' she'd sneered at him and the ugly look of utter disdain on her face had woken him with a sickening jolt.
It had been so vivid and had drawn such a strong emotional response from him he felt exhausted now, as if he'd not slept a wink all night.
Remembering with relief that it was the weekend, he allowed himself to lie in bed and read the news on an app on his phone, determinedly pushing the unsettling echoes of the dream out of his mind, before finally dragging himself out of bed for a shower then heading down to the kitchen for breakfast.
He'd just got the coffee machine set up and running when the buzzer for the entry phone at the front gate went.
Frowning to himself and wondering who the hell would be so uncouth as to turn up uninvited at this hour on a Saturday morning, he picked up the handset and snapped, 'Yes?'
'Mr McQueen? This is Samuel Pinker. I've been employed by your great-aunt's estate to visit you and your wife at home in accordance with her will.'
Heat rushed across his skin, quickly followed by a wave of cold panic that made his hair stand on end. So they were being checked up on already?
'Okay. Well, you'd better come in,' he said, forcing his tone to sound jovial and upbeat. 'Solitaire is still in bed, I'm afraid. It's a little early for her, so you'll need to give me a few moments to rouse her.'
'No problem,' the voice of Pinker said over the phone. 'I have a quick call to make so I'll park in your driveway and give you a knock in five minutes, if that suits you?'
'That would be fine,' Xavier said steadily.
'Thank you,' the man replied.
Xavier pressed the button to release the gate, then dashed out of the kitchen and down the corridor to Soli's room, where he banged hard on the door.
It was a full minute before he was able to rouse her.
'Xavier? What's the matter?' she asked, hiding a yawn behind her hand and looking a little bewildered to see him standing there.
'There's someone here from my great-aunt's solicitor's office to check up on us. You need to get dressed. Quickly,' he said, determinedly trying not to notice how appealing she looked, all bed-rumpled and sleepy.
Her eyes sprang open at that and she did a little nervous jiggle on the spot. 'Should I get dressed?' she asked, her voice wobbling with panic.
'No,' he said, thinking it would seem more realistic if it looked as though she'd just got out of bed. 'But perhaps put a robe over your pyjamas?'
'I don't have one!' she said, flustered.
'There's one on the back of my bedroom door. Go and put that on. It'll look better if you come from upstairs anyway. As soon as you hear me open the door, come down. Okay?'
She looked slightly terrified now. 'What am I meant to say to him? We haven't worked it out yet!'
'Don't worry. Let me do the talking. Just try and make it look as though you like me and find me attractive,' he said, flipping her a wry smile.
She nodded jerkily. 'Okay. I can do that.'
There was a strange, zingy tension suddenly between them, but he brushed it off, not having the time to consider what it might mean.
Two minutes after she'd dashed upstairs, the doorbell went and Xavier took a deep, steadying breath before going to answer it.
Their visitor was a portly man, with a shock of russet hair and a cheerful smile.
'Samuel Pinker,' he said, holding out a hand, which Xavier shook firmly.
'Good to meet you. Please come in.'
He guided the man into the hallway, just as Soli made a timely appearance at the top of the stairs.
The two men looked up at her as she descended and Pinker once again held out his hand in greeting as Soli reached the bottom stair.
Soli, to her credit, appeared to be totally relaxed as she shook his hand and introduced herself, giving Pinker a warm smile that lit up her whole face.
Xavier's stomach did an odd swoop as he once again thought how attractive she looked with her wild curls framing her pretty face and her cheeks pink and a little sleep-creased. The robe she'd found in his bedroom was far too big for her, but it only enhanced how feminine and delicate she was.
He had the strangest compulsion to wrap his arms around her, to protect her from Pinker's searching gaze, but he held back, not wanting to spook her and give the two of them away.
To his surprise, Soli appeared to have the same urge to touch him and he gave a small involuntary grunt of surprise as she walked straight up to him and slid her arms around his middle, pulling herself close so they were chest to chest. He looked down to see she was gazing adoringly into his eyes.
His heart did a three-hundred-and-sixty-degree turn.
Remembering the exercise they'd done only the night before, he maintained eye contact with her, immediately recalling that same close connection he'd experienced, so didn't even blink when she stood on tiptoe to plant a light, soft kiss on his mouth.
The sweet, sleep-warmed scent of her invaded his consciousness, making his senses reel and his body instinctively tighten with lust.
Ah, hell. If she pressed herself any closer to him there was a good chance this could turn into a very embarrassing moment indeed.
Luckily, she pulled away before he completely lost control of the situation and turned to smile at Pinker, who was watching them with a focused sort of smile on his face.
'Can I get you a cup of coffee, Mr Pinker?' she asked him. 'We were just about to have one.'
To Xavier's surprise, Pinker shook his head. 'No, thank you, Mrs McQueen. I only popped in so I could tick the "living together" box on the paperwork. It's very clear you both live here, what with me turning up at such an ungodly hour at the weekend and finding you in residence. I do apologise for any inconvenience caused—it's my job, you know?'
'Of course! We totally understand,' Soli said, with warmth in her voice.
'Well, I'll let you good people enjoy the rest of your morning,' Pinker said, tipping Soli a courteous nod and offering Xavier a smile. 'No doubt I'll see the two of you again at some point. Until then...' He made for the door, giving Xavier a nod of thanks when he opened it for him, and vanished outside.
Xavier waited until he saw Pinker's car pull out of the driveway before shutting the door with a sigh of relief and turning to face Soli.
'Well done,' he said, walking towards where she still stood by the stairs. 'I think we convinced him.'
Soli smiled back, looking a bit sheepish now. 'Sorry for kissing you like that without warning, but I thought it would look more authentic. It's the sort of thing I'd do if we were really married.'
'No problem,' he said, lifting a hand to touch his lips where the ghost of Soli's kiss still lingered. 'It's a good job you insisted on that eye-gazing thing yesterday—it made it a lot easier for me to look at you.'
'Thanks. I think,' Soli said, flashing him a rueful grin.
'I didn't mean...' He shook his head, irritated with himself. 'I meant it made it easier to be immediately intimate with each other.'
'I knew what you meant,' Soli said, widening her grin, evidently enjoying winding him up.
He couldn't help but smile back at her, a surge of relief-filled happiness appearing out of nowhere. Clearly he'd made the right choice in Soli for his pretend wife. Her quick reaction today gave him confidence they'd actually be able to pull this thing off at the party this evening.
'Oh, wow!' Soli said loudly.
'What?' he asked, startled.
'Your smile. It's incredible. You should do that more often.'
'Uh, thanks.' He smiled again in a show of nonchalant acceptance but her words had made a tingle shoot straight up his spine. No one had complimented him on his smile in a very long time. Perhaps because he hadn't had much to smile about.
They stood there, just looking at each other for a couple of beats, and once again Xavier had the weirdest feeling that Soli was someone he could really trust and rely on. Strangely—considering she was still practically a stranger to him—she was one of the few people he'd ever felt that about.
She looked back at him with curiosity in her eyes. And something else. Something that made his skin heat and his body tense with arousal.
He suddenly wanted to kiss her again, just to experience that same thrilling feeling of being connected.
No. No. Not a good idea.
Obviously it had been much too long since he'd been physically involved with a woman if he was contemplating messing with this precariously balanced business relationship he'd negotiated so carefully.
'You know, it occurs to me that I didn't ask anything about you when we did that getting to know each other exercise last night,' he said, taking a deliberate step back away from her. 'Want to chat now over coffee?'
The look of surprise on her face made him feel equal parts amused and guilty and a hot sort of discomfort trickled through him. Apparently she'd not expected him to take any interest in her as a person.
'Just in case we're caught out like that again,' he added quickly, not wanting her to read too much into his offer of friendship.
'Er...yes, sure. If you like,' she said, folding her arms around her middle and looking suddenly a little uncomfortable about standing in front of him in her nightwear.
'Feel free to get dressed first,' he said with a reassuring smile, 'but only if you want to. Don't do it for me.'
'Nah, I'm fine,' she said, letting her arms swing down to her sides and pushing back her shoulders. 'Let's do it now.'
He really liked her self-confidence.
So they started off by chatting about simple things like their favourite books and films and music, then they moved on to which countries they'd like to visit and why.
'So it sounds like you haven't travelled much yet,' he said as she reeled off the long list of places to visit on her bucket list.
'I've not had much chance,' she said with a sad smile. 'Ever since my dad died I've had to spend all my time working and looking after my mum and sister.'
'Yes, of course,' Xavier said, chastened. He'd almost forgotten how much she'd had to deal with during her relatively short life. 'That must have been pretty tough.'
'Yeah, it was at the beginning,' she said, hunching her shoulders, but maintaining her sunny smile. 'My mum got very depressed after we lost him and started really struggling with her Parkinson's, and Domino was too young to help out. She still needed someone to look after her and I was the only one available. It didn't leave a lot of time for me.'
'No. I bet,' was all he could muster in response. It made him realise how easy he'd had it being an only child with family money behind him. Glancing at his watch, he gave a start of surprise. He'd been enjoying chatting with her so much he'd not noticed how the time had flown. 'Hey, it's nearly lunchtime already.'
'Oh!' she said, looking slightly panicked. 'I'd better get dressed and grab some lunch. I have a hair appointment at two o'clock.'
Once she'd dashed off and changed her clothes they reconvened in the kitchen and ate their fill of the delicious food that Soli had loaded the fridge with—which mostly consisted of Mediterranean-inspired fare like brightly coloured salads, a cold meat platter and a range of healthy grains—before she excused herself to go to her appointment, leaving him on his own.
Experiencing a strange surge of energy once she'd gone, he took himself off for a long, hard session in the gym, followed by a lengthy swim in the pool.
Finally feeling as if he'd got past the odd edgy tension that had kept him moving, he went for a scorching hot shower, coming down from his bedroom to find a wonderful smell wafting from the direction of the kitchen.
Striding in, he found the room empty, though it was clear Soli had been in here recently because there were mixing bowls in the sink and a dusting of flour on the work surfaces. The smell seemed to be coming from the large range oven and he peered through the glass to see a large cake rising inside its tin.
Despite having stuffed himself at lunchtime, he heard his stomach give a growl of hunger. He'd told her he didn't have much of a sweet tooth, but he'd make an exception for something that smelt that good.
'I thought we could take it to the party as a gift for your friends,' came a soft voice behind him and he spun around to see Soli standing in the doorway with a tentative smile on her face.
There was a strange rising sensation in his chest when he noticed how her wild curls had been tamed into sleek blonde waves, making her look a good few years older than she was. Not that he didn't like her usual hairstyle. In fact, he probably preferred her hair au naturel, but he appreciated the effort she'd gone to for the party.
'Your hair looks nice,' he croaked through a suddenly dry throat.
It must be the heat from the oven getting to him.
She gave him a wide, delighted smile. 'Thanks, I'm glad you like it.'
'Are you going to be wearing that later?' he asked jokily to distract himself from the inappropriate way his body seemed to be responding to her now, pointing to the bathrobe he'd loaned her that morning, which she had wrapped tightly around her.
She smiled back. 'I wasn't planning on it, no. I bought a dress especially for the occasion. I hope it's the right sort of thing.' He noticed her jaw twitch and realised she was probably as nervous about going tonight as he was.
'Don't worry, they're a friendly crowd, on the whole.' He took a stabilising breath. 'I probably ought to warn you that there's a good chance my ex, Harriet, might be there, and there could be a bit of tension.' He frowned, wondering how best to explain this without having to go into too much embarrassing detail. 'We didn't part on great terms.'
'Oh. I'm sorry to hear that. So it wasn't a mutual split, then?'
He looked at her steadily for a second, weighing up whether or not to answer that, before shaking his head. 'No. It was her choice. But I'm over it now.'
Soli's eyes narrowed as if suspicious of this bold statement. 'How long were you with her?'
'Four years. We met in our last year of university.'
The usual wave of hurt flooded through him at the thought of Harriet and all that had happened between them, but he was determined not to let seeing her again ruin his night tonight. He needed to concentrate on getting through this thing successfully with Soli and convincing his friends they were a real married couple in order to avoid any more personal humiliation. That had to be his top priority.
'Is there anything specific I need to know? So I don't make a faux pas if we meet her?' Soli asked, her expression open and her voice so kind it reached right inside him and tugged at his heart.
He almost told her everything, right there and then, but decided against it at the last second. She didn't need to know all the sordid details. It wasn't as if everyone would still be gossiping about it now. Surely they'd all moved on.
'No. I think it's probably best if we just avoid her. I don't want to cause a scene, especially not in front of Hugo and Veronica, who are friends with both of us,' he said stiffly.
'Okay. Well, thanks for the heads up.' She was looking at him now with a concerned expression, as if she suspected there was more to it than he was telling her.
Nothing, it seemed, got past Soli.
There was an awkward beat of silence where they both smiled at each other and he couldn't help but think how pretty she looked. Her pupils seemed to dilate as she continued to maintain eye contact with him and she drew in a soft, breathy gasp, opening her lips a fraction as if she couldn't quite get enough air.
His gaze immediately moved to her mouth and he had to force himself not to start wondering what it would feel like to kiss those soft, inviting lips again.
'I should get the cake out of the oven before it burns,' she said a little over-brightly, jarring him out of his lascivious trance.
'Yes. Okay. You do that,' he said, a little rattled by his body's instinctive response to her. 'I'd better go and get ready for the party. We need to leave in about an hour.'
At the door he turned back to watch her as she busied herself around the kitchen, drawing in a great lungful of the delicious smell of the cake as she opened the oven door and bent down to lift it out.
'There's a good chance that won't make it out of the house,' he joked. 'It's altogether too tempting.'
She turned to give him a startled look, quickly recovering her composure when she realised he was talking about the cake. 'I thought you didn't like sweet things?'
He raised an eyebrow, tamping down on his amusement about the misinterpreted innuendo. It wasn't really appropriate to flirt with her when they were on their own. 'For that,' he nodded to the tin she had in her oven-gloved hands, 'I'll make an exception. It smells incredible.'
Her answering blush brought a smile to his lips and he allowed himself to flip her one last grin before walking away to prepare for the party, hyper-aware of his blood pumping hard in his ears.
CHAPTER EIGHT
Cluedo (UK)/Clue (US)—uncover the culprit with deduction and guile.
IT WAS MORE of a garden party than a posh evening do, as it turned out, which suited Soli just fine. It meant she was less likely to melt into a sweaty, nervous puddle as they mingled with the large throng of elegant, sophisticated guests.
Xavier was received with a mixture of exaggerated bonhomie and friendly curiosity and he seemed, to Soli, to be completely at ease as they moved from group to group, giving a nod here and there and occasionally stopping to introduce her before politely excusing them and moving swiftly on.
The comforting weight and heat of Xavier's arm around her waist kept her grounded as they circulated around the party and after a while she started to relax and chat with the people who asked kind, but slightly bemused questions of her, particularly about how she'd met Xavier. Once she'd given her answer about meeting him through the business she quickly moved the focus of the conversation onto the other person, making sure to ask them lots of questions about themselves.
They'd been there for about twenty minutes without catching sight of the host or hostess, who according to one guest were either down in the wine cellar stocking up on booze or settling a tantrum-throwing child, when there was a shout from behind them. 'Xavier McQueen, where have you been hiding?'
They both turned away from the couple they were chatting with to see a tall, slim woman wearing a flowing cornflower-blue silk cocktail dress striding purposefully towards them. She pushed her long, sleek black hair away from her face as she got nearer and gave them a huge grin, her dark eyes sparkling with delight.
'Veronica. Lovely to see you,' Xavier said, taking a step towards her. Soli could tell from the smile in his voice that he was genuinely pleased to see her. But if that was the case, why had he not seen his friends for so long? He couldn't have been that busy with work, surely.
'I'm so pleased you came! I had a bet on with Hugo that you would. He owes me a fiver, oh, he of little faith!' She raised a playful eyebrow at Soli. 'My darling husband thought the two of you might be a bit too busy "being newlyweds".' She turned to give Xavier a playful wink. 'But I knew you wouldn't be so mean as to deny us your wonderful presence at such an important do. Five years we've been married, can you believe it?'
Without waiting for a response from him she turned to look at Soli with wide, discerning eyes. 'Now, who is this delightful creature? Introduce me to your cute-as-a-button new wife, Xavier.' She held out a manicured hand, which Soli shook, a little surprised by the strength of the woman's grip.
'This is Solitaire—Soli for short,' Xavier said. The sound of her name on his lips gave Soli a strange little shiver of delight. He had such a wonderful deep, gravelly voice—it always did something to her whenever he spoke.
'It's so wonderful to meet you,' Veronica said, pulling her in for an enveloping hug. 'What a gorgeous name!'
Soli believed she really meant it too—she didn't seem at all patronising or false; her smile and touch were too warm for that.
'Lovely to meet you too. Thanks so much for inviting us,' Soli said, returning Veronica's smile. 'What a beautiful garden you have,' she said, gesturing towards what must have been half an acre of land, with its vibrant borders of summer flowers in full bloom and its springy grass that her heels kept sinking into, rooting her to the spot.
'Oh! Thank you! We love being out here in the summer and it's great for the children when they need to burn off some steam. Which is all the time!' She laughed, then glanced at Xavier, her smile faltering as an expression that Soli couldn't quite read flashed across her face. But only for a second. Had she felt as if she'd put her foot in it with him for some reason?
'I probably should mention that we've invited Harriet and her partner here today,' Veronica said, her expression a bit strained now.
'I expected you would,' Xavier said in a neutral tone, though Soli could have sworn she felt him stiffen beside her.
'I just thought I'd warn you so it wouldn't be a shock if you saw her here. I understand you haven't been in touch since...' She broke off, glancing quickly at Soli before returning her gaze to him. 'I just mean I can understand why you kept a low profile after what happened,' Veronica finished in a sympathetic tone.
Soli stood motionless, listening intently, wondering what the heck she was missing here.
Xavier shrugged. 'I thought it'd be better to let Harriet feel as if she could still see her friends without me being around all the time,' he said gruffly. 'You were her friends first, after all.'
Was she imagining it, or was there a hint of pain in his voice? The very idea made her shiver with horror and she experienced the strongest urge to wrap her arms around him, to let him know she was there for him and on his side should he need her. He froze for a second as she gave in to her instinct and slid her arm around his waist, but quickly relaxed again, pulling her closer in to his side.
Veronica smiled at the two of them, seeming to decide that Xavier must be over it. 'Well, I'm delighted that you have this gorgeous creature by your side now.' She gave Soli a warm smile, before moving her gaze back to Xavier.
'You know, there's a real spark between the two of you—something I never saw between you and Harriet. I always thought she was a bit too straight for you. In retrospect I'm sure getting married wasn't the right thing for either of you at the time. It was probably for the best that she called it off.'
Soli's heart gave an extra-hard thump in her chest. So he'd nearly married Harriet, but she'd called it off?
There was an uncomfortable pause before Xavier cleared his throat and said, 'Yes, perhaps so. It would have been nice to have more than an hour's notice that she'd changed her mind though.'
Soli's breath caught in her throat. So Harriet had dumped him at the altar! How humiliating for him. She couldn't imagine anything worse for someone as proud as Xavier. No wonder he'd been so determined not to get married again—and that his great-aunt had felt she'd had to go to such extremes to make him even consider it.
Not that he'd married her for love.
The thought made her stomach do a sickening sort of lurch.
'Oh, darling, I know.' It was Veronica's turn to sound uncomfortable now. 'It was a terrible thing to do to you, but I think she just panicked. She'd been feeling wobbly about it for a while but Hugo and I persuaded her it was just pre-wedding jitters. I'm so sorry we didn't talk to you about it beforehand. Still, it's done now and you're happy with your lovely new wife,' Veronica said brightly, as if sensing the need to move the conversation onto safer ground. 'By the way, where did you two meet? I forgot to ask.'
'Through the business,' Xavier said in a confident tone that even had Soli convinced for a second.
Veronica smiled indulgently. 'You know, Xavier, I don't think I've ever seen you looking so relaxed. The two of you are obviously good for each other.'
Soli stood stock-still, her heart beating rapidly in her throat, wondering what on earth Xavier would say to that. She hoped with all her heart it would be something nice. It felt incredibly important at that moment that it was.
'Soli is really good for me,' he said, to her surprise. 'The first thing I noticed about her was how vibrant and upbeat she is. I find her positivity really inspiring, to be honest, because, as we know, I'm not always the sunniest of people.' The look of genuine approval on his face when he turned to look at her made Soli's tummy flutter with exhilaration.
Veronica gave a tinkling laugh. 'Don't do yourself down. You're a wonderful person and a great friend—when you deign to accept invitations from us.' She laughed again, but it was plain from her tone that she was genuinely joking around with him.
At that moment a small child ran up to them, shouting, 'Mummy, Mummy!' and did a spectacular trip-dive just before reaching them, splashing her beaker full of blackcurrant juice all over Veronica's shoes.
There were a few seconds of noisy disruption as the child began to wail and Veronica half admonished, half comforted her.
Turning to them with a tearful child pressed up against her, she said, 'I'm so sorry, will you excuse me for a second? I'd better take her inside and get myself cleaned up. Don't run off, now, will you? I want to chat more later. It's been so long! I want to hear all about what you've been up to in the last few years, including all the details about your wedding.'
As soon as she was out of earshot Soli turned to look at Xavier, who was standing stiffly by her side, watching Veronica walk away with a frown on his face. 'So it wasn't just your parents that let you down, it was your fiancée too,' she said quietly.
She waited, looking up at him expectantly, willing him to trust her and tell her the whole story.
His frown deepened, then he gave a curt nod.
So that was all she was getting? Her pulse rate soared as frustration shot through her.
'Why didn't you tell me Harriet stood you up at the altar?' she blurted into the silence, unable to hold on to her patience any longer.
'Why do you think?' he shot back gruffly. 'It's not something I generally like to shout about.'
She sighed. 'Okay, I get that it must have been humiliating and not an easy thing to talk about, but I could have done with knowing about it before coming here and having to find out about it from your friends.'
He took a step away and ran a hand through his hair, looking frustrated with himself. 'Yes, okay. You're right, I should have told you. I'm sorry.'
'It's fine. It's just—it makes it hard for me to react appropriately when I don't have all the information.' She frowned, not wanting to turn this into a fight, but determined to make her feelings very clear. 'I want to get this right, Xavier. For both our sakes.'
Xavier smiled back ruefully, his facial muscles tense. 'I appreciate that. I'll try to be a bit more open and honest with you from now on, I promise.'
'Thank you,' she said, giving him a nod of appreciation.
He looked back at her, his eyes seeming to grow darker as their gazes locked.
'You know, I'm really impressed with how you've handled yourself here today,' he murmured. 'I wasn't sure how it was going to go, but everyone we've spoken to has clearly really liked you.'
'That's probably because I asked them so many questions about themselves and actually listened to their answers,' she joked, a little uncomfortable with his praise.
'No. It's not just that. People really respond to you. You have a real charm about you and you're clearly very intelligent, judging by the way you followed the different conversations you were thrust into.'
She swallowed, feeling a bubble of pride rise through her chest. 'Thanks. That's really nice to hear. I've never thought of myself as intelligent.' She glanced away as heat began to creep up her neck. 'I wasn't great at learning at school—it bored me, to be honest—but I've read a lot since I've been working in the café and I talk to such a wide variety of people in there every day I hear all sorts of interesting things. I guess it's made me good at general-knowledge subjects. In fact, I probably should warn you, I'm killer at Trivial Pursuit.'
He smiled at that and his whole face lit up.
Soli dragged in a breath as her body flooded with heat in response to it. He was such a beautiful man, even more so when he relaxed a little and let himself show his emotions on his face.
She swallowed as he took a step closer to her and reached out to pick a small leaf out of her hair. 'You look lovely today, by the way. I don't think I told you that back at the house. And I really appreciate you making so much effort for the party.'
'You're welcome,' she murmured through lips that were now having trouble forming actual words. He was standing so close to her, all she was aware of was his tantalising scent.
'We should probably slip away home now, before Veronica or Hugo come back out here,' he said, so quietly she was forced to lean in even closer to hear him. Her pulse throbbed hard in her veins as she felt the masculine heat of him radiate towards her. 'Veronica seemed intent on getting all the details about our marriage and I don't think I'm quite in the headspace to make up convincing enough lies right now,' he went on, apparently unaware of how he was turning her to jelly.
'Sure. Whatever you want,' she managed to say, forcing her frozen facial muscles into a smile.
Xavier was facing the house and he glanced towards it, seemingly to check if it was safe to make a sneaky exit, but he must have seen something, or someone, that alarmed him because the smile dropped from his face and his whole body stiffened. He appeared to pale as he continued to stare in shock at whatever had caught his attention.
'What's wrong?' Soli asked, turning to look in the direction he was gazing, a slow, heavy feeling of foreboding sinking through her.
'It's Harriet,' he replied in a tense voice.
'Your ex?' she asked, turning to watch a strikingly beautiful woman walking slowly towards them.
Soli frowned, wondering why she seemed to be moving so awkwardly, then as she looked down she realised the woman was heavily pregnant and the weight of the baby was making her waddle in her heels across the spongy grass.
'Yes,' was all Xavier had time to reply before Harriet was upon them, holding out her arms to Xavier in greeting.
'Xavier! I heard you were here with your new wife, so I thought it'd be all right to come over and say hello. It's wonderful to see you happy.' She offered Soli a friendly smile, the warmth in it only increasing her beauty. 'I'd heard you were becoming something of a confirmed bachelor-stroke-playboy.' She grinned affectionately at Xavier and Soli's stomach gave a sickening twist at how excluded she suddenly felt. These two clearly had some serious history between them.
And Harriet was exactly the sort of woman she'd expect Xavier to be married to. Elegant, intelligent and classically beautiful—the woman seemed to radiate vivacity. By stark contrast, Xavier's face looked stonier than ever.
'Oh, come on, Xave, you know I'm only teasing,' Harriet cooed, giving him a playful slap on the arm. This seemed to wake him up somehow and the corners of his mouth actually turned up for a moment. Not that the smile reached his eyes.
'I'm really happy for you,' he said in a voice that sounded as though he was having to force the words through his throat. 'You're actually glowing. I thought that was just an expression. I didn't realise pregnancy really did that to a woman.'
'So I've been told,' she said with a kind smile. 'It's all those hormones rushing around my system. I'm sure I won't look like this once the baby's born though. I've never been good with lack of sleep.'
'No. I remember,' Xavier said.
Soli's stomach twisted even harder, making her feel a bit sick. Now she was imagining the two of them in bed together, waking up and smiling at each other in mutual adoration before rolling together and—
No. Not going there right now.
'Well, I'd better get back to Neil—he'll be sending out a search party if I don't show myself soon. He's turned into something of a worrier since I fell pregnant. Calls me umpteen times a day to make sure I haven't gone into labour unexpectedly and forgotten to phone him!'
She shook her head, but from the way she was beaming Soli could tell she didn't mind this show of over-the-top possessiveness one little bit.
She could understand that. It must be wonderful for someone to love and worry about you that much.
'When are you due?' she asked, to cover the slightly awkward atmosphere that throbbed between them all now.
'Three weeks. I can't wait, to be honest—I feel like an elephant lumbering around!'
She really didn't look like one though; she was carrying all at the front, so if you saw her from the back Soli bet you wouldn't even know she was pregnant.
'Oh, by the way,' she said, turning to Xavier, 'happy birthday for Wednesday. Are you doing anything special for it?' She glanced between the two of them with interest.
Soli tensed as an intense desire to protect Xavier from the hurt this woman had caused him mixed with a sting of her own hurt that he hadn't even mentioned his birthday to her.
'Yes, of course, but it's a big surprise,' she lied quickly, giving her a covert wink in the hope she'd believe the bluff. No way was she having this woman thinking she wasn't as good a partner to Xavier as Harriet had been in the past.
But then, Soli hadn't shredded his heart and humiliated him in front of the people he loved and respected most in the world, so she was already one up on her.
'Sweetheart, I'm so sorry to drag you away early but this migraine seems set to stay,' she said to Xavier, linking her fingers through his and feeling him squeeze her hand in silent thanks. 'It was good to meet you, Harriet. Good luck with the baby,' she said, giving the woman one last smile and a nod, then pulling subtly on Xavier's hand.
'Bye, Harriet,' Xavier said, falling into step with her as they walked back towards the house. They made a swift exit through the huge throng of people who were now milling around the large garden and made it back to the car without encountering any objections to their leaving.
'Thanks,' was all Xavier said as he pulled the car out of the parking space and set off back to their house.
Soli waited until they were driving down the main road into Hampstead before she asked, 'Are you okay?'
He didn't answer her immediately, just kept staring ahead at the road. 'Yeah, I'm fine,' he said eventually, though he really didn't sound it.
'I take it you didn't know she was pregnant?'
There was a heavy pause in which he stared at the road ahead again. 'No, I didn't.'
'How did it make you feel?' she asked tentatively.
He turned to glance at her and the expression in his eyes made her stomach sink to the floor. 'Not great, to be honest.' He dragged in a sharp breath. 'The main reason she gave for not wanting to marry me was that I wanted to have kids but she didn't. She thought having a family would mean she wouldn't be able to focus on her career—that it'd hold her back.' His jaw clenched tight before he spoke again, as if he was fighting with his emotions. 'But I guess she did want kids after all—just not with me.'
Soli's stomach dropped even lower, making her feel nauseous on his behalf. 'Oh, Xavier, I'm so sorry.'
He shrugged. 'She probably had it right. I'd have been a terrible father and a selfish husband. I've been so focused on work and building my company I probably wouldn't have given my relationships priority—as I'm sure you've noticed,' he added with a grimace. There was a heavy pause before he cleared his throat and said, 'It was for the best that we didn't get married.'
She wanted to reassure him, to tell him that he had it all wrong, but she couldn't. She didn't know him well enough to be able to say something like that and he knew it.
Her heart went out to him. No wonder he was so unwilling to get close to someone again. Anybody would be wary of allowing themselves to love and trust if they'd been through what he had.
She wanted to help somehow, so much it made her chest physically ache.
But do what?
Taking a breath, she gave him a supportive smile as an idea began to form in her head.
She might not be able to do anything about the past, but she could definitely do something about the future.
Acknowledging a small but insistent voice in her head reminding her not to get too emotionally involved—because she knew, deep down, that if she set out on that path it would be almost impossible to turn back—she assured herself she would only do it to make their time together more enjoyable, and perhaps to say thank you to Xavier for giving her the chance to make a better life for herself and her family.
Yes. She would do it for him because she could.
But, perhaps even more importantly, because she knew that nobody else would.
CHAPTER NINE
Mr & Mrs—discover things about your partner you didn't know.
IT HAD BEEN quite an eye-opener for Xavier, seeing Harriet for the first time in five years. The pregnancy had knocked him off his game to begin with, but once they'd started talking he'd soon regained his equilibrium. Meeting her again had been something he'd been dreading, but now it had happened he realised it hadn't affected him in the way he'd expected.
In fact he'd been surprised, after talking to Soli about it all and reflecting on it later that evening, by the startling realisation that he wasn't angry with Harriet any more, despite the pregnancy she'd sworn she'd never want. Thinking about what he'd had with her just conjured a sense of bittersweet nostalgia now, but that was all it was. It seemed he'd finally moved on from the hurt and bitterness he'd felt about her rejecting him.
It was actually quite liberating. He'd been walking around with this sense of loss and inadequacy for so long, it was a huge relief to finally feel it lifting.
He had a strong suspicion that Soli's steadfast support and compassion had had a hand in breaking the grief he'd been living with too. Her tackling the subject head-on had made him realise how he'd bottled his feelings up, and how unhealthy that had been. He'd not talked to anyone about what had happened to him before, worried it would make him seem weak, and had cut everyone who knew about it out of his life so that he hadn't had to face it. But he knew now he couldn't hide for ever. It was time to get past his hang-ups.
He had an inkling that Soli wouldn't let him carry on the way he'd been going anyway.
She was a real force of nature.
They'd spent the next day in the house together, with her reading a book in the garden and him moving between his home office and joining her on the terrace to eat, where they'd chatted about inconsequential things. He'd appreciated her allowing him some personal space and leaving the subject of Harriet alone, but also making it clear she was happy to talk if he wanted to.
It seemed he'd made a good move in choosing her to be his stand-in wife. He found he was actually enjoying her company now and could imagine them getting on fine for the next few months. Until it was time to call a halt to it all.
Strangely, the thought of that brought with it a heavy, tense feeling, so he pushed it to the back of his mind. There was no point in dwelling on what would happen when their arrangement came to an end. It was months away yet.
* * *
Walking into the kitchen on the morning of his birthday to grab a quick cup of coffee, he stopped in amazement as he saw that the table was set for two and there was a plate loaded with Scotch pancakes covered in maple syrup and two large mugs of coffee sitting on it. Soli was standing at the stove, attending to what looked like poached eggs.
'Happy birthday!' she said brightly when she noticed him standing there, open-mouthed. 'I thought I'd make you breakfast. You like eggs, right?'
'Er...yes. I love them.'
'Great. Take a seat and dig in to the pancakes. I made them fresh. There's more coffee if you need a refill after that one too.'
'You didn't have to do this—' he began to protest, but she waved his words away.
'Of course I did! It's your birthday. Everyone should get special treatment on their birthday.'
Never having had 'special treatment' like this before, Xavier went to object, but snapped his mouth shut at the last second, feeling it would be rude and unkind to contradict her. Just because no one else had done it for him, it didn't mean he couldn't accept it from her. Tingly heat rushed across his skin as he made the conscious decision to accept her indulging him today. It would actually be pretty nice to celebrate his birthday for once.
'Well, I appreciate the thought,' he said, sitting down at the table. 'I'll wait for you before I eat though.'
'Okay,' she said, shooting him a warm smile.
A minute later, she came to join him, laying down plates of buttered wholemeal toast with two poached eggs balanced on top.
'So! What are you going to do with your day?' she asked, sitting down opposite him and picking up her cutlery.
He frowned at her. 'Go to work.'
She looked aghast. 'Really? Can't you take the day off for once?'
'I wasn't planning to. There's a lot going on at the minute.'
'But you have a large staff working for you.'
'Yes.'
'So let them do the work today. You're the boss, right? Take the day off and hang out with me. Treat yourself.'
He considered this for a moment, feeling a throb of unease about not going in to work at such short notice. Though, if he thought about it, he'd not had a day off in five years, so it was probably overdue. Soli was right too—his staff were more than capable of getting on with what needed doing without him for one day.
'Well, I suppose I could—'
'Great!' She grinned at this and began tucking into her breakfast.
After they'd finished the delicious meal, Soli ordered him to go and sit in the garden and read the papers that she'd nipped out and bought earlier while she cleared up. She wouldn't hear of him helping her, even though he pointed out he should be the one to do it since she'd cooked.
They spent a lovely morning looking out across the garden, reading and chatting about current affairs, which she seemed impressively clued up on—another by-product of working in the café, he supposed.
Just before lunchtime, Soli stood up and brushed down her skirt. 'Right! It's time for a walk on the heath, then lunch. I've already put together a picnic. You'll need your swimming trunks, your trainers and your tennis racket,' she called over her shoulder, as she walked away into the house.
He stared after her, dumbstruck, his body rushing with endorphins, as it occurred to him what she'd done. She'd planned the 'perfect day' he'd told her about for his birthday.
Heat pooled in his belly and he was horrified to find his eyes had welled with tears.
No. No! He couldn't allow himself to feel sentimental about this. She was just fulfilling some obligation she felt she had as his wife, that was all.
Still, a little voice told him, she hadn't needed to do it, and from the glee he'd heard in her voice he suspected she was actually enjoying it.
* * *
A couple of hours later they were stretched out on picnic blankets, groaning happily after wolfing down the fabulous lunch she'd put together, which included slices of her amazing—and apparently legendary—chocolate fudge cake.
'I can see why that's so popular in your café,' Xavier said, nodding towards the now empty container that had held the cake. 'I don't think I've ever tasted anything like it.'
'An "orgasm in food form", one of my friends calls it,' she said with a grin.
There was a small pause, where they didn't look at each other for a moment and pretended to watch a couple of squirrels running up a tree trunk instead.
Xavier cleared his throat, reminding himself what a bad idea it would be to act on the kinds of thoughts that had just popped into his head following that comment.
'So tell me what your perfect day would consist of,' he said, attempting to circumnavigate the strange atmosphere now zinging between them.
Her brow furrowed as she appeared to think about this for a moment.
'Well, if I could choose anything, I'd probably go for spending the day on a Mediterranean island. I'd eat lunch on the beach and go for a swim in the sea. Then I'd spend the evening having someone servicing my every need.'
'That sounds good,' he said, smiling at her and noticing how her cheeks had flushed an adorable shade of pink. 'I think I could probably manage to enjoy a day like that too.'
'Well, maybe we'll do it for your next birthday,' she quipped, her face falling as it obviously occurred to her that they wouldn't be together for his next birthday. Their time, and their marriage, would be over by then.
'When was the last time you had a holiday abroad?' he asked hurriedly, attempting to sweep past the awkwardness.
'Er...' She thought about this for a second, looking relieved to have the conversational diversion.
'I guess it was about four years ago, just before my dad died. We went to Brittany on the ferry. It was a great holiday. We were all really happy.' He saw grief flash across her face and a sudden and acute instinct to make her happy again overwhelmed him.
'You know, it's probably been that long for me too. I've been so focused on building the business, I've not stopped to take a proper break.' He dragged in a breath, throwing caution to the wind. 'You know, I own a holiday property on Corsica. It's just been renovated and they've completed the fit-out early, so it's just sitting empty at the moment. We could go and stay for a week.' He shrugged, trying to appear nonchalant. 'It would do us both good to get away.' He smiled. 'And there wouldn't be any solicitors there, checking up on us.'
Her eyes had lit with excitement as he'd talked and as soon as he finished speaking she blurted, 'That would be amazing! I've always wanted to go there. It sounds like such a beautiful place.'
'Okay, then,' he said, satisfaction coursing through him at being able to bring back her smile. 'I'll arrange for us to go this Saturday. I'll just need to let my colleagues know.' He paused and smiled to himself. 'They're going to wonder what's happened to me. I never take time off.'
'I guess they'll just assume you're having a honeymoon with your new wife,' she pointed out with a grin.
'Yes, of course.' He blinked, realising it was a detail he'd overlooked and feeling pleased it would still fit their story. That solidified it as a really good practical excuse to go.
Satisfied with his decision, he lay back and stared up at the fluffy clouds floating slowly across the sky above him.
Yes. Getting some sun on their faces, eating good food and relaxing would be a great thing for both of them.
* * *
Soli's stomach had jumped with excitement as they'd stepped off the plane that Xavier had chartered into the balmy Corsican air. She'd not been able to believe it when he'd led them to a private lounge at the airport and announced in a really matter-of-fact voice that their pilot would be there to greet them soon and they would be in Corsica within a couple of hours.
Soli's only experience of being on a plane had been on a family holiday to Spain when they'd gone with a budget airline. This trip had been entirely different, what with the separate leather seats, à la carte food and total lack of general public.
And then when he'd led them over to a soft-top classic car and loaded their bags into the boot her insides had done a little dance of excitement. The trip to his hillside house had been a thing of joy.
The cliff-top cottage where they were to stay for the next week was quite a bit smaller than the Hampstead house, but no less impressive. It had been newly decorated in fresh Mediterranean colours and filled with French-style furniture, all of which looked incredibly expensive and dauntingly classy.
'It's a rental aimed at the high-end market,' Xavier said conversationally as they walked inside and Soli gasped in delight at the amazing view of the sea through panoramic bi-folding doors. 'We have our own private cove down there,' he said, walking to the doors and opening them back so the sweet, briny smell of the sea below them rushed inside and tickled her nose.
'It's a bit of a walk down a narrow path in the cliff, but well worth it.'
Soli followed him onto the deck, which housed a couple of comfortable-looking loungers as well as a small dining table with a canopy flapping gently in the warm breeze, and turned to give him a smile that felt as if it might split her face in two. 'I love it!' she said, her voice raspy with delight. 'I can't wait to explore.'
'Let me show you your room first,' he said, strolling back inside.
She followed him through the living area and down a short corridor to where two doors stood opposite each other.
'You're in here,' he said, opening the door on the right. They walked inside and stood by the king-sized bed that was dressed with white linen sheets and an abundance of pillows in a soft blue-grey colour.
'There's an en-suite bathroom over there,' he said, gesturing to a small door on the right.
'Oh, Xavier, it's wonderful!' she said, turning to give him a grateful smile. It was a long time since she'd been this excited about being somewhere and her whole body felt jittery with adrenaline. Or was it that she was nervous about being here with him? The whole place shouted Romance!
He smiled back at her with the genuine, wide grin that always made her tummy flutter whenever he decided to trust her with it. 'I'm glad you like it.' He huffed out a breath, frowning down at the floor before looking back into her eyes. 'I wanted to say thank you—for all the support you've given me recently. It was above and beyond the call of duty and I want you to know I really appreciate it.'
She suddenly felt a bit shy. 'It was my pleasure. Anything to help.'
The air between them felt prickly with emotion. Or was she imagining it?
Something in his expression made her breath catch in her throat.
'You know, you're the most genuinely caring person I've ever met,' he murmured, taking a step closer to her.
Soli was suddenly intensely aware of how alone the two of them were, and how close to the enormous, inviting bed.
Heat throbbed through her, making her head swim.
'Thank you,' she breathed through numbed lips. She suddenly wanted to kiss him so badly her whole body ached with it.
The look in his eyes mesmerised her, keeping her suspended in a state of such intense longing she knew she had no hope of escaping from it. It wouldn't take much to be close enough to touch him. Just one more step. She could feel the heat radiating from his body and his enticing scent filled her nose, making her mouth water.
His Adam's apple bobbed in his throat as he swallowed and her gaze flashed down to it for a second.
The break in eye contact seemed to shock him out of whatever state he was in and he gave a rough cough and took a step backwards, sweeping his hand towards the door.
'Okay, well, I'll grab the bags from the car and you can get settled in. We should go out for dinner. There's a lovely tavern in the next town with magnificent views of the harbour. It's quite a sight.'
It was as if he'd brought the shutters down on his emotions.
'That sounds wonderful,' Soli said quietly, her stomach sinking with disappointment at his sudden change in demeanour.
'Good,' he said, giving her a jerky nod and swiftly leaving the room.
She flopped down onto the bed and took a few deep, steadying breaths.
Perhaps this holiday wasn't going to be as relaxing as she'd hoped after all.
* * *
The tavern was as magnificent as Xavier had described and Soli had a hard time choosing what to eat from the comprehensive seafood menu.
'I don't usually eat fish. I've never learnt to cook it because Domino has an extreme aversion to the smell,' she said, her mouth watering at the thought of the meal she'd just ordered. She'd been hesitant to ask for the lobster—she'd always wanted to try it—but Xavier seemed to have sensed her worry and told her to order whatever she wanted. So lobster it was, with a large glass of crisp Sancerre, which he'd recommended.
She was in heaven.
'So, I thought tomorrow we could explore some of the coves. Maybe take a picnic lunch with us. I have a small motorboat moored in our cove which we can take along the coast,' Xavier said, putting down his wine glass and fixing her with his tingle-inducing gaze.
'Oh, wow! I'd love that!' she said. This whole set-up was like something from the movies.
The smile he gave her made his eyes twinkle and she took a hurried gulp of her own wine to calm her suddenly racing pulse.
She needed to remember that this wasn't real. They were just pretending.
At least he was.
She wasn't so sure about herself any more.
* * *
They slept in late the next morning on Xavier's insistence and ate a light breakfast before strolling down the winding path to their cove and climbing into the boat.
The turquoise sea glittered in the late morning sunlight and Soli laughed with joy as they sped through the gentle waves to the other side of the island and one of the other secluded, and usually deserted, coves that Xavier knew about.
Once there, he brought the boat in to shore and tied it to a large rock, making sure it wouldn't float away before helping Soli out. Her hand was warm and firm in his and for a moment he contemplated not letting go of her and pulling her towards him instead.
To do what, he wasn't sure.
No. He was. But he couldn't allow himself to think like that. They'd made a pact and he needed to keep to it. For his own sanity.
After exploring the small cove, where Soli scrambled over the rocks and stood on the largest one with her hands on her hips and her curls blowing in the wind—eliciting in him the strangest feeling of lightness he'd ever experienced—they settled down onto the picnic blanket he'd brought and tucked into the food his housekeeper had stocked the fridge with.
Once they'd finished he lay back on the blanket with his hands behind his head and gazed up at the azure-blue sky, letting the heat of the day wash over him. It had been a very long time since he'd felt this relaxed—even though an undercurrent of crackling energy still pulsed through him. Something he couldn't quite identify, though he had his suspicions.
'Phew! It's so hot!' Soli said next to him. 'I think I'll go for a swim. Want to join me?' She looked at him expectantly.
'Not right now,' he said, unwilling to move from his comfortable position. 'You go ahead.'
He tried not to watch as she took off the light summer dress she'd been wearing to uncover a cherry-red bikini.
Tried and failed.
He also tried to keep his gaze off her as she skipped down the golden sand to the water and tentatively dipped her toes in.
That was a lost hope too.
'It's warm,' she called back, sounding delighted with the discovery.
He gave up and turned to rest on his side, watching her splash around, grinning from ear to ear as she went deeper into the water.
Her body was lithe, but curvaceous, imperfect in places—and wonderfully real for it.
It had to be the most alluring sight he'd ever seen.
She was magnificent, standing tall and proud in the waves as they gently lapped at her thighs. The very picture of female beauty.
A sense of contentment washed over him, taking him by surprise.
Being here with Soli felt so very right.
He was both amazed and pleased by how much he'd found he enjoyed sharing things with her. Being an only child, he wasn't used to sharing, so he was surprised to find how much he actually liked it. Soli took such joy in being introduced to new things too and she seemed to genuinely enjoy them. The women he usually dated were well used to the high life and didn't tend to comment much on the places he took them or the gifts he bought, so it had been wonderful to see Soli's genuine appreciation. It made him feel as though he was making a valuable contribution to her happiness, and her life. Something he'd never experienced before with a partner.
She was so full of wonder and enchantment with the world, despite the trials she'd been through in her life.
He'd never met such a positive person and it thrilled him to his core.
What was he doing, anyway, just lying there alone? He should take her lead and go and enjoy himself in the water.
With her.
Pushing himself up to standing, he walked swiftly across the hot sand, feeling acute pleasure as the cool sea rushed over his feet and lapped against his legs. He waded towards where she was bobbing about in the water and when she turned and saw him coming her face broke into a smile.
Euphoria swelled in the pit of his stomach, then rushed up through his chest, taking his breath away.
'The water feels incredible against your hot skin,' she called, moving forwards to meet him.
All he could think about at that moment was what it would feel like to touch her slick skin and feel her incredible body pressed against him.
Her smile faltered as he got closer and he realised that the longing he was feeling must be showing on his face.
'Are you okay?' she asked, reaching out to him and putting her hand on his shoulder. Before he knew what was happening a particularly strong wave had pushed her towards him and their bodies connected, their mouths only inches apart now. The feeling of her pressed against him tipped him over the edge and without another thought he dropped his mouth to hers, sliding his hands around her waist to steady them both.
Everything else faded away as the sensation of her soft lips under his took over his entire awareness, plunging him into a state of desire so intense it left him breathless. Instinctively he deepened the kiss and felt her respond, opening her mouth so he could slide his tongue inside. She tasted just as sweet as he remembered, but also salty from the seawater, and he let out a groan of pleasure deep in his throat. Her body was slippery and smooth against his and he pulled her closer to him, losing himself in pure physical indulgence.
He'd wanted to do this for so damn long.
His body responded as his desire deepened, hardening and pressing into her with obvious intent.
'Oh!' she gasped against his mouth and the sound of her surprise shocked him out of his erotic daze. He pulled away abruptly, suddenly furious with himself for allowing his need to override his common sense. What was he doing? He couldn't kiss her like this and expect everything to stay the same between them.
But did he really want it to?
Yes. They couldn't afford to let their agreement slip; it wasn't sensible.
'Sorry,' he muttered, drawing away from her and noting with a sting of regret how shell-shocked she seemed by his sudden withdrawal. 'I shouldn't have done that.'
'No. Okay,' she rasped, her confusion clear on her face.
'I'm going for a swim, then we should think about getting back to the house,' he said gruffly, diving underneath the cool water before she had a chance to answer him, praying it would bring him back to his senses and extinguish his driving need to swim right back and kiss her again.
* * *
'You know, you're very mature for your age. I guess that comes from taking on so much responsibility when you were so young,' he said to her later, back at the house, while they sat on the terrace gazing out over the water, sipping their second large glasses of wine, trying to pretend that the kiss hadn't really happened.
She turned to shoot him a startled smile. 'Do you think so? Most of the time I feel like I have no idea how to be an adult.'
He gave her a wry smile back. 'I think most people feel like that. We're all just making it up as we go along.'
'That's good to hear.'
There was a small pause where they just looked at each other.
'I'm...er...just going to get a glass of water. This wine's going straight to my head,' she said, standing up quickly and stumbling as she caught her foot on the chair leg.
Xavier shot out a hand to steady her, rising from his chair so he could take her weight.
She gazed up into his face, an expression of embarrassment clouding her face. 'Sorry,' she whispered.
'No need to be sorry,' he said roughly, hyper-aware of his pulse throbbing in his throat. She smelt wonderful, like sunshine and fresh air and her own distinctive sweetness, and he drew in a great lungful of her scent, his chest swelling with a longing so powerful his whole body rushed with it.
'Soli...' He wasn't sure what he wanted to say. A thousand thoughts flashed through his head but words wouldn't form in his mouth.
So he kissed her again. Hard.
She let out a small gasp of surprise, but didn't pull away. Instead she wrapped her arms around his back and pulled him closer to her, forcing their bodies together.
And then his hands were in her hair and his body was responding in all sorts of inappropriate ways, but he couldn't stop himself this time. Didn't want to.
From the desperate sounds in her throat, it seemed as though Soli was having the exact same problem.
'Let's go inside,' she murmured when he pulled back a fraction to breathe, her grip tightening on him as if she was afraid he might change his mind again and let her go.
Through his haze of desire, Xavier couldn't remember any of the reasons why they should stop any more, so without another word he slid his hands beneath her thighs and lifted her up against him and carried her into the house, still kissing her as he went, unwilling to break the connection.
In the bedroom they discarded their clothes quickly and he rolled on top of her, luxuriating in the feeling of her soft skin against his.
Through the fug of feeling a terrible thought struck him like a blow. 'I don't have any condoms,' he bit out, suddenly furious with himself for overlooking such a simple thing. His thoughts swirled wildly as he tried to reconcile all the concerns that were now jabbing at the edges of his brain.
'It's okay, I'm on the Pill,' she muttered back, gazing up into his eyes with a look of such genuine honesty he knew deep down that she was telling him the truth. Because she always told him the truth.
'Thank God for that,' he said, letting out a breath and feeling relief radiate from the pit of his stomach. Sliding his hands back into her hair, he took a moment to marvel at the sensation of the silky strands running through his fingers. 'Thank God,' he repeated huskily, before losing himself in the wonderfulness of her again.
CHAPTER TEN
Dominoes—set them up with a steady hand, then see them fall with the slightest knock.
XAVIER SAT ON the terrace cradling a large cup of coffee in his hands and staring out over the sea as the sun rose in the distance.
He'd left Soli sleeping peacefully in his bed, with her wild curls spread out around her head, having to force himself to leave the room and not slide back in next to her and pull her soft, welcoming body against him again.
What the hell did he think he was doing, messing with their no-sex agreement like that? All day yesterday he'd fought the urge to get closer to her, keeping his physical distance and his tone light and friendly. He'd deliberately not flirted in any way, strictly disciplining himself against giving in to the desire that had swelled and throbbed inside him.
And then it had all gone to hell.
It had been the heady mixture of sunshine and lust and then being forced to touch her again when she'd stumbled that had been his final undoing. The expression of open hunger in her eyes when she'd looked up at him had been impossible to resist.
The sex had been great too. Fun and surprising in its intensity. They'd fitted together so well, delighting in exploring each other's bodies for the first time but also seeming to know intuitively what made the other tick.
They had been around each other a lot recently, of course, so maybe that was why it had felt so comfortable being with her.
Not that he was going to let it happen again. No. He'd be a fool to indulge his base desires and potentially ruin the good thing they had going. The last thing he needed was for this business relationship to take a wrong turn and for the solicitors to get wind and it put his inheritance in jeopardy.
That house was much more important to him than sating his physical desires for the next few months.
When Soli got up he'd discuss that with her in a straight and honest manner. He didn't want her to think he'd taken advantage of her last night, but he wanted to make it clear that it had been a mistake on his part and that they should move their relationship back to their being friends only.
Surely she'd understand the necessity of that. She might even wake up this morning regretting what had happened last night too.
Just as he thought this there was movement by the terrace doors and Soli emerged into the sunlight wearing one of the robes from the room.
He swallowed hard and gave her a friendly but controlled smile, not wanting to make her feel awkward, but needing to set the right businesslike tone for the conversation that was to come.
'Good morning,' he said as she moved slowly towards him, appearing to stumble a little on the smooth tiled floor. 'Are you okay?' he asked, feeling a sting of concern as she got closer to him.
She really didn't look good. Her skin was a strange grey-green colour and her eyes looked a little unfocused.
'I don't feel too great actually.'
'Here.' He stood up and pulled out a chair for her. 'Sit down.'
She came towards him and put her hand on the back of the chair, as if not entirely convinced it was what she wanted to do.
'I felt a bit light-headed last night but I thought it was because of the...um...wine.'
And the sexual tension, he filled in silently for her.
There was a sudden flash of panic in her eyes and she took a hasty step backwards. 'Sorry. Excuse me for a second—' and with that she dashed back inside, the door to her bedroom slamming loudly behind her.
Panic inched through him. He had no idea what he should do. Should he go into the bedroom to check if she was okay? If she was being sick she might prefer that he leave her alone. Or was she hoping he'd go in there and hold her hair and rub her back? No. He couldn't imagine her wanting that. Especially with their relationship being so...confused right now. So he stayed at the table and waited, with his pulse beating hard in his head, until he heard her bedroom door open again.
Striding into the living area, he saw her standing in the doorway, seemingly holding herself up by propping her hands against the door jambs.
'You should go straight back to bed,' he said, noting how she was trembling now.
'I don't think I drank too much last night, I think I'm ill,' she murmured, looking at him with bloodshot eyes. 'I must have picked up a tummy bug from somewhere, or perhaps it was something I ate—'
'I agree. Straight back to bed for you.' She nodded and turned around, heading back into her room, where she climbed shakily under the fresh cotton sheets of her bed.
Good that we used my bed last night so her sheets are clean, he thought as she snuggled into the pillows and let out a small sigh of relief. At least she'd be able to lie there without being constantly reminded of what happened between them last night. He didn't want her associating it with feeling ill.
Not that that should be his priority right now, he reminded himself sternly. 'I'll go and fetch you a glass of water,' he said, heading quickly out of the room and into the kitchen.
He filled a glass for her and searched out a small bowl from the cleaning cupboard, which he then took into her bedroom.
'Here you go, just in case you feel ill again and don't think you can make it to the bathroom. Not that it'll be a problem,' he added quickly, not wanting her to think he'd be angry if she made a mess.
'Thank you,' she whispered, clearly trying to smile at him, but not quite managing it.
'Try and sleep,' he said quietly, backing out of the room. 'If you need me, shout. I'll be right next door.'
He saw her give him a small nod before he turned around and headed back to the living area.
It looked as if the conversation about what had happened last night was going to have to wait.
* * *
Soli slept fitfully, weaving in and out of consciousness, sometimes aware of Xavier's presence in the room, other times thinking she must have imagined it.
In the middle of the night she rolled over and stretched out her arm to see whether he'd got into bed with her, but had found the other side cold and empty.
Not that she could blame him. Who'd want to sleep next to someone who had to get up to vomit every half an hour? She hoped to goodness he wasn't sick too—that would really make her feel wretched. He'd been really kind, looking after her, coming in to check on her regularly and refreshing her water. Luckily she'd not needed the bowl he'd brought in. The thought of him having to deal with that had given her the horrors.
She couldn't bring herself to think about what had happened between them the night before though. As wonderful as it had been, it made her stomach tie into knots just thinking about it, so she pushed it to the back of her mind. It would be something she'd mull over once she was feeling better—which would hopefully be soon.
She hated being ill.
Especially when she was meant to be on holiday.
And with Xavier right next door.
* * *
She was still sick the following day and only got out of bed to dry heave before crawling back under the covers to sleep, but the next morning she woke up feeling as if the worst of it had lifted and tentatively sat up in bed to check the time.
Ten o'clock.
She'd been in bed for nearly two days.
Poor Xavier—he must have been bored out of his mind.
Slowly, carefully, she swung her legs out of bed and tested how she'd feel standing up, finding she wasn't as dizzy as she'd been the day before, and instead of lurching, her stomach gave a growl of hunger.
Thank goodness for that.
It seemed it was only a forty-eight-hour bug. Still, as she pulled on her robe and made her way to the door she had a moment where she thought she might pass out.
Better take it easy, Soli; no need to rush.
After pressing her hands to the wall and taking a few deep breaths to give her head a chance to catch up with the movement, she walked slowly out of the room and into the living area.
It appeared to be deserted, though there was a smell of coffee in the air and a plate and a mug next to the sink.
'How are you feeling this morning?' came a voice from outside and she turned to see that Xavier was sitting on the terrace, looking at her with concern in his eyes.
'A lot better, thanks,' she said, walking slowly out onto the terrace and over to where he was sitting on one of the loungers with a tablet on his lap.
'You still look a bit pale. Are you sure you're okay to be up?' he asked, his gaze raking her face.
She cursed herself for not even looking in the mirror and flattening down her hair before leaving her room, but decided just to brazen it out now she was here. He certainly didn't look disgusted by her appearance. If anything his expression was one of friendly concern.
'A bit of fresh air will probably do me the world of good,' she said, sitting down carefully on the lounger next to him and offering him a tentative smile. 'I'm so sorry for ruining your holiday.'
He waved a hand at her. 'Don't be ridiculous. It's not as if you got ill on purpose.' He sat up and swung his legs to the side, putting the tablet onto a coffee table next to him. 'Could you eat?'
'Actually, yes. I think I could.'
He nodded. 'Okay. Stay there. I'll fetch you some breakfast.'
Before she could object, he stood up and strode swiftly and purposefully back into the house, returning a few minutes later with a mug of coffee and a plate of wholemeal toast.
'Here,' he said, handing her the food, then putting the drink onto the table next to his tablet. 'I thought something quite plain would be good to start with.'
'Thank you,' she said, gratefully accepting the plate and sitting up on the lounger to eat.
She felt him watching her as she took her first tentative bite, then when it was obvious she wasn't going to be sick again he went back into the house, returning a couple of minutes later with a mug of coffee for himself.
'I guess you were hungry,' he said, gesturing to the now empty plate she had on her lap.
'I guess I was,' she agreed, giving him a grateful smile.
'Listen. We don't need to rush off anywhere today, if you're worried about that,' he said, his expression sincere. 'I'll be very happy to hang out here until you feel more up to going out. Perhaps you could teach me how to play a board game or something.'
'That would be great,' she said, immediately perking up at the thought. Lazing around and recuperating whilst playing one of her favourite games would be her idea of heaven right now. 'Except I didn't bring one with me.'
'No problem,' Xavier said with a twitch of his eyebrows. 'We have some here in the house. Apparently they're popular with holidaymakers who want to switch off their phones and get back to a screen-free existence for the duration of their stay, or so the interior designer told me the other day. It made me think of you actually.'
She blinked in surprise. 'Well, that's great. I guess the lure of board games isn't dead after all.'
'I guess not,' he agreed, the corner of his mouth turning up. 'I'll go and fetch them and we can have a game now if you feel up to it.'
'Great,' she said, 'but we should move to the table. I take my playing very seriously.'
'I'd expect nothing less,' he said with a smile.
So he went to the fetch the games that he'd had stocked in a cupboard in the living area for guests—as a reaction to her love for them? she wondered hopefully—while she went to get dressed.
They played happily for the next couple of hours, sipping coffee and feeling the warm sea breeze on their faces as they alternated between asking questions and answering them, each taking great pleasure in challenging the other's knowledge of random facts.
'I really appreciate you looking after me. It's kind of you,' Soli said after they'd finally called a halt to the game. She'd been intensely aware of all the things they hadn't said to each other since 'the other night' and wanted to clear the air so she could breathe properly again. Even if it meant dealing with something she didn't want to hear. Which, from Xavier's careful avoidance of touching her or maintaining eye contact for too long, she suspected might be that he believed they'd made a mistake in sleeping together.
Nausea rose from her stomach as she waited for his response, only this time it came about through nerves.
'Well, I did agree to be there for you in sickness and in health, so I guess I'm just fulfilling my promise,' he quipped, though she couldn't help but notice how his whole body had tensed.
She shot him a sheepish smile. 'Still, it's good of you.'
He shrugged, clearly uncomfortable with her praising him. 'You deserve to be looked after too,' he said gruffly.
Letting out a low breath, she leant back in her chair and crossed her arms. 'I guess I'm just not used to it any more. It's been a while.'
'Since your dad died?' he asked, their gazes locking.
The sudden swell of emotion she experienced made her suck in a breath.
'Yeah,' she said, telling herself to pull it together. This was no time to go to pieces.
Xavier seemed to sense her discomfort and gave her an understanding nod. 'Hey, I forgot to ask—how's your sister doing at Oxford?'
'Great! It sounds like she's really enjoying herself,' she said, grateful for the change in conversation.
'Good for you for making that happen. I'm seriously impressed. I'm not sure many siblings would go to the lengths you did to make sure their sister was set up like that.'
She shrugged, but her face heated at the compliment. 'It would have been a crime not to let her go to university. She's so smart. The world is going to thank me one day.'
'I'm sure it is.' He sat back and studied her for a moment. 'I guess I should thank you too. You saved me from losing my home. And perhaps my sanity.' His smile was wry.
'Well, you've been very generous. You've given me more than I asked for.'
He laughed, perhaps to cover his unease. 'We seem to have turned into the Mutual Admiration Society.'
She smiled. 'Yes, I guess we have.'
'Well, why not? We are pretty amazing people, are we not?'
'We are, I suppose.'
'No! No "suppose" about it. We are.'
Their gazes locked and Soli got lost in a sudden flash of memory as she thought of how he'd looked into her eyes while they were making love.
'Listen. About what happened the other night,' he said in a careful tone that made her spirits plummet.
'Yes?'
'I don't think it's a good idea to let that happen again. Especially since we still have rather a long time to live together. It could make things tricky.' He didn't say if we end up falling out but it was implicit in his manner.
'Oh. Er...okay.'
She tried not to sound disappointed, but obviously failed because he said, 'It's for the best, Soli,' in a tone that brooked no argument.
'I'm not looking for you to fall in love with me or anything, you know,' she said without stopping to think too hard about what she was saying. 'We could keep it as a purely physical thing. No strings.'
'I'm not sure those arrangements ever work,' he said, his expression radiating extreme scepticism.
Even though her brain was telling her he might have a point, her libido didn't want to listen. They'd crossed the line now and there was no real way back, it pointed out. In fact, it actually felt to her as if she'd turned a corner in her life after sleeping with Xavier—that she'd finally stepped into adulthood. Now she'd pushed herself over the ledge and found her heart had survived it, she was confident she was mature enough to keep sex and love separate.
'We'd make it work. We both know we're not in this for the long haul. It's just temporary. And it'd make living together much more fun and much less frustrating.' She flashed him a smile, which he returned. 'Especially since we agreed not to see other people during the time we're married,' she added.
There was a short pause while he appeared to consider this.
'No. I don't think it's a good idea,' he said eventually and she could tell from the resolute look in his eyes that he'd made his mind up and was unlikely to change it.
The businessman was back.
Her heart sank.
'Okay, whatever you think,' she said brightly to cover her frustration. Perhaps he was right. It might make things more complicated.
But it didn't stop her from hoping he'd change his mind.
* * *
The journey home was just as luxurious and comfortable as the one out there, except for a strange sort of stiffness between them that hadn't been there previously. They'd been really careful around each other ever since they'd had that conversation about not having sex again, as if they were tiptoeing around a bombshell that could trigger the second they relaxed and took a misstep and it was making both of them act in an over-the-top, super-polite way towards each other.
So by the time they walked in through the front door to the Hampstead house, Soli was completely and utterly exhausted from nervous tension.
So much for having a relaxing holiday.
Xavier immediately excused himself and she found herself alone again in the kitchen, fixing herself a meal for one, wondering whether her life would ever be normal again.
She came to the conclusion that it was very unlikely. Especially now she knew how wonderful it could be to be wanted by Xavier McQueen. She'd so enjoyed being able to get close to him. They'd been good together in bed. He was certainly a lot better than the men—well, boys really—that she'd slept with in the past. He'd known exactly what to do to give her the most pleasure and had been incredibly attentive to her needs. She'd never experienced anything like it. And she wanted more. Much more.
The idea of living with him until the year was up, with this sexual tension throbbing between them the whole time, and not being allowed to act on it, made her stomach turn over with restlessness. They'd go insane, surely.
They'd be much better to give in to their physical urges and ride things out till the marriage was up. They'd probably be bored with each other by then anyway.
Refusing to listen to a niggling little voice that told her not to bank on that, she decided to keep herself open to the chance it could happen, but not push for it in any way. Xavier was definitely the kind of man who needed to feel in control of his decisions—and his destiny—so if it was going to happen it would need to come from him.
It would be fine with her either way.
Absolutely fine.
To keep herself occupied, so she wouldn't go crazy thinking about it all day, she went into the café to check everything had run smoothly without her.
After being away for a while she found to her shock and distress that the place seemed shabby and cluttered to her now, and the warm, cosy atmosphere that she'd been hanging on to as the excuse not to change a thing about it was sadly lacking. It had all been in her head. A phantom of the past.
She'd been desperately hanging on to her father's vision for the place, to try and keep a part of him alive, but it was actually holding the business back. Destroying it, in fact.
He never would have wanted that.
She knew now, with absolute clarity, that it was time to let go. She needed to stop being afraid of the future and allow herself to finally move on.
It was time to make some changes.
* * *
It took two more days of Xavier making himself scarce in the evenings and acting all stiff and formal with her again before Soli's resolve to be cool and indifferent about how their relationship would go from now on snapped.
'Are we really back to you treating me like a piece of furniture again?' she bit out in frustration the morning of the third day when Xavier swept into the kitchen, poured himself a coffee and gave her a polite nod before starting to retreat outside with it.
He turned back, then carefully put his mug down on the nearest work surface, the coffee slopping over the edge as if his hand had been trembling.
'That's not what I'm doing, Soli,' he said quietly, his expression surprisingly tortured.
Her stomach flipped at the sight of it, but she held her nerve. They really needed to address this and the sooner the better.
'Look, I get that you don't want to have a physical relationship with me, but I'd really prefer it if you at least acknowledged my presence in the house. We can be friends, surely?'
The muscle in his jaw was working overtime. 'It's not that I don't want a physical relationship with you, Soli,' he ground out, taking a step towards her, his shoulders rigid and his eyes flashing with frustration, 'it's that I do.'
'You do?' she whispered, shocked by the passion in his voice and the intensity in his eyes.
'Yes,' he said, letting out a low, frustrated-sounding breath.
'Oh. Okay. I see.'
'And seeing you every day but not being able to touch you, when I know how good we can be together, is driving me insane.' His chest rose and fell in rapid movements as if he was fighting for control and barely keeping it together.
The idea that she was doing that to him thrilled her to her core. Without thinking, she took a step closer to him, unable to fight the instinct to push him a little bit further to see what he'd do.
They stared into each other's eyes, their breath coming fast and their bodies tense.
Xavier made the first move, closing the gap between them and sliding his hand into her hair, then pulling her towards him so her mouth met his. The kiss was rough and full of the longing they'd clearly both been battling since the last time they'd given in to the need to touch each other.
He let out a low growl in the base of his throat as she wound her arms around him and pressed herself closer to him, feeling exactly how much he wanted her right then.
She pulled back from the kiss to look him in the eyes again, wanting to make sure they were both thinking the same way before this went any further so they didn't end up in an even more awkward position later.
'I thought you said—'
'I know what I said, but I decided not to listen to myself for once,' he ground out, his brow pinched in a half-relieved, half-frustrated sort of frown. His eyes were alive with pleasure though, which her body responded to by sending a throb of pure desire through her.
She couldn't help but grin at him. 'Good decision. Let's be wild. It's a healthy life choice.'
'I don't know about healthy...' His words tickled her lips.
'Liberating, then. We're grown-ups. We can handle it. Both of us know the score and we're not looking for anything in addition to our agreement.'
'You mean that, right?' His expression was deadly serious now.
'Yes,' she said, matching his seriousness with her own. 'I absolutely do. I promise.'
His hands had been tight on her back, holding her against him, but they relaxed a little as something seemed to occur to him. 'Still, we need to be sensible, so we should use condoms too from now on, just to be extra-safe,' he said.
She nodded. 'Fine by me.'
And it was fine, because the last thing she needed right now was an added complication in her life.
CHAPTER ELEVEN
Battleship—sink your opponent with a blow to the heart.
XAVIER WAS LATE for work, for the first time ever, after taking Soli to bed then having a lot of trouble forcing himself to get out of it again and leave her there, all warm and sexily rumpled.
It had definitely been worth it though. More than worth it. All the pent-up frustration and jagged need that had been stressing him out—to the point where he was having trouble concentrating at work—completely evaporated the moment he let himself give in and do what he'd been desperate to do again ever since that incredible night in Corsica.
Ever since they'd crossed that line it had been inevitable they'd not be able to go back to their platonic existence.
He felt pretty confident it would be okay though. They'd had another talk about sex not changing anything about their deal and had both agreed they were still happy with that. Their physical relationship would only last as long as the marriage; after that they'd both walk away.
A clean break.
Now the thought of having Soli at home, waiting for him every evening, had a very different impact on his mood.
He even started coming home from work earlier, looking forward to spending his evenings with her, eating her wonderful food, or seeing the look of astonished approval when he insisted on cooking for her instead.
Life was pleasingly satisfying and straightforward, with his rapport with Soli on an even keel, both knowing exactly where they stood with each other.
There was no pretence and no underlying conflict of interest.
It was the most simple and rewarding relationship he'd ever had.
She seemed to know just how to pull him out of one of his funks with a few choice words and a smile, and she charmed him with her positivity and humble joy. For those few weeks he felt as if he was finally living the life he'd always wanted.
A few days after they decided to ignore their no-sex agreement Soli moved into his bedroom—which he'd suggested, saying it was ridiculous for her to return to her own room every night.
He loved waking up with her beside him. It had been a long time since he'd shared his bed like this and he was surprised to find how much he'd missed it.
Setting off for the office bright and early one morning, he turned back from opening his car door to see Soli standing on the doorstep, still wearing the short silky pyjamas she looked so damned alluring in, blinking in the sunshine.
'Where are you off to so early?' she asked, carefully picking her way over the gravel towards him in her bare feet.
He met her halfway and lifted her into his arms so she could wrap her legs around his middle and save the soles of her feet from the sharp stones.
Walking back to the car, he put her down gently onto the bonnet and leant in to kiss her, breathing in the enticing, sleep-warmed scent of her.
'I'm off to the office. I want to get everything that needs doing today finished in good time so I can come home early and take you out for dinner. I've booked that new place on Hampstead High Street that you mentioned the other day.'
'Really?' she said, beaming at him. 'That's so thoughtful of you.'
He smiled back. 'It's mostly a selfish move on my part. I really fancied checking it out too.'
Leaning back in, she kissed him hard and he felt her smile against his lips.
'What are you going to do today?' he asked when they broke apart.
'I'm going in to the café. The marketing I've been doing seems to be paying off and we've seen a real increase in business recently.' She took a breath and he felt her fingers dig into his back as she tightened her grip on him. 'And I've decided it's time to give the place a new fit-out.' Her brow pinched in a frown as if she found this idea troubling. 'I can't keep clinging on to my father's vision for the place. It's looking so shabby now and needs modernising.' Her eyes welled with tears. 'Time to let go of the past and look to the future.'
'I think that's a good decision,' he said, running a finger gently under her eye to brush away a tear. His heart gave an extra-hard beat in his chest as it suddenly occurred to him that she was talking about a future he'd have no part in. His time with her would quickly slip away and at some point soon so would she.
'You know, I think my time with you has really made me grow up and look at life from a new perspective. So thank you for that,' she added with a sad sort of smile.
'I'm glad to have helped,' he said roughly, leaning in again and kissing her hard to try and disguise the troubling emotions that were now raging through him. Emotions he didn't know what to do with.
She let out a small sigh of contentment and pressed herself harder against him.
His mind went mercifully blank. All he wanted to think about right now was the feel of her soft, supple body against his and the luscious, honeyed taste of her in his mouth.
So it was a few seconds before he realised there was someone standing on the driveway with them, clearing his throat to politely get their attention.
Pulling away from Soli, he turned his head to see Samuel Pinker, his great-aunt's spy, looking back at him with a slightly sheepish look on his face.
'Sorry to interrupt you. The front gate was open so I thought it'd be okay to come right in. I have some documents from your great-aunt's solicitor to drop off and as I was in the area...'
They nodded politely, all of them patently aware that it had actually been another ruse to check up on the two of them, as per Aunt Faith's decree.
'That's quite all right, Mr Pinker,' Soli said with a grin in her voice. 'I was just sending my husband off to work in the best possible way I know.'
Mr Pinker cleared his throat again as a red flush appeared to creep up his neck and brighten his cheeks. 'Your husband's a very lucky man, Mrs McQueen,' he replied, 'and could I just say, I think the two of you make a lovely couple?' He turned to look at Xavier. 'Your great-aunt would be very pleased to see you so happy, Mr McQueen.'
And with those incisive words hanging in the air, he handed the envelope he was carrying to Xavier, gave them both a friendly nod of goodbye and strolled away back towards the open gates.
Soli couldn't help giggling. 'I don't think we need to worry about him thinking we're not behaving like a real married couple any more,' she said with a grin.
Something about the way she said this, on top of Pinker's parting shot, made a shiver of discomfort rush across his skin. Were they acting like a married couple? He wouldn't have said so. They were just enjoying each other's company whilst they were forced to live together. In order to be convincingly married there would have to be a palpable emotional connection between them as well as a physical one. Which he didn't think they had.
Did they?
He mentally shook himself. It wasn't important. They both knew the score. At least, he was pretty sure Soli did. Though from what she'd just said he was a little worried now that she was allowing herself to become more emotionally attached to him than was wise.
That wasn't what he wanted at all.
Was it?
No. That wasn't what he'd signed up for.
His chest suddenly felt tight.
He needed to get out of there and off to work before his concern became apparent to Soli and potentially started a conflict he really didn't want to deal with right then.
'Anyway, I'll see you later,' he said, forcing himself to give her an unconcerned smile.
'Looking forward to it,' she replied, not seeming to notice the sudden tension in the air. 'Have a good day, darling!'
* * *
Soli strolled back into the house, closed the front door and stopped abruptly in the hallway as a sudden horrifying knowledge that she was going to be sick assaulted her.
Her stomach lurched and churned as she dashed to the nearest bathroom, only just making it there in time.
The bug she'd had in Corsica couldn't have got her again, surely. She'd only had it a couple of weeks ago.
Was that right?
No. Hang on, it couldn't have been that short a time ago.
Life had been such a whirl; she'd barely had time to notice how the weeks had flown by. It seemed like only yesterday that they'd been in Corsica.
But it wasn't. It was at least six weeks ago.
A cold sensation trickled down her spine as something alarming occurred to her.
She should have had her period by now. She remembered spotting a bit a couple of weeks ago, which wasn't unusual because of the pill she was on, but she'd forgotten to write down when her last proper period was.
Her heart gave a painful thump in her chest and she dragged in a breath as she felt suddenly dizzy.
She couldn't be pregnant though. Could she? They'd used double protection, apart from that time in Corsica when they'd just relied on her pill.
Which she'd not taken for the following two days after they'd had sex because she'd been so ill.
But there was no point panicking. She needed to be sensible and keep a cool head about this.
Pulling herself together, she got dressed then went straight out to the nearest chemist for a pregnancy test, using it as soon as she got home. Just because some things pointed towards her being pregnant, it didn't mean she actually was, she reminded herself as she waited for the results to show. It could be another bug, or stress or—something else that her whirling brain couldn't think of right now.
The lines appeared in the little window.
It wasn't anything else.
She was pregnant.
Sitting slumped on the bathroom floor, staring at the little white stick in her hand, she felt hot tears press at the backs of her eyes as her stomach dropped to the floor.
This wasn't supposed to happen. This wasn't the plan.
What was Xavier going to say? He'd been so definite about this thing between them being a sex-only arrangement.
But perhaps he'd actually be happy about it, a determined little voice in her head whispered. He'd told her that he'd wanted to have kids before Harriet had left him, after all.
She took some deep breaths and forced herself to calm down.
It could be a wonderful thing for both of them. They were good together. They worked, both in and out of the bedroom. He made her laugh and feel good about herself. She seemed to bring out the best in him. He'd even told her that a couple of times in the last few weeks—perhaps not in those words, but he'd certainly alluded to it. He enjoyed her company, and she his.
But could she really have a child with someone who didn't love her?
Someone who didn't love her as much as she loved him?
Because she knew, in that moment, without a doubt in her mind that there was no point denying it to herself any more.
She was in love with him.
Completely and utterly.
She let out a cynical, broken laugh. So much for being mature enough to deal with a sex-only arrangement. She'd been kidding herself this whole time.
She'd not wanted to admit it to herself before now because she'd been terrified that he didn't feel the same—he was so guarded with his emotions it was difficult to know how he really felt sometimes.
But she knew what she really wanted.
She loved him and wanted them to raise this baby together. For them to stay married after the year was up.
But would he feel the same way?
Could he?
She got up from the floor and brushed herself down with trembling hands.
There was only one way to find out.
* * *
Soli was in the sitting room, anxiously biting her thumbnail, when she finally heard Xavier's car pull into the driveway that evening.
Getting up from the sofa, she went to meet him in the hall on trembling legs. Her heart was racing so hard she felt light-headed.
The door opened and, almost in a dream, she watched Xavier stride inside.
Her stomach flipped right over when he flashed her the smile she loved so much.
'Hi,' he said, dumping his laptop case by the hall table. When he straightened up to look at her the smile fell from his face and was replaced with a concerned frown.
'Are you okay? You look a bit pale. You're not ill again, are you?' He walked towards her and put his hand on her forehead. 'You don't feel hot.'
The compassion of this gesture gave her the confidence to blurt out what she needed to tell him.
'I'm not ill, I'm pregnant.'
It seemed to take him a couple of seconds to process what she'd said, then, shockingly, he took a swift step away from her.
'What?' he said, his voice as hard as stone. 'How? We've been using condoms.' The expression in his eyes was wild as he stared at her in angry denial.
Her heart plummeted, provoking a fresh wave of nausea.
That was not the joyful reaction she'd been hoping for.
She swallowed hard, telling herself he was just in shock and to give him a minute to process it. 'I did a test. It came out positive.' She took it out of her back pocket to show him. 'I think it might have been that night in Corsica.' She dragged in a shaky breath. 'Or maybe one of the condoms failed and my pill wasn't working properly again after I'd been ill.'
He shoved his hand roughly through his hair. 'This wasn't part of the deal, Soli,' he bit out, his voice icy cold and the expression in his eyes so hard it made her shiver.
A feeling of pure dread flooded through her body.
'I take it you want to keep the baby?' he asked.
'Of course I do,' she gasped, horrified that he'd even think for one second that she'd consider not doing so. 'But if you don't want to be involved I'll find a way to look after it on my own,' she shot back angrily, wounded by his harsh reaction.
'The hell you will,' he snapped.
'What's that supposed to mean?'
'It means there's no way I'm going to abandon my child. Or its mother.'
'But you've been so sure about not wanting to stay married after the year's up,' she pointed out, unable to keep the hurt out of her voice.
He just shrugged. 'We'll work something out. I can speak to Russell and get him to make some amendments to the contract,' he said, his stiff practicality sending little shocks of dismay through her.
'You want us to continue having a contract?' she said, appalled by the thought of it. Had it really come to this? Was he going to keep treating her—and now their child—like a commodity? Just as his parents had done to him?
'Isn't that what every mother wants?' he said coolly. 'To know that she and her child will never have money troubles again?'
His dispassion shocked her. She'd seen him flip into business mode before but this was a step way beyond that.
'No! That's not what I want! I want a partner who loves me. Who wants to be with me for me. Not because he feels he has to because we have a child together.' She wrapped her arms around her body, aware that she was shaking now. 'We'll only end up resenting each other. It'll be hell living in the same house and I can't let my child grow up in that sort of toxic environment. I can't spend the rest of my life with you if you don't love me back, Xavier. Because I love you!'
He wasn't looking at her now, but down at the floor instead, with a muscle flicking in his jaw.
'Do you love me?' she whispered brokenly, her heart thumping painfully in her throat.
He didn't answer her right away, but even before he opened his mouth to speak she could tell what he was going to say from the tension in his body language.
'I'm not sure I'm capable of loving anyone any more.'
The expression in his eyes was as hard as marble when he finally looked at her.
Grief squeezed her chest, stealing her breath away.
'If you can't love me, I can't stay here with you any longer,' she said, her voice barely making it past her lips. 'I'll make sure this child has all the love it needs and a happy life, but I'll do it without you and your money.'
She went to sweep past him but he put out a hand to stop her. 'Soli, don't be ridiculous—'
Her whole body was suddenly hot now. Burning with anger. 'Don't call me ridiculous! I'm sick of hearing that from you. I know you think I'm just some naïve idiot with money problems, but know this: I'm capable of being happier than you'll ever be because I let people into my heart and treat them with genuine respect—like an equal. If you can't get past that block you have, if you're not willing to, then there's no hope for you, Xavier McQueen. No matter how many houses you own or how much money you have, you'll never be happy if you can't learn to let go and fall in love again, to trust and share your life with someone else. With me!' Her voice broke on the last word but she fought back the tears, determined not to cry.
Letting out a loud, painful-sounding sigh, he sat down on the edge of the hall table and looked up at her with such cold resentment she felt a shiver run down her spine.
'Where do we go from here?' she whispered, panic rising in her chest.
He frowned and shook his head. 'I don't know, Soli. I just don't know. I don't think I can give you what you want.'
She saw his throat move as he swallowed but his gaze remained implacable.
After what felt like a lifetime of silence, where it became plain he wasn't going to tell her what she needed to hear, she realised the only thing she could do now was get out of there, just so she could calm down and think straight again. She needed to figure out what the hell she was going to do without him.
'Okay. Well, it's clear how you really feel about me.' She coughed to ease the painful tension in her throat. 'So I'm going to go home,' she said shakily, willing herself to hold it together, just for a few more minutes.
He didn't say anything as she turned and walked away. He didn't follow her up to the room they'd shared so happily for the last few weeks and he wasn't at the front door when she let herself out with a small bag she'd packed.
In short, he made it crystal clear that he wasn't going to stop her from leaving.
CHAPTER TWELVE
Game of Life—spin the wheel of fate for a chance at the life you want.
IT WAS VERY QUIET in the house once she'd gone.
Xavier paced the floor for what felt like hours, trying to reconcile his thoughts about what Soli had said to him. And what he'd said to her.
Her out-of-the-blue revelation had blindsided him—his harsh words a reaction to the crushing fear that he'd been wrong to relax around her. He'd been totally unprepared for how to handle her unexpected and shocking news, and something dark and instinctive had reared up from deep inside him, making him lash out at her. To protect himself.
The whole time they'd been sleeping together he'd been afraid of this happening, but he'd pushed it away, not wanting to dwell on it, telling himself he was worrying for no reason. He thought he'd been so careful, so clever, using extra protection and making sure they were both on the same page. But he'd forgotten about Corsica. The one weak spot in the whole game.
A restless sort of dread lay heavily in his stomach as he paced around, his veins on fire with adrenaline and terror.
He couldn't give himself fully to her, not in the way she wanted. He'd protected himself from loving anyone else ever since Harriet had torn his heart to shreds and he'd seriously believed he wasn't cut out for that sort of relationship with anyone ever again. He was his parents' son, after all.
And he'd managed to keep his feelings under control and his heart protected, until Soli had come along and turned his world upside down.
Letting her in had been such a gradual process, he'd barely noticed it. He thought they were just having fun, but from the way he was feeling now that she'd walked out on him it was clear she'd worked her way firmly into his affections.
He spent a rough night, barely sleeping a wink, Soli's sweet scent on his sheets haunting his dreams when he did sleep.
Getting up groggy and tired the next day, with his head heavy and tight with stress, he made his way down to the kitchen in the hope that a strong cup of coffee would help him think straight.
Just the sight of the empty room where he and Soli had spent so much time enjoying each other's company made him want to punch a wall in frustrated regret.
What was he going to do? How would he handle this? He suddenly desperately wanted her to be there to talk to—to see her kind, reassuring smile again. Supporting him. Caring for him.
Not that she was ever likely to be doing that again after the way he'd treated her.
Did he really believe he couldn't love her?
He didn't know any more—his head felt as if it was stuffed with cotton wool, and his blood was like sludge in his veins.
Dropping his head into his hands, he let out a loud, frustrated sigh and sank back against the work surface, feeling the hard ridge of it digging in to his spine, but he didn't move; instead he revelled in the pain it caused him, glad of the distraction from the more problematic pain in his heart.
* * *
The next few days were hell.
Soli kept her word, staying away from the house, and from him.
He'd thought it would be okay, that it would be hard but he'd be able to cope without her there, but he felt sick every time he came downstairs and found she still hadn't returned. The house was so silent and dark without her, as if she'd taken all the life and colour of the place with her when she left.
Most distressingly, the house no longer felt like his home.
There was a constant, tight ache in his chest, which he accepted, when he finally allowed himself to acknowledge it, was because he missed her.
He missed her like crazy.
The following Saturday he woke up early, the sense of doom he'd been carrying around with him since she'd gone weighing on him more heavily than ever.
Once downstairs he found he couldn't settle to anything. It was too quiet, too still in the house, so he grabbed his jacket and walked to Hampstead Village in search of something to distract him.
He was just passing some seating outside a coffee house when he heard someone calling his name. Turning to see who it was, he felt his pulse leap as he saw Harriet sitting in one of the chairs, cradling a tightly swaddled baby.
'Harriet. How are you?' he asked, his voice a little unsteady with a strange kind of yearning that had swelled in him at the sight of the child in her arms.
'I'm really well, thanks. Meet Harry, my son,' she said, beaming down towards the baby.
'Hello, Harry,' Xavier said, bending down to look more closely at the tiny human in her arms, fighting back an intense surge of emotion that was threatening to engulf him.
Would his and Soli's child be a boy? he wondered wildly, his heart thumping hard at the thought.
'Where's Soli?' Harriet asked, as if sensing his turmoil.
'She's at home,' he lied, standing up straight again, pain throbbing hard through his chest at the sound of her name.
'I guess the two of you might be having your own little bundle of joy soon,' Harriet said with a twinkle in her eye. 'I always thought you'd make a fabulous father, despite what I said all those years ago.' She appeared to swallow and blink, as if suddenly uncomfortable. 'Listen, I just wanted to say sorry for the way I treated you back then,' she went on before he could respond. 'It was an incredibly selfish way to behave.' Her smile was full of what looked like genuine regret now. 'I was afraid you didn't really love me for me, you just thought I ticked all the boxes for the sort of woman you thought you should be marrying.' She hugged her son a little tighter to her. 'But I guess it all worked out for the best. You and Soli looked so happy together when I saw you at the party.'
'We were,' he said, realising with a shock that he was actually speaking the truth now. He had been happy then. 'And I forgive you for what happened with us. You were right; we wouldn't have been good for each other. We probably would have made each other miserable.'
And he really meant that too.
Because he recognised now that Soli was right for him. She'd brought him back from the brink of despair and helped him realise that he wasn't like his parents at all; that he was capable of loving someone other than himself. Soli made him happy because she truly cared about him, for him, just as Great-Aunt Faith had. The warmth that the two most important women in his life had bestowed on him had instinctively made him feel secure. Wanted.
Loved.
Which was what he'd really been trying to hang on to all along. Not the house, but the security he'd thought it had represented.
He turned away from Harriet to stare at the empty space beside him, remember how he'd grown used to turning to find Soli smiling at him and how it had felt like being given a shot of adrenaline straight to his heart.
His life was empty without her.
Dragging in a deep, fortifying breath, he turned to glance in the opposite direction, towards the rental units he owned on the High Street, suddenly so clearly knowing exactly what he wanted.
He wanted Soli back. And he wanted their child too. He wanted them to be a family—something he'd always longed for, but had previously accepted he'd never have. Until now. Until her.
So what the hell was he doing still standing there?
Blood pulsing hard through his body, he said goodbye to Harriet and started off in the direction of the board game café. He was going to talk this thing out with Soli once and for all. Exactly what he was going to say, he wasn't sure, but he felt confident it would come to him as soon as he set eyes on her.
He knew some sort of grand gesture was in order if he had any chance of winning her back though. He'd done too much damage with his selfish silence to just expect her to listen to his demands.
She was too self-possessed for that.
He needed to find a way to prove to her that he was genuinely sorry and that he meant it when he told her he couldn't live without her. That he wanted them to be a real family, something he'd never had before—had never felt worthy of—but had ached for his whole life.
But what if she turned him down? What if she just laughed in his face?
His pace slowed as the idea rattled through him and he stopped and leant a hand against the window of the nearest shop as a dark kind of fear seized him.
That was probably what he deserved after the way he'd treated her, but could he really put himself through that sort of humiliation again? It had nearly killed him the last time it had happened to him.
On the other hand, was he really prepared to lose everything he'd ever wanted because he was too afraid he might lose it later? That made no sense at all. His damn stupid pride was getting in the way of his happiness, and Soli's too, and he couldn't allow that to happen. She was the only person that had never let him down.
She loved him. That was abundantly clear.
And he knew without a doubt that she'd make an incredible mother to his child and the most loving, caring partner he could ever hope for.
Even if he struggled to get the loving thing right, he knew that Soli would be there the whole way, backing him up and evening him out. They would make a brilliant team.
He wanted her back. The woman who had helped him live again and appreciate everything he had for more than just its monetary worth.
The woman who made him happy.
The woman he loved.
He knew with absolute clarity now that the house meant nothing to him if she wasn't in it. It was just bricks and mortar, full of ghosts and regrets—an empty shell without her.
She, and her love and affection, was what he'd really needed all along.
The house was his past, but she was his future. She and their baby.
With unwavering resolve making his heart thump hard in his chest, he started walking swiftly towards the café again, prepared to do everything in his power to save their marriage.
He was in love with Soli and he was going to do whatever it took to prove it to her.
* * *
Soli was practically dead on her feet.
She'd not slept properly since leaving Xavier's house—the place she'd previously begun to consider her home too until she'd been made to feel supremely unwelcome there—and it was really beginning to show.
Even her mother had noticed her lack of energy and positivity in the last few days.
Xavier's initial, shocked reaction to the pregnancy had been understandable—she recognised that—but to reject her love, then let her leave and not contact her again, was just plain heartless.
To be perfectly honest, it had broken her heart.
But then, she'd not contacted him either, she accepted with a thud of despair. She'd been too afraid to in case she saw that look of cool reproach on his face again.
Sighing loudly, she picked up a mug from a tray of newly washed crockery and rubbed the tea towel over it. It was all very well holing herself up here, but she was going to have to face him again eventually so they could work this thing out between them. She'd signed a contract to stay married to him for the next few months after all and there was no way she was going to welch on their deal. She wasn't a quitter.
She was just finishing this thought when a familiar and stomach-twisting scent hit her senses.
Looking up, she found herself staring into the dark intensity of Xavier's gaze.
'Hi, Soli,' he said quietly.
'What are you doing here?' she blurted, completely losing any poise she'd coached herself to exude when next confronted with him.
'I came to talk to you.'
'Here?' she said, glancing around her, unable to keep the incredulity out of her voice.
The corner of his mouth twitched up. 'Yes. I figured it's as good a place as any.'
'It's not exactly private,' she said, gesturing round at the tables full of customers and at her sister and mother, who were currently drinking coffee at the other end of the bar, celebrating a successful end to Domino's first term at university. The two women were now watching them, their interest clearly piqued.
'That's my sister and mother over there,' Soli hissed under her breath so they wouldn't hear her.
'Have you told them about us?' he asked.
'No. Not yet.'
He gave a sharp nod, but didn't move away from the counter. 'Well, we could go somewhere a bit quieter if you like.'
Suddenly she didn't want to move away from behind the counter. She wanted to make him pay for his indifference towards her by forcing him to say whatever he had to say in front of all these people. To hell with his manly pride. She didn't need to take his feelings into consideration any more, especially when he obviously had no intention of protecting hers.
'No. Whatever you have to say to me you can do right here. I'm busy doing my job—I don't have time to leave just so you can reject me again.' She folded her arms and stared him down, determined not to let him win this time.
She might love him, much more than she was willing to admit right now, but she was damned if she was going to let him hurt her any more than he already had.
She and the baby would be fine without him and his money. She had enough love in her to count for two parents.
Out of the corner of her eye she noticed her sister was still staring at them both.
'Soli, did this guy break your heart or something? Is that why you've been moping around looking so broken since I got back?' Domino called out.
Obviously they weren't being as discreet as she'd hoped.
She swallowed and was about to answer when Xavier spoke over her.
'Yes, I'm afraid I did. And I came here to apologise and tell your sister how much I love and miss her.'
He turned back to face her and looked directly into her eyes.
She just stood there dumbstruck, with blood rushing in her ears, wondering if she'd misheard him.
'I'm sorry,' he said, with such emotion in his voice she knew immediately that he genuinely meant it. 'I'm so sorry for hurting you.' He put his hands on the counter and leant in closer to her. 'I don't work properly without you. You've changed me, changed the way I think about life and what love really is. I don't deserve you, Soli, I know that, but I want you back. I want us all to be a family.'
'Really?' was all she could think to reply to that, her scrambled brain slow to catch up with everything he was saying.
'Yes. Really. I got spooked when you told me about the baby because I'd promised myself I wouldn't fall for you and it made me realise that I had. That I wanted you more than I should. I've been a complete idiot. You're the best thing that's ever happened to me—you and the baby.'
There was a loud gasp and a squeal from the direction of her mother and sister, but Soli ignored it, her attention focused solely on Xavier now.
'You really want me back?' she asked shakily.
'Yes,' he said with more determination in his voice than she'd ever heard before. He moved round to the opening in the counter and took her hand in his, gently tugging on it to ask her to follow him out to the floor of the café.
When they were both standing in front of the now mostly silent room—since the whole of the clientele appeared to have stopped playing their games to stare at them instead—Xavier dropped to one knee and looked up at her with such intensity in his eyes her heart nearly flew out of her body with excitement. His hand was shaking as he held onto hers tightly, their palms pressed hard together, and she realised with a shock that he was scared. He was afraid she would reject him, but he was still going through with this, in front of all these people.
He wasn't hiding any more.
He swallowed, looking as though he was having trouble starting his sentence, but his voice was loud and clear when he spoke. 'As you're aware, I find it really hard to trust people, but I know—I guess I've always known, deep down—that I can trust you, Soli. I love the idea of you being the mother of my child. You're the most wonderful, caring, compassionate person I've ever met and it would be a huge privilege to spend the rest of my life with you.'
There was a low murmur from the crowd, but he didn't let it distract him.
'I promise you, I will do everything in my power to make you happy. I'll give you everything I have, everything, if you'll come back home with me.'
She could barely breathe as the reality of his words began to sink in.
'I love you, Solitaire McQueen. Will you marry me.' He paused and smiled before adding, 'For real this time?'
Soli thought her heart might explode with happiness.
Xavier loved her and he wanted them all to be a family. A real family.
'I love you too,' she sobbed, unable to control her emotions any longer. 'And yes,' she took a steadying breath, 'I'll marry you.' She grinned through her tears. 'Again.'
He kissed her hand before standing up and pulling her into his arms, where he held her to him so tightly she could barely breathe, before finally drawing away just far enough to kiss her deeply, with the love and emotion she'd always known he had in him.
'We'll make one hell of a team, Mrs McQueen,' he murmured against her lips, and she knew, right then, that there would be no more games.
She'd won the ultimate prize.
EPILOGUE
Bingo—hold your nerve for the ultimate prize.
Five years later
'FOUND YOU!' CAME the excited voice of four-year-old Faith McQueen as she pulled back the curtain to reveal her mother crouched on the window seat behind it.
'Darn! And I thought I had the perfect hiding place,' Soli said with a grin, ruffling the hair of her eldest daughter and pulling her in for a kiss.
'Mum, let go—I still need to find Dad and Joy,' Faith said, struggling out of her arms, but not before dabbing one last wet kiss on Soli's cheek.
'Okay, go ahead,' Soli said, unfolding herself from her cramped position with a sigh of relief and heading back to the kitchen, where she still had some last-minute preparations to do for Xavier's birthday meal.
She'd made one of her famous chocolate fudge cakes for dessert and she needed to get it out of the oven now so it'd be absolutely perfect.
It wasn't long before Xavier came striding in through the door with two-year-old Joy on his shoulders and Faith clinging around his waist with her feet on top of his, doing the penguin walk.
'You seem to have a new hat and pair of trousers,' she joked. 'Did you get them for your birthday?'
She laughed as Xavier shot her a rueful look while he tried, and failed, to prise his children off him.
He was amazing with them, as Soli had always suspected he would be, and they'd both become real Daddy's girls to prove her right. He clearly adored them and had even hinted he'd be very happy to have more kids if she thought she might too.
'Where's everyone else?' Faith asked, her eyes round with excitement when she saw the kitchen table laden with the special birthday meal.
She was talking, in part, about Soli's mother, who had moved in with them and taken the downstairs bedroom when her Parkinson's had worsened and it had become impossible for her to manage stairs. It was a great arrangement because it meant the girls could spend lots of time with their grandmother, and she absolutely adored having them around her. She often said to Soli that they kept her young.
After acing her degree at Oxford, Domino had gone on to do a pure maths MSc, and was starting a PhD at King's College in London, so had moved in with them too last year, latterly bringing with her her boyfriend, who was another PhD in her department and deeply devoted to Soli's brilliant sister. If the two of them had kids, Soli suspected they'd be so smart they'd go on to solve all six of the remaining mathematical Millennium prize projects or something.
The house was no longer the silent show home it had been when she'd first moved in. It was now always full of life and laughter, which Xavier happily asserted would have delighted his great-aunt Faith, and openly admitted that he loved too.
Soli had sold the business on to another board game enthusiast when she fell pregnant for the second time and had decided she wanted to be able to spend more time at home with both the children and her mother. She didn't regret it for a second. It had always been her father's dream really, and it had been high time she'd chased her own.
To that end, she'd enrolled in college to study fashion design, something she'd really enjoyed at school, and was now in the process of setting up her own clothing line, which she'd decided to call Solitaire.
It was all up for grabs, which was exactly the way she'd come to like it.
'Speaking of birthday presents—' she said to Xavier once he'd finally freed himself from his daughters' vice-like grip.
'You know I didn't want you to get me anything.' He gave her a mock-stern frown, then pulled her hard against his body, wrapping his arms around her. 'Just this,' he said, dropping his mouth to hers for a long, sweet kiss.
'Well, I'm afraid I ignored your instructions,' she teased, when they finally broke apart, flashing him a smile.
He let out a long, fake sigh. 'Why am I not surprised?'
'I thought you might like this though,' she said, pulling out a long, flat package.
He frowned, taking it from her and pulling off the wrapping paper to reveal a pregnancy testing kit.
'You're pregnant again?' he asked, his eyes lighting up with excitement.
'I am,' she said, grinning as he pulled her back against him and planted little kisses all over her face.
'Well, that's the best birthday present I've ever had,' he said gruffly, grinning from ear to ear.
'You realise there are going to be even more years of hide-and-seek to get through now,' she joked, laughing as he began to kiss down her throat.
'That works for me,' he said, his voice muffled against her skin. 'I know all the best hiding places in this house now.'
She smiled, thinking how much he'd changed since she'd first met him. He was so full of love and happiness with his place in the world now she barely recognised the withdrawn, emotionally scarred man he'd been then.
Contentment swelled in her chest as she looked at her family gathered around her.
Right at that moment she couldn't have been happier. She had everyone she loved right there with her and an exciting future stretching ahead with Xavier by her side.
A future she was no longer afraid to face.
She was proud of the brave, strong and confident person she'd become, thanks to Xavier's belief in her.
In fact, they'd made a pact that there would be no more hiding from life—for either of them—ever again.
He'd been absolutely right: they made the perfect team.
* * * * *
If you enjoyed this story, check out these other great reads from
Christy McKellen
HIS MISTLETOE PROPOSAL
THE UNFORGETTABLE SPANISH TYCOON
A COUNTESS FOR CHRISTMAS
ONE WEEK WITH THE FRENCH TYCOON
All available now!
Keep reading for an excerpt from UNLOCKING THE MILLIONAIRE'S HEART by Bella Bucannon.
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Unlocking the Millionaire's Heart
by Bella Bucannon
CHAPTER ONE
NATE THORNTON SHOOK the rain from his hair with vigour before entering the towering central office block in Sydney's city centre. He'd had to reschedule planned video meetings to make the train trip from Katoomba at Brian Hamilton's insistence, and he'd been further frustrated by his evasive remarks.
'It has to be Thursday the ninth. I think I've found a resolution for your hero and heroine interaction problem. And there's a publisher who's interested in seeing a revised copy of your book.'
Late-night research had shown Brian Hamilton to be one of the best literary agents in Australia. After initial contact he had asked for, and read, Nate's synopsis and first three chapters, then requested the full manuscript. His brutal honesty on its marketability had convinced Nate he was the contract negotiator he wanted.
Attempts to rewrite the scenes he'd specified, however, had proved that particular aspect wasn't his forte. And when he'd been tempted to suggest cutting them out, the feeling in his gut had told him it wasn't that simple, and to ask if the agent could find a better solution.
It wasn't the possibility of income that drove Nate to his computer. Astute investment of an inheritance and a significant part of his earnings while working abroad meant he was financially secure for years. Or, as his brother claimed, 'filthy rich'—a phrase he detested. Although he envied Sam the satisfaction he'd achieved as a pilot in the air force, currently stationed at Edinburgh, north of Adelaide.
His compulsion to write had been driven by the need to put the hardships and traumas he'd witnessed as an international reporter where they belonged—in his past. Those harrowing images of man's inhumanity to man were still in his head, though for the most part he managed to keep them buried.
There was nothing he could do regarding the way he now viewed life and interacted with new acquaintances. The walls he'd built for his own emotional protection were solid and permanent.
Frowning at the number of floors all six lifts had to descend before reaching him, he punched the 'up' button and tapped his fingers on his thigh. Okay, so he wasn't so hot on the touchy-feely sentimental stuff. Hell, the rest of his hundred thousand words were damn good, and his target readers weren't romantic females.
No disrespect intended.
The street doors sliding open drew his attention. The woman who came in brushing raindrops from her hair held it. He had a quick impression of black tights, then a flash of blue patterned fabric under a beige raincoat as she unbuttoned and shook it.
His mind registered long brown hair, a straight nose and red lips above a cute chin—great descriptive characterisation for an author, Thornton—then, as their eyes met, he felt a distinct jolt in his stomach.
Dark blue eyes framed by thick lashes stared, then blinked. Her smooth brow furrowed, and she swung away abruptly to study the board on the wall. He huffed in wry amusement at having been dismissed as un-noteworthy—not his usual first reaction from women.
The lift pinged and he moved aside to allow an exiting couple room. Another quick appraisal of the stationary figure of the woman, and he stepped inside.
* * *
Brian's personal assistant had notified Brian of Nate's arrival, and in less than the time it took her to hang his damp jacket on a stand in the corner the agent was greeting him with enthusiasm.
'Punctual as always.' He peered over Nate's shoulder, as if expecting someone else. 'Come on in. Coffee?'
'Yes—if it's going to be rough and take that long.'
Brian laughed. 'It all depends on how determined you are to have a successful publication.'
He followed Nate into the well-appointed corner office, waved at the four comfy leather armchairs round a long low table and went to the coffee machine on a built-in cabinet.
'Strong and black, right?'
'Please.'
Nate sat and studied the view of nearby commercial buildings: hundreds of glass pane eyes, letting in sunlight while hiding the secrets of the people behind them. He'd need to be a heap of floors higher to get even a smidgeon of a harbour view.
'How was the journey down? Ah, excuse me, Nate.' Brian walked over to answer the ring from his desk phone, said 'Thank you, Ella,' then hung up and went to the door.
'I won't be a moment, then we can get started. Your coffee should be ready.'
Spooning sugar into the mug, Nate added extra, figuring he was going to need it. He heard Brian's muted voice, and a quiet female answer. Distracted by the sounds, he drank too soon, letting out a low curse when the hot brew burnt his tongue. This day wasn't getting any better.
'Come in—there's someone I'd like you to meet.'
A second later he was experiencing the same reaction as he had a few moments ago on the ground floor. The woman who'd caused it stood in the doorway, her stunning eyes wide with surprise. And some other, darker emotion.
The absence of her raincoat—presumably hanging up with his jacket—revealed a slender form in a hip-length, blue-patterned, long-sleeved garment with no fastening at the front. The black tights drew his gaze to shapely legs and flat black laced shoes.
This close, he appreciated the smoothness of her lightly tanned skin, the blue of her irises and the perfect shape of her full lips. Not so acceptable was her hesitation and the glance behind her. An action that allowed him to make out the nuances of colour in her hair—shades of his teak table at home.
One look at his agent's satisfied expression and his brain slammed into full alert. This young woman seemed more likely to be a problem for his libido than a resolution for his fictional characters' relationship. What the hell did Brian have in mind?
* * *
With Brian urging her in, Jemma Harrison had no choice but to enter the room, pressing the tips of her left-hand fingers into her palm. The man from the lobby seemed no more pleased to see her than she felt about him. Down there, with the length of the foyer between them, his self-assured stance and the arrogant lift of his head had proclaimed his type. One she recognised, classified and avoided.
She'd dismissed the blip in her pulse as their eyes met, swinging away before her mind could process any of his features. Now, against her will, it memorised deep-set storm-grey eyes with dark lashes, thick, sun streaked brown hair and a stubborn jaw. Attractive in an outdoor, man-of-action way. The tan summer sweater he wore emphasised impressive pecs and broad shoulders. He'd teamed it with black chinos and sneakers, and she knew her socialite sister, Vanessa, would rate him as 'cool.'
'Jemma, meet Nate Thornton. Nate, Jemma Harrison.' Brian grinned, as if he'd pulled off an impossible coup.
Jemma stepped forward as Nate placed his mug on the bench and did the same. His cool eyes gave no indication of his thoughts, and his barely there smile vanished more quickly than it had formed.
For no fathomable reason her body tensed as he shook her hand, his grip gentle yet showing underlying strength. A man you'd want on your side in any battle. A man whose touch initiated tremors across her skin and heat in the pit of her stomach. A man she hoped lived a long way from her home town.
'Hello, Jemma. From your expression, I assume Brian didn't tell you I'd be here, so we're both in the dark.'
Against her will, she responded to the sound of his voice—firm and confident, deep and strong, with a hint of abrasion. The kind of voice that would stir sensations when whispering romantic phrases in a woman's ear.
Oh, heck, now she was thinking like one of her starry-eyed heroines, and feeling bereft as he let go and moved away.
'Brian invited me to come in any time I was in Sydney. He didn't mention anyone else being here today.'
'I'll explain once you have a drink,' Brian said. 'Coffee, tea or cold?'
'Flat white coffee with sugar, please.'
She settled into one of the chairs. Nate retrieved his mug and dropped into the one alongside. She was aware of his scrutiny as she scanned the office she'd been too nervous to admire during her first appointment here. It was furnished to give the impression of success with moderation—very apt for the occupant himself.
Average in appearance, and normally mild-mannered, Brian let his passion surface when speaking of books, of guiding authors on their journey to publication and the joy of sharing their triumphs. In assessment he was never condescending, highlighting the positives before giving honest evaluation of the low points, and offering suggestions for improvement.
Why had he invited Nate Thornton to join them? She'd bet he had no idea of the romance genre, and wouldn't appreciate any relevant cover if she held it up in front of his face.
Brian placed a mug in front of her, sat down with his and smiled—first at her, then towards Nate.
'We have here an agent's dilemma: two writers with great potential for literary success, both with flaws that prohibit that achievement.'
Jemma turned her head to meet Nate's appraising gaze and raised eyebrows and frowned. Why wasn't he as surprised as she was at this announcement?
Brian regained her attention and continued.
'Discussions and revision attempts haven't been successful for either of you. But, as they say in the game, I had a lightbulb moment after Jemma told me she was coming to Sydney.'
He took a drink before going on, and Jemma's stomach curled in anticipation—or was it trepidation? She wasn't sure she wanted to hear any solution which meant involvement with this stranger by her side.
'Nate has a talent for action storytelling—very marketable in any media. Regrettably, the interaction between his hero and heroine is bland and unimaginative.'
That was hard to believe. Any man as handsome as he would have no trouble finding willing women to date and seduce. She'd seen the macho flare in his eyes when they'd been introduced, and her body's response had been instinctive.
'Jemma's characters and their interaction make for riveting reading. But the storyline between the extremely satisfying emotional scenes has little impact and won't keep pages turning. So, as a trial, I'm proposing we combine your strengths in Nate's manuscript.'
* * *
Nate's protest drowned out the startled objections coming from the woman on his right. It took supreme effort not to surge to his feet and pace the room—a lifelong habit when agitated or problem solving.
'Oh, come on, Brian. You know the hours and the effort—physical and mental—that I've put into that book. I can understand bringing someone else in...could even accept an experienced author...'
He struggled for words. Huh, so much for being a great writer.
'You expect me to permit an unproven amateur to mess with my manuscript? Her hearts and flowers characters will never fit.'
'Isn't your "amateur status" the reason you're here too, Mr Thornton? I doubt you've ever held a romance novel, let alone read the blurb on the back.'
The quiet, pleasant voice from minutes ago now had bite. He swung round to refute her comment, so riled up its intriguing quality barely registered.
'Wrong, Jemma. Every single word of one—from the title on the front cover to the ending of that enlightening two-paragraph description—to win a bet. Can't say I was impressed.'
Her chin lifted, her dark blue eyes widened in mock indignation and her lips, which his errant brain was assessing as decidedly kissable, curled at the corners. Her short chuckle had his breath catching in his throat, and his pulse booting up faster than his top-of-the-range computer.
'Let me guess. It was selected by a woman—the one who claimed you wouldn't make it through the first chapter, let alone to the happy ending.'
Shoot! His stomach clenched as if he'd been sucker-punched. Baited and played by his sister, Alice, he'd read every page of that badly written, highly sexed paperback to prove a point.
Brian cut in, so his plans for sibling payback had to be shelved for the future.
'Relax, Nate. Your hero and heroine's action stories are absorbing and believable. It's their relationship that won't be credible to the reader. I'm convinced Jemma can rectify that.'
'You're asking me to give her access? Let her delete and make changes to suit her reading preferences?'
No way. Not now. Not ever.
'No.'
'No!'
Their denials meshed.
Brian was the one who negated his outburst.
'No one's suggesting such a drastic measure. To start with I'd like the two of you to have lunch. Get to know each other a little. If you can reach a truce, we could start with a trial collaboration on two or three chapters.'
Lunch? Food and table talk with a woman who'd shown an adverse reaction to him on sight?
He sucked in air, blew it out and shrugged his shoulders. What did he have to lose? A book contract, for starters.
He matched the challenge in Jemma's eyes, nodded and forced a smile.
'Would you care to have lunch with me, Jemma?'
'It will be my pleasure, Nate.'
Her polite acceptance and return smile alleviated his mood a tad, though the option he'd been given still rankled. He disliked coercion—especially if it meant having a meal with an attractive woman who was somehow breaching the barriers he'd built for mental survival. Another reason for not entering into a working relationship with her.
He avoided entanglements. One heart-ripping experience had been enough, and was not to be chanced again. It was only his fact-finding skill that had prevented his being conned out of a fortune as well. Any woman he met now had to prove herself worthy of his trust before it was given.
Brian had been straight and honest with him from the start. And Jemma had shown spirit, so she might be good company. He'd enjoy a good meal, and then...
Well, for starters he'd be spending a lot of time reading writing manuals until he'd mastered the art of accurately describing a relationship.
* * *
It was warming up as Jemma exited the building with Nate. The rain had cleared, leaving the pavements wet and steamy and the air clammy. With a soft touch to her elbow he steered her to the right and they walked in silence, each lost in their own thoughts.
She was mulling over the recent conversation between the two of them and Brian, and assumed he was doing the same. Agreeing to Brian's proposition would mean being in frequent contact—albeit via electronic media—with a man whose innate self-assurance reminded her of her treacherous ex-boyfriend and her over-polite and social-climbing brother-in-law.
But unlike those two Nate also had an aura of macho strength and detachment. The latter was a plus for her—especially with her unexpected response when facing him eye to eye and having her hand clasped in his. Throughout the meeting she'd become increasingly aware of his musky aroma with its hint of vanilla and citrus. Alluring and different from anything she'd ever smelt, it had had her imagining a cosy setting in front of a wood fire.
Other pedestrians flowed around them, eager to reach their destinations. Nate came to a sudden stop, caught her arm and drew her across to a shop window. Dropping his hand, he regarded her for a moment with sombre eyes, his body language telling her he'd rather be anywhere else, with anyone else.
'Any particular restaurant you fancy?' Reluctance resonated in his voice.
'I haven't a clue.' She arched her head to stare beyond him. An impish impulse to razz him for his hostile attitude overrode her normal discretion and she grinned. 'How about that one?'
He followed her gaze to the isolated round glass floor on the communications tower soaring above the nearby buildings. His eyebrows arched, the corner of his mouth quirked, and something akin to amusement flashed like lightning in his storm-grey eyes.
'The Sydney Tower? Probably booked out weeks ahead, but we can try.'
'I was joking—it's obviously a tourist draw. If we'd been a few steps to the right I wouldn't even have seen it. You decide.'
'You're not familiar with Sydney, are you?'
His voice was gentler, as if her living a distance away was acceptable.
'Basic facts from television and limited visits over many years—more since some of my friends moved here.'
'Darling Harbour's not too far, and there's a variety of restaurants there. We'll take a cab.'
'Sounds good.' She'd have been content to walk—she loved the hustle and bustle of the crowds, the rich accents of different languages and the variety of personal and food aromas wafting through the air. Tantalising mixtures only found in busy cities.
She followed him to the kerb, trying to memorise every detail while he watched for a ride. Once they were on their way her fingers itched to write it all down in the notepad tucked in the side pocket of her shoulder bag—an essential any time she left home.
As a writer, he might understand. As a man who'd been coerced into having lunch with her, who knew how he'd react?
Erring on the side of caution, she clasped her hands together and fixed the images in her mind.
Copyright © 2018 by Harriet Nichola Jarvis
ISBN-13: 9781488089534
A Contract, A Wedding, A Wife?
First North American publication 2018
Copyright © 2018 by Christy McKellen
All rights reserved. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this ebook on-screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereafter invented, without the express written permission of the publisher, Harlequin Enterprises Limited, 22 Adelaide St. West, 40th Floor, Toronto, Ontario M5H 4E3, Canada.
This is a work of fiction. Names, characters, places and incidents are either the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events or locales is entirely coincidental. This edition published by arrangement with Harlequin Books S.A.
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Biografia
Gli esordi musicali e teatrali
Joanna Pacitti, di origini italo-irlandesi, è nata il 6 ottobre del 1984 da Stella e Giuseppe Pacitti. Joanna si appassionò al mondo della musica all'età di cinque anni quando iniziò a cantare ai clienti del salone di barbiere gestito da suo padre. Successivamente si iscrisse a danza classica e grazie all'impegno di sua madre ha iniziato a competere con alcuni talenti nei locali di ballo e a partecipare ad alcuni concorsi. Nel 1996, all'età di soli 12 anni fu scelta per il 20º anniversario del musical Annie dopo aver partecipato ad un concorso sponsorizzato dal centro commerciale Macy's.
Joanna Pacitti partecipò a ben 106 performance nel tour nazionale, ottenendo contrastanti critiche. Due settimane prima dello show era stata costretta ad atterrare a Broadway. Successivamente si ammalò di bronchite, ma dopo essere stata sostituita ritornò subito in scena e iniziò a venire pubblicizzata in modo massiccio. Sono in molti a ritenere che questa mossa non sia stata altro che programmata dai produttori al fine di promuoverne ulteriormente il debutto a Broadway. Successivamente Joanna venne sponsorizzata e pubblicizzata ovunque: su ABC World News Tonight, Good Morning America, American Journal e Turning Point. Dopo questa esperienza ebbe poi l'opportunità di pubblicare libri e di partecipare a show televisivi.
Carriera musicale
All'età di 16 anni riuscì a firmare un contratto con la A&M di Ron Fair, l'uomo che lanciò artisti come Christina Aguilera o Vanessa Carlton. Joanna Pacitti registrò alcune colonne sonore, tra cui il brano Watch Me Shine per Legally Blonde nel 2001, contenuta anche in un CD promozionale di 5 tracce pubblicato nel 2002. Il brano Watch Me Shine venne successivamente registrato gruppo taiwanese S.H.E che ne incise una cover.
Nel maggio del 2006 venne pubblicato il suo primo singolo "Let It Slide" tramite la Geffen Records e venne diffuso nelle radio e nelle televisioni verso la fine di giugno. Il suo album di debutto, This Crazy Life è stato pubblicato il 15 agosto del 2006.
L'album ottenne un successo piuttosto scarso, il flop portò la Geffen Records a licenziare la cantante. Dal settembre del 2007 Joanna Pacitti non è stata più sotto contratto con la Geffen.
Successivamente Joanna incise altre colonne sonore e si mise a lavorare ad un nuovo album, pur essendo priva di contratto.
Joanna però dopo essere stata concorrente nell'ottava edizione di American Idol, dove ha superato anche un'audizione, ha dichiarato di stare lavorando al secondo album che si trova già in fase di registrazione.
Discografia
Album
Colonne sonore
2001 Legally Blonde Soundtrack – Watch Me Shine
2003 My Scene's Chelsea Mix – Just When You're Leaving
2004 First Daughter Soundtrack - Fall
2007 Nancy Drew Soundtrack - Pretty Much Amazing
2007 Bratz Soundtrack - Out From Under
Filmografia
What I Like About You (2005)
Altri progetti
Collegamenti esterni
Attori italoamericani
Irlando-americani | {
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\section{Introduction} \label{sec:intro}
Young massive star clusters (YMCs -- ages $\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}$ 50Myr, masses $\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}
10^4$M\mbox{$_{\normalsize\odot}$}) have relevance to many areas of astrophysics. They contain
large numbers of massive stars, whose high temperatures, high
luminosities, dense winds, and supernova explosions make YMCs a
considerable source of mechanical energy, ionizing radiation and
chemically processed ejecta. The effect that they have on their
surroundings is profound, clearing away the remains of their natal
molecular cloud, whilst revealing and triggering subsequent
generations of star formation
\citep[e.g.][]{Gonzalez-Delgado00,Smith06_census,Danks-paper,DeMarchi11}. Their
large populations of massive stars make them ideal natural
laboratories with which to study the evolution of massive stars up to
supernova and beyond
\citep[e.g.][]{Martins07,Martins08,Bibby08,RSGC1paper,SGR1900paper}. Finally,
they can dominate the radiative output of their host galaxies, either
through direct ultraviolet and optical emission, or through
reprocessed emission in the form of ionised gas or heated dust
\citep{Alonso-Herrero02}.
Our knowledge of our own Galaxy's population of YMCs is extremely
incomplete, in stark contrast to external galaxies
\citep[e.g.][]{Bastian05,Konstantopoulos09}. The high levels of
interstellar extinction in the plane of the Galaxy have meant that
until recently very few clusters were known beyond a distance of
$\sim$1kpc. Most known clusters beyond this distance were found by
targeted searches of, for example, the Galactic Centre
\citep{Cotera96,Figer99_GC}, giant H{\sc ii}-regions
\citep{Blum99,Blum00,Blum01}, fields around newly-born neutron stars
\citep{Fuchs99,Vrba00}, or simply because the foreground extinction
was low enough to detect the cluster at optical wavelengths
\citep{Westerlund87}. However, recent infrared (IR) surveys of the Galactic
plane, beginning with 2MASS \citep{Skrutskie06}, and more recently
Spitzer/GLIMPSE \citep{Benjamin03} and VVV \citep{VVV}, are at last
helping us to uncover the cluster population of the Galactic disk, and
affording the opportunity to search the Galactic Plane for clusters in
a systematic way.
By-eye and algorithmic searches of survey data
\citep[e.g.][]{Ivanov02,Dutra03,Mercer05,Froebrich07,Borissova11} have
yielded over 1,000 candidates for newly discovered star
clusters. However, such catalogues inevitably contain large numbers of
false positives, due to chance alignments of stars, patchy foreground
extinction, and spatially extended emission incorrectly classified as
unresolved star clusters. Therefore, these catalogues of candidates
must be analysed carefully using multiwavelength survey data and,
ultimately, follow-up spectroscopic observations before their
distances and physical properties may be derived
\citep[e.g.][]{Kurtev07,Messineo09,Hanson10}. Only then can they be
placed in the framework of the Galaxy's recent star-forming history.
One such candidate is object \#81 from the catalogue of
\citet{Mercer05}, known hereafter as Mc81\footnote{Objects in this
catalogue are referred to by some authors with the prefix GLIMPSE,
e.g.\ {\it GLIMPSE81}}. This object was found in an algorithmic
search of the GLIMPSE survey, by looking for spatial groups of stars
with similar photometric properties. The clues to its nature, however,
come from cross-correlation with other data in the literature. The
object appears to be at the centre of the H{\sc ii}-region G338.4+0.1, a
bubble of warm dust and ionized gas, visible in the SUMSS 843MHz
\citep{SUMSS} and {\it MSX}\ 8$\mu$m\ \citep{MSX} images, and in more detail
in the GLIMPSE 8$\mu$m\ image (\fig{fig:wfim}). The object's location is
close to a supernova remnant (SNR) by \citet{Green04} from the `shell'
morphology of continuum radio emission, but with no spectral index
measurement the object could also be a wind-blown bubble.
Close to the centre of the SNR is a high-energy TeV gamma-ray source,
HESS~J1640-465 \citep{Aharonian05}, at a distance of
8$^{\prime}$\ (a linear distance of 22pc, if the complex is at a distance
of 11kpc -- see Sect.\ \ref{sec:extinct}). This source, as with many other TeV
sources, is thought to be associated with a pulsar wind nebula
\citep{Funk07}, indicative of recent SN activity. Indeed, TeV emission
can be a useful tracer of massive star formation in the Galactic Plane
as it is unaffected by absorption, and there are other young massive
star clusters in the literature that are known to be associated with
such sources -- RSGC1 \citep{Figer06,RSGC1paper}, Cl~1813-178
\citep{Messineo08}, and Westerlund~2 \citep{Aharonian07}. Finally, in
a follow-up of HESS~J1640-465, a number of hard X-ray sources were
detected in the field of Mc81, with one source being perfectly aligned
with the cluster \citep[][ see right panel of
\fig{fig:wfim}]{Landi06}. Such emission could be explained by either
a recently-formed neutron star or a colliding wind binary, with both
explanations being indicators of youth \citep[for an analysis of the
X-ray emission from another young star cluster, Westerlund~1,
see][]{Clark08}.
Based on this evidence we have extensively followed-up Mc81 with
near-infrared (NIR) photometry and spectroscopy, with a view to
confirming that the object is indeed a young star cluster, and
ultimately to determine the cluster's physical properties.
We begin in Sect.\ \ref{sec:obs} with a description of the
observations, data reduction and analysis steps. In
Sect.\ \ref{sec:res} we describe our results, and show that the object
is indeed a highly extincted star cluster, and estimate its age and
mass. We summarize our results in Sect.\ \ref{sec:summary}.
\begin{figure*}
\centering
\includegraphics[width=18cm,bb=60 15 955 449,clip]{wfim.eps}
\caption{Wide-field images of Mc81 and its surroundings. {\it Left}:
the MSX 8$\mu$m\ image, which shows the diffuse nebula surrounding
the cluster. The locations of {\it Swift} X-ray sources are
indicated \citep{Landi06}, as well as the location of the TeV
emission source HESS J1640-465. The contours indicate the
morphology of the 843MHz emission \citep{SUMSS}, and the dashed
green box shows the field-of-view of the right-hand panel. {\it
Right}: a higher resolution 8$\mu$m\ image from the GLIMPSE
survey. The brightest stars of the cluster are coincident with the
X-ray source Sw~2. The NICMOS cluster and control fields of view
are illustrated by the dashed green boxes.}
\label{fig:wfim}
\end{figure*}
\section{Observations \& data reduction} \label{sec:obs}
\subsection{Imaging}
Images were obtained with HST/NICMOS on 22 October 2008, as part of
observing programme \#11545 (PI: B.\ Davies). We used the NIC3 camera
which has a field-of-view of 51.2$^{\prime \prime}$$\times$51.2$^{\prime \prime}$\ and a
pixel scale of 0.2$^{\prime \prime}$. We observed the cluster through filters
F160W and F222M, as well as the narrow-band filters F187N and F190N
which are centred on P$\alpha$ and the neighbouring continuum
respectively. In addition to the cluster we observed a nearby control
field through the F160W and F222M filters in order to characterize the
foreground population. The fields of observation are indicated in
Fig.\ \ref{fig:wfim}.
Our observations used a spiral dither pattern with six pointings with
offset distance was set to 5.07$^{\prime \prime}$. This sub-pixel dithering
technique was designed to minimise the impact of non-uniform
intra-pixel sensitivity on our photometry. The MULTIACCUM read modes
were used, with the sampling sequences and total integration times
that are listed in Table \ref{tab:samp}.
Our reduction procedure followed the guidelines of the NICMOS Data
Handbook v7.0. The standard reduction steps of bias subtraction,
dark-current correction and flat-fielding were performed using {\sc
calnica}. Before mosaicing, each dithered observation was subsampled
onto a 3$\times$ finer grid using bi-linear intepolation to account
for the sub-pixel dithering.
Photometry was extracted from the images using the {\sc starfinder}
package which run within IDL \citep{STARFINDER}, in conjunction with
point-spread functions (PSFs) which were computed for each filter
using {\sc tinytim}. {\sc starfinder} uses these PSFs to locate stars
within each image, and we employed two iteration cycles to fine-tune
the astrometry and photometry. Since our fields of observation are not
very crowded, we did not use the deblending algorithm, as this was
found to produce many false detections. Uncertainties and completeness
limits were determined by inserting fake stars into each image and
measuring the recovery rate.
\begin{figure*}
\centering
\includegraphics[width=18cm,bb=15 8 672 280,clip]{colfig_2.eps}
\caption{NICMOS images of the cluster. {\it Left}: image of the
cluster taken through the F222M filter. The two stars for which we
have spectra, as well as the other emission-line stars, are
indicated by the blue triangles. {\it Right}: the
difference image (F187N-F190N), which highlights the emission-line
stars. The arrows in the top-right of each image indicate the
orientation. }
\label{fig:colfig2}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=18cm,bb=34 22 436 336,clip]{colfig_3col.eps}
\caption{Three-colour image of the cluster from the
NICMOS data, with colours as follows -- R=F222M, G=F160W,
B=(F187N-F190N). Therefore, the highly reddened cluster stars
appear red/orange, and the cluster's emission-line stars appear
pink/megenta.}
\label{fig:3col}
\end{figure*}
\begin{table}
\centering
\caption{Read-sequences and total integration times employed for each filter
during the NICMOS observations. }
\begin{tabular}{lccccc}
\hline
\hline
Filter & SAMP-SEQ & NSAMP & $T_{\rm int}$ (s) \\
\hline
F160W & STEP2 & 15 & 144 \\
F222M & STEP8 & 12 & 336 \\
F187N & STEP8 & 10 & 240 \\
F190N & STEP8 & 10 & 240 \\
\hline \\
\end{tabular}
\label{tab:samp}
\end{table}
\subsection{Spectroscopy}
Spectroscopic data were taken on the nights of 2009-4-11 and 2009-5-4
as part of ESO observing programme 083.D-0765(A) (PI: E.~Puga), using
the ISAAC spectrograph on the VLT \citep{Moorwood98} in `medium'
resolution mode. Our targets were the stars labelled `1' and `3' in
\fig{fig:colfig2}, which were determined to be likely cluster members
based on their photometric properties (see later). We used the
0.8$^{\prime \prime}$\ slit at three different central wavlengths: 1.71$\mu$m,
2.09$\mu$m, and 2.21$\mu$m, providing a spectral resolution of $\Delta
\lambda / \lambda \sim$4,000. The DIT$\times$NDIT$\times$NINT
combination for each wavelength setting was (8$\times$8$\times$30s),
(8$\times$8$\times$32s), (16$\times$8$\times$24s) respectively. The
observations were taken in a ABBA pattern to isolate and subtract sky
emission features. In addition to the target stars we observed the
B9\,{\sc v} stars Hip090248 and Hip091286 as measures of the telluric
absorption, as well as the usual observations of flat-fields, dark
frames and arcs for wavelength calibration. To characterize the
spatial distortion, a bright field star was stepped along the slit and
re-observed multiple times.
The data reduction procedure began with subtraction of nod pairs to
remove sky emission, and division by a normalized flat-field. The 2-D
frames were then rectified onto an orthogonal grid, using the
stepped-star and arc frames to characterize the distortion in the
spatial and dispersion directions respectively. This process also
wavelength calibrates the data, with r.m.s.\ residuals which were found to
be less than a tenth of a resolution element ($\sim$10\,km~s$^{-1}$). After
rectification, the spectra were extracted and combined.
The telluric standard spectra had their intrinsic H\,{\sc i}\ absoption
removed by fitting the lines with Voigt profiles. The telluric spectra
were then cross-correlated with the target spectra in the region of
isolated telluric features to correct for any residual sub-pixel
shifts, before the target spectra were divided through by the telluric
spectra.
\begin{figure*}
\centering
\includegraphics[width=18cm,bb=0 10 730 369,clip]{cmd.eps}
\caption{Colour-magnitude diagrams for Mc81 from the NICMOS
photometry.{\it Left}: The stars within 15$^{\prime \prime}$\ of the cluster
centre, which we define as the position of Mc81-2, compared to
stars in a nearby control field of the same angular size. {\it
Centre}: the P$\alpha$ emission of stars within 15$^{\prime \prime}$\ of
the cluster centre, as determined from the colour ($m_{187} -
m_{190}$). Stars with significant emission are marked with green
circles. {\it Right}: the same as the left panel, after
the cluster field has been decontaminated of foreground stars
using the control field observations. The long-dashed lines in
each figure show the 50\% completeness levels.}
\label{fig:cmd}
\end{figure*}
\section{Results \& Analysis} \label{sec:res}
\subsection{Photometry} \label{sec:photom}
The NICMOS images of Mc81 can be seen in \fig{fig:colfig2}. The left
panel shows the F222M image of the cluster. In the right panel, we
show the difference image [F187N-F190N], which clearly highlights 9
stars with significant P$\alpha$\ emission\footnote{It is possible that
He\,{\sc ii}\ emission contributes to the flux in this band, especially if
the stars are WRs.}. This strongly suggests that these are hot stars
with strong winds, and are therefore likely to have ages $\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}$10Myr.
In \fig{fig:3col} we show a 3-colour RGB image of F222M (red), F160W
(green), and [F187N-F190N] (blue). Around the coordinate centre, there
is a clear group of highly reddened stars which form the putative
cluster. In this colour scheme, the emission line stars, which are
also heavily reddened, show up as pink/magenta. From these data there
is already strong evidence for a young, highly reddened cluster of
stars in the field of Mc81. The four bright stars to the south of the
cluster have a green/yellow colour, indicating that they are in the
foreground. Ironically, it is likely that these four stars in part
triggered the cluster detection algorithm used by
\citet{Mercer05}. These authors claim that they detected an
association of 65 stars, though the positions of these stars are not
listed, so it is not possible for us to check whether the cluster
detected by Mercer et al.'s algorithm bears any relation to the
cluster we describe here. For the sake of clarity and consistency, we
continue to refer to the cluster studied in this work as Mc81.
The results of the NICMOS photometry are shown in \fig{fig:cmd}. The
left panel shows a colour-magnitude diagram of the stars within
18$^{\prime \prime}$\ of the cluster centre, which we define as the location of
the brightest emission-line star. Also shown on the plot are data from
an area in the control field of identical size. There is an obvious
difference between the two regions, with the `cluster' field showing
an excess of stars at ($m_{160}-m_{222}$)$\simeq$2.3. The centre panel
shows the P$\alpha$\ excess stars. Those stars with ($m_{187}-m_{190}$)
colours $<$0.3 and $m_{222} >$13 are defined as `P$\alpha$\ emitters', and
are indicated on the plot. The locations and photometry of these stars
are listed in Table \ref{tab:emit}.
Finally, the right-hand panel of \fig{fig:cmd} shows the same as the
left, after the cluster field has been decontaminated of field stars
using the data from the control-field. We do this by eliminating stars
from the cluster field which have a control-field star close by in
colour-magnitude space. We set this limiting distance to be the larger
of either the cluster field star's 1-$\sigma$ photometric errors, or
0.15mags in colour and 0.1mags in magnitude. We also set the
restriction that no control-field star can eliminate more than one
cluster field star. After decontamination, the cluster sequence at
($m_{160}-m_{222}$)$\simeq$2.3 becomes clearer, though there is still
some scatter.
\begin{table}
\centering
\caption{Astrometry and photometry of Mc81-1, plus the emission-line
stars. Astrometry is taken from the HST observations, and
comparisons to 2MASS indicate that it is accurate to
$\sim$1$^{\prime \prime}$. }
\begin{tabular}{lccccc}
\hline \hline
ID & RA-DEC (J2000) & $m_{160}$ & $m_{222}$ & $m_{187}$ & $m_{190}$\\
\hline
1 & 16 40 29.83 -46 23 33.9 & 11.14 & 8.90 & 9.67 & 9.75 \\
2 & 16 40 29.65 -46 23 29.1 & 13.25 & 10.59 & 10.69 & 11.63 \\
3 & 16 40 30.08 -46 23 11.4 & 12.97 & 10.82 & 10.75 & 11.60 \\
4 & 16 40 29.65 -46 23 28.7 & 13.44 & 10.95 & 11.15 & 11.91 \\
5 & 16 40 28.35 -46 23 25.6 & 14.13 & 11.42 & 11.67 & 12.45 \\
6 & 16 40 29.32 -46 23 11.6 & 13.46 & 11.42 & 11.56 & 12.14 \\
7 & 16 40 29.60 -46 23 25.6 & 13.55 & 11.45 & 11.56 & 12.31 \\
8 & 16 40 28.94 -46 23 27.1 & 14.27 & 11.46 & 11.95 & 12.65 \\
9 & 16 40 29.32 -46 23 38.2 & 13.76 & 11.82 & 12.16 & 12.54 \\
10 & 16 40 30.05 -46 23 24.3 & 15.08 & 12.78 & 13.41 & 13.70 \\
\hline \\
\end{tabular}
\label{tab:emit}
\end{table}
\begin{figure*}
\centering
\includegraphics[width=18cm,clip]{m81_s1_2_spec.eps}
\caption{Spectra of the two stars observed. The wiggles seen at
$\sim$2.05$\mu$m\ are due to fringing on the detector, which we were
unable to remove. }
\label{fig:spec}
\end{figure*}
\subsection{Spectroscopy}
We now present the results of the spectroscopic observations. To
classify the spectra, we refer to the spectral atlases of
\citet{Hanson05}, \citet{Figer97}, as well as \citet{Crowther-Wd106}.
The spectra of the two brightest stars in Mc81 are shown in
\fig{fig:spec}. The brighter star, Mc81-1, has relatively weak
spectral features. The absorption lines of the Hydrogen Brackett
series are seen, as well as some faint Mg\,{\sc ii}\ emission. We see no
evidence of He~I in either absorption or emission. This absence of
He~I suggests a spectral type later than $\sim$B5, while the emission
lines of Mg\,{\sc ii}\ are indicative of a substantial stellar wind, more
typical of supergiants. The `ripples' seen between
2.05-2.08$\mu$m\ are due to poor cancellation of the CO$_2$
absorption. For now, we assign a loose spectral type to this star of
late-B/early-A supergiant. Taking the distance and extinction derived
in Section \ref{sec:extinct}, as well as the bolometric corrections
tabulated by \citet{Blum00} for spectral types B7-A2, we estimate the
luminosity of the star to be in the range $\log(L/L_{\odot}) =
$5.4--5.8, placing the star close to the empirical stellar luminosity
limit at $\log(L/L_{\odot}) \simeq$5.9 \citep{H-D79}. Such stars are
also seen in Westerlund~1 \citep{Clark05}. The star's proximity to the
Humphreys-Davidson limit suggests that the star may be in an unstable
phase of evolution, such as a Luminous Blue Variable or Yellow
Hypergiant phase, though further spectroscopic and photometric
monitoring would be required to verify this.
In contrast to Mc81-1, Mc81-3 has a spectrum rich in strong, broad
emission lines. These lines can be attributed to H\,{\sc i}, He\,{\sc i}, He\,{\sc ii}\ and
N\,{\sc iii}. The ratio of the He\,{\sc ii}\ 2.189$\mu$m\ to the complex at 2.115$\mu$m,
as well as the absorption of He\,{\sc i}\ 2.189$\mu$m, allow us to tightly
constrain the spectral type of this star to be WN7-8.
\subsection{Extinction and distance} \label{sec:extinct}
To calculate the extinction, we first determine the reddening of the
cluster sequence from the right-hand panel of \fig{fig:cmd}. The
average colour of the stars in the decontaminated cluster field is
($m_{160}-m_{222}$)=2.3$\pm$0.3. If we make the approximation that all
main-sequence stars that we detect should have colours of
approximately zero, the observed average colour is due to
extinction. We can then determine the extinction from the following
relation,
\begin{equation}
A_{\lambda_{2}} = \frac{ E_{\lambda_{1}-\lambda_{2}} }
{ (\lambda_{1}/\lambda_{2})^{\alpha} - 1 }
\label{equ:extinct}
\end{equation}
\noindent with $\lambda_{1} = $1.60$\mu$m\ and $\lambda_{2} = $2.22$\mu$m,
i.e. the wavelengths of the NICMOS F160W and F222M filters. The
parameter $\alpha$ has been studied by numerous authors in recent
years \citep[see e.g.][ and references therein]{Stead-Hoare09}, with
the most contemporary measurements converging on $\alpha = -2.0 \pm
0.1$. This therefore implies that the extinction towards Mc81 is
$A_{2.22} = 2.5 \pm 0.5$. Extrapolating this extinction to the optical
is known to be highly uncertain, but we estimate $A_{V} = 45 \pm
15$\footnote{Using the value of $\alpha = -1.53$ from \citet{R-L85},
we find $A_{2.22} = 3.5 \pm 0.5$, and $A_{V} = 30 \pm 4$}. Mc81 is
therefore one of the most heavily reddened clusters known, with an
extinction comparable to that of the Galactic Centre.
We estimate the distance to the cluster from the radial velocity of
the surrounding molecular cloud. \citet{Caswell-Haynes87} studied the
radio recombination line emission of the two clouds of ionized gas
either side of the cluster, G338.4+0.2 and G338.4+0.1 (see
\fig{fig:wfim}), finding velocities relative to the local standard of
rest of $v_{\rm lsr}$=-29\,km~s$^{-1}$\ and -37\,km~s$^{-1}$\ respectively. In addition, a
number of massive young stellar objects (YSOs) and compact
H{\sc ii}-regions have been detected in the region by the {\it Red MSX
Source (RMS) Survey} \citep{Hoare05,Urquhart07a,Urquhart07b}, with
an average radial velocity of $v_{\rm lsr}$\ = -35.1$\pm$2.8 \,km~s$^{-1}$.
Comparing this average $v_{\rm lsr}$\ to the Galactic rotation curve of
\citet{B-B93}, using a Galacto-centric distance of 7.6$\pm$0.3\,kpc
and a rotational velocity of 214$\pm$7\,\,km~s$^{-1}$\ \citep[][ and references
therein]{K-D07}, we find near and far kinematic distances of 3.8kpc
and 11kpc. To resolve the near/far ambiguity, we refer to the {\it
Southern Galactic Plane Survey (SGPS)} of H{\sc i}. These data were
recently analysed by \citet{Lemiere09}, who found that for the two
large H{\sc ii}-regions G338.4+0.2 and G338.4+0.1, H{\sc i} absorption
components could be seen at several velocities up to the tangent point
velocity of 130\,km~s$^{-1}$. This indicates the H{\sc ii}-regions, and by
association the YSOs and the central star cluster, lay {\it beyond}
the tangent point. From this we conclude that the clusters lay at the
far-side distance of 11\,kpc. At this distance, the uncertainty is
dominated by the systematic uncertainties in the Galactic rotation
curve, which by its nature is difficult to quantify. However, if we
assume that the system may have a peculiar velocity of up to
$\pm$20\,\,km~s$^{-1}$\ \citep{Russeil03}, this gives an uncertainty on this
distance of $\pm$2\,kpc.
We can check this distance by calculating the absolute brightness of
the WNL star and comparing to similar objects. Using the extinction
calculated in the previous section and a distance of 11$\pm$2\,kpc, we
find an absolute magnitude for Mc81-2 of $M_{222} = -7.1 \pm 0.6$. By
comparison, Galactic WNL stars are typically found to have $M_{K} =
5.9 \pm 1.0$ \citep{Crowther-Wd106}. Mc81-2 is therefore somewhat
luminous for its spectral type, though it is within the errors for
other Galactic WNL stars.
\begin{figure}
\centering
\includegraphics[bb=10 0 566 453,width=8.5cm]{morph.eps}
\caption{An illustration of the spatial extent of the cluster. Red
crosses denote all stars in the NICMOS field brighter than the
50\% detection limit; while blue circles indicate stars with
colours of {$m_{160} - m_{222} > 2.0$}. The contours indicate
where the stellar density is 25\%, 50\% and 75\% of the
maximum. As the data were smoothed to make the contours, the
circle in the bottom left shows the size of a resolution element. }
\label{fig:rad}
\end{figure}
\subsection{Cluster size}
In \fig{fig:rad}, we illustrate the physical extent of the
cluster. The figure shows the locations of all stars in the NICMOS
field-of-view which are brighter than the 50\% completeness limit
($m_{222} < 17$), overlayed with those stars which have colours
consistent with the cluster (i.e.\ $m_{160} - m_{222} > 2.0$). The
figure shows that there is a clear overdensity of stars at the
coordinate centre (defined as the position of star Mc81-2).
To measure the size and morphology of this overdensity, we first made
a map of the stellar density by dividing the field up into square bins
of size 3$^{\prime \prime}$$\times$3$^{\prime \prime}$\ and counting the number of stars per
bin. To reduce noise, this map was then smoothed, such that the
effective resolution of the map was 6$^{\prime \prime}$\ (illustrated in the
bottom corner of \fig{fig:rad}). This resolution size was chosen as a
trade-off between spatial resolution and signal strength, though our
results were robust to changes in this parameter. The ambient stellar
density was found by computing the background level of this map using
the GSFC IDL routine {\sc sky}. Finally, we computed isodensity
contours in this map at percentiles of the maximum stellar density.
We defined the size of the cluster where the stellar density drops to
50\% of its maximum value, which we deem to be roughly equivalant to
its half-light radius\footnote{Ideally, to measure the half-light
radius one would measure the cumulative surface brightness out to a
distance where it becomes asymptotic. However, the field-of-view of
our observations is too small to do this.}. Once deconvolved with
the effective spatial resolution, we find that the cluster has major
and minor axes of 29$^{\prime \prime}$~$\times$~18$^{\prime \prime}$. At a distance of
11kpc, this corresponds to 1.5$\times$1.0\,pc, and so is comparable to
other young Galactic clusters which typically have diameters between
1-2pc \citep[e.g. Trumpler~14, Westerlund~1;][]{Figer08}.
\begin{table}
\label{tab:cmfgen}
\centering
\caption{Best fitting model atmosphere parameters for Mc81-2.}
\begin{tabular}{lc}
\hline
\hline
Parameter & Value \\
\hline
$T_{\rm eff}$ (K) & 36000 $\pm$ 1000 \\
$T_{\tau = 20}$ (K) & 38000 $\pm$ 000 \\
$v_{\infty}$ (\,km~s$^{-1}$) & 1350 $\pm$ 100 \\
$\beta$ & 1.25 \\
A(H/He) & 0.75 $\pm$ 0.25 \\% \smallskip\\
$\log (L /$$L$\mbox{$_{\normalsize\odot}$}) & 6.3 $\pm$ 0.4 \\
$\log(\dot{M}/$$M$ \mbox{$_{\normalsize\odot}$}\rm{yr}$^{-1}$) & -4.2 $\pm$0.2 \smallskip\\
\hline \\
\end{tabular}
\end{table}
\begin{figure}
\centering
\includegraphics[width=8.5cm,bb=20 20 600 470,clip]{logl_hhe_color.eps}
\caption{Geneva evolutionary tracks from \citet{Mey-Mae00}. The $x$
and $y$ axes plot the H/He mass fraction and luminosity
respectively, while the colour of the track indicates the
effective temperature at that point in the evolution. The initial
mass of each track is indicated at the top. Our derived parameters
for Mc81-2 are indicated by the position and colour of the star
symbol. Though the symbol intercepts the 120M\mbox{$_{\normalsize\odot}$}\ track in
(H/He)-$L$ space, the temperature of the star means that the best
fit is actually with the 60M\mbox{$_{\normalsize\odot}$}\ track (see also
Fig.\ \ref{fig:prob}).}
\label{fig:logl-hhe}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8.5cm,bb=10 5 605 475,clip]{prob_wnl_r.eps}
\caption{Probability map derived from fitting stellar mass tracks to
the luminosity, temperature and H/He of the WNL star Mc81-2. The age
and initial mass of the best fitting model are indicated by the
cross, and iso-probability contours are drawn at 67\% and 50\%. }
\label{fig:prob}
\end{figure}
\subsection{Quantitative spectral analysis} \label{sec:cmfgen}
To model the WNL star in Mercer81 and estimate its physical
parameters, we proceed as in \citet{Najarro04,Najarro09}. Briefly,
we have used CMFGEN, the iterative, non-LTE line blanketing method
presented by \citet{H-M98} which solves the radiative transfer
equation in the co-moving frame and in spherical geometry for the
expanding atmospheres of early-type stars. The model is prescribed by
the stellar radius, \hbox{$R_*$}, the stellar luminosity, \hbox{$L_*$}, the
mass-loss rate, \hbox{$\dot {M}$}, the velocity field, $v(r)$ (defined by the
terminal wind speed \hbox{$v_\infty$}\ and the wind acceleration parameter
$\beta$), the volume filling factor characterizing the clumping of the
stellar wind, {\it f(r)}, and elemental
abundances. \citet{H-M98,H-M99} present a detailed discussion of
the code. For the present analysis, we have assumed the atmosphere to
be composed of H, He, C, N, O, Si, S, Fe and Ni. Observational
constraints are provided by the H, K-band spectra of the stars and the
dereddened F166W, F190N and F222M magnitudes.
Given the extreme sensitivity of the $H$ and $K$-Band He\,{\sc i}\ and
He\,{\sc ii}\ line profiles ratios in this parameter domain to changes in
temperature, we estimate our errors in the temperature to be below
1000~K. Likewise, the relative strengths between the H and He lines
constrain the H/He ratio to be within 0.5 and 1.0 by mass. The error
on \hbox{$R_*$}\, and hence on \hbox{$L_*$}\ and \hbox{$\dot {M}$}\ is dominated by those in
the assumed distance and the slope of the extinction law.
The best-fitting model is overplotted in \fig{fig:spec}. The model
provides a good fit to the features of the observed spectrum, with the
exception of the Br10 line, which is blended with emission from
N\,{\sc iv}/C\,{\sc iv}/O\,{\sc iv}. This discrepancy is due primarily to the deficiencies
in the CNO {\sc iv} model atoms, the correction of which is beyond the
scope of the current work. The model's physical parameters are given
in Table \ref{tab:cmfgen}, and are typical for a late-type WN
star. For completeness, we list the temperature at an optical depth of
$\tau = 2/3$, which is comparable to the star's effective temperature;
and the temperature at $\tau = 20$ which is comparable to the
hydrostatic temperature of stellar structure models. In the following
Section we use these results to estimate the age of Mc81-3, and
therefore of the cluster itself.
\subsection{Cluster age}
In order to contrain the age of the cluster, we make a quantitative
comparison between the physical properties of the WNL star Mc81-3 and
the predictions of stellar evolutionary models. Under the assumption
that the star and the host cluster are coeval, we can then estimate
the cluster's age.
For this analysis, the models we have chosen are those of
\citet{Mey-Mae00} which are optimized for massive stars. In our
method, we linearly interpolate these mass tracks at intervals of
1M\mbox{$_{\normalsize\odot}$}\ and $10^5$yrs. For each point on each interpolated mass track,
we then calculate the probability that there is a match between the
mass track and Mc81-3, based on the star's luminosity $L_{\star}$,
temperature $T$, and H/He ratio $A(\rm He)$ derived in the previous
Section. For the star's temperature, we use the temperature at an
optical depth of $\tau=20$, since this is more comparable to the
hydrostatic temperature calculated by \citet{Mey-Mae00}.
\Fig{fig:logl-hhe} shows the inter-related behaviour of the three
variables $L$\mbox{$_{\star}$}, $T$ and H/He for models with a range of initial
stellar masses. We also plot the derived physical parameters of
Mc81-3. The plot shows that, although the star's luminosity and H/He
ratio place it on the 120M\mbox{$_{\normalsize\odot}$}\ mass track, the temperature of the
star (illustrated by the colour of the plotting symbol) more closely
matches the 60M\mbox{$_{\normalsize\odot}$}\ track.
We assume that the errors on $L_{\star}$, $T$ and $A(\rm He)$ are
gaussian, and therefore the probability $p$ of a match between Mc81-3
and the mass track of initial mass $m$ at time $t$ is given by,
\begin{equation}
p(m,t) = \prod_{i} \exp\displaystyle\left(- \frac{(\mathcal{M}_{i} -
\mathcal{O}_{i})^2}{2\sigma_{\mathcal{O}_{i}}^{2}} \right)
\label{equ:prob}
\end{equation}
\noindent where $\mathcal{O}$ is the observed quantity (either
$L_{\star}$, $T$ or $A(\rm He)$), $\sigma_{\mathcal{O}}$ is its
associated uncertainty, and $\mathcal{M}$ is the corresponding
quantity predicted by the model mass track. Each term is therefore
weighted by its associated uncertainty.
In \fig{fig:prob} we plot how the probability varies across the 2-D
plane of mass and stellar age for the rotating models of
\citet{Mey-Mae00}. The maximum probability ($p=0.75$) is obtained for
an initial mass of $M_{\star}=$62M\mbox{$_{\normalsize\odot}$}\ and an age of 3.7Myr. The
morphology of the iso-probability contours are highly non-gaussian, so
for the experimental uncertainty we cannot simply compute the standard
deviation. Instead, we take the iso-probability contour at 50\%
and determine the minimum and maximum values of mass and age for that
contour. In this way, for Mc81-3 we find $M_{\star} = 62 ^{+6}_{-7}$M\mbox{$_{\normalsize\odot}$}\
and an age of $3.7^{+0.4}_{-0.5}$Myr. Using the non-rotating versions
of the same stellar evolution models produces a slightly different
morphology to the probability distribution, with a reduced
probability, but with a best-fitting age and mass that do not differ
significantly from that derived using the rotating models.
The value we obtain for Mc81-3's age can be understood through a
simple qualitative analysis of the star's parameters. The high
luminosity clearly favours high initial masses, and therefore a young
age. In addition, the He enrichment indicates an object which is in an
advanced evolutionary state, and so older than $\sim$2Myr, but younger
than the total lifetime of a high-mass star, and so therefore younger
than $\sim$5Myr.
\subsubsection{The impact of binary evolution on our derived cluster age}
Throughout this analysis we have assumed that Mc81-3 has evolved as a
single star. However, there is the possibility that the star is in an
interacting binary: both \citet{Landi06} and \citet{Funk07} have
detected X-ray emission from the centre of the cluster, which could be
evidence of a colliding wind binary system. As shown by
\citet{Eldridge09} for the case of $\gamma^2$~Vel, including the
effects of binary evolution in the analysis can alter the derived age
of a star.
In the case of Mc81-3, the star's high luminosity places a strong
constraint on the initial mass of the star, and hence on the upper
limit of its age. Though binary evolution can affect the surface
abundances and temperature of a star, and prolong its lifetime, it is
unlikely to increase the star's maximum luminosity by more than
$\sim$0.1dex, which is governed primarily by the initial stellar mass
(all other parameters being equal) \citep{Eldridge08}. An exception to
this would be if two stars merged to produce a completely rejeuvenated
and more massive star. In the absence of any evidence for such an
event in the history of Mc81-3, we maintain that the upper limit to
the star's age is that derived in the previous paragraph.
The lower limit to the cluster age may be reduced if Mc81-3 is in an
interacting binary system. Mass transfer from the primary to the
secondary star may speed up the rate at which H is depleted from the
primary's surface. In this case, we would underestimate the stellar
(and hence cluster) age by using single star evolution models in our
analysis.
However, a simple morphological analysis of the nebula surrounding
Mc81, and a comparison to similar systems, serves as a sanity check on
our age estimate. The cluster is located at the centre of a cavity,
which was presumably evacuated by the winds, ionizing radiation and
SNe explosions of the most massive stars in the cluster. At the
periphery of the cavity evidence of further generations of star
formation is seen, which may or may not have been triggered by
feedback from the cluster. This morphology is reminiscent of other
cluster + nebula systems such as G305, NGC~3603 and NGC~346 to name
but a few. Ages of these other systems are commonly found to be 2-4Myr
\citep{Danks-paper,Harayama08,Bouret03}, and therefore are consistent
with our estimate for Mc81.
\subsection{Cluster mass}
For the cluster mass, it is difficult to make an accurate estimate
without further spectroscopy of the stars in the cluster. We can
however make a rough estimate of the cluster mass from the emission
line stars. If we assume that {\it all} the nine strong P$\alpha$\ emitters
listed in Table \ref{tab:emit} are WRs, then since the age we derive
for Mc81 is roughly the same as that of Westerlund~1 (Wd1)
\citep[3-5Myr, ][]{Crowther-Wd106,Brandner08} which has 27 WRs, this
suggests that Mc81 may be a factor of $\sim$3 less massive than
Wd1. As most estimates of Wd1's mass are around
$10^5$M\mbox{$_{\normalsize\odot}$}\ \citep{Clark05,Brandner08} this implies that the mass of
Mc81 is a few $\times 10^4$M\mbox{$_{\normalsize\odot}$}. We stress however that this is only
an order-of-magnitude estimate; a more precise measurement of the
cluster mass awaits further analysis of its stellar population.
\section{Discussion}
\begin{figure}
\centering
\includegraphics[width=8.5cm,bb=0 0 651 651]{gal_arms.eps}
\caption{Top down view of the Galaxy, showing the locations of young
star clusters. The colour of the plotting symbols indicate the
extinction of each cluster, derived from their near-infrared
colours and assuming an extinction law slope of -2.0. The spiral
arms are those defined by \citet{C-L02}. Sites of multiple star
clusters are indicated by a filled cicle surrounded by an open
circle. }
\label{fig:gal-arms}
\end{figure}
\begin{table}
\label{tab:clust}
\caption{Young star clusters in the Galactic Plane. The visual
extinction $A_V$ has been determined homogeneously from each
cluster's infrared colour excess $E(H-K)$, using the relation
$A_{V} \simeq 19 \times E(H-K)$, which follows from a NIR
extinction law with slope $\alpha = -2.0$. See the referenced
papers for detailed error analysis on the distances. }
\begin{tabular}{lcccc}
\hline
Cluster & $l$ (\degr) & $D$ (kpc) & $A_{V}$ & Ref. \\
\hline \hline
Cl1806-20 & 10.0 & 8.7 & 30.3 & 1\\
W31S & 10.1 & 4.5 & 20.9 & 2\\
Cl1813-17 & 12.7 & 4.7 & 9.5 & 3\\
M17 & 15.0 & 2.4 & 9.5 & 4\\
Mc9 & 22.8 & 4.2 & 19.0 & 5\\
W42 & 25.4 & 2.7 & 9.5 & 6\\
RSGC1..5, Quartet & 26.0 & 6.0 & 19.0 & 7,8,9,10,11,12\\
W43 & 30.8 & 6.2 & 39.8 & 13\\
Cl1900+14 & 43.0 & 12.5 & 13.3 & 14\\
Mc20 & 44.2 & 3.5 & 17.1 & 12\\
Mc23 & 53.7 & 6.5 & 6.6 & 15\\
CygOB2 & 80.2 & 1.4 & 1.9 & 16\\
h+$\chi$ Per & 135.0 & 2.3 & 1.9 & 17\\
$[$DBS2003$]$45 & 283.9 & 4.5 & 7.6 & 18\\
Westerlund 2 & 284.2 & 8.0 & 7.6 & 19\\
Trumpler 14 & 287.0 & 2.5 & 2.5 & 20\\
NGC3603 & 291.6 & 6.0 & 4.7 & 21\\
Mc30 & 298.8 & 7.2 & 10.5 & 22\\
Danks 1 \& 2 & 305.0 & 4.0 & 9.5 & 23\\
Mc81 & 338.4 & 11.0 & 41.7 & This work.\\
Westerlund 1 & 339.5 & 3.9 & 9.5 & 24\\
NGC6231 & 343.5 & 1.8 & 3.8 & 25\\
$[$DBS2003$]$179 & 347.6 & 9.0 & 15.2 & 26, This work.\\
\hline \\
\end{tabular}
References: 1:~\citet{Bibby08}; 2:~\citet{Blum01};
3:~\citet{Messineo08}; 4:~\citet{Hanson96}; 5:~\citet{Messineo10};
6:~\citet{Blum00}; 7:~\citet{RSGC1paper}; 8:~\citet{RSGC2paper};
9:~\citet{RSGC3paper}; 10:~\citet{RSGC4paper}; 11:\citet{RSGC5paper};
12:~\citet{Messineo09}; 13:~\citet{Blum99}; 14:~\citet{SGR1900paper};
15:~\citet{Hanson10}; 16:~\citet{Hanson03}; 17:~\citet{Currie10};
18:~\citet{Zhu09}; 19:~\citet{Rauw07}; 20:~\citet{Ascenso07};
21:~\citet{Harayama08}; 22:~\citet{Kurtev07};
23:~\citet{Danks-paper}; 24:~\citet{K-D07};
25:~\citet{Raboud97}; 26:~\citet{Borissova08}. \\
\end{table}
\subsection{Location in the Galaxy}
With the many recent discoveries of young star clusters in the Galaxy,
we can now begin to build up a picture of the Galaxy's recent cluster
formation. In \fig{fig:gal-arms} we plot the locations of all known
young Galactic clusters with distances from the Sun greater than
$\sim$2\,kpc. All clusters in the plot are thought to have masses in
excess of $10^3$M\mbox{$_{\normalsize\odot}$}\ and ages $\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$<$}}}$20Myr. The references for each
data-point are listed in Table \ref{tab:clust}. We have colour-coded
each data-point according to its visual extinction, which we have
calculated in a homogeneous way from each cluster's $E(H-K)$, measured
either from the references listed in \ref{tab:clust} or from 2MASS
photometry, and an extinction law slope of $\alpha = -2.0$ (see
Sect.\ \ref{sec:extinct}).
\Fig{fig:gal-arms} shows that there are now a significant number of
reddened star clusters known at Galactic longitudes between 10$\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}} l
\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}}$50. This line-of-sight corresponds to the tangent of the
Scutum-Crux arm, as well as the near end of the Galactic Bar
\citep{Benjamin05}. Since one would expect the star formation rate to
be comparatively high in this location of the Galaxy, it is also
reasonable to expect it to be rich in young star clusters. Indeed, the
region hosts five known YMCs, plus a substantial field population of
Red Supergiants, indicating a starburst episode of
$\sim10^6$M\mbox{$_{\normalsize\odot}$}\ around 20Myr ago
\citep{Garzon97,Lopez-Corredoira99,Figer06,RSGC2paper,RSGC3paper,RSGC4paper,RSGC5paper}.
However, less is known about the opposite side of the Galactic Centre
and far end of the Bar. There are two possible reasons for this:
firstly, the larger distance, high extinction and larger number of
foreground stars (due to the intervening Galactic Bulge) make it more
difficult to pick out clusters in by-eye searches. Indeed, it is
unlikely that Mc81 would have been found were it not for the four
bright foreground stars (see Sect.\ \ref{sec:photom}). Secondly, the
fact that no star clusters are known in this direction means that
investigators are less likely to search this region. This is in
contrast to the near end of the Bar, where the initial discovery of
the cluster RSGC1 in this region by \citet{Figer06} led
to the subsequent discoveries of a further 4 clusters within the same
complex
\footnote{Though
RSGC1 and RSGC2 were first recognized as associations of stars by
\citet{Bica03} and \citet{Stephenson90} respectively, the nature of
each cluster was not understood until later.}.
From our distance estimate of Mc81, we can place this cluster close to
where we suppose the far-end of the Bar may be, assuming an azimuthal
angle of 44\degr\ and a bar length of 4.4\,kpc
\citep{Benjamin05}. Another cluster nearby, which may too trace the
end of the Bar, is [DBS2003]179. The distance to this cluster is not
well known, and is based on spectro-photometric distance estimates for
stars with unknown luminosity classes \citep{Borissova08}. We have
reassesed the distance to this cluster using a similar methodology
that we have presented here for Mc81. Specifically, we assume that the
cluster is physically associated to its nearby molecular cloud, and
use the massive YSOs detected in the cloud to determine the systemic
radial velocity. We then use the SGPS survey
\citep{McClure-Griffiths05} to measure the velocity spectrum of the
intervening H\,{\sc i}\ gas to resolve the distance ambiguity. The average
radial velocities of the YSOs ($v_{\rm lsr}$=-36$\pm$3\,km~s$^{-1}$), combined with the
H\,{\sc i}\ absorption which is seen up to tangent-point velocities of
130\,km~s$^{-1}$, give a far-side kinematic distance of 9\,kpc, again with a
$\pm$2\,\,km~s$^{-1}$\ uncertainty to allow for deviations from the Galactic
rotation curve (see Sect.\ \ref{sec:extinct}). This is within the
errors of the spectro-photometric distance of 7.9\,kpc derived by
\citet{Borissova08}.
These two clusters -- Mc81 and [DBS2003]179 -- are then the first
young star clusters to be discovered in this region of the Galaxy. In
addition to these clusters a group of giant H{\sc ii}-regions, of which
G338.4+0.1 is one \citep{Russeil03}, suggest that this region may be
an active star-formation site, similar to the region of the
Scutum-Crux tangent at the opposite end of the Bar. As such, future
targeted surveys of this region may unearth a number of other such
objects.
\subsection{Association with HESS~1640-465}
As was noted in the introduction, the Mc81 cluster is located only a
few arcminutes from the TeV source HESS~1640-465, which is likely to
be powered by a neutron star. If the two were associated, it would
allow us to estimate the initial mass of the neutron star's
progenitor. Here we discuss the possible association between the two
objects.
HESS~1640-465 is known to have a counterpart source at GeV energies
\citep{Slane10}, and in the X-ray, detected by {\it Swift}, {\it XMM}
and {\it Chandra} \citep{Landi06,Funk07,Lemiere09}. Analysis of the
{\it XMM} X-ray spectrum yielded a column density of $n_{\rm H} =
6.1^{+2.1}_{-0.6} \times 10^{22}$cm$^{-2}$ or $3.6^{+1.1}_{-0.8}
\times 10^{22}$cm$^{-2}$, depending on whether a power-law or absorbed
black-body model was used \citep{Funk07}. However, \citet{Lemiere09}
analysed the {\it Chandra} data and found that they required a higher
column density of $1.4 \times 10^{23}$cm$^{-2}$ to fit the
data. Assuming a standard calibration between $n_{\rm H}$ and optical
extinction $A_{V}$ of $n_{\rm H} = 1.8 \times 10^{21}
A_{V}$\,cm$^{-2}$ \citep{Predehl-Schmitt95}, this implies an visual
extinction of between 20 and 70 mags, depending on the model for the
X-ray emission. Though the errors are large, this is consistent with
our measurment of the extinction to the cluster of $A_{V} = 45 \pm
15$.
From this evidence, it seems likely that HESS~1640-465 is associated
with the G338.4+0.1 H{\sc ii}-region surrounding Mc81. However, its
connection with the cluster itself is not clear. If the central source
of HESS~1640-465 is a neutron star (as seems likely), and the
progenitor {\it was} born with the cluster but was ejected, there are
two possibilities: either the progenitor was dynamically ejected from
the cluster; or it received a kick from the supernova (SN). The
location of HESS~1640-465 at the centre of SNR~338.3+0.0
\citep{Green04} provides circumstantial evidence against the latter
explanation, since this suggests that the progenitor exploded close to
its present location. If the progenitor formed with the cluster but
was ejected at a time $t_{\rm ej}$ ago, the ejection velocity is
$\simeq 20 (t_{\rm ej}/{\rm Myr})$\,km~s$^{-1}$ (assuming a projected distance
of 22\,pc). Therefore, it is entirely plausible that the progenitor
star formed along with the rest of the stars in Mc81 and was
dynamically ejected during the formation of the cluster.
Finally, there is the possibility that HESS~1640-465 formed out of the
same molecular cloud as Mc81, and at a similar time, but that the two
formed independently of one another. The morphology of the G338.4
region suggests inside-out star-formation, with the $\sim$3Myr old
cluster in the centre and a series of YSOs and UC-H{\sc ii}\ regions at the
periphery of the surrounding cavity, which have ages of a few $\times
10^5$yrs \citep{simgal}. The location of HESS~1640-465 does not fit
this picture, since the progenitor star must have formed at least 2Myr
ago, which is before the first SNe occured in Mc81. However, there are
other known instances of `multi-seeded' star formation, where collapse
occurs at multiple causally-unrelated sites across the host GMC
\citep[e.g.\ W51,][]{Clark09}.
In summary, we conclude that HESS~1640-465 is likely associated with
the star-formation region of G338.4+0.1. However, we are unable to
make a definitive association with the star cluster Mc81, and so we
are unable to use the age of the cluster to estimate the mass of the
neutron star's progenitor, as we were in the cases of e.g.\ RSGC1 and
Cl~1900+14 \citep{RSGC1paper,SGR1900paper}.
\section{Summary} \label{sec:summary}
We have provided a near-infrared photometric and spectroscopic
investigation of the candidate star cluster Mercer~81 (Mc81). We find
that that a highly extincted ($A_{V} = 45 \pm 15$) cluster exists in
the field identified by Mercer et al.\ (2005), but that the bright
four stars at the centre of the field are unrelated foreground
objects. The cluster is located at the centre of a cavity in a large
H{\sc ii}-region in the direction of G338.4+0.1, with evidence of on-going
star formation in the periphery of the cloud. Our analysis of the
cluster has revealed nine stars with strong P$\alpha$\ emission, one of
which we identify spectroscopically as a late-type N-rich Wolf-Rayet
star (WNL), in addition to a luminous early A-type supergiant. Via
detailed modelling of the WNL star's spectrum we estimate an age for
the cluster of $3.7^{+0.4}_{-0.5}$Myr. Under the assumption that the
stars with strong line emission are WRs, we have made an
order-of-magnitude estimate of the cluster's mass of $\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$}}}\hbox{$>$}}} 10^4$M\mbox{$_{\normalsize\odot}$}.
From a kinematic analysis of the host cloud, we obtain a distance to
the host star-forming complex of 11$\pm$2\,kpc. Our distance estimate
therefore places the G338.4+0.1 complex in the same region of the
Galaxy as the far end of the Galactic Bar. The recent detection of
another star cluster close to this location, as well as other giant
H{\sc ii}-regions, suggest that this region of the Galaxy may be as active
in star-formation as the opposite end of the Bar, where a $\sim
10^6$M\mbox{$_{\normalsize\odot}$}\ starburst event is known to have occurred in the last
$\sim$20Myr. A targeted search of the far end of the Bar will likely
uncover many more young star clusters, though the high extinction,
large distance and dense stellar field of the intervening Galactic
Bulge will mean that such a search will require considerable
observational effort.
\section*{Acknowledgments}
We thank the anonymous referee for comments and suggestions which
helped us improve the paper. BD is supported by a fellowship from the
Royal Astronomical Society. This work is in part based on observations
made with the NASA/ESA Hubble Space Telescope, obtained at the Space
Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS
5-26555. These observations are associated with program
\#11545. Support for program \#11545 was provided by NASA through a
grant from the Space Telescope Science Institute, which is operated by
the Association of Universities for Research in Astronomy, Inc., under
NASA contract NAS 5-26555. This work is in part based on observations
collected at the European Organisation for Astronomical Research in
the Southern Hemisphere, Chile, under programme number
083.D-0765(A). Financial support from the Spanish Ministerio de
Ciencia e Innovaci\'on under projects AYA2008-06166-C03-02 and
AYA2010-21697-C05-01 is acknowledged.
\bibliographystyle{/fat/Data/bibtex/aa}
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Chicago P.D. 3×07 "A Dead Kid, a Notebook, and a Lot of Maybes" Recap
Spoilers Ahead
This episode featured some of the best performances we've seen all season leaving me completely enamored while reminding me why I love this series so much.
Episode Summary: When Colin's body is found next to a homemade bomb the team believes it's a suicide, but when they're given his journal of personal drawings pointing to a revenge scheme, they're led to another kid, Ethan Jones, who reveals a lot more than any of them imagined. Jay's connection with Ethan allows him to open up about his past. Adam and Kim's date night is crashed by Olinsky and Michelle who need to place to stay. And Roman is understandably taking Andrew's death hard.
Review | Analysis: I've always said that the cases closely involving members of the Intelligence unit end up being the most riveting. Frankly, if there's one thing that'll always keep me coming back to Chicago P.D. it's the unbelievably talented group of actors and actresses that continuously leave me floored. Jesse Lee Soffer was at the top of his game this week — we've seen him do some commendable work, but in "A Dead Kid, a Notebook and a Lot of Maybes", Soffer's work is indescribable. This week's episode also gave us some great developments between other members of the units and it's always intriguing to see the differences in relationships on the series.
It's fantastic that the episode's theme once again focused on protecting children reminding us that sometimes, a hero can be anyone. And if that's not enough, what most characters have done this week is given their all to make sure someone else feels safe enough to be lean on them.
First of all, I've not voiced many concerns with the storyline between Olinsky and Michelle because up until this moment, it has not felt forced. However, it continues to make me wonder that something's bound to happen to her that's going to set a lot of things in motion. This can't all be for the sake of dividing Olinsky and his family again. We've been there and it'd be redundant. Olinsky signing the paperwork makes complete sense — his heart is enormous, and he'd never leave her out on the streets or in a foster home that could potentially harm her. Nevertheless, I do feel that he could've fought harder with his wife. Elias Koteas puts on some of the most beautiful performances and I wanted to see more of that fight. If we are to believe he cares this deeply for a girl he's only known for a while, then we need to see him try to get through to Lexi as well. He needs to fight for her as well and remind not only the audience, but his family that she too means a great deal to him. That said, while I'm glad he feels comfortable enough to confide in Adam this way, I'm not exactly on board this particular storyline at the moment. I keep saying there has to be something more because there's way too much focus on her.
Speaking of Adam, I'm so ridiculously fond of him when he's showcasing just how deeply infatuated he is with Kim. It's what makes it so easy to love them. He reveals this side of him at the most inappropriate times sometimes, but if he didn't, he wouldn't exactly be Adam. His timing's got to be a tad off for it to make sense. It was lovely to see him try to persuade her to come over so he'd cook for her — it's almost as though they're still dating, still falling for each other, and still in the first steps of developing a romance. And it's exactly what makes them so interesting to watch because even though they're engaged, there's still a deep adoration between them that almost makes them seem like kids falling in love for the first time. It's what's essentially so heartbreaking because I cannot imagine what the series is about to do with them for the sake of "drama", but I'm such a fan of their moments in the locker room. It's where they've comforted one another numerous times, made out like they were in hiding, gotten engaged in, and shared some hilarious moments establishing them so uniquely as a couple.
While we're on the topic of locker rooms, I'm so thrilled we got to hear Erin talk about Nadia again. She chose to reach out and comfort Roman reminding him that even though he doesn't want to talk about it at the moment, he has people he can lean on. She knows what it's like to stay quiet after a death that's hit hard, and she knows that it can lead to a dangerous place thereby, having her be the one to reach out to him this way felt fitting. It's exactly what the audience needed to understand more of her struggle — sometimes, talking about things like death is what makes it real. It's almost as though if you ignore it and push it aside, it hasn't happened. You can forget about it. You can choose not to feel. And she knows what doing that is like and doesn't want to see anyone else go through that pain of isolation and heartache that's not easy to recover from. Reminding him of the fact that they're all there for him is exactly what he needs to feel safe enough to grieve properly. He needs to know that when he falls, he's got an entire group willing to lend him a shoulder. And sometimes reminding someone that they aren't alone is the most important thing you can do for them.
It was also nice to see Erin bond with Atwater because that's something we've seen too often, but ever since she's returned she's slowly made her way around being there for her fellow detectives. It's interesting that she was there to talk to him about Voight and his faith in detectives, but what's even more interesting is that I don't exactly feel Atwater's ties with Captain Whitaker will go as smoothly as they appear to be in the end.
Ultimately, tonight's episode was exclusively about this group of individuals showcasing their selfless hearts by giving someone their sincere attention in order to help them come to terms with what's hurting them inside. And while the little moments we got with other characters were wonderful, Jay Halstead takes the crown this week.
"A Dead Kid, a Notebook and a Lot of Maybes" finally gave viewers the opportunity to understand some of the horrors that layer Halstead — reminding us of why this job means so much to him and why cases like this essentially break him the most. When we first see Ethan in the interrogation room, Jay begins to deconstruct his character revealing that he too used to be just like him: a loner and perhaps at that times even bullied. At this point, it's no surprise that Jay's especially sensitive when it comes to cases dealing with sexual abuse. And although he didn't know Ethan as well, it doesn't change the fact that he was not only able to empathize with him, but he was able to make him feel safe enough to open up about what really happened. Heroic acts are in my opinion at least, anything that's done out of selflessness. It's what Jay's done with Ethan, and it's what Ethan's done by speaking up.
I love that the chief theme of this week's episode showcases the importance of attentively and compassionately reminding lost souls of the courage that's inside of them. Falling into a hopeless, fearful, or heartbroken state is inevitable — it happens to even the toughest people, but it's essential to note that sometimes, having someone be strong for you isn't a sign of weakness but rather a spectacle of heroism. Ethan had to be brave enough to tell the truth in order to save other kids from the same fate he faced. Jay had to be brave enough to stay patient and kindhearted in order to make Ethan feel safe enough to be honest. Jay also had to be brave enough to open up about his past in order to reveal that no matter how alone Ethan feels, he understands, and he's willing to listen without passing any judgment. I've said it countless times by now, but when it comes to delivering raw emotional scenes, Jesse Lee Soffer is at his strongest. Although I'll save most of my commending for This Week's Most Noteworthy Performance review, the work Soffer's done this week have surpassed anything he's delivered in the past. The heartrending reactions and sheer vulnerability visible clearly in his expressiveness have left me astounded. Both Soffer and Elijah Marcano did excellent work in the interrogation room selling the emotions they're feeling in that moment remarkably. It's not often young actors do such a prestigious job of delivering emotional scenes like this, and while it was undoubtedly heartbreaking, it's what made the scene that much more realistic.
It's necessary to point out that Jay's choice to treat Ethan like an adult was an excellent decision because it's what gave him the encouragement to open up and feel safe. Sometimes when we treat people younger than us as though they're somehow inferior, it makes them feel less significant and it makes them doubt themselves, their versatilities, and their emotions. Perhaps it is understanding the lasting effects being bullied leaves on a person, but Jay's choice to comfort him as somewhat of an equal shows great wisdom on his behalf. When a person's at their weakest, the most important thing that can be done is to remember that there's no age or gender when speaking to someone about their innermost struggles. And Jay's tendency to always project compassion towards those who need it most is truly heroic.
I was recently reminded of the fact that while some soldiers can speak up about the experiences they've had, others have a much more difficult time doing so. And because we haven't seen Jay open up in the past, it's safe to presume it's not the easiest thing for him to talk about. However, it took a great deal of courage to tell Ethan about the fact that he was a soldier — he's seen indescribable terrors that have undeniably scarred him and made him the man he is today. And when Ethan asked if he'd lost friends, Jay's response about having lost many followed by sharing his coping mechanism was perhaps the loveliest moment. It's never easy losing someone, but Jay telling Ethan he's able to move forward knowing that in doing so he honors the courageous sacrifice his friends made is a gorgeous way of helping Ethan grieve for his father. You never stop missing those who've passed, but there comes a point where you remind yourself that they'd want you to be happy, free, and living the life they couldn't. And that almost always becomes the key in helping us move forward.
One of my favorite things to write about whether it's for Chicago P.D. or any of the other series I review, it's the fact that vulnerability requires tremendous strength. There's a misconception out there that when humans are at a vulnerable state, then they're weak, but what's not taken into consideration is the fact that it takes admirable strength to be open. In order to let our walls down, we have to be brave enough to show a side of ourselves that may be judged wrongfully by some. Because that'll always happen — someone will always mistake your vulnerability for either a sign of weakness or a plea for attention. The reality however, is that when a person's in a susceptible state, their walls are not only down but their heart is completely bare, and that requires a kind of strength that's not nearly as praised enough as it should be. Jay's not afraid of being vulnerable, but he's also not someone who shows that side of him freely. and his choice to reveal parts of his past with both Voight and Erin this week said a great deal about how much this case has distraught him. Anyone who's served has seen traumatizing events some of us can't even bear to imagine, and even though he doesn't speak about his past often, this is something everyone's fully aware of. However, telling them about a time in his childhood where he's felt out of place felt so incredibly sincere in that moment because as Jay's looking through the glass, Soffer makes it clear that Jay's been mentally and emotionally transported back to a time in his childhood. And I loved how Voight too was given the chance to see a part of Halstead that's yet to be revealed. As stated above, it's always been evident that cases like this make his blood boil the most, but the team has yet to see him resonate with someone as closely as he did with Ethan. This was his chance to not only keep history from repeating itself, but the opportunity to make sure a kid feels safe in the world again. Batman said it best in The Dark Knight Rises when he tells Det. Gordon: "a hero can be anyone. Even a man doing something as simple and reassuring as putting a coat around a young boy's shoulders to let him know his world hadn't ended." Police officers do this constantly. Roman's done it with Andrew. Erin's done it for Nadia. Voight's done it for Erin. Jay's done it numerous times as well, but the courage he showcased this week by opening up a part of himself he probably never imagined he'd have to is beautifully admirable and worth commending more than once. I was so effortlessly moved with Soffer's meticulous acting choices this week — placing his hand on Ethan to show him both physical and emotional security showcased that not everyone would hurt him. He certainly wouldn't and thereby, this made it easier for Ethan to open up and feel safe with Jay. It's why he wanted Jay to be the one to tell his mom because the character he's shown has not only inspired Ethan, but it's what has made him trust in another male figure after everything he's been through. That's anything but easy to do after a traumatic experience trusting in someone who took advantage of your situation and character, but Jay's sincerity restored humanity back into Ethan deeming him a hero in more ways than one. Jay's choice to tell him that his father would not only be proud of his bravery to speak up but because it's all over now made for the perfect ending. It gave Ethan something bold to hold onto and the strength to move forward knowing justice has been rightfully served. And when a hero thinks you're a hero, well, there's no praise quite as higher. Jay's perpetually changed and inspired Ethan revealing once more that he's without a doubt the most honorable. At least in my book.
Jay and Erin's ridiculous banter will always be entertaining and a huge part of what makes them special, but nothing fortifies a couple's bond the way they comfort one another in times of sheer vulnerability. Throughout the episode, Sophia Bush did an exquisite job of subtly exhibiting the apprehension Erin's feeling because of what Jay's going through. The occasional glances towards him as they listened on the interrogations happening followed by pulling him aside to make sure he's okay were fantastic moments of development for the two. We've seen her in positions of complete vulnerability around him, and it's always been evident she's the only person he feels comfortable with to show a side of him not many see. It's why he wanted her to be the one who came down when he was kidnapped. There's no need to wear an armor around one another and now that they're officially together, it's even easier to lean on each other in times of despondency. I write about it so often in other reviews, and it brings me great joy to finally be able to talk about it with Erin and Jay: love is strength. And though the two haven't exchanged those words yet, the feelings that have been growing from the very beginning are strong enough to be considered a form of adoration. Nevertheless, the most incomparable part about having a partner to go through life with is the fact that being with your best friend makes every part of the day easier. It's what Jay and Erin have always been for each other: strength. Their partnership has always extended outside of the field, but it took them a while to get to a place where they can explore their true feelings for one another.
They've been each other's strength numerous times before in the past, but because they're now together, it's easier to give everything they've got. Whether physically or emotionally, it's easier to be open. And gentle physical gestures are always a beautiful exhibition of adoration. Sometimes all it takes to really comfort someone is a hand on the shoulder — just as Jay's done with Ethan, Erin's done the same for Jay. It's anything but easy for Jay to be vulnerable, but in this moment, it's comforting to be. It then becomes simpler for him to ask Erin for a favor because even that takes great strength; leaning on someone else and asking for help requires a great deal of bravery. Assuring him that she's got him was perhaps one of the most incredible things she's said because while I've said before, we've always known where Jay's heart has been, but Erin's has been more closed off. She's not shown her feelings as evidently. Although they've each had relationships in the past, anything we've seen with Erin hasn't appeared to have been something that's made her believe in the possibility of a lasting love. She's never known someone who wouldn't give up on her until Jay came along. This was her moment to remind him of the fact that just as he's shown time and time again that he's by her side through it all, she can be the strength he needs in this moment of absolute frustration, anger, and heartache. The English nerd in me that often geeks out over word choices is having an absolute ball with "I got you." He's hers, and she does a gorgeous job of assuring him that he's safe with her. She's got him mentally, physically, and emotionally thereby, giving him the courage to move forward knowing he's not enduring all of this on his own — his heart's not only in good hands, but his girl is by side in case he comes undone.
Sharing burdens is just as important as sharing moments of pure bliss, and when two people are able to do both, then their relationship is at a solid place. That's what Erin's done gorgeously throughout the episode — she's shown him in every way she could that he doesn't have to carry the weight on his shoulder alone. She's bearing it all with him just as he's done for her in the past and will continue to do in the future when needed. I'll gladly take a 100 moments like this one for the rest of the season because they're often what illuminate just how deeply adoration connects two people. When one of them is hurting, the other is aching just as badly, but because it's important to remain strong for one another, a true partnership requires selflessness. A relationship is all about carrying each other through life. It's about being there for the good, the bad, and the ugly all while ceaselessly supporting and adoring each other.
What are your thoughts on this week's episode? Make sure you check back here Sunday afternoon for that's when we'll post a performance review focused on Jesse Lee Soffer's outstanding work in more detail. Remember if there's anything you'd like us to discuss agreements/disagreements let us know in the comments section below and we'll get right back to you. As always, we welcome opposing opinions as long as we can be adults in our discussion but hateful comments will be blocked and ignored.
— @GissaneSophia
November 6, 2015 mgcirclesmedia adam ruzek, alvin olinsky, burzek, chicago pd, cpd, cpdr, erin lindsay, hank voight, jay halstead, jesse lee soffer, kim burgess, linstead, sean roman, sophia bush
2 thoughts on "Chicago P.D. 3×07 "A Dead Kid, a Notebook, and a Lot of Maybes" Recap"
Your reviews are phenomenal. Always look forward to reading them. I love how you explain the characters especially linstead. Thank you for taking the time to do this every week.
mgcirclesmedia says:
Aww, thank you so much for the kind words. We're glad you enjoy them! | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,777 |
\section*{ Appendix A: Transition rates }
Laser cooling is realized by the Hamiltonian
\begin{equation}
\hat H_{las}=
\frac12\Omega
\int d^3r~
e^{i\vec k_{L}\vec r}
\hat\psi_{e}^{\dagger}(\vec r)
\hat\psi_{-}(\vec r) ~+~
{\rm h.c.}~,
\end{equation}
driving coherent oscillations with Rabi frequency $\Omega$ between the
excited state $e$ and the state $-$, together with
the spontaneous emission
superoperator
\begin{eqnarray}
{\cal L}~\hat\rho&=&
\gamma\int d\varphi~d\cos\theta~{\cal W}(\theta,\varphi)
\int d^3r_1~d^3r_2~
e^{ i \vec k ( \vec r_1 - \vec r_2 ) }
\nonumber\\
&\times&
\left[
2
\hat\psi_-^{\dagger}(\vec r_2)\hat\psi_e(\vec r_2)
\hat\rho~
\hat\psi_e^{\dagger}(\vec r_1)\hat\psi_-(\vec r_1)~-~
\hat\psi_e^{\dagger}(\vec r_1)\hat\psi_-(\vec r_1)
\hat\psi_-^{\dagger}(\vec r_2)\hat\psi_e(\vec r_2)
~\hat\rho~-~
~\hat\rho~
\hat\psi_e^{\dagger}(\vec r_1)\hat\psi_-(\vec r_1)
\hat\psi_-^{\dagger}(\vec r_2)\hat\psi_e(\vec r_2)
\right]~,
\end{eqnarray}
with an effective spontaneous emission rate of $2\gamma$
and the fluorescence dipole pattern ${\cal W}(\theta,\varphi)$.
Expansion in the eigenmodes $w_l(\vec r)$ of the harmonic trap
\begin{equation}
\hat\psi_e(\vec r)=\sum_l \hat e_l~w_l(\vec r)~,
\end{equation}
and the Bogoliubov transformation (see main text) give
\begin{equation}
\hat H_{las}=
\frac12\Omega
\sum_{ml}
\left[
\hat b_{m,-} ~u_{ml}(\vec k_L) ~+~
\hat b_{m,+}^{\dagger} ~v^*_{ml}(\vec k_L)
\right]~
\hat e_l^{\dagger}~+~
{\rm h.c.}~.
\end{equation}
Here the (generalized) Frank-Condon factors are
\begin{eqnarray}
&&
u_{ml}(\vec k)=
\int d^3r~
e^{i\vec k \vec r}
w_l^*(\vec r)
u_m(\vec r)~,
\nonumber\\
&&
v^*_{ml}(\vec k)=
\int d^3r~
e^{i\vec k \vec r}
w_l^*(\vec r)
v^*_m(\vec r)~.
\end{eqnarray}
In a similar way, and after rotating wave approximation for $e$-atoms
and the Bogoliubov quasiparticles, the spontaneous emission term becomes
\begin{eqnarray}
{\cal L}~\hat\rho&=&
\gamma \sum_{ml} U_{ml}
\left(
2\hat b_{m,-}^{\dagger}\hat e_l~\rho~\hat e_l^{\dagger}\hat b_{m,-} -
\hat e_l^{\dagger}\hat e_l \hat b_{m,-}\hat b_{m,-}^{\dagger}~\rho -
\rho~\hat e_l^{\dagger}\hat e_l \hat b_{m,-}\hat b_{m,-}^{\dagger}
\right)~+
\nonumber\\
&&
\gamma \sum_{ml} V_{ml}
\left(
2\hat b_{m,+}\hat e_l~\rho~\hat e_l^{\dagger}\hat b_{m,+}^{\dagger} -
\hat e_l^{\dagger}\hat e_l \hat b_{m,+}^{\dagger}\hat b_{m,+}~ \rho -
\rho~ \hat e_l^{\dagger}\hat e_l \hat b_{m,+}^{\dagger}\hat b_{m,+}
\right)~.
\end{eqnarray}
Here the spontaneous emission rates are
\begin{eqnarray}
&&
U_{ml}=
\int d\varphi d\cos\theta~ {\cal W}(\theta,\varphi)~
|u_{ml}(\vec k)|^2~,
\nonumber\\
&&
V_{ml}=
\int d\varphi d\cos\theta~ {\cal W}(\theta,\varphi)~
|v_{ml}(\vec k)|^2~.
\end{eqnarray}
Adiabatic elimination of the excited state $e$, similar as in
Ref.\cite{AE}, results in the following evolution equations for
the occupation numbers
\begin{eqnarray}
\frac{d N_{m,-}}{dt} &=&
\sum_n~
\Gamma^{(-)}_{m\leftarrow n}
\left( 1 - N_{m,-} \right) N_{n,-} -
\Gamma^{(-)}_{n\leftarrow m}
\left( 1 - N_{n,-} \right) N_{m,-} +
C_{mn} \left( 1 - N_{m,-} \right) \left( 1 - N_{n,+} \right) -
A_{nm} N_{m,-} N_{n,+} ~,
\\
\frac{d N_{m,+}}{dt} &=&
\sum_n~
\Gamma^{(+)}_{m\leftarrow n}
\left( 1 - N_{m,+} \right) N_{n,+} -
\Gamma^{(+)}_{n\leftarrow m}
\left( 1 - N_{n,+} \right) N_{m,+} +
C_{nm} \left( 1 - N_{m,+} \right) \left( 1 - N_{n,-} \right) -
A_{mn} N_{m,+} N_{n,-} ~.
\end{eqnarray}
There are contributions from four different processes:
\begin{itemize}
\item relaxation of $-$ quasiparticles
\begin{equation}
\Gamma^{(-)}_{m\leftarrow n}=
\frac{\Omega^2}{2\gamma}
\sum_l
\frac{
\gamma^2
U_{ml}
|u_{nl}(\vec k_L)|^2
}
{
\left|
\delta-\omega^e_l+
\omega_n+
i\gamma \sum_s~
U_{sl} ( 1 - N_{s,-} + \delta_{s,n} )+
V_{sl} N_{s,+}
\right|^2
}~,
\end{equation}
\item relaxation of $+$ quasiparticles
\begin{equation}
\Gamma^{(+)}_{m\leftarrow n}=
\frac{\Omega^2}{2\gamma}
\sum_l
\frac{
\gamma^2
V_{nl}
|v_{ml}(\vec k_L)|^2
}
{
\left|
\delta-\omega^e_l-
\omega_m+
i\gamma \sum_s~
U_{sl} ( 1 - N_{s,-} )+
V_{sl} ( N_{s,+} + \delta_{m,s} )
\right|^2
}~,
\end{equation}
\item creation of pairs of $+$ and $-$ quasiparticles
\begin{equation}
C_{mn}=
\frac{\Omega^2}{2\gamma}
\sum_l
\frac{
\gamma^2
U_{ml}
|v_{nl}(\vec k_L)|^2
}
{
\left|
\delta-\omega^e_l-
\omega_n+
i\gamma \sum_s~
U_{sl} ( 1 - N_{s,-} ) +
V_{sl} ( N_{s,+} + \delta_{n,s} )
\right|^2
}~,
\end{equation}
\item annihilation of pairs of $+$ and $-$ quasiparticles
\begin{equation}
A_{mn}=
\frac{\Omega^2}{2\gamma}
\sum_l
\frac{
\gamma^2
V_{ml}
|u_{nl}(\vec k_L)|^2
}
{
\left|
\delta-\omega^e_l+
\omega_n+
i\gamma \sum_s~
U_{sl} ( 1 - N_{s,-} + \delta_{n,s} ) +
V_{sl} N_{s,+}
\right|^2
}~.
\end{equation}
\end{itemize}
\section*{ Appendix B: Spherical symmetry }
Simulation of laser cooling with a non-vanishing pairing function $P(\vec
r)$, and a Hartree potential $g_0\rho(\vec r)$ is a numerically hard
problem.
The
main difficulty is that after every short period of laser cooling it is
necessary to reiterate Bogoliubov-de Gennes equations in order to obtain a
new self-consistent pairing function, Hartree potential, and the
Bogoliubov modes $(\omega_m,u_m,v_m)$. With the new Bogoliubov modes, the
transition rates $A_{mn},C_{mn},\Gamma^{\pm}_{m\leftarrow n}$ have to be
calculated again, and this is a very time-consuming operation. This
numerical effort is reduced a lot with assumption of spherical symmetry.
With spherically symmetric $\Delta(r)$ and $\rho(r)$ the Bogoliubov modes
can be decomposed into spherical and radial parts
\begin{eqnarray}
&&
u_{ln}(r)~Y_{lm}(\theta,\varphi)~,
\\
&&
v_{ln}(r)~Y_{lm}(\theta,\varphi)~.
\end{eqnarray}
In a similar way harmonic oscillator modes become
$W_{ln}(r)Y_{lm}(\theta,\varphi)$. Quasiparticle energies $\omega_{ln}$ do
not depend on $m$. We assume fast equilibration within each degenerate
energy shell $nl$ so that all occupation numbers in a given shell are the
same $N_{ln,\pm}$. This assumption greatly simplifies the rate equations
\begin{eqnarray}
\frac{d}{dt} N_{l_1n_1,-}
&=\sum_{l_2n_2}&
\gamma^{(-)}_{l_1n_1\leftarrow l_2n_2}
\left( 1 - N_{l_1n_1,-} \right) N_{l_2n_2,-} -
\gamma^{(-)}_{l_2n_2\leftarrow l_1n_1}
\left( 1 - N_{l_2n_2,-} \right) N_{l_1n_1,-} +
\nonumber\\
&&
c_{l_1n_1l_2n_2}
\left( 1 - N_{l_1n_1,-} \right)
\left( 1 - N_{l_2n_2,+} \right) -
a_{l_2n_2l_1n_1}
N_{l_1n_1,-} N_{l_2n_2,+} ~,
\\
\frac{d}{dt} N_{l_1n_1,+}
&=\sum_{l_2n_2}&
\gamma^{(+)}_{l_1n_1\leftarrow l_2n_2}
\left( 1 - N_{l_1n_1,+} \right) N_{l_2n_2,+} -
\gamma^{(+)}_{l_2n_2\leftarrow l_1n_1}
\left( 1 - N_{l_2n_2,+} \right) N_{l_1n_1,+} +
\nonumber\\
&&
c_{l_2n_2l_1n_1}
\left( 1 - N_{l_1n_1,+} \right)
\left( 1 - N_{l_2n_2,-} \right) -
a_{l_1n_1l_2n_2} N_{l_1n_1,+} N_{l_2n_2,-} ~.
\end{eqnarray}
Here $\gamma^{(\pm)},a,c$ are averaged transition rates, for example
\begin{equation}
\gamma^{(-)}_{l_1n_1\leftarrow l_2n_2}~=~
\frac{1}{2l_1+1}
\sum_{m_1m_2}
\Gamma^{(-)}_{l_1n_1m_1\leftarrow l_2n_2m_2}~.
\end{equation}
This equation combined together with the definition of $\Gamma^{(-)}$
gives
\begin{equation}
\gamma^{(-)}_{l_1n_1\leftarrow l_2n_2}~=~
\frac{\gamma\Omega^2}{2(2l_1+1)}
\sum_{m_1m_2}
\sum_{l_en_em_e}
\frac{
U_{l_1n_1m_1l_en_em_e}
|u_{l_2n_2m_2l_en_em_e}(\vec k_L)|^2
}
{
\left[ \delta-\omega(l_e+2n_e-\mu)+\omega_{l_2n_2}\right]^2+
\gamma^2 R_{l_en_e}^2
}~.
\end{equation}
Here the frequency of the intermediate excited state
$\omega(l_e+2n_e-\mu)$ is measured with respect to the chemical potential
$\mu$ because the quasiparticle energies $\omega_{l_2n_2}$ are also
defined with respect to $\mu$. For a laser beam with a $\vec k_L$
along the $\hat z$-axis the indices $m_e$ and $m_2$ must be the same.
Furthermore, it follows from the properties of Clebsch-Gordan
coefficients that the sum $\sum_{m_1}U_{l_1n_1m_1l_en_em_e}$
does not depend on $m_e$ so that we can set e.g. $m_e=0$ and
\begin{eqnarray}
\gamma^{(-)}_{l_1n_1\leftarrow l_2n_2}&=&
\frac{\gamma\Omega^2}{2(2l_1+1)}
\sum_{l_en_e}
\frac{
\left( \sum_{m_1} U_{l_1n_1m_1l_en_e0} \right)
\left( \sum_{m_2} |u_{l_2n_2m_2l_en_em_2}(\vec k_L)|^2 \right)
}
{
\left[ \delta-\omega(l_e+2n_e-\mu)+\omega_n \right]^2+
\gamma^2 R_{l_en_e}^2
}~\equiv
\nonumber\\
&&
\frac{\gamma\Omega^2}{2(2l_1+1)}
\sum_{l_en_e}
\frac{
SU_{l_1n_1l_en_e}~~
su_{l_2n_2l_en_e}
}
{
\left[ \delta-\omega(l_e+2n_e-\mu)+\omega_n \right]^2+
\gamma^2 R_{l_en_e}^2
}.
\end{eqnarray}
The last form shows that the transition rate $\gamma^{(-)}$ can be
constructed out of matrices $SU,su$ and a vector $R$. These
elements can be expressed through even more elementary building blocks
\begin{eqnarray}
&&
SU_{l_1n_1l_en_e}~=~
\sum_l
(jWu)^2_{ll_en_el_1n_1}~
S_{ll_el_1}~,
\nonumber\\
&&
su_{l_2n_2l_en_e}~=~
\sum_m
\left|
\sum_l
s_{ll_el_2m}
(jWu)_{ll_en_el_2n_2}
\right|^2~.
\nonumber
\end{eqnarray}
The more elementary building blocks are
\begin{eqnarray}
&&
(jWu)_{ll_en_el_1n_1}=
\int r^2dr~j_l(k r)W_{l_en_e}(r)u_{l_1n_1}(r)~,
\nonumber\\
&&
S_{ll_el_1}=
(2l+1)
\frac{2l_e+1}{2l_1+1}
\sum_m
\langle l,l_e,m,0 | l_1,m \rangle^2
\langle l,l_e,0,0 | l_1,0 \rangle^2~,
\nonumber\\
&&
s_{ll_el_2m}=
i^l(2l+1)
\sqrt{\frac{2l_e+1}{2l_2+1}}
\langle l,l_e,0,m|l_2,m \rangle
\langle l,l_e,0,0|l_2,0 \rangle~.
\end{eqnarray}
The line-width of an excited state $l_e,n_e$ is approximately given by
\begin{equation}
R_{l_en_e}=
\sum_{ln}
(1-N_{ln})
SU_{lnl_en_e} +
N_{ln}
SV_{lnl_en_e}~.
\end{equation}
The matrix $SV$ is obtained from the matrix $SU$
by a substitution $u\to v$. As the
eigenfunction $u_{ln}(r)$ evolves in the process of laser cooling
it is more efficient to express the matrix $(jWu)$ through a static
matrix $(jWW)$ build out of harmonic oscillator modes
\begin{equation}
(jWW)_{ll_en_el_1n_1}=
\int r^2dr~j_l(k r)W_{l_en_e}(r)W_{l_1n_1}(r)~.
\end{equation}
The static matrices $s,S,jWW$ were prepared once, and stored then on a
disk.
They facilitate calculation of the temperature dependent matrices
$SU,su,R$ every time the wave functions $u,v$ are updated.
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,902 |
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